arXiv:astro-ph/0603494v2 29 May 2006LAPTH-1131/05, IFIC/05-59Massive neutrinos and cosmologyJulien Lesgourgues a and Sergio Pastor baLaboratoire de Physique The´orique LAPTH (CNRS-Universite´ de Savoie)B.P. 110, F-74941 Annecy-le-Vieux Cedex, FrancebInstituto de F´ısica Corpuscular (CSIC-Universitat de Vale`ncia)Ed. Institutos de Investigacio´n, Apdo. 22085, E-46071 Valencia, SpainAbstractThe present experimental results on neutrino flavour oscillations provide evidencefor non-zero neutrino masses, but give no hint on their absolute mass scale, whichis the target of beta decay and neutrinoless double-beta decay experiments. Crucialcomplementary information on neutrino masses can be obtained from the analysis ofdata on cosmological observables, such as the anisotropies of the cosmic microwavebackground or the distribution of large-scale structure. In this review we describein detail how free-streaming massive neutrinos affect the evolution of cosmologicalperturbations. We summarize the current bounds on the sum of neutrino massesthat can be derived from various combinations of cosmological data, including themost recent analysis by the WMAP team. We also discuss how future cosmologicalexperiments are expected to be sensitive to neutrino masses well into the sub-eVrange.Key words: Neutrino masses, Cosmology, Dark matterPACS: 14.60.Pq, 95.35.+d, 98.80.EsContents1 Introduction 32 Neutrino oscillations and absolute neutrino mass searches 53 The cosmic neutrino background 93.1 Basics on relic neutrinos, including neutrino decoupling 93.2 Extra radiation and the effective number of neutrinos 13Email addresses: lesgourg@lapp.in2p3.fr (Julien Lesgourgues),pastor@ific.uv.es (Sergio Pastor).Preprint submitted to Elsevier Science 3 February 20083.3 Massive neutrinos as dark matter 154 Massive neutrinos and cosmological perturbations 164.1 Observables targets: definition of the power spectra 184.2 Background evolution 204.3 Gauge transformations and Einstein equations 214.4 Linear perturbation theory in a neutrinoless Universe (pure ΛCDM) 254.5 Linear perturbation theory in presence of neutrinos (ΛMDM) 384.6 Summary of the neutrino mass effects 605 Current observations and bounds 675.1 Statistical methods for parameter inference from cosmological data 675.2 CMB anisotropies 715.3 Galaxy redshift surveys 735.4 Lyman-α forest 765.5 Summary of current bounds 775.6 Extra parameters 795.7 Non-standard relic neutrinos 816 Future sensitivities and new experimental techniques 826.1 Fisher matrix 826.2 Future CMB experiments 836.3 Future galaxy redshift surveys 926.4 CMB weak lensing 976.5 Galaxy weak lensing (cosmic shear) 1036.6 Galaxy cluster surveys 1077 Conclusions 108References 11121 IntroductionNeutrino cosmology is a fascinating example of the fecund interaction betweenparticle physics and astrophysics. At the present time, the researchers workingon neutrino physics know that the advances in this field need a combined effortin the two areas.From the point of view of cosmologists, the idea that massive neutrinos couldplay a significant role in the history of the Universe and in the formation ofstructures has been discussed for more than thirty years, first as a pure spec-ulation. However, nowadays we know from experimental results on flavourneutrino oscillations that neutrinos are massive. At least two neutrino stateshave a large enough mass for being non-relativistic today, thus making up asmall fraction of the dark matter of the Universe. At a stage in which cos-mology reaches high precision thanks to the large amount of observationaldata, it is unavoidable to take into account the presence of massive neutrinos.In particular, the observable matter density power spectrum is damped onsmall scales by massive neutrinos. This effect can range from a few per centfor 0.05− 0.1 eV masses, the minimal values of the total neutrino mass com-patible with oscillation data, up to 10− 20% in the limit of three degeneratemasses.From the point of view of particle physicists, fixing the absolute neutrino massscale (or, equivalently, the lightest neutrino mass once the data on flavour os-cillations are taken into account) is the target of terrestrial experiments suchas the searches for neutrinoless double beta decay or tritium beta decay ex-periments, a difficult task in spite of huge efforts and very promising sched-uled experiments. At the moment, the best bounds come from the analysisof cosmological data, from the requirement that neutrinos did not wash outtoo much of the small-scale cosmological structures. Recently the cosmolog-ical limits on neutrino masses progressed in a spectacular way, in particularthanks to the precise observation of the Cosmic Microwave Background (CMB)anisotropies by the Wilkinson Microwave Anisotropy Probe (WMAP) satel-lite, and to the results of a new generation of very deep galaxy redshift surveys.Neutrino physicists look with anxiety at any new development in this field,since in the next years many cosmological observations will be available withunprecedented precision.At this very exciting moment, which might be preceding a breaking discoverywithin a few years, our aim is to present here a clear and comprehensive re-view of the role of massive neutrinos in cosmology, including a description ofthe underlying theory of cosmological perturbations, a summary of the cur-rent bounds and a review of the sensitivities expected for future cosmologicalobservations. Our main goal is to address this manuscript simultaneously to3particle physicists and cosmologists: for each aspect, we try to avoid uselessjargon and to present a self-contained summary. Compared to recent shortreviews on the subject, such as Refs. [1,2,3,4], we tried to present a more de-tailed discussion. At the same time, we will focus on the case of three flavourneutrinos with masses in accordance with the current non-cosmological data.A recent review on primordial neutrinos can be found in [5], while for manyother aspects of neutrino cosmology we refer the reader to [6]. Finally, a moregeneral review on the connection between particle physics and cosmology canbe found in [7].We begin in Sec. 2 with an introduction to flavour neutrino oscillations, andtheir implications for neutrino masses in the standard three-neutrino scenario.We also briefly review the limits on neutrino masses from laboratory exper-iments. Then, in Sec. 3, we describe some basic properties of the CosmicNeutrino Background. According to the Big Bang cosmological model, each ofthe three flavour neutrinos were in thermal equilibrium in the Early universe,and then decoupled while still relativistic. We review the consequences of thissimple assumption for the phase-space distribution and number density of therelic neutrinos, and we summarize how this standard picture is confirmed byobservations from Big Bang Nucleosynthesis (BBN) and cosmological pertur-bations. Finally we explain why neutrinos are expected to constitute a fractionof the dark matter today.In Sec. 4, we describe the impact of massive neutrinos on cosmological pertur-bations. This section is crucial for understanding the rest, since current andfuture bounds rely precisely on the observation of cosmological perturbations.Although the theory of cosmological perturbations is a very vast and technicaltopic, we try here to be accessible both to particle physicists willing to learnthe field, and to cosmologists willing to understand better the specific role ofneutrinos. We introduce the minimal number of concepts, technicalities andequations for understanding the main results related to neutrinos. Still thissection is quite long because we wanted to make it self-contained, and to ex-plain many intermediate steps that are usually hidden in most works on thesubject.In Sec. 4.1, we define the quantities that can be observed, i.e. those that wewant to compute. In Sec. 4.2, we describe the evolution of homogeneous quan-tities and in Sec. 4.3 we define the general setup for the theory of cosmologicalperturbations. Then in Sec. 4.4 we described in a rather simplified way whatthe evolution of perturbations would look like in absence of neutrinos. For acosmologist, these four sections are part of common knowledge and can beskipped. In Sec. 4.5, we present a detailed description of the impact of neu-trinos on cosmological perturbations. This long section is the most technicalpart of this work, so the reader who wants to avoid technicalities and to knowthe results can go directly to Sec. 4.6, for a comprehensive summary of the4effects of neutrino masses on cosmological observables.Then, in Sec. 5 we review the current existing bounds on neutrino masses,starting from those involving only CMB observations, and then adding differ-ent sets of data on the distribution of the Large Scale Structure (LSS) of theUniverse, which include those coming from galaxy redshift surveys and theLyman-α forest. We explain why a unique cosmological bound on neutrinomasses does not exist, and why there are significant variations from paper topaper, depending on the data included and the assumed cosmological model.Finally, in Sec. 6 we describe the prospects for the next ten or fifteen years.We summarize the forecasts which can be found in the literature concerningthe sensitivity to neutrino masses of future CMB experiments, galaxy redshiftsurveys, weak lensing observations (either from the CMB or from galaxy el-lipticity) and galaxy cluster surveys. We conclude with some general remarksin Sec. 7.2 Neutrino oscillations and absolute neutrino mass searchesNeutrinos have played a fundamental role in the understanding of weak inter-actions since they were postulated by W. Pauli in 1930 to safeguard energyconservation in beta decay processes. These chargeless leptons are masslessin the framework of the succesful Standard Model (SM) of particle physics.However, this is an accidental prediction of the SM, and there are many well-motivated extended models where neutrinos acquire mass and other non-trivialproperties (see e.g. [8,9,10,11]). Thus the measurement of neutrino massescould give us some hints on the new fundamental theory, of which the SM isjust the low-energy limit, and in particular on new energy scales.It was realized by B. Pontecorvo in 1957 that if neutrinos were massive therecould exist processes where the neutrino flavour is not conserved, that we callneutrino oscillations. For small neutrino masses, these oscillations actuallytake place on macroscopic distances and can be measured if we are able todetect neutrinos from distant sources, identify their flavour and compare theresults with the theoretical predictions for the initial neutrino fluxes. Actually,after decades of experimental efforts in underground facilities, neutrinos havebeen detected from various natural or artificial sources. The former include theneutrinos produced in the nuclear reactions in the Sun, as secondary particlesfrom the interaction of cosmic rays in the atmosphere of the Earth and evena few neutrinos were detected from a supernova explosion (SN1987A). Wehave also measured the neutrino fluxes originated in artificial sources such asnuclear reactors or accelerators.5∆m 2atm∆m 2sun∆m 2sun∆m 2atm112233m mNORMAL INVERTEDFig. 1. The two neutrino schemes allowed if ∆m2atm ≫ ∆m2sun: normal hierarchy(NH) and inverted hierarchy (IH).Nowadays there exist compelling evidences for flavour neutrino oscillationsfrom a variety of experimental data on solar, atmospheric, reactor and accel-erator neutrinos. These are very important results, because the existence offlavour change implies that neutrinos mix and have non-zero masses, whichin turn requires particle physics beyond the SM. There are many excellentreviews on neutrino oscillations and their implications, to which we refer thereader for more details (see e.g. the recent ones [13,14]).We know that the number of light neutrinos sensitive to weak interactions(flavour or active neutrinos) equals three from the analysis of the invisible Z-boson width at LEP, Nν = 2.994± 0.012 [12], and the three flavour neutrinos(νe, νµ, ντ ) are linear combinations of states with definite mass νi, where i isthe number of massive neutrinos.In a three-neutrino scenario flavour and mass eigenstates are related by themixing matrix U , parametrized as [12,14]c12c13 s12c13 s13e−iδ−s12c23 − c12s23s13eiδ c12c23 − s12s23s13eiδ s23c13s12s23 − c12c23s13eiδ −c12s23 − s12c23s13eiδ c23c13 . (1)Here cij = cos θij and sij = sin θij for ij = 12, 23 or 13, and δ is a CP-violatingphase. Together with the three masses, in total there are seven flavour param-eters in the neutrino sector. This would be all if neutrinos were Dirac particles,like the charged leptons, and the total lepton number is conserved. Instead, ifneutrinos are Majorana particles (i.e. a neutrino is its own antiparticle), thematrix U is multiplied by a diagonal matrix of phases that can be taken asdiag(1, eiφ2/2, ei(φ3+2δ)/2). These two phases do not show up in neutrino oscil-lations, but appear in lepton-number-violating processes such as neutrinolessdouble beta decay, as discussed later.6 0.001 0.01 0.1 1 0.05 0.1 0.3 0.5 1mi (eV)Σmi (eV)12331, 2InvertedNormal 0.03 0.1 0.3 1 3 0.001 0.01 0.1 1Σmi (eV)lightest mν (eV)InvertedNormalFig. 2. Expected values of neutrino masses according to the values in Eq. (2). Left:individual neutrino masses as a function of the total mass for the best-fit values ofthe ∆m2. Right: ranges of total neutrino mass as a function of the lightest statewithin the 3σ regions (thick lines) and for a future determination at the 5% level(thin lines).Oscillation experiments can measure the differences of squared neutrino masses∆m221 = m22−m21 and ∆m231 = m23−m21, the relevant ones for solar and atmo-spheric neutrinos, respectively. As a reference, we take the following 3σ rangesof mixing parameters from an update of ref. [13],∆m221= (7.9+1.0−0.8)× 10−5 eV2 |∆m231| = (2.2+1.1−0.8)× 10−3 eV2s212=0.30+0.10−0.06 s223 = 0.50+0.18−0.16 s213 ≤ 0.043 (2)Unfortunately oscillation experiments are insensitive to the absolute scale ofneutrino masses, since the knowledge of ∆m221 > 0 and |∆m231| leads to the twopossible schemes shown in Fig. 1, but leaves one neutrino mass unconstrained(see e.g. the discussion in the reviews [14,15,16,17,18]). These two schemesare known as normal (NH) and inverted (IH) hierarchies, characterized bythe sign of ∆m231, positive and negative, respectively. For small values of thelightest neutrino mass m0, i.e. m1 (m3) for NH (IH), the mass states followa hierarchical scenario, while for masses much larger than the differences allneutrinos share in practice the same mass and then we say that they aredegenerate. In general, the relation between the individual masses and thetotal neutrino mass can be found numerically, as shown in Fig. 2.It is also possible that the number of massive neutrino states is larger thanthe number of flavor neutrinos. In such a case, in order to not violate theLEP results the extra neutrino states must be sterile, i.e. singlets of the SMgauge group and thus insensitive to weak interactions. At present, the resultsof the Liquid Scintillator Neutrino Detector (LSND) [19], an experiment that7has measured the appearance of electron antineutrinos in a muon antineutrinobeam, constitute an independent evidence of neutrino conversions at a largermass difference than those in Eq. 2. In such a case, a fourth sterile neutrinois required with mass of O(eV) [13]. The LSND results will be checked by theongoing MiniBoone experiment [20], whose first data are expected for 2006. Inthis review, we will mainly consider the three-neutrino scenario, but we brieflycomment on the cosmological bounds on the four-neutrino mass schemes thatinclude the LSND results in Sec. 5.6.As we discuss in the next sections, cosmology is at first order sensitive tothe total neutrino mass if all states have the same number density, provid-ing information on m0 but blind to neutrino mixing angles or possible CPviolating phases. Thus cosmological results are complementary to terrestrialexperiments such as beta decay and neutrinoless double beta decay, which arerespectively sensitive to the effective massesmβ =(∑i|Uei|2m2i)1/2= (c212c213m21 + s212c213m22 + s213m23)1/2,mββ =∣∣∣∣∣∑iU2eimi∣∣∣∣∣ = |c212c213m1 + s212c213m2 eiφ2 + s213m3 eiφ3 | . (3)The search for a signal of non-zero neutrino masses in a beta decay exper-iment is, in principle, the best strategy for measuring directly the neutrinomass, since it involves only the kinematics of electrons (see [21] for a recentreview). The current limits from tritium beta decay apply only to the range ofdegenerate neutrino masses, so that mβ ≃ m0 (if unitarity of U is assumed).The bound at 95% CL is m0 < 2.05− 2.3 eV from the Troitsk [22] and Mainz[23] experiments, respectively. This value is expected to be improved by theKATRIN project [24] to reach a discovery potential for 0.3− 0.35 eV masses(or a sensitivity of 0.2 eV at 90% CL).The neutrinoless double beta decay (Z,A)→ (Z+2, A)+2e− (in short 0ν2β)is a rare nuclear processes where lepton number is violated and whose obser-vation would mean that neutrinos are Majorana particles. If the 0ν2β processis mediated by a light neutrino, the results from neutrinoless double beta de-cay experiments are converted into an upper bound or a measurement of theeffective mass mββ in Eq. (3). A recent work [25] gives the following upperbounds at 99% CL|mββ| < (0.44− 0.62) hN eV (4)where hN is a parameter that characterizes the uncertainties on the corre-sponding nuclear matrix elements (see [26] for a recent discussion and [15]8for a review). The above range corresponds to the results of recent doublebeta decay experiments based on 76Ge, such as Heidelberg-Moscow [27] andIGEX [28], or 130Te such as Cuoricino [29] (see e.g. [30] for results from otherisotopes). In addition, there exists a claim of a positive 0ν2β signal whichwould correspond to the approximate range 0.1 < |mββ|/eV < 0.9 [31]. Fu-ture 0ν2β projects will improve the current sensitivities down to values ofthe order |mββ| ∼ 0.01 − 0.05 eV [15]. An experimental detection of 0ν2β,in combination with results from beta decay or cosmology, will also help usto discriminate between the two neutrino mass spectra and to pin down thevalues of the neutrino flavour parameters, as discussed for instance in [32,33].There are other ways to obtain information on the absolute scale of neutrinomasses, such as the measurement of the time-of-flight dispersion of a supernovaneutrino signal (at most sensitive to masses of order some eV, see [34] andreferences therein), but to conclude this section let us note that the sum ofneutrino masses is restricted to the approximate range0.056 (0.095) eV <∼∑imi <∼ 6 eV (5)where the upper limit comes exclusively from tritium beta decay results andthe lower limit reflects the minimum values of the total neutrino mass in thenormal (inverted) hierarchy.3 The cosmic neutrino background3.1 Basics on relic neutrinos, including neutrino decouplingThe existence of a relic sea of neutrinos is a generic prediction of the standardhot big bang model, in number only slightly below that of relic photons thatconstitute the CMB. Produced at large temperatures by frequent weak interac-tions, cosmic neutrinos were kept in equilibrium until these processes becameineffective in the course of the expansion. While coupled to the rest of theprimeval plasma, neutrinos had a momentum spectrum with an equilibriumFermi-Dirac form with temperature T ,feq(p) =[exp(p− µνT)+ 1]−1. (6)Here we have included a neutrino chemical potential µν that would exist in thepresence of a neutrino-antineutrino asymmetry, but it was shown in [35,36,37]that the stringent BBN bounds on µνe apply to all flavours, since neutrino9oscillations lead to approximate flavour equilibrium before BBN. The currentbounds on the common value of the neutrino degeneracy parameter ξν ≡ µν/Tare −0.05 < ξν < 0.07 at 2σ [38] (relaxed to −0.13 < ξν < 0.3 in presence ofextra relativistic degrees of freedom [39,40]). Thus the contribution of a relicneutrino asymmetry can be safely ignored.As the universe cools, the weak interaction rate Γν falls below the expansionrate given by the Hubble parameter H and one says that neutrinos decouplefrom the rest of the plasma. An estimate of the decoupling temperature canbe found by equating the thermally averaged value of the weak interactionrateΓν = 〈σν nν〉 , (7)where σν ∝ G2F is the cross section of the electron-neutrino processes with GFthe Fermi constant and nν is the neutrino number density, with the expansionrateH =√8πρ3M2P, (8)where ρ is the total energy density and MP is the Planck mass. If we ap-proximate the numerical factors to unity, with Γν ≈ G2FT 5 and H ≈ T 2/MP ,we obtain the rough estimate Tdec ≈ 1 MeV. More accurate calculations giveslightly higher values of Tdec which are flavour dependent since electron neu-trinos and antineutrinos are in closer contact with e±, as shown e.g. in [6].Although neutrino decoupling is not described by a unique Tdec, it can be ap-proximated as an instantaneous process. The standard picture of instantaneousneutrino decoupling is very simple (see e.g. [41,42]) and reasonably accurate.In this approximation, the spectrum in Eq. (6) is preserved after decoupling,since both neutrino momenta and temperature redshift identically with theuniverse expansion. In other words, the number density of non-interactingneutrinos remains constant in a comoving volume since the decoupling epoch.We have seen in Sec. 2 that active neutrinos cannot possess masses muchlarger than 1 eV, so they were ultra-relativistic at decoupling. This is thereason why the momentum distribution in Eq. (6) does not depend on theneutrino masses, even after decoupling, i.e. there is no neutrino energy in theexponential of feq(p).When calculating quantities related to relic neutrinos, one must consider thevarious possible degrees of freedom per flavour. If neutrinos are massless orMajorana particles, there are two degrees of freedom for each flavor, one for10neutrinos (one negative helicity state) and one for antineutrinos (one posi-tive helicity state). Instead, for Dirac neutrinos there are in principle twicemore degrees of freedom, corresponding to the two helicity states. However,the extra degrees of freedom should be included in the computation only ifthey are populated and brought into equilibrium before the time of neutrinodecoupling. In practice, the Dirac neutrinos with the “wrong-helicity” statesdo not interact with the plasma at temperatures of the MeV order and havea vanishingly small density with respect to the usual left-handed neutrinos(unless neutrinos have masses close to the keV range, as explained in Sec. 6.4of [6], but such a large mass is excluded for active neutrinos). Thus the relicdensity of active neutrinos does not depend on their nature, either Dirac orMajorana particles.Shortly after neutrino decoupling the photon temperature drops below theelectron mass, favouring e± annihilations that heat the photons. If one as-sumes that this entropy transfer did not affect the neutrinos because theywere already completely decoupled, it is easy to calculate the ratio betweenthe temperatures of relic photons and neutrinos Tγ/Tν = (11/4)1/3 ≃ 1.40102.Any quantity related to relic neutrinos can be calculated at later times withthe spectrum in Eq. (6) and Tν . For instance, the number density per flavouris fixed by the temperature,nν =311nγ =6ζ(3)11π2T 3γ , (9)which leads to a present value of 113 neutrinos and antineutrinos of eachflavour per cm3. Instead, the energy density for massive neutrinos should inprinciple be calculated numerically, with two well-defined analytical limits,ρν(mν ≪ Tν)= 7π2120(411)4/3T 4γ ,ρν(mν ≫ Tν)=mνnν . (10)Thus we see that the contribution of massive neutrinos to the energy densityin the non-relativistic limit is a function of the mass (or the sum of masses ifall neutrino states have mi ≫ Tν).In a more accurate analysis of neutrino decoupling, the standard picture de-scribed above is modified: the processes of neutrino decoupling and e± an-nihilations are sufficiently close in time so that some relic interactions be-tween e± and neutrinos exist. These relic processes are more efficient forlarger neutrino energies, leading to non-thermal distortions in the neutrinospectra and a slightly smaller increase of the comoving photon temperature,as noted in a series of works (see the full list given in the review [6]). Aproper calculation of the process of non-instantaneous neutrino decoupling11 1 1.01 1.02 1.03 1.04 1.05 1.06 0 2 4 6 8 10 12Frozen f ν(y) / feq(y)y = p aνeνµ,τNo oscMixed-5e-05 0 5e-05 0.0001 0.00015 0.0002 0.00025 0 2 4 6 8 10 12y2/pi2 (f ν-f eq)y = p aνeνµ,τNo oscMixedFig. 3. The upper (lower) lines are the frozen distortions of the electron (muon ortau) neutrino spectra as a function of the comoving momentum y, calculated with(solid) and without (dotted) the effect of flavour oscillations. Left: real neutrinodistribution functions normalized to the equilibrium one. Right: contribution of thedistortions to the comoving number density. Here the scale factor was normalizedso that a(t)→ 1/Tγ at large temperatures.demands solving the Boltzmann equations for the neutrino spectra, a set ofintegro-differential kinetic equations that are difficult to solve numerically. Themomentum-dependent calculations were carried out in refs. [43,44,45], whilethe inclusion of finite temperature QED corrections to the electromagneticplasma was done in [46].A recent work [47] has considered the effect of flavour neutrino oscillationson the neutrino decoupling process (see also [48]). The frozen values of theneutrino distributions are shown in Fig. 3, for the cases calculated with andwithout the effect of flavour oscillations. One can see that the distortions growwith the neutrino momentum, and their contribution to the number densityof neutrinos is maximal around y = 4. For the best-fit values of the mixingparameters in Eq. (2), Ref. [47] found an increase in the neutrino energy den-sities of 0.73% and 0.52% for νe’s and νµ,τ ’s, respectively. At the same time,after e± annihilations the comoving photon temperature is a factor 1.3978larger, instead of 1.40102 in the approximation of instantaneous decoupling.Note, however, that for any cosmological epoch when neutrino masses can berelevant one should consider the spectra of the neutrino mass eigenstates ν1,2,3instead of flavour eigenstates νe,µ,τ . In principle, for numerical calculations ofthe cosmological perturbation evolution such as those done by the Boltzmanncodes cmbfast [49] or camb [50], one should include the full distributionfunction of each mass eigenstate, including the distortions shown in Fig. 3.It is only when the neutrinos are relativistic or when one is interested in anobservable that does not depend on neutrino masses (such as the number den-sity) that the effect of the distortion can be simply integrated over momentum.12For instance, the contribution of relativistic relic neutrinos to the total energydensity is taken into account just by using Neff = 3.046 as defined later inEq. (11). In practice, the distortions calculated in Ref. [47] only have smallconsequences on the evolution of cosmological perturbations, and for manypurposes they can be safely neglected.3.2 Extra radiation and the effective number of neutrinosNeutrinos fix the expansion rate during the cosmological era when the Universeis dominated by radiation. Their contribution to the total radiation contentcan be parametrized in terms of the effective number of neutrinos Neff [51,52],through the relationρR =[1 +78(411)4/3Neff]ργ , (11)where ργ is the energy density of photons, whose value today is known from themeasurement of the CMB temperature. This equation is valid when neutrinodecoupling is complete and holds as long as all neutrinos are relativistic.In Sec. 2 we saw that from accelerator data the number of active neutrinosis three, while in the previous subsection we learned from the analysis ofneutrino decoupling these three active neutrinos contribute as Neff = 3.046.Any departure of Neff from this last value would be due to non-standardneutrino features or to the contribution of other relativistic relics.A detailed discussion of cosmological scenarios where Neff is not fixed to threecan be found in the reviews [6,53], while the particular case of active-sterileneutrino mixing was recently analyzed in [54]. Since in the present work wefocus on the standard case of three active neutrinos, here we only give a briefreview of the most recent bounds on Neff from cosmological data.The value of Neff is constrained at the BBN epoch from the comparison oftheoretical predictions and experimental data on the primordial abundancesof light elements, which also depend on the baryon-to-photon ratio ηb = nb/nγ(or baryon density). The main effect of Neff is to fix the Hubble expansion ratethrough its contribution to the total energy density. This in turn changes thefreezing temperature of the neutron-to-proton ratio, therefore producing adifferent abundance of 4He.The BBN bounds on Neff have been recently reanalyzed taking in input thevalue of the baryon density derived from the WMAP first year data [55]ηCMB = 6.14 ± 0.25. In Fig. 4 we show the results from [40], where the al-130.016 0.018 0.02 0.022 0.024 0.026 0.028 0.03ωb12345678Nffe!!Fig. 4. Bounds on Neff from BBN including D and4He (contours at 68% and95% CL) and from a combined analysis of BBN (D only) and CMB data. Hereωb = Ωbh2 ≃ 1010(ηb/274). This Figure is taken from Ref. [40].lowed range Neff = 2.5+1.1−0.9 (95% CL) was inferred from data on light elementabundances (see also [56,57,58] and [59] for a recent review on BBN). Thisrange is perfectly compatible with the standard prediction of 3.046. However,the reader should be cautious in the interpretation of the BBN allowed rangefor Neff such as that in Fig. 4. It is well-known that the main problem whenderiving the primordial abundances from observations in astrophysical sourcesis the existence of systematics not accounted for, in particular for 4He (a dis-cussion of the observational situation can be found in Ref. [60]). For instance,the detailed BBN analysis in [61] gives the allowed range 1.90 < Neff < 3.77for the 4He “box” range in [60].Independent bounds on the radiation content of the universe at a later epochcan be extracted from the analysis of the power spectrum of CMB anisotropies.We will describe later the neutrino effects on the CMB spectrum. Assum-ing a minimal set of cosmological parameters and a flat universe, Ref. [62]found the value Neff = 3.5+3.3−2.1 (95% CL) (see also [56,57,63]) from the com-bination of WMAP first year data with other CMB and LSS data. More re-cently, it has been shown that the addition of Supernovae and new CMB dataleads to a reduction of the allowed range: Neff = 4.2+1.2−1.7 [64] (with the latestBOOMERANG data) and Neff = 3.3+0.9−4.4 [65] (with WMAP three year data).143.3 Massive neutrinos as dark matterA priori, massive neutrinos are excellent candidates for contributing to thedark matter density, in particular because we are certain that they exist, incontrast with other candidate particles. Together with CMB photons, relicneutrinos can be found anywhere in the Universe with a number density of339 neutrinos and antineutrinos per cm3. Depending on the value of the mass,this density is enhanced when neutrinos cluster into gravitational potentialwells, although recent analyses show that the overdensity is limited to smallfactors (see e.g. [66]).In addition, it is easy to have a neutrino contribution of order unity to thepresent value of the energy density of the Universe, just by considering eVneutrino masses. In such a case, we saw before that all neutrinos should ap-proximately share the same mass m0 and their energy density in units of thecritical value of the energy density (see Eq. (21)) is 1Ων =ρνρc=∑imi93.14 h2 eV, (12)where h is the present value of the Hubble parameter in units of 100 km s−1Mpc−1 and∑imi = 3m0. Even if the three neutrinos are non-degenerate inmass, Eq. (12) can be safely applied. Indeed, we know from neutrino oscillationdata that at least two of the neutrino states are non-relativistic today, sinceboth (∆m231)1/2 ≃ 0.047 eV and (∆m221)1/2 ≃ 0.009 eV are larger than thetemperature Tν ≃ 1.96 K ≃ 1.7× 10−4 eV. If the third neutrino state is verylight and still relativistic, its relative contribution to Ων is negligible and Eq.(12) remains an excellent approximation of the total density.If we demand that neutrinos should not be heavy enough to overclose theUniverse (Ων < 1), we obtain an upper bound m0 <∼ 15 eV for the absoluteneutrino mass scale m0 =∑imi/3 (fixing h = 0.7). This argument was usedmany years ago by Gershtein and Zeldovich [67] (see also [68]). Since frompresent analysis of cosmological data we know that the approximate contri-bution of matter is Ωm ≃ 0.3, the neutrino masses should obey the strongerbound m0 <∼ 5 eV.Dark matter particles with a large velocity dispersion such as that of neutrinosare called hot dark matter (HDM). The role of neutrinos as HDM particleshas been widely discussed since the 1970s, and the reader can find a historicalreview in Ref. [69]. It was realized in the mid-1980s (see e.g. [70,71,72]) that1 For high precision, the relation between (m1,m2,m3) and Ωνh2 must be evaluatednumerically using the distorted distributions described in [47].15HDM affects the evolution of cosmological perturbations in a particular way:it erases the density contrasts on wavelengths smaller than a mass-dependentfree-streaming scale (we will discuss in more detail the effects of massive neu-trinos on the evolution of cosmological perturbations in Secs. 4.5 and 4.6). In auniverse dominated by HDM, this suppression is in contradiction with variousobservations. For instance, large objects such as superclusters of galaxies formfirst, while smaller structures like clusters and galaxies form via a fragmen-tation process. This top-down scenario is at odds with the fact that galaxiesseem older than clusters.Given the failure of HDM-dominated scenarios, the attention then turned tocold dark matter (CDM) candidates, i.e. particles which were non-relativisticat the epoch when the universe became matter-dominated, which provided abetter agreement with observations. Still in the mid-1990s it appeared thata small mixture of HDM in a universe dominated by CDM fitted better theobservational data on density fluctuations at small scales than a pure CDMmodel (see e.g. [73]). However, within the presently favoured ΛCDM modeldominated at late times by a cosmological constant (or some form of darkenergy) there is no need for a significant contribution of HDM. Instead, one canuse the available cosmological data to find how large the neutrino contributioncan be. In Sec. 4, we will explain the effect of neutrino masses on cosmologicalobservables. In Sec. 5, we will review the upper bounds on the sum of allneutrino masses which have been derived from current data.4 Massive neutrinos and cosmological perturbationsLet us start with some generalities on the theory of cosmological perturba-tions. This field has been thoroughly investigated over the past thirty years,and many excellent reviews have been written on the subject (see for instanceRefs. [42,74,75,76,77]). Many equations in this section are reminiscent of thosein Ma & Bertschinger [74] and Bertschinger [77] – however, with several signdifferences, since we choose the metric signature to be (+,−,−,−). This sig-nature tends to be the most popular nowadays in cosmology.In our Universe, the metric and the energy-momentum tensor are inhomoge-neous. Their perturbations, given byδgµν(x, t)= gµν(x, t)− g¯µν(t) , (13)δTµν(x, t)=Tµν(x, t)− T¯µν(t) , (14)are known to be small in the early Universe, typically 105 times smaller thanthe background quantities, as shown by CMB anisotropies. As we shall see in16this section, after photon decoupling, the matter perturbations grow by grav-itational collapse and reach the non-linear regime, starting with the smallestscales. However, the linear perturbation theory is a good tool both for describ-ing the early Universe at any scales, and the recent universe on the largestscales. The most reliable observations in cosmology are those involving mainlylinear (or quasi-linear) perturbations. In particular, current cosmological neu-trino mass bounds are based on such observations. Therefore, our goal in thissection is to describe the evolution of linear cosmological perturbations. Thegreat advantage of linear theory is, as usual, to obtain independent equationsof evolution for each Fourier mode. Note that the Fourier decomposition mustbe performed with respect to the comoving coordinate system: so, the quantity(2π/k) is the comoving wavelength of a perturbation of wavevector k, whilethe physical wavelength is given byλ = a(t)2πk, (15)where a(t) is the scale factor of the Universe. For each mode k, the perturba-tion amplitudes evolve under some equations of motion (which depend only onthe modulus k, since the background is isotropic), and on top of this evolution,the physical wavelength is stretched according to the Universe expansion.For concision, we will restrict ourselves to the main-stream standard cosmo-logical model: the inflationary ΛCDM scenario, whose background evolutionis reviewed in Sec. 4.2. In Sec. 5 we will briefly comment on some resultsbased on more exotic scenarios. However, the theory of cosmological pertur-bations is so rich that in order to present a short and pedagogical summary, itis necessary to start from strong assumptions concerning the underlying cos-mological background. Therefore, we will assume that at early times (e.g. atBBN) the Universe contains a stochastic background of Gaussian, adiabatic 2and nearly scale-invariant primordial perturbations, as predicted by inflation,and as strongly suggested by CMB observations. Since the perturbations arestochastic and Gaussian, what we will call “amplitude” should often be un-derstood as “variance”: the linear equations of evolution give the evolution ofeach individual mode δfk of a perturbation δf , and also, more interestingly,that of the root mean square 〈|δf |2〉1/2k , obtained by averaging over all modesof fixed wavenumber k. It is this last quantity which is the target of mostobservations, and which carries information on the cosmological parameters.The perturbations defined in Eqs. (13) contain many degrees of freedom: the2 If we relax the assumption of adiabaticity, i.e. if we allow for isocurvature modesin the early Universe, the neutrino background can play a very specific role andprovides some non-trivial initial conditions for the evolution of cosmological per-turbations. In this review, we will not cover this possibility and refer the interestedreader to Refs. [78,79,80,81,82,83,84].17homogeneity and isotropy of the background implies that g¯µν and T¯µν arediagonal, but in general this is not true at the level of perturbations. However,we will see in Sec. 4.3 that some of these degrees of freedom are just artifactsof the relativistic perturbation theory set-up; moreover, only a fraction ofthe physical degrees of freedom contribute significantly to the CMB and LSSobservables. So, we will see at the end of Sec. 4.3 that the problem can bereduced to the integration of a small number of linear equations of evolution.In order to perform this integration, one must specify the properties of eachfluid contributing to the energy density. Apart from the cosmological constant,the standard ΛCDM scenario includes contributions from photons, neutrinos,baryons and cold dark matter. In Sec. 4.4, we will review the behavior of themost relevant perturbations under the assumption that neutrinos are absent,in order to better identify the specific role of neutrinos in Sec. 4.5. In Sec. 4.6,we will summarize the effects of neutrino masses on cosmological observables,which are important for understanding the present bounds. Thus, the readeralready familiar with cosmological perturbation theory in ΛCDM and ΛMDM(Λ Mixed Dark Matter) models can go directly to the last subsection.4.1 Observables targets: definition of the power spectraOur goal is to compute observable quantities like• the CMB temperature anisotropy power spectrum, defined as the angulartwo-point correlation function of CMB maps δT/T¯ (nˆ) (nˆ being a directionin the sky). This function is usually expanded in Legendre multipoles〈δTT¯(nˆ)δTT¯(nˆ′)〉=∞∑l=0(2l + 1)4πClPl(nˆ · nˆ′) , (16)where Pl(x) are the Legendre polynomials. So, for Gaussian fluctuations, allthe information is encoded in the multipoles Cl which probe correlations onangular scales θ = π/l. Since for each neutrino family the mass is alreadyknown to be at most of the order of 1 eV (see Sec. 5), the transition tothe non-relativistic regime is expected to take place after the time of re-combination between electrons and nucleons, i.e. after photon decoupling.The shape of the CMB spectrum is related mainly to the physical evolutionbefore recombination. Therefore, the CMB will only be marginally affectedby the neutrino mass (we will see later that there is only an indirect ef-fect through the modified background evolution). For this reason, we willonly mention in this review some essential aspects of the CMB temperaturespectrum, without entering into details. For the effects on CMB of heavierneutrinos (with masses of a few eV), we refer the reader to Ref. [85].18• the CMB polarization anisotropy power spectra bring interesting comple-mentary information to the temperature one, but for the same reasons asabove, plus the fact that current polarization measurements are still ratherimprecise, we will not discuss polarization in this section.• the matter power spectrum, observed with various techniques described inthe next section (directly or indirectly, today or in the near past), probesthe current Large Scale Structure of the Universe. It is defined as the two-point correlation function of non-relativistic matter fluctuations in FourierspaceP (k, z) = 〈|δm(k, z)|2〉 , (17)where δm = δρm/ρ¯m. In the case of several fluids (e.g. CDM, baryons andnon-relativistic neutrinos), the total matter perturbation can be expandedasδm =∑i ρ¯i δi∑i ρ¯i. (18)Since the energy density is related to the mass density of non-relativisticmatter through E = mc2, δm represents indifferently the energy or masspower spectrum. When the redshift z is not explicitly written, we assumethat P (k) refers to the matter power spectrum evaluated today (at z = 0).The shape of the matter power spectrum is the key observable for constrain-ing small neutrino masses with cosmological methods. Therefore the mainpurpose of this section is to review the physics responsible for its shape,in a pedagogical but sufficiently detailed way, starting with linear structureformation in absence of neutrinos (Sec. 4.4), and then including the impactof free-streaming massive neutrinos (Secs. 4.5, 4.6).• finally, one can build some cross-correlation power spectra (which measurethe angular or spatial correlation between two different types of observ-ables). These correlations are in principle very interesting to extract fromobservations, since they provide some additional information beyond thatcontained in the self-correlation power spectra Cl or P (k). For instance, thecross-correlation between CMB temperature and polarization maps is non-zero, since the two signals are affected by the same physical inhomogeneitiesclose to the last-scattering surface. The CMB temperature maps are alsocorrelated to some extent to the matter distribution at low redshift, due tothe late integrated Sachs-Wolfe effect (that we will briefly introduce in Sec.4.4.7). Finally, different measurements of the matter distribution in differ-ent overlapping redshift ranges or with different techniques are likely to becorrelated with each other. The measurement of various cross-correlationpower spectra is expected to be crucial in the future, but since nowadays itonly plays a marginal role in neutrino mass determinations [86], we will not19introduce the corresponding formalism in this review.4.2 Background evolutionIn most of this section, we will use the conformal time τ instead of theproper time t. The two are related through dτ = dt/a(t). A dot will de-note a derivative with respect to τ . For instance, the Hubble parameter readsH = d(ln a)/dt = a˙/a2. In absence of spatial curvature, the background Fried-mann metric reduces tods2 = a(τ)2[dτ 2 − δijdxidxj ] , (19)and the Friedmann equation readsH2 =8πG3(ργ + ρcdm + ρb + ρν + ρΛ) , (20)where the homogeneous density of photons ργ scales like a−4, that of non-relativistic matter (ρcdm for CDM and ρb for baryons) like a−3, that of neu-trinos ρν interpolates between the two behaviors at the time of the non-relativistic transition, and finally the cosmological constant density ρΛ is ofcourse time-independent. At any time, the critical density ρc is defined asρc = 3H2/8πG, and the current value H0 (or reduced value h) of the Hubbleparameter gives the critical density todayρ0c = 1.8788× 10−29h2 g cm−3 . (21)The normalization of the scale factor a is arbitrary. If we first neglect neu-trino masses, the photon and neutrino densities can be combined into a totalradiation density ρr = ργ + ρν scaling like a−4, while the CDM and baryonscombine into a matter density ρm = ρcdm + ρb scaling like a−3. Then, we canchoose the normalization of a such that ρm = 3/(8πGa3) in order to obtaina particularly simple Friedmann equation valid during radiation domination(RD) and matter domination (MD)(a˙)2 = a + aeq , (22)where aeq is the scale factor at equality, when ρm = ρr. There is a simplesolution,a =τ 24+√aeq τ =τ4(τ + 2(√2 + 1)τeq), (23)20106103 110-3 110-310-610-910-31103106ρ i1/4 (eV)a/a0Tν (eV)BBNγ dec.cdmbγΛν3ν2ν1 1e-04 0.001 0.01 0.1 1 110-310-610-91.95103106109Ωia/a0Tν (K)Fig. 5. Evolution of the background densities from the time when Tν = 1 MeV (soonafter neutrino decoupling) until now, for each component of a flat ΛMDM modelwith h = 0.7 and current density fractions ΩΛ = 0.70, Ωb = 0.05, Ων = 0.0013 andΩcdm = 1 − ΩΛ − Ωb − Ων . The three neutrino masses are distributed accordingto the Normal Hierarchy scheme (see Sec. 2) with m1 = 0, m2 = 0.009 eV andm3 = 0.05 eV. On the left plot we show the densities to the power 1/4 (in eV units)as a function of the scale factor. On the right plot, we display the evolution of thedensity fractions (i.e., the densities in units of the critical density). We also show onthe top axis the neutrino temperature (on the left in eV, and on the right in Kelvinunits). The density of the neutrino mass states ν2 and ν3 is clearly enhanced oncethey become non-relativistic. On the left plot, we also display the characteristictimes for the end of BBN and for photon decoupling or recombination.where τeq = 2(√2− 1)√aeq is the value of the conformal time at equality. Atlow redshift (typically z < 0.5), the cosmological constant density takes over,causing a departure from the above solution, with an acceleration of the scalefactor. Finally, if we include the effect of small neutrino masses, the solutionis also slightly modified, since the non-relativistic transition of each neutrinospecies amounts in converting a fraction of radiation into matter. This canbe seen in Fig. 5, where we plot the evolution of background densities for aΛMDMmodel in which the three neutrino masses follow the Normal Hierarchyscheme (see Sec. 2) with m1 = 0, m2 = 0.009 eV and m3 = 0.05 eV.4.3 Gauge transformations and Einstein equationsIn the real Universe all physical quantities (densities, curvature...) are func-tions of time and space. Thanks to the covariance of general relativity, theycan be described in principle in any coordinate system, without changing thephysical predictions. The problem is that in order to obtain simple equa-tions of evolution, we wish to use a linear perturbation theory, in which thetrue physical quantities are artificially decomposed into a homogeneous back-ground and some small perturbations. This is artificial because the homoge-neous quantities are defined as spatial averages over hypersurfaces of simul-taneity: f¯(t) = 〈f(t,x)〉x. Any change of coordinate system which(1) mixes time and space (therefore, redefining hypersurfaces of simultaneity,and changing the way to perform spatial averages), and(2) remains small everywhere, so that the differences between true quantitiesand spatial averages are still small perturbations,gives a new set of perturbations (new equations of evolution, new initial con-ditions), although the physical quantities (i.e., the total ones) are the same.This ambiguity is called the gauge freedom in the context of relativistic per-turbation theory.Of course, using a linear perturbation theory is only possible when there existsat least one system of coordinates in which the Universe looks approximatelyhomogeneous. We know that this is the case at least until the time of photondecoupling: in some reference frames, the CMB anisotropies do appear as smallperturbations. It is a necessary condition for using linear theory to be in sucha frame; however, this condition is vague and leaves a lot of gauge freedom, i.e.many possible ways to slice the spacetime into hypersurfaces of simultaneity.We can also notice that the definition of hypersurfaces of simultaneity is notambiguous at small distances, as long as different observers can exchange lightsignals in order to synchronize their clocks. Intuitively, we see that the gaugefreedom is an infrared problem, since on very large distances (larger than theHubble distance) the word “simultaneous” does not have a clear meaning. Thefact that the gauge ambiguity is only present on large scales emerges naturallyfrom the mathematical framework describing gauge transformations.Formally, a gauge transformation is described by a quadrivector field ǫµ(x, t)(see e.g. Ref. [74]). When the latter is infinitesimal, the Lorentz scalars, vectorsand tensors describing the perturbations are shifted by the Lie derivative alongǫ,δAµν...(x, t)→ δAµν...(x, t) + Lǫ[δAµν...(x, t)] . (24)Since there are four degrees of freedom (d.o.f.) in this transformation - the fourcomponents of ǫµ - we see that among the ten d.o.f. of the perturbed Einsteinequation δGµν = 8πG δTµν , four represent gauge modes, and six representphysical degrees of freedom.22In addition, it can be shown that this equation contains three decoupled sec-tors. In other words, when the metric and the energy-momentum tensor areparametrized in an adequate way, the ten equations can be decomposed intothree systems independent of each other:(1) four equations relate four scalars in the perturbed metric δgµν to fourscalars in δTµν ,(2) four equations relate two transverse 3-vectors in the perturbed metric (4d.o.f. in total) to two transverse 3-vectors in δTµν ,(3) two equations relate one transverse traceless 3×3-tensor in the perturbedmetric (2 d.o.f. in total) to a similar tensor in δTµν .Moreover, this decomposition is left invariant by gauge transformations. Thesethree types of variables are called scalar, vector and tensor modes; physically,they describe respectively the generalization of Newtonian gravity, gravito-magnetism, and gravitational waves. It can be shown that two scalar d.o.f.and two vector d.o.f. are gauge modes, so each of the three sectors containsonly two physical d.o.f. In principle, all modes can contribute to the CMBanisotropy maps, but CMB observations show that the vector and tensor con-tributions are negligible (at least for temperature anisotropies). As for theLSS of the Universe, it is related to the mass/energy density distribution ofnon-relativistic matter, i.e., again to the scalar sector. Therefore, in this re-port, we will focus only on scalar perturbations, not because the cosmologicalbackgrounds of vector/tensor modes are insensitive to neutrino properties, butbecause they are not likely to be observed with great precision in a near future.It is possible to build some gauge-invariant combinations, and to reduce theEinstein equation into a set of gauge-invariant equations. This is not the mosteconomic way to proceed: one can simply choose an arbitrary gauge-fixing con-dition, i.e., a prescription that will limit the number of effective degrees of free-dom to that of physical modes only, and make all calculations into that gauge.When the same problem is studied in two different gauges, the solutions canlook very different on large wavelength; for instance, the total energy densityperturbation δρ(t, k) for a given time and wavenumber can appear as grow-ing in one gauge, and constant in another gauge, although the two solutionsdescribe the same Universe. However, physical observables - like the matterdensity perturbations probed by galaxy surveys or temperature/polarizationanisotropies probed by CMB experiments - are always limited to small scales,at most of the order of the Hubble length. On those scales, the predictionsarising from different gauge choices always coincide with each other.Throughout our discussion, we choose to work in the longitudinal gauge, whichis probably the most popular for studying cosmological perturbations. In this23gauge, one requires that the non-diagonal metric perturbations vanish. Thiseliminates two scalar degrees of freedom, and the remaining ones are definedin such a way that Eq. (19) readsds2 = gµνdxµdxν = a2(τ)[(1 + 2φ)dτ 2 − (1− 2ψ)δijdxidxj ] . (25)For the energy-momentum tensor, the four scalar degrees of freedom can beidentified asδT 00 = δρ , (26)δT 0i =(ρ¯+ p¯)v||i , (27)δT ij =−δp δij + Σi||j , (28)where δρ is the energy density perturbation, δp the pressure perturbation, v||i =∂iv˜ the longitudinal component of the velocity field, and Σi||j = (∂i∂j− 13δij∇2)σ˜the traceless and longitudinal-divergence component of the 3×3 tensor δT ij .For these last two degrees of freedom, we could write the equations in termsof the two potentials v˜ and σ˜. However, it is more conventional to deal withthe two quantities (θ, σ) which represent respectively the velocity divergenceand the shear stress (or anisotropic stress)θ≡∑i∂ivi = ∇2v˜ , (29)(ρ¯+ p¯)∇2σ≡−∑i,j(∂i∂j − 13∇2δij)Σi||j = −23∇4σ˜ . (30)Let us recall that the four scalar components of the energy-momentum tensorare not gauge-invariant: we define the above quantities in the longitudinalgauge, but in another gauge, for instance, δT 00 could behave differently thanthe above δρ, especially on super-Hubble scales.We can now write the perturbed Einstein equation for the scalar sector in thelongitudinal gauge,δG00=2a−2{−3(a˙a)2φ− 3 a˙aψ˙ +∇2ψ}= 8πG δρ , (31)δG0i =2a−2∂i{a˙aφ+ ψ˙}= 8πG (ρ¯+ p¯) vi , (32)δGij =−2a−2{[(2a¨a−(a˙a)2)φ+a˙a(φ˙+ 2ψ˙) + ψ¨ +13∇2(φ− ψ)]δij−12(∂i∂j − 13∇2δij)(φ− ψ)}(33)24=8πG (−δp δij + Σij) . (34)Using the variables δ ≡ δρ/ρ¯, δp, θ, σ, and switching to comoving Fourierspace, we obtain− 3(a˙a)2φ− 3 a˙aψ˙ − k2ψ=4πG a2 ρ¯ δ , (35)−k2(a˙aφ+ ψ˙)=4πG a2 (ρ¯+ p¯) θ , (36)(2a¨a−(a˙a)2)φ+a˙a(φ˙+ 2ψ˙) + ψ¨ − k23(φ− ψ)= 4πG a2 δp , (37)k2(φ− ψ)= 12πG a2 (ρ¯+ p¯) σ .(38)Through the Bianchi identities, the Einstein equation implies the conservationof the total energy-momentum tensor. Actually, the energy-momentum tensorof each uncoupled fluid is conserved, and obeys the continuity equationδ˙ = (1 + w)(θ + 3ψ˙) , (39)and the Euler equationθ˙ =a˙a(3w − 1)θ − w˙1 + wθ − k2φ− k2σ − w1 + wk2δ , (40)where we assumed an equation of state p(x, t) = wρ(x, t) for the fluid, so thatw = p¯/ρ¯ = δp/δρ.4.4 Linear perturbation theory in a neutrinoless Universe (pure ΛCDM)4.4.1 Perfect fluidsIntuitively, in perfect fluids, microscopic interactions enforce a “collective be-havior” of the particles. More precisely, they guarantee that the stress tensorTij is isotropic and diagonal, and that the energy-momentum tensor can besimply described in terms of functions of time and space: the density ρ, thepressure p and the bulk velocity Uµ,T µν = −p gµν + (ρ+ p)UµUν . (41)Here, Uµ = dxµ/√ds2 is the 4-velocity. As long as the bulk 3-velocity vi =dxi/dτ remains first-order in perturbations, we can approximate the 4-velocity25by Uµ = dxµ/[a(1 + φ)dτ ] and obtain:Uµ =(a−1[1− φ], a−1vi), T 00 = ρ , Ti0 = vi , T ii = −p , (42)where T ii is not summed over the repeated index. In a pure ΛCDM model, theUniverse contains photons, baryons and cold dark matter. Strictly speaking,the perfect fluid approximation does not apply to this situation, since on theone hand photons become collisionless after decoupling, and on the other handcold dark matter is collisionless at any time (so it cannot even be called a fluid).However, we will see in the next paragraphs that we can use Eqs. (41, 42) inan effective way.First, after its own decoupling time, the cold dark matter component is alwaysnon-relativistic and collisionless. As long as the bulk motion of CDM particlescan be described by a single-valued flow, the CDM energy-momentum tensor isformally identical to that of a perfect fluid with density ρm and zero pressure.Second, let us discuss the case of photons and baryons. If our purpose wasto review the CMB physics, it would be crucial to introduce separately theexact form of the photon and baryon energy-momentum tensors. However,in this report, we only wish to provide a simplified description of large scalestructure formation. For this purpose, we can view the baryon and photoncomponents before recombination as forming a single tightly-coupled fluid,obeying to Eq. (41) with the equation of state of ordinary radiation: pr =13ρr(this assumes that the baryon density is negligible with respect to the photondensity, which is only a crude approximation around the time of equality andrecombination). The local value of ρr is related to the local value of the photonblackbody temperature. The bulk velocity of this fluid remains small enoughto use Eqs. (42). After recombination, the photons are irrelevant for structureformation, and we will simply discard this component; the baryons behave likea non-relativistic collisionless medium, but for simplicity we will neglect theirdensity with respect to that of cold dark matter.Finally, since we are only interested in scalar perturbations, we can reduce thebulk velocities of radiation and CDM into irrotational vector fields, and definetwo scalars θr and θm as in Eq. (29).In summary, we can solve the Einstein equations with the following perturbedenergy-momentum tensorδT 00 = δρr + δρm , (43)∂i(δT 0i )= (ρ¯r + p¯r)θr + ρ¯mθm =43ρ¯rθr + ρ¯mθm , (44)26δT ii =−δpr = −13δρr . (45)4.4.2 Jeans lengthIn a spatially flat Friedmann Universe, a physical process switched on at atime ti and propagating at a velocity v along radial geodesics (v dt = a(t) dx)can only affect wavelengths smaller than a causal horizon, defined asd(ti, t) = a(t)t∫tidx = a(t)t∫tiv dt′a(t′). (46)This horizon is simply the maximal physical distance on which the signal canpropagate between ti and t (sometimes, is it defined with a factor two biggerthan above). The so-called particle horizon dH obeys this definition in theparticular case of a signal traveling at the speed of light, v = 1 (we adoptedunits in which c = 1). If both ti and t are chosen during the matter or radiationdominated stage, and if t≫ ti, it is straightforward to show that the particlehorizon can be approximated by the Hubble length RH(t) = 1/H(t), up toa numerical factor of order one. Indeed, for a power-law expansion a(t) ∝ tnwith n < 1, one hasRH(t) =tn, dH(t≫ ti) ≃ t1− n . (47)Similarly, the characteristic velocity cs under which acoustic perturbationspropagate before photon decoupling defines a characteristic length ds(ti, t),called the sound horizon. For a constant sound speed, and under the conditionsdescribed above for ti and t, the sound horizon is obviously given (up to anumerical factor of order one) by the ratio cs/H(t), which is called the Jeanslength. The exact definition of the Jeans wavenumber and Jeans length isusuallykJ(t) =(4πGρ¯(t)a2(t)c2s(t))1/2, λJ(t) = 2πa(t)kJ(t)= 2π√23cs(t)H(t), (48)where the numerical factors –which are unimportant for the purpose of un-derstanding the physics– are simply dictated by the particular form of theNewtonian equation of evolution of the density contrast of a single uncoupledperfect fluid with constant sound speed,δ¨ +a˙aδ˙ + (k2 − k2J) c2s δ = 0 , (49)27which derives from the combination of the continuity, Euler and Poisson equa-tions.Our physical expectation is that for a fluid of sound speed cs, modes withk > kJ will oscillate with a pulsation ω = kcs, due to the competition betweenthe gas pressure and the gravitational compression. These modes are saidto be Jeans stable. On the other hand, for modes k < kJ , pressure cannotcausally resist to gravitational compression, and density perturbations shouldin principle grow monotonically. This Jeans instability explains some basicphenomena in the inhomogeneous Universe: before recombination, the photon-baryon fluid has a sound speed of order cs ∼ c/√3 and oscillates on scalessmaller than λJ ; after recombination, cs and λJ become vanishingly small, kJgrows to infinity and structures form.4.4.3 Radiation dominationDeep inside the radiation era, we have seen that the radiation component canbe approximated by a perfect fluid of sound speed cs, due to the tight couplingbetween baryon and photons. This self-gravitating fluid oscillates inside theJeans length (or sound horizon). The cold dark matter evolves as a test fluidin this background, until the time (close to equality) at which its gravitationalback-reaction becomes important. However, for simplicity, we will work at firstorder in the expansion parameter ρ¯m/ρ¯r.In this limit, it is possible to derive an analytic expression for the evolutionof δr and δm. The absence of anisotropic stress implies that the two metricfluctuations are equal, φ = ψ. The full continuity, Euler and Einstein equationsfor the radiation and matter fluid readδ˙m = θm + 3φ˙ , δ˙r =43θr + 4φ˙ , (50)θ˙m = − a˙aθm − k2φ , θ˙r = −k24δr − k2φ , (51)−3 a˙aφ˙−(3(a˙a)2+ k2)φ = 4πGa2 (ρ¯mδm + ρ¯rδr) , (52)−k2(φ˙+a˙aφ)= 4πGa2(ρ¯mθm +43ρ¯rθr), (53)φ¨+ 3a˙aφ˙+(2a¨a−(a˙a)2)φ = 4πGa2(13ρ¯rδr), (54)and only five out of these seven equations are independent. In addition, oneout of the five independent equations is just a constraint equation, so thesystem admits four independent solutions. In order to find them, let us firstnotice that since we have two fluids, we can have differences in the number28density contrasts, usually parametrized by the entropy perturbationη =34δr − δm . (55)The pressure perturbation δp = δpr can be uniquely decomposed in the basisof the total density perturbation δρ = δρr + δρm and entropy perturbation η,δp =[3(1 +3ρ¯m4ρ¯r)]−1(δρ+ ρ¯m η) . (56)This defines the effective sound speed in the two-fluid asc2eff =[3(1 +3ρ¯m4ρ¯r)]−1=˙¯p˙¯ρ. (57)We can reduce the system (50)-(54) to a pair of second-order coupled equa-tions, obtained from the combinations [(54)−c2eff(52)] and [k2(52)−3a˙/a(53)],as followsφ¨+ 3a˙a(1 + c2eff)φ˙+1a[3(1 +ρ¯rρ¯m)c2eff −ρ¯rρ¯m]φ+ k2c2effφ=32ac2effη , (58)13c2effη¨ +a˙aη˙ +k24ρ¯mρ¯rη=k46aρ¯mρ¯rφ .(59)Once the four independent solutions have been found, the density contrastscan be obtained e.g. from combining Eqs. (54) and (58)δr=3c2eff4πGa2ρ¯r(32aη − 3 a˙aφ˙−(3(a˙a)2+ k2)φ), (60)δm=9c2eff16πGa2ρ¯r(32aη − 3 a˙aφ˙−(3(a˙a)2+ k2)φ)− η , (61)and the velocity gradients follow from Eqs. (51). The evolution of the scalefactor can be taken from equation (23). Deep inside the radiation era, wecan write a, c2eff and the two equations [τ2 × (58)] and (59) at first order inρ¯m/ρ¯r = τ/τeq,τ 2φ¨+ 4τφ˙+ (τ 2ω2)φ = 0 , η¨ +1τη˙ =3ω4τ 22φ , (62)where ω = kceff = k/√3. In this limit, the first equation decouples: this justconfirms that the radiation fluid is self-gravitating, and the CDM behaves as29a test fluid. Two independent solutions of the system, both satisfying η(τ →0) = 0, readφ = (ωτ)−2{C(k)(sinωτωτ− cosωτ)+D(k)(− sinωτ − cosωτωτ)},η = −3C(k) ωτ∫0cosx− 1xdx− 12(cosωτ − 1)+D(k) ωτ∫0sin xxdx+12sinωτ . (63)The solution proportional to C(k) represents the growing adiabatic (or isen-tropic) mode, the other one is the decaying adiabatic mode. There are twoother obvious solutions (φ, η) = (0, A(k)) and (φ, η) = (0, B(k) ln τ) whichstand respectively for the isocurvature growing and decaying mode. For thegrowing adiabatic mode, we get from Eqs. (60) and (61) that the radiationand matter density contrasts evolve likeδr=−2C(k){2 sinωτ − ωτ cosωτωτ− 2sinωτ − ωτ cosωτ(ωτ)3},δm=3C(k)−sinωτωτ +sinωτ − ωτ cosωτ(ωτ)3+12+ωτ∫0cos x− 1xdx . (64)The integral is given byωτ∫0cosx− 1xdx = Ci(ωτ)− C − ln(ωτ) , (65)where Ci(x) is the cosine integral and C ≃ 0.5772 is the Euler constant. Forx > 1, the function Ci(x) describes damped oscillations around zero. Letus summarize the physics described by these solutions. During the radiationera and far outside the sound horizon, all quantities appear as frozen in thelongitudinal gauge, withδm =34δr = −32φ = −12C(k) . [RD, ωτ = kτ/√3≪ 1] (66)Then, inside the sound horizon, the evolution of the radiation and matterperturbations are radically different from each other. The self-gravitating ra-diation component oscillates on scales smaller than its Jeans length, and Eq.(64) represents a first-order approximation to the acoustic oscillations ob-served in the CMB anisotropy spectrum. Instead, the cold dark matter has30a considerably smaller Jeans length and does not fluctuate on its own. Thegravitational driving force only induces damped oscillations on δm, on top of anet gravitational clustering effect. This clustering is not very efficient, becausethe gravitational potential fluctuations are damped by the acoustic oscillationsand do not feel significant back-reaction from the CDM component. In aver-age, the growth of the CDM perturbations is proportional to the logarithm ofconformal timeδm ≃ 3C(k)[1/2 + C + ln(ωτ)] . [RD, ωτ = kτ/√3≫ 1] (67)4.4.4 Matter dominationDeep into the matter domination era, when a ∝ τ 2, the effective sound speedis vanishingly small and Eq. (58) simply reduces toφ¨+ 3a˙aφ˙ = 0 , (68)with solutionsφ = C˜(k) + τ−5D˜(k) . (69)The matter density contrast then follows from Eq. (52), which simplifies into−(12τ 2+ k2)φ− 6τφ˙ =3τ 2δm , (70)and we finally obtainδm = −(4 +k2τ 23)C˜(k) +(k2τ 23− 6)τ−5D˜(k) . (71)We can identify the solution proportional to C˜(k) (resp. D˜(k)) with the adia-batic growing (resp. decaying) mode. A detailed and precise matching betweenthe solutions (66), (67) –valid deep inside the radiation era– and the solution(71) –valid deep inside the matter era– would require some significant work inorder to take into account various effects close to the time of equality. How-ever, a crude matching is sufficient for catching the essential behavior of thematter power spectrum. From (66), (67) and (71) we obtain that at any timeτ belonging to the MD stage,δm(k, τ) =(4 +k2τ 23)δm(k, τeq)4 + k2τ 2eq/3, (72)31where δm(k, τeq) is equal to Eq. (64) evaluated at τ = τeq. Since we are inter-ested only in modes which are inside the Hubble radius in the recent Universe,we can take the limit k2τ 2 ≫ 1. During matter domination, it is also possibleto replace τ by 2/(aH). Then, we obtainδm(k, τM) =k23(aH)2× δm(k, τeq)1 + k2τ 2eq/12. (73)4.4.5 Dark energy dominationAfter the matter dominated era, the dark energy component becomes thedominant contribution to the energy density of the Universe. In this era, theevolution of the gravitational potential is still given by Eq. (54), but now theeffective mass term does not cancel. When the dark energy corresponds tothe contribution of a cosmological constant with equation of state parameterw = −1 (PΛ = −ρΛ),φ¨+ 3a˙aφ˙+ (8πGa2ρ¯Λ)φ = 0 , (74)and the potential decays proportionally to a scale-independent damping factorg(τ) = φ(τ)/φ(τM), where τM can be any time during matter domination. Thisfactor can be found by numerical integration. In the case of a cosmologicalconstant, a fairly good analytic approximation of g(τ0) – where τ0 is the presenttime – is provided in Ref. [87]g(τ0) ≃ Ω0.2m /[1 + 0.003(ΩΛ/Ωm)4/3] . (75)The modes probed by LSS experiments are all deep inside the Hubble ra-dius during the end of matter domination and during Λ domination. Onthose scales, the Poisson equation of Newtonian gravity (which correspondsto Eq. (52) with k ≫ aH) gives a simple relation between the gravitationalfluctuations and the matter perturbations,∆φ = 4πGρ¯mδm ⇐⇒ δm = − k2φ4πGa2ρ¯m= − 2k2φ3Ωm(aH)2. (76)So, δm is proportional to aφ, which proves that the decay of the gravitationalpotential slows down the growth of matter fluctuations as a function of thescale factor. Also, we immediately obtain a relation between δm during Λdomination and matter dominationδm(τ) =[a(τM )H(τM)a(τ)H(τ)]2g(τ)Ωm(τ)δm(τM) . (77)32Finally, Eqs. (64), (72) and (77) give us the expression of matter perturbationson sub-Hubble scales during Λ domination,δm(k, τ) =g(τ)k2C(k)Ωm(τ)[a(τ)H(τ)]2× (78)[−(sin kx)/(kx) + (sin kx− kx cos kx)/(kx)3 + 12+ Ci(kx)− C − ln(kx)][1 + (kx)2/12],where x ≡ τeq/√3. The matter power spectrum is defined as P (k, τ) =〈|δm(k, τ)|2〉. The previous equation allows us to relate it to the primordialpower spectrum, usually parametrized by a power-lawk3〈|φ(k, τ → 0)|2〉 = k39〈|C(k)|2〉 ≡ Akns−1 , (79)where the index ns is the scalar tilt. We can compute the two asymptotesfor the modes which were far outside/inside the sound horizon at the time ofequality,P (k, τ)≃ g(τ)2Ωm(τ)2[a(τ)H(τ)]4× Akns4, kτeq/√3≪ 1 , (80)P (k, τ)≃ g(τ)2Ωm(τ)2[a(τ)H(τ)]4× 9 [12]2Akns [ln kx]2[kx]4, kτeq/√3≫ 1. (81)Because of the crude approximations performed around the time of equality,this solution is not accurate, in particular for the global amplitude and forthe behavior near k ∼ √3/τeq. However, the shape of the two asymptotesreflects the behavior of the true power spectrum obtained by a full numericalsimulation, which is close to the BBKS [88] fitting formulaP (k, τ)≃ g(τ)2Ωm(τ)2[a(τ)H(τ)]49Akns25T (q)2 , (82)T (q)≡ ln(1 + 2.34 q)2.34q[1 + 3.89 q + (16.1 q)2 + (5.46 q)3 + (6.71 q)4]−1/4 ,q≡ kΩmh2 exp(−2Ωb) Mpc .Note also that the cosmological constant not only damps the perturbationsδρm but also implies a smaller fraction Ωm. The net effect is that today and forfixed a0H0, the power spectrum normalization is enhanced by the cosmologicalconstant (i.e. g(τ0)/Ωm(τ0) > 1 if ΩΛ > 0).3310-1 10-2 10-3 106 104 102 110-410-510-610-7- k3/2 φHubble crossingR/M equalityk (h/Mpc) a0/a10-1 10-2 10-3 10-6 10-4 10-2 1 110-110-210-310-410-510-6k3/2 δmHubble crossingR/M equalityk (h/Mpc) a/a0Fig. 6. Evolution of the longitudinal metric perturbations φ = ψ (top) and the mat-ter density perturbations δm = δρm/ρ¯m (bottom) in a neutrinoless ΛCDM model,obtained numerically with a Boltzmann code, as a function of the scale factor a andFourier wavenumber k. In order to get a better view, we chose opposite time axes inthe two plots: time is evolving from back to front in the upper plot, and from frontto back in the lower plot. The initial condition was set arbitrarily to k3/2φ = −10−5.4.4.6 Numerical resultsWe show in Fig. 6 the evolution of ψ = φ and δm for a neutrinoless ΛCDMmodel, obtained numerically by solving the exact Einstein and conservationequations. We run the public code cmbfast [49] based on the synchronousgauge, and convert the output into the longitudinal gauge variables usedthroughout this section. We start from a flat primordial spectrum with ampli-tude k3/2φ = 10−5 (let us recall that the “amplitudes” calculated by the codeshould be interpreted in the real Universe as the variance of random Gaussian34 60005000400030002000100001000100102l(l+1) C lT / 2pi (µK)2lreferenceωm decreaseΩΛ increaseωb increase 10410310-110-210-3P(k) (Mpc/h)3k (h/Mpc)referenceωm decreaseΩΛ increaseωb increaseFig. 7. (Red solid lines) The power spectrum of CMB temperature anisotropiesin multipole space (left), and the LSS matter power spectrum evaluated today inFourier space (right), for a reference neutrinoless ΛCDM model with ωm = 0.13,ΩΛ = 0.74, ωb = 0.022, ns = 0.95, τ = 0.08. (Dashed and dotted lines) Samepower spectra for three models in which we keep all parameters fixed but one:either ωm = 0.12, or ΩΛ = 0.80, or ωb = 0.026. The changes in the power spectraillustrate the effects labeled from (1) to (4) in Sec. 4.4.7, as explained in details atthe end of this subsection.perturbations). We show in Fig. 7 the CMB temperature anisotropy and LSSpower spectra for the same cosmological model (but now with the primordialspectrum amplitude preferred by current data)The global evolution is consistent with the analytical solutions found in theprevious subsections. Indeed, for metric perturbations ψ = φ (upper plot,time evolving from back to front), we see that during radiation domination,super-Hubble modes are constant, and sub-Hubble modes are damped (withsmall oscillations). During matter domination, all modes are constant, so theshape imprinted near the time of equality survives until today (at least, aslong as perturbations remain in the linear regime). During Λ domination,there is a small scale-independent damping of all perturbations, hardly seenon the figure. For matter density perturbations δm (lower plot, time evolvingfrom front to back), we see that during radiation domination, super-Hubblemodes are constant, and sub-Hubble modes grow slowly (asymptotically likethe logarithm of a). During matter domination, super-Hubble modes are stillconstant, while sub-Hubble modes grow linearly with a. The smallest scalesare the first ones to reach the condition of non-linearity δm ∼ 1, and largescales are still linear today. During Λ domination, δm grows a bit more slowlyas a function of a.4.4.7 Parameter dependenceA minimal flat ΛCDM models without neutrinos can be described with sixparameters: the cosmological constant fraction ΩΛ, the total non-relativisticmatter density ωm = Ωmh2, the baryon density ωb = Ωbh2, the primordialspectrum amplitude A and tilt ns, the optical depth to reionization τ . These35parameters control various physical effect which are responsible for the shapeof the observable power spectra:(1) The time of Radiation/Matter equality. In the above parameterbasis, the time of equality between ρm and ρr is fixed by ωm only, since(aeq/a0) = ρ0r/ρ0m (the script0 means “evaluated today”), and ρ0r is fixedby the CMB temperature. A late equality implies more metric pertur-bation damping for modes entering the Hubble radius during radiationdomination (see Fig. 6). For the LSS matter power spectrum, the con-sequence is the following: since on sub-Hubble scales δm grows more ef-ficiently during MD than during RD (because ψ does not decay), it isclear that when ωm decreases (late equality) the matter power spectrumis suppressed on small scales relatively to large scales. Also, the globalnormalization increases, because of the factor ω−2m in Eqs. (80,81). So,the net effect of postponing equality is to amplifies P (k) only for smallk. For the CMB, the effect is opposite and counter-intuitive: small-scaleperturbations are boosted. This is related to the dilation effect beforerecombination, and to the early Integrated Sachs-Wolfe effect just afterrecombination. In simple words, the rapid decay of ψ(k, τ) just after Hub-ble crossing during RD tends to boost the beginning of the first acousticoscillation through gravitational redshift effects. So, a later equality in-duces higher CMB peaks, especially for the first one.(2) The time of Matter/Λ equality. If the cosmological constant is larger,equality between matter and Λ takes place earlier. We have seen that af-ter that time, ψ decays and δm grows more slowly. Thus the matter powerspectrum normalization is slightly suppressed by a long Λ domination.The CMB spectrum is affected in a more subtle way: the time-variationof the metric perturbations leads to a net redshifting of the CMB photonstraveling across a gravitational potential well at small z (late IntegratedSachs-Wolfe effect). Therefore, the nearby distribution of galaxy clusterleaves an imprint in the large-scale CMB spectrum: the small-l multipolesare enhanced (the larger is Λ, the more significant is the enhancement).This effect is difficult to measure experimentally due to cosmic variance(on large scales, we see only a few realization of the stochastic mode am-plitudes, and we expect a large scattering of the data points around thetheoretically predicted variance).(3) The physical scale of the sound horizon at equality, and theangular scale of the sound horizon at recombination. From theprevious sections it is clear that the characteristic scale in the matterpower spectrum is fixed by the sound horizon at the time of equality(i.e., essentially, the Hubble scale RH(τeq) divided by√3). Since thisscale can be computed by extrapolating the present value of the Hub-ble parameter back in time, it depends on h, ΩΛ, and on the time of36equality. In our parameter basis, it means that changing any of the twoparameters ωm, ΩΛ shifts the matter power spectrum horizontally. Thecharacteristic scale of the oscillations in the CMB power spectrum is setby the sound horizon ds(τrec) at recombination. The time τrec is more orless fixed by thermodynamics, but the sound horizon is an integral overcsdt/a(t) between 0 and τrec. This integral depends on the time of equal-ity, and on the baryon density at late times (through cs). In addition, theactual observable quantity is the angular scale of this sound horizon onthe last scattering surface, which is given, for fixed ds(τrec), by h and ΩΛ.We conclude that the observed angular scale of the peaks constrains acombination of the three parameters ΩΛ, ωm and ωb.(4) The balance between gravity and pressure in the tightly-coupledphoton-baryon fluid. In this section, for simplicity, we assumed thatbefore recombination the energy density of the tightly-coupled photon-baryon fluid was dominated by that of photons. Actually, when the baryondensity becomes significant (near the time of equality), the fluid is moreaffected by the gravitational compression –which is more efficient fornon-relativistic baryons than for relativistic photons– relatively to thecompeting photon pressure. Inside a gravitational potential well of givenamplitude, the zero-point of oscillations (i.e. the instantaneous value ofδT/T for which gravity and pressure exactly cancel each other) is dis-placed. The main effect in the CMB spectrum is an increase of the firstand third peaks, but not of the second one. As far as the matter powerspectrum is concerned, we should remember that P (k) probes the totalcontribution from baryonic and cold dark matter perturbations. In thelimit of a small CDM contribution (ωb/ωm → 1), the small-scale matterpower spectrum has a low amplitude, because it does not benefit fromthe slow growth of CDM perturbations before equality on sub-Hubblescales (visible in Fig. 6). Also, in this limit, the oscillatory structure of δbat the time of recombination survives until now, and the matter powerspectrum exhibits visible wiggles. In the opposite limit ωb/ωm → 0, whichwe adopted in our analytical approximate solution, the small-scale powerspectrum is higher, and the baryonic oscillations in P (k) are small (still,they are visible in Fig. 7 between k = 0.05 h−1 Mpc and k = 0.2 h−1Mpc, and they have actually been observed recently, see Ref. [89]).(5) The amplitude of primordial perturbations (A), defined in Eq. (79),obviously fixes the global normalization of all spectra.(6) The tilt of primordial perturbations (ns), defined in Eq. (79), fixesthe balance between large and small scale amplitudes. Together withthe previous parameter it provides interesting constraints on inflationarymodels.37(7) The optical depth to reionization (τ) is a rather simplified way ofparameterizing the reionization of the Universe at redshift of order 10to 20 (depending on the model). This parameter has no effect on thematter power spectrum, but it damps the CMB temperature fluctuationsfor scales which are inside the Hubble horizon at the time of reionization,leaving small-l temperature multipoles unchanged.In Fig. 7, we illustrate graphically the consequence of varying either ωm, ΩΛor ωb, while keeping all the other cosmological parameters fixed. When wedecrease ωm, the relevant effects are (1) and (3): equality is postponed, boost-ing the first CMB peak as well as P (k) for small k; simultaneously, there isa tiny horizontal shift of both spectra. When we increase ΩΛ, we can see theeffects (2) and (3): the smallest CMB temperature multipoles increase, theP (k) normalization decreases and both spectra are shifted to the left. Whenωb is increased, the effects (3) and (4) are taking place: the first CMB peakincreases, while the second one decreases slightly; P (k) is slightly suppressedon small scales; finally, the scale of CMB peaks is slightly modified.We have seen that (A, ns, τ) have very specific effects, while (ΩΛ, ωm, ωb) haveintricate effects. Nevertheless, since there are seven effects for six parameters,it is in principle easy to measure all of them from CMB and LSS data, assuminga flat ΛCDM model. However, the neutrinoless model described in this sectionfor pedagogical purposes is unable to provide a good fit to the data: it is timenow to include neutrinos.4.5 Linear perturbation theory in presence of neutrinos (ΛMDM)We will now restore the neutrino component, which is expected to behaveat late times (after the non-relativistic transition) as an extra sub-dominantdark matter component: the standard cosmological scenario should actuallybe called Λ Mixed Dark Matter (ΛMDM).The impact of neutrinos on cosmological perturbations was first clearly ex-plained in 1980 by Bond, Efstathiou and Silk [70]. Later, during the 1980’s,a very large number of papers investigated this issue in more details, andit would be difficult to provide an exhaustive list of references (see e.g. thosegiven in [6,90]). Despite their interest, many of these papers are not of practicaluse nowadays because they generally assumed a neutrino-dominated universewith no CDM or Λ components. For instance, the interested reader can takea look at Refs. [71,72] and go through the historical summary of Ref. [69].Let us first introduce some notations and formalism –which are very closeto those of Ma & Bertschinger [74], but have been adapted to our metricsignature (+,−,−,−). The momentum of individual neutrinos can be labeled38by Pi, which is the canonical conjugate of the comoving coordinate xi. It differsfrom the proper momentum pi measured by a comoving observer (with fixedxi coordinates) byPi = a(1− ψ)pi . (83)Note that in absence of metric perturbations, Pi would remain constant, whilepi would decrease with the expansion like a−1. Usually, the formulas are conve-niently expressed in terms of the third variable qi ≡ api, which can be viewedas the proper momentum “corrected” from the effect of homogeneous expan-sion. The proper energy of a single neutrino measured by a comoving observeris equal to (p2 +m2)1/2. We define the related quantity ǫǫ = a(p2 +m2)1/2 = (q2 + a2m2)1/2 . (84)From PµPµ = m2 and Pi = (1− ψ)qi, we get P0 = (1 + φ)ǫ. Finally, we recallthat by definition the phase-space distribution f gives the number of particlesin an infinitesimal phase-space volume: dN = f(xi, Pj, τ) d3xi d3Pj.4.5.1 Collisionless fluidsIn the approximation of a homogeneous Universe, the phase-space distributionof neutrinos would be perfectly isotropic,f(xi, Pj , τ) = f0(P, τ) , (85)where f0 is the Fermi-Dirac distribution already introduced in Eq. (6). Then,the energy-momentum tensor would be diagonal with an isotropic pressuretermT 00 = ρ¯ν =4πa4∫q2dq ǫ f0(q) , (86)T ii = −p¯ν =−4π3a4∫q2dqq2ǫf0(q) . (87)Note that f0 as a function of q is time-independent,f0(q) =1eq/aTν + 1, (88)because after neutrino decoupling the product of the neutrino temperature Tνby the scale factor a is a constant number.39Spatial perturbations in the metric will induce variations in the neutrinophase-space distribution depending on time, space and momentum. The energy-momentum tensor will be perturbed accordingly. With respect to a perfectfluid, the main difference is is that, in absence of microscopic interactions,there is no reason for the stress tensor δTij to be isotropic at first order inperturbations. Therefore, the scalar sector of δTµν contains an extra degree offreedom, the anisotropic stress σ.This can be understood in physical terms. First, strongly-interacting fluidshave bulk motions: the coarse-grained fluid has a unique velocity in a givenpoint. In the local rest-frame, an observer can measure the pressure e.g. byenclosing some particles in a small rubber balloon. At first order in pertur-bation, the balloon will expand or contract isotropically, because anisotropiesarise from velocity gradients which can be shown to be second order in per-turbations. In contrast, in collisionless fluids, there are no bulk motions: at agiven point, there can be flows of particles in all directions. Inhomogeneitiesand anisotropies are described perturbatively by a function Ψ≪ 1 such thatf(xi, Pj , τ) = f0(q)[1 + Ψ(xi, qj, τ)] with Pj = (1−Ψ)qj , (89)and anisotropic pressure can now appear at first order in perturbations. In-deed, the energy-momentum tensor can be computed from the phase-spacedistribution f and the 4-momentum PµTµν =∫dP1dP2dP3 (−g)−1/2 PµPνP 0f(xi, Pj, τ) , (90)where (−g)−1/2 = a−4(1− φ+ 3ψ). At first order in perturbations, this givesT 00 (xi)= a−4∫q2dq dΩ ǫ f0(q) [1 + Ψ(xi, qnˆj, τ)] ,T 0i (xi)= a−4∫q2dq dΩ q nˆi f0(q)Ψ(xi, qnˆj , τ) , (91)T ij (xi)=−a−4∫q2dqdΩq2ǫnˆinˆj f0(q) [1 + Ψ(xi, qnˆj , τ)] ,where dΩ is the differential of the momentum direction nˆj = qj/q. It is thenstraightforward to obtain the perturbed components of the energy-momentumtensorδρν = a−4∫q2dq dΩ ǫf0Ψ , δPν =13a−4∫q2dq dΩq2ǫf0Ψ , (92)δT 0i ν = a−4∫q2dq dΩ qnˆi f0Ψ , Σij ν = −a−4∫q2dq dΩq2ǫ(nˆinˆj − 13δij) f0Ψ .404.5.2 Free-streamingWe have seen that in perfect fluids, sound waves can propagate at the soundspeed on scales smaller than the sound horizon. Sound waves cannot prop-agate in a collisionless fluid, but the individual particles free-stream with acharacteristic velocity –for neutrinos, in average, the thermal velocity vth. So,it is possible to define an horizon as the typical distance on which particlestravel between time ti and t. During MD and RD and for t≫ ti, this horizonis, as usual, asymptotically equal to vth/H , up to a numerical factor of orderone (see section 4.4.2). Exactly as we defined the Jeans length, we can definethe free-streaming length by taking Eq. (48) and replacing cs by vthkFS(t) =(4πGρ¯(t)a2(t)v2th(t))1/2, λFS(t) = 2πa(t)kFS(t)= 2π√23vth(t)H(t). (93)As long as neutrinos are relativistic, they travel at the speed of light and theirfree-streaming length is simply equal to the Hubble radius. When they becomenon-relativistic, their thermal velocity decays likevth ≡ 〈p〉m≃ 3Tνm=3T 0νm(a0a)≃ 150(1 + z)(1 eVm)km s−1 , (94)where we used for the present neutrino temperature T 0ν ≃ (4/11)1/3T 0γ andT 0γ ≃ 2.726 K. This gives for the free-streaming wavelength and wavenumberduring matter or Λ dominationλFS(t)= 7.71 + z√ΩΛ + Ωm(1 + z)3(1 eVm)h−1Mpc , (95)kFS(t)= 0.82√ΩΛ + Ωm(1 + z)3(1 + z)2(m1 eV)hMpc−1, (96)where ΩΛ (resp. Ωm) is the cosmological constant (resp. matter) density frac-tion evaluated today. So, after the non-relativistic transition and during mat-ter domination, the free-streaming length continues to increase, but only like(aH)−1 ∝ t1/3, i.e. more slowly than the scale factor a ∝ t2/3. Therefore, thecomoving free-streaming length λFS/a actually decreases like (a2H)−1 ∝ t−1/3.As a consequence, for neutrinos becoming non-relativistic during matter dom-ination, the comoving free-streaming wavenumber passes through a minimumknr at the time of the transition, i.e. when m = 〈p〉 = 3Tν and a0/a = (1+z) =2.0× 103(m/1 eV). This minimum value is found to beknr ≃ 0.018 Ω1/2m(m1 eV)1/2hMpc−1 . (97)41The physical effect of free-streaming is to damp small-scale neutrino densityfluctuations: neutrinos cannot be confined into (or kept outside of) regionssmaller than the free-streaming length, for obvious kinematic reasons. Wewill see that the metric perturbations are also damped on those scales bygravitational back-reaction. On scales much larger than the free-streamingscale, the neutrino velocity can be effectively considered as vanishing, and afterthe non-relativistic transition the neutrino perturbations behave like cold darkmatter perturbations. In particular, modes with k < knr are never affected byfree-streaming and evolve like in a pure ΛCDM model.4.5.3 Vlasov equationThe neutrino equation of evolution is the collisionless Boltzmann equation,or Vlasov equation, which just expresses the conservation of the number ofparticles along the phase-space trajectoriesDfdτ=∂f∂τ+dxidτ∂f∂xi+dqdτ∂f∂q+dnidτ∂f∂ni= 0 . (98)We can expand f like in Eq. (89) and keep only first-order perturbations. Usingthe fact that dxi/dτ = P i/P 0 = −Pi/P0 = −qi/ǫ at order zero, that dq/dτ =q ψ˙+ ǫ nˆi ∂iφ at order one (from the geodesic equation P0P˙ 0+Γ0µνPµP ν = 0),and that dni/dτ = 0 at order zero (in absence of metric perturbations, themomenta keep a fixed orientation), we obtainf0Ψ˙− qiǫf0 ∂iΨ+ (q ψ˙ + ǫ nˆi ∂iφ)∂f0∂q= 0 , (99)or in Fourier space and after dividing by f0Ψ˙− i qǫ(~k · nˆ)Ψ = −(ψ˙ + iǫq(~k · nˆ)φ)∂ ln f0∂ ln q. (100)This equation gives the response of the neutrino phase-space distribution tometric perturbations.4.5.4 Neutrino perturbations during the relativistic regimeWhen neutrinos are ultra-relativistic (ǫ = q), the Vlasov equation reduces toΨ˙− i (~k · nˆ)Ψ = −(ψ˙ + i (~k · nˆ)φ) ∂ ln f0∂ ln q. (101)42The homogeneous equation does not involve q, while the source term has adefinite q-dependence, fixed once and for all by the shape of the homogeneousdistribution. This simply reflects the fact that neutrinos at a given point andin a given direction are all redshifted or blueshifted by the same amount, sothat their spectrum always remain that of a blackbody, with a temperatureshift δTν(xi, nˆj, τ) at a given point and in a given directionf(xi, q, nˆj, τ)=(exp[qa(Tν + δTν)]+ 1)−1= f0(q) +∂f0∂Tν(q) δTν(xi, nˆj, τ) . (102)Since f0 is a function of q/aTν , one has ∂f0/∂Tν = −(q/Tν) ∂f0/∂q, andf(xi, q, nˆj, τ) = f0(q) − ∂f0∂ ln q(q)δTνTν(xi, nˆj , τ) . (103)Identifying this result with the definition of the phase-space perturbation Ψ,we getΨ(xi, q, nˆj, τ) = −∂ ln f0∂ ln q(q)δTνTν(xi, nˆj, τ) . (104)We now see explicitly that the presence of the function ∂ ln f0∂ ln qin the Vlasovequation means precisely that the spectrum remains Planckian. Since for rel-ativistic neutrinos the local q-dependence is trivial, we only need to computethe evolution of the mean momentum, which is a function of position, directionand time. One can define the quantity Fν(xi, nˆj, τ) asFν(xi, nˆj, τ) ≡∫q2dq qf0(q) [1 + Ψ(xi, nˆj, τ)]∫q2dq qf0(q)− 1 , (105)for which the Vlasov equation reads in Fourier space (after integrating bypart)F˙ν − i (~k · nˆ)Fν = 4(ψ˙ + i (~k · nˆ)φ). (106)We can further reduce the dimensionality of the problem by noticing that theevolution depends only on the direction nˆ through the angle ~k · nˆ, which isa natural consequence of the isotropy of the homogeneous background. Weperform a Legendre expansion with respect to this angle,Fν(~k, nˆ, τ) =∞∑l=0(−i)l(2l + 1)Fν l(~k, τ)Pl(kˆ · nˆ) . (107)43By comparing Eqs. (92) and (93) (with ǫ = q) with the above definitions, itis straightforward to show that the density contrast (as well as δpν =13δρν)is given by the monopole: δν = Fν 0, the velocity gradient by the dipole: θν =34kFν 1, and the shear (or anisotropic stress) by the quadrupole: σν = −12Fν 2.The Vlasov equation now reduces to its most “concentrated” form: an infinitehierarchy of coupled equation for the multipoles Fν l(~k, τ)δ˙ν =43θν + 4ψ˙ ,θ˙ν =−k24δν − k2σν − k2φ ,F˙ν l=k2 l + 1[(l + 1)Fν (l+1) − l Fν (l−1)], l ≥ 2 . (108)Note that the first two equations consistently reproduce the continuity andEuler equations (39)-(40).After introducing all this formalism, it is possible to understand the physicalevolution of relativistic neutrino perturbations. We assume isentropic (adi-abatic) initial conditions during radiation domination and on super-Hubblescales: so, we start from δν = δγ and from time-independent metric fluctua-tions. On super-Hubble scales, the neutrino distribution is static, with a localdensity (and temperature) perturbation proportional to the metric fluctuation.An observer would see locally a quasi-isotropic temperature distribution: thedipole is suppressed by a factor k/aH = kτ , and the l-th multipole by (kτ)l.At leading order in (kτ), the Einstein equations (35) and (36) immediatelygiveδν = δγ = −2φ = 4δTνTν, θν = θγ = −12k2τφ , (109)while the expressions of ψ and σν require more work (actually, the usual wayto obtain them is to solve the Einstein equations in the long wavelength limitin the synchronous gauge, and then to convert the result into longitudinalgauge variables). The result is found to be (see Ref. [74])ψ =(1 +25Rν)φ , σν = − 115(kτ)2φ , with Rν ≡ ρ¯νρ¯ν + ρ¯γ. (110)For three standard non-relativistic neutrinos, on has Rν ≃ 0.405 and ψ ≃1.162φ. After Hubble crossing, i.e. on scales smaller than the free-streaminglength, the various multipoles are populated one after each other, at the ex-pense of the low multipoles which get reduced. Physically, this corresponds tothe fact that an observer would see locally the superposition of a growing num-ber of neutrino flows, coming from all directions. Since the flows come from441 10-1 10-2 10-3 1 10-2 10-4 10-6 10-4 5.10-5 2.10-50-5.10-5 -10-4k3/2 δνHubble crossingR/M equalityk (h/Mpc) a/a01 10-1 10-2 10-3 1 10-2 10-4 10-6 10-4 5.10-5 2.10-50-5.10-5 -10-4k3/2 δγHubble crossingR/M equalityk (h/Mpc) a/a0Fig. 8. Evolution of (longitudinal gauge) massless neutrino density perturbationsδν = δρν/ρ¯ν (top), compared with those of the photon density δγ = δργ/ρ¯γ (bot-tom), in a ΛCDM model with three massless neutrinos, as a function of the scalefactor a and Fourier wavenumber k, obtained numerically with a Boltzmann code.Time is evolving from left (a/a0 = 10−6) to right (a/a0 = 1), and the initial condi-tion is set arbitrarily to k3/2φ = −10−5. Unlike the photon density contrast, whichoscillates between Hubble crossing and equality, the neutrino density contrast isimmediately damped after the first half-oscillation. After equality, δν and δγ re-main constant: they are preserved from gravitational collapse by free-streaming andrelativistic pressure.a random distribution of over- and under-densities, the local density contrasttends to average to zero.451.0 10-50.8 10-50.6 10-50.4 10-50.2 10-50.0 10-5-0.2 10-5110-110-210-310-410-5a/a0k3/2 σνk3/2 Fν5k3/2 Fν10k3/2 Fν15Fig. 9. Evolution of some of the first neutrino multipoles Fνl as a function of thescale factor, for the mode k = 0.1hMpc−1 which enters into the Hubble radiusclose to the end of radiation domination, when a/a0 = 2.5 × 10−5. We imposed aninitial condition k3/2φ = −10−5. We see that after Hubble crossing, the multipolesare populated one after each other, reflecting the fact that an observer would seelocally a superposition of a growing number of neutrino flows, coming from alldirections.It is useful for pedagogical purpose to compare the evolution equation of neu-trinos with that of photons before recombination. For this purpose, we combinethe continuity and Euler equations into a second-order equationδ¨ν = −13k2δν − 43k2φ+ 4ψ¨ − 43k2σν . (111)In absence of shear, this equation would be similar to the photon equation ofevolution before recombination and in the limit of small baryon-to-photon ra-tio Rb = ρ¯b/ρ¯γ . The terms are easy to identify: −13k2δν is the force arising fromrelativistic pressure, −43k2φ the gravitational force, and 4ψ¨ the dilation effect:a local increase of ψ is equivalent to a local decrease of the scale factor and ofthe photon/neutrino wavelength, and hence to an increase of the blackbodytemperature and density. In the case of photons, in the tight-coupling limit,the microscopic interactions guarantee that σγ vanishes (even if Rb ≪ 1), andthe equivalent of the above equation is that of an harmonic oscillator drivenby metric fluctuations. In the case of neutrinos, the shear becomes important46inside the Hubble length, and couples the above equation to the whole multi-pole hierarchy; qualitatively, the shear acts like viscosity in a fluid; instead ofexperiencing driven oscillations, the neutrino overdensity δν is damped, whilethe power is transfered to higher multipoles. This behavior is illustrated inFigs. 8 and 9 which derive from exact numerical simulations (using the codecmbfast [49]). In particular, one can see explicitly on Fig. 8 that in the re-gion where the photon density contrast experiences acoustic oscillations, theneutrino density contrasts is damped.It is possible to gain even more intuition by going back to the relativisticVlasov equation (101), for which the full solution can be decomposed into thoseof the homogeneous and inhomogeneous differential equations. The generalsolution of the homogeneous equation is actually trivialΨ(~k, nˆ, q, τ) = − δTνTν(~k, nˆ) ei(~k·nˆ)τ ∂ ln f0∂ ln q. (112)It corresponds to the free propagation of plane waves of temperature pertur-bations, i.e. to free-streaming. In the Fν l basis, this yields Fν l(~k, τ) ∝ jl(kτ),where the jl’s are the spherical Bessel functions of the first kind, and the con-tribution to δν is equal to [sin(kτ)/(kτ)] times a constant of integration. Asexpected, this contribution tends to zero after Hubble crossing (because ofthe averaging between various flows), while higher multipoles are populatedone after each other, with a peak around τ ∼ l/k (the numerical solution ofFig. 9 looks slightly more complicated than plain Bessel functions: this is dueto the small time-variation of φ and ψ caused by neutrino shear). On top ofthese terms, the full solution receives a contribution from the solution of theinhomogeneous equation. During radiation domination, ψ and φ decay, andthis solution is complicated. However, it is clear that it should also decay.Instead, during matter domination, ψ and φ are frozen, and the solution isobvious: Ψ = φ ∂ ln f0∂ ln q. In the Fν l basis, this term only contributes to Fν0 = δν ,for which the full solution readsδν = −4φ(k) + α(k) sin(kτ)/(kτ) , τ > τeq , (113)where the constant of integration α(k) could be found my matching withthe radiation-dominated solution. The first term corresponds to the station-ary equilibrium configuration for which, in any potential well, the relativisticpressure compensates exactly the gravitational force. This behavior can beclearly seen on Fig. 8 (note that after decoupling, the photons free-stream likethe relativistic neutrinos, so δγ also tends towards −4φ).In conclusion, during the relativistic regime, the combined effects of free-streaming and relativistic pressure result in a suppression of the neutrinodensity contrast inside the Hubble radius: during radiation domination, δν is47driven to zero instead of oscillating like δγ and δb, and during matter domi-nation, it remains constant instead of growing like δb and δcdm.4.5.5 Neutrino perturbations during the non-relativistic regimeAfter the non-relativistic transition, the q-dependence of the phase-space dis-tribution becomes non-trivial. The geodesic equationdqdτ= qψ˙ + (q2 + a2m2)1/2nˆi ∂iφ (114)shows that the amount of blueshift or redshift dq/q experienced by a neutrinowhen it travels over metric fluctuations depends on how non-relativistic it is.In a given point and direction (xi, nˆj), there is a superposition of neutrinosof different momenta which are more or less deep inside the non-relativisticregime. Lowest and highest energy neutrinos experience different redshifts, andthe thermal spectrum of Eqs. (102) and (103) is inevitably distorted at firstorder in perturbation (but not at zero order, since the distortion is inducedby metric perturbations and not by the background geometry). Therefore, itis impossible to eliminate the q-dependence from the problem, as we did inthe relativistic regime. However, it is still possible to get rid partially of thedirection dependence by expanding Ψ in multipole spaceΨ(~k, q, nˆ, τ) =∞∑l=0(−i)l(2l + 1)Ψl(~k, q, τ)Pl(kˆ · nˆ) . (115)Then, the Vlasov equation becomesΨ˙0=q kǫΨ1 − ψ˙ ∂ ln f0∂ ln q,Ψ˙1=q k3 ǫ(Ψ2 − 2Ψ0) + ǫ k3 qφ∂ ln f0∂ ln q, (116)Ψ˙l=q k(2l + 1) ǫ[(l + 1)Ψl+1 − lΨl−1] , l ≥ 2 .The perturbed energy density, pressure, energy flux divergence and shear stressin k-space are given byδρν =4πa−4∫q2dq ǫf0(q)Ψ0 ,δpν =4π3a−4∫q2dqq2ǫf0(q)Ψ0 ,48110-110-210-310-410-510-6110-110-210-310-410-510-6a/a0pν / ρνδpν / δρνσν / δνFig. 10. Evolution of the neutrino isotropic pressure perturbation δpν andanisotropic stress σν in units of the density perturbation (for the modek = 0.1hMpc−1 which enters into the Hubble radius when a/a0 = 2.5×10−5) com-pared to the evolution of the background equation of state parameter p¯ν/ρ¯ν . Thethin curves show the case of massless neutrinos for which p¯ν/ρ¯ν = δpν/δρν = 1/3and σν/δν decays slowly after Hubble crossing. The thick lines correspond to neutri-nos with mν = 0.1 eV, which become non-relativistic around a/a0 = 5×10−3. Afterthe transition, pressure and shear perturbations become negligible with respect todensity perturbations, just like for cold dark matter.(ρ¯ν + p¯ν)θν =4πka−4∫q2dq qf0(q)Ψ1 ,(ρ¯ν + p¯ν)σν =−8π3a−4∫q2dqq2ǫf0(q)Ψ2 . (117)When for a given family the neutrinos are well inside the non-relativisticregime, i.e. when most of the momenta for which f0(q) is non-negligible satisfyq ≪ ǫ ∼ am, the above equations show that both δpν and (ρ¯ν + p¯ν)σν aresuppressed with respect to δρν (as can be checked in Fig. (10)). As expected,the continuity and Euler equations (39) and (40) become gradually identicalto those of ordinary non-relativistic matterδ˙ν = θν + 3ψ˙ , (118)θ˙ν =− a˙aθν − k2φ . (119)49Like for CDM (and baryons after recombination), this system leads to thesecond-order equation of evolutionδ¨ν +a˙aδ˙ν = −k2φ+ 3(ψ¨ + a˙aψ˙) . (120)We are interested in neutrinos becoming non-relativistic during matter dom-ination, when φ and ψ are approximately constant and a ∝ τ 2. In this case,the solution of (120) readsδν =A ln τ +B − (kτ)26φ (121)= A˜ ln a+ B˜ − 23(kaH)2φ , (122)where A, B (or A˜, B˜) are some constants of integration. The last term (thesolution of the inhomogeneous equation) grows like τ 2, i.e. like the scale factor.This solution has a simple interpretation:• first, for k > knr, i.e. for modes which are inside the Hubble radius atthe time of the non-relativistic transition, the density contrast δν is muchsmaller than δcdm at that time (as a consequence of free streaming), andthen grows faster than δcdm ∝ a because of the term (A˜ ln a). At some pointthe first and third terms in Eq. (122) will be equal, and then asymptoticallyδν will be dominated byδν −→ −23(kaH)2φ ∝ a , (123)which corresponds exactly to the solution of the Poisson equation in amatter-dominated Universe (see Eq. (76) with Ωm=1). Indeed, this asymp-totic value is simply the one imposed to all non-relativistic species by New-tonian gravitation, and after an infinite amount of time δν would be equal toδcdm on all scales. The same is also true for baryons after photon decoupling,but since baryons did not experience free-streaming, the equality betweenδb and δcdm is reached quickly after τ = τdec. Instead, for neutrinos withmasses of order of 1 eV or smaller, the asymptote is far from being reachedtoday, except for scales very close to k = knr. On smaller scales, δν remainsmuch smaller than δcdm ≃ δb.• second, for k < knr, i.e. for modes which are outside the Hubble radius atthe time of the non-relativistic transition, the density contrast δν is time-independent and of the same order as (φ, δcdm) on super-Hubble scales. Afterhorizon crossing, the solution is dominated again by (123) and becomesrapidly equal to δcdm.50The conclusion of this section is that if we could picture the three densitycontrasts at a fixed time near the end of matter domination, we would expectδν = δcdm = δb for a0H0 < k < knr, while for k > knr the ratio δν/δcdm = δν/δbwould be smaller than one, and tending towards zero as k goes to infinity.The recent stage of Λ domination does not change this conclusion: for a0H0 knr. Therefore, we should study how theevolution of δcdm = δb is affected by non-relativistic neutrinos deep in thematter-dominated regime.How neutrinos slow down the growth of matter perturbations.We are arriving to the most important point of this section, since we are goingto derive the dominant observable effect of neutrino masses. Before getting intothe equations, let us try to describe the effect qualitatively. For this purpose,we first need to come back to the issue of linear clustering of CDM and baryonsduring matter domination in absence of neutrinos. In Sec. 4.4, we stated thatφ = ψ remains constant during matter domination, basing ourselves on Eq.(58) with a zero sound speed (or equivalently on the perturbed Einstein Eqs.(37) and (38) with zero pressure perturbation and shear on the right-handside). Then, we used the Poisson equation− k2a2ψ = 4πGδρ , (125)to show that inside the Hubble radius, the density contrast δcdm = δb growslike [a2(ρ¯cdm + ρ¯b)]−1, i.e. like a linear function of the scale factor. The factthat linear structure formation corresponds to a constant gravitational poten-tial sounds counter-intuitive at first sight. Actually, it results from an exactcompensation between clustering and the stretching of spacetime. During grav-itational collapse, the non-relativistic matter density perturbations δρ do notgrow with time, as one would naively expect from a common intuition basedon a static Universe. Instead, δρ decreases, but only like a−2 and not as fastas the dilution factor a−3, so that the density contrasts δρ/ρ¯ actually grows52like a. The behavior δρ ∝ a−2 corresponds precisely to a static potential, sincethe gradient ∆ψ = −(k/a)2ψ also decreases like a−2 due to the stretching ofspace-time.Let us restore now the neutrino background. During matter domination and onscales smaller than the free-streaming scale, the neutrino perturbations δν donot contribute to gravitational clustering. Indeed, we have seen in Secs. 4.5.4and 4.5.5 that free-streaming leads to δν ≪ δcdm, and since ρ¯ν < ρ¯cdm we seethat δρ¯ν ≪ δρ¯cdm: so, neutrinos can be simply omitted from the Poisson equa-tion. On the other hand, they do contribute to the homogeneous expansionthrough Friedmann equation. Therefore the exact compensation between clus-tering and expansion described in the previous paragraph is slightly shifted:the balance is displaced in favour of the expansion effect, and we expect φ = ψto decay slowly, while δcdm = δb should grow not as fast as the scale factor.This mechanism, first described and quantified in Ref. [70], belongs to the firstof the two categories defined at the beginning of this subsection: physically, itis a pure background effect, which leads to a modified evolution of metric andmatter perturbations.Let us estimate the magnitude of this effect in the same fashion as in thepioneering paper [70]. At any time we can combine the continuity and Eulerequations of CDM perturbations (39) and (40) intoδ¨cdm +a˙aδ˙cdm = −k2φ+ 3(ψ¨ + a˙aψ˙) , (126)but well-inside the Hubble radius the source term is dominated by the co-moving gradient −k2φ, which is approximately equal to −k2ψ as soon as theneutrino shear can be neglected. This gradient is given by the Poisson equa-tion, and we obtainδ¨cdm +a˙aδ˙cdm = 4πGa2 δρ , (127)where δρ is the total density perturbation. In absence of neutrinos and deepinside the matter-dominated regime, we would have δcdm = δb, a total densityperturbation δρ = (ρ¯cdm + ρ¯b) δcdm, an expansion rate given by 3 (a˙/a)2 =8πGa2(ρ¯cdm + ρ¯b) ∝ a−1, a scale factor a ∝ τ 2, and the equation would readδ¨cdm +2τδ˙cdm − 6τ 2δcdm = 0 , (128)with two solutions δcdm ∝ τ 2 and δcdm ∝ τ−3. Neglecting the decaying modewe recover the standard result δcdm ∝ a. Now, let us consider instead the casewith massive neutrinos, still deep inside the matter-dominated regime and53on scales k ≫ knr, so that δρν does not contribute to the Poisson equation:δρ = (ρ¯cdm + ρ¯b) δcdm, while the neutrino background density does contributeto the expansion rate: 3 (a˙/a)2 = 8πGa2(ρ¯cdm + ρ¯b + ρ¯ν). Let us assume thatρ¯ν is dominated by non-relativistic neutrinos, so that it decays approximatelylike a−3, and the numberfν ≡ ρν(ρcdm + ρb + ρν)=ΩνΩm(129)remains approximately constant. Then, the scale factor still evolves like τ 2and the equation of evolution readsδ¨cdm +2τδ˙cdm − 6τ 2(1− fν) δcdm = 0 . (130)Looking for solutions in δcdm ∝ τ 2p, we find two rootsp± =−1±√1 + 24(1− fν)4, (131)and we conclude that the growing solution for the CDM density contrast readsδcdm ∝ ap+ ≃ a1− 35fν , (132)where in the last step we assumed fν ≪ 1. As expected, the growth of δcdm isreduced due to the fact that one of the component in the Universe contributesto the homogeneous expansion rate but not to the gravitational clustering.The Poisson equation gives− k2ψ ∝ ap+−1 ≃ a− 35fν , (133)showing that for the same reason the gravitational potential slowly decaysduring matter domination.At the end of matter domination and during Λ domination, we have alreadyseen that in absence of neutrinos φ decays like g(a) and δcdm grows like a g(a)(we recall that the damping factor g(a) is normalized to g = 1 for a ≪ aΛ).The combined effect of Λ and of neutrinos on the growth of δcdm can be wellapproximated by [93]δcdm ∝ [a g(a)]p+ ≃ [a g(a)]1− 35fν . (134)Matter power spectrum for massive versus massless neutrinos.Let us try to predict analytically the difference between the power spectrum54of two cosmological models with respectively massive and massless neutrinos(ΛMDM versus ΛCDM) assuming that they share the same values of (ωm, ωb,ΩΛ, A, ns, τ). The difference between the two models lies in the values of ωνand ωcdm = ωm − ωb − ων, and can be conveniently parametrized by fν .(1) for very large scales k < knr, neutrino perturbations were never affectedby free streaming, and in the non-relativistic regime they become indis-tinguishable from CDM perturbations. In particular, inside the Hubbleradius, neutrinos contribute both to the Poisson equation with δν = δcdmand to the background evolution, so the density contrast grows like a. So,for fixed ωm, this branch of the spectrum is completely insensitive to thevalue of fν and of the neutrino mass.(2) for very small scales with k ≫ knr and k ≫ keq, the two spectra will differfor various reasons. First, equality does not take place for the same valueof the scale factor in the two models, since one always has (aeq/a0) =ωr/(ωb + ωcdm) = (1 − fν)−1ωr/ωm, where ωr includes the density ofphotons and of three effectively massless neutrinos. Since the two modelsshare the same value of ωr and ωm, the ratio between the two values ofthe scale factor at equality is afνeq/afν=0eq = (1 − fν)−1. At equality, andmore generally at any time before the non-relativistic transition, the twomodels are rigorously equivalent, modulo a shift in the scale factor by theabove factor. So, the identityδfνcdm[a] = δfν=0cdm [(1− fν) a] (135)is valid as long as a ≤ anr. After a = anr and in the massive model,the mechanism described in Eqs. (130)-(133) will take place. So, in theapproximation of a time-independent fraction fν for a ≥ anr (which is notstrictly valid in the case of several massive non-degenerate neutrinos), thegrowth of CDM perturbations is given by Eq. (134), i.e. by a constantlogarithmic slope (1− 35fν) for the scale factor (or for the product [a g(a)]during dark energy domination),δfνcdm[a0] =(a0 g(a0)anr)1− 35fνδfνcdm[anr] . (136)In the massless case, everything is identical to the massive model untilthe time at which a = (1− fν) anr (i.e. until the value of the scale factorfor which neutrinos become non-relativistic in the massive case “shiftedback” to the massless case according to Eq.(135)). Therefore, at this time,the logarithmic slope of δfν=0cdm must also be equal to (1− 35fν), but thenit will increase progressively until unity, so that δfν=0cdm becomes a linear55function of a or [a g(a)]. In a crude approximation, we can writeδfν=0cdm [a0] ≃(a0 g(a0)(1− fν) anr)δfν=0cdm [(1− fν) anr] , (137)but this tends to overestimate the growth of perturbations in the masslesscase: it assumes that right after a = anr the logarithmic slope is equal toone, which is not true immediately. Indeed, a comparison with numericalresults shows that the total growth factor is a bit smaller,δfν=0cdm [a0] ≃(a0 g(a0)(1− fν)1/2anr)δfν=0cdm [(1− fν) anr] . (138)Using this semi-analytic result, we find that the ratio between the presentvalue of δcdm in the two models readsδfνcdm[a0]δfν=0cdm [a0]= (1− fν)1/2(a0 g(a0)anr)− 35fν. (139)According to Eq. (124) this means that the total matter power spectrumis reduced byP (k)fνP (k)fν=0= (1− fν)3(a0 g(a0)anr)− 65fν. (140)Finally, we can replace (a0/anr) by 2000mν/(1 eV) and, assuming thatthe mass mν is shared by a number Nν of families, we can use mν =(ων/Nν) 93.2 eV. We obtain an expression that depends only on (fν , Nν ,ωm, ΩΛ)P (k)fνP (k)fν=0= (1− fν)3[1.9× 105 g(a0)ωm fν/Nν]− 65fν. (141)We show in Fig. 12 that this semi-analytic expression is a very goodapproximation of the exact numerical result, and also that for plausiblevalues of (ωm, Nν ,ΩΛ) and for fν < 0.07, it can be approximated by thewell-known linear expression [94]P (k)fνP (k)fν=0≃ −8 fν . (142)In conclusion –and as confirmed by the numerical simulations in the nextsubsection– the ratio P (k)fν/ P (k)fν=0 computed for fixed parameters (ωm,ΩΛ) smoothly interpolates between one for k < knr and a plateau given by Eq.(141) for k ≫ keq.56110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 10-3 h/Mpc-k3/2φ -k3/2ψ k3/2δcdmk3/2δb k3/2δγ k3/2δν 110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 10-3 h/Mpcanr-k3/2φ -k3/2ψ k3/2δcdmk3/2δb k3/2δγ k3/2δν 110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 10-2 h/Mpc110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 10-2 h/Mpcanr110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 10-1 h/Mpc110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 10-1 h/Mpcanr110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 1 h/Mpc110-110-210-310-410-510-610-7110-110-210-310-410-510-6 aeq aΛk=aHk = 1 h/MpcanrFig. 11. Evolution of the metric and density perturbations as a function of thescale factor (normalized to a0 = 1 today), in the longitudinal gauge, for modes10−3 hMpc−1 < k < 1hMpc−1 (from top to bottom), and for two cosmologicalmodels: ΛCDM (left) and ΛMDM (right), both with ωm = 0.147 and ΩΛ = 0.7. Theintegration has been performed with the code cmbfast starting from the initialcondition k3/2φ = −10−5. The ΛMDM model has three degenerate neutrinos withmν = 0.46 eV, corresponding to fν = 0.1.574.5.7 Numerical resultsIn Fig. 11, we tried to summarize all the features described in this section byshowing the numerical solutions for the metric and density perturbations (φ,ψ, δcdm, δb, δγ , δν) at different wavenumbers, for a ΛCDM (left) and ΛMDM(right) a model. The two models share common values of (ωm, ωb, ΩΛ, τ)and the initial amplitudes are the same. In the ΛCDM model, the three neu-trino families are massless, while in the ΛMDM model they share a commonvalue of the mass corresponding to fν = 0.1. At the bottom of each plot, weshow the value of the scale factor at the time of Hubble entry (k = aH),radiation/matter equality (aeq) and matter/Λ equality (aΛ). Decoupling takesplace soon after equality, around a = 9×10−4. These solutions have been com-puted with the public code cmbfast [49] in the synchronous gauge, and thentranslated into the longitudinal gauge. Let us briefly summarize the behaviorof each quantity.For a ΛCDM model (left plots in Fig. 11), one has that• the metric perturbations are essentially constant on large scales k < keq dur-ing radiation and matter domination. Instead, on small scales k > keq, theydecay between Hubble entry and decoupling, damped by the propagation ofacoustic waves. On all scales, they are also damped during Λ domination.Initially the difference ψ−φ is constant and non-zero, due to neutrino shear.After equality, the difference becomes negligible.• the photon perturbations undergo acoustic oscillations between Hubble en-try and decoupling. Then, during matter domination, they free-stream,maintaining and average density contrast δγ = −4φ, corresponding (as wehave seen) to gravity vs. pressure equilibrium.• the relativistic neutrino perturbations are damped during radiation domina-tion after Hubble crossing, because of free-streaming. During matter dom-ination free-streaming goes on but δν maintains itself around −4φ like forphotons.• the CDM perturbations always grow inside the Hubble radius. During radi-ation domination this growth starts abruptly, but then becomes only loga-rithmic; during matter domination it is linear; finally during Λ domination itslows down. In the bottom left plot for k = 1 hMpc−1, the solid red line is alinear function of a normalized in order to fit δcdm during matter domination.• the baryon perturbations follow closely photon perturbations until decou-pling, experiencing the same acoustic oscillations. After decoupling, theygrow very quickly towards δcdm, and after a ∼ 0.01 the baryon perturba-tions remain equal to the CDM ones.58 0 0.2 0.4 0.6 0.8 1 0 0.02 0.04 0.06 0.08 0.1P(k *)f ν / P(k *)f ν=0fνnumerical Nν=1analytical Nν=1numerical Nν=3analytical Nν=3- 8 fνFig. 12. Ratio of the matter power spectrum including Nν = 1 or 3 degeneratemassive neutrinos to that with three massless neutrinos, computed at the scalek∗ = 5hMpc−1 for fixed parameters (ωm, ΩΛ) = (0.147, 0.70), and plotted as afunction of the density fraction fν . The numerical result is compared with thesemi-analytical approximation of Eq. (141) and with the linear approximation −8 fν .For a ΛMDM model (right plots in Fig. 11), the main differences with respectto the previous case are(1) the fact that equality takes place a bit later, by a fraction (1−fν)−1 = 1.1.(2) the behavior of δν after the time of the non-relativistic transition, whichis shown at the bottom of the plots by the label “anr”. For a > anr, the neu-trino density perturbations track δcdm = δb instead of δγ . For large scales withk < knr (upper plot), the non-relativistic transition takes place when the wave-length is larger than the Hubble radius, so δcdm and δν are not very differentwhen a = anr, and soon after δν reaches δcdm. For slightly smaller scales likek = 10−2hMpc−1 (right plot in second row), the neutrino perturbations aremuch smaller than the CDM ones when a = anr, but then they have timeto grow and reach δcdm before the present time. For k ≥ 10−1hMpc−1 (lowerplots), they do not have time to do so and remain subdominant today.(3) the growth of CDM perturbations is suppressed in the ΛMDM case for tworeasons: first because equality takes place a bit later, and more importantlybecause δcdm does not grow as fast as the scale factor for a > anr. This can beseen very clearly on the lower right plot, where we show exactly the same linearfunction as in the lower left one. The change in the time of equality shifts theδcdm normalization down, while the effect of non-relativistic neutrinos lowersthe value of the slope as predicted analytically in the last subsection.59 0 0.2 0.4 0.6 0.8 1 1.2110-110-210-310-4P(k)fν / P(k)f ν=0k (h/Mpc)knrFig. 13. Ratio of the matter power spectrum including three degenerate massiveneutrinos with density fraction fν to that with three massless neutrinos. The pa-rameters (ωm, ΩΛ) = (0.147, 0.70) are kept fixed, and from top to bottom the curvescorrespond to fν = 0.01, 0.02, 0.03, . . . , 0.10. The individual masses mν range from0.046 eV to 0.46 eV, and the scale knr from 2.1×10−3hMpc−1 to 6.7×10−3hMpc−1as shown on the top of the figure. keq is approximately equal to 1.5× 10−2hMpc−1.Looking now at all wavenumbers, we plot in Fig. 12 the ratio of the matterpower spectrum for ΛMDM over that of ΛCDM, for different values of fν ,but for fixed parameters (ωm, ΩΛ). Here again, the ΛMDM model has threedegenerate massive neutrinos. As expected from the analytical results, thisratio is a step-like function, equal to one for k < knr and to a constant fork ≫ keq. The value of the small-scale suppression factor is plotted in Fig. 13as a function of fν and of the number Nν of degenerate massive neutrinos, stillfor fixed (ωm, ΩΛ). The numerical result is found to be in excellent agreementwith the analytical prediction of Eq. (141). For simplicity, the growth factorg(a0) ≃ 0.8 can even be replaced by one in Eq. (141) without changing theresult significantly. The well-known formula P (k)fν/P (k)fν=0 ≃ −8 fν is areasonable first-order approximation for 0 < fν < 0.07.4.6 Summary of the neutrino mass effects4.6.1 Effects on CMB and LSS power spectra for fixed (ωm, ΩΛ) and degen-erate massesIn Fig. 14, we show CTl and P (k) for two models: ΛCDM with fν = 0 andΛMDM with Nν = 3 massive neutrinos and a total density fraction fν = 0.1.We also display for comparison the neutrinoless model of Sec. 4.4.6. In all6060005000400030002000100001400120010008006004002002l(l+1) C l / 2pi (µK)2lno ν’sfν=0fν=0.110410310210-110-210-3P(k) (Mpc/h)3k (h/Mpc)no ν’sfν=0fν=0.1Fig. 14. CMB temperature anisotropy spectrum CTl and matter power spectrumP (k) for three models: the neutrinoless ΛCDM model of section 4.4.6, a more re-alistic ΛCDM model with three massless neutrinos (fν ≃ 0), and finally a ΛMDMmodel with three massive degenerate neutrinos and a total density fraction fν = 0.1.In all models, the values of (ωb, ωm, ΩΛ, As, n, τ) have been kept fixed.60005000400030002000100001400120010008006004002002l(l+1) C l / 2pi (µK)2lfν=0fν=0.1, fixed (ωc, ΩΛ)fν=0.1, fixed (ωc, h) 10410310210-110-210-3P(k) (Mpc/h)3k (h/Mpc)fν=0fν=0.1, fixed (ωc, ΩΛ)fν=0.1, fixed (ωc, h) Fig. 15. CMB temperature anisotropy spectrum CTl and matter power spectrumP (k) for three models: the same ΛCDM model as in the previous figure, with threemassless neutrinos (fν ≃ 0); and two models with three massive degenerate neutri-nos and a total density fraction fν = 0.1, sharing the same value of ωb and ωcdm asthe massless model, which implies a shift either in h (green dashed) or in ΩΛ (bluedotted).models, the values of (ωb, ωm, ΩΛ, As, n, τ) have been kept fixed, with theincrease in ων being compensated by a decrease in ωcdm. There is a cleardifference between the neutrinoless and massless neutrino cases, caused by alarge change in the time of equality and by the role of the neutrino energy-momentum fluctuations in the perturbed Einstein equation [91]. However ourpurpose is to focus on the impact of the mass, i.e. on the difference betweenthe solid (red) and thick dashed (green) curves in Fig. 14.Impact on the CMB temperature spectrum. For fν ≤ 0.1, the threeneutrino species are still relativistic at the time of decoupling, and the di-rect effect of free-streaming neutrinos on the evolution of the baryon-photon61acoustic oscillations is the same in the ΛCDM and ΛMDM cases. Therefore,the effect of the mass is indirect, appearing only at the level of the backgroundevolution: the fact that the neutrinos account today for a fraction Ων of thecritical density implies some change either in the present value of the spatialcurvature, or in the relative density of other species. In this section, we chooseto maintain a flat Universe with fixed (ωm, ωb,ΩΛ) –other options would bepossible, as discussed in the next section. Thus, while Ωb and ΩΛ are con-stant, Ωcdm is constrained to decrease as Ων increases. The main effect on theCMB anisotropy spectrum results from a change in the time of equality. Sinceneutrinos are still relativistic at decoupling, they should be counted as radia-tion instead of matter around the time of equality, which is found by solvingρb + ρcdm = ργ + ρν . This givesaeq =ωrωb + ωcdm, (143)where ωr stands for the radiation density extrapolated until today assumingthat all neutrinos would remain massless, given by Eq. (11) with Neff = 3.04.So, when fν increases, aeq increases proportionally to [1 − fν ]−1: equality ispostponed. We already saw in Sec. 4.4.7 that this produces an enhancement ofsmall-scale perturbations, especially near the first acoustic peak. Also, post-poning the time of equality increases slightly the size of the sound horizon atrecombination. These two features explain why in Fig. 14 the acoustic peaksare slightly enhanced and shifted to the left in the ΛMDM case.If neutrinos were heavier than a few eV, they would already be non-relativisticat decoupling. This case would have more complicated consequences for theCMB, as described in [85]. However, we will see in section 5 that this situationis disfavoured by current upper bounds on the neutrino mass.Impact on the matter power spectrum. This issue has been discussedin details in the previous subsection: the combined effect of the shift in thetime of equality and of the reduced CDM fluctuation growth during matterdomination produces an attenuation of small-scale perturbations for k > knr,obeying to Eq. (141) in the large k limit. Various studies, including Refs.[93,95,96,97,98,99], have derived some analytical approximation to the fullMDM or ΛMDM matter power spectrum, valid for arbitrary scales and red-shifts. However, the Λ 6= 0 case is only considered in Refs. [93,98].4.6.2 Degeneracy with other cosmological parametersPossible parameter degeneracy in the CMB temperature spectrum.Since the effect of the neutrino mass on CMB fluctuations is indirect and ap-pears only at the background level, one could think that by changing the value62of other cosmological parameters it would be possible to cancel exactly this ef-fect. For the simplest ΛMDM model with only seven cosmological parameters,this is impossible: one cannot vary the neutrino mass while keeping fixed aeqand all other quantities governing the CMB spectrum, that we already listedin Sec. 4.4.7. Let us illustrate this with a particular example. Since the timeof equality is related to (ωb + ωcdm) rather than ωm, we could try to maintainthis sum fixed. In this case, it is now ωm that will increase like [1− fν ]−1. Westill have some freedom concerning other parameters: we could choose eitherto maintain a constant ΩΛ, in which case h will increase as [1 − fν ]−1/2; orto maintain a constant h, in which case (1 − ΩΛ) will increase as [1 − fν ]−1.The result of these two possibilities is displayed on Fig. 15. Since the valueof the scale factor at equality is now more or less fixed, the change in peakamplitude is smaller than in Fig. 14. The counterpart is a larger shift in peaklocation: since we now have variations of either h or ΩΛ, the angular scale ofthe sound horizon at decoupling cannot be kept constant.We conclude that for a seven parameter ΛMDM model, there is no parameterdegeneracy in the CMB spectrum. Therefore, it is possible to constrain theneutrino mass using CMB experiments alone (although neutrinos are still rel-ativistic at decoupling). We will see in the next sections how this applies tocurrent and future experiments.This conclusion can be altered in more complicated models with extra cos-mological parameters. For instance, allowing for an open Universe, it wouldbe possible to increase fν with fixed (ωcdm, ωb, ΩΛ), and to tune the spatialcurvature in order to shift back the peaks to their original location. One couldalso vary the number of relativistic degrees of freedom in order to play with thetime of equality (see e.g. [100]). In such extended models the CMB alone is notsufficient for constraining the mass, but fortunately the LSS power spectrumcan lift the degeneracy.Possible parameter degeneracy in the matter power spectrum. Thepossibility of a parameter degeneracy in the matter power spectrum dependson the interval [kmin, kmax] in which it can be accurately measured. Ideally,if we could have kmin ≤ 10−2hMpc−1 and kmax ≥ 1 hMpc−1, the effect ofthe neutrino mass would be non-degenerate, because of its very characteristicstep-like effect. In contrast, parameters like the scalar tilt or the tilt runningchange the spectrum slope on all scales, while the cosmological parameterswhich determine the shape of the matter power spectrum below the scale ofequality change the slope for all wavenumbers k ≥ keq.The problem is that usually, the matter power spectrum can only be accu-rately measured in the intermediate region where the mass effect is neithernull nor maximal: in other words, many experiments only have access to thetransition region in the step-like transfer function. In this region, the neutrino63mass affects the slope of the matter power spectrum in a way which can beeasily confused with the effect of the scalar tilt, tilt running, or many othercosmological parameters. Because of these parameter degeneracies, the LSSdata alone cannot provide significant constraints on the neutrino mass, andit is necessary to combine them with CMB data. The latter usually lift alldegeneracies. For instance, the CMB probes primordial fluctuations on a verybroad range of scales and makes a very good determination of the tilt andtilt running: so, degeneracies with ns or dns/dk are usually not a problem. Wewill see however in Sec. 5.6 that when exotic models with e.g. extra relativisticdegrees of freedom, a constant equation-of-state parameter of the dark energydifferent from −1 or a non-power-law primordial spectrum are introduced, theneutrino mass bound can become significantly weaker.4.6.3 Effects of non-degenerate neutrino massesIt is often said that cosmological perturbations only probe the total neutrinomass∑imi = m1 +m2 +m3 (from now on denoted as Mν). Let us scrutinizethis point.We saw in the previous subsection that small neutrino masses only have anindirect background effect on the CMB anisotropy spectrum, related to thecurrent density ων , and therefore to the total mass Mν (see Eq. 12). So, invery good approximation, CMB anisotropies probe the total mass with noinformation on the splitting between different mass eigenstates.As far as LSS is concerned, the situation is a bit more subtle. In Sec. 4.5,we always assumed for simplicity that there was one, two or three massiveneutrino species with a common mass. In this case, we have seen that theposition of the break in the power spectrum P (k) is related to knr, i.e. tothe time of the non-relativistic transition which depends on the individualmass mν . Instead, the amplitude of the break is related both to fν and anr,which depend respectively on Mν and mν . However, the results in Fig. 12present a much stronger dependence on fν than on Nν : thus like for the CMB,the dominant effect is that of the total mass, but the individual masses arenot completely irrelevant. In conclusion, we expect various scenarios with thesame total mass Mν and different numbers of massive species Nν = 1, 2, 3 toshare the same CMB spectrum, but not the same matter power spectrum,with small variations in the scale and amplitude of the damping caused byneutrino free-streaming. For the mass schemes shown in Fig. 1, the normalhierarchy (NH) and the inverted hierarchy (IH), the differences are expectedto be even more subtle, because there are three distinct values of mν , anr andknr.This problem has been studied numerically in Ref. [101], using a modified6411.051.11.151.210-4 10-3 10-2 10-1 1ρ / ρ(m0,m0,m0)(a/a0)12IHNH0.990.99511.0051.0110-4 10-3 10-2 10-1P(k) / P(k)(m0,m0,m0)k (h/Mpc)12IHNHFig. 16. (Left) Evolution of the total neutrino energy density as a function of thescale factor of the Universe for models where the same total mass Mν = 0.12 eVis distributed among the three species in various ways. Each line corresponds tothe energy density in one of four different cases (only one massive eigenstates (1),two degenerate massive states (2), normal hierarchy (NH), inverted hierarchy (IH))normalized to the case with three degenerate massive states. (Right) Comparisonof the matter power spectrum obtained for the same models, divided each time bythat with three degenerate massive states. Differences in the mass splitting affectthe position and amplitude of the break in the power spectrum.version of cmbfast with three independent neutrino sectors 3 . Fig. 16 is takenfrom this reference. First, for a fixed total neutrino mass Mν = 0.12 eV, wesee on the left plot the ratio of the total neutrino density ρ¯ν(a) in the NH,IH, Nν = 1 or degenerate Nν = 2 scenario over that of the degenerate Nν = 3scenario. The bump, more or less pronounced depending on the case, is relatedto the different epochs of transition to the non-relativistic regime. In the rightplot, we can see the ratio between the matter power spectrum P (k) in thesame cases over that of the degenerate Nν = 3 scenario. The bump aroundk = 10−2 hMpc−1 is caused by the shift in the break scale, and the large-kasymptotic values differ from one because of the different break amplitudes.Generally speaking, the NH scenario interpolates between the Nν = 1 and thedegenerate Nν = 3 cases, depending on the value of the lightest neutrino; theIH case interpolates between the degenerate Nν = 2 and Nν = 3 cases. But thedifference between the NH and IH power spectra always remain very small:for instance, in the case shown in Fig. 16 with Mν = 0.12 eV, the two P (k)differ at most by 0.1%. It is very unlikely that this effect can be measured oneday (see however section 6).3 See also the recent discussion in [102], as well as the study in [103], where cambwas modified in order to perform calculations with two different neutrino masses.65 0.7 0.8 0.9 1 1.1 1.2 1.3 0.01 0.1 1[(1+z)/g(z)]2n P(k,z) / [1/g(0)]2n P(k,0)k (h/Mpc)0 < z < 10n = 1n = 1- 3/5 fνFig. 17. Shape distortion of the matter power spectrum P (k, z) at different redshifts.Upper solid curves: the power spectra are divided by [a g(a)]2 = [g(z)/(1 + z)]2,which accounts for the evolution on large scales. From top to bottom, the curvescorrespond to z = 10, 8, 6, 5, 4, 3, 2, 1, 0. Lower dashed curves: the power spectraare divided instead by [a g(a)]2−6/5 fν = [g(z)/(1 + z)]2−6/5 fν , which approximatesthe power spectrum evolution on small scales. From top to bottom, the curvescorrespond to z = 0, 1, 2, 3, 4, 5, 6, 8, 10. All cosmological parameters are kept fixedand the neutrino density fraction is fν = 0.1.4.6.4 Massive neutrinos and redshift dependence of the matter power spec-trumIn absence of neutrinos, we have already seen that the shape of the powerspectrum during matter and Λ domination is redshift-independent for modeswell inside the Hubble radius, while the global power spectrum amplitudegrows like [a g(a)]2, where g(a) is a damping factor normalized to g = 1 fora≪ aΛ (here aΛ is the scale factor at matter/Λ equality). In terms of redshiftevolution, this givesP (k, z)fν=0 =(g(z)(1 + z) g(0))2P (k, 0)fν=0 , k ≫ aH . (144)The reason for this shape invariance is simply that the matter fluctuationsobey to a scale-independent equation of evolution (see for instance Eq. (128),which can easily be generalized to the case of Λ domination).In presence of massive neutrinos, the shape of the power spectrum is not in-variant with redshift. Indeed, we are effectively dealing with a two-componentsystem (baryons and CDM on the one hand, neutrinos on the other) in whichthe largest scales satisfy δν = δcdm ∝ [a g(a)], while for the smallest scalesδν ≪ δcdm ∝ [a g(a)]1−3/5 fν , as explained in Sec. 4.5.6. Therefore, the power66spectrum has a different redshift dependence on large and small scales, obeyingP (k, z) =(g(z)(1 + z)g(0))2P (k, 0) for aH < k < knr ,(g(z)(1 + z)g(0))2−6/5 fνP (k, 0) for k ≫ knr .(145)We illustrate this behavior in Fig. 17. Had we plotted the ratio P (k, z)/P (k, 0),we would have seen both the shape and amplitude evolution of the matterpower spectrum. Instead, we want to visualize only the shape distortion. So,we divide all power spectra either by [g(z)/(1 + z)]2 or [g(z)/(1 + z)]2−6/5 fν .As expected, in the first case, the largest scales are corrected for their timeevolution, and the ratio remains equal to one for k < 0.01 hMpc−1; in thesecond case, the smallest scales are corrected for their time-evolution and theratio is equal to one for k > 2 hMpc−1. Between these two limits, the neutrinoperturbations are just about to reach the δν −→ δcdm asymptotic behavioraround the present time (see Sec. 4.5.7). For fν = 0.1, the maximal shapedistortion of the matter power spectrum is as large as 6% for z = 1 and 15%for z = 3.5 Current observations and boundsIn this section we review how the available cosmological data can be usedto get information on the absolute scale of neutrino masses, complementaryto laboratory experiments. After a brief introduction to the statistical meth-ods used in cosmology (in particular, Bayesian inference), we describe theimportance of each set or combination of cosmological data with respect toneutrino masses, and explain how the bounds depend on the data used andthe underlying cosmological model. We summarize the current limits in Tables1–3, where the upper bounds on neutrino masses are given at 95% C.L. aftermarginalization over all free cosmological parameters. We briefly discuss howthe current bounds on neutrino masses can be relaxed or even avoided whenneutrinos possess non-standard properties.5.1 Statistical methods for parameter inference from cosmological dataThe theory of cosmological perturbations is a stochastic theory. For a givenmodel, corresponding to some parameter values θi, the theory predicts theprobability of any particular realization, i.e. of any Universe with given per-turbations. Therefore, it can also predict the probability of obtaining a given67data set. The likelihood function L(xn|θi) is defined as the probability of ob-serving the data xn given the model parameters θi. Such a function can bebuilt for any data which has been properly modeled, i.e. for which the instru-mental noise properties have been fully understood, and the level of correlationbetween different data points has been estimated.The likelihood function can be used as a tool for solving the inverse problem,i.e. measuring the goodness-of-fit of a model given the data, or deriving con-fidence limits on the model parameters θi. This highly non-trivial task can beperformed in different ways; most particle physicists follow the so-called fre-quentist approach, while until now, cosmologists usually prefer the Bayesianapproach (with very few exceptions, see e.g. [104,105,106]). None of thesemethods is better in absolute; they are not directly comparable, because theyaddress different questions. The selection of one method rather than the othershould depend of the particular goal of the analysis, on how much the theoryis constrained and understood, and on how much we trust the data modeling(for a detailed comparison of the two methods, see [107] or [108] and referencestherein).The neutrino mass bounds derived in the next subsections are all based on theBayesian inference method, which has presumably no secrets for readers withsome experience in cosmology. On the other hand, many particle physicistsare so familiar with the frequentist philosophy that cosmological bounds leavethem perplex. In the rest of this subsection, we briefly summarize the methodused for deriving Bayesian confidence limits. Then, we emphasize some prop-erties of Bayesian inference which are radically different from their frequentistcounterpart.In any Bayesian parameter extraction, the first step is to assume a theoreticalmodel, with a set of N free parameters θi, and N probability distributionsΠ(θi) representing our knowledge on each parameter a priori, before usingthe data. These priors can be simply flat distributions over the interval inwhich the parameter is meaningful. For instance, in the case of the neutrinofraction fν , it seems reasonable to define Π(fν) = 1 for 0 < fν < 1 andΠ(fν) = 0 otherwise: such a prior is called a top-hat. Priors can also containsome information from external data sets. For instance, if one of the freeparameters is the current expansion rate H0, one can choose to derive itsprior from the results of the Hubble Space Telescope (HST) Key Project, anddefine Π(H0) as a Gaussian probability with central value 72 km s−1Mpc−1and standard deviation σ = 8 km s−1Mpc−1 [109].Given the likelihood L(xn|θi) and the priors Π(θi), one can compute the prob-ability distribution of any parameter θj , or the joint probability for any subsetof parameters. Such a distribution is called a posterior, since it represents theparameter probability a posteriori, after including the full data. Posterior dis-68 0 1 2 3 4 5 6 7 0 0.1 0.2 0.3 0.4 0.5 0.6L1(θ)L2(θ)L1(θ) x L2(θ)-2-1 0 1 2 1 2L(θ1,1)L(θ1,θ2)P(θ1)θ1θ2Fig. 18. Situations which are possible in Bayesian inference, while they would beparadoxical in a frequentist analysis. (Left) Adding more data does not always leadto stronger bounds. We consider here a model with one free parameter θ with flatprior. A first data set has a Gaussian likelihood L1(θ) which predicts θ = 0.3 ± 0.1at the 68% confidence level (this interval corresponds to the solid vertical lines). Asecond data set has a non-Gaussian likelihood L2(θ). The product L1(θ) × L2(θ)leads to a wider allowed range, corresponding to the dotted vertical lines. (Right)Adding more free parameters does not always lead to weaker bounds. We study herea two-dimensional model with likelihood L(θ1, θ2) and flat priors. If there are goodphysical motivations to fix θ2 = 1, one can also study the one-dimensional modelwith likelihood L(θ1, 1) (the thick line on the plot). We choose the likelihood in suchway that∫dθ1L(θ1, 1) = 1: the thick line can be directly interpreted as the posteriordistribution for θ1 in the one-dimensional model. For the two-dimensional model, thedashed line shows the posterior P (θ1) obtained by marginalizing over θ2, and nor-malizing to one. Given the very specific form of the likelihood, the two-dimensionalmodel leads to a narrower posterior and to stronger bounds on θ1.tributions are obtained by marginalizing over the remaining free parameters.In practice, this amounts in integrating the product of the likelihood timesthe priors over these parameters. The result is a function of one or severalparameters θj , which can be normalized to one. One finally obtains a proba-bility distribution P(θj), from which it is straightforward to derive confidencelimits: the n% confidence interval I is defined as the region in which(1) the cumulative probability is equal to n/100:∫Idθj P(θj) = n/100, and(2) the probability is maximal: the value of P(θj) for any point inside I islarger than that for any point outside I.The Bayesian inference method is well-defined, efficient, and has several ad-vantages. However, Bayesian results cannot be interpreted in the same way asfrequentist ones, and there are significant differences that should be kept inmind:• Bayesian inference always produce some confidence intervals, even if the69assumed model is unable to provide good fits to the data. The results of aBayesian analysis contain no statements on the goodness-of-fit of the model.Therefore, most authors provide some complementary information, like themaximum value of the likelihood Lmax (or equivalently, the minimum valueof the effective χ2, which is defined – even for a non-Gaussian likelihood –as χ2eff = −2 lnL). Then, it is possible to estimate in the frequentist wayhow good is the best fit 4 .• for each parameter, the best-fit value θbi generally differs from the most likelyvalue θli. Indeed, θbi corresponds to the best-fit model, while θli maximizesthe marginalization integral. They can be quite different, for instance if forθi fixed to θbi there are many bad fits and very few excellent fits, while forθi fixed to θli there are many good ones. In a future analysis, it would beconceivable that the best-fit value of the total neutrino mass be zero, whilethe most likely value be strictly positive. This is a natural outcome of theBayesian approach, which is based on probability distributions in parame-ter space, and therefore favours regions in which models are good in average.• the exact meaning of a Bayesian confidence interval is: “given a particularunderlying model and given the data, it is likely to be in this region, and un-likely to be outside”. One should be careful not to over-interpret a Bayesianconfidence interval. It is risky to say that a parameter value sitting outsideof the interval is excluded. First, it is possible that with such a value onecan find a very good fit to the data (in the frequentist sense): the Bayesiananalysis could penalize this point just because the well-fitting region has atiny volume in the subspace of parameters to be marginalized over. Second,one could possibly assume another model (with extra free parameters) inwhich the same parameter value would be comfortably allowed. In summary,Bayesian confidence intervals are well-defined, but they are attached to aparticular underlying model. It makes no sense to quote a Bayesian upperlimit, e.g. on the neutrino masses, without explicitly specifying the modelassumed.• in the frequentist approach, adding more data always lead to strongerbounds on each parameter, while adding more free parameters can onlyenlarge the allowed region. In the Bayesian approach, this is true in generalbut not always: it is easy to build counter-examples, as we did in Fig. 18.Again, Bayesian inference offers many advantages for cosmological parameterextraction, and the previous items should not be regarded as criticisms, but4 However, comparing the merits of two models is a non-trivial task. A simple com-parison of the two maximum likelihood values is insufficient. Following the Bayesianphilosophy, the most rigorous approach consists in computing the so-called BayesianEvidence [110,111].70rather as a list of warnings for correctly interpreting the results quoted in thissection.5.2 CMB anisotropiesThe experimental situation of the measurement of the CMB anisotropieschanged radically with the first year release of WMAP data (WMAP1), thatincludes the temperature × temperature (TT) [112] and temperature × E-polarization (TE) [113] correlation functions. The recent third-year release(WMAP3) improves the TT and TE power spectra [114], and adds a detectionof the E-polarization self-correlation spectrum (EE) [115]. The WMAP exper-iment measured with high precision the TT spectrum on large angular scales(l <∼ 900), while at similar or smaller scales we have results from experimentsthat are either ground-based such as the Arcminute Cosmology Bolometer Ar-ray Receiver (ACBAR) [116], the Very Small Array (VSA) [117], the CosmicBackground Imager (CBI) [118] and the Degree Angular Scale Interferome-ter (DASI) [119], or balloon-borne such as ARCHEOPS [120], BOOMERANG(TT [121], TE [122] and EE [123]) and MAXIMA [124]. When using data fromdifferent CMB experiments, one should take into account that they overlap insome multipole region, and not all data are uncorrelated.Small neutrino masses only have a small influence on the power spectrumof CMB anisotropies, since for mν smaller than approximately 0.5 eV theneutrinos are still relativistic at the time of recombination, so that their directeffect on the CMB perturbations is identical to that of massless neutrinos.However, the signature of a mass smaller than 0.5 eV does not vanish: werecalled in Sec. 4.6 that there is always a background effect, proportional toΩν , which changes some characteristic times and scales in the evolution of theUniverse, and affects mainly the amplitude of the first acoustic peak as wellas the location of all the peaks, as shown in Figs. 14 and 15. Therefore, it ispossible to constrain neutrino masses using CMB experiments only, down tothe level at which this background effect is masked by instrumental noise, or bycosmic variance, or by parameter degeneracies in the case of some cosmologicalmodels beyond the minimal Λ Mixed Dark matter (ΛMDM) framework.We list in Table 1 the bounds on the sum of neutrino masses that were obtainedin analyses of CMB data after the first WMAP results appeared. In theseworks, a minimal ΛMDM scenario was assumed: the total neutrino mass wasthe only additional parameter with respect to a flat ΛCDM cosmological modelcharacterized by 6 parameters 5 . However, there were some differences in theanalysis procedure.5 This will be the case for the bounds reviewed in this section, unless specifiedotherwise.71Table 1Upper bounds on Mν (eV, 95% CL) from recent analyses of CMB data only. Notethat the results of Refs. [127,129,130,131] are consistent with each other (see thetext for details).Ref. Bound Data used[125] 10.6 WMAP1[127] 2.0 WMAP1[65] 2.0 WMAP3[129] 2.1 WMAP1, VSA, ACBAR & CBI[130] 1.6 WMAP1, VSA, ACBAR & CBI[131] 3.1 WMAP1, BOOMERANG03, VSA, ACBAR,CBI, DASI & MAXIMAThe analysis in Ref. [125] used a Monte Carlo Markov Chain (MCMC) methodfor the minimization, concluding that WMAP data are consistent with neutri-nos making 100% of the dark matter component (see also [126]), thus givingthe loose bound on the total neutrino mass Mν < 10.6 eV. This would meanthat eV masses are basically indistinguishable from CDM at the epoch of lastscattering, and therefore have little impact on the CMB scales measured byWMAP.However, a very different result was discussed in ref. [127] (see also [128]),where an upper limit of 2 eV was found from the analysis of WMAP1 onlyusing a grid-based method. The authors of this last analysis ascribe the dif-ference with respect to [125] to a better marginalization procedure. However,using the WMAP1 data and an MCMC method, we find the same result as in[127]. Finally, the 2 eV bound is now confirmed by the analysis of the WMAP3data in Ref. [65].In Table 1, we also show the results from two recent works [129,131] basedon the combination of WMAP1 with other recent CMB experiments. We alsoshow the result of a very recent analysis [130] in a cosmological model with 11parameters (similar to that in Table 4 of Ref. [132], but in a flat Universe). The2σ bound obtained by [129,130] is still close to 2 eV. The result of [131], whichinclude more data from BOOMERANG, DASI and MAXIMA, is larger by50%. We saw in Sec. 5.1 that in a Bayesian analysis, it is possible to add moredata and obtain weaker bounds. In the present case, a careful examinationof Fig. 8 in [131] shows that their analysis based on WMAP1 only is in verygood agreement with [127,129]; however, at least one of the DASI or MAXIMAdata set tends to push the bound upwards, because it is better fitted with anon-zero neutrino mass.72In conclusion, many analyses support the conclusion that a sensible bound onneutrino masses exists, of order of 2−3 eV at 2σ. This is an important result,since it does not depend on the uncertainties from LSS data (see Sec. 5.3).The only way to relax this bound would be to complicate the cosmologicalscenario assumed for parameter extraction, but this issue will be discussed inSecs. 5.6 and 5.7.However, these results do not take advantage of the most significant observablesignature of massive neutrinos: the suppression of small-scale matter fluctua-tions induced by neutrino free-streaming. In order to get improved bounds onneutrino masses it is crucial to combine CMB data with independent large-scale structure observations.5.3 Galaxy redshift surveysWe have seen in Sec. 4.6 that free-streaming of massive neutrinos produces adirect effect on the formation of cosmological structures. As shown in Fig. 13,the presence of neutrino masses leads to an attenuation of the linear matterpower spectrum on small scales.In a seminal paper, Hu and collaborators [94] showed that an efficient wayto probe neutrino masses of order eV was to use data from large redshiftsurveys, which measure the distance to a large number of galaxies, giving usa three-dimensional picture of the universe (modulo the effect of unknownpeculiar velocities on top of the Hubble flow, which can be corrected onlystatistically). At present, we have data from two large projects: the 2 degreeField (2dF) galaxy redshift survey [133,134], whose final results were obtainedfrom more than 220,000 galaxy redshifts, and the Sloan Digital Sky Survey(SDSS) [135,89], which will be completed soon with data from one milliongalaxies 6 .One of the main goals of galaxy redshift surveys is to reconstruct the powerspectrum of matter fluctuations on very large scales, whose cosmological evo-lution is described entirely by linear perturbation theory. However, the linearpower spectrum must be reconstructed from individual galaxies, i.e. from ob-jects which underwent a strongly non-linear evolution. A simple analytic modelof structure formation suggests that on large scales, the galaxy-galaxy corre-lation function should be, not equal, but proportional to the linear matterdensity power spectrum, up to a constant factor that is called the light-to-mass bias. This parameter, denoted by b, can be obtained from independentmethods (e.g. high-order statistics beyond the two-point correlation function).6 An extension of this experiment, SDSS-II was recently funded (see the SDSSwebpage http://www.sdss.org)73Table 2Upper bounds on Mν (eV, 95% CL) from recent analyses combining CMB, galaxyredshift surveys (shape of the matter power spectrum only) and other data.Ref. Bound Data used (in addition to WMAP1 except for Ref. [65])[56] 1.2 other CMB (pre-WMAP), 2dF-gal1.0 previous + HST, SNIa[125] 1.74 SDSS-gal[139] 0.75 other CMB (pre-WMAP), 2dF-gal, SDSS-gal, HST[100] 1.0 ACBAR, 2dF-gal, SDSS-gal0.6 previous + HST, SNIa[132] 0.96 VSA, 2dF-gal[137] 1.54 SDSS-gal, SNIa[140] 1.4 other CMB , 2dF-gal, HST, SNIa[131] 1.2 other CMB , 2dF-gal, SDSS-gal[129] 1.27 other CMB , SDSS-gal1.16 other CMB , 2dF-gal[65] 0.87 WMAP3, 2dF-galCurrently, these methods tend to confirm that the linear biasing assumptionis correct, at least in first approximation.A conservative way to use the measurements of galaxy-galaxy correlations inan analysis of cosmological data is to take the bias as a free parameter, i.e. toconsider only the shape of the matter power spectrum at the correspondingscales and not its amplitude. In this case, we will refer to the correspondingdata of the 2dF and SDSS surveys as 2dF-gal and SDSS-gal, respectively.Alternatively, one can add a measurement of b, as recently done in three anal-yses. The value b = 1.04± 0.11(1σ) [2dF-bias] [136] from the 2dF bispectrumanalysis was used in [55], while [137] adopted the estimate b = 0.99±0.11(1σ)[SDSS-bias] from galaxy-galaxy lensing from SDSS [138]. Finally, in Ref. [131]the intermediate estimate b = 1.0± 0.10(1σ) was used.We summarize in Table 2 the bounds on neutrino masses that were obtainedin combined analyses of CMB and LSS data, using galaxy clustering data inthe range of wavenumbers (0.015 − 0.02) h/Mpc< k < (0.15 − 0.20) h/Mpc(including non-linear corrections for the smallest wavelengths [141]). The re-sults in this Table were obtained leaving the bias as a free parameter. Oneconcludes from Table 2 that the addition of LSS data improves the boundswith respect to the numbers in Table 1, with an upper limit on Mν between740.9 and 1.7 eV for SDSS-gal and/or 2dF-gal added to CMB data. In general,one obtains weaker bounds on neutrino masses using preliminary SDSS resultsinstead of 2dF data, but this conclusion is expected to change after the nextSDSS releases. The improvement with respect to previous results is evident:in 2001, ref. [142] found Mν < 4.2 eV (95% CL) from the combination of thePSCz survey [143] with pre-WMAP CMB data, while the analysis in [144]used early 2dF data to obtain Mν < 2.2 eV with priors on the values of Ωmand h.There exist other measurements of cosmological parameters that are oftenadded to CMB and LSS data, as shown in Table 2. For instance, the presentvalue of the Hubble parameter was measured by the Key Project of the Hub-ble Space Telescope (HST), giving h = 0.72± 0.08 (1σ) [109], which excludeslow values of h and leads to a stronger upper bound on the total neutrinomass. In addition, one can include the constraints on the current density ofthe dark energy component deduced from the redshift dependence of type Iasupernovae (SNIa) luminosity, which measures the late evolution of the ex-pansion rate of the Universe. For a flat Universe with a cosmological constant,these constraints can be translated into bounds for the matter density Ωm,which range from the conservative value Ωm = 0.28± 0.14 (1σ) from [145] tothe more recent Ωm = 0.29+0.05−0.03 (1σ) [146] from the analysis of SNIa with theHST (other recent works include [147,148]).As shown in Table 3, the bounds on neutrino masses are more stringent whenthe amplitude of the matter power spectrum is fixed with a measurement ofthe bias [55,131,137], instead of leaving it as a free parameter. The upperlimits on Mν are reduced to values of order 0.5− 0.9 eV (95% CL), althoughsome analyses also add Lyman-α data (see next subsection).Finally, a galaxy redshift survey performed in a large volume can also besensitive to the imprint created by the baryon acoustic oscillations (BAO) atlarge scales on the power spectrum of non-relativistic matter, that we discussedin Sec. 4.4 (see Fig. 7). Since baryons are only a subdominant component ofthe non-relativistic matter, the BAO feature is manifested as a small singlepeak in the galaxy correlation function in real space that was recently detectedfrom the analysis of the SDSS luminous red galaxy (LRG) sample [89] at aseparation of 100 h−1 Mpc. The observed position of this baryon oscillationpeak provides a way to measure the angular diameter distance out to thetypical LRG redshift of z = 0.35, which in turn can be used to constrain theparameters of the underlying cosmological model. The SDSS measurement ofthe angular diameter distance at z = 0.35 [SDSS-BAO] was included in theanalysis of Ref. [152] to get a bound of 0.44 eV (95% CL) on the total neutrinomass Mν , as shown in Table 3.75Table 3Upper bounds on Mν (eV, 95% CL) from recent analyses combining CMB, LSS(including bias and/or Lyα) and other cosmological data.Ref. Bound Data used (in addition to WMAP1 except for Ref. [65])[55] 0.63 ACBAR, CBI, 2dF-gal, 2dF-bias0.68 previous + Lyα [149,150][137] 0.54 SDSS-gal, SNIa, SDSS-Lyα0.42 previous + SDSS-bias[140] 0.47 other CMB, 2dF-gal, HST, SNIa, SDSS-Lyα[151] 0.65 SDSS-gal, HST, Lyα [149,150][131] 0.48 other CMB, 2dF-gal, SDSS-gal, bias b = 1.0 ± 0.10[152] 0.44 other CMB, 2dF-gal, SDSS-gal, HST, SNIa, SDSS-BAO0.30 previous + SDSS-Lyα[65] 0.91 WMAP3, SDSS-gal, SDSS-bias0.68 previous + other CMB, 2dF-gal, SNIa5.4 Lyman-α forestThe matter power spectrum on small scales can also be inferred from data onthe so-called Lyman-α forest. This corresponds to the Lyman-α absorption ofphotons traveling from distant quasars (z ∼ 2 − 3) by the neutral hydrogenin the intergalactic medium. As an effect of the Universe expansion, photonsare continuously red-shifted along the line of sight, and can be absorbed whenthey reach a wavelength of 1216 A˚ in the rest-frame of the intervening medium.Therefore, the quasar spectrum contains a series of absorption lines, whoseamplitude as a function of wavelength traces back the density and temperaturefluctuations of neutral hydrogen along the line of sight. It is then possible toinfer the matter density fluctuations in the linear or quasi-linear regime (seee.g. [153,154,155,156]).In order to use the Lyman-α forest data, one needs to recover the matter powerspectrum from the spectrum of the transmitted flux, a task that requires theuse of hydro-dynamical simulations for the corresponding cosmological model.This matter power spectrum is again sensitive to the suppression of growthof mass fluctuations caused by massive neutrinos. In 1999, Ref. [157] obtainedan upper bound of mν < 5.5 eV (for one massive neutrino species) from thecombined analysis of Lyman-α data and other cosmological measurements.Given the various systematics involved in the analysis pipeline, the robustness76of Lyman-α forest data is still a subject of intense discussion between ex-perts. Among these systematics, we can cite the uncertainty existing on: themean absorption by the intergalactic medium (the effective optical depth), thetemperature-density relation within the absorbing gas, the effect of galacticwinds, or the proper use and accuracy of hydro-dynamical simulations in themildly non-linear regime. Different authors tend to describe or to deal withthese uncertainties in slightly different ways [149,150,158,159], an indicationthat the field might not be fully mature at the moment. In addition, the way inwhich neutrinos are implemented in hydro-dynamical simulations is generallyrather simplistic.Some of the results shown in Table 3 were obtained adding Lyman-α datato CMB and other LSS data. For a free bias, one finds that Lyman-α datahelp to reduce the upper bounds on the total neutrino mass: using the SDSSLyman-α forest [159,160], Ref. [137] finds Mν < 0.54 eV (95% CL), whileRef. [151] quotes 0.65 eV using Lyman-α data from [149,150]. At least tworecent analyses combined results on the bias and the Lyman-α forest: in Ref.[137] the addition of SDSS-bias allowed to reduced the upper bound on Mνto 0.42 eV, while the first year analysis of WMAP collaboration [55] found aslightly weaker bound of 0.68 eV when adding 2dF-bias and Lyman-α datafrom [149,150]. Finally, Ref. [152] found the upper bound Mν < 0.30 eV (95%CL) adding simultaneously SDSS-Lyα and SDSS-BAO data.Thus, at present, the impact of Lyman-α data on the neutrino mass boundsis comparable to that of galaxy bias measurements, if not weaker. We canconclude that both the Lyman-α data and the light-to-mass bias determinationare powerful tools for getting information on neutrino masses, but due tothe many involved systematics, the current bounds obtained with them areprobably not as robust as those in Tables 1 and 2.5.5 Summary of current boundsThe results shown in Tables 1-3 are representative of an important fact: asingle cosmological bound on neutrino masses does not exist. A graphicalsummary is presented in Fig. 19, where the cosmological bounds correspondto the vertical bands, which were grouped according to the ranges quoted inTables 1, 2 and 3, respectively. The thickness of these bands roughly describethe spread of values obtained from similar cosmological data: 2 − 3 eV forCMB only, 0.9− 1.7 eV for CMB and 2dF/SDSS-gal or 0.3− 0.9 eV with theinclusion of a measurement of the bias and/or Lyman-α forest data and/orthe SDSS measurement of the baryon oscillation peak.One can see from Fig. 19 that current cosmological data probe the region of77 0.03 0.1 0.3 1 3 0.001 0.01 0.1 1Total neutrino mass Mν (eV)lightest mν (eV)Present boundsCMB only+ 2dF/SDSS-gal+ bias and/or Ly-α and/or SDSS-BAOInvertedNormalFig. 19. Current upper bounds (95%CL) from cosmological data on the sum ofneutrino masses, compared to the values in agreement at a 3σ level with neutrinooscillation data in Eq. (2).neutrino masses where the 3 neutrino states are degenerate, with a massMν/3.This mass region is conservatively bounded to values below approximately 1 eVfrom CMB results combined only with galaxy clustering data from 2dF and/orSDSS (i.e. the shape of the matter power spectrum for the relevant scales). Theaddition of further data leads to an improvement of the bounds, which reachthe lowest values when data from Lyman-α and/or the SDSS measurement ofthe baryon oscillation peak are included or the bias is fixed. In such cases thecontribution of a total neutrino mass of the order 0.3-0.6 eV seems alreadydisfavoured, in particular covering the recent positive 0ν2β signal from [31](of course with the corresponding caveats, see e.g. the discussion in [14]).There exist other measurements of cosmological parameters that were notincluded in the analyses we have reviewed so far, although they could playa role for neutrino mass determination. In particular, we have seen that anymeasurement that fixes the amplitude of the matter power spectrum, suchas the bias, would be very important. For instance, the parameter σ8 –whichgives the linear rms mass fluctuations in spheres of radius 8 h−1Mpc– can bemeasured, for instance, from the determination of cluster masses through X-ray observations (see e.g. [161]) or from weak lensing (see e.g. [162]). If variousmethods happened to converge towards an accurate determination of σ8, thedetermination of neutrino masses would improve significantly, as discussed inRef. [163]. Since σ8 probes the amplitude of matter fluctuations on scales of78the order a few Mpc which are deeply affected by neutrino free-streaming,a large value of σ8 would push down the upper bound on Mν , while a lowvalue would potentially bring evidence for non-zero masses. However, at themoment, the ensemble of σ8 determinations is rather spread (see Table 5 in[125]), suggesting that not all systematics errors are properly modelled.One of the first analysis of neutrino mass bounds from cosmology used dataon σ8 from cluster abundance to obtain an upper bound of order a few eV[164]. Another possibility was recently explored in [165], using data on theredshift-evolution of the number density of massive galaxy clusters to find thebound Mν < 2.4 eV (95% C.L.). Ref. [166] used another probe of the small-scale power spectrum amplitude from X-ray cluster data, and found somemarginal evidence for a non-zero neutrino total mass, Mν = 0.56+0.30−0.26 eV at68% C.L. This result is related to the fact that the value of σ8 preferred bythese X-ray data is as small as to contradict most other determinations ofthis parameter, for instance from weak lensing. This example shows that thesensitivity of cosmological observations to neutrino masses is a powerful tool,but its implications should not be extracted without care.5.6 Extra parametersAssuming that the relic neutrinos are standard, we have seen that the limitsmainly depend on the cosmological data used in the analysis, although minordifferences on the quoted bounds exist even when using similar data. Thesedifferences arise because not all the analyses in the Tables used the same set ofcosmological parameters and priors (such as the assumption of a flat Universe).It is thus important to test whether the impressive cosmological bounds onneutrino masses change much if additional cosmological parameters, beyondthose included in the minimal ΛCDM, are allowed. This could be the casewhenever a new parameter degeneracy with the neutrino masses arises.An interesting case of degeneracy between cosmological parameters is that be-tween neutrino masses and the radiation content of the universe (parametrizedvia the effective number of neutrinos Neff). The extra radiation partially com-pensates the effect of neutrino masses, provided that other cosmological pa-rameters such as Ωm and h are varied, leading to a less stringent bound onMν [56,100,167,168]. This parameter degeneracy is shown in Fig. 20, takenfrom Ref. [100]. For instance, when one considers a model with four insteadof three species of massive neutrinos, the upper bound found from CMB and2dF/SDSS-gal is relaxed from 0.8 to 1.2 eV when one of the states is muchheavier than the others. These results are interesting for the 4-neutrino massschemes that also incorporate the results of the LSND experiment [19], thatwe discussed in Sec. 2. At present, the LSND regions in the space of oscillation79Neff(a) (b)(c)M (eV)(d)1234567890 0.5 1 1.5 21234567890 0.5 1 1.5 21234567890 0.5 1 1.5 21234567890 0.5 1 1.5 2Fig. 20. Two-dimensional likelihood in (Neff ,Mν) space, marginalized over the othercosmological parameters of the model. We plot the 1σ (green / dark) and 2σ (yellow/ light) allowed regions. Here we used CMB (WMAP1 & ACBAR) and LSS (2dF& SDSS) data, adding extra priors on h (HST) and Ωm (SN99 or SN03) as follows:(a) no priors, (b) HST, (c) HST+SN99, (d) HST+SN03. For details, see ref. [100].parameters are not yet completely disfavoured 7 by cosmological data.Recently a degeneracy between neutrino masses and the parameter w, thatcharacterizes the equation of state of the dark energy component X (pX =wρX), was studied in [170]. For constant values w < −1, in the so-calledphantom energy regime (which could describe particular scenarios based onstring theory or modified gravity), it was shown that the neutrino mass boundcan be relaxed to Mν < 1.48 eV (95% C.L.), compared with Mν < 0.65 eVfor fixed w = −1. Similar conclusions were obtained by the WMAP team (seeFig. 18 in [65]). Ref. [170] studied the effect of this parameter degeneracy onthe analysis of future cosmological data (see also [86] for models with a time-varying equation of state of dark energy). Finally, Ref. [152] showed that data7 However, a very recent analysis [169] showed that in a 4-neutrino scenario, theheavier neutrino mass is bounded to values smaller than 0.55 eV (95% CL) whendata from CMB, LSS and Lyman-α are included.80on baryon acoustic oscillations (see Sec. 5.3) helps in breaking the mν − wdegeneracy.Finally, the neutrino mass bounds could certainly be relaxed by assumingthat the primordial power spectrum has a non-trivial shape or Broken ScaleInvariance (BSI). This eventuality cannot be excluded, for instance if a phasetransition takes place during inflation. Such an assumption opens so manydegrees of freedom in the parametrization of cosmological models that a sys-tematic study is almost impossible. However, some authors [171] used thispossibility in combination with a non-zero neutrino mass for building a cos-mological scenario without any form of dark energy, that could fit most of thedata available at the time of publication, except for SNIa data. This modelis now ruled out by the combination of WMAP with the SDSS galaxy-galaxycorrelation function, but it illustrates the fact that if one is ready to add moredegrees of freedom in cosmological scenarios (in the case of Ref. [171], theywere actually not so many), then there is most probably a way to evade thebounds presented in this review.5.7 Non-standard relic neutrinosFinally, let us remind the reader that the cosmological implications of neutrinomasses could be very different if the spectrum or evolution of the cosmic neu-trino background was non-standard. Our aim here is not to give an exhaustivelist of all possibilities (many results on non-standard neutrinos were reviewedin [6]), but only to briefly summarize some recent works.For instance, the bounds on neutrino masses would be modified if relic neutri-nos presented a momentum spectrum significantly different from the equilib-rium Fermi-Dirac distribution of Eq. (6). For instance, neutrinos could violatethe Pauli exclusion principle and obey Bose-Einstein statistics with importantcosmological and astrophysical implications [172,173]. Non-equilibrium neu-trino spectra can also be produced in low-reheating scenarios [174,175,176,177,178,179],from the decay of massive neutrinos into relativistic products (see e.g. [180,181])or as the result of active-sterile neutrino oscillations after decoupling (see[182,183] and references therein). A recent analysis of cosmological data in-cluding large non-thermal corrections in the neutrino spectra was done in Ref.[184].We conclude giving two examples of models where the cosmological boundson neutrino masses that we have discussed can be evaded. In the first case,massive neutrinos are strongly coupled to a light scalar field, in such a waythat they annihilate as soon as they become non-relativistic and do not affectthe formation of LSS [185]. The viability of this kind of models depends onthe cosmological data included in the analysis and on the number of neutrinospecies strongly interacting (see the discussions in Refs. [186,187,188]).The second possibility, described in [189,190] and many other recent works (seee.g. the discussion in [191]), concerns neutrinos coupled to the dark energy sothat the dark energy density is a function of the neutrino mass (Mass-VaryingNeutrinos or MaVaNs). Then, neutrino masses are only relevant for the veryrecent stages of the Universe, so that no bounds can be placed from high-redshift cosmological data. Note, however, that this is not true for all MaVaNsscenarios, see for instance [192].6 Future sensitivities and new experimental techniquesIf the characteristics of future experiments are known with some precision,it is possible to assume a “fiducial model”, i.e. a cosmological model thatwould yield the best fit to future data, and to estimate the error bar on aparticular parameter that will be obtained after marginalizing the hypothet-ical likelihood distribution over all the other free parameters (see Sec. 5.1).Technically, the simplest way to forecast this error is to compute a Fishermatrix, as explained in the next subsection. This technique has been widelyused in the literature, for many different models and hypothetical datasets.Here we will focus on the results for σ(Mν), the forecast 68% CL error onthe total neutrino mass, assuming various combinations of future observations(CMB anisotropies, galaxy redshift surveys, weak lensing surveys, . . . ) andvarious cosmological models with fiducial parameter values. Many papers de-rived some predictions for σ(Mν); we will summarize here the main resultsand see whether the expected sensitivities will be sufficient for detecting thetotal neutrino mass, even if it is close to the minimum values guaranteed bypresent data on flavour neutrino oscillations.6.1 Fisher matrixIn a Bayesian analysis, the probability distribution in the space of cosmologicalparameters is inferred from the likelihood L(xn|θi) of the data xn given atheoretical model described by a set of parameters θi (see Sec. 5.1). If weassume that the maximum likelihood Lmax is obtained for a particular modelθ0i (the “fiducial model”), and if we approximate the likelihood function as amultivariate Gaussian in the vicinity of the maximum, the symmetrical error82on each parameter can be inferred from the Fisher matrix, defined asFij ≡ −∂2 lnL∂θi∂θj∣∣∣∣∣θ0i. (146)The 1σ (or 68% confidence) error bar on a particular parameter θj assumingthat all other parameters are fixed to the fiducial values θ0i would be givenby (Fjj)−1/2. However, the really interesting quantity is the error on θj as-suming that the other parameters are unknown (and simply marginalized outby integrating over the likelihood). In that case, the error can be shown tobe [193]σ(θj) = (F−1)1/2jj . (147)Geometrically, and in the multivariate Gaussian approximation, the Fishermatrix gives the coefficients of the ellipsoids Fijθiθj = f(c) corresponding togiven confidence levels c in parameter space. Therefore, the study of its eigen-vectors and eigenvalues gives an idea of the linear parameter combinationswith the best and poorest determination (the latter correspond to parameterdegeneracies).In many cases one refers to data points xi (i = 1, ..., n) with a Gaussianprobability distribution centered in zeroL(~x|~θ)= [(2π)n|C|]−1/2e− 12~xtC~x , (148)where the “data covariance matrix” C contains the variance expected for thedata, C ≡ 〈~x~xt〉, after a detailed modelling of all possible contributions. Typ-ically, the data covariance matrix will be the sum of the theoretical modelprediction (for parameters ~θ) and of the noise variance estimated from theinstrumental characteristics. In general, the data covariance matrix is non-diagonal and the data points are correlated with each other. A purely alge-braic calculation shows that the Gaussian likelihood of Eq. (148) inserted inEq. (146) leads to the following expression for the Fisher matrix [193]Fij =12Trace[C−1∂C∂θiC−1∂C∂θj]. (149)6.2 Future CMB experimentsThe quality of the data from the WMAP satellite [55,65], complemented bythe results of other experiments at smaller angular scales such as those listed83in section 5.2, has shown the importance of CMB data as a probe of cosmo-logical parameters. There are several other projects in operational or designstage, aimed mainly at improving the sensitivity towards small scales and/orpolarization anisotropies. The neutrino mass is not the primary target of theseexperiments, because as we have already seen in Sec. 4.6, small masses onlyhave a modest signature on the CMB. But still, we know that there is a back-ground effect proportional to ων which renders the primary CMB anisotropiessensitive –at least in principle– to very small values of the total neutrinomass. Instead, typical targets are the scalar tilt and the tensor-to-scalar ratio,in order to better constrain the predictions of inflationary theories. However,in order to measure with precision the neutrino masses, it will be useful tocombine some large scale structure observations with the best possible CMBdataset in order to reduce as much as possible parameter degeneracies. In thissection we review the Fisher matrix computation and expected sensitivity offuture CMB experiments alone; in the next sections the CMB contribution tothe Fisher matrix will be combined with that from other hypothetical obser-vations.The CMB temperature maps can be decomposed in multipoles∆TT(θ, φ) =∑l,maTlmYlm(θ, φ) , (150)and similarly the polarization map are described by multipoles aElm and aBlm.The data covariance matrix C = 〈aXlmaX′l′m′〉 receives contributions from:• primary anisotropies, for which the expected variance is predicted by thecosmological perturbation theory: 〈aXlmaX′l′m′〉 = δll′δmm′CXX′l . For parity rea-sons the non-vanishing power spectra are CTTl , CEEl , CBBl and CTEl .• secondary anisotropies imprinted at redshifts z ≪ 1100, like the late Inte-grated Sachs-Wolfe (ISW) effect and the weak lensing induced by nearbyclusters, as well as various foreground contaminations. The late ISW effectis usually included in the theoretical power spectrum computation, whilethe lensing effect is small enough to give a negligible contribution to C(excepted for the small-scale B-mode variance). Here we will neglect thelensing distortions, which will be the object of Sec. 6.4. Most of the fore-ground contamination is expected to be accurately removed from the rawdata using a multi-frequency analysis (since most foreground signals do nothave a blackbody distribution).• experimental noise, which is usually parametrized by two quantities: theangular resolution and the sensitivity per pixel (here, a pixel represents thesmallest area in a map on which an independent measurement is performed).Intuitively, a given angular resolution implies that the error-bar will grow84dramatically below a given scale, and the finite sensitivity per pixel impliesthat even in absence of a primary signal, the covariance matrix would getsome non-zero contribution at a given level. Technically, the angular resolu-tion is expressed as an angle θb, the Full-Width at Half-Maximum (FWHM)of the instrument beam. The sensitivity (or noise squared variance) per pixel∆2X (for X = T,E,B) is usually given in units of a squared temperature,and the dimensionless sensitivity per pixel ∆¯2X is obtained dividing by thesquare of T0 ≃ 2.726 K. The solid angle of a pixel can be approximated byθ2b , so that the dimensionless sensitivity per steradians reads (θ2b∆¯2X). Forone instrument, the noise variance can be shown to be [194]〈aXlmaX′l′m′〉 = δll′δmm′δXX′NXXl , NXXl ≡ (θ2b∆¯2X)B−2l , (151)where the Gaussian beam window function Bl ≡ exp[−l(l + 1)θ2b / (16 ln 2)]has the effect of cutting the sensitivity for l ≫ θ−1b . Note that for X 6= X ′the variance vanishes, since the noise in one map (for mode T , E or B) isexpected to be uncorrelated with the noise in another map. Finally, mostexperiments consist of various detectors, for which a global sensitivity canbe built by summing the inverse variances〈aXlmaX′l′m′〉 =(∑i[θ−2b ∆¯−2X B2l]i)−1. (152)Detectors are generally grouped in channels, i.e. ensembles of detectorsworking at a given frequency. An efficient foreground extraction requiresa large number of channels, at the expense of decreasing the number ofdetectors per channel, and hence increasing the noise per channel.The theoretical and noise variance are usually assumed to be uncorrelated,which allows to express the covariance matrix in a rather simple way. If werepresent the data as a vector ~x = xlm = (aT , aE, aB)lm, where the indexes(l,m) run over all multipoles, the full data covariance matrix can be decom-posed in diagonal blocks Clm representing the covariance 〈xlmxtlm〉 for a given(l,m)Clm = Cl =CTTl +NTTl CTEl 0CTEl CEEl +NEEl 00 0 CBBl +NBBl . (153)The index m does not appear in this matrix, since all multipoles (l,m) withthe same l share the same variance. If we consider the data as Gaussian (whichis a very good approximation except for small-scale B-mode multipoles), wecan use Eq. (149). The first step of the calculation is to estimate the numberof multipoles (l,m) with a fixed l. For full-sky experiments, m runs from −l85to l and this number is (2l+1). For experiments probing a fraction fsky of thesky, the number of independent multipole measurement at a given l is closeto (2l + 1)fsky. This leads toFCMBij =12lmax∑l=2(2l + 1)fskyTrace[C−1l∂Cl∂θiC−1l∂Cl∂θj], (154)which can be further reduced to an expression of the type [195]FCMBij =lmax∑l=2∑α,β∂Cαl∂θi(Covl)−1αβ∂Cβl∂θj, (155)where α and β run over TT , EE, TE, BB and Covl is the “spectra covariancematrix”. For space reason we do not give here the full expression of Covl (seee.g. Ref. [196]). As an example, the first coefficient of the matrix reads(Covl)TT,TT =2(2l + 1)fsky(CTTl +NTTl )2 . (156)The factor (2l+1)fsky which appears in the denominator of Covl accounts forthe effect of cosmic variance: for large l values, an experiment makes manyindependent measurements (corresponding to several m’s) and the variancein absence of noise is much smaller than the power spectrum CXXl of a singlemultipole aXlm. For small l, there are only few independent measurement andthe variance is of the same order of magnitude as CXXl .Let us apply this machinery to concrete cases. In Table 4, we present theexpected sensitivity of a non-exhaustive list of future CMB experiments. Thedesign of projected experiments is subject to constant evolution, and this com-pilation should be regarded as indicative only. We did not take the numbersfrom some official documentation edited by each experiment, but from thepapers cited below, which worked out some neutrino mass error forecasts onthe basis of the numbers presented here. We refer to these papers for moredetails and references concerning the sensitivity parameters.After WMAP, we can expect some very interesting results from a large numberof ground-based experiments, mapping the anisotropies only in a small regionsof the sky but usually with excellent sensitivity and resolution. Most of theseexperiments are optimized for measuring E-polarization or even B-polarizationanisotropies, which will still be poorly constrained after the completion ofWMAP, and even after Planck in the case of B-polarization. Among thesecollaborations, we can cite the South Pole Telescope 8 (SPT, in construction)8 http://spt.uchicago.edu/86Table 4Experimental parameters of a few CMB projects, used in the neutrino mass forecastpapers mentioned in this section. Here fsky is the observed fraction of the sky, ν isthe channel frequency in GHz, θb measures the FWHM of the beam in arc-minutesand ∆T,P is the square root of the sensitivity per pixel in µK for temperature andpolarization.Experiment fsky ν θb ∆T ∆PSPT [197] 0.1 217 0.9’ 12 17BICEP [198] 0.03 100 60’ 0.33 0.47150 42’ 0.35 0.49QUaD [198] 0.025 100 6.3’ 3.5 5.0150 4.2’ 4.6 6.6BRAIN [198] 0.03 100 50’ 0.23 0.33150 50’ 0.27 0.38220 50’ 0.40 0.56ClOVER [198] 0.018 100 15’ 0.19 0.30143 15’ 0.25 0.35217 15’ 0.55 0.76Planck [198] 0.65 30 33’ 4.4 6.244 23’ 6.5 9.270 14’ 9.8 13.9100 9.5’ 6.8 10.9143 7.1’ 6.0 11.4217 5.0’ 13.1 26.7353 5.0’ 40.1 81.2545 5.0’ 401 ∞857 5.0’ 18300 ∞SAMPAN [198] 0.65 100 42’ 0.13 0.18143 30’ 0.16 0.22217 20’ 0.26 0.37CMBpol [197] 0.65 217 3.0’ 1 1.4Inflation Probe 0.65 70 6.0’ 0.29 0.41(hypothetical) [198] 100 4.2’ 0.42 0.59150 2.8’ 0.63 0.88220 1.9’ 0.92 1.3087[199], the Atacama Cosmology Telescope 9 (ACT, funded in January 2004),and two complementary experiments: Background Imaging of Cosmic Extra-galactic Polarization 10 (BICEP) [200], designed for large angular scale, andQUest at DASI 11 (QUaD) [201], designed for small angular scales. The secondexperiment, which is already collecting data, is composed of the Q and U Ex-tragalactic Sub-mm Telescope (QUEST) instrument mounted on the structureof the DASI experiment. A second set of experiments is scheduled in Antarc-tica at the French-Italian Concordia station and in the Atacama plateau inChile, for unprecedented precision measurements of the B-mode for l < 1000(which is particularly useful for probing the primordial gravitational wavesfrom inflation): the B-modes Radiation measurement from Antarctica with abolometric INterferometer 12 (BRAIN) [202] instrument for measuring largescales, and the Cl ObserVER 13 (ClOVER) [203] instrument for intermediatescales. ClOVER was approved for funding by PPARC in late 2004 and couldbe operational by 2008.At that time, a CMB satellite (the third one after COBE and WMAP) shouldbe already collecting data: the Planck 14 satellite [204] has already been builtand should be launched in late 2007 or early 2008 by the European SpaceAgency (ESA). The temperature sensitivity of Planck will be so good thatit can be thought as the “ultimate” observation in the sector of temperatureanisotropies. However, it will still be possible to improve the measurement ofE-polarization on small angular scales, and B-polarization will be poorly con-strained by Planck. On intermediate scales, the ground-based experimentsshould be quite efficient, but progress will still be needed on very large scales(requiring full-sky coverage) and very small scales (requiring both high reso-lution and excellent sensitivity). Beyond Planck, at least two space projectsare under investigation. The mini-satellite SAMPAN (SAtellite to Measurethe Polarized ANisotropies) [205] is targeted for large scales, and aims atimproving the ClOVER measurement of the B-mode for l < 1000, thanksto its full-sky coverage and slightly better sensitivity. The NASA wishes tolaunch a more ambitious satellite project in order to make the “ultimate”measurement of E-polarization, like Planck for temperature, and a very goodmeasurement of B-polarization on all scales, with for the first time an instru-mental noise smaller than the B-mode cosmic variance up to l ∼ 1000 or evenmaybe l ∼ 1500. The generic name of the NASA call for projects is Infla-tion Probe15 . Its design is still very uncertain. A very preliminary project9 http://www.hep.upenn.edu/∼angelica/act/act.html10 http://www.astro.caltech.edu/∼lgg/bicep front.html11 http://www.astro.cf.ac.uk/groups/instrumentation/projects/quad/12 http://apc-p7.org/APC CS/Experiences/Brain/index.phtml13 http://www-astro.physics.ox.ac.uk/∼act/clover.html14 http://sci.esa.int/science-e/www/area/index.cfm?fareaid=1715 http://universe.gsfc.nasa.gov/program/inflation.html88was called CMBpol; this name is mentioned in some works presented there-after [197,206,101], with sketchy characteristics mentioned in Table 4. Thereare now more advanced projects under investigation. Nobody knows whichproject and which design will be eventually approved. We mention in theTable some hypothetical characteristics for Inflation Probe suggested inRef. [198], assuming a satellite project with a bolometer array and a passivelycooled telescope of 3-4 m of aperture. This is one of the various possibilitieswhich are being discussed, and it should be regarded as purely indicative.Some predictions concerning the sensitivity of these experiments (not com-bined with other observations) to the neutrino mass are published in Refs.[196,207,101,198] for Planck and in Ref. [198] for the combinations BICEP+QUaD, BRAIN+ClOVER, as well as SAMPAN alone, Planck+SAMPANand the hypothetical version of Inflation probe (see also [208,209] for fore-casts including a large neutrino asymmetry, now disfavoured as we saw inSec. 3.1). Like for current bounds, forecast errors are not straightforward tocompare from paper to paper, because different authors use• different fiducial models with different fiducial values for the cosmologicalparameters. In Ref. [196], the fiducial model which is closer to the currentconcordance ΛCDM model is the one in Table II of that paper; here, wewill only quote results from this table. There are eleven free parameters,which are the usual six of the ΛCDM model, the mass of a single massiveneutrino, the tensor-to-scale ratio T/S, the scalar tilt running α, the cur-vature fraction Ωk and the ionized helium fraction YHe. The fiducial valueof Mν is very close to zero (so that the fiducial CMB spectra are indistin-guishable from those with massless neutrinos). In Ref. [207], the numberof free parameters is reduced to eight, by removing (T/S, α, YHe) whichare fixed respectively to (0, 0, 0.24). There is a single massive neutrino withfiducial mass 0.07 eV. Next, in Table III of Ref. [101], there are also eightfree parameters, but the last one is YHe instead of Ωk. There are three mas-sive neutrinos with total mass Mν = 0.3 eV distributed according to thenormal hierarchy scheme. Finally, in Table II of Ref. [198], there authorsmarginalize either over eight parameters (the usual six of ΛCDM model, themass of a single massive neutrino and YHe) or over eleven parameters (thesame ones, plus the scalar tilt running α, the parameter w of the equationof state of dark energy, and the effective number Neff of extra relativisticdegrees of freedom). The fiducial value of the neutrino mass isMν = 0.1 eV.• different experimental sensitivities. For instance, the modelling of Planckin Refs. [196], [101] and [198] are roughly equivalent, although in [101,198] itwas updated to the current instrument design. Instead, in Ref. [207], the au-thor replaces the expression of the noise variance NTTl , NEEl [see Eq. (151)]by two step functions equal to zero below some critical multipoles (lTmax,lEmax)=(2500,1500) and to infinity above. This is equivalent to assuming a89Table 5Forecast error σ(Mν) in eV using Planck alone: compared results from the liter-ature. The third column shows the free parameters over which the final result ismarginalized (“7” stands for six usual parameters of the minimal ΛCDM model,plus total neutrino mass Mν). The fourth column gives the assumed fiducial valueof the mass in eV. In the last two lines, the two numbers correspond to optimistic orpessimistic assumptions concerning the foregrounds contamination of the primaryCMB signal.Ref. σ(Mν) parameters fiducial Mν Planck sensitivity[196] 0.3 7+{α, T/S, Ωk, YHe} 0 slightly optimistic[207] 0.07 7+{Ωk} 0.07 very optimistic[101] 0.3 7+{YHe} 0.3 up-to-date[198] 0.45-0.49 7+{YHe} 0.1 up-to-date[198] 0.51-0.56 7+{α, w, Neff , YHe} 0.1 up-to-dateslightly over-optimistic version of Planck.• different parameter basis and different finite steps for the computation ofthe derivatives ∂CXX′l /∂θi. The likelihood L(~x|~θ)can be close to a multi-variate Gaussian in one basis, and more complicated in another basis - inwhich case Eq. (155) may give only a poor approximation of the Fishermatrix. The possible non-gaussianity also implies that the numerical valueof the derivative ∂CXX′l /∂θi depends on the step ∆θi used for its calculation.However, authors usually optimize their choice of basis and step size [seeRef. [196])] in order to obtain robust results.In order to illustrate how these different assumptions are translated into differ-ent forecasts from paper to paper, we summarize in Table 5 the predictions forPlanck published by the four references [196,207,101,198]. In the case of Ref.[198] we quote two numbers, which represent some optimistic or pessimisticassumptions concerning the subtraction of astrophysical foregrounds from theobserved CMB maps: these two values, which are not very different from eachother, are supposed to bracket the true realistic 1σ error. The difference be-tween σ(Mν) = 0.3 eV in [196] and σ(Mν) = 0.07 eV in [207] (taken fromFig. 3 of this reference) probably comes from the sketchy and very optimisticmodelling of the noise in the second reference. The results of [196] and [101]seem to be consistent with each other, since in the latter reference the numberof parameters is smaller but the experimental characteristics are more real-istic. The difference between σ(Mν) = 0.3 eV in [101] and σ(Mν) ≃ 0.45 eVin [198], for identical cosmological models and experimental characteristics,comes from variations in the fiducial value of the total neutrino mass, whichdecreases from 0.3 to 0.1 eV. Indeed, Ref. [101] explores the dependence of90Table 6Forecast error σ(Mν) in eV using various CMB experiments or combinations ofthem. These numbers are all taken from Ref. [198], which assumes either an eightparameter model (the usual six ones of ΛCDM plus Mν and YHe), or an elevenparameter model (adding α, w andNeff). The fiducial value of the total mass is takento be Mν = 0.1 eV, and in each case the two numbers correspond to optimistic orpessimistic assumptions concerning the foregrounds contamination of the primaryCMB signal.Experiment 8 parameters 11 parametersBICEP+QUaD 1.3 - 1.6 1.5 - 1.9BRAIN+ClOVER 1.5 - 1.8 1.7 - 2.0Planck 0.45 - 0.49 0.51 - 0.56SAMPAN 0.34 - 0.40 0.37 - 0.44Planck+SAMPAN 0.32 - 0.36 0.34 - 0.40Inflation probe 0.14 - 0.16 0.25 - 0.26σ(Mν) on the fiducial valueMν (see Fig. 22), and find that whenMν decreasesfrom approximately 0.5 to 0.1 eV, σ(Mν) is multiplied roughly by a factor two.Finally, the comparison of the two results obtained in Ref. [198] shows thatthe determination of the neutrino mass by Planck alone does not suffer sig-nificantly from parameter degeneracies which appear when α, w or Neff areincluded as extra free parameters. Similarly, the comparison of [196] and [101]suggests that this also applies for the possible inclusion of T/S and Ωk.For other CMB experiments than Planck, not combined with other type ofobservations, the only available reference is [198], whose results are summa-rized in Table 6. We see in this table that ground based experiments alone arenot helpful for constraining the neutrino mass: this is due to that fact thatwith a small sky coverage, and in spite of their excellent sensitivity, they haverelatively large error-bars (with respect to Planck) for small and intermedi-ate values of l. We will see in Sec. 6.4 how this conclusion changes when oneassumes that the results of these experiments are analyzed with the “lensingextraction” technique.In summary, the satellites Planck, SAMPAN or the combination of bothwould reach a 1σ error in the range σ(Mν) = 0.3− 0.4 eV, depending on thecase. This is of the same order of magnitude as the current neutrino massbounds presented in section 5, when all available CMB and LSS data arecombined with each other. Finally, if the Inflation probe mission can actuallyreach the sensitivity assumed in Table 4, it will provide a sensitivity rangingfrom σ(Mν) = 0.15 eV to 0.25 eV depending on the number of marginalizedparameters.916.3 Future galaxy redshift surveysAs already discussed in Sec. 5.3, Galaxy Redshift Surveys (GRS) allow fora reconstruction of the matter power spectrum modulo a light-to-mass biasfactor. The analytic models for non-linear structure formation show that therelation between the reconstructed power spectrum and the total matter lin-ear power spectrum should be scale-independent, Pobs(k) ≡ b2P (k), at leastup to a wavenumber kmax where non-linear corrections start to induce scale-dependent biasing. At redshift zero and for the concordance ΛCDM model,this wavenumber is expected to be around kmax = 0.15hMpc−1. The usualapproach is to discard any information for k > kmax in order to avoid manycomplications related to biasing, non-Gaussian statistics and uncertainties inthe predictions of the non-linear matter power spectrum.A Fisher matrix can be derived for future surveys following the lines of Sec.6.1. It was shown by Tegmark [210] that the Fisher matrix takes a very simpleform under a couple of assumptions.• For k ≤ kmax, one assumes that the power spectrum reconstructed fromthe data, Pobs(k), is really a tracer of the total matter power spectrum,not significantly affected by non-linear corrections or by redshift-space dis-tortions. We recall that what is measured is the redshift of each galaxy,not its physical distance; therefore, strictly speaking, the power spectrum isnot reconstructed in three-dimensional real space but in redshift space; theredshift-space and real-space power spectra do not coincide exactly, they arerelated through a function which depends on the cosmological parameters,and this complication is ignored here in first approximation.• One assumes that the survey is volume-limited rather than brightness-limited, i.e. the spectrum is reconstructed from data within a volume Vin which the selection function n(r), which represents the number den-sity of observed galaxies, is approximately constant (instead of being sup-pressed near the edges due to the fact that only the brightest galaxies wouldbe seen). Moreover, one assumes that the density is large enough so that1/n(r) ≤ P (k) for all relevant wavenumbers kmin < k < kmax, which meansthat the galaxy correlation function really probes cosmological informationrather than shot noise due to insufficient sampling.Under these assumptions, the Fisher matrix takes a form which is analogousto its CMB counterpart of Eq. (155):FGRSij =12kmax∫kmin∂Pobs(k)∂ ln θi∂Pobs(k)∂ ln θj4πw(k)P 2obs(k)d ln k , (157)92where the weight function w(k) of the survey is defined asw(k) =Vλ3=V(2π/k)3. (158)In Eq. (157), the factor 4πw(k)d lnk accounts for cosmic (or sampling) vari-ance: it gives the number of independent measurements with wavenumberin the range [k, k + dk] within the survey volume. This factor is negligiblefor wavenumbers smaller than kmin, the smallest wavenumber probed by thesurvey: so, for practical purposes, the integral can be performed starting fromzero. The result of Ref. [210] is actually more general than above, and includesthe case where shot noise is not negligible and where the selection function isnot constant inside the volume V . In these cases, one must replace V in Eqs.(157) and (158) by an effective volume Veff(k) defined asVeff(k) =∫d3r[Pobs(k)1/n(r) + Pobs(k)]2. (159)However, for the future surveys discussed in this section, we can systematicallyassume that the power spectrum reconstruction will be performed inside avolume V chosen in such way that throughout this volume 1/n(r) is smallerthan Pobs(k) for all kmin < k < kmax, so that Veff(k) can be replaced by V .For instance, for the completed SDSS Bright Red Galaxy (BRG) survey, thisvolume will include galaxies up to a distance λ ≃ 1 h−1Gpc with a sky cov-erage fsky = 0.25. So, one can take V ∼ fsky(4/3)πλ3 = 1 (Gpc/h)3 [196].Beyond SDSS, there are various plans for larger surveys, like for instance theSDSS-II LEGACY project, the Advanced Large, Homogeneous Area MediumBand Redshift Astronomical survey 16 (ALHAMBRA) [211], the Dark EnergySurvey 17 (DES) [212,213,214], or the Kilo-Aperture Optical Spectrograph 18(KAOS). Some of these surveys will go to such high redshift that it will bepossible, first, to compare the observed power spectrum with the linear oneup to wave-numbers significantly larger than the usual kmax ∼ 0.15h Mpc−1(since the non-linear evolution affects larger and larger modes as time passesby); and second, to do some tomography, i.e. to reconstruct P (k) in variousredshift bins, corresponding to different times in the evolution of the Uni-verse, in order to measure the variations of the linear growth factor and toget some good handle on the dark energy variables. Tomography is also idealfor neutrino mass extraction, since massive neutrinos induce a very pecu-liar redshift-dependence on the matter power spectrum, as we have seen in16 http://alhambra.iaa.es:8080/17 http://www.darkenergysurvey.org/18 http://www.noao.edu/kaos93Sec. 4.6.4. For instance, KAOS could build two catalogs centered around red-shifts z = 1 and z = 3, corresponding roughly to kmax ∼ 0.2 h Mpc−1 andkmax ∼ 0.48 h Mpc−1 respectively, instead of kmax ∼ 0.1 h Mpc−1 for currentsurveys (conservative values). In both catalogs, the number density would besuch that 1/n ∼ P (kmax), and the effective volume of the two samples closeto V ∼ 0.5 (Gpc/h)3 and V ∼ 0.6 (Gpc/h)3 respectively 19 .In order to forecast the sensitivity of a galaxy survey combined with a CMBexperiment, one can simply add the two Fisher matrices and invert their sum.The sensitivity of large redshift surveys to the neutrino mass was fist studiedin Ref. [94]; however this pioneering paper was rather crude in the way to com-bine GRS with CMB experiments. This work was updated and generalized in[196], and later in [207,101,102]. In Table 7, we quote from these papers theforecast error σ(Mν) obtained assuming that the completed SDSS survey iscombined either with the WMAP or Planck data. These results are based onTable II in [196], Figs. 4 and 5 in [207], and Table III in [101], for a commonconservative value kmax = 0.1hMpc−1. With respect to the results quoted inthe previous subsection, the number of free parameters has increased by oneunit, since there is now an unknown free bias parameter b = [Pobs(k)/P (k)]1/2which is marginalized out. We see that SDSS alone gives essentially no informa-tion on Mν given the various parameter degeneracies. For the SDSS+WMAPcombination, there is an excellent agreement between [196] and [207], andthe forecast error σ(Mν) = 0.3 eV is three times smaller than the currenterror based on incomplete WMAP+SDSS results σ(Mν) = 0.9 eV [135]. ForSDSS+Planck, it seems again that the Planck noise modelling in [207] ismuch too optimistic, while Refs. [196] and [101] agree on σ(Mν) = 0.2 eV. Ifwe compare this with the result obtained in [101] under the same assumptionsfor Planck alone, σ(Mν) = 0.3 eV, we see that SDSS allows for a significantimprovement.Concerning experiments beyond SDSS, Ref. [207] and later Ref. [101] presentforecast errors in a very general way, by simply showing σ(Mν) as a functionof two free parameters V and kmax. Instead Ref. [102] assumes some detailedcharacteristics for three high-redshift surveys, each of them operating in threeredshift bins. Here we will summarize the results of Ref. [101], assuming thatan hypothetical future redshift survey with given (V, kmax) is combined witha CMB experiment that could be SPTpol, Planck, CMBpol (see Table 4for the value of assumed sensitivity parameters), or an “ideal CMB experi-ment” that would be limited only by cosmic variance up to l = 2500 (bothfor temperature and polarization). The main difficulty for reaching this goalwould be to subtract accurately small-scale foregrounds (point-like sources,19 The characteristics of KAOS are taken from the “Purple Book” available on-lineat http://www.noao.edu/kaos.94Table 7Forecast error σ(Mν) in eV for the completed SDSS survey, with volume V =1(Gpc/h)3, eventually combined with WMAP or Planck data: compared resultsfrom the literature, assuming the conservative value kmax = 0.1hMpc−1.Ref. SDSS +WMAP +Planck assumptions[196] 9 0.3 0.2 12 free parameters,optimistic Planck characteristics[207] – 0.3 0.06 9 free parameters,very optimistic WMAP/Planck charact.[101] 7 – 0.2 9 (other) free parameters,up-to-date Planck characteristicsdusty galaxies, etc.) but even with current technology such an ideal experi-ment is not unconceivable. For these four cases, Fig. 21 shows the predicted2σ error on the total neutrino mass Mν . In these plots the fiducial value ofthe total mass was fixed to Mν = 0.11 eV, and distributed according to theNH scheme. For SDSS (or for any survey with z < 1) we expect the relevantvalue of kmax to be around 0.15 h Mpc−1. However, depending on the over-all amplitude of the matter power spectrum (often parametrized by σ8, andstill poorly constrained) and on future improvements in our understanding ofnon-linear corrections, this value might appear to be either too optimistic ortoo pessimistic: this is the reason why it is interesting to leave it as a free pa-rameter. The figure shows the importance of employing high-volume surveys,which have the potential to improved the forecast errors even for the bestCMB experiments. Keeping kmax = 0.15 h Mpc−1, we see that a two-sigma de-tection of the minimal mass in the IH scenario would require V = 40 (Gpc/h)3after Planck, V = 12 (Gpc/h)3 after CMBpol, while the “ideal CMB exper-iment” alone would suffice. A two-sigma detection of the minimal mass in theNH scenario seems to be unreachable for any plausible survey volume.As in the previous subsection, we should mention that the results for σ(Mν)depend significantly on the assumed fiducial value Mν . This is illustrated inFig. 22, taken from Ref. [101]. For the combination of either Planck withSDSS, or of the “ideal CMB experiment” with a larger hypothetical survey,we can see that the 2σ error is typically twice larger for Mν ∼ 0.1 eV than forMν ∼ 0.5 eV.It is legitimate to wonder whether the most precise combination of experimentswould be sensitive to different neutrino mass splittings between the threefamilies, for a fixed value of the total mass. Indeed, we have seen in Sec. 4.6that unlike CMB observables, the power spectrum of large scale structureis in principle sensitive both to the total and the individual values of the951100 0.1 0.2 0.31100 0.1 0.2 0.31100 0.1 0.2 0.3Veff / Veff (SDSS) min IH0.30.20.050.10.150.20.250.30.350.40.450.5min IH0.30.2min IHmin NH0.2kmax (h Mpc-1)0.050.10.150.20.250.30.350.40.450.51100 0.1 0.2 0.3min IHmin NHFig. 21. Results from Ref. [101] on the neutrino mass sensitivity of future redshiftsurveys, combined with future CMB experiment. The level of gray gives the 2σ ex-pected errors on the total massMν (eV) for a fiducial value of 0.11 eV, as a functionof the largest wavenumber kmax used in the analysis [in units of hMpc−1] and ofthe survey volume V [in units of VSDSS = 1 (Gpc/h)3]. The vertical lines indicatethe cut-off wavenumber kmax for the linear matter power spectrum at the conser-vative (optimistic) value 0.15(0.2)h Mpc−1. The thin contours are (from bottom totop) for 0.3 and 0.2 eV, while the thick contours correspond to the minimum valuesof Mν in the IH (lower lines) and NH (upper lines) schemes, assuming a squaredmass difference ∆m2atm = 2.6× 10−3 eV2 (thick solid lines). The thick dashed linescorrespond to the same limits for a value ∆m2atm = 3.7 × 10−3 eV2 which is nowessentially ruled out by current data on neutrino oscillations.three masses. The analysis of Ref. [101] demonstrates however that the tinydifference between the power spectrum obtained in the NH and IH cases fora fixed value of the total mass (close to 0.1 eV) is far too small for beingdetected even with the combination of a huge redshift survey of volume V =40 (Gpc/h)3 with the “ideal CMB experiment”.The issue of degeneracies between the neutrino mass and other parameters,such as those describing the dark energy evolution, probably needs furtherinvestigation. Assuming that dark energy has a constant equation-of-state pa-9600.10.20.30.40.50.60.70.12σ bounds on M (eV)fiducial M (eV)∆m2atm= 2.6 x 10-3 eV2 (best fit)0.05 0.2 0.5NHIH00.10.20.30.40.50.60.70.12σ bounds on M (eV)fiducial M (eV)∆m2atm= 2.6 x 10-3 eV2 (best fit)0.05 0.2 0.5NHIHFig. 22. Predicted 2σ error on the total neutrino mass as a function of the as-sumed fiducial value of Mν , using: (left) Planck+SDSS, (right) an ideal CMBexperiment (limited only by cosmic variance up to l = 2500, both for temperatureand polarization) and a redshift survey covering 75% of the sky up to z ≃ 0.8[Veff = 40 (Gpc/h)3]. Both plots assume a limit kmax = 0.15h Mpc−1 and a squaredmass difference ∆m2atm = 2.6×10−3 eV2. In each case, we show the results assumingeither NH or IH. Figure from Ref. [101].rameter w, Ref. [170] showed that future analyses of combined CMB and LSSdata could be affected by a significant degeneracy between Mν and w, a resultwhich was not found in Ref. [101]. The authors of Ref. [86] focused on thisdegeneracy, including a possible dependence of w on redshift. They point outthat in order to help resolving the degeneracy, it is useful to take into ac-count the fact that the CMB and LSS data are not statistically independent,as assumed for simplicity in all the works cited in this section. Instead, it iswell-known that due to the late Integrated Sachs-Wolfe (ISW) effect (see Sec.4.4.7), the power spectrum of temperature anisotropies encodes some infor-mation on neighboring structures (like galaxy clusters). So, there is a non-zerocross-correlation between temperature and galaxy maps, which has actuallybeen already measured [215,216], still with a rather low level of significance. Inthe future, precise measurements of this cross-correlation could be very usefulfor neutrino mass extraction.6.4 CMB weak lensing6.4.1 PrincipleIt was recently realized that the CMB anisotropies encode even more cos-mological information than expected, because it should be possible in a nearfuture to measure the deflection field caused by the weak lensing of CMB pho-tons by the large scale structure of the neighboring universe up to a redshiftz ∼ 3 [217,218,219] (see [220] for a recent review on CMB lensing). The power97spectrum of the deflection field encodes some information concerning struc-ture formation mainly in the linear or quasi-linear regime, and is thereforeextremely useful for measuring parameters like the total neutrino mass (orthe dark energy equation-of-state) which mildly affect the primary anisotropy[197]. So, the next generation of CMB experiments could output for free a LSSpower spectrum, without suffering from the systematics induced by mass-to-light bias and by strong non-linear corrections on small scales at z ≤ 0.2, asis the case for galaxy redshift surveys.More precisely, weak lensing produces a deflection of light rays parametrizedby a two-dimensional deflection field d(n) such that an object located in adirection n will appear in a direction n+d(n). At leading order [221,222], thedeflection field can be written as the gradient of a lensing potential, d = ∇φ.The lensing potential contains information on the gravitational potential alongthe photon trajectory, convolved by a window function which depends on thedistance to the sourceφ(nˆ; rs) =rs∫0dr W (r; rs) Φ(rnˆ, η = −r) , (160)where r is the comoving coordinate along the line of sight, rs is the sourcecoordinate, Φ the gravitational potential, and η the conformal time definedby dt = −a dη and η = 0 today (with this definition, the photons emitted bythe source and reaching us today travel according to r(η) = −η). Using thegeodesic equation, it is possible to show that the window function W (r; rs) isequal toW (r; rs) = 2rs − rr rs. (161)In the present case, the source is the last scattering surface and rs stands forits comoving radius. For a given cosmological model, the primordial spectrum〈|Φ|2〉 and its time evolution can be calculated; the same numerical code whichcomputes the CMB anisotropy spectrum CXX′l and the matter power spectrumP (k) can also easily compute the deflection field power spectrum Cddl , whichencodes some information on LSS at redshifts of order z ∼ 3 or so. Hence, ifwe could map the CMB deflection field d, we could measure Cddl and compareit with theoretical predictions in order to constrain cosmological parameters.There are several methods on the market for extracting the deflection map[221,223,224,225], all based on the non-Gaussianity induced by lensing [217].These methods start from the assumption that both the primary anisotropiesand the deflection field are Gaussian; they also assume that the noise presentin the temperature and polarization maps is Gaussian and uncorrelated with98the signal. None of these assumptions is exactly true. Ref. [226] estimated towhich extent the lensing extraction will be biased, first, by the non-Gaussianityof the lensing potential caused by the non-linear growth of matter perturba-tions on small scales, and second, by the imperfect cleaning of the CMB mapsfrom the kinetic Sunyaev-Zel’dovich effect, which also has a blackbody spec-trum, induces non-Gaussianity, and features spatial correlations with manyof the structures responsible for the lensing. Both effects were found to berelevant (i.e. to induce a significant bias in the estimators). However, they aresmall enough to preserve the validity of the method. Ref. [227] estimated theimpact on lensing extraction of a small level of non-Gaussianity produced inthe Early Universe by inflation, reheating or some other mechanisms (for areview, see [228]), and concluded that given the current bounds on primor-dial non-gaussianity inferred from CMB maps, this non-gaussianity cannot belarge enough to threaten lensing extraction.6.4.2 Quadratic estimator methodThis field is rather technical, but we can stress the main ideas without enteringinto many details. For simplicity, we will write the main formulas in the flat-sky approximation, which amounts in replacing the expansion of CMB mapsin harmonic space by an expansion in two-dimensional Fourier space, and touse two-dimensional vectors l instead of pairs of integers (l,m). In the flat-skyapproximation the Fourier modes are fully described by the power spectra C˜ablwhere a and b belong to the basis {T,E,B} of temperature, T -mode polariza-tion and E-mode polarization. Weak lensing correlates the lensed multipoles[229,217] according to〈a(l)b(l′)〉CMB = (2π)2δ(l+ l′)C˜abl + fab(l, l′)φ(l+ l′) (162)where the average holds over different realizations (or different Hubble patches)of a given cosmological model with fixed primordial spectrum and backgroundevolution (i.e. fixed cosmological parameters). In this average, the lensing po-tential φ is also kept fixed by convention, which makes sense because the CMBanisotropies and the LSS that we observe in our past light-cone are statisti-cally independent, at least as long as we neglect the integrated Sachs-Wolfeeffect. The above functions fab are defined in [224]. For instance, for ab = TTone hasfTT (l, l′)C = CTTl (l+ l′) · l + CTTl′ (l+ l′) · l′ . (163)Let us introduce the basic principle of the quadratic estimator method of Hu& Okamoto [223,224,225] (which is equivalent in terms of precision to thealternative iterative estimator method of Hirata & Seljak [221] as long as99CMB experiments will make noise-dominated measurements of the B-mode,i.e. at least for the next decade). By inverting Eq. (162), one builds a quadraticcombination of the temperature and polarization observed Fourier modesdab(L) =iLAabLL2∫d2l1(2π)2a(l1)b(l2)gab(l1, l2) (164)where l2 = L− l1, and in which the normalization conditionAabL = L2[d2l1(2π)2fTT (l1, l2)gab(l1, l2)]−1(165)ensures that dab is an unbiased estimator of the lensing potential〈dab(L)〉CMB = iLφ(L) = d(L) . (166)Note that, so far, the coefficients gab(l1, l2) are still arbitrary. From the ob-served temperature and polarization maps, one could compute each mode ofdab and obtain various estimates of the deflection modes, precise up to cosmicvariance and experimental errors. In order to quantify the total error, it isnecessary to compute the power spectra of the quadratic estimators〈dab∗(L)dab(L)〉 = (2π)2δ(L− L′)Cdd(ab)L (167)where the average is now taken over both CMB and LSS realizations, sinceφ(L) is also a stochastic quantity. In this definition, the power spectra are writ-ten with a superscript dd(ab) in order to be distinguished from the actual powerspectrum of the true deflection field. These spectra feature the four-point cor-relation function of the observed (lensed) Fourier modes 〈a(l1)b(l2)a(l3)b(l4)〉,which should be expanded at order two in φ(L) in order to catch the leadingnon-Gaussian contribution.The four-point correlation functions are composed as usual of a connected andan unconnected piece. The connected piece is by definition a function of thepower spectra Cabl in which we now include all sources of variance: cosmicvariance, lensing contribution and experimental noise. The unconnected pieceis a function of the same spectra plus the deflection spectrum Cddl , and as usualit can be decomposed in three terms corresponding to the different pairingsof the four indexes [232]: (l1, l2), (l3, l4) or (l1, l3), (l2, l4) or (l1, l4), (l3, l2).The first term leads to considerable simplifications when it is plugged intothe expression of the quadratic estimator power spectrum, and the result issimply Cddl , as one would expect naively from squaring Eq. (166). The other100terms lead to more complicated expressions that we will write as a noise termCdd(ab)L = CddL + [Nc]abL +NabL , (168)which represent respectively the contribution from the connected piece andfrom the two non-trivial terms of the unconnected piece [230,232]. In orderto get an efficient estimator, one should adopt the set of coefficients gab(l1, l2)which minimize the noise terms. It is actually much easier to minimize theconnected term only, which leads to the simple resultsgaa(l1, l2) =faa(l, l′)2Caal Caal′and [Nc]aaL = AaaL (169)for a = b (for a 6= b see [224]). With such a choice, the unconnected piececontribution NabL can be shown to be smaller than AaaL , but not completelynegligible [230].The various estimators dab can be constructed for each pair of modes, exceptfor the pair BB, because the spectrum CBBl is dominated by lensing at least onsmall scales, which invalidates the present method. Therefore, the quadraticestimator technique would not be optimal for long-term CMB experimentswith cosmic-variance-dominated measurement of the B mode [221,231]. For anexperiment of given sensitivity, the five other estimators can be combined intoa final minimum variance estimator, which gives the best possible estimate ofthe deflection field by weighing each estimator accordingly to its noise level.The sensitivity of the Planck satellite is slightly above the threshold forsuccessful lensing extraction, but only at intermediate angular scales, andwith essentially all the signal coming from the dTT estimator. The followinggeneration of experiments – such as the CMBpol or Inflation probe project –should obtain the lowest noise level from the dEB estimator [224].6.4.3 Neutrino mass from CMB lensing extraction.The authors of Refs. [197] and [198] studied the limits that could be obtainedon neutrino mass by using future CMB experiments only, assuming that thelensing power spectrum could be extracted using the quadratic estimator tech-nique. In this way, it is possible to combine information on CMB acousticoscillations and on the surrounding LSS of the Universe at redshifts of orderz ≤ 3, avoiding many of the complicated features of galaxy redshift surveysrelated to the mass-to-light bias and to the strongly non-linear evolution ofsmall-scale perturbations at small redshift.The principle of Refs. [197,198] is to treat the power spectrum of primordialanisotropies (for temperature and polarization) and of the deflection field as101two independent information on the cosmological parameters. The expectederror-bars on the temperature and polarization multipoles are estimated fromthe characteristics of a given CMB experiment, while those on on the lens-ing multipoles are taken to be the dominant term [Nc]abL in Eq. (168), whichcan be computed from the same experimental characteristics using essentiallyEqs. (169) and (165). Then, the authors assume a fiducial model and per-form a Fisher matrix analysis in the same fashion as described in the previoussubsection.The fiducial model of Ref. [197] includes ten free parameters: the usual sixparameters of the minimal ΛCDM model, the neutrino mass Mν in the caseof three degenerate neutrinos, the equation of state parameter w, the runningof the tilt α and the primordial Helium abundance YHe. The authors concludethat the Planck satellite will output a 1σ error-bar σ(Mν) = 0.15 eV, whichshows that the use of the lensing extraction technique should improve theglobal sensitivity in a rather spectacular way at least for this parameter. Inthe case of the CMBpol project an impressive error of σ(Mν) = 0.044 eV ispredicted.This study was repeated in Ref. [206], with one extra parameter representingthe variation of the dark energy equation-of-state with respect to the scalefactor. We saw before that this parameter could potentially introduce extradegeneracies involving the neutrino mass, but the forecast errors are foundto be approximately the same: σ(Mν) = 0.16 eV and σ(Mν) = 0.046 eV forPlanck and CMBpol, respectively.Finally, the authors of [198] assumed two possible fiducial models, one witheighth free parameters (the same as in Ref. [197] with the exception of αand w) and one with eleven free parameters (same as in Ref. [197] plus theeffective number of extra relativistic degrees of freedomNeff). They consider sixexperiments or combinations of experiments as listed in Table 8, and for eachcase they quote two numbers, corresponding either to optimistic or pessimisticassumptions concerning the subtraction of astrophysical foregrounds from theobserved CMB maps.It can be seen in Table 8 that the combination QUaD+BICEP benefits a lotfrom lensing extraction, since the error σ(Mν) decreases from approximately1.5 eV to at least 0.4 eV. Thus, with QUaD+BICEP it should be possibleto reach in a near future –using CMB only– the same precision that we havetoday combining many observations of different types (galaxy-galaxy correla-tion function, Lyman-α forest) which are affected by various systematics. Thesituation is almost the same for BRAIN+ClOVER, which should also achieveσ(Mν) ∼ 0.4 eV using lensing extraction. Planck should make a decisiveimprovement, lowering the error to σ(Mν) ∼ 0.15 eV, in excellent agreementwith the results of Ref. [197]. Note that without lensing extraction the error102Table 8Forecast error σ(Mν) in eV using various CMB experiments or combinations ofthem, and assuming that the lensing map is extracted following the quadratic esti-mator method. These numbers are all taken from Ref. [198], which assumes eitheran eight parameter model (the usual six of ΛCDM plus Mν and YHe), or an elevenparameter model (the same ones plus α, w, Neff). The fiducial value of the totalmass is taken to be Mν = 0.1 eV, and in each case the two numbers correspond tooptimistic or pessimistic assumptions concerning the foregrounds contamination ofthe primary CMB signal.Experiment 8 parameters 11 parametersBICEP+QUaD 0.31 - 0.36 0.36 - 0.40BRAIN+ClOVER 0.34 - 0.43 0.42 - 0.51Planck 0.13 - 0.14 0.15 - 0.15SAMPAN 0.10 - 0.17 0.12 - 0.18Planck+SAMPAN 0.08 - 0.10 0.10 - 0.12Inflation Probe 0.032 - 0.036 0.035 - 0.039would be multiplied by three (by four in the case with extra free parameters).No significant difference between the forecast errors in the eight and elevenparameter models was found in Ref. [198]. SAMPAN alone is slightly moreefficient than Planck, and the combination Planck+SAMPAN is the firstone to reach σ(Mν) ∼ 0.1 eV, even in the pessimistic case of large foregroundresiduals and extra free parameters. Finally, the version of the InflationProbe satellite considered in [198], which is slightly more ambitious thanthe CMBpol assumption of Ref. [197], is able to reach σ(Mν) ∼ 0.035 eVboth in the eight and eleven parameter cases. In any case, the results seemto be relatively robust against pessimistic assumptions concerning foregroundcontamination.6.5 Galaxy weak lensing (cosmic shear)6.5.1 PrincipleWeak lensing changes the apparent shape of galaxies in various ways. Amongothers, it changes slightly their apparent ellipticity: i.e. galaxies which wouldbe apparently spherical without weak lensing can look in fact elliptical, stret-ched in one direction and squeezed in the orthogonal direction. This effect,which is coherent over the angular size of the lensing gravitational field, i.e.potentially over many sources, is called cosmic shear. It can be detected pro-vided one has a very dense sample of galaxies with enough resolution in order103to measure each individual shape. In order to be able to reconstruct the lens-ing gravitational field, it is also necessary to know either the redshift of eachsource galaxy, or at least their coarse-grained number density g(z, nˆ) in red-shift space and in direction nˆ. Let us assume for simplicity that we have anisotropic sample with a spherical number density g(z) (or g(r), where r is thecomoving radius). We can divide the sample in small solid angles or pixels.Since intrinsic ellipticities are randomly distributed, the average ellipticity ineach pixel will give a measurement of the average cosmic shear, up to a shotnoise term proportional to N−1/2pix , where Npix is the number of galaxies perpixel. Therefore, the size of the pixels must be chosen optimally in a trade-offbetween high angular resolution and high signal-to-noise.The cosmic shear field reconstructed by this method derives from the samelensing potential φ as the deflection field discussed in the previous subsection.The relation between the lensing potential φ and the gravitational potentialΦ along the line of sight is the same as in Eqs. (160) and (161), except thatone must now average over all individual sources of comoving distance rs ∈[0, rmaxs ]. In the continuous limit and for an ensemble of sources described bythe density function g(r), this givesφ(nˆ; g) =rmaxs∫0drs g(rs) φ(nˆ; rs)rmaxs∫0drs g(rs)=rmaxs∫0dr W (r; g) Φ(rnˆ, η = −r) , (170)withW (r; g) ≡rmaxs∫rdrs g(rs) W (r; rs)rmaxs∫0drs g(rs). (171)Like for CMB anisotropies, once we have the lensing potential map φ(nˆ; g)we can compute the power spectrum Cφφ,gl , which gives us an informationon the surrounding large scale structure projected along the line of sight.Therefore, cosmic shear surveys offer another opportunity to measure the two-point correlation function of total matter fluctuations, complementary to thethree-dimensional Fourier power spectrum P (k) obtained from galaxy redshiftsurveys (probing lower redshifts) and to the CMB lensing power spectrum(probing higher redshifts).If a shear survey contains a large enough number of galaxies with known104redshifts, it is even possible to refine this technique by splitting the sourcesinto redshift (or distance) bins, from 0 < rs < r1 to rn−1 < rs < rn. It isthen possible to measure n lensing potential maps φ(nˆ, gi), such that the bindistribution gi(rs) vanishes outside of the range ri−1 < rs < ri. From thesemaps one can obtain n power spectra Cφφ,gil and n(n− 1)/2 cross-correlationspectra Cφφ,gigjl (cross-correlations do not vanish because the same structurecan contribute to the lensing of sources in different bins). Since the highestredshift bin probes structures at larger redshift, this method allows to someextent for a three-dimensional reconstruction of the gravitational fluctuations.This technique is called tomography, and brings some hope of tracking theevolution of P (k, z) as a function of redshift caused by dark energy [233] andby neutrino masses (see Sec. 4.6.4).6.5.2 Neutrino mass from cosmic shear surveysThere are various ongoing and planned cosmic shear surveys, like for instancethe Canada-France-Hawaii Telescope Legacy Survey 20 (CFHTLS) [162]; theDark Energy Survey 21 (DES) [212,213,214]; the SuperNova Acceleration Probe 22(SNAP); the Panoramic Survey Telescope And Rapid Response System 23(Pan-STARRS); or the Large Synoptic Survey Telescope 24 (LSST) [234,235].The sensitivity of this type of observation to neutrino masses has been ad-dressed first in [236], and later in [237,206,238]. Here we will review the re-sults of [206], where four hypothetical experiments are considered. The firsttwo, called S300 and S1000, stand for future satellite surveys with an averagesource density of 100 galaxies per squared arc-minute and a sky coverage of300 square degrees (fsky = 0.0073) and 1000 squared degrees (fsky = 0.024),respectively. The SNAP lensing survey could be close to one of these two as-sumptions. The last two, called G2π and G4π, stand for future ground-basedsurveys with an average source density of 65 galaxies per square arc-minuteand a sky coverage of respectively fsky = 0.5 (as expected for LSST) andfsky = 1 (as expected for Pan-STARRS). The authors make some realisticassumption concerning the redshift distribution of the sources (given experi-mental limitations) and perform a Fisher matrix analysis for the combinationof the cosmic shear surveys with either Planck or CMBpol (see Table 4),including in each case information on the CMB lensing, extracted with thequadratic estimator method described in Sec. 6.4. They assume a tomographicanalysis of the galaxy-ellipticity data with eight redshifts bins. Thanks to thequadratic estimator method, the CMB experiments also provide a measure-20 http://www.cfht.hawaii.edu/Science/CFHLS/21 http://www.darkenergysurvey.org/22 http://snap.lbl.gov23 http://pan-starrs.ifa.hawaii.edu/24 http://www.lsst.org/lsst home.shtml105Table 9Forecast error σ(Mν) in eV from reference [206], using some hypothetical cosmicshear surveys (S300, S1000, G2pi and G4pi) described in the text, combined witheither Planck or CMBpol (including, in each case, information on the CMB lensing,extracted with the quadratic estimator method).S300 S1000 G2pi G4piPlanck 0.12 0.090 0.052 0.045CMBpol 0.047 0.045 0.031 0.027ment of the cosmic shear power spectrum, sensitive to higher redshifts: herethe information from CMB lensing plays the role of a ninth redshift bin.There are eleven free cosmological parameters in the analysis of Ref. [206]:the usual six ΛCDM parameters, plus the total neutrino mass (attributedto a single family), the primordial helium fraction, the running of the scalartilt, and two parameters describing the dark energy equation of state: w(a) =w0 + (a0 − a)wa. The results are summarized in Table 9, from which oneconcludes that cosmic shear surveys constitute an extremely powerful toolfor constraining the total neutrino mass. The sensitivities obtained with thismethod are the absolute best forecasts derived so far. This is not surprising,given the ideal properties of weak lensing observations: precise, high redshiftmeasurement of the matter power spectrum over a wide range of scales, andtomography.Following Ref. [206] and Table 9, the most ambitious hypothetical experimentG4π combined with the Planck data would reach σ(Mν) ≃ 0.045 eV, i.e.the same sensitivity as CMBpol alone (see Sec. 6.4.3); while G4π combinedwith CMBpol would reach σ(Mν) ≃ 0.027 eV, which means that a 2σ detec-tion of the neutrino mass would occur even for the smallest Mν in the NHscheme, of order (∆m2atm)1/2 ≃ 0.05 eV. The results of Table 9 are relativelyconservative in the sense that in the Fisher matrix computation, the cosmicshear power spectrum was limited to the angular scales θ > 0.18 deg (in amultipole expansion, l < 1000) below which non-linearities render the powerspectrum reconstruction more difficult. However, the authors of Ref. [206] re-fer to a method for extracting information on smaller scales proposed by Refs.[239,240]. Ref. [206] speculates that using this method, one could gain a fac-tor three (two) with respect to the results of Table 9 for the combinationof Planck (CMBpol) with G2π or G4π. This would imply a real precisionmeasurement of the total neutrino mass with cosmological observables.Finally, the recent analysis in Ref. [238] considers the combination of Planck(no lensing extraction) with two lensing surveys inspired by the SNAP andLSST projects, using up to five tomography bins. The results are consistentwith those of Ref. [206].1066.6 Galaxy cluster surveysThe last method that we will discuss is expected to bring some extra inde-pendent information, and could further improve the bounds discussed in theprevious subsection.In the past years, the study of the cluster abundance evolution (the functionN(z), where N is the number density of galaxy clusters and z the redshift) haslead to interesting constraints on the normalization of the matter power spec-trum at small scales σ8 (defined in Sec. 5.5), and on the matter density fractionΩm. For a given cosmological model, the function N(z) can be predicted fromnon-linear simulations, and compared with experimental data from galaxycluster surveys. The non-linear simulations generally introduce systematic er-rors, but since cluster scales are not too deep inside the non-linear regime eventoday, the method can be kept under control.The functionN(z) –or better, its derivative dN/dz– is affected by the evolutionof the linear growth factor on cluster scales. Therefore, as emphasized recentlyin [241], future precise measurements of cluster abundances will be usefulprobes of the neutrino free-streaming effect.On the observational side, the prospects for galaxy cluster surveys are verypromising, since catalogs will be derived from various techniques, by mappingthe X–ray emission of the hot cluster gas, or the scattering of CMB photons inthis gas (Sunyaev–Zel’dovich effect, SZE), or the cosmic shear measured fromgalaxy weak lensing (as discussed in the previous subsection). We have alreadyseen that ambitious projects are being planned for galaxy weak lensing; this isalso the case for X–ray and SZE observations. The full data will then provide:first, a measurement of the matter power spectrum P (k, z) (modulo somebias and redshift-space distortion effects), which can be measured in variousredshift bins, like for weak lensing tomography; and second, an estimate ofthe abundance evolution dN/dz, also at various redshifts.An expression of the Fisher matrix describing future P (k, z) and dN/dz(z)observations has been proposed in [242]; this forecast method was applied toneutrino mass extraction in [241], using the expected instrumental sensitivityof the following projects: for CMB temperature and polarization anisotropies,Planck (no lensing extraction); for X-ray observations, the Dark UniverseObservatory (DUO) survey (see [243] for the description of a similar, previ-ously proposed survey), which should see ∼ 11, 500 clusters in 6,150 deg2;for SZE observations, the South Pole Telescope (SPT) survey [199], yielding∼ 20, 000 clusters in 4,000 deg2; and for galaxy weak lensing, the LSST project(already introduced in Sec. 6.5.2), assuming ∼ 200, 000 clusters in 18,000 deg2.Among the three data sets, Ref. [241] finds that the most powerful should be107 0.03 0.1 0.3 1 3 0.001 0.01 0.1 1Σmi (eV)lightest mν (eV)2σ sensitivitieswithout weak lensingPLANCKPLANCK+SDSSCMBpol orCMBpol+SDSSInvertedNormal 0.03 0.1 0.3 1 3 0.001 0.01 0.1 1Σmi (eV)lightest mν (eV)2σ sensitivities includingCMB lensing / cosmic shear surveysPLANCK lensingCMBpol lensingPLANCK lensing+ S300/S1000CMBpol lensing+ G2pi/G4piInvertedNormalFig. 23. Forecast 2σ sensitivities to the total neutrino mass from future cosmologicalexperiments, as described in Sec. 6, compared to the values in agreement withpresent neutrino oscillation data in Eq. (2) (assuming a future determination atthe 5% level). Left: sensitivities expected for future CMB experiments (withoutlensing extraction), alone and combined with the completed SDSS galaxy redshiftsurvey. Right: sensitivities expected for future CMB experiments including lensinginformation, alone and combined with future cosmic shear surveys. Here CMBpolrefers to a hypothetical CMB experiment roughly corresponding to the InflationProbe mission.the LSST lensing survey: the forecast error σ(Mν) is typically 50% larger forSPT, and 60 to 100% larger for DUO. For the combination Planck+LSST,it is interesting to note that the inclusion of dN/dz improves the neutrinomass determination by a factor two: using only information from the CMBand from P (k, z), the authors find σ(Mν) ≃ 0.09 eV, while adding dN/dz theyget σ(Mν) ≃ 0.04 eV. Including the DUO and SPT data marginally improvesthis bound.7 ConclusionsCosmology has played an important role in constraining neutrino properties,providing information on these elusive particles that complements the efforts oflaboratory experiments, and neutrino cosmology will remain an active researchfield in the next years.In this review, we were interested in the connection between neutrino massesand cosmology. We have described in detail how the evolution of cosmologicalperturbations is modified by massive neutrinos, and how their effects leave animprint in the different cosmological observables. We focused on the standard108case of three active neutrino species, known to be massive thanks to the ex-perimental results on flavour neutrino oscillations which, however, do not fixthe absolute neutrino mass scale.We saw how the analysis of cosmological observables can provide either a pos-itive indication or an upper bound on a total neutrino mass in the range from0.05 to a few eV indicated by oscillation data and other laboratory results.We discussed why a unique cosmological bound does not exist: the valuesdepend on the assumed cosmological model and the considered cosmologicaldata. Within a minimal ΛMDM model, the present cosmological bounds at95% CL are given in Tables 1-3, ranging from 2-3 eV with CMB data only tomore stringent values when LSS data are added: 0.9-1.7 eV considering onlythe shape of the matter power spectrum from 2dF or SDSS data, or 0.3-0.9eV fixing the light-to-mass bias and/or adding Lyman-α forest data and/orthe SDSS measurement of the baryon oscillation peak. These values limit theneutrino masses in the so-called degenerate region, as presented in Fig. 19. Wealso discussed how these cosmological bounds could be relaxed or even dis-appear when adding some extra cosmological parameters or in scenarios withnon-standard relic neutrinos. It is interesting to note that these mass boundsalso apply for other thermal relic particles, such as axions with masses in theeV range as shown in [244].Finally, we have discussed how the sensitivity to neutrino masses will improvewith new cosmological data, such as new CMB experiments or larger galaxyredshift surveys. In particular we have reviewed the excellent prospects offuture weak lensing measurements, either from CMB data or with cosmic shearsurveys. We saw how these future data will be sensitive to a total neutrinomass well below 1 eV, even when considering extra cosmological parameters.We give a graphical summary of the forecast sensitivities to neutrino massesof different cosmological data in Fig. 23, compared to the allowed values ofneutrino masses in the two possible 3-neutrino schemes. One can see from thisfigure that there are very good prospects for testing neutrino masses in thedegenerate and quasi-degenerate mass regions above 0.2 eV or so. A detectionat a significant level of the minimal value of the total neutrino mass in theinverted hierarchy scheme will demand the combination of future data fromCMB lensing and cosmic shear surveys, whose more ambitious projects willprovide a 2σ sensitivity to the minimal value in the case of normal hierarchy(of order 0.05 eV). The combination of CMB observations with future galaxycluster surveys (derived from the same weak lensing observations, as well asX-ray and Sunyaev–Zel’dovich surveys) should yield a similar sensitivity.The information on neutrino masses from analyses of cosmological data iscomplementary and can not replace the efforts in terrestrial projects such astritium beta decay and neutrinoless double beta decay experiments (and vice109versa). In particular, a positive result from one of these three possibilities,which as reviewed in Sec. 2 do not measure the same combination of neutrinomasses and mixing parameters, should be cross-checked by the others. Ofcourse any information on the absolute neutrino mass scale will be a veryimportant input for theoretical models of particle physics beyond the StandardModel.Note that light massive neutrinos could also play a role in the generation ofthe baryon asymmetry of the Universe from a previously created lepton asym-metry. In these leptogenesis scenarios, one can also obtain quite restrictivebounds on light neutrino masses, which are however model-dependent (see[245] for a recent review).This review was entirely devoted to the case of light neutrinos with massesof order eV or smaller. However, we would like to mention the case of a ster-ile neutrino with a mass of the order of a few keV’s and a small mixingwith the flavour neutrinos. It has been shown that these keV neutrinos couldalso have interesting cosmological consequences: they would play the role ofdark matter and replace the usual CDM component (see [246,247,248,249]for an incomplete list of references) . Because of their large thermal velocity(slightly smaller than that of active neutrinos), they would behave as WarmDark Matter and erase small-scale cosmological structures. Their mass canbe bounded from below using Lyman-α forest data from quasar spectra, andfrom above using X-ray observations [250,251,252,253,254]. This scenario, al-ready disfavoured according to the results in [254], is currently under carefulexamination.Note addedA few new analyses of cosmological data including non-zero neutrino massesappeared after this review was accepted for publication. In Ref. [255] theauthors used data on the Lyman-α forest from SDSS (improved thanks tothe inclusion of the mean flux constraints derived from a principal componentanalysis of quasar spectra), SDSS and 2dF galaxy clustering, SNIa and CMBpower spectra (WMAP3 and others) to get an upper bound of Mν < 0.17 eV(at 95% CL) on the total mass of the three active neutrinos (Mν < 0.26 eVfor four thermalized species). Instead, in Ref. [256] the limit Mν < 2.0 eV forthree neutrinos was found using WMAP3 data only, in agreement with theresult of the WMAP team [65].110AcknowledgmentsWe would like to thank Elena Pierpaoli for many useful comments, as well asAlessandro Cuoco, Alexander Dolgov, Kazuhide Ichikawa, Gianpiero Mangano,Laurence Perotto, Georg Raffelt, Rafael Rebolo, Jose´ A. Rubin˜o-Mart´ın, SubirSarkar and Pasquale Serpico for interesting feedback on this manuscript. Thiswork was supported by a MEC-IN2P3 agreement and by the European Net-work of Theoretical Astroparticle Physics ILIAS/N6 under contract numberRII3-CT-2004-506222. SP was supported by the Spanish grants FPA2005-01269, BFM2002-00345 and GV/05/017 of Generalitat Valenciana, as well asby a Ramo´n y Cajal contract of MEC.References[1] S. Hannestad, New J. Phys. 6 (2004) 108 [hep-ph/0404239].[2] Ø. Elgarøy and O. Lahav, New J. Phys. 7 (2005) 61 [hep-ph/0412075].[3] S. Hannestad, Prog. Part. Nucl. Phys. 57 (2006) 309 [astro-ph/0511595].[4] M. Tegmark, in Proceedings of Nobel Symposium 2004, Enko¨ping, Sweden(August 2004), hep-ph/0503257.[5] S. Hannestad, hep-ph/0602058.[6] A.D. Dolgov, Phys. Rep. 370 (2002) 333 [hep-ph/0202122].[7] M. Kamionkowski and A. Kosowsky, Ann. Rev. Nucl. Part. Sci. 49 (1999) 77[astro-ph/9904108].[8] M.C. Gonza´lez-Garc´ıa and Y. Nir, Rev. Mod. Phys. 75 (2003) 345[hep-ph/0202058].[9] M. Hirsch and J.W.F. Valle, New J. Phys. 6 (2004) 76 [hep-ph/0405015].[10] G. Altarelli and F. Feruglio, New J. Phys. 6 (2004) 106 [hep-ph/0405048].[11] R.N. Mohapatra and A.Yu. Smirnov, hep-ph/0603118.[12] S. Eidelman et al [Particle Data Group], Phys. Lett. B 592 (2004) 1.[13] M. Maltoni, T. Schwetz, M.A. To´rtola and J.W.F. Valle, New J. Phys. 6 (2004)122 [hep-ph/0405172].[14] G.L. Fogli, E. Lisi, A. Marrone and A. Palazzo, Prog. Part. Nucl. Phys., in press[hep-ph/0506083].[15] S.R. Elliott and P. Vogel, Ann. Rev. Nucl. Part. Sci. 52 (2002) 115[hep-ph/0202264].111[16] S.M. Bilenky, C. Giunti, J.A. Grifols and E. Masso´, Phys. Rep. 379 (2003) 69[hep-ph/0211462].[17] V. Barger, D. Marfatia and K. Whisnant, Int. J. Mod. Phys. E 12 (2003) 569[hep-ph/0308123].[18] R.D. McKeown and P. Vogel, Phys. Rep. 394 (2004) 315 [hep-ph/0402025].[19] A. Aguilar et al [LSND Coll.], Phys. Rev. D 64 (2001) 112007 [hep-ex/0104049].[20] A. Aguilar-Arevalo et al [MiniBooNE coll.], The MiniBooNE Run Plan (2003),available at http://www-boone.fnal.gov/publicpages/runplan.ps.gz[21] K. Eitel, Nucl. Phys. Proc. Suppl. 143 (2005) 197.[22] V.M. Lobashev, Nucl. Phys. A 719 (2003) 153.[23] C. Kraus et al, Eur. Phys. J. C 40 (2005) 447 [hep-ex/0412056].[24] A. Osipowicz et al, hep-ex/0109033. KATRIN experiment homepage athttp://www-ik.fzk.de/katrin/[25] A. Strumia and F. Vissani, Nucl. Phys. B 726 (2005) 294 [hep-ph/0503246].[26] V.A. Rodin, A. Faessler, F. Sˇimkovic and P. Vogel, Nucl. Phys. A 766 (2006)107 [nucl-th/0503063].[27] H.V. Klapdor-Kleingrothaus et al, Eur. Phys. J. A 12 (2001) 147[hep-ph/0103062].[28] C.E. Aalseth et al [IGEX Coll.], Phys. Rev. C 59 (1999) 2108.[29] C. Arnaboldi et al [Cuoricino Coll.], Phys. Rev. Lett. 95 (2005) 142501[hep-ex/0501034].[30] R. Arnold et al [NEMO Coll.], Phys. Rev. Lett. 95 (2005) 182302[hep-ex/0507083].[31] H.V. Klapdor-Kleingrothaus, I.V. Krivosheina, A. Dietz and O. Chkvorets,Phys. Lett. B 586 (2004) 198 [hep-ph/0404088].[32] S.T. Petcov, New J. Phys. 6 (2004) 109.[33] S. Pascoli, S.T. Petcov and T. Schwetz, Nucl. Phys. B 734 (2006) 24[hep-ph/0505226].[34] E. Nardi and J.I. Zuluaga, Nucl. Phys. B 731 (2005) 140 [hep-ph/0412104].[35] A.D. Dolgov et al, Nucl. Phys. B 632 (2002) 363 [hep-ph/0201287].[36] Y.Y.Y. Wong, Phys. Rev. D 66 (2002) 025015 [hep-ph/0203180].[37] K.N. Abazajian, J.F. Beacom and N.F. Bell, Phys. Rev. D 66 (2002) 013008[astro-ph/0203442].112[38] P.D. Serpico and G.G. Raffelt, Phys. Rev. D 71 (2005) 127301[astro-ph/0506162].[39] V. Barger et al, Phys. Lett. B 569 (2003) 123 [hep-ph/0306061].[40] A. Cuoco et al, Int. J. Mod. Phys. A 19 (2004) 4431 [astro-ph/0307213].[41] E.W. Kolb and M.S. Turner, The Early Universe (Addison-Wesley, 1990).[42] S. Dodelson, Modern Cosmology (Academic Press, 2003).[43] S. Hannestad and J. Madsen, Phys. Rev. D 52 (1995) 1764 [astro-ph/9506015].[44] A.D. Dolgov, S.H. Hansen and D.V. Semikoz, Nucl. Phys. B 503 (1997) 426[hep-ph/9703315].[45] S. Esposito et al, Nucl. Phys. B 590 (2000) 539 [astro-ph/0005573].[46] G. Mangano, G. Miele, S. Pastor and M. Peloso, Phys. Lett. B 534 (2002) 8[astro-ph/0111408].[47] G. Mangano et al, Nucl. Phys. B 729 (2005) 221 [hep-ph/0506164].[48] S. Hannestad, Phys. Rev. D 65 (2002) 083006 [astro-ph/0111423].[49] U. Seljak and M. Zaldarriaga, Astrophys. J. 469 (1996) 437 [astro-ph/9603033].See also the webpage http://cmbfast.org[50] A. Lewis, A. Challinor and A. Lasenby, Astrophys. J. 538 (2000) 473[astro-ph/9911177]. See also the webpage http://camb.info[51] V.F. Shvartsman, Pisma Zh. Eksp. Teor. Fiz. 9 (1969) 315 [JETP Lett. 9 (1969)184].[52] G. Steigman, D.N. Schramm and J.R. Gunn, Phys. Lett. B 66 (1977) 202.[53] S. Sarkar, Rep. Prog. Phys. 59 (1996) 1493 [hep-ph/9602260].[54] M. Cirelli, G. Marandella, A. Strumia and F. Vissani, Nucl. Phys. B 708 (2005)215 [hep-ph/0403158].[55] D.N. Spergel et al [WMAP Coll.], Astrophys. J. Suppl. 148 (2003) 175[astro-ph/0302209].[56] S. Hannestad, JCAP 0305 (2003) 004 [astro-ph/0303076].[57] V. Barger et al, Phys. Lett. B 566 (2003) 8 [hep-ph/0305075].[58] R.H. Cyburt, B.D. Fields, K.A. Olive and E. Skillman, Astropart. Phys. 23(2005) 313 [astro-ph/0408033].[59] G. Steigman, Int. J. Mod. Phys. E 15 (2006) 1 [astro-ph/0511534].[60] K.A. Olive and E.D. Skillman, Astrophys. J. 617 (2004) 29 [astro-ph/0405588].[61] P.D. Serpico et al, JCAP 0412 (2004) 010 [astro-ph/0408076].113[62] P. Crotty, J. Lesgourgues and S. Pastor, Phys. Rev. D 67 (2003) 123005[astro-ph/0302337].[63] E. Pierpaoli, Mon. Not. Roy. Astron. Soc. 342 (2003) L63 [astro-ph/0302465].[64] S. Hannestad, JCAP 0601 (2006) 001 [astro-ph/0510582].[65] D.N. Spergel et al [WMAP Coll.], astro-ph/0603449.[66] A. Ringwald and Y.Y.Y. Wong, JCAP 0412 (2004) 005 [hep-ph/0408241].[67] S.S. Gershtein and Ya.B. Zeldovich, JETP Lett. 4 (1966) 120 [Pisma Zh. Eksp.Teor. Fiz. 4 (1966) 174].[68] R. Cowsik and J. McClelland, Phys. Rev. Lett. 29 (1972) 669.[69] J.R. Primack, SLAC Beam Line 31N3 (2001) 50 [astro-ph/0112336].[70] J.R. Bond, G. Efstathiou and J. Silk, Phys. Rev. Lett. 45 (1980) 1980.[71] J.R. Bond and A.S. Szalay, Astrophys. J. 274 (1983) 443.[72] S.D.M. White, C.S. Frenk and M. Davis, Astrophys. J. 274 (1983) L1.[73] R. Valdarnini, T. Kahniashvili and B. Novosyadlyj, Astron. Astrophys. 336(1998) 11 [astro-ph/9804057].[74] C.P. Ma and E. Bertschinger, Astrophys. J. 455 (1995) 7 [astro-ph/9506072].[75] W.T. Hu, Ph.D. thesis (Univ. of California at Berkeley, 1995),astro-ph/9508126.[76] A.R. Liddle and D.H. Lyth, Cosmological inflation and large-scale structure(Cambridge Univ. Press, 2000).[77] E. Bertschinger, Proceedings of Cosmology 2000, Lisbon, Portugal (July 2000),astro-ph/0101009.[78] M. Bucher, K. Moodley and N. Turok, Phys. Rev. D 62 (2000) 083508[astro-ph/9904231].[79] R. Trotta, A. Riazuelo and R. Durrer, Phys. Rev. Lett. 87 (2001) 231301[astro-ph/0104017].[80] D.H. Lyth, C. Ungarelli and D. Wands, Phys. Rev. D 67, 023503 (2003)[astro-ph/0208055].[81] P. Crotty, J. Garc´ıa-Bellido, J. Lesgourgues and A. Riazuelo, Phys. Rev. Lett.91 (2003) 171301 [astro-ph/0306286].[82] M. Bucher et al, Phys. Rev. Lett. 93 (2004) 081301 [astro-ph/0401417].[83] K. Moodley et al, Phys. Rev. D 70 (2004) 103520 [astro-ph/0407304].[84] M. Beltra´n, J. Garc´ıa-Bellido, J. Lesgourgues and A. Riazuelo, Phys. Rev. D70 (2004) 103530 [astro-ph/0409326].114[85] S. Dodelson, E. Gates and A. Stebbins, Astrophys. J. 467 (1996) 10[astro-ph/9509147].[86] K. Ichikawa and T. Takahashi, astro-ph/0510849.[87] L.A. Kofman, N.Y. Gnedin and N.A. Bahcall, Astrophys. J. 413 (1993) 1.[88] J.M. Bardeen, J.R. Bond, N. Kaiser and A.S. Szalay, Astrophys. J. 304 (1986)15.[89] D.J. Eisenstein et al, Astrophys. J. 633 (2005) 560 [astro-ph/0501171].[90] M.Yu. Khlopov, Cosmoparticle physics (World Scientific, 1999)[91] S. Bashinsky and U. Seljak, Phys. Rev. D 69 (2004) 083002 [astro-ph/0310198].[92] R. Trotta and A. Melchiorri, Phys. Rev. Lett. 95 (2005) 011305[astro-ph/0412066].[93] W. Hu and D.J. Eisenstein, Astrophys. J. 498 (1998) 497 [astro-ph/9710216].[94] W. Hu, D.J. Eisenstein and M. Tegmark, Phys. Rev. Lett. 80 (1998) 5255[astro-ph/9712057].[95] J.A. Holtzman, Astrophys. J. Suppl. 71 (1989) 1.[96] D.Y. Pogosyan and A.A. Starobinsky, Astrophys. J. 447 (1995) 465[astro-ph/9409074].[97] C.P. Ma, Astrophys. J. 471 (1996) 13 [astro-ph/9605198].[98] D.J. Eisenstein and W. Hu, Astrophys. J. 511 (1997) 5 [astro-ph/9710252].[99] B. Novosyadlyj, R. Durrer and V.N. Lukash, Astron. Astrophys. 347 (1999)799 [astro-ph/9811262].[100] P. Crotty, J. Lesgourgues and S. Pastor, Phys. Rev. D 69 (2004) 123007[hep-ph/0402049].[101] J. Lesgourgues, S. Pastor and L. Perotto, Phys. Rev. D 70 (2004) 045016[hep-ph/0403296].[102] M. Takada, E. Komatsu and T. Futamase, Phys. Rev. D 73 (2006) 083520[astro-ph/0512374].[103] A. Slosar, astro-ph/0602133.[104] R. Trotta, A. Riazuelo and R. Durrer, Phys. Rev. D 67 (2003) 063520[astro-ph/0211600].[105] A. Upadhye, M. Ishak and P.J. Steinhardt, Phys. Rev. D 72 (2005) 063501[astro-ph/0411803].[106] C. Ye`che et al, Astron. Astrophys. 448 (2006) 831 [astro-ph/0507170].[107] G. Zech, Eur. Phys. J. direct C 4 (2002) 12 [hep-ex/0106023].115[108] R. Trotta, Ph.D. thesis (Univ. of Geneva, 2004), astro-ph/0410115.[109] W.L. Freedman et al, Astrophys. J. 553 (2001) 47 [astro-ph/0012376].[110] M. Beltra´n et al, Phys. Rev. D 71 (2005) 063532 [astro-ph/0501477].[111] R. Trotta, astro-ph/0504022.[112] G. Hinshaw et al, Astrophys. J. Suppl. 148 (2003) 135 [astro-ph/0302217].[113] A. Kogut et al, Astrophys. J. Suppl. 148 (2003) 161 [astro-ph/0302213].[114] G. Hinshaw et al [WMAP Coll.], astro-ph/0603451.[115] L. Page et al [WMAP Coll.], astro-ph/0603450.[116] C.L. Kuo et al [ACBAR coll.], Astrophys. J. 600 (2004) 32 [astro-ph/0212289].[117] C. Dickinson et al, Mon. Not. Roy. Astron. Soc. 353 (2004) 732[astro-ph/0402498].[118] A.C.S. Readhead et al, Astrophys. J. 609 (2004) 498 [astro-ph/0402359].[119] N.W. Halverson et al, Astrophys. J. 568 (2002) 38 [astro-ph/0104489].[120] M. Tristram et al, Astron. Astrophys. 436 (2005) 785 [astro-ph/0411633].[121] W.C. Jones et al, astro-ph/0507494.[122] F. Piacentini et al, astro-ph/0507507.[123] T.E. Montroy et al, astro-ph/0507514.[124] S. Hanany et al, Astrophys. J. 545 (2000) L5 [astro-ph/0005123].[125] M. Tegmark et al, Phys. Rev. D 69 (2004) 103501 [astro-ph/0310723].[126] Ø. Elgarøy and O. Lahav, JCAP 0304 (2003) 004 [astro-ph/0303089].[127] K. Ichikawa, M. Fukugita and M. Kawasaki, Phys. Rev. D 71 (2005) 043001[astro-ph/0409768].[128] M. Fukugita, contribution to the proceedings of NuFact’05, Frascati (Italy),June 2005 [hep-ph/0511068][129] A.G. Sa´nchez et al, Mon. Not. Roy. Astron. Soc. 366 (2006) 189[astro-ph/0507583].[130] R. Rebolo and J.A. Rubin˜o-Mart´ın, private communication.[131] C.J. MacTavish et al, astro-ph/0507503.[132] R. Rebolo et al, Mon. Not. Roy. Astron. Soc. 353 (2004) 747[astro-ph/0402466].[133] W.J. Percival et al [The 2dFGRS Coll.], Mon. Not. Roy. Astron. Soc. 327(2001) 1297 [astro-ph/0105252].116[134] S. Cole et al [The 2dFGRS Coll.], Mon. Not. Roy. Astron. Soc. 362 (2005) 505[astro-ph/0501174].[135] M. Tegmark et al [SDSS Coll.], Astrophys. J. 606 (2004) 702[astro-ph/0310725].[136] L. Verde et al, Astrophys. J. Suppl. 148 (2003) 195 [astro-ph/0302218].[137] U. Seljak et al, Phys. Rev. D 71 (2005) 103515 [astro-ph/0407372].[138] U. Seljak et al, Phys. Rev. D 71 (2005) 043511 [astro-ph/0406594].[139] V. Barger, D. Marfatia and A. Tregre, Phys. Lett. B595 (2004) 55[hep-ph/0312065].[140] G.L. Fogli et al, Phys. Rev. D 70 (2004) 113003 [hep-ph/0408045].[141] R.E. Smith et al [The Virgo Consortium Coll.], Mon. Not. Roy. Astron. Soc.341 (2003) 1311 [astro-ph/0207664].[142] X. Wang, M. Tegmark and M. Zaldarriaga, Phys. Rev. D 65 (2002) 123001[astro-ph/0105091].[143] W. Saunders et al, Mon. Not. Roy. Astron. Soc. 317 (2000) 55[astro-ph/0001117].[144] Ø. Elgarøy et al, Phys. Rev. Lett. 89 (2002) 061301 [astro-ph/0204152].[145] S. Perlmutter et al, Astrophys. J. 517 (1999) 565 [astro-ph/9812133].[146] A.G. Riess et al, Astrophys. J. 607 (2004) 665 [astro-ph/0402512].[147] J.L. Tonry et al, Astrophys. J. 594 (2003) 1 [astro-ph/0305008].[148] P. Astier et al, Astron. Astrophys. 447 (2006) 31 [astro-ph/0510447].[149] R.A.C. Croft et al, Astrophys. J. 581 (2002) 20 [astro-ph/0012324].[150] N.Y. Gnedin and A.J.S. Hamilton, Mon. Not. Roy. Astron. Soc. 334 (2002)107 [astro-ph/0111194].[151] S. Hannestad, in Proceedings of SEESAW25, Paris, France (June 2004)[hep-ph/0409108].[152] A. Goobar, S. Hannestad, E. Mortsell and H. Tu, astro-ph/0602155.[153] H. Bi, Astrophys. J. 405 (1993) 479.[154] M. Viel et al, Mon. Not. Roy. Astron. Soc. 329 (2002) 848 [astro-ph/0105233].[155] M. Zaldarriaga, R. Scoccimarro and L. Hui, Astrophys. J. 590 (2003) 1[astro-ph/0111230].[156] L. Hui, N.Y. Gnedin and Y. Zhang, Astrophys. J. 486 (1997) 599[astro-ph/9608157].117[157] R.A.C. Croft, W. Hu and R. Dave´, Phys. Rev. Lett. 83 (1999) 1092[astro-ph/9903335].[158] M. Viel, M.G. Haehnelt and V. Springel, Mon. Not. Roy. Astron. Soc. 354(2004) 684 [astro-ph/0404600].[159] P. McDonald et al, Astrophys. J. Suppl. 163 (2006) 80 [astro-ph/0405013].[160] P. McDonald et al, Astrophys. J. 635 (2005) 761 [astro-ph/0407377].[161] E. Pierpaoli, S. Borgani, D. Scott and M.J. White, Mon. Not. Roy. Astron.Soc. 342 (2003) 163 [astro-ph/0210567].[162] E. Semboloni et al, astro-ph/0511090.[163] E. Pierpaoli, Proceedings of the ”Multiwavelength Cosmology” conference,Mykonos Island, Greece (June 2003). Ed. by M. Plionis (Kluwer AcademicPub., 2004) p. 93[164] M. Fukugita, G.C. Liu and N. Sugiyama, Phys. Rev. Lett. 84 (2000) 1082[hep-ph/9908450].[165] T. Kahniashvili, E. von Toerne, N.A. Arhipova and B. Ratra, Phys. Rev. D71 (2005) 125009 [astro-ph/0503328].[166] S.W. Allen, R.W. Schmidt and S.L. Bridle, Mon. Not. Roy. Astron. Soc. 346(2003) 593 [astro-ph/0306386].[167] S. Hannestad and G. Raffelt, JCAP 0404 (2004) 008 [hep-ph/0312154].[168] M. Lattanzi, R. Ruffini and G.V. Vereshchagin, Phys. Rev. D 72 (2005) 063003[astro-ph/0509079].[169] S. Dodelson, A. Melchiorri and A. Slosar, astro-ph/0511500.[170] S. Hannestad, Phys. Rev. Lett. 95 (2005) 221301 [astro-ph/0505551].[171] A. Blanchard, M. Douspis, M. Rowan-Robinson and S. Sarkar, Astron.Astrophys. 412 (2003) 35 [astro-ph/0304237].[172] A.D. Dolgov and A.Yu. Smirnov, Phys. Lett. B 621 (2005) 1 [hep-ph/0501066].[173] A.D. Dolgov, S.H. Hansen and A.Yu. Smirnov, JCAP 0506 (2005) 004[astro-ph/0503612].[174] M. Kawasaki, K. Kohri and N. Sugiyama, Phys. Rev. Lett. 82 (1999) 4168[astro-ph/9811437].[175] M. Kawasaki, K. Kohri and N. Sugiyama, Phys. Rev. D 62 (2000) 023506[astro-ph/0002127].[176] G.F. Giudice et al, Phys. Rev. D 64 (2001) 043512 [hep-ph/0012317].[177] P. Adhya, D.R. Chaudhuri and S. Hannestad, Phys. Rev. D 68 (2003) 083519[astro-ph/0309135].118[178] S. Hannestad, Phys. Rev. D 70 (2004) 043506 [astro-ph/0403291].[179] K. Ichikawa, M. Kawasaki and F. Takahashi, Phys. Rev. D 72 (2005) 043522[astro-ph/0505395].[180] S. Hannestad, Phys. Rev. D 59 (1999) 125020 [astro-ph/9903475].[181] M. Kaplinghat, R.E. Lopez, S. Dodelson and R.J. Scherrer, Phys. Rev. D 60(1999) 123508 [astro-ph/9907388].[182] D. Kirilova, Int. J. Mod. Phys. D 13 (2004) 831 [hep-ph/0209104].[183] K. Abazajian, N.F. Bell, G.M. Fuller and Y.Y.Y. Wong, Phys. Rev. D 72(2005) 063004 [astro-ph/0410175].[184] A. Cuoco, J. Lesgourgues, G. Mangano and S. Pastor, Phys. Rev. D 71 (2005)123501 [astro-ph/0502465].[185] J.F. Beacom, N.F. Bell and S. Dodelson, Phys. Rev. Lett. 93 (2004) 121302[astro-ph/0404585].[186] S. Hannestad, JCAP 0502 (2005) 011 [astro-ph/0411475].[187] S. Hannestad and G. Raffelt, Phys. Rev. D 72 (2005) 103514 [hep-ph/0509278].[188] N.F. Bell, E. Pierpaoli and K. Sigurdson, Phys. Rev. D 73 (2006) 063523[astro-ph/0511410].[189] P.Q. Hung, hep-ph/0010126.[190] R. Fardon, A.E. Nelson and N. Weiner, JCAP 0410 (2004) 005[astro-ph/0309800].[191] E.J. Copeland, M. Sami and S. Tsujikawa, hep-th/0603057.[192] A.W. Brookfield, C. van de Bruck, D.F. Mota and D. Tocchini-Valentini, Phys.Rev. D 73 (2006) 083515 [astro-ph/0512367].[193] M. Tegmark, A. Taylor and A. Heavens, Astrophys. J. 480 (1997) 22[astro-ph/9603021].[194] M. Tegmark, Phys. Rev. D 56 (1997) 4514 [astro-ph/9705188].[195] G. Jungman, M. Kamionkowski, A. Kosowsky and D.N. Spergel, Phys. Rev.Lett. 76 (1996) 1007 [astro-ph/9507080].[196] D.J. Eisenstein, W. Hu and M. Tegmark, Astrophys. J. 518 (1998) 2[astro-ph/9807130].[197] M. Kaplinghat, L. Knox and Y.S. Song, Phys. Rev. Lett. 91 (2003) 241301[astro-ph/0303344].[198] J. Lesgourgues, L. Perotto, S. Pastor and M. Piat, Phys. Rev. D 73 (2006)045021 [astro-ph/0511735].119[199] J. Ruhl et al [SPT Coll.], in Astronomical Structures and MechanismsTechnology, ed. by J. Antebi and D. Lemke, proceedings of the SPIE 5498(2004) 11.[200] B.G. Keating et al, in Polarimetry in Astronomy, ed. by S. Fineschi,Proceedings of the SPIE 4843 (2003) 284.[201] S. Church et al [QUaD Coll.], New Astron. Rev. 47 (2003) 1083.[202] M. Piat et al [BRAIN-CLOVER Coll.], SF2A 2004 proceedings (2004)[astro-ph/0412590].[203] B. Maffei et al [BRAIN-ClOVER Coll.], EAS Publications Series 14 (2005)251.[204] J.A. Tauber, Adv. Space Res. 34 (2004) 491.[205] F.R. Bouchet et al, SF2A 2005 proceedings (2005) [astro-ph/0510423].[206] Y.S. Song and L. Knox, Phys. Rev. D 70 (2004) 063510 [astro-ph/0312175].[207] S. Hannestad, Phys. Rev. D 67 (2003) 085017 [astro-ph/0211106].[208] J. Lesgourgues, S. Pastor and S. Prunet, Phys. Rev. D 62 (2000) 023001[hep-ph/9912363].[209] L. Popa, C. Burigana, F. Finelli and N. Mandolesi, Astron. Astrophys. 363(2000) 825 [astro-ph/0009417].[210] M. Tegmark, Phys. Rev. Lett. 79 (1997) 3806 [astro-ph/9706198].[211] M. Moles et al, astro-ph/0504545.[212] B. Flaugher [Dark Energy Survey Coll.], Int. J. Mod. Phys. A 20 (2005) 3121.[213] J. Annis et al, astro-ph/0510195.[214] T. Abbott et al [Dark Energy Survey Coll.], astro-ph/0510346.[215] P. Fosalba, E. Gaztan˜aga and F. Castander, Astrophys. J. 597 (2003) L89[astro-ph/0307249].[216] R. Scranton et al [SDSS Coll.], astro-ph/0307335.[217] F. Bernardeau, Astron. Astrophys. 324 (1997) 15. [astro-ph/9611012].[218] M. Zaldarriaga and U. Seljak, Phys. Rev. D 58 (1998) 023003[astro-ph/9803150].[219] U. Seljak and M. Zaldarriaga, Phys. Rev. Lett. 82 (1999) 2636[astro-ph/9810092].[220] A. Lewis and A. Challinor, Phys. Rep. 429 (2006) 1 [astro-ph/0601594].[221] C.M. Hirata and U. Seljak, Phys. Rev. D 68 (2003) 083002 [astro-ph/0306354].120[222] A. Cooray, M. Kamionkowski and R.R. Caldwell, Phys. Rev. D 71 (2005)123527 [astro-ph/0503002].[223] W. Hu, Astrophys. J. 557 (2001) L79 [astro-ph/0105424].[224] W. Hu and T. Okamoto, Astrophys. J. 574 (2002) 566 [astro-ph/0111606].[225] T. Okamoto and W. Hu, Phys. Rev. D 67 (2003) 083002 [astro-ph/0301031][226] A. Amblard, C. Vale and M.J. White, New Astron. 9 (2004) 687[astro-ph/0403075].[227] J. Lesgourgues, M. Liguori, S. Matarrese and A. Riotto, Phys. Rev. D 71(2005) 103514 [astro-ph/0412551].[228] N. Bartolo, E. Komatsu, S. Matarrese and A. Riotto, Phys. Rep. 402 (2004)103 [astro-ph/0406398].[229] U. Seljak, Astrophys. J. 463 (1996) 1 [astro-ph/9505109].[230] A. Cooray and M. Kesden, New Astron. 8 (2003) 231 [astro-ph/0204068].[231] K.M. Smith, W. Hu and M. Kaplinghat, Phys. Rev. D 70 (2004) 043002[astro-ph/0402442].[232] W. Hu, Phys. Rev. D 64 (2001) 083005 [astro-ph/0105117].[233] W. Hu, Astrophys. J. 522 (1999) L21 [astro-ph/9904153].[234] J.A. Tyson, D.M. Wittman, J.F. Hennawi and D.N. Spergel, Nucl. Phys. Proc.Suppl. 124 (2003) 21 [astro-ph/0209632].[235] J.A. Tyson [the LSST Coll.], Proc. SPIE Int. Soc. Opt. Eng. 4836 (2002) 10[astro-ph/0302102].[236] A.R. Cooray, Astron. Astrophys. 348 (1999) 31 [astro-ph/9904246].[237] K.N. Abazajian and S. Dodelson, Phys. Rev. Lett. 91 (2003) 041301[astro-ph/0212216].[238] S. Hannestad, H. Tu and Y.Y.Y. Wong, astro-ph/0603019.[239] B. Jain and A. Taylor, Phys. Rev. Lett. 91 (2003) 141302 [astro-ph/0306046].[240] G.M. Bernstein and B. Jain, Astrophys. J. 600 (2004) 17 [astro-ph/0309332].[241] S. Wang et al, Phys. Rev. Lett. 95 (2005) 011302 [astro-ph/0505390].[242] S. Wang, J. Khoury, Z. Haiman and M. May, Phys. Rev. D 70 (2004) 123008[astro-ph/0406331].[243] K. Jahoda [The DUET Coll.], Astron. Nachr. 324 (2003) 132[astro-ph/0211287][244] S. Hannestad, A. Mirizzi and G. Raffelt, JCAP 0507 (2005) 002[hep-ph/0504059].121[245] W. Buchmu¨ller, P. Di Bari and M. Plu¨macher, Annals Phys. 315 (2005) 305[hep-ph/0401240].[246] S. Dodelson and L.M. Widrow, Phys. Rev. Lett. 72 (1994) 17[hep-ph/9303287].[247] A.D. Dolgov and S.H. Hansen, Astropart. Phys. 16, 339 (2002)[hep-ph/0009083].[248] K. Abazajian, Phys. Rev. D 73, 063506 (2006) [astro-ph/0511630].[249] A. Boyarsky et al, astro-ph/0603660.[250] K. Abazajian, G.M. Fuller and W.H. Tucker, Astrophys. J. 562 (2001) 593[astro-ph/0106002].[251] S.H. Hansen, J. Lesgourgues, S. Pastor and J. Silk, Mon. Not. Roy. Astron.Soc. 333, 544 (2002) [astro-ph/0106108].[252] M. Viel et al, Phys. Rev. D 71, 063534 (2005) [astro-ph/0501562].[253] K. Abazajian, Phys. Rev. D 73 (2006) 063513 [astro-ph/0512631].[254] U. Seljak, A. Makarov, P. McDonald and H. Trac, astro-ph/0602430.[255] U. Seljak, A. Slosar and P. McDonald, astro-ph/0604335.[256] M. Fukugita, K. Ichikawa, M. Kawasaki and O. Lahav, astro-ph/0605362.122

Neutrinos, Cosmology and Astrophysics

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