Ultra-local models of modified gravity without kinetic term

40 pages, 23 figures, updated referencesInternational audienceWe present a class of modified-gravity theories which we call ultra-local models. We add a scalar field, with negligible kinetic terms, to the Einstein-Hilbert action. We also introduce a conformal coupling to matter. This gives rise to a new screening mechanism which is not entirely due to the non-linearity of the scalar field potential or the coupling function but to the absence of the kinetic term. As a result this removes any fifth force between isolated objects in vacuum. The predictions of these models only depend on a single free function, as the potential and the coupling function are degenerate, with an amplitude given by a parameter $\alpha \lesssim 10^{-6}$, whose magnitude springs from requiring a small modification of Newton's potential astrophysically and cosmologically. This singles out a redshift $z_{\alpha} \sim \alpha^{-1/3} \gtrsim 100$ where the fifth force is the greatest. The cosmological background follows the $\Lambda$-CDM history within a $10^{-6}$ accuracy, while cosmological perturbations are significantly enhanced (or damped) on small scales, $k \gtrsim 2 h {\rm Mpc}^{-1}$ at $z=0$. The spherical collapse and the halo mass function are modified in the same manner. We find that the modifications of gravity are greater for galactic or sub-galactic structures. We also present a thermodynamic analysis of the non-linear and inhomogeneous fifth-force regime where we find that the Universe is not made more inhomogeneous before $z_\alpha$ when the fifth force dominates, and does not lead to the existence of clumped matter on extra small scales inside halos for large masses while this possibility exists for masses $M\lesssim 10^{11} M_\odot$ where the phenomenology of ultra-local models would be most different from $\Lambda$-CDM