Bose–Einstein condensate haloes embedded in dark energy

Context. We have studied clusters of self-gravitating collisionless Newtonian bosons in their ground state and in the presence of the cosmological constant to model dark haloes of dwarf spheroidal (dSph) galaxies. Aims. We aim to analyse the influence of the cosmological constant on the structure of these systems. Observational data of Milky Way dSph galaxies allow us to estimate the boson mass. Methods. We obtained the energy of the ground state of the cluster in the Hartree approximation by solving a variational problem in the particle density. We have also developed and applied the virial theorem. Dark halo models were tested in a sample of 19 galaxies. Galaxy radii, 3D deprojected half-light radii, mass enclosed within them, and luminosity-weighted averages of the square of line-of-sight velocity dispersions are used to estimate the particle mass. Results. Cosmological constant repulsive effects are embedded in one parameter ξ. They are appreciable for ξ > 10−5. Bound structures appear for ξ ≤ ξc = 1.65 × 10−4, what imposes a lower bound for cluster masses as a function of the particle mass. In principle, these systems present tunnelling through a potential barrier; however, after estimating their mean lifes, we realize that their existence is not affected by the age of the Universe. When Milky Way dSph galaxies are used to test the model, we obtain $3.5_{-1.0}^{&#x002B;1.3}\times 10^{-22}$ eV for the particle mass and a lower limit of $5.1^{&#x002B;2.2}_{-2.8}\times 10^6\,{M_{\odot}}$ for bound haloes. Conclusions. Our estimation for the boson mass is in agreement with other recent results which use different methods. From our particle mass estimation, the treated dSph galaxies would present dark halo masses ~5–11 ×107 M⊙. With these values, they would not be affected by the cosmological constant (ξ 10−5) would already feel their effects. Our model that includes dark energy allows us to deal with these dark haloes. Assuming quantities averaged in the sample of galaxies, 10−5 < ξ ≤ ξc dark haloes would contain stars up to ~8–15 kpc with luminosities ~9–4 ×103 L⊙. Then, their observation would be complicated. The comparison of our lower bound for dark halo masses with other bounds based on different arguments, leads us to think that the cosmological constant is actually the responsible of limiting the halo mass