Effects of the cosmological constant on cold dark matter clusters

¹ Departamento de Física TeóricaUniversidad de Zaragoza, 50009 Zaragoza, Spain
e-mail: membrado@unizar.es
² BIFI, Instituto de Biofísica y Física de Sistemas Complejos, Universidad de Zaragoza, 50009 Zaragoza, Spain
e-mail: amalio@unizar.es

Received: 28 February 2014
Accepted: 2 May 2014

Abstract

Context. Cold dark matter inhomogeneities are considered in a homogeneous background of matter, radiation, and the cosmological constant in a flat universe.

Aims. We investigate the influence of the cosmological constant on the non-linear collapse of cold dark matter clusters.

Methods. For simplicity, a spherical infall model has been used to describe the collapse of non-relativistic mass shells; besides, an average distribution of density around a cluster of galaxies has been taken. Boundary conditions are imposed by the solution of the linearized equation for the growth of matter perturbations and by the cold dark matter power spectrum.

Results. For an average cluster, the radii of shells and masses enclosed by them have been obtained at their zero proper acceleration (ZA) redshifts, at their turn-around (TA) redshifts and at their virialization (VIR) redshifts. According to our results at present, the shell that reaches its turn-around point shows [r^TA] ₀ = 6.85 Mpc and [ℳ^TA] ₀ = 6.76 × 10¹⁴ ℳ_⊙. The virializing shell fulfills [r^TA] ₀ = 4.57 [r^VIR] ₀ and [ℳ^TA] ₀ = 1.95 [ℳ^VIR] ₀. These results differ appreciably from those derived from a model with cosmological constant equal to zero in a flat universe: [r^TA(Λ = 0)] ₀ = 6.62 [r^VIR(Λ = 0)] ₀ and [ℳ^TA(Λ = 0)] ₀ = 5.26 [ℳ^VIR(Λ = 0)] ₀; this discrepancy could be considered as a new independent proof of the existence of dark energy. The shell with zero proper acceleration presents [r^ZA] ₀ = 1.59 [r^TA] ₀ and [ℳ^ZA] ₀ = 1.63 [ℳ^TA] ₀. We have found that there is a limit to the mass of the average cluster, which is able to virialize; its value is { ℳ^VIR } _MAX = 8.1 × 10¹⁴ M_⊙. As expected, we found that shells present null proper acceleration at redshift values that are smaller than 0.755.

Conclusions. We have noticed that the cosmological constant imposes an upper limit for the mass enclosed by shells, which are able to reach zero proper velocity. Hence, this mass is the maximum mass of the virialized core, { ℳ^VIR } _MAX. For the average cluster addressed in this work, the value is 2.34 times the mass of the virialized core at present. Shells enclosing masses ℳ > { ℳ^VIR } _MAX achieve zero proper acceleration and speed up, moving away from the virialized core, and never reach a turn-around point. Shells with ℳ ≫ { ℳ^VIR } _MAX show zero proper aceleration at redshifts close to that at which the universe background acceleration is null. Finally, we have found that the relation between shell proper velocities and their radii can be adjusted by a straight line at z = 0 and from approximately 20 up to 40 Mpc; however, this line does not intercept the origin as velocities due to the Hubble flux do.