Relativistic approach to the kinematics of large-scale peculiar motions

Abstract

We consider the linear kinematics of large-scale peculiar motions in a perturbed Friedmann universe. In so doing, we take the viewpoint of the “real” observers that move along with the peculiar flow, relative to the smooth Hubble expansion. Using relativistic cosmological perturbation theory, we study the linear evolution of the peculiar velocity field, as well as the expansion/contraction, the shear and the rotation of the bulk motion. Our solutions show growth rates considerably stronger than those of the earlier treatments, which were mostly Newtonian. On scales near and beyond the Hubble radius, namely at the long-wavelength limit, peculiar velocities are found to grow as $$a^2$$a2, in terms of the scale factor, instead of the Newtonian $$a^{1/2}$$a1/2-law. We attribute this to the fact that, in general relativity, the energy flux, triggered here by the peculiar motion of the matter, also contributes to the local gravitational field. In a sense, the bulk flow gravitates, an effect that has been bypassed in related relativistic studies. These stronger growth-rates imply faster peculiar velocities at horizon crossing and higher residual values for the peculiar-velocity field. Alternatively, one could say that our study favours bulk peculiar flows larger and faster than anticipated.

The peculiar Jeans length

“…(2.3.1) of [3] (or Eq. (10.101) of [4]), in the bulk-flow frame, gives [11][12][13] Δ a = −Z a − 3a Hṽ a − 3a H 2ṽ a + aD aθ , (…”

“…There is no flux contribution to the Newtonian gravitational field. This extra input to Einstein's equations feeds into the conservation laws and eventually into the relativistic formulae governing the evolution of peculiar velocity perturbations (see Appendix A here for a comparison to the Newtonian study and also [12,13] for further discussion).…”

Non-comoving cold dark matter in a $$\varLambda $$CDM background

“…As shown in Appendix C (see also [22]), the "pancake models" of [22] described above rely on the fact that quasiplane Szekeres-II model with metric (7) and the spatially flat ΛCDM model with metric (C.13) admit a smooth matching along an arbitrary number of hypersurfaces Z i = 0, i = 1..n parametrized in the cylindrical coordinates of (C.13) as x a = [t, w i 0 , r, φ] where w i 0 are arbitrary constants. The resulting configurations are sequences of arbitrary numbers of Szekeres-II and ΛCDM patches separated by matching hypersurfaces w = w i 0 , i = 1..n. The junction conditions for these matchings in cylindrical coordinates are (see Appendix C)…”

“…It is important to remark that the connection between the Szekeres-II model and a ΛCDM background that we are considering is based on performing smooth matchings between different Szekeres-II regions with metric (7) and ΛCDM regions of metric (C.13) as discussed above (based on [22] and illustrated by Fig. 1), with the parameters of the Szekeres-II regions only restricted by fulfilling the matching conditions (36).…”

“…Different congruences of observers define different Hubble flows characterized by the kinematic quantities associated with their 4-velocities. In particular, Tsagas has examined the effect of considering peculiar velocities under this approach on the interpretation of cosmological observations (see [6,7] and references therein). Another example of non-comoving matter can be found in [8], which examined the evolution of cosmic voids formed by baryons and CDM.…”