Self-Similarity Breaking of Cosmological Solutions with Collisionless Matter

Abstract

In this paper we consider the Einstein-Vlasov system with Bianchi VII0 symmetry. Under the assumption of small data we show that self-similarity breaking occurs for reflection symmetric solutions. This generalizes the previous work concerning the non-tilted fluid case [13] to the Vlasov case, and we obtain detailed information about the late-time behaviour of metric and matter terms. * holee@khu.ac.kr † ernesto.nungesser@icmat.es

On almost Ehlers-Geren-Sachs theorems

“…One could argue that these results are special since the matter model is a perfect fluid, but more recently the massive case for reflection symmetric solutions to the Einstein-Vlasov system with Bianchi VII 0 symmetry was covered in [7]. Since the microwave background comes from massless particles it is of special interest to treat this case and in the present paper we obtain a similar result for the massless case assuming small data.…”

“…and that one can express J i in terms of the metric, second fundamental form and the structure constants. Using (7) and considering the Bianchi A case where C l ml = 0, the last equation turns to…”

“…Since Σ a b is trace free sometimes it is convenient to work with Σ + and Σ − as was done in [7] which are defined by…”

“…For the reflection symmetric massless Bianchi VII 0 case we have the same equations for the curvature variables as in the massive case [7]. Let (ijk) denote a cyclic permutation of (123) and let us suspend the Einstein summation convention for the next three formulas.…”

“…The relevant equations for Σ + , M , X and ψ are obtained from equations ( 23)-( 26) of [7] by making the following replacements in terms of the relevant quantities we use here:…”