Structure formation in the Lemaître-Tolman model

Abstract

Structure formation within the Lemaître-Tolman model is investigated in a general manner. We seek models such that the initial density perturbation within a homogeneous background has a smaller mass than the structure into which it will develop, and the perturbation then accretes more mass during evolution. This is a generalisation of the approach taken by Bonnor in 1956. It is proved that any two spherically symmetric density profiles specified on any two constant time slices can be joined by a Lemaître-Tolman evolution, and exact implicit formulae for the arbitrary functions that determine the resulting L-T model are obtained. Examples of the process are investigated numerically.

Backreaction in the Lemaître–Tolman–Bondi model

“…Another possibility could be a quasi-perturbative approach where one keeps to first order perturbation theory for scales which are in the linear regime, but applies a non-perturbative solution for scales which have gone non-linear. Since the LTB model can describe the formation of clusters and voids smoothly starting from small perturbations [28], it is ideal for this purpose. (A simple version of embedding LTB solutions into a FRW universe was done in [36], and the idea goes back to [23].)…”

“…In particular, when t 0 is constant, (9) reduces to the FRW case, ρ = 4M 2 P l /(3τ 2 ). It is possible to specify a LTB solution by giving the energy density on initial and final hypersurfaces [28], instead of giving E and t 0 as done here.…”

“…The LTB model can be viewed as an Einstein-de Sitter universe with a single spherically symmetric perturbation which can be arbitrarily large, as opposed to the linearly perturbed FRW universe which has an ensemble of small perturbations. The LTB model can describe the collapse of an overdensity or the formation of a void in an expanding universe [28], and it has also been used to describe the local inhomogeneous universe [29]. As a model for the entire universe, it is a toy model which is useful because the backreaction problem can be studied quantitatively and without any approximations, as has been previously done in [30] (averaging in the LTB model has also been discussed in [31]; see also [32]).…”

Probing spatial homogeneity with LTB models: a detailed discussion

“…(4) and can only be computed numerically. We use the parametric solutions from Appendix A in combination with the TOMS 748 root-bracketing algorithm from Alefeld et al (1995) to determine E(r) as a function of t 0 , r and M(r), see Krasiński & Hellaby (2002) for details on choosing the initial bracket. We reject mass models that require E(r) ≤ −1/2 because this would cause singular lineelements (cf.…”

Gravitational lensing by a structure in a cosmological background

“…We will start with a toy model expressible in analytic functions and then continue to construct a more realistic model using the procedure proposed in [15,16].…”