Quasi-spherical collapse of matter in $\Lambda$CDM

TL;DR

This paper derives new exact analytical solutions for quasi-spherical matter collapse in $\Lambda$CDM cosmology, demonstrating earlier collapse times compared to purely spherical models using a Lagrangian fluid approach.

Contribution

It introduces a convergent Taylor-series method for analyzing quasi-spherical collapse in $\Lambda$CDM, extending previous spherical models to more realistic cosmological conditions.

Findings

Collapse occurs earlier than in spherical models.

The Taylor-series approach converges until shell-crossing.

Method applies to both CDM and $\Lambda$CDM$ universes.

Abstract

We report the findings of new exact analytical solutions to the cosmological fluid equations, namely for the case where the initial conditions are perturbatively close to a spherical top-hat profile. To do so we enable a fluid description in a Lagrangian-coordinates approach, and prove the convergence of the Taylor-series representation of the Lagrangian displacement field until the time of collapse ("shell-crossing"). This allows the determination of the time for quasi-spherical collapse, which is shown to happen generically earlier than in the spherical case. For pedagogical reasons, calculations are first given for a spatially flat universe that is only filled with a non-relativistic component of cold dark matter (CDM). Then, the methodology is updated to a $\Lambda$CDM Universe, with the inclusion of a cosmological constant $\Lambda>0$.