Quasi-spherical collapse of matter in ΛCDM

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 $\Lambda>0$.