How smooth are particle trajectories in a $\Lambda$CDM Universe?

TL;DR

This paper proves that in a flat $ m f ext{Lambda}$CDM universe, particle trajectories are analytically expressible in terms of the scale factor until a finite time after decoupling, using a novel Lagrangian formulation and recursion relations.

Contribution

It introduces explicit all-order recursion relations for the Lagrangian displacement field, establishing the temporal analyticity of particle trajectories in $ m f ext{Lambda}$CDM models.

Findings

Trajectories are analytic in the scale factor until a finite time after decoupling.

Analyticity depends on $ m f ext{Lambda}$ and initial density smoothness.

Analyticity is preserved with curvature but not with primordial radiation.

Abstract

It is shown here that in a flat, cold dark matter (CDM) dominated Universe with positive cosmological constant ($\Lambda$), modelled in terms of a Newtonian and collisionless fluid, particle trajectories are analytical in time (representable by a convergent Taylor series) until at least a finite time after decoupling. The time variable used for this statement is the cosmic scale factor, i.e., the "$a$-time", and not the cosmic time. For this, a Lagrangian-coordinates formulation of the Euler-Poisson equations is employed, originally used by Cauchy for 3-D incompressible flow. Temporal analyticity for $\Lambda$CDM is found to be a consequence of novel explicit all-order recursion relations for the $a$-time Taylor coefficients of the Lagrangian displacement field, from which we derive the convergence of the $a$-time Taylor series. A lower bound for the $a$-time where analyticity is guaranteed and shell-crossing is ruled out is obtained, whose value depends only on $\Lambda$ and on the initial spatial smoothness of the density field. The largest time interval is achieved when $\Lambda$ vanishes, i.e., for an Einstein-de Sitter universe. Analyticity holds also if, instead of the $a$-time, one uses the linear structure growth $D$-time, but no simple recursion relations are then obtained. The analyticity result also holds when a curvature term is included in the Friedmann equation for the background, but inclusion of a radiation term arising from the primordial era spoils analyticity.