How smooth are particle trajectories in a ΛCDM Universe?

Abstract

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