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Abstract

Very. Indeed, 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.