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