  The critical properties characterizing the formation of the Floquet time
crystal are investigated analytically in the periodically driven $O(N)$ model.
In particular, we focus on the critical line separating the trivial phase with
period synchronized dynamics and absence of long-range spatial order from the
non-trivial phase where long-range spatial order is accompanied by
period-doubling dynamics. In the vicinity of the critical line, with a
combination of dimensional expansion and exact solution for $N\to\infty$, we
determine the exponent $\nu$ that characterizes the divergence of the spatial
correlation length of the equal-time correlation functions, the exponent
$\beta$ characterizing the growth of the amplitude of the order-parameter, as
well as the initial-slip exponent $\theta$ of the aging dynamics when a quench
is performed from deep in the trivial phase to the critical line. The exponents
$\nu, \beta, \theta$ are found to be identical to those in the absence of a
drive. In addition, the functional form of the aging is found to depend on
whether the system is probed at times that are small or large compared to the
drive period. The spatial structure of the two-point correlation functions,
obtained as a linear response to a perturbing potential in the vicinity of the
critical line, is found to show algebraic decays that are longer ranged than in
the absence of a drive, and besides being period-doubled, are also found to
oscillate in space at the wave-vector $\omega/(2 v)$, $v$ being the velocity of
the quasiparticles, and $\omega$ being the drive frequency.
