  Using the Boltzmann transport equation (BTE), we study the evolution of
nonequilibrium carrier distributions in simple ($sp$) metals, assumed to have
been instantaneously excited by an ultrafast laser pulse with photon energy $h
\nu$. The mathematical structure of the BTE scattering integrals reveals that
$h \nu$ is a natural energy scale for describing the dynamics. Normalizing all
energy quantities by $h \nu$ leads to a set of three unitless parameters --
$\beta / \delta$, $\gamma$, and $\alpha$ -- that control the relaxation
dynamics: $\beta / \delta$ is the normalized ratio of electron-phonon to
electron-electron scattering strengths, $\gamma$ is the normalized phonon
(lattice) temperature, and $\alpha$ is the normalized absorbed energy density.
Using this theory, we systematically investigate relaxation times for the
high-energy part of the distribution ($\tau_H$), energy transfer to the phonon
subsystem ($\tau_E$), and intracarrier thermalization ($\tau_{th}$). In the
linear region of response (valid when $\alpha$ is sufficiently small), we offer
heuristic descriptions of each of these relaxation times as functions of $\beta
/ \delta$ and $\gamma$. Our results as a function of excitation level $\alpha$
show that many ultrafast experimental investigations lie in a transition region
between low excitation (where the relaxation times are independent of $\alpha$)
and high excitation (where the two-temperature model of carrier dynamics is
valid). Approximate boundaries that separate these three regions are described
by simple expressions involving the normalized parameters of our model.
