  In its simplest statistical-mechanical description, a granular fluid can be
modeled as composed of smooth inelastic hard spheres (with a constant
coefficient of normal restitution $\alpha$) whose velocity distribution
function obeys the Enskog-Boltzmann equation. The basic state of a granular
fluid is the homogeneous cooling state, characterized by a homogeneous,
isotropic, and stationary distribution of scaled velocities, $F(\mathbf{c})$.
The behavior of $F(\mathbf{c})$ in the domain of thermal velocities ($c\sim 1$)
can be characterized by the two first non-trivial coefficients ($a_2$ and
$a_3$) of an expansion in Sonine polynomials. The main goals of this paper are
to review some of the previous efforts made to estimate (and measure in
computer simulations) the $\alpha$-dependence of $a_2$ and $a_3$, to report new
computer simulations results of $a_2$ and $a_3$ for two-dimensional systems,
and to investigate the possibility of proposing theoretical estimates of $a_2$
and $a_3$ with an optimal compromise between simplicity and accuracy.
