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The homogeneous cooling state (HCS) of a granular gas described by theinelastic Boltzmann equation is reconsidered. As usual, particles are taken asinelastic hard disks or spheres, but now the coefficient of normal restitution$\alpha$ is allowed to take negative values $\alpha\in[-1,1]$, a simple way ofmodeling more complicated inelastic interactions. The distribution function ofthe HCS is studied at the long-time limit, as well as for intermediate times.At the long-time limit, the relevant information of the HCS is given by ascaling distribution function $\phi_s(c)$, where the time dependence occursthrough a dimensionless velocity $c$. For $\alpha\gtrsim -0.75$, $\phi_s$remains close to the gaussian distribution in the thermal region, its cumulantsand exponential tails being well described by the first Sonine approximation.On the contrary, for $\alpha\lesssim -0.75$, the distribution function becomesmultimodal, its maxima located at $c\ne 0$, and its observable tails algebraic.The latter is a consequence of an unbalanced relaxation-dissipationcompetition, and is analytically demonstrated for $\alpha\simeq -1$ thanks to areduction of the Boltzmann equation to a Fokker-Planck-like equation. Finally,a generalized scaling solution to the Boltzmann equation is also found$\phi(c,\beta)$. Apart from the time dependence occurring through thedimensionless velocity, $\phi(c,\beta)$ depends on time through a new parameter$\beta$ measuring the departure of the HCS from its long-time limit. It isshown that $\phi(c,\beta)$ describes the time evolution of the HCS for almostall times. The relevance of the new scaling is also discussed.

Generalized time evolution of the homogeneous cooling state of a granular gas with positive and negative coefficient of normal restitution