Q-ball formation in the wake of Hubble-induced radiative corrections

We discuss some interesting aspects of the $\mathrm{Q}$-ball formation during the early oscillations of the flat directions. These oscillations are triggered by the running of soft $(\mathrm{m}\mathrm{a}\mathrm{s}\mathrm{s}{)}^2$ stemming from the nonzero energy density of the Universe. However, this is quite different from the standard $\rm Q$-ball formation. The running in presence of gauge and Yukawa couplings becomes strong if $m_{1\mathrm{/}2}\mathrm{/}{m}_0$ is sufficiently large. Moreover, the $\rm Q$-balls which are formed during the early oscillations constantly evolve, due to the redshift of the Hubble-induced soft mass, until the low-energy supersymmtery breaking becomes dominant. For smaller $m_{1\mathrm{/}2}\mathrm{/}{m}_0$, $\rm Q$-balls are not formed during early oscillations because of the shrinking of the instability band due to the Hubble expansion. In this case the $\rm Q$-balls are formed only at the weak scale, but typically carry smaller charges, as a result of their amplitude redshift. Therefore, the Hubble-induced corrections to the flat directions give rise to a successful $\mathrm{Q}$-ball cosmology.