Growth and dissolution of spherical density enhancements in SCDEW cosmologies

Strongly Coupled Dark Energy plus Warm dark matter (SCDEW) cosmologies are based on the finding of a conformally invariant (CI) attractor solution during the early radiative expansion, requiring then the stationary presence of $\sim 1\, \%$ of coupled-DM and DE, since inflationary reheating. In these models, coupled-DM fluctuations, even in the early radiative expansion, grow up to non-linearity, as shown in a previous associated paper. Such early non-linear stages are modelized here through the evolution of a top-hat density enhancement. As expected, its radius $R$ increases up to a maximum and then starts to decrease. Virial balance is reached when the coupled-DM density contrast is just 25-26 and DM density enhancement is $\cal O$$(10\, \%)$ of total density. Moreover, we find that this is not an equilibrium configuration as, afterwards, coupling causes DM particle velocities to increase, so that the fluctuation gradually dissolves. We estimate the duration of the whole process, from horizon crossing to dissolution, and find $z_{horizon}/z_{erasing} \sim 3 \times 10^4$. Therefore, only fluctuations entering the horizon at $z \lesssim 10^9$-$10^{10}$ are able to accrete WDM with mass $\sim 100\,$eV -as soon as it becomes non-relativistic- so avoiding full disruption. Accordingly, SCDEW cosmologies, whose WDM has mass $\sim 100\,$eV, can preserve primeval fluctuations down to stellar mass scale.