Revisiting coupled CDM-massive neutrino perturbations in diverse cosmological backgrounds

Massive neutrinos are well-known to cause a characteristic suppression in the growth of structures at scales below the neutrino free-streaming length. A detailed understanding of this suppression is essential in the era of precision cosmology we are entering into, enabling us to better constrain the total neutrino mass and possibly probe (beyond)-$\Lambda$CDM cosmological model(s). Instead of the usual N-body simulation or Boltzmann solver, in this article we consider a two-fluid framework at the linear scales, where the neutrino fluid perturbations are coupled to the CDM (+ baryon) fluid via gravity at redshifts of interest. Treating the neutrino mass fraction $f_\nu$ as a perturbative parameter, we find solutions to the system with redshift-dependent neutrino free-streaming length in $\Lambda$CDM background via two separate approaches. The perturbative scale-dependent solution is shown to be in excellent agreement with numerical solution of the two-fluid equations valid to all orders in $f_{\nu}$, and also agrees with results from {\texttt{CLASS}} to a good accuracy. We further generalize the framework to incorporate different evolving dark energy backgrounds and found sub-percent level differences in the suppression, all of which lie within the observational uncertainty of BOSS-like surveys. We also present a brief discussion on the prospects of the current analysis in the context of upcoming missions.