The Effective Field Theory of Large-Scale Structure in the presence of Massive Neutrinos

We develop a formalism to analytically describe the clustering of matter in the mildly non-linear regime in the presence of massive neutrinos. Neutrinos, whose free streaming wavenumber ($k_{\mathrm{f}\mathrm{s}}$) is typically longer than the non-linear scale ($k_{\mathrm{N}\mathrm{L}}$) are described by a Boltzmann equation coupled to the effective fluid-like equations that describe dark matter. We solve the equations expanding in the neutrino density fraction $(f_{\nu })$ and in $k/ k_{\rm NL}$, and add suitable counterterms to renormalize the theory. This allows us to describe the contribution of short distances to long-distance observables. Equivalently, we construct an effective Boltzmann equation where we add additional terms whose coefficients renormalize the contribution from short-distance physics. We argue that neutrinos with $k_{\rm fs}\gtrsim k_{\rm NL}$requireanadditionalcountertermsimilartothespeedofsound($c_s$) for dark matter. We compute the one-loop total-matter power spectrum, and find that it is roughly equal to $16f_{\nu }$ times the dark matter one for $k$’s larger that the typical $k_{\mathrm{f}\mathrm{s}}$. It is about half of that for smaller $k$’s. The leading contribution results from the back-reaction of the neutrinos on the dynamics of the dark matter. The counterterms contribute in a hierarchical way: the leading ones can either be computed in terms of $c_s$, or can be accounted for by shifting $c_s$ by an amount proportional to $f_{\nu }$.