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

Abstract

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 ($k_{\mathrm{fs}}$) is typically longer than the non-linear scale ($k_{\mathrm{NL}}$) 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 $k/k_{\mathrm{NL}}$, 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 $k_{\mathrm{fs}}\gtrsim k_{\mathrm{NL}}$ require an additional counterterm similar to the speed of sound ($c_s$) 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 $k_{\mathrm{fs}}$. 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 $f_{\nu }$.

67 pages, 23 figures; v2: minor corrections

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