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

TL;DR

This work extends the Effective Field Theory of Large-Scale Structure (EFTofLSS) to include massive neutrinos by treating neutrinos with a Boltzmann equation coupled to dark-matter-like fluid dynamics. The authors solve perturbatively in the neutrino fraction $f_\nu$ and in $k/k_{\rm NL}$, introducing counterterms to renormalize short-distance physics and distinguishing fast ($k_{fs}$) and slow neutrinos, the latter requiring a speed-of-sound–like counterterm. They compute the one-loop total-matter power spectrum, finding $\Delta P_{\rm total} \sim -16 f_\nu P_{\rm dm,dm}$ for $k$ above the free-streaming scale and about half that for lower $k$, with the leading contribution from the back-reaction of neutrinos on dark matter; many counterterms resemble a rescaled dark-matter $c_s^2$, making a simple $c_s^2 \to c_s^2 + f_\nu \Delta c_s^2$ approach viable. The framework provides a controlled analytic path to incorporate neutrinos in LSS predictions, enabling neutrino-mass inference from cosmological data and guiding comparisons with simulations and other analytic models.

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 ($k_{\rm fs}$) is typically longer than the non-linear scale ($k_{\rm NL}$) 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 $(f_ν)$ 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 $k_{\rm fs}\gtrsim k_{\rm NL}$ 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 $16f_ν$ times the dark matter one for $k$'s larger that the typical $k_{\rm fs}$. 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 $f_ν$.

Figures (23)

• Figure 1: Perturbative solution for the neutrino distributions. Dashed arrows represent the neutrinos, while continuous line represent the dark matter field. The circle represents the background initial neutrino distribution $f^{[0]}$, while the boxes represent the initial dark matter fields, of which we take correlations to create power-spectrum diagrams later on. All these diagrams involve the linearly evolved dark matter field.
• Figure 2: Perturbative solution for the neutrino distributions that use non-linear dark matter fields at second order. First on the right, we have $f^{[1,2]}$, where a second order solution is obtained by using the second order dark matter field on the $f^{[1]}$ solution for neutrinos. Similarly, $f^{[2,12]}$ and $f^{[2,21]}$ represent third order solutions obtained using $f^{[2]}$ and the second order dark matter potential. The fourth diagram from the left represents a perturbative solution for the neutrino distributions that use non-linear dark matter fields at third order, and that we denote by $f^{[1,3]}$. For later convenience, we also represent the perturbative solutions for dark matter, $\delta_{\rm dm}^{\{(1),(2),(3)\}}$.
• Figure 3: Left: One-loop diagram obtained contracting $f^{[1,1]}$ with $\delta_{\rm dm}^{(3)}$. Dotted line means that we take the expectation value over the initial conditions. Right: One-loop diagram obtained contracting $f^{[1,2]}$ with $\delta_{\rm dm}^{(2)}$.
• Figure 4: Left: One-loop diagram obtained contracting $f^{[1,3]}$ with $\delta_{\rm dm}^{(1)}$. Right: One-loop diagram obtained contracting $f^{[2,11]}$ with $\delta_{\rm dm}^{(2)}$.
• Figure 5: Left: One-loop diagram obtained contracting $f^{[2,12]}$ with $\delta_{\rm dm}^{(1)}$. We do not show the analogous diagram build using $f^{[2,21]}$. Right: One-loop diagram obtained contracting $f^{[3,111]}$ with $\delta_{\rm dm}^{(1)}$.