The Effective Field Theory of Cosmological Large Scale Structures

TL;DR

This work develops an effective field theory for cosmological large-scale structure by smearing the collisionless dark matter Boltzmann equation into an IR fluid with a small set of parameters (notably the effective speed of sound and viscosities). These parameters encode UV physics and are calibrated from N-body simulations, enabling a convergent perturbative expansion and a one-loop power spectrum that matches nonlinear spectra to percent precision up to k ~ 0.24 h/Mpc. The EFT framework both explains the damping of power relative to standard perturbation theory and provides a controlled path to higher-precision predictions by including additional terms and higher loops. Overall, the approach offers a scalable, predictive description of dark matter clustering that complements numerical simulations and can be extended to broader cosmologies and observables.

Abstract

Large scale structure surveys will likely become the next leading cosmological probe. In our universe, matter perturbations are large on short distances and small at long scales, i.e. strongly coupled in the UV and weakly coupled in the IR. To make precise analytical predictions on large scales, we develop an effective field theory formulated in terms of an IR effective fluid characterized by several parameters, such as speed of sound and viscosity. These parameters, determined by the UV physics described by the Boltzmann equation, are measured from N-body simulations. We find that the speed of sound of the effective fluid is c_s^2 10^(-6) and that the viscosity contributions are of the same order. The fluid describes all the relevant physics at long scales k and permits a manifestly convergent perturbative expansion in the size of the matter perturbations δ(k) for all the observables. As an example, we calculate the correction to the power spectrum at order δ(k)^4. The predictions of the effective field theory are found to be in much better agreement with observation than standard cosmological perturbation theory, already reaching percent precision at this order up to a relatively short scale k \sim 0.24 h/Mpc.

Figures (11)

• Figure 1: Diagrammatic representation of $P_{22}$ (top left), $P_{13}$ (top right), and $P_{13,\; c^2_{\rm comb}}$ (bottom). Continuous green lines represent Green's functions, red dashed lines represent free fields, and red crosses circled by a dotted blue line represent correlation among free fields.
• Figure 2: Time dependence of $c^2_{\rm\,comb}$ as inferred using the correct time dependence from $P_{13}$ and instead using the approximate time dependence derived from the growth functions (see App. \ref{['app:approx_perturbation_theory']}). Starting from very early times, we see that $c^2_{\rm\,comb}$ grows as a functions of time, peaks at about $a\simeq 0.7$, and then decreases near the present epoch, probably as due to the onset of dark energy. $c^2_{\rm\,comb}$ is positive, implying that this term tends to slow down the collapse of structures.
• Figure 3: Running of $c_{\rm\,comb}^2$ as a function of $\Lambda$. The purple band contains the region for the values of $c_{\rm\,comb}^2$ as inferred from matching with the non-linear power spectrum from CAMB at the renormalization scale $k=0.1h$ Mpc$^{-1}$ and $k=0.18h$ Mpc$^{-1}$. The dependence on the renormalization scale is a measure of the importance of higher loops. We see that as $\Lambda\to\infty$, $c_{\rm\,comb}^2$ decreases as more and more modes are included within the regime of validity of the EFT. However, the fact that as $\Lambda=\infty$, $c_{\rm\,comb}^2\neq 0$ is an indication of the fact that the fundamental theory is not described by a pressureless ideal fluid, but by indeed freely streaming dark matter particles. Data points with $1\,\sigma$ error bars represent the value obtained from $N$-body numerical simulations using the methods described in Sec. \ref{['sec:simulation_results']} using two different smoothing lengths $\Lambda^{-1}$. Given the error bars from numerical evaluation, the measured values are in remarkable agreement with what inferred from renormalizing using the power spectrum.
• Figure 4: In this plot we present to values obtained for $c^2_{\rm\,comb}$ as a function of the external momentum used in (\ref{['eq:cs_running']}). We see that only as $k_{\rm ext}\to0$, $c^2_{\rm\,comb}$ becomes $k_{\rm ext}$ independent. This is so because at high $k$ any higher derivative terms suppressed by powers of $k/k_{NL}$ are important.
• Figure 5: Measurement of ${c^2_{\rm comb}}$ in the UV with $\Lambda=1/3~ h$ Mpc$^{-1}$ and $\Lambda=1/6~ h$ Mpc$^{-1}$.