The One-Loop Matter Bispectrum in the Effective Field Theory of Large Scale Structures

TL;DR

This work demonstrates that the equal-time dark matter bispectrum at one loop is accurately predicted by the EFTofLSS using only the counterterm fixed from the power spectrum, with good agreement to $N$-body data up to $k\simeq 0.25\,h\,\mathrm{Mpc}^{-1}$ and modest configuration-dependent deviations (worst in equilateral triangles). The authors implement an IR-safe one-loop integrand and include linear and quadratic EFT counterterms, showing that no additional free parameter is required for the bispectrum at this order. They validate their approach against the Millennium-XXL simulation, estimate errors, and quantify the EFT reach relative to standard perturbation theory, finding a substantial improvement (about 65% in $k$-range) with only one fitted parameter. The results provide strong evidence that EFTofLSS can unify predictions across observables (power spectrum, momentum spectrum, and bispectrum) and motivate advancing to two-loop calculations with IR resummation to unlock more cosmological information from upcoming surveys.

Abstract

Given the importance of future large scale structure surveys for delivering new cosmological information, it is crucial to reliably predict their observables. The Effective Field Theory of Large Scale Structures (EFTofLSS) provides a manifestly convergent perturbative scheme to compute the clustering of dark matter in the weakly nonlinear regime in an expansion in $k/k_{\rm NL}$, where $k$ is the wavenumber of interest and $k_{\rm NL}$ is the wavenumber associated to the nonlinear scale. It has been recently shown that the EFTofLSS matches to $1\%$ level the dark matter power spectrum at redshift zero up to $k\simeq 0.3 h\,$Mpc$^{-1}$ and $k\simeq 0.6 h\,$Mpc$^{-1}$ at one and two loops respectively, using only one counterterm that is fit to data. Similar results have been obtained for the momentum power spectrum at one loop. This is a remarkable improvement with respect to former analytical techniques. Here we study the prediction for the equal-time dark matter bispectrum at one loop. We find that at this order it is sufficient to consider the same counterterm that was measured in the power spectrum. Without any remaining free parameter, and in a cosmology for which $k_{\rm NL}$ is smaller than in the previously considered cases ($σ_8=0.9$), we find that the prediction from the EFTofLSS agrees very well with $N$-body simulations up to $k\simeq 0.25 h\,$Mpc$^{-1}$, given the accuracy of the measurements, which is of order a few percent at the highest $k$'s of interest. While the fit is very good on average up to $k\simeq 0.25 h\,$Mpc$^{-1}$, the fit performs slightly worse on equilateral configurations, in agreement with expectations that for a given maximum $k$, equilateral triangles are the most nonlinear.

Figures (12)

• Figure 1: Ratios of various terms in the bispectrum prediction to the tree-level expression $B_\text{tree}$, plotted with $k_1=0.1h\,$Mpc$^{-1}$ and in terms of $x_2\equiv k_2/k_1$ and $x_3\equiv k_3/k_1$. To avoid redundancy, we only plot configurations with $x_2\leq x_3$, while the triangle inequality restricts physical configurations to satisfy $1-x_3\leq x_2$. Each term is strongest on equilateral triangles ($x_2=x_3=1$), becoming relatively weaker for other configurations such as squeezed ($x_2\to 0$) or flat ($x_2+x_3=1$). This implies that configurations where three short modes interact are more nonlinear than configurations involving one or more longer modes and one short mode---in some sense, of all triangles with $k_1$ fixed, equilateral triangles are "closest" to the nonlinear scale. As we let $k_1$ grow, all terms grow in size relative to $B_{\rm tree}$ but the shape remains quite unaltered.
• Figure 2: Same as Fig. \ref{['fig:shape-plots']}, but for the 1-loop contribution for $k_1=0.1 h\,$Mpc$^{-1}$ on the left and $k_1=0.3 h\,$Mpc$^{-1}$ on the right.
• Figure 3: One-loop SPT and EFT predictions for the matter power spectrum, normalized to measurements from the simulations described in Sec. \ref{['sec:simulations']}. The parameter $c_{s (1)}^2$ in the EFT prediction has been determined by fitting the IR-resummed curve over the range $0.02h\,$Mpc$^{-1} \leq k \leq 0.1h\,$Mpc$^{-1}$. The resulting value for $c_{s (1)}^2$ can then be used in the EFT prediction for the bispectrum, without re-fitting. The black dashed line corresponds to cosmic variance plus an assumed 1% systematic error in the simulations, added in quadrature with a 1% accuracy goal for the prediction. The shaded bands show the uncertainty in the EFT predictions from the 1$\sigma$ uncertainty in the value of $c_{s (1)}^2$.
• Figure 4: Top: $P$-values corresponding to comparisons of various theory curves to nonlinear data, as described in the main text, as a function of the maximum side length ($k_{\rm comp}$) of the triangles used to compute the $p$-value. The solid, short-dashed, and long-dashed lines correspond to one-loop EFT, one-loop SPT, and tree-level SPT or EFT respectively. For the one-loop EFT prediction, the value of $c_{s (1)}^2$ has been fixed by fitting the EFT prediction for the matter power spectrum, so there is no free parameter. Middle: reduced $\chi^2$ for the various predictions. (It has the same information of the top panel, but presented in a different way.) Bottom: same as top, but with $k^3_{\rm comp}$ on the $x$-axis. Since the number of available modes grows as $k^3_{\rm comp}$, the bottom plot gives a pictorial representation of the gain in information that is obtained by reaching higher $k$'s.
• Figure 5: The shape of the dark matter bispectrum given by the EFTofLSS, with $k_1=0.1h\,$Mpc$^{-1}$ on the left, and $k_1=0.3h\,$Mpc$^{-1}$ on the right.