The Bispectrum in the Effective Field Theory of Large Scale Structure

TL;DR

The paper develops and implements the one-loop bispectrum in the Effective Field Theory of Large Scale Structure (EFTofLSS), showing that perturbative short-scale dynamics can be encoded by a local-in-time effective stress tensor with a small set of parameters. It derives the next-to-leading order contributions, demonstrates renormalization of UV divergences by these EFT counterterms, and establishes a close link between power-spectrum and bispectrum corrections. A key result is a zero-parameter (or minimally parameterized) bispectrum formula obtained by fixing the leading EFT coefficient from the power spectrum, achieving accurate predictions up to $k_{ m max} \approx 0.22\, h\mathrm{Mpc}^{-1}$ at $z=0$, roughly doubling the valid range relative to one-loop SPT. The work validates the EFT framework against $N$-body simulations and outlines a diagrammatic interpretation of renormalization, with clear paths for extending to higher orders and joint analyses of multiple observables.

Abstract

We study the bispectrum in the Effective Field Theory of Large Scale Structure, consistently accounting for the effects of short-scale dynamics. We begin by proving that, as long as the theory is perturbative, it can be formulated to arbitrary order using only operators that are local in time. We then derive all the new operators required to cancel the UV-divergences and obtain a physically meaningful prediction for the one-loop bispectrum. In addition to new, subleading stochastic noises and the viscosity term needed for the one-loop power spectrum, we find three new effective operators. The three new parameters can be constrained by comparing with N-body simulations. The best fit is precisely what is suggested by the structure of UV-divergences, hence justifying a formula for the EFTofLSS bispectrum whose only fitting parameter is already fixed by the power spectrum. This result predicts the bispectrum of N-body simulations up to $k \approx 0.22\, h\, \text{Mpc}^{-1}$ at $z=0$, an improvement by nearly a factor of two as compared to one-loop standard perturbation theory.

Figures (18)

• Figure 1: SPT vertex.
• Figure 2: Tree-level and one-loop power spectrum.
• Figure 3: Tree-level and one loop-bispectra.
• Figure 4: The one-loop diagramgs $B_{222}$ (solid red), $B_{321}^I$ (dashed red), $|B_{321}^{II}|$ (solid blue), $|B_{411}|$ (dashed blue) and the sum of all four (dashed black) is shown for three special configurations of $k_1$, $k_2$ and $k_3$ on a logarithmic scale. The solid black curve is the tree-level contribution. In the squeezed limit, we set $\Delta k= 0.013 \; h \hbox{Mpc}^{-1}$. We plot the absolute value of the diagrams $|B_{321}^{II}|$ and $|B_{411}|$ since they are negative. Our figures agree well with the ones in Ref. Sefusatti2010 and, as can be seen, there are no large cancellations among the diagrams.
• Figure 5: The shapes of the tree-level and one-loop bispectrum divided by $\Sigma_0$ are plotted as a function of $x_2 = k_2/k_1$ and $x_3=k_3/k_1$ for a fixed $k_1 = 0.2 \, h\, \text{Mpc}^{-1}$. The shape is restricted to the range of $x_2 \geq x_3 \geq 1-x_2$.