One-Loop Galaxy Bispectrum: Consistent Theory, Efficient Analysis with COBRA, and Implications for Cosmological Parameters

TL;DR

This work develops an efficient, consistent EFT-based pipeline for the one-loop galaxy bispectrum in redshift space using the COBRA basis, enabling rapid inference with minimal cosmology-dependent computation. The authors derive a comprehensive theory model including fourth-order bias, redshift-space distortions, UV counterterms, and stochastic contributions, together with infrared resummation, and implement it in a COBRA-FFTLog framework that reduces loop integrals to tensor operations. Validation on the PTChallenge simulations shows sub-percent accuracy for the COBRA decomposition and unbiased recovery of $\omega_{\rm cdm}$, $H_0$, and $\sigma_8$ up to $k_{ m max}\approx0.15\,h\mathrm{Mpc}^{-1}$, with one-loop bispectrum providing stronger cosmological and PNG constraints than tree-level analyses. The results indicate that one-loop bispectrum analyses will be a powerful tool for DESI, Euclid, and similar surveys, while highlighting the need to control discreteness effects and covariance approximations when pushing to smaller scales.

Abstract

We present an efficient and accurate pipeline for the analysis of the redshift-space galaxy bispectrum multipoles at one-loop order in effective field theory (EFT). We provide a systematic theory derivation based on power counting, which features the first comprehensive treatment of stochastic EFT contributions -- these are found to significantly improve the match to data. Our computational pipeline utilizes the COBRA technique that expands the linear matter power spectrum over a basis of principal components based on a singular value decomposition, allowing the cosmology dependence to be captured to sub-permille accuracy with just eight templates. This transforms the problem of computing the one-loop EFT bispectrum to a simple tensor multiplication, reducing the computation time to around a second per cosmology with negligible loss of accuracy. Using these tools, we study the cosmological information in the bispectrum by analyzing PTChallenge simulations, whose gigantic volume provides the most powerful test of the one-loop EFT bispectrum so far. We find that the one-loop prediction provides an excellent match to the bispectrum data up to $k_{\rm max}=0.15~h$ Mpc$^{-1}$, as evidenced by the precise recovery of the dark matter density $ω_\text{cdm}$, Hubble constant $H_0$, and mass fluctuation amplitude $σ_8$ parameters, and the amplitude of equilateral primordial non-Gaussianity (PNG) $f_{\rm NL}^{\rm equil}$. Combined with the power spectrum, the COBRA-based one-loop bispectrum multipoles yield tighter constraints than the tree-level bispectrum monopole, with the posteriors on $ω_{\text{cdm}}$, $H_0$, and $σ_8$ shrinking by 41\%, 25\%, and 19\%, respectively. This suggests that the COBRA-based bispectrum analysis will be an important tool in the interpretation of data from ongoing redshift surveys such as DESI and Euclid.

Figures (10)

• Figure 1: Accuracy of the https://www.flickr.com/photos/198816819@N07/54686938553/in/dateposted-public/ decomposition applied to infrared-resummed power spectra at $z=0.61$, using $N_{\rm COBRA}=8$. We plot the $95$th percentile of the errors across $120$ test cosmologies, drawn from the default $\Lambda$CDM range discussed in Bakx:2024zgu. The basis functions are computed using an SVD applied to a grid of infrared-resummed power spectra, computing using a fixed damping exponent, $\Sigma_{\rm Planck}$. The dashed red curve shows the error under the same approximation (analyzing power spectra computed with fixed damping), while the solid blue curve indicates the true error, projecting the full resummed spectra (with cosmology-dependent $\Sigma=\Sigma_{\rm true}$) onto the fixed-cosmology basis. We find subpercent precision in all cases, justifying our choice of basis and $N_{\rm COBRA}$.
• Figure 2: Discreteness weights for the bispectrum quadrupole and hexadecapole moments. These are equal to the ratio of the bispectrum computed on a finite grid of ${\bm k}$-modes to that binned assuming a continuous distribution. Here, we show results obtained from a box of side-length $L=3800~h^{-1}{\text{Mpc}}$, matching the PTChallenge simulations, for all $k's$ in range $0.01~h{\text{Mpc}}^{-1}\leq k\leq 0.08~h{\text{Mpc}}^{-1}$, the site length $L=2000~h^{-1}{\text{Mpc}}$ for bins with the largest wavenumber $0.08~h{\text{Mpc}}^{-1}< k \lesssim 0.11 ~h{\text{Mpc}}^{-1}$, and $L=1000~h^{-1}{\text{Mpc}}$ for bins with the largest wavenumber $0.11~h{\text{Mpc}}^{-1} \lesssim k$. The smaller box sizes are chosen to reduce computational cost. The bins correspond to triangles with $0.01~h{\text{Mpc}}^{-1}\leq k\leq 0.12~h{\text{Mpc}}^{-1}$ used the bispectrum multipole analysis. The discreteness weights for the monopole are close to 1 for the entire range of scales used in the analysis ($0.01~h{\text{Mpc}}^{-1}\leq k\leq 0.20~h{\text{Mpc}}^{-1}$), and therefore are thus omitted both here and in the analysis.
• Figure 3: Example of the bispectrum multipoles computed using our pipeline. The solid red, blue, and green lines show (respectively) the tree-level bispectrum monopole, quadrupole, and hexadecapole, with the dotted lines including also the deterministic one-loop contributions. The color-scale ranges indicates the maximum $k$ of each bispectrum bin ranging from $k=0.01~h{\text{Mpc}}^{-1}$ (lightest) to $k = 0.15~h{\text{Mpc}}^{-1}$ (darkest). For reference, we highlight the equilateral configurations, which are connected by the almost horizontal lines. To form this plot, we assume a Planck-like cosmology at $z=0.61$ and fix the bias parameters to the PTChallenge best-fit results.
• Figure 4: Accuracy of the https://www.flickr.com/photos/198816819@N07/54686938553/in/dateposted-public/ decomposition applied to the one-loop galaxy bispectrum multipoles. We compare the https://www.flickr.com/photos/198816819@N07/54686938553/in/dateposted-public/ bispectra (which are the main novelty of this work) to directly computed bispectra for 10 choices of primordial power spectrum, which are overplotted on the figure. In all cases, use the same bias parameters as Figure \ref{['fig: bispectrum-model']} and assume $N_{\rm COBRA}=8$. The left panel compares the error to the total (tree-plus-one-loop) bispectrum, finding subpercent agreement in all cases, with largest differences for the hexadecapole. The right panel compares the error to the expected bispectrum errorbar, obtained by rescaling the PTChallenge covariance to the cosmology of interest by a factor $\prod_{i=1}^3\sqrt{P(k_i)/P_{\rm fid}(k_i)}$. Despite the huge volume of PTChallenge ($V=566~h^{-3}\mathrm{Gpc}^3$, much larger than any current or upcoming survey), we find $\lesssim 0.2\sigma$ agreement for all triangles with just $N_{\rm COBRA}=8$.
• Figure 5: PTChallenge constraints on cosmological parameters obtained from analyzing the redshift-space power spectrum and the bispectrum monopole. As in Table \ref{['tab:tab1']}, we report deviations of the cosmological parameters from their true values as well as bias parameters normalized to the values obtained from real-space bispectrum analyses (or the low-$k$ limit of the galaxy-matter cross-spectrum for $b_1$). While the power spectrum likelihood is the same in all analyses, we consider five different variants of the bispectrum monopole likelihood: tree-level with $k_{\rm max}^{B_0}=0.08~h{\text{Mpc}}^{-1}$ (gray), one-loop with ${k_\text{max}}=0.12~h{\text{Mpc}}^{-1}$ (green), $k_{\rm max}^{B_0}=0.15~h{\text{Mpc}}^{-1}$ (blue), $k_{\rm max}^{B_0}=0.17~h{\text{Mpc}}^{-1}$ (red) and finally $k_{\rm max}^{B_0}=0.20~h{\text{Mpc}}^{-1}$ where also the power spectrum cutoff is increased to $k_{\rm max}^{P_\ell} = 0.20~h{\text{Mpc}}^{-1}$ (yellow). (Note that in the first four analyses we also use $Q_0$ with $0.16\leq k/(h{\text{Mpc}}^{-1})< 0.2$.) For the bias parameters, the light-dashed lines indicate best-fit values obtained from the PTChallenge real-space power spectrum and bispectrum analysis which we use as proxies for the true values of the bias parameters. We find consistent results on cosmological parameters for $k_{\rm max}^{B_0}\lesssim 0.15~h{\text{Mpc}}^{-1}$ (which are significantly tighter than the tree-level results) but find significant deviations for larger $k_{\rm max}^{B_0}$.