1-loop Perturbative Corrections to the Matter and Galaxy Bispectrum with non-Gaussian Initial Conditions

TL;DR

This paper develops a 1-loop perturbative framework to compute the matter and galaxy bispectra in the presence of primordial non-Gaussianity (NG), focusing on local bias and two NG shapes: local and equilateral. Using Eulerian perturbation theory, it derives the tree-level and 1-loop corrections to the matter bispectrum and trispectrum, and extends the approach to galaxy clustering under a local bias model, including NG contributions from the initial bispectrum $B_0$ and trispectrum $T_0$. The main findings are twofold: (i) for matter, NG 1-loop corrections are typically only a few percent of the gravity-driven component at mildly nonlinear scales; (ii) for galaxies, nonlinear bias terms amplified by NG can dominate the large-scale bispectrum, with primordial $T_0$-driven and $B_0$-driven terms producing strong scale- and shape-dependent signals, including $1/k^4$ and $1/k^2$ scalings in certain configurations. The results imply that the galaxy bispectrum is a powerful, complementary probe of primordial NG (including trispectrum) and motivate quantitative tests against simulations and future redshift surveys.

Abstract

We derive the expressions for the 1-loop corrections in cosmological, Eulerian, perturbation theory to the matter bispectrum and to the galaxy bispectrum, assuming local galaxy bias, in presence of non-Gaussian initial conditions. We compute them explicitly for the particular case of non-vanishing initial bispectrum and trispectrum in the local model and for a non-vanishing initial bispectrum alone for the equilateral model of primordial non-Gaussianity. While the primordial contribution to the matter bispectrum for values compatible with CMB observations is dominant over the component due to gravitational instability at large scales, 1-loop perturbative corrections due to non-Gaussian initial conditions correspond to just a few percent of the gravity-induced bispectrum at mildly non-linear scales, similarly to what happens for the matter power spectrum. However, in the perturbative expansion for the galaxy bispectrum, 1-loop diagrams arising from non-linear bias are responsible for significant large-scale contributions, indeed exceeding the primordial component, both for the local and equilateral model. We study the peculiar dependence on scale and on the shape of the triangular configurations of such additional terms, similar in their origin to the large-scale corrections to the halo and galaxy power spectra that raised significant interest in the recent literature.

Figures (10)

• Figure 1: Elements of the diagrammatic representation of perturbative corrections of matter correlators and diagrams representing the tree-level ($P_{11}$), 1-loop ($P_{12}$, $P_{22}^I$ and $P_{13}^I$) and 2-loop ($P_{22}^{II}$ and $P_{13}^{II}$) contributions to the matter power spectrum up to terms of order ${\mathcal{O}}(\delta_0^5)$. Lines connecting two points correspond to the linear power spectrum $P_0(k)$, the three connected lines in the $P_{12}$ term corresponds to the initial bispectrum $B_0$ and the four connected lines in $P_{22}^{II}$ and $P_{13}^{II}$ correspond to the initial trispectrum $T_0$.
• Figure 2: Diagrams representing the tree-level and 1-loop contributions to the matter bispectrum with generic non-Gaussian initial conditions.
• Figure 3: Equilateral configurations of the matter bispectrum, $B(k,k,k)$, for non-Gaussian initial conditions. Upper panels: local model with $f_{NL}^{\rm loc.}=50$ and $g_{NL}=10^4$ at redshifts $z=1$ ( left) and $z=3$ ( right). Lower panels: equilateral model with $f_{NL}^{\rm loc.}=100$ at redshifts $z=1$ ( left) and $z=3$ ( right). The main effect due to non-Gaussian initial conditions is represented by the primordial component $B^{(3)}$.
• Figure 4: Reduced matter bispectrum $Q(k_1,k_2,k_3)$ with fixed $k_1$ and $k_2=1.5 k_1$ as a function of the angle $\theta$ between ${\bf k}_1$ and ${\bf k}_2$ for local non-Gaussian initial conditions with $f_{NL}=50$ and $g_{NL}=10^4$. Upper panels assume $k_1=0.01\, h \, {\rm Mpc}^{-1}$ while lower panels assume $k_1=0.05\, h \, {\rm Mpc}^{-1}$. Predictions in the left and right panels are evaluated respectively at redshift $z=1$ and $z=3$. As in Fig. \ref{['fig:bsmeq']}, $Q^{(-1)}$ ( short-dashed line), which includes the primordial component $B^{(3)}$, represents the main non-Gaussian contribution, which assumes the largest values for $\theta\simeq \pi$, corresponding to the squeezed limit for this set of configurations.
• Figure 5: Same as Fig. \ref{['fig:bsmaLc']} but for the equilateral non-Gaussian model with $f_{NL}=100$. In this case, values of $\theta\simeq 0.7\pi$ correspond to the equilateral triangle limit for this set of configurations.