Angular bispectrum of matter number counts in cosmic structures

TL;DR

This work computes the full-sky angular bispectrum of matter-number counts at tree level in second-order perturbation theory without relying on the Limber approximation, incorporating finite redshift-bin windowing, Newtonian dynamics, projection effects, general-relativistic dynamics, and radiation corrections. By extending the Assassi et al. formalism with FFTLog-based kernels and tomographic binning, the authors evaluate 14 second-order contributions across 8 physical channels, delivering a numerically tractable, publicly available code that facilitates robust comparisons with relativistic simulations. They find that Newtonian terms dominate on small scales, while projection, GR, and radiation effects become relevant on large scales and in squeezed configurations, with radiation often dominating the relativistic signal in the squeezed limit at high redshift. The results agree reasonably with relativistic simulations for the total bispectrum, though numerical artefacts and resolution limitations in simulations complicate clean disentanglement of dynamical relativistic effects. This framework provides a practical, Limber-free pathway to exploit future wide-area surveys for probing gravity and early-Universe physics through the angular bispectrum of galaxy number counts.

Abstract

The bispectrum of galaxy number counts is a key probe of large-scale structure, offering insights into the initial conditions of the Universe, the nature of gravity, and cosmological parameters. In this work, we present the first full-sky computation of the angular bispectrum in second-order perturbation theory without invoking the Limber approximation, and formulated for finite redshift bins via window functions. To our knowledge, even the Newtonian part within this setup is novel. Building on this, we also include, up to second order in perturbation theory, the dynamical general relativistic and radiation effects, together with the leading relativistic projection effects. For simplicity, we neglect tracer bias and line-of-sight integrated contributions, however note that in particular the former can be straightforwardly incorporated within our framework. We evaluate the bispectrum contributions for two redshift bins, $1.75 \leq z \leq 2.25$ and $0.55 \leq z \leq 0.65$, and compare our theoretical prediction against relativistic light-cone simulations, with line-of-sight integral effects removed so as to enable direct consistency checks. As expected, we find that the Newtonian contributions are typically one or more orders of magnitudes larger than the relativistic signal across the entire spectrum for both redshifts. At $z=2$, we find that projection and dynamical relativistic effects have comparable amplitudes on large scales; somewhat unexpectedly, however, radiation effects dominate the relativistic signal in the squeezed limit. At $z=0.6$, the expected hierarchy is recovered, though dynamical corrections remain non-negligible -- only a factor of 2-3 smaller than projection effects. Our theoretical results agree fairly well with simulation measurements for the total bispectrum. To facilitate future applications and reproducibility, we make the corresponding code publicly available.

Figures (9)

• Figure 1: We show the angular bispectrum for two redshift bins: $1.75 \leq z \leq 2.25$ on the first row and $0.55 \leq z \leq 0.65$ on the second row. The first column, we have fixed $\ell_1=514$ and vary $\ell_2=\ell_3$ on the $x$-axis. The middle column shows equilateral configurations. Note the reversed $x$-axis which ensure the continuity of the curves between the columns. In the right column, we have fixed $\ell_1=2$ and increase $\ell_2=\ell_3$. The grey dotted line represents the sum of all the terms, including all relativistic terms, while the black line only contains the Newtonian terms indicated in the legend with a "N" subscript. The relativistic projection effects are shown in red while the pure dynamical GR (radiation) effects are shown in brown (violet). In grey and pink, we show the coupling between projection relativistic effects with GR and radiation, respectively. Dashed lines indicates negative values.
• Figure 1: We show each contribution to the Newtonian quadratic term $P^{\rm N}_2$ according to \ref{['eq:QN']} for the redshift bin $1.75 \leq z \leq 2.25$. The axis are the same as Fig. \ref{['fig:term_comparison']}.
• Figure 2: We plot the ratio between the relativistic contributions and the total bispectrum shown in Fig. \ref{['fig:term_comparison']}. The structure of the panel is the same as Fig. \ref{['fig:term_comparison']}. We can see that the relativistic contributions reach almost $10 \%$ of the total amplitude at $1.75 \leq z \leq 2.25$ for large-scale equilateral configurations and the squeezed limit. At $0.55 \leq z \leq 0.65$, the relative amplitude reaches at most $2\%$.
• Figure 3: We show each contribution to the Newtonian quadratic term $P^{\rm N}_2$ according to \ref{['eq:QN']} for the redshift bin $1.75 \leq z \leq 2.25$. The axis are the same as Fig. \ref{['fig:term_comparison']}.
• Figure 4: We show each contribution to the general relativistic term $D^{\rm GR}_2$ according to the first equation of \ref{['eq:gr']} for the redshift bin $1.75 \leq z \leq 2.25$. We consider here only the dominant terms, hence neglecting the coupling of projection effects with GR terms. The axes are the same as Fig. \ref{['fig:quadratic']}.