Angular bispectrum of matter number counts in cosmic structures

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, i.e., using tomographic spherical harmonics. 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 (from nonlinear evolution) 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.