Angular bispectrum of matter number counts in cosmic structures

Technical Summary: Angular Bispectrum of Matter Number Counts in Cosmic Structures

Problem Statement
The two-point correlation function (power spectrum) remains the primary tool for constraining the standard $\Lambda$CDM model. However, as large-scale structure (LSS) evolves into the weakly nonlinear regime, higher-order statistics, specifically the three-point correlation function and its Fourier counterpart, the bispectrum, become essential for probing Fourier phase correlations and mode couplings induced by nonlinear gravitational evolution and primordial non-Gaussianity (PNG).

Current analyses of galaxy bispectra rely heavily on the local flat-sky approximation, the Limber approximation, and Newtonian dynamics. These approximations break down for next-generation surveys (e.g., Euclid, DESI, SPHEREx) that cover wide areas and utilize finite redshift bins (tomography). Furthermore, while relativistic effects (projection, dynamical General Relativity, and radiation) are theoretically known to exist, their precise impact on the angular bispectrum in a finite redshift bin setup without the Limber approximation has not been fully computed. Additionally, extracting these subtle relativistic signals from simulations is challenging due to numerical artifacts and the dominance of Newtonian terms.

Methodology
The authors present a full-sky computation of the angular bispectrum for matter number counts using second-order perturbation theory. The methodology is characterized by the following technical choices:

1. Framework: The calculation utilizes "tomographic spherical harmonics" (TSH), employing window functions to define finite redshift bins ($1.75 \le z \le 2.25$ and $0.55 \le z \le 0.65$). This approach avoids the Limber approximation and the flat-sky assumption.
2. Perturbation Expansion: The number count perturbation $\Delta$ is expanded to second order. The authors organize terms according to the weak-field expansion parameter $H/k$:
   • Newtonian Terms ($O((H/k)^0)$): Includes linear density, redshift-space distortions (RSD), and quadratic terms (density-density, density-velocity couplings).
   • Relativistic Projection Effects ($O(H/k)$): Leading-order projection effects (Doppler, Sachs-Wolfe, etc.) up to first order in the weak-field expansion.
   • Dynamical GR and Radiation Effects ($O((H/k)^2)$ and higher): Includes nonlinear dynamical GR corrections and radiative corrections (following Tram et al. 2016).
   • Coupling Terms: Includes couplings between projection and dynamical effects up to $O((H/k)^3)$.
3. Exclusions: For simplicity and to enable direct consistency checks, the authors neglect tracer bias, line-of-sight integrated effects (lensing, Integrated Sachs-Wolfe), and observer-position terms. They also omit next-to-leading projection effects at second order due to complexity and lack of consensus in the literature.
4. Numerical Implementation: The authors extend the efficient mathematical framework of Assassi et al. (2017), utilizing FFTLog decompositions and hyper-geometric function evaluations to reduce the problem to one-dimensional integrals. They provide a public code, ang_bispec, to compute the 14 distinct tree-level bispectrum contributions.
5. Validation: Theoretical predictions are compared against relativistic $N$-body simulations (Montandon et al. 2023). To ensure a fair comparison, the authors remove integrated projection effects from the simulation data and apply binning and Gaussian smoothing to the theoretical predictions to match the simulation estimator (Bucher et al. 2016).

Key Contributions

• First Full-Sky Computation: This work provides the first full-sky calculation of the angular bispectrum in second-order perturbation theory without the Limber approximation, formulated for finite redshift bins.
• Novel Newtonian Calculation: Even the Newtonian part of the bispectrum within this specific setup (finite bins, no Limber) is reported as novel.
• Comprehensive Relativistic Inclusion: The authors include, up to second order, all dynamical GR effects, radiation corrections, and leading relativistic projection effects within a unified framework.
• Code Release: The ang_bispec code is made publicly available, extending previous computations to all dominant Newtonian terms and relativistic effects.

Results
The authors evaluated the bispectrum for two redshift bins and compared the contributions of various terms:

1. Dominance of Newtonian Terms: As expected, Newtonian contributions (density, RSD, quadratic) are typically one or more orders of magnitude larger than relativistic signals across the entire spectrum for both redshifts.
2. Relativistic Hierarchy at $z \approx 2$:
   • Projection and dynamical GR effects have comparable amplitudes on large scales.
   • Unexpected Finding: In the squeezed limit, radiation effects dominate the relativistic signal, exceeding other relativistic terms by an order of magnitude.
3. Relativistic Hierarchy at $z \approx 0.6$: The expected hierarchy is recovered where projection effects dominate. However, dynamical corrections (from nonlinear evolution) remain non-negligible, being only a factor of 2–3 smaller than projection effects.
4. Simulation Comparison:
   • Total Bispectrum: The theoretical results agree fairly well with simulation measurements for the total bispectrum, except at small scales where simulation resolution limits the power.
   • Relativistic Signal: When comparing the "pure" relativistic signal (sum of GR and radiation terms) to simulation measurements (derived by subtracting Newtonian from Relativistic simulations), the authors find a discrepancy. The theoretical sum is smaller by a factor of $\sim 5$ than the simulation measurement at high redshift.
   • Interpretation of Discrepancy: The authors attribute this gap to numerical artifacts in the simulations, specifically residual gauge effects (mismatch between Poisson and N-body gauges) and backscaling effects (how radiation is included in Newtonian runs). They conclude that these numerical errors are comparable in amplitude to the physical dynamical relativistic effects.
   • Smoothing Effects: The analysis highlights that smoothing (necessary for detecting tiny relativistic signals) can introduce systematic biases by leaking small-scale numerical errors to large scales, potentially explaining offsets observed in low-redshift comparisons.

Significance and Claims
The paper claims to establish a robust theoretical pipeline for analyzing the angular bispectrum of matter number counts in photometric surveys, bridging the gap between lensing and galaxy clustering observables.

• Validation of Amplitudes: The work confirms the amplitude estimates of relativistic effects derived from simulations by Montandon et al. (2023), while providing a term-by-term decomposition that simulations cannot easily isolate.
• Relevance of Dynamical Effects: Contrary to the assumption that projection effects are the sole significant relativistic contribution, the authors demonstrate that dynamical GR and radiation effects are of the same order of magnitude as projection effects at high redshift and remain significant (within a factor of 2) at lower redshifts.
• Caution on Simulation Extraction: A significant finding is the identification of numerical artifacts (gauge and backscaling) in current relativistic simulations that are as large as the physical signals being sought. This suggests that extracting "pure" relativistic signals from current simulation subtraction methods requires extreme caution.
• Future Utility: The authors posit that their framework is essential for future next-generation surveys (like Euclid) to disentangle primordial non-Gaussianity from intrinsic non-Gaussianity and to test theories of gravity in the nonlinear sector, provided that simplifications like galaxy bias and line-of-sight integrals are eventually incorporated.