Constraints from CMB lensing tomography with projected bispectra

Technical Summary: Constraints from CMB Lensing Tomography with Projected Bispectra

Problem and Motivation
The analysis of projected probes of large-scale structure (LSS), such as galaxy clustering and Cosmic Microwave Background (CMB) lensing, has traditionally relied on two-point statistics (power spectra). While the "2×2-point" analysis (combining galaxy-galaxy and galaxy-convergence power spectra) has successfully constrained cosmological parameters with percent-level precision, it fails to capture non-Gaussian features generated by non-linear gravitational collapse. To recover this lost information, higher-order statistics like the bispectrum are required. However, bispectrum analysis faces significant hurdles: computational complexity in estimating optimal estimators, the proliferation of "triangle configurations" spreading information, difficulties in deriving robust covariance estimates, and the challenge of modeling the bispectrum from first principles without extensive simulations.

Methodology
This paper addresses these challenges by applying the "filtered-squared bispectrum" (FSB) estimator, previously introduced by Harscouet et al. (2025) (H25), to a new context: cross-correlations between multiple fields.

• The FSB Estimator: The authors generalize the FSB to handle cross-correlations between a filtered-squared field ($a^2_L$) and a third field ($f$). This estimator probes the cross-bispectrum $b_{aaf}$. The method involves filtering a field in harmonic space, squaring it in real space, and computing the angular power spectrum of the result against a third field. This approach allows the use of efficient power spectrum techniques (specifically the NaMaster pseudo-$C_\ell$ framework) to handle mode-coupling effects from survey masks.
• Covariance Estimation: A key advantage of the FSB is that its covariance matrix (including cross-covariance with power spectra) can be estimated analytically and model-independently using data-driven methods, avoiding the need for thousands of simulations. The authors derive the necessary analytical expressions for the multi-field case ($N_{222}$ and $N_{32}$ terms).
• Data: The analysis utilizes photometric Luminous Red Galaxies (LRGs) from the DESI Legacy Imaging Survey (9th data release) across four redshift bins ($0.4 \lesssim z \lesssim 1$) and CMB lensing convergence ($\kappa$) maps from the Planck experiment (PR3).
• Theoretical Modeling: The authors employ a tree-level perturbation theory model to predict the projected bispectra ($g g g$ and $g g \kappa$). They use a linear bias relation for the power spectrum and include non-linear bias terms ($b_2, b_s$) for the bispectrum, linking them to the linear bias $b_1$ via empirical "coevolution relations." The model accounts for shot noise and magnification bias (for power spectra only).
• Inference: A Gaussian likelihood is constructed using the combined data vector of power spectra ($C_{gg}, C_{g\kappa}$) and FSBs ($\Phi_{ggg}, \Phi_{gg\kappa}$). The analysis is performed within a simplified parameter space ($\sigma_8, \Omega_m, b_1$), with other cosmological parameters fixed to Planck best-fit values. Conservative scale cuts ($k_{max} \approx 0.05 \, h \text{Mpc}^{-1}$ for bispectra) are applied to ensure the validity of the tree-level approximation.

Key Contributions

1. Generalization of FSB: The paper extends the FSB estimator to multi-field cross-correlations, enabling the measurement of the galaxy-galaxy-convergence ($g g \kappa$) bispectrum alongside the galaxy auto-bispectrum ($g g g$).
2. First Detection of Projected $g g \kappa$ Bispectrum: The authors report the first detection of the projected galaxy-galaxy-convergence bispectrum in real data.
3. Rudimentary Cosmological Analysis: This work presents the first cosmological analysis combining projected power spectra and bispectra under a unified model, testing the ability of the bispectrum to break parameter degeneracies and improve constraints on $\sigma_8$ and $\Omega_m$.

Results

• Detection Significance:
  • The galaxy auto-bispectrum ($g g g$) is detected with very high significance ($>30\sigma$) in all four redshift bins.
  • The galaxy-galaxy-convergence bispectrum ($g g \kappa$) is detected above $5\sigma$ in the three highest-redshift bins, though the signal-to-noise ratio is too low to include it in the final cosmological constraints.
• Model Agreement: The measured bispectra are reasonably well described by the tree-level theoretical prediction using linear bias values derived from the power spectrum, particularly on large scales.
• Cosmological Constraints:
  • Degeneracy Breaking: The combination of the $g g g$ bispectrum with power spectra ($C_{gg}$ or $C_{g\kappa}$) successfully breaks the degeneracy between the clustering amplitude $\sigma_8$ and the linear bias $b_1$.
  • Parameter Recovery: Combinations involving the bispectrum recover constraints on $\sigma_8$ and $\Omega_m$ that are consistent with the traditional "2×2-point" ($C_{gg} + C_{g\kappa}$) analysis across all redshift bins.
  • Precision Improvement: Including the $g g g$ bispectrum with both power spectra ($g g g + g g + g \kappa$) leads to a modest improvement of approximately 10% in the constraint on $\sigma_8$ compared to the standard 2×2-point analysis.
  • Systematics: The $g g + g g g$ combination (without lensing data) yields constraints on $\sigma_8$ that are consistent with other methods but slightly lower, highlighting the potential for independent checks free from CMB lensing systematics. However, the authors note these results are sensitive to the assumed bias relations.
  • Tension: All data combinations yield values of $\sigma_8(z)$ that are systematically lower than the Planck CMB prediction, a trend consistent with previous CMB lensing tomography studies on the same dataset.

Significance and Claims
The paper claims that the projected bispectrum, measured via the FSB estimator, is a viable and robust cosmological probe. It demonstrates that:

1. The FSB approach allows for the efficient measurement of projected bispectra and their covariances in real data.
2. The bispectrum provides complementary information to the power spectrum, effectively breaking parameter degeneracies (specifically between $\sigma_8$ and $b_1$) and tightening constraints on cosmological parameters.
3. The results are consistent with standard analyses, validating the robustness of the method despite the use of simplified tree-level models and conservative scale cuts.

The authors emphasize that these results are a "first tentative" exploration. They acknowledge that the current model is simplistic (tree-level only, empirical bias relations) and that the observed preference for lower $\sigma_8$ values may be influenced by these assumptions. They state that a more rigorous analysis, including 1-loop power spectrum modeling and self-consistent bias parameter estimation, is required for definitive cosmological conclusions and is the subject of future work.