Improving constraints on primordial non-Gaussianity from Quaia with a new cosmological observable: angular redshift fluctuations

Technical Summary: Improving Constraints on Primordial Non-Gaussianity from Quaia with Angular Redshift Fluctuations

Problem and Motivation
The detection of primordial non-Gaussianity (PNG), parameterized by $f_{\text{NL}}$, is a critical test for inflationary models, where a non-zero value would indicate deviations from Gaussian initial conditions. While the tightest constraints currently come from the Cosmic Microwave Background (CMB) bispectrum (e.g., Planck 2018), Large Scale Structure (LSS) offers a complementary probe through the scale-dependent bias effect. This effect, predicted by Dalal et al. (2008), introduces a characteristic $1/k^2$ signature in the galaxy bias for local PNG. However, measuring this from LSS is challenging due to the need for precise control of observational systematics on the largest scales.

Recent studies have utilized 2D angular power spectra and cross-correlations with CMB lensing to constrain $f_{\text{NL}}$. This work addresses the potential to improve these constraints by incorporating a new cosmological observable: Angular Redshift Fluctuations (ARFs). ARFs measure the 2D angular deviations of the average redshift of a matter tracer within a given shell. Theoretically, ARFs are sensitive to galaxy bias and thus to PNG, yet their application to real data for $f_{\text{NL}}$ constraints had not been previously demonstrated.

Methodology
The authors analyzed the Quaia catalog, a full-sky sample of $\sim1.3$ million quasars (QSOs) derived from Gaia DR3 and unWISE data, covering a large cosmological volume. The analysis pipeline involved the following steps:

1. Data Preparation: The Quaia sample was split into two redshift bins ($\bar{z} \approx 0.97$ and $\bar{z} \approx 2.10$). HEALPix maps ($N_{\text{side}}=128$) were generated for both galaxy number counts (density) and ARFs. The ARF field was defined as the weighted average redshift fluctuation per pixel, utilizing a wide Gaussian window equivalent to the survey's intrinsic redshift distribution.
2. Cross-Correlations: The study combined the Quaia density and ARF maps with Planck PR4 CMB lensing convergence maps. This yielded four observables: the auto-power spectra of density ($C_{gg}^\ell$) and ARFs ($C_{zz}^\ell$), and their cross-correlations with CMB lensing ($C_{\kappa g}^\ell$ and $C_{\kappa z}^\ell$).
3. Theoretical Modeling: A modified version of the CAMB Boltzmann code (ARFCAMB) was used to compute theoretical power spectra. This code incorporates the ARF kernel and accounts for photometric redshift uncertainties ($\sigma_z \approx 0.06(1+z)$). The model included the scale-dependent bias induced by $f_{\text{NL}}$ under the universality relation (assuming $p=1$), alongside a flexible parameterization for the redshift evolution of the galaxy bias ($b(z)$).
4. Systematics and Covariance: To address observational systematics, the authors employed systematics deprojection using linear regression against Quaia templates. Scale cuts were determined by comparing data shifts against mock realizations; specifically, $\ell_{\text{min}}=14$ for density and $\ell_{\text{min}}=10$ for ARF auto-spectra were applied to remove low-$\ell$ offsets incompatible with statistical fluctuations. A covariance matrix was derived from 1,000 correlated simulations, with corrections applied to mock auto-spectra to match the excess power observed in the real data at low multipoles.
5. Parameter Inference: A Gaussian likelihood was constructed using the measured power spectra and the covariance matrix. Markov Chain Monte Carlo (MCMC) sampling (emcee) was used to constrain $f_{\text{NL}}$ and bias parameters ($b_0, \alpha$).

Key Results
The analysis yielded the following constraints on the local non-Gaussianity parameter $f_{\text{NL}}$ (assuming the universality relation with $p=1$):

• ARFs alone: Combining ARF auto-correlation and ARF-CMB lensing cross-correlation yielded $f_{\text{NL}} = -43^{+50}_{-44}$.
• Density alone: Combining density auto-correlation and density-CMB lensing cross-correlation yielded $f_{\text{NL}} = -25^{+19}_{-17}$, consistent with previous Quaia analyses (Fabbian et al. 2026).
• Joint Analysis: Combining all four observables ($C_{gg}^\ell, C_{\kappa g}^\ell, C_{zz}^\ell, C_{\kappa z}^\ell$) resulted in $f_{\text{NL}} = -3 \pm 14$ at 68% confidence.
• Goodness-of-Fit Optimization: The authors noted mild tensions in the $\chi^2$ when including the ARF-CMB cross-correlation ($C_{\kappa z}^\ell$). Excluding this specific cross-correlation to improve the model's goodness-of-fit resulted in $f_{\text{NL}} = -5^{+16}_{-15}$.

The joint constraint represents an improvement of approximately 25% over the previous Quaia measurement using only density and CMB lensing. This result stands as the second tightest constraint on $f_{\text{NL}}$ using LSS two-point statistics to date (after DESI DR1 spectroscopic data) and the most precise achieved using projected 2D summary statistics.

Significance and Robustness
The paper claims that the inclusion of ARFs as a cosmological observable provides a measurable improvement in $f_{\text{NL}}$ constraints using real data. The authors verified the robustness of their results through several tests:

• Bias Modeling: The results remained stable when varying the fiducial galaxy bias evolution model (testing both Piccirilli et al. 2024 and Laurent et al. 2017 models) and when allowing for independent bias parameters for density and ARF fields.
• Scale Cuts: Constraints were found to be stable against variations in the minimum multipole ($\ell_{\text{min}}$) cuts, although some tension in bias parameters was observed, attributed to the ARF kernel's higher sensitivity to nonlinear scales ($k$-modes) compared to the density kernel.
• Universality Relation: When relaxing the assumption of $p=1$ to $p=1.6$ (a value suggested for QSOs in some literature), the constraint shifted to $f_{\text{NL}} = 2^{+20}_{-21}$, or $b_\phi f_{\text{NL}} = -63^{+79}_{-89}$ in a model-independent approach.

The authors conclude that while internal tensions exist between observables—likely due to the need for improved modeling of nonlinear scales in ARFs—the addition of ARFs to 2D analyses is a promising avenue. They advocate for the inclusion of ARFs in future analyses of large surveys such as DESI, Euclid, and LSST, noting that ARFs could offer a robust 2D alternative to standard 3D clustering, particularly for photometric surveys where redshift uncertainties are non-negligible.