Euclid preparation: Testing multi-field inflation with galaxy power spectrum and bispectrum

Technical Summary: Testing Multi-Field Inflation with Galaxy Power Spectrum and Bispectrum

Problem Statement
Primordial non-Gaussianity (PNG), particularly of the local type, serves as a critical probe for distinguishing between single-field and multi-field models of inflation. While the cosmic microwave background (CMB) has placed tight constraints on local PNG (nearly cosmic-variance limited), Stage-IV galaxy surveys like Euclid promise to surpass these bounds by measuring 2-point (power spectrum) and 3-point (bispectrum) clustering statistics over vast cosmological volumes. However, extracting robust constraints on the PNG parameter $f_{\text{NL}}$ from galaxy clustering requires a rigorous joint analysis framework that accounts for redshift-space distortions (RSD), non-linear gravitational evolution, and galaxy bias. A primary challenge lies in the degeneracy between $f_{\text{NL}}$ and the PNG-induced bias parameter $b_\phi$, which is difficult to break without informative priors or a comprehensive treatment of theoretical systematics.

Methodology
The authors validate a theoretical modeling and inference pipeline designed for the Euclid spectroscopic H$\alpha$ galaxy sample. The work utilizes:

• Simulations: A suite of Abacus-PNG N-body simulations with local-type PNG initial conditions ($f_{\text{NL}} = \{0, 30\}$) and Gaussian counterparts. These are populated with galaxies using a Halo Occupation Distribution (HOD) tuned to reproduce the clustering properties of the Euclid Flagship 2 (FS2) simulation. An abundance matching (AM) procedure is applied to reconcile differences in halo mass definitions between the FS2 and Abacus simulations.
• Theoretical Modeling: The analysis employs the Effective Field Theory of Large Scale Structure (EFTofLSS). The redshift-space power spectrum is modeled at the one-loop order, while the bispectrum is modeled at tree level. The framework includes:
  • Renormalized bias expansion extended to include PNG operators.
  • Redshift-space distortions modeled via a velocity-moment expansion.
  • Counterterms to absorb ultraviolet sensitivities and model small-scale velocity effects (Fingers of God), specifically adopting a counterterm motivated by the velocity-difference generator (VDG) framework for the bispectrum.
  • Infrared resummation and Alcock–Paczynski effects.
• Inference Pipeline: A new code, CosmoSIS-gClust, is introduced to perform joint likelihood analyses of the power spectrum multipoles ($P_0, P_2, P_4$) and bispectrum multipoles ($B_0, B_2$). The analysis marginalizes over nuisance parameters (counterterms, shot noise) and tests various priors on the PNG bias parameter $b_\phi$.
• Metrics: The validity of the model is assessed using the Figure of Bias (FoB) to check for systematic errors and the Figure of Merit (FoM) to quantify information content. Scale cuts ($k_{\text{max}}$) are optimized to ensure unbiased recovery of cosmological parameters.

Key Contributions

1. Pipeline Validation: The paper provides a thorough validation of a joint power spectrum and bispectrum analysis pipeline for local PNG in redshift space, specifically tailored for Euclid-like data.
2. Bias Prior Strategy: The authors demonstrate that a "prior-agnostic" approach (using wide flat priors on $b_\phi$) leads to biased or unconstrained $f_{\text{NL}}$ estimates due to the strong degeneracy with $b_\phi$. They propose and validate a physically motivated Gaussian prior on the parameter $p_{b_\phi}$ (where $b_\phi = 2\delta_c(b_1 - p_{b_\phi})$) that brackets both Universal Halo Mass Function (UHMF) and Galaxy Formation Model (GFM) predictions. This choice yields unbiased constraints while accounting for theoretical uncertainty.
3. Scale Cut Optimization: The study determines optimal small-scale cuts ($k_{\text{max}}$) for both statistics. For the power spectrum, one-loop models remain unbiased up to $k \approx 0.30 \, h \text{ Mpc}^{-1}$. For the bispectrum, tree-level models remain unbiased up to $k \approx 0.12 \, h \text{ Mpc}^{-1}$ (lower at higher redshifts), beyond which the lack of loop corrections introduces bias.
4. Null Testing: A null test on Gaussian initial condition simulations ($f_{\text{NL}}=0$) confirms that the pipeline does not spuriously detect PNG or bias standard $\Lambda$CDM parameters.

Results

• Detection Significance: In a 'prior-agnostic' setup, high-significance detection of the dominant PNG term ($f_{\text{NL}} b_\phi$) in single redshift bins is difficult. However, with the proposed informative prior on $b_\phi$, the joint analysis recovers the fiducial $f_{\text{NL}} = 30$ with less than $1\sigma$ bias.
• Information Gain:
  • At fixed redshift and scale cuts, the bispectrum alone reduces the error on $f_{\text{NL}}$ ($\sigma(f_{\text{NL}})$) by approximately 29–46% relative to the power spectrum alone.
  • The joint power spectrum-bispectrum analysis tightens constraints further by an additional 8–13% compared to the bispectrum alone.
  • The cumulative gain in precision from $z=0.8$ to $z=1.7$ is a factor of $\sim 2.3$ for the joint case.
• Multipole Importance: The bispectrum quadrupole ($B_2$) is identified as a key contributor, breaking degeneracies between $f_{\text{NL}}$, $\omega_c$, and bias parameters that limit monopole-only analyses.
• Redshift Dependence: Constraints improve significantly with redshift. The strongest results are achieved at $z=1.7$, yielding a $1.9\sigma$ detection for $f_{\text{NL}} b_\phi$ (prior-agnostic) and a $2.35\sigma$ detection for $f_{\text{NL}}$ (prior-based).
• Large-Scale Sensitivity: The analysis highlights the critical importance of large-scale modes; excluding the first few $k$-bins degrades constraints on $f_{\text{NL}}$ and introduces significant biases, particularly for the bispectrum.

Significance and Claims
The authors state that this work serves as a preparation for the analysis of the Euclid spectroscopic sample, focusing on validating theoretical systematics rather than providing final forecasts for the full survey. The paper claims that joint analyses of the power spectrum and bispectrum offer "strong prospects for testing multi-field inflation," provided that end-to-end validation with observational systematics and realistic survey geometries is performed. The results confirm that with robust modeling and appropriate priors on PNG biases, Stage-IV surveys can effectively constrain local PNG, potentially surpassing current CMB limits. The authors explicitly note that the reported constraints are conservative, as they utilize a subset of the survey volume and do not yet incorporate full observational systematics or cross-correlations with other tracers.