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

Abstract: Primordial non-Gaussianity (PNG) is a powerful probe of the origin of cosmic structure. Stage-IV surveys like \Euclid will measure galaxy $2$- and $3$-point clustering at high signal-to-noise, whose exploitation requires robust joint analysis. We prepare for Euclid's spectroscopic sample by validating a redshift-space power-spectrum and bispectrum pipeline (one-loop $P_\ell$, tree-level $B_\ell$) on Euclid-like mocks from Abacus-PNG $N$-body simulations with Gaussian and local-PNG initial conditions, using a halo occupation distribution (HOD) tuned to Euclid Flagship 2. We stress-test analysis choices -- PNG-bias parametrisation, priors, and scale cuts -- and perform null tests without PNG. In a `prior-agnostic setup', detection of the dominant PNG term $\propto f_{\rm NL} \, b_φ$ in single redshift bins is difficult; nevertheless, the bispectrum provides constraints on other PNG combinations that partially lift degeneracies. We propose a physically motivated prior on $b_φ$ that yields unbiased $f_{\rm NL}$ while accounting for theory uncertainty, and determine scale cuts that give unbiased $Λ$CDM and $f_{\rm NL}$. With $V_{\rm eff}=16\,h^{-3}\,{\rm Gpc}^3$ across four snapshots ($0.8\le z\le1.7$), our likelihood analyses recover $<1σ$ bias in $f_{\rm NL}$ and $Λ$CDM. At fixed cuts, $B_\ell$ alone reduces $σ({f_{\rm NL}})$ by $\sim29$--$46\%$ relative to $P_\ell$, and joint power spectrum-bispectrum analysis tightens a further $\sim8$--$13\%$; the cumulative gain from $z=0.8$ to $1.7$ is $\sim2.3$ for the joint case. The bispectrum quadrupole is key. Our strongest results are at $z=1.7$: $1.9σ$ for $f_{\rm NL} \, b_φ$ (prior-agnostic) and $2.35σ$ for $f_{\rm NL}$ (prior-based). Joint analyses thus offer strong prospects for testing multi-field inflation, pending end-to-end validation in the full Euclid geometry with observational systematics.

• The paper demonstrates that combining power spectrum and bispectrum analyses significantly reduces error on fₙₗ, improving constraints by up to 53% compared to single-probe estimates.
• It employs advanced EFTofLSS modeling, rigorous redshift-space and bias treatments, and synthetic galaxy mocks to validate methodology across multiple redshift bins.
• The study underscores the importance of physically motivated priors and scale cuts, laying groundwork for robust multi-field inflation tests in future Stage-IV spectroscopic surveys.

Joint Power Spectrum and Bispectrum Analysis for Local-type Primordial Non-Gaussianity with Euclid: Validation, Systematics, and Parameter Constraints

Introduction and Motivation

Primordial non-Gaussianity (PNG) provides a critical window into the physics of the early universe, especially the dynamics of inflation. Local-type PNG, encapsulated by the parameter $f_{\rm NL}$, is especially sensitive to multi-field inflation models, with the single-field consistency relation predicting a nearly vanishing amplitude for this shape. Current constraints from the CMB are cosmic variance limited, making the three-dimensional large-scale structure (LSS) from future spectroscopic surveys like Euclid a compelling avenue for improved sensitivity.

While the power spectrum has been the primary observable for constraining $f_{\rm NL}$ via its scale-dependent bias signature, the bispectrum offers complementary information due to its sensitivity to multiple biasing and primordial operators, and the different triangle configurations that can be probed. The challenge lies in the proper modeling (including redshift-space distortions and bias), robust treatment of analysis systematics (priors, scale cuts), and joint likelihood inference given high-dimensional data vectors. Addressing these, the authors validate a redshift-space pipeline for the joint analysis of the galaxy power spectrum and bispectrum, focusing on Euclid-like synthetic mocks from N-body simulations.

Theoretical Modelling: Power Spectrum and Bispectrum in Redshift Space with PNG

The perturbative modelling is based on the EFTofLSS formalism. For the power spectrum, a one-loop model is employed, extended to include the impact of local PNG. The bispectrum is modeled at tree-level but incorporates a counterterm motivated by the velocity-difference generator (VDG) to improve modeling of nonlinear redshift-space effects and extend the perturbative reach.

Redshift-space distortions (RSDs) are modeled via the velocity-moment expansion, retaining velocity contributions up to fourth order. The galaxy bias expansion is built to include not only operators of the matter density and tidal field but additional terms sourced by the primordial potential, per the requirements of PNG bias theory. PNG influences both the galaxy power spectrum, primarily through the scale-dependent bias from $b_\phi$, and the bispectrum via contributions from $f_{\rm NL}$, $f_{\rm NL}b_\phi$, and $f_{\rm NL}b_{\phi\delta}$.

Synthetic Galaxy Catalogs and Likelihood Analysis

Galaxy mocks are constructed from the Abacus N-body simulation suite, spanning both Gaussian and local-PNG initial conditions (with $f_{\rm NL}=30$ as the primary non-Gaussian case). Halo occupation distribution (HOD) parameters are calibrated to the Euclid Flagship 2 (FS2) mock, then applied after abundance matching the halo mass function to minimize discrepancies arising from different halo finders between simulations.

The analysis pipeline builds a Gaussian likelihood for a data vector including the three lowest multipoles of the power spectrum and bispectrum ($\ell=0,2,4$ for $P_\ell$ and up to $\ell=2$ for $f_{\rm NL}$0 in practice, due to noise in the bispectrum hexadecapole), with analytic marginalisation of linear nuisance parameters for computational efficiency. The adopted scale cuts and priors are extensively validated to avoid bias. Gaussian covariances are assumed, noting that in real survey applications, non-Gaussian covariance and cross-statistic correlations need further treatment.

Figure 1: Measured HOD from the Euclid FS2 simulation and its fit at $f_{\rm NL}$1, showcasing central and satellite occupation fits crucial for accurate HOD-based mock galaxy assignment.

Figure 2: Abundance matching process bringing the AbacusSummit halo mass function into agreement with the Flagship 2 reference for robust bias calibration.

Prior Choices and Systematic Impact on $f_{\rm NL}$2 Inference

A key systematic is the prior on $f_{\rm NL}$3, the bias parameter entering the leading $f_{\rm NL}$4 signal in the power spectrum. The authors show that a broad, uninformative prior results in biased estimates, effectively defaulting to prior-volume effects that drive the marginalized posterior of $f_{\rm NL}$5 toward zero. The shape and degeneracy structure in $f_{\rm NL}$6 make informative, physically motivated priors essential.

Two competing theoretical models are tested for the mapping $f_{\rm NL}$7: the universal halo mass function (UHMF, $f_{\rm NL}$8) and a galaxy formation motivated value (GFM, $f_{\rm NL}$9) inspired by results from hydrodynamical simulations. A Gaussian prior with mean $b_\phi$0 and width $b_\phi$1 encompasses both, suppressing projection biases and leading to accurate and stable $b_\phi$2 inference across redshift bins.

Figure 3: Posterior distributions for power spectrum and bispectrum from a prior-agnostic analysis, illustrating the dominant degeneracy directions and the limitation of unconstrained $b_\phi$3.

Figure 4: Sensitivity of $b_\phi$4 posterior to the prior on $b_\phi$5, exemplifying the necessity of physically informed prior choices.

Measurements of $b_\phi$6 using separate-universe simulations further validate the chosen prior's adequacy over the redshift range considered.

Scale Cut Selections and Robustness

Extensive scale-cut tests demonstrate that one-loop power spectrum modeling is unbiased up to $b_\phi$7Mpc at $b_\phi$8–$b_\phi$9, while the tree-level bispectrum with VDG counterterm is valid to $f_{\rm NL}$0Mpc at $f_{\rm NL}$1 and $f_{\rm NL}$2Mpc at $f_{\rm NL}$3. The information content (FoM) and bias metrics (FoB) confirm that these choices are efficient and conservative. Removing large-scale modes ($f_{\rm NL}$4) quickly degrades $f_{\rm NL}$5 constraints, particularly for the bispectrum, underscoring the necessity of maximal volumetric coverage for PNG science.

Figure 5: Marginal posterior distributions across redshifts and scale cuts for key cosmological parameters, mapping the bias and variance trade-off.

Figure 6: Impact of removing large-scale modes on $f_{\rm NL}$6 error and bias, demonstrating the disproportionate role of the largest accessible scales.

Main Numerical Results: Joint Constraints on $f_{\rm NL}$7

Application of the validated pipeline yields unbiased $f_{\rm NL}$8 recovery in PNG simulations (fiducial $f_{\rm NL}$9), with null tests on Gaussian initial conditions accurately returning $f_{\rm NL}b_\phi$0 within $f_{\rm NL}b_\phi$1. For singly-analyzed redshift bins with $f_{\rm NL}b_\phi$2, the joint power spectrum-bispectrum analysis reduces the marginalized $f_{\rm NL}b_\phi$3 error by $f_{\rm NL}b_\phi$4, $f_{\rm NL}b_\phi$5, $f_{\rm NL}b_\phi$6, and $f_{\rm NL}b_\phi$7 at $f_{\rm NL}b_\phi$8, $f_{\rm NL}b_\phi$9, $f_{\rm NL}b_{\phi\delta}$0, and $f_{\rm NL}b_{\phi\delta}$1 respectively compared to power spectrum alone; bispectrum alone accounts for $f_{\rm NL}b_{\phi\delta}$2–$f_{\rm NL}b_{\phi\delta}$3 of the gain over $f_{\rm NL}b_{\phi\delta}$4. The detection significance for $f_{\rm NL}b_{\phi\delta}$5 rises from $f_{\rm NL}b_{\phi\delta}$6 at $f_{\rm NL}b_{\phi\delta}$7 to $f_{\rm NL}b_{\phi\delta}$8 at $f_{\rm NL}b_{\phi\delta}$9.

Figure 7: Marginalized posterior distributions for $f_{\rm NL}=30$0 and cosmological parameters for power spectrum (blue), bispectrum (green), and combined (red) analyses for each redshift bin.

Figure 8: $f_{\rm NL}=30$1 error on $f_{\rm NL}=30$2 for fiducial $f_{\rm NL}=30$3 comparing power spectrum, bispectrum, and joint constraints at each redshift.

Figure 9: Relative information content of various multipole combinations for $f_{\rm NL}=30$4, highlighting the key added value of the bispectrum quadrupole.

Higher-order multipoles, particularly the bispectrum quadrupole, substantially improve constraints by breaking degeneracies with bias and amplitude parameters and by providing sensitivity to RSD-modulated primordial structures.

Implications and Outlook

The results demonstrate that Stage-IV spectroscopic surveys can meaningfully constrain local-type PNG and, by extension, multi-field inflation scenarios, provided that theoretical modeling, prior selection, and data analysis practices are rigorously validated. The bispectrum, especially its quadrupole, is essential for realizing these gains—not just as an auxiliary observable, but as the primary driver of $f_{\rm NL}=30$5 precision improvement.

While these results are obtained in periodic-box simulations with idealized selection, several steps are necessary for application to real survey data:

• Extension beyond tree-level bispectrum and comprehensive non-Gaussian covariance modeling.
• Full survey geometry, selection, and observational systematic effects (e.g., completeness, fiber collisions).
• Joint analyses across all redshift bins, with optimal weighting or multi-tracer strategies to maximize available information.
• Incorporation of cross-observables (CMB lensing, photometric samples) and field-level analyses to reduce sample variance.

Rapid advancements in theoretical modeling (e.g., fast loop algorithms, emulator-based inference), observational techniques, and high-performance computing are making high-fidelity joint LSS analyses feasible for upcoming data releases.

Conclusion

The study provides a comprehensive, end-to-end validation of joint galaxy power spectrum and bispectrum analyses for constraining local-type PNG in the context of Euclid. It demonstrates that the bispectrum, properly modeled and combined with the power spectrum, can reduce the error bar on $f_{\rm NL}=30$6 by up to a factor of two relative to traditional analyses, with essential sensitivity residing in the bispectrum quadrupole. Robustness to prior and scale selection is ensured through exhaustive validation on synthetic data. These results lay essential groundwork for extracting the maximal inflationary information from future Stage-IV spectroscopic survey data and present a clear methodology for systematic, bias-free inference of primordial statistics in LSS.

Figure 10: Comparison of best-fit models with measured spectra for power spectrum and bispectrum multipoles, with residuals indicating model fidelity across scales and configurations.