Euclid preparation. XLI. Galaxy power spectrum modelling in real space

TL;DR

This paper benchmarks real-space galaxy power-spectrum modelling for Euclid against two approaches: a state-of-the-art Eulerian EFTofLSS bias expansion and a hybrid Lagrangian bias model emulator. Using four large Flagship I Hα-mock catalogs spanning $z=0.9$–$1.8$, the authors quantify model validity, scale cuts, and parameter degeneracies via FoB, GoF, and FoM, revealing robust cosmological inferences for $h$ and $\omega_c$ up to $k_{\max}=0.45\,h\mathrm{Mpc}^{-1}$. They show that both models are competitive, with EFT delivering higher constraining power when all nuisance parameters are free, while the emulator remains competitive at smaller scales; volume rescaling and emulator theory-error considerations further shape the practical limits. The work validates real-space full-shape analyses as a solid stepping stone for Euclid's 3×2-point program, and outlines a path toward redshift-space and bispectrum extensions. These results support applying perturbative galaxy bias models to upcoming Euclid data with careful treatment of bias degeneracies and model systematics.

Abstract

We investigate the accuracy of the perturbative galaxy bias expansion in view of the forthcoming analysis of the Euclid spectroscopic galaxy samples. We compare the performance of an Eulerian galaxy bias expansion, using state-of-art prescriptions from the effective field theory of large-scale structure (EFTofLSS), against a hybrid approach based on Lagrangian perturbation theory and high-resolution simulations. These models are benchmarked against comoving snapshots of the Flagship I N-body simulation at $z=(0.9,1.2,1.5,1.8)$, which have been populated with H$α$ galaxies leading to catalogues of millions of objects within a volume of about $58\,h^{-3}\,{\rm Gpc}^3$. Our analysis suggests that both models can be used to provide a robust inference of the parameters $(h, ω_{\rm c})$ in the redshift range under consideration, with comparable constraining power. We additionally determine the range of validity of the EFTofLSS model in terms of scale cuts and model degrees of freedom. From these tests, it emerges that the standard third-order Eulerian bias expansion can accurately describe the full shape of the real-space galaxy power spectrum up to the maximum wavenumber $k_{\rm max}=0.45\,h\,{\rm Mpc}^{-1}$, even with a measurement precision well below the percent level. In particular, this is true for a configuration with six free nuisance parameters, including local and non-local bias parameters, a matter counterterm, and a correction to the shot-noise contribution. Fixing either tidal bias parameters to physically-motivated relations still leads to unbiased cosmological constraints. We finally repeat our analysis assuming a volume that matches the expected footprint of Euclid, but without considering observational effects, as purity and completeness, showing that we can get consistent cosmological constraints over this range of scales and redshifts.

Figures (21)

• Figure 1: Halo occupation dsitribution profiles of the eight H$\alpha$ samples employed in this analysis. Individual panels show the profiles at different redshift (increasing from the top left to the bottom right panel) for both Model 1 (blue) and 3 (orange). Solid and dashed lines identify the average number of central and satellite galaxies, respectively, in halos of mass $M_{\rm h}$.
• Figure 2: Galaxy power spectrum measurements and uncertainties obtained from the Flagship I comoving snapshots. Top: Measurements of the Model 1 samples. The colour gradient identifies the different redshifts of the samples, as shown in the legend. Dashed horizontal lines correspond to the amplitude of the Poisson shot-noise term $P^{\,\rm sn}$ -- obtained as the inverse of the number density specified in the last column of Table \ref{['tab:hod_samples']} -- for the different redshifts. Centre: Same but for the Model 3 samples. Bottom: Error-to-measurement ratios, assuming a Gaussian covariance matrix as in Eq. (\ref{['eq:gaussian_cov']}). The coloured solid lines are obtained using the Poisson noise-subtracted power spectra, while the dashed black line highlights the linear relationship from Eq. (\ref{['eq:gaussian_cov']}), i.e. $2/N_k$. Grey bands mark the $1\%$, $0.5\%$, and $0.1\%$ limit.
• Figure 3: Performance metrics ( in the top row and in the bottom row) extracted from the Model 3 samples as a function of the maximum wave mode $k_{\rm max}$ of the fit, assuming the rescaled covariance matrices matching the four spectroscopic redshift bins described in Sect. \ref{['sec:vol_resc']}. Different curves correspond to different models, as described in the legend. The panels are normalised in units of the reference , corresponding to the one of the EFT model with all parameters free at $k_{\rm max}=0.1\, h \, {\rm Mpc}^{-1}$. The grey bands in the panels represent the $68\%$ and $95\%$ percentiles of the corresponding distribution, as explained in Sect. \ref{['sec:figure_of_bias']}.
• Figure 4: Same as Fig. \ref{['fig:compare_baccoemu_scaled']} but assuming a covariance matrix corresponding to the full simulation volume. The additional middle row corresponds to the averaged $\chi^2$ normalised by the total number of degrees of freedom. Similarly to the panels, the grey shaded bands in the $\chi^2$ panels mark the 68th and 95th percentile of the corresponding $\chi^2$ distribution with the corresponding number of degrees of freedom.
• Figure 5: Comparison between the marginalised constraints on the linear bias parameter $b_1$ and the shot-noise parameter $\alpha_{P,1}$ obtained at fixed cosmology, and the fiducial values listed in Table \ref{['tab:fiducial_b1_aP1']} obtained using only the large scale-limit of Eq. (\ref{['eq:two_pars_model']}). The first two and last two rows show results for the Model 1 and Model 3 samples, respectively. In both cases, the upper panels show constraints on the linear bias $b_1$, while the bottom ones show constraints on the constant shot-noise parameter $\alpha_{P,1}$. Different colours correspond to different assumptions on the total number of free bias parameters, as shown in the legend. Star symbols highlight the position of the maximum-likelihood for the case with all bias parameters free to vary. Dashed grey lines and shaded bands mark the fiducial value and $1\sigma$ confidence interval from Table \ref{['tab:fiducial_b1_aP1']}.