Testing template-fitting models for the multipoles of the two-point clustering of galaxy clusters

Abstract: The \textit{Euclid} satellite will deliver a catalogue of optically-selected galaxy clusters spanning from around $2000$ deg$^2$ in Data Release (DR) 1 to around $14\,000$ deg$^2$ in DR3. We assess the validity of cluster clustering (CC) models for template-fitting, which complements the full-shape methodology in providing cosmological information from the anisotropy of the redshift-space two-point correlation function (2PCF). Both will be used to analyse the cluster 2PCF multipoles in \textit{Euclid}. We analyse the multipoles of the two-point redshift-space clustering of galaxy clusters simulated with the semi-analytic \code{PINOCCHIO} code using third-order Lagrangian perturbation theory, assuming a \textit{Euclid} DR1-like footprint of 500 deg$^2$ in the Northern Hemisphere and 1400 deg$^2$ in the Southern Hemisphere. We estimate the first three even multipoles of the 2PCF and associated covariance matrix from 1000 DR1-like synthetic catalogues. We study the impact of the modelling of nonlinearities, halo bias, and photometric redshift uncertainties on the 2PCF. We apply three clustering models to the mock catalogues at $0<z\<2$ and virial mass $M_{\rm vir}\>10^{14}\;h^{-1}\,M_\odot$ under realistic and optimistic photometric redshift uncertainty scenarios. We formulate a set of permissive and conservative criteria that should be fulfilled by the multipole cut-off scales and validate against 100 mock catalogues via inference of the growth rate times the matter power spectrum normalisation parameter, $fσ_8$. We test the dispersion, Scoccimarro, and Taruya--Nishimichi--Saito models. We find that the simplest of the three -- the dispersion model -- yields unbiased inferences on $fσ_8$ from CC down to $10$ $h^{-1}$ Mpc in a DR1-like setting. All clustering models provide very similar goodness-of-fit metrics in the presence of DR1-like cluster redshift uncertainties.

• The paper demonstrates that for scales ≥10 h⁻¹ Mpc, the dispersion model yields unbiased fσ₈ inference under DR1-like conditions.
• It employs simulated PINOCCHIO lightcones to systematically compare dispersion, Scoccimarro, and TNS models under varying photometric redshift uncertainties.
• The study provides clear pipeline recommendations for Euclid DR1 and outlines limitations at higher redshift and multipoles that require improved nonlinear bias modeling.

Template Fitting for Multipole Clustering Analysis of Galaxy Clusters in Euclid DR1

Introduction and Context

The accurate modeling of the two-point correlation function (2PCF) of galaxy clusters in redshift space is critical for constraining the growth of structure and extracting robust cosmological information in forthcoming Stage-IV surveys such as Euclid. The work "Testing template-fitting models for the multipoles of the two-point clustering of galaxy clusters" (2604.25762) addresses the problem of model validation for template-based inference on the multipoles of 2PCFs, focusing on the practicalities required by the Euclid DR1 cluster sample. The study employs a suite of simulated catalogues from the PINOCCHIO code to systematically test different redshift-space clustering models, evaluate the impact of photometric redshift uncertainties and nonlinear/nonlocal corrections, and define statistical criteria for cut-off scales required in cosmological inference.

Simulated Data and Survey Footprint

The analysis uses 1000 lightcones generated with third-order Lagrangian perturbation theory as implemented by the PINOCCHIO algorithm, ensuring comprehensive statistical coverage while maintaining computational efficiency. The adopted mock DR1 footprint comprises $500~\mathrm{deg}^2$ in the Northern Galactic hemisphere and $1400~\mathrm{deg}^2$ in the Southern, selecting halo masses $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$ and extending up to $z=2$. Completeness and purity are, for this baseline study, idealized to 100% in all relevant bins, optimizing statistical power for model discrimination.

Figure 1: Footprint of the PINOCCHIO cluster mock catalogue before and after DR1-like angular cuts.

2PCF Multipole Estimation and Redshift-Space Modeling

The 2PCF multipoles—monopole, quadrupole, and, for specialized analysis, the hexadecapole—are estimated in bins from $10$ to $150~h^{-1}\mathrm{Mpc}$, using a Landy–Szalay-type estimator sampled across 1000 independent realizations. Statistical covariances are determined empirically, justifying by Kolmogorov-Smirnov tests the use of Gaussian likelihoods for parameter inference.

The paper tests three classes of template fitting models for the redshift-space power spectrum:

• Dispersion Model \citep{1994MNRAS.267.1020P}: Extends the linear Kaiser model by introducing an empirical Gaussian (or Lorentzian) damping factor parameterized by the velocity dispersion scale $\sigma_v$. For clusters, with low internal velocity dispersions, the dominant source of this damping is photometric redshift uncertainty.
• Scoccimarro Model \citep{2004PhRvD..70h3007S}: Incorporates cross terms between the matter and velocity divergence fields, improving accuracy at mildly nonlinear scales.
• TNS Model \citep{2010PhRvD..82f3522T}: Employs a perturbative expansion including higher-order corrections, calibrated for the galaxy case.

All models are linear in bias and share the matter power spectrum (fixed to Planck 2013 $\Lambda$CDM) as input. Photometric redshift uncertainties are modeled as Gaussian, parameterized as $\sigma_z = \sigma_0(1+z)$, with both optimistic ($\sigma_0 = 0.005$) and realistic ($1400~\mathrm{deg}^2$0) scenarios explored.

Figure 2: Performance comparison of the dispersion, Scoccimarro, and TNS models at ground-truth parameters, under varying photometric redshift uncertainties.

Model Validation and Cut-off Scale Statistical Criteria

The principal goal is to determine the smallest scale (largest wavenumber) at which the models provide unbiased inference for $1400~\mathrm{deg}^2$1, the key parameter encoding the linear growth rate and power spectrum normalization. The authors introduce two sets of statistical criteria:

1. Permissive Tests:
   • Model is accepted if the reduced $1400~\mathrm{deg}^2$2 of fits is consistent with unity within statistical fluctuations.
   • Ensures the model provides an adequate fit to DR1-like data but does not strictly test for statistical consistency of posteriors.
2. Conservative Tests:
   • Additional constraints on posterior bias and coverage in ensembles of mocks (using metrics such as $1400~\mathrm{deg}^2$3 and highest posterior density (HPD) coverage), monotonicity conditions, and catalog-level robustness.
   • These stricter requirements are intended to serve as baseline for higher-fidelity DR2/3 analyses, where increased S/N will render systematics more prominent.

The grid search in minimum scale cut-offs for both monopole and quadrupole is performed in each bin, and posterior distributions for $1400~\mathrm{deg}^2$4 are rigorously characterized.

Main Results

• Model Performance: All three tested models (dispersion, Scoccimarro, TNS) are statistically indistinguishable in the presence of DR1-like photometric redshift uncertainties; the simple dispersion model is sufficient at scales $1400~\mathrm{deg}^2$5 for both monopole and quadrupole.
• Bias Characterization: For informative bins (i.e., ones where posteriors are not strongly prior-dominated), the inferred $1400~\mathrm{deg}^2$6 bias is $1400~\mathrm{deg}^2$7 for all tested scenarios, supporting the statistical adequacy of the dispersion model at the adopted scale cuts.

  Figure 3: Monopole and quadrupole measurements in \code{PINOCCHIO} mocks versus redshift-space model predictions under DR1-like redshift uncertainties, with ensemble error analysis.

• Redshift and Multipole Dependence: The scale at which model deviations become significant increases with redshift and multipole order, an effect traced to increasing halo bias and nonlinearity—not merely failures of redshift-space modeling, but also the limitations of the linear bias assumption, particularly significant at $1400~\mathrm{deg}^2$8.
• Hexadecapole: The inclusion of the $1400~\mathrm{deg}^2$9 term (hexadecapole) in DR1-like analyses provides negligible improvement to $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$0 constraints; this could change for future DRs with improved S/N.

  Figure 4: Posterior distributions for $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$1 as a function of redshift bin and photometric redshift error; dashed line is ground truth, dotted line the prior.

Implications for Euclid and Cosmological Inference

• Pipeline Recommendations: For DR1, the dispersion model, used with a conservative minimum scale of $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$2, is sufficient for unbiased template fitting on the 2PCF monopole and quadrupole; more complex models provide no improvement given DR1 photo-z uncertainty and sample variance.
• Limiting Factors and Future Directions: For higher-redshift analysis or future Euclid data releases (DR2/3), especially with reduced photometric redshift uncertainty or improved mass calibration, nonlinear and nonlocal bias effects will become the principal limiting systematics. Without improved halo bias prescriptions and higher-fidelity $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$3-body–based mocks (see [2025A&A...697A...5E]), small-scale clustering modeling cannot be reliably extended to $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$4 or to higher-order multipoles.

  Figure 5: Impact of including the hexadecapole on $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$5 posteriors in the optimal redshift bin for realistic redshift uncertainties.

Theoretical and Practical Impact

This study establishes a robust, statistically calibrated framework for the use of template-fitting models in cluster 2PCF multipole analysis in the photometric regime, forming the foundation for the cosmological likelihood infrastructure for the Euclid survey. The defined scale cuts and statistical testing protocols will directly shape the implementation of cluster clustering cosmology pipelines and error budgets. The results emphasize the current sufficiency of the simplest analytic models in the photo-z dominated regime, but also highlight the necessity of improved theory, particularly for the cluster bias function, to fully exploit the cosmological constraining power of future, larger cluster samples and improved data.

Conclusion

The research provides a comprehensive validation of 2PCF multipole template-fitting procedures for photometric galaxy cluster surveys in the context of Euclid DR1. The key findings are that the dispersion model suffices for robust inference on $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$6 down to $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$7 under DR1-like conditions, with minimal bias in the informative regime. Extending cosmological analysis to higher redshift or smaller scales requires both improved modeling of nonlinear bias and the use of simulations with $M_\mathrm{vir} > 10^{14}~h^{-1}M_\odot$8-body accuracy. This work is pertinent for the architecture of large-scale structure analysis pipelines in next-generation cosmological surveys (2604.25762).