Probing gravity with non-linear clustering in redshift space

Problem Statement
The observed accelerated expansion of the universe is typically attributed to dark energy or a cosmological constant ($\Lambda$), but alternative theories propose modifications to General Relativity (GR) on cosmological scales. A primary challenge in testing these theories is breaking the degeneracy between modified gravity (MG) models and dark energy models, particularly given that screening mechanisms in MG theories (like the chameleon mechanism in $f(R)$ gravity) suppress deviations in high-density regions like the Solar System. The $E_G$ estimator was proposed as a scale-independent probe of gravity on large scales, defined as the ratio of the curvature potential (related to weak lensing) to the velocity field (related to redshift-space distortions, RSD). However, accurately measuring $E_G$ requires precise, unbiased determinations of the growth rate of structure ($f$) and the linear galaxy bias ($b_1$). Current analyses often assume a scale-independent growth rate and rely on linear theory, which may be insufficient for detecting subtle deviations in MG models where the growth rate becomes scale-dependent. Furthermore, the ability of $E_G$ to distinguish between viable MG models (such as the Hu & Sawicki $f(R)$ model with $|f_{R0}| = 10^{-5}$, denoted as F5) and GR in the presence of non-linear clustering remains an open question.

Methodology
The authors conduct an end-to-end cosmological analysis using high-fidelity simulated galaxy mocks derived from two twin N-body simulations: one based on GR and the other on the F5 Hu & Sawicki $f(R)$ model. Both simulations share identical initial conditions and cosmological parameters ($\Omega_m=0.3089$, $\Omega_\Lambda=0.6911$, etc.) but differ in the gravitational force law.

1. Data Generation: Galaxy mocks are generated using a combination of Halo Occupation Distribution (HOD) and Sub-Halo Abundance Matching (SHAM) applied to the simulations. The study utilizes a full-sky lightcone output with spectroscopic redshifts, selecting six distinct galaxy samples (full, central, red, blue, bright, and faint) across three redshift bins ($z=0.35, 0.55, 0.8$).
2. Estimator Construction: The authors adopt the refined $E_G$ estimator described in Ref. [106], which utilizes angular power spectra ($C_\ell$) in configuration space. The estimator is formulated as:
   $$\hat{E}_G^\ell(z_{eff}) \approx \Gamma(z_{eff}) \frac{C_{\kappa g^*}^\ell}{\beta C_{gg}^\ell}$$
   where $C_{gg}^\ell$ is the galaxy auto-power spectrum, $C_{\kappa g^*}^\ell$ is the weighted convergence-galaxy cross-power spectrum, and $\beta = f/b_1$ is derived from RSD analysis.
3. Growth Rate and Bias Estimation:
   • Linear Bias ($b_1$): Estimated independently from the galaxy angular auto-correlation using the code pyCCL, providing a Gaussian prior for subsequent RSD fits.
   • Growth Rate ($f$): Estimated by fitting the monopole and quadrupole of the 2-point correlation function in configuration space. The authors employ the COMET-VDG emulator, which implements the Velocity Difference Generating function model to capture non-linear RSD effects (including the Finger-of-God effect) more accurately than standard Effective Field Theory (EFT) approaches at small scales.
   • Scale Splitting: To investigate scale dependence, the analysis splits the data into "large" (quasi-linear, $50-140$ Mpc/$h$) and "small" ($20-50$ Mpc/$h$) scales.
4. MG Boosting: Since COMET does not natively support MG cosmologies, the authors apply the power spectrum boost from the e-mantis emulator (specific to $f(R)$) to the matter power spectrum before generating the theoretical multipoles.
5. Statistical Analysis: A Markov Chain Monte Carlo (MCMC) approach using MultiNest is used to fit the model parameters. The covariance matrix is estimated via 100 Jackknife resampling regions. The final $E_G$ distribution is computed using a ratio distribution of the observables to account for non-Gaussianity.

Key Contributions

• End-to-End Pipeline: The study presents a comprehensive pipeline for calculating $E_G$ using realistic galaxy mocks in modified gravity, incorporating state-of-the-art non-linear modeling (COMET-VDG) and scale-dependent growth rate estimation.
• Scale-Dependent Analysis: Unlike previous works that often assume scale independence, this paper explicitly tests the $E_G$ estimator across different scale ranges (small vs. large) to probe the specific signatures of $f(R)$ gravity.
• Null Test Proposal: The authors propose a "null test" of gravity based on the 2-point clustering statistics in redshift space. This test compares the ratio of data vectors (F5 vs. GR) directly, bypassing the complexities of the $E_G$ estimator to assess the intrinsic discriminative power of small-scale clustering.
• Robustness Checks: The analysis includes extensive robustness tests regarding galaxy sample selection (color, magnitude, centrality), covariance matrix choices (Jackknife vs. theoretical vs. diagonal), and prior assumptions on non-linear bias parameters.

Results

• $E_G$ Discriminatory Power: Even for an ideal, all-sky, systematics-free survey, the $E_G$ estimator cannot clearly distinguish between the F5 and GR models. The error bars are too large, and the predicted differences between the models (approx. 5%) are comparable to the statistical uncertainties.
• Scale Dependence: The estimated $E_G$ values show a trend opposite to theoretical predictions when split by scale (over-predicted at small scales, under-predicted at large scales), likely due to modeling limitations and projection effects in the non-linear regime.
• Growth Rate Bias: The growth rate $f$ is recovered with some bias at small scales, particularly for the F5 model, where it is biased low. However, this bias is partially compensated by the clustering ratio term in the $E_G$ calculation.
• Null Test Success: In contrast to the $E_G$ estimator, the proposed null test based on the ratio of 2-point correlation function multipoles (monopole and quadrupole) shows significant discriminatory power. The data vectors for F5 and GR differ at a significance level of $\sim 3\sigma$ to $4\sigma$ across the redshift bins, indicating that the information content exists in the data but is obscured by the specific construction of the $E_G$ estimator and current modeling limitations.
• Modeling Limitations: The high $\chi^2_\nu$ values observed in the fits, particularly at small scales, suggest that current perturbative models (VDG/EFT) struggle to accurately describe the clustering on scales below $\sim 10-20$ Mpc/$h$, where the strongest deviations from GR are expected.

Significance and Claims
The paper modestly claims that while the $E_G$ estimator is a theoretically robust, bias-independent test of gravity, its practical utility for distinguishing between viable MG models (like F5) and GR is limited by current non-linear modeling capabilities and statistical noise, even in idealized scenarios. The authors emphasize that the failure to distinguish models via $E_G$ is not necessarily a failure of the data, but rather a limitation of the estimator's construction, which averages out the scale-dependent signals where MG effects are strongest.

The study concludes that future progress requires:

1. Improved Modeling: Developing MG-consistent RSD templates and emulators (e.g., BACCO) that can accurately model clustering on smaller scales ($<10$ Mpc/$h$).
2. Alternative Probes: Utilizing the full information content of redshift-space clustering (as demonstrated by the null test) or combining multiple observables (e.g., 3x2pt analyses) to break degeneracies between nuisance parameters (like galaxy bias) and cosmological parameters.
3. Realistic Forecasts: The results presented are upper bounds under ideal conditions; real surveys will face additional systematics (baryonic physics, intrinsic alignments, photometric errors) that will further degrade the discriminatory power of $E_G$.

Ultimately, the work serves as a benchmark for the limitations of current $E_G$ methodologies and motivates the development of more sophisticated, non-linear modeling frameworks to fully exploit the potential of next-generation galaxy surveys (like Euclid, DESI, and LSST) in testing gravity.