Probing Gravity at Large Scales with kSZ-Reconstructed Velocities and CMB Lensing

TL;DR

This work develops a novel $\widehat{V}_G$ estimator to measure the $E_G$ statistic by combining CMB lensing with kSZ-velocity reconstruction, providing a new probe of gravity on linear cosmological scales that does not rely on redshift-space distortions. The framework uses the $\mu$–$\gamma$ (or $\mu$–$\Sigma$) parametrization to predict MG signatures, deriving $E_G^{\mathrm{MG}}(k,z)=\frac{\Omega_{m,0}\,\Sigma(k,z)}{f(k,z)}$ and highlighting that GR yields a scale-independent $E_G(z)=\frac{\Omega_{m,0}}{f(z)}$. The estimator computes $\widehat{V}_G(\ell,z_{\mathrm{eff}})=\left(\frac{2c}{3H_0^2}\right)\frac{C_\ell^{\kappa g}}{\widetilde{C_\ell^{vg^{\dagger}}}}$, combining a CMB lensing–galaxy cross-spectrum with a kSZ-derived velocity cross-spectrum tied to the same galaxy sample, and accounts for effective redshifts through a reweighted galaxy sample. Forecasts for ACT DR6 and SO with three DESI galaxy samples predict cumulative detections $S/N\sim20$–$55$, enabling discrimination between GR and representative MG models such as Hu-Sawicki $f(R)$ and Chameleon theories, particularly at low redshift. The approach carefully addresses potential systematics, including optical-depth degeneracy and velocity bias, and emphasizes the need for covariance validation via simulations as data quality improves, marking a new pathway to test gravity on the largest observable scales.

Abstract

We present a new method for measuring the $E_G$ statistic that combines two CMB secondaries -- the kinematic Sunyaev-Zeldovich (kSZ) effect and CMB lensing -- for the first time to probe gravity on linear scales. The $E_G$ statistic is a discriminating tool for modified gravity theories, which leave imprints in lensing observables and peculiar velocities. Existing $E_G$ measurements rely on redshift space distortions (RSD) to infer the velocity field. Here, we employ kSZ velocity-reconstruction instead of RSD, a complementary technique that constrains the largest-scale modes better than the galaxy survey it uses. We construct a novel $\widehat{V}_G$ estimator that involves a ratio between cross-correlations of a galaxy sample with a CMB convergence map and that with a 3D kSZ-reconstructed velocity field. We forecast for current and upcoming CMB maps from the Atacama Cosmology Telescope (ACT) and the Simons Observatory (SO), respectively, in combination with three spectroscopic galaxy samples from the Dark Energy Spectroscopic Instrument (DESI). We find cumulative detection significances in the range $S/N \sim 20-55$, which can robustly test the scale-independent $E_G$ prediction under general relativity (GR) at different effective redshifts of the galaxy samples ($z\approx 0.73, 1.33, 1.84$). In particular, the SO$\times$DESI LRG measurement would be able to distinguish between GR and certain modified gravity models, including Hu-Sawicki $f(R)$ and Chameleon theories, with high confidence. The proposed $\widehat{V}_G$ estimator opens up a new avenue for stress-testing gravity and the $Λ$CDM+GR model at the largest observable scales.

Figures (4)

• Figure 1: Forecasted signal-to-noise ratios as a function of multipole $(\mathrm{SNR}(\ell) \equiv E_G(\ell, z_{\mathrm{eff}})/\sigma\,[\widehat{V}_G(\ell, z_{\mathrm{eff}})])$ for measuring the $E_G$ statistic at $z_{\mathrm{eff}}$ using the proposed $\widehat{V}_G$ estimator. Results for the three DESI galaxy samples considered (Section \ref{['survey']}): LRG, ELG, and QSO, are shown as red, blue, and pink curves, respectively, by combining them with CMB data (a) from ACT DR6, and (b) from SO. For each galaxy sample, the highest multipole we consider is given by $= (k_{\max}\chi_{\mathrm{eff}})$, with a default assumed value of $k_{\max}=0.1 \, \text{Mpc}^{-1}$ (the 'squeezed' limit Smith2018); a more stringent possible scale-cut giri2020 of $k_{\max}=0.035 \, \text{Mpc}^{-1}$ is also depicted here for reference by dotted lines.
• Figure 2: Predictions of the $E_G$ statistic as a function of redshift: from GR (black line), from the Hu-Sawicki $f(R)$ model ($f_{R0} = 10^{-5}$; violet curve), and from two representative Chameleon gravity models (green curves), obtained at each $z$ by averaging over the corresponding range of considered scales ($20 \leq \ell \lesssim k_{\max}\chi(z))$. The gray shaded region denotes the current associated uncertainty of the GR prediction. We also show error bars corresponding to the cumulative SNR of $\widehat{V}_G$ measurements using CMB data from (a) ACT DR6 and (b) SO, when combined with the DESI LRG (red), ELG (blue), and QSO (pink) galaxy samples, at their respective effective redshifts.
• Figure 3: Comparing predictions of the $E_G$ statistic as a function of scale at $z_{\mathrm{eff}} = 0.73$ (corresponding to the DESI LRG sample) from GR (black line), the Hu-Sawicki $f(R)$ model ($f_{R0} = 10^{-5}$; violet curve), and a few representative Chameleon gravity models (green curves). The gray shaded region denotes the current associated uncertainty of the GR prediction. For reference, we also show the forecasted error bars of a $\widehat{V}_G$ measurement with SO$\times$DESI LRG (red) centered on the GR prediction, where the multipole range is split into six linearly spaced $\ell-$bins.
• Figure 4: Prediction of the $E_G$ statistic as a function of redshift from GR (black line), obtained at each $z$ by averaging over the corresponding range of considered scales ($20 \leq \ell \lesssim k_{\max}\,\chi(z))$). The gray shaded region denotes the current associated uncertainty of the GR prediction. We show error bars corresponding to the cumulative SNR of $\widehat{V}_G$ measurements using CMB data from (a) ACT DR6 and (b) SO, when combined with the DESI LRG (red), ELG (blue), and QSO (pink) galaxy samples, at their respective effective redshifts. The solid error bars are computed with a scale-cut of $k_{\max}= 0.1 \ \text{Mpc}^{-1})$ (as shown in Fig. \ref{['fig:EG_z']} too). These are compared against the larger, dashed error bars (and lower cumulative SNR) that are obtained if a more stringent scale-cut of $k_{\max} = 0.035 \ \text{Mpc}^{-1}$ is used instead.