A step towards testing general relativity using weak gravitational lensing and redshift surveys

TL;DR

This work develops a model-independent framework to test General Relativity on cosmological scales by deriving GR-based consistency relations within linear perturbation theory and combining multiple observables. It constructs a web of tests—energy-momentum, metric, dynamical, and Poisson consistency—that relate matter perturbations, velocities, and metric potentials, enabling cross-checks with galaxy clustering, redshift-space distortions, and weak lensing. It then details practical strategies to trace matter perturbations with galaxies, addressing redshift uncertainties and galaxy bias, and introduces a method to reconstruct the lensing signal from galaxy data, plus a tomographic estimator for testing the Poisson equation via an alpha parameter. The study demonstrates feasibility with spectroscopic and photometric surveys, showing how galaxy bias and photo-z biases can be mitigated and highlighting that GR-violating models like certain $f(R)$ theories would fail these tests, thereby offering a robust, data-driven path for upcoming surveys to probe gravity on cosmological scales.

Abstract

Using the linear theory of perturbations in General Relativity, we express a set of consistency relations that can be observationally tested with current and future large scale structure surveys. We then outline a stringent model-independent program to test gravity on cosmological scales. We illustrate the feasibility of such a program by jointly using several observables like peculiar velocities, galaxy clustering and weak gravitational lensing. After addressing possible observational or astrophysical caveats like galaxy bias and redshift uncertainties, we forecast in particular how well one can predict the lensing signal from a cosmic shear survey using an over-lapping galaxy survey. We finally discuss the specific physics probed this way and illustrate how $f(R)$ gravity models would fail such a test.

Figures (7)

• Figure 1: Web of cosmological tests of GR (see an analogous plot in Fig. 1 of Uzan:2006mf).
• Figure 2: Projected galaxy over-density angular power spectrum as a function of multipole $\ell$ as defined in Eq. \ref{['eq:cl_gg']} and Eq. \ref{['eq:il']} (solid line) and its limber approximation as defined in Eq. \ref{['eq:cl_gg_limber']} (dashed line). The various curves correspond to various bin width. Obviously, the wider the redshift bin the better is the Limber approximation but it is not a very accurate one except at smaller scales.
• Figure 3: Left panel, top:$n_g$ distribution plotted for three various $\sigma_z$, i.e. $3\times10^{-4},3\times10^{-3},3\times10^{-2}$ and the underlying galaxy distribution. Bottom : Corresponding galaxy overdensity angular power spectrum, $C_\ell^{gg}$. Photometric redshift error entails an important bias. Right panel: Relative difference between $C_l^{gg}$ power spectra with or without photo-$z$ errors considering $\sigma_z=0.03$. Solid, long dash, short dash and dotted curves correspond respectively to $C_l^{gg}$ at $z$=0.3, 0.5, 1.5 and 2.5.
• Figure 4: Left panels: The top panel show the fractional errors for the reconstructed $P_{\Theta_g\Theta_g}^{}$ using the Fisher matrix formalism written in Eq. \ref{['eq:fish_gg_tt']}. The bottom plot shows the corresponding fractional errors for $P_{gg}^{}$. Right panels: 68% CL contour plots in the $w-w_a$ plane. The top panel shows the constraints obtained using $P_{\Theta_g\Theta_g}$ (no bias marginalization) only and $P_{\Theta_g\Theta_g}^{}$+$P_{gg}^{}$ (with bias marginalization). The bottom panel shows the constraints using $P_{\Theta_g\Theta_g}^{}$+$P_{gg}^{}$ and various value of the parameters $\sigma_{th}$ that quantifies the accuracy of the modeling of the Finger of God effect (see Eq. 15 of White:2008jy for details).
• Figure 5: Relative uncertainties for the reconstructed $C_l^{dd}$. The dash line corresponds to the statistical uncertainties when measuring the bias using a large spectroscopic survey (as in Sec. \ref{['sec:bias_spectro']}) and the dotted line corresponds to the statistical uncertainties when using a photometric survey (as in Sec. \ref{['sec:bias_photo']}). Other systematic bias are illustrated in Fig. \ref{['fig:cl_wl_wl_error']}.