Testing General Relativity Using Large-scale Structure Photometric Redshift Surveys and the Cosmic Microwave Background Lensing Effect

The EG statistic provides a valuable tool for evaluating the predictions of general relativity (GR) by probing the relationship between gravitational potential and galaxy clustering on cosmological scales within the observable Universe. In this study, we constrain the EG statistic using photometric redshift data from the Dark Energy Survey (DES) MagLim sample in combination with the Planck 2018 cosmic microwave background (CMB) lensing map. Unlike spectroscopic redshift surveys, photometric redshift measurements are subject to significant redshift uncertainties, making it challenging to constrain the redshift distortion parameter β with high precision. We adopt a new definition for this parameter, β(z) = fσ8(z)/bσ8(z). In this formulation, we reconstruct the growth rate of structure, fσ8(z), using the artificial neural network method, while simultaneously utilizing model-independent constraints on the parameter bσ8(z), directly obtained from the DES collaboration. After obtaining the angular power spectra Cℓgg (galaxy–galaxy) and Cℓgκ (galaxy–CMB lensing) from the combination of DES photometric data and Planck lensing, we derive new measurements of the EG statistic: EG = 0.354 ± 0.146, 0.452 ± 0.092, 0.414 ± 0.069, and 0.296 ± 0.069 (68% confidence limit) across four redshift bins z = 0.30, 0.47, 0.63, and 0.80, respectively, which are consistent with the predictions of the standard Lambda cold dark matter model. Finally, we forecast the EG statistic using future photometric redshift data from the China Space Station Telescope, combined with lensing measurements from the CMB-S4 project, indicating an achievable constraint on EG of approximately 1%, improving the precision of tests for GR on cosmological scales.