Constraining gravity with the decay rate of cosmological gravitational potential

A key task in cosmology is to test the validity of general relativity (GR) at cosmological scales and, therefore, to distinguish between dark energy and modified gravity (MG) as the driver of the late-time cosmic acceleration. The decay rate ($DR$) of cosmological gravitational potential, being sensitive to gravity and being immune to various astrophysical uncertainties, enables GR tests independent to other structure growth probes. Recently we have measured $DR$ at $0.2\leq z\leq 1.4$, combining the DR9 galaxy catalog from the DESI imaging surveys and Planck cosmic microwave background maps \citep{arXiv:2411.12594}. Here we use this measurement to test gravity, and restrict the analysis to one-parameter extensions to the standard $\Lambda$CDM cosmology. We consider four one-parameter MG parameterizations. One is $f(a)=\Omega_m^\gamma(a)$. The other three adopt the gravitational slip parameter $\eta=1$ and consider variations in the effective gravitational constant $G_{\rm eff}/G$ with the parameterization $\Sigma(a)=\Sigma_\Lambda \Omega_\Lambda(a)/\Omega_\Lambda$, $\Sigma(a)=\Sigma_1 a$ or $\Sigma(a)=\Sigma_2 a^2$. We find $\gamma=0.47^{+0.22}_{-0.15}$, consistent with the GR prediction $\gamma\simeq 0.55$. We also find $\Sigma_\Lambda=0.018^{+0.052}_{-0.053}$, $\Sigma_1=0.020^{+0.065}_{-0.062}$, and $\Sigma_2=0.027^{+0.067}_{-0.069}$, fully consistent with the GR case of $\Sigma=0$, regardless of parameterizations of $\Sigma(a)$. The constraining power is already competitive, while a factor of 2 further improvement is expected for the upcoming full-sky galaxy surveys.