Synergy between the gravitational potential decay rate and other structure growth probes in testing gravity

We test gravity by exploiting the synergy between the gravitational potential decay rate ($\mathit{DR}$) and complementary structure-growth probes: these observables respond to MG parameters with different degeneracy directions, so their combination yields stronger constraints than any single probe. We adopt the tomographic $\mathit{DR}$ measurements reported in \citep{2025ApJ...982...99D} and combine them with CMB-lensing-tomography $\Sigma_8$ constraints and $f\sigma_8$ measurements from DESI DR1 full-shape analyses and the DESI peculiar-velocity field. We apply this joint data vector to two representative frameworks: phenomenological parameterizations and the Effective Field Theory (EFT) $\alpha$-basis. For the phenomenological form $P_{\rm MG}(a)=1+P_{{\rm MG},0}\,\Omega_{\rm DE}(a)/\Omega_{\rm DE}(0)$, where $P_{\rm MG}$ denotes $\mu$, $\eta$, or $\Sigma$, we obtain $\mu_0=0.09\pm0.35$ and $\Sigma_0=0.01\pm0.06$. Compared to the measurements combination $\Sigma_8+f\sigma_8$, including $\mathit{DR}$ tightens the constraint on $\Sigma_0$ by a factor of $\sim2$. For the $(\mu_0,\eta_0)$ case we find $\mu_0=0.06^{+0.17}_{-0.23}$ and $\eta_0=-0.03^{+0.36}_{-0.46}$; relative to $\Sigma_8+f\sigma_8$, adding $\mathit{DR}$ improves the constraints on both parameters by a factor of $\sim1.5$. In the EFT $\alpha$-basis, adopting the parameterization $\alpha_i(a)=c_i\,\Omega_{\rm DE}(a)$ with $i\in\{{\rm M,B}\}$, we find $c_{\rm M}=0.64^{+0.32}_{-0.72}$ and $c{\rm B}=0.31^{+0.19}_{-0.29}$. The corresponding EFT uncertainties are about a factor of $\sim2$ smaller than those reported in \citep{2025JCAP...09..053I}, which combined DESI full-shape and BAO measurements with DES-SN5YR and CMB data. These results demonstrate the capability of $\mathit{DR}$ and the necessity of including the $\mathit{DR}$ measurements in testing gravity.