Joint Curvature and Growth Rate Measurements with Supernova Peculiar Velocities and the CMB

Type Ia supernova (SN) magnitudes present correlations due to the fact that their peculiar velocities are sourced by the large-scale structure of the Universe. This can be used to constrain properties related to the distribution and growth of matter perturbations. We analyze both Pantheon+ and Dark Energy Survey (DES-Y5) SN catalogues in combination with CMB data from Planck PR4 to constrain $\sigma_8$, optionally including both curvature and a modified growth index $\gamma$. We show that SN and CMB datasets are highly complementary and capable of measuring $\sigma_8$, $\gamma$ and $\Omega_k$ simultaneously.  Using only SN, we find $\sigma_8 = 0.73 \pm 0.22$ ($0.87 \pm 0.31$) for Pantheon+ (DES-Y5) for the flat $\Lambda$CDM model. Interestingly, allowing for free $\gamma$ and $\Omega_k$, we find hints of positive curvature: $\Omega_k = -0.011 \pm 0.006$ $(-0.014 \pm 0.005)$, which exclude flatness at 2.2$\sigma$ (3.0$\sigma$), for the combination of CMB with Pantheon+ (DES-Y5). Such hints do not degrade if we also include a modified amplitude of CMB lensing $A_L$. We find that $\gamma = 0.519^{+0.061}_{-0.099}$ ($0.461^{+0.085}_{-0.069}$), consistent with the predictions of General Relativity. In terms of $f\sigma_8(z)$, we find $f\sigma_8(0.024)=0.461^{+0.066}_{-0.035}$ ($f\sigma_8(0.038) = 0.498^{+0.045}_{-0.050}$) for CMB + Pantheon+ (DES-Y5). Finally, the strong degeneracy between all three $\Omega_k$, $\gamma$ and $H_0$ results in a broader CMB $H_0$ posterior. However, if we include SH0ES $H_0$ data, which is in known strong tension with the CMB in flat $\Lambda$CDM, we find that the $H_0$ tension is recast in terms of a significantly negative curvature and suppressed growth of structures.