A Test of the Standard Cosmological Model with Geometry and Growth

We perform a general test of the $\Lambda{\rm CDM}$ and $w {\rm CDM}$ cosmological models by comparing constraints on the geometry of the expansion history to those on the growth of structure. Specifically, we split the total matter energy density, $\Omega_M$, and (for $w {\rm CDM}$) dark energy equation of state, $w$, into two parameters each: one that captures the geometry, and another that captures the growth. We constrain our split models using current cosmological data, including type Ia supernovae, baryon acoustic oscillations, redshift space distortions, gravitational lensing, and cosmic microwave background (CMB) anisotropies. We focus on two tasks: (i) constraining deviations from the standard model, captured by the parameters $\Delta\Omega_M \equiv \Omega_M^{\rm grow}-\Omega_M^{\rm geom}$ and $\Delta w \equiv w^{\rm grow}-w^{\rm geom}$, and (ii) investigating whether the $S_8$ tension between the CMB and weak lensing can be translated into a tension between geometry and growth, i.e. $\Delta\Omega_M \neq 0$, $\Delta w \neq 0$. In both the split $\Lambda{\rm CDM}$ and $w {\rm CDM}$ cases, our results from combining all data are consistent with $\Delta\Omega_M = 0$ and $\Delta w = 0$. If we omit BAO/RSD data and constrain the split $w {\rm CDM}$ cosmology, we find the data prefers $\Delta w<0$ at $3.6\sigma$ significance and $\Delta\Omega_M>0$ at $4.2\sigma$ evidence. We also find that for both CMB and weak lensing, $\Delta\Omega_M$ and $S_8$ are correlated, with CMB showing a slightly stronger correlation. The general broadening of the contours in our extended model does alleviate the $S_8$ tension, but the allowed nonzero values of $\Delta\Omega_M$ do not encompass the $S_8$ values that would point toward a mismatch between geometry and growth as the origin of the tension.