Spatial curvature and cosmological tests of general relativity

It is well-known that allowing for spatial curvature affects constraints on cosmological parameters such as the dark energy equation of state parameters. Here we study the effect of curvature on constraints on parameters used to test general relativity (GR) at cosmological scales, commonly known as modified growth (MG) parameters, as while current data taken in the context of the $\Lambda$CDM model points to a universe that is flat or very close to it, this constraint may not hold in modified theories of gravity. Using the latest cosmological data sets we find that MG parameters are correlated with the curvature parameter $\Omega_k$ and the constraints on the MG parameters are weakened compared to when $\Omega_k$ is not included in the parameter analysis. We next use various future simulated data sets, including cosmic microwave background, weak lensing, and Integrated Sachs-Wolfe-galaxy cross-correlations, where the fiducial model is spatially curved but we assume a flat model when fitting the MG parameters. We find the assumption of a spatially flat model on a spatially curved universe does indeed cause an artificial shift in the constraints on the MG parameters, in some cases even producing an apparent deviation from GR in the MG parameter space. For our simulated data, tension with GR begins to manifest itself for fiducial models with $\abs{\Omega_k} \geq 0.02$ and apparent deviations appear for $\abs{\Omega_k} \geq 0.05$. We find that for negatively curved models the apparent deviation is more significant. The manifestation of this apparent deviation from GR due to the assumption of spatial flatness above leads one to conclude that, when using future high-precision data to perform these tests, spatial curvature must be included in the parameter analysis along with the other core cosmological parameters and the MG parameters.