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. While current data taken in the context of the $\Lambda \mathrm{CDM}$ model points to a universe that is spatially flat, this constraint does not necessarily hold in modified gravity theories or even in relativistic inhomogeneous cosmological models. 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 $|{\Omega }_k|\ge 0.02$ and apparent deviations appear for $|{\Omega }_k|\ge 0.05$. 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.

Figure 1

We plot the 2D confidence contours for ${\Omega }_k$ and the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 from using traditional bins for $k$ and $z$. As seen in Table  this evolution method has the most correlation between the MG parameters and ${\Omega }_k$ of the three evolution methods.

Figure 2

We plot the 2D confidence contours for ${\Omega }_k$ and the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 from the hybrid method to evolve the MG parameters. As seen in Table  this evolution method has a moderate amount of correlation between the MG parameters and ${\Omega }_k$.

Figure 3

We plot the 2D confidence contours for ${\Omega }_k$ and the MG parameters $Q_0$ and ${\mathcal{D}}_0$, from using the functional form to evolve the MG parameters. As seen in Table  this evolution method overall has the least amount of correlations between the MG parameters and the ${\Omega }_k$ of the three evolution methods.

Figure 4

Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used but the actual underlying universe has ${\Omega }_k=0.01$. TOP: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the traditional binning method is used. All parameter values are pulled to smaller values. MIDDLE: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the hybrid evolution method is used. Most of the parameter contours are pulled to smaller values. BOTTOM: Confidence contours for the MG parameters $Q_0$ and ${\mathcal{D}}_0$ and $R_0$ when the functional form binning method is used. The $Q_0-{\mathcal{D}}_0$ contour is pulled noticeably toward smaller parameter values.

Figure 5

Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used but the actual underlying universe has ${\Omega }_k=-0.02$. TOP: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the traditional binning method is used. All parameter values are pulled to larger values. MIDDLE: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the hybrid evolution method is used. Most of the parameter contours are pulled to larger values. BOTTOM: Confidence contours for the MG parameters $Q_0$ and ${\mathcal{D}}_0$ and $R_0$ when the functional form binning method is used. The $Q_0-{\mathcal{D}}_0$ contour is pulled noticeably toward larger parameter values.

Figure 6

Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used but the actual underlying universe has ${\Omega }_k=0.05$. TOP: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the traditional binning method is used. All parameter values are pulled to smaller values and indicate a deviation from general relativity. MIDDLE: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the hybrid evolution method is used. Most of the parameter contours are pulled to smaller values and indicate a deviation from general relativity. BOTTOM: Confidence contours for the MG parameters $Q_0$ and ${\mathcal{D}}_0$ and $R_0$ when the functional form binning method is used. A strong deviation (due to the assumption of spatial flatness) from general relativity is present in both contours.

Figure 7

Plotted are the 68% and 95% 2D confidence contours for the MG parameters when a spatially flat model is used but the actual underlying universe has ${\Omega }_k=-0.1$. TOP: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the traditional binning method is used. All parameter values are pulled to larger values and indicate a deviation from general relativity. MIDDLE: Confidence contours for the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1,\dots ,4$ when the hybrid evolution method is used. Most of the parameter contours are pulled to larger values and indicate a deviation from general relativity. BOTTOM: Confidence contours for the MG parameters $Q_0$ and ${\mathcal{D}}_0$ and $R_0$ when the functional form binning method is used. The $Q_0-{\mathcal{D}}_0$ shows a deviation from general relativity, while the $Q_0-R_0$ contour shows some tension with the GR point.

Figure 8

Here we plot the best-fit points for the parameters $Q$ and $\mathcal{D}$ for the various evolution methods versus the underlying ${\Omega }_k$ fiducial values while a spatially flat universe is assumed. Most of the trends have a negative slope, which is expected from looking at Eqs. (8, 10) (note that this trend is consistent with the positive correlations between the MG parameters and the curvature parameter as explained in Sec. 4c). TOP LEFT: Plots for the parameters $Q_i$, $i=1$, 2, 3, 4 for the traditional binning method. TOP MIDDLE: Plots for the parameters $Q_i$, $i=1$, 2, 3, 4 for the hybrid evolution method. TOP RIGHT: Plots for the parameters $Q_0$ for the functional form evolution method. BOTTOM LEFT: Plots for the parameters ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 for the traditional binning method. BOTTOM MIDDLE: Plots for the parameters ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 for the hybrid evolution method. BOTTOM RIGHT: Plots for the parameters ${\mathcal{D}}_0$ for the functional form evolution method.