Testing general relativity at cosmological scales: Implementation and parameter correlations

The testing of general relativity at cosmological scales has become a possible and timely endeavor that is not only motivated by the pressing question of cosmic acceleration but also by the proposals of some extensions to general relativity that would manifest themselves at large scales of distance. We analyze here correlations between modified gravity growth parameters and some core cosmological parameters using the latest cosmological data sets including the refined Cosmic Evolution Survey 3D weak lensing. We provide the parametrized modified growth equations and their evolution. We implement known functional and binning approaches, and propose a new hybrid approach to evolve the modified gravity parameters in redshift (time) and scale. The hybrid parametrization combines a binned redshift dependence and a smooth evolution in scale avoiding a jump in the matter power spectrum. The formalism developed to test the consistency of current and future data with general relativity is implemented in a package that we make publicly available and call ISiTGR (Integrated Software in Testing General Relativity), an integrated set of modified modules for the publicly available packages CosmoMC and CAMB, including a modified version of the integrated Sachs-Wolfe-galaxy cross correlation module of Ho et al. and a new weak-lensing likelihood module for the refined Hubble Space Telescope Cosmic Evolution Survey weak gravitational lensing tomography data. We obtain parameter constraints and correlation coefficients finding that modified gravity parameters are significantly correlated with ${\sigma }_8$ and mildly correlated with ${\Omega }_m$, for all evolution methods. The degeneracies between ${\sigma }_8$ and modified gravity parameters are found to be substantial for the functional form and also for some specific bins in the hybrid and binned methods indicating that these degeneracies will need to be taken into consideration when using future high precision data.

Figure 1

MG parameter evolution in redshift and scale modeled using a new hybrid method. We plot here a 3D representation for an example of the new hybrid binned evolution for the MG parameter $Q(k,a)$ as given by our Eqs. (12, 13) for the parameters $Q(k,a)$ with $Q_1=1.20$, $Q_2=1.15$, $Q_3=1.05$, $Q_4=1.10$, $z_{\mathrm{TGR}}=2$, and $k_c=0.01$. We can see along the $z$ axis how the binned aspect can allow for different best fit values for the MG parameters in the redshift space, while along the $k$ axis we can see the monotonic evolution in $k$ evolving from some large scale (small $k$) value to a small scale (large $k$) value exponentially. The hybrid parametrization combines the $z$-binning method that was shown to be robust with a smooth evolution in $k$ space.

Figure 2

We compare the effect the evolution of the MG parameters has on the matter power spectrum using both the binning and the new hybrid methods. The solid black line is the matter power spectrum produced using the best fit WMAP7 parameters. The blue dashed line was produced using the traditional binning method, and the red dash-dotted line was produced using the new hybrid method. The hybrid method produces a more realistic, smooth matter power spectrum and thus is physically well motivated. These plots were produced using parameters $Q_1=1.20$, $Q_2=0.88$, $Q_3=1.52$, $Q_4=0.95$, ${\mathcal{D}}_1=1.03$, ${\mathcal{D}}_2=0.95$, ${\mathcal{D}}_3=0.99$, ${\mathcal{D}}_4=0.93$, $z_{\mathrm{TGR}}=2$, and $k_c=0.01$. The amplitudes of the two modified matter power spectra are normalized to the GR amplitude at $k\sim 7\times {10}^{-2}$.

Figure 3

We plot the 68% and 95% C. L. constraints on the MG parameters $Q_0$, $R_0$, and ${\mathcal{D}}_0$ from evolving the parameters with a functional form. We use all available data sets included in ISiTGR: SN, BAO, AGE, $H_0$, CMB, MPK, ISW, and WL. All constraints using this method are fully consistent with GR; however, as we show further, the correlations between these parameters and some cosmological parameters are significant.

Figure 4

Left panel: We plot the 68% and 95% C. L. constraints on the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 from using traditional bins for $k$ and $z$. $z$ bins are $0<z\le 1$ and $1<z\le 2$ with GR assumed for $z>2$ and $k$ bins are $k\le 0.01$ and $k>0.01$. These constraints come from using all available data sets included in ISiTGR: SN, BAO, AGE, $H_0$, CMB, MPK, ISW, and WL. While all constraints are consistent with GR, bin 2 seems to be indicating some tensions with GR values just within the 95% C. L. Right panel: We plot the 68% and 95% C. L. constraints on the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 from using the new hybrid binning method. $z$ bins are still $0<z\le 1$ and $1<z\le 2$ with GR assumed for $z>2$, while the transition scale for $k$ evolution is $k_c=0.01$. Again, we use all available data sets included in ISiTGR: SN, BAO, AGE, $H_0$, CMB, MPK, ISW, and WL. All constraints using this method are fully consistent with GR but as we show further, the correlations between these parameters and some cosmological parameters are significant.

Figure 5

We plot here the 2D confidence contours for ${\Omega }_m$ and ${\sigma }_8$ 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 most amount of correlations between the MG parameters and the two cosmological parameters.

Figure 6

We plot here the 2D confidence contours for ${\Omega }_m$ and ${\sigma }_8$ 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 least amount of correlation between the MG parameters and the two cosmological parameters of the three evolution methods, but as discussed earlier suffers from the appearance of a jump in the matter power spectrum due to the rapid transition between scale bins.

Figure 7

We plot here the 2D confidence contours for ${\Omega }_m$ and ${\sigma }_8$ and the MG parameters $Q_i$ and ${\mathcal{D}}_i$, $i=1$, 2, 3, 4 from using the new hybrid method to evolve the MG parameters. As seen in Table  this evolution method has only a moderate amount of correlation between the MG parameters and the two cosmological parameters.