Confronting general relativity with further cosmological data

Deviations from general relativity in order to explain cosmic acceleration generically have both time and scale-dependent signatures in cosmological data. We extend our previous work by investigating model-independent gravitational deviations in bins of redshift and length scale, by incorporating further cosmological probes such as temperature-galaxy and galaxy-galaxy cross-correlations, and by examining correlations between deviations. Markov Chain Monte Carlo likelihood analysis of the model-independent parameters fitting current data indicates that at low redshift general relativity deviates from the best fit at the 99% confidence level. We trace this to two different properties of the CFHTLS weak lensing data set and demonstrate that COSMOS weak lensing data does not show such deviation. Upcoming galaxy survey data will greatly improve the ability to test time and scale-dependent extensions to gravity and we calculate the constraints that the BigBOSS galaxy redshift survey could enable.

Figure 1

1D marginalized probability of the post-GR parameter $\varpi$ in the redshift bin $1<z<2$. The narrower, dashed (black) distribution fixes the other post-GR parameter $\mu =1$, making it more difficult to fit the data (consistent with GR) by compensating one parameter with another. The wider, solid (red) distribution includes a simultaneous fit for $\mu$. All other cosmological parameters, including $\varpi$ and $\mu$ in the lower and higher redshift bins, are marginalized over.

Figure 2

2D joint probability contours at 95% C.L. between the post-GR functions $\varpi$ and $\mu$, for three independent redshift bins. Values within the redshift bins are consistent with each other and with GR [denoted by the cross at (0,0)]. The solid, black curve, motivated by a modified Poisson equation, closely follows the degeneracy direction and suggests a more insightful parametrization using variables along and perpendicular to the curve.

Figure 3

68% and 95% confidence limit contours for $\mathcal{V}-1$ and $\mathcal{G}-1$ are given for $2\times 2$ binning in redshift and $k$ space, using WMAP7 [13], Union2 [6], and CFHTLS [14] data. The diagonal, dotted-dashed line denotes values of $\mathcal{V}$ and $\mathcal{G}$ for which $\mu =0$ and gravity vanishes [see Eq. (3)]. The x’s denote GR values.

Figure 4

The square of the aperture mass (see Eq. 5 of 14) is plotted for different cosmological models in comparison to data from the CFHTLS survey. The solid, black curve shows the results from the $\Lambda \mathrm{CDM}$ concordance model in GR. One can match the small-angle behavior by suppressing growth through decreasing the gravitational coupling $\mathcal{G}$, but increasing growth by increasing ${\Omega }_m$. Exploring the larger angular scales, the dashed, red curve shows the effect of changing $\mathcal{G}(k>0.01{\mathrm{Mpc}}^{-1};z<1)$ while compensating ${\Omega }_m$. The dotted-dashed, green curve shows the case for $\mathcal{G}(k>0.1{\mathrm{Mpc}}^{-1};z<1)$ as taken in Fig. 8 of 9. Because this parametrization divides $k$ bins in the midst of the scales probed by the data, this curve fits better the (possibly spurious) bump in $⟨M_{ap}^2⟩$ seen in the data between $60\mathrm{arcmin}<\theta <180\mathrm{arcmin}$. Data is taken from Table B2 of 14.

Figure 5

A view of the data in Fig. 4 with a log scale in $\theta$ to zoom in on small angles. The theory curves use bins divided at $k_{\mathrm{bin}}=0.01{\mathrm{Mpc}}^{-1}$, and each is generated with identical background cosmology parameters, fixing the amplitude of the primordial scalar perturbations, so that different post-GR parameter values give different values of ${\sigma }_8$. Labeled values of $\mathcal{V}-1$ are set in the high $k$–high $z$ bin. Values of $\mathcal{G}-1$ in the high $k$–low $z$ bin are then given by the approximate degeneracy relation $\mathcal{G}-1=-0.2(\mathcal{V}-1)+0.06$. All other post-GR parameters are set to zero. To fit the rise at small angles, much steeper than in GR, requires very negative $\mathcal{V}$ and hence low ${\sigma }_8$. Even raising the primordial perturbation amplitude (dashed red curve) cannot bring ${\sigma }_8$ into the usual range. Values of ${\chi }^2$ reported in the legend are calculated naively assuming a diagonal covariance matrix using the error bars shown. The four smallest-scale data points are excluded from the ${\chi }^2$ calculation.

Figure 6

The 68% and 95% C.L. contours in ${\Omega }_m\mathrm{\text{−}}{\sigma }_8$ space for WMAP7, Union2, and CFHTLS data in the case of our post-GR parametrization (solid, green contour) and in the case of GR (dashed, blue contour). The inclusion of post-GR parameters seems to eliminate the degeneracy evident in the GR case and pulls the contours to lower values of ${\sigma }_8$. This is due to the influence of the steeply rising small-scale CFHTLS data, as illustrated in Fig. 5.

Figure 7

(Top) 95% C.L. contours in $\mathcal{V}\mathrm{\text{−}}\mathcal{G}$ space for the high $k$–high $z$ bin are compared for three different combinations of data sets. The solid (black) contour shows the results from Fig. 3c using CMB, supernovae, and weak lensing data. The thin-dashed (red) contour adds temperature-galaxy (Tg) cross-correlation data from 20. The thick-dashed (green) contour further adds galaxy-galaxy (gg) correlation data from 25. The diagonal, dotted-dashed line gives the $\mu =0$ boundary. As data sets are added, the contours close in on GR parameter values (the magenta x). Current galaxy correlation data is not yet sensitive enough though to put a meaningful constraint on $\mathcal{V}$. (Bottom) Addition of data sets can sometimes shift rather than tighten the contours, as shown here in $\mathcal{G}(1<z<2)\mathrm{\text{−}}\mathcal{G}(z<1)$ space for the low $k$ bin. Note that with the additional data GR now lies on the edge of the 95% C.L. region.

Figure 8

68% and 95% C.L. constraints on $\mathcal{V}-1$ and $\mathcal{G}-1$ are plotted for the two redshift and two wave number bins, using CMB, supernovae, weak lensing, Tg, and gg data. Foreground (blue) contours use CFHTLS weak lensing data. Background (yellow) contours use COSMOS weak lensing data. The dotted contours reproduce the 95% C.L. contours without Tg or gg data from Figs. 3. The diagonal, dotted-dashed line gives the $\mu =0$ boundary, from which the low $k$ contours at least have now pulled away. The x’s denote GR values. Both $k$ bins at low $z$ exhibit some preference for non-GR parameter values when using CFHTLS, but not when using COSMOS, weak lensing data.

Figure 9

68% and 95% C.L. constraints on the correlations of $\mathcal{V}-1$ (top) and $\mathcal{G}-1$ (bottom) between redshift bins, using CMB, supernovae, weak lensing (CFHTLS), Tg, and gg data. Contours are labeled according to $k$ binning. The x’s denote GR values. The high $k$ case of $\mathcal{V}$ exhibits a deviation from GR, corresponding to a different growth amplitude.

Figure 10

68% and 95% C.L. constraints on the correlations of $\mathcal{V}-1$ and $\mathcal{G}-1$ between redshift bins, using CMB, supernovae, weak lensing, Tg, and gg data. Substituting COSMOS (background) for CFHTLS (foreground) eliminates the apparent exclusion of GR. The x’s denote GR values. We do not plot the high $k$ bin correlations for $\mathcal{G}-1$ as there are no interesting correlations or deviations there.

Figure 11

68% and 95% C.L. constraints on $\mathcal{V}-1$ and $\mathcal{G}-1$ are plotted for the two redshift and two wave number bins using mock future data. Foreground (blue) contours use mock BigBOSS, Planck, and JDEM supernova data. Background (yellow) contours use only mock Planck and JDEM supernova data. The dotted contours recreate the 95% C.L. current data contours from Figs. 8 (using CFHTLS) to illustrate the expected improvement in constraints. The x’s denote the fiducial GR values.