Testing general relativity with current cosmological data

Deviations from general relativity, such as could be responsible for the cosmic acceleration, would influence the growth of large-scale structure and the deflection of light by that structure. We clarify the relations between several different model-independent approaches to deviations from general relativity appearing in the literature, devising a translation table. We examine current constraints on such deviations, using weak gravitational lensing data of the CFHTLS and COSMOS surveys, cosmic microwave background radiation data of WMAP5, and supernova distance data of Union2. A Markov chain Monte Carlo likelihood analysis of the parameters over various redshift ranges yields consistency with general relativity at the 95% confidence level.

Figure 1

CMB anisotropy spectra are plotted as a function of the parameters ${\varpi }_0$ and ${\mu }_0$ in Eqs. (18). As in 34, the post-GR effects all occur in the low-$\ell$ multipoles. The CMB anisotropy is more sensitive to variations in ${\mu }_0$ than to variations in ${\varpi }_0$. See Fig. 2 for more on this point and on varying ${\varpi }_0$ and ${\mu }_0$ simultaneously.

Figure 2

The change in quadrupole power relative to the value in GR is plotted as a function of ${\varpi }_0$ and ${\mu }_0$. The blue, dot-dashed curve shows the effects of varying ${\varpi }_0$ with fixed ${\mu }_0=0$. The red, dashed curve shows the effects of varying ${\mu }_0$ with fixed ${\varpi }_0=0$. One can mimic the unmodified GR CMB spectrum over a much wider range of post-GR parameter values by simultaneously varying ${\varpi }_0$ and ${\mu }_0$ in opposite directions, as shown in the black, solid curve using ${\mu }_0=2/(2+{\varpi }_0)-1$. The horizontal dotted line denotes perfect agreement with GR.

Figure 3

We plot the matter power spectrum (normalized to $k=1{\mathrm{Mpc}}^{-1}$) generated by varying the parameters ${\varpi }_0$ and ${\mu }_0$. Unlike under $\varpi \mathrm{CDM}$ [34], even our scale-independent parametrization has scale-dependent effects due to the $k^2$ factor in the Poisson equation. The bottom panel shows the residuals of the top panel, i.e. the deviation relative to GR when varying ${\varpi }_0$ (the ${\mu }_0$ case looks similar), to highlight the scale-dependent regime at low-$k$ and scale-independent regime at high-$k$.

Figure 4

We plot the ratio of the $E$ mode of the weak lensing shear two-point correlation function (Eq. 8 of 4) to the same statistic calculated in GR, with all parameters but either ${\varpi }_0$ or ${\mu }_0$ fixed, to see the influence of the non-GR parameters. For the most part, post-GR parameters serve to renormalize the correlation function. As with the CMB anisotropy and matter power spectra, the effect is more sensitive to changes in ${\mu }_0$ than to changes in ${\varpi }_0$.

Figure 5

Marginalized probabilities of the post-GR parameters ${\varpi }_{0a,b,c}$ defined in high, medium, and low redshift bins, respectively. The parameter $\mu$ has been fixed to $\mu =1$, consistent with general relativity. Green (dot-dashed) curves are constraints determined from the WMAP 5 yr [57] and supernova Union2 [60] data sets only. Red (dashed) curves also include the COSMOS weak lensing tomography data [3]. Black (solid) curves use measurements of the aperture mass taken from the CFHTLS weak lensing survey [4] in addition to COSMOS, WMAP5, and Union2.

Figure 6

Marginalized probabilities of the post-GR parameters ${\mu }_{0a,b,c}$ defined in high, medium, and low redshift bins, respectively. The parameter $\varpi$ has been fixed to $\varpi =0$, consistent with general relativity. All curves show constraints derived using data from the WMAP 5 yr release [57], supernova Union2 set [60], and COSMOS [3], plus CFHTLS [4] weak lensing data.

Figure 7

Confidence contours of 68% (inner) and 95% (outer) for modified gravity parameters are plotted in the ${\varpi }_0\mathrm{\text{−}}{\mu }_0$ plane (all other parameters marginalized), where ${\varpi }_0$ and ${\mu }_0$ are defined as in Eq. (18). Blue (background) contours use the WMAP 5 yr, supernova Union2, and COSMOS data sets. Red (foreground) contours use WMAP 5 yr, supernova Union2, COSMOS, and CFHTLS data. The black curve plots the degeneracy direction ${\mu }_0=2/(2+{\varpi }_0)-1$ from Fig. 2. The yellow x denotes GR parameter values.