Complementarity of weak lensing and peculiar velocity measurements in testing general relativity

We explore the complementarity of weak lensing and galaxy peculiar velocity measurements to better constrain modifications to General Relativity. We find no evidence for deviations from General Relativity on cosmological scales from a combination of peculiar velocity measurements (for Luminous Red Galaxies in the Sloan Digital Sky Survey) with weak lensing measurements (from the Canadian France Hawaii Telescope Legacy Survey). We provide a Fisher error forecast for a Euclid-like space-based survey including both lensing and peculiar velocity measurements and show that the expected constraints on modified gravity will be at least an order of magnitude better than with present data, i.e. we will obtain $\simeq 5\%$ errors on the modified gravity parametrization described here. We also present a model-independent method for constraining modified gravity parameters using tomographic peculiar velocity information, and apply this methodology to the present data set.

Figure 1

The constraints on ${\Sigma }_s$ and ${\mu }_s$ from the latest observational data; $s=1$ linear model on the left and the $s=3$ cubic model on the right. The inner dark-shaded contours are the 68% confidence region, while the outer lighter-shaded contours are the 95% confidence region. The horizontal green bands show the peculiar velocity constraints, while the magenta bands show the weak lensing constraints. The central blue contours show the constraints possible when the two data sets are analyzed together. The cross symbol shows the expected parameterization for General Relativity.

Figure 2

Contour plots of ${\Sigma }_s$ and ${\mu }_s$ calculated from our Fisher forecast analysis. The left panel shows the $s=1$ case, while the right panel shows the $s=3$ case. Solid curves represent the scenario in which all other cosmological parameters except ${\Sigma }_s$ and ${\mu }_s$ are fixed, while the dashed curves represent the scenario in which $w=-1$ is assumed, and the dotted curves represent the scenario where all the cosmological parameters are allowed to vary including $w$.

Figure 3

Constraints from PV data on $\mu$ as a function of redshift. We show the results of using 2 redshift bins, with $\mu ={\mu }_A$ within $[z_1,z_2]$, and $\mu ={\mu }_B$ within $[z_2,z_s]$. It is assumed that $\mu =1$ above $z_s=2$ (left) and above $z_s=10$ (right). In addition we display (grey box) results for constant $\mu$ below redshift $z_s=2$ (left) and below $z_s=10$ (right) when PV is combined with CMB shift parameter and SNe. Technically, the grey box could be extended to $z=0$, since the corresponding model assumes $\mu =\mathrm{const}$ within $[0,z_s]$ However, the measurements we used do not provide information about the value of $\mu$ at $z<0.25$. The lower panels show the linear combination of ${\mu }_A$ and ${\mu }_B$ that diagonalize the covariance matrix.