Probing Gravity at Cosmological Scales by Measurements which Test the Relationship between Gravitational Lensing and Matter Overdensity

The standard cosmology is based on general relativity (GR) and includes dark matter and dark energy and predicts a fixed relationship between the gravitational potentials responsible for gravitational lensing and the matter overdensity. Alternative theories of gravity often make different predictions. We propose a set of measurements which can test this relationship, thereby distinguishing between dark energy or matter models and models in which gravity differs from GR. Planned surveys will be able to measure $E_G$, an observational quantity whose expectation value is equal to the ratio of the Laplacian of the Newtonian potentials to the peculiar velocity divergence, to percent accuracy. This will easily separate alternatives such as the cold dark matter model with a cosmological constant, Dvali-Gabadadze-Porrati, TeVeS, and $f(R)$ gravity.

Figure 1

$E_G$ as a smoking gun of gravity. Error estimation is based on $\Lambda \mathrm{CDM}$ and error bars are centered on the $\Lambda \mathrm{CDM}$ prediction (black solid straight line). We only show those $k$ modes well in the linear regime. For clarity, we shift the error bars of $\mathrm{LAMOST}/\mathrm{AS}2+\mathrm{LSST}$ and $\mathrm{ADEPT}+\mathrm{LSST}$ slightly rightward. Irregularities in the error bars are caused by irregularities in the available discrete $\mathbf{k}$ modes of redshift distortion. Dotted lines are the results of a flat DGP model with ${\Omega }_0=0.2$. Dashed lines are for $f(R)=-{\lambda }_1{H}_0^2\exp (-R/{\lambda }_2{H}_0^2)$ with ${\lambda }_2=100$. Differences in expansion histories of these models are of percent level at $z<2$ and are not the main cause of differences in $E_G$. Solid lines with wiggles are for TeVeS with $K_B=0.08$, 0.09, 0.1, where the lines with most significant wiggles have $K_B=0.1$.