Cosmologists are exploring two possible sets of explanations for the remarkable observation of cosmic acceleration: dark energy fills space or general relativity fails on cosmological scales. We define a null test parameter $ϵ(k,a)\equiv {\Omega }_m^{-\gamma }d\ln D/d\ln a-1$, where $a$ is the scale factor, $D$ is the growth rate of structure, ${\Omega }_m(a)$ is the matter density parameter, and $\gamma$ is a simple function of redshift. We show that it can be expressed entirely in terms of the bias factor, $b(a)$, measured from cross correlations with cosmic microwave background (CMB) lensing, and the amplitude of redshift-space distortions, $\beta (k,a)$. Measurements of the CMB power spectrum determine ${\Omega }_{m0}{H}_0^2$. If dark energy within general relativity is the solution to the cosmic acceleration problem, then the logarithmic growth rate of structure $d\ln D/d\ln a={\Omega }_m^{\gamma }$. Thus, $ϵ(k,a)=0$ on linear scales to better than 1%. We show that in the class of modified gravity models known as $f(R)$, the growth rate has a different dependence on scale and redshift. By combining measurements of the amplitude of $\beta$ and of the bias, $b$, redshift surveys will be able to determine the logarithmic growth rate as a function of scale and redshift. We estimate the predicted sensitivity of the proposed SDSS III (BOSS) survey and the proposed ADEPT mission and find that they will test structure growth in general relativity to the percent level.

• Received 14 March 2008

DOI:https://doi.org/10.1103/PhysRevD.78.043514