Testing Gravity Against Early Time Integrated Sachs-Wolfe Effect

A generic prediction of general relativity is that the cosmological linear density growth factor $D$ is scale independent. But in general, modified gravities do not preserve this signature. A scale dependent $D$ can cause time variation in gravitational potential at high redshifts and provides a new cosmological test of gravity, through early time integrated Sachs-Wolfe (ISW) effect-large scale structure (LSS) cross correlation. We demonstrate the power of this test for a class of $f(R)$ gravity, with the form $f(R)=-\lambda_1 H_0^2\exp(-R/\lambda_2H_0^2)$. Such $f(R)$ gravity, even with degenerate expansion history to $\Lambda$CDM, can produce detectable ISW effect at $z\ga 3$ and $l\ga 20$. Null-detection of such effect would constrain $\lambda_2$ to be $\lambda_2>1000$ at $>95%$ confidence level. On the other hand, robust detection of ISW-LSS cross correlation at high $z$ will severely challenge general relativity.