Testing gravity against the 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 H-2(0)2). Such f(R) gravity, even with degenerate expansion history to Lambda CDM, can produce detectable ISW effect at z greater than or similar to 3 and l greater than or similar to 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.