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)=-λ₁H₀²exp(-R/λ₂H₀²). Such $f(R)$ gravity, even with degenerate expansion history to ΛCDM, can produce detectable ISW effect at z3 and l20. Null-detection of such effect would constrain λ₂ to be λ₂>1000 at $>95%$ confidence level. On the other hand, robust detection of ISW-LSS cross correlation at high z will severely challenge general relativity.