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 $f(R)=-{\lambda }_1{H}_0^2\exp (-R/{\lambda }_2{H}_0^2)$. Such $f(R)$ gravity, even with degenerate expansion history to $\Lambda \mathrm{CDM}$, can produce detectable ISW effect at $z\gtrsim 3$ and $l\gtrsim 20$. 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.

Figure 1

The $H(z)\mathrm{\text{−}}z$ relation and structure growth in the exponential $f(R)$ gravity. Top left panel: $H\mathrm{\text{−}}z$. ${\lambda }_2\to \infty$ corresponds to $\Lambda \mathrm{CDM}$ cosmology. Top right panel: $Q(k,a)\propto k^2$, which describes the main effect of $f(R)$ gravity to structure formation. We plot the result of $k=0.01\mathrm{h}/\mathrm{Mpc}$. Bottom left panel: $f_R(a)$, which determines the effective Newton’s constant $G_{\mathrm{eff}}=G/(1+f_R)$. For ${\lambda }_2\gtrsim 100$, its effect to structure formation can be neglected. Bottom right panel: $D(k,a)/a$ (${\lambda }_2=1000$), where the linear density growth factor $D$ is normalized such that $D\to a$ when $a\to 0$.

Figure 2

The ISW effect. ${\lambda }_2=1000$ is adopted. Top left panel: $D/a-dD/da$, which determines the sign and amplitude of the ISW effect. $D$ is normalized such that $D\to a$ when $a\to 0$. Bottom left panel: the ISW effect. Bottom right panel: Cumulative $S/N$ of the ISW-LSS cross correlation measurements.