Testing Gravity Against Early Time Integrated Sachs-Wolfe Effect

Abstract

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 $f(R)=-λ_1 H_0^2\exp(-R/λ_2H_0^2)$. Such $f(R)$ gravity, even with degenerate expansion history to $Λ$CDM, can produce detectable ISW effect at $z\ga 3$ and $l\ga 20$. Null-detection of such effect would constrain $λ_2$ to be $λ_2>1000$ at $>95%$ confidence level. On the other hand, robust detection of ISW-LSS cross correlation at high $z$ will severely challenge general relativity.

Figures (2)

• Figure 1: The $H(z)$-$z$ relation and structure growth in the exponential $f(R)$ gravity. Top left panel: $H$-$z$. $\lambda_2\rightarrow \infty$ corresponds to $\Lambda$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 $k=0.01h/$Mpc. Bottom left panel: $f_R(a)$, which determines the effective Newton's constant $G_{\rm eff}=G/(1+f_R)$. 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 $D\rightarrow a$ when $a\rightarrow 0$.
• 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 $D\rightarrow a$ when $a\rightarrow 0$. Bottom left panel: the ISW effect. Bottom right panel: Cumulative S/N of the ISW-LSS cross correlation measurements.