Weak gravity on a $\hat{I}›$CDM background

Abstract

We consider Horndeski modified gravity models obeying stability, velocity of gravitational waves $c_T$ equals $c$ and quasistatic approximation (QSA) on subhorizon scales. We assume further a $\hat{I}›$CDM background expansion and a monotonic evolution on the cosmic background of the $\hat{I}\pm$ functions as $\hat{I}{\pm }_i=\hat{I}{\pm }_{i0}{a}^s$ where $i=M,B$, $a$ is the scale factor and $\hat{I}{\pm }_{i0}$ ($\hat{I}{\pm }_{M0},\hat{I}{\pm }_{B0}$), $s$ are arbitrary parameters. We show that the growth and lensing reduced (dimensionless) gravitational couplings $\hat{I}¼\equiv G_{\mathrm{growth}}/G$, $\hat{I}\pounds \equiv G_{\mathrm{lensing}}/G$ exhibit the following generic properties today: $\hat{I}{\pounds }_0<1$ for all viable parameters, $\hat{I}{¼}_0<1$ (weak gravity today) is favored for small $s$ while $\hat{I}{¼}_0>1$ is favored for large $s$. We establish also the relation $\hat{I}¼\ge \hat{I}\pounds$ at all times. Taking into account the $f\ddot{I}{ƒ}_8$ and $E_G$ data constrains the parameter $s$ to satisfy $s\lesssim 2$. Hence these data select essentially the weak gravity regime today ($\hat{I}{¼}_0<1$) when $s<2$, while $\hat{I}{¼}_0>1$ subsists only marginally for $s\approx 2$. At least the interval $0.5\lesssim s\lesssim 2$ would be ruled out in the absence of screening. We consider further the growth index ${\hat{I}}^3(z)$ and identify the $(\hat{I}{\pm }_{M0},\hat{I}{\pm }_{B0},s)$ parameter region that corresponds to specific signs of the differences $γ_0-γ_0^{ΛCDM}$, and $γ_1-γ_1^{ΛCDM}$, where ${\hat{I}}_0^3\equiv {\hat{I}}^3{|}_{z=0}$ and ${\hat{I}}_1^3\equiv \frac{\mathrm{d}{\hat{I}}^3}{\mathrm{dz}}{|}_{z=0}$. In this way important information is gained on the past evolution of $\hat{I}¼$. We obtain in particular the signature $γ_0>γ_0^{ΛCDM}$ for $s<2$ in the selected weak gravity region.

14 pages, 2 figures