Weak gravity on a $\Lambda$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 $\Lambda$CDM background expansion and a monotonic evolution on the cosmic background of the $\alpha$ functions as ${\alpha }_i={\alpha }_{i0}{a}^s$ where $i=M,B$, $a$ is the scale factor and ${\alpha }_{i0}$ (${\alpha }_{M0},{\alpha }_{B0}$), $s$ are arbitrary parameters. We show that the growth and lensing reduced (dimensionless) gravitational couplings $\mu \equiv G_{\mathrm{growth}}/G$, $\Sigma \equiv G_{\mathrm{lensing}}/G$ exhibit the following generic properties today: ${\Sigma }_0<1$ for all viable parameters, ${\mu }_0<1$ (weak gravity today) is favored for small $s$ while ${\mu }_0>1$ is favored for large $s$. We establish also the relation $\mu \ge \Sigma$ at all times. Taking into account the $f{\sigma }_8$ and $E_G$ data constrains the parameter $s$ to satisfy $s\lesssim 2$. Hence these data select essentially the weak gravity regime today (${\mu }_0<1$) when $s<2$, while ${\mu }_0>1$ 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 $\gamma (z)$ and identify the $({\alpha }_{M0},{\alpha }_{B0},s)$ parameter region that corresponds to specific signs of the differences ${\gamma }_0-{\gamma }_0^{\Lambda CDM}$, and ${\gamma }_1-{\gamma }_1^{\Lambda CDM}$, where ${\gamma }_0\equiv \gamma {|}_{z=0}$ and ${\gamma }_1\equiv \frac{\mathrm{d}\gamma }{\mathrm{dz}}{|}_{z=0}$. In this way important information is gained on the past evolution of $\mu$. We obtain in particular the signature ${\gamma }_0>{\gamma }_0^{\Lambda CDM}$ for $s<2$ in the selected weak gravity region.

14 pages, 2 figures

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