Weak gravity on a $\Lambda$CDM background
  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 $\mu\equiv G_{\rm growth}/G$, $\Sigma\equiv G_{\rm lensing}/G$
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 $\mu\geq
\Sigma$ 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 $\gamma_0\equiv \gamma\bigl|_{z=0}$
and $\gamma_1\equiv \frac{{\rm d}\gamma}{\rm d z}\bigl|_{z=0}$. 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.
