Consistency of Modified Gravity with a decreasing $G_{\rm eff}(z)$ in a $Λ$CDM background

Abstract

Recent analyses \cite{Nesseris:2017vor,Kazantzidis:2018rnb} have indicated that an effective Newton's constant $G_{\rm eff}(z)$ decreasing with redshift may relieve the observed tension between the Planck15 best fit $Λ$CDM cosmological background ({\it i.e.} Planck15/$Λ$CDM) and the corresponding $Λ$CDM background favored by growth $fσ_8$ and weak lensing data. We investigate the consistency of such a decreasing $G_{\rm eff}(z)$ with some viable scalar-tensor models and $f(R)$ theories. We stress that $f(R)$ theories generically can not lead to a decreasing $G_{\rm eff}(z)$ for any cosmological background. For scalar-tensor models we deduce that in the context of a $Λ$CDM cosmological background, a decreasing $G_{\rm eff}(z)$ is not consistent with a large Brans-Dicke parameter $ω_{BD,0}$ today. This inconsistency remains and amplifies in the presence of a phantom dark energy equation of state parameter ($w < -1$). However it can be avoided for $w >-1$. We also find that any modified gravity model with the required decreasing $G_{\rm eff}(z)$ and $G_{{\rm eff},0}=G$, would have a characteristic signature in its growth index $γ$ with $0.61\lesssim γ_0\lesssim 0.69$ and large slopes $γ_0'$, $0.16\lesssim γ_0'\lesssim 0.4$, which is a characteristic signature of a decreasing (with $z$) $G_{\rm eff}(z)<G$ on small redshifts. This is a substantial departure today from the quasi-static behaviour in $Λ$CDM with $(γ_0,γ_0')\approx (0.55,-0.02)$.