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

Abstract

Recent analyses \cite{Nesseris:2017vor,Kazantzidis:2018rnb} have indicated that an effective Newton's constant $G_{\mathrm{eff}}(z)$ decreasing with redshift may relieve the observed tension between the Planck15 best fit $\Lambda$CDM cosmological background ({\it i.e.} Planck15/$\Lambda$CDM) and the corresponding $\Lambda$CDM background favored by growth $f{\sigma }_8$ and weak lensing data. We investigate the consistency of such a decreasing $G_{\mathrm{eff}}(z)$ with some viable scalar-tensor models and $f(R)$ theories. We stress that $f(R)$ theories generically can not lead to a decreasing $G_{\mathrm{eff}}(z)$ for any cosmological background. For scalar-tensor models we deduce that in the context of a $\Lambda$CDM cosmological background, a decreasing $G_{\mathrm{eff}}(z)$ is not consistent with a large Brans-Dicke parameter ${\omega }_{BD,0}$ 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 $G_{\mathrm{eff}}(z)$ and $G_{\mathrm{eff},0}=G$, would have a characteristic signature in its growth index $\gamma$ with $0.61\lesssim {\gamma }_0\lesssim 0.69$ and large slopes ${\gamma }_0^{\prime }$, $0.16\lesssim {\gamma }_0^{\prime }\lesssim 0.4$, which is a characteristic signature of a decreasing (with $z$) $G_{\mathrm{eff}}(z)<G$ on small redshifts. This is a substantial departure today from the quasi-static behaviour in $\Lambda$CDM with $({\gamma }_0,{\gamma }_0^{\prime })\approx (0.55,-0.02)$.

10 pages, 3 figures

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