Consistency of modified gravity with a decreasing Geff (z) in a ΛcDM background

Recent analyses [S. Nesseris et al., Phys. Rev. D 96, 023543 (2017)PRVDAQ2470-001010.1103/PhysRevD.96.023543; L. Kazantzidis and L. Pervolaropoulos, Phys. Rev. D 97, 103503 (2018)PRVDAQ2470-001010.1103/PhysRevD.97.103503] have indicated that an effective Newton's constant Geff(z) decreasing with redshift may relieve the observed tension between the Planck15 best fit ΛCDM cosmological background (i.e., Planck15/ΛCDM) and the corresponding ΛCDM background favored by growth fσ8 and weak lensing data. We investigate the consistency of such a decreasing Geff(z) with some viable scalar-tensor models and f(R) theories. We stress that f(R) theories generically cannot lead to a decreasing Geff(z) for any cosmological background. For scalar-tensor models we deduce that in the context of a ΛCDM cosmological background, a decreasing Geff(z) is not consistent with a large Brans-Dicke parameter ωBD,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 Geff(z) and Geff,0=G would have a characteristic signature in its growth index γ with 0.61γ00.69 and large slopes γ0′, 0.16γ0′0.4, which is a characteristic signature of a decreasing (with z) Geff(z)<G on small redshifts. This is a substantial departure today from the quasistatic behavior in ΛCDM with (γ0,γ0′)≈(0.55,-0.02).