Scale-Dependent Gravity in Quantum Cosmology II: Phenomenological Consistency Tests from BBN, Structure Growth, and the CMB

We examine phenomenological consequences of a scale-dependent gravity model in which the gravitational coupling runs with the cosmological scale factor as G(a)=G0an. In the companion paper, this running was shown to modify the Friedmann equation through effective starred couplings and to produce corresponding changes in the Wheeler–DeWitt minisuperspace potential. For values of n, we ask whether the same running can survive into observationally tested epochs. Applying three consistency tests—the expansion rate during Big-Bang nucleosynthesis, the growth of linear matter perturbations, and the CMB acoustic scale—we find that BBN and CMB consistency require n to be very small if the power-law form persists to nucleosynthesis or recombination. These tests constrain the epoch over which the pure power-law form can remain valid. The power-law running is best considered as an early-universe scaling regime that must shut off before nucleosynthesis — the question is not whether, but when and how. Any realistic extension of the strong-running regime must cut off, screen, or smoothly transition the gravitational coupling back to standard gravity before the nucleosynthesis era.