Do Newton’s $G$ and Milgrom’s $a_0$ vary with cosmological epoch?

In the scalar-tensor gravitational theories Newton’s constant $G_N$ evolves in the expanding universe. Likewise, it has been speculated that the acceleration scale ${\mathfrak{a}}_0$ in Milgrom’s modified Newtonian dynamics is tied to the scale of the cosmos, and must thus evolve. With the advent of relativistic implementations of the modified dynamics, one can address the issue of variability of the two gravitational “constants” with some confidence. Using TeVeS, the tensor-vector-scalar gravitational theory, as an implementation of Milgrom’s modified Newtonian dynamics, we calculate the dependence of $G_N$ and ${\mathfrak{a}}_0$ on the TeVeS parameters and the coeval cosmological value of its scalar field, ${\varphi }_c$. We find that $G_N$, when expressed in atomic units, is strictly nonevolving, a result fully consistent with recent empirical limits on the variation of $G_N$. By contrast, we find that ${\mathfrak{a}}_0$ depends on ${\varphi }_c$ and may thus vary with cosmological epoch. However, for the brand of TeVeS which seems most promising, ${\mathfrak{a}}_0$ variation occurs on a time scale much longer than Hubble’s, and should be imperceptible back to redshift unity or even beyond it. This is consistent with emergent data on the rotation curves of disk galaxies at significant redshifts.