Constraints on the fractional changes of the fundamental constants at a look-back time of 2.5 Myrs

Abstract: The quantum nature of gravity remains one of the greatest mysteries of modern physics, with many unified theories predicting variations in fundamental constants across space and time. Here we present precise measurements of these variations at galactic dynamical timescales - a critical but previously unexplored regime. Using simultaneous observations of H \textsc{i} and OH lines in M31, we probe potential variations of fundamental constants at a lookback time of 2.5 million years. We obtained $\Delta(\mu\alpha^2g_p^{0.64})/(\mu\alpha^2g_p^{0.64}) < 3.6 \times 10^{-6}$, with complementary constraints on $\Delta(\mu\alpha^2)/(\mu\alpha^2) < 4.6 \times 10^{-3}$, and $\Delta g_p/g_p < 7.2 \times 10^{-3}$, where $\alpha$ is the fine structure constant, $\mu$ is the proton-electron mass ratio, and $g_p$ is the proton $g$-factor. These results bridge the gap between laboratory tests and cosmological observations, providing unique insights into the coupling between local dynamics and fundamental physics. Our findings challenge theories predicting significant variations over galactic timescales, while demonstrating a powerful new probe of quantum gravity models.