The Milky Way and M31 rotation curves in Yukawa gravity: phenomenology and Bayesian analysis

Abstract: Yukawa gravity provides a generalized framework for modeling gravity modification. We investigate the rotation curve profiles of spiral galaxies under Yukawa-like theories governed by the coupling strength $\beta$ and the interaction range $\lambda$. We develop a unified analytical and numerical framework to calculate rotational velocities under Yukawa gravity, which includes contributions from all major galactic components: stellar bulge, disk, dark matter (DM) halo, and central supermassive black hole. The calculations show that $\beta$ and $\lambda$ strongly influence velocity distributions, shifting peaks, creating double-peak structures, or enhancing dark matter dominance in the bulge or disk. To assess observational implications, we perform Bayesian analyses using data from the Milky Way (MW) and Andromeda (M31), which offer complementary characteristics: MW provides precise velocity profiles across multiple scales, while M31 includes broader morphological constraints. We examine four scenarios: Yukawa gravity without dark matter, dark matter with non-trivial coupling, fully modified gravity, and standard Newtonian gravity. Results show that MW models with $\lambda < 1$ kpc yield high Bayes factors but risk overfitting, as dark matter mimics baryonic kinematics, while M31's photometric priors from conjugate observations mitigate this, yielding robust parameter estimates. However, in M31, Bayes factors favor Newtonian gravity, suggesting that current data lack the precision to resolve more complex models. This finding highlights two key needs: (i) realistic, physically or empirically informed priors to avoid biased constraints, and (ii) high-precision data with independent photometry to guard against overfitting. Our framework offers a scalable approach for testing gravity with large galactic rotation curve datasets.