Test of conformal gravity as an alternative to dark matter from the observations of elliptical galaxies

As an alternative gravitational theory to General Relativity (GR), the Conformal Gravity (CG) has recently been successfully verified by observations of Type Ia supernovae (SN Ia) and the rotation curves of spiral galaxies. The observations of galaxies only pertain to the non-relativistic form of gravity. In this context, within the framework of the Newtonian theory of gravity (the non-relativistic form of GR), dark matter is postulated to account for the observations. On the other hand, the non-relativistic form of CG predicts an additional potential: besides the Newtonian potential, there is a so-called linear potential term, characterized by the parameter $\gamma^*$, as an alternative to dark matter in Newtonian gravity. To test CG in its non-relativistic form, much work has been done by fitting the predictions to the observations of circular velocity (rotation curves) for spiral galaxies. In this paper, we test CG with the observations from elliptical galaxies. Instead of the circular velocities for spiral galaxies, we use the velocity dispersion for elliptical galaxies. By replacing the Newtonian potential with that predicted by non-relativistic form of CG in Hamiltonian, we directly extend the Jeans equation derived in Newtonian theory to that for CG. By comparing the results derived from the ellipticals with that from spirals, we find that the extra potential predicted by CG is not sufficient to account for the observations of ellipticals. Furthermore, we discover a strong correlation between $\gamma^*$ and the stellar mass $M^*$ in dwarf spheroidal galaxies. This finding implies that the variation in $\gamma^*$ violates a fundamental prediction of Conformal Gravity (CG), which posits that $\gamma^*$ should be a universal constant.