There is growing interest in testing alternative gravity theories using the subtle gravitational redshifts in clusters of galaxies. However, current models all neglect a transverse Doppler redshift of similar magnitude, and some models are not self-consistent. An equilibrium model would fix the gravitational and transverse Doppler velocity shifts to be about $6{\sigma }^2/c$ and $3{\sigma }^2/2c$ in order to fit the observed velocity dispersion $\sigma$ self-consistently. This result comes from the virial theorem for a spherical isotropic cluster, and is insensitive to the theory of gravity. A gravitational redshift signal also does not directly distinguish between the Einsteinian and $f(R)$ gravity theories, because each theory requires a different dark halo mass function to keep the clusters in equilibrium. When this constraint is imposed, the gravitational redshift has no sensitivity to theory. Indeed, our $N$-body simulations show that the halo mass function differs in $f(R)$, and that the transverse Doppler effect is stronger than analytically predicted due to nonequilibrium.

• Received 22 June 2012

DOI:https://doi.org/10.1103/PhysRevD.88.043013