If a spacetime is homogenous and isotropic can I say that $\nabla_\mu R =0$?
I was reading this paper https://arxiv.org/abs/astro-ph/0610483 and, I think that is the justification for the authors setting $\nabla_\mu R =0$ (just below Eq. (3)). (Am I correct here?)
I found some references (such as section 5.1 of Wald) that state a spacelike surface that is homogenous and isotropic will have constant curvature, but what bothers me is that the space like surface of curvature scalar K is not the same curvature scalar for a 3+1 universe with curvature scalar R.
But if this is not the case, I don't understand the author's justification for setting $\nabla_\mu R =0$ in the paper above.