Let $M$ be a complete Riemannian manifold, and $p(t, x, y)$ denotes its heat kernel. I am trying to find sufficient conditions for when the following holds: $$ p(t, x, y) \leq Ct^{-n/2}, \forall x, y, t > 0.$$

In particular, I am interested in the following question: does lower Ricci bounds imply the above heat kernel bounds?