  We prove the existence and uniqueness of a weighted analogue of the
Fefferman-Graham ambient metric for manifolds with density. We then show that
this ambient metric forms the natural geometric framework for the singular
Ricci flow: given a singular gradient Ricci flow spacetime in the Kleiner-Lott
sense, we construct a unique global ambient half-space from it. We also prove
the converse, that every global ambient space contains a singular gradient
Ricci flow spacetime, thereby completing the correspondence. Our main
application is the construction of infinite families of fully non-linear
analogues of Perelman's $\mathcal{F}$ and $\mathcal{W}$ functionals. We extend
Perelman's monotonicity result to these two families of functionals under
several conditions, including for shrinking solitons and Einstein manifolds. We
do so by constructing a "Ricci flow vector field" in the ambient space, which
may be of independent research interest. We also prove that the weighted GJMS
operators associated with the weighted ambient metric are formally
self-adjoint, and that the associated weighted renormalized volume coefficients
are variational.
