furthermore we assume that the $O(x^k, g_0)$ term is *polyhomogeneous conormal*, a regularity assumption defined precisely in Section \[sec:mwc\] below, which roughly speaking means that the coefficients have an asymptotic expansion at $x = 0$ analogous to a Taylor expansion but with non-integer powers and with precise derivative bounds on the error terms. Metrics satisfying these assumptions are what we refer to henceforth as **incomplete cusp edge metrics**. (Note that the assumptions on $g$ are stronger than merely assuming that $g \in {\operatorname{Sym}}^{0,2}({{}^{ice}T^{*}}(M))$, as the latter space contains e.g. $x (x^k dz \otimes_{sym} dx)$, which does not obey the error bound.)

As is shown in [@MS2015] (see the introduction for further discussion) with previous results for example in [@Wolpert-WP-geometry; @Yamada] the Weil-Petersson metric on moduli space takes the form near the interior of a divisor and satisfies the polyhomogeneity assumption.

As an example of such a metric, one can take an *exact* incomplete cusp edge metric, constructed as follows. Let $h_{1} \in
{\operatorname{Sym}}^{0, 2}(y)$ be a metric on the base, and $h = \pi^{*}h_{1}$ its $x$-independent pullback via the projection $\pi \colon {\partial}M
{\longrightarrow}Y$ extended to our tubular neighborhood $\mathcal{U} \simeq {\partial}M
\times [0, 1)_x$ in the obvious way. The 