In Mazzeo-Alexakis, there is a brief discussion that if $(M^3,g) \sim \mathbb{H}^3/\Gamma$ (for $\Gamma$ convex cocompact), then the metric can be expanded near the topological boundary as

$$g = \frac{dx^2 + h_0 + x^2 h_2 + x^4 h_4}{x^2}$$

Here, $x$ is a boundary defining function and $h_{2k}$ are tensors on $\partial M$. I'm looking for more references on this expansion. I've also been told that $h_2$ or $h_4$ may have connections to distinguishing what element of moduli space (or Teichmuller space?) $M^3$ is, with potential connections to McMullen's quasi-fuchsian reciprocity.

My background is more in conformal geometry and less so the theory of hyperbolic $3$-manifolds so apologies for any in the above that is incorrect. References to any of the above would be deeply appreciated!