The moduli space of a rational map is Carathéodory hyperbolic

Let $f$ be a rational map of degree $d\ge 2$. The moduli space ${\mathcal{M}}_f$, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of $f$. For $f$ that is not flexible Lattès, we show that there is a normal affine variety $X_f$ of dimension $2d-2$ and a holomorphic injection $i:{\mathcal{M}}_f\to X_f$ such that $i({\mathcal{M}}_f)$ is precompact in $X_f$. In particular ${\mathcal{M}}_f$ is Carathéodory hyperbolic (i.e. bounded holomorphic functions separate points in ${\mathcal{M}}_f$), provided that $f$ is not flexible Lattès. This solves a conjecture of McMullen. When $d\ge 4$, we give a concrete construction of $X_f$ as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.