The moduli space of a rational map is Carath\'eodory hyperbolic

Let f be a rational map of degree d≥ 2. The moduli space Mf, introduced by McMullen and Sullivan, is a complex analytic space consisting all quasiconformal conjugacy classes of f. For f that is not flexible Latt\`es, we show that there is a normal affine variety Xf of dimension $2d-2$ and a holomorphic injection i:Mf→ Xf such that i(Mf) is precompact in Xf. In particular Mf is Carath\'eodory hyperbolic (i.e. bounded holomorphic functions separate points in Mf), provided that f is not flexible Latt\`es. This solves a conjecture of McMullen. When d≥ 4, we give a concrete construction of Xf as the normalization of the Zariski closure of the image of the reciprocal multiplier spectrum morphism.