A computability theoretic equivalent to Vaught’s conjecture

We prove that, for every theory TT which is given by an Lω1,ωLω1,ω sentence, TT has less than 2ℵ02ℵ0 many countable models if and only if we have that, for every X∈2ωX∈2ω on a cone of Turing degrees, every XX-hyperarithmetic model of TT has an XX-computable copy. We also find a concrete description, relative to some oracle, of the Turing-degree spectra of all the models of a counterexample to Vaught’s conjecture.