Integrals of $\psi$-classes on twisted double ramification cycles and spaces of differentials

We prove a closed formula for the integral of a power of a single $\psi$-class on strata of $k$-differentials. In many cases, these integrals correspond to intersection numbers on twisted double ramification cycles. Then we conjecture an expression of a refinement of double ramification cycles according to the parity of spin structures. Assuming that this conjecture is valid, we also compute the integral of a single $\psi$-class on the even and odd components of strata of $k$-differentials. As an application of these results we give a closed formula for the Euler characteristic of components of minimal strata of abelian differentials.