Curvature of higher direct image sheaves

Given a family $(F,h)\to X\times S$ of Hermite-Einstein bundles on a compact K"ahler manifold $(X,g)$ we consider the higher direct image sheaves $R^q{p}_{\ast }\mathcal{O}(F)$ on $S$, where $p:X\times S\to S$ is the projection. On the complement of an analytic subset these sheaves are locally free and carry a natural metric, induced by the $L_2$ inner product of harmonic forms on the fibers. We compute the curvature of this metric which has a simpler form for families with fixed determinant and families of endomorphism bundles. Furthermore, we discuss the metric for moduli spaces of stable vector bundles.