Noncontractible periodic orbits in cotangent bundles and Floer homology

For every nontrivial free homotopy class $\alpha$ of loops in every closed connected Riemannian manifold $M$, we prove existence of a noncontractible 1-periodic orbit, for every compactly supported time-dependent Hamiltonian on the open unit cotangent bundle which is sufficiently large over the zero section. The proof shows that the Biran-Polterovich-Salamon capacity is finite for every closed connected Riemannian manifold and every free homotopy class of loops. This implies a dense existence theorem for periodic orbits on level hypersurfaces and, consequently, a refined version of the Weinstein conjecture: Existence of closed characteristics (one associated to each nontrivial $\alpha$) on hypersurfaces in $T^*M$ which are of contact type and contain the zero section.