Linear viscoelastic dynamics in holography

We study the mechanical response under time dependent sources of a simple class of holographic models that exhibit viscoelastic features. The ratio of viscosity over elastic modulus defines an intrinsic relaxation timescale---the so-called Maxwell relaxation time τM, which has been identified traditionally with the relaxation timescale. We compute explicitly the relaxation time in our examples and find that it differs from τM. At high temperatures τM overestimates the actual relaxation time, although not by much and moreover it still captures reasonably well the temperature behavior. At sufficiently low temperatures the situation is reversed: τM underestimates the actual relaxation time, in some cases quite drastically. Moreover, when τM underestimates the real-time response exhibits an overshoot phenomenon before relaxation. We comment on the $T=0$ limit, where the relaxation is power law because our models exhibit criticality.