Fluctuations in non-conformal holographic plasma at criticality

We continue the study initiated in [arXiv:1708.02252] of the fluctuations of a strongly-coupled non-conformal plasma described holographically by Einstein gravity coupled to a dilaton with an exponential potential. The plasma approaches a critical point of a continuous phase transition in a specific limit, where the metric becomes a linear-dilaton background. This results to an analytic description of the quasi-normal mode spectrum, that can be extended perturbatively in the deviation away from the critical point. In the previous paper we showed that at criticality the quasinormal frequencies coalesce into a branch cut on the real axis. In this paper we give a more extended and complete discussion of these results. We compare in detail the numerical and analytical approximations in order to confirm their validity; we study (numerically and in a WKB approximation) the momentum dependence of the modes, in order to determine the cross-over scale that limits the validity of the hydrodynamic approximation, and which becomes arbitrarily low at the critical point; and we discuss in detail the procedure we use to complete the theory in the UV by gluing a slice of AdS geometry, and the extent to which it should provide a good approximation to a smooth UV-complete situation.