  This paper presents a systematic approach to derive physical bounds for
Positive Real (PR) functions directly in the Time-Domain (TD). The theory is
based on Cauer's representation of an arbitrary PR function together with
associated sum rules (moments of the measure) and exploits the unilateral
Laplace transform to derive rigorous bounds on the TD response of a passive
system. The existence of useful sum rules and related physical bounds relies
heavily on an assumption about the PR function having a low- or high-frequency
asymptotic expansion at least of odd order 1. As a canonical example, we
explore the time-domain dielectric step response of a passive material, either
with or without a given pulse raise time. As a particular numerical example, we
consider here the electric susceptibility of gold (Au) which is commonly
modeled by well established Drude or Brendel Bormann models. An explicit
physical bound on the early-time step response of the material is then given in
terms of a quadratic function in time which is completely determined by the
plasma frequency of the metal.
