On the population of remnant FRII radio galaxies and implications for
  radio source dynamics
  The purpose of this work is two-fold: (1) to quantify the occurrence of
ultra-steep spectrum remnant FRII radio galaxies in a 74 MHz flux limited
sample, and (2) perform Monte-Carlo simulations of the population of active and
remnant FRII radio galaxies to confront models of remnant lobe evolution, and
provide guidance for further investigation of remnant radio galaxies. We find
that fewer than 2$\%$ of FRII radio galaxies with S$_{ \rm74~MHz} > 1.5$ Jy are
candidate ultra-steep spectrum remnants, where we define ultra-steep spectrum
as $\alpha_{\rm 74~MHz}^{\rm 1400~MHz} > 1.2$. Our Monte-Carlo simulations
demonstrate that models involving Sedov-like expansion in the remnant phase,
resulting in rapid adiabatic energy losses, are consistent with this upper
limit, and predict the existence of nearly twice as many remnants with normal
(not ultra-steep) spectra in the observed frequency range as there are
ultra-steep spectrum remnants. This model also predicts an ultra-steep remnant
fraction approaching 10$\%$ at redshifts $z < 0.5$. Importantly, this model
implies the lobes remain over-pressured with respect to the ambient medium well
after their active lifetime, in contrast with existing observational evidence
that many FRII radio galaxy lobes reach pressure equilibrium with the external
medium whilst still in the active phase. The predicted age distribution of
remnants is a steeply decreasing function of age. In other words young remnants
are expected to be much more common than old remnants in flux limited samples.
For this reason, incorporating higher frequency data $\gtrsim 5$ GHz will be of
great benefit to future studies of the remnant population.
