Refined modelling of the radio SZ signal: kinematic terms, relativistic
  temperature corrections and anisotropies in the radio background
  A significant cosmological radio background will inevitably lead to a radio
Sunyaev-Zeldovich (SZ) effect. In the simplest limit, the combined signal from
the scattered radio and cosmic microwave background exhibits a null at around
$\nu \simeq 735$ MHz. Here, we show that kinematic and relativistic temperature
corrections to this radio SZ signal are easily calculable. We treat both the
cluster and observer motion, and the scattering of anisotropies in the radio
background, highlighting how the spectrum of the radio SZ effect is affected in
each case. Although relativistic temperature corrections only enter at the
level of a few percent, our expressions allow high-precision modelling of these
terms. By measuring the SZ signal around the radio null, one is in principle
able to place constraints on the properties of a cosmological radio background.
A combination with standard SZ measurements from large cluster samples could
provide a promising avenue towards breaking degeneracies between different
contributions. Stacking analyses can reduce the effect of kinematic corrections
and dipolar anisotropies in the radio background, thereby providing a way to
constrain the redshift dependence of the average radio background. Our
qualitative discussion is meant to give an analytic understanding of the
various effects and also motivate further studies with the aim to obtain
quantitative forecasts of their observability. At this stage, a detection of
the corrections seems rather futuristic, but the advent of large SZ and X-ray
cluster samples could drastically improve our ability to disentangle various
effects.
