  Dozens of stellar tidal disruption events (TDEs) have been identified at
optical, UV and X-ray wavelengths. A small fraction of these, most notably
Swift J1644+57, produce radio synchrotron emission, consistent with a powerful,
relativistic jet shocking the surrounding circumnuclear gas. The dearth of
similar non-thermal radio emission in the majority of TDEs may imply that
powerful jet formation is intrinsically rare, or that the conditions in
galactic nuclei are typically unfavorable for producing a detectable signal.
Here we explore the latter possibility by constraining the radial profile of
the gas density encountered by a TDE jet using a one-dimensional model for the
circumnuclear medium which includes mass and energy input from a stellar
population. Near the jet Sedov radius of 10$^{18}$ cm, we find gas densities in
the range of $n_{18} \sim$ 0.1$-$1000 cm$^{-3}$ across a wide range of
plausible star formation histories. Using one- and two-dimensional relativistic
hydrodynamical simulations, we calculate the synchrotron radio light curves of
TDE jets (as viewed both on and off-axis) across the allowed range of density
profiles. We find that bright radio emission would be produced across the
plausible range of nuclear gas densities by jets as powerful as Swift J1644+57,
and we quantify the relationship between the radio luminosity and jet energy.
We use existing radio detections and upper limits to constrain the energy
distribution of TDE jets. Radio follow up observations several months to
several years after the TDE candidate will strongly constrain the energetics of
any relativistic flow.
