  We explore the detection limits of the phase modulation (PM) method of
finding binary systems among multi-periodic pulsating stars. The method is an
attractive way of finding non-transiting planets in the habitable zones of
intermediate mass stars, whose rapid rotation inhibits detections via the
radial velocity (RV) method. While oscillation amplitudes of a few mmag are
required to find planets, many $\delta$ Scuti stars have these amplitudes. In
sub-optimal cases where the signal-to-noise of the oscillations is lower,
low-mass brown dwarfs ($\sim$13 M$_{\rm Jup}$) are detectable at orbital
periods longer than about 1 yr, and the lowest mass main-sequence stars
(0.1-0.2 M$_{\odot}$) are detectable at all orbital periods where the PM method
can be applied. We use purpose-written Markov chain Monte Carlo (MCMC) software
for the calculation of the PM orbits, which offers robust uncertainties for
comparison with RV solutions. Using Kepler data and ground-based RVs, we verify
that these two methods are in agreement, even at short orbital periods where
the PM method undersamples the orbit. We develop new theory to account for the
undersampling of the time delays, which is also necessary for the inclusion of
RVs as observational data in the MCMC software. We show that combining RVs with
time delays substantially refines the orbits because of the complementarity of
working in both the spatial (PM) and velocity (RV) domains simultaneously.
Software outputs were tested through an extensive hare and hounds exercise,
covering a wide range of orbital configurations including binaries containing
two pulsators.
