  We propose a novel source of gravitational wave emission: the inspirals of
compact fragments inside primordial supermassive stars (SMSs). Such systems are
thought to be an essential channel in the as-yet little understood formation of
supermassive black holes (SMBHs). One model suggests that high accretion rates
of $0.1$-1 M$_\odot$/yr attainable in atomically-cooled primordial halos can
lead to the formation of a nuclear-burning SMS. This will ultimately undergo
collapse through a relativistic instability, leaving a massive BH remnant.
Recent simulations suggest that supermassive stars rarely form in isolation,
and that companion stars and even black holes formed may be captured/accreted
and inspiral to the SMS core due to gas dynamical friction. Here, we explore
the GW emission produced from such inspirals, which could probe the formation
and evolution of SMS and seeds of the first supermassive black holes. We use a
semi-analytic gas-dynamical friction model of the inspirals in the SMS to
characterize their properties. We find such sources could potentially be
observable by upcoming space-born GW-detectors at their formation redshifts
with the benefit of gravitational lensing. Mergers within closely-related
quasi-stars may produce a much stronger signal, though disambiguating such
events from other high-z events may prove challenging.
