Can LISA See Inside the Little Red Dots?

Abstract

The Little Red Dots (LRDs) are compact, broad-line sources that appear to contain accreting black holes (BHs) embedded in dense gas. Proposed models for their central environments (including quasi-stars, radiative BH envelopes and super-Eddington accretion discs) predict gas densities at the radii probed by an extreme-mass-ratio inspiral that differ by many orders of magnitude. We compute the inspiral of a $10$--$30\Msun$ BH for a range of LRD interior models, including environmental effects from the gas, and test with a mismatch criterion whether the resulting waveforms can be distinguished from vacuum and from one another. Dense interiors (Bondi-like free-fall cavities and radiative envelopes) produce measurable deviations from vacuum in essentially every detectable event. Convective quasi-star interiors are measurable for $\MBH\lesssim10^5\Msun$ and $z\lesssim0.6$, while a co-rotating disc leaves no measurable imprint. In favorable cases the models are also mutually distinguishable. None of the 14 JWST LRDs at $z>3.9$ would be detectable with a $10\Msun$ companion, but 30 of 31 LRDs at $z<0.9$ would be if their black holes are $\sim100$ times less massive than their virial estimates, and in many of them the interior's imprint would be measurable. Across the LRD population to $z=6$, a 4-yr LISA mission would yield $0.08$--$0.8$ detectable events for $10^5\Msun$ hosts and $10\Msun$ companions ($1.5$--$15$ for $30\Msun$) if LRD nuclei host inspirals at the rate of ordinary galactic nuclei. In dense clusters, where captures are plunges, this falls to $\lesssim10^{-4}$.