Probing dark-matter effects with gravitational waves using the parameterized post-Einsteinian framework

A massive black hole can develop a dark-matter overdensity, and the dark matter changes the evolution of a stellar-mass compact object inspiraling around the massive black hole through the dense dark-matter environment. Specifically, dynamical friction speeds up the inspiral of the compact object and causes feedback on the dark-matter distribution. These intermediate mass-ratio inspirals with dark matter are a source of gravitational waves (GWs), and the waves can dephase significantly from an equivalent system in vacuum. Prior work has shown that this dephasing needs to be modeled to detect the GWs from these systems with LISA (the Laser Interferometer Space Antenna); it also showed that the density and distribution of dark matter can be inferred from a GW measurement. In this paper, we study whether the parametrized post-Einsteinian (ppE) framework can be used to infer the presence of dark matter in these systems. We confirm that if vacuum waveform templates are used to model the GWs from an inspiral in a dark-matter halo, then the resulting parameter estimation is biased. We then apply the ppE framework to determine whether it can reduce the parameter-estimation biases, and we find that adding one ppE phase term to a waveform template eliminates the parameter-estimation biases (statistical errors become larger than the systematic ones), but the effective post-Newtonian order in the ppE framework must be specified without uncertainties. When the post-Newtonian order has uncertainty, we find that the systematic errors on the ppE and the binary's parameters exceed the statistical errors. Thus, the simplest ppE framework would not give unbiased results for these systems, and a further extension of it, or dedicated parameter estimation with gravitational waveforms that include dark-matter effects would be needed.