Probing dark matter halo profiles with multi-band observations of gravitational waves

In this paper, we evaluate the potential of multiband gravitational wave observations from a deci-Hz space-based detector and third-generation ground-based gravitational wave detectors to constrain the properties of dark matter spikes around intermediate-mass ratio inspirals. The presence of dark matter influences the orbital evolution of the secondary compact object through dynamic friction, which leads to a phase shift in the gravitational waveform compared to the vacuum case. Our analysis shows that the proposed Indian space-based detector GWSat, operating in the deciHz frequency band, provides the most stringent constraints on the dark matter spike parameters, as IMRIs spend a significant portion of their inspiral phase within its sensitivity range. While third-generation ground-based detectors such as the Einstein Telescope and Cosmic Explorer offer additional constraints, their contribution is somewhat limited, particularly for higher-mass systems where the signal duration in their frequency bands is shorter. However, for systems with detector-frame total masses $M_z<400{\mathrm{M}}_{\odot }$, Cosmic Explorer and Einstein Telescope could improve the estimation of the chirp mass, symmetric mass ratio, luminosity distance, and dark matter spike power-law index by more than $15\mathrm{\%}$. Nonetheless, their impact on the constraint of spike density is minimal. These results highlight the crucial role of deciHz space-based detectors in probing dark matter interactions with gravitational wave sources.