Tidal Love numbers and quasi-normal modes of the ECO in a Dark Matter halo

It is well-known that exotic compact objects (ECOs) are a class of objects categorised as Black Hole (BH) mimickers. ECOs have been shown to possess signatures distinguishing them from BHs. However, in our universe, no object exists in complete isolation. Consequently, any compact object, whether a BH or an ECO, must reside within some environment that inevitably influences the surrounding spacetime geometry due to back-reaction. In this paper, we investigate a scenario where an ECO is embedded in an environment of dark matter (DM). In this work, we assume two different models of the DM halo profile. We compute the Love numbers and GW echoes of this composite system to assess the impact of the surrounding dark matter halo. To analyze the echoes, we focus on odd-parity perturbations, while for calculating the tidal Love numbers, we consider both even and odd parity perturbations. We aim to understand how the DM properties couple to the ECO signatures in the Love number or the GW-echo signal, both of which have a strong bearing as observables in future-generation detectors.