Distinctive GWBs from eccentric inspiraling SMBH binaries with a DM spike

Recent detections of a low-frequency gravitational wave background (GWB) from various pulsar-timing-array (PTA) observations have renewed the interest in the inspiraling supermassive black hole binaries (SMBHBs), whose population is believed to be the most promising candidate with possible generalizations from including either orbital eccentricity or dark matter (DM) spike. In this paper, we show that the inclusion of both can further display distinctive features detectable in future PTA observations. With a typical initial eccentricity $e_0\sim\mathcal{O}(0.1)$ for the inspiraling SMBHBs, even a shallow DM spike can easily drive the orbital eccentricity close to $1$, leaving behind a large turnover eccentricity when GWs begin to dominate the orbital circularization. In particular, the DM spike index $\gamma_\mathrm{sp}$ universally manifests itself in the characteristic strain by $h_c\sim f^{7/6-\gamma_\mathrm{sp}/3}$ in the far infrared and features a novel oscillation structure at low frequencies. Future PTA detection of such characteristics would be the smoking gun for the DM spike and even reveal the nature of DM.