On equation of state of dark matter around massive black holes

The nature of dark matter (DM) remains mysterious despite the substantial evidence from astrophysical and cosmological observations. While the majority of DM in our universe is non-relativistic, collisionless and its equation of state (EoS) is approximately pressureless $p\simeq~0$, DM becomes relativistic near the massive black holes (BHs) in galactic center. Yet its EoS is seldom discussed in the relativistic regime. Here we initially explore the possible EoS for DM in the vicinity of Schwarzschild BHs. We work in a spherical and quasi-static background spacetime, and describe DM as a perfect fluid in equilibrium. By numerically solving the Tolman-Oppenheimer-Volkoff equations with physical spatial initial conditions, we show that DM can have static profiles near BHs and its pressure should be negative in order to support the viable density profiles $\rho$. We illustrate with two simple general EoSs, namely the polytropic-like $p~\propto~\rho^\gamma$ and the radius-dependent $p~\propto~r\cdot~\rho$, and compare them with the observations of the Milky Way. Our findings provide insights into the model-building of DM, which should incorporate the possibility of negative pressure in the relativistic regime around BHs if such shallower DM profiles are probed by future gravitational-wave detectors in space.