### 3.6 Fermionic-bosonic stars

The possibility of compact stellar objects made with a mixture of bosonic and fermionic matter was
studied in Refs. [111, 112]. In the simplest case, the boson component interacts with the fermionic
component only via the gravitational field, although different couplings were suggested in [112] and have
been further explored in [66, 180]. Such a simple interaction is, at the very least, consistent with models of
a bosonic dark matter coupling only gravitationally with visible matter, and the idea that such a bosonic
component would become gravitationally bound within fermionic stars is arguably a natural
expectation.
One can consider a perfect fluid as the fermionic component such that the stress-energy tensor takes the
standard form

where is the energy density, is the pressure of the fluid and its four-velocity. Such a fluid
requires an equation of state to close the system of equations (see Ref. [85] for more about fluids in
relativity). In most work with fermionic-boson stars, the fluid is described by a degenerate, relativistic
Fermi gas, so that the pressure is given by the parametric equation of state of Chandrasekhar
where and the mass of the fermion. The parameter is given by
where is the maximum value of the momentum in the Fermi distribution at radius .
The perfect fluid obeys relativistic versions of the Euler equations, which account for the conservation of
the fluid energy and momentum, plus the conservation of the baryonic number (i.e., mass conservation).
The complex scalar field representing the bosonic component is once again described by the
Klein–Gordon equation. The spacetime is computed through the Einstein equations with a
stress-energy tensor, which is a combination of the complex scalar field and the perfect fluid

After imposing the harmonic time dependence of Eq. (33) on the complex scalar field, assuming a static
metric as in Eq. (34) and the static fluid , one obtains the equations describing equilibrium
fermion-boson configurations
These equations can be written in adimensional form by rescaling the variables by introducing the
following quantities

By varying the central value of the fermion density and the scalar field , one finds
stars dominated by either bosons or fermions, with a continuous spectrum in between.
It was shown that the stability arguments made with boson stars can also be applied to
these mixed objects [121]. The existence of slowly rotating fermion-boson stars was shown
in [65], although no solutions were found in previous attempts [136]. Also see [73] for unstable
solutions consisting of a real scalar field coupled to a perfect fluid with a polytropic equation of
state.