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

Taflbuid = (μ + p)ua ub + p gab, (72 )
where μ is the energy density, p is the pressure of the fluid and ua 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
[ ( ) ] K t μ = K (sinh t − t) p = -3- sinh t − 8 sinh 2- + 3 t , (73 )
where K = m4n∕(32π2 ) and mn the mass of the fermion. The parameter t is given by
⌊ ( ( ) )1∕2⌋ po po 2 t(r) = 4log⌈ m---+ 1 + m--- ⌉ , (74 ) n n
where po is the maximum value of the momentum in the Fermi distribution at radius r.

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

ϕ fluid Tab = T ab + Tab . (75 )
After imposing the harmonic time dependence of Eq. (33View Equation) on the complex scalar field, assuming a static metric as in Eq. (34View Equation) and the static fluid ua = 0, one obtains the equations describing equilibrium fermion-boson configurations
da a { 1 [( ω2 ) ]} ---= -- -(1 − a2) + 4π Gr --2 + m2 a2 ϕ2(r) + Φ2 (r) + 2a2 μ dr 2 { r [ (α ) ]} d-α α- 1- 2 ω2- 2 2 2 2 2 dr = 2 r (a − 1 ) + 4π Gr α2 − m a ϕ (r) + Φ (r) + 2a p d-ϕ = Φ (r) dr ( ) dΦ 2 ω2 2 [ 2 2 2( 2 2 )] Φ --- = m − -2- a ϕ − 1 + a − 4π Ga r m ϕ + μ − p -- dr α ′ r dp- α-- dr = − (μ + p)α .

These equations can be written in adimensional form by rescaling the variables by introducing the following quantities

√ ----- x ≡ mr , σ(x) ≡ 4π Gϕ (0,r), Ω ≡ ω ∕m2 , ¯μ ≡ (4 πG ∕m2 )μ , ¯p ≡ (4π G∕m2 )p . (76 )
By varying the central value of the fermion density μ(r = 0) and the scalar field ϕ (r = 0), 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.

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