2.4 Mini-boson stars

The concept of a star entails a configuration of matter which remains localized. One, therefore, looks for a localized and time-independent matter configuration such that the gravitational field is stationary and regular everywhere. As shown in [89], such a configuration does not exist for a real scalar field. But since the stress-energy tensor depends only on the modulus of the scalar field and its gradients, one can relax the assumption of time-independence of the scalar while retaining a time-independent gravitational field. The key is to assume a harmonic ansatz for the scalar field
ϕ(r,t) = ϕ0(r)eiωt, (33 )
where ϕ0 is a real scalar which is the profile of the star and ω is a real constant denoting the angular frequency of the phase of the field in the complex plane.

We consider spherically symmetric, equilibrium configurations corresponding to minimal energy solutions while requiring the spacetime to be static. In Schwarzschild-like coordinates, the general, spherically symmetric, static metric can be written as

2 2 2 2 2 2 2 ds = − α (r) dt + a (r) dr + r dΩ , (34 )
in terms of two real metric functions, α and a. The coordinate r is an areal radius such that spheres of constant r have surface area 2 4πr. For this reason, these coordinates are often called polar-areal coordinates.

The equilibrium equations are obtained by substituting the metric of Eq. (34View Equation) and the harmonic ansatz of Eq. (33View Equation) into the spherically symmetric EKG system of Eqs. (27View Equation32View Equation) with β = 0,b = 1, resulting in three first order partial differential equations (PDEs)

∂ a = − -a-(a2 − 1) + 4π Gra2 τ (35 ) r 2r α-( 2 ) 2 r ∂rα = 2r a − 1 + 4π Gr αa S r (36 ) [ ] Φ ( ω2 dV ) ∂rΦ = − 1 + a2 + 4π Gr2a2 (Srr − τ) --− --2 − ----2 a2ϕ0 . (37 ) r α d|ϕ|
Notice that these equations hold for any stress-energy contributions and for a generic type of self-potentials 2 V(|ϕ| ). In order to close the system of Eqs. (35View Equation37View Equation), we still have to prescribe this potential. The simplest case admitting localized solutions is the free field case of Eq. (12View Equation) for which the potential describes a field with mass m and for which the equations can be written as
a { a2 − 1 [ (ω2 ) ] } ∂ra = -- − ------ +4π Gr --2 + m2 a2ϕ20 + Φ2 , (38 ) 2 { r [( α ) ]} α- a2 −-1 ω2- 2 2 2 2 ∂rα = 2 r +4 πGr α2 − m a ϕ0 + Φ , (39 ) ( 2 ) { 2 2 2 2 2} Φ- ω-- 2 2 ∂rΦ = − 1 + a − 4π Gr a m ϕ 0 r − α2 − m ϕ0a . (40 )
In order to obtain a physical solution of this system, we have to impose the following boundary conditions,
ϕ0 (0) = ϕc, (41 ) Φ (0) = 0, (42 ) a(0) = 1, (43 ) lim ϕ0(r) = 0, (44 ) r→ ∞ -1-- lri→m∞ α (r) = lri→m∞ a(r) , (45 )
which guarantee regularity at the origin and asymptotic flatness. For a given central value of the field {ϕc}, we need only to adjust the eigenvalue {ω } to find a solution which matches the asymptotic behavior of Eqs. (44View Equation45View Equation). This system can be solved as a shooting problem by integrating from r = 0 towards the outer boundary r = r out. Eq. (39View Equation) is linear and homogeneous in α and one is, therefore, able to rescale the field consistent with Eq. (45View Equation). We can get rid of the constants in the equations by re-scaling the variables in the following manner
√ ----- &tidle;ϕ ≡ 4π Gϕ , r&tidle;≡ m r, &tidle;t ≡ ω t, &tidle;α ≡ (m ∕ω)α . (46 ) 0 0

Notice that the form of the metric in Eq. (34View Equation) resembles Schwarzschild allowing the association a2 ≡ (1 − 2 M ∕r)−1, where M is the ADM mass of the spacetime. This allow us to define a more general mass aspect function

r ( 1 ) M (r,t) = -- 1 − -2----- , (47 ) 2 a (r,t)
which measures the total mass contained in a coordinate sphere of radius r at time t.

In isotropic coordinates, the spherically symmetric metric can be written as

2 2 2 4( 2 2 2) ds = − α (R ) dt + ψ (R ) dR + R dΩ , (48 )
where ψ is the conformal factor. A change of the radial coordinate R = R(r) can transform the solution obtained in Schwarzschild coordinates into isotropic ones, in particular
[( √ -)2 ] R (rmax) = 1-+---a r- (49 ) 2 a rmax dR-= a R-, (50 ) dr r
where the first condition is the initial value to integrate the second equation backwards, obtained by imposing that far away from the boson star the spacetime resembles Schwarzschild solution.

As above, boson stars are spherically symmetric solutions of the Eqs. (38View Equation40View Equation) with asymptotic behavior given by Eqs. (41View Equation45View Equation). For a given value of the central amplitude of the scalar field ϕ (r = 0) = ϕ 0 c, there exist configurations with some effective radius and a given mass satisfying the previous conditions for a different set of n discrete eigenvalues ω (n). As n increases, one obtains solutions with an increasing number of nodes in ϕ0. The configuration without nodes is the ground state, while all those with any nodes are excited states. As the number of nodes increases, the distribution of the mass as a function of the radius becomes more homogeneous.

As the amplitude ϕc increases, the stable configuration has a larger mass while its effective radius decreases. This trend indicates that the compactness of the boson star increases. However, at some point the mass instead decreases with increasing central amplitude. Similar to models of neutron stars (see Section 4 of [59]), this turnaround implies a maximum allowed mass for a boson star in the ground state, which numerically was found to be 2 Mmax = 0.633 M Planck∕m. The existence of a maximum mass for boson stars is a relativistic effect, which is not present in the Newtonian limit, while the maximum of baryonic stars is an intrinsic property.

Solutions for a few representative boson stars in the ground state are shown in Figure 2View Image in isotropic coordinates. The boson stars becomes more compact for higher values of ϕ c, implying narrower profiles for the scalar field, larger conformal factors, and smaller lapse functions, as the total mass increases.

View Image

Figure 2: Profiles characterizing static, spherically symmetric boson stars with a few different values of the central scalar field (top left). Reprinted with permission from [141].

  Go to previous page Go up Go to next page