Spontaneous scalarization of boson stars

While spherically symmetric, asymptotically flat electro-vacuum black holes cannot be spontaneously scalarized in the $\alpha \phi^2 {\cal R}$ scalar-tensor gravity model, we show that this is possible for spherically symmetric, globally regular and asymptotically flat boson stars. These compact objects are made of a complex valued scalar field that has harmonic time dependence, while their spacetime is static and they can reach densities and masses similar to that of supermassive black holes. We find that boson stars can be scalarized for both signs of the scalar-tensor coupling $\alpha$ and for sufficiently large values of $\vert\alpha\vert$. Our results demonstrate that the scalarization is a distinguishing feature for spherically symmetric, asymptotically flat compact objects with static space-times in the $\alpha \phi^2 {\cal R}$ scalar-tensor gravity model.


Introduction
"Compact objects" is the name given to solutions in General Relativity (or alternative models of gravity) that are typically so dense that the curvature of space-time around them has detectable effects. A "measure" of the strength of the gravitational field of a body of mass M and radius R for r > R can be given by comparing the dimensionless ratio 2GM/(c 2 R) to unity, where G is Newton's constant and c the speed of light. For a white dwarf [1] and neutron star (for a recent review see e.g. [2]), this ratio is on the order of 2GM WD /(c 2 R WD ) ∼ O(10 −8 ) and 2GM NS /(c 2 R NS ) ∼ O(10 −1 ), respectively, but objects exist that are so dense, that R ≡ r h = c 2 /(2GM). These objects are called black holes [3], because the surface associated to r = r h is a surface of infinite redshift for an asymptotic observer, called event horizon. This means that any signal emitted from r ≤ r h cannot reach an asymptotic observer.
While neutron stars have gravitational fields comparable to that of black holes, limits on their upper mass exist that are on the order of a few solar masses. Hence, they cannot account for the objects in the centre of galaxies that possess millions of solar masses in a spatially small region. It seems that the only viable and standard option are supermassive black holes [4]. However, it has to be stated that the observation of the identifying characteristic of a black hole, the event horizon, has not been observationally achieved to this day ( see also [5] for a recent discussion about evidence for the existence of a supermassive black hole in the Milky Ways centre). An alternative, albeit more exotic, are compact objects made out of bosonic fields, the simplest example being the boson star made of a complex valued, massive scalar field [6][7][8][9][10][11][12]. The global U(1) symmetry in the model leads to a locally conserved Noether current and a globally conserved Noether charge Q, where the latter can be interpreted as the number of scalar bosons of a given mass that form the boson star in a self-gravitating, bound system. And, indeed, these objects have early on been discussed as viable alternatives for the centres of galaxies [13,14]. Since the discovery of a fundamental scalar field in nature, the Higgs field [15], and the fact that scalar fields are extensively used in diverse areas of physics, these objects can certainly be considered to have a probability to exist in nature or, at least, to effectively describe very dense states of matter. The study of compact objects as dense as black holes and boson stars has interest in its own right, but is also very important from another perspective : since these objects have very strong gravitational fields, they are also an ideal testing ground for alternative models of gravity and/or the limits of General Relativity, which works extremely well in the weak regime, but has not been explored in full detail in the regime of very strong gravity. Because of the reasons mentioned above for the re-newed interest in scalar fields, scalar-tensor gravity models are discussed extensively currently, in particular in the context of the primordial universe and its exponential expansion very shortly after the Planck era, referred to as inflation. Horndeski scalar-tensor gravity models [16] lead like General Relativity to equations of motion that are maximally of second order in derivatives [17,18]. Black holes have been studied in models that possess a shift symmetry for the scalar field [19,20] as well as in models that allow spontaneous scalarization of black holes in the sense that black holes carry scalar hair for sufficiently large values of the non-minimal coupling constants. In general, the models contain terms of the form f (φ)I(g µν ; Σ), where f (φ) is a function of the scalar field and I depends on the metric g µν and/or other fields Σ. Models have been studied with f (φ) ∼ φ 2 [21][22][23][24] with different other forms of f (φ) with a single term in f (φ) [25][26][27] as well as a combination of different powers of φ [28]. In all these cases, I has been chosen to be the Gauss-Bonnet (GB) term G. In [29] a model combining the original shift symmetric scalar field and a quadratic scalar field coupled to the GB term has been studied bridging between shift symmetry and spontaneous scalarization. All the above mentioned studies considered uncharged black holes, but charge can be included and leads to new effects. The first studies of this type have been done in the conformally coupled scalar field case [21] as well as for the f (φ)G case [30]. The studies were extended to discuss the approach to the extremal limit in [31] and it was demonstrates that scalarized black holes can exist for both signs of the scalar-tensor coupling due to the fact that G becomes negative close to the horizon when approaching extremality. Since the electromagnetic field can source the scalar field when non-minimally coupled, charged black holes can also carry scalar hair without scalar-tensor coupling [32,33]. In this paper, we study a different type of compact object in a scalar-tensor gravity model a non-rotating boson star, which is a globally regular solution to gravity models coupled to a complex valued scalar field. The space-time of these solutions is static and outside the boson stars core is very similar to that of the exterior Schwarzschild solution. In Section 2, we give the model, while in Section 3 we discuss the tachyonic instability appearing in the system which leads to the conclusion that boson stars can, indeed, be scalarized. In Section 4, we discuss the full backreacted system, and Section 5 contains are conclusions. In the following, we will use units such that c ≡ 1.

The model
The model we are studying here is a scalar-tensor gravity model that contains a non-minimal coupling between the square of a real scalar field and the Ricci scalar R : where L Ψ corresponds to the Lagrangian density of a complex valued, massive scalar field : m is the mass of the scalar field and G Newton's constant, respectively. Note that the action possesses a global U(1) symmetry of the form Ψ → exp(iχ)Ψ, where χ ∈ R is a constant, but arbitrary phase. This leads to the existence of a globally conserved Noether charge Q, which can be interpreted as the number of bosonic particles of mass m that form a self-gravitating, bound system which is referred to as a boson star.
The point of this paper is to show that a boson star can be spontaneously scalarized in a scalartensor model in which spontaneous scalarization of spherically symmetric, electro-vacuum black holes is not possible. To understand how this works, let us point out that the energy-momentum tensor of the Ψ-field reads : and has (in general) non-vanishing trace T (Ψ) ≡ g µν T (Ψ) µν and hence, even in the perturbative limit of the scalar field φ in the background space-time of the boson star, we would expect a tachyonic instability to be possible through scalar-tensor coupling of the φ 2 R type since in this limit R = −8πGT (Ψ) . Note that this is fundamentally different to the case of spherically symmetric, asymptotically flat electro-vacuum black holes for which R = 0. In this latter case, it is thence necessary to couple the scalar field differently, e.g. to the Gauss-Bonnet term, in order to achieve spontaneous scalarization.
In the following we will demonstrate that this in, indeed, possible for spherically symmetric, nonrotating boson stars. For the metric and scalar field, we choose the following Ansatz : while the complex scalar field forming the boson star reads : Variation of the action (1) with respect to the metric, the gravity scalar φ and the boson star scalar ψ leads to a set of coupled, non-linear, ordinary differential equations that have to be solved with the appropriate boundary conditions. We will use units such that 8πG ≡ 1.
The boson stars will be characterized by their mass M and Noether charge Q. The former can be read off from the behaviour of the metric function N(r) at infinity : while the Noether charge in our coordinate system reads : The boundary conditions for the system are determined by the requirement of finite energy and global regularity of the functions. These read at the origin r = 0, where ψ 0 is a real constant that is related to the value of ω, albeit not necessarily in one-to-one correspondence. At infinity, the functions have the following behaviour : where Q φ denotes the charge of the scalar field φ.
In order to explain the spontaneous scalarization of boson stars, let us first recall the pattern of solutions in standard Einstein gravity. For that, note that the asymptotic behaviour of the scalar

A tachyonic instability for boson stars
Similar to the case of a scalar-Gauss-Bonnet coupling (see [22]), the spontaneous scalarization of a black hole in a model containing a non-minimal coupling between the scalar field and the Ricci scalar R can be related to the appearance of a tachyonic instability of the scalar field φ. We assume the background space-time of the boson star to be fixed and study the scalar field equation in this background for the lowest angular mode in θ, i.e. for ℓ = 0. With our ansatz, the relevant scalar field equation has the form : 1 where now and in the following the prime denotes the derivative with respect to r. Since the background space-time is fixed and a solution to the Einstein equation G µν = T (ψ) µν , we can use the identity R = −T (ψ) to replace the Ricci scalar R in terms of the trace of the energy momentum tensor of the scalar field ψ, T (ψ) : The equation (10) can be solved with the boundary conditions for the field φ (see (8) and (9), respectively). Note also that the equation is linear in φ, so we can choose φ(0) = 1 without loss of generality. For completeness let us remark that (10) can be put into the form of a Schrödinger-like equation : where r * is related to r via dr/dr * = σN and χ(r) := rφ(r). The tachyonic instability corresponds to the existence of bound states of the Schrdinger-like equation. In the following, we will demonstrate that these bound states do exist. Obviously, we would not expect a non-trivial solution for φ for generic values of α. We have solved the equation (10) numerically in the background of the (numerically given) boson star solutions and find that boson stars can be scalarized for sufficiently large values of |α|.
In Fig. 1, we show the effective potential V eff (r) as given in (12) Table 1). We also give the corresponding bound state solution φ(r) for these same values of ψ 0 for positive α (dashed). For values of α see Table 1. φ(r) with the largest (smallest) value for r ∈ [0 : 1] corresponds to ψ 0 = 1.0 (ψ 0 = 0.5, respectively). . We find that for ψ 0 sufficiently large, two different solutions to (10) exist: one for positive α (purple solid) and one for negative α (green dashed).
function σ(r) at r = 0, of the background boson star as well as the value of α > 0, for which bound states exist, are given in Table 1. The value of the effective potential V eff (r = 0) increases with increasing ψ 0 or equivalently decreasing σ(0). We observe that the larger the value of ψ 0 , the larger we have to choose the value of α in order to obtain bound state solutions. In Fig. 2 we give the value of the scalar-tensor coupling α for which a non-trivial solution of (10) exists in dependence of the value ψ 0 (left side of Fig.  2) and in dependence of the frequency ω/m (right side of Fig. 2) of the boson star, respectively. For comparison, we have also plotted the mass M of the background boson star. As both figures indicate, we find that for sufficiently large values of ψ 0 and for ω approximately in the interval [ω min : ω cr,1 ], i.e. the interval of existence of the second branch of boson star solutions, a solution for positive α and one for negative α exists, respectively. This is related to the fact that the effective potential V eff (see (12)) becomes negative close to the origin of the coordinate system Fig.1 clearly demonstrates this for this range of frequencies and hence a tachyonic instability exists for α < 0. To give an idea on the difference between the solutions, we show the profile of φ(r) for a boson star background with ψ 0 = 1.468 (which corresponds to ω = 0.85) for the two possible solutions in Fig. 3 . Clearly, the solution with α = −34.85 is non-trivial much closer to the center of the boson star and is monotonically decreasing, while the α = 4.16 solution has a maximum roughly at radius of order unity, which corresponds more or less to the outer radius of the boson star, as the profile of the boson star scalar field ψ(r) indicates.

Influence of backreaction
The results obtained in the previous section indicate that scalarization of boson stars is possible. In the following, we demonstrate that, indeed, scalarized boson stars exist when taking the full backreaction of the space-time into account. To study the effect, we have fixed the value of the frequency ω and studied the dependence of the parameters describing the solution in function of α. We have concentrated on the main branch of solutions, choosing ω = 0.96 and ω = 0.85, respectively, which corresponds to ψ 0 ≈ 0.05 and ψ 0 ≈ 0.28 with φ 0 small, for which the non-backreacted case is a good approximation. Using these solutions we have then increased the value of φ 0 to study the influence of backreaction. Our results for α > 0 are show in Fig. 4 . Here, we give the mass M and the Noether charge Q (left) as well as the value of ψ 0 and the value of the gravity scalar φ(r) at the origin, φ(r = 0) ≡ φ 0 (right), respectively. The figure . We also give the value of ψ 0 (purple dashed) and φ 0 (solid green) as function of α for these two different frequencies.
demonstrates that scalarized boson stars exist only for α sufficiently large. When increasing α, the mass M and Noether charge Q decrease with the ratio M/Q decreasing indicating that the gravity scalar leads to a stronger binding, which seems sensible since it acts attractively. This is true for both ω = 0.85 and ω = 0.96. Moreover, for both values of the frequency ω, the value of the gravity scalar φ(r) at the center of the boson star, φ 0 , first increases to a maximal possible value and from there decreases when increasing the value of α. Moreover, we find that the value of the scalar field forming the boson star at the centre of the boson stars, ψ 0 , decreases when increasing α. In order to characterise the scalarized boson stars, we have also computed the scalar charge Q φ associated to the fall-off of the gravity scalar φ at infinity, see (9). The dependence of this charge on the ratio between mass and Noether charge, M/Q, is shown in Fig. 5. This figure demonstrates that for a fixed value of M/Q, the scalar charge Q φ characterises the scalarized boson star uniquely. With increasing M/Q, the scalar charge increases up to a maximal value and then decreases sharply to zero at the maximal possible value of M/Q for these solutions. Note that M/Q ≤ 1 for all scalarized boson stars for this choice of parameters. We have also studied solutions with larger values of ψ 0 that correspond -in the non-backreacted case -to solutions on the second and further branches, respectively. As demonstrated above, the effective potential in (12) becomes negative when increasing ψ 0 . Hence, our results indicate that the branch of backreacted scalarized boson stars for positive α becomes smaller in extend and very likely disappears for sufficiently large ψ 0 . However, scalarized boson stars are possible for α < 0 and we have constructed these solutions numerically. More details will be given elsewhere.

Conclusions
Spontaneous scalarization of black holes in scalar-tensor gravity models has been discussed extensively recently. Since static, electro-vacuum, asymptotically flat black holes fulfill R ≡ 0, these black holes cannot be scalarized in a αφ 2 R model. In this paper, we have demonstrated that ultra-compact objects that can account for the supermassive centres of galaxies, i.e. black holes and boson stars, behave differently in a scalar-tensor gravity model with a direct coupling between the gravity scalar and the Ricci scalar. While spherically symmetric, electro-vacuum black holes cannot be scalarized, non-rotating boson stars can carry long-range scalar fields in this model. These solutions can be uniquely characterised by the scalar charge Q φ as well as by the mass to Noether charge ratio M/Q. In contrast to the other distinguishing feature between a black hole and a boson star the existence of an event horizon the scalar charge Q φ can be measured far away from these objects since the scalar field has a power law fall-off at infinity. Let us remark that the existence of branches for both values of the scalar-tensor coupling, albeit in the case of a coupling between the gravity scalar and the GB term, is also present in the case of static, charged black holes [31] as well as for static, spherically symmetric neutron stars that are a solution of the Tolman-Oppenheimer-Volkoff equations [22]. This can be traced back to the fact that the effective potential in the scalar field equation (see e.g. (12)) becomes negative close to the event horizon or center of the compact object, respectively.