Boson Star Superradiance

Recently, it has been realized that in some systems internal space rotation can induce energy amplification for scattering waves, similar to rotation in real space. Particularly, it has been shown that energy extraction is possible for a Q-ball, a stationary non-topological soliton that is coherently rotating in its field space. In this paper, we generalize the analysis to the case of boson stars, and show that the same energy extraction mechanism still works for boson stars.

The term superradiance was originally coined by Dicke [38] in the study of emission enhancement of radiation in a coherent medium.Subsequently, it has been widely employed to describe a diverse array of phenomena characterized by enhanced radiation.One of the prominent examples is (real space) rotational superradiance, which refers to the ability of a rotating body to transfer its energy and angular momentum to the surrounding environment via its scattering of the waves impinging on it [39,40].As Kerr black holes are just such rotating bodies, black hole superradiance has been widely studied in the field of gravitational physics and astrophysics over the past decades .Superradiance in rotating stars has recently been discussed in [65][66][67][68][69]. Reviews about superradiance can be found in [70,71].
These intriguing structures may arise naturally during the early stages of the universe [76][77][78][79][80]. Furthermore, Qballs have also been postulated as plausible candidates for dark matter [81][82][83][84][85], and can have composite nonlinear structures [86][87][88].Recently, Ref [89] has discovered a novel mechanism to extract energy from a Q-ball by incident waves, which was referred to as Q-ball superradiance.The energy extraction is possible owing to the Q-ball's coherent internal rotation in field space and the coupling of the two inherent scalar modes.While Ref [89] defined the energy amplification with the total energy in a far-away region and identified the amplification criteria accordingly, Ref [90] pointed out that this does not coincide with the energy amplification defined using the flux to and from infinity, due to the difference between the group velocities of the two coupled modes [90].Indeed, with the flux definition, a Zel'dovich-like amplification criterion can be found when there is only one ingoing mode.The difference between the two definitions presents an intriguing feature of such a system.Ref [90] has also argued that generic time-periodic solitons such as Newtonian boson stars are prone to energy extraction.In view of the resemblance between Q-balls and (relativistic) boson stars, it is reasonable to anticipate a similar mechanism applies to boson stars as well.The purpose of this paper is to confirm this anticipation and explicitly demonstrate that the internal rotation of a (relativistic) boson star also allows the same kind of energy extraction.
The paper is organized as follows.Section II presents the model we will focus on, setting up the stage.In Section III, we compute the background boson star solutions by formulating it in a manner suitable for solving using a 1D shooting method.In Section IV, we investigate the superradiant amplification of scattering waves impinging on a boson star and delineate the criteria for the energy extraction to occur.We summarize in Section V.

II. MODEL
As mentioned above, for our novel energy extraction mechanism to work, we need at least two interacting de-grees of freedom.So we consider a complex scalar field Φ minimally coupled to the Einstein-Hilbert action: where the potential V is to be specified later.Depending on convenience, we shall also use the reduced Planck mass M P = (8πG) −1/2 .The equations of motion for this system are given by where |Φ| 2 = Φ † Φ and the energy-momentum tensor of the bosonic field Φ is given by Here the symmetrization is defined as

III. BOSON STAR
The energy extraction mechanism results from rotation in the internal space, so it suffices to demonstrate it in spherical symmetry in real space.To this end, we adopt the following ansatz for the scalar and the metric: where dΩ 2 = dϑ 2 + sin 2 ϑdϕ 2 .φ B (r), v(r) and u(r) are three real radial functions to be solved with the field equations.With this ansatz, the field equations read where , a prime denotes a derivative with respect to r, and the energy density ρ Φ and radial pressure p Φ are given respectively by Defining a radius-dependent mass quantity M(r) via e −u = 1 − 2GM(r)/r, it is easy to see Eq. ( 7) can be re-written as dM(r) dr = 4πr 2 ρ Φ .
So one may loosely refer to M(R) = R 0 4πr 2 drρ Φ as the energy of the boson star within radius R. From this definition, we see that the total energy of the boson star is simply the ADM mass of the asymptotically flat spacetime Now let us parametrize the scalar potential to be used in this paper.We assume that the potential can be expanded perturbatively around Φ = 0. Due to the U(1) symmetry, the leading terms of the effective potential are as follows: A preliminary numerical survey seems to suggest that significant energy amplification is possible if λ < 0, a strong resemblance to the case of Q-balls.Although its effects on the energy extraction is negligible, a small |Φ| 6 term can prevent the vacuum from decaying quantum mechanically.We shall neglect the higher order terms in the following explicit computations.For a given ω B , Eqs. (7-9) with appropriate boundary conditions can be numerically solved by viewing it as an "initial" value problem with r playing the role of "time".That is, one can start at a small r with appropriate "initial" conditions and integrate to a large r to satisfy certain "final" conditions.However, a twodimensional shooting method is needed to directly solve this system, corresponding to choosing appropriate values of φ B and v at r = 0. Fortunately, the time scaling symmetry of the solutions allows us to cast Eqs.(7-9) into a system that can be solved by a one-dimensional shooting method, which is much easier to implement.That is, if a solution is given by Eqs.(5)(6), simultaneously scaling t → λt and e v → λ 2 e v gives another solution.This allows us to eliminate ω B from the equations of motion.
To this end, we shall replace the v variable with the dimensionless variable where Together with the dimensionless variables the equations of motion (7-9) can be written as  17).
where a dot ˙denotes a derivative with respect to x and we have defined The appropriate boundary conditions for an asymptotically flat boson star solution are In this formulation, we only need to shoot to find the correct value of f 0 .The boundary value of σ(+∞) does not require a shooting procedure, as it merely gives the internal rotational frequency of the complex scalar of the boson star: σ(+∞) → 1/w B .In Figure 1, we have computed the relation between the boson star mass M and the internal space rotational frequency of the boson star w B for two example sets of potential parameters.

IV. SCATTERING OFF A BOSON STAR
Having found the boson star solution, we now proceed to scatter waves on to the boson star and examine how the energy of the outgoing waves changes, compared to the ingoing waves.To this end, we view the scattering waves as a small perturbation Θ(t, r) around the boson star background Φ B (t, r) = φ B (r)e −iω B t : To leading order, the equation of motion for Θ is given by where = e −u (∂ rr We shall neglect the back-reaction of the scattering waves on the background geometry.Due to the Θ † term, the two real modes of the perturbative complex scalar are coupled, which, as we will see, is fundamental for our energy extraction mechanism to work.For the potential ( 14), we have where we have defined So we see that the coherent internal rotation of the boson star provides a driving factor e −2iω B t in Eq. ( 25), another important ingredient for the energy amplification mechanism to work.Let us first consider the simplest case of a quadratic potential, The quadratic potential can only support mini-boson stars, whose masses are limited by the Kaup bound M Kaup ≈ 15.9M 2 P /µ [1].Also, mini-boson stars can only achieve relatively low compactness.Much more massive and compact boson stars can be obtained with interacting potentials.The free quadratic potential can be obtained by setting g = h = 0 in Eq. ( 25), which leads to W = 0 and Θ satisfying In this case, there is no mode mixing, so the new energy amplification mechanism is not at work here, which we have confirmed with explicit numerical computations.For the generic case of an interacting scalar field theory, which supports more massive boson stars and thus is more interesting phenomenologically, scattering waves can extract energy from a boson star.To see this, let us consider the minimal case of two scattering modes: where The equations of motion for η ± are Defining FIG. 2. Waves scattering on and off a boson star (cf.Eq. ( 36)).
The frequencies of the two modes are w± = wB ± w.We choose wB > 0 (thus a positively charged boson star) and use solid blue (dashed red) lines to represent positive (negative) charges.
and using the dimensionless variables defined around Eq. ( 17), Eq. ( 32) becomes where again a dot is a derivative with respect to x and we have defined Asymptotically at large x, the scattering waves take the form where we have defined k ± = (w 2 ± −1) 1/2 .This means that for propagating waves, we need |w ± | = |w B ±w| > 1, with the 1 coming from the scalar mass µ 2 in dimensionful variables.Without loss of generality, we shall assume w B > 0, and thus we have a mass gap for the frequencies As we have chosen k ± ≥ 0, whether A ± and B ± represent incoming or outgoing waves depends on the sign of w; see Figure 2 for a quick reference.Before numerically solving Eq. ( 34) for scattering waves with the asymptotical form (36), we would like to point out an important property of this kind of scattering, which is that the "particle number" is conserved in the process.To see this, we define Y = xe q/4 χ + , Z = xe q/4 χ † − (38) and and then Eq. ( 34) and its complex conjugate can be written as It is easy to see that the equations of motion are invariant under a global U(1) transformation where α is a real constant.This U(1) symmetry on the perturbative field is inherited from the U(1) symmetry of the full field.The Noether charge for this symmetry is given by and we have dN /dx = 0.At x = 0, it is easy to see that N = 0. Asymptotically, as x → +∞, we have so conservation of N implies that in the scattering the "particle number" of the ingoing waves is the same as that of the outgoing waves: which we have confirmed numerically.The reason why the above equation corresponds to the particle number conservation in the scattering can be understood as follows.The radial flux of the Noether current at infinity over a unit area and averaged over time is . By identifying a unit charge as a particle and recalling the meanings of these coefficients as in Figure 2, we find that the number of the outgoing particles is equal to that of the ingoing ones.(Recall that the flux of a spherical wave of the form Ce ikr−iωt is −2k|C| 2 .) We also want to define the energy amplification factor for the asymptotic waves.Since the system contains two modes with different group velocity, there are two ways to define the energy amplification.One is to take into account all the scalar energy in an asymptotic spherical shell with interior radius r 1 and exterior radius r where r 2 − r 1 includes at least a full spatial oscillation of the longest wave, denotes time average over a few oscillations and we have kept the leading order in 1/r.Using this total energy measure, the energy amplification factor can be defined as the ratio of energy in the outgoing modes against the ingoing modes, For this definition of amplification and the "conservation" law (44), one can immediately see that the criterion that delineates the amplification and attenuation of the total average energy is as this is the point when A ρ E = 1.Another measure one may use, which was first pointed out in [90], is to compute the amplification factor for the energy fluxes to and from spatial infinity: With this energy flux, we can define another amplification factor For this flux definition, if there is only one mode ingoing (but two modes outgoing), one can find a clear Zel'dovich-like amplification criterion [90].That is, if there is only the + mode ingoing, A F E will be greater than 1 if w + = w B + w < w B , i.e., w < 0; if there is only the − mode ingoing, A F E will be greater than 1 if w − = w B − w < w B , i.e., w > 0.
For the case of a single mode (both ingoing and outgoing), the two definitions of amplification would coincide.However, for the generic case where two modes are involved, they approximately agree with each other far away from the mass gap, but differ near the gap (i.e., when |w ± | = (k 2 ± + 1) 1/2 is close to 1).As we will see, if there is only one mode ingoing (but two modes outgoing), A ρ E is typically greater than A F E for frequencies on one side of the mass gap and less than A F E on the other side of the mass gap.Now we numerically solve Eq. ( 34), viewing it as an "initial" value problem.First, let us clarifiy its "initial" conditions.The χ ± solutions should be regular near the origin r = 0, which requires χ ± (ω, r → 0) → F ± .Also, thanks to the linearity of Eq. (34), if (χ + , χ − ) is a solution, then (ζχ + , ζ * η− ) is also a solution, ζ being a complex constant.Therefore, we can fix the overall scale of the solutions by scaling (say) F + to unity, which gives us the following "initial" condition as r → 0 : The system can be solved with standard ODE solvers in Matlab, for example.Figure 3 and Figure 4 respectively depict how the energy amplification factor A ρ E and A F E change with the frequency w when the ingoing waves only have the η + mode.We have numerically confirmed the superradiant amplification criteria for both of the two definitions with the total and flux scalar energy.In these figures, we have chosen a small h, compared to g, and reducing h further (say h = 0) only changes the plots very slightly.
For the case of a single ingoing mode, we see that the two definitions of energy amplification differ significantly near the mass gap.As mentioned, this is due to the fact that the two modes have different group velocities.Compared to the energy flux at infinity, the total energy in a spherical shell gets enhanced on one side of the mass gap but suppressed on the other side, which is an interesting feature for a system with multiple frequency modes.If the shell-energy in a far-away region is more than the energy flux at infinity, it means that, apart from the energy radiated to infinity, there is also energy localized in far-away regions, as a result of the waves scattering on the boson star.It seems that this kind of energy could also be harvested with appropriate means.
Generically, there can be two ingoing modes.In Figure 5 and Figure 6, we plot the energy amplification factor A ρ E and A F E respectively for various w B and w with both the + and − ingoing modes.The two modes are parameterized by the complex F − parameter, which characterizes the behaviors of the modes near the origin.We see that, compared to the single ingoing mode case, the maximum energy amplification factors can be enhanced for certain combinations of the two ingoing modes.
V. SUMMARY In Ref [89], it has been found that waves scattering on a Q-ball can extract energy from the Q-ball.Due to the similarity between Q-balls and boson stars, it has been postulated that the same energy extraction mechanism works for a boson star.In this paper, we have confirmed this conjecture for the case of relativistic boson stars.The possibility of a similar energy amplification mechanism for a Newtonian boson star has been pointed out by [90].
For an interacting scalar field, there are generically two coupled modes in the problem of waves scattering on a boson star.(If one prepares one ingoing mode, generically there will be two outgoing modes.)It can be shown that the particle number of the outgoing waves is exactly the same as the ingoing ones, so energy amplification is possible thanks to the redistribution of particles between the two modes in the scattering.We have shown that energy amplification is possible with only one ingoing mode (and two outgoing modes), but the maximal energy amplification is typically achieved with two ingoing modes.
We have computed two ways of defining the energy amplification, one with the total scalar energy and another with the scalar energy flux at infinity.The two definitions differ from each other near the mass gap in the frequency band of the scattering waves.
We would like to emphasize that, apart from the coherent internal rotation, the form of the scalar potential is also important for the energy extraction here.For Q-balls, only a restricted class of potentials allows for them to form in the first place, but due to the gravitational attraction boson stars can form for more generic classes of scalar potentials.Nevertheless, a preliminary survey of the potentials seems to suggest that sizable energy amplifications can be achieved when the potential is of the Q-ball type.Therefore, internal time-periodic oscillations are only a necessary condition for the amplification mechanism to work, and appropriate interactions are needed to re-distribute the energy among different modes.
We have referred to the energy extraction mechanism as superradiance or superradiant amplification, although the energy of an individual mode never gets enhanced in the scattering.The term superradiance, as coined by Dicke, originally meant an enhancement of energy in a physical process, and Dicke superradiance is still in ac-tive research nowadays (see for example [91]).For the Q-ball system, since it contains two inseparable modes, it is natural to count all the incoming and outgoing modes to compute the amplification factors.Also, a boson star is a Bose-Einstein condensate within which particles are tightly entangled and oscillates coherently, which is similar to Dicke's original scenario where superradiance is made possible by a medium of coherent atoms.It is in these contexts that we use the term superradiance, not requiring the energy of an individual mode to be enhanced in the scattering as a necessary qualifier.
We have focused on the simplest case of spherically symmetric boson stars, which are rotating in the (scalar) internal space but not in the real space.A natural extension of this work is to consider rotating boson stars and to investigate how the spatial rotation affect the boson star superradiance [92].Of course, it remains to be shown whether a boson star is subject to superradiant instabilities, which needs to satisfy a stronger criterion than the existence of superradiant amplification that we have demonstrated.Or, one may ask whether there are extra mechanisms to turn the energy amplification into superradiant instabilities.We leave these for future work.

12 FIG. 1 .
FIG.1.Relation between the boson star mass M and internal rotational frequancy wB.The potential parameters g and h are defined in Eq. (17).

FIG. 3 .
FIG. 3. Energy amplification factorA ρ E (defined with the total scalar energy) with only the + ingoing modes.wB is the frequency of the boson star internal rotation.The + and − scattering waves have frequency w+ = wB + w and w− = wB − w respectively.The gap around w = 0 is due to the fact that the scalar has a mass.The potential parameters are g = −10 and h = 0.5.

2 FIG. 4 .
FIG.4.Same as Figure3but with the energy amplification factor A F E (defined with the flux scalar energy).

FIG. 5 .
FIG. 5. Energy amplification factor A ρ E (defined with the total scalar energy) for various wB and w with both the + and − ingoing modes.The potential parameters are g = −10 and h = 0.5.

FIG. 6 .
FIG. 6.Energy amplification factor A F E (defined with the flux scalar energy) for various wB and w with both + and − ingoing modes.The potential parameters are g = −10 and h = 0.5.
The action is invariant under a global U (1) symmetry Φ → Φe iκ , κ being a real constant, which leads to a conserved Noether current