Superradiant Amplification by Stars and Black Holes

Superradiant ampliﬁcation by stars and black holes

Vitor Cardoso1,2 , Richard Brito1, Jo˜ao L. Rosa1, 1 CENTRA, Departamento de F´ısica, Instituto Superior T´ecnico – IST, Universidade de Lisboa – UL, Avenida Rovisco Pais 1, 1049 Lisboa, Portugal and 2 Perimeter Institute for Theoretical Physics Waterloo, Ontario N2J 2W9, Canada

It has long been known that dissipation is a crucial ingredient in the superradiant amplification of wavepackets off rotating objects. We show that, once appropriate dissipation mechanisms are included, stars are also prone to superradiance and superradiant instabilities. In particular, ultralight dark matter with small interaction cross section with the star material or self-annihilation can trigger a superradiant instability. On long timescales, the instability strips the star of most of its angular momentum. Whether or not stationary hairy configurations exist, remains to be seen.

PACS numbers: 04.70.-s,04.80.-y,12.60.-i,11.10.St

I. INTRODUCTION long as dissipation is included [23]. The purpose of this work is to compute explicitly the amplification factors when a toy model for dissipation is used, and to show Rotating bodies with internal degrees of freedom – that, within this toy model, stars are superradiantly- where energy can be dumped into – display superradi- unstable against massive scalar-field fluctuations. ance. This means that the scattering of low-frequency Our results make possible the existence of hairy stars, waves extracts rotational energy away from the body and supported solely by superradiance. For dark matter mod- is used to amplify incoming radiation [1–3]. When the els comprised of light degrees of freedom, our results open object is a black hole, dissipation is intrinsically built the door to constrain the interaction cross section with in the problem under the form of a one-way membrane: ordinary matter or the self-annihilation cross-section. the event horizon. Black hole superradiance was in fact shown to exist and to trigger interesting phenomena. These include black hole bombs, once the black hole is II. SETUP surrounded by some confining medium, giving rise to ex- ponentially growing modes [4–8]. The scattering of massive fields belongs to this category: the mass term effec- We are interested in describing ultra-light bosonic de- tively confines the field giving rise to floating orbits [9] grees of freedom, described by a scalar minimally cou- or to superradiant instabilities, which extract rotational pled to gravity, and interacting with a star. Dissipation energy away from the black hole [10–16]. The instabil- can be modelled in several – and complex – ways. For ity leads to a depletion of mass and angular momen- instance, one can let the scalar couple to spacetime de- tum from the black hole, resulting in observationally- grees of freedom, or one can consider the coupling to mi- interesting “holes” in the Regge-plane [17–19]. As a croscopic degrees of freedom, describing the interaction consequence, the observation of supermassive black holes of self-annihilation of – for example – dark matter with alone can impose stringent bounds on the mass of ultra- a star. The description of a dissipative system within light bosons [13, 15]. Finally, in multi-scalar theories or full General Relativity for systems (such as stars) with theories with a complex scalar, superradiance is the re- many internal degrees of freedom is a very complex prob- sponsible for new hairy black hole solutions [20] (whether lem [24]. or not these solutions are generically approached as the Instead, we follow Zeldovich’s seminal work and con- end-state of the superradiant-instability is discussed in sider a toy model for absorption, by modifying the Klein- Ref. [18]). For a Review, we refer the reader to Ref. [3]. Gordon equation inside the star and in a frame co- rotating with the star as [21] Given the rich phenomenology of black hole superradiance, one might ask whether those results carry through ∂Φ ✷Φ+ α = µ2Φ . (1) to other self-gravitating objects. Ideal stars do not dis- ∂t play superradiance and none of the associated instabilities (except for possible ergoregions instabilities [3]). But Here, µ~ is the mass of the scalar field. The α term is realistic stars do dissipate energy and they have several added to break Lorentz invariance, and describes absorp- channels available for it. The original arguments by Zel- tion on a timescale τ 1/α, in a frame co-rotating with dovich already made clear that any classical system with the star. The constant∼ 1/α has dimensions of time, or dissipation will be prone to superradiant scattering [1, 21] length in geometrical units. Contact with particle physics (see also Ref. [22] who studied the correspondence be- can be made by considering the particle associated with tween superradiance and tidal friction on viscous Newto- the scalar field and identifying 1/α with the mean free nian anisotropic stars). In fact, stars in full General Rel- path ℓ of the particle inside the star. In other words, for a ativity were recently shown to display superradiance, as particle with cross section σ to decay into other particles 2 via interaction with say a neutron of mass mN , then when transforming to a frame at rest with the star, and using (1). α = nσ , (2) where n is the nucleon number density in the star. One can also express the dissipation parameter α with the III. SUPERRADIANT AMPLIFICATION help of the star’s total mass M, in the dimensionless com- FACTORS OF MASSLESS FIELDS bination

GM ρσ -2 Mα = 2 10 c mN -3 10 R=50M ρ M σ -4 10 0.92 15 −3 −41 2 . (3) ∼ 10 Kg m M⊙ 10 cm -5

-1 10 2

| -6 Thus, the dimensionless combination can be of order one in 10 A R=10M -7 for interesting dark matter candidates [25–27] (and ref- / | 10 2 erences therein). The constant α could also be used sim- | -8

out 10 A ilarly to describe dark matter self-annihilation. | -9 10 R=5M It is easy to see, following Zeldovich, that if the fre- -10 10 numerical quency in the accelerated frame is ω and the field behaves -11 analytical −iωt+imϕ 10 as e then in the inertial frame the azimuthal -12 coordinate is ϕ = ϕ′ Ωt, and hence the frequency is 10 0 0.002 0.004 0.006 0.008 0.010 ω ω′ = ω mΩ. In other− words, the effective damping pa- M rameter−αω′ becomes negative in the superradiant regime and the medium amplifies – rather than absorbing– radi- FIG. 1. Superradiant amplification of a scalar field by a rotat- ation [1, 3, 21]. This procedure can be used to derive the ing star, where dissipation is modeled through equation (1). wave equation of a scalar field in a rotating black hole Here, Mα =0.1, MΩ=0.01. We find that the amplification background (up to first order in rotation), starting from factor scales with the dissipation parameter α for small Mα. that of a scalar field in a Schwarzschild geometry. As we For non-compact, slowly-rotating stars, the amplification fac- will shortly see, it also allows to predict the amplification tor (solid lines) is well described by the analytic result (8), factors of scalar fields in a Kerr background. Thus, while shown as a dashed curve in the plot. only a phenomenological approach to the problem, the inclusion of the Lorentz-violating term is well motivated To solve the Klein-Gordon equation (1), we use the and consistent with known results. In Appendix A we following ansatz for the field, work out another model for dissipation, inspired in the theory behind dissipation in fluids [28]. The results are Ψ(r) Φ(t, r, ϑ, ϕ)= e−iωt+imϕP (cos ϑ) , (6) perfectly consistent with the toy model (1) adopted in r l the main body of this work. As is clear from the previous discussion, superradi- where Pl are Legendre functions. To solve the equation ance requires only rotation and dissipation. The par- inside the star, we first transform to a frame co-rotating ticular background geometry or material is only relevant with the star (which amounts to replacing ϕ ϕ Ωt, or ω ω mΩ), impose regularity at the center→ and− then to quantitatively describe the phenomena. We will focus → − exclusively on backgrounds describing a constant density, match the radial function and its derivative. For now we perfect fluid star in General Relativity as a proxy for the focus on zero field mass, µ = 0. At large distances, the generic case: solution takes the form 2 iωr −iωr 2 2Φ 2 dr 2 2 Ψ= Aoute + Aine , (7) ds = e dt + + r dΩ , (4) − 1 2m(r)/r − where the subscripts denote an out-going and an in-going with piece (due to the time dependence of the ansatz for the 4π 1 2Mr2/R3 1 2M/R field). Thus, the asymptotic solution allows one to frame m = ρr3 , P = ρ − − − , 2 3 this in a context of a scattering experiment. We define 3 3 1 2M/R 1 2Mr /R ! 2 2 p − − p− the amplification factor as Aout / Ain 1. Positive 3 1 | | | | − eΦ = 1 2M/R p1 2Mr2/R3p. (5) amplification factors correspond to superradiant ampli- 2 − − 2 − fication from the star. Negative factors correspond to Here, M,Rp are the star’sp mass and radius, and are related overall absorption by the star. 4π 3 to the density ρ via M = 3 ρR . Our numerical results are summarized in Fig. 1 for Notice that we take an exact general-relativistic solu- Mα = 0.1, MΩ = 0.01. Superradiant amplification does tion describing a non-rotating star. Although not self- exist, as was shown generically in Ref. [23], and the am- consistent, the effects of General Relativity are subdomi- plification can be significant. In fact, we find that it nant. The leading effect of rotation is taken into account scales linearly with Mα for small Mα and it increases 3 significantly as the star’s surface velocity increases. Am- V. DISCUSSION plification factors can be of order of those around rotating Kerr black holes or higher. For non-relativistic, Newtonian configurations Neutron stars or other highly compact stars with dis- (M/R, ΩR 1) the wave equation can be solved sipative channels may be interesting probes of physics analytically≪ inside and outside the star in terms of beyond the Standard Model, specially of bosonic fields Bessel functions. We find a simple analytic expression with a Compton wavelength of the order of the star size. for the amplification factor, Our working model is extremely crude and more sophisti- cated ways of including dissipation should be sought for. 2 2l+1 Nevertheless, our work certainly shows that there may be Aout 4αR (RΩ ωR)(ωR) | |2 1= − . (8) a payoff to these efforts: highly spinning compact stars Ain − (2l + 1)!!(2l + 3)!! | | develop instabilities against massive fields which may be As can be seen from Fig. 1, this relation gives a very used (see e.g. Refs. [3, 18]) to further constrain ultralight good approximation to the numerical results for a good bosonic fields, while the bosonic clouds formed during the fraction of the parameter space, including relatively com- process might emit very distinctive gravitational-wave pact stars. This relation is also interesting, as it allows signals within the optimal sensitivity band of ground- one to predict the amplification factor for rotating black based detectors [19]. It is even possible, but it remains holes. For black holes, R 2M and 1/α = M is the to be proven in a concrete theory with dissipation, that only possible timescale in the∼ problem. For example, the hairy stars exist, which would be the counterpart to hairy above relation predicts that slowly rotating black holes black hole solutions recently studied in theories with a in General Relativity amplify l = 1 scalar fields with minimally coupled massive scalar [20, 30].

2 Aout 16 3 | |2 1= M (Ω ω) (2Mω) . (9) Ain − 45 − | |

On the other hand, a matched-asymptotic expansion cal- ACKNOWLEDGMENTS culation in full General Relativity yields approximately the same result (the coefficient turns out to be 2/9 0.22 ∼ instead of 16/45 0.355 [3, 29]). V.C. acknowledges financial support provided under ∼ the European Union’s FP7 ERC Starting Grant “The dynamics of black holes: testing the limits of Einstein’s IV. SUPERRADIANT INSTABILITY AGAINST theory” grant agreement no. DyBHo–256667, and FP7 MASSIVE FIELDS ERC Consolidator Grant “Matter and strong-field gravity: New frontiers in Einstein’s theory” grant agree- As soon as the field is allowed to be massive, superradi- ment no. MaGRaTh–646597. R.B. acknowledges finan- ance can trigger new phenomena, in particular instabili- cial support from the FCT-IDPASC program through ties: the mass confines the field within a distance 1/µ2 of the Grant No. SFRH/BD/52047/2012, and from the the star, whereas superradiance amplifies it. This effect Funda¸cão Calouste Gulbenkian through the Programa was shown to occur for black holes [3, 10–16], we now Gulbenkian de Est´ımulo àInvestiga¸cão Cient´ıfica. This show that it occurs generically. research was supported in part by the Perimeter Institute For massive fields, the asymptotic behavior at spatial for Theoretical Physics. Research at Perimeter Institute 2 2 infinity is of the form e±√µ −ω r. Thus, it is possible is supported by the Government of Canada through In- dustry Canada and by the Province of Ontario through that some eigenvalues ω = ωR + iωI exist for which the the Ministry of Economic Development & Innovation. field is regular at infinity (for ωR <µ) and for which the This work was supported by the NRHEP 295189 FP7- mode is unstable (for ωI > 0, cf. Eq. (6)). We have used a direct integration approach to search for these modes. PEOPLE-2011-IRSES Grant. Figure 2 summarizes our results for the instability of massive scalars coupled to dissipative stars. For a fixed star radius and mass (R, M) and dissipation parameter α, there are unstable modes. At very small values of Mµ, the instability timescale decreases when the scalar mass µ increases. We find that at low masses µ our results η are well described by ωI (ωR mΩ)(Mµ) with η ∝ − ∼ 9 2. However, the real part of the mode ωR µ also increases.± Thus, for µ Ω the m = 1 mode ceases∼ to be unstable. This is clearly∼ seen in Fig. 2, which shows that the transition from instability to stability coincides with a change in sign of ωR mΩ. − 4

-6 10 10

-7 10 9 | I

ω -8 10

|M 8 -9 )

10 8 7 x 10 (

I 6 -2 10 ω )| M

Ω -3

- 5

R 10

ω -4 10 4 |M ( -5 10 -6 3 10 0.14 0.16 0.18 0.2 0.22 0 10 20 30 40 50 60 70 80 90 100 Mµ αM

FIG. 2. Superradiant instability of a massive scalar ﬁeld coupled to a rotating star, where dissipation is modeled through equation (1). Here, R/M = 4, MΩ=0.2, Mα = 20 (left panel) and Mµ = 0.17 (right panel). The transition from instability ωI > 0 to stability ωI < 0 coincides with a change in sign of ωR − mΩ. We ﬁnd that MωI scales linearly with the dissipation parameter α for small Mα.

Appendix A: Superradiant amplification factors in a when the dissipation inside the star is modeled through viscous gravity analogue (A1). We are here concerned only with rotational superradiance in flat spacetime, and we therefore take 2 to be the flat-space Laplacian in spherical coordinates.∇ -7 10 The process of transforming to a co-rotating frame and the imposition of regularity at the center are identical to -8 10 that of the model studied in the main text. The amplification factors were numerically obtained and are shown -1 2

| -9 in Fig. 3. Note that for low frequencies it is possible in 10 A to map this analogue model into the previous toy model. / | 2 | -10 Equation (A1) can be written as

out 10 A |

-11 4 4 10 ✷ 2 numerical 1 iων φ + iων∂t φ = 0, (A2) analytical − 3 3 -12 10 0 0.1 0.2 0.3 0.4 0.5 ωR which in the limit ω 1 is the same as Eq. (1), for 4 2 ≪ α = 3 νω . This map between α and ν allows us to FIG. 3. Superradiant amplification of a scalar field by a rotat- obtain the non-relativistic analytical expression for the ing star, where dissipation is modeled through equation (A1). 4 amplification factors from equation (8): Here, 3R ν =0.002, RΩ=0.5.

2 2 3 2l+1 A different way to model dissipation in a gravitational Aout 16νR (Ω ω) (ωR) | |2 1= − . (A3) Ain − 3(2l + 1)!!(2l + 3)!! system is to use the methods of analogue gravity. For | | example, linear perturbations in an inviscid, irrotational flow propagate as fields on a curved space-time. Thus, It is worth noting that, if one were to try the naive one natural option to model dissipative phenomena is black hole limit R = 2M with ν of order M, the above through the inclusion of a kinematic viscosity. For a fluid result would never recover the correct behavior for the at rest, the wave equation for the perturbations is given amplification of waves by rotating black holes. The rea- by [28] son, it seems, is that dissipation in non-rotating black holes can be best described through (1) rather than this 2 2 4 2 ∂t φ = φ + ν∂t φ. (A1) viscous analogue. This is possibly due to the fact that ∇ 3 ∇ the dispersion relation of Eq. (A1) in the eikonal limit Thus, we will here briefly describe how our results change leads to an energy-dependent dissipative term [28]. 5

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