Stringent constraints on the light boson model with supermassive black hole spin measurements
SStringent constraints on the light boson model with supermassive black hole spin measurements
Lei Zu , , Lei Feng , ∗ , Qiang Yuan , , † , Yi-Zhong Fan , ‡ Key Laboratory of Dark Matter and Space Astronomy,Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210033, China School of Astronomy and Space Science, University of Science and Technology of China, Hefei, Anhui 230026, ChinaJoint Center for Particle, Nuclear Physics and Cosmology,Nanjing University – Purple Mountain Observatory, Nanjing 210093, China Center for High Energy Physics, Peking University, Beijing 100871, China
Massive bosons, such as light scalars and vector bosons, can lead to instabilities of rotating black holes by thesuperradiance e ff ect, which extracts energy and angular momentum from rapidly-rotating black holes e ff ectively.This process results in spinning-down of black holes and the formation of boson clouds around them. In thiswork, we used the masses and spins of supermassive black holes measured from the ultraviolet / optical or X-rayobservations to constrain the model parameters of the light bosons. We find that the mass range of light bosonsfrom 10 − eV to 10 − eV can be largely excluded by a set of supermassive black holes (including also theextremely massive ones OJ 287, Ton 618 and SDSS J140821.67 + ∼ × − eV to 10 − eV with a decay constant f a > GeV can be excluded, which convincingly eliminate the QCD axions at these masses.
PACS numbers: 14.80.Va,95.35. + d,97.60.Lf,05.30.Jp I. INTRODUCTION
The scattering between a rotating black hole (BH) and lightbosons can extract energy and angular momentum of the BH,which is the so-called superradiance e ff ect [1–12]. This phe-nomenon takes place when the superradiance condition is sat-isfied, i.e., 0 < ω < mw + , (1)where ω is the frequency corresponding to the light boson, m is the magnetic quantum number, and w + is the angular veloc-ity of the BH horizon which reads w + = r g a ∗ + (cid:112) − a ∗ , (2)where a ∗ = a / r g is the dimensionless spin parameter of theBH with a = J / M bh being the spin-to-mass ratio and r g = GM bh being the gravitational radius.When the wavelength of the light boson is comparable tothe size of the BH, the number of bosons surrounding theBH grows exponentially to form a boson cloud. The self-interaction of the bosons would lead to the collapse of thecloud when reaching a critical size, which is known as “bosen-ova” [10, 13]. The superradiance process extracts energy andangular momentum from the BH, which enables us to con-strain the parameters of the light boson if the masses and spinsof BHs have been reasonably measured [10, 14–16].Supermassive black holes (SMBH) at the centers of galax-ies have been found with masses ∼ − M (cid:12) . To mea-sure the spin of the SMBHs is somehow challenging. By ∗ [email protected] † [email protected] ‡ [email protected] means of the first direct imaging of the SMBH in the centerof M 87 with the Event Horizon Telescope (EHT), the massof the SMBH was determined to be ∼ . × M (cid:12) [17, 18],and it was inferred to be likely highly spining [19, 20]. TheEHT measurement was then applied to exclude the ultralightbosons with mass between 8 . × − eV and 4 . × − eVfor a given BH time scale τ bh ∼ years [14]. For mostof SMBHs, there are lack of direct imaging of the shadows,and the spin parameters are instead estimated through fittingthe X-ray and / or UV-optical spectral energy distributions [21–23].In this work we employ the currently available sampleof high-spin SMBHs based on the UV-optical / X-ray spec-troscopy method [22, 24], to constrain the ultralight bosonmodel. The sample of SMBHs with di ff erent masses can covera wide mass range of the light bosons. Moreover, with afew extremely massive BHs, we probe the fuzzy dark mat-ter (FDM) scenario with boson mass of (1 ∼ × − eV[25–27] which is almost out of reach via only the EHT obser-vations of M 87 [14]. II. SUPERRADIANCE
Superradiance is a kind of Penrose process related to waves.A massive boson field in the Kerr background may causean unstable solution with an imaginary part of the frequency[10, 28, 29]. It leads to an exponential growth of the numberof bosons, forming a “gravitational atom” with energy levels( c = (cid:126) = (cid:15) (cid:39) µ (cid:32) − α n (cid:33) , (3)where µ is the mass of the boson, α = r g µ , and ¯ n = n + l + n and l being the principal and orbital quantum numbers,respectively [10, 29]. a r X i v : . [ a s t r o - ph . H E ] J u l The leading contribution of the imaginary part of frequencyis (i.e., the fastest-superradiating mode of the small- α analyt-ical approximation) [6, 30] Γ s = a ∗ r g µ , (4) Γ v = a ∗ r g µ , (5)where Γ s and Γ v are the superradaince rates related to scalarand vector bosons. In this work, we use the leading term ofthe analytical solution, which is consistent with the numericalresults [29, 30], to constrain the masses of the bosons.Without taking into account self-interactions, the numberevolution is simply dNdt = Γ N . (6)For a BH with given spin, the maximally allowed size ofthe boson cloud is N max (cid:39) GM µ ∆ a ∗ , (7)where ∆ a ∗ is the di ff erence between the initial and final spinsof the BH. The corresponding boson parameter space is sim-ply ruled out if the superradiance process is e ffi cient enoughthat the BH lose too much angular momentum within its life-time τ bh , i.e., Γ τ bh > ln N max , (8)which in turn sets a bound on the boson parameters.As the size of the cloud keeps growing, the self-interactione ff ect becomes important. When it grows up to a critical size, N bosenova , the cloud collapses which is known as “bosenova”.For scalar particles we have N bosenova (cid:39) c n α (cid:32) M bh M (cid:12) (cid:33) (cid:32) f a M pl (cid:33) , (9)where f a is the decay constant for the scalar bosons, M pl = × GeV is the Planck energy, and c ∼ N bosenova < N max ), the scalar cloud collapses when its size reaches N bosenova . Thus this process would repeat N max / N bosenova timesat most and the maximally allowed size of the boson cloudwould be replaced by N bosenova . Therefore the exclusion con-dition Eq. (8) can be revised as [8] Γ τ bh ( N bosenova / N max ) > ln N bosenova . (10) III. THE SMBH SAMPLE AND CONSTRAINTS
A widely adopted way to estimate the spin of an SMBH isto fit the radiation spectrum in UV-optical or X-ray bands (i.e.,to model the AGN continuum emission or relativistic X-ray reflection). We summarize some SMBHs with inferred highspins with this method in Table I [22, 24]. There are someeven more massive black holes. OJ 287 has the dynamicallymeasured mass M bh = (1 . ± . × M (cid:12) and spin a ∗ = . ± .
004 [32]. The masses of Ton 618 and SDSSJ140821.67 + . × M (cid:12) [33]and 1 . × M (cid:12) [34], respectively. The allowed param-eter space of the BH mass and spin within the thin disc ac-cretion model has been examined in Ref. [35], giving lowerlimits of the spin of about 0.6 for Ton 618 and 0.97 for SDSSJ140821.67 + τ Salpeter ∼ . × years [36], which is theEddington limit for the accreting material.We firstly ignore the self-interaction of light bosons [14, 16,30]. In this case the constraints can be obtained through thecombination of Eq. (1) and Eq. (8). We further restrict ourdiscussion in the m = ff erent sources are shown in Fig. 1.The green (red) bands correspond to the exclusion regions forthe scalar (vector) bosons by the SMBHs summarized in Ta-ble I. The orange (blue) bands are those for the most massiveSMBHs OJ 287, Ton 618, and SDSS J140821.67 + − ∼ − eV can be e ff ectively constrained by thesesources, particularly for the vector boson scenario due to itsfaster superradiance rate compared with the scalar scenario(see Eq. (4) and Eq. (5)). There is a lack of constraints for µ ∼ − eV. We expect that additional SMBHs with massesaround 10 M (cid:12) will be helpful to close the mass windowaround 10 − eV. Fig. 2 shows the exclusion capabilities ofSDSS J140821.67 + a ∗ ∼ .
4, the constraints are still e ff ective.It is interesting to note that the most massive SMBHs OJ287, SDSS J140821.67 + − ∼ − eV [27]) if it consists of vectorbosons. For the scalar case, the constraint is less stringent.The exclusion regions also depend on the value of the BHlifetime τ bh . Longer lifetime would give wider exclusion re-gions. In Fig. 1, we also show the constraints from M 87, with M bh = (6 . ± . × M (cid:12) and a ∗ = . ± .
1. Here we take τ bh ∼ . × years rather than ∼ years as in Ref. [14].This is why our constraint is not as strong as that in Ref. [14]for this specific object. Anyhow, our enlarged SMBH samplewith diverse masses and spins are clearly more powerful inconstraining the light boson model.If the self-interaction is strong enough ( N bosenova < N max ),the boson cloud collapses when growing up to the critical size N bosenova , after which the superradiance process restarts andthe cycle repeats. The exclusion condition is now Eq. (1) andEq. (10). Generally speaking, including the self-interaction of TABLE I: SMBHs with masses and spins measured with various method.Object M bh (10 M (cid:12) ) Spin Refs.X-ray 3C 120 0 . + . − . > .
95 [37, 38]MCG 6-30-15 0 . + . − . > .
98 [39, 40]Mrk 110 0 . + . − . . + . − . [37, 41]Mrk 335 0 . + . − . > .
91 [37, 42]NGC 3783 0 . + . − . > .
98 [37, 43]NGC 4051 0 . + . − . > .
99 [37, 44]NGC 4151 0 . + . − . > .
90 [45, 46]NGC 5506 0 . + . − . . + . − . [47, 48]UV / optical J1152 + . + . − . . + . − . [22]J1158 − . + . − . . + . − . [22]J0941 + . + . − . . + . − . [22]J0303 + . + . − . . + . − . [22]J0927 + . + . − . . + . − . [22]Mostmassive OJ 287 183 . ± .
08 0 . ± .
004 [32]Ton 618 660 > .
60 [33]SDSS J140821.67 + > .
97 [34] bosons would introduce one more free parameter, the decayconstant f a , and relax somehow the constraints. As an illus-tration, we discuss the scalar case in this work. The bosenovaprocess for vector bosons with self-interactions is more com-plicated, and may need more dedicated studies in future. Herewe use the numerical solution to the superradiance rate Γ forscalars such as axions in Ref. [29]. The excluded parameterspace in the mass-coupling plane of the scalars is shown bythe shaded regions in Fig. 3, assuming τ bh = . × years.The mass range from ∼ × − eV to 10 − eV with a decayconstant f a > GeV can be excluded. It is very interestingto see that the theoretical prediction of the mass-coupling re-lation for QCD axions [8] has been excluded in a wide massrange with our sample. Note also that the spin method con-strains the axion parameters with large f a , which is comple-mentary to the polarization method [49]. IV. SUMMARY AND DISCUSSION
Superradiance leads to an e ff ective extraction of angularmomentum from a rapidly-rotating BH. Therefore the SMBHswith measured masses and spins serve as powerful probes ofthe presence of light bosons around rotating BHs. Using asample of high-spin SMBHs, inferred from the UV-opticalor X-ray spectroscopy, we constrain the model parameters oflight bosons in this work. The boson mass in the range of(10 − ∼ − ) eV are e ff ectively constrained. The most massive BHs OJ 287, SDSS J140821.67 + ff ectively extend the constraints to the FDM massrange. We also consider to include the self-interactions forscalar bosons, and exclude a large part of the parameter spacefor 3 × − < µ/ eV < − and f a > GeV.We are aware that there are several uncertainties of the re-sults, such as the systematical uncertainties of the parameterestimates of the BH masses and spins, and the calculation ofthe superradiance rate of bosons. Furthermore, the inclusionof other modes of the boson cloud would also change some-how the quantitative results derived in this work. We leavesuch improvements / refinements in future works. We expectthat more precise determinations of the masses and spins ofSMBHs in future will provide significantly improved and ro-bust constraints on the light bosons in a wide mass region. Acknowledgements
This work is supported by the National Key Re-search and Development Program of China (Grant No.2016YFA0400200), the National Natural Science Founda-tion of China (Grants No. 11525313, No. 11722328,No. 11773075, No. U1738210, No. U1738136, and No.U1738206), the 100 Talents Program of Chinese Academy ofSciences, and the Youth Innovation Promotion Association ofChinese Academy of Sciences (Grant No. 2016288). [1] R. Penrose, Nuovo Cimento Rivista Serie , 252 (1969).[2] J. M. Bardeen, W. H. Press, and S. A. Teukolsky, Astrophys. J. , 347 (1972). [3] C. W. Misner, Phys. Rev. Lett. , 994 (1972).[4] W. H. Press and S. A. Teukolsky, Nature , 211 (1972).[5] W. H. Press and S. A. Teukolsky, Astrophys. J. , 649 (1973).
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22 21 20 19 18 17 log ( /eV)FDM (b)combined bounds
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