Probing the string axiverse by gravitational waves from Cygnus X-1
PProg. Theor. Exp. Phys. , 00000 (8 pages)DOI: 10.1093 / ptep/0000000000 Probing the string axiverse by gravitationalwaves from Cygnus X-1
Hirotaka Yoshino and Hideo Kodama Institute of Particle and Nuclear Studies, KEK, Tsukuba, Ibaraki, 305-0801, Japan ∗ E-mail: [email protected] Department of Particle and Nuclear Physics, Graduate University for AdvancedStudies, Tsukuba 305-0801, Japan . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
In the axiverse scenario, a massive scalar field (string axion) forms a cloud arounda rotating black hole (BH) by superradiant instability and emits continuous gravi-tational waves (GWs). We examine constraints on the string axion parameters thatcan be obtained from GW observations. If no signal is detected in a targeted searchfor GWs from Cygnus X-1 in the LIGO data taking account of axion nonlinear self-interaction effects, the decay constant f a must be smaller than the GUT scale in the massrange 1 . × − eV < µ < . × − eV. Possibility of observing GWs from invisibleisolated BHs is briefly discussed. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Subject Index E31, E02, C15, B29 Introduction
The second-generation ground-based gravitational wave (GW) detectorswill begin operations within a few years and provide us with a new eye to discover various newphenomena, which include those caused by fundamental fields in hidden sector. Promisingcandidates for such hidden sector objects are string axions with tiny masses [1–3]. In stringtheories, a variety of moduli appear when the extra dimensions get compactified, and pertur-batively, some of them are predicted to behave as massless pseudo-scalar fields due to shiftsymmetry from the four-dimensional perspective. Because of nonperturbative effects, thesemassless fields are expected to acquire small masses. The axion masses are naturally expectedto be uniformly distributed in the logarithmic scale in the range − (cid:46) log ( µ [eV]) (cid:46) − µ . Then, arounda rotating black hole (BH) with mass M , the axion field is known to extract the BH rotationenergy through the superradiant instability and forms an axion cloud around the BH, if M µ is O (1) in the natural units c = G = (cid:126) = 1. Such an axion cloud causes rich phenomena dueto its self-interaction, and also emits GWs [4, 5]. In particular, it always emits continuousGWs with frequency ˜ ω ≈ µ .Searches for continuous GWs have been already done for the data of the LIGO science runs,assuming that their sources are rotating distorted neutron stars (see [6] for a recent reportand references for other searches). An important feature of the continuous wave search is thatsensitivity can be improved with the increase in the observation time T obs because the lower c (cid:13) The Author(s) 2012. Published by Oxford University Press on behalf of the Physical Society of Japan.This is an Open Access article distributed under the terms of the Creative Commons Attribution License(http://creativecommons.org/licenses/by-nc/3.0), which permits unrestricted use,distribution, and reproduction in any medium, provided the original work is properly cited. a r X i v : . [ g r- q c ] A p r imit on detectable GW amplitudes is given by h ∼ O (100) (cid:112) S n /T obs with S n being thenoise spectral density, when the frequency width of the GW is smaller than 1 /T obs . Utilizingthis feature, the LIGO team derived the strong upper limit on the amplitude, h UL ∼ − ,from the null detection in the observation data of order one year.The purpose of the present paper is to point out that the same method can be appliedto continuous GWs from the well-known stellar-mass BH in Cygnus X-1 to obtain definiteconstraints on string axion parameters by the LIGO data and the future data from thesecond-generation detectors. We also discuss the possibility to apply a similar idea to nearbyinvisible isolated BHs. BH-axion system
In this paper, the field Φ is assumed to be real and to obey theSine-Gordon equation, ∇ ϕ − µ sin ϕ = 0 , (1)where ϕ := Φ /f a is the amplitude normalized by the decay constant f a . The potential termin this equation naturally arises by the nonperturbative instanton effect for the QCD axion,and a similar mechanism is expected for string axions [7]. Although f a and µ are related toeach other in the QCD axion case, they are treated as independent parameters for stringaxions. When ϕ is small, the Sine-Gordon equation is well approximated by the Klein-Gordonequation, ∇ ϕ − µ ϕ = 0, while the nonlinear effect becomes relevant for ϕ ∼ ω = ω R + iω I . (2)If the discrete real part ω R satisfies the superradiant condition ω R < m Ω H , where m is theazimuthal quantum number and Ω H is the angular velocity of the horizon, the energy fluxacross the horizon becomes negative and ω I becomes positive. This indicates that the scalarfield amplitude grows exponentially. The typical time scale of this superradiant instability is T SR (cid:38) M , which is around one minute for a solar-mass BH. In the case of M µ (cid:28)
1, eigenstates can be obtained by the method of matched asymptotic expansion [8]. Inthis approximation, a solution for Φ in a distant region is obtained from a solution tothe nonrelativistic Schr¨odinger equation for the hydrogen atom by replacing e with M µ ,and each mode is labeled by the angular quantum numbers (cid:96) and m with − m ≤ (cid:96) ≤ m and the principal quantum number n . Here, the principal quantum number is defined as n = (cid:96) + 1 + n r in terms of the radial quantum number n r = 0 , , , ... that characterizes theoscillatory behavior of the mode function in the radial direction. The unstable mode functionwith (cid:96) = m = 1 and n = 2 readsΦ ≈ ( M µ ) (cid:114) E a πM ( kr ) e − kr sin θ cos( ωt − φ ) , (3)where E a is the total energy of the axion cloud and k := M µ /
2. The angular frequency forthis state is ω ≈ µ [1 − ( M µ ) / ω ≈ µ holds for M µ (cid:28) ϕ becomes larger, the nonlinear effect becomes important. In our previous paper [4],we studied this phase by numerical simulations of the axion cloud in the (cid:96) = m = 1 modewith M µ = 0 .
4. At some point with ϕ ≈ .
67, a new mode is suddenly excited, and it carries osenova Bosenova T sr ∼ M Scalar field amplitude Bosenova
Burst GWs
Superradiant instability Superradiant instability Superradiant instabilitySuperradiant instability TimeContinuous Gravitational Waves
Burst GWs Burst GWs ϕ ≈ . T BN ∼ M Fig. 1
Schematic picture for time evolution of the scalar field amplitude and emittedGWs. See text for details.positive energy to the horizon and to the far region terminating the superradiant instability.We call this phenomenon “bosenova”. The typical time scale of the bosenova is ∼ M , andabout 5% of the axion cloud energy falls into the BH. Then, the system again settles to thesuperradiant phase. In a long time simulation, the system was observed to alternate betweenthe bosenova and the superradiant phase. A schematic picture for the time evolution of thefield amplitude is shown in Fig. 1.Burst GWs are generated by the infall of the axion cloud energy during the bosenova. Inour order estimate [4], the GW frequency is within the observation bands of the ground-based detectors, but its amplitude may be marginal to be detected by the second-generationdetectors in the case an axion field with the decay constant f a ≈ GeV causes a bosenovaat Cygnus X-1.In addition to burst GWs from bosenovae, an axion cloud continuously emits GWs bythe level transition of axions and the two-axion annihilation [2]. The former process canbe calculated by the quadrupole approximation [2], while the latter process requires directcalculations of a perturbation equation [5]. Among these two, the two-axion annihilation isthe primary process, and we discuss its observational consequence in this paper. In this pro-cess, the energy-momentum tensor T µν fluctuating with the angular frequency 2 ω generatesmonochromatic GWs with the same frequency˜ ω ≈ µ. (4)From an axion cloud in the (cid:96) = m = 1 mode, GWs in the ˜ (cid:96) = ˜ m = 2 mode are radiated. InRef. [5], we found the approximate formula for M µ (cid:28) h ≈ (cid:114) C n(cid:96) (cid:18) E a M (cid:19) ( M µ ) (cid:18) Md (cid:19) , (5)where d is the distance to the BH. Here, the functional form of Eq. (5) is reliable exceptfor a factor (the value of C n(cid:96) cannot be determined within this approximation). We also irectly calculated the GW radiation rate numerically for a Kerr background, and checkedthat Eq. (5) holds with C n(cid:96) ≈ − . Our conclusion of Ref. [5] is that the energy loss rate bythe GW radiation is smaller than the energy extraction rate of the axion cloud, and hence,the axion cloud grows until the bosenova happens. Note that we have ignored the nonlinearself-interaction effects and used the solution for the quasibound state of the linear Klein-Gordon field in estimating the GW amplitude (5). This is a rather strong approximation,and we will come back to this point later.Since continuous waves from a distorted neutron star are also in the ˜ (cid:96) = ˜ m = 2 mode in thequadrupole approximation, GWs from the (cid:96) = m = 1 axion cloud share the similar features(the angular pattern and the ratio between the plus and cross modes) with GWs from aneutron star. Therefore, the same method of the continuous wave search can be applied toGWs from axion clouds. Method for constraining string axion models.
The frequency region where contin-uous waves have been analyzed is 50 Hz ≤ f ≤ . × − eV ≤ µ ≤ . × − eV . (6)This covers a certain range of the possible mass values of string axions. For the BH mass M ≈ M (cid:12) to be considered in this paper, the parameter M µ is in the range 0 . ≤ M µ ≤ . (cid:96) = m = 1 mode with n = 2. Although other unstable modes may also grow later, we ignoretheir contribution because the GW emission from these modes is much smaller [5]. Then, wecan use the approximate formula for the emitted GW amplitude, Eq. (5).We determine the value of E a /M as follows. As mentioned in the previous section, thesuperradiant instability of an axion cloud is saturated around ϕ := Φ /f a ≈ .
67. Therefore,the energy content in this situation is given by the formula ϕ max ≈ exp( − √ π (cid:114) E a M (cid:18) f a M pl (cid:19) − ( µM ) ≈ . . (7)Substituting E a /M determined by this equation into Eq. (5), we derive the condition h ≈ . × − (cid:18) f a GeV (cid:19) (cid:16) µ − eV (cid:17) (cid:18) M M (cid:12) (cid:19) (cid:18) d (cid:19) − < h UL . (8)Here, the left-hand side is the amplitude expressed in the axion parameters ( µ, f a ) andthe BH parameters ( d, M ), and h UL on the right-hand side is the upper bound on the GWamplitude derived from observations. Fixing the BH parameters ( d, M ), this inequality givesa constraint on the axion parameters ( µ, f a ).Before applying the above argument to Cygnus X-1, we note some subtleties. Cygnus X-1 isknown to have a large spin parameter, a/M (cid:38) . M and J . Therefore, the consistency with the observedspin parameter must be checked. Recently, the adiabatic evolution of the BH parameters wasstudied for a system of a BH wearing a Klein-Gordon field (without nonlinear self-interaction) µM (cid:28)
1. When µM becomes important, the scalar field extracts theBH angular momentum and the spin parameter a/M drops until the superradiant conditionbecomes marginally satisfied, µ ≈ m Ω H . After that, the spin parameter gradually increasesto unity approximately keeping the marginal superradiant condition. Here, we point out thatthe evolution depends on the value of the decay constant f a if the nonlinear self-interactionis present, because the bosenova occurs and the growth of the axion cloud effectively stopswhen Φ ≈ f a (Fig. 1). If f a is order of or smaller than the GUT scale, the bosenova typicallyhappens much before the axion cloud significantly decreases the spin parameter: See Ref. [4]and condition (9) below. For this reason, we assume that the axion cloud scarcely decreasesthe BH spin parameter and the high spin parameter of Cygnus X-1 does not contradict theexistence of the axion cloud.Another subtlety is that although we have treated the scalar field as a test field in Refs. [4,5], its gravity becomes strong as the axion total energy is increased. In Ref. [17], it wasargued that the gravitational backreaction is not important for a small M µ because theaxion cloud spreads over a large scale. But since there remains a possibility that the gravityof an axion cloud may affect the estimate on the BH parameters by changing the propertiesof the accretion disk, we adopt the region where E a /M < .
05 is satisfied. Using Eq. (7),this criterion can be expressed as (cid:18) f a GeV (cid:19) < . × (cid:18) M M (cid:12) (cid:19) (cid:16) µ − eV (cid:17) . (9) Expected constraints from Cygnus X-1
Now we apply the above argument to CygnusX-1. The Cygnus X-1 is in binary with a companion star, and accretion of matter from acompanion star makes it possible to observe the phenomena around the BH. The recentobservation [13–16] determines the distance from the earth, the mass and the spin param-eter as d = 1 . +0 . − . kpc, M = 14 . ± . M (cid:12) , and a/M (cid:38) . i = 27 . ± . d = 1 .
86 kpc and M = 15 M (cid:12) into theinequality (8), we have6 . × − (cid:18) f a GeV (cid:19) (cid:16) µ − eV (cid:17) < h UL . (10)Figure 2 shows the expected constraints in the parameter space ( µ, f a ) that come fromthe observations by the LIGO and the Advanced LIGO (aLIGO). The upper one of thetwo monotonically decreasing curves is the border line of the inequality (10) for the LIGOobservation. Here, we have substituted the upper limit on the continuous wave amplitudederived by LIGO’s all-sky search [6] into h UL . Note that the constraint given in this waymust be interpreted as a theoretical forecast, because the authors of [6] looked for continuouswaves from isolated neutron stars and their result cannot be applied to GWs from binarieslike Cygnus X-1 in which the binary motion causes the frequency modulations by the Dopplershift. In order to obtain a reliable value for h UL , a targeted search with matched filtering forGWs from Cygnus X-1 has to be carried out. The border line of the criteria (9) is depictedby the monotonically increasing curve, above which the gravity effect of the axion cloudmay become important. These two curves intersect at µ ≈ . × − eV, and therefore,the border line of the condition (10) is not reliable in the region µ < . × − eV as xcludedAllowed LIGOaLIGO (cid:72) ? (cid:76) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) (cid:180) (cid:45) Μ (cid:64) eV (cid:68) L og (cid:72) f a (cid:64) G e V (cid:68) (cid:76) Cygnus X (cid:45) Fig. 2
Expected constraints for the mass µ and the decay constant f a of string axionsderived by continuous GWs from Cygnus X-1. The cases for the LIGO and aLIGO detectorsare shown. Also shown is the curve above which the gravity effect of the axion cloud becomesimportant. The constraints may not be reliable above this line as indicated by “(?)”.indicated by “(?)”. In contrast, for µ > . × − eV, the parameters on the border linesatisfy the condition (9) and the curve is reliable. Since the GW amplitude is expected tobe a monotonically increasing function of f a for a fixed µ , we exclude all of the region abovethis border line. In particular, the decay constant f a ≈ GeV, which seems one of thenatural choices [1], is excluded in the mass range 1 . × − eV < µ < . × − eV.The lower one of the two decreasing curves is for the aLIGO observation (the curves fromthe other second-generation detectors are similar). Here, we assumed h UL to be given by ≈ (cid:112) S n /T obs with √ S n the design sensitivity presented in [22] and the observation time T obs = 5000 hours. The two curves intersect at µ ≈ . × − eV. Since the sensitivity ofthe aLIGO detector is 10 times better than that of the LIGO detector, the constraint on f a can be improved by a factor of three. In particular, the value of f a of 0 . × GUT scale isexcluded in the range 0 . × − (cid:46) µ (cid:46) . × − .On the other hand, if continuous GWs from Cygnus X-1 are detected, we can determine thevalues of ( µ, f a ). Note that continuous GWs from Cygnus X-1 can be clearly distinguishedfrom other continuous GWs from distorted neutron stars for the following reason. Due to theDoppler shift caused by the binary motion, continuous GWs from Cygnus X-1 has frequencymodulation with the same period as the orbital period. On the other hand, continuous GWsfrom distorted neutron stars do not have such frequency modulation if they are isolated,and have different periods of frequency modulation if they are in binary. Therefore, theinformation unique to Cygnus X-1 is encoded in the wave forms. Since the sensitivity tosignals of continuous GWs highly depends on the phase behavior, the distinction is possibleif a targeted search is carred out taking account of such frequency modulation. r * ( p ea k ) t / M Fig. 3
The time dependence of the radial position (in the tortoise coordinate r ∗ ) of the fieldpeak of the axion cloud in the simulation for M µ = 0 . ϕ = 0 . M µ = 0 . ϕ ≈ .
60 in our simulation [4]. Since the axion cloud oscillates in theradial direction, the GW frequency is expected to modulate by a factor of few % due tothe gravitational redshift effect. This point requires a careful check by direct numericalcalculations of GWs in the presence of axion nonlinear self-interaction. If the frequencymodulation with the amplitude ∆ ω is present, quadratic estimators cannot improve thesignal-to-noise ratio even if we make the observation period longer than 1 / ∆ ω . Therefore,data analyses using accurate emprical wave forms as templates are necessary to enhance thesensitivity. This point will be discussed elsewhere. Summary and Discussion.
In this paper, we have discussed how to constrain thestring axion parameters ( µ, f a ) by observations of continuous GWs from an axion cloudaround a BH. The expected constraints from Cygnus X-1 are shown in Fig. 2 for the LIGOand aLIGO detectors. A targeted search for continuous waves from Cygnus X-1 is requiredin order to detect the signal or derive the upper bound on the GW amplitude in the condi-tion (10). Such a targeted search should be possible, since similar analyses have been donealready for neutron stars in binary systems [23, 24].There are other solar-mass BHs in binaries for which the system parameters have beenobservationally studied, and similar constraints can be obtained from these BHs. But wehave to be careful about the BH spin parameters a/M , as they take various values fromzero to unity (see Table 1 of Ref. [25]). If we use a moderately spinning BH, a constraintcan be imposed in a smaller range of the axion mass compared to the rapidly spinningcase, because an axion cloud in the (cid:96) = m = 1 mode forms only when the superradiantcondition µ ≈ ω ≤ Ω H is satisfied. For the spin parameter a/M = 0 .
7, e.g., the (cid:96) = m = 1mode is unstable for M µ (cid:46) .
2, and hence, the constraint can be discussed only in the range µ (cid:46) . × − eV for the BH mass M = 15 M (cid:12) . Except for this point, the same methodholds as well. Although the growth rate of the superradiant instability becomes smaller as a/M is decreased, this is not a problem because its time scale is still much shorter than the ge of the BH and there is enough time for the formation of an axion cloud. Also, the waveform of continuous GWs scarcely changes with the value of the spin parameter.Finally, we discuss the possiblity of detecting continuous GWs from an axion cloud aroundan isolated BH. In addition to visible BHs, 10 –10 isolated BHs are expected to exist in ourgalaxy [26, 27]. Since the averaged distance between two neighboring BHs is ∼
10 pc in thisestimate, the detection may be easier compared to the case of Cygnus X-1. But in this case,we have to explore the method to distinguish GWs from an axion cloud and those from adistorted neutron star, since the recent theoretical studies suggest that neutron stars couldgenerate detectable GWs as well [28–30]. For this purpose, more detailed modeling of waveforms including the axion self-interaction effect is necessary. The frequency modulation dueto radial oscillation of an axion cloud should provide a smoking gun for GWs of axion origin.
Acknowledgements
We thank Keith Riles for discussions and various suggestions. We thank the Yukawa Institute forTheoretical Physics at Kyoto University for hospitality during the YITP-T-14-1 workshop on “Holo-graphic vistas on Gravity and Strings,” where part of this work has been done. This work wassupported by the Grant-in-Aid for Scientific Research (A) (Numbers 22244030 and 26247042) fromJapan Society for the Promotion of Science (JSPS).
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