Eur. Phys. J. C (2021) 81:769 https://doi.org/10.1140/epjc/s10052-021-09555-1 Regular Article - Theoretical Physics Constrains on the electric charges of the binary black holes with GWTC-1 events Hai-Tang Wang1,2,a , Peng-Cheng Li3,4, Jin-Liang Jiang1,2, Guan-Wen Yuan1,2,Yi-MingHu5, Yi-Zhong Fan1,2 1 Key Laboratory of Dark Matter and Space Astronomy, Purple Mountain Observatory, Chinese Academy of Sciences, Nanjing 210023, People’s Republic of China 2 School of Astronomy and Space Science, University of Science and Technology of China, Hefei 230026, Anhui, People’s Republic of China 3 Center for High Energy Physics, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, People’s Republic of China 4 Department of Physics, State Key Laboratory of Nuclear Physics and Technology, Peking University, No. 5 Yiheyuan Rd, Beijing 100871, People’s Republic of China 5 TianQin Research Center for Gravitational Physics and School of Physics and Astronomy, Sun Yat-sen University (Zhuhai Campus), Zhuhai 519082, People’s Republic of China Received: 20 May 2021 / Accepted: 16 August 2021 © The Author(s) 2021 Abstract Testing black hole’s charged property is a fasci- case, the discharge process is much slower than for ordinary nating topic in modified gravity and black hole astrophysics. plasma so the charged BHs are viable. So it is meaningful to In the first Gravitational-Wave Transient Catalog (GWTC- study the effect on gravitational wave if electric charges are 1), ten binary black hole merger events have been formally presented in these special theories. reported, and these gravitational wave signals have signifi- At present, using the shadow of supermassive BH to esti- cantly enhanced our understanding of the black hole. In this mate its charge has been developed and widely practiced paper, we try to constrain the amount of electric charge with [11], for example Sgr A* and M87* measured by VLBI [12– the parameterized post-Einsteinian framework by treating the 15]. These proposals are either far from being accurate or electric charge as a small perturbation in a Bayesian way. We model-dependent. Therefore, developing new technique to find that the current limits in our work are consistent with model-dependently test the charged property is needed, and the result of Fisher information matrix method in previous gravitational wave has always been expected, which encodes works. We also develop a waveform model considering a the BH’s information [16]. Fortunately, after decades of hard leading order charge effect for binary black hole inspiral. work, LIGO-Virgo announced the first detection of the grav- itational wave (GW) signal GW150914 in 2015, generated by the merger of a binary BH [17]. To characterize GW, 1 Introduction we often use inspiral, merger and ringdown to describe the whole coalescence of a binary BH, where the inspiral stage is The prominent black hole (BH) no-hair theorems [1,2]imply generally described by post-Newtonian theory [18], merger vacuum astrophysical black hole can be described by a spe- stage is generally approximated by the numerical simulation cial case of Kerr-Newman metric, which could be character- [19], and ringdown stage is described by the BH perturbation ized by its mass, spin and charge only [3–5]. Although astro- theory [20,21]. Some waveform models, such as IMRPhe- physical BHs are usually considered as electrically neutral nomPv2 (IMR) waveform model, have incorporated these due to charge neutralization by astrophysical plasma, quan- three stages [22]. tum discharge effects, and electron-positron pair production The GW waveform will be affected if the two BHs are elec- [6–8], an accurate upper limit on the amount of the charges of trically charged. For simplicity, consider the electric dipole BHs is still absent. Therefore, it is important and meaningful radiation during their orbital motions, which will be reflected to give a quantitative measurement on the amount of charges in the phase of inspiral stage, and thus provides us a new trick of BHs. Besides, many novel charging mechanisms has been to detect charged binary BH. Depending on the amount of the discussed theoretically, such as primordial black hole [9], charge, there are two different treatments. When the charge and BHs formed by millicharge dark matter [10]. In the later of the black hole is small, the dipole radiation can be taken as a perturbation or correction to the phase of the inspiral a e-mail: [email protected] (corresponding author) stage of the GW waveform, which can be incorporated into 0123456789().: V,-vol 123 769 Page 2 of 7 Eur. Phys. J. C (2021) 81:769 the parameterized post-Einsteinian (ppE) framework [16]. In parameter entering at −1 post-Newtonian order, similar to this case, the correction due to the electric dipole radiation the dipole radiation from the scalar–tensor theory [23]. is completely described by the coefficient of the −1 post- Assuming the early-inspiral stage of the IMR waveform i − Newtonian (PN) order in the waveform, the rest part of the model is reduced to the form of he−ins( f ) = Ae−ins( f )e e ins , waveform is the same as that of the accurate waveform model then the waveform model due to modifications from different describing the coalescence of two neutral black holes, such modified gravity effects can be written as as the well-known phenomenological waveform model [22]. ie−ins+i This is similar to the dipole radiation caused by the scalar he−ins( f ) = Ae−ins( f )e (1) charges carried by black holes in the scalar-tensor theory [23]. However, if the charge of the black hole is large, the In ppE framework [25,26], ppE framework may not be applicable anymore, as the latter b/3 requires the expansion in PN is always linear, which may not ppE = β(πGM f ) , (2) be assured as a prior. We also consider only the leading order gravitational where β is the amplitude coefficient and b is the exponent quadrupole radiation and the electric dipole radiation, based coefficient. Note that the modification enters at (b +5)/2PN 3/5 on which we call such waveform the leading order charged order for a related modified gravity. M = (m1m2) /(m1 + 1/5 (LOC) one. It is apparent that the LOC waveform has a bad m2) is the chirp mass with component masses m1 and m2, accuracy when applied to the analysis of GW data unless the and f is the frequency of the GW. inspiral duration is long enough and the orbit of the two black The study in Tahura et al. [27] shows that at least in theo- holes decays very slow. Despite this shortcoming, the LOC ries where the leading corrections enter at negative PN orders, waveform provides us a toy model such that the effect of the the phase-only analyzes can produce sufficiently accurate charge can be demonstrated explicitly without the assump- constraints. In this analysis, we only consider the correc- tion that the charge of the black hole is small. The detailed tion on the phase and neglect the correction to the ampli- calculation of the LOC waveform is shown in Sect. 5. tude. The frequency at the end of this stage is given by In this work, based on IMR waveform model in LIGO- fc = 0.018/[G(m1 + m2)] [26], above which this model Virgo Algorithm Library [24], we use the Bayesian method is calibrated with numerical-relativity data and can not be to test the dipole radiation of GW signals [25,26] with the applied to ppE formalism. ppE framework. As the main conclusion, we find no visible The effect of the charge on the waveform can be taken as charge taking by astrophysical BH. a perturbation term in the phase [10,28], The rest of this paper is organized as follows. In Sect. 2, we introduce the ppE framework. In Sect. 3, we briefly intro- 5 2/5 2 2 −7/3 q =− η ζ κ (πκGM f ) (3) duce the Bayesian method for the GW data processing and 3584 the results are presented in Sect.4. In order to verify the con- 2 sistency of the results obtained from ppE method, in Sect. 5, where η = m1m2/(m1 + m2) is symmetric mass ratio, ζ we introduce the toy model, the LOC waveform, to study represents the difference between the charges√ of the binary the effect of the amount of the charge on the waveform. In BH and is√ defined by ζ =|λ1 − λ2|/ 1 − λ1λ2, where Sect. 6, we summarize the calculation results and present the λi = qi /( Gmi ) is the charge-to-mass ratio and qi is the conclusion, and discuss the deficiencies of this work as well electric charge of a BH. Here κ = 1 − λ1λ2 ≈ 1 since we as what can be done further. We assume c = 1 throughout consider the effects of λ1 and λ2 as small perturbations. In β =− 5 η2/5ζ 2 =− the paper unless otherwise specified. this case, 3584 and b 7, corresponding to Eq. (2). In TAble I of Yunes et al. [25], obtained constraint |β| 2 Effects on gravitational wave signals 1.6 × 10−4 at −1 PN order for GW150914. The relationship between β and ζ is: β =−5/3584η2/5ζ 2. Thus, one can In this section, we introduce the waveform models adopted derive the corresponding constraint ζ 0.45. There are two in this work. We treat the charge effect as a small pertur- ways to enhance the constraint on the charge. Firstly, one bation, which can be well described by the parameterized may expect the increase of the sensitivities of GW detectors, post-Einsteinian framework.