Measurement of the central Galactic black hole by extremely large mass-ratio inspirals

In the Galaxy, extremely large mass-ratio inspirals(X-MRIs) composed of brown dwarfs and the massive black hole at the Galactic Center are expected to be promising gravitational wave sources for space-borne detectors. In this work, we simulate the gravitational wave signals from twenty X-MRI systems by an axisymmetric Konoplya-Rezzolla-Zhidenko metric with varied parameters. We find that the mass, spin, and deviation parameters of the Kerr black hole could be determined accurately ( $\sim 10^{-5} - 10^{-6}$ ) with only one X-MRI event with a high signal-to-noise ratio. The measurement of the above parameters could be improved with more X-MRI observations.


Introduction
The first observations of gravitational waves(GWs) from binary black hole mergers and 10 binary neutron star inspirals ushered in a new era of GW physics and astronomy [1,2]. Since 11 then, the ground-based detectors have detected 90 GW events [3][4][5]. The detectable frequency 12 band of current ground-based GW detectors such as Advanced LIGO [6], Advanced Virgo [7], 13 and KAGRA [8] ranges from 10 to 10,000 Hz, which makes ground-based GW detectors unable 14 to detect any GWs with frequencies less than 10 Hz, while abundant sources are emitting GWs 15 in the low-frequency band [9]. The space-borne GW detectors such as LISA [10], Taiji [11], and 16 TianQin [12], which will be launched in the 2030s, will open GW windows from 0.1 mHz to 1 17 Hz, and are expected to probe the nature of astrophysics, cosmology, and fundamental physics. 18 One of the most essential and promising GW sources for space-borne GW detectors is 19 the extreme-mass ratio inspiral (EMRI), which is formed when a massive black hole (MBH) 20 captures a small compact object. [9,13]. The word "inspiral" here means the inspiralling process 21 that the relatively lighter object gradually spirals in toward the MBH due to the emission of 22 GWs. The small object should be compact to keep it from being tidally disrupted by the MBH 23 so that it is unlikely to be a main-sequence star. The possible candidate could be a stellar-mass 24 black hole(BH), neutron star, white dwarf, or other compact objects. The designed space-borne 25 detectors will be sensitive to EMRIs that contain MBHs with the mass 10 4 − 10 7 M and small 26 compact objects with stellar mass, and the fiducial mass ratio will be 10 3 − 10 6 [14]. 27 Moreover, a special kind of EMRI, extremely large mass-ratio inspirals (X-MRIs) with 28 a mass ratio of q ∼ 10 8 also are potential sources for space-borne GW detectors [15,16]. The 29 X-MRI system is formed when an MBH captures a brown dwarf (BD) with mass ∼ 10 −2 M . 30 Brown dwarfs are substellar objects with insufficient mass to sustain nuclear fusion and become 31 main-sequence stars [17]. Brown dwarfs are denser than main-sequence stars, and their Roche 32 limit is closer to the horizon of MBH [15,18]. Therefore, brown dwarfs could survive very close 33 to the MBH. 34 The mass of BD is relatively tiny, so space-borne GW detectors like LISA could only 35 observe X-MRIs nearby, especially X-MRIs at the Galactic Center(GC) [15,16]. The MBH of these 36 X-MRIs, Sgr A*, is 8 kpc from the solar system, and its mass is about 4 × 10 6 M [19][20][21][22]. A 37 typical X-MRI at the GC covers ∼ 10 8 cycles, which last millions of years in the LISA band [16]. 38 Such X-MRI could have a relatively high SNR (more than 1000), and dozens of X-MRIs might 39 be observed during the LISA mission period [16]. Therefore, the X-MRIs at the GC offers a 40 natural laboratory for studying the properties of BH and testing theories of gravity. 41 In this paper, we simulate the GW signals of X-MRIs at GC to show how and to what 42 extent the fine structure of Sgr A* could be figured. In general relativity(GR), according to 43 the no-hair theorem, BHs are characterized by their masses, spins, and electric charges, and 44 the Kerr metric is believed to be the metric that describes the space-time of BH. However, 45 alternative theories of gravity predict hairy black holes [23] and other metrics that describe the 46 space-time of BH[24]. The parameterized metrics are proposed to describe the space-time of 47 non-Kerr black holes. In this paper, to describe the space-time of X-MRIs at GC, we use a model-48 independent parameterization metric, Konoplya-Rezzolla-Zhidenko metric(KRZ metric)[24], 49 which can describe metrics that is generic stationery and axisymmetric.

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This paper is organized as follows, in section 2, we review the KRZ parametrization. In 51 section 3, we introduce the "kludge" waveforms used in our work and simulate the GW signals 52 emitted by X-MRIs at GC. In section 4, based on the simulated GW signals, we apply the Fisher 53 matrix to these GWs and present the accuracy of parameter estimation of Sgr A* for future 54 space-borne GW detectors. The conclusion and outlook are given in section5. Throughout this 55 letter, we use natural units (G = c = 1), greek letters (µ, ν, σ, ...) stand for space-time indices, 56 and Einstein summation is assumed.  58 GR is the most accurate and concise theory of gravity by far [25]. While in practice, there 59 are quite a few other theories of gravity, whose predictions resemble general relativity's, to 60 be tested. In the framework of GR, the Schwarzschild or Kerr metric describes the space-time 61 of uncharged BH. However, in modified and alternative theories of gravity, there are other 62 possible solutions for the description of the space-time of BHs[26-31]. The predictions of 63 different theories of gravity are different, so a universal and reasonable theory about the GWs 64 of X-MRI should be model-independent.

KRZ prametrized metric
In order to deal with numerous metrics of non-Kerr black holes, one may use the pa-66 rameterized metric to describe the space-time of non-Kerr black holes. There are several 67 model-independent frameworks, one of which parametrizes the most generic black hole ge-68 ometry through a finite number of adjustable quantities and is known as Johannsen-Psaltis 69 parametrization (J-P metric) [32]. The J-P metric expresses deviations from general relativity 70 in terms of a Taylor expansion in powers of M/r, where M is the mass of BH and r is the 71 radial coordinate.The J-P parametrization is widely adopted, but it is not a robust and generic 72 parametrization for rotating black holes [24,33]. Notably, the parametric axisymmetric J-P 73 metric obtained from the Janis-Newman algorithm [34] does not cover all deviations from Kerr 74 space-time.

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Another model-independent parameterization metric [24,35], KRZ metric, is based on a 76 double expansion in both the polar and radial directions of a generic stationary and axisym-77 metric metric.The KRZ metric is effective in reproducing the space-time of three commonly 78 used rotating black holes (Kerr, rotating dilation [36], and Einstein-dilaton-Gauss-Bonnet black 79 holes [37]) with finite parameters (see Ref. [24] for more details). According to KRZ param-80 eterization, the space-time of any axisymmetric black hole with total mass M and rotation 81 parameter a could be expressed in the following form[24]: where [38] 83 S = Σ N, B, W,and K are the functions of the radial and polar coordinates (expanded in term cos θ), with where x = 1 − r 0 /r, and r 0 is the radius of the black hole horizon in the equatorial plane. 86 The metric (1) is characterized by the order of expansion in radial and polar directions. The 87 parameters a ij , b ij , ω ij , k ij (here i = 0, 1, 2, 3..., j = 1, 2, 3...) are effectively independent. This is 88 because one of these functions, A i (x), B i (x), W i (x) and K i (x), is fixed by coordinate choice [38]. 89 In the following, we present the parameterized metric with first-order radial expansion 90 and second-order polar direction, which describe the space-time of a deformed Kerr black 91 hole [33,38]: The radius of the horizon and the Kerr parameter are where J is the total angular momentum. For simplicity, here M has one unit, i.e. M = 93 1. One can obtain related variables and parameters from dimensionless quantity by scale 94 transformations [39][40][41][42][43][44]: tM → t, rM → r, etc. The coefficient r 0 , a 20 , a 21 , 0 , k 00 , k 21 and ω 00 in 95 the KRZ metric can be expressed as follows [33,45] hereã = a/M stands for the spin parameter. The deformation parameters δ j (j = 1, 2, ..., 8) 97 represent the deviations from the Kerr metric. The physical meaning of these parameters 98 could be summarized as follows: δ 1 is related to deformation of g tt ; δ 2 , δ 3 are related to the 99 rotational deformation of the metric; δ 4 , δ 5 are related to deformation of g rr and δ 6 is related 100 to the deformation of the event horizon (see Ref. [24] for more details). The KRZ metric is 101 an appropriate tool to measure the potential deviations from the Kerr metric. As a first order 102 approximation, in this work we mainly consider δ 1 and δ 2 .

Waveform model for KRZ black holes
104 Several waveform models can simulate the signal of EMRI [14,[46][47][48][49][50]. Among these 105 models, the kludge model can generate waveforms quickly and have a 95% accuracy compared 106 with the Teukolsky-based waveforms [49]. The kludge waveforms may be essential in searching 107 for EMRIs/X-MRIs for future space-borne GW detectors. We employ the kludge waveforms 108 to simulate X-MRI waveforms [45]. Before presenting the results, we would like to review the 109 structure and logic of the calculation. The calculation of waveforms can be summarized in the 110 following steps: First, to consider the brown dwarf of the X-MRI as a point particle.

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• Second, to use the given metric to calculate the particle's trajectory by integrating the 113 geodesic equations that contain the radiation flux.

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• Finally, to use the quadrupole expression to get the GWs emitted from the system of the 115 X-MRI.

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To get the trajectory of the particle, we start by calculating the geodesics using the following 117 equations: where x µ is the coordinate of the particle, u µ is the 4-velocity, which satisfies and Γ µ ρσ are the Christoffel symbols. For stable bounded geodesics, the orbital eccentricity e and semi-latus rectum p can be defined by periastron r p and apastron r a , and the inclination angle ι is defined in the Keplerian convention by: where θ min is the minimum of θ along the geodesic. The geodesic may be specified by the 119 parameters (r a , r p , θ min ), which fully describe the range of motion in the radial and polar 120 coordinates. In this paper, we define (e, p, ι) from (r a , r p , θ min ) by the numerically generated 121 trajectory.

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In the background of Kerr metric, the geodesic can be described by the orbital energy E, 123 the z component of the orbital angular momentum L z , and the Carter constant Q[45]. E and L z 124 still exist in the KRZ background, and take the form While the orbital constants (E, L z , Q) in the above geodesic setup do not vary with time, it is 126 convenient to work with alternative parametrizations of (E, L z , Q). The relationship between 127 (r a , r p , θ min ) and (E, L z , Q) is given by [14] 128 Because of the extreme mass ratio of X-MRI, the deviations from the geodesics due to radiation reaction should be small. While in this work, for accuracy, we consider the effect of radiation reaction, which is included by replacing the Eq. (31) with the following one where the radiation force F µ is connected with the adiabatic radiation fluxes (Ė,L z ,Q) as Eq. (42) can be deduced by taking derivatives with respect to proper time in Eqs. (35)- (37). 130 Integrating the geodesic equations that contain the radiation flux is crucial for calculating 131 the particle's trajectory. In this paper, due to the short integration time, we use the Runge-132 Kutta method. There are also several geometric numerical integration methods for integrating 133 the equations of geodesics. Such as manifold correction schemes[53-55], extended phase 134 space methods [56][57][58][59], explicit and implicit combined symplectic methods[60-62], and explicit 135 symplectic integrators [39][40][41][42][43][44]. For situations such as the long-term evolution of Hamiltonian 136 systems [55], geometric numerical integration methods can be helpful.

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Finally, after generating the trajectory, we turn to the third step -to calculate the gravita-138 tional waveforms. We start from transforming the Boyer-Lindquist coordinates (t, r, φ, θ) into 139 Cartesian coordinates (t, x, y, z) using the relations: Then we calculate the quadrupole expression (see Ref. [49]) where I jk (t ) is the source's mass quadrupole moment, T 00 is component of the energymomentum tensor T µν (t , x ), andh µν = h µν − 1 2 η µν η ρσ h ρσ is the trace-reversed metric perturbation. Then we transform the waveform into the transverse-traceless gauge (see Ref. [49] for more details) Now we get the plus and cross components of the waveform observed at latitudinal angle Θ 143 and azimuthal angle Φ The strength of the signal in a detector could be characterized by the signal-to-noise ratio (SNR). The SNR of the signals can be defined as [63] ρ := h|h , where ·|· is the standard matched-filtering inner product between two data streams. The 145 inner product between signal a(t) and template b(t) is whereã( f ) is the Fourier transform of the time series signal a(t),ã * ( f ) is the complex conjugate 147 ofã( f ) and S n ( f ) is the power spectral density of the GW detectors' noise. Throughout this 148 paper, the power spectral density is taken to be the noise level of LISA.

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In this work, to quantify the differences between GW signals and the templates, we use maximized fitting factor (overlap) If we include the time shift t s and the phase shift φ s , the fitting factor reads the maximized fitting factor is defined as

Data analysis 150
In this section, we first specify the main parameters values we used in this work. Then 151 we use XSPEG, a software for generating GWs in the KRZ metric, provided by the authors of 152 Ref.
[45] to calculate the gravitational waveforms and do some analysis. Finally, we employ 153 the Fisher information matrix to evaluate the parameter estimation accuracy for LISA-like GW 154 detectors.

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Based on the parameters above, we first simulate the GW signals of twenty X-MRIs at 162 the GC (see Table 1 ). The mass ratio q ranges from 5 × 10 7 to 4.0 × 10 8 , the orbit eccentricity e 163 ranges from 0.1 to 0.8, the semi-latus rectum p ranges from 10.6 to 50.0, the inclination angle ι 164 ranges from −2π/3 to π/3, and the duration of above signals is one year. Then, we calculate 165 the overlaps between above GW signals and many GW series with varying parameters. Finally, 166 we use the Fisher information matrix to provide the uncertainties of parameter estimations. Suppose the GW signal and corresponding GW template overlaps are above 0.97 [68]. In 169 that case, we would find neither the deviations from GR nor the unusual parameters of X-MRIs, 170 which is called the confusion problem [68]. The confusion problem can prevent us from getting 171 accurate parameter estimation of the X-MRIs. To make sure there is no confusion in our study, 172 we calculate the overlaps between different gravitational waveforms of twenty X-MRIs with 173 varying parameters λ i (λ i = a, M, δ 1 , δ 2 , e, p, ι). Here (a, M) are the parameters of the Sgr A*, 174 (δ 1 , δ 2 ) are the deformation parameters of the space-time from the Kerr solution, and (e, p, ι) 175 are the parameters of orbit (eccentricity, semi-latus rectum, inclination).

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Because a, M, δ 1 , and δ 2 are the intrinsic parameters of Sgr A* and present the nature 177 of MBH directly, we pay more attention to these four parameters. The Figs. 1-4      overlaps between the original waveforms and the waveforms with varying parameters a, M, δ 1 179 and δ 2 . As these figures show, the overlap tends to decrease while the increment of λ increases. 180 Taking the overlap value 0.97 as a criterion would give the constraints on λ. Specifically, 181 to get the constraints on δλ i by the GWs of X-MRI, we first keep the other parameters fixed 182 and generate several waveforms with varying λ i . Then we calculate the overlaps between the 183 original waveform and the waveforms with varying λ i . Finally, the corresponding value of λ i 184 when overlap equals 0.97 can be regarded as the limit of λ i . From these figures, we observe the 185 parameter constraint ability for different X-MRI varies. The SNRs of the X-MRI GW signals is high enough to apply the Fisher information 188 matrix to estimate the accuracy of parameter estimation. We present the accuracy of parameter 189 estimation for Sgr A* in this part using the Fisher information matrix. To better estimate the 190 distance between Sgr A* and the solar system, we take account of the external parameter R p 191 and constrain it by the gravitational waveforms of the X-MRIs in Table 1. 192 The Fisher information matrix Γ for a GW signal h parameterized by λ is given by (See Ref [69] for details) where λ i = (a, M, δ 1 , δ 2 , e, p, ι, R p ) is one of the parameters of the X-MRI system. The parameter estimation uncertainty ∆λ due to Gaussian noise has the normal distribution N (0, Γ −1 ) in the case of high SNR, so the root-mean-square uncertainty in the general case can be approximated as For parameter estimation uncertainty ∆λ i , ∆λ j (i = j), the corresponding likelihood is [69][70][71].
For an X-MRI with eight parameters, we can get a Fisher matrix (Γ i,j ) 8×8 by applying the 193 results of these parameters' preliminary constraints to equation (63). Element Γ i,j (i = j) in 194 the Fisher matrix is the result of the combination of parameter λ i and parameter λ j . With the 195 Fisher matrix, absolute uncertainty ∆λ i of any parameter λ i can be estimated by calculating 196 the equation (64). Here we focus on the estimations of Sgr A*'s parameters (a, M, δ 1 , δ 2 , R p ).

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By using the Fisher matrix, the parameter estimation accuracy of (a, M, δ 1 , δ 2 , R p ) for the 198 twenty X-MRI signals is shown in Table 1. Different X-MRI systems have different abilities to 199 estimate the uncertainty accuracy of the same parameter. For the spin of Sgr A*, the relative 200 uncertainty ∆a/a estimated by X-MRI 01, X-MRI 02, X-MRI 03, and X-MRI 10 reach a very 201 high precision ∼ 10 −6 . While ∆a/a estimated by X-MRI 15 is only ∼ 10 −2 . For the mass 202 of Sgr A*, its relative uncertainty ∆M/M estimated by X-MRI 01, X-MRI 02, X-MRI 12, and 203 X-MRI 20 reach ∼ 10 −7 , and ∆M/M estimated by X-MRI 15 is ∼ 10 −4 . For the space-time 204 deformation around Sgr A*, ∆δ 1 and ∆δ 2 estimated by X-MRI 01 reach ∼ 10 −6 , while the 205 relative uncertainty of these deformation parameters estimated by X-MRI 15 is only ∼ 10 −2 . 206 For the distance R p , its relative uncertainty ∆R p /R p estimated by X-MRI 01 reaches ∼ 10 −4 , 207 while the accuracy of ∆R p /R p estimated by X-MRI 06, X-MRI 07, X-MRI 11, and X-MRI 19 is 208 only ∼ 10 −2 . From the above analysis, we find that X-MRI 01 has stringent constraints for the 209 five parameters (a, M, δ 1 , δ 2 , R p ). Therefore, we take X-MRI 01 as an example to present its 210 likelihoods calculated by Eqs. 63-65. As shown in Figs. 5-7, it is obvious that the parameter 211 estimation for X-MRI 01 may be affected by any other parameter. Thus, it is reasonable to 212 consider the parameters of one X-MRI signal to estimate any parameter. 213 We further study the influence of the combination of GW signals on the parameter estimation accuracy. Here we take parameter α as an example to present the data processing. Firstly, we assume that there are n X-MRI systems at the GC. Then, we calculate the Fisher matrices of all these signals to determine the diagonal element Γ α,α . Sort the value of Γ α,α by the order of size, and the corresponding matrix will be Γ α1 , Γ α2 , ..., Γ αn . Then we add these matrices to get the matrix Γ α , with Γ α , we get the estimation of absolute uncertainty from the equation We repeat the steps of the estimation for ∆α, and calculate the absolute uncertainty of a, M, δ 1 , δ 2 , R p .

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Then we will get the relative uncertainty. The results are shown in Figs. 8. The accuracy gets 215 better as the number of X-MRI increases. With all twenty X-MRI systems in Table 1, the esti-216 mation accuracy for these parameters all reach higher precision. ∆a/a reaches the accuracy 217 ∼ 10 −7 . ∆M/M reaches the accuracy ∼ 10 −8 . ∆δ 1 reaches the accuracy ∼ 10 −6 . ∆δ 2 reaches 218 the accuracy ∼ 10 −6 . ∆R p /R p reaches the accuracy ∼ 10 −4 . The observation number of 219 X-MRI systems does make sense for parameter estimation. Finally, we must emphasize that the 220 parameter estimation results predicted by the Fisher information matrix here only stand for the 221 ideal situation, in the actual parameter estimation practice, because of all kinds of noise, the 222 results would not be that kind of good.

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Sgr A* is the closest MBH for the Solar system. It is therefore an ideal laboratory to study 225 the properties of black holes and to test alternative theories of gravity. To investigate the 226 structure of Sgr A*, we simulate the GW signals for twenty X-MRI systems using the KRZ 227 metric and the kludge waveform. We then apply the Fisher information matrix method to 228 these GW signals. With a single GW X-MRI event detected, we were able to obtain a relatively 229 accurate estimate of spin a, mass M, and deviation parameters δ 1 , δ 2 . More X-MRI observations 230 would improve the measurement of the above parameters.