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Let us start with the action for EdGB gravity [20, 21],
S=∫d4x√−g16π(R−12∂μϕ∂μϕ+αGB4eϕR2GB)+Sm,
(1) where
Sm is the matter sector, g denotes the determinant of the metric, R is the Ricci scalar, ϕ is a dynamical scalar field,αGB is the coupling parameter with the dimensions of a quadratic length, andR2GB is the Gauss-Bonnet invariant expressed asR2GB=RμνρσRμνρσ−4RμνRμν+R2.
(2) When EdGB gravity was proposed, the construction of black hole solutions garnered considerable interest [39−42]. In general, a small-coupling regime [20, 41] is adopted to simplify the calculation as one confronts the problem of clumsy equations in the analytical solution. For spinning EdGB black holes, the field equation can be solved by expanding the spin
χf≪1 and dimensionless deviating parameterαGBM2≪1 to the ideal order. In comparison with Kerr black holes, they possess a minimal mass and larger angular momentum [40, 43−45].To simplify the notation, the dimensionless deviating parameter is given by
ζ=αGBM2,
(3) where M is the mass of the black hole.
Furthermore, QNMs have been extensively studied in EdGB gravity [36, 46−48]. For our purpose, we only focus on the gravitational-led QNMs of extremely slowly-rotating black holes with first order in the spin. The resultant QNMs can be modeled as [36]
ωnlm(χf,ζ)=ωnl0(ζ)+χfmωnl1(ζ).
(4) Here,
ωnl0 denotes the QNMs of a non-rotating black hole in the polar sector [47],ωnl1 represents the QNMs of spin corrections,χf is the dimensionless spin andχf=J/M2 , where J is the spin angular momentum of the black hole, and m is the azimuthal number. Specifically, for the lowest-lying QNMs with the multipole numberl=2,3 , the analytic fitting formula reads as [36, 46]Mω0l0=(1+f1ζ2+f2ζ3+f3ζ4)ω0ls,Mω0l1=q1+q2ζ2+q3ζ3+q4ζ4+q5ζ5+q6ζ6.
(5) Here,
ω0ls denotes the QNMs of Schwarzschild spacetime. For the modes withl=2 ,Mω02s≈0.37370−0.08896i . For the modes withl=3 ,Mω03s≈0.5994−0.0927i . The coefficientsfi andqi are listed in Table 1. As the dimensionless deviating parameter tends to zero, the QNMs of EdGB gravity fall into Kerr spacetime [36, 48, 49].(l,m) QNMs f1 f2 f3 q1 q2 q3 q4 q5 q6 (2,2) ωR −0.03135 −0.09674 0.23750 0.06290 −0.01560 −0.00758 −0.06440 0.26800 −0.60300 ωI 0.04371 0.17940 −0.29470 0.00099 −0.00110 0.01864 −0.17271 0.56422 −0.81190 (3,3) ωR −0.09911 −0.04907 0.09286 0.06740 −0.02910 0.02510 −0.32090 1.17030 −1.33410 ωI 0.07710 0.13990 −0.34500 0.00065 0.00023 0.02330 −0.28320 1.32300 −2.44200 The dependence of the relative departures between the QNMs of Kerr black holes on the dimensionless deviating parameter ζ for different spins
χf is illustrated in Fig. 1, whereδωnlmR,I=(ωnlmR,I(ζ)−ωnlmR,I(0))/ωnlmR,I(0) . We note that as the spin increases, the relative departures of the real and imaginary parts of the QNMs increase, which implies that the correction of QNMs in EdGB gravity can be magnified by the spin. -
The ringdown waves from a distorted black hole consist of the plus component
h+ and cross componenth× , which are dominated by the formsh+(t)=MDL∑l,mAlmYlm+(ι)e−t/τlmcos(ωlmt−mϕ0),h×(t)=−MDL∑l,mAlmYlm×(ι)e−t/τlmsin(ωlmt−mϕ0),
(6) for
t≥t0 , andh+,×(t)=0 fort<t0 , wheret0 is the initial time of ringdown. Here,M is the mass of the final black hole,DL is the luminosity distance to the source, l and m are the harmonic indices,Alm,ϕ0 are the amplitude and initial phase of the QNMs,τlm,ωlm are the damping time and oscillation frequency of the QNMs determined by Eq. (4) in EdGB gravity, andι∈[0,π] is the inclination angle of the remnant. The functionYlm+,×(ι) indicates the total of−2 weighted spin spheroidal harmonics, which can be expressed as [50]Ylm+(ι)=−2Ylm(ι,0)+(−1)l−2Yl−m(ι,0),Ylm×(ι)=−2Ylm(ι,0)−(−1)l−2Yl−m(ι,0).
(7) To construct the ringdown waveform, we focus only on two dominant modes,
l=m=2,3 , in EdGB gravity. More specifically,Y22+(ι)=√54π1+cos2ι2,Y22×(ι)=√54πcosι,Y33+(ι)=−√218π1+cos2ι2sinι,Y33×(ι)=−√218πsinιcosι.
(8) Moreover,
Alm is well fitted as [51, 52]A22(ν)=0.864ν,A33(ν)=0.44(1−4ν)0.45A22(ν).
(9) Here,
ν=m1m2/(m1+m2)2 is the symmetric mass ratio, andm1,m2 are the masses of two separated black holes before coalescence.Now, we exploit the first generation TDI Michelson combination X [38] to obtain the frequency-domain ringdown signals, which can be written as
h(f)=∑A=+,×12(1−e−2iu)(DAuT(u,ˆn⋅ˆu)−DAvT(u,ˆn⋅ˆv))hA(f),
(10) where
h+,×(f) are the frequency-domain ringdown signals after the Fourier transformation ofh+,×(t) ,D+u=[cos2θcos2(ϕ−π/6)−sin2(ϕ−π/6)]cos2ψ−cosθsin(2ϕ−π/3)sin2ψ,D×u=−cosθsin(2ϕ−π/3)cos2ψ−[cos2θcos2(ϕ−π/6)−sin2(ϕ−π/6)]sin2ψ,D+v=[cos2θcos2(ϕ+π/6)−sin2(ϕ+π/6)]cos2ψ−cosθsin(2ϕ+π/3)sin2ψ,D×v=−cosθsin(2ϕ+π/3)cos2ψ−[cos2θcos2(ϕ+π/6)−sin2(ϕ+π/6)]sin2ψ.
(11) The frequency-dependent transfer function
T isT(u,ˆn⋅ˆu)=12e−iu[e−iu(1−ˆn⋅ˆu)/2sinc(u(1+ˆn⋅ˆu)/2)+eiu(1+ˆn⋅ˆu)/2sinc(u(1−ˆn⋅ˆu)/2)].
(12) Here,
u=2πfLc , where L is the arm length of the detector, and c is the speed of light,sinc(z)=sinzz ,ˆn=(sinθcosϕ,sinθsinϕ,cosθ) is the orientation of the source, and the unit vectors with respect to the detector's arms,ˆu,ˆv , areˆu=(cosπ6,sinπ6,0),ˆv=(cosπ6,−sinπ6,0).
(13) For convenience, Fig. 2 shows the detector coordinate system adopted in this study. The origin is placed at spacecraft 1.
(ˆp,ˆq,ˆk) are the basis vectors of the canonical reference frame, whereˆk denotes the direction of propagation of gravitational waves,(ˆϕ,ˆθ,ˆn) are the basis vectors of the observational reference frame, and ψ is the polarization angle.To test EdGB gravity with space-based gravitational wave detectors, it is necessary to evaluate the capability of LISA, TaiJi, and TianQin. Here, we adopt the noise power spectral density and average response functions of tensor polarizations for the Michelson combination X [53]:
SN(u)X=4sin2uu2[s2aL2u2c4(3+cos2u)+u2s2xL2],
(14) R(u)X=sin2u2u2[(−7sinu+2sin2u)/u+(−4+5cosu−4cos2u)/u2+(−5sinu+4sin2u)/u3+(5+cos2u)/3−6cos2u(Ciu−2Ci2u+Ci3u+ln4/3)+4(Ciu−Ci2u+log2)−6sin2u(Siu−2Si2u+Si3u)],
(15) where SinIntegral
Si(z)=∫z0(sint/t)dt , CosIntegralCi(z)=−∫∞z(cost/t)dt ,Sa is the residual acceleration noise, andSx is the displacement noise. For LISA,Sa=3×10−15ms−2/√Hz ,Sx=1.5×10−11m/√Hz , andL=2.5×109m [54]. For TaiJi,Sa=3×10−15ms−2/√Hz ,Sx=8×10−12m/√Hz , andL=3×109m [55]. For TianQin,Sa=1×10−15ms−2/√Hz ,Sx=1×10−12m/√Hz , andL=√3×108m [56]. The sky-averaged sensitivity is defined as [57]Sn(f)=SN(f)R(f).
(16) In particular, the galactic confusion noise mainly generated by abundant double white dwarf binaries plays a non-negligible role in the detection of gravitational waves. For LISA and TaiJi, the galactic confusion noise takes the form [57]
Sc(f)=αf−7/3e−fβ+γfsin(ηf)[1+tanh(λ(fc−f))]Hz−1.
(17) For TianQin, the galactic confusion noise can be modeled as [58]
Sctq(f)=106∑i=0ai(log(f/103))i.
(18) Provided that the detector scenario is operated for four years, the corresponding coefficients are
α=9×10−45 ,β=0.138 ,γ=−221 ,η=521 ,λ=1680 ,fc=0.0013 ,a0=−18.6,a1=−1.43,a2=−0.687,a3=0.24,a4=−0.15, a5=−1.8 anda6=−3.2 . Thus, the full sensitivity curve is derived by adding the galactic confusion noise toSn(f) .Figure 3 shows the sensitivity curves for LISA, TaiJi, and TianQin, from which we can see that TianQin is more sensitive to gravitational wave signals at higher frequencies, whereas TaiJi and LISA are reliable in detecting signals for lower frequencies. TaiJi is better than LISA in detecting gravitational wave signals at higher frequencies because the target displacement noise of TaiJi is better than LISA in cases where both the residual acceleration noise and arm length of the detector are similar. Obviously, the galactic confusion noise provokes a small rise in the sensitivity value in the low-frequency range while TianQin is less affected than TaiJi and LISA.
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The inner product weighted by the detector noise spectral density of two frequency-domain signals
h1(f),h2(f) is defined as(h1|h2)=2∫fhighflowh1∗(f)h2(f)+h1(f)h2∗(f)SN(f)df,
(19) where we choose
flow to be half of the smallest oscillation frequency andfhigh to be twice the highest oscillation frequency to prevent "junk" radiation in the Fourier transformation [9]. The sky-averaged SNR (denoted by ρ) based on the definition of the inner product can be expressed asρ=√(h|h).
(20) Supposing that the probability distribution for the measurement errors of parameters is Gaussian in the limit of large SNR [59, 60], the measurement errors on parameters
θi can be derived from the Fisher information matrix:Δθi≈√(Γ−1)ii.
(21) Here, the Fisher information matrix is given by
Γij=(∂h∂θi∣∂h∂θj),
(22) where
→θ is a seven-dimensional parameter space in the ringdown signals, namely,→θ={M,χf,DL,ν,ϕ0,ι,ζ} .First, the SNR varying with the mass of the black hole for LISA, TaiJi, and TianQin is calculated, as shown in Fig. 4, from which we plot two dominant QNMs in the ringdown signals. The total SNR is heavily dominated by the strongest
(2,2) mode. Overall, as the mass increases, the SNR increases until reaching the maximum and then decreases gradually for all detectors. After comparing the SNRs of LISA, TaiJi, and TianQin, it is found that TianQin is more sensitive to gravitational signals with smaller masses, whereas TaiJi and LISA are more reliable in detecting signals for more massive black holes. In particular, the galactic confusion noise serves as a catalyst for a small dip appearing around a mass of6×106M⊙ <∼x M <∼x 108M⊙ for TaiJi and LISA, whereas for TianQin, this effect is trivial. Its implications may be negligible for less massive black holes.Figure 4. (color online) SNRs of LISA, TaiJi, and TianQin with the change in the mass for the ringdown signal. The calculations are performed with
χf=0.01, DL=2.5 Gpc, ν=2/9, ϕ0=0, ι=π/3, ζ=0.2 .Next, we estimate the measurement errors for the dimensionless deviating parameter ζ (denoted by
Δζ ) via available tools. Before introducing the standard parameters of space-based gravitational wave detectors, we roughly evaluate the influence of the arm length of the detector on the dimensionless deviating parameter, where the residual acceleration noise is fixed and the displacement noise is proportional to the arm length for gravitational wave detection. Fig. 5 shows the measurement errors as a function of arm length for different sources of gravitational waves, where the maximum error of ζ is shown by a cyan dotted horizontal line, and the real arm lengths of LISA, TaiJi, and TianQin are shown by green dashed vertical lines. The test of EdGB gravity will be affected by sources of different masses. The arm lengths of TaiJi and LISA are more appropriate to test EdGB gravity for more massive black holes, whereas for less massive black holes, a more practical arm length can be provided by TianQin. Because the above are qualitative analyses, only the parameters related to LISA are used here to evaluate the ability to test EdGB gravity on the whole. For the specific calculation below, the standard parameters of space-based gravitational wave detectors are considered.Figure 5. (color online) Dependence of parameter estimation accuracy
Δζ on the arm length of the detector L for different sources of gravitational waves. Here, we choose the basic parameters of LISA as a reference. The residual acceleration noise is fixed asSa=3×10−15ms−2/√Hz . The displacement noise is proportional to the arm lengthSx∼αL , whereα=1.5×10−112.5×109 . The cyan dotted horizontal line represents the maximum error of ζ, and the green dashed vertical lines denote the real arm lengths of LISA, TaiJi, and TianQin. The calculations are performed withχf=0.01, ν=2/9, ϕ0=0, ι=π/3, ζ=0.2 .In addition, to explore how several source parameters, such as the mass
M , the luminosity distanceDL , the spin of the remnant black holeχf , and the symmetric mass ratio ν, affectΔζ , we show the variation inΔζ with related parameters in Fig. 6. The top left plot of Fig. 6 shows the measurement errors as a function of the mass, where we assume thatχf=0.01, DL=2.5 Gpc, ν=2/9 . As expected,Δζ first decreases with increasingM until arriving at the minimum value but soon increases with the accumulation ofM for all detectors. For specific massive black holes, a slight bulge emerges in the measurement errors owing to the galactic confusion noise, which implies that galactic confusion noise plays a negative role in constraining EdGB gravity. It is clear that TianQin can constrain ζ more accurately for smaller masses, whereas TaiJi and LISA perform well for more massive black holes, which is also reflected by the sensitivity of the detectors. The measurement errors as a function of the luminosity distance are plotted in the top right plot of Fig. 6, where we fixM=107M⊙, χf=0.01, ν=2/9 . As shown,Δζ increases with increasingDL , which is obvious because the SNR increases monotonically with decreasing distance. This is intuitively reflected in the ringdown waveform containing the term1/DL ; hence, there is a1/DL2 in the error function after Fisher analysis. The bottom left plot of Fig. 6 presents the dependence of measurement errors on the spin of the remnant black hole, where we setM=107M⊙, DL=2.5 Gpc, ν=2/9 . Note thatΔζ decreases extremely slowly asχf increases. It is implied that the spin parameter of the black hole has little effect on measurement errors for EdGB gravity by detectors. The dependence of measurement errors on the symmetric mass ratio is illustrated in the bottom right plot of Fig. 6, where we chooseM=107M⊙, DL=2.5 Gpc, χf=0.01 . We observe thatΔζ first decreases with increasing ν and then increases abruptly as ν approaches0.25 . This is because radiated energy increases with large symmetric mass ratios, and as ν approaches0.25 , the amplitude of the QNMs for(3,3) is zero. To estimate the associated parameters better, it is necessary to avoid selecting ν in this limit or replace it with other dominant modes.Figure 6. (color online) Dependence of parameter estimation accuracy
Δζ on the mass M (top left), the luminosity distanceDL (top right), the spin of the remnant black holeχf (bottom left), and the symmetric mass ratio ν (bottom right). The black, magenta, and blue curves represent the errors detected by LISA, TaiJi, and TianQin, respectively. Other parameters used areϕ0=0, ι=π/3, ζ=0.2 .In particular, we pay attention to how the measurement errors
Δζ varies with ζ, which is illustrated in Fig. 7. As shown, the effect ofΔζ on ζ is dynamic because ζ is nonlinear in Eqs. (4) and (5). Owing to the relatively large coefficients in Table 1, higher-order corrections are not discarded here. Hence, the result of the covariance matrix contains variable ζ, and specifically,Δζ decreases with increasing ζ. When the dimensionless deviating parameter is small, it is difficult to distinguish EdGB gravity from general relativity owing to the larger uncertainty of ζ. We emphasize this problem by tracing the maximum error of ζ in EdGB gravity (denoted byΔζmax ), namely, the solution of the equationΔζ=ζ , which is treated as the maximum constraint of the detector by probing a particular wave source, marked as red dots. Only in the area below that do space-based gravitational wave detectors have the potential to tell the difference between EdGB gravity and general relativity. Because there is no significant difference in the standard parameters of the space-based gravitational wave detector between TaiJi and LISA, for comparison purposes, the remainder of our paper focuses only on the results of LISA and TianQin. Furthermore, to investigate the maximum constraint on the dimensionless deviating parameter for EdGB gravity with LISA and TianQin, we show the density plots ofΔζmax in thelg(DL/Gpc)−lg(M/M⊙) plane in Fig. 8. We observe that ζ can be best constrainted with LISA forM∼5.5×106M⊙ and TianQin forM∼3×106M⊙ . Moreover,Δζ decreases with increasing SNR, which is demonstrated in Fig. 9. For a considerably smaller deviation from general relativity, we must count on detectors with larger SNRs to obtain the maximum constraint. The increase in the required SNR is not linear, and as the accuracy of the constraints increases, the SNR increases dramatically. Compared with TianQin, LISA is noticeably more likely to give an optimal constraint in the future.Figure 7. (color online) Dependence of parameter estimation accuracy
Δζ on the dimensionless deviating parameter ζ. The green dashed line denotesΔζ=ζ . The red intersections areΔζmax for each detector. The profiles are obtained withM=107M⊙, χf=0.01, DL=2.5 Gpc, ν=2/9, ϕ0=0, ι=π/3 for different space-based gravitational wave detectors. -
To verify the results calculated in the Fisher information matrix analysis, we employ the Bayesian inference method to analyze the simulated source. The Bayesian inference method is widely used to estimate the probability distribution of unknown parameters from sampled data containing signals and noise, which is instrumental in astrophysical and cosmological analyses. Unlike Fisher information matrix analysis, which is limited to large SNRs, Bayesian analysis is applicable to a wider range of situations. Furthermore, Bayesian posterior probability distributions provide more information. The disadvantage is that it is more computationally intensive. In essence, Bayesian statistics involves creating a likelihood to associate unknown and measurement parameters. Then, the probability distribution of the unknown parameters will be updated through the distribution of measurement data. This process is based on Bayes' theorem:
P(→θ|d)=π(→θ)L(d|→θ)p(d),
(23) where
P(→θ|d) is the posterior probability of the set of free parameters,d=h(→θ0)+n represents the measurement data, which collects the gravitational-wave signal modeled by all the true parameters→θ0 and detector noise n modeled by the noise power spectra,π(→θ) is the prior probability of→θ ,p(d) is a normalization constant, which is also called the evidence of d, andL(d|→θ) is the likelihood, which can be written asL(d|→θ)=exp[−12(h(→θ)−d|h(→θ)−d)].
(24) The amplitudes of the strain data for ringdown and the noise power spectra for LISA and TianQin are illustrated in Fig. 10. As shown, the highest peak corresponds to mode
(2,2) . By comparison, the noise power spectrum of TianQin reaches the lowest level in the present frequency band.Figure 10. (color online) Amplitudes of the strain data d for ringdown and the noise power spectra for LISA and TianQin. The results are obtained using the parameters
M=6×106M⊙, χf=0.01, DL=2.5 Gpc, ν=2/9, ϕ0=0, ι=π/3, ζ=0.2. After generating simulation data, we utilize the Bayesian inference method to obtain the probability distributions of the source parameters, including
lg(M/M⊙) ,χf ,DL/Gpc , ν,ϕ0 , ι, and ζ. Fig. 11 shows the posterior distribution for the ringdown signal with LISA, where the true parameters for the ringdown signal are set asM=106.5M⊙, χf=0.1, DL=10 Gpc, ν=2/9, ϕ0=0, ι=π/3, ζ=0.2 . The priors of the corresponding parameters are set as uniform distributions within the ranges[6,7] ,[0.001,0.2] ,[9,14] ,[0,1/4] ,[0,2π] ,[0,π] ,[0,0.4] , respectively. For comparison, the results for the same wave source with TianQin are shown in Fig. 12. As shown, the probability distribution forDL is relatively poor, even if we set a narrower prior. If a broad priori is set, the estimation accuracy ofDL is even worse. Its estimation affects the estimation accuracy of other parameters. Fortunately, the ringdown signal is usually spotted after the inspiral and merger, which will provide an estimate of the luminosity distance. Now that we focus only on the probability distribution for ζ, Fig. 13 shows the posterior possibility of ζ for LISA and TianQin with different luminosity distances to the source. As shown on the left side of Fig. 13, the luminosity distance is10 Gpc , and the dimensionless deviating parameter cannot be distinguished from 0. The 95% credible upper limit given by different detectors is 0.385 for LISA and 0.387 for TianQin. For the right subplot, all parameters remain the same, except the luminosity distance is reduced to0.5 Gpc . For such a signal with an extremely large SNR (over 1000), space-based detectors can distinguish between the dimensionless deviating parameter and 0 by the ringdown signal. The estimate for the dimensionless deviating parameter with both detectors is0.1943+0.0287−0.0216 for LISA and0.1727+0.0402−0.0292 for TianQin. It is clear that LISA can provide the best limit. The posterior possibility of ζ at shorter luminosity distances is found to be better than that at longer luminosity distances, which implies that possible constraints to ζ become more accurate at high SNRs. The uncertainty of the dimensionless deviating parameter for different detectors matches the results of the Fisher information matrix.Figure 11. (color online) Posterior distribution for the ringdown signals with LISA. The true parameters are set as
M=106.5M⊙, χf=0.1, DL=10 Gpc, ν=2/9, ϕ0=0, ι=π/3, ζ=0.2 .
Parameter estimation for Einstein-dilaton-Gauss-Bonnet gravity with ringdown signals
- Received Date: 2023-06-14
- Available Online: 2023-10-15
Abstract: Future space-based gravitational-wave detectors will detect gravitational waves with high sensitivity in the millihertz frequency band, providing more opportunities to test theories of gravity than ground-based detectors. The study of quasinormal modes (QNMs) and their application in gravity theory testing have been an important aspect in the field of gravitational physics. In this study, we investigate the capability of future space-based gravitational wave detectors, such as LISA, TaiJi, and TianQin, to constrain the dimensionless deviating parameter for Einstein-dilaton-Gauss-Bonnet (EdGB) gravity with ringdown signals from the merger of binary black holes. The ringdown signal is modeled by the two strongest QNMs in EdGB gravity. Considering time-delay interferometry, we calculate the signal-to-noise ratio of different space-based detectors for ringdown signals to analyze their capabilities. The Fisher information matrix is employed to analyze the accuracy of parameter estimation, with particular focus on the dimensionless deviating parameter for EdGB gravity. The impact of the parameters of gravitational wave sources on the estimation accuracy of the dimensionless deviating parameter is also studied. We find that the constraint ability of EdGB gravity is limited because the uncertainty of the dimensionless deviating parameter increases with a decrease in the dimensionless deviating parameter. LISA and TaiJi offer more advantages in constraining the dimensionless deviating parameter to a more accurate level for massive black holes, whereas TianQin is more suited to less massive black holes. The Bayesian inference method is used to perform parameter estimation on simulated data, which verifies the reliability of the conclusion.