Post-Keplerian waveform model for binary compact object as sources of space-based gravitational wave detector and its implications

Binary stars are unique astrophysical laboratories for the study of fundamental physics and cosmology. Such laboratories have been realized through pulsar timing detection of binary pulsar Kramer et al. (2021); Lorimer and Kramer (2004) and gravitational wave detection of binary coalescence The LIGO Scientific Collaboration et al. (2021). But pulsar timing detection and current gravitational wave detection concern two different life stages of the binary system. Pulsar timing detection is about the early inspiral stage. While current gravitational wave detection by ground based detectors is about the stage near merger. In the coming future the space-based gravitational wave detectors will change this situation. Then gravitational wave detection can also provides information about early inspiral stage Willems et al. (2008); Huang et al. (2020). For example, LISA may resolve $\sim 10^{4}$ double white dwarfs (DWDs) Nelemans et al. (2001); Seto (2022).

Waveform model is important for gravitational wave data analysis and for extracting the information of the sources. For the merger stage of a binary, extremal general relativity (GR) condition makes numerical relativity calculation necessary to construct the waveform model Cao and Han (2017); Liu et al. (2020, 2022). For the early inspiral stage, the gravitational interaction is much weaker. And Newtonian approximation is valid. The binary compact objects including stellar massive binary black hole, binary neutron star, binary white dwarf and mixture of these compact objects belong to such early inspiral binaries for the future space-based gravitational wave detectors. The simplest waveform model for such sources is based on quadrupole formula for monochromatic approximation Huang et al. (2020). When the GR leading order of inspiral behavior is taken into consideration, the chirp waveform model has been widely used Takahashi and Seto (2002). Besides the GR effect, the mass transfer between the two compact objects Kremer et al. (2017) and the tidal deformation of the two objects Benacquista (2011); Wolz et al. (2020) can also contribute to the chirp signal. Recently, the Ref. Willems et al. (2008) used Newtonian approximation to construct a waveform model and focused on the effect of white dwarf interior on the periastron precession. These different waveform models focus on different physical mechanism and introduce different waveform parameters consequently. Such diversity of waveform models make data analysis and source property study hard. In order to catch weak GW signals, ones have to analyze the same data many many times with different specific waveform models. Even worse, some signals may be lost due to the uncomplete analysis. As promising laboratory to test gravity theories, it is interesting and important to construct waveform model for different gravity theories other than GR. Unfortunately such waveform models have not been studied yet.

In contrast to the aforementioned specific waveform models for different physical situations, an unified waveform model will be useful. Such an unified waveform model will definitely facilitate the data analysis for the space-based detectors. With such an unified waveform model, ones can not only catch the signal much easier but also distinguish different physical situations and even constraint different gravity theories. We will propose such an unified waveform model in the current paper.

Since the component objects of a binary in early inspiral stage are well separated, the orbital dynamics can be well described by a perturbation to a Kepler orbit. Such perturbation may come from gravity theory, environment, tidal interaction and mass transfer between the two components. All kinds of effects on the binary dynamics can be included. Consequently, the parameterized post-Keplerian formalism Damour and Taylor (1992); Moore et al. (2016) is a good framework to describe the system. This unified waveform model uses post-Keplerian parameters instead of component masses and other physical parameters. This unified waveform model is consequently valid for any kinds of gravity theories and any detailed complicated matter dynamics. If ones can measure the post-Keplerian parameters accurately, gravity theory and the theory of stellar structure and evolution can be tested and investigated Kramer et al. (2021); Willems et al. (2008). We will show later in the current paper that this is exactly true for the current planning space-based gravitational wave detectors including LISA, Taiji and Tianqin.

The arrangement of the rest of this paper is as following. We will present the post-Keplerian waveform model in the next section. Then we relate the involved post-Keplerian parameters to general relativity (GR) effect, gravity theory effect beyond GR, tidal interaction and the mass transfer effect between the two components of the binary. When we construct such relation we also give guide lines to use post-Keplerian parameters to distinguish these different effects. After that we use Fisher matrix technique to estimate the measurement accuracy of the post-Keplerian parameters based on LISA, Taiji and Tianqin respectively. We find that the involved post-Keplerian parameters can be well determined. And consequently the detail matter dynamics of the binary can be studied then. For binary with purely gravitational interactions, gravity theory can be constrained well. That is to say, equipped with our post-Keplerian waveform model, the future space-based gravitational wave detectors can do good science for compact object binary observations. Throughout this paper we will use units $c=G=1$.

The general bound Keplerian orbit is an eccentric orbit. The binary moves along an eccentric orbit and the corresponding gravitational wave form respect to the detector can be written as Takahashi and Seto (2002); Barack and Cutler (2004); Willems et al. (2008)

where $J_{i}(x)$ is the $i$-th order Bessel function of the first kind. The amplitude $A$ depends on the gravity theory and the specific dynamics of the binary. $n$ corresponds to higher harmonics excited by the eccentric orbit. In Liu et al. (2022), we call it tones to distinguish with the spherical harmonics. $\phi(t)$ is the Doppler shifted orbital phase

with initial phase $\phi_{0}$.

Perturbations to the above Kepler orbit may result in changing of orbital frequency $f$, inclination angle $\iota$, orbit eccentricity $e$, periastron direction $\gamma$. In all we have adiabatic changing

Suitable coordinate choice can eliminate $\gamma_{0}=0$. The investigation in Willems et al. (2008) neglected parameters $\dot{\iota}_{0}$ and $\dot{e}_{0}$. The authors in Wolz et al. (2020) only considered the changing of $f$ but to second order.

On the first glance, our waveform model has nothing to do with the masses of the two components of the binary. That is because the information of mass and spin are coded in post-Keplerian parameters like $\dot{f}_{0}$. And just because of this fact, our post-Keplerian waveform model is quite generic for stellar binary sources of space detector.

Besides the above intrinsic parameters, the pattern function $F_{+,\times}$ will introduce three more extrinsic parameters Cutler (1998)

The above pattern functions depends on the source right ascension $\alpha$ and declination $\delta$, and the wave polarization angle $\psi$.

Including both the intrinsic and extrinsic parameters we have 12 parameters $(A,f_{0},\dot{f}_{0},\iota_{0},\dot{\iota}_{0},e_{0},\dot{e}_{0},\dot{\gamma}_{0},\phi_{0},\alpha,\delta,\psi)$ in all. Source localization can be analyzed as Kocsis et al. (2007); Baibhav et al. (2020); Wang et al. (2020); Ruan et al. (2021); Hu et al. (2021); Zhang et al. (2021). Since we do not care about the source localization in the current work and the source location can not be determined by a single detector, we instead fix $\alpha=\psi=0$ and $\delta=\pi/2$ in the current work. So we need only consider 9 parameters $(A,f_{0},\dot{f}_{0},\iota_{0},\dot{\iota}_{0},e_{0},\dot{e}_{0},\dot{\gamma}_{0},\phi_{0})$ and the pattern functions become

Eqs. (1)-(11) correspond to the post-Keplerian waveform model. The approximated pattern function shown above is only for simplified discussion involved in the next sections. Such simplification does not affect the general conclusion. And the post-Keplerian waveform model can be straight forwardly applied to any realistic situations.

Assuming the masses of the two components of the binary are $M_{1,2}$, the chirp mass, symmetric mass ratio and total mass are given by

In order to make the mass transfer between the two compact objects stable till merger, the mass ratio should be limited in $(0,0.24)$ Verbunt and Rappaport (1988); Han and Webbink (1999); Yu and Jeffery (2010).

In this section we will use Fisher information matrix to estimate the measurement accuracy of the post-Keplerian parameters

where $p_{i}$ represents the $i$-th parameter among the 9 parameters $(A,f_{0},\dot{f}_{0},\iota_{0},\dot{\iota}_{0},e_{0},\dot{e}_{0},\dot{\gamma}_{0},\phi_{0})$. Inversing the Fisher matrix we can estimate the measurement error as

The inner product in the above equation is defined as

where $f_{0}$ corresponds to the post-Keplerian parameter and the adiabatic approximation has been used Wolz et al. (2020).

Regarding to the sensitivity of a space-based detector, we use the following approximation (Eq. (13) of Robson et al. (2019))

For LISA Robson et al. (2019) we have

For Taiji Wang et al. (2020) we have

For Tianqin we have Luo et al. (2016)

Due to the relation (71), the measurement accuracy for different detector is similar only upto the factor $1/S_{n}(f_{0})$. Since this fact we calculate the measurement accuracy for LISA first and deduce the accuracy for other detectors accordingly in the following. As Huang et al. (2020), we take the observation time as $T=5$ years.

We list the typical measurement accuracies of post-Keplerian parameters in Table. 1. Four typical post-Keplerian parameters are investigated. Each set corresponds to a typical binary system. The corresponding waveforms of these four groups parameters are shown in Fig. 3.

The first group corresponds to a typical binary white dwarf with pure GR effect. We can see all three detectors including LISA, Taiji and Tianqin can accurately measure almost parameters except $\dot{f}_{0}$ and $\dot{e}_{0}$. Here when we say ‘accurately’ we mean the relative error is less than one percent. Regarding to $\phi_{0}$, it’s value takes order 0.1 as the angle $\iota_{0}$ although the assumed value is 0 here. Regarding to $\dot{\iota}_{0}$, if the effect like (54) comes in, its value will be order $10^{-8}$. Relative to these values, the measurement error bar is small.

The well estimated $\dot{\gamma}_{0}$ can be used to judge whether the matter effect is strong or not based on (56). If the matter effect is ignorable, the well estimated $\dot{\gamma}_{0}$ can be used to constrain gravity theory beyond GR based on (43). Note that $\frac{\Delta\dot{\gamma}_{0}}{\dot{\gamma}_{0}}\lesssim 10^{-4}$, we can constrain

If we can further make sure the source is two black holes, we can get

If everything indicates that only GR dominates, we can plug $\mathcal{M}$ and $D_{L}$ into the the Eqs. (18)-(27) and do the parameters estimation again. Then the chirp mass $\mathcal{M}$ can be well estimated Willems et al. (2008); Wolz et al. (2020). With the information of $\dot{\gamma}_{0}$, $e_{0}$ and $f_{0}$, the Eq. (27) gives us the information of total mass $M$. Combining $M$ and $\mathcal{M}$ we can deduce mass ratio. Consequently the individual mass can be well estimated. And more we can use detail waveform model according to specific gravity theory and plug in component masses at this stage. Then the involved parameters may be constrained more strictly.

If the matter of the compact stars is very stiff, the two components may be very close. Possibly the frequency $f_{0}$ may be as high as $10^{-2}$Hz. We investigate such case in the second group of Table. 1. This set of parameters also corresponds to stellar origin binary black hole or binary neutron star. If the mass transfer and the tidal interaction between two white dwarfs are very strong, the binary’s frequency may also fall in this group. Firstly we can see LISA and Tianqin get roughly the same measurement accuracy. This is because the sensitivity for LISA and Tianqin roughly equals at $10^{-2}$Hz. Compared to the first group, the measurement accuracy improves roughly one order. This is due to that the detector sensitivity at $10^{-2}$Hz is about 10 times better than that at $10^{-3}$Hz. Consequently the afore mentioned parameter constrain and gravity theory constrain may be improved one order based on this kind of sources.

Compared to the first group, now we consider more eccentric binary with $e_{0}=0.6$ in the third group. Along with the increasing of the eccentricity, the waveform becomes more complicated. As shown in Fig. 3, higher tones appear clearly Han et al. (2017); Liu et al. (2022). As expected, higher eccentricity improves much the measurement accuracy Sun et al. (2015); Ma et al. (2017); Pan et al. (2019); Ma and Yunes (2019); Yang et al. (2022, 2023). Compared to the first group, the measurement accuracy improves one order. This means more eccentric binary even facilitate more the study of the gravity theory and the nuclear matter property.

In the fourth group we consider higher $\dot{f}_{0}$, $\dot{e}_{0}$ and $\dot{\gamma}_{0}$ compared to the first group. This may happen if the tidal interaction and the mass transfer between the two binary components take effect. We find that the measurement accuracy is roughly the same to the ones got in the first group. This result indicates only $A$, $f_{0}$ and $e_{0}$ affect the measurement accuracy strongly. At the same time we can conclude that the current planning space-based gravitational wave detectors can determine the post-Keplerian parameters as well as that shown in the first group of Table. 1. If the signal is stronger (larger $A$), and/or the frequency is higher (larger $f_{0}$), and/or the orbit is more eccentric (larger $e_{0}$), we can get even better parameter estimation. Consequently we can constrain gravity theory tight and give deep insight to the nuclear matter.

Compact object binary provides ones a valuable laboratory for fundamental physics and astronomy. Pulsar timing detection has made a good use of binary pulsars, Existing gravitational wave detection has made a good use of coalescence of binary black hole, binary neutron star and neutron star-black hole binary. In the near future, the space-based gravitational wave detectors will facilitate us to make early inspiraling compact object binaries also become good laboratories. To do so, feasible waveform model is important.

In the current work, we propose a post-Keplerian waveform model for early inspiraling compact object binaries. Such a waveform model takes all kinds of gravity theories and all kinds of matter dynamics involved in the binary into consideration. This post-Keplerian waveform model can be looked as a unification of all previous existing waveform models for early inspiraling compact object binaries.

The post-Keplerian expansion parameters of the binary orbit are the waveform model parameters instead of binary component mass and other matter parameters. Based on this waveform model we can constrain gravity theory without the assumption of specific gravity theory. This trick is quite similar to that used in pulsar timing experiments. Based on the post-Keplerian expansion, we have constructed a ready-to-use unified waveform model of binary compact object for space-based detectors.

We have also checked the parameter estimation accuracy for the post-Keplerian parameters. Based on the current planning space-based gravitational wave detectors including LISA, Taiji and Tianqin, we find that the post-Keplerian parameters can be determined accurately. Consequently gravitational interaction dominating or matter dynamics dominating can be well distinguished. Following that the model independent constrain on gravity theory can be well done for gravitational dominated binary. And the nuclear matter properties can be well studied for matter dynamics dominated binary. Detail parameter estimation accuracy has been shown in Table. 1. Using our waveform model, ones can not only catch the GW signal much easier but also distinguish different physical situations and even constraint different gravity theories based on LISA, Taiji or Tianqin detectors. When specific gravity theory or matter dynamics is determined, more specific waveform model can be used to fine tune the involved parameters.

Phenomenologically the effects of spin precession and orbital eccentricity will result in similar waveforms. Like event GW190521, it is hard to distinguish spin precession and orbital eccentricity for ground-based detectors Romero-Shaw et al. (2023). In the GW190521 case, we are limited by the small number of observed GW cycles. For the space-based detectors we considered in the current work, this will not be an issue. This is because the observation time is 5 years and many wave cycles will be accumulated.

In conclusion, our post-Keplerian waveform model provides a good toolkit for the future space-based gravitational wave detectors to study early inspiraling compact object binaries.

This work was supported in part by the National Key Research and Development Program of China Grant No. 2021YFC2203001 and the CAS Project for Young Scientists in Basic Research YSBR-006.