Gravitational waveforms and radiation powers of the triple system PSR J0337+1715 in modified theories of gravity

TL;DR

This study analyzes gravitational waveforms and radiation powers of the triple system PSR J0337+1715 across three gravity theories, revealing differences in polarization modes and potential for constraining Einstein-aether theory via future gravitational wave observations.

Contribution

It provides the first detailed comparison of gravitational wave signatures in GR, Brans-Dicke, and Einstein-aether theories for a relativistic triple system using post-Newtonian approximations.

Findings

Tensor modes are similar across theories with peaks at twice the orbital frequencies.

Einstein-aether theory predicts additional polarization modes, including breathing and longitudinal.

Dipole emission can be comparable to quadrupole emission in Einstein-aether theory.

Abstract

In this paper, we study the gravitational waveforms, polarizations and radiation powers of the first relativistic triple systems PSR J0337 + 1715, observed in 2014, by using the post-Newtonian approximations to their lowest order. Although they cannot be observed either by current or next generation of the detectors, they do provide useful information to test different theories of gravity. In particular, we carry out the studies in three different theories, general relativity (GR), Brans-Dicke (BD) gravity, and Einstein-aether ($\ae$) theory. The tensor modes $h_{+}$ and $h_{\times}$ exist in all three theories and have almost equal amplitudes. Their frequencies are all peaked at two locations, $ f^{(+, \times)}_1 \simeq 0.0686656 \mu$Hz and $ f^{(+, \times)}_2 \simeq 14.2138 \mu$Hz, which are about twice of the outer and inner orbital frequencies of the triple system, as predicted in GR. In $\ae$-theory, all the six polarization modes are different from zero, but the breathing $h_b$ and longitudinal $h_L$ modes are not independent and also peaked at two frequencies, but at the frequencies, $ f^{(b, L, \ae)}_{1} \simeq 0.0457771 \mu$Hz and $f^{(b, L, \ae)}_{ 2} \simeq 7.09545 \mu$Hz. A similar phenomenon is also observed in BD gravity, in which only the three modes $h_{+},\; h_{\times}$ and $h_{b}$ exit. We also study the radiation powers, and find that the quadrupole emission in each of the three theories has almost the same amplitude, but the dipole emission can be as big as the quadrupole emission in $\ae$-theory. This can provide a very promising window to obtain severe constraints on $\ae$-theory by the multi-band gravitational wave astronomy.

Figures

Equations

$${\ddot{\zeta }}_i=-R_{0i0j}{\zeta }^j,$$

$${\ddot{\zeta }}_i=-R_{0i0j}{\zeta }^j=\frac{1}{2}{\ddot{h}}_{ij}^{TT}{\zeta }^j,$$

$$h_{ij}^{TT}(t,\mathbf{x})=\frac{2G_N}{R}{\ddot{Q}}_{ij}^{TT}(t-R),$$

$${\Lambda }_{ijkl}(\mathbf{N})=P_{ik}{P}_{jl}-\frac{1}{2}{P}_{ij}{P}_{kl},$$

$${\mathbf{e}}_X$$

$${\mathbf{e}}_Y$$

$${\mathbf{e}}_Z$$

$$h_{+}^{GR}=\frac{G_N}{R}{\ddot{Q}}_{kl}{e}_{+}^{kl},h_{\times }^{GR}=\frac{G_N}{R}{\ddot{Q}}_{kl}{e}_{\times }^{kl},$$

$${\tilde{h}}_A^n=\frac{1}{N}\sum _{m=0}^{N-1}{h}_A^m{e}^{-i2\pi nm/N},$$

$$f_1^{+,\times }$$

$$f_2^{+,\times }$$

$$f_o=0.035\mu \text{mbox}Hz,f_i=7.103\mu \text{mbox}Hz.$$

$${\ddot{\zeta }}_i=-R_{0i0j}{\zeta }^j\equiv \frac{1}{2}{\ddot{\mathcal{P}}}_{ij}{\zeta }^j,$$

$${\mathcal{P}}_{ij}$$

$$h_{+}^{\text{ae}}$$

$$h_{\times }^{\text{ae}}$$

$$h_b^{\text{ae}}$$

$$h_L^{\text{ae}}$$

$$h_X^{\text{ae}}$$

$$h_Y^{\text{ae}}$$

$$Q_{ij}$$

$$I_{ij}$$

$$c_1$$

$$c_3$$

$$h_X^{\text{ae}}=h_Y^{\text{ae}}=0,(c_{13}=0).$$

$$\frac{h_{+,\times }^{\text{ae}}}{h_{+,\times }^{GR}}=1-\frac{1}{2}{c}_{14}.$$

$$\Delta h_{+,\times }^{\text{ae}}\equiv \frac{h_{+,\times }^{\text{ae}}-h_{+,\times }^{GR}}{h_{+,\times }^{GR}}=-\frac{1}{2}{c}_{14}.$$

$$\Delta |{\tilde{h}}_{+,\times }^{\text{ae}}|\equiv \left(|{\tilde{h}}_{+,\times }^{\text{ae}}|-|{\tilde{h}}_{+,\times }^{GR}|\right)/|{\tilde{h}}_{+,\times }^{GR}|.$$

$$f_1^{b,L}$$

$$f_2^{b,L}$$

$$h_b^{\text{ae}}$$

$$c_1$$

$$c_3$$

$$c_1$$

$$c_3$$

$${\theta }^{ij}$$

$${\phi }^{BD}$$

Full text

I Introduction

The concept of gravitational waves (GWs) was first developed by Einstein in 1916 right following his general theory of relativity (GR). He proposed that GWs are the ripples of spacetimes that are propagating with the speed of light Einstein . A century passed, the Laser Interferometer Gravitational-Wave Observatory (LIGO) first verified his GW theory by directly detecting the GW signal on Sep. 14, 2015 GW150914 . After this, ten more GWs have been detected by LIGO Scientific Collaboration and Virgo Collaboration GW151226 ; GW170104 ; GW170608 ; GW170814 ; GW170817 ; GWs . The sources of these eleven GW events are all binary systems of black holes, except for the event GW170817, which is a binary system of neutron stars GW170817 .

In fact, there are about $13\%$ of low-mass stellar systems containing three or more stars FC17 , and $96\%$ of low-mass binaries with periods shorter than 3 days which are part of a larger hierarchy Tok06 . Recently, a realistic triple system was observed, named as PSR J0337 + 1715 (J0337) Ransom14 , which consists of an inner binary and a third companion. The inner binary consists of a pulsar with mass $m_{1}=1.44M_{\bigodot}$ and a white dwarf (WD) with mass $m_{2}=0.20M_{\bigodot}$ in a 1.6 day orbit. The outer binary consists of the inner binary and a second dwarf with mass $m_{3}=0.41M_{\bigodot}$ in a 327 day orbit. The two orbits are very circular with its eccentricities $e_{I}\simeq 6.9\times 10^{-4}$ for the inner binary and $e_{O}\simeq 3.5\times 10^{-2}$ for the outer orbit. The two orbital planes are remarkably coplanar with an inclination $\lesssim 0.01^{o}$ [See Fig. 1].

A triple system is an ideal place to test the strong equivalence principle Shao16 . Remarkably, after 6-year observations, recently it was found that the accelerations of the pulsar and its nearby white-dwarf companion differ fractionally by no more than $2.6\times 10^{-6}$ Archibald18 , which provides the most severe constraint on the violation of the strong equivalence principle.

In this paper, we investigate this triple system in three different theories of gravity, General Relativity (GR), Einstein-aether theory (${\ae}$-theory) and Brans-Dicke (BD) gravity, by using the post-Newtonian approximations to their lowest order. We shall pay particular attention on the differences predicted by these theories. Although neither the current generation of detectors nor the next one can detect the GWs emitted by this system 222The frequency of the GWs emitted by this system is about $10^{-8}$ Hz. With this frequency, only the pulsar timing array (PTA), such as IPTA and SKA, can detect such GWs. But, the amplitudes of these GWs are far below the sensitivities of these detectors., it serves well as a realistic example to show clearly the different predictions from each of these theories. In particular, we shall study, gravitational waveforms, their polarizations, Discrete Fourier transfom (DFT) of polarizations as well as the radiation powers. Among the modified theories of gravity, ${\ae}$-theory locally breaks the Lorentz symmetry by introducing a globally time-like unit vector field (the ${\ae}$ther) Jacobson , while in BD gravity the gravitational interaction is mediated by both a scalar and a tensor fields Brans .

Specifically, the paper is organized as follows: In Sec. II, we study the gravitational waveforms, polarization modes and their DFTs in GR, $\ae$-theory, and BD gravity respectively, while in Sec. III we investigate the radiation powers of the GW in each theory. Due to the presence of the extra vector and scalar fields in $\ae$-theory, and the extra scalar filed in BD gravity, the total emission power is different in each of the three theories. In particular, we find that the dipole emission in $\ae$-theory can be as large as the quadrupole emission in GR, which can provide a very promising window to obtain severe constraints on $\ae$-theory by the multi-band gravitational wave astronomy AS16 . There is also an appendix, in which we present a very brief introduction to ${\ae}$-theory Jacobson .

Before turning to the next section, we would like to note that, in the framework of æ-theory, Foster Foster07 and Yagi et al Yagi14 derived the metric and equations of motion to the 1PN order for a N-body system. Recently, Will applied them to study the 3-body problem and obtained the accelerations of a 2-body system in the presence of the third body at the quasi-Newtonian order Will18 . For nearly circular coplanar orbits, he also calculated the “strong-field” Nordtvedt parameter $\hat{\eta}_{N}$. For triple system J0337, ignoring the sensitivities of the two white-dwarf companions, Will found that $\hat{\eta}_{N}$ is given by $\hat{\eta}_{N}=s_{1}/(1-s_{1})$, where $s_{1}$ denotes the sensitivity of the pulsar.

In this paper, we will adopt the following conventions: All the repeated indices $i,j,k,l\;(i,j,k,l=1,2,3)$ will be summed over regardless of their vertical poition, while repeated indices $a,b,c\;(a,b,c=1,2,3)$ will not be summed over unless the summation is explicitly indicated. We set the speed of light equal to one ($c=1$). The metric signature is $(-,+,+,+)$. Since J0337 is a triple system, there exists no analytical orbit, we are using the numerical orbit supplied by Dr. Lijing Shao Shao16 .

II Gravitational Waveforms and their Polarizations

When a GW passes two test masses, the distance between them will be changed. Assuming that $\zeta^{i}$ denotes the spatial coordinates between these two test masses in the Minkowski coordinates $x^{\mu}=(t,x,y,z)$, the equations of the geodesic deviation in the weak-field approximations read,

[TABLE]

where $R_{\mu\nu\alpha\beta}$ denotes the linearized Riemann tensor, which is determined by the field equations of a given theory. Different theory yield different components of $R_{0i0j}$. Therefore, in the following we shall consider GR, æ-theory and BD gravity separately.

II.1 Gravitational Waveforms and Their Polarizations in GR

In GR, the equations of the geodesic deviation take the form MM ,

[TABLE]

where

[TABLE]

with $R\equiv|{\bf x}|$ denoting the distance from the observer to the source and $G_{N}$ denoting the Newtonian constant. In the above equations, $TT$ represents the transverse-traceless part of the tensor, which can be obtained by applying $TT$ operator $\Lambda_{ijkl}$,

[TABLE]

where $P_{ij}({\bf N})=\delta_{ij}-N_{i}N_{j}$. Let’s introduce a set of orthorgonal bases, (${\bf e}_{X},{\bf e}_{Y},{\bf e}_{Z}$), where ${\bf e}_{Z}\equiv\bf{N}$ denotes the propagation direction of the GW from the source to the observers. Thus, (${\bf e}_{X},{\bf e}_{Y}$) forms a plane orthogonal to the propagation direction $\bf{N}$ of the GW. In the $(t,x,y,z)$ coordinates, they are given by,

[TABLE]

where $\vartheta$ and $\varphi$ are the two spherical angular coordinates. In the case of J0337, $\vartheta$ and $\varphi$ are 0∘ and 270∘, respectively Ransom14 . In GR, $h_{ij}^{TT}$ has only two degrees of freedom corresponding to the plus (“+”) and cross (“$\times$”) polarizations, which are given by,

[TABLE]

where $e_{+}^{kl}\equiv e_{X}^{k}e_{X}^{l}-e_{Y}^{k}e_{Y}^{l}$, $e_{\times}^{kl}\equiv e_{X}^{k}e_{Y}^{l}+e_{Y}^{k}e_{X}^{l}$. In Figs. 2 and 3, we plot these two polarizations, from figures it can be seen that the amplitudes of both polarizations are about $10^{-22}$, which is in the range of the designed sensitivity of the current generation of the ground-based detectors, such as, LIGO, Virgo and KAGRA, but not their frequencies (See, e.g. Moore15 ). This is because the orbital frequency of J0337 is out of the observational bands of the current detectors. You can find the frequencies of these two polarizations easily from Figs. 4 and 5.

To obtain the Fourier transform for each polarization mode, instead of using the continuous Fourier transform, $\tilde{h}_{A}(f)=(2\pi)^{-1}\int{h_{A}(t)e^{-i2\pi ft}dt}$, we adopt the discrete Fourier transform (DFT). First, the time interval of signal $h_{A}(t)$ is divided into $N-1$ subintervals, $t=\left\{t_{0},t_{1},...,t_{N-1}\right\}$, $t_{m}=m\Delta t$, $\Delta t$ is the length of subinterval. Then we have a list of dicrete values $h_{A}^{m}\equiv h_{A}(t_{m}),$ and we get the DFT accodrding to the following formula

[TABLE]

where $\tilde{h}_{A}^{n}\approx\tilde{h}_{A}(f)$, $f=\left\{f_{0},f_{1},...,f_{N-1}\right\}$, $f_{n}=n/(N\Delta t)$. In this paper, we shall use the built-in function of software Mathematica to calculate DFT of waves. With this in mind, in Figs. 4 and 5 we plot $\tilde{h}_{+}^{GR}(f)$ and $\tilde{h}_{\times}^{GR}(f)$, respectively. In these figures, there are two peaks and the corresponding frequencies are

[TABLE]

where $f_{o}$ and $f_{i}$ represent outer and inner orbital frequencies of the triple system J0337 Ransom14 ,

[TABLE]

Thus, $f^{+,\times}_{1}$ and $f^{+,\times}_{2}$ are about twice of the outer and inner orbital frequencies of the triple system, but not exactly. In GR, for a binary system the GW frequency is exactly equal to two times of their orbital frequency MM . However, it must be noted that here the difference is due to the presence of the third component of the triple system.

II.2 Gravitational Waveforms and Their Polarizations in æ-theory

In $\ae$-theory, the equation of the geodesic deviation reads Lin ,

[TABLE]

where

[TABLE]

and $\phi_{ij},\;\Psi^{\mathrm{I}}_{i}$ and $\Phi^{\mathrm{II}}$ are, respectively, the gauge-invariant quantities of the tensor, vector and scalar modes defined in Lin . The quantities $c_{V}$ and $c_{S}$ denote the speeds of the vector and scalar modes, respectively. Due to their presence, the polarizations of a GW have five independent modes which are given respectively by Lin

[TABLE]

where $c_{-}\equiv c_{1}-c_{3}$, $G_{\ae}=G_{N}(1-c_{14}/2)$, ${\cal{P}}_{XY}\equiv{\cal{P}}_{ij}e^{i}_{X}e^{j}_{Y}$, and so on. We have also defined,

[TABLE]

where $x^{i}_{a}$ is the location of the $a$-th body and $I\equiv I_{kk}$. For more details, see Lin . From the above expressions, in addition to the usual plus and cross polarization modes, $\ae$-theory predicts three extra independent modes, $h_{b}^{\ae},\;h_{X}^{\ae}$ and $h_{Y}^{\ae}$. Comparing to $h_{+}^{\ae}$ and $h_{\times}^{\ae}$, these extra modes are suppressed, respectively, by a factor, $c_{14}\lesssim 10^{-5}$ and $c_{13}\lesssim 10^{-15}$ Lin . The longitudinal mode $h_{L}^{\ae}$ is proportional to the breathing mode $h_{b}^{\ae}$.

In the following, let us first consider the case,

[TABLE]

which satisfy all the theoretical and observational constraints given by Eq.(A.11) (See also OMW18 ). Note that for this choice we have $c_{13}=0$, and then the two modes $h_{X}^{\ae}$ and $h_{Y}^{\ae}$ vanish identically,

[TABLE]

So, in the rest of this paper, we shall not consider them any further.

In Figs. 2 and 3, we plot the two polarization modes $h_{+}^{\ae}$ and $h_{\times}^{\ae}$, while in Figs. 4 and 5, we plot their DFTs. From these figures, it can be seen clearly that these two modes are almost identical to the ones given in GR, after all the constraints of $\ae$-theory are taken into account. In fact, we have

[TABLE]

The differences between æ-theory and GR are determined by $c_{14}$,

[TABLE]

Recall that $c_{14}\lesssim 2.5\times 10^{-5}$. Therefore, the signals of these two modes in ${\ae}$-theory and GR are overlapping, their frequencies are precisely the same, as shown in Figs. 4 and 5. Similarly, the differences in frequency domain can be defined as

[TABLE]

In Fig. 6, we plot $h_{b}^{\ae}$ and $h_{L}^{\ae}$, which are about three orders lower than $h_{+}^{\ae}$ and $h_{\times}^{\ae}$. In Fig. 7, we plot the DFT of the $h_{b}$ and $h_{L}$. It is remarkable that now the two peaked frequencies are approxiamtely equal to the outer and inner orbital frequencies.

[TABLE]

where $f_{o}$ and $f_{i}$ are given by Eq.(2.9). $f^{b,L}_{1}$ and $f^{b,L}_{2}$ are almost equal to the outer and inner orbital frequencies of the triple system.

Recently, we find that, for binary systems in $\ae$-theory, the polarization modes $h_{b}$ and $h_{L}$ all contain two frequencies, one is equal to the binary’s orbital frequency and the other is twice the orbital frequency Zhao19 . This confirms the above result since J0337 can be considered a hierarchical system consisting of two binaries. Again, the reason that $f^{b,L}_{1}$ and $f^{b,L}_{2}$ are not exactly equal to the outer and inner orbital frequencies of the triple system is due to the influence of the third component of the triple system, rather than the predictions of the $\ae$-theory itself.

To see further the dependence of $h_{b,L}^{\ae}$ on $c_{i}$’s, let us first note that

[TABLE]

where $c_{14}/c_{2}\lesssim 1$, as it can be seen from Eq.(A.11).

From above equation, we see that $h_{b,L}^{\ae}$ are dependent on $c_{2}$ and $c_{14}$ mainly. To see the effects explicitly, in the following we consider two more cases.

In the first one, $c_{2}$ is chosen to be different from Eq. (II.2), now we have

[TABLE]

In Figs. 8 and 9 we plot out the breathing ($h_{b}^{\ae}$) and longitudinal ($h_{L}^{\ae}$) polarization modes for this case. Comparing them, respectively, with Figs. 6 and 7, we find that the amplitudes of $h_{b}^{\ae}$ and $h_{L}^{\ae}$ decrease when $c_{2}$ increases, whereas their frequencies remain the same.

On the other hand, $c_{14}$ is chosen to be different from Eq. (II.2), the parameters are

[TABLE]

we plot out the corresponding $h_{b}^{\ae}$ and $h_{L}^{\ae}$ modes in Figs. 10 and 11. Comparing them with Figs. 6 and Fig. 7, we find that the amplitudes of $h_{b}^{\ae}$ and $h_{L}^{\ae}$ decrease when $c_{14}$ decreases, whereas their frequencies stay the same, as it is expected from Eq.(2.25).

II.3 Gravitational Waveforms and Their Polarizations in BD Gravity

The metric perturbation and scalar field perturbation are given by Will ,

[TABLE]

where $\phi_{0}$ is the value of the BD scalar field in the Minkowski background, which satisfies $\phi_{0}=(4+2\omega_{BD})/[(3+2\omega_{BD})G_{N}]$ Zhang17 , and

[TABLE]

where $\omega_{BD}$ is the BD parameter of the theory. In this paper, we choose sensitivities such that $s_{1}$ (for pulsar) $=0.2$, $s_{2}$ (for inner WD) $=0$, $s_{3}$ (for outer WD) $=0$ and the coupling constant $\omega_{BD}=10^{5}$ Archibald18 . Note that in writing the above expressions, we had dropped the non-propagating terms in $\varphi^{BD}$. Then, the components $R_{0i0j}$ of the Riemann tensor can be cast in the form,

[TABLE]

Then, it can be shown that there are only three independent polarization modes, given, respectively, by

[TABLE]

which are plotted out, respectively, in Figs. 2, 3 and 6, while their DFTs are plotted out in Figs. 4, 5 and 7. From these figures, it can be seen that the two polarization modes $h_{+}^{BD}$ and $h_{\times}^{BD}$ are overlapping with those in GR and $\ae$-theory, due to the observational constraints on the $\omega_{BD}$. In fact, we have

[TABLE]

The differences between BD gravity and GR are determined by $\omega_{BD}$,

[TABLE]

Recall that $\omega_{BD}\sim{\cal O}(10^{5})$. Therefore, the signals of these two modes in BD gravity and GR are overlapping, their frequencies are precisely the same, as shown in Figs. 4 and 5. Similarly the differences in frequency domain can be defined as

[TABLE]

From Fig. 6, the breathing mode ($h_{b}^{BD}$) is different from $\ae$-theory. From Fig. 7, its DFT has also two peaked frequencies and are equal to those of $\ae$-theory, i.e., breathing mode in BD gravity only has first harmonics of orbital phase. In contrast to the polarization modes $h_{+}$ and $h_{\times}$, which have second harmonics of orbital phase.

III Radiation Power

In GR, the total radiation power is given by MM ; Dmitra ,

[TABLE]

where $Q_{ij}$ is the mass quadrupole moment defined in Eq.(II.2) and the angular brackets denote the time average. 333However, since we are considering periodic GWs, we will not take this time average in the relevant plots. Otherwise, it will be zero for such periodic waves. The same will also apply to the cases of $\ae$-theory and BD gravity.. Note that in this section we shall not distinguish the time $t$ and its corresponding retarded time. Strictly speaking, all the quantities should be evaluated at the retarded time. However, it is not necessary for our current purpose.

The reference frame is chosen such that the inclination is $39.25^{\circ}$, where the inclination is the angle of the orbital plane relative to the ($x,y$)-plane perpendicular to the line-of-sight from Earth to the pulsar. In Fig. 12, we plot the radiation power in GR for about 500 days, where the inserted image shows the details from day 12 to day 26.

In $\ae$-theory, from Foster ; Lin we find that

[TABLE]

where $\Sigma_{i}$ is defined as

[TABLE]

and

[TABLE]

with Foster06 ,

[TABLE]

Here $v^{i}_{a}\equiv\dot{x}^{i}_{a}$ is the velocity of $a$-th body along $x^{i}$-direction, $\Omega_{a}$ is the binding energy of $a$-th body. For J0337 Ransom14 , we have $\Omega_{1}$ (for pulsar) =$-2.56955\times 10^{46}$ J, $\Omega_{2}$ (for inner WD)=$-9.75554\times 10^{40}$ J, $\Omega_{3}$ (for outer WD)=$-2.12650\times 10^{42}$ J. where $c_{T}$, $c_{V}$ and $c_{S}$ are the speeds of the tensor, vector and scalar modes, given by Eq.(Appendix A: Einstein-aether Theory) in Appendix A, in which the definitions $c_{ijk}\equiv c_{i}+c_{j}+c_{k}$ and $c_{ij}\equiv c_{i}+c_{j}$ are given.

In Fig. 13, we plot the radiation power in ${\ae}$-theory of the parts ${\cal A}$, ${\cal B}$ and ${\cal C}$ separately, for about 500 days. Again, the inserted images are from day 12 to day 26. Note that at every moment during the 252 days in the plot, the ${\cal A}$ part of ${\ae}$-theory is quite close to that of GR with the relative difference proportional to $c_{14}$ Lin ,

[TABLE]

From this figure, it is also clear that the dipole part ${\cal C}$ has almost the same amplitude as that of the quadrupole part ${\cal A}$, while the monopole part ${\cal B}$ is suppressed by a factor $c_{14}$ Lin . The large magnitude of the dipole contribution $\cal{C}$ seemingly contradicts to the analysis given in Lin . In particular, Eq.(3.13) in Lin shows that ${\cal{W}}_{\cal{C}}^{\mbox{NS}}/{\cal{W}}_{\cal{A}}^{\mbox{NS}}\simeq 10^{-2}$, where

[TABLE]

where ${\cal{A}},\;{\cal{C}}$ and $\Sigma$ are all given explicitly in Lin . However, in deriving ${\cal{W}}_{\cal{C}}^{\mbox{NS}}/{\cal{W}}_{\cal{A}}^{\mbox{NS}}\simeq 10^{-2}$ we assumed that ${\cal{O}}\left(v^{2}\right)\simeq 10^{-5}$, while in the case of J0337 we find that the relative velocities of the inner binary system are of the order of ${\cal{O}}\left(v^{2}\right)\simeq 10^{-7}$. After this is taken into account, we find that ${\cal{W}}_{\cal{C}}/{\cal{W}}_{\cal{A}}\simeq{\cal{O}}(1)$ for the current triple system.

It is remarkable to note that, with the multi-band gravitational wave astronomy AS16 , joint observations of GW150914-like by LIGO/Virgo/KAGRA and LISA will improve bounds on dipole emission from black hole binaries by six orders of magnitude relative to current constraints BYY18 . Thus, it is very promising that the third generation of detectors, both space-borne and ground-based, could provide severe constraints on $\ae$-theory.

In BD gravity, following Will we obtain

[TABLE]

where

[TABLE]

where ${M}_{1}^{i}$ and ${M}_{2}^{ij}$ are defined by Eq.(II.3). Note that in writing down the above expressions, we had dropped the non-propagating terms.

In Fig. 14, we plot the radiation power in BD gravity for about 500 days, where the inserted image shows the details only from day 12 to day 26. Note that at every moment during the 500 days, the first part of BD is quite close to that given in GR. In fact, we find that

[TABLE]

IV Conclusions

In this paper, we studied the gravitational waveforms, their polarizations and Fourier transforms, as well as the radiation powers of the relativistic triple systems PSR J0337 + 1715, observed in 2014 Ransom14 . This system consists of an inner binary and a third companion. The inner binary consists of a pulsar with mass $m_{1}=1.44M_{\bigodot}$ and a white dwarf with mass $m_{2}=0.20M_{\bigodot}$ in a 1.6 day orbit. The outer binary consists of the inner binary and a second dwarf with mass $m_{3}=0.41M_{\bigodot}$ in a 327 day orbit. The two orbits are very circular with its eccentricities $e_{I}\simeq 6.9\times 10^{-4}$ for the inner binary and $e_{O}\simeq 3.5\times 10^{-2}$ for the outer orbit. The two orbital planes are remarkably coplanar with an inclination $\lesssim 0.01^{o}$ [See Fig. 1].

Our studies were carried out in three different theories, GR, BD gravity, and $\ae$-theory. In GR, only the tensor mode exists, so a GW has only two polarization modes, the so-called, plus ($h_{+}$) and cross ($h_{\times}$) modes. Their amplitudes and Fourier transforms are plotted in Figs. 2 - 4, from which it can be seen that their amplitude is about $10^{-23}$, while their frequencies are peaked in two locations, $f^{+,\times}_{1}=0.068658\mu$Hz (for the outer orbit) and $f^{+,\times}_{2}=14.212\mu$Hz (for the inner orbit), respectively. These are about two times of the inner and outer orbital frequencies of the triple system, and agree well with the GR predictions MM .

In $\ae$-theory, all the six polarization modes are different from zero, but the beating ($h_{b}$) and longitudinal ($h_{L}$) modes are not independent and are related to each other by Eq.(2.12). In comparing with $h_{+}$ and $h_{\times}$, however, they are suppressed by a factor $c_{14}$ which is observationally restricted to $c_{14}\lesssim 10^{-5}$ OMW18 . A somehow surprising result is that the two peaked frequencies are not two times of those of the inner and outer orbital frequencies of the triple system. Instead, they are about equal to them, $f^{b,L}_{1}=0.045772\mu$Hz (for the outer orbit) and $f^{b,L}_{2}=7.0947\mu$Hz (for the inner orbit) [Cf. Fig. 7].

The effects of the parameters $c_{i}$’s on $h_{b}^{\ae}$ and $h_{L}^{\ae}$ were also studied in detial, and in particular we found that their amplitudes are weekly dependent on the choices of these parameters, while the frequencies of their Fourier transforms remain the same.

The other two independent polarization modes in $\ae$-theory are the vector modes, $h_{X}$ and $h_{Y}$, which are all proportional to $c_{13}$. The current observations from GW170817 GW170817 and GRB 170817A GRB170817 on the speed of the tensor mode requires $c_{13}\lesssim 10^{-15}$. Therefore, these two modes are highly restricted by the limit of the speed of the tensor mode.

We also studied the radiation power due to the tensor, vector and scalar modes in $\ae$-theory, and three different parts were plotted in Fig. 9. The amplitude of the quadrupole part (${\cal{A}}$), contributed from all of these three modes, tensor, vector and scalar Foster ; Lin is quite comparable with that of GR. But, the monopole (${\cal{B}}$) part has contributions only from the scalar mode, while the dipole (${\cal{C}}$) part has contributions from both the scalar and vector modes, but does not have any contributions from the tensor mode, as expected Foster ; Lin . The monopole part is suppressed by a factor $c_{14}\lesssim{\cal{O}}\left(10^{-5}\right)$, but the dipole part is almost in the same order of the quadrupole part. With the arrival of the multi-band gravitational wave astronomy AS16 , joint observations of GW150914-like by LIGO/Virgo/KAGRA and LISA will improve serve constraints on the dipole emission BYY18 . Thus, the multi-band gravitational wave astronomy will provide a very promising direction to constrain $\ae$-theory.

We also carried out similar studies in BD gravity, and the relevant quantities were plotted in Figs. 2 - 7 and 10. Due to the severe observational constraint on the BD parameter $\omega_{BD}\gtrsim 10^{5}$, we did not find significant deviations from GR, except that the frequency of the breathing mode $h_{b}$ is also peaked in two locations, $f^{b}_{1}=0.045772\mu$Hz and $f^{b}_{2}=7.0947\mu$Hz, which are about the outer and inner orbital frequencies of the triple system, quite similar to that in $\ae$-theory [Cf. Fig. 7] but with a low amplitude.

Acknowledgments

We would like to thank Dr. Lijing Shao for providing us the original data of PSR J0337+1715. This work is supported in part by the National Natural Foundation of China (NNSFC) with the grant numbers: Nos. 11603020, 11633001, 11173021, 11322324, 11653002, 11421303, 11375153, 11675145, 11675143, 11105120, 11805166, 11835009, 11690022, 11375247, 11435006, 11575109, 11647601, and No. 11773028.

Appendix A: Einstein-aether Theory

In this appendix, we shall give a brief introduction to Einstein-aether Theory. For detail, we refer readers to Jacobson’s review Jacobson , and Lin for recent development. In æ-theory, the fundamental variables are JM01 ,

[TABLE]

where $g_{\mu\nu}$ is the four-dimensional metric of the space-time, $u^{\mu}$ is the aether four-velocity, and $\lambda$ is a Lagrangian multiplier, which guarantees that the aether four-velocity is always timelike. The general action of æ-theory is given by Jacobson ,

[TABLE]

where $S_{m}$ denotes the action of matter, and $S_{\ae}$ the gravitational action of the $\ae$-theory, given by

[TABLE]

Here $\psi$ collectively denotes the matter fields, $R$ and $g$ are, respectively, the Ricci scalar 444Note that $R$ here is different from the one used in Sec.II.A which denotes the distance. and determinant of $g_{\mu\nu}$, and

[TABLE]

where $D_{\mu}$ denotes the covariant derivative with respect to $g_{\mu\nu}$, and $M^{\alpha\beta}_{~{}~{}~{}~{}\mu\nu}$ is defined as

[TABLE]

The four coupling constants $c_{i}$’s are all dimensionless, and $G_{\ae}$ is related to the Newtonian constant $G_{N}$ via the relation CL04 ,

[TABLE]

where $c_{ij}\equiv c_{i}+c_{j}$.

It is easy to show that the Minkowski spacetime is a solution of $\ae$-theory, in which the aether is aligned along the time direction, $\bar{u}_{\mu}=\delta^{0}_{\mu}$. Then, the linear perturbations around the Minkowski background show that the theory in general possess three types of excitations, scalar, vector and tensor modes JM04 , with their squared speeds given, respectively, by

[TABLE]

where $c_{ijk}\equiv c_{i}+c_{j}+c_{k}$.

Recently, the combination of the gravitational wave event GW170817 GW170817 , and the event of the gamma-ray burst GRB 170817A GRB170817 provides a remarkably stringent constraint on the speed of the spin-2 mode,

[TABLE]

which, together with Eq.(Appendix A: Einstein-aether Theory), implies that

[TABLE]

Imposing further the following conditions: (a) the theory is free of ghosts; (b) the squared speeds $c_{I}^{2}\;(I=S,V,T)$ must be non-negative; (c) $c_{I}^{2}-1$ must be greater than $-10^{-15}$ or so, in order to avoid the existence of the vacuum gravi-Čerenkov radiation by matter such as cosmic rays EMS05 ; and (d) the theory must be consistent with the current observations on the primordial helium abundance $\left|G_{cos}/G_{N}-1\right|\lesssim 1/8$, where $G_{cos}\equiv G_{\ae}/(1+(c_{13}+3c_{2})/2)$ CL04 , together with Eq.(A.9) and the conditions,

[TABLE]

from the Solar System observations Will06 , it was found that the parameter space of the theory is restricted to OMW18 ,

[TABLE]

Finally, we note that the theoretical and observational constraints of $\ae$-theory and gravitational waves were also studied in GHLP18 .