Constraining alternative theories of gravity by gravitational waves from precessing eccentric compact binaries with LISA

We calculate how strongly one can put constraints on alternative theories of gravity such as Brans-Dicke and massive graviton theories with LISA. We consider inspiral gravitational waves from a compact binary composed of a neutron star and an intermediate mass black hole in Brans-Dicke (BD) theory and that composed of a super massive black hole in massive graviton theories. We use the restricted second post-Newtonian waveforms including the effects of spins. We also take both precession and eccentricity of the orbit into account. For simplicity, we set the fiducial value for the spin of one of the binary constituents to zero so that we can apply the approximation called simple precession. We perform the Monte Carlo simulations of ${10}^4$ binaries, estimating the determination accuracy of binary parameters including the BD parameter ${\omega }_{\mathrm{BD}}$ and the Compton wavelength of graviton ${\lambda }_g$ for each binary using the Fisher matrix method. We find that including both the spin-spin coupling $\sigma$ and the eccentricity $e$ into the binary parameters reduces the determination accuracy by an order of magnitude for the Brans-Dicke case, while it has less influence on massive graviton theories. On the other hand, including precession enhances the constraint on ${\omega }_{\mathrm{BD}}$ only 20% but it increases the constraint on ${\lambda }_g$ by several factors. Using a $(1.4+1000)M_{\odot }$ neutron star/black hole binary of $\mathrm{SNR}=\sqrt{200}$, one can put a constraint ${\omega }_{\mathrm{BD}}>6944$, while using a $({10}^7+{10}^6)M_{\odot }$ black hole/black hole binary at 3 Gpc, one can put ${\lambda }_g>3.10\times {10}^{21}\mathrm{cm}$, on average. The latter is 4 orders of magnitude stronger than the one obtained from the solar system experiment. These results are consistent with previous results within uncontrolled errors and indicate that the effects of precession and eccentricity must be taken carefully in the parameter estimation analysis.

Figure 1

We use two types of coordinates: (i) a barred barycentric frame $(\overline{x},\overline{y},\overline{z})$ tied to the ecliptic and centered in the solar system barycenter, (ii) an unbarred detector frame $(x,y,z)$, centered in the barycenter of the triangle and attached to the detector.

Figure 2

The Thomas precession phase $\delta \varphi (t)$ is the angle from the vector $\widehat{\mathit{u}}$ to the principal axis $\widehat{\mathit{N}}\times \widehat{\mathit{L}}$. We define $\delta {\varphi }^{\prime }$ as the angle from the vector $\widehat{\mathit{L}}\times \widehat{\mathit{u}}$ to the one $\widehat{\mathit{N}}\times \widehat{\mathit{L}}$. Notice that these two vectors always lie on the orbital plane. $\delta {\varphi }^{\prime }$ equals to $\delta \varphi +\pi /2$ up to $O(|\delta \mathit{L}|/L)$.

Figure 3

The noise spectral density for LISA.

Figure 4

The curve showing 68% confidence level on the $\Delta e^2\mathrm{\text{−}}\Delta \overline{\omega }$ plane for a $(1.4+400)M_{\odot }$ NS/BH binary with $\rho =10$, such that the fiducial values lie at the center of the ellipse.

Figure 5

The curve showing 68% confidence level on the $\Delta e^2\mathrm{\text{−}}\Delta {\beta }_g$ plane for a $({10}^7+{10}^7)M_{\odot }$ SMBH/BH binary at 3 Gpc. The fiducial values lie at the center of the ellipse as in Fig. 4.

Figure 6

The histograms showing the probability distribution of the lower bound of ${\omega }_{\mathrm{BD}}$ obtained from the Monte Carlo simulations with $(1.4+1000)M_{\odot }$ NS/BH binaries with $\rho =\sqrt{200}$ in Brans-Dicke theory. The (black) dotted thin one represents the estimate without precession and $I_e$ is not taken into account. The (blue) solid thin one shows the one without precession but $I_e$ is taken into parameters. The (purple) dotted thick one includes precession but does not include $I_e$. The (red) solid thick curve shows the one including both precession and $I_e$. Here the prior information $I_e>0$ is not used to estimate the lower bound of ${\omega }_{\mathrm{BD}}$.

Figure 7

The histograms showing the probability distribution of the error estimates of the chirp mass $\Delta \mathcal{M}/\mathcal{M}$, the reduced mass $\Delta \mu /\mu$, the spin-orbit coupling $\Delta \beta$, and the spin-spin coupling $\Delta \sigma$ in Brans-Dicke theory. These are obtained from the Monte Carlo simulations with $(1.4+1000)M_{\odot }$ NS/BH binaries with $\rho =\sqrt{200}$. The meaning of different types of curves is the same as in Fig. 6.

Figure 8

The histograms showing the probability distribution of the error estimates of the luminosity distance $\Delta D_L/D_L$ and the angular resolution $\Delta {\Omega }_S$ in Brans-Dicke theory. These are obtained from the Monte Carlo simulations with $(1.4+1000)M_{\odot }$ NS/BH binaries with $\rho =\sqrt{200}$. The meaning of different types of curves is the same as in Fig. 6.

Figure 9

${10}^4$ plots of the lower bounds of ${\omega }_{\mathrm{BD}}$ against $\widehat{\mathit{L}}·\widehat{\mathit{N}}$. This result is obtained from the Monte Carlo simulations with $(1.4+1000)M_{\odot }$ NS/BH binaries with $\rho =\sqrt{200}$ in Brans-Dicke theory. $I_e$ is taken into account though the precessional effect is not included.

Figure 10

${10}^4$ plots of the lower bounds of ${\omega }_{\mathrm{BD}}$ against ${\overline{\theta }}_{\mathrm{S}}$ and ${\overline{\varphi }}_{\mathrm{S}}$, as in Fig. 9.

Figure 11

The same plot as Fig. 9 except we do not include the Doppler phase ${\phi }_D(t)$ in this case.

Figure 12

The same plot as Fig. 10 except we do not include the Doppler phase ${\phi }_D(t)$ in this case.

Figure 13

The histograms showing the probability distribution of the lower bound of ${\lambda }_g$ obtained from the Monte Carlo simulations with $({10}^7+{10}^6)M_{\odot }$ BH/BH binaries at 3 Gpc in massive graviton theories. The (black) dotted thin one represents the estimate without precession and $I_e$ is not taken into account. The (purple) dotted thick one shows the one without precession but $I_e$ is taken into parameters. The (blue) solid thin one includes precession but does not take $I_e$ into account. The (red) solid thick one shows the one including precession and $I_e$ is also taken into account.

Figure 14

The histograms showing the probability distribution of the error estimates of the chirp mass $\Delta \mathcal{M}/\mathcal{M}$, the reduced mass $\Delta \mu /\mu$, the spin-orbit coupling $\Delta \beta$, and the spin-spin coupling $\Delta \sigma$ in massive graviton theories. These are obtained from the Monte Carlo simulations with $({10}^7+{10}^6)M_{\odot }$ BH/BH binaries at 3 Gpc. The meaning of different types of curves is the same as in Fig. 13.

Figure 15

The histograms showing the probability distribution of the error estimates of the luminosity distance $\Delta D_L/D_L$ and the angular resolution $\Delta {\Omega }_S$ in massive graviton theories. These are obtained from the Monte Carlo simulations with $({10}^7+{10}^6)M_{\odot }$ BH/BH binaries at 3 Gpc. The meaning of different types of curves is the same as in Fig. 13. The accuracy of the luminosity distance with and without precession are shown separately.

Figure 16

The histograms showing the probability distribution of the error estimates of the spin-orbit coupling $\Delta \beta$ in Brans-Dicke theory. These are obtained from the Monte Carlo simulations with $(1.4+1000)M_{\odot }$ NS/BH binaries with $\rho =\sqrt{200}$. Eccentricity is not included into the binary parameters. The (black) dotted thin one represents the estimate without spin precession and the fiducial value for ${\chi }_{\mathrm{BH}}$ is set to 0, while the (black) solid one represents the one with ${\chi }_{\mathrm{BH}}=0.5$. The (purple) dotted thick one represents the one including spin precession and the fiducial value for ${\chi }_{\mathrm{BH}}$ is set to 0.5.

Figure 17

The unit orbital angular momentum vector $\widehat{\mathit{L}}$ precesses around a fixed vector ${\widehat{\mathit{J}}}_0$. The opening angle of the precession cone is given by ${\lambda }_{\mathrm{L}}$ and the precession angle is denoted as $\alpha$.