Detectability of bigravity with graviton oscillations using gravitational wave observations

The gravitational waveforms in the ghost-free bigravity theory exhibit deviations from those in general relativity. The main difference is caused by graviton oscillations in the bigravity theory. We investigate the prospects for the detection of the corrections to gravitational waveforms from coalescing compact binaries due to graviton oscillations and for constraining bigravity parameters with the gravitational wave observations. We consider the bigravity model discussed by the De Felice-Nakamura-Tanaka subset of the bigravity model, and the phenomenological model in which the bigravity parameters are treated as independent variables. In both models, the bigravity waveform shows strong amplitude modulation, and there can be a characteristic frequency of the largest peak of the amplitude, which depends on the bigravity parameters. We show that there is a detectable region of the bigravity parameters for the advanced ground-based laser interferometers, such as Advanced LIGO, Advanced Virgo, and KAGRA. This region corresponds to the effective graviton mass of $\mu \gtrsim {10}^{-17}{\mathrm{cm}}^{-1}$ for $\tilde{c}-1\gtrsim {10}^{-19}$ in the phenomenological model, while $\mu \gtrsim {10}^{-16.5}{\mathrm{cm}}^{-1}$ for $\kappa {\xi }_c^2\gtrsim {10}^{0.5}$ in the De Felice-Nakamura-Tanaka subset of the bigravity model, respectively, where $\tilde{c}$ is the propagation speed of the massive graviton and $\kappa {\xi }_c^2$ corresponds to the corrections to the gravitational constant in general relativity. These regions are not excluded by existing solar system tests. We also show that, in the case of $1.4-1.4M_{\odot }$ binaries at the distance of 200 Mpc, $\log {\mu }^2$ is determined with an accuracy of $\mathcal{O}(0.1)\%$ at the $1\sigma$ level for a fiducial model with ${\mu }^2=10^{-33}{\mathrm{cm}}^{-2}$ in the case of the phenomenological model.

Figure 1

The frequency-domain gravitational waves $h(f)$ for different values of the model parameter sets of $({\mu }^2,\tilde{c}-1)$. The curves are plotted for (a) GR [solid (blue)] and for the bigravity models with (b) $({\mu }^2,\tilde{c}-1)=(10^{-33.2}{\mathrm{cm}}^{-2},10^{-17.8})$ [dot-dashed (green)], (c) $(10^{-33}{\mathrm{cm}}^{-2},10^{-18})$ [long-dashed (red)], and (d) $(10^{-32.8}{\mathrm{cm}}^{-2},10^{-18.2})$ [dashed (black)], respectively, at fixed $\kappa {\xi }_c^2=100$. Here we consider BNS at the distance, $D_L=200\mathrm{Mpc}$. The SNR and the fitting factor between the GR waveform and each waveform in this figure become as follows: $(\mathrm{SNR},\mathrm{FF})=$ (a) (8.7,1.0), (b) (31,0.50), (c) (26,0.47), (d) (21,0.53). The definition of FF is given in Eq. (23).

Figure 2

The time-domain gravitational waveform $h(t)$. The coalescence time $t_c$ is set to 0. The parameters and the definitions of the curves are the same as those of Fig. 1.

Figure 3

The same as Fig. 1 but for different values of $\kappa {\xi }_c^2$ in the case of $({\mu }^2,\tilde{c}-1)=(10^{-33}{\mathrm{cm}}^{-2},10^{-18})$. The curves are for (a) GR [solid (blue)] and the bigravity model with (b) $\kappa {\xi }_c^2=50$ [dot-dashed (green)], (c) $\kappa {\xi }_c^2=100$ [long-dashed (red)], and (d) $\kappa {\xi }_c^2=1000$ [dashed (black)], respectively. Each curve corresponds to $(\mathrm{SNR},\mathrm{FF})=$ (a) (8.7,1.0), (b) (19,0.58), (c) (26,0.47), (d) (34,0.41).

Figure 4

The time-domain gravitational waveform $h(t)$. The parameters are the same as those of Fig. 3.

Figure 5

The detectable region of the bigravity corrections to the waveforms in the case $(m_1,m_2)=(1.4M_{\odot },1.4M_{\odot })$ and $\kappa {\xi }_c^2=100$. Curves correspond to the distance to the source at $D_L=200\mathrm{Mpc}$ (solid) and 100 Mpc (dashed). The detectable region is upper and right-hand side of these curves. The detectable region is defined as the region where $\mathrm{SNR}>{\mathrm{SNR}}_{\text{req}}$ is satisfied. The false-alarm probability is set to $F=10^{-4}$.

Figure 6

A plot similar to Fig. 5 but for the waveforms from BNS with $(m_1,m_2)=(1.4M_{\odot },1.4M_{\odot })$ at 200 Mpc (solid), NSBH with $(m_1,m_2)=(1.4M_{\odot },10M_{\odot })$ at 416 Mpc (dashed), and BBH with $(m_1,m_2)=(10M_{\odot },10M_{\odot })$ at 980 Mpc (dot-dashed), respectively. We set $\kappa {\xi }_c^2=100$. The detectable region is upper and right-hand side of these curves. SNR of the gravitational waves from these systems in GR limit are 8.7.

Figure 7

A plot similar to Fig. 5, but for $\kappa {\xi }_c^2=50$ (dashed), 100 (solid), and 1000 (dot-dashed), respectively. The masses are $(m_1,m_2)=(1.4M_{\odot },1.4M_{\odot })$ and the distance is 200 Mpc.

Figure 8

Contour plots of the fitting factor between the GR and bigravity waveforms in the $({\mu }^2,\tilde{c}-1)$ parameter space. Here we adopt the model $\kappa {\xi }_c^2=100$. Curves correspond to contours of $\mathrm{FF}=0.9$ (solid), $\mathrm{FF}=0.95$ (dashed), and $\mathrm{FF}=0.99$ (dotted). We assume BNS at $D_L=200\mathrm{Mpc}$.

Figure 9

Contour plots of the SNR of bigravity waveforms in the $({\mu }^2,\tilde{c}-1)$ parameter space. The parameters are the same as those of Fig. 5. Curves correspond to contours of $\mathrm{SNR}=8.75$ (solid), $\mathrm{SNR}=10$ (dashed), $\mathrm{SNR}=18$ (dotted), and $\mathrm{SNR}=25$ (dot-dashed). We assume BNS at $D_L=200\mathrm{Mpc}$.

Figure 10

Projected $1\sigma$ error contours on the (${\mu }^2$, $\tilde{c}-1$) plane. The results are obtained from the Fisher matrix with 8-parameters, $\log {\mu }^2,\log (\tilde{c}-1),\kappa {\xi }_c^2,\log D_L,\mathcal{M},\eta ,t_c$, and ${\mathrm{\Phi }}_c$, and marginalized over 6 parameters other than $\log {\mu }^2$ and $\log (\tilde{c}-1)$. The fiducial model is $({\mu }^2,\tilde{c}-1)=(10^{-33}{\mathrm{cm}}^{-2},10^{-18})$, for BNS at $D_L=200\mathrm{Mpc}$ (solid) and at 100 Mpc (dashed). SNR is renormalized to $\mathrm{SNR}=10$.

Figure 11

Same as Fig. 10 but for the fiducial model, $({\mu }^2,\tilde{c}-1)=(10^{-32}{\mathrm{cm}}^{-2},10^{-19})$. SNR is renormalized to $\mathrm{SNR}=10$.

Figure 12

The frequency-domain gravitational waves $h(f)$ for DFNT subset of the bigravity model for different values of the average density in the galaxies ${\rho }_{\text{gal}}$, where GWs are generated. The curves are plotted for (a) GR (solid (blue)) and for the DFNT subset of the bigravity model with (b) ${\rho }_{\text{gal}}=10^{5.5}{\rho }_{\mathrm{c}}$ (dot-dashed (green)), (c) $10^5{\rho }_{\mathrm{c}}$ (long-dashed (red)), and (d) $10^{4.5}{\rho }_{\mathrm{c}}$ (dashed (black)), respectively, at fixed $({\mu }^2,\kappa {\xi }_c^2)=(10^{-32}{\mathrm{cm}}^{-2},100)$. Here we consider BNS at the distance, $D_L=200\mathrm{Mpc}$. The SNR and the fitting factor between GR waveform and each waveform in this figure become as follows. $(\mathrm{SNR},\mathrm{FF})=$ (a) (8.7,1.0), (b) (26,0.71), (c) (24,0.72), (d) (19,0.73).

Figure 13

The detectable region of the bigravity corrections to the waveforms for DFNT subset of the bigravity model in the case $(m_1,m_2)=(1.4M_{\odot },1.4M_{\odot })$ and $D_L=200\mathrm{Mpc}$. Curves correspond to the average density in the galaxies ${\rho }_{\text{gal}}=10^{5.5}{\rho }_{\mathrm{c}}$ (dashed), $10^5{\rho }_{\mathrm{c}}$ (solid), and $10^{4.5}{\rho }_{\mathrm{c}}$ (dot-dashed).

Figure 14

A plot similar to Fig. 13, but for $D_L=100\mathrm{Mpc}$ (dashed) and 200 Mpc (solid), respectively. The masses are $(m_1,m_2)=(1.4M_{\odot },1.4M_{\odot })$. We set ${\rho }_{\text{gal}}=10^5{\rho }_{\mathrm{c}}$.