Probing for massive stochastic gravitational-wave background with a detector network

In a general metric theory of gravitation in four dimensions, six polarizations of a gravitational wave are allowed: two scalar and two vector modes, in addition to two tensor modes in general relativity. Such additional polarization modes appear due to additional degrees of freedom in modified gravity theories. Also, graviton mass, which could be different in each polarization, is another characteristic of modification of gravity. Thus, testing the existence of additional polarization modes and graviton mass can be a model-independent test of gravity theories. Here we extend the previous framework of correlation analysis of a gravitational-wave background to the massive case and show that a ground-based detector network can probe for massive stochastic gravitational waves with its mass around $\sim {10}^{-14}\mathrm{eV}$. We also show that more than three detectors can cleanly separate the mixture of polarization modes in detector outputs and determine the graviton mass.

Figure 1

Coordinate system on Earth for a detector pair.

Figure 2

Overlap reduction functions for real-detector pairs on Earth. The frequency corresponding to graviton mass is set to $f_g=40\mathrm{Hz}$ ($m_g\approx 2\times {10}^{-14}\mathrm{eV}$) (vertical solid line) for illustration. The characteristic frequency $f_c$, which is relevant for a massless overlap reduction function, is also shown (vertical dashed line). Each curve shows the tensor mode (red, solid), vector mode (green, dotted), and scalar mode (blue, dashed). The overlap reduction functions for the massless graviton case are also shown with monochromatic curves just for reference.

Figure 3

SNR and the parameter estimation accuracy as a function of a fiducial value of $f_g$ in Hz in the presence of a single polarization mode. The fiducial values of the other parameters are $h_0^2{\Omega }_{\mathrm{gw},0}={10}^{-7}$ and $n_t=0$. The top left is the SNR, the top right is $\Delta {\Omega }_{\mathrm{gw},0}/{\Omega }_{\mathrm{gw},0}$, the bottom left is $\Delta f_g/f_g$, and the bottom right is $\Delta n_t$. The red, green, and blue curves are the tensor, vector, and scalar modes, respectively.

Figure 4

Detectable parameter region of $m_g$ and ${\Omega }_{\mathrm{gw},0}$ when $n_t^A=0$. The red, green, and blue curves are the detection thresholds for the tensor, vector, and scalar modes, respectively. The region above the curves is detectable with a detector network.

Figure 5

Measurement accuracy of graviton mass when all polarization signals are mixed and each mode has different graviton mass. In the legend, fgVSeq V and fgVSeq S are the measurement accuracies of $f_g^V$ and $f_g^S$ when both have equal masses $f_g=f_g^V=f_g^S$. While fgV40Sxx S is the measurement accuracy of $f_g^S$ when the vector graviton mass is fixed to $f_g^V=40\mathrm{Hz}$ and $f_g=f_g^S$. The dashed curve is the one we obtained in the previous section in the presence of a scalar polarization mode. In all cases, the fiducial amplitude is $h_0^2{\Omega }_{\mathrm{gw},0}^A={10}^{-7}$ for all polarization modes.

Figure 6

Measurement accuracy of graviton mass when all polarization signals are mixed and each mode has a different GWB amplitude. Vector and scalar graviton masses are set to $f_g=f_g^V=f_g^S$. In the legend, 777 V and 777 S represent the measurement accuracies of $f_g^V$ and $f_g^S$ when the fiducial amplitude is $h_0^2{\Omega }_{\mathrm{gw},0}^A={10}^{-7}$ for all polarization modes; 788 V and 788 S are those when the fiducial amplitudes are $h_0^2{\Omega }_{\mathrm{gw},0}^T={10}^{-7}$ and $h_0^2{\Omega }_{\mathrm{gw},0}^V=h_0^2{\Omega }_{\mathrm{gw},0}^S={10}^{-8}$.