Exact amplitudes of six polarization modes for gravitational waves

The exact amplitudes of six polarization modes of gravitational waves are constructed in terms of both the small metric perturbations and the Newman-Penrose scalars. The obtained formulas are applicable to any metric-compatible gravity theories whose gravitational waves propagate along either the null or non-null geodesics. Once a gravity theory (specifically, its linearized wave equation) is written, comparison to the observed data of the laser interferometer experiments is direct.

Figure 1

The six polarization modes: (a) breathing mode $p_1^{(b)}$, (b) longitudinal mode $p_6^{(l)}$, (c) $\text{vector}\text{−}x$ mode $p_2^{(x)}$, (d) $\text{vector}\text{−}y$ mode $p_3^{(y)}$, (e) plus mode $p_4^{(+)}$, and (f) cross mode $p_5^{(\times )}$. Here we added the superscript of every corresponding polarization mode to $p_n$ to clearly show its geometrical description. The red, green, and blue curves indicate scalar, vector, and tensor modes, respectively. The circled dot in panels (b), (e), and (f) indicates that the wave is propagating out of the page, and the right-pointing arrow in panels (a), (c), and (d) indicates that the wave is propagating in the $z$ direction.

Figure 2

The behavior of the exact (solid lines) and approximate (dashed lines) polarization amplitudes are compared by choosing a mass parameter $m_{\mathrm{g}}=7.6\times {10}^{-20}\mathrm{eV}/{\mathrm{c}}^2$ of the dispersion relation $\omega =\sqrt{m_{\mathrm{g}}^2+k^2}$. The mode amplitude appearing for the time-like geodesic and that for the null geodesic are also compared by the black and blue curves.

Figure 3

The ratio between the approximate and exact mode amplitudes in the low-frequency region is shown for the weakest model-independent graviton mass bound. The more that the mode amplitude appearing for the time-like geodesic is enhanced, the greater the deviation of the approximate amplitude from the exact one grows.