Frequency space derivation of linear and nonlinear memory gravitational wave signals from eccentric binary orbits

The memory effect in gravitational wave (GW) signals is the phenomenon, wherein the relative position of two inertial GW detectors undergoes a permanent displacement owing to the passage of GWs through them. Measurement of the memory signal is an important target for future observations as it establishes a connection between observations with field-theoretic results like the soft-graviton theorems. Theoretically, the memory signal is predicted at the leading order quadrupole formula for sources like binaries in hyperbolic orbits. This can be in the realm of observations by Advanced LIGO, Einstein-Telescope, or LISA for black holes with masses $\sim O({10}^3{M}_{\odot })$ scattered by the supermassive black hole at the galactic center. Apart from the direct memory component there is a nonlinear memory signal in the secondary GW emitted from the primary GW chirp-signals emitted by coalescing binaries. In this paper, we compute the gravitational wave signals and their energy spectrum using the field-theoretic method by computing the scattering amplitudes for eccentric elliptical and hyperbolic binary orbits. The field theoretic calculation gives us the gravitational waveforms of linear and nonlinear memory signals directly in the frequency space. The frequency domain templates are useful for extracting signals from the data. We compare our results with other calculations of linear and nonlinear memory signals in literature and point out novel features we find in our calculations like the presence of $\log (\omega )$ terms in the linear memory from hyperbolic orbits.

Figure 1

The power spectrum given in Eq. (27), in frequency space, of gravitational wave radiation from binaries in hyperbolic orbits with varying eccentricities. For this plot we have taken the following parameters $m_1=m_2=30M_{\odot }$, $a=0.01\mathrm{AU}$. The power spectrum is nonzero at zero frequency which represents the energy radiated as the memory signal.

Figure 2

Plots displaying the characteristics of $h_{+}(t)$ and $h_{\times }(t)$ for different eccentricities. Here, $\theta =0$, $\varphi =\frac{\pi }{12}$, and $h^{\prime }=\iota \frac{4G}{r}(\frac{2\mu a^2{\omega }_0}{e^2})$ is the dimensionful part of the memory waveform.

Figure 3

Schematic representation of a hyperbolic encounter between two black holes of masses $m_1$ and $m_2$ ($m_1<m_2$), in the rest frame of the heavier body. Here, $v_0$ denotes the asymptotic incoming velocity of the lighter body, $b$ and ${\mathrm{\Theta }}_s$ correspond to the impact parameter and the scattering angle and $r_m$ represents the distance of the closest approach.

Figure 4

The axis of rotation of the binary designated by $\overrightarrow{L}$ makes an angle $i$ with the $z$-axis and therefore lies in the $y-z$ plane. The primary graviton emits a secondary graviton at ${\overrightarrow{r}}^{\prime }=r^{\prime }{\widehat{n}}^{\prime }=r^{\prime }(\sin {\theta }^{\prime }\cos {\varphi }^{\prime },\sin {\theta }^{\prime }\sin {\varphi }^{\prime },\cos {\theta }^{\prime })$ and the secondary graviton travels to the earth located at $\overrightarrow{r}=r\widehat{n}=(0,0,1)$.

Figure 5

The variation in ${\overline{h}}_{+}^{\mathrm{mem}}$ with respect to the eccentricity of the orbit, plotted for different choices of the angle between the line of sight of observation and the binary axis.

Figure 6

The variation in ${\overline{h}}_{+}^{\mathrm{mem}}$ with respect to the time interval $\overline{t}-{\overline{t}}_c$ plotted for different choices of the angle between the line of sight of observation and the binary axis.

Figure 7

The variation in ${\overline{h}}_{+}^{\mathrm{mem}}$ with respect to the dimensionless frequency parameter $\overline{\nu }$, defined in Eq. (91), plotted for different choices of the angle between the line of sight of observation and the binary axis.

Figure 8

The variation in $h_{+}^{\mathrm{mem}}$ against time plotted for different choices of the angle $i$ between the line of sight of observation and the binary axis.