# Frequency space derivation of linear and non-linear memory gravitational
wave signals from eccentric binary orbits

Arpan Hait<EMAIL_ADDRESS>Indian Institute of Technology Kanpur,
Kalyanpur, Kanpur 208016, India
Subhendra Mohanty<EMAIL_ADDRESS>Theoretical Physics Division, Physical
Research Laboratory, Ahmedabad - 380009, India
Suraj Prakash<EMAIL_ADDRESS>Indian Institute of Technology Kanpur,
Kalyanpur, Kanpur 208016, India

###### Abstract

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 stellar mass black-holes scattered by the
super-massive black-hole at the galactic center. Apart from the direct memory
component there is a non-linear 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 non-linear 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 non-linear 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.

##  I Introduction

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 [1, 2, 3]. The
memory effect is categorised broadly according to the nature of the source of
GW. Linear memory arises from sources which have unbound components, like
hyperbolic binary orbits [4, 5], neutrinos from supernova [6, 7, 8, 9, 10, 11,
12], gamma-ray bursts [13, 14, 15] and exotic objects like cosmic strings
[16].

Another type of memory signal is the non-linear memory [17, 18, 19, 20], where
secondary gravitational waves are produced by the primary gravitational waves
from sources like coalescing binaries. The importance of non-linear memory is
that it will be an experimental proof of graviton-graviton coupling or the
non-linear nature of the gravitational field equations in Einsteinian gravity.

The memory signal has a significance in field theory as it follows from the
soft-graviton theorems [21, 22, 23] where the amplitude of a low energy
graviton emission from a scattering process can be related, by a
multiplicative kinematic factor, to the hard scattering amplitude without the
graviton emission. The zero-graviton frequency amplitude of the soft-amplitude
has a pole in the frequency space which in Fourier space is a step function in
time which is the characteristic of the memory signal. Weinberg’s soft theorem
amplitude has been generalised to include higher order terms in graviton
momenta using the gravitational gauge invariance an angular momentum
conservation [24]. Calculations of graviton emission amplitudes show that
there are non-analytic logarithmic terms in graviton frequencies even in tree
level scattering [25, 26, 27].

The non-linear memory signal, for binaries in a quasi-circular orbit, already
occurs at the 0-PN order [18, 29]. In [30, 31] the non-linear memory for the
quasi-circular orbit was computed at the 3-PN (Post-Newtonian) order. The non-
linear memory signal from eccentric binary orbits was calculated at 3-PN order
in [32] and the 3-PN calculation for eccentric orbits including the tail
contributions is done in [33].

Binaries (like the Hulse-Taylor) can have large initial eccentricities but by
the time their frequencies enter the threshold of detectors like Advanced LIGO
with a threshold of $\sim 10$ Hz, they lose their eccentricities due to
gravitational radiation reaction. However, there can be other initial
configurations of binary star orbits which could give rise to large
eccentricity orbits by the time the frequency enters the Advanced LIGO
threshold [34, 35]. An eccentric binary can be formed by capture from an
unbound orbit [36, 37]. An initial three body system may eject one of the
bodies and result in an eccentric binary at coalescence [38]. A three body
system in which the orbit of one body is at a larger radius, may cause the
distant body to perturb the orbit of the tight binary system. This
perturbation may give rise to Kozai-Lidov oscillations [39, 40] in the binary
orbit which can drive the eccentricity to large values [41, 42]. Constructing
gravitational wave templates for coalescent eccentric binaries is therefore of
importance for observations [35] at future detectors like Advanced-LIGO. A
determination of eccentricity of the orbits of the coalescent binary will be
possible at Advanced-LIGO by improving the low frequency sensitivity [43].
Experimental determination of both the memory signal and the eccentricity of
the orbit affect the signal near the low-frequency threshold of the detector,
therefore construction of memory signal templates must accurately take into
account the eccentricity of the orbit.

The non-linear gravitational wave memory from binary mergers may be
discernible through next generation ground based detectors like Advanced LIGO
[44] and Advanced Virgo [45] by improving the sensitivity in the 5-10 Hz band
[49, 47, 46, 48]. The memory effect for individual binaries may be resolvable
by forthcoming interferometers such as LISA [50], Cosmic Explorer [51] and
Einstein Telescope [52]. Gravitational-wave memory from binary mergers may
also be seen in Advanced LIGO by combining the signals from multiple events
[53]. The memory signal from several unresolved binary event mergers may be
observed as a cumulative change over time in the pulsar timing residuals and
may be observed in pulsar timing arrays (PTAs) [54, 55, 56, 57, 58]. Ground-
based detectors are sensitive to the frequency of the GW in the $5-10^{3}$ Hz
band. Space-based detector LISA will probe frequencies as low as $10^{-3}$ Hz,
and pulsar timing arrays may measure frequency as low as $10^{-9}$ Hz.

Black-holes in hyperbolic orbits in the gravitational field of super-massive
black-holes at the galactic center radiate bursts of gravitational waves [59,
60, 61, 62, 63, 64, 65] which may be detectable at Advanced LIGO or Einstein
Telescope [66, 67].

For hyperbolic orbits the nonlinear memory appears at the 2.5-PN level in the
waveforem. This is contrary to the case of elliptical and quasicircular
orbits, where the nonlinear memory appears at the Newtonian (0-PN) order [18,
32]. This is because the radiation reaction effects accumulate over time in
the closed orbits while for open orbits the radiation reaction is maximum at
the closest approach and is zero at asymptotic past and future times.

The Baysian inference of specific signals from the data is most efficiently
achieved by match filtering of signals in the frequency domain with the data
[70, 71]. With this aim, we compute the wave-forms for the memory signals for
eccentric elliptical and hyperbolic orbits in the frequency domain. We compute
the waveforms in using the tree level graviton emission amplitude with the
stress tensor of the binary orbits as sources. Using this graviton emission
amplitude we compute the frequency spectrum of the energy radiated following
[72, 73, 74, 74, 75]. For elliptical orbits, the energy radiation spectrum is
the source of the non-linear gravitational memory which we thus obtain
directly in the frequency space.

The anatomy of this article can be described as follows: In section II, we
outline how the gravitational waveform can be constructed in a field-theoretic
approach and we also highlight the computation of the rate of energy loss
associated with gravitational wave radiation. In section III, we compute
components of the stress tensor, in frequency domain, for binaries in
hyperbolic orbits. This is followed by the evaluation of the same quantities
in the limit of vanishing frequency and the associated linear memory in
section IV. Next, we elucidate the general formalism for the non-linear memory
associated with the radiation from binaries in closed orbits in section V. We
delve into the details for the special case of elliptical orbits in section
VI, where we compute the transverse-traceless memory waveform. We summarize
our conclusions in section VII.

##  II Gravitational waveform and energy radiated from scattering amplitudes

The probability amplitude of emitting a graviton of polarisation
$\epsilon^{\lambda}_{\mu\nu}(\vec{n})$ from a source with stress tensor (in
the momentum space ) ${\widetilde{T}}^{\mu\nu}(k)$ is given by

$\displaystyle{\cal
A}_{\lambda}(k_{0},\vec{n}k_{0})=-\iota\,\cfrac{\kappa}{2}\,\epsilon_{\mu\nu}^{*\lambda}(\vec{n})\,{\widetilde{T}}^{\mu\nu}(k_{0},\vec{n}\,k_{0})\,.$
(1)

We can express the gravitational wave metric observed at the detector in terms
of the probability amplitude of a graviton emission by a source at a distance
$r$ as,

$\displaystyle h_{\alpha\beta}(\vec{x},t)=\cfrac{1}{4\pi
r}\,\int\,\cfrac{dk_{0}}{(2\pi)}\,\sum_{\lambda=1}^{2}\,\epsilon_{\alpha\beta}^{\lambda}(\vec{n})\,{\cal
A}_{\lambda}(k_{0},\vec{n}\,k_{0})\,e^{-\iota k_{0}(t-r)}\,.$ (2)

The graviton field in Eq. (2) is a canonical spin-2 field with mass dimension
1 as it is defined as an expansion of the metric
$g_{\mu\nu}=\eta_{\mu\nu}+\kappa\,h_{\mu\nu}$ where $\kappa=\sqrt{32\pi G}$.
The metric perturbation identified as gravitational wave is the dimensionless
quantity $\widetilde{h}_{\mu\nu}\equiv
g_{\mu\nu}-\eta_{\mu\nu}=\kappa\,h_{\mu\nu}$. The expression for the
dimensionless gravitational wave in terms of the amplitude is therefore from
Eq. (2) given by

$\displaystyle\widetilde{h}_{\alpha\beta}(\vec{x},t)=\cfrac{\kappa}{4\pi
r}\,\int\,\cfrac{dk_{0}}{(2\pi)}\,\sum_{\lambda=1}^{2}\,\epsilon_{\alpha\beta}^{\lambda}(\vec{n})\,{\cal
A}_{\lambda}(k_{0},\vec{n}\,k_{0})\,e^{-\iota k_{0}(t-r)}\,.$ (3)

This relates the waveform at the detector to the probability amplitude of
graviton emission by the source. To relate the waveform at the detector to the
source stress tensor we substitute in Eq. (3), the expression for the
amplitude given in Eq. (1), to obtain

$\displaystyle\widetilde{h}_{\alpha\beta}(\vec{x},t)$ $\displaystyle=$
$\displaystyle-\cfrac{\kappa^{2}}{8\pi
r}\,\int\,\cfrac{dk_{0}}{2\pi}\,\sum_{\lambda=1}^{2}\,\epsilon_{\alpha\beta}^{\lambda}(\vec{n})\,\epsilon_{\mu\nu}^{*\lambda}(\vec{n})\,{\widetilde{T}}^{\mu\nu}(k_{0},\vec{n}\,k_{0})\,e^{-\iota
k_{0}(t-r)}\,$ (4) $\displaystyle=$
$\displaystyle-\cfrac{4G}{r}\,\int\,\cfrac{dk_{0}}{2\pi}\,\left(\widetilde{T}_{\alpha\beta}(k_{0},\vec{n}\,k_{0})-\cfrac{1}{2}\,\eta_{\alpha\beta}\,{\widetilde{T}^{\mu}}_{\mu}(k_{0},\vec{n}\,k_{0})\right)\,e^{-\iota
k_{0}(t-r)}\,.$

where we made use of the completeness relation

$\displaystyle\sum_{\lambda=1}^{2}\,\epsilon_{\mu\nu}^{\lambda}(k)\,\epsilon_{\alpha\beta}^{*\lambda}(k)=\cfrac{1}{2}\,(\eta_{\mu\alpha}\,\eta_{\nu\beta}+\eta_{\mu\beta}\,\eta_{\nu\alpha})-\cfrac{1}{2}\,\eta_{\mu\nu}\,\eta_{\alpha\beta}.$
(5)

To obtain the propagating degrees of freedom, we need to project the
transverse traceless (TT) components of the wavefunction constructed in Eq.
(4).

$\displaystyle\Big{[}\widetilde{h}_{ij}\Big{]}^{\text{TT}}(\vec{x},t)=-\cfrac{4G}{r}\,\Lambda_{ij,kl}(\vec{n})\,\int\,\cfrac{dk_{0}}{2\pi}\,\left(\widetilde{T}_{kl}(k_{0},\vec{n}\,k_{0})-\cfrac{1}{2}\,\eta_{kl}\,{\widetilde{T}^{\mu}}_{\,\,\,\,\mu}(k_{0},\vec{n}\,k_{0})\right)\,e^{-\iota
k_{0}(t-r)}.$ (6)

where, $\Lambda_{ij,kl}(\vec{n})$ is the transverse-traceless projection
operator defined with respect to the direction, $\hat{n}$, of the emitted
gravitational wave. The explicit form of the TT projection operator is

$\displaystyle\Lambda_{ij,kl}(\hat{n})$ $\displaystyle=$ $\displaystyle
P_{ik}(\hat{n})P_{jl}(\hat{n})-\cfrac{1}{2}P_{ij}(\hat{n})P_{kl}(\hat{n})=\left(\delta_{ik}-n_{i}n_{k}\right)\left(\delta_{jl}-n_{j}n_{l}\right)-\cfrac{1}{2}\left(\delta_{ij}-n_{i}n_{j}\right)\left(\delta_{kl}-n_{k}n_{l}\right).$
(7)

Since $\Lambda_{ij,kl}\,\,\eta_{kl}=0$, the
${\widetilde{T}^{\mu}}_{\,\,\,\,\mu}$ term vanishes and we obtain the simpler
result,

$\displaystyle\Big{[}\widetilde{h}_{ij}\Big{]}^{\text{TT}}(\vec{x},t)=-\cfrac{4G}{r}\,\Lambda_{ij,kl}(\vec{n})\,\int\,\cfrac{dk_{0}}{2\pi}\,T_{kl}(k_{0},\vec{n}\,k_{0})\,e^{-\iota
k_{0}(t-r)}.$ (8)

In frequency space, the observed gravitational wave and the stress-tensor of
the source can be related as,

$\displaystyle\Big{[}\widetilde{h}_{ij}\Big{]}^{\text{TT}}(\vec{x},k_{0})=-\cfrac{4G}{r}\,\Lambda_{ij,kl}(\vec{n})\,T_{kl}(k_{0},\vec{n}\,k_{0}).$
(9)

We shall use Eq. (9) to compute the gravitational waveform from various
sources, like compact binaries in bound and unbound orbits, by computing the
stress-tensor of the source in frequency space.

###  II.1 Power spectrum of gravitational wave in field theoretic approach

The rate of graviton emission is given by the Fermi Golden Rule, and is the
amplitude squared summed over the final state graviton polarisation and
integrated over the phase space volume,

$\displaystyle\Gamma$ $\displaystyle=$
$\displaystyle\sum_{\lambda}\,\int\,\cfrac{|S_{fi}|^{2}}{T}\,\cfrac{d^{3}\vec{k}}{(2\pi)^{3}\,2\omega}=\sum_{n}\,\sum_{\lambda}\,\int\,\big{|}{\cal
A}_{\lambda}(\omega,\omega^{\prime}_{n})\big{|}^{2}\,(2\pi)\,\delta(\omega-\omega_{n}^{\prime})\,\cfrac{d^{3}\vec{k}}{(2\pi)^{3}\,2\omega}$
(10) $\displaystyle=$
$\displaystyle\cfrac{\kappa^{2}}{4}\,\sum_{n}\,\sum_{\lambda}\,\int\,\big{|}T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})\,\epsilon_{\lambda}^{*\mu\nu}(\vec{k})\big{|}^{2}\,(2\pi)\,\delta(\omega-\omega_{n}^{\prime})\,\cfrac{d^{3}\vec{k}}{(2\pi)^{3}\,2\omega_{k}}.$

Here, $\lambda$ accounts for the possible states of polarization and $n$
denotes the harmonics corresponding to the emission. The rate of energy
radiated is obtained from the probability of radiation given above by
including an extra factor of $\omega_{k}=|\vec{k}|$ in the integral,

$\displaystyle\cfrac{dE_{\text{gw}}}{dt}$ $\displaystyle=$
$\displaystyle\cfrac{\kappa^{2}}{4}\,\sum_{n}\sum_{\lambda}\,\int\,\big{|}T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})\,\epsilon_{\lambda}^{*\mu\nu}(\vec{k})\big{|}^{2}\,(2\pi)\,\delta(\omega-\omega_{n}^{\prime})\,\,\,\omega_{k}\,\,\cfrac{d^{3}\vec{k}}{(2\pi)^{3}\,2\omega}.$
(11)

We can open the modulus squared piece of the integrand in Eq. (11) and
simplify using the polarisation sum relation, see Eq. (5), as follows:

$\displaystyle\sum_{\lambda}\,\big{|}T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})\,\epsilon_{\lambda}^{*\mu\nu}(\vec{k})\big{|}^{2}$
$\displaystyle=$
$\displaystyle\sum_{\lambda}\,\left(T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})\,T_{\alpha\beta}^{*}(\vec{k},\omega_{n}^{\prime})\right)\left(\epsilon_{\lambda}^{*\mu\nu}(\vec{k})\,\epsilon_{\lambda}^{\alpha\beta}(\vec{k})\right)$
(12) $\displaystyle=$ $\displaystyle
T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})\,T^{*\nu\mu}(\vec{k},\omega_{n}^{\prime})-\cfrac{1}{2}\,\big{|}{T^{\mu}}_{\mu}(\vec{k},\omega_{n}^{\prime})\big{|}^{2}\,.$

The $T^{00}$ and $T^{i0}$ components can be expressed in terms of the $T^{ij}$
ones by utilizing the conserved current relation, $k_{\mu}T^{\mu\nu}=0$. This
allows us to write

$T_{0j}=-\hat{k^{i}}\,T_{ij},\hskip
14.22636ptT_{00}=\hat{k^{i}}\,\hat{k^{j}}\,T_{ij}.$ (13)

Using these relations, we can rewrite Eq. (12) as

$\displaystyle\big{|}T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})\big{|}^{2}-\cfrac{1}{2}\,\big{|}T^{\mu}{}_{\mu}(\vec{k},\omega_{n}^{\prime})\big{|}^{2}$
$\displaystyle=$ $\displaystyle
T_{ij}T^{*ji}+T_{00}T^{*00}+T_{0i}T^{*i0}+T_{i0}T^{*0i}-\cfrac{1}{2}\left({T^{0}}_{0}+{T^{i}}_{i}\right)\left({T^{*0}}_{0}+{T^{*j}}_{j}\right)$
(14) $\displaystyle=$
$\displaystyle\left(T_{ij}T^{*ji}-\cfrac{1}{2}\,T^{i}_{i}T^{*j}_{j}\right)+\cfrac{1}{2}\,\hat{k^{i}}\hat{k^{j}}\hat{k^{l}}\hat{k^{m}}T_{ij}T^{*}_{lm}-\left(\hat{k^{l}}\,\hat{k^{m}}\,T_{il}T^{*}_{mi}+\hat{k^{l}}\,\hat{k^{m}}\,T_{il}T^{*}_{mi}\right)$
$\displaystyle\qquad\,+\,\cfrac{1}{2}\,\left(\hat{k^{l}}\,\hat{k^{m}}\,T^{*}_{lm}T^{i}_{i}+\hat{k^{l}}\,\hat{k^{m}}\,T_{lm}T^{*j}_{j}\right).$

In the quadrupole approximation of the source, for sources smaller in size
than the wavelength of the GWs, $\vec{k}\cdot\vec{x}\ll 1$, the stress tensor
in momentum space $T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})$ has no explicit
$\vec{k}$ dependence. Therefore, after substituting the contents of Eq. (14)
in Eq. (11), one can perform the angular integrations using the following
relations:

$\displaystyle\int d\Omega_{k}=4\pi,\hskip 25.6073pt\int
d\Omega_{k}\,\hat{k^{i}}\,\hat{k^{j}}=\cfrac{4\pi}{3}\,\delta_{ij},\hskip
19.91684pt\int
d\Omega_{k}\,\hat{k^{i}}\,\hat{k^{j}}\,\hat{k^{l}}\,\hat{k^{m}}=\cfrac{4\pi}{15}\,\left(\delta_{ij}\,\delta_{lm}+\delta_{il}\,\delta_{jm}+\delta_{im}\,\delta_{jl}\right).$
(15)

to obtain

$\int\,d\Omega_{k}\,\Big{[}|T_{\mu\nu}(\vec{k},\omega_{n}^{\prime})|^{2}-\cfrac{1}{2}\,|T^{\mu}{}_{\mu}(\vec{k},\omega_{n}^{\prime})|^{2}\Big{]}=\cfrac{8\pi}{5}\,\Big{(}T_{ij}(\omega_{n}^{\prime})\,T^{*}_{ji}(\omega_{n}^{\prime})-\cfrac{1}{3}\,|T^{i}{}_{i}(\omega_{n}^{\prime})|^{2}\Big{)},$
(16)

Finally, using the result of Eq. (16), the expression for the energy radiated
by a source in terms of the source stress tensor can be obtained as a
modification of Eq. (11) as

$\displaystyle\cfrac{dE_{\text{gw}}}{dt}=\cfrac{\kappa^{2}}{4}\,\sum_{n}\,\int\,\cfrac{8\pi}{5}\,\Big{(}T_{ij}(\omega_{n}^{\prime})T^{*}_{ji}(\omega_{n}^{\prime})-\cfrac{1}{3}\,|T^{i}{}_{i}(\omega_{n}^{\prime})|^{2}\Big{)}\,{\omega}^{2}\,2\pi\,\delta(\omega_{n}^{\prime}-\omega)\cfrac{d\omega}{(2\pi)^{3}\,2\omega}.$
(17)

We make use of this expression for computing the energy radiated by binaries
in elliptical orbits which is the source of the non-linear memory signal. We
will also show using this expression that in the memory signal from hyperbolic
orbits, there is non-zero energy radiated around the zero-frequency band.

##  III Hyperbolic Orbits

For the case of unbounded orbits, we consider a black-hole of mass $m_{2}$ in
a hyperbolic orbit around a larger black-hole of mass $m_{1}$. Similar to the
case of the elliptical orbits, in the centre of mass frame, the motion is
equivalently described by a single body having the reduced mass
$\mu=\cfrac{m_{1}\,m_{2}}{m_{1}+m_{2}}$. The following quantities describe the
system:

  * •

The coordinates parametrized as,

$\displaystyle x(\xi)=a\,(e-\cosh\xi),\qquad y(\xi)=b\,\sinh\xi,\qquad
z(\xi)=0,\qquad\cfrac{\omega^{\prime}}{\nu}\,t=\Omega\,t=(e\sinh\xi-\xi),$
(18)

where, $a$ and $b\,\,(=a\sqrt{e^{2}-1},\,\,e>1)$ denote the semi-major and
semi-minor axes and the variables $e$ and $\xi\in(-\infty,\infty)$ refer to
the eccentricity and the hyperbolic anomaly of the orbit respectively.

  * •

The angular frequency $\omega^{\prime}$ is proportional to the fundamental
frequency $\Omega$, as $\omega^{\prime}=\nu\,\Omega$, where $\nu\in(0,\infty)$
is a non negative real number and
$\Omega=\left(\cfrac{G\,(m_{1}+m_{2})}{a^{3}}\right)^{1/2}$.

We start with the computation of the stress-tensor components in frequency
space. Focussing first on the $xx$ component, the calculation proceeds as
follows:

$\displaystyle T_{xx}(\omega^{\prime})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dt\,e^{i\omega^{\prime}t}\,\mu\,\dot{x}^{2}\,=\,-\iota\,\mu\,\omega^{\prime}\int_{-\infty}^{\infty}dt\,e^{i\omega^{\prime}t}\,\,x\,\dot{x}\,=\,-\iota\,\mu\,\omega^{\prime}\int_{-\infty}^{\infty}d\xi\,e^{i\nu(e\sinh\xi-\xi)}\,\,x\,\cfrac{dx}{d\xi},$
(19)

where, the second equality is obtained after implementing integration by parts
and neglecting a term proportional to $\ddot{x}(t)$. The final expression is
obtained after incorporating a change of variables $t\rightarrow\xi$. Since,
the integration is over $\xi$, the result will simply be a function of $\nu$.
Also, since $\omega^{\prime}=\nu\Omega$, it makes more sense to express
$T_{ij}$ as a function of $\nu$ instead of $\omega^{\prime}$. Thus, we obtain

$\displaystyle T_{xx}(\nu)$ $\displaystyle=$
$\displaystyle\,\iota\,\mu\,\nu\,\Omega\,a^{2}\int_{-\infty}^{\infty}d\xi\,e^{\iota\nu(e\sinh\xi-\xi)}\,\,\sinh\xi\,\,(e-\cosh\xi)\,$
(20) $\displaystyle=$
$\displaystyle\mu\,\nu\,\Omega\,a^{2}\pi\,\left[\,\cfrac{\iota}{\nu
e^{2}}\,H^{(1)}_{\iota\nu}(ie\nu)-\left(e-\cfrac{1}{e}\right){H_{\iota\nu}^{(1)}}^{\prime}(\iota
e\nu)\,\right].$

We replaced $\sinh\xi$, $\cosh\xi$ by their exponential counterparts.
Subsequently, we identified and replaced the integrals with Hankel functions,
i.e.,

$\displaystyle
H_{p}^{(1)}(q)=\cfrac{1}{\iota\pi}\int\limits_{-\infty}^{\infty}d\xi\,\,e^{q\sinh\xi-p\xi}\,\,.$
(21)

Then, we utilized the following recurrence relations for Hankel functions to
simplify the expression:

$\displaystyle H_{p-1}^{(1)}(q)+H_{p+1}^{(1)}(q)$ $\displaystyle=$
$\displaystyle\cfrac{2\,p}{q}\,H_{p}^{(1)}(q)\,\qquad\text{and}\qquad
H_{p-1}^{(1)}(q)-H_{p+1}^{(1)}(q)=2\,{H_{p}^{(1)}}^{\prime}(q).$ (22)

The remaining non-zero components can similarly be evaluated as:

$\displaystyle T_{yy}(\omega^{\prime})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dt\,e^{\iota\omega^{\prime}t}\,\mu\,\dot{y}^{2}\,=\,-\iota\,\mu\,\omega^{\prime}\int_{-\infty}^{\infty}dt\,e^{\iota\omega^{\prime}t}\,\,y\,\dot{y}\,=\,-\iota\,\mu\,\omega^{\prime}\int_{-\infty}^{\infty}d\xi\,e^{\iota\nu(e\sinh\xi-\xi)}\,\,y\,\cfrac{dy}{d\xi}$
$\displaystyle T_{yy}(\nu)$ $\displaystyle=$
$\displaystyle\mu\,\nu\,\Omega\,a^{2}(e^{2}-1)\,\pi\bigg{[}\cfrac{\iota}{\nu
e^{2}}\,{H_{\iota\nu}^{(1)}}(\iota
e\nu)+\cfrac{1}{e}\,{H_{\iota\nu}^{(1)}}^{\prime}(\iota e\nu)\bigg{]},$
$\displaystyle T_{xy}(\omega^{\prime})$ $\displaystyle=$
$\displaystyle\int_{-\infty}^{\infty}dt\,e^{\iota\omega^{\prime}t}\,\mu\,\dot{x}\,\dot{y}\,=\,-\iota\,\mu\,\omega^{\prime}\int_{-\infty}^{\infty}dt\,e^{\iota\omega^{\prime}t}\,\,y\,\dot{x}\,=\,-\iota\,\mu\,\omega^{\prime}\int_{-\infty}^{\infty}d\xi\,e^{\iota\nu(e\sinh\xi-\xi)}\,\,x\,\cfrac{dx}{d\xi}$
$\displaystyle T_{xy}(\nu)$ $\displaystyle=$
$\displaystyle-\mu\,\nu\,\Omega\,a^{2}\,\sqrt{e^{2}-1}\,\pi\left[\left(\cfrac{1}{e^{2}}-1\right)\,{H_{\iota\nu}^{(1)}}(\iota
e\nu)+\cfrac{\iota}{\nu e}\,{H_{\iota\nu}^{(1)}}^{\prime}(\iota e\nu)\right].$
(23)

###  III.1 Rate of emission of gravitational wave energy

Using the expressions for $T_{xx}$, $T_{yy}$ and $T_{xy}$ derived in Eqs. (20)
and (III) we obtain,

$\displaystyle
T_{ij}(\nu)\,T^{*}_{ji}(\nu)-\cfrac{1}{3}\,|\,T^{i}{}_{i}(\nu)\,|^{2}=\mu^{2}\,\nu^{2}\,\Omega^{2}\,a^{4}\,\pi^{2}\,\Big{[}\,f_{1}(\nu,e)\,\left(H_{\iota\nu}^{(1)}(\iota
e\nu)\right)^{2}+f_{2}(\nu,e)\,\left({H_{\iota\nu}^{(1)}}^{\prime}(\iota
e\nu)\right)^{2}\,\Big{]}.$ (24)

Here,

$\displaystyle
f_{1}(\nu,e)=\cfrac{2}{e^{4}}\,(e^{2}-1)^{3}+\cfrac{6-6e^{2}+2e^{4}}{3\,\nu^{2}\,e^{4}}\qquad\text{and}\qquad
f_{2}(\nu,e)=2\,\left(\cfrac{e^{2}-1}{e}\right)\left(\cfrac{1}{\nu^{2}\,e}+\cfrac{e^{2}-1}{e}\right).$
(25)

After evaluating the integral in Eq. (17), we get:

$\displaystyle P(\nu)$
$\displaystyle=\cfrac{dE_{\text{gw}}}{dt}=\cfrac{\kappa^{2}}{20}\,\,\mu^{2}\,\Omega^{4}\,\nu^{4}\,a^{4}\,\pi\,\Big{[}\,\underbrace{f_{1}(\nu,e)\,\left(H_{\iota\nu}^{(1)}(\iota
e\nu)\right)^{2}+f_{2}(\nu,e)\,\left({H_{\iota\nu}^{(1)}}^{\prime}(\iota
e\nu)\right)^{2}}_{f(\nu,e)}\,\Big{]},$ (26)

where $f_{1,2}(\nu,e)$ are given in Eq. (25). A plot depicting the features of
$P(\nu)$ for different values of eccentricity has been shown in Fig. 1. The
total energy loss can be obtained from the power spectrum and is given by:

$\displaystyle\Delta
E=\int_{-\infty}^{\infty}dt\,P(t)=\cfrac{1}{\pi}\int_{0}^{\infty}d\nu\,P(\nu).$
(27)

Explicit integration over products of Hankel functions is difficult,
especially when the order $p$ of the function $H^{(1)}_{p}(q)$ also depends on
the integration variable. Therefore, we have presented an estimate by doing a
numerical integration with various choices of $\nu_{\text{max}}$ (the upper
limit of the integral). The results for different eccentricities have been
plotted in Fig. 2. We notice that, in each case, the curves become parallet to
the $x$-axis and approach a constant, for fairly small values of
$\nu_{\text{max}}$.

Figure 1: The power spectrum, in frequency space, of gravitational wave
radiation from binaries in hyperbolic orbits with varying eccentricities.
Figure 2: Numerical estimate of the total energy loss due to gravitational
wave radiation from binaries in hyperbolic orbits. Individual curves
correspond to varying eccentricities. The horizontal axis corresponds to the
upper limit in the integral $\Delta
E=\cfrac{1}{\pi}\int\limits_{0}^{\nu_{\text{max}}}d\nu\,P(\nu)$. It is evident
that even for small enough values of $\nu_{\text{max}}$, the energy loss
attains a constant value. This suggests that with a large $\nu_{\text{max}}$
the numerical estimate is a good approximation.

##  IV Linear memory from hyperbolic orbits

###  IV.1 Power emitted at zero frequency

Computing the zero frequency limits of the stress tensors is equivalent to
obtaining approximations for them in the $\nu$ $\rightarrow$ 0 limit. As $\nu
e$ $\rightarrow$ 0, the Hankel function and its first derivative assume the
following form:

$\displaystyle H_{\iota\nu}^{(1)}(\iota e\nu)\simeq\cfrac{2\iota}{\pi}\ln(\nu
e),\hskip 34.14322ptH_{\iota\nu}^{(1)\prime}(\iota e\nu)\simeq\cfrac{2}{\pi\nu
e}.$ (28)

Substituting the above in the expressions for $T_{xx}$, $T_{yy}$ and $T_{xy}$,
in Eqs. (20) and (III) yields,

$\displaystyle T_{xx}$ $\displaystyle=$
$\displaystyle-\cfrac{2\,\mu\,a^{2}\,\Omega}{e^{2}}\bigg{[}\ln(\nu
e)+(e^{2}-1)\bigg{]},\hskip 11.38092pt\hskip
42.67912ptT_{yy}=-\cfrac{2\,\mu\,a^{2}\,\Omega}{e^{2}}\left(e^{2}-1\right)\bigg{[}\ln(\nu
e)-1\bigg{]},$ $\displaystyle T_{xy}$ $\displaystyle=$ $\displaystyle
2\,\iota\,\mu\,\nu\,\Omega\,a^{2}\,\sqrt{e^{2}-1}\left[\cfrac{(e^{2}-1)}{e^{2}}\ln(\nu
e)-\cfrac{1}{\nu^{2}\,e^{2}}\right]$ (29)

Using these, we can rewrite,

$\displaystyle
T_{ij}(\nu)\,T^{*}_{ji}(\nu)-\cfrac{1}{3}\,|\,T^{i}{}_{i}(\nu)\,|^{2}=4\,\mu^{2}\,\nu^{2}\,a^{4}\left[\widetilde{f}_{1}(\nu,e)(\ln(\nu
e))^{2}\,+\,\widetilde{f}_{2}(\nu,e)\ln(\nu
e)\,+\,\widetilde{f}_{3}(\nu,e)\right].$ (30)

with

$\displaystyle\widetilde{f}_{1}(\nu,e)$ $\displaystyle=$
$\displaystyle\cfrac{2}{e^{4}}(e^{2}-1)^{3}+\cfrac{6-6e^{2}+2e^{4}}{3\,\nu^{2}\,e^{4}},\qquad\qquad\widetilde{f}_{2}(\nu,e)=\cfrac{2\,(e^{2}-1)}{\nu^{2}\,e^{4}}-\cfrac{6\,(e^{2}-1)^{2}}{\nu^{2}\,e^{4}},$
$\displaystyle\widetilde{f}_{3}(\nu,e)$ $\displaystyle=$
$\displaystyle\cfrac{2\,(e^{2}-1)}{\nu^{4}\,e^{4}}+\cfrac{2\,(e^{2}-1)^{2}}{\nu^{2}\,e^{4}}.$
(31)

The expression for the power spectrum of the gravitational wave, see Eq. (26),
now becomes,

$\displaystyle
P(\nu)=\cfrac{dE}{dt}=\cfrac{\kappa^{2}}{5}\,\pi\,\mu^{2}\,a^{4}\,\Omega^{4}\,\nu^{4}\,f(\nu,\,e).$
(32)

In the limit $\nu\rightarrow 0$,

$\displaystyle\lim_{\nu\to
0}\nu^{4}\,f(\nu,\,e)=\cfrac{2\,(e^{2}-1)}{\pi^{2}\,e^{4}},$ (33)

which is finite and different from zero, except for $e=1$ and
$e\rightarrow\infty$. Then the power radiated by the GW of zero frequency is
given by,

$\displaystyle
P(\nu=0)=\cfrac{64\,G}{5}\,\mu^{2}\,a^{4}\,\Omega^{4}\,\cfrac{(e^{2}-1)}{e^{4}}.$
(34)

This result agrees with the expression derived in [62].

###  IV.2 The memory waveform

The gravitational wave amplitude corresponding to polarisation $\lambda$,
measured by a detector located at a distance $r$, is given (in the frequency
domain) by

$\displaystyle
h_{\lambda}(\omega^{\prime},r)=\frac{4G}{r}\epsilon^{ij}_{\lambda}(\vec{n})\,T_{ij}(n,\omega^{\prime})$
(35)

The waveforms for the $+$ and $\times$ polarisations are

$\displaystyle h_{+}(\omega^{\prime},r)$ $\displaystyle=$
$\displaystyle\iota\cfrac{4G}{r}\epsilon^{ij}_{+}(\vec{n})T_{ij}(\vec{n},\omega^{\prime})=\iota\cfrac{4G}{r}\left(\vec{e}_{\theta
i}\vec{e}_{\theta j}-\vec{e}_{\phi i}\vec{e}_{\phi
j}\right)T_{ij}(\vec{n},\omega^{\prime})$ $\displaystyle=$
$\displaystyle\iota\frac{4G}{r}\Big{(}T_{xx}(\cos^{2}\phi-\sin^{2}\phi\cos^{2}\theta)+T_{yy}(\sin^{2}\phi-\cos^{2}\phi\cos^{2}\theta)-T_{zz}\sin^{2}\theta$
$\displaystyle\qquad-T_{xy}\sin 2\phi(1+\cos^{2}\theta)+T_{xz}\sin\phi\sin
2\theta+T_{yz}\cos\phi\sin 2\theta\Big{)},$ $\displaystyle
h_{\times}(\omega^{\prime},r)$ $\displaystyle=$
$\displaystyle\iota\frac{4G}{r}\epsilon^{ij}_{\times}(\vec{n})T_{ij}(\vec{n},\omega^{\prime})=\iota\frac{4G}{r}\left(\vec{e}_{\theta
i}(\vec{e}_{\phi})_{j}+(\vec{e}_{\phi})_{i}(\vec{e}_{\theta})_{j}\right)T_{ij}(\vec{n},\omega^{\prime})$
(36) $\displaystyle=$
$\displaystyle\iota\frac{4G}{r}\Big{(}(T_{xx}-T_{yy})\sin
2\phi\cos\theta+2T_{xy}\cos
2\phi\cos\theta)-2T_{xz}\cos\phi\sin\theta+2T_{yz}\sin\phi\sin\theta\Big{)}\,.$

We have used the spherical coordinates to describe the polarisation of a GW
traversing in the radial $\vec{n}$ direction. The time domain waveforms can be
obtained after evaluating the Fourier transforms of $h_{+}(\omega^{\prime},r)$
and $h_{\times}(\omega^{\prime},r)$. This can be accomplished numerically and
the results have been presented in Fig. 3

Figure 3: Plots displaying the characteristics of $h_{+}(t)$ and
$h_{\times}(t)$ for different eccentricities. Here, $\phi$ = $\theta$ = 0.

###  IV.3 Comparison with soft-theorems

According to the soft-graviton theorems, the amplitude of a graviton emission
from the scattering of $n$-particles of momenta $p_{a}$ can be written as the
product of a kinematical factor and the amplitude ${\cal A}_{n}(p_{a})$ of the
n-particle scattering as [24, 26],

$\displaystyle{\cal
A}_{n+1}(p_{a},q)=\frac{\kappa}{2}\epsilon^{*}_{\lambda\mu\nu}\sum_{a=1}^{n}\Big{(}\frac{p_{a\mu}p_{a\nu}}{p_{a}\cdot
q}-\iota\frac{p_{a}^{\mu}q_{\beta}J_{a}^{\beta\nu}}{p_{a}\cdot
q}-\frac{q_{\alpha}q_{\beta}J_{a}^{\alpha\mu}J_{a}^{\beta\nu}}{p_{a}\cdot
q}\Big{)}{\cal A}_{n}(p_{a}).$ (37)

Here,
$J_{a}^{\alpha\beta}=x_{a}^{\alpha}\,p_{a}^{\beta}-x_{a}^{\beta}\,p_{a}^{\alpha}+S_{a}^{\alpha\beta}$
describes the total angular momentum of particle $``a"$. The series of the
soft factors are at the same order in the gravitational coupling but in
increasing powers of the graviton frequency $q_{0}=\omega$. The leading term
goes as $\omega^{-1}$ while the sub-leading terms go as $\sim\omega^{0}$ and
$\sim\omega^{1}$ respectively. The gravitational waveform at distance $r$ of
the radiated soft graviton of momentum $q$ is given by

$\displaystyle
h_{\mu\nu}(r,q)=\frac{4G}{r}\Big{(}\frac{p_{a\mu}p_{a\nu}}{p_{a}\cdot
q}-\iota\frac{p_{a}^{\mu}q_{\beta}J_{a}^{\beta\nu}}{p_{a}\cdot
q}-\frac{q_{\alpha}q_{\beta}J_{a}^{\alpha\mu}J_{a}^{\beta\nu}}{p_{a}\cdot
q}\Big{)}.$ (38)

There are logarithmic corrections to the leading order terms suppressed by $G$
which give a tail contribution to the linear memory signal (which as a
function of time goes as $1/t$), [25, 26, 27, 28]. The low frequency graviton
signal from a generic hard scattering can be written as [25],

$\displaystyle
h_{ij}(\omega)=\iota\omega^{-1}A_{ij}+B_{ij}\ln\omega^{-1}+\cdots,$ (39)

where the coefficients $A_{ij}$ and $B_{ij}$ can be obtained in terms of
incoming and outgoing momenta [25, 26]. Using the relations $\omega=\nu\Omega$
with $\Omega=(GM/a^{3})^{1/2}$, where $M$ and $a$ are the total mass and the
semi-major axis of the hyperbolic orbit respectively, the memory waveforms
obtained in Eqs. (IV.1) can be rewritten as

$\displaystyle h_{xx}$ $\displaystyle=$
$\displaystyle-\frac{4G}{r}\cfrac{2\,\mu\,a^{2}\,\Omega}{e^{2}}\bigg{[}\ln\left(\frac{\omega
e}{\Omega}\right)+(e^{2}-1)\bigg{]},$ $\displaystyle h_{yy}$ $\displaystyle=$
$\displaystyle-\frac{4G}{r}\cfrac{2\,\mu\,a^{2}\,\Omega}{e^{2}}\left(e^{2}-1\right)\bigg{[}\ln\left(\frac{\omega
e}{\Omega}\right)-1\bigg{]},$ $\displaystyle h_{xy}$ $\displaystyle=$
$\displaystyle\frac{4G}{r}\,\frac{2\,\iota\,\mu\,a^{2}\,\Omega}{e^{2}}\,\sqrt{e^{2}-1}\left[\cfrac{(e^{2}-1)\omega}{\Omega}\ln\left(\frac{\omega
e}{\Omega}\right)-\cfrac{\Omega}{\omega\,}\right]\,.$ (40)

From the above equation, we see that the memory signal in frequency space has
both the $1/\omega$ and $\ln\omega$ terms as predicted from the general soft-
graviton amplitude calculation, see Eq. (39). To compare the coefficients
$A_{ij}$ and $B_{ij}$ of the general result [25] with the particular case of
hyperbolic orbits we express the memory waveform components in Eq. (IV.3) in
terms of the initial velocity $\vec{v}$, ($|\vec{v}|=v_{0}$) and impact
parameter $b$ as shown in Fig. 4.

Figure 4: 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 $\Theta_{s}$ correspond to the impact parameter and the
scattering angle and $r_{m}$ represents the distance of the closest approach.

In the Keplerian orbit there are two conserved quantities $E=\frac{1}{2}\mu
v^{2}$ and $L=\mu bv$ (where $\vec{v}$ is the velocity of the reduced mass).
The semi-major axis $a$ is given by $a=GM/v_{0}^{2}$. The impact parameter $b$
is related to the eccentricity as $e=(1+b^{2}/a^{2})^{1/2}=(1+L^{2}/(\mu
v_{0}a)^{2}))^{1/2}$. Doing an expansion of $h_{ij}$ in powers of the angular
momentum $L$ and retaining terms to the leading order in $L$, the coefficient
of the $\iota\omega^{-1}$ term follows from Eq. (IV.3) as

$A_{xy}=-\cfrac{4G}{r}2\mu v_{0}^{2}\,\cfrac{Lv_{0}}{\mu
GM}\left(1-\left(\frac{Lv_{0}}{\mu GM}\right)^{2}\right),\quad
A_{xx}=0\,,\quad A_{yy}=0\,.$ (41)

and the coefficient of the $\ln\omega^{-1}$ terms in the memory signal are

$\displaystyle
B_{xx}=\frac{4G}{r}2\mu\frac{GM}{v_{0}}\left(1-\left(\frac{Lv_{0}}{\mu
GM}\right)^{2}\right),\quad
B_{yy}=\frac{4G}{r}2\mu\frac{GM}{v_{0}}\left(\frac{Lv_{0}}{\mu
GM}\right)^{2}\,,\quad B_{xy}=0.$ (42)

With suitable substitutions, we have verified that the general expressions for
the components of the energy-momentum tensor $T_{ij}$ computed in [28] lead to
the same results as presented in Eqs. (IV.1) for the specific case of a
hyperbolic encounter. Consequently, there is an agreement between our result
for the linear memory waveform and the waveform constructed using soft
theorems [25, 26, 27, 28].

##  V Non-linear memory

The non-linear memory is due to the secondary gravitational waves which are
emitted by the primary gravitational waves from an oscillating source, such
as, a coalescing binary [3, 17, 18, 30, 31, 79].

The stress tensor of gravitational waves is related to the energy radiated as

$\displaystyle\tau_{ij}^{\text{gw}}=\cfrac{dE_{\text{gw}}}{dt\,d\Omega}\,n_{i}\,n_{j},$
(43)

and the non-linear memory waveform is given by [3, 17, 18],

$\displaystyle
h_{ij}(t,r\,\hat{n})=\cfrac{4G}{r}\,\int_{-\infty}^{t-r}\,dt^{\prime}\,d\Omega^{\prime}\,\cfrac{dE_{\text{gw}}}{dt^{\prime}\,d\Omega^{\prime}}\,\left[\cfrac{n^{\prime}_{i}\,n^{\prime}_{j}}{1-\vec{n}^{\prime}\cdot\vec{n}}\right]^{\rm
TT}.$ (44)

Here $\vec{n}^{\prime}$ is the unit vector from the source to the solid angle
denoted by $d\Omega^{\prime}$ and $\vec{n}$ is the unit vector along the line
of sight from the source to the detector.

We outline the derivation of these results in this section. The source term of
the non-linear gravitational waves is the Landau-Lifshitz stress tensor of the
primary gravitational waves given by

$\displaystyle\tau^{\text{gw}}_{ij}=\cfrac{1}{32\pi
G}\,\langle\partial_{i}h^{ab}\partial_{j}h_{ab}\rangle,$ (45)

where the angular brackets encode averaging over time longer than the time
period and volumes larger than the wavelengths of the source gravitational
waves $h_{ab}$. The source gravitational waves travel outward radially with
speed of light and are functions of $(t-r)$, i.e
$h_{ab}(t,\vec{x})=h_{ab}(t-r,\Omega)$. This implies we can relate their
spatial and time derivatives as,
$\partial_{i}\,h_{ab}(t-r)=-n_{i}\,\partial_{0}\,h_{ab}(t-r)$, where
$n_{i}=x_{i}/r$. Therefore we can write

$\displaystyle\tau^{\text{gw}}_{ij}=\cfrac{1}{32\pi
G}\,\langle\partial_{i}h^{ab}\partial_{j}h_{ab}\rangle=n_{i}n_{j}\tau^{\text{gw}}_{00}\,.$
(46)

We can model the energy density $\tau^{\text{gw}}_{00}$ produced from a source
and propagating radially outwards on null-rays as

$\displaystyle\tau^{\text{gw}}_{00}(t,\vec{x})=\cfrac{1}{r^{2}}\,\cfrac{dE_{\text{gw}}(t-r,\Omega)}{dt\,d\Omega}.$
(47)

where $dE_{\text{gw}}/dt$ is the luminosity of the source in gravitational
waves and $dE_{\text{gw}}/d\Omega$ denotes the angular distribution of the
source luminosity. Therefore we can write $\tau_{ij}^{\text{gw}}$ in terms of
the energy flux as

$\displaystyle\tau_{ij}^{\text{gw}}=n_{i}n_{j}\tau_{00}^{\text{gw}}=n_{i}n_{j}\,\cfrac{1}{r^{2}}\,\cfrac{dE_{\text{gw}}(t-r,\Omega)}{dt\,d\Omega}\,.$
(48)

The secondary gravitational waves sourced by the gravitational wave stress
tensor will obey the in-homogenous wave equation:

$\displaystyle\square\,h_{ij}=-16\pi G\,\tau^{\text{gw}}_{ij}.$ (49)

whose solution assumes the following form,

$\displaystyle h^{\text{mem}}_{ij}(t,\vec{x})=4G\int
dt^{\prime}\,d^{3}x^{\prime}\,\,\tau^{\text{gw}}_{ij}(t^{\prime},\vec{x}^{\prime})\,\cfrac{\delta\left(t^{\prime}-(t-|\vec{x}-\vec{x}^{\prime}|)\right)}{|\vec{x}-\vec{x}^{\prime}|}.$
(50)

We can express the source term, Eq. (46), in terms of the null coordinate
$u=t^{\prime}-r^{\prime}$ as follows

$\displaystyle\tau^{\text{gw}}_{ij}(t^{\prime},\vec{x}^{\prime})$
$\displaystyle=$
$\displaystyle\cfrac{n^{\prime}_{i}n^{\prime}_{j}}{{r^{\prime}}^{2}}\,\cfrac{dE_{\text{gw}}(t^{\prime}-r^{\prime},\Omega^{\prime})}{dt^{\prime}d\Omega^{\prime}}\,=\,\int
du\,\,\cfrac{n^{\prime}_{i}n^{\prime}_{j}}{{r^{\prime}}^{2}}\,\,\delta(u-(t^{\prime}-r^{\prime}))\,\cfrac{dE_{\text{gw}}(u,\Omega)}{dt^{\prime}d\Omega^{\prime}}.$
(51)

Substituting the above in Eq. (50) we obtain

$\displaystyle h^{\text{mem}}_{ij}(t,\vec{x})=4G\,\int
du\,dt^{\prime}\,dr^{\prime}\,{r^{\prime}}^{2}\,d\Omega^{\prime}\,\cfrac{n^{\prime}_{i}n^{\prime}_{j}}{{r^{\prime}}^{2}}\,\cfrac{\delta\left(t^{\prime}-(t-|\vec{x}-\vec{x}^{\prime}|)\right)}{|\vec{x}-\vec{x}^{\prime}|}\,\,\delta(u-(t^{\prime}-r^{\prime}))\,\cfrac{dE_{\text{gw}}(u,\Omega)}{dt^{\prime}d\Omega^{\prime}}.$
(52)

Since the distance to the observer is much larger than the source size, $r\gg
r^{\prime}$, we take the approximations

$\displaystyle\cfrac{1}{|\vec{x}-\vec{x}^{\prime}|}\simeq\cfrac{1}{r\,(1-\vec{n}^{\prime}\cdot\vec{n})}\,,\qquad\qquad\delta\left(t^{\prime}-(t-|\vec{x}-\vec{x}^{\prime}|)\right)\simeq\delta(t^{\prime}-(t-r)).$
(53)

Now we can perform the integral over $r^{\prime}$ using the second delta
function in Eq. (52) and then do the $t^{\prime}$ integration using the
remaining delta function to obtain

$\displaystyle
h^{\text{mem}}_{ij}(t,\vec{x})=\cfrac{4G}{r}\,\int_{-\infty}^{t-r}\,du\int_{4\pi}\,d\Omega^{\prime}\,\,\cfrac{dE_{\text{gw}}(u,\Omega)}{du\,d\Omega^{\prime}}\,\,\cfrac{n^{\prime}_{i}n^{\prime}_{j}}{(1-\vec{n}^{\prime}\cdot\vec{n})}.$
(54)

Expression for the corresponding transverse-traceless wavefunction is obtained
after multiplication by the projection operator $\Lambda_{ij,kl}(\vec{n})$,
defined in Eq. (7),

$\displaystyle\Big{[}h^{\text{mem}}_{ij}\Big{]}^{\text{TT}}(t,\vec{x})=\cfrac{4G}{r}\,\int_{-\infty}^{t-r}du\,\int_{4\pi}\,d\Omega^{\prime}\,\,\cfrac{dE_{\text{gw}}(u,\Omega)}{du\,d\Omega^{\prime}}\,\,\cfrac{\Lambda_{ij,kl}(\vec{n})\,n^{\prime}_{k}n^{\prime}_{l}}{(1-\vec{n}^{\prime}\cdot\vec{n})},$
(55)

###  V.1 Non-linear memory from circular orbits

We will now compute the non-linear memory from binary orbits. The case of
elliptical orbits is important from the point of observations as the non-
linear memory signal arises at the same Newtonian order as the oscillatory
signals. For the case of hyperbolic orbits the non-linear term is suppressed
by a factor $(v/c)^{5}$ compared to the quadrupole signal [18].

We will start by computing the non-linear memory from binaries in a circular
orbit. The expression for rate of energy radiated in the direction
$d\Omega^{\prime}$ is given by

$\displaystyle\cfrac{dE_{\text{gw}}}{dt^{\prime}\,d\Omega^{\prime}}=\cfrac{\kappa^{2}}{4}\int\,\Big{(}T_{ij}(\omega^{\prime})\,T^{*}_{kl}(\omega^{\prime})\,\Lambda_{ij,kl}(\vec{n}^{\prime})\Big{)}\,{\omega^{\prime}}^{3}\,2\pi\,\delta(\omega^{\prime}-2\omega_{0})\,\cfrac{d\omega^{\prime}}{(2\pi)^{3}\,2\omega^{\prime}}.$
(56)

where $\omega_{0}=\sqrt{G(m_{1}+m_{2})/a^{3}}$ is the angular frequency of the
Kepler orbit.

Figure 5:  The axis of rotation of the binary designated by $\vec{L}$ makes an
angle $i$ with the $z$-axis and therefore lies in the $y$-$z$ plane. The
primary graviton emits a secondary graviton at
$\vec{r}^{\prime}=r^{\prime}\hat{n}^{\prime}=r^{\prime}(\sin\theta^{\prime}\cos\phi^{\prime},\sin\theta^{\prime}\sin\phi^{\prime},\cos\theta^{\prime})$
and the secondary graviton travels to the earth located at
$\vec{r}=r\hat{n}=(0,0,1)$.

The stress-energy tensor $T_{ij}$, where $i,j=x,y,z$, can be written in matrix
form as:

$\displaystyle\textsf{T}\,(\omega_{0})=\cfrac{\mu\,a^{2}\omega_{0}^{2}}{2}\,\begin{pmatrix}1&&\iota&&0\\\
\iota&&-1&&0\\\ 0&&0&&0\end{pmatrix}.$ (57)

We consider the orientation defined in Fig. 5, where the axis of rotation of
the binary system lies in the $y$-$z$ plane and is counter-clockwise rotated,
by an angle $i$, with respect to the $z$ axis, i.e. $\vec{L}=(0,-\sin i,\cos
i)$. Rotation of the axis impacts the form of the stress-tensor matrix T as
follows:

$\displaystyle\textsf{T}\rightarrow\textsf{T}^{\prime}=\mathcal{R}\,\textsf{T}\,\mathcal{R}^{\text{T}},\hskip
5.69046pt\text{with}\hskip 5.69046pt\mathcal{R}=\begin{pmatrix}1&&0&&0\\\
0&&\cos i&&-\sin i\\\ 0&&\sin i&&\cos i\end{pmatrix}\hskip
2.84544pt\Rightarrow\hskip
2.84544pt\textsf{T}^{\prime}=\cfrac{\mu\,a^{2}\omega_{0}^{2}}{2}\begin{pmatrix}1&&\iota\cos
i&&\iota\sin i\\\ \iota\cos i&&-\cos^{2}i&&-\cos i\sin i\\\ \iota\sin i&&-\cos
i\sin i&&-\sin^{2}i\end{pmatrix}$ (58)

The product $T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\vec{n}^{\prime})$ which can
be rewritten entirely as a product of $T_{ij}$’s and $\hat{n}^{\prime}$, using
Eq. (7), as follows:

$\displaystyle T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\hat{n}^{\prime})$
$\displaystyle=$ $\displaystyle
T_{ij}\,T^{*}_{ji}-2\,T_{ij}\,T^{*}_{jl}\,n_{i}^{\prime}\,n_{l}^{\prime}+\cfrac{1}{2}\,T_{ij}\,T^{*}_{kl}\,n_{i}^{\prime}\,n_{j}^{\prime}\,n_{k}^{\prime}\,n_{l}^{\prime}+\cfrac{1}{2}\,\left(T_{ii}\,T^{*}_{kl}\,n_{k}^{\prime}\,n_{l}^{\prime}+T_{ij}\,T^{*}_{kk}\,n_{i}^{\prime}\,n_{j}^{\prime}-T_{ii}\,T^{*}_{kk}\right).$
(59)

For the circular case both the stress-tensor matrices $\mathsf{T}$ and
$\mathsf{T}^{\prime}$ are already traceless. Therefore, the last three terms,
within the parentheses, on the right hand side of Eq. (59) vanish. The
remaining terms can be evaluated as products of matrices as shown below:

$\displaystyle T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\hat{n}^{\prime})\,$
$\displaystyle\equiv$
$\displaystyle\,\text{Tr}\,[\mathsf{T}^{\prime}\,\mathsf{T}^{\prime\dagger}]-2\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime}\right)\cdot\,\left(\mathsf{T}^{\prime\dagger}\hat{n}^{\prime}\right)+\cfrac{1}{2}\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime}\hat{n}^{\prime}\right)\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime\dagger}\hat{n}^{\prime}\right)$
(60)

Substituting for $\mathsf{T}^{\prime}$, $\hat{n}^{\prime}$ in Eq. (60) and
evaluating the matrix products and traces yields:

$\displaystyle T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\hat{n}^{\prime})\,$
$\displaystyle\equiv$
$\displaystyle\,\cfrac{\mu^{2}a^{4}\omega_{0}^{4}}{4}\,\cfrac{1}{2}\,\Big{(}1+6\cos^{2}\theta+\cos^{4}\theta\Big{)}\,.$
(61)

Here, $\theta$ is the angle between the axis of rotation of the binary
$\vec{L}=\left(0,-\sin i,\cos i\right)$ and the direction where the primary
graviton emits the second graviton:
$\hat{n}^{\prime}=\left(\sin\theta^{\prime}\cos\phi^{\prime},\sin\theta^{\prime}\sin\phi^{\prime},\cos\theta^{\prime}\right)$,
i.e.,

$\displaystyle\cos\theta=-\sin i\sin\theta^{\prime}\sin\phi^{\prime}+\cos
i\cos\theta^{\prime}.$ (62)

Eq. (56) can now be expressed, using the result of Eq. (61) as

$\displaystyle\cfrac{dE_{\text{gw}}}{dt^{\prime}\,d\Omega^{\prime}}=\cfrac{G}{2\pi}\,\omega_{0}^{6}\,\mu^{2}a^{4}\,\Big{(}1+6\cos^{2}\theta+\cos^{4}\theta\Big{)}\,.$
(63)

The Kepler orbital frequency can be written in terms of the semi-major axis
$a$, i.e.$\omega_{0}=(GM/a^{3})^{1/2}$ ($M=m_{1}+m_{2}$). This allows us to
rewrite the expression for the rate of energy radiated as,

$\displaystyle\cfrac{dE_{\text{gw}}}{dt^{\prime}d\Omega^{\prime}}=\cfrac{G^{4}}{2\pi}\cfrac{\mu^{2}M^{3}}{a^{5}}\Big{(}1+6\cos^{2}\theta+\cos^{4}\theta\Big{)}\,.$
(64)

Substituting the above in Eq. (54), we obtain the following expression for the
memory signal:

$\displaystyle h_{ij}^{\text{mem}}(t,\vec{x})$ $\displaystyle=$
$\displaystyle\cfrac{4G}{r}\int_{-\infty}^{t-r}dt^{\prime}\int_{4\pi}d\Omega^{\prime}\,\cfrac{dE_{\text{gw}}(t^{\prime},\Omega)}{dt^{\prime}d\Omega^{\prime}}\cfrac{n^{\prime}_{i}n^{\prime}_{j}}{(1-\hat{n}^{\prime}\cdot\hat{n})}$
(65) $\displaystyle=$
$\displaystyle\cfrac{4G}{r}\int_{-\infty}^{t}dt^{\prime}\,\cfrac{G^{4}}{2\pi}\,\cfrac{\mu^{2}M^{3}}{a^{5}}\underbrace{\int_{4\pi}d\Omega^{\prime}\,\Big{(}1+6\cos^{2}\theta+\cos^{4}\theta\Big{)}\cfrac{n^{\prime}_{i}n^{\prime}_{j}}{(1-\hat{n}^{\prime}\cdot\hat{n})}}_{\textsf{A}_{ij}}.$

where we have replaced $t-r$ by $t$ in the limit of the integrand as the waves
are detected at a fixed $r$ and the dependence in $u$ then becomes a
dependence on time of observation $t$.

To obtain the transverse-traceless part of $h^{\text{mem}}_{ij}$ we apply the
TT projection operator, through multiplication with
$\Lambda_{ij,kl}(\vec{n})$, see Eq. (55),

$\displaystyle\Big{[}h_{ij}^{\text{mem}}\Big{]}^{\text{TT}}(t,\vec{x})$
$\displaystyle=$ $\displaystyle\cfrac{4G^{5}}{2\pi
r}\,\mu^{2}M^{3}\int_{-\infty}^{t}dt^{\prime}\cfrac{1}{a^{5}}\,\underbrace{\int_{4\pi}d\Omega^{\prime}\,\Big{(}1+6\cos^{2}\theta+\cos^{4}\theta\Big{)}\cfrac{\Lambda_{ij,kl}(\hat{n})\,n^{\prime}_{k}n^{\prime}_{l}}{(1-\hat{n}^{\prime}\cdot\hat{n})}}_{\big{[}\textsf{A}_{ij}\big{]}^{\text{TT}}}.$
(66)

The tensor product within the angular integral can be further expressed as:

$\displaystyle\Lambda_{ij,kl}(\hat{n})\,n^{\prime}_{k}n^{\prime}_{l}=\left[\left(n^{\prime}_{i}n^{\prime}_{j}-\cfrac{1}{2}\,\delta_{ij}+\cfrac{1}{2}\,n_{i}n_{j}\right)-\left(n_{i}n^{\prime}_{j}+n^{\prime}_{i}n_{j}\right)\left(\hat{n}^{\prime}\cdot\hat{n}\right)+\cfrac{1}{2}\,\left(\delta_{ij}+n_{i}n_{j}\right)\left(\hat{n}^{\prime}\cdot\hat{n}\right)^{2}\right].$
(67)

Since, $\hat{n}=(0,0,1)$, $\hat{n}^{\prime}\cdot\hat{n}=\cos\theta^{\prime}$
and substituting for the components of $\hat{n}$, $\hat{n}^{\prime}$ and
$\delta_{ij}$, the angular integrals can be evaluated, for $i,j=1,2$ as
follows:

$\displaystyle\Big{[}\textsf{A}_{11}\Big{]}^{\text{TT}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}\sin\theta^{\prime}\,d\theta^{\prime}\,\int_{0}^{2\pi}d\phi^{\prime}\,\left(1+6\cos^{2}\theta+\cos^{4}\theta\right)\,\times\,\cfrac{1}{2}\left(1+\cos\theta^{\prime}\right)\,\cos
2\phi^{\prime}=-\cfrac{2\pi}{15}\,\left(17+\cos^{2}i\right)\sin^{2}i,$
$\displaystyle\Big{[}\textsf{A}_{12}\Big{]}^{\text{TT}}$ $\displaystyle=$
$\displaystyle\int_{0}^{\pi}\sin\theta^{\prime}\,d\theta^{\prime}\,\int_{0}^{2\pi}d\phi^{\prime}\,\left(1+6\cos^{2}\theta+\cos^{4}\theta\right)\,\times\,\cfrac{1}{2}\left(1+\cos\theta^{\prime}\right)\,\sin
2\phi^{\prime}=0.$ (68)

The other components are:
$\Big{[}\textsf{A}_{22}\Big{]}^{\text{TT}}=-\Big{[}\textsf{A}_{11}\Big{]}^{\text{TT}}$
and
$\Big{[}\textsf{A}_{12}\Big{]}^{\text{TT}}=\Big{[}\textsf{A}_{21}\Big{]}^{\text{TT}}=0$.
The transverse-traceless gravitational wavefunction can be decomposed into $+$
and $\times$ modes:

$\displaystyle\Big{[}\textsf{h}^{\text{mem}}_{ij}\Big{]}^{\text{TT}}=h_{I}\,\epsilon^{I}_{ij}=h^{\text{mem}}_{+}\,\epsilon^{+}_{ij}+h^{\text{mem}}_{\times}\,\epsilon^{\times}_{ij}.$
(69)

Based on this we can make the identification:

$\displaystyle h_{+}^{\text{mem}}(t,\vec{x})$ $\displaystyle=$
$\displaystyle\cfrac{4G^{5}}{2\pi
r}\,\,\mu^{2}M^{3}\int_{-\infty}^{t-r}dt^{\prime}\cfrac{1}{a^{5}}\,\times\,\left[\cfrac{2\pi}{15}\,\left(17+\cos^{2}i\right)\sin^{2}i\right],\qquad
h_{\times}^{\text{mem}}(t,\vec{x})=0.$ (70)

The time dependence of the integrand in Eq. (65) is due to the change in the
radius of the orbit which occurs due to the energy loss of the orbit by the
primary gravitational waves. This change in the radius is given as
$da/dt=(dE/dt)\,(da/dE)$, which using $E=-(1/2)\,G\mu M/a$ is

$\displaystyle\cfrac{da}{dt}=-\cfrac{64\,G^{3}}{5}\,\cfrac{\mu
M^{2}}{a^{3}}\,.$ (71)

Solving this equation gives us the time dependence of the separation distance
$a(t)$

$\displaystyle a(t)=a_{c}\left[1+\cfrac{256}{5}\,\cfrac{G^{3}\mu
M^{2}}{a_{c}^{4}}(t_{c}-t)\right]^{1/4}.$ (72)

where $t_{c}$ is the time of coalescence of the black-holes. The growth of
memory signal stops at the time of the coalescence $t_{c}$ when the black-hole
separation becomes smaller than the innermost stable circular orbit radius
$a_{c}=6\,GM$. The frequency $f=\omega_{0}/\pi=(1/\pi)\sqrt{(GM)/a^{3}}$
increases till the time of coalescence as

$\displaystyle f(t^{\prime})$ $\displaystyle\simeq$
$\displaystyle\left(\cfrac{5}{256}\right)^{3/8}\cfrac{1}{\pi}\,(G{\mathcal{M}_{c}})^{-5/8}\left[\cfrac{5}{256}\,\cfrac{a_{c}^{4}}{G^{3}\mu
M^{2}}+(t_{c}-t^{\prime})\right]^{-3/8}$ (73)

where ${\cal M}_{c}=\mu^{3/5}M^{2/5}$ is the chirp mass of the binary pair.

Using Eq. (72) to substitute for $a(t)$ in Eq. (74), we obtain the the non-
linear memory wavefunction of binaries, whose rotation axis makes an angle $i$
with respect to the earth-source direction, as

$\displaystyle h^{\text{mem}}_{+}(t)$ $\displaystyle=$
$\displaystyle\cfrac{1}{192}\cfrac{G}{r}\left(5G\mu^{3}M^{2}\right)^{1/4}\,\frac{1}{\left[5\left(\frac{3}{2}\right)^{4}\frac{GM^{2}}{\mu}+(t_{c}-t)\right]^{1/4}}\,{\sin^{2}i\left(17+\cos^{2}i\right)},$
(74) $\displaystyle=$ $\displaystyle
h_{\text{circ}}\,\cfrac{1}{\left[1+(t^{\prime}_{c}-t^{\prime})\right]^{1/4}}\,{\sin^{2}i\left(17+\cos^{2}i\right)},$

where $h_{\text{circ}}=\frac{2}{3}\,\frac{1}{192}\,\frac{G\mu}{r}$ and
$(t^{\prime}_{c}-t^{\prime})=(t_{c}-t)/T^{\prime}$ with $T^{\prime}$ being
$\left(\frac{3}{2}\right)^{4}\frac{5GM^{2}}{\mu}$. The memory effect will be
strongest in edge on binaries $i=\pi/2$ and will be zero in the face on
binaries $i=0$. This is demonstrated in Fig. 6.

Figure 6: The variation in $h_{+}^{\text{mem}}$ against time plotted for
different choices of the angle $i$ between the line of sight of observation
and the binary axis.

##  VI Elliptical Orbits

For eccentric elliptical orbits, Peters and Mathews [82, 83] calculated the
average energy and angular momentum emission rates at Newtonian order. Their
calculation has been improved to the 3-PN level [84, 85], including nonlinear
tail-effect (which arises from the scattering of gravitational waves by the
near-field potential) at 3-PN [86].

In this section we compute the frequency spectrum of the energy radiated
following [72, 73, 74, 74, 75]. The radiated energy acts as the source term
for secondary gravitational waves which carry the non-linear memory signal.

To describe a compact binary system, comprised of stars having masses $m_{1}$
and $m_{2}$, in an elliptical orbit, the following quantities are of
relevance:

  * •

Since the orbital motion is around the common centre of mass, it is useful to
describe the motion in terms of the reduced mass
$\mu=\cfrac{m_{1}\,m_{2}}{m_{1}+m_{2}}$ and the total mass $M=(m_{1}+m_{2})$.

  * •

The elliptical Keplerian orbit can be parametrized as:

$\displaystyle\quad x(\xi)=a\,(\cos\xi-e),\qquad y(\xi)=b\,\sin\xi,\qquad
z(\xi)=0,\qquad\cfrac{\omega_{n}^{\prime}}{n}\,t=\Omega\,t=(\xi-e\sin\xi).$
(75)

Here, $a$ and $b\,\,(=a\sqrt{1-e^{2}},\,\,e<1)$ denote the semi-major and
semi-minor axes respectively. The variables $e$ and $\xi\in(0,2\pi)$ refer to
the eccentricity of the orbit and the eccentric anomaly respectively.

  * •

The angular frequency corresponding to the $n^{th}$ harmonic is denoted as
$\omega_{n}^{\prime}=n\,\Omega$, with $n=\\{0,1,2,....\\}$ being a non-
negative integer and the fundamental frequency $\Omega$ can be related to the
semi-major axis and the total mass of the system as:
$\quad\Omega=\left(\cfrac{G\,(m_{1}+m_{2})}{a^{3}}\right)^{1/2}$.

The first step in the computation of the rate of energy radiated involves the
evaluation of the stress-tensor components in frequency space. Once again
starting with the $xx$ component, the calculation proceeds as follows:

$\displaystyle T_{xx}(\omega_{n}^{\prime})$ $\displaystyle=$
$\displaystyle\cfrac{\mu}{T}\,\int_{0}^{T}\,dt\,\dot{x}^{2}(t)e^{\iota\omega_{n}^{\prime}t}=\cfrac{-\iota\,\mu\,\omega_{n}^{\prime}}{T}\,\int_{0}^{T}\,dt\,\dot{x}(t)\,x(t)\,e^{\iota\omega_{n}^{\prime}t}.$
(76)

In the above, we have used integration by parts. Next, the parametric form of
the orbit coordinates, shown in Eq. (75), allows us to write,

$\displaystyle\dot{x}\,dt=\cfrac{dx}{d\xi}\,d\xi=-a\sin\xi\,d\xi.$ (77)

Using the above transformation and also substituting for $x$ and
$\omega_{n}^{\prime}\,t$ in terms of functions of $\xi$, we get

$\displaystyle T_{xx}(\omega_{n}^{\prime})$ $\displaystyle=$
$\displaystyle\cfrac{\iota\,\mu a^{2}\omega_{n}^{\prime 2}}{2\pi
n}\int_{0}^{2\pi}d\xi\,\sin\xi\,(\cos\xi-e)\,e^{\iota
n(\xi-e\sin\xi)}=-\cfrac{\mu a^{2}\omega_{n}^{\prime
2}}{n}\left[\left(\cfrac{1-e^{2}}{e}\right)J_{n}^{\prime}(ne)-\cfrac{1}{ne^{2}}J_{n}(ne)\right].$
(78)

To arrive at the second line of Eq. (78), we replaced the trigonometric
functions by the corresponding exponential functions and identified the
integral form of Bessel functions of first kind,

$\displaystyle J_{n}(z)=\cfrac{1}{2\pi}\,\int\limits_{0}^{2\pi}\,e^{\iota
n(\xi-e\sin\xi)}d\xi.$ (79)

The final expression in terms of $J_{n}(ne)$ and $J^{\prime}_{n}(ne)$ is
obtained by utilizing the recurrence relations given below:

$\displaystyle J_{n-1}(z)+J_{n+1}(z)=\cfrac{2\,n}{z}\,J_{n}(z),\hskip
28.45274ptJ_{n-1}(z)-J_{n+1}(z)=2\,J^{\prime}_{n}(z).$ (80)

The other non-zero components, i.e $T_{yy}$ and $T_{xy}$ can similarly be
obtained as:

$\displaystyle T_{yy}(\omega_{n}^{\prime})$ $\displaystyle=$
$\displaystyle\cfrac{\mu}{T}\int_{0}^{T}dt\,\dot{y}^{2}(t)e^{\iota\omega_{n}^{\prime}t}=\cfrac{\mu\,\omega_{n}^{\prime
2}\,a^{2}(1-e^{2})}{n}\left[\cfrac{1}{e}J_{n}^{\prime}(ne)-\cfrac{1}{ne^{2}}J_{n}(ne)\right],$
$\displaystyle T_{xy}(\omega_{n}^{\prime})$ $\displaystyle=$
$\displaystyle\cfrac{\mu}{T}\,\int_{0}^{T}\,dt\,\dot{x}(t)\dot{y}(t)e^{\iota\omega_{n}^{\prime}t}=\cfrac{\iota\mu\,\omega_{n}^{\prime
2}\,a^{2}\sqrt{(1-e^{2})}}{n}\left[-\left(\cfrac{1-e^{2}}{e^{2}}\right)J_{n}(ne)+\cfrac{1}{ne}J_{n}^{\prime}(ne)\right].$
(81)

###  VI.1 Rate of gravitational radiation from binaries in elliptical orbits

Substituting for $T_{xx}$, $T_{yy}$ and $T_{xy}$ using Eqs. (78) and (VI)
gives,

$\displaystyle
T_{ij}(\omega^{\prime})\,T^{*}_{ji}(\omega^{\prime})-\cfrac{1}{3}\,|\,T^{i}{}_{i}(\omega^{\prime})\,|^{2}=g(n,e).$
(82)

where we have defined $g(n,e)$ as,

$\displaystyle g(n,e)$ $\displaystyle=$ $\displaystyle
J_{n}(ne)^{2}\left[\cfrac{2n^{2}}{e^{4}}(1-e^{2})^{3}+\cfrac{6-6e^{2}+2e^{4}}{3e^{4}}\right]+J_{n}^{\prime}(ne)^{2}\left[\cfrac{2n^{2}}{e^{2}}(1-e^{2})^{2}+\cfrac{2(1-e^{2})}{e^{2}}\right]$
(83)
$\displaystyle+\,J_{n}(ne)J_{n}^{\prime}(ne)\left[\cfrac{(-8+14e^{2}-6e^{4})n}{e^{3}}\right].$

In this case, the energy loss due to gravitational radiation as given in Eq.
(17) becomes,

$\displaystyle\cfrac{dE_{\text{gw}}}{dt}(e)$ $\displaystyle=$
$\displaystyle\cfrac{\kappa^{2}}{8(2\pi)^{2}}\sum_{n=0}\cfrac{8\pi}{5}\left[T_{ij}(\omega_{n}^{\prime})T_{ji}^{*}(\omega_{n}^{\prime})-\cfrac{1}{3}|T^{i}{i}(\omega_{n}^{\prime})|\right]\omega_{n}^{\prime
2}=\cfrac{32\,G}{20}\,\Omega^{6}\mu^{2}a^{4}\sum_{n=0}^{\infty}n^{2}g(n,e).$
(84)

The series sum over products of Bessel functions and their derivative,
weighted by powers of $n$, can be expressed completely as functions of $e$,
using the identities derived in the appendix of [82] and reproduced in Eq.
(VIII). This enables the identification of the eccentricity dependent part of
the energy loss as

$\displaystyle\widetilde{g}(e)=\sum_{n=0}^{\infty}n^{2}\,g(n,e)=\cfrac{4}{(1-e^{2})^{7/2}}\left(1+\cfrac{73}{24}\,e^{2}+\cfrac{37}{96}\,e^{4}\right).$
(85)

Therefore, the energy radiated as gravitational waves from the binary can be
written as:

$\displaystyle\cfrac{dE_{\text{gw}}}{dt}=\cfrac{32\,G}{5}\,\Omega^{6}\left(\cfrac{m_{1}m_{2}}{m_{1}+m_{2}}\right)^{2}a^{4}\cfrac{1}{(1-e^{2})^{7/2}}\left(1+\cfrac{73}{24}\,e^{2}+\cfrac{37}{96}\,e^{4}\right).$
(86)

###  VI.2 Non-linear memory from elliptical orbits

Rate of energy radiated with respect to time and solid angle is described
using the following formula:

$\displaystyle\cfrac{dE_{\text{gw}}}{dt^{\prime}\,d\Omega^{\prime}}$
$\displaystyle=$
$\displaystyle\cfrac{\kappa^{2}}{8(2\pi)^{2}}\sum_{n=0}\bigg{[}T_{ij}(\omega_{n}^{\prime})\,T^{*}_{kl}(\omega_{n}^{\prime})\,\Lambda_{ij,kl}(\hat{n}^{\prime})\bigg{]}\,\omega_{n}^{\prime
2}$ (87)

The stress-energy tensor $T_{ij}$, where $i,j=x,y,z$, can be written in matrix
form as:

$\displaystyle\textsf{T}\,(n,e)=\mu\,a^{2}\,\Omega^{2}\,\begin{pmatrix}q_{1}(n,e)&&\iota\,q_{2}(n,e)&&0\\\
\iota\,q_{2}(n,e)&&q_{3}(n,e)&&0\\\ 0&&0&&0\end{pmatrix}.$ (88)

The non-zero elements of the matrix are functions of $n$, corresponding to the
$n$th harmonic, and $e$, the eccentricity of the orbit and these are given as

$\displaystyle q_{1}(n,e)$ $\displaystyle=$
$\displaystyle-\cfrac{n\,\left(1-e^{2}\right)}{e}\,J_{n}^{\prime}(ne)+\cfrac{1}{e^{2}}\,J_{n}(ne),$
$\displaystyle q_{2}(n,e)$ $\displaystyle=$
$\displaystyle\cfrac{\left(1-e^{2}\right)^{1/2}}{e}\,J_{n}^{\prime}(ne)-\cfrac{n\,\left(1-e^{2}\right)^{3/2}}{e^{2}}\,J_{n}(ne),$
$\displaystyle q_{3}(n,e)$ $\displaystyle=$
$\displaystyle\cfrac{n\,\left(1-e^{2}\right)}{e}\,J_{n}^{\prime}(ne)-\cfrac{\left(1-e^{2}\right)}{e^{2}}\,J_{n}(ne).$
(89)

We follow the orientation for the system as shown in Fig. 5. The stress-energy
matrix for the rotated system is given as
$\textsf{T}^{\prime}=\mathcal{R}\,\textsf{T}\,\mathcal{R}^{\text{T}}$, with
$\mathcal{R}$ being the rotation matrix defined in Eq. (58). This allows us to
write,

$\displaystyle\textsf{T}^{\prime}=\mu\,a^{2}\,\Omega^{2}\begin{pmatrix}q_{1}(n,e)&&\iota\,q_{2}(n,e)\,\cos
i&&\iota\,q_{2}(n,e)\,\sin i\\\ \iota\,q_{2}(n,e)\,\cos
i&&q_{3}(n,e)\,\cos^{2}i&&q_{3}(n,e)\,\cos i\sin i\\\ \iota\,q_{2}(n,e)\,\sin
i&&q_{3}(n,e)\,\cos i\sin i&&q_{3}(n,e)\,\sin^{2}i\end{pmatrix}.$ (90)

Expanding the matrix product
$T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\hat{n}^{\prime})$ in terms of products
of $T_{ij}$ and $n^{\prime}_{i}$, yields the same result as Eq. (59),

$\displaystyle T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\hat{n}^{\prime})$
$\displaystyle=$ $\displaystyle
T_{ij}\,T^{*}_{ji}-2\,T_{ij}\,T^{*}_{jl}\,n_{i}^{\prime}\,n_{l}^{\prime}+\cfrac{1}{2}\,T_{ij}\,T^{*}_{kl}\,n_{i}^{\prime}\,n_{j}^{\prime}\,n_{k}^{\prime}\,n_{l}^{\prime}+\cfrac{1}{2}\,\left(T_{ii}\,T^{*}_{kl}\,n_{k}^{\prime}\,n_{l}^{\prime}+T_{ij}\,T^{*}_{kk}\,n_{i}^{\prime}\,n_{j}^{\prime}-T_{ii}\,T^{*}_{kk}\right).$
(91)

Unlike the case of circular orbits, none of the terms on the right hand side
of Eq. (91) vanish. Therefore, the expression when rewritten as a sum over
matrix products appears as,

$\displaystyle T_{ij}\,T^{*}_{kl}\,\Lambda_{ij,kl}(\hat{n}^{\prime})\,$
$\displaystyle\equiv$
$\displaystyle\,\text{Tr}\,[\mathsf{T}^{\prime}\,\mathsf{T}^{\prime\dagger}]-2\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime}\right)\cdot\,\left(\mathsf{T}^{\prime\dagger}\hat{n}^{\prime}\right)+\cfrac{1}{2}\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime}\hat{n}^{\prime}\right)\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime\dagger}\hat{n}^{\prime}\right)+$
(92)
$\displaystyle\cfrac{1}{2}\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime}\hat{n}^{\prime}\right)\,\text{Tr}\,[\mathsf{T}^{\prime\dagger}]+\cfrac{1}{2}\,\text{Tr}\,[\mathsf{T}^{\prime}]\,\left(\hat{n}^{\prime\text{T}}\,\mathsf{T}^{\prime\dagger}\hat{n}^{\prime}\right)-\cfrac{1}{2}\,\text{Tr}\,[\mathsf{T}^{\prime}]\,\text{Tr}\,[\mathsf{T}^{\prime\dagger}].$

Since, $T_{ij}$’s are functions of $n$ and $e$, while $\hat{n}^{\prime}$ is
parametrized in terms of the angles $\theta^{\prime}$, $\phi^{\prime}$, the
matrix product can be expressed as an overall function of $n$, $e$,
$\theta^{\prime}$, and $\phi^{\prime}$

$\displaystyle
T_{ij}(n,e)\,T^{*}_{kl}(n,e)\,\Lambda_{ij,kl}(\theta^{\prime},\phi^{\prime})$
$\displaystyle=$
$\displaystyle\mu^{2}a^{4}\,\Omega^{4}\,\,\mathcal{I}(n,e,\theta^{\prime},\phi^{\prime})$
(93)

with

$\displaystyle\mathcal{I}(n,e,\theta^{\prime},\phi^{\prime})$ $\displaystyle=$
$\displaystyle\cfrac{1}{2}\,q^{2}_{1}(n,e)\left[1-p_{1}^{2}(\theta^{\prime},\phi^{\prime})\right]^{2}+\cfrac{1}{2}\,q^{2}_{3}(n,e)\left[1-p_{2}^{2}(\theta^{\prime},\phi^{\prime})\right]^{2}-2\,q_{1}(n,e)\,q_{3}(n,e)$
(94)
$\displaystyle+\,2\,q^{2}_{2}(n,e)\left[1-p_{1}^{2}(\theta^{\prime},\phi^{\prime})\right]\left[1-p_{2}^{2}(\theta^{\prime},\phi^{\prime})\right]+q_{1}(n,e)\,q_{3}(n,e)\left[1+p_{1}^{2}(\theta^{\prime},\phi^{\prime})\right]\left[1+p_{2}^{2}(\theta^{\prime},\phi^{\prime})\right].$

Here,

$\displaystyle
p_{1}(\theta^{\prime},\phi^{\prime})=\sin\theta^{\prime}\,\cos\phi^{\prime}\quad\text{and}\quad
p_{2}(\theta^{\prime},\phi^{\prime})=\cos
i\sin\theta^{\prime}\sin\phi^{\prime}+\sin i\cos\theta^{\prime}$ (95)

Computation of the transverse-traceless wavefunction involves (i) an angular
integral over $\theta^{\prime}$, $\phi^{\prime}$ and (ii) a sum over the
orders ($n$) of the Bessel functions. This will ultimately yield an
eccentricity dependent function:

$\displaystyle\Big{[}\textsf{A}_{ij}\Big{]}^{\text{TT}}(e)=\sum_{n=0}^{\infty}\,\,\int_{4\pi}d\,\Omega^{\prime}\,\mathcal{I}(n,e,\theta^{\prime},\phi^{\prime})\,\cfrac{\Lambda_{ij,kl}(\hat{n})\,n^{\prime}_{k}n^{\prime}_{l}}{(1-\hat{n}^{\prime}\cdot\hat{n})}.$
(96)

The tensor product within the angular integral can be further expanded as:

$\displaystyle\Lambda_{ij,kl}(\hat{n})\,n^{\prime}_{k}n^{\prime}_{l}=\left[\left(n^{\prime}_{i}n^{\prime}_{j}-\cfrac{1}{2}\,\delta_{ij}+\cfrac{1}{2}\,n_{i}n_{j}\right)-\left(n_{i}n^{\prime}_{j}+n^{\prime}_{i}n_{j}\right)\left(\hat{n}^{\prime}\cdot\hat{n}\right)+\cfrac{1}{2}\,\left(\delta_{ij}+n_{i}n_{j}\right)\left(\hat{n}^{\prime}\cdot\hat{n}\right)^{2}\right].$
(97)

Since, $\hat{n}=(0,0,1)$, $\hat{n}^{\prime}\cdot\hat{n}=\cos\theta^{\prime}$
and substituting for the components of $\hat{n}$, $\hat{n}^{\prime}$ and
$\delta_{ij}$, the angular integrals can be evaluated, for $i,j=1,2$ as
follows:

$\displaystyle\Big{[}\textsf{A}_{11}\Big{]}^{\text{TT}}(e)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\,\,\int_{0}^{\pi}\sin\theta^{\prime}\,d\theta^{\prime}\,\int_{0}^{2\pi}d\phi^{\prime}\,\mathcal{I}(n,e,\theta^{\prime},\phi^{\prime})\,\times\,\cfrac{1}{2}\left(1+\cos\theta^{\prime}\right)\,\cos
2\phi^{\prime}$ (98) $\displaystyle=$
$\displaystyle\cfrac{2\pi}{15}\,\sum_{n=0}^{\infty}\,\bigg{[}\Big{(}6\,q_{1}(n,e)\,q_{3}(n,e)-3\,q^{2}_{1}(n,e)-8\,q^{2}_{2}(n,e)\Big{)}$
$\displaystyle\qquad\qquad+\Big{(}8\,q^{2}_{2}(n,e)+2\,q^{2}_{3}(n,e)-6\,q_{1}(n,e)\,q_{3}(n,e)\Big{)}\cos^{2}i+q^{2}_{3}(n,e)\cos^{4}i\bigg{]}$
$\displaystyle=$
$\displaystyle\cfrac{2\pi}{15}\left(C_{0}(e)+C_{2}(e)\cos^{2}i+C_{4}(e)\cos^{4}i\right)$

The sum over $n$ can be evaluated, using the identities given in Eq. (VIII),
to obtain the coefficients $C_{i}(e)$, $i=0,1,2$:

$\displaystyle C_{0}(e)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\,\bigg{[}\cfrac{(6e^{2}-9)}{e^{4}}\,n^{2}\left[J_{n}(ne)\right]^{2}-\cfrac{8(1-e^{2})^{3}}{e^{4}}\,n^{4}\left[J_{n}(ne)\right]^{2}-\cfrac{8(1-e^{2})}{e^{2}}\,n^{2}\left[J^{\prime}_{n}(ne)\right]^{2}-\cfrac{9(1-e^{2})^{2}}{e^{2}}\,n^{4}\left[J^{\prime}_{n}(ne)\right]^{2}$
$\displaystyle\qquad\qquad+\cfrac{(1-e^{2})}{e^{3}}(34-22e^{2})\,n^{3}\left[J_{n}(ne)\,J^{\prime}_{n}(ne)\right]\bigg{]}$
$\displaystyle=$
$\displaystyle-\cfrac{1}{4e^{2}(1-e^{2})^{7/2}}\,\bigg{[}68\,e^{2}+\cfrac{1565}{8}\,e^{4}+\cfrac{1071}{64}\,e^{6}\bigg{]},$
$\displaystyle C_{2}(e)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\,\bigg{[}\cfrac{2(4-5e^{2}+e^{4})}{e^{4}}\,n^{2}\left[J_{n}(ne)\right]^{2}+\cfrac{8(1-e^{2})^{3}}{e^{4}}\,n^{4}\left[J_{n}(ne)\right]^{2}+\cfrac{8(1-e^{2})}{e^{2}}\,n^{2}\left[J^{\prime}_{n}(ne)\right]^{2}+\cfrac{8(1-e^{2})^{2}}{e^{2}}\,n^{4}\left[J^{\prime}_{n}(ne)\right]^{2}$
$\displaystyle\qquad\qquad+\cfrac{2(1-e^{2})}{e^{3}}(-16+13e^{2})\,n^{3}\left[J_{n}(ne)\,J^{\prime}_{n}(ne)\right]\bigg{]}$
$\displaystyle=$
$\displaystyle\cfrac{1}{4e^{2}(1-e^{2})^{7/2}}\bigg{[}64\,e^{2}+\cfrac{399}{2}\,e^{4}+\cfrac{101}{4}\,e^{6}\bigg{]},$
$\displaystyle C_{4}(e)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\,\cfrac{(1-e^{2})^{2}}{e^{4}}\bigg{[}n^{2}\left[J_{n}(ne)\right]^{2}+e^{2}n^{4}\left[J^{\prime}_{n}(ne)\right]^{2}-2e\,n^{3}\left[J_{n}(ne)\,J^{\prime}_{n}(ne)\right]\bigg{]}$
(99) $\displaystyle=$
$\displaystyle\cfrac{1}{4e^{2}(1-e^{2})^{9/2}}\bigg{[}-1-\cfrac{15}{4}\,e^{2}+\cfrac{83}{4}\,e^{4}+\cfrac{21}{8}\,e^{6}-\cfrac{37}{64}\,e^{8}\bigg{]}.$

The cross-component $\big{[}\textsf{A}_{12}\big{]}^{\text{TT}}(e)$ can
similarly be evaluated as:

$\displaystyle\Big{[}\textsf{A}_{12}\Big{]}^{\text{TT}}(e)$ $\displaystyle=$
$\displaystyle\sum_{n=0}^{\infty}\,\,\int_{0}^{\pi}\sin\theta^{\prime}\,d\theta^{\prime}\,\int_{0}^{2\pi}d\phi^{\prime}\,\mathcal{I}(n,e,\theta^{\prime},\phi^{\prime})\,\times\,\cfrac{1}{2}\left(1+\cos\theta^{\prime}\right)\,\sin
2\phi^{\prime}=0.$ (100)

The other components can be obtained using the above as:
$\big{[}\textsf{A}_{22}\big{]}^{\text{TT}}=-\big{[}\textsf{A}_{11}\big{]}^{\text{TT}}$
and
$\big{[}\textsf{A}_{12}\big{]}^{\text{TT}}=\big{[}\textsf{A}_{21}\big{]}^{\text{TT}}=0$.
The decomposition of the transverse-traceless wavefunction into $+$ and
$\times$ polarization modes, see Eq. (69), allows us to identify:

$\displaystyle h_{+}^{\text{mem}}(t,\vec{x})$ $\displaystyle=$
$\displaystyle\cfrac{4G}{r}\int_{-\infty}^{t-r}dt^{\prime}\cfrac{1}{\pi}\cfrac{G^{4}\mu^{2}M^{3}}{a(t^{\prime})^{5}}\,\times\,\cfrac{2\pi}{15}\,\Big{(}C_{0}(e)+C_{2}(e)\cos^{2}i+C_{4}(e)\cos^{4}i\Big{)},\qquad
h_{\times}^{\text{mem}}(t,\vec{x})=0.$ (101)

Within the integrand, the time-dependence is encoded in both the semi-major
axis $a$ and the eccentricity $e$ of the orbit. This is on account of the fact
that energy loss due to primary gravitational waves alters the features of the
orbit. The explicit time-dependence for $a(t)$ and $e(t)$ can be obtained by
solving the following system of non-linear differential equations [83]:

$\displaystyle\cfrac{da}{dt}=-\cfrac{64}{5}\,\cfrac{G^{3}\mu
M^{2}}{a^{3}\,(1-e^{2})^{7/2}}\,\left(1+\cfrac{73}{24}\,e^{2}+\cfrac{37}{96}\,e^{4}\right),$
(102) $\displaystyle\cfrac{de}{dt}=-\cfrac{304}{15}\,\cfrac{G^{3}\mu
M^{2}}{a^{4}\,(1-e^{2})^{5/2}}\,\left(1+\cfrac{121}{304}\,e^{2}\right).$ (103)

The above can be used to eliminate $t$ and obtain a differential equation
involving $a$ and $e$, i.e.,

$\displaystyle\cfrac{da}{de}=\cfrac{12}{19}\,\cfrac{a}{e\,(1-e^{2})}\,\cfrac{1+(73/24)\,e^{2}+(37/96)\,e^{4}}{1+(121/304)\,e^{2}},$
(104)

whose solution expresses $a$ in terms of $e$,

$\displaystyle
a(e)=c_{0}\,\cfrac{e^{12/19}}{(1-e^{2})}\left(1+\cfrac{121}{304}\,e^{2}\right)^{870/2299},$
(105)

where
$c_{0}=a_{0}\,e_{0}^{-12/19}(1-e_{0}^{2})(1+(121/304)\,e_{0}^{2})^{-870/2299}$
and $a_{0}=a(e_{0})$ defines the initial condition necessary for solving the
differential equation. The time evolution can be given as a function of
eccentricity as

$\displaystyle t-t_{c}=-\cfrac{15}{304}\,\cfrac{c_{0}^{4}}{G^{3}\mu
M^{2}}\,\cfrac{19}{48}\,e^{48/19}\,F_{1}\left(\cfrac{24}{19}\,,-\cfrac{1181}{2299}\,,\cfrac{3}{2}\,,\cfrac{43}{19}\,;-\cfrac{121}{304}\,e^{2},e^{2}\right),$
(106)

where $t_{c}$ corresponds to the instant of coalescence of the black-holes,
i.e., when $e$ becomes close to zero. $F_{1}$ is the hypergeometric Appell
function with the following integral representation:

$\displaystyle
F_{1}(a,b_{1},b_{2},c;x,y)=\cfrac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}dt,\,\,\,\,\,\,\,\,\mathfrak{R}c>\mathfrak{R}a>0.$
(107)

Eq. (106) can be expressed entirely in terms of dimensionless quantities by
noting that $\overline{T}=c_{0}^{4}/(G^{3}\mu M^{2})$ has the dimensions of
time. Therefore,

$\displaystyle\cfrac{t-t_{c}}{\overline{T}}=\overline{t}-\overline{t}_{c}=-\cfrac{15}{304}\,\cfrac{19}{48}\,e^{48/19}\,F_{1}\left(\cfrac{24}{19}\,,-\cfrac{1181}{2299}\,,\cfrac{3}{2}\,,\cfrac{43}{19}\,;-\cfrac{121}{304}\,e^{2},e^{2}\right),$
(108)

The wavefunction in Eq. (101) can be rewritten, after a change of integration
variable, from $t^{\prime}$ to $e^{\prime}$ as,

$\displaystyle h_{+}^{\text{mem}}(e,\vec{x})$ $\displaystyle=$
$\displaystyle\cfrac{4G}{r}\int_{e}^{0}\cfrac{de^{\prime}}{\dot{e}(e^{\prime})}\cfrac{1}{\pi}\cfrac{G^{4}\mu^{2}M^{3}}{a(e^{\prime})^{5}}\,\cfrac{2\pi}{15}\left[C_{0}(e^{\prime})+C_{2}(e^{\prime})\cos^{2}i+C_{4}(e^{\prime})\cos^{4}i\right],$
(109) $\displaystyle=$ $\displaystyle\cfrac{4G}{r}\cfrac{15}{304}\cfrac{G\mu
M}{c_{0}}\,\cfrac{2}{15}\left[C_{h}^{(0)}(e)+C_{h}^{(2)}(e)\cos^{2}i+C_{h}^{(4)}(e)\cos^{4}i\right],$

or in terms of dimensionless quantities, we can write

$\displaystyle\cfrac{h_{+}^{\text{mem}}(e,\vec{x})}{h_{0}}$ $\displaystyle=$
$\displaystyle\overline{h}_{+}^{\text{mem}}(e,\vec{x})=\cfrac{1}{152}\left[C_{h}^{(0)}(e)+C_{h}^{(2)}(e)\cos^{2}i+C_{h}^{(4)}(e)\cos^{4}i\right].$
(110)

Here, $h_{0}=(4\,G^{2}\mu M)/(r\,c_{0})$. The eccentricity dependent part is
constituted of

$\displaystyle C_{h}^{(0)}(e)$ $\displaystyle=$ $\displaystyle
17\,\cfrac{19\,e^{-12/19}}{12}\,F_{1}\left(-\cfrac{6}{19}\,,\cfrac{3169}{2299}\,,0\,,\cfrac{13}{19}\,;-\cfrac{121}{304}\,e^{2}\,,0\right)-\cfrac{1565}{32}\,\cfrac{19\,e^{26/19}}{26}\,F_{1}\left(\cfrac{13}{19}\,,\cfrac{3169}{2299}\,,0\,,\cfrac{32}{19}\,;-\cfrac{121}{304}\,e^{2}\,,0\right)$
$\displaystyle-\cfrac{1071}{256}\,\cfrac{19\,e^{64/19}}{64}\,F_{1}\left(\cfrac{32}{19}\,,\cfrac{3169}{2299}\,,0\,,\cfrac{51}{19}\,;-\cfrac{121}{304}\,e^{2}\,,0\right),$
$\displaystyle C_{h}^{(2)}(e)$ $\displaystyle=$
$\displaystyle-16\,\cfrac{19\,e^{-12/19}}{12}\,F_{1}\left(-\cfrac{6}{19}\,,\cfrac{3169}{2299}\,,0\,,\cfrac{13}{19}\,;-\cfrac{121}{304}\,e^{2}\,,0\right)+\cfrac{399}{8}\,\cfrac{19\,e^{26/19}}{26}\,F_{1}\left(\cfrac{13}{19}\,,\cfrac{3169}{2299}\,,0\,,\cfrac{32}{19}\,;-\cfrac{121}{304}\,e^{2}\,,0\right)$
$\displaystyle+\cfrac{101}{16}\,\cfrac{19\,e^{64/19}}{64}\,F_{1}\left(\cfrac{32}{19}\,,\cfrac{3169}{2299}\,,0\,,\cfrac{51}{19}\,;-\cfrac{121}{304}\,e^{2}\,,0\right),$
$\displaystyle C_{h}^{(4)}(e)$ $\displaystyle=$
$\displaystyle\cfrac{1}{4}\,\cfrac{19\,e^{-50/19}}{50}\,F_{1}\left(-\cfrac{25}{19}\,,\cfrac{3169}{2299}\,,1\,,-\cfrac{6}{19}\,;-\cfrac{121}{304}\,e^{2}\,,e^{2}\right)+\cfrac{15}{16}\,\cfrac{19\,e^{-12/19}}{12}\,F_{1}\left(-\cfrac{6}{19}\,,\cfrac{3169}{2299}\,,1\,,\cfrac{13}{19}\,;-\cfrac{121}{304}\,e^{2}\,,e^{2}\right)$
(111)
$\displaystyle+\cfrac{83}{16}\,\cfrac{19\,e^{26/19}}{26}\,F_{1}\left(\cfrac{13}{19}\,,\cfrac{3169}{2299}\,,1\,,\cfrac{32}{19}\,;-\cfrac{121}{304}\,e^{2}\,,e^{2}\right)+\cfrac{21}{32}\,\cfrac{19\,e^{64/19}}{64}\,F_{1}\left(\cfrac{32}{19}\,,\cfrac{3169}{2299}\,,1\,,\cfrac{51}{19}\,;-\cfrac{121}{304}\,e^{2}\,,e^{2}\right)$
$\displaystyle-\cfrac{37}{256}\,\cfrac{19\,e^{102/19}}{102}\,F_{1}\left(\cfrac{51}{19}\,,\cfrac{3169}{2299}\,,1\,,\cfrac{70}{19}\,;-\cfrac{121}{304}\,e^{2}\,,e^{2}\right),$

The variation in the memory waveform with time can be obtained by exploiting
the relation between the waveform and eccentricity of the orbit, as described
in Eq. (110) and the implicit relation between time and eccentricity
highlighted in Eq. (108). A plot depicting such variation is shown in Fig. 7.

Figure 7:  The variation in $\overline{h}_{+}^{\text{mem}}$ with respect to
the time interval $\overline{t}-\overline{t}_{c}$ plotted for different
choices of the angle between between the line of sight of observation and the
binary axis. The first figure displays the $\overline{h}_{+}^{\text{mem}}$
values on a logarithmic scale, whereas the second figure presents the same
information but with the regular scaling of the axes.

An alternative description can be obtained in terms of the frequency
$\nu=(2\pi)^{-1}\,m^{1/2}\,a^{-3/2}$, by noting that the relation between
$\nu$ and $e$ can be obtained by solving the following differential equation:

$\displaystyle\cfrac{d\nu}{de}=-\cfrac{18}{19}\,\cfrac{\nu}{e\,(1-e^{2})}\,\cfrac{1+(73/24)\,e^{2}+(37/96)\,e^{4}}{1+(121/304)\,e^{2}}.$
(112)

The solution can be evaluated as,

$\displaystyle\nu(e)=\widetilde{c}_{0}\,e^{-18/19}\,\left(1-e^{2}\right)^{3/2}\left(1+\cfrac{121}{304}\,e^{2}\right)^{-1305/2299},$
(113)

where
$\widetilde{c}_{0}=\nu_{0}\,e_{0}^{-18/19}(1-e_{0}^{2})^{-3/2}(1+(121/304)\,e_{0}^{2})^{1305/2299}$
with $\nu_{0}=\nu(e_{0})$ defining the initial condition. In terms of
dimensionless quantities, the same equation can be rewritten as,

$\displaystyle\cfrac{\nu(e)}{\widetilde{c}_{0}}=\overline{\nu}(e)=e^{-18/19}\,\left(1-e^{2}\right)^{3/2}\left(1+\cfrac{121}{304}\,e^{2}\right)^{-1305/2299},$
(114)

The variation in the memory signal with respect to the frequency can be
obtained based on their mutual dependence on eccentricity, as highlighted in
Eqs. (110) and (114). A plot depicting such a relation is shown in Fig. 8.

Figure 8: The variation in $\overline{h}_{+}^{\text{mem}}$ with respect to the
dimensionless frequency parameter $\bar{\nu}$, defined in Eq. (114), plotted
for different choices of the angle between between the line of sight of
observation and the binary axis.

##  VII Conclusions

In this paper we have computed the linear memory signal for eccentric
hyperbolic encounters in both frequency and time domain. We performed a field
theoretic calculation of the amplitude of graviton emission from a classical
stress tensor and related this amplitude to the gravitational wave signal as a
function of frequency. We took the zero-frequency limit of the waveform to
identify the memory signals which we computed in the frequency space and
taking the Fourier transform also in the time domain. We performed these
calculations for hyperbolic orbits with large eccentricity while retaining
terms of all orders in eccentricity. We find that in the eccentric hyperbolic
orbits the low-frequency memory component has terms which vary with the
frequency $\omega=\nu\Omega$ of the gravitational waves as $\log(\omega
e/\Omega)$. This $\log$ dependence is due to the eccentric hyperbolic orbit
and is different from the tail terms which occur for any unbound scattering at
${\cal O}(G^{2})$ . Our results are in agreement with the expressions obtained
for the stress tensors and the memory waveforms based on the soft graviton
theorems [25, 26, 27, 28], once appropriate substitutions are made in the
latter to describe the specific case of a hyperbolic encounter.

Hyperbolic encounters of supermassive black-holes of mass $10^{7}M_{\odot}$
with closest approach of 1pc will produce gravitational waves with peak
frequency $\sim 0.2$ mHz and encounters between blackholes of mass
$10^{3}M_{\odot}$ will produce gravitational waves at peak frequency $0.04$
mHz . The time window for the observations is of the order of 1 day which
makes such observations possible for third generation gravitational detectors.

The non-linear memory which occurs from GW radiated by GW is equally important
as it would prove the non-linear nature of graviton-graviton interactions. We
have computed the non-linear memory for eccentric elliptical orbits. Highly
eccentric orbits are possible if the initial system has three bodies of which
one is ejected [38] or by capture from an unbound orbit [36, 37]. We compute
the frequency spectrum energy radiated by the binaries using the field theory
technique which gives the energy spectrum directly in the frequency space.
This is used as the source term for the secondary gravitational waves, which
results in the memory waveform in the frequency space. We use the
instantaneous eccentricity as the parameter for tracking the change of
frequency and semi-major axis in time due to radiation reaction. We thus
obtain the non-linear memory as a function of the instantaneous eccentricity.
We then exploited the change in eccentricity with respect to both time and
frequency to obtain the non-linear waveform as a function of both time and
frequency. The calculation of the non-linear memory has been done up to all
orders in eccentricity. These signal templates in frequency and time domain
may be useful for extracting the memory signal from data by the future
detectors like Advanced-LIGO, Einstein Telescope and LISA.

##  VIII Appendix

For evaluating the sum over $n$ for elliptical binaries, we have used
following formulae [82]:

$\displaystyle\sum_{n=0}^{\infty}\,n^{2}J_{n}^{2}(ne)=\cfrac{e^{2}}{4(1-e^{2})^{7/2}}\left(1+\cfrac{e^{2}}{4}\right),$
$\displaystyle\sum_{n=0}^{\infty}\,n^{2}[J_{n}^{\prime}(ne)]^{2}=\cfrac{e^{2}}{4(1-e^{2})^{5/2}}\left(1+\cfrac{3e^{2}}{4}\right),$
$\displaystyle\sum_{n=0}^{\infty}\,n^{3}J_{n}(ne)J_{n}^{\prime}(ne)=\cfrac{e}{4(1-e^{2})^{9/2}}\left(1+3e^{2}+\cfrac{3}{8}e^{4}\right),$
$\displaystyle\sum_{n=0}^{\infty}\,n^{4}J_{n}^{2}(ne)=\cfrac{e^{2}}{4(1-e^{2})^{13/2}}\left(1+\cfrac{37}{4}e^{2}+\cfrac{59}{8}e^{4}+\cfrac{27}{64}e^{6}\right),$
$\displaystyle\sum_{n=0}^{\infty}\,n^{4}[J_{n}^{\prime}(ne)]^{2}=\cfrac{1}{4(1-e^{2})^{11/2}}\left(1+\cfrac{39}{4}e^{2}+\cfrac{79}{8}e^{4}+\cfrac{45}{64}e^{6}\right).$
(115)

###### Acknowledgements.

The authors would like to thank Ashoke Sen and Biswajit Sahoo for helpful
discussions. They would also like to thank Diptarka Das for hosting S.M. at
the Indian Institute of Technology Kanpur, which was supported through the Max
Planck Partner Group grant MAXPLA/PHY/2018577. S.P. is supported by the MHRD,
Government of India, under the Prime Minister’s Research Fellows (PMRF)
Scheme, 2020. A.H. would like to thank the MHRD, Government of India for
research fellowship.

## References

  * [1] Y. B. Zel’dovich and A. G. Polnarev, ”Radiation of gravitational waves by a cluster of superdense stars”, Soviet Astr.18, 17 (1974); https://adsabs.harvard.edu/full/1974SvA….18…17Z
  * [2] L. P. Grishchuk and A. G. Polnarev, “Gravitational wave pulses with ’velocity coded memory.’,” Sov. Phys. JETP 69, 653-657 (1989)
  * [3] V. B. Braginskii and K. S. Thorne, “Gravitational-wave bursts with memory and experimental prospects,” Nature 327 123–125 (1987).
  * [4] M. Turner,”Gravitational radiation from point-masses in unbound orbits - Newtonian results” Ap J., 216, 610-619 (1977).
  * [5] M. Turner and C.M. Will, ”Post-Newtonian gravitational bremsstrahlung” Ap J., 220, 1107-1124 (1978).
  * [6] R. Epstein, “The Generation of Gravitational Radiation by Escaping Supernova Neutrinos,” Astrophys. J. 223, 1037-1045 (1978) doi:10.1086/156337
  * [7] M.S. Turner,”Gravitational radiation from supernova neutrino bursts”, Nature 274 565 ( 1978).
  * [8] K. Kotake, K. Sato and K. Takahashi, “Explosion mechanism, neutrino burst, and gravitational wave in core-collapse supernovae,” Rept. Prog. Phys. 69, 971-1144 (2006) doi:10.1088/0034-4885/69/4/R03 [arXiv:astro-ph/0509456 [astro-ph]].
  * [9] H. Andresen, B. Müller, E. Müller and H. T. Janka, “Gravitational Wave Signals from 3D Neutrino Hydrodynamics Simulations of Core-Collapse Supernovae,” Mon. Not. Roy. Astron. Soc. 468, no.2, 2032-2051 (2017) doi:10.1093/mnras/stx618 [arXiv:1607.05199 [astro-ph.HE]].
  * [10] V. Morozova, D. Radice, A. Burrows and D. Vartanyan, “The gravitational wave signal from core-collapse supernovae,” Astrophys. J. 861, no.1, 10 (2018) doi:10.3847/1538-4357/aac5f1 [arXiv:1801.01914 [astro-ph.HE]].
  * [11] D. Vartanyan and A. Burrows, “Gravitational Waves from Neutrino Emission Asymmetries in Core-collapse Supernovae,” Astrophys. J. 901, no.2, 108 (2020) doi:10.3847/1538-4357/abafac [arXiv:2007.07261 [astro-ph.HE]].
  * [12] M. Mukhopadhyay, C. Cardona and C. Lunardini, “The neutrino gravitational memory from a core collapse supernova: phenomenology and physics potential,” JCAP 07, 055 (2021) doi:10.1088/1475-7516/2021/07/055 [arXiv:2105.05862 [astro-ph.HE]].
  * [13] N. Sago, K. Ioka, T. Nakamura and R. Yamazaki, “Gravitational wave memory of gamma-ray burst jets,” Phys. Rev. D 70, 104012 (2004) doi:10.1103/PhysRevD.70.104012 [arXiv:gr-qc/0405067 [gr-qc]].
  * [14] G. Urrutia, F. De Colle, C. Moreno and M. Zanolin, “Gravitational Waves from Long Gamma-Ray Bursts and Supernovae,” [arXiv:2208.00129 [astro-ph.HE]].
  * [15] T. Piran, “Jet Gravitational Waves,” [arXiv:2210.02740 [astro-ph.HE]].
  * [16] A. C. Jenkins and M. Sakellariadou, “Nonlinear gravitational-wave memory from cusps and kinks on cosmic strings,” Class. Quant. Grav. 38, no.16, 165004 (2021) [arXiv:2102.12487 [gr-qc]].
  * [17] D. Christodoulou, “Nonlinear nature of gravitation and gravitational wave experiments,” Phys. Rev. Lett. 67 , 1486–1489 (1991).
  * [18] A. G. Wiseman and C. M. Will, “Christodoulou’s nonlinear gravitational wave memory: Evaluation in the quadrupole approximation,” Phys. Rev. D44 no. 10, R2945–R2949 (1991) .
  * [19] L. Blanchet and T. Damour, “Hereditary effects in gravitational radiation,” Phys. Rev. D 46, 4304-4319 (1992) doi:10.1103/PhysRevD.46.4304
  * [20] K. S. Thorne, “Gravitational-wave bursts with memory: The Christodoulou effect,” Phys. Rev. D 45, no.2, 520-524 (1992) doi:10.1103/PhysRevD.45.520
  * [21] S. Weinberg, ”Photons and Gravitons in Perturbation Theory: Derivation of Maxwell’s and Einstein’s Equations”, Phys. Rev. 138, B988 (1965).
  * [22] S. Weinberg, “Infrared photons and gravitons,” Phys. Rev. 140, B516 (1965)
  * [23] S. Weinberg,“The Quantum theory of fields. Vol. 1: Foundations,” Cambridge, UK: Univ. Pr. (1995).
  * [24] F. Cachazo and A. Strominger, “Evidence for a New Soft Graviton Theorem,” [arXiv:1404.4091 [hep-th]].
  * [25] A. Laddha and A. Sen, “Logarithmic Terms in the Soft Expansion in Four Dimensions,” JHEP 10, 056 (2018) doi:10.1007/JHEP10(2018)056 [arXiv:1804.09193 [hep-th]].
  * [26] B. Sahoo and A. Sen, “Classical and Quantum Results on Logarithmic Terms in the Soft Theorem in Four Dimensions,” JHEP 02, 086 (2019) [arXiv:1808.03288 [hep-th]].
  * [27] A. Laddha and A. Sen, “Observational Signature of the Logarithmic Terms in the Soft Graviton Theorem,” Phys. Rev. D 100, no.2, 024009 (2019) doi:10.1103/PhysRevD.100.024009 [arXiv:1806.01872 [hep-th]].
  * [28] A. P. Saha, B. Sahoo and A. Sen, JHEP 06 (2020), 153 doi:10.1007/JHEP06(2020)153 [arXiv:1912.06413 [hep-th]].
  * [29] D. Kennefick, ”Prospects for detecting the Christodoulou memory of gravitational waves from a coalescing compact binary and using it to measure neutron-star radii”, Phys. Rev. D 50, 3587 (1994)
  * [30] M. Favata, “Post-Newtonian corrections to the gravitational-wave memory for quasi-circular, inspiralling compact binaries,” Phys. Rev. D 80, 024002 (2009) [arXiv:0812.0069 [gr-qc]].
  * [31] M. Favata, Class. Quant. Grav. 27 (2010), 084036 doi:10.1088/0264-9381/27/8/084036 [arXiv:1003.3486 [gr-qc]].
  * [32] M. Favata, “The Gravitational-wave memory from eccentric binaries,” Phys. Rev. D 84, 124013 (2011) [arXiv:1108.3121 [gr-qc]].
  * [33] M. Ebersold, Y. Boetzel, G. Faye, C. K. Mishra, B. R. Iyer and P. Jetzer, “Gravitational-wave amplitudes for compact binaries in eccentric orbits at the third post-Newtonian order: Memory contributions,” Phys. Rev. D 100, no.8, 084043 (2019) doi:10.1103/PhysRevD.100.084043 [arXiv:1906.06263 [gr-qc]].
  * [34] J. Samsing, “Eccentric Black Hole Mergers Forming in Globular Clusters,” Phys. Rev. D 97, no.10, 103014 (2018) doi:10.1103/PhysRevD.97.103014 [arXiv:1711.07452 [astro-ph.HE]].
  * [35] A. Tucker and C. M. Will, “Residual eccentricity of inspiralling orbits at the gravitational-wave detection threshold: Accurate estimates using post-Newtonian theory,” Phys. Rev. D 104, no.10, 104023 (2021) doi:10.1103/PhysRevD.104.104023 [arXiv:2108.12210 [gr-qc]].
  * [36] G. D. Quinlan and S. L. Shapiro, ”The collapse of dense star clusters to supermassive black holes - Binaries and gravitational radiation”, Astrophys. J. 321, 199 (1987).
  * [37] R. M. O’Leary, B. Kocsis and A. Loeb, “Gravitational waves from scattering of stellar-mass black holes in galactic nuclei,” Mon. Not. Roy. Astron. Soc. 395, no.4, 2127-2146 (2009) doi:10.1111/j.1365-2966.2009.14653.x [arXiv:0807.2638 [astro-ph]].
  * [38] D. C. Heggie, ”Binary evolution in stellar dynamics”, Mon. Not. R. Astron. Soc. 173, 729 (1975).
  * [39] Y. Kozai, ”Secular perturbations of asteroids with high inclination and eccentricity”, Astron. J. 67, 591 (1962).
  * [40] M. L. Lidov, ”The evolution of orbits of artificial satellites of planets under the action of gravitational perturbations of external bodies”,Planetary Sp. Sci.9, 719 (1962)
  * [41] L. Wen, “On the eccentricity distribution of coalescing black hole binaries driven by the Kozai mechanism in globular clusters,” Astrophys. J. 598, 419-430 (2003) doi:10.1086/378794 [arXiv:astro-ph/0211492 [astro-ph]].
  * [42] S. Naoz, B. Kocsis, A. Loeb and N. Yunes, “Resonant Post-Newtonian Eccentricity Excitation in Hierarchical Three-body Systems,” Astrophys. J. 773, 187 (2013) doi:10.1088/0004-637X/773/2/187 [arXiv:1206.4316 [astro-ph.SR]].
  * [43] M. Favata, C. Kim, K. G. Arun, J. Kim and H. W. Lee, “Constraining the orbital eccentricity of inspiralling compact binary systems with Advanced LIGO,” Phys. Rev. D 105, no.2, 023003 (2022) doi:10.1103/PhysRevD.105.023003 [arXiv:2108.05861 [gr-qc]].
  * [44] J. Aasi et al. [LIGO Scientific], “Advanced LIGO,” Class. Quant. Grav. 32, 074001 (2015) doi:10.1088/0264-9381/32/7/074001 [arXiv:1411.4547 [gr-qc]].
  * [45] F. Acernese et al. [VIRGO], “Advanced Virgo: a second-generation interferometric gravitational wave detector,” Class. Quant. Grav. 32, no.2, 024001 (2015) doi:10.1088/0264-9381/32/2/024001 [arXiv:1408.3978 [gr-qc]].
  * [46] H. Yu, D. Martynov, S. Vitale, M. Evans, D. Shoemaker, B. Barr, G. Hammond, S. Hild, J. Hough and S. Huttner, et al. “Prospects for detecting gravitational waves at 5 Hz with ground-based detectors,” Phys. Rev. Lett. 120, no.14, 141102 (2018) doi:10.1103/PhysRevLett.120.141102 [arXiv:1712.05417 [astro-ph.IM]].
  * [47] H. Yang and D. Martynov, “Testing Gravitational Memory Generation with Compact Binary Mergers,” Phys. Rev. Lett. 121, no.7, 071102 (2018) doi:10.1103/PhysRevLett.121.071102 [arXiv:1803.02429 [gr-qc]].
  * [48] C. Talbot, E. Thrane, P. D. Lasky and F. Lin, “Gravitational-wave memory: waveforms and phenomenology,” Phys. Rev. D 98, no.6, 064031 (2018) [arXiv:1807.00990 [astro-ph.HE]].
  * [49] A. D. Johnson, S. J. Kapadia, A. Osborne, A. Hixon and D. Kennefick, “Prospects of detecting the nonlinear gravitational wave memory,” Phys. Rev. D 99, no.4, 044045 (2019) doi:10.1103/PhysRevD.99.044045 [arXiv:1810.09563 [gr-qc]].
  * [50] K. G. Arun et al. [LISA], “New horizons for fundamental physics with LISA,” Living Rev. Rel. 25, no.1, 4 (2022) doi:10.1007/s41114-022-00036-9 [arXiv:2205.01597 [gr-qc]].
  * [51] B. P. Abbott et al. [LIGO Scientific], “Exploring the Sensitivity of Next Generation Gravitational Wave Detectors,” Class. Quant. Grav. 34, no.4, 044001 (2017) doi:10.1088/1361-6382/aa51f4 [arXiv:1607.08697 [astro-ph.IM]].
  * [52] S. Hild et al., ”Sensitivity studies for third-generation gravitational wave observatories”, Class. Quant. Grav. 28, 094013 (2011).
  * [53] P. D. Lasky, E. Thrane, Y. Levin, J. Blackman and Y. Chen, “Detecting gravitational-wave memory with LIGO: implications of GW150914,” Phys. Rev. Lett. 117, no.6, 061102 (2016) doi:10.1103/PhysRevLett.117.061102 [arXiv:1605.01415 [astro-ph.HE]].
  * [54] M. Maiorano, F. De Paolis and A. A. Nucita, “Principles of Gravitational-Wave Detection with Pulsar Timing Arrays,” Symmetry 13, 2418 (2021) doi:10.3390/sym13122418 [arXiv:2112.08064 [astro-ph.GA]].
  * [55] K. Aggarwal et al. [NANOGrav], “The NANOGrav 11 yr Data Set: Limits on Gravitational Wave Memory,” Astrophys. J. 889, 38 (2020) doi:10.3847/1538-4357/ab6083 [arXiv:1911.08488 [astro-ph.HE]].
  * [56] S. Burke-Spolaor, S. R. Taylor, M. Charisi, T. Dolch, J. S. Hazboun, A. M. Holgado, L. Z. Kelley, T. J. W. Lazio, D. R. Madison and N. McMann, et al. “The Astrophysics of Nanohertz Gravitational Waves,” Astron. Astrophys. Rev. 27, no.1, 5 (2019) doi:10.1007/s00159-019-0115-7 [arXiv:1811.08826 [astro-ph.HE]].
  * [57] P. A. Rosado, A. Sesana and J. Gair, “Expected properties of the first gravitational wave signal detected with pulsar timing arrays,” Mon. Not. Roy. Astron. Soc. 451, no.3, 2417-2433 (2015) doi:10.1093/mnras/stv1098 [arXiv:1503.04803 [astro-ph.HE]].
  * [58] M. S. Pshirkov, D. Baskaran and K. A. Postnov, “Observing gravitational wave bursts in pulsar timing measurements,” Mon. Not. Roy. Astron. Soc. 402, 417 (2010) doi:10.1111/j.1365-2966.2009.15887.x [arXiv:0909.0742 [astro-ph.CO]].
  * [59] S. Capozziello, M. De Laurentis, F. De Paolis, G. Ingrosso and A. Nucita, “Gravitational waves from hyperbolic encounters,” Mod. Phys. Lett. A 23, 99-107 (2008) doi:10.1142/S0217732308026236 [arXiv:0801.0122 [gr-qc]].
  * [60] L. De Vittori, P. Jetzer and A. Klein, “Gravitational wave energy spectrum of hyperbolic encounters,” Phys. Rev. D 86, 044017 (2012) [arXiv:1207.5359 [gr-qc]].
  * [61] L. De Vittori, A. Gopakumar, A. Gupta and P. Jetzer, “Gravitational waves from spinning compact binaries in hyperbolic orbits,” Phys. Rev. D 90, no.12, 124066 (2014) doi:10.1103/PhysRevD.90.124066 [arXiv:1410.6311 [gr-qc]].
  * [62] J. García-Bellido and S. Nesseris, “Gravitational wave energy emission and detection rates of Primordial Black Hole hyperbolic encounters,” Phys. Dark Univ. 21, 61-69 (2018) doi:10.1016/j.dark.2018.06.001 [arXiv:1711.09702 [astro-ph.HE]].
  * [63] J. Garcia-Bellido and S. Nesseris, “Gravitational wave bursts from Primordial Black Hole hyperbolic encounters,” Phys. Dark Univ. 18, 123-126 (2017) doi:10.1016/j.dark.2017.10.002 [arXiv:1706.02111 [astro-ph.CO]].
  * [64] M. Gröbner, P. Jetzer, M. Haney, S. Tiwari and W. Ishibashi, “A note on the gravitational wave energy spectrum of parabolic and hyperbolic encounters,” Class. Quant. Grav. 37, no.6, 067002 (2020) doi:10.1088/1361-6382/ab6be2 [arXiv:2001.05187 [gr-qc]].
  * [65] G. Cho, A. Gopakumar, M. Haney and H. M. Lee, “Gravitational waves from compact binaries in post-Newtonian accurate hyperbolic orbits,” Phys. Rev. D 98, no.2, 024039 (2018) doi:10.1103/PhysRevD.98.024039 [arXiv:1807.02380 [gr-qc]].
  * [66] S. Mukherjee, S. Mitra and S. Chatterjee, “Gravitational wave observatories may be able to detect hyperbolic encounters of black holes,” Mon. Not. Roy. Astron. Soc. 508, no.4, 5064-5073 (2021) doi:10.1093/mnras/stab2721 [arXiv:2010.00916 [gr-qc]].
  * [67] E. Codazzo, M. Di Giovanni, J. Harms, M. Dall’Amico and M. Mapelli, “Study on the detectability of gravitational radiation from single-binary encounters between black holes in nuclear star cluster: the case of hyperbolic flybys,” [arXiv:2207.01326 [astro-ph.HE]].
  * [68] J. García-Bellido, S. Jaraba and S. Kuroyanagi, “The stochastic gravitational wave background from close hyperbolic encounters of primordial black holes in dense clusters,” Phys. Dark Univ. 36, 101009 (2022) doi:10.1016/j.dark.2022.101009 [arXiv:2109.11376 [gr-qc]].
  * [69] D. Bini and A. Geralico, “Frequency domain analysis of the gravitational wave energy loss in hyperbolic encounters,” Phys. Rev. D 104, no.10, 104019 (2021) doi:10.1103/PhysRevD.104.104019 [arXiv:2108.02472 [gr-qc]].
  * [70] S. Khan, S. Husa, M. Hannam, F. Ohme, M. Pürrer, X. Jiménez Forteza and A. Bohé, “Frequency-domain gravitational waves from nonprecessing black-hole binaries. II. A phenomenological model for the advanced detector era,” Phys. Rev. D 93, no.4, 044007 (2016) doi:10.1103/PhysRevD.93.044007 [arXiv:1508.07253 [gr-qc]].
  * [71] C. García-Quirós, M. Colleoni, S. Husa, H. Estellés, G. Pratten, A. Ramos-Buades, M. Mateu-Lucena and R. Jaume, “Multimode frequency-domain model for the gravitational wave signal from nonprecessing black-hole binaries,” Phys. Rev. D 102, no.6, 064002 (2020) [arXiv:2001.10914 [gr-qc]].
  * [72] S. Mohanty and P. Kumar Panda, “Particle physics bounds from the Hulse-Taylor binary,” Phys. Rev. D 53, 5723-5726 (1996) doi:10.1103/PhysRevD.53.5723 [arXiv:hep-ph/9403205 [hep-ph]].
  * [73] T. Kumar Poddar, S. Mohanty and S. Jana, “Constraints on ultralight axions from compact binary systems,” Phys. Rev. D 101, no.8, 083007 (2020) [arXiv:1906.00666 [hep-ph]].
  * [74] T. Kumar Poddar, S. Mohanty and S. Jana, “Vector gauge boson radiation from compact binary systems in a gauged $L_{\mu}-L_{\tau}$ scenario,” Phys. Rev. D 100, no.12, 123023 (2019) doi:10.1103/PhysRevD.100.123023 [arXiv:1908.09732 [hep-ph]].
  * [75] T. K. Poddar, S. Mohanty and S. Jana, “Gravitational radiation from binary systems in massive graviton theories,” JCAP 03, 019 (2022) doi:10.1088/1475-7516/2022/03/019 [arXiv:2105.13335 [gr-qc]].
  * [76] A. Addazi, M. Bianchi and G. Veneziano, “Glimpses of black hole formation/evaporation in highly inelastic, ultra-planckian string collisions,” JHEP 02, 111 (2017) [arXiv:1611.03643 [hep-th]].
  * [77] A. Addazi, M. Bianchi and G. Veneziano, “Soft gravitational radiation from ultra-relativistic collisions at sub- and sub-sub-leading order,” JHEP 05, 050 (2019) [arXiv:1901.10986 [hep-th]].
  * [78] A. Addazi and K. A. Zeng, “Soft gravitational radiation from multi-body collisions,” JHEP 11, 193 (2021) [arXiv:2110.01194 [hep-th]].
  * [79] A. Tolish and R. M. Wald, Phys. Rev. D 89 (2014) no.6, 064008 doi:10.1103/PhysRevD.89.064008 [arXiv:1401.5831 [gr-qc]].
  * [80] A. Tolish and R. M. Wald, “Cosmological memory effect,” Phys. Rev. D 94, no.4, 044009 (2016) [arXiv:1606.04894 [gr-qc]].
  * [81] A. Kehagias and A. Riotto, “BMS in Cosmology,” JCAP 05, 059 (2016) [arXiv:1602.02653 [hep-th]].
  * [82] P. C. Peters and J. Mathews, “Gravitational radiation from point masses in a Keplerian orbit,” Phys. Rev. 131, 435-439 (1963)
  * [83] P. C. Peters, “Gravitational Radiation and the Motion of Two Point Masses,” Phys. Rev. 136, B1224-B1232 (1964)
  * [84] L. Blanchet, T. Damour, G. Esposito-Farese and B. R. Iyer, “Gravitational radiation from inspiralling compact binaries completed at the third post-Newtonian order,” Phys. Rev. Lett. 93, 091101 (2004) doi:10.1103/PhysRevLett.93.091101 [arXiv:gr-qc/0406012 [gr-qc]].
  * [85] K. G. Arun, L. Blanchet, B. R. Iyer and M. S. S. Qusailah, “Inspiralling compact binaries in quasi-elliptical orbits: The Complete 3PN energy flux,” Phys. Rev. D 77, 064035 (2008) doi:10.1103/PhysRevD.77.064035 [arXiv:0711.0302 [gr-qc]].
  * [86] K. G. Arun, L. Blanchet, B. R. Iyer and M. S. S. Qusailah, “Tail effects in the 3PN gravitational wave energy flux of compact binaries in quasi-elliptical orbits,” Phys. Rev. D 77, 064034 (2008) doi:10.1103/PhysRevD.77.064034 [arXiv:0711.0250 [gr-qc]].