#  Interaction of Electromagnetic field with a Gravitational wave in Minkowski
and de-Sitter space-time.

Akash Patel  Arundhati Dasgupta  Physics and Astronomy, University of
Lethbridge, 4401 University Drive, Lethbridge, Canada T1K 3M4.
<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

It is important to re-examine some of the interactions of gravitational waves
with matter due to the recent discovery of the waves in LIGO. In a previous
paper with Morales, interaction of gravitational waves was studied with scalar
and neutrino fields in Minkowski space-time. In this paper, we find that
Electromagnetic waves have a similar interaction with gravitational wave in a
flat background, and we comment on the implications of this for a
gravitational wave detector. Primordial gravitational waves were generated in
the early universe and thus it is important to study interaction of the
gravitational waves in cosmological backgrounds. In this paper we discuss the
interaction of electromagnetic waves and gravitational waves in a de-Sitter
background. We find that the Electromagnetic wave is perturbed and modulated.
This result is new and will be useful in studying inhomogeneities in Cosmic
Microwave Background.

##  I Introduction

In the recent years gravitational wave detection with LIGO ligo  has opened a
new window for the study of the universe. Gravitational waves from distant
events are arriving on earth and are being studied and analyzed using the LIGO
detector. The Space Laser Interferometer will also be functional in the near
future. In this circumstance, it is important to study the interaction of
gravitational waves with matter, and predict new observations. Further,
primordial gravitational waves have not been detected yet and new interactions
might shed some light on the stochastic relics from early universe cosmology.
For recent observations on the background gravitational waves see the NanoGrav
experiment nano . In this paper we analyze and confirm a perturbation for
electromagnetic (EM) waves induced by gravitational waves, previously observed
for scalars and neutrinos in morales  and in tsagas ; chou . We discuss
resonant interactions, and if this new ‘frequency’ mode as predicted can be
detected in current detectors. Although our results are similar to tsagas ;
chou , the analysis of resonance is different. We find no interactions when
the gravitational waves and the EM waves are propagating in the same
direction. When the waves are anti-parallel or at an angle, solutions are
obtained for the perturbation, and we discuss the resultant EM wave. We find
that the initial plane wave of the EM radiation changes direction and flows at
an angle to the initial wave. The shape of the wavefront is also perturbed.
However, the perturbation is very weak compared to the initial wave. The
perturbation shows the ‘phase shift’ which can be identified with that causing
the interference pattern observed in LIGO. In addition we predict a new mode,
which can be observed if the frequency of the EM wave is of the order of the
frequency of the gravitational wave. In the current detectors, the frequency
of the EM laser is about $10^{14}$ Hz and the frequency of the gravitational
wave $\sim 1-100$ Hz. Therefore, the EM frequency dominates in the
observations. The work on designing an experiment to detect the mode is in
progress with Maher maher .

In this paper we also discuss the interaction of gravitational waves with EM
waves in a conformally flat cosmological background. We solve for the
Maxwell’s equation in the background of conformal de-Sitter space-time. We
take this particular cosmology as this is a candidate space-time for early
universe inflationary scenario, and also for maximally symmetric space-times
with a positive cosmological constant. We find that the EM perturbations grow
quadratically with conformal time, and change in direction. We comment on the
plausible detection of this perturbation as a inhomogeneity in the current
cosmic microwave background.

In the next section we discuss the interaction of the EM waves with
gravitational waves in a flat space-time. We find the perturbations and
analyze the Poynting vector. We discuss the interactions when the waves are
perpendicular to each other, and when they are in the same or opposite
direction. In the third section we discuss the interaction of the EM waves
with a gravitational wave in de-Sitter space background. We show that the
resultant perturbations grow quadratically in conformal time whereas the
zeroeth order gauge fields grow linearly in time. The fourth section is a
conclusion. In this paper we work in the units of Newton’s constant $G$ and
speed of light $c$, $G=c=1$.

##  II Interaction of Electromagnetic Waves with Gravitational Waves in
Minkowski space-time

In this section we study the interaction of a gravitational wave travelling in
the z-direction, and an electromagnetic wave in the x-direction. Due to the
perpendicular nature of the direction of propagation, in linear analysis these
waves should propagate independently of each other. However, due to the nature
of the interactions of the electromagnetic wave and the gravitational wave, we
find using a perturbation analysis that the electromagnetic wave gets non-
trivially changed. We refrain from studying the back reaction of the
electromagnetic wave on the gravitational wave, but that can also be found. At
this time, we simply find the perturbation of the EM wave and predict how this
perturbation might be detected at the LIGO detector. Initially we assume that
there is a gravitational wave $h_{\mu\nu}$ with polarizations
$h_{+},h_{\times}$ propagating at a frequency $\omega_{g}$ in the z-direction.
The amplitudes are taken as $A_{+}$ and $A_{\times}$. The gravitational wave
metric is taken as liv

$\left(\begin{array}[]{cccc}-1&0&0&0\\\
0&1+A_{+}\cos(\omega_{g}(z-t))&A_{\times}\cos(\omega_{g}(z-t))&0\\\
0&A_{\times}\cos(\omega_{g}(z-t))&1-A_{+}\cos(\omega_{g}(z-t))&0\\\
0&0&0&1\end{array}\right)$ (1)

The electromagnetic wave is taken to be in the x -direction such that the
magnetic field is

$\vec{B}=B_{0y}\ Re\\{e^{i\omega_{e}(x-t)}\ \\}\hat{y}$ (2)

where $B_{0y}$ is its amplitude, and $\omega_{e}$ is its frequency. In the
Lorenz Gauge the gauge field is found to be

$\vec{A}=-\frac{1}{\omega}_{e}B_{0y}\sin(\omega_{e}x-\omega_{e}t)\ \hat{z}.$
(3)

The field strength is ($\mu,\nu=0,1,2,3$)

$F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu}.$ (4)

The non-zero components of the EM tensor is $F_{03}$ and (we use the notation
t,x,y,z and 0,1,2,3 to represent the same coordinate directions
interchangeably according to the convenience of the equations.)

$F_{13}=-B_{0y}\cos(\omega_{e}(x-t)).$ (5)

We solve the Maxwell’s equation in the background of the gravitational wave
which is as following:

$\frac{1}{\sqrt{-g}}\partial_{\mu}\left(\sqrt{-g}g^{\mu\alpha}g^{\nu\beta}F_{\alpha\beta}\right)=0.$
(6)

Using $g^{\mu\nu}=\eta^{\mu\nu}-h^{\mu\nu}$ , the inverted metric, we get from
the above

$\eta^{\mu\alpha}\eta^{\nu\beta}\partial_{\mu}F_{\alpha\beta}-\partial_{\mu}(h^{\mu\alpha})\eta^{\nu\beta}F_{\alpha\beta}-\partial_{\mu}(h^{\nu\alpha})\eta^{\mu\beta}F_{\alpha\beta}-h^{\mu\alpha}\eta^{\nu\beta}\partial_{\mu}F_{\alpha\beta}-\eta^{\mu\alpha}h^{\nu\beta}\partial_{\mu}F_{\alpha\beta}=0.$
(7)

We then assume a perturbation of the initial gauge field
$A_{\mu}=\bar{A}_{\mu}+\tilde{A}_{\mu}$ (where $\bar{A}$ represents the wave
without the gravitational wave and $\tilde{A}$ is the perturbation) and keep
terms to linear order in $\tilde{A}$ and $h$ in Equation(7). Using the Lorenz
gauge conditions $\partial_{\mu}A^{\mu}=0$ and
$\partial^{\mu}h_{\mu\alpha}=0$; the perturbation equations are

$\displaystyle\Box\tilde{A}_{0}$ $\displaystyle=$ $\displaystyle 0$ (8)
$\displaystyle\Box\tilde{A}_{1}$ $\displaystyle=$
$\displaystyle(\partial_{3}h^{11})F_{31}=-A_{+}B_{0y}\
\omega_{g}\sin(\omega_{g}z-\omega_{g}t)\cos(\omega_{e}x-\omega_{e}t)$ (9)
$\displaystyle\Box\tilde{A}_{2}$ $\displaystyle=$
$\displaystyle\partial_{3}(h^{21}F_{13})=A_{\times}B_{0y}\
\omega_{g}\sin(\omega_{g}z-\omega_{g}t+\delta)\cos(\omega_{e}x-\omega_{e}t)$
(10) $\displaystyle\Box\tilde{A}_{3}$ $\displaystyle=$ $\displaystyle
h^{11}\partial_{1}F_{13}=A_{+}B_{0y}\
\omega_{e}\sin(\omega_{e}x-\omega_{e}t)\cos(\omega_{g}z-\omega_{g}t).$ (11)

($\square A_{\mu}$ is the relativistic d’Alembertian acting on the gauge
field.)
Note that the inhomogeneous differential equations in the linearized
approximation is natural, as we are solving for Maxwell’s equation in curved
geometry. The curvature of the geometry provides the ‘driving force’ for the
perturbations. The inhomogeneous terms can be interpreted as a ‘current
source’ proportional to the $B_{0y}A_{+}$, product of the EM wave and the
gravitational wave. This also signifies that if the initial EM wave is zero,
there is no perturbation at all. The linearized approximation has been
discussed in monta , in analogy of a circuit, however, we differ from their
analysis. In monta  it has been claimed that there has to be an ‘external
source of emf’ to justify the inhomogeneous terms in the above equations.
However, circuits are known to have the capacity to generate their own emf,
e.g. ‘self inductance’ . In a similar analogy, we can see that the there can
be self generation of emf or ‘back emf’ in the circuit due to the curvature of
the background. Therefore linearized approximations which give source terms
for the perturbations can be used to solve for interactions, and some of the
answers obtained in monta  using Eikonal approximation is similar
qualitatively to the results of this paper. It remains that experimental
detection of the new results will confirm correctness of the approximations.

The perturbation solution is obtained using the methods of morales  and as
outlined in the Appendix1. The x-component is found to be

$\displaystyle\tilde{A}_{x}$ $\displaystyle=$
$\displaystyle-\frac{B_{0y}A_{+}}{4\omega_{e}\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}}\left[(\omega_{e}+\omega_{g})\cos(\omega_{e}x+\omega_{g}z)\sin(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})-\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}\
\sin(\omega_{e}x+\omega_{g}z)\cos(t\sqrt{\omega_{g}^{2}+\omega_{e}^{2}})\right.$
(12) $\displaystyle+$
$\displaystyle(\omega_{e}-\omega_{g})\cos(\omega_{e}x-\omega_{g}z)\sin(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})-\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}\
\sin(\omega_{e}x-\omega_{g}z)\cos(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})$
$\displaystyle+$ $\displaystyle\left.\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}\
\sin(\omega_{e}x+\omega_{g}z-t(\omega_{g}+\omega_{e}))+\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}\
\sin(\omega_{e}x-\omega_{g}z-t(\omega_{e}-\omega_{g}))\right].$

The solution for $\tilde{A}_{y}$ would be the same as this, except for
$-A_{+}\rightarrow A_{\times}$ and $\omega_{g}z\rightarrow\omega_{g}z+\delta$
. The solution for $\tilde{A}_{z}$ will also be the similar; which we can
obtain by using phase shifts of $\omega_{e}x\rightarrow\omega_{e}x+\pi/2$ and
$\omega_{g}z\rightarrow\omega_{g}z+\pi/2$ and in the coefficient
$-B_{0y}A_{+}/(4\omega_{e}\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})\rightarrow
B_{0y}A_{+}/(4\omega_{g}\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})$. Note that
these solutions are obtained with the boundary condition that the
perturbations are zero at time $t=0$.

It is interesting to see that there are terms of the form
$\sin(\omega_{e}x\pm\omega_{g}z-t(\omega_{g}\pm\omega_{e}))$ which can be
interpreted as causation of ‘phase shifts’, and would lead to interference
patterns, when incident on a detector with another EM wave. But there are also
terms of the form
$\cos(\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}\leavevmode\nobreak\ t)$ which see a
new frequency ‘mode’ in the solution. These terms have been observed
previously in tsagas ; chou . Question is why haven’t they been observed in
the LIGO? The simple answer is that the LASER frequency used in the LIGO or
$\omega_{e}\sim 10^{14}$ Hz, and the gravitational wave frequency is of the
order of $1-100$ Hz. Therefore the frequency of the new mode
$\omega\sim\omega_{e}$ and therefore one has to modify the details of the
experiment to be able to see the mode. This is under investigation in maher .
Note that the above perturbations amplitude is inversely proportional to
$\omega_{e}$ and thus bringing down the frequency of the EM wave in the
interferometer will increase the strength of the signal. The z-component of
the gauge potential, which represents a EM wave perturbation over the original
has an amplitude which is proportional to $1/\omega_{g}$ and therefore gives
rise to a bigger amplitude wave than the other components. Methods to detect
the perpendicular component of the Poynting vector is under investigation in
maher . The perturbed $\vec{E}$ and the $\vec{B}$ components of the fields
from the above gauge potentials can be derived by computing

$\vec{\tilde{E}}=-\frac{\partial\vec{\tilde{A}}}{\partial t}$ (13)

and we can see that the perturbed field has components in all three
dimensions. E.g. the x-component is (the y and z components can similarly be
found)

$\displaystyle\tilde{E}_{x}$ $\displaystyle=$
$\displaystyle\frac{B_{0y}A_{+}}{4\omega_{e}}\left[(\omega_{e}+\omega_{g})\cos(\omega_{e}x+\omega_{g}z)\cos(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})+\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}\
\sin(\omega_{e}x+\omega_{g}z)\sin(t\sqrt{\omega_{g}^{2}+\omega_{e}^{2}})\right.$
(14) $\displaystyle+$
$\displaystyle(\omega_{e}-\omega_{g})\cos(\omega_{e}x-\omega_{g}z)\cos(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})+\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}\
\sin(\omega_{e}x-\omega_{g}z)\sin(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})$
$\displaystyle-$ $\displaystyle\left.(\omega_{g}+\omega_{e})\
\sin(\omega_{e}x+\omega_{g}z-t(\omega_{g}+\omega_{e}))-(\omega_{e}-\omega_{g})\
\sin(\omega_{e}x-\omega_{g}z-t(\omega_{e}-\omega_{g}))\right].$

The Magnetic Field is:

$\vec{B}=\vec{\nabla}\times\vec{A}.$ (15)

The newly generated z-component of the $\vec{B}$ field is hence given by

$\tilde{B}_{z}=\frac{\partial\tilde{A}_{y}}{\partial x}$ (16)
$\displaystyle\tilde{B}_{z}$ $\displaystyle=$
$\displaystyle\frac{B_{0y}A_{\times}}{4\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}}\left[-(\omega_{e}+\omega_{g})\sin(\omega_{e}x+\omega_{g}z+\delta)\sin(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})\right.$
(17) $\displaystyle-$ $\displaystyle\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}\
\cos(\omega_{e}x+\omega_{g}z+\delta)\cos(t\sqrt{\omega_{g}^{2}+\omega_{e}^{2}})$
$\displaystyle-$
$\displaystyle(\omega_{e}-\omega_{g})\sin(\omega_{e}x-\omega_{g}z+\delta)\sin(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})-\sqrt{\omega_{g}^{2}+\omega_{e}^{2}}\
\cos(\omega_{e}x-\omega_{g}z+\delta)\cos(t\sqrt{\omega_{e}^{2}+\omega_{g}^{2}})$
$\displaystyle+$ $\displaystyle\left.\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}\
\cos(\omega_{e}x+\omega_{g}z+\delta-t(\omega_{g}+\omega_{e}))+\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}\
\cos(\omega_{e}x-\omega_{g}z+\delta-t(\omega_{e}-\omega_{g}))\right].$

The x,y components can be similarly derived.

To find the propagation of the perturbed EM field, we calculate the Poynting
vector of the perturbation solution, as the $T^{0i}$ components of the Energy
Momentum tensor. The EM energy momentum tensor is given by

$T^{\mu\nu}=-\frac{1}{4\pi}\left(g^{\mu\alpha}F_{\alpha\lambda}F^{\lambda\nu}+\frac{1}{4}g^{\mu\nu}F_{\alpha\beta}F^{\alpha\beta}\right).$
(18)

This formula is taken from jkson  but differs from that by a minus sign as we
use a metric of signature (-,+,+,+) instead of (+,-,-,-) as in that book.
Setting specific indices, one gets the components as:

$\displaystyle T^{01}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left((1-h_{+})((1+h_{+})F_{02}F_{21}+F_{31}F_{03})\right)$
(19) $\displaystyle T^{02}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left((1-h_{+})(1+h_{+})F_{12}F_{01}+(1+h_{+})F_{32}F_{02}\right)$
(20) $\displaystyle T^{03}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\left((1-h_{+})F_{13}F_{01}+(1+h_{+})F_{23}F_{02}\right).$
(21)

At the linearized level, the energy momentum non-zero components are in the
original x-direction, and in the z-direction. If we compute them explicitly
then

$\displaystyle T^{0x}$ $\displaystyle=$
$\displaystyle\frac{1}{4\pi}\bigl{[}B_{0y}^{2}\cos^{2}(\omega_{e}x-\omega_{e}t)(1-A_{+}\cos(\omega_{g}z-\omega_{g}t))\bigr{.}$
(22) $\displaystyle+$
$\displaystyle\frac{B^{2}_{0y}A_{+}\cos(\omega_{e}x-\omega_{e}t)}{4\omega_{e}\omega_{g}}\left\\{\right.\left(\frac{\omega_{g}^{3}-2\omega_{e}^{3}-\omega_{e}^{2}\omega_{g}}{||\omega||}\right)\sin(\omega_{e}x+\omega_{g}z)\sin(||\omega||t)$
$\displaystyle-$
$\displaystyle\left(\frac{\omega_{e}^{2}\omega_{g}-2\omega_{e}^{3}-\omega_{g}^{3}}{||\omega||}\right)\sin(\omega_{e}x-\omega_{g}z)\sin(||\omega||t)+\left(\omega_{g}^{2}-2\omega_{e}^{2}-\omega_{g}\omega_{e}\right)\cos(\omega_{e}x+\omega_{g}z)\cos(t||\omega||)$
$\displaystyle-$
$\displaystyle\left(\omega_{g}^{2}-2\omega_{e}^{2}+\omega_{g}\omega_{e}\right)\cos(\omega_{e}x-\omega_{g}z)\cos(t||\omega||)+\left(\omega_{g}^{2}-2\omega_{e}^{2}+\omega_{g}\omega_{e}\right)\cos(\omega_{e}x-\omega_{g}z-t\
\tilde{\omega})$ $\displaystyle-$
$\displaystyle\left(\omega_{g}^{2}-2\omega_{e}^{2}-\omega_{e}\omega_{g}\right)\cos(\omega_{e}x+\omega_{z}z-t\
\bar{\omega})\left.\left.\right\\}\right]$

Where we have used the notation
$||\omega||=\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}$ and
$\bar{\omega}=\omega_{e}+\omega_{g}$ and
$\tilde{\omega}=\omega_{e}-\omega_{g}$. The other non-zero component is

$\displaystyle T^{0z}$ $\displaystyle=$
$\displaystyle-\frac{B_{0y}^{2}A_{+}}{16\pi\omega_{e}}\cos(\omega_{e}x-\omega_{e}t)\left[\bar{\omega}\cos(\omega_{e}x+\omega_{g}z)\cos(||\omega||t)+\tilde{\omega}\cos(\omega_{e}x-\omega_{g}z)\cos(||\omega||t)\right.$
(23) $\displaystyle+$
$\displaystyle||\omega||\left(\sin(\omega_{e}x+\omega_{g}z)+\sin(\omega_{e}x-\omega_{g}z)\right)\sin(||\omega||t)-\bar{\omega}\cos(\omega_{e}x+\omega_{g}z-t\bar{\omega})-\tilde{\omega}\cos(\omega_{e}x-\omega_{g}z-t\tilde{\omega})\left.\right].$

We make a field plot of the Poynting vector perturbations in Figure (1) to
show that there should be significant deviation from the initial direction if
one is able to adjust the wave frequencies accordingly, and measure the
perturbations. We again take the frequencies to be $\omega_{e}=4{\rm
Hz},\omega_{g}=3{\rm Hz}$. The plots are of various instants of the field flow
of the EM field momentum.

Figure 1: Plot of the Poynting vector in time

The plot of the time snapshots as shown above Figure (1) of the perturbation
Poynting vector shows that the field fluctuates in direction. To see if the
fluctuations average to zero, we take a time integral of the momentum over the
time period of the ‘frequency mode $\sqrt{\omega_{e}^{2}+\omega_{g}^{2}}$. The
time average Poynting vector $\bar{T}^{0i}=\int_{0}^{2\pi/||\omega||}T^{0i}\
dt$ plot in Figure (2) shows that there is an over all bending of the EM wave
at an angle which can be detected in a Maser detector maher .

Figure 2: Time Average Poynting vector Field

We further examine propagation of waves in the same direction, and find that
there is no interaction. We take a magnetic field of the form
$\vec{B}=B_{0y}\cos(\omega_{e}z-\omega_{e}t)\hat{y}$ and the corresponding
$A_{x}$ and $E_{x}$ fields, and the equation (7) shows that there is no source
for the perturbation fields. A study of the interaction of the waves in anti-
parallel directions shows that there is interaction. For this we take the
magnetic field to be of the form
$\vec{B}=B_{0y}\cos(\omega_{e}z+\omega_{e}t)\hat{y}$.

For the example of waves travelling in the opposite direction we have this
form of the perturbation:

$\displaystyle\tilde{A_{x}}$ $\displaystyle=$ $\displaystyle\frac{A_{+}B_{0y}\
\omega}{2}\left[\frac{1}{\bar{\omega}+\tilde{\omega}}\left(\frac{1}{\bar{\omega}}\sin(\bar{\omega}t+\bar{\omega}z)+\frac{1}{\tilde{\omega}}\sin(\tilde{\omega}t+\tilde{\omega}z)\right)\right.$
(24) $\displaystyle+$
$\displaystyle\frac{1}{\bar{\omega}-\tilde{\omega}}\left(\frac{1}{\bar{\omega}}\sin(-\bar{\omega}t+\bar{\omega}z)-\frac{1}{\tilde{\omega}}\sin(-\tilde{\omega}t+\tilde{\omega}z)\right)$
$\displaystyle-$
$\displaystyle\frac{1}{(\bar{\omega}+\tilde{\omega})}\left(\frac{1}{\bar{\omega}}\sin(\bar{\omega}z-\tilde{\omega}t)+\frac{1}{\tilde{\omega}}\sin(\tilde{\omega}z-\bar{\omega}t)\right)$
$\displaystyle+$
$\displaystyle\left.\frac{1}{\bar{\omega}-\tilde{\omega}}\left(\frac{1}{\tilde{\omega}}\sin(\tilde{\omega}z-\bar{\omega}t)-\frac{1}{\bar{\omega}}\sin(\bar{\omega}z-\tilde{\omega}t)\right)\right].$

where
$\bar{\omega}=\omega_{g}+\omega_{e},\tilde{\omega}=\omega_{g}-\omega_{e}$ and
the $\tilde{A}_{y},{\tilde{A}}_{z}$ component can also be similarly derived.
When $\omega_{g}=\omega_{e}=\omega$ the exact solution gives us

$\tilde{A}_{x}=A_{+}B_{0y}\frac{1}{8\omega}\left[4(\omega t-\sin(2\omega
t))+\sin(2\omega(z+t))+\sin(2\omega(z-t))-2\sin(2\omega z)\right]$ (25)

There is a linear term in the perturbed mode, but we can trust the solution as
far as the approximation $\tilde{A}_{x}\ll\bar{A}_{x}$ is true. Also as the
$\vec{E}$ and ${\vec{B}}$ fields do not see the linear mode, the ‘resonance’
which is characterized by runaway amplitude does not exist for the EM wave
interaction with the gravitational wave.

##  III Interaction in de-Sitter background

It is expected that gravitational waves were generated in the early universe.
The remnants of the EM radiation from that time are manifest in the cosmic
microwave background (CMB) which is thermalized. Primordial gravitational
waves are also there in the background, but too weak to be detected yet nano ,
indirect evidence of the waves are expected in the B-modes of the CMB.
However, the study of interaction of gravitational waves and matter in
cosmological background is yet to be explored in details. In this paper we
study the interaction of EM fields with gravitational waves in the background
of conformal de-Sitter background. For this particular study we take the
cosmological metric with a flat spatial slice, and study the system in a
conformal time coordinate frame. In this frame, the metric of space-time is
given by

$ds^{2}=a^{2}(\tau)\left(-d\tau^{2}+dx^{2}+dy^{2}+dz^{2}\right)$ (26)

where the scale factor is $a(\tau)$ and $\tau$ is conformal time. The
conformal time is related to co-moving time $t$ as $-(1/H)\exp(-Ht)=\tau$,
where $H$ is the Hubble constant. The scale factor is $a(\tau)=-1/H\tau$. Note
that when $t\rightarrow\infty$ ; $\tau=0$ therefore conformal time is negative
$-\infty<\tau<0$ as $-\infty<t<\infty$. However, for a careful analysis of the
cosmology; one must include the volume factor which is a function of the scale
factor, and the scale factor goes from $0$ to $\infty$ as expected from a big-
bang scenario. $a(\tau\rightarrow-\infty)\rightarrow 0$ beginning ; and
expanding $a(\tau\rightarrow 0^{-})\rightarrow\infty$ as expected from the
cosmological arrow of time . We begin with solving the EM waves in the
cosmological metric. This gives the solutions as given they are of the form
$\bar{A}_{\mu}(\tau,\vec{x})=\bar{A}_{\mu}(\tau)\
e^{i\vec{k_{e}}\cdot\vec{x}}$. Where $\vec{k_{e}}$ is the momentum wave vector
of the EM field and $k_{e}=\sqrt{k_{ex}^{2}+k_{ey}^{2}+k_{ez}^{2}}$. We use
Lorenz gauge and the solutions to the Maxwell’s equations in the de-Sitter
background are as follows:

$\displaystyle\bar{A}_{0}(\tau)$ $\displaystyle=$
$\displaystyle\left(\frac{2}{\pi k_{e}}\right)^{1/2}\tau\sin(k_{e}\tau)$
$\displaystyle\bar{A}_{1}(\tau)$ $\displaystyle=$ $\displaystyle
P_{1}\sin(k_{e}\tau)+Q_{1}\cos(k_{e}\tau)-ik_{e1}\left(\frac{2}{\pi
k_{e}^{3}}\right)^{1/2}\tau\cos(k_{e}\tau)$ $\displaystyle\bar{A}_{2}(\tau)$
$\displaystyle=$ $\displaystyle
P_{2}\sin(k_{e}\tau)+Q_{2}\cos(k_{e}\tau)-ik_{e2}\left(\frac{2}{\pi
k_{e}^{3}}\right)^{1/2}\tau\cos(k_{e}\tau)$ $\displaystyle\bar{A}_{3}(\tau)$
$\displaystyle=$ $\displaystyle
P_{3}\sin(k_{e}\tau)+Q_{3}\cos(k_{e}\tau)-ik_{e3}\left(\frac{2}{\pi
k_{e}^{3}}\right)^{1/2}\tau\cos(k_{e}\tau)$ (27)

$P_{i},Q_{i}$ are constants, fixed by the boundary conditions. In presence of
the gravitational wave the background is of the form:

$ds^{2}=a^{2}(\tau)\left(-d\tau^{2}+dx^{a}dx^{b}+h_{ab}dx^{a}dx^{b}\right).$
(28)

The gravitational waves are of the same polarizations as in flat space-time,
the $h_{+}$ and the $h_{\times}$. The exact form of the wave functions are
different. If we take a wave propagating in the z-direction, then the

$h_{+}(z,\tau)=A_{+}\left({\omega_{g}\tau}-i\right)e^{-i\omega_{g}\tau}e^{i\omega_{g}z}$
(29)

where $\omega_{g}$ is the frequency of the gravitational wave. For this paper,
we discuss the effect of the $h_{+}$ mode on the EM fields. For details of the
solution for the gravitational wave in de-Sitter background see allen1 ;
allen2 .

The perturbation equations for the gauge fields in the cosmological wave
background with a gravitational wave are (as the Maxwell’s equation are
conformally invariant, one can take the equations from the previous section):

$\eta^{\alpha\mu}\eta^{\beta\nu}\partial_{\beta}F_{\mu\nu}=\eta^{\alpha\mu}h^{\beta\nu}\partial_{\beta}F_{\mu\nu}+h^{\alpha\mu}\eta^{\beta\nu}\partial_{\beta}F_{\mu\nu}+\eta^{\beta\nu}F_{\mu\nu}\partial_{\beta}h^{\alpha\mu}$
(30)

In addition to the above one has to impose the Lorenz gauge condition for the
EM gauge field $\nabla^{\mu}A_{\mu}=0$. The equation for the 0th component of
the perturbation can be obtained by setting $\alpha=0$. This gives from
Equation (30)

$\eta^{00}\eta^{\beta\nu}\partial_{\beta}\tilde{F}_{0\nu}=\eta^{00}h^{\beta\nu}\partial_{\beta}{\bar{F}}_{0\nu}.$
(31)

(where $\tilde{F}$ is the perturbation and $\bar{F}$ is the background.)
Setting $\alpha=i=1,2,3$ one gets for the LHS of Equation (30):

$\displaystyle\eta^{ii}\eta^{\beta\nu}\partial_{\beta}F_{i\nu}$
$\displaystyle=$
$\displaystyle\eta^{ii}\eta^{00}\partial_{i}\partial_{0}A_{0}-\eta^{ii}\left(\eta^{00}\partial_{0}^{2}+\eta^{jj}\partial_{j}^{2}\right)A_{i}+\eta^{ii}\eta^{jj}\partial_{i}\partial_{j}A_{j}$
(32) $\displaystyle=$
$\displaystyle\eta^{ii}\eta^{00}\partial_{i}\partial_{0}A_{0}-\eta^{ii}(\square
A_{i})+\eta^{ii}\partial_{i}\left(2A_{0}{\cal
H}+\partial_{0}A_{0}-h_{+}(\partial_{2}A_{2}-\partial_{1}A_{1})+h_{\times}(\partial_{1}A_{2}+\partial_{2}A_{1})\right)$
(33)

In the second equation we have used the Lorenz gauge condition
$\nabla_{\mu}A^{\mu}=0$. This gives for Equation (30) (${\cal
H}=\frac{\dot{a}}{a}$ is the Hubble’s parameter)

$-\eta^{ii}\square A_{i}+\eta^{ii}2{\cal
H}\partial_{i}A_{0}=\eta^{ii}\partial_{i}(h_{+}(\partial_{2}A_{2}-\partial_{1}A_{1}))-\eta^{ii}\partial_{i}(h_{\times}(\partial_{1}A_{2}+\partial_{2}A_{1}))+\eta^{ii}h^{\beta\nu}\partial_{\beta}F_{i\nu}+h^{i\mu}\eta^{\beta\nu}\partial_{\beta}F_{\mu\nu}+\eta^{\beta\nu}F_{\mu\nu}\partial_{\beta}h^{i\mu}.$
(34)

Using the example of the EM field where the constants $P_{i},Q_{i}$ are zero,
we get the non-zero components of the Electric field strength to be

$\bar{F}_{0i}=-i\frac{k_{ei}}{k_{e}}\left(\frac{2}{\pi
k_{e}}\right)^{1/2}\cos(k_{e}\tau)e^{i\vec{k_{e}}\cdot\vec{x}}.$

This boundary condition of this form, produces spontaneous ‘EM radiation’.
Using that one gets for the perturbations in the gravitational wave background
as (without the $h_{\times}$) :

For the zeroeth component one has

$\displaystyle\square\tilde{A}_{0}-2\dot{\cal H}\tilde{A}_{0}-2{\cal
H}\partial_{0}\tilde{A}_{0}$ $\displaystyle=$
$\displaystyle\partial_{0}h_{+}(\partial_{1}\bar{A}_{1}-\partial_{2}\bar{A}_{2})+h_{+}(\partial^{2}_{1}-\partial_{2}^{2})\bar{A}_{0}$
(36) $\displaystyle=$ $\displaystyle
A_{+}e^{-i\omega_{g}\tau}(\omega_{g}\tau)e^{i\omega_{g}z}(-i\omega_{g})(k_{e1}^{2}-k_{e2}^{2})\left(\frac{2}{\pi
k_{e}^{3}}\right)^{1/2}\tau\cos(k_{e}\tau)e^{i\vec{k_{e}}\cdot\vec{x}}$
$\displaystyle-
A_{+}e^{-i\omega_{g}\tau}\left(\omega_{g}\tau-i\right)e^{i\omega_{g}z}(k_{e1}^{2}-k_{e2}^{2})\left(\frac{2}{\pi
k_{e}}\right)^{1/2}\tau\sin(k_{e}\tau)e^{i\vec{k_{e}}\cdot\vec{x}}$

We assume motivated from the RHS of the above equation the
$\tilde{A}(\tau,\vec{x})=\tilde{A}(\tau)\ e^{i\vec{\tilde{k}}\cdot x}$ where
$\vec{\tilde{k}}=\vec{k}_{g}+\vec{k_{e}}$. Plugging this we get an equation
for $\tilde{A}_{0}$ which is a pure function of $\tau$ of the form

$\frac{d^{2}{\tilde{A}_{0}(\tau)}}{d\tau^{2}}-\frac{2}{\tau}\frac{d{\tilde{A}_{0}(\tau)}}{d\tau}+(\tilde{k}^{2}+\frac{2}{\tau^{2}})\tilde{A}_{0}(\tau)=-g(\tau)$
(37)

where

$\displaystyle g(\tau)$ $\displaystyle=$ $\displaystyle
A_{+}e^{-i\omega_{g}\tau}\omega_{g}\tau(-i\omega_{g})(k_{e1}^{2}-k_{e2}^{2})\left(\frac{2}{\pi
k_{e}^{3}}\right)^{1/2}\tau\cos(k_{e}\tau)$ (38) $\displaystyle-
A_{+}e^{-i\omega_{g}\tau}\omega_{g}\tau\left(1-\frac{i}{\omega_{g}\tau}\right)(k_{e1}^{2}-k_{e2}^{2})\left(\frac{2}{\pi
k_{e}}\right)^{1/2}\tau\sin(k_{e}\tau).$

Using MAPLE we obtain the solution to the above equation as

$\tilde{A}_{0}=C_{0}\tau\cos(\tilde{k}\tau)+D_{0}\tau\sin(\tilde{k}\tau)-\frac{\tau}{\tilde{k}}\left(\sin(\tilde{k}\tau)\int\frac{\cos(\tilde{k}\tau)g(\tau)}{\tau}\
d\tau-\cos(\tilde{k}\tau)\int\frac{\sin(\tilde{k}\tau)g(\tau)}{\tau}\
d\tau\right)$ (39)

where $C_{0},D_{0}$ are constants. We find the re-appearance of the ‘new mode’
frequencies of $\tilde{k}=\sqrt{k_{g}^{2}+k_{e}^{2}}$ and the integrals can be
obtained using MAPLE, and the answer is obtained in rather complicated form
given in the Appendix (53). Using MAPLE plot as a function of $\tau$ (using
$C_{0}=D_{0}=0$) we find that the gravitational wave produces modulation over
the shape of the unperturbed wave. Note this perturbation is proportional to
$A_{+}$ and therefore much weaker than the original wave. We expect to study a
way to make these modulations detectable, and its effect on the CMB in the
near future.

For the plot we use $k_{e1}=4,k_{e2}=0,\omega_{g}=3$ and the frequency of
output mode as $\tilde{k}=5$. The solution from Equation (53) takes the
following form:

${\tilde{A}}_{0}(\tau)\sim
C_{0}\tau\cos(5\tau)+D_{0}\tau\sin(5\tau)-\frac{4}{\sqrt{2\pi}}A_{+}\tau
e^{-3i\tau}\left(\left(2i\tau+\frac{4}{3}\right)\cos(4\tau)+\frac{(-i+3)}{2}\sin(4\tau)\right)$
(40)

We have used the similarity sign as there can be some normalization constants.
We then find the mode for the propagation of the $\tilde{A}_{1}$ component.
The Equation for the $\tilde{A}_{1}$ is found as

$\square\tilde{A}_{1}=2{\cal
H}\partial_{1}\tilde{A}_{0}+h_{+}\partial_{1}^{2}\bar{A}_{1}-h_{+}\partial_{0}\bar{F}_{01}-\bar{F}_{01}\partial_{0}h_{+}$
(41)

Using the same method as for the $\tilde{A}_{0}$ we find that the time
dependent part of the equation can be obtained as:

$\partial_{0}^{2}\tilde{A}_{1}(\tau)+\tilde{k}^{2}\tilde{A}_{1}(\tau)=-\frac{2}{\tau}(ik_{e1})\tilde{A}_{0}(\tau)-A_{+}\frac{ik_{e1}}{k_{e}}\left(\frac{2}{\pi
k_{e}}\right)^{1/2}\left[\left(k_{e1}^{2}(\omega_{g}\tau-i)+i\omega_{g}^{2}\right)\tau\cos(k_{e}\tau)+(\omega_{g}\tau-i)k_{e}\sin(k_{e}\tau)\right]e^{-i\omega_{g}\tau}$
(42)

As previously we take the example of $k_{e1}=4,k_{e2}=0,\omega_{g}=3$ and find
the following solution for $\tilde{A}_{1}(\tau)$ gives

$\tilde{A}_{1}(\tau)\sim
C_{1}\cos(5\tau)+D_{1}\sin(5\tau)-\frac{2A_{+}}{\sqrt{2\pi}}\left(\left(\frac{1}{3}(1-i)+i\tau^{2}+\left(\frac{13}{16}+\frac{i}{3}\right)\tau\right)\sin(4\tau)+\frac{1}{4}\left(i\tau-\frac{25}{48}+\frac{2i}{3}\right)\cos(4\tau)\right)$
(43)

If we set $k_{e2}=0$ then

$\square{\tilde{A}}_{2}=0$ (44)

if we keep $k_{e2}$ then the equation is similar to that for $\tilde{A}_{1}$.
Further:

$\displaystyle\square\tilde{A}_{3}(\tau)$ $\displaystyle=$
$\displaystyle-2{\cal
H}\partial_{3}\tilde{A}_{0}+\partial_{3}h_{+}(\partial_{2}\bar{A}_{2}-\partial_{1}\bar{A}_{1})\sim-2\frac{i\omega_{g}}{\tau}\tilde{A}_{0}+i\omega_{g}e^{-i\omega_{g}\tau}(\omega_{g}\tau-i)(k_{e2}^{2}-k_{e1}^{2})\tau\cos(k_{e}\tau)$
(45)

We can solve for $\tilde{A}_{3}$ easily using MAPLE and we get:

$\tilde{A}_{3}(\tau)\sim
C_{3}\cos(5\tau)+D_{3}\sin(5\tau)-\frac{2A_{+}}{\sqrt{2\pi}}e^{-3i\tau}\left[\left(-\frac{3\tau}{4}+\frac{i}{4}+\frac{3}{8}\right)\cos(4\tau)+\left(-\frac{3}{2}\tau^{2}+\frac{3}{16}+i\tau\right)\sin(4\tau)\right]$
(46)

Figure 3: The perturbed Gauge field (real part) as a function of increasing
conformal time, which show oscillation in time as a response to the
gravitational wave.

As expected, the fields grow, but unlike the zeroeth order solution, the
perturbed gauge potential grows as $\tau^{2}$. This is a significant result,
and as the perturbation grows faster than the unperturbed Gauge field, the
physics of the perturbation will be important very quickly, and the
perturbation approximation will break down. In the above analysis however, we
are simply comparing the functional form of the perturbations with the zeroeth
order, where the gauge field grows linearly with $\tau$. In the plots the
arrow of increasing conformal time is not the same as the increase in
cosmological scale factor. The cosmological arrow of time is different, as a
function of the conformal time $a(\tau)=-1/H\tau$, and the behaviour of the
gauge fields will be interpreted accordingly. However, what is interesting is
that there is a clear distinct pattern which is repetitive in the plots. The
pattern can be identified by modding out the $\tau^{2}$ growth. A typical
sample of the perturbation of $\tilde{A}_{1}$ is given in the Figure (4). The
pattern period is approximately $\sim 6.27s(\pm.02s)$ for each of the gauge
fields. The above can be identified with a frequency of
$\omega=2\pi/6.27Hz\approx 1.001Hz$.

Figure 4: $\tilde{A}_{1}/\tau^{2}$ which identifies the time period of the
pattern as 6.27 s, The plot is for $\tau=20s..60s$.

This is not surprising as this is the same as the difference in frequency of
the EM and the gravitational waves, and this is a natural phenomena in the
superposition of two waves. Here of course there is more to the story as the
interaction causes the amplitude to grow. The pattern is also different from
that of a simple sinusoidal wave, and this frequency represents the profile of
the wave and its propagation in time. If we analyze this for realistic
situation then the typical EM frequency far exceeds the Gravitational wave
frequency. To find any tangible effect one has to use the radio frequency
range of the CMB spectrum as in the flat space example. Further, as there are
no external sources for the EM fields due to the boundary condition
$P_{i}=Q_{i}=0$, the EM perturbations are generated spontaneously from the
gravitational waves and the cosmological background. This phenomena when
quantized will correspond to particle creation in the cosmological background.
Note that the exact nature of the perturbation should be evaluated using the
Green’s function for the EM field in de-Sitter background as in morales . The
Green’s function can be found in woodard , however, the calculations are much
more complicated than that obtained using Fourier decomposition as in the
above. In this paper we have shown that the perturbations on a EM wave induced
by a gravitational wave can be calculated, however we have to study the system
in a more realistic situation, with the study of observables like the Poynting
vector. Further one has to study the effect of this interaction on the actual
thermalization process and emergence of the CMB if the inflationary scenario
is true. This is work in progress.

##  IV Conclusion

In this paper, we studied the interaction of the EM wave with the
gravitational wave and obtained some interesting observations about this in
two different backgrounds (i) Flat space-time, and (ii) De-sitter space. Both
of the calculations were done in the linearized approximations. In flat space-
time, we find that the EM wave deviates from its original direction when the
waves have a perpendicular component, and the perturbed EM wave has a new
frequency. To detect both of these results we have to re-design an experiment
which is work in progress maher . Further, in this paper, the boundary
conditions as required in a realistic interferometric set up chrus  have to be
discussed. However, as we observe to measure the new ‘frequency’ in an earth
experiment one has to use Masers and this is being investigated maher . The
work does however clear one aspect of the EM and gravitational wave
interaction, and that is the fact that when the waves are in the same
direction, there is no interaction. When the waves are in opposite directions,
there is an interaction, which again is measurable in an interferometer. In
the example of the EM and gravitational field interaction in de-sitter space,
we solve for the Fourier mode of the perturbation. We find that the Fourier
mode grows quadratically in conformal time which is different from the
functional behaviour of the zeroeth order gauge field which grows linearly.
The frequency of the perturbed wave is affected by the gravitational wave, and
this should be observable when the frequencies are comparable. The remnants of
this interactions have to be found in the CMB. We just have to find a way to
access them in the current day observations.
Aknowledgements: Note that most of the work appears in AP’s M. Sc. thesis. We
would like to thank the supervisory committee members Prof. Faraoni and Prof.
Seyed-Mahmoud for their positive comments on the work. *

##  Appendix A Appendix1

Here we describe the solution of a inhomogeneous wave equation as described in
morales  Given a wave equation for a unknown function $\phi(t,x)$

$\square\phi(t,x)=h(t,x)$ (47)

with the boundary conditions $\phi(0,x)=0$ and
$\frac{\partial\phi(0,x)}{\partial t}=0$ , the solution using Duhamel’s
principle and Kirchoff’s theorem is

$\phi(t,x)=-\frac{1}{4\pi}\int\frac{h(t^{\prime},x^{\prime})}{r}\
d^{3}x^{\prime}$ (48)

where $r=||\vec{x}-\vec{x^{\prime}}||$, and using the causality restriction
$t^{\prime}=t-r$ (note the speed of light c=1). In the derivations of the
solutions as in Equation(12) the integrals which have been used are

$\int\frac{\sin(\omega_{g}z^{\prime}-\omega_{g}t^{\prime})\cos(\omega_{e}x^{\prime}-\omega_{e}t^{\prime})}{r}d^{3}x^{\prime}$
(49)

writing the exponential forms of the sines and cosines, we get integrals of
the form

$I(x,t)=\int\exp(i(\omega_{g}z^{\prime}-\omega_{g}t^{\prime}))\exp(i(\omega_{e}x^{\prime}-\omega_{e}t^{\prime}))rdr\sin\theta
d\theta d\phi$ (50)

where we have taken the spherical measure. We then write
$z^{\prime}=r\cos\theta+z,x^{\prime}=r\sin\theta\cos\phi+x,t^{\prime}=t-r$ in
the exponents and compute the integrals.

$\displaystyle I(x,t)$ $\displaystyle=$ $\displaystyle
e^{i\omega_{g}(z-t)}e^{i\omega_{e}(x-t)}\int_{r=0}^{t}\int_{\theta=0}^{\pi}\int_{\phi=0}^{2\pi}e^{i(\omega_{g}+\omega_{e})r+i\omega_{g}r\cos\theta}e^{i\omega_{e}r\sin\theta\cos\phi}rdr\sin\theta
d\theta d\phi$ (51) $\displaystyle=$ $\displaystyle 2\pi
e^{i\omega_{g}(z-t)}e^{i\omega_{e}(x-t)}\int_{r=0}^{t}\int_{\theta=0}^{\pi}e^{i(\omega_{g}+\omega_{e})r+i\omega_{g}r\cos\theta}J_{0}(\omega_{e}r\sin\theta)r\
dr\sin\theta d\theta$ $\displaystyle=$ $\displaystyle
e^{i\omega_{g}(z-t)}e^{i\omega_{e}(x-t)}\int_{r=0}^{r=t}e^{i(\omega_{g}+\omega_{e})r}\frac{\sin{||\omega||r}}{||\omega||}dr$
$\displaystyle=$ $\displaystyle
e^{i\omega_{g}(z-t)}e^{i\omega_{e}(x-t)}\frac{2\pi}{i\omega_{g}\omega_{e}}\left[-1+e^{i(\omega_{g}+\omega_{e})t}\left(\cos(||\omega||t)-i\frac{(\omega_{g}+\omega_{e})}{||\omega||}\sin(||\omega||t)\right)\right]$
(52)

In the above we have used Gradshteyn Rhyzik (pages 491 & 722) grad  to compute
the $\phi$ and the $\theta$ integral. Computing all the terms in the integral
of the exponential form, we get as the Real part of the solution Equation(12).

##  Appendix B Appendix2

The explicit formulas for the Gauge fields solved in the cosmological
background.

$\displaystyle\tilde{A}_{0}$ $\displaystyle\approx$
$\displaystyle\left(\frac{2}{\pi
k_{e}}\right)^{1/2}\frac{A_{+}\tau\exp(-i\omega_{g}\tau)k_{e}}{((\tilde{k}-k_{e}+\omega_{g})^{2}(\tilde{k}-k_{e}-\omega_{g})^{2}(\tilde{k}+k_{e}+\omega_{g})^{2}(\tilde{k}+k_{e}-\omega_{g})^{2})}\left[k_{e}\left((\omega_{g}\tau-1)k_{e}^{6}\right.\right.$
(53) $\displaystyle+$
$\displaystyle\left((-\tau+2)\omega_{g}^{3}+7i\omega_{g}^{2}-3{\tilde{k}}^{2}\omega_{g}\tau+3i\tilde{k}^{2}\right)k_{e}^{4}$
$\displaystyle+$
$\displaystyle((-\tau-4)\omega_{g}^{5}-3i\omega_{g}^{4}-2\tilde{k}^{2}(\tau+2)\omega_{g}^{3}-2i{\tilde{k}}^{2}\omega_{g}^{2}+3\tilde{k}^{4}\omega_{g}\tau-3i\tilde{k}^{4})k_{e}^{2}$
$\displaystyle+$
$\displaystyle(\tilde{k}-\omega_{g})^{2}((\tau+2)\omega_{g}^{3}-3i\omega_{g}^{2}-\tilde{k}^{2}i)(\tilde{k}+\omega_{g})^{2}\left.\right)\sin(k_{e}\tau)+\left(-i(2\tau+1)k_{e}^{6}+\left((4\tau+1)i\omega_{g}^{2}+8\omega_{g}\right.\right.$
$\displaystyle+$ $\displaystyle
4i{\tilde{k}}^{2}\left(\tau-\frac{3}{4}\right)k_{e}^{4}-2(\tilde{k}^{2}-\omega_{g}^{2})\left((-ix+\frac{1}{2})\omega_{g}^{2}-4\omega_{g}+i(\tau+\frac{3}{2}){\tilde{k}}^{2}\right)k_{e}^{2}+(\tilde{k}^{2}-\omega_{g}^{2})^{3}i\left.\right)\omega_{g}^{2}\cos(k_{e}\tau))\left.\right]$

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