# Correlation analysis for isotropic stochastic gravitational wave backgrounds
with maximally allowed polarization degrees

Hidetoshi Omiya<EMAIL_ADDRESS>Naoki Seto
<EMAIL_ADDRESS>Department of Physics, Kyoto University, Kyoto
606-8502, Japan

###### Abstract

We study correlation analysis for monopole components of stochastic
gravitational wave backgrounds, including the maximally allowed polarization
degrees. We show that, for typical detector networks, the correlation analysis
can probe virtually five spectra: three for the intensities of the tensor,
vector, and scalar modes and two for the chiral asymmetries of the tensor and
vector modes. The chiral asymmetric signal for the vector modes has been left
untouched so far. In this paper, we derive the overlap reduction function for
this signal and thus complete the basic ingredients required for widely
dealing with polarization degrees. We comprehensively analyze the geometrical
properties of all the five overlap reduction functions. In particular, we
point out the importance of reflection transformations for configuring
preferable networks in the future.

##  I Introduction

A stochastic gravitational wave background is one of the important
observational targets in the near future. It can be generated by cosmological
processes Starobinsky (1979); Easther et al. (2007); Kamionkowski et al.
(1994); Caprini et al. (2008), and has a potential to probe unknown physics in
the early universe (see Maggiore (2000); Romano and Cornish (2017);
Christensen (2019); Kuroyanagi et al. (2018) for reviews). When searching for
a cosmological background, our primary target would be its isotropic
(monopole) component.

The polarization states are basic characteristics of a background, and it
would be interesting to discuss how well we can observationally extract
related information. In fact, gravitational waves can take at most six
polarization patterns (see Will (1993) for the geometrical explanation of the
patterns). In General Relativity (GR), we only have the two tensor (T) modes
known as the $+$ and $\times$ patterns. However, alternative theories of
gravity allow the existence of the additional four modes: the two vector (V)
modes (the $x$ and $y$ patterns) and the two scalar (S) modes (the $b$ and $l$
patterns). By detecting these vector and scalar modes of a background, we can
probe a violation of GR (see for example Nishizawa et al. (2009, 2010);
Cornish et al. (2018); Abbott et al. (2019)). For the tensor and vector
sectors, we could introduce the circular polarization bases, instead of the
familiar linear ones. The asymmetry between the right- and left-handed
patterns of a background would be a strong evidence for a parity violation
process Lue et al. (1999); Seto (2006); Kato and Soda (2016); Smith and
Caldwell (2017); Domcke et al. (2020); Belgacem and Kamionkowski (2020) (see
also Alexander et al. (2006); Satoh et al. (2008); Adshead and Wyman (2012);
Kahniashvili et al. (2005); Ellis et al. (2020) for generations).

The correlation analysis is a powerful approach for detecting weak stochastic
background signals under the presence of detector noises Christensen (1992);
Flanagan (1993); Allen and Romano (1999). When considering only the monopole
components, under the low frequency approximation, we can show that the
correlation analysis can virtually probe the five spectra
($I_{T},I_{V},I_{S},W_{T},W_{V}$) characterizing the polarization states. The
three spectra $(I_{T},I_{V},I_{S})$ represent the total intensities of the
tensor, vector, and scalar modes, respectively. The other two spectra
$(W_{T},W_{V})$ show the chiral asymmetries of the tensor and vector modes
which are usually referred to as the Stokes “$V$” parameter. In this paper, we
used the notation $W$ instead of the conventional one “$V$”, not to confuse
with the abbreviation $V$ which exclusively represents the vector modes.

Here, we should notice that the chiral spectra $(W_{T},W_{V})$ change their
signs with respect to the parity transformation. On the other hand, the total
intensities ($I_{T},I_{V},I_{S}$) are invariant under the parity
transformation. Therefore, we contrastively call the formers by the parity odd
spectra and later by the parity even spectra.

The overlap reduction functions (ORFs) describe the correlated responses of
the two detectors to the five spectra. We denote them by
$\gamma^{I_{P}}(P=T,V,S)$ and $\gamma^{W_{P}}(P=T,V)$. These ORFs play key
roles in the correlation analysis. The analytic expressions for four functions
($\gamma^{I_{T}},\gamma^{I_{V}},\gamma^{I_{S}},\gamma^{W_{T}}$) can be found
in the literatures Flanagan (1993); Allen and Romano (1999); Nishizawa et al.
(2009, 2010); Seto and Taruya (2008). However, the parity asymmetric vector
spectrum $W_{V}$ and the associated ORF $\gamma^{W_{V}}$ have been left
untouched so far.

In this paper, we derive the analytic expression for the remaining function
$\gamma^{W_{V}}$ and make comparative discussions on the five ORFs, paying
special attention to behaviors under the parity and reflection
transformations. We will see that these transformations are crucial for
optimizing the sensitivity to the asymmetric components $(W_{T},W_{V})$ and
their isolation from the symmetric components ($I_{T},I_{V},I_{S}$). These
results would be useful for designing future detector networks, including
space interferometers.

This paper is organized as follows. In Sec. II, we review the possible
polarization states of a stationary and isotropic gravitational wave
background. We will argue that the monopole pattern of a background is
effectively characterized by the five spectra
$(I_{T},I_{V},I_{S},W_{T},W_{V})$. In Sec. III, we discuss the correlation
analysis of a stochastic gravitational wave background and newly derive the
analytic expression for $\gamma^{W_{V}}$. In Sec. IV, we focus on the networks
which are insensitive to the parity even spectra ($I_{T},I_{V},I_{S}$) for
solely detecting the odd ones $(W_{T},W_{V})$. In Sec. V, we summarize our
results and discuss possible applications of our study.

##  II Polarization of a stochastic gravitational wave background

First, we describe the polarization decomposition of a stochastic
gravitational wave background, specifically focusing on the vector modes.
Since our universe is highly isotropic and homogeneous, we set an isotropic
background as our primary target. In addition, considering that the observed
propagation speed $v_{g}$ of gravitational waves is nearly the same as the
speed of light $c$ Abbott et al. (2017), we hereafter assume $v_{g}=c$.

We formally apply the plane wave expansion for the metric perturbation
$h_{ij}$ induced by a gravitational wave background as

$\displaystyle\begin{aligned} h_{ij}(t,\bm{x})=&\sum_{P}\int df\int
d\bm{\Omega}\\\
&\times\tilde{h}_{P}(f,\bm{\Omega})\bm{e}_{P,ij}(\bm{\Omega})e^{-2\pi
if(t-\bm{\Omega}\cdot\bm{x}/c)}~{}.\end{aligned}$ (1)

Here, $\bm{\Omega}$ is the unit vector on the two sphere parametrized by

$\displaystyle\bm{\Omega}$
$\displaystyle=\left(\begin{array}[]{c}\sin\theta\cos\phi\\\
\sin\theta\sin\phi\\\ \cos\theta\end{array}\right)$ (5)

in the cartesian coordinate. Note that we normalized the integral measure
$d\bm{\Omega}$ by $\int d\bm{\Omega}=4\pi$.

In Eq. (1), the expression $e_{ij,P}(\Omega)$ represents the polarization
tensor and $\tilde{h}_{P}$ is the mode coefficient. The subscript $P$ denotes
the polarization modes of a gravitational wave and takes the following six
patterns $P=+,\times,x,y,b,$and $l$ in the most general case Will (1993). The
$+$\- and $\times$-patterns are the usual tensor (T) modes predicted by GR. On
the other hand, the remaining modes do not appear in GR. The $x$\- and
$y$-patterns are called the vector (V) modes and the $b$\- and $l$-patterns
are the scalar (S) modes. The corresponding polarization tensors are given by

$\displaystyle\begin{aligned}
\bm{e}_{+}&=\bm{m}\otimes\bm{m}-\bm{n}\otimes\bm{n}~{},&\bm{e}_{\times}&=\bm{m}\otimes\bm{n}+\bm{n}\otimes\bm{m}~{},\\\
\bm{e}_{x}&=\bm{\Omega}\otimes\bm{m}+\bm{m}\otimes\bm{\Omega}~{},&\bm{e}_{y}&=\bm{\Omega}\otimes\bm{n}+\bm{n}\otimes\bm{\Omega}~{},\\\
\bm{e}_{b}&=\sqrt{3}(\bm{m}\otimes\bm{m}+\bm{n}\otimes\bm{n})~{},&\bm{e}_{l}&=\sqrt{3}(\bm{\Omega}\otimes\bm{\Omega})~{},\end{aligned}$
(6)

where the unit transverse vectors $\bm{m}$ and $\bm{n}$ are given by

$\displaystyle\bm{m}$
$\displaystyle=\left(\begin{array}[]{c}\cos\theta\cos\phi\\\
\cos\theta\sin\phi\\\ -\sin\theta\end{array}\right)~{},$ $\displaystyle\bm{n}$
$\displaystyle=\left(\begin{array}[]{c}-\sin\phi\\\ \cos\phi\\\
0\end{array}\right)~{}.$ (13)

Here, the unconventional factor $\sqrt{3}$ of the scalar modes are chosen for
normalizing the strain fluctuations (see appendix A of Omiya and Seto (2020)).

In Eq. (1), the mode coefficients $\tilde{h}_{P}$ are random quantities and
their statistical properties are characterized by the power spectrum
densities. In concrete terms, we first discuss the vector modes. Because the
vector modes (the $x$\- and $y$\- patterns) can be regarded as the massless
spin-1 particles Will (1993), we can characterize their polarization
properties similarly to the electromagnetic waves Rybicki and Lightman (1979).
Therefore, the $2\times 2$ matrix (for $P,P^{\prime}=x,y$) for their power
spectra is given by

$\displaystyle\begin{aligned}
\braket{\tilde{h}_{P}(f,\bm{\Omega})\tilde{h}_{P^{\prime}}^{*}(f^{\prime},\bm{\Omega^{\prime}})}&=\frac{1}{2}\delta_{\Omega\Omega^{\prime}}\delta(f-f^{\prime})\\\
&\times\left(\begin{array}[]{cc}I_{V}+Q_{V}&U_{V}-iW_{V}\\\
U_{V}+iW_{V}&I_{V}-Q_{V}\end{array}\right)~{},\end{aligned}$ (14)

with the ensemble average $\braket{\cdots}$ (omitting the $(f,\bm{\Omega})$
dependence existing in the right-hand-side). Here, $I_{V},W_{V},Q_{V}$, and
$U_{V}$ are the Stokes parameters Rybicki and Lightman (1979). As already
mentioned, we used $W$ instead of the conventional choice $V$ (representing
“vector” in this paper).

The physical meaning of $I_{V}$ and $W_{V}$ can be seen more clearly in the
circular polarization bases $(\bm{e}^{V}_{R},\bm{e}^{V}_{L})$ rather than the
linear polarization bases $(\bm{e}_{x},\bm{e}_{y})$. We can relate them by

$\displaystyle\bm{e}^{V}_{R}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bm{e}_{x}+i\bm{e}_{y}\right)~{},$
$\displaystyle\bm{e}^{V}_{L}$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\bm{e}_{x}-i\bm{e}_{y}\right)~{}.$ (15)

Using these relations, the corresponding mode coefficients in the circular
polarization bases are given by

$\displaystyle\tilde{h}^{V}_{L}(f,\bm{\Omega})$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\tilde{h}_{x}(f,\bm{\Omega})+i\tilde{h}_{y}(f,\bm{\Omega})\right)~{},$
(16) $\displaystyle\tilde{h}^{V}_{R}(f,\bm{\Omega})$
$\displaystyle=\frac{1}{\sqrt{2}}\left(\tilde{h}_{x}(f,\bm{\Omega})-i\tilde{h}_{y}(f,\bm{\Omega})\right)~{}.$
(17)

Then, from Eqs. (14), (16), and (17) we obtain

$\displaystyle\begin{aligned}
I_{V}&=\braket{\tilde{h}^{V}_{R}\tilde{h}_{R}^{V*}}+\braket{\tilde{h}^{V}_{L}\tilde{h}_{L}^{V*}}~{},\\\
W_{V}&=\braket{\tilde{h}^{V}_{R}\tilde{h}_{R}^{V*}}-\braket{\tilde{h}^{V}_{L}\tilde{h}_{L}^{v*}}~{},\\\
Q_{V}&=\braket{\tilde{h}^{V}_{R}\tilde{h}_{L}^{V*}}+\braket{\tilde{h}_{L}\tilde{h}_{R}^{V*}}~{},\\\
U_{V}&=i(\braket{\tilde{h}_{R}^{V}\tilde{h}_{L}^{V*}}-\braket{\tilde{h}_{L}^{V}\tilde{h}_{R}^{V*}})~{},\end{aligned}$
(18)

by omitting the delta functions and the $(f,\bm{\Omega})$ dependence. From
these expressions, we see that $I_{V}$ and $W_{V}$ respectively characterize
the total and asymmetry of the amplitudes of the right and the left handed
waves. We can also confirm that the combinations $Q_{V}\pm iU_{V}$
characterize the linear polarization Rybicki and Lightman (1979).

As commented earlier, the polarization patterns of the vector modes are
essentially the same as the electromagnetic waves. Therefore, if we rotate the
transverse coordinate around the propagation direction $\bm{\Omega}$ by the
angle $\alpha$, the left- and right- polarization modes transform as

$\displaystyle\tilde{h}^{V}_{L}$ $\displaystyle\to
e^{-i\alpha}\tilde{h}^{V}_{L}~{},$ (19) $\displaystyle\tilde{h}^{V}_{R}$
$\displaystyle\to e^{+i\alpha}\tilde{h}^{V}_{R}~{}.$ (20)

From Eq. (18), we correspondingly obtain

$\displaystyle\begin{aligned} &I_{V}\to I_{V}~{},W_{V}\to W_{V}~{},\\\
&Q_{V}\pm iU_{V}\to e^{\pm 2i\alpha}(Q_{V}\pm iU_{V})~{}.\end{aligned}$ (21)

We observe that $I_{V}$ and $W_{V}$ are spin-0 and $Q_{V}\pm iU_{V}$ are
spin-2. Below, we focus our study on the isotropic and stationary background
with no preferred spatial direction and orientation. Then, for the correlation
between the mode coefficients, we only need to keep the monopole components of
the spin-0 combinations. This is because a higher spin combination (e.g.
$Q_{V}\pm iU_{V}$) introduces a specific orientation and should vanish for an
isotropic background. Accordingly, we hereafter put
$I_{V}(f,\bm{\Omega})=I_{V}(f)$ and $W_{V}(f,\bm{\Omega})=W_{V}(f)$, ignoring
the angular pattern.

For the tensor modes, the power spectra are analogous to the vector modes.
Indeed, the $2\times 2$ matrix (for $P,P^{\prime}=+,\times$) for the tensor
power spectra is given by

$\displaystyle\begin{aligned}
\braket{\tilde{h}_{P}(f,\bm{\Omega})\tilde{h}_{P^{\prime}}^{*}(f^{\prime},\bm{\Omega^{\prime}})}&=\frac{1}{2}\delta_{\Omega\Omega^{\prime}}\delta(f-f^{\prime})\\\
&\times\left(\begin{array}[]{cc}I_{T}+Q_{T}&U_{T}-iW_{T}\\\
U_{T}+iW_{T}&I_{T}-Q_{T}\end{array}\right)~{}\end{aligned}$ (22)

Seto and Taruya (2008). But here, we should recall that the tensor modes are
spin-2. Then, the Stokes parameters are transformed similarly to the vector
modes, as shown in Eq. (21) with the factor $e^{\pm 4i\alpha}$ for the linear
polarization ($Q_{T}\pm iU_{T}$). Therefore, we keep only
$I_{T}(f,\bm{\Omega})=I_{T}(f)$ and $W_{T}(f,\bm{\Omega})=W_{T}(f)$ for an
isotropic background, as in the case of the vector modes.

The mode coefficients for the scalar modes are transformed as spin-0
particles. We put their covariance matrix ($P,P^{\prime}=b,l$) as

$\displaystyle\begin{aligned}
\braket{\tilde{h}_{P}(f,\bm{\Omega})\tilde{h}_{P^{\prime}}^{*}(f^{\prime},\bm{\Omega^{\prime}})}=&\frac{1}{2}\delta_{\Omega\Omega^{\prime}}\delta(f-f^{\prime})\\\
&\times\left(\begin{array}[]{cc}I_{b}&C_{S}\\\
C_{S}^{*}&I_{l}\end{array}\right)\end{aligned}$ (23)

taking into account the possible off-diagonal (correlation) terms. However, as
long as the low frequency approximation is valid, the two modes are
observationally degenerated (as shown in the next section). As a result, the
correlation analysis can probe only the mean amplitude defined by

$\displaystyle
I_{S}\equiv\frac{1}{2}\left(I_{b}+I_{l}-C_{S}-C_{S}^{*}\right)~{}.$ (24)

The mechanism behind this expression will be explained also in the next
section.

One might be interested in the correlation between different polarization
modes, such as the $T-V$, $T-S$, and $V-S$ pairs. However, they cannot produce
spin-0 combinations and should vanish for an isotropic background.

To summarize, there are at most five spectra ($I_{T},I_{V},I_{S},W_{T}$ and
$W_{V}$) that effectively characterize a stationary and isotropic
gravitational wave background (under the low frequency approximation). The
spectra $I_{T},I_{V},$ and $I_{S}$ represent the total intensity of the
tensor, vector, and scalar modes respectively. In contrast, the spectra
$W_{T}$ and $W_{V}$ are the odd parity component and probe the parity
violation process for the tensor and vector polarizations.

##  III Overlap reduction functions

The correlation analysis is a powerful approach for detecting a gravitational
wave background Christensen (1992); Flanagan (1993); Allen and Romano (1999).
The ORFs are its key elements and characterize the correlated response of two
detectors to an isotropic background. As mentioned earlier, we have, in total,
the five monopole spectra. The ORF for $I_{T}$ was first discussed in
Christensen (1992); Flanagan (1993), for $W_{T}$ in Seto (2007); Seto and
Taruya (2007), and for $I_{V}$ and $I_{S}$ in Nishizawa et al. (2009, 2010).
However, the function for the remaining one $W_{V}$ had not been studied so
far.

In this section, we discuss the ORFs, paying special attention to the
unexplored one $W_{V}$ in relation to the analog one $W_{T}$. For
systematically handling intermediate calculations, we will also introduce the
new orthogonal tensorial bases, utilizing the underlying geometrical symmetry.

###  III.1 General formulation

Let us consider the interaction of a detector $A$ with a gravitational wave
background. Under the low frequency approximation valid for $f\ll(2\pi
L/c)^{-1}$ with the detector arm length $L$, the response of the detector can
be modeled as Forward (1978)

$\displaystyle h_{A}(f)=D_{A}^{ij}\tilde{h}_{ij}(f,\bm{x}_{A})~{}.$ (25)

Here, $\tilde{h}_{ij}(f,\bm{x}_{A})$ is the metric perturbation of the
background at the detector. We also defined the detector tensor $\bm{D}_{A}$
by

$\displaystyle\bm{D}_{A}=\frac{\bm{u}_{A}\otimes\bm{u}_{A}-\bm{v}_{A}\otimes\bm{v}_{A}}{2}$
(26)

where $\bm{u}_{A}$ and $\bm{v}_{A}$ are the unit vectors representing the two
arm directions.

By cross-correlating two noise independent detectors, one can distinguish a
stochastic background from random detector noises. We define the expectation
value of the correlated signal for two detectors $A$ and $B$ as

$\displaystyle C_{AB}(f)\equiv\braket{h_{A}(f)h_{B}^{*}(f)}~{},$ (27)

again omitting the apparent delta function. Using Eqs. (1), (14), (22), (23),
and (25), and leaving only the monopole components, we obtain

$\displaystyle\begin{aligned}
C_{AB}(f)=\frac{4\pi}{5}&D_{A}^{ij}D_{B}^{kl}\left(\sum_{P=T,V,S}\Gamma^{I_{P}}_{ijkl}(f)I_{P}(f)\right.\\\
&\left.+\sum_{P=T,V}\Gamma^{W_{P}}_{ijkl}(f)W_{P}(f)\right)~{},\end{aligned}$
(28)

with

$\displaystyle\Gamma^{I_{T}}_{ijkl}$ $\displaystyle\equiv\frac{5}{8\pi}\int
d\bm{\Omega}(e_{+,ij}e_{+,kl}+e_{\times,ij}e_{\times,kl})e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}~{},$
(29) $\displaystyle\Gamma^{I_{V}}_{ijkl}$
$\displaystyle\equiv\frac{5}{8\pi}\int
d\bm{\Omega}(e_{x,ij}e_{x,kl}+e_{y,ij}e_{y,kl})e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}~{},$
(30) $\displaystyle\Gamma^{I_{S}}_{ijkl}$
$\displaystyle\equiv\frac{5}{8\pi}\int
d\bm{\Omega}(e_{b,ij}e_{b,kl}+e_{l,ij}e_{l,kl})e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}~{},$
(31) $\displaystyle\Gamma^{W_{T}}_{ijkl}$
$\displaystyle\equiv-\frac{5i}{8\pi}\int
d\bm{\Omega}(e_{+,ij}e_{\times,kl}-e_{\times,ij}e_{+,kl})e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}~{},$
(32) $\displaystyle\Gamma^{W_{V}}_{ijkl}$
$\displaystyle\equiv-\frac{5i}{8\pi}\int
d\bm{\Omega}(e_{x,ij}e_{y,kl}-e_{y,ij}e_{x,kl})e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}~{}.$
(33)

In these expressions, we introduced the unit vector
$\hat{\bm{d}}=(\bm{x}_{A}-\bm{x}_{B})/d$ with $d\equiv|\bm{x}_{A}-\bm{x}_{B}|$
and $y=2\pi fd/c$. The tensors $\Gamma_{ijkl}^{I_{P},W_{P}}$ should be
completely determined by $y$ and $\hat{d}$.

Here we comment on the degeneracy between the two scalar patters ($b$ and
$l$). Note that, the sum of the polarization tensor for these two patterns is

$\displaystyle e_{b,ij}+e_{l,ij}=\sqrt{3}\delta_{ij}~{}.$ (34)

Since the detector tensor $D_{ij}$ is traceless, we identically have

$\displaystyle e_{b,ij}D_{ij}=-e_{l,ij}D_{ij}~{}.$ (35)

With the identity Eq. (35), the cross correlation between the scalar modes can
be evaluated as

$\displaystyle\braket{h_{A}(f)h_{B}^{*}(f)}|_{\rm scalar}$
$\displaystyle=D_{A,ij}D_{B,kl}\int
d\bm{\Omega}\left(e_{b,ij}e_{b,kl}\frac{I_{b}}{2}+e_{l,ij}e_{l,kl}\frac{I_{l}}{2}+e_{b,ij}e_{l,kl}\frac{C_{S}}{2}+e_{l,ij}e_{b,kl}\frac{C_{S}^{*}}{2}\right)e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}$
(36) $\displaystyle=\frac{1}{2}D_{A,ij}D_{B,kl}\left(\int
d\bm{\Omega}\left(e_{b,ij}e_{b,kl}+e_{l,ij}e_{l,kl}\right)e^{iy\bm{\Omega}\cdot\hat{\bm{d}}}\right)\frac{I_{b}+I_{l}-C_{S}-C_{s}^{*}}{2}~{}.$
(37)

As a result, under the low frequency approximation, only the combination
$I_{S}$ in Eq. (24) appears for the cross correlation.

By contracting the tensors $\Gamma_{ijkl}^{I_{P}}$ and $\Gamma_{ijkl}^{W_{P}}$
with detector tensors $D_{A}^{ij}$ and $D_{B}^{ij}$, we obtain the formal
expression of the ORFs as

$\displaystyle\gamma^{I_{P}}_{AB}$ $\displaystyle(f)\equiv
D_{A}^{ij}D_{B}^{kl}\Gamma^{I_{P}}_{ijkl}~{},$ (38)
$\displaystyle\gamma^{W_{P}}_{AB}$ $\displaystyle(f)\equiv
D_{A}^{ij}D_{B}^{kl}\Gamma^{W_{P}}_{ijkl}~{}.$ (39)

Then the cross correlation (28) can be written as

$\displaystyle\begin{aligned}
C_{AB}(f)=\frac{4\pi}{5}&\left(\sum_{P=T,V,S}\gamma^{I_{P}}_{AB}(f)I_{P}(f)\right.\\\
&\left.+\sum_{P=T,V}\gamma^{W_{P}}_{AB}(f)W_{P}(f)\right)~{}.\end{aligned}$
(40)

The functions $\gamma^{I_{P}}_{AB}$ and $\gamma^{W_{P}}_{AB}$ clearly
characterize the correlated response of the detectors to the corresponding
background spectra. Following the classification of the spectra, we call
$\gamma^{I_{P}}_{AB}(P=T,V,S)$ by the parity even ORFs and
$\gamma^{W_{P}}_{AB}(P=T,V)$ by the parity odd ones.

From Eqs. (29) - (33), we can identify the symmetric properties of the tensors
$\Gamma_{ijkl}^{I_{P},W_{P}}$. From the definition of the polarization bases
(see Eq. (6)), the tensor $\Gamma_{ijkl}^{I_{P},W_{P}}$ are symmetric under
exchange of indices,

$\displaystyle\Gamma^{I_{P},W_{P}}_{ijkl}=\Gamma^{I_{P},W_{P}}_{jikl}=\Gamma^{I_{P},W_{P}}_{ijlk}$
(41)

Also, it is easy to confirm that the parity even ones ((29) - (31)) satisfy

$\displaystyle\Gamma^{I_{P}}_{ijkl}$ $\displaystyle=\Gamma^{I_{P}}_{klij}~{},$
(42)

while we have

$\displaystyle\Gamma^{W_{P}}_{ijkl}$ $\displaystyle=-\Gamma^{W_{P}}_{klij}~{}$
(43)

for the odd ones ((32) and (33)). In the next subsection (and appendix A), we
extensively use these properties for deriving analytic expressions of the
ORFs.

###  III.2 Symmetries to Transformations

At this point, we briefly discuss responses of various quantities to the
three-dimensional parity transformation. As commented earlier, it interchanges
the right- and left-handed waves, keeping the scalar modes invariant. We have
the correspondences for the spectra

$\displaystyle I_{P}^{\prime}{}=I_{P},~{}~{}W_{P}^{\prime}{}=-W_{P}$ (44)

with the prime attached to the transformed quantities. Since the correlation
product is unchanged with the parity transformation, we also have

$\displaystyle C_{AB}^{\prime}{}(f)=C_{AB}(f),$ (45)

and resultantly obtain

$\displaystyle{\gamma_{AB}^{I_{P}}}^{\prime}{}(f)={\gamma_{AB}^{I_{P}}}(f),~{}~{}{\gamma_{AB}^{W_{P}}}^{\prime}{}(f)=-{\gamma_{AB}^{W_{P}}}(f).$
(46)

Meanwhile, we can easily confirm

$\displaystyle\bm{D}_{A}^{\prime}{}=\bm{D}_{A},~{}~{}\bm{D}_{B}^{\prime}{}=\bm{D}_{B}.$
(47)

Then, we have

$\displaystyle{\Gamma^{I_{P}}_{ijkl}}^{\prime}{}=\Gamma^{I_{P}}_{ijkl},~{}~{}{\Gamma^{W_{P}}_{ijkl}}^{\prime}{}=-\Gamma^{W_{P}}_{ijkl}.$
(48)

Next, we discuss a mirror transformation (reflection) with respect to a single
plane. In fact, the parity transformation is generated by the consecutive
operations of a mirror transformation and an associated spatial rotation (e.g.
reflection at the $yz$-plane and rotation around the $x$-axis by the angle
$180^{\circ}$). Since we are dealing with an isotropic background, a spatial
rotation plays no role in correlation analysis. Therefore, for a mirror
transformation at an arbitrary plane, we have the correspondences identical to
(44)-(46) (but not generally (47) and (48)).

###  III.3 Construction of Parity Odd ORFs

Now we derive analytic expressions for the parity odd ORFs
$\gamma^{W_{T}}_{AB}$ and $\gamma^{W_{V}}_{AB}$. Our basic strategy is to
apply the irreducible decomposition on the rank-4 tensors
$\Gamma^{W_{P}}_{ijkl}$. This is a systematic extension of the approach
applied in Flanagan Flanagan (1993), but has not been used in the literature.
In appendix A, we discuss the parity even ORFs, following a similar procedure.

First, we specify the possible tensors which can be used for composing
$\gamma^{W_{T}}_{AB}$. As already mentioned, the unit vector $\hat{\bm{d}}$
should be the unique candidate for the vector. In addition, since the
polarization tensors transform under the special orthogonal group ${\rm
SO}(3)$, the tensors $\Gamma^{W_{P}}_{ijkl}$ should also transform under ${\rm
SO}(3)$. Correspondingly, in addition to $\hat{\bm{d}}$, we may use
$\delta_{ij}$ and $\epsilon_{ijk}$. Therefore, the basic building blocks of
the $\Gamma_{ijkl}$ must be the three tensors below

$\displaystyle\delta_{ij}~{},$ $\displaystyle M^{0}_{ij}$
$\displaystyle\equiv\hat{d}_{i}\hat{d}_{j}-\delta_{ij}/3~{},$
$\displaystyle\omega_{ij}$ $\displaystyle\equiv\epsilon_{ijk}\hat{d}_{k}~{}.$
(49)

Note that $M^{0}$ is traceless and symmetric, while $\omega$ is anti-
symmetric.

Following the general procedure for the irreducible decomposition of ${\rm
SO}(N)$ tensor Hamermesh (1989), we construct the tensorial bases for the
rank-4 tensors satisfying Eqs. (41) and (43). The relevant tensors should be
the following ones:

$\displaystyle\tilde{H}^{0}_{ijkl}=\frac{1}{2}\left(\delta_{ij}M^{0}_{kl}-M^{0}_{ij}\delta_{kl}\right)~{},$
(50)
$\displaystyle\tilde{K}^{0}_{ijkl}=\frac{1}{2\sqrt{10}}\left(\delta_{ik}\omega_{jl}+\delta_{il}\omega_{jk}+\delta_{jk}\omega_{il}+\delta_{jl}\omega_{ik}\right)~{},$
(51) $\displaystyle\begin{aligned}
\tilde{F}^{0}_{ijkl}=&\frac{1}{4}\sqrt{\frac{5}{2}}\left(\omega_{ik}M^{0}_{jl}+\omega_{il}M^{0}_{jk}+\omega_{jk}M^{0}_{il}+\omega_{jl}M^{0}_{ik}\right)\\\
&+\frac{1}{3}\tilde{K}^{0}_{ijkl}~{},\end{aligned}$ (52)

These tensors are orthonormal in the following sense

$\displaystyle\tilde{F}^{0}_{ijkl}\tilde{H}^{0}_{ijkl}=\tilde{F}^{0}_{ijkl}\tilde{K}^{0}_{ijkl}=\tilde{H}^{0}_{ijkl}\tilde{K}^{0}_{ijkl}=0~{},$
(53)
$\displaystyle\tilde{F}^{0}_{ijkl}\tilde{F}^{0}_{ijkl}=\tilde{H}^{0}_{ijkl}\tilde{H}^{0}_{ijkl}=\tilde{K}^{0}_{ijkl}\tilde{K}^{0}_{ijkl}=1~{}.$
(54)

Note that $\tilde{F}^{0}$ is traceless

$\displaystyle\tilde{F}^{0}_{iijk}=\tilde{F}^{0}_{ijik}=\dots=\tilde{F}^{0}_{jkii}=0~{},$
(55)

which is required by the irreducible decomposition.

Using the tensors (50),(51), and (52), we can expand $\Gamma^{W_{P}}$ as

$\displaystyle\Gamma^{W_{P}}_{ijkl}$
$\displaystyle=\tilde{\rho}^{P}_{\tilde{F}}\tilde{F}^{0}_{ijkl}+\tilde{\rho}^{P}_{\tilde{K}}\tilde{K}^{0}_{ijkl}+\tilde{\rho}^{P}_{\tilde{H}}\tilde{H}^{0}_{ijkl}~{}.$
(56)

The orthonormal nature of the basis allows us to obtain the expansion
coefficients by simply contracting the tensors with $\Gamma^{W_{P}}_{ijkl}$:

$\displaystyle\tilde{\rho}^{P}_{\tilde{F}}$
$\displaystyle=\tilde{F}^{0}_{ijkl}\Gamma^{W_{P}}_{ijkl}~{},$ (57)
$\displaystyle\tilde{\rho}^{P}_{\tilde{H}}$
$\displaystyle=\tilde{H}^{0}_{ijkl}\Gamma^{W_{P}}_{ijkl}~{},$ (58)
$\displaystyle\tilde{\rho}^{P}_{\tilde{K}}$
$\displaystyle=\tilde{K}^{0}_{ijkl}\Gamma^{W_{P}}_{ijkl}~{}.$ (59)

After some elementary calculations, we obtain the coefficients

$\displaystyle(\tilde{\rho}^{T}_{\tilde{F}},\tilde{\rho}^{T}_{\tilde{K}},\tilde{\rho}^{T}_{\tilde{H}})$
$\displaystyle=\sqrt{10}(j_{3}(y),2j_{1}(y),0)~{},$ (60)
$\displaystyle(\tilde{\rho}^{V}_{\tilde{F}},\tilde{\rho}^{V}_{\tilde{K}},\tilde{\rho}^{V}_{\tilde{H}})$
$\displaystyle=\sqrt{10}(-2j_{3}(y),j_{1}(y),0)~{},$ (61)

where $j_{n}(y)$ are the spherical Bessel functions. Notice that we
identically have $\rho^{P}_{\tilde{H}}=0$. This straightforwardly follows from
the oddness of the tensors $\Gamma^{W_{P}}_{ijkl}$ with respect to parity
transformation. We must have an odd power of $\hat{\bm{d}}$, in contrast to
$\tilde{H}^{0}_{ijkl}$.

Using the expressions presented so far, we have the analytic expressions of
$\gamma^{W_{T}}$ and $\gamma^{W_{V}}$ as

$\displaystyle\gamma^{W_{T}}_{AB}$
$\displaystyle=\sqrt{10}\left(D^{\tilde{F}}_{AB}j_{3}(y)+2D^{\tilde{K}}_{AB}j_{1}(y)\right)~{},$
(62) $\displaystyle\gamma^{W_{V}}_{AB}$
$\displaystyle=\sqrt{10}\left(-2D^{\tilde{F}}_{AB}j_{3}(y)+D^{\tilde{K}}_{AB}j_{1}(y)\right)~{}.$
(63)

Here, we defined

$\displaystyle D^{\tilde{F}}_{AB}$ $\displaystyle\equiv
D_{A,ij}D_{B,kl}\tilde{F}^{0}_{ijkl}~{},$ (64) $\displaystyle
D^{\tilde{K}}_{AB}$ $\displaystyle\equiv
D_{A,ij}D_{B,kl}\tilde{K}^{0}_{ijkl}~{}.$ (65)

We present the ready-to-use expressions for networks composed by two ground-
based detectors in appendix B. In Fig. 1, as typical examples, we show the
parity odd ORFs for the VIRGO-LIGO Hanford network.

Figure 1: The parity odd ORFs for the VIRGO-LIGO Hanford network. The black
solid and red dashed lines correspond to the tensor and vector modes,
respectively. The distance between two detectors is $d\sim 8.2\times
10^{3}{\rm km}$.

The asymptotic behaviors of the ORFs can be easily understood from Eqs. (62)
and (63). For the large frequency regime ($y=2\pi fd/c\gg 1$), the spherical
Bessel functions behave as

$\displaystyle j_{1}(y)\underset{y\to\infty}{\to}-\frac{\cos y}{y}~{},$ (66)
$\displaystyle j_{3}(y)\underset{y\to\infty}{\to}\frac{\cos y}{y}~{}.$ (67)

Then we have

$\displaystyle\gamma^{W_{T}}_{AB}\underset{y\to\infty}{\to}\sqrt{10}\left(D^{\tilde{F}}_{AB}-2D^{\tilde{K}}_{AB}\right)\frac{\cos
y}{y}~{},$ (68)
$\displaystyle\gamma^{W_{V}}_{AB}\underset{y\to\infty}{\to}\sqrt{10}\left(-2D^{\tilde{F}}_{AB}-D^{\tilde{K}}_{AB}\right)\frac{\cos
y}{y}~{}.$ (69)

Thus, they oscillate with the frequency interval $c/d$ and the envelope
$\propto 1/f$, as in Fig. 1.

In the small frequency regime ($y\ll 1$), the spherical Bessel functions can
be expanded as

$\displaystyle j_{1}(y)\underset{y\to 0}{\to}\frac{y}{3}~{},$ (70)
$\displaystyle j_{3}(y)\underset{y\to 0}{\to}\frac{y^{3}}{105}~{}.$ (71)

We then have

$\displaystyle\gamma^{W_{T}}_{AB}\underset{y\to
0}{\to}\frac{2\sqrt{10}}{3}D^{\tilde{K}}_{AB}\,y~{},$ (72)
$\displaystyle\gamma^{W_{V}}_{AB}\underset{y\to
0}{\to}\frac{\sqrt{10}}{3}D^{\tilde{K}}_{AB}\,y~{}.$ (73)

Therefore, the ORFs approach to zero in the small frequency regime as shown in
Fig. 1.

These asymptotic behaviors indicate the blindness of the coincident detectors
($d=0$ or equivalently $y=0$) to the parity odd components. We can understand
this from the oddness of the function $\gamma^{W_{P}}_{AB}$ with respect to
the parity transformation (essentially the same as the discussion on
$\rho^{P}_{\tilde{H}}=0$ above). In the next section, we discuss the responses
of the parity odd ORFs to a mirror transformation (reflection).

##  IV Asymmetric networks

As shown in Eq. (38), the expectation value of the correlation product is a
linear combination of the even and odd parity spectra. Since the latter are
closely related to the parity violation process, we would like to detect them
without contamination by the even spectra. In this section, using a mirror
transformation, we discuss how to realize the desired network with
$\gamma_{AB}^{I_{p}}=0$ ($P=T,V$ and $S$). In the following, to simplify our
expressions, we omit the subscript $AB$ (labels for detectors) and put
$\gamma^{I_{P}}_{AB}=\gamma^{I_{P}}$ and $\gamma^{W_{P}}_{AB}=\gamma^{W_{P}}$.

###  IV.1 General consideration

For the isolation of the odd spectra, our basic strategy here is to
geometrically identify the networks that have the correspondence

$\displaystyle{\gamma^{I_{P}}}^{\prime}{}=-{\gamma^{I_{P}}}$ (74)

with respect to a mirror transformation. Since we identically have
${\gamma^{I_{P}}}^{\prime}{}={\gamma^{I_{P}}}$ for an arbitrarily mirror
transformation (see Sec. III.2), we obtain ${\gamma^{I_{P}}}=0$ for a network
with Eq. (74).

As shown in Fig. 2, we take the $z$-axis parallel to the direction vector
$\hat{\bm{d}}$, and consider the mirror transformation at the $yz$-plane. As
in the case of $\hat{\bm{d}}$, The four rank tensors $\Gamma^{I_{P}}_{ijkl}$
are invariant with this transformation
${\Gamma^{I_{P}}_{ijkl}}^{\prime}{}=\Gamma^{I_{P}}_{ijkl}$, and we have

$\displaystyle{\gamma^{I_{P}}}^{\prime}{}={{\Gamma^{I_{P}}_{ijkl}}^{\prime}{}{D_{A}^{ij}}}^{\prime}{}{D_{B}^{kl}}^{\prime}{}=\Gamma^{I_{P}}_{ijkl}{D_{A}^{ij}}^{\prime}{}{D_{B}^{kl}}^{\prime}{}$
(75)

in comparison to the original one
${\gamma^{I_{P}}}=\Gamma^{I_{P}}_{ijkl}{D_{A}^{ij}}{D_{B}^{kl}}$. Therefore,
we can realize the desired condition
${\gamma^{I_{P}}}^{\prime}{}=-{\gamma^{I_{P}}}$ by using two detectors $A$ and
$B$ transformed as

$\displaystyle\bm{D}_{A}^{\prime}{}=-\bm{D}_{A},~{}~{}~{}\bm{D}_{B}^{\prime}{}=\bm{D}_{B}.$
(76)

Figure 2: The configuration of the detector network under consideration. We
fix the positions of the two detectors and examine their detector tensors with
respect to the reflection (mirror transformation) at the $yz$-plane.

Looking back at the arguments so far, one would consider that our networks
with Eq. (76) would be just a subset of the networks with the desired property
${\gamma^{I_{P}}}=0$. But, after various examinations, we deduced that our
requirements (76) actually cover the whole network geometries satisfying the
identity ${\gamma^{I_{P}}}=0$. Hereafter, we assume that this is really the
case and call our networks the asymmetric networks.

###  IV.2 Detector Tensors

We now identify detector tensors which are transformed as Eq. (76) with
respect to the reflection at the $yz$-plane.

####  IV.2.1 flipped one

First, we examine the flipped one with $\bm{D}_{A}^{\prime}{}=-\bm{D}_{A}$.
Since a detector tensor is formally given by
$\bm{D}_{A}=({\bm{u}_{A}\otimes\bm{u}_{A}-\bm{v}_{A}\otimes\bm{v}_{A}})/{2}$,
we can simply compose the flipped one by using two vectors interchanged as
$\bm{u}_{A}^{\prime}{}=\bm{v}_{A}$ and $\bm{v}_{A}^{\prime}{}=\bm{u}_{A}$.
They are mutually mirrored images and parameterized as

$\displaystyle\begin{aligned}
\bm{u}_{A}&=\frac{1}{\sqrt{2}}(1,\sin\theta_{0},\cos\theta_{0})~{},\\\
\bm{v}_{A}&=\frac{1}{\sqrt{2}}(-1,\sin\theta_{0},\cos\theta_{0})~{}\end{aligned}$
(77)

with the angle $\theta_{0}$ between the bisecting vector and the $z$-axis.
(see Fig. 3)

Note that the detector tensor $\bm{D}_{A}$ is given as quadratures of unit
vectors, and we can multiply $-1$ to $\bm{u}_{A}$ and/ or $\bm{v}_{A}$ in Eq.
(77), keeping the relation $\bm{D}_{A}^{\prime}{}=-\bm{D}_{A}$. In this
manner, we can make totally $2\times 2$ equivalent pairs of unit vectors. We
call this reduplication the “multiplicity of vector signs”. Correspondingly,
the detector tensor with the angle $\theta_{0}=\alpha+\pi$ is essentially the
same as that with $\theta_{0}=\alpha$.

Figure 3: The two arm directions for the flipped detector tensor (see Eq.
(77)). The arms are mirror symmetric with respect to the $yz$-plane. We define
$\theta_{0}$ as the angle between their bisector and the $z$-axis.

####  IV.2.2 invariant ones

Figure 4: (Left) The configuration of the type-I invariant detector tensor
(see Eq. (79)). The vector $\bm{u}_{B}$ is on the $x$-axis and $\bm{v}_{B}$ is
on the $yz$-plane with the angle $\theta_{1}$ relative to the $y$-axis.
(Right) The configuration of the type-II invariant detector tensor (see Eq.
(80)). The two vectors are on the $yz$-plane with the bisecting angle
$\theta_{2}$.

Next we discuss the detector tensors with $\bm{D}_{B}^{\prime}{}=\bm{D}_{B}$.
We can compose it with the unit vectors transformed as

$\displaystyle\bm{u}_{B}^{\prime}{}$
$\displaystyle=\pm\bm{u}_{B},~{}~{}~{}\bm{v}_{B}^{\prime}{}=\pm\bm{v}_{B}.$
(78)

After all, we can find two independent types of solutions. The first one (type
I) is parameterized as

$\displaystyle\bm{u}_{B}$ $\displaystyle=(1,0,0)~{},$
$\displaystyle\bm{v}_{B}$ $\displaystyle=(0,\cos\theta_{1},\sin\theta_{1})~{}$
(79)

with the transformations $\bm{u}_{B}^{\prime}{}=\bm{u}_{B}$ and
$\bm{v}_{B}^{\prime}{}=-\bm{v}_{B}$. We still have the multiplicity of vector
signs, and the detector with the angle $\theta_{1}=\beta+\pi$ is essentially
the same as that with $\theta_{1}=\beta$.

The second one (type II) is parameterized as

$\displaystyle\begin{aligned}
\bm{u}_{B}&=(0,\cos\left(\theta_{2}-\frac{\pi}{4}\right),\sin\left(\theta_{2}-\frac{\pi}{4}\right))~{},\\\
\bm{v}_{B}&=(0,\cos\left(\theta_{2}+\frac{\pi}{4}\right),\sin\left(\theta_{2}+\frac{\pi}{4}\right))\end{aligned}$
(80)

with $\bm{u}_{B}^{\prime}{}=\bm{u}_{B}$ and
$\bm{v}_{B}^{\prime}{}=\bm{v}_{B}$. In this case, due to the“multiplicity of
vector signs”, the detector with $\theta_{2}=\gamma+\pi/2$ is essentially the
same as $\theta_{2}=\gamma$. We should notice that the phase offset $\pi/2$ is
different from $\pi$ associated with $\theta_{0}$ and $\theta_{1}$ in Eqs.
(77) and (79).

###  IV.3 Parity Odd ORFs

Figure 5: (Left) The type-I network formed by combining one flipped detector
and one type-I invariant detector. This network is characterized by two angles
$\theta_{0}$ and $\theta_{1}$ (see Figs. 3 and 4 for their definitions).
(Right) The type-II network formed by combining one flipped detector and one
type-II invariant detector. This network is characterized by the angular
parameters $\theta_{0}$ and $\theta_{2}$ (see Figs. 3 and 4 for their
definitions).

As mentioned in Sec. IV.1, an asymmetric network is insensitive to the parity
even spectra and allows us to exclusively probe the parity odd spectra. Such a
network can be formed by combining one invariant detector and one flipped
detector, as shown in Eq. (76). Following the classification of the invariant
detector, we divide the asymmetric networks into the types I and II (see Fig.
5).

We now evaluate the parity odd ORFs $\gamma^{W_{P}}$ ($P=T$ and $V$) for the
asymmetric networks. We deal with the two types separately in the following
subsections. Based on the interest in realizing good sensitivities, we examine
the maximums of the absolute values $|\gamma^{W_{p}}|$. From our experience in
mathematical analysis, we expect that the maximum values will be obtained for
highly symmetric network geometries.

####  IV.3.1 type I network

We first derive the analytic expressions of the functions $\gamma^{W_{p}}$ for
the type I network. With Eqs. (77) and (79) for the orientations of the arms,
we obtain the detector tensors (28) as

$\displaystyle\bm{D}_{A}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{ccc}0&\sin\theta_{0}&\cos\theta_{0}\\\
\sin\theta_{0}&0&0\\\ \cos\theta_{0}&0&0\end{array}\right)~{},$ (84)
$\displaystyle\bm{D}^{\rm I}_{B}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{ccc}1&0&0\\\
0&-\cos^{2}\theta_{1}&-\frac{1}{2}\sin 2\theta_{1}\\\ 0&-\frac{1}{2}\sin
2\theta_{1}&-\sin^{2}\theta_{1}\end{array}\right)~{}.$ (88)

Then, using Eqs. (62) and (63), we have

$\displaystyle\begin{aligned} \gamma^{W_{T}}_{\rm
I}(y;\theta_{0},\theta_{1})=&-\frac{1}{8}(3+\cos
2\theta_{1})\sin\theta_{0}\left(4j_{1}-j_{3}\right)\\\
&{\color[rgb]{1,0,0}-}\frac{1}{2}\sin
2\theta_{1}\cos\theta_{0}(j_{1}+j_{3})~{},\end{aligned}$ (89)
$\displaystyle\begin{aligned} \gamma^{W_{V}}_{\rm
I}(y;\theta_{0},\theta_{1})=&-\frac{1}{4}(3+\cos
2\theta_{1})\sin\theta_{0}(j_{1}+j_{3})\\\
&{\color[rgb]{1,0,0}-}\frac{1}{4}\sin
2\theta_{1}\cos\theta_{0}\left(j_{1}-4j_{3}\right)~{}.\end{aligned}$ (90)

We comment on basic properties of these expressions. First, there exist the
periodicities $\theta_{0}\to\theta_{0}+2\pi$ and
$\theta_{1}\to\theta_{1}+\pi$. In addition, for the absolute values of these
functions, the periodicities reduce to $\theta_{0}\to\theta_{0}+\pi$ and
$\theta_{1}\to\theta_{1}+\pi$, as expected from the “multiplicity of vector
signs” pointed out in Sec. III.3.

Figure 6: The highly symmetric configurations of the type-I network resulting
in Eq. (91). All the five ORFs vanish for these networks. (Left) The
configuration with the angular parameters $(\theta_{0},\theta_{1})=(0,0)$. The
upper detector (red lines) is on the $xz$-plane and the lower detector (blue
lines) is on the $xy$-plane. (Right) The configuration with the angular
parameters $(\theta_{0},\theta_{1})=(0,\pi/2)$. Both upper detector (red
lines) and lower detector (blue lines) are on the $xz$-plane.

Secondly, the two ORFs identically become

$\displaystyle\gamma^{W_{V}}_{\rm I}=\gamma^{W_{T}}_{\rm I}=0$ (91)

at $(\theta_{0},\theta_{1})=(0,0)$ and $(0,\pi/2)$ (see Fig. 6 for the
corresponding configurations). In these highly symmetric configurations, the
network is parity even with respect to the reflection at the $xz$-plane. Then,
following an argument similar to Sec. IV.1, we can readily confirm Eq. (91)
(see also Seto and Taruya (2008) for an earlier discussion on the tensor
modes). We should notice that, the networks in Fig. 6 are still parity odd for
the reflection at the $yz$-plane and all of the five ORFs vanish in Eq. (40).

We can also find that the ORFs (89) and (90) are invariant under the following
transformation

$\displaystyle\begin{cases}\theta_{0}\to\pi-\theta_{0}~{},\\\
\theta_{1}\to\pi-\theta_{1}~{}.\end{cases}$ (92)

Geometrically, this corresponds to taking the reflection at the $xz$-plane and
subsequently interchanging two arms of the upper detector (detector A) in the
left panel in Fig. 5. The overall signs of the ORFs are changed twice and we
recover the original forms.

Figure 7: (Left) The contour plot of the function $F_{\rm
I}^{W_{T}}(\theta_{0},\theta_{1})$ defined in Eq. (93). The minimum value is 0
and the maximum value is $0.840$. (Right) The contour plot of the function
$F_{\rm I}^{W_{V}}(\theta_{0},\theta_{1})$ ranging from 0 to 0.520. Figure 8:
Highly symmetric network configuration with
$(\theta_{0},\theta_{1})=(\pi/2,0)$ corresponding to the maximums in Fig. 7.
Both upper detector (red lines) and lower detector (blue lines) are parallel
to the $xy$-plane. This network is parity odd for the reflections at the
$xz$\- and $yz$-planes.

Now we examine the network configurations that realize large absolute values
$|\gamma^{W_{P}}_{\rm I}(y,\theta_{0},\theta_{1})|$ for the two odd modes
$P=T$ and $V$. To begin with, for given angular variables
$(\theta_{0},\theta_{1})$ and an index $P$, we numerically identify the
parameters $y=y^{P}_{m,{\rm I}}(\theta_{0},\theta_{1})$ that maximize the
functions $|\gamma^{W_{P}}_{\rm I}(y,\theta_{0},\theta_{1})|$. We then define
the resultant maximum values by

$\displaystyle F_{\rm I}^{P}(\theta_{0},\theta_{1})\equiv|\gamma_{\rm
I}^{W_{P}}(y^{P}_{m,{\rm
I}}(\theta_{0},\theta_{1});\theta_{0},\theta_{1})|~{}.$ (93)

In Fig. 7, we show their contour plots. We can easily confirm the properties
mentioned earlier, such as the periodicities, the invariance under
transformation (92), and $\gamma^{W_{T}}_{\rm I}=\gamma^{W_{V}}_{\rm I}=0$ at
$(\theta_{0},\theta_{1})=(0,0)$ and $(0,\pi/2)$.

The global maximum values of the functions $F_{\rm I}^{P}$ are commonly at
$(\theta_{0},\theta_{1})=(\pi/2,0)$ with the combinations

$\displaystyle F_{\rm I}^{T}(\pi/2,0)\simeq$ $\displaystyle 0.840~{}$
$\displaystyle{\rm}~{}~{}y_{m,T}(\pi/2,0)$ $\displaystyle\simeq 2.000~{},$
(94) $\displaystyle F_{\rm I}^{V}(\pi/2,0)\simeq$ $\displaystyle 0.520~{}$
$\displaystyle{\rm}~{}~{}y_{m,V}(\pi/2,0)$ $\displaystyle\simeq 2.501~{}.$
(95)

In Fig. 8, we show the network configuration with
$(\theta_{0},\theta_{1})=(\pi/2,0)$. We can find that this network is parity
odd with respect to both the $xz$\- and $yz$-planes. This also supports our
naive expectation that sensitivity is maximized for a highly symmetric
configuration.

In comparison, for the even parity modes, the maximum values of the ORFs are
simply given by
$\gamma^{I_{T}}(y=0)=\gamma^{I_{V}}(y=0)=\gamma^{I_{S}}(y=0)=1$ for two co-
aligned detectors (with $\gamma^{W_{T}}(y)=\gamma^{W_{V}}(y)=0$).

####  IV.3.2 type II network

Next, we consider the type II network. The analysis here is parallel to the
previous subsubsection for the type I network. The geometrical difference
between the two types is the orientation of the invariant detector ($B$ in
Fig. 5). Its detector arms are given by Eq. (80) with the detector tensor

$\displaystyle\bm{D}^{\rm II}_{B}$
$\displaystyle=\frac{1}{2}\left(\begin{array}[]{ccc}0&0&0\\\ 0&\sin
2\theta_{2}&-\cos 2\theta_{2}\\\ 0&-\cos 2\theta_{2}&-\sin
2\theta_{2}\end{array}\right)~{}.$ (99)

Then we obtain the analytic expressions for the ORFs as

$\displaystyle\begin{aligned} \gamma^{W_{T}}_{\rm
II}=&{\color[rgb]{1,0,0}-}\frac{1}{4}\sin
2\theta_{2}\sin\theta_{0}\left(-4j_{1}+j_{3}\right)\\\ &-\cos
2\theta_{2}\cos\theta_{0}(j_{1}+j_{3})~{},\end{aligned}$ (100)
$\displaystyle\begin{aligned} \gamma^{{\color[rgb]{1,0,0}W_{V}}}_{\rm
II}=&{\color[rgb]{1,0,0}+}\frac{1}{2}\sin
2\theta_{2}\sin\theta_{0}(j_{1}+j_{3})\\\ &+\frac{1}{2}\cos
2\theta_{2}\cos\theta_{0}\left(-j_{1}+4j_{3}\right)~{}.\end{aligned}$ (101)

The basic properties of these functions are similar to those already mentioned
for the type I network. More specifically, we have the identical periodicities
$\theta_{0}\to\theta_{0}+2\pi~{}$ and $\theta_{2}\to\theta_{2}+\pi$. In
contrast, for their absolute values, the periods result in
$\theta_{0}\to\theta_{0}+\pi$ and $\theta_{2}\to\theta_{2}+\frac{\pi}{2}$ with
the factor 2 difference for the latter (reflecting the “multiplicity of vector
signs” noted in Sec. III.3).

Figure 9: The highly symmetric configurations of the type-II network resulting
in Eq. (102). All the five ORFs vanish for these networks. (Left) The
configuration with the angular parameters $(\theta_{0},\theta_{2})=(0,\pi/4)$.
The upper detector (red lines) is on the $xz$-plane and the lower detector
(blue lines) is on the $yz$-plane. (Right) The configuration with the angular
parameters $(\theta_{0},\theta_{2})=(\pi/2,\pi/2)$. The upper detector (red
lines) is parallel to the $xy$-plane and the lower detector (blue lines) is on
the $yz$-plane with bisector of the two arms on the $z$ axis.

In addition, we have the identities

$\displaystyle\gamma_{\rm II}^{W_{T}}=\gamma_{\rm II}^{W_{V}}=0~{},$ (102)

at $(\theta_{0},\theta_{2})=(0,\pi/4)$ and $(\pi/2,0)$ (see Fig. 9 for the
corresponding configurations), again reflecting the parity evenness with
respect to the reflection at the $xz$-plane.

In contrast to the invariance of the type I network under transformation (92),
the functions (100) and (101) change the overall signs under the simultaneous
transformations

$\displaystyle\begin{cases}\theta_{0}\to\pi-\theta_{0}~{},\\\
\theta_{2}\to\pi-\theta_{2}~{}.\end{cases}$ (103)

The geometrical interpretation is almost the same as the previous case, except
for the additional interchange of the two arms of the lower detector B in the
right panel of Fig. 5. This additional operation results in the extra minus
signs, compared to the case with Eq. (92).

Figure 10: (Left) The contour plot of the function $F_{\rm
II}^{W_{T}}(\theta_{0},\theta_{2})$ defined in Eq. (104). The minimum value is
0 and the maximum value is $0.520$. (Right) The contour plot of the function
$F_{\rm II}^{W_{V}}(\theta_{0},\theta_{2})$ ranging from 0 to 0.508.

Now we study the network configuration which maximizes the absolute values
$|\gamma^{W_{P}}_{\rm II}|$. Following the same strategy as before, we define
the two functions ($P=T$ and $V$)

$\displaystyle F_{\rm II}^{P}(\theta_{0},\theta_{2})\equiv|\gamma_{\rm
II}^{W_{P}}(y^{P}_{m,{\rm
II}}(\theta_{0},\theta_{2});\theta_{0},\theta_{2})|~{},$ (104)

where $y^{P}_{m,{\rm II}}(\theta_{0},\theta_{2})$ maximize
$\gamma^{W_{P}}_{\rm II}(y;\theta_{0},\theta_{2})$ for fixed
$(\theta_{0},\theta_{2})$. In Fig. 10, we show the contour plots of
$F^{T}_{\rm II}$ and $F^{V}_{\rm II}$. We can confirm the basic properties of
the ORFs mentioned earlier.

Figure 11: Highly symmetric network configuration with
$(\theta_{0},\theta_{2})=(0,0)$ corresponding to the maximums in Fig. 10. The
upper detector (red lines) is on the $xz$-plane and lower detector (blue
lines) is on the $yz$-plane with bisector of two arms on the $y$ axis. This
network is parity odd for the reflections at the $xz$\- and $yz$-planes.

The global maximums of the type II detectors are realized at
$(\theta_{0},\theta_{2})=(0,0)$ with

$\displaystyle F_{\rm II}^{T}(0,0)$ $\displaystyle\sim 0.520~{},$
$\displaystyle y_{m,{\rm II}}^{T}$ $\displaystyle\sim 2.501~{},$ (105)
$\displaystyle F_{\rm II}^{V}(0,0)$ $\displaystyle\sim 0.508~{},$
$\displaystyle y_{m,{\rm II}}^{V}$ $\displaystyle\sim 4.921~{}.$ (106)

Fig. 11 shows the corresponding configuration with
$(\theta_{0},\theta_{2})=(0,0)$. Again, this configuration is parity odd for
the reflections both at the $yz$\- and $xz$\- planes. However, the maximum
values are smaller than Eqs. (94) and (95).

###  IV.4 Relation to the ground-based networks

Figure 12: The relative configuration of two ground based detectors A and B.
The two detectors are on the blue planes which are tangential to the Earth
sphere. The angle $\beta$ shows the separation of two detectors, measured from
the center of the Earth. The two angles $\sigma_{A}$ and $\sigma_{B}$ describe
the orientation of the bisectors of the two detector arms (dotted lines) in a
counterclockwise manner, relative to the great circle connecting the two
detectors. The angular parameters $\Delta$ and $\delta$ are defined by
$\Delta\equiv(\sigma_{A}+\sigma_{B})/2)$ and
$\delta\equiv(\sigma_{A}-\sigma_{B})/2$.

Until now, we have discussed the ORFs for highly symmetric configurations
purely from geometrical viewpoints. Here, for the odd parity ORFs, we point
out the connection of the type I network to a network composed of two ground
based detectors that are tangential to the Earth sphere.

As presented in appendix B, we have the expression for the latter as

$\displaystyle\gamma^{W_{P}}$ $\displaystyle=\Xi^{P}(y,\beta)\sin 4\Delta~{},$
$\displaystyle(P$ $\displaystyle=T,V)$ (107)

with the two angular parameters $\beta$ and $\Delta$ (see Fig. 12 for their
definitions). The first one $\beta$ is the angular separation between two
detectors measured from the center of the Earth. The second one $\Delta$ is
determined by the orientations of two detectors as

$\displaystyle\Delta=\frac{\sigma_{A}+\sigma_{B}}{2}~{}.$ (108)

By taking $|\sin 4\Delta|=1$ we can maximize
$|\gamma^{W_{P}}|=|\Xi^{P}(y,\beta)|$.

In fact, the function $|\Xi^{P}(y,\beta)|$ is identical to the odd parity ORFs
$|\gamma_{\rm I}^{W_{p}}|$ given in Eqs. (89) and (90) for the type I network,
after tuning its angular parameters $\theta_{0}$ and $\theta_{1}$. More
specifically, we impose the relation

$\displaystyle|\theta_{1}-\theta_{0}|=\pi/2~{},$ (109)

corresponding to the condition that two detectors A and B in Fig. 5 are
tangential to a sphere. Under the relation (109), the separation angle $\beta$
is given by

$\displaystyle\beta$
$\displaystyle=\begin{cases}2\theta_{0}~{},&(0<\theta_{0}<\frac{\pi}{2})~{},\\\
2\pi-2\theta_{0}~{},&(\frac{\pi}{2}<\theta_{0}<\pi)~{},\end{cases}$ (110)

and we obtain $|\Xi^{P}(y,\beta)|=|\gamma_{\rm
I}^{W_{p}}(y,\theta_{0},\theta_{1})|$ ($P=T,V$).

##  V Summary and discussion

In this paper, we studied the correlation analysis for detecting various
polarization modes of a stationary and isotropic stochastic gravitational wave
background. We pointed out that, as long as the low frequency approximation is
valid, we can probe the five spectra $I_{T},I_{V},I_{S},W_{T},$ and $W_{V}$.
The three spectra $I_{P}(P=T,V,S)$ represent the intensities of the tensor,
vector, and scalar modes. While the remaining two spectra $W_{P}(P=T,V)$ show
the chiral asymmetries of the tensor and vector modes. Other correlations,
such as the tensor-vector pair start from the higher multipole components and
thus do not contribute to the monopole components.

When performing the correlation analysis, the ORFs play key roles and
characterize the correlated responses of detectors to the five spectra. In
this paper, we newly derived the function $\gamma^{W_{V}}$ for the parity odd
vector modes and completed all the ORFs required for generally analyzing
polarization states of an isotropic background. For its derivation, we applied
a systematic method, explicitly respecting the rotational symmetry (SO(3)) of
the system. We also paid special attention to the parity transformation of the
system. These help us to understand the symmetrical structure of the ORFs and
their building blocks.

Furthermore, we examine two detector networks with respect to reflection
transformations which are closely related to the parity transformation. For a
reflection, the ORFs have the same parity signatures as the corresponding
spectra. This property allows us to easily design detector networks that are
sensitive to either even or odd spectra of a background. Such networks are
particularly interesting for clearly isolating the two parity. In Fig. 5, we
illustrated the two types of network geometries that are insensitive to the
even ones.

Then we examined the odd ORFs $\gamma^{W_{P}}(P=T,V)$ specifically for the two
geometrical types. When we tune the networks to maximize the sensitivity, they
take a highly symmetric configuration which is parity odd simultaneously to
the two different planes. In contrast, for other highly symmetric
configurations with angular parameters
$(\theta_{0},\theta_{1})=(0,0),(0,\pi/2)$ and
$(\theta_{0},\theta_{2})=(0,\pi/4),(\pi/2,\pi/2)$ (see Figs. 6 and 9), we
identically have $\gamma^{I_{P}}=0$ $(P=T,V,S)$ and $\gamma^{W_{P}}=0$
$(P=T,V)$, and thus the detector networks become blind to all the five
monopole components.

In this paper, we have concentrated on the basic properties of the ORFs. Now
we comment on potential applications of our results. One of the immediate
studies would be the prospect for the ongoing ground-based detector network
including LIGO-Hanford, LIGO-Livingston Aasi et al. (2015), Virgo Acernese et
al. (2015), and KAGRA Aso et al. (2013). We need at least five pairs of
detectors to algebraically separate the five spectra Seto and Taruya (2008).
Using the four detectors listed above, we can make ${}_{4}C_{2}=6$ pairs in
total and can separate the five spectra. But their estimation errors would be
strongly correlated. By adding LIGO-India Bala et al. (2011) to the network,
we will have ${}_{5}C_{2}=10$ pairs of detectors, and the noise correlation
would be considerably reduced. Also, it might be interesting to examine third
generation detectors Amalberti et al. (2021).

Another application would be a case study for the space detectors, such as the
LISA-Taiji network Amaro-Seoane et al. (2017); Hu and Wu (2017); Wang and Han
(2021); Wang et al. (2021); Seto (2020a); Orlando et al. (2021); Omiya and
Seto (2020). However, because of the existing geometrical symmetry, we will
have only three independent data combinations and cannot separate the five
components completely Omiya and Seto (2020). If Tian-Qin Luo et al. (2016) is
additionally available, we can solve the degeneracy in principle. But, its
optimal frequency is higher than LISA and Taiji, and the overall performance
of the correlation analysis would be limited Seto (2020b).

Throughout this paper, we applied the low frequency approximation for
responses of individual detectors. This is an efficient approximation for most
observational situations, but we have the degeneracy between the two scalar
modes. In some cases, we need to carefully deal with the finiteness of the arm
length (see e.g. Seto (2007); Amalberti et al. (2021)). It might be
interesting to study the possibility of resolving the degeneracy.

###### Acknowledgements.

We are sincerely grateful to the referee for carefully reading the draft and
pointing out many errors in our expressions. This work is supported by JSPS
Kakenhi Grant-in-Aid for Scientific Research (Nos. 17H06358 and 19K03870).

##  Appendix A Parity even Overlap reduction functions with orthogonal tensors

In this appendix, we derive the ORFs for the even parity spectra
($I_{T},I_{V},$ and $I_{S}$) with the method that we applied for the odd
parity ones (see Sec. III.3). The procedure for the irreducible decomposition
is almost the same. But, for parity even ones, we have the symmetry (42)
instead of Eq. (43). After some calculations, we find that the following five
tensors form the orthonormal basis for the decomposition:

$\displaystyle H_{ijkl}=\frac{1}{3}\delta_{ij}\delta_{kl}~{},$ (111)
$\displaystyle
K_{ijkl}=\frac{1}{2\sqrt{5}}\left(\delta_{ik}\delta_{jl}+\delta_{il}\delta_{jk}\right)-\frac{1}{\sqrt{5}}H_{ijkl}~{},$
(112) $\displaystyle
H^{0}_{ijkl}=\frac{1}{2}\left(\delta_{ij}M^{0}_{kl}+M^{0}_{ij}\delta_{kl}\right)~{},$
(113) $\displaystyle\begin{aligned}
K^{0}_{ijkl}=&\frac{3}{2\sqrt{14}}\left(\delta_{ik}M^{0}_{jl}+\delta_{il}M^{0}_{jk}+\delta_{jk}M^{0}_{il}+\delta_{jl}M^{0}_{ik}\right)\\\
&-\frac{4}{\sqrt{14}}H^{0}_{ijkl}~{},\end{aligned}$ (114) $\displaystyle
F^{0}_{ijkl}=\frac{1}{2}\sqrt{\frac{35}{2}}M^{0}_{ij}M^{0}_{kl}-\frac{\sqrt{5}}{3}K^{0}_{ijkl}-\frac{1}{3}\sqrt{\frac{7}{2}}K_{ijkl}~{}.$
(115)

Note that $F^{0}$ satisfies the traceless property

$\displaystyle F^{0}_{iijk}=F^{0}_{ijik}=\dots=F^{0}_{jkii}=0~{}.$ (116)

Using these basis tensors, the even parity functions (Eq. (29) - (31)) are
expanded as

$\displaystyle\Gamma^{I_{P}}_{ijkl}=$
$\displaystyle\rho^{P}_{F^{0}}F^{0}_{ijkl}+\rho^{P}_{H^{0}}H^{0}_{ijkl}$
$\displaystyle+\rho^{P}_{K^{0}}K^{0}_{ijkl}+\rho^{P}_{H}H_{ijkl}+\rho^{P}_{K}K_{ijkl}~{}.$
(117)

The orthonormality of the basis tensors allows us to obtain the coefficients
as

$\displaystyle\rho^{P}_{F^{0}}$
$\displaystyle=F^{0}_{ijkl}\Gamma^{I_{P}}_{ijkl}~{},$ (118)
$\displaystyle\rho^{P}_{H^{0}}$
$\displaystyle=H^{0}_{ijkl}\Gamma^{I_{P}}_{ijkl}~{},$ (119)
$\displaystyle\rho^{P}_{K^{0}}$
$\displaystyle=K^{0}_{ijkl}\Gamma^{I_{P}}_{ijkl}~{},$ (120)
$\displaystyle\rho^{P}_{H}$ $\displaystyle=H_{ijkl}\Gamma^{I_{P}}_{ijkl}~{},$
(121) $\displaystyle\rho^{P}_{K}$
$\displaystyle=K_{ijkl}\Gamma^{I_{P}}_{ijkl}~{}.$ (122)

After some elementary integral, we obtain

$\displaystyle(\rho^{T}_{F^{0}},\rho^{T}_{H^{0}},\rho^{T}_{K^{0}},\rho^{T}_{H},\rho^{T}_{K})$
$\displaystyle=2\sqrt{5}(\frac{1}{\sqrt{14}}j_{4},0,\sqrt{\frac{10}{7}}j_{2},0,j_{0})~{},$
(123)
$\displaystyle(\rho^{V}_{F^{0}},\rho^{V}_{H^{0}},\rho^{V}_{K^{0}},\rho^{V}_{H},\rho^{V}_{K})$
$\displaystyle=2\sqrt{5}(-2\sqrt{\frac{2}{7}}j_{4},0,-\sqrt{\frac{5}{14}}j_{2},0,j_{0})~{},$
(124)
$\displaystyle(\rho^{S}_{F^{0}},\rho^{S}_{H^{0}},\rho^{S}_{K^{0}},\rho^{S}_{H},\rho^{S}_{K})$
$\displaystyle=2\sqrt{5}(3\sqrt{\frac{2}{7}}j_{4},\frac{\sqrt{5}}{2}j_{2},-\sqrt{\frac{10}{7}}j_{2},5\frac{\sqrt{5}}{{\color[rgb]{1,0,0}4}}j_{0},j_{0})~{}.$
(125)

By contracting $\Gamma^{I_{P}}_{ijkl}$ with the detector tensors, we have the
explicit form of the ORF as

$\displaystyle\gamma^{I_{T}}_{AB}$
$\displaystyle=2\sqrt{5}\left(\frac{1}{\sqrt{14}}D^{F}_{AB}j_{4}(y){\color[rgb]{1,0,0}+}\sqrt{\frac{10}{7}}D^{K^{0}}_{AB}j_{2}(y)+D^{K}_{AB}j_{0}(y)\right)~{},$
(126) $\displaystyle\gamma^{I_{V}}_{AB}$
$\displaystyle=2\sqrt{5}\left(-2\sqrt{\frac{2}{7}}D^{F}_{AB}j_{4}(y)-\sqrt{\frac{5}{14}}D^{K^{0}}_{AB}j_{2}(y)+D^{K}_{AB}j_{0}(y)\right)~{},$
(127) $\displaystyle\gamma^{I_{S}}_{AB}$
$\displaystyle=2\sqrt{5}\left(3\sqrt{\frac{2}{7}}D^{F}_{AB}j_{4}(y)-\sqrt{\frac{10}{7}}D^{K^{0}}_{AB}j_{2}(y)+D^{K}_{AB}j_{0}(y)\right)~{}.$
(128)

Here, we defined

$\displaystyle D^{F}_{AB}$ $\displaystyle\equiv
D_{A,ij}D_{B,kl}F^{0}_{ijkl}~{},$ (129) $\displaystyle D^{K^{0}}_{AB}$
$\displaystyle\equiv D_{A,ij}D_{B,kl}K^{0}_{ijkl}~{},$ (130) $\displaystyle
D^{K}_{AB}$ $\displaystyle\equiv D_{A,ij}D_{B,kl}K_{ijkl}~{}.$ (131)

The coefficients $\rho^{S}_{2}$ and $\rho^{S}_{4}$ do not contribute to the
ORFs, because the contraction with $H_{ijkl}$ and $H^{0}_{ijkl}$ is
identically zero

$\displaystyle D_{A,ij}D_{B,kl}H_{ijkl}=0~{},$ (132) $\displaystyle
D_{A,ij}D_{B,kl}H^{0}_{ijkl}=0~{},$ (133)

which obey from the traceless property of the detector tensor.

##  Appendix B Explicit formulae for ground-based detector network

In this section, we give the explicit formulae of the five ORFs
$\gamma^{I_{P}}$ and $\gamma^{W_{P}}$ for a network composed by two ground-
based detectors. A ground-based detector is virtually tangential to the
Earth’s surface that can be regarded as a sphere. Therefore, the relative
geometry of two arbitrary detectors is characterized by the three angles
$(\beta,\delta,\Delta)$ (see Fig. 1 and Eq. (21) of Seto and Taruya (2008) for
their definitions). After some algebra, for Eqs. (64)-(65) and (129)-(131), we
find

$\displaystyle
D^{\tilde{F}}_{AB}=-\frac{7+3\cos\beta}{8\sqrt{10}}\sin\left(\frac{\beta}{2}\right)\sin
4\Delta~{},$ (134) $\displaystyle
D^{\tilde{K}}_{AB}=\frac{1}{\sqrt{10}}\sin^{3}\left(\frac{\beta}{2}\right)\sin
4\Delta~{}.$ (135) $\displaystyle\begin{aligned}
D^{F}_{AB}=&\frac{3}{16\sqrt{70}}\cos^{4}\left(\frac{\beta}{2}\right)\cos
4\delta\\\ &-\frac{169+108\cos\beta+3\cos 2\beta}{128\sqrt{70}}\cos
4\Delta~{},\end{aligned}$ (136) $\displaystyle
D^{K^{0}}_{AB}=\frac{1}{2\sqrt{14}}\cos^{4}\left(\frac{\beta}{2}\right)\cos
4\delta+\frac{5+\cos\beta}{4\sqrt{14}}\sin^{2}\left(\frac{\beta}{2}\right)\cos
4\Delta~{},$ (137) $\displaystyle
D^{K}_{AB}=\frac{1}{2\sqrt{5}}\cos^{4}\left(\frac{\beta}{2}\right)\cos
4\delta-\frac{1}{2\sqrt{5}}\sin^{4}\left(\frac{\beta}{2}\right)\cos
4\Delta~{}.$ (138)

Substituting these coefficients to Eqs. (62), (63), (126), (127), and (128),
we obtain

$\displaystyle\gamma^{W_{P}}$ $\displaystyle=\Xi^{P}(y,\beta)\sin 4\Delta~{},$
$\displaystyle(P$ $\displaystyle=T,V)~{},$ (139) $\displaystyle\gamma^{I_{P}}$
$\displaystyle=\Theta_{\Delta}^{P}(y,\beta)\cos
4\Delta+\Theta_{\delta}^{P}(y,\beta)\cos 4\delta~{},$ $\displaystyle(P$
$\displaystyle=T,V,S)~{}.$ (140)

Here, the coefficients $\Xi^{P},\Theta_{\Delta}^{P},$ and
$\Theta_{\delta}^{P}$ are given by

$\displaystyle\Xi^{T}(y,\beta)$
$\displaystyle=\sin\left(\frac{\beta}{2}\right)\left((1-\cos\beta)j_{1}(y)-\frac{7+3\cos\beta}{8}j_{3}(y)\right)~{},$
(141) $\displaystyle\Xi^{V}(y,\beta)$
$\displaystyle=\frac{1}{2}\sin\left(\frac{\beta}{2}\right)\left((1-\cos\beta)j_{1}(y)+\frac{7+3\cos\beta}{2}j_{3}(y)\right)~{},$
(142) $\displaystyle\Theta^{T}_{\Delta}(y,\beta)$
$\displaystyle=-\sin^{4}\left(\frac{\beta}{2}\right)j_{0}(y)-\frac{5}{56}(-9+8\cos\beta+\cos
2\beta)j_{2}(y)-\frac{1}{896}(169+108\cos\beta+3\cos 2\beta)j_{4}(y)~{},$
(143) $\displaystyle\Theta^{V}_{\Delta}(y,\beta)$
$\displaystyle=-\sin^{4}\left(\frac{\beta}{2}\right)j_{0}(y)+\frac{5}{112}(-9+8\cos\beta+\cos
2\beta)j_{2}(y)+\frac{1}{224}(169+108\cos\beta+3\cos 2\beta)j_{4}(y)~{},$
(144) $\displaystyle\Theta^{S}_{\Delta}(y,\beta)$
$\displaystyle=-\sin^{4}\left(\frac{\beta}{2}\right)j_{0}(y)+\frac{5}{56}(-9+8\cos\beta+\cos
2\beta)j_{2}(y)-\frac{3}{448}(169+108\cos\beta+3\cos 2\beta)j_{4}(y)~{},$
(145) $\displaystyle\Theta^{T}_{\delta}(y,\beta)$
$\displaystyle=\cos^{4}\left(\frac{\beta}{2}\right)\left(j_{0}(y)+\frac{5}{14}j_{2}(y)+\frac{3}{112}j_{4}(y)\right)~{},$
(146) $\displaystyle\Theta^{V}_{\delta}(y,\beta)$
$\displaystyle=\cos^{4}\left(\frac{\beta}{2}\right)\left(j_{0}(y)-\frac{5}{14}j_{2}(y)-\frac{3}{28}j_{4}(y)\right)~{},$
(147) $\displaystyle\Theta^{S}_{\delta}(y,\beta)$
$\displaystyle=\cos^{4}\left(\frac{\beta}{2}\right)\left(j_{0}(y)-\frac{5}{7}j_{2}(y)+\frac{9}{56}j_{4}(y)\right)~{}.$
(148)

These expressions (except for the newly derived $\gamma^{W_{V}}$) are
essentially the same as those in the literature Flanagan (1993); Seto and
Taruya (2008); Nishizawa et al. (2009).

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