# Antipodal Angular Correlations of
Inflationary Stochastic Gravitational Wave Background

Zhen-Yuan Wu  Graduate School of Sciences and Technology for Innovation,
Yamaguchi University, Yamaguchi 753-8512, Japan  Ryo Saito  Graduate School of
Sciences and Technology for Innovation, Yamaguchi University, Yamaguchi
753-8512, Japan  Kavli Institute for the Physics and Mathematics of the
Universe (WPI), University of Tokyo Institutes for Advanced Study, University
of Tokyo, Chiba 277-8583, Japan  Nobuyuki Sakai  Graduate School of Sciences
and Technology for Innovation, Yamaguchi University, Yamaguchi 753-8512, Japan

###### Abstract

The measurement of the inflationary stochastic gravitational-wave background
(SGWB) is one of the main goals of future GW experiments. In direct GW
experiments, an obstacle to achieving it is the isolation of the inflationary
SGWB from the other types of SGWB. In this paper, as a distinguishable
signature of the inflationary SGWB, we argue the detectability of its
universal property: antipodal correlations, i.e., correlations of GWs from the
opposite directions, as a consequence of the horizon re-entry. A phase-
coherent method has been known to be of no use for detecting the angular
correlations in SGWB due to a problematic phase factor that erases the signal.
We thus investigate whether we can construct a phase-incoherent estimator of
the antipodal correlations in the intensity map. We found that the conclusion
depends on whether the inflationary GWs have statistical isotropy or not. In
the standard inflationary models with statistical homogeneity and isotropy,
there is no estimator that is sensitive to the antipodal correlations but does
not suffer from the problematic phase factor. On the other hand, it is
possible to find a non-vanishing estimator of the antipodal correlations for
inflationary models with statistical anisotropy. SGWB from anisotropic
inflation is distinguishable from the other components.

##  I  Introduction

Inflation [1, 2, 3, 4] is a strong candidate for the mechanism to seed the
structure of our universe. According to the standard paradigm, the accelerated
expansion during inflation stretches the microscopic quantum fluctuations of
the inflaton field to superhorizon scales, which are converted to the
primordial density fluctuations in the post-inflationary universe. An
inevitable prediction is that inflation also generates the primordial
gravitational waves (GWs) from tensor-type quantum fluctuations of spacetime
[5, 6, 7]. Thus, the detection of the inflationary GWs gives strong evidence
for inflation.

In direct-detection experiments, the inflationary GWs are observed as a
stochastic gravitational-wave background (SGWB), i.e., GWs coming from all
directions in the sky. The detection of the inflationary SGWB is challenging
because it has a tiny amplitude in typical inflationary models. As well as
improving the sensitivity of GW detectors, we need to isolate it from SGWB
generated by the other sources: a superposition of GWs from many unresolvable
astrophysical and cosmological sources (see, e.g., Refs. [8, 9, 10, 11, 12,
13]). Numerous studies have been carried out on methods for separating the
astrophysical components in SGWB: spectral separation [14, 15, 16, 17, 18,
19], subtraction [20, 21, 22, 23, 24], anisotropies [25, 26, 27, 28, 29, 30,
31, 32], polarization [33, 34, 35, 36, 37, 38, 39], and so on. These methods
work well to place upper limits on the inflationary SGWB. However, in these
methods, it is impossible to guarantee that the remaining exotic component is
of inflationary origin without a priori assumptions on an inflationary model
as well as on the other cosmological sources. For example, although the slow-
roll inflation predicts the spectral density $S_{h}(f)\propto
f^{-\alpha}~{}(\alpha\simeq 3)$, it is not a universal prediction of
inflation. Inflation can predict a wide variety of spectra, especially in
models generating SGWB detectable by upcoming experiments [40, 41, 12].

The main purpose of this paper is to investigate whether the inflationary SGWB
is distinguishable from the other components in direct-detection experiments
without any a priori assumptions, focusing on a unique and universal
prediction of inflation: the generation of superhorizon modes. 111Several
alternatives of inflation have been proposed to generate the superhorizon
modes (see, e.g., Sec. 6.5 in Ref. [12]). To be exact, our argument is also
applied to these scenarios. Superhorizon modes are generated by inflation but
not by any causal mechanism in the post-inflationary universe. In consequence,
the inflationary GWs have a standing-wave nature after the horizon re-entry
[42, 43, 44, 45, 46]. As reviewed in section III.2, the standing-wave nature
is observed as unusual properties of SGWB, most notably, correlations between
GWs from opposite directions. Although this property has been already noticed
in the literature, we would like to emphasize it as a unique prediction of
inflation and name it antipodal correlations. The astrophysical SGWB, or any
type of SGWB from localized sources, will not have such correlations because
GWs from distant sources are uncorrelated with each other. Therefore, it is a
unique signature for the inflationary SGWB.

About 20 years ago, however, Allen et al. [43] showed that the unusual
properties of the inflationary SWGB above cannot be detected in the stain
correlation analysis. The antipodal correlations rapidly oscillate due to
interference between GWs from opposite directions. It is inevitably smoothed
out by averaging over frequencies unresolvable because of the finite
observation time. Moreover, it was recently pointed out in Margalit et al.
[47] that metric perturbations along the line-of-sight randomize the GW
phases. This effect reduces the detectability of the antipodal correlations
because the observed quantity is the strain smoothed over the sky with the
finite angular resolution of a detector. The above two effects have been also
pointed out for the three-point correlation function in Refs. [48, 49, 50]. As
noted in Refs. [47, 50], we need to use phase-incoherent methods such as the
intensity map [51, 52] to avoid these problems of interference. In this paper,
we thus investigate whether we can construct a phase-incoherent estimator of
the intensity map to detect the antipodal correlations. We found that the
conclusion depends on whether the inflationary GWs have statistical isotropy
or not. In the standard inflationary models with statistical homogeneity and
isotropy, there is no estimator that is sensitive to the antipodal
correlations but does not suffer from the problematic phase factor. On the
other hand, it is possible to find a non-vanishing estimator of the antipodal
correlations for inflationary models with statistical anisotropy. SGWB from
anisotropic inflation is distinguishable from the other components.

This paper is organized as follows. In section II, we briefly review possible
properties of SGWB and define the antipodal correlation. In section III, we
show the standing-wave nature of the inflationary GWs and how it leads to the
antipodal correlations in SGWB. In section IV, after reviewing the
detectability of the antipodal correlations in the strain correlation
approach, we consider the intensity correlation approach. Our conclusions are
summarized in V. In Appendix A, we also discuss the detectability of the
antipodal correlations in the time domain analysis.

##  II  Stochastic Gravitational Wave Background

In this section, we shortly review how SGWB can be characterized with emphasis
on the unusual statistical properties of the inflationary SGWB.

The stochastic gravitational wave background is defined by a superposition of
GWs from all directions of the sky. In the transverse-traceless gauge, it can
be expanded as222In this paper, we denote a stochastic quantity with a hat.

$\displaystyle\hat{h}_{ij}(t,{\bm{x}})=$
$\displaystyle\quad\sum_{A}\int_{-\infty}^{\infty}{\rm d}f\int{\rm
d}^{2}{\bm{n}}\ \hat{h}_{A}(f,{\bm{n}})e_{ij}^{A}({\bm{n}})e^{-2\pi
if(t-{\bm{n}}\cdot{\bm{x}})}\,,$ (1)

in terms of plane waves with a frequency $f$ and a propagating direction
${\bm{n}}$. 333We follow the notations in Maggiore’s book [53]. In some
literature, ${\bm{n}}$ is used for a direction on the sky, which is opposite
to the propagating direction. The tensors $e_{ij}^{A}({\bm{n}})$ are the
polarization tensors for the two GW polarization states with normalization
$e_{ij}^{A}({\bm{n}})e^{B,ij}({\bm{n}})=2\delta^{AB}$. To simplify the
expressions below, we specialize to a circular polarization basis
$e_{ij}^{\pm}({\bm{n}})$: they are related to the “plus-cross” polarization
vectors as $e_{ij}^{\pm}({\bm{n}})=e_{ij}^{\text{P.}}({\bm{n}})\pm
i\,e_{ij}^{\text{C.}}({\bm{n}})$ and thus satisfy
$e_{ij}^{\pm}(-{\bm{n}})=e_{ij}^{\mp}({\bm{n}})$. We define $t=0$ as the start
time of observation.

The Fourier amplitudes $\hat{h}_{A}(f,{\bm{n}})$ are random variables and
their statistical distribution characterizes the stochastic background.
Usually, we make the following assumptions on the statistical distribution:

  1. (a)

Gaussianity: all the statistical information in SGWB can be characterized by
the two-point correlation function
$\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle$.

  2. (b)

Isotropy: the correlation functions are invariant under the rotation on the
celestial sphere, i.e.,

$\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle$
depends on ${\bm{n}}_{1}$ and ${\bm{n}}_{2}$ only through
${\bm{n}}_{1}\cdot{\bm{n}}_{2}$.

  3. (c)

No angular correlations 444Note that the statistical isotropy (b) does not
forbid the angular correlations as is the case with the temperature map of
cosmic microwave background (CMB). The properties (b) and 4 are independent
assumptions. : GWs from different directions are not correlated with each
other, i.e.,

$\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle\propto\delta^{2}({\bm{n}}_{1},{\bm{n}}_{2})$.

  4. (d)

Stationarity: the correlation functions are invariant under the time
translation, i.e.,

$\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle\propto\delta(f_{1}-f_{2})$.

  5. (e)

Unpolarized: different polarization modes are independent and have the same
statistics, i.e.,

$\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle\propto\delta_{AB}$,

and the coefficient is independent of the polarizations.

When all these assumptions are satisfied, SGWB is characterized as

$\displaystyle\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle=$
$\displaystyle\hskip
56.9055pt\frac{S_{h}^{\text{(D)}}(f_{1})\delta_{AB}}{4\pi}\delta(f_{1}-f_{2})\delta^{2}({\bm{n}}_{1},{\bm{n}}_{2})\,,$
(2)

with a (double-sided) spectral density $S^{\text{(D)}}_{h}(f)$.

As shown in Refs. [42, 43, 44, 45, 46], the inflationary SGWB does not satisfy
the two assumptions 4 and (d) as well as the last assumption (e):555The second
term in Eq. (II) is proportional to $\delta_{AB}$ in Ref. [43]. As we will
show in the next section, it should be replaced by $\delta_{A(-B)}$. the
correlation function has an additional component as

$\displaystyle\langle\hat{h}^{\dagger}_{A}(f_{1},{\bm{n}}_{1})\hat{h}_{B}(f_{2},{\bm{n}}_{2})\rangle=$
$\displaystyle\hskip
28.45274pt\quad\frac{S_{h}^{\text{(D)}}(f_{1})\delta_{AB}}{4\pi}\delta(f_{1}-f_{2})\delta^{2}({\bm{n}}_{1},{\bm{n}}_{2})$
$\displaystyle\hskip
28.45274pt+\frac{A_{h}^{\text{(D)}}(f_{1})\delta_{A(-B)}}{4\pi}\delta(f_{1}+f_{2})\delta^{2}({\bm{n}}_{1},-{\bm{n}}_{2})\,.$
(3)

Here, we have defined $S^{\text{(D)}}_{h}(f)$ and $A_{h}^{\text{(D)}}(f)$ as
double-sided quantities. The corresponding single-sided spectral densities are
defined by

$\displaystyle S_{h}(f)$ $\displaystyle\equiv
S_{h}^{\text{(D)}}(f)+S_{h}^{\text{(D)}}(-f)=2S_{h}^{\text{(D)}}(f)\,,$ (4)
$\displaystyle A_{h}(f)$ $\displaystyle\equiv
A_{h}^{\text{(D)}}(f)+A_{h}^{\text{(D)}}(-f)=2{\rm
Re}[A_{h}^{\text{(D)}}(f)]\,,$ (5)

respectively. The second new term in Eq. (II) shows that GWs from the opposite
directions are correlated, i.e., it corresponds to the antipodal correlations.
In the next section, we will show that inflation universally predicts such
correlations.

##  III  Antipodal correlations

In this section, we show how the inflationary GWs cause the antipodal
correlations, i.e., the correlations between GWs from the opposite directions,
in the observed SGWB. Although most arguments in this section have been
presented in the literature [42, 43, 44, 45, 46], we rederive them in terms of
realizations instead of statistically-averaged quantities for the later
arguments on the construction of the estimator in Sec. IV.

###  III.1  Traveling/Standing-wave nature of stochastic gravitational wave
background

The expansion (1) can be derived from the Fourier transform of GWs (see, e.g.,
Sec. 1.2 of Ref. [53]):

$\displaystyle\hat{h}_{ij}(t,{\bm{x}})=\sum_{A=\pm}\int\frac{{\rm
d}^{3}k}{(2\pi)^{3}}~{}\hat{h}_{A}(t,{\bm{k}})e_{ij}^{A}({\bm{n}}_{{\bm{k}}})e^{i{\bm{k}}\cdot{\bm{x}}}\,,$
(6)

where $\hat{h}^{\dagger}_{\pm}(t,{\bm{k}})=\hat{h}_{\pm}(t,-{\bm{k}})$. The
vector ${\bm{n}}_{{\bm{k}}}$ is the unit vector along ${\bm{k}}$:
${\bm{n}}_{{\bm{k}}}\equiv{\bm{k}}/|{\bm{k}}|$. In the local universe,
$\hat{h}_{ij}(t,{\bm{x}})$ satisfies the wave equation $\square\hat{h}_{ij}=0$
in a good approximation and therefore $\hat{h}_{A}(t,{\bm{k}})$ can be
expanded into the positive and negative frequency modes as

$\displaystyle\hat{h}_{A}(t,{\bm{k}})=\hat{\cal
A}^{{\rm(p)}}_{A}({\bm{k}})e^{-ikt}+\hat{\cal
A}^{{\rm(n)}}_{A}({\bm{k}})e^{ikt}\,.$ (7)

Here, the coefficients $\hat{\cal A}^{{\rm(p)}}_{A}({\bm{k}})$ and $\hat{\cal
A}^{{\rm(n)}}_{A}({\bm{k}})$ are the integration constants and satisfy
$\hat{\cal A}^{{\rm(n)}}_{\pm}({\bm{k}})=[\hat{\cal
A}^{{\rm(p)}}_{\pm}(-{\bm{k}})]^{\dagger}$ from the reality condition of
$\hat{h}_{ij}(t,{\bm{x}})$. In Eq. (7), the first (second) term represents a
plane wave moving along ${\bm{n}}={\bm{n}}_{{\bm{k}}}$
($-{\bm{n}}_{{\bm{k}}}$) with the frequency $f=k/2\pi$ ($-k/2\pi$). Therefore,
the amplitude $\hat{h}_{A}(t,{\bm{n}})$ in Eq. (1) is read as

$\displaystyle\hat{h}_{\pm}(f,{\bm{n}})=\begin{cases}f^{2}\hat{\cal
A}^{{\rm(p)}}_{\pm}(2\pi|f|{\bm{n}})\quad\text{for}~{}f>0\,,\\\\[6.0pt]
f^{2}\hat{\cal
A}^{{\rm(n)}}_{\mp}(-2\pi|f|{\bm{n}})\quad\text{for}~{}f<0\,.\end{cases}$ (8)

Note that the relation
$[\hat{h}_{\pm}(f,{\bm{n}})]^{\dagger}=\hat{h}_{\mp}(-f,{\bm{n}})$ is
satisfied as expected from the reality condition of
$\hat{h}_{ij}(t,{\bm{x}})$.

The coefficients $\hat{\cal A}^{{\rm(p)}}_{A}({\bm{k}})$ and $\hat{\cal
A}^{{\rm(n)}}_{A}({\bm{k}})$ are determined by the initial conditions and
characterize the GW sources. When all of them are independent, the GW
background (1) is given by the superposition of independent traveling waves.
This is expected for SGWB from localized sources. However, this is not the
only possibility even when the statistical homogeneity is assumed [45]: the
statistical homogeneity forbids the correlations between $\hat{\cal
A}^{{\rm(p)}}_{A}({\bm{k}})\,,\hat{\cal A}^{{\rm(n)}}_{A}({\bm{k}})$ with
different values of ${\bm{k}}$ but not those between $\hat{\cal
A}^{{\rm(p)}}_{A}({\bm{k}})$ and $\hat{\cal A}^{{\rm(n)}}_{A}({\bm{k}})$ with
the same value of ${\bm{k}}$:

$\displaystyle\left\langle\left[\hat{\cal
A}^{{\rm(p)}}_{A}({\bm{k}})\right]^{\dagger}\hat{\cal
A}^{{\rm(n)}}_{A}({\bm{k}})\right\rangle\neq 0\,.$ (9)

From Eq. (8), this leads to the correlation between GWs with opposite
frequencies and directions, i.e., the $A_{h}$ term in Eq. (II). 666This
argument shows that SGWB can only have the antipodal correlations as angular
correlations when the statistical isotropy and homogeneity are assumed for
$\hat{h}_{ij}(t,{\bm{x}})$. It can be also shown that the polarization
dependence is restricted by imposing the invariance under the rotation around
${\bm{n}}_{{\bm{k}}}$, for which the circular polarization basis is
transformed as $e_{ij}^{\pm}({\bm{n}}_{{\bm{k}}})\to e^{\pm
2i\psi}e_{ij}^{\pm}({\bm{n}}_{{\bm{k}}})$. In the next subsection, we will
show that $\hat{\cal A}^{{\rm(p)}}_{A}({\bm{k}})$ and $\hat{\cal
A}^{{\rm(n)}}_{A}({\bm{k}})$ have almost the same magnitude for the
inflationary GWs. This means that the inflationary GWs have a standing-wave
nature [42, 43, 44, 45, 46].

###  III.2  Propagation in the homogeneous universe

To make the basic idea clearer, let us consider the propagation of the
inflationary GWs in an idealistic homogeneous universe,

$\displaystyle{\rm d}s^{2}=a^{2}(\eta)\left[-{\rm
d}\eta^{2}+(\delta_{ij}+h_{ij}){\rm d}x^{i}{\rm d}x^{j}\right]\,,$ (10)

where we have introduced the conformal time $\eta$ and the scale factor
$a(\eta)$. The scale factor is normalized as $a(\eta_{0})=1$ for the start
time of observation $\eta=\eta_{0}$. Thus, the comoving wavenumber below can
be identified with the physical wavenumber in Eq. (6). Inhomogeneities of the
universe have large effects on the GW phases [48, 49, 47]. However, this does
not change our conclusion on the detectability in the next section IV as we
will give comments there.

Inflation generates stochastic GWs from vacuum fluctuations. A remarkable
point is that inflation can generate the superhorizon modes with $k\eta\ll 1$
whereas the other causal mechanism in the post-inflationary universe cannot.
On superhorizon scales, the solutions of the evolution equation in the
expanding universe

$\displaystyle\hat{h}_{A}^{\prime\prime}+2{\cal
H}\hat{h}_{A}^{\prime}+k^{2}\hat{h}_{A}=0\,,$ (11)

are constituted by constant and decaying modes. Here, ${\cal H}$ is the
conformal Hubble parameter: ${\cal H}\equiv a^{\prime}/a$. The prime ′ denotes
the derivative with respect to the conformal time $\eta$. Shortly after the
horizon crossing during inflation, the amplitude of the decaying mode
decreases quickly and thus the Fourier amplitude $\hat{h}_{A}(\eta,{\bm{k}})$
only contains a single statistical variable:

$\displaystyle\hat{h}_{A}(\eta,{\bm{k}})\to\chi_{k}(\eta)\hat{h}^{\text{(prim)}}_{A,{\bm{k}}}\,,$
(12)

where $\chi(k)$ is the transfer function with $\chi_{k}(\eta)\to 1$ for
$k\eta\ll 1$. In the standard inflationary scenario, the primordial amplitudes
$\hat{h}^{\text{(prim)}}_{A,{\bm{k}}}$ are Gaussian random variables with
statistical homogeneity and isotropy:

$\displaystyle\langle~{}[h^{(\text{prim})}_{A,{\bm{k}}_{1}}]^{\dagger}\hat{h}^{\text{(prim)}}_{B,{\bm{k}}_{2}}~{}\rangle=\delta_{AB}P_{h}(k_{1})\delta^{3}({\bm{k}}_{1}-{\bm{k}}_{2})\,.$
(13)

Matching the local solution (7) to the superhorizon solution (12), we find
that the two amplitudes $\hat{\cal A}^{{\rm(p)}}_{A}({\bm{k}})$, $\hat{\cal
A}^{{\rm(n)}}_{A}({\bm{k}})$ should be correlated with each other for the
inflationary GWs.

We can estimate the correlations between the positive and negative frequency
modes by solving the following evolution equation for the transfer function,

$\displaystyle\chi_{k}^{\prime\prime}+2{\cal
H}\chi_{k}^{\prime}+k^{2}\chi_{k}=0\,.$ (14)

In the subhorizon regime, it has the WKB solutions $\chi_{k}(\eta)\propto
e^{\pm ik\eta}/a$. Imposing the initial condition $\chi_{k}(\eta)\to 1$ in the
superhorizon regime, the subhorizon solution has both positive and negative
frequency modes with the same amplitude because $\chi_{k}$ should be real:

$\displaystyle\chi_{k}(\eta)=\frac{\alpha_{k}e^{-ik\eta}+\alpha_{k}^{\ast}e^{ik\eta}}{a(\eta)}\,,$
(15)

where $\alpha_{k}$ is a constant. We can find an analytic solution,

$\displaystyle\chi_{k}(\eta)=\frac{e^{ik\eta}-e^{-ik\eta}}{2ik\eta}\,,$ (16)

in the radiation-dominated era, where the relevant modes for the GW
interferometers re-enter the horizon. This solution can be rewritten as

$\displaystyle\chi_{k}(\eta)=\frac{a(\eta_{k})}{a(\eta)}\left(\frac{e^{ik\eta}-e^{-ik\eta}}{2i}\right)\,,$
(17)

introducing the horizon re-entry time $\eta_{k}$ by $k\eta_{k}=1$. The
coefficients $\alpha_{k}$ in Eq. (15) at the present time are obtained by
connecting this solution to the late-time universe. Unless a nonadiabatic
transition occurs, the solution at the present time is given in the form (17)
(see Refs. [54, 55] for a more accurate transfer function).

Comparing Eq. (17) with Eq. (7), we find

$\displaystyle\begin{aligned} \hat{\cal
A}^{{\rm(p)}}_{A}({\bm{k}})&=\overline{\cal
T}_{k}e^{-i(k\eta_{0}-\frac{\pi}{2})}\hat{h}^{\text{(prim)}}_{A,{\bm{k}}}\,,\\\
\hat{\cal A}^{{\rm(n)}}_{A}({\bm{k}})&=\overline{\cal
T}_{k}e^{i(k\eta_{0}-\frac{\pi}{2})}\hat{h}^{\text{(prim)}}_{A,{\bm{k}}}\,,\end{aligned}$
(18)

with

$\displaystyle\overline{\cal
T}_{k}\equiv\frac{1}{2}\frac{a(\eta_{k})}{a(\eta_{0})}\,.$ (19)

Here, we have rewritten the conformal time $\eta$ in terms of the cosmic time
$t$ as $\eta\simeq\eta_{0}+t/a(\eta_{0})$. 777We have estimated the cosmic
time as $t=\int_{\eta_{0}}^{\eta}a(\eta^{\prime}){\rm d}\eta^{\prime}\simeq
a(\eta_{0})(\eta-\eta_{0})$ by neglecting the evolution of $a(\eta)$ during
the observation. The damping factors and phase shifts in Eq. (18) are
geometrically determined. As expected, the two amplitudes $\hat{\cal
A}^{{\rm(p)}}_{A}({\bm{k}})$, $\hat{\cal A}^{{\rm(n)}}_{A}({\bm{k}})$ are
represented by the single statistical variable
$\hat{h}^{\text{(prim)}}_{A,{\bm{k}}}$ and thus are correlated with each
other. Substituting these results to Eq. (8), we find

$\displaystyle\hat{h}_{\pm}(f,{\bm{n}})=$
$\displaystyle\quad\begin{cases}f^{2}\overline{\cal T}_{2\pi|f|}e^{-2\pi
if\eta_{0}+\frac{i\pi}{2}}\hat{h}^{\text{(prim)}}_{\pm,2\pi
f{\bm{n}}}\quad\text{for}~{}f>0\,,\\\\[6.0pt] f^{2}\overline{\cal
T}_{2\pi|f|}e^{-2\pi
if\eta_{0}-\frac{i\pi}{2}}\hat{h}^{\text{(prim)}}_{\mp,2\pi
f{\bm{n}}}\quad\text{for}~{}f<0\,,\end{cases}$ (20)

and thus the following relation:

$\displaystyle h_{\pm}(f,{\bm{n}})=-e^{4\pi
if\eta_{0}}h_{\mp}(-f,-{\bm{n}})\,.$ (21)

This relation shows that:

  1. (i)

There is a one-to-one correspondence between realizations of SGWB with the
opposite frequencies, directions, and circular polarizations.

  2. (ii)

Their amplitudes are the same.

  3. (iii)

Their phase difference is huge and proportional to the frequency $f$.

Figure 1: Propagation of the inflationary GWs in the homogeneous universe
(vertical: the time direction, horizontal: the spatial direction parallel to
${\bm{k}}$). The point $O$ represents the observer’s position. The points
$\overline{P}_{L}$ and $\overline{P}_{R}$ represent the points where right-
and left-moving null geodesics cross the constant-time hypersurface
$\eta=\eta_{k}$, respectively.

These results can be easily understood from Fig. 1. The inflationary GWs
induce coherent standing waves on the constant-time hypersurface
$\eta=\eta_{k}$; left- and right-moving modes are emitted with the same
amplitude and the definite phase difference at each point. The positive
(negative) frequency modes at the observer’s position $O$ are the right (left)
moving modes coming from the point $\overline{P}_{L}$ ($\overline{P}_{R}$).
The amplitudes of the positive and negative frequency modes are the same
because the right and left moving modes are damped by the cosmic expansion at
the same rate. Since the phase is conserved along the null geodesic, the phase
difference is given by the number of cycles between $\overline{P}_{L}$ and
$\overline{P}_{R}$, $2k\eta_{0}$, with a small correction from the intrinsic
phase difference between the right- and left-moving modes on the hypersurface
$\eta=\eta_{k}$.

Using Eq. (21), it is easy to find the relation

$\displaystyle A_{h}^{\text{(D)}}(f)=-S_{h}^{\text{(D)}}(f)e^{4\pi
if\eta_{0}}\,,$ (22)

between the two spectral densities in Eq. (II). Therefore, inflation predicts
large antipodal correlations.

##  IV  (Un)Detectability of the antipodal correlations

###  IV.1  The argument in Allen et al. (1999)

We review the argument in Allen et al. (1999) [43] on the undetectability of
the $A_{h}$ term in Eq. (II). The $A_{h}$ term in Eq. (22) is a highly
oscillating function of $f$. Its period is of the order of $1/T_{\rm age}$ for
the age of the Universe $T_{\rm age}\sim\eta_{0}$. This oscillation has a
clear physical interpretation: it is interference between GWs from the
antipodal points $\overline{P}_{L}$ and $\overline{P}_{R}$ in Fig. 1.

The point of the argument is that $\hat{h}_{A}(f,{\bm{n}})$ is not an
observable: our frequency resolution is fundamentally limited by the
observation time $T$ as $\Delta f\sim 1/T\gg 1/T_{\rm age}$. To take into
account the finite frequency resolution, we introduce the smoothed quantity,

$\displaystyle\hat{h}_{A;T}(f,{\bm{n}})\equiv\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}~{}W_{T}(f-f^{\prime})\hat{h}_{A}(f^{\prime},{\bm{n}})\,,$ (23)

and consider it as observable. Here, $W_{T}(f)$ is the window function with
the width $\Delta f\sim 1/T$. For example, when we use the short-time Fourier
transform,

$\displaystyle\hat{h}_{A;T}(f,{\bm{n}})=\int_{0}^{T}\\!{\rm
d}t~{}\hat{h}_{A}(t,{\bm{n}})e^{2\pi ift}\,,$ (24)

the window function is given by

$\displaystyle W_{T}(f)=\frac{e^{i\pi fT}\sin(\pi fT)}{\pi f}\,.$ (25)

Computing the antipodal correlations for the smoothed quantity (23), we find

$\displaystyle\langle\hat{h}^{\dagger}_{\pm;T}(f,{\bm{n}})\hat{h}_{\mp;T}(-f,-{\bm{n}})\rangle=$
$\displaystyle\hskip 56.9055pt\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}~{}|W_{T}(f-f^{\prime})|^{2}A_{h}^{\text{(D)}}(f^{\prime})\,,$
(26)

and thus

$\displaystyle\langle\hat{h}^{\dagger}_{\pm;T}(f,{\bm{n}})\hat{h}_{\mp;T}(-f,-{\bm{n}})\rangle=$
$\displaystyle\hskip 28.45274pt-\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}~{}|W_{T}(f-f^{\prime})|^{2}S_{h}^{\text{(D)}}(f^{\prime})e^{4\pi
if^{\prime}\eta_{0}}\,,$ (27)

using the relation (22). Therefore, even when we take the best resolution
$\Delta f\sim 1/T$, the $A_{h}$ term is erased by smoothing over the
unresolvable frequencies in eq. (IV.1) unless the spectral density
$S_{h}^{\text{(D)}}(f)$ has a very sharp peak with a width much less than
$1/T_{\rm age}$. The situation becomes worse when we take into account the
inhomogeneities. The inhomogeneities introduce the ${\bm{n}}$-dependent phase
$e^{2\pi if\eta_{0}\hat{\phi}({\bm{n}})}$ in eq. (22) with the function
$\hat{\phi}({\bm{n}})$ written in terms of the gravitational potential along
the line-of-sight [48, 49, 47]. Thus, the $A_{h}$ term also rapidly oscillates
for the direction ${\bm{n}}$ and vanishes when smoothed over ${\bm{n}}$. In
conclusion, provided that the spectral density $S_{h}(f)$ slowly varies with
respect to $f$, the correlation function for the smoothed field becomes

$\displaystyle\langle\hat{h}^{\dagger}_{A;T}(f_{1},{\bm{n}}_{1})\hat{h}_{B;T}(f_{2},{\bm{n}}_{2})\rangle=$
$\displaystyle\hskip
56.9055pt\frac{S_{h}^{\text{(D)}}(f_{1})\delta_{AB}}{8\pi}\delta_{T}(f_{1}-f_{2})\delta^{2}({\bm{n}}_{1},{\bm{n}}_{2})\,,$
(28)

where

$\displaystyle\delta_{T}(f_{1}-f_{2})\equiv\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}~{}W_{T}(f_{1}-f_{2}+f^{\prime})W^{\ast}_{T}(f^{\prime})\,,$ (29)

and indistinguishable from a non-inflationary SGWB (II).

The root of the cancellation is the fact that the phase difference between
$h_{\pm}(f,{\bm{n}})$ and $h_{\mp}(-f,-{\bm{n}})$ is rapidly oscillating with
respect to the frequency $f$ (see Eq. (21)). This motivates us to use the
intensity map,

$\displaystyle\hat{I}_{A}(f,{\bm{n}})\equiv|\hat{h}_{A}(f,{\bm{n}})|^{2}\,,$
(30)

to detect the antipodal correlations. From Eq. (21), it is easy to see that
there is a coincidence between the realizations of the intensity with the
opposite directions:

$\displaystyle\hat{I}_{\pm}(f,{\bm{n}})=\hat{I}_{\pm}(f,-{\bm{n}})\,,$ (31)

by using the reality condition
$[\hat{h}_{\pm}(f,{\bm{n}})]^{\dagger}=\hat{h}_{\mp}(-f,{\bm{n}})$. This
relation is not modified much even when we take into account the propagation
through the inhomogeneous universe, because the modification is the order of
the cosmological perturbations [56]. Since there is no problematic phase
factor in Eq. (31), the intensity map would work to detect the antipodal
correlations. In the next subsection, we will discuss this possibility.

###  IV.2  Antipodal correlations in the intensity map

In this subsection, we discuss whether the antipodal correlations can be
detected by using the intensity map,

$\displaystyle\hat{I}_{A}(f,{\bm{n}})\equiv|\hat{h}_{A}(f,{\bm{n}})|^{2}\,.$
(32)

To take into account the finite frequency resolution, we introduce the
intensity of the smoothed quantity (23) by

$\displaystyle\hat{I}_{A;T}(f,{\bm{n}})\equiv|\hat{h}_{A;T}(f,{\bm{n}})|^{2}\,,$
(33)

and investigate whether the antipodal relation (31) can be confirmed through
it. We would like to remark that the quantity (33) is not the smoothing of the
intensity (32):

$\displaystyle\hat{I}_{A;T}(f,{\bm{n}})\neq\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}|W_{T}(f-f^{\prime})|^{2}\hat{I}_{A}(f^{\prime},{\bm{n}})\,,$ (34)

while it is true when the ensemble average is taken:

$\displaystyle\langle\hat{I}_{A;T}(f,{\bm{n}})\rangle=\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}|W_{T}(f-f^{\prime})|^{2}\langle\hat{I}_{A}(f^{\prime},{\bm{n}})\rangle\,.$
(35)

By using the relations (III.2), we can rewrite the smoothed intensity (33) as

$\displaystyle\hat{I}_{A;T}(f,{\bm{n}})=$
$\displaystyle\qquad\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime\prime}~{}W^{\ast}_{T}(f-f^{\prime})W_{T}(f-f^{\prime\prime})f^{\prime
2}f^{\prime\prime 2}\overline{\cal T}_{2\pi|f^{\prime}|}\overline{\cal
T}_{2\pi|f^{\prime\prime}|}~{}e^{-2\pi
i(f^{\prime\prime}-f^{\prime})\eta_{0}}[\hat{h}^{\text{(prim)}}_{A^{\prime},2\pi
f^{\prime}{\bm{n}}}]^{\dagger}\hat{h}^{\text{(prim)}}_{A^{\prime\prime},2\pi
f^{\prime\prime}{\bm{n}}}\,,$ (36)

where $A^{\prime}$ and $A^{\prime\prime}$ are $+A$ for
$f^{\prime},f^{\prime\prime}>0$ and $-A$ for $f^{\prime},f^{\prime\prime}<0$.
We can see that the problematic phase factor $e^{-2\pi
i(f^{\prime\prime}-f^{\prime})\eta_{0}}$ in the smoothed intensity
$\hat{I}_{A;T}(f,{\bm{n}})$ remains unless the other factor in the integrand
has a sharp peak at $f^{\prime\prime}=f^{\prime}$ with the width
$|f^{\prime\prime}-f^{\prime}|\ll 1/T_{\rm age}$. This is not the case for the
realization (33). Therefore, unlike the unsmoothed intensity (32), the
antipodal relation (31) does not hold for the realizations of the smoothed
intensity $\hat{I}_{A;T}(f,{\bm{n}})$:

$\displaystyle\hat{I}_{\pm;T}(f,{\bm{n}})\neq\hat{I}_{\pm;T}(f,-{\bm{n}})\,.$
(37)

The different phase factor $e^{+2\pi i(f^{\prime\prime}-f^{\prime})\eta_{0}}$
appears for $\hat{I}_{\pm;T}(f,-{\bm{n}})$ instead of $e^{-2\pi
i(f^{\prime\prime}-f^{\prime})\eta_{0}}$ in Eq. (IV.2).

Here, we discuss whether it is possible to test the antipodal relation (31) by
constructing an appropriate estimator. First, we can see that the higher-order
statistics is of no use for this purpose. This becomes clear by decomposing
the smoothed intensity $\hat{I}_{A;T}(f,{\bm{n}})$ into the ensemble average
and the deviation from it:

$\displaystyle\hat{I}_{A;T}=\langle\hat{I}_{A;T}\rangle+\delta\hat{I}_{A;T}\,.$
(38)

These two terms $\langle\hat{I}_{A;T}\rangle$ and $\delta\hat{I}_{A;T}$
correspond to the contributions from $f^{\prime\prime}=f^{\prime}$ and
$f^{\prime\prime}\neq f^{\prime}$ respectively in the integral (IV.2) because
$\langle[\hat{h}^{\text{(prim)}}_{A,2\pi
f^{\prime}{\bm{n}}}]^{\dagger}\hat{h}^{\text{(prim)}}_{A,2\pi
f^{\prime\prime}{\bm{n}}}\rangle$ contains
$\delta(f^{\prime\prime}-f^{\prime})$. Therefore, the problematic phase factor
remains in the deviation $\delta\hat{I}_{A;T}$ and spoils the antipodal
relation. In fact, we can show that the antipodal contribution in the two-
point function vanishes with assuming the Gaussianity of
$\hat{h}_{A}(f,{\bm{n}})$:

$\displaystyle
C_{I}({\bm{n}}_{1},{\bm{n}}_{2})\equiv\langle\delta\hat{I}_{A;T}(f_{1},{\bm{n}}_{1})\delta\hat{I}_{B;T}(f_{2},{\bm{n}}_{2})\rangle\,,$
(39)

can be rewritten in terms of the correlation functions of
$\hat{h}_{A;T}(f,{\bm{n}})$ as

$\displaystyle
C_{I}({\bm{n}}_{1},{\bm{n}}_{2})=|\langle\hat{h}^{\dagger}_{A;T}(-f_{1},{\bm{n}}_{1})\hat{h}_{B;T}(f_{2},{\bm{n}}_{2})\rangle|^{2}$
$\displaystyle\hskip
56.9055pt+|\langle\hat{h}^{\dagger}_{A;T}(f_{1},{\bm{n}}_{1})\hat{h}_{B;T}(f_{2},{\bm{n}}_{2})\rangle|^{2}\,,$
(40)

where the reality condition
$\hat{h}^{\dagger}_{A;T}(f,{\bm{n}})=\hat{h}_{A;T}(-f,{\bm{n}})$ has been
used. Using the expression (IV.1) for the correlation functions of
$\hat{h}_{A;T}(f,{\bm{n}})$, we can find

$\displaystyle
C_{I}({\bm{n}}_{1},{\bm{n}}_{2})\propto\delta^{2}({\bm{n}}_{1},{\bm{n}}_{2})\,,$
(41)

and the coefficient is written only in terms of the spectral density
$S_{h}(f)$. From similar arguments, we can show that higher-point functions
are of no use for testing the antipodal relation (31).

The remaining possibility is a one-point function. The problematic phase
factor in Eq. (IV.2) is erased in the ensemble average

$\displaystyle\langle\hat{I}_{A;T}(f,{\bm{n}})\rangle\equiv
I_{A;T}(f,{\bm{n}})\,,$ (42)

because $\langle[\hat{h}^{\text{(prim)}}_{A,2\pi
f^{\prime}{\bm{n}}}]^{\dagger}\hat{h}^{\text{(prim)}}_{A,2\pi
f^{\prime\prime}{\bm{n}}}\rangle$ contains
$\delta(f^{\prime\prime}-f^{\prime})$. This quantity is used for mapping SGWB
in the literature (e.g., Refs. [57, 51]). However, the averaging
simultaneously erases the directional dependence in the intensity when the
statistical isotropy is assumed: introducing the anisotropies

$\displaystyle\Delta I_{A;T}(f,{\bm{n}})\equiv
I_{A;T}(f,{\bm{n}})-\bar{I}_{A;T}(f)\,,$ (43)

with the angular average in the sky $\bar{I}_{A;T}(f)$,

$\displaystyle\Delta I_{A;T}(f,{\bm{n}})=0\,.$ (44)

Therefore, we cannot find an estimator of the intensity that is sensitive to
the antipodal correlations but does not suffer from the problematic phase
factor in the standard inflationary modes with the statistical homogeneity and
isotropy (13). We have illustrated the situation in Fig. 2.

Figure 2: The angular correlations in the intensity map. The circle of the
dotted line represents the intersection between the past light cone and the
hypersurface $\eta=\eta_{k}$. The direction and length of the arrows indicate
the moving direction and intensity of GWs, respectively. The intensity of GWs
is the same for modes with parallel moving directions and uncorrelated between
modes with nonparallel moving directions (see Eq. (13)). The radius of the
shaded circles represents the expectation values of the intensity. It is
independent of the moving direction and the emission point in the standard
inflationary models with statistical homogeneity and isotropy as depicted in
the figure.

The situation changes for inflationary models with statistical anisotropy,
i.e., hypothesis (b) in the section II is broken (see, e.g., Refs. [58, 59,
60, 61] for concrete models)

$\displaystyle\langle~{}[h^{(\text{prim})}_{A,{\bm{k}}_{1}}]^{\dagger}\hat{h}^{\text{(prim)}}_{B,{\bm{k}}_{2}}~{}\rangle=\delta_{AB}P_{h}({\bm{k}}_{1})\delta^{3}({\bm{k}}_{1}-{\bm{k}}_{2})\,.$
(45)

In this case, the anisotropies in the averaged intensity are not erased,

$\displaystyle\Delta I_{A;T}(f,{\bm{n}})\neq 0\,,$ (46)

while the sharp peak $\delta(f^{\prime}-f^{\prime\prime})$ still appears due
to the statistical homogeneity and thus the problematic phase factor
disappears:

$\displaystyle I_{A;T}(f,{\bm{n}})=$
$\displaystyle\qquad\int_{-\infty}^{\infty}\\!{\rm
d}f^{\prime}~{}|W_{T}(f-f^{\prime})|^{2}f^{\prime 4}\overline{\cal
T}_{2\pi|f^{\prime}|}^{2}P_{h}(2\pi f^{\prime}{\bm{n}})\,.$ (47)

The intensity (IV.2) satisfies the antipodal relation for the anisotropies

$\displaystyle\Delta I_{\pm;T}(f,{\bm{n}})=\Delta I_{\pm;T}(f,-{\bm{n}})\,,$
(48)

as a consequence of the standing-wave nature of the inflationary GWs (18).
Therefore, the inflationary SGWB can be distinguished from the other
components if we detect (i) non-vanishing anisotropies $\Delta
I_{A;T}(f,{\bm{n}})$ and (ii) their antipodal relation (48).

Let us also comment on the case when hypothesis (e) on polarization is broken
[62, 63, 64, 65, 66, 67, 68, 69]. In this case, we can show antipodal
relations for all the Stokes parameters through the relation (III.2). With the
same arguments above, these antipodal relations are undetectable in the
isotropic case and detectable in the anisotropic case. In the detectable case,
they will give more evidence for the inflationary GWs.

Before closing this section, it might be noteworthy to mention a difference
from CMB. In contrast to GWs, electromagnetic waves (EMWs) are scattered many
times by electrons in the early universe. Therefore, the angular correlations
intrinsic in EMWs are erased, and there is no counterpart of the antipodal
correlations in the CMB anisotropies. Instead, the CMB angular correlations
are a tracer of the inhomogeneous background: the intensity is spatially
modulated in the vicinity of an emission point by long-wavelength
perturbations, and EMWs in these regions are scattered into the line-of-sight
direction. The statistical isotropy at each emission point is locally broken
by the long-wavelength perturbations. Moreover, the long-wavelength
perturbations also break the statistical homogeneity among the emission points
in Fig. 2. Therefore, the argument of Eq. (44) is not applied to this type of
angular correlation.

##  V  Conclusion

The measurement of the inflationary SGWB is one of the main goals of future GW
experiments. One obstacle to achieving it is the isolation of the inflationary
SGWB from the other components generated by the unresolvable astronomical and
cosmological GW sources. In this paper, we argued the detectability of a
unique and universal property of the inflationary SGWB: antipodal
correlations, i.e., correlations of GWs from opposite directions.

It was argued in Allen et al. [43] that the conclusion is negative when we use
a phase-coherent method, i.e., the standard strain correlation analysis, due
to the phase oscillation unresolvable in the observation time. We thus
investigate whether we can construct a phase-incoherent estimator of the
intensity map to detect the antipodal correlations. We found that the
conclusion depends on whether the inflationary GWs have statistical isotropy
or not. Under the standard assumption of statistical homogeneity and isotropy,
it is impossible to find an observable that is sensitive to the antipodal
correlations but does not suffer from the problematic phase factor: the
intensity constructed from the observed GW strain still has the annoying phase
factor that erases the antipodal correlations. The ensemble average can get
rid of the phase factor but simultaneously drops the angular information due
to statistical isotropy. However, the latter argument is not applied to the
inflationary models with statistical anisotropy. We can find a non-vanishing
observable for the antipodal correlations and thus conclude that SGWB from
anisotropic inflation is distinguishable from the other components.

Our argument can be applied to other types of angular correlations. The
problematic phase factor erases any types of angular correlations in the
strain and non-averaged intensity. The ensemble average can get rid of the
problematic phase factor but simultaneously drops the angular information
under statistical isotropy. On the other hand, we can measure the angular
correlations in CMB even under statistical isotropy. A natural question is
thus whether we can find a way to measure the angular correlations in SGWB
888The Boltzmann approach [70] is widely used to estimate the SGWB
anisotropies as well as the CMB anisotropies. Our argument implies that we
need to carefully discuss how an estimator of the distribution function
(intensity, energy density) should be defined.  and what kind of angular
correlations are detectable. As we have remarked in Sec. IV.2, the local
violation of statistical isotropy and homogeneity by long-wavelength
perturbations is crucial for the detectability of the CMB angular
correlations. Because the CMB angular correlations are detectable, it would be
possible to find an estimator for the SGWB angular correlations induced by
long-wavelength perturbations, e.g., through propagation and long-short
wavelength mode couplings [56, 71, 29, 30]. In a subsequent paper, we will
discuss how we should define the estimator to get rid of the problematic phase
factor with (partially) keeping the angular information.

###### Acknowledgements.

We thank the anonymous referee for the helpful suggestion to discuss the
anisotropic case. This research was supported by the JSPS Grant-in-Aid for
Scientific Research (No. 17K14286, No. 19H01891, No. 20H05860) and JST SPRING
(No. JPMJSP2111).

##  Appendix A  Correlation analysis in the time domain

In the main text, we have shown that the antipodal correlations cannot be
detected with the maps of the Fourier amplitude. In both methods, the root of
the undetectability is the fundamental limitation in frequency resolution due
to the finite observation time. In this Appendix, we will show the same fact
for the original signal (1) without taking its finite-time Fourier transform
(24) to confirm that the undetectability discussed in section IV.1 is not a
result of the limitation of the Fourier analysis.

We compute the following correlation functions in the time domain:

$\displaystyle\langle\hat{h}_{A}(t-\tau/2,{\bm{n}}_{1})\hat{h}_{B}(t+\tau/2,{\bm{n}}_{2})\rangle\,,$
(49)

where

$\displaystyle\hat{h}_{A}(t,{\bm{n}})\equiv\int_{-\infty}^{\infty}{\rm d}f\
\hat{h}_{A}(f,{\bm{n}})e^{-2\pi ift}\,.$ (50)

The correlation functions of the filtered signals and the intensity map can be
written in terms of them.

For the standard contribution ${\bm{n}}_{1}={\bm{n}}_{2}(={\bm{n}})$, the
correlation functions (49) are computed as

$\displaystyle\langle\hat{h}_{A}(t-\tau/2,{\bm{n}})\hat{h}_{B}(t+\tau/2,{\bm{n}})\rangle$
$\displaystyle\hskip 56.9055pt=\delta_{AB}\int_{-\infty}^{\infty}{\rm d}f\
S^{\text{(D)}}_{h}(f)e^{2\pi if\tau}\,.$ (51)

The result is independent of $t$. Therefore, we can use
$\hat{h}_{A}(t-\tau/2,{\bm{n}})\hat{h}_{B}(t+\tau/2,{\bm{n}})$ for different
values of $t$ as samples to estimate

$\displaystyle C_{S}(\tau)\equiv\int_{-\infty}^{\infty}{\rm d}f\
S^{\text{(D)}}_{h}(f)e^{2\pi if\tau}\,.$ (52)

The function $C_{S}(\tau)$ is not small for sufficiently small values of
$\tau$ because

$\displaystyle C_{S}(0)=\int_{-\infty}^{\infty}{\rm d}f\
S^{\text{(D)}}_{h}(f)\,,$ (53)

and the spectral density $S^{\text{(D)}}_{h}(f)$ is positive semi-definite. We
can estimate the spectral density $S^{\text{(D)}}_{h}(f)$ by taking the short-
time Fourier transform of $C_{S}(\tau)$. This corresponds to the fact shown in
the previous section. To further increase the sensitivity, in the standard
correlation analysis, we usually apply the optimal filter $Q_{S}(\tau)$ to the
estimator of $C_{S}(\tau)$ assuming the shape of $S^{\text{(D)}}_{h}(f)$ (e.g.
Ref. [53]):

$\displaystyle\int_{0}^{T}{\rm d}\tau\
C_{S}(\tau)Q_{S}(\tau)=\int_{-\infty}^{\infty}{\rm d}f\
S^{\text{(D)}}_{h}(f)Q_{S;T}(f)\,,$ (54)

where $Q_{S;T}(f)$ is the short-time Fourier transform of $Q_{S}(\tau)$. 999To
be exact, the integration domain for $\tau$ is not $[0,T]$. However, we do not
need to care about it because the optimal filter $Q_{A}(\tau)$ decays quickly
as $\tau$ increases. The positive semi-definiteness of $S^{\text{(D)}}_{h}(f)$
ensures that the signal (54) for the overall amplitude of the spectrum can be
enhanced compared to the noise by choosing the filter function $Q_{S}(\tau)$
appropriately.

For the antipodal contribution ${\bm{n}}_{1}=-{\bm{n}}_{2}(={\bm{n}})$, the
correlation functions (49) are computed as

$\displaystyle\langle\hat{h}_{A}(t-\tau/2,{\bm{n}})\hat{h}_{B}(t+\tau/2,-{\bm{n}})\rangle$
$\displaystyle\hskip 56.9055pt=\delta_{A(-B)}\int_{-\infty}^{\infty}{\rm d}f\
A^{\text{(D)}}_{h}(f)e^{-4\pi ift}\,.$ (55)

The result is independent of $\tau$. Therefore, we can use
$\hat{h}_{A}(t-\tau/2,{\bm{n}})\hat{h}_{B}(t+\tau/2,-{\bm{n}})$ for different
values of $\tau$ as samples to estimate 101010Note that the subscript $A$ of
$C_{A}$ is not the index of the polarization but represents that it is a
quantity for the antipodal correlations.

$\displaystyle C_{A}(t)$ $\displaystyle\equiv\int_{-\infty}^{\infty}{\rm d}f\
A^{\text{(D)}}_{h}(f)e^{-4\pi ift}$
$\displaystyle=-\int_{-\infty}^{\infty}{\rm d}f\ S^{\text{(D)}}_{h}(f)e^{4\pi
if(\eta_{0}-t)}\,.$ (56)

Here, we have used Eq. (22) in the second line. The discussion seems to be
parallel to the standard one. However, the problem is that $C_{A}(t)$ is
extremely small compared to $C_{S}(\tau)$. Due to the phase factor $e^{4\pi
if\eta_{0}}$, the large contributions to $C_{A}(t)$ come from the modes with
$f\lesssim 1/\eta_{0}\sim 1/T_{\rm age}$. Moreover, when we take into account
the ${\bm{n}}$-dependent phase from the inhomogeneities, $e^{2\pi
if\eta_{0}\hat{\phi}({\bm{n}})}$, it will effectively work as the overlap
reduction function: it suppresses contributions other than low-frequency modes
with $f\lesssim 1/(\epsilon\eta_{0})$. Here, $\epsilon$ represents the typical
magnitude of $\hat{\phi}({\bm{n}})$, i.e., the order of the scalar
perturbations. However, the signal $h_{A}(t,{\bm{n}})$ does not contain such
low-frequency modes because any detector cannot detect GWs that do not vary
over the observation time $T$. Therefore, $C_{A}(t)$ is extremely small and
almost impossible to be detected. In Appendix A.1, we explicitly give
$C_{A}(t)$ for some examples of $S^{\text{(D)}}_{h}(f)$. We can also see that
a filter for $t$, $Q_{A}(t)$, is not effective in increasing the sensitivity.
Applying the filter $Q_{A}(t)$, we obtain

$\displaystyle\int_{0}^{T}{\rm d}t\ C_{A}(t)Q_{A}(t)=$ $\displaystyle\hskip
28.45274pt-\int_{-\infty}^{\infty}{\rm d}f\ S^{\text{(D)}}_{h}(f)e^{4\pi
if\eta_{0}}Q_{A;T}(-2f)\,,$ (57)

where $Q_{A;T}(f)$ is the short-time Fourier transform of $Q_{A}(t)$. For any
choice of the filter function $Q_{A}(t)$, the support of $Q_{A;T}(f)$ has a
width larger than $1/T\gg 1/\eta_{0}$. It is also impossible to cancel the
phase factor $e^{4\pi if\eta_{0}}$ by $Q_{A;T}(f)$. To achieve the
cancellation, the inverse Fourier transform of $Q_{A;T}(f)$ should have a
sharp peak at $t\simeq\eta_{0}\sim T_{\rm age}$ but $t\sim T_{\rm age}$ is not
included in the support of the inverse Fourier transform of $Q_{A;T}(f)$:
$T_{\rm age}\notin[0,T]$. Therefore, the signal cannot be greatly enhanced for
any filter function $Q_{A}(t)$.

###  A.1  Some examples of $C_{A}(t)$

Here, we will compute $C_{A}(t)$,

$\displaystyle C_{A}(t)$ $\displaystyle=-\int_{-\infty}^{\infty}{\rm d}f\
S^{\text{(D)}}_{h}(f)e^{4\pi if(\eta_{0}-t)}$
$\displaystyle=-\int_{0}^{\infty}{\rm d}f\ S_{h}(f)\cos[4\pi
f(\eta_{0}-t)]\,,$ (58)

for some examples of $S_{h}(f)$. Here, we have rewritten the integral in terms
of the single-sided spectral density (4).

####  A.1.1 Power-law spectrum

First, we consider the power-law spectrum $S_{h}(f)\propto f^{-\alpha}$ as
usually assumed for the inflationary SGWB with $\alpha\simeq 3$. As we have
commented in Sec. A, the signal does not contain the low-frequency modes with
$f\lesssim f_{\rm min}\equiv 1/T$ for the observation time $T$. Moreover, the
spectrum should have an upper cut-off frequency $f_{\rm max}$ or the spectral
index should satisfy $\alpha>3$ in order that the total GW energy density is
finite. Therefore, we consider the following integral,

$\displaystyle\int_{f_{\rm min}}^{f_{\rm max}}{\rm d}f\
f^{-\alpha}e^{ifT_{\ast}}\,;\quad T_{\ast}\equiv 4\pi(\eta_{0}-t)\,,$ (59)

whose real part gives $C_{A}(t)$. Because the phase $fT_{\ast}$ is very large
in the integral domain, we can estimate the integral by using the method of
steepest descent. By deforming the path to $[f_{\rm min},f_{\rm
min}+i\infty]\cup[f_{\rm min}+i\infty,f_{\rm max}+i\infty]\cup[f_{\rm
max}+i\infty,f_{\rm max}]$, we can evaluate the integral (59) as

$\displaystyle\int_{f_{\rm min}}^{f_{\rm max}}{\rm d}f\
f^{-\alpha}e^{ifT_{\ast}}=$ $\displaystyle\qquad\int_{0}^{\infty}dp\ (f_{\rm
min}+ip)^{-\alpha}e^{if_{\rm min}T_{\ast}-T_{\ast}p}$
$\displaystyle\qquad\quad-\int_{0}^{\infty}dp\ (f_{\rm
max}+ip)^{-\alpha}e^{if_{\rm max}T_{\ast}-T_{\ast}p}\,,$ (60)

where we have dropped the contribution from the path $[f_{\rm
min}+i\infty,f_{\rm max}+i\infty]$ because
$\lim_{p\to\infty}(f+ip)^{-\alpha}e^{-T_{\ast}p}=0$. Since the damping factor
$e^{-T_{\ast}p}$ suppresses contributions other than $p<1/T_{\ast}\ll f_{\rm
min}<f_{\rm max}$, we obtain

$\displaystyle\int_{f_{\rm min}}^{f_{\rm max}}{\rm d}f\
f^{-\alpha}e^{ifT_{\ast}}\simeq$ $\displaystyle\hskip 56.9055pt\frac{f_{\rm
min}^{-\alpha}e^{if_{\rm min}T_{\ast}}-f_{\rm max}^{-\alpha}e^{if_{\rm
max}T_{\ast}}}{T_{\ast}}\,.$ (61)

Therefore, $C_{A}(t)$ is estimated to be

$\displaystyle C_{A}(t)=$ $\displaystyle\quad-\frac{S_{h}(f_{\rm
min})\sin(f_{\rm min}T_{\ast})-S_{h}(f_{\rm max})\sin(f_{\rm
max}T_{\ast})}{T_{\ast}}\,,$ (62)

with $T_{\ast}\equiv 4\pi(\eta_{0}-t)$. On the other hand, $C_{S}(0)$ given in
Eq. (53) is estimated to be

$\displaystyle C_{S}(0)=\frac{S_{h}(f_{\rm max})f_{\rm max}-S_{h}(f_{\rm
min})f_{\rm min}}{1-\alpha}\,.$ (63)

Comparing Eq. (A.1.1) with Eq. (63), we find that the antipodal contribution
is suppressed at least by the small factor $1/(f_{\rm min}T_{\ast})\sim
T/T_{\rm age}={\cal O}(10^{-10})$ compared to the standard one.

####  A.1.2 Gaussian spectrum

Next, we consider the Gaussian spectrum
$S_{h}(f)\propto\exp[-(f-f_{\ast})^{2}/2\sigma_{f}^{2}]$ as an example of a
spectrum with a peak. We consider the following integral,

$\displaystyle\int_{-\infty}^{\infty}{\rm d}f\
e^{-\frac{(f-f_{\ast})^{2}}{2\sigma_{f}^{2}}}e^{ifT_{\ast}}\,;\quad
T_{\ast}\equiv 4\pi(\eta_{0}-t)\,,$ (64)

where we have extended the integral domain by assuming that the peak is
sufficiently sharp. This integral can be estimated as

$\displaystyle\sqrt{2\pi}\sigma_{f}e^{if_{\ast}T_{\ast}}e^{-\frac{\sigma_{f}^{2}T_{\ast}^{2}}{4}}\,,$
(65)

and thus $C_{A}(t)$ is

$\displaystyle
C_{A}(t)=-\sqrt{2\pi}\sigma_{f}S_{h}(f_{\ast})\cos(f_{\ast}T_{\ast})e^{-\frac{\sigma_{f}^{2}T_{\ast}^{2}}{4}}\,,$
(66)

with $T_{\ast}\equiv 4\pi(\eta_{0}-t)$. On the other hand, $C_{S}(0)$ given in
Eq. (53) is estimated to be

$\displaystyle C_{S}(0)=\sqrt{2\pi}\sigma_{f}S_{h}(f_{\ast})\,.$ (67)

Comparing Eqs. (66) and (67), we find that the antipodal contribution is
suppressed by the small factor $\exp(-\sigma_{f}^{2}T_{\ast}^{2}/4)$ compared
to the standard one. Therefore, the antipodal contribution is extremely small
unless the peak width $\sigma_{f}$ is much less than $1/T_{\ast}\sim 1/T_{\rm
age}$.

## References

  * Starobinsky [1980] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91, 99 (1980).
  * Kazanas [1980] D. Kazanas, Dynamics of the Universe and Spontaneous Symmetry Breaking, Astrophys. J. Lett. 241, L59 (1980).
  * Sato [1981] K. Sato, First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon. Not. Roy. Astron. Soc. 195, 467 (1981).
  * Guth [1981] A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23, 347 (1981).
  * Grishchuk [1974] L. P. Grishchuk, Amplification of gravitational waves in an istropic universe, Zh. Eksp. Teor. Fiz. 67, 825 (1974).
  * Starobinsky [1979] A. A. Starobinsky, Spectrum of relict gravitational radiation and the early state of the universe, JETP Lett. 30, 682 (1979).
  * Rubakov  _et al._ [1982] V. A. Rubakov, M. V. Sazhin, and A. V. Veryaskin, Graviton Creation in the Inflationary Universe and the Grand Unification Scale, Phys. Lett. B 115, 189 (1982).
  * Regimbau [2011] T. Regimbau, The astrophysical gravitational wave stochastic background, Res. Astron. Astrophys. 11, 369 (2011), arXiv:1101.2762 [astro-ph.CO] .
  * Abbott  _et al._ [2017] B. P. Abbott _et al._ (LIGO Scientific, Virgo), Upper Limits on the Stochastic Gravitational-Wave Background from Advanced LIGO’s First Observing Run, Phys. Rev. Lett. 118, 121101 (2017), [Erratum: Phys.Rev.Lett. 119, 029901 (2017)], arXiv:1612.02029 [gr-qc] .
  * Amaro-Seoane  _et al._ [2017] P. Amaro-Seoane, H. Audley, S. Babak, J. Baker, E. Barausse, P. Bender, E. Berti, P. Binetruy, M. Born, D. Bortoluzzi, _et al._ , Laser Interferometer Space Antenna, arXiv e-prints , arXiv:1702.00786 (2017), arXiv:1702.00786 [astro-ph.IM] .
  * Christensen [2019] N. Christensen, Stochastic Gravitational Wave Backgrounds, Rept. Prog. Phys. 82, 016903 (2019), arXiv:1811.08797 [gr-qc] .
  * Caprini and Figueroa [2018] C. Caprini and D. G. Figueroa, Cosmological Backgrounds of Gravitational Waves, Class. Quant. Grav. 35, 163001 (2018), arXiv:1801.04268 [astro-ph.CO] .
  * Abbott  _et al._ [2019] B. P. Abbott _et al._ (LIGO Scientific, Virgo), Search for the isotropic stochastic background using data from Advanced LIGO’s second observing run, Phys. Rev. D 100, 061101 (2019), arXiv:1903.02886 [gr-qc] .
  * Ungarelli and Vecchio [2004] C. Ungarelli and A. Vecchio, A Family of filters to search for frequency dependent gravitational wave stochastic backgrounds, Class. Quant. Grav. 21, S857 (2004), arXiv:gr-qc/0312061 .
  * Adams and Cornish [2014] M. R. Adams and N. J. Cornish, Detecting a Stochastic Gravitational Wave Background in the presence of a Galactic Foreground and Instrument Noise, Phys. Rev. D 89, 022001 (2014), arXiv:1307.4116 [gr-qc] .
  * Parida  _et al._ [2016] A. Parida, S. Mitra, and S. Jhingan, Component Separation of a Isotropic Gravitational Wave Background, JCAP 04, 024, arXiv:1510.07994 [astro-ph.CO] .
  * Pieroni and Barausse [2020] M. Pieroni and E. Barausse, Foreground cleaning and template-free stochastic background extraction for LISA, JCAP 07, 021, [Erratum: JCAP 09, E01 (2020)], arXiv:2004.01135 [astro-ph.CO] .
  * Boileau  _et al._ [2021] G. Boileau, N. Christensen, R. Meyer, and N. J. Cornish, Spectral separation of the stochastic gravitational-wave background for LISA: Observing both cosmological and astrophysical backgrounds, Phys. Rev. D 103, 103529 (2021), arXiv:2011.05055 [gr-qc] .
  * Poletti [2021] D. Poletti, Measuring the primordial gravitational wave background in the presence of other stochastic signals, JCAP 05, 052, arXiv:2101.02713 [gr-qc] .
  * Regimbau  _et al._ [2017] T. Regimbau, M. Evans, N. Christensen, E. Katsavounidis, B. Sathyaprakash, and S. Vitale, Digging deeper: Observing primordial gravitational waves below the binary black hole produced stochastic background, Phys. Rev. Lett. 118, 151105 (2017), arXiv:1611.08943 [astro-ph.CO] .
  * Pan and Yang [2020] Z. Pan and H. Yang, Probing Primordial Stochastic Gravitational Wave Background with Multi-band Astrophysical Foreground Cleaning, Class. Quant. Grav. 37, 195020 (2020), arXiv:1910.09637 [astro-ph.CO] .
  * Sharma and Harms [2020] A. Sharma and J. Harms, Searching for cosmological gravitational-wave backgrounds with third-generation detectors in the presence of an astrophysical foreground, Phys. Rev. D 102, 063009 (2020), arXiv:2006.16116 [gr-qc] .
  * Martinovic  _et al._ [2021a] K. Martinovic, P. M. Meyers, M. Sakellariadou, and N. Christensen, Simultaneous estimation of astrophysical and cosmological stochastic gravitational-wave backgrounds with terrestrial detectors, Phys. Rev. D 103, 043023 (2021a), arXiv:2011.05697 [gr-qc] .
  * Sachdev  _et al._ [2020] S. Sachdev, T. Regimbau, and B. S. Sathyaprakash, Subtracting compact binary foreground sources to reveal primordial gravitational-wave backgrounds, Phys. Rev. D 102, 024051 (2020), arXiv:2002.05365 [gr-qc] .
  * Cusin  _et al._ [2017] G. Cusin, C. Pitrou, and J.-P. Uzan, Anisotropy of the astrophysical gravitational wave background: Analytic expression of the angular power spectrum and correlation with cosmological observations, Phys. Rev. D 96, 103019 (2017), arXiv:1704.06184 [astro-ph.CO] .
  * Adshead  _et al._ [2021] P. Adshead, N. Afshordi, E. Dimastrogiovanni, M. Fasiello, E. A. Lim, and G. Tasinato, Multimessenger cosmology: Correlating cosmic microwave background and stochastic gravitational wave background measurements, Phys. Rev. D 103, 023532 (2021), arXiv:2004.06619 [astro-ph.CO] .
  * Malhotra  _et al._ [2021] A. Malhotra, E. Dimastrogiovanni, M. Fasiello, and M. Shiraishi, Cross-correlations as a Diagnostic Tool for Primordial Gravitational Waves, JCAP 03, 088, arXiv:2012.03498 [astro-ph.CO] .
  * Bartolo  _et al._ [2022] N. Bartolo _et al._ (LISA Cosmology Working Group), Probing Anisotropies of the Stochastic Gravitational Wave Background with LISA, (2022), arXiv:2201.08782 [astro-ph.CO] .
  * Dimastrogiovanni  _et al._ [2020] E. Dimastrogiovanni, M. Fasiello, and G. Tasinato, Searching for Fossil Fields in the Gravity Sector, Phys. Rev. Lett. 124, 061302 (2020), arXiv:1906.07204 [astro-ph.CO] .
  * Dimastrogiovanni  _et al._ [2022a] E. Dimastrogiovanni, M. Fasiello, A. Malhotra, P. D. Meerburg, and G. Orlando, Testing the early universe with anisotropies of the gravitational wave background, JCAP 02 (02), 040, arXiv:2109.03077 [astro-ph.CO] .
  * Dimastrogiovanni  _et al._ [2022b] E. Dimastrogiovanni, M. Fasiello, and L. Pinol, Primordial stochastic gravitational wave background anisotropies: in-in formalization and applications, JCAP 09, 031, arXiv:2203.17192 [astro-ph.CO] .
  * Orlando [2022] G. Orlando, Probing parity-odd bispectra with anisotropies of GW V modes, JCAP 12, 019, arXiv:2206.14173 [astro-ph.CO] .
  * Seto [2007] N. Seto, Quest for circular polarization of gravitational wave background and orbits of laser interferometers in space, Phys. Rev. D 75, 061302 (2007), arXiv:astro-ph/0609633 .
  * Seto [2006] N. Seto, Prospects for direct detection of circular polarization of gravitational-wave background, Phys. Rev. Lett. 97, 151101 (2006), arXiv:astro-ph/0609504 .
  * Smith and Caldwell [2017] T. L. Smith and R. Caldwell, Sensitivity to a Frequency-Dependent Circular Polarization in an Isotropic Stochastic Gravitational Wave Background, Phys. Rev. D 95, 044036 (2017), arXiv:1609.05901 [gr-qc] .
  * Domcke  _et al._ [2020] V. Domcke, J. Garcia-Bellido, M. Peloso, M. Pieroni, A. Ricciardone, L. Sorbo, and G. Tasinato, Measuring the net circular polarization of the stochastic gravitational wave background with interferometers, JCAP 05, 028, arXiv:1910.08052 [astro-ph.CO] .
  * Seto [2020] N. Seto, Measuring Parity Asymmetry of Gravitational Wave Backgrounds with a Heliocentric Detector Network in the mHz Band, Phys. Rev. Lett. 125, 251101 (2020), arXiv:2009.02928 [gr-qc] .
  * Orlando  _et al._ [2021] G. Orlando, M. Pieroni, and A. Ricciardone, Measuring Parity Violation in the Stochastic Gravitational Wave Background with the LISA-Taiji network, JCAP 03, 069, arXiv:2011.07059 [astro-ph.CO] .
  * Martinovic  _et al._ [2021b] K. Martinovic, C. Badger, M. Sakellariadou, and V. Mandic, Searching for parity violation with the LIGO-Virgo-KAGRA network, Phys. Rev. D 104, L081101 (2021b), arXiv:2103.06718 [gr-qc] .
  * Guzzetti  _et al._ [2016] M. C. Guzzetti, N. Bartolo, M. Liguori, and S. Matarrese, Gravitational waves from inflation, Riv. Nuovo Cim. 39, 399 (2016), arXiv:1605.01615 [astro-ph.CO] .
  * Bartolo  _et al._ [2016] N. Bartolo _et al._ , Science with the space-based interferometer LISA. IV: Probing inflation with gravitational waves, JCAP 12, 026, arXiv:1610.06481 [astro-ph.CO] .
  * Grishchuk [1993] L. P. Grishchuk, Quantum effects in cosmology, Class. Quant. Grav. 10, 2449 (1993), arXiv:gr-qc/9302036 .
  * Allen  _et al._ [1999] B. Allen, E. E. Flanagan, and M. A. Papa, Is the squeezing of relic gravitational waves produced by inflation detectable?, Phys. Rev. D 61, 024024 (1999), arXiv:gr-qc/9906054 .
  * Gubitosi and Magueijo [2017] G. Gubitosi and J. Magueijo, The phenomenology of squeezing and its status in non-inflationary theories, JCAP 11, 014, arXiv:1706.09065 [gr-qc] .
  * Gubitosi and Magueijo [2018] G. Gubitosi and J. a. Magueijo, Primordial standing waves, Phys. Rev. D 97, 063509 (2018), arXiv:1711.05539 [gr-qc] .
  * Contaldi and Magueijo [2018] C. R. Contaldi and J. a. Magueijo, Unsqueezing of standing waves due to inflationary domain structure, Phys. Rev. D 98, 043523 (2018), arXiv:1803.03649 [astro-ph.CO] .
  * Margalit  _et al._ [2020] A. Margalit, C. R. Contaldi, and M. Pieroni, Phase decoherence of gravitational wave backgrounds, Phys. Rev. D 102, 083506 (2020), arXiv:2004.01727 [astro-ph.CO] .
  * Bartolo  _et al._ [2019a] N. Bartolo, V. De Luca, G. Franciolini, A. Lewis, M. Peloso, and A. Riotto, Primordial Black Hole Dark Matter: LISA Serendipity, Phys. Rev. Lett. 122, 211301 (2019a), arXiv:1810.12218 [astro-ph.CO] .
  * Bartolo  _et al._ [2019b] N. Bartolo, V. De Luca, G. Franciolini, M. Peloso, D. Racco, and A. Riotto, Testing primordial black holes as dark matter with LISA, Phys. Rev. D 99, 103521 (2019b), arXiv:1810.12224 [astro-ph.CO] .
  * Bartolo  _et al._ [2019c] N. Bartolo, D. Bertacca, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, and G. Tasinato, Anisotropies and non-Gaussianity of the Cosmological Gravitational Wave Background, Phys. Rev. D 100, 121501(R) (2019c), arXiv:1908.00527 [astro-ph.CO] .
  * Renzini and Contaldi [2018] A. I. Renzini and C. R. Contaldi, Mapping Incoherent Gravitational Wave Backgrounds, Mon. Not. Roy. Astron. Soc. 481, 4650 (2018), arXiv:1806.11360 [astro-ph.IM] .
  * Renzini and Contaldi [2019] A. I. Renzini and C. R. Contaldi, Gravitational Wave Background Sky Maps from Advanced LIGO O1 Data, Phys. Rev. Lett. 122, 081102 (2019), arXiv:1811.12922 [astro-ph.CO] .
  * Maggiore [2007] M. Maggiore,  _Gravitational Waves. Vol. 1: Theory and Experiments_ , Oxford Master Series in Physics (Oxford University Press, 2007).
  * Watanabe and Komatsu [2006] Y. Watanabe and E. Komatsu, Improved Calculation of the Primordial Gravitational Wave Spectrum in the Standard Model, Phys. Rev. D 73, 123515 (2006), arXiv:astro-ph/0604176 .
  * Saikawa and Shirai [2018] K. Saikawa and S. Shirai, Primordial gravitational waves, precisely: The role of thermodynamics in the Standard Model, JCAP 05, 035, arXiv:1803.01038 [hep-ph] .
  * Laguna  _et al._ [2010] P. Laguna, S. L. Larson, D. Spergel, and N. Yunes, Integrated Sachs-Wolfe Effect for Gravitational Radiation, Astrophys. J. Lett. 715, L12 (2010), arXiv:0905.1908 [gr-qc] .
  * Mitra  _et al._ [2008] S. Mitra, S. Dhurandhar, T. Souradeep, A. Lazzarini, V. Mandic, S. Bose, and S. Ballmer, Gravitational wave radiometry: Mapping a stochastic gravitational wave background, Phys. Rev. D 77, 042002 (2008), arXiv:0708.2728 [gr-qc] .
  * Kanno  _et al._ [2008] S. Kanno, M. Kimura, J. Soda, and S. Yokoyama, Anisotropic Inflation from Vector Impurity, JCAP 08, 034, arXiv:0806.2422 [hep-ph] .
  * Watanabe  _et al._ [2009] M.-a. Watanabe, S. Kanno, and J. Soda, Inflationary Universe with Anisotropic Hair, Phys. Rev. Lett. 102, 191302 (2009), arXiv:0902.2833 [hep-th] .
  * Kanno  _et al._ [2010] S. Kanno, J. Soda, and M.-a. Watanabe, Anisotropic Power-law Inflation, JCAP 12, 024, arXiv:1010.5307 [hep-th] .
  * Soda [2012] J. Soda, Statistical Anisotropy from Anisotropic Inflation, Class. Quant. Grav. 29, 083001 (2012), arXiv:1201.6434 [hep-th] .
  * Lue  _et al._ [1999] A. Lue, L.-M. Wang, and M. Kamionkowski, Cosmological signature of new parity violating interactions, Phys. Rev. Lett. 83, 1506 (1999), arXiv:astro-ph/9812088 .
  * Contaldi  _et al._ [2008] C. R. Contaldi, J. Magueijo, and L. Smolin, Anomalous CMB polarization and gravitational chirality, Phys. Rev. Lett. 101, 141101 (2008), arXiv:0806.3082 [astro-ph] .
  * Takahashi and Soda [2009] T. Takahashi and J. Soda, Chiral Primordial Gravitational Waves from a Lifshitz Point, Phys. Rev. Lett. 102, 231301 (2009), arXiv:0904.0554 [hep-th] .
  * Sorbo [2011] L. Sorbo, Parity violation in the Cosmic Microwave Background from a pseudoscalar inflaton, JCAP 06, 003, arXiv:1101.1525 [astro-ph.CO] .
  * Maleknejad and Sheikh-Jabbari [2011] A. Maleknejad and M. M. Sheikh-Jabbari, Non-Abelian Gauge Field Inflation, Phys. Rev. D 84, 043515 (2011), arXiv:1102.1932 [hep-ph] .
  * Anber and Sorbo [2012] M. M. Anber and L. Sorbo, Non-Gaussianities and chiral gravitational waves in natural steep inflation, Phys. Rev. D 85, 123537 (2012), arXiv:1203.5849 [astro-ph.CO] .
  * Adshead  _et al._ [2013] P. Adshead, E. Martinec, and M. Wyman, Gauge fields and inflation: Chiral gravitational waves, fluctuations, and the Lyth bound, Phys. Rev. D 88, 021302 (2013), arXiv:1301.2598 [hep-th] .
  * Dimastrogiovanni  _et al._ [2017] E. Dimastrogiovanni, M. Fasiello, and T. Fujita, Primordial Gravitational Waves from Axion-Gauge Fields Dynamics, JCAP 01, 019, arXiv:1608.04216 [astro-ph.CO] .
  * Contaldi [2017] C. R. Contaldi, Anisotropies of Gravitational Wave Backgrounds: A Line Of Sight Approach, Phys. Lett. B 771, 9 (2017), arXiv:1609.08168 [astro-ph.CO] .
  * Alba and Maldacena [2016] V. Alba and J. Maldacena, Primordial gravity wave background anisotropies, JHEP 03, 115, arXiv:1512.01531 [hep-th] .