# Hemispherical asymmetry of primordial power spectra

K. Sravan Kumar1<EMAIL_ADDRESS>João Marto2<EMAIL_ADDRESS>1Department of Physics, Tokyo Institute of Technology 1-12-1 Ookayama, Meguro-
ku, Tokyo 152-8551, Japan  2Departamento de Física e Centro de Matemática e
Aplicações, Universidade da Beira Interior, Rua Marquês D’Ávila e Bolama
6200-001 Covilhã, Portugal

###### Abstract

Although the cosmic microwave background (CMB) is largely understood to be
homogeneous and isotropic, the hemispherical power asymmetry anomaly seems to
breakdown the isotropy, since the difference between the power spectrum in the
two hemispheres of the CMB is of the order of $10^{-2}$ at large angular
scales. We argue that the existence of an anisotropic power spectrum can
simply be explained by considering the existence of two distinct power spectra
in the two hemispheres of the CMB. We achieve this by proposing a double
vacuum structure for (single field) inflationary quantum fluctuations based on
discrete spacetime transformations ($\mathcal{P}\mathcal{T}$) in a
gravitational context, first in de Sitter and finally in quasi de Sitter. As a
result we obtain inflationary quantum fluctuations that are produced in pairs
with which we are able to reproduce the amplitude of the observed dipolar
asymmetry at different scales of $10^{-4}{\rm Mpc^{-1}}\lesssim k\lesssim
1{\rm Mpc^{-1}}$ fixing the pivot scale $k=0.05\,{\rm Mpc}^{-1}$ for $N=55$
e-foldings of inflation. We also predict that a similar HPA should arise for
the primordial gravitational waves (PGWs) as well and we compute the power
asymmetry of PGW spectra at various wave numbers. In our framework we do not
introduce any new parameters.

##  I Introduction

The inflationary paradigm in the early Universe lays a strong foundation for
successful description of the CMB and large scale structure (LSS) of the
Universe so far Starobinsky (1980, 1979); Akrami  _et al._ (2018) with the
predictions of spectral index $n_{s}=0.9649\pm 0.0042$ for $50-60$ e-foldings
of quasi-de Sitter expansion. Besides its apparent success, some CMB anomalies
seem to arise and challenge any current model of inflation. Understanding the
observed anomalies in the latest Planck’s CMB data could further elucidate
some fundamental aspects of the quantum nature of the fluctuations which are
believed to seed the large scale structure (LSS) of the Universe. Concretely,
one of the main CMB anomalies is the presence of a hemispherical power
asymmetry (HPA) at large angular scales Akrami  _et al._ (2019) which has been
seen from the NASA’s Wilkinson Microwave Anisotropy Probe (WMAP) to the latest
Planck data and its significance is standing now at $3.3\sigma$ Akrami  _et
al._ (2014). In short, the HPA is an asymmetry in the two point temperature
correlations on opposite hemispheres of the sky, with slightly higher power in
the southern ecliptic hemisphere and slightly lower power in the northern
ecliptic hemisphere Creswell and Naselsky (2021). This is a significant
challenge to our standard model of (inflationary) cosmology, which predicts
isotropy Yeung and Chu (2022); Aluri  _et al._ (2022). This anomaly seems to
appear significantly at low-multipoles $\ell=2-64$ Schwarz  _et al._ (2016).
The HPA is phenomenologically represented by the following form of the
position dependent primordial power spectrum parametrized as Lyth (2013, 2015)

$\mathcal{P}_{\mathcal{R}}\left(k,\,\hat{\boldsymbol{n}}\right)\simeq\mathcal{P}_{\mathcal{R}\>\>iso}(k)\left(1+2A(k)\hat{\textbf{p}}\cdot\hat{\textbf{n}}\right)\,,$
(1)

where $\mathcal{P}_{\mathcal{R}\>\>iso}(k)$ is the statistically homogeneous
and isotropic power spectrum, $A(k)$ is the amplitude of the observed dipolar
asymmetry which is constrained as $|A|=0.066\pm 0.021$ ($3.3\sigma$) at the
large angular scales $\ell<64$ or at the wave numbers $k\lesssim
0.0045\textrm{ Mpc}^{-1}$. The HPA seems to be present even up to the
multipoles $~{}600$ as seen by the Planck 2013, Hansen _et al._ and Aiola _et
al._ Ade  _et al._ (2014); Hansen  _et al._ (2009); Aiola  _et al._ (2015).
Here $\hat{\boldsymbol{p}}$ is the direction of maximal symmetry and
$\hat{\textbf{n}}=\frac{\boldsymbol{x}}{x_{\text{ls}}}$ is the line of sight
from earth and $x_{\text{ls}}=14,000\textrm{Mpc}^{-1}$ is the co-moving
distance to the surface of last scattering. Following (1) we can derive Lyth
(2013)

$A(k)=\frac{\mathcal{P}_{\mathcal{R}}\left(k,\,\hat{\textbf{n}}\right)-\mathcal{P}_{\mathcal{R}}\left(k,\,-\hat{\textbf{n}}\right)}{4\mathcal{P}_{\mathcal{R}\>\>iso}}\,.$
(2)

The best explanation for the HPA, so far, invoked an additional modulated
superhorizon fluctuation with a wavelength larger than the last scattering
surface, whose interactions with the short wavelength curvature perturbations
introduces large scale dependent primordial non-Gaussianities Schmidt and Hui
(2013); Erickcek  _et al._ (2008a, b); Byrnes and Tarrant (2015); Byrnes  _et
al._ (2016a); Namjoo  _et al._ (2014). It was first noted in Lyth (2013, 2015)
that both the HPA and and the power suppression (PS) at low-multipoles Akrami
_et al._ (2019) could be addressed in a single framework in the context of
scale dependent primordial non-Gaussianities and curvaton-type scenarios. This
later has been generalized and phenomenologically studied in model independent
way Adhikari  _et al._ (2016); Byrnes  _et al._ (2016b). In addition, the HPA
has been addressed via a spatial dependent parametrization of the
$\alpha$-vacua Bogoliubov coefficients of de Sitter space Ashoorioon and
Koivisto (2016), in the context of anisotropic inflationary models involving
gauge fields Maleknejad  _et al._ (2013) and also with spatial dependent
initial conditions for the inflaton field Donoghue  _et al._ (2009). So far
all the studies ascertained that the HPA cannot be explained by the standard
single-field slow-roll inflationary scenario Akrami  _et al._ (2014); Hansen
_et al._ (2009).

Despite the above mentioned had hoc explanations for the HPA Akrami  _et al._
(2019), Planck data actually hints at it for both 2-point and 3-point
correlations (even the equilateral shape for which three wave numbers of the
modes are equal). This means that the anomaly seems to be there in all the
$n$-point correlations, which begs for a fundamental explanation. Therefore,
the answer must be there somewhere in the description of inflationary quantum
fluctuations.

The main goal of this paper is to investigate whether the HPA can be explained
through a new understanding of the quantization of the inflationary
perturbations without introducing any new parameters and any new degrees of
freedom. Thus we only focus on single field slow-roll inflation and propose a
new vacuum for inflationary quantum fluctuations. Being more precise, we pay
careful attention to the discrete symmetries such as
$\mathcal{C}\mathcal{P}\mathcal{T}$ which is expected to be spontaneously
broken in a dynamical spacetime. Our scheme of quantization exactly exploits
this last aspect leading to a promising explanation for the HPA anomaly, with
additional prediction for primordial in gravitational wave spectrum as well.

Throughout the paper, we use the metric signature
$\left(-,\,+,\,+,\,+\right)$, overdot and ′ denotes derivative with respect to
cosmic time (t) and conformal time ($\tau$) respectively, overbar denotes
background values for flat Friedmann-Lemaître-Robertson-Walker (FLRW)
spacetime described by the metric $ds^{2}=-dt^{2}+a^{2}(t)d\textbf{x}^{2}$,
$H=\frac{\dot{a}}{a}$ denotes the Hubble parameter. Four-dimensional indices
are labelled by small Greek letters and three dimensional quantities are
denoted by $i,j=1,2,3$. We set $\hbar=c=1$ and $M_{p}=\frac{1}{\sqrt{8\pi
G}}=1$.

##  II Brief review of inflationary quantum fluctuations in single field
inflation

In this section, we review standard inflationary quantum fluctuations and the
corresponding derivation of observables Martin (2016); Baumann (2018). In
standard single field inflation, we start with a classical action for a
canonical scalar field with a suitable choice of potential.111Note that
inflationary scenarios like Starobinsky and Higgs inflation can be brought to
this form via a conformal transformation Kehagias  _et al._ (2014).

$S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{p}^{2}}{2}R-\frac{1}{2}(\partial_{\mu}\phi)(\partial^{\mu}\phi)-V\left(\phi\right)\right]\,,$
(3)

To implement inflationary framework of expansion, we first solve the equations
of motion of (3) in flat FLRW spacetime and obtain a quasi-de Sitter solution
described by the following conditions on the slow-roll parameters:

$\epsilon=-\frac{\dot{H}}{H^{2}}\ll
1,\quad\eta=\frac{\dot{\epsilon}}{H\epsilon}\ll 1\,\text{,}$ (4)

which can be computed from the Friedmann and Raychaudhuri equations given by

$H^{2}=\frac{1}{3}\left(\frac{\dot{\phi}^{2}}{2}+V(\phi)\right),\quad\dot{H}=-\frac{\dot{\phi}^{2}}{2}\,.$
(5)

The next step is to study the scalar and tensor perturbations around the
obtained inflationary background solution. When it comes to the scalar part we
have perturbations arising from metric fluctuations
($g_{\mu\nu}=\bar{g}_{\mu\nu}+\delta g_{\mu\nu}$) and also the scalar field
fluctuations ($\phi=\bar{\phi}+\delta\phi$). The metric fluctuations can be
represented by the Arnowitt-Deser-Misner (ADM) metric of the form

$ds^{2}=-a^{2}\left(\tau\right)\Big{(}N^{2}d\tau^{2}+\gamma_{ij}\left(dx^{i}+N^{i}d\tau\right)\left(dx^{j}+N^{j}d\tau\right)\Big{)}\,.$
(6)

where $d\tau=\frac{dt}{a}$ is conformal time, $N$ and $N_{i}$ are the lapse
and shift functions respectively. In the unitary gauge, we fix $\delta\phi=0$
then we linearly expand the ADM metric as

$N=1+\delta N,\quad
N_{i}=a\partial_{i}\chi,\quad\gamma_{ij}=\left(1+2\zeta\right)\delta_{ij}+h_{ij}\,,$
(7)

where $\chi$ is a scalar function of spacetime, $\zeta$ is the curvature
perturbation , $h_{ij}$ is the transverse and traceless spin-2 fluctuation.
From the linear perturbed equations of motion we obtain the following
constraints

$\displaystyle\delta N$ $\displaystyle=\frac{\dot{\zeta}}{H}$ (8)
$\displaystyle\partial^{2}\chi$
$\displaystyle=-\frac{\partial^{2}\zeta}{H}+\epsilon a^{2}\dot{\zeta}$

Substituting (6) into the action (3) and expanding to second order in the
fluctuations, we find the second order action for the scalar perturbation:

$\delta^{(2)}S_{s}=\frac{1}{2}\int d\tau d^{3}xa^{2}\frac{\phi^{\prime
2}}{\mathcal{H}^{2}}\Bigg{[}\zeta^{\prime
2}-\left(\partial\zeta\right)^{2}\Bigg{]}\,\text{,}$ (9)

where $\mathcal{H}=\frac{a^{\prime}}{a}=aH$. Similarly we obtain the following
second order action for tensor fluctuations:

$\delta^{(2)}S_{h}=\frac{1}{8}\int d\tau
d^{3}xa^{2}\Bigg{[}h_{ij}^{\prime}h_{ij}^{\prime}-\partial_{k}h_{ij}\partial^{k}h^{ij}\Bigg{]}\,\text{.}$
(10)

What we have so far is totally classical and we haven’t promoted any fields
and conjugate momenta to operators. To quantize the fluctuations we first
redefine the fields as

$v=a\frac{\dot{\phi}}{H}\zeta,\quad u_{ij}=\frac{a}{2}h_{ij}\,,$ (11)

The second order actions in these new variables (which are famously known as
Mukhanov-Sasaki variables) become

$\delta^{(2)}S_{s}=\frac{1}{2}\int d\tau d^{3}x\Big{[}v^{\prime
2}-\left(\partial_{i}v\right)^{2}-\left(\frac{\nu_{s}^{2}-\frac{1}{4}}{\tau^{2}}\right)v^{2}\Big{]},$
(12)

where

$\nu_{s}\approx\frac{3}{2}+\epsilon+\frac{\eta}{2}$ (13)
$\delta^{(2)}S_{s}=\int d\tau d^{3}x\Big{[}u_{ij}^{\prime
2}-\left(\partial_{i}u_{ij}\right)^{2}-\left(\frac{\nu_{t}^{2}-\frac{1}{4}}{\tau^{2}}\right)u_{ij}^{2}\Big{]},$
(14)

and

$\nu_{t}\approx\frac{3}{2}+\epsilon\,.$ (15)

We now promote the fields $v,\,u_{ij}$ and their corresponding conjugate
momenta $\pi_{s},\,\pi_{ij}$ to operators and define the canonical commutation
relations as

$\displaystyle\Big{[}\hat{v}\left(\tau,\,\textbf{x}\right),\,\hat{\pi}_{s}\left(\tau,\,\textbf{x}^{\prime}\right)\Big{]}$
$\displaystyle=i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right)$ (16)
$\displaystyle\Big{[}\hat{u}_{ij}\left(\tau,\,\textbf{x}\right),\,\hat{\pi}_{ij}\left(\tau,\,\textbf{x}^{\prime}\right)\Big{]}$
$\displaystyle=i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right)$

We expand the field $\hat{v}\left(\tau,\,\textbf{x}\right)$ in terms of
creation and annihilation operators
$a_{\textbf{k}},\,a^{\dagger}_{\textbf{k}}$ as

$\displaystyle\hat{v}\left(\tau,\,\textbf{x}\right)$
$\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}}\int
d^{3}k\hat{v}_{\textbf{k}}\left(\tau\right)e^{i\textbf{k}\cdot\textbf{x}}$
(17) $\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}}\int
d^{3}k\Bigg{[}a_{\textbf{k}}{v}_{k}\left(\tau\right)e^{i\textbf{k}\cdot\textbf{x}}+a_{\textbf{k}}^{\dagger}{v}_{k}^{\ast}e^{-i\textbf{k}\cdot\textbf{x}}\Bigg{]}\,$

where $v_{k}$ is the mode function which satisfies Mukhanov-Sasaki (MS)
equation

$v_{k}^{\prime\prime}+\left(k^{2}-\frac{v_{s}^{2}-\frac{1}{4}}{\tau^{2}}\right)v_{k}=0\,.$
(18)

and
$\left[a_{\textbf{k}},\,a^{\dagger}_{\textbf{k}^{\prime}}\right]=\delta\left(\textbf{k}-{\textbf{k}}^{\prime}\right)$
The generic solution of (18) by treating $\nu_{s}\approx\,{\rm const}$ is

$\displaystyle v_{k}\left(\tau\right)$ (19)
$\displaystyle=\dfrac{\sqrt{\pi}}{2}\left(-\tau\right)^{1/2}e^{i\left(2\nu_{s}+1\right)\pi}\Bigg{[}A_{k}H^{(1)}_{\nu_{s}}\left(-k\tau\right)+B_{k}H^{(2)}_{\nu_{s}}\left(-k\tau\right)\Bigg{]}.$

where $A_{k}$ and $B_{k}$ are the Bogoliubov coefficients. Due to the
commutation relations (16), the mode functions must satisfy

$v_{k}v^{\ast\prime}_{k}-v_{k}^{\ast}v^{\prime}_{k}=i$ (20)

which implies

$|A_{k}|^{2}-|B_{k}|^{2}=1\,.$ (21)

From the above relation we can carry a parametrization $A_{k}=\cosh\alpha$ and
$B_{k}=\sinh\alpha$. The case $\alpha\neq 0$ corresponds the well-known
$\alpha-$ vacuum in the limit $\left(\epsilon,\,\eta\right)\to 0$ Allen
(1985). The case $\alpha=0$ corresponds to the Bunch-Davies (BD) vacuum in
which case the mode functions behave like Minkowski-one in the limit
$\tau\to-\infty$ when they are deep inside the horizon i.e., when $k\gg aH$
the mode function of the BD vacuum becomes

$v_{k}\simeq\sqrt{\frac{1}{2k}}e^{-ik\tau}$ (22)

Thus the BD vacuum is given by

$a_{\textbf{k}}|0\rangle=0\,.$ (23)

In the limit $\left(\epsilon,\,\eta\right)\to 0$ the mode functions can be
approximated as

$v_{k}\left(\tau\right)=\sqrt{\frac{1}{2k}}e^{-ik\tau}\left(1-\frac{i}{k\tau}\right)$
(24)

The two point correlations of the fields can be computed as

$\langle
0|\hat{v}\left(\textbf{x},\,\tau\right)\hat{v}\left(\textbf{x}+\boldsymbol{\xi},\,\tau\right)|0\rangle=\frac{4\pi}{\left(2\pi\right)^{3}}\int\frac{dk}{k}\frac{\sin
k\xi}{k\xi}k^{3}|v_{k}|^{2}\,.$ (25)

The power spectrum of the canonical scalar field $v$ is defined as

$\displaystyle\langle
0|\hat{v}_{\textbf{k}}\hat{v}_{\textbf{k}^{\prime}}|0\rangle$
$\displaystyle=|v_{k}|^{2}\delta\left(\textbf{k}+\textbf{k}^{\prime}\right)\,$
(26)
$\displaystyle=P_{v}\left(k\right)\delta\left(\textbf{k}+\textbf{k}^{\prime}\right)$

The power spectrum of curvature perturbation is obtained by pulling back the
rescaling we have performed in (11)

$\langle\zeta_{\textbf{k}}\zeta_{\textbf{k}^{\prime}}\rangle=\frac{1}{2a^{2}\epsilon}\times\langle
v_{\textbf{k}}v_{\textbf{k}^{\prime}}\rangle\,=\frac{2\pi^{2}}{k^{3}}P_{\zeta}\left(k\right)\delta\left(\textbf{k}+\textbf{k}^{\prime}\right)\,.$
(27)

The curvature perturbation $\zeta$ freezes out on super-horizon scales and
therefore we evaluate the spectrum of curvature perturbation at the time of
horizon exit $k=aH$. Thus, the dimensionless power spectrum of curvature
perturbation becomes222Since the curvature perturbation is conversed on super-
horizon scales the (scale dependent) magnitude of power spectrum is given by
(28) which is evaluated at Horizon exit.

$\displaystyle P_{\zeta}\left(k\right)\Big{|}_{k=aH}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{2a^{2}\epsilon}P_{v}$ (28)
$\displaystyle=\frac{H^{2}}{8\pi^{2}\epsilon}\Big{|}_{k=aH}\,.$

In the above derivation we substituted $\tau\approx-\frac{1}{aH}$. The tilt of
the power spectrum $P_{\zeta}$ can be computed as

$n_{s}-1=\frac{d\ln P_{\zeta}}{d\ln k}\approx-\frac{d\ln
P_{\zeta}}{dN}\approx-2\epsilon-\eta\,.$ (29)

Note that the above result (28) and (29) remains the same even without
performing the approximation $\nu_{s}\approx
3/2\implies\left(\epsilon,\,\eta\right)\to 0$ Kinney (2009). Since the power
spectrum is obtained by evaluating it at $k=aH$, the small corrections to the
parameter $\nu_{s}$ give subdominant results. In addition, the spectral index
$n_{s}$ and $n_{t}$ are often written in terms of $\nu_{s},\,\nu_{t}$, but
such definitions are not precisely correct as it is explained in Kinney
(2009). One must indeed evaluate the power spectrum at $k=aH$ because beyond
this point curvature perturbation is frozen. So, the super-horizon amplitude
of the power spectrum can be written in terms of quantities evaluated at the
horizon exit Leach and Liddle (2001) by fixing the normalization factors
Kinney (2009). Similarly, we can work out the mode function for the tensor
fluctuations and imposing the BD vacuum we obtain

$\displaystyle
u_{ij}=e_{ij}\dfrac{\sqrt{\pi}}{2}\left(-\tau\right)^{1/2}e^{i\left(2\nu_{t}+1\right)\pi}H^{(1)}_{\nu_{t}}\left(-k\tau\right)$
(30)

We can perform similar calculations for the tensor power spectrum and tilt and
as a result we obtain

$P_{T}=\frac{2H^{2}}{\pi^{2}}\Bigg{|}_{k=aH},\quad n_{t}=-\frac{d\ln
P_{T}}{d\ln N}\Bigg{|}_{k=aH}=-2\epsilon\,.$ (31)

##  III New scheme of quantizing inflationary power spectra

In the previous section, we have discussed the standard quantization scheme of
inflationary fluctuations. In this section, we enlist some open questions in
the context of quantization of inflationary fluctuations. Based on these, we
propose a new scheme of quantization of the inflationary fluctuations, which
not only target some of the open questions, but also leads to some observable
features in the CMB and primordial gravitational wave spectra. Below we
discuss point by point some of the aspects of standard calculations of
inflationary quantum fluctuations in relation with our understanding of
spacetime in quantum theory and general relativity.

  * •

By quantizing inflationary fluctuations we actually deal with quantum field
theory (QFT) in curved spacetime. But in standard QFT we have particles that
propagate forward and backward in time (anti-particle). A natural question is
what happens to particles and anti-particle states in a curved spacetime. This
implies one should consistently define the role of time reversal in a curved
spacetime.

  * •

QFT in Minkowski spacetime is mainly built on discrete symmetries such as
$\mathcal{C}\mathcal{P}\mathcal{T}$ or $\mathcal{P}\mathcal{T}$ (in the
context of real scalar field), but in curved spacetime we often do not pay
attention to the notion of discrete symmetries. The open question here is
whether $\mathcal{C}\mathcal{P}\mathcal{T}$ holds or not in curved spacetime?

  * •

In classical GR time is a coordinate and in quantum theory time is a parameter
Donoghue  _et al._ (2017); Donoghue and Menezes (2019). In the case of
inflationary cosmology we usually quantize both gravitational and matter
degrees of freedom Martin (2005); Baumann (2018); Kinney (2009) taking a
classical notion of time. However, the concept of time in quantum theory is
very different from the classical one Rovelli (2004).

  * •

Since we quantize gravitational degrees of freedom, we can interpret
inflationary quantum fluctuations as features of (linearized) quantum gravity
Martin (2005). Note that, in the canonical quantum gravity that emerges
through Wheeler-de Witt equation, time does not appear explicitly which
indicates a difficulty to define positive/negative frequencies, like we
usually do following the standard Schrödinger equation. This is famously known
as the problem of time in quantum cosmology Kiefer (2007); Rovelli (2004).

  * •

Quantum theory is always time symmetric. The arrow of time emerges only after
we specify initial and final states Hartle (2014). This implies that we could
first formulate QFT in curved spacetime, in a time symmetric way, and then
impose initial conditions.

Our new formalism for the inflationary quantum fluctuations is highly
motivated from the above questions. Before we go into details, we write down
here few guidelines

  * •

We identify the expansion or contraction of Universe with the shrinking or
growing size of the co-moving horizon (in the context of de Sitter and quasi
de Sitter). Change of size of the horizon defines our classical or
thermodynamical arrow of time.

  * •

We build the quantum theory based on discrete transformations of spacetime
within the co-moving horizon. We completely detach ourselves from the notions
of observers, Penrose diagrams and the corresponding regions of spacetime when
we do quantization because all such concepts are not well-defined according to
quantum theory. In fact, classical notion of region of spacetime cannot have
any consistent meaning from the quantum mechanical point of view Donoghue and
Menezes (2021).

  * •

The way forward is to first understand quantum mechanics in a dynamical
spacetime and then take the classical limit. To achieve this goal, we take a
small step by distinguishing classical notion of time with the quantum
mechanical notion of time. This small step allow us to create all possible
quantum states in the given spacetime geometry.

  * •

In our formalism, quantum fluctuations always come in pair from two different
vacua described by $\mathcal{P}\mathcal{T}$ or
$\mathcal{C}\mathcal{P}\mathcal{T}$ transformations. By this scheme we have
the quantum fluctuations that propagate forward in time as well as backward in
time and both can exist within the horizon. As the horizon changes, both of
these type of fluctuations either exit or enter the horizon depending on
whether the spacetime is expanding or contracting.

  * •

These pairs of quantum fluctuations within the horizon evolve independently of
each other which means the commutator of corresponding operators always
vanish. This means even though we can create quantum states non-locally we
still do not violate locality and causality in our formulation of QFT.

##  IV Quantum fluctuations in an expanding de Sitter spacetime

In this section, we focus on elucidating our proposal in the context of exact
de Sitter (dS) spacetime in FLRW co-ordinates. Since inflationary spacetime is
near dS there are several concepts we can take forward from the understanding
of dS. To make our analysis simpler we consider quantization of a massless
scalar field in dS spacetime ignoring the issues with IR singularities and
defining dS invariant vacuum Higuchi and Lee (2009); Allen (1985).

The dS spacetime in FLRW coordinates looks like

$\displaystyle ds^{2}$ $\displaystyle=-dt^{2}+a(t)^{2}d\textbf{x}^{2}$
$\displaystyle=\frac{1}{H^{2}\tau^{2}}\left(-d\tau^{2}+d\textbf{x}^{2}\right)\,.$
(32)

where $d\tau=\frac{dt}{a}$ is the conformal time and the scale factor is given
by

$a(t)=e^{Ht},\quad H^{2}=\left(\frac{1}{a}\frac{da}{dt}\right)^{2}={\rm
const}\,.$ (33)

Note that dS metric (32) is
$\mathcal{P}\mathcal{T}:\tau\to-\tau,\,\textbf{x}\to-\textbf{x}$ symmetric.
The scalar curvature of dS spacetime is $R=12H^{2}$ which does not tell
whether the Hubble parameter is positive or negative. Each point in dS is
surrounded by a co-moving horizon given by the radius

$r_{H}=\Big{|}\frac{1}{aH}\Big{|}$ (34)

One very simple observation we can make from (32) is that the spatial size of
the Universe $a(t)^{2}$ grows with coordinate time $t$ when $H>0$. We then
interpret the metric (32) as modelling an expanding universe. In this case the
scale factor grows with time $t$ and the size of the horizon shrinks. We note
here that this observation stems from the convention that $t$ runs from
$-\infty$ to $+\infty$ (which means conformal time $-\infty<\tau<0$). Defining
$\bar{t}=-t$, which corresponds to $t$ running from $+\infty$ to $-\infty$ or
$+\infty>\tau>0$ we find the scale factor $a(\tau)$ growing with $\bar{t}$ for
$H<0$, corresponding to expansion. Therefore, the expanding dS space can only
characterized by shrinking horizon rather than fixing the notion of time being
positive or negative. Since time is a parameter in quantum theory, we can have
quantum states that are labeled by $\tau<0$ and/or $\tau>0$. In the case of
standard QFT in dS one usually make a biased choice of either $\tau<0$ or
$\tau>0$ which actually breaks the time reversal nature of dS spacetime and
also standard scheme of quantization completely ignores existence of states
that evolve backward in time. Therefore, in our scheme of quantization in dS
spacetime, we can have quantum states that propagate forward in time
$t:-\infty\to\infty$ with $H>0$ and states that propagate backward in time
following an arrow of time $t:\infty\to-\infty$ with $H<0$. Note that both
type of quantum states can co-exist in a single expanding Universe. We stress
again, the expanding dS Universe is defined by the following arrows of time

$\tau:\pm\infty\to 0$ (35)

where $-$ sign corresponds to a state propagating forward in time and $+$sign
corresponds to a state propagating backward in time. In our new picture
quantization of a (massless) scalar field in dS spacetime leads to a pair of
quantum fields that are produced out of two vacua. Being more precise we
define total vacuum as the direct sum of two vacua

$|0\rangle=\frac{1}{\sqrt{2}}\Bigg{(}|0\rangle_{\rm I}\oplus|0\rangle_{\rm
II}\Bigg{)}\,.$ (36)

In the vacuum $|0\rangle_{\rm I}$ we create quantum fields at the position x
that evolve forward in time ($\tau<0$) and in the vacuum $|0\rangle_{\rm II}$
we create quantum fields at $-\textbf{x}$ which evolve backward in time
($\tau>0$). The action of a massless scalar field in dS space is given

$S_{\phi}=-\frac{1}{2}\int d\tau
d^{3}xa^{2}\phi\left(\partial_{\tau}^{2}+2\mathcal{H}\partial_{\tau}+k^{2}\right)\phi\,.$
(37)

First thing we can notice here is that the above action is invariant under
$\mathcal{P}\mathcal{T}$ which means $\textbf{x}\to-\textbf{x}$ and

$t\to-t\quad\left(-\tau\to\tau\right)\implies
H\to-H\quad\left(\mathcal{H}\to-\mathcal{H}\right)$ (38)

Rescaling the field $\phi\to a\phi$ gives the following action which leads to
Mukhanov-Sasaki equation as

$S_{\phi}=\frac{1}{2}\int d\tau d^{3}x\Big{[}\phi^{\prime
2}-\left(\partial_{i}\phi\right)^{2}-\frac{2}{\tau^{2}}\phi^{2}\Big{]}\,.$
(39)

The above Lagrangian is again symmetric under
$\mathcal{P}\mathcal{T}:\textbf{x}\to-\textbf{x},\,\tau\to-\tau$. When we
quantize this scalar field we propose that the scalar field operator
$\hat{\phi}\left(\tau,\,\textbf{x}\right)$ is the direct sum of the pair of
fields as

$\hat{\phi}\left(\tau,\,\textbf{x}\right)=\frac{1}{\sqrt{2}}\hat{\varphi}_{\rm
I}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{\varphi}_{\rm
II}\left(-\tau,\,-\textbf{x}\right)\,.$ (40)

which can be expanded in terms of different creation and annihilation
operators in the following way

$\displaystyle\hat{\varphi}_{\rm I}\left(\tau,\,\textbf{x}\right)$ (41)
$\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}}\int d\tau
d^{3}k\Bigg{[}a_{\textbf{k}}{\varphi}_{\rm
I\,k}\left(\tau\right)e^{i\textbf{k}\cdot\textbf{x}}+a_{\textbf{k}}^{\dagger}{\varphi}^{\ast}_{\rm
I\,k}\left(\tau\right)e^{-i\textbf{k}\cdot\textbf{x}}\Bigg{]}\,$

where the creation and annihilation operators
$a_{\textbf{k}},\,a^{\dagger}_{\textbf{k}}$ satisfy the canonical commutation
relations and

$a_{\textbf{k}}|0\rangle_{\rm I}=0\,.$ (42)

Here the mode function $\varphi_{\rm I\,,k}$ is the solution of Mukhanov-
Sasaki equation (that comes from varying (39)) given by

$\varphi_{\rm
I,\,k}=\alpha_{1k}\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)+\beta_{1k}\frac{e^{ik\tau}}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right)\,,$
(43)

where the Bogoliubov coefficients can be fixed as
$\left(\alpha_{1,\,k},\,\beta_{1,\,k}\right)=\left(1,\,0\right)$, which is
compatible with the Wronskian condition $\varphi_{\rm
I,\,k}\varphi^{\prime\ast}_{\rm I,\,k}-\varphi^{\ast}_{\rm
I,\,k}\varphi^{\prime}_{\rm I,\,k}=i$ that corresponds to the canonical
commutation relation

$\big{[}\hat{\varphi}_{\rm I}\left(\tau,\,\textbf{x}\right),\,\hat{\pi}_{\rm
I}\left(\tau,\,\textbf{x}^{\prime}\right)\big{]}=i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right).$
(44)

The choice $\left(\alpha_{\rm I,\,k},\,\beta_{\rm
I,\,k}\right)=\left(1,\,0\right)$ defines the vacuum and $\hat{\varphi}_{\rm
I}\left(\tau,\,\textbf{x}\right)|0\rangle$ corresponds to the positive
frequency modes in the limit $\tau\to-\infty$. Quantum mechanically
$\hat{\varphi}_{\rm I}\left(\tau,\,\textbf{x}\right)|0\rangle$ is the postive
energy state that propagate forward in time at x. Similarly, we can expand the
second field operator as

$\displaystyle\hat{\varphi}_{\rm II}\left(-\tau,\,-\textbf{x}\right)$ (45)
$\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}}\int d\tau
d^{3}k\Bigg{[}b_{\textbf{k}}\varphi_{\rm
II\,k}\left(-\tau\right)e^{-i\textbf{k}\cdot\textbf{x}}+b_{\textbf{k}}^{\dagger}{\varphi}^{\ast}_{\rm
II\,k}\left(-\tau\right)e^{i\textbf{k}\cdot\textbf{x}}\Bigg{]}\,$

where

$\varphi_{\rm
II,\,k}=\alpha_{2,\,k}\frac{e^{ik\tau}}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right)+\beta_{2,\,k}\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)\,,$
(46)

where the Bogoliubov coefficients can be fixed as
$\left(\alpha_{2,\,k},\,\beta_{2,\,k}\right)=\left(1,\,0\right)$ which is
compatible with the Wronskian condition $\varphi_{\rm
II,\,k}\varphi^{\prime\ast}_{\rm II,\,k}-\varphi^{\ast}_{\rm
II,\,k}\varphi^{\prime}_{\rm II,\,k}=-i$ that corresponds to the canonical
commutation relation Donoghue and Menezes (2019)

$\big{[}\hat{\varphi}_{\rm
II}\left(-\tau,\,-\textbf{x}\right),\,\hat{\pi}_{\rm
II}\left(-\tau,\,-\textbf{x}^{\prime}\right)\big{]}=-i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right).$
(47)

which describe the quantum fields that propagate backward in time. Here the
second vacuum is defined by

$b_{\textbf{k}}|0\rangle_{\rm II}=0\,.$ (48)

Here $\hat{\varphi}_{\rm II}\left(-\tau,\,-\textbf{x}\right)|0\rangle_{\rm
II}$ corresponds to positive energy state that evolves backward in time at -x.
We demand that these two quantum fluctuations evolve independently. This is
manifest by

$[\hat{\varphi}_{\rm I}\left(\tau,\,\textbf{x}\right),\,\hat{\varphi}_{\rm
II}\left(-\tau,\,-\textbf{x}\right)]=0\,.$ (49)

which implies that the respective creation and annihilation operators commute

$\big{[}a_{\textbf{k}},\,b_{\textbf{k}^{\prime}}\big{]}=0,\quad\big{[}a^{\dagger}_{\textbf{k}},\,b^{\dagger}_{\textbf{k}^{\prime}}\big{]}=0$
(50)

Since dS spacetime is perfectly $\mathcal{P}\mathcal{T}$ symmetric we have
that the quantum fields $\hat{\varphi}_{\rm
I}\left(\tau,\,\textbf{x}\right)|0\rangle_{\rm I}$ and $\hat{\varphi}_{\rm
II}\left(-\tau,\,-\textbf{x}\right)|0\rangle_{\rm II}$ behave identically,
which can be seen from the fact that their equal time correlations are the
same

$\displaystyle\frac{1}{a^{2}}{}_{\rm I}\langle 0|\hat{\varphi}_{\rm
I}\left(\tau,\,\textbf{x}\right)\hat{\varphi}_{\rm
I}\left(\tau,\,\textbf{x}^{\prime}\right)|0\rangle_{\rm I}=$ (51)
$\displaystyle\frac{1}{a^{2}}{}_{\rm II}\langle 0|\hat{\varphi}_{\rm
II}\left(-\tau,\,-\textbf{x}\right)\hat{\varphi}_{\rm
II}\left(-\tau,\,-\textbf{x}^{\prime}\right)|0\rangle_{\rm
II}=\frac{H^{2}}{4\pi^{2}k^{3}}\,.$

This implies the correlations of quantum fields related by
$\mathcal{P}\mathcal{T}$ transformations are identical in the case of dS
spacetime. This indicates our formulation of QFT in dS spacetime might be
$\mathcal{P}\mathcal{T}$ or $\mathcal{C}\mathcal{P}\mathcal{T}$ (if we include
charged fields) symmetric.333We note that the result in (51) is similar to
what was derived in the context of elliptic dS spacetime Sánchez (1987); Erwin
Schrodinger (1956). However, in our scheme of quantization we do not make any
(classical) antipodal identification of spacetime and therefore we
significantly deviate conceptually from the proposal of elliptic dS made by
Erwin Schrödinger Erwin Schrodinger (1956).

However, this may not be true in a spacetime with less symmetries such as
quasi-dS. Indeed, in the context of inflationary quantum fluctuations, it is
often assumed, in the effective field theory (EFT) approach, that Lorentz
symmetry (in other words $(\mathcal{C})\mathcal{P}\mathcal{T}$) must be
spontaneously broken Cheung  _et al._ (2008). However, it is not known how
$\mathcal{C}\mathcal{P}\mathcal{T}$ symmetry can be broken in a curved
spacetime. In the next section, we provide a framework of quantization where
we can explicitly see that $(\mathcal{C})\mathcal{P}\mathcal{T}$ is
spontaneously broken in the sense that the qunatum fields defined by our new
notion of $\mathcal{P}\mathcal{T}$ transformations evolve differently.

##  V Double vacuum inflationary spacetime and the power spectra

In the previous section, we have studied the quantization of a massless scalar
field in dS. Here we quantize the inflationary quantum fluctuations. There is
a crucial difference here. In the previous section we quantized an arbitrary
scalar field in dS spacetime which is completely fixed but in the context of
inflation we quantize metric and matter degrees of freedom in order to find an
effective quantum correction to the classical quasi-dS (inflationary)
spacetime. To be more precise, the classical inflationary spacetime is given
by an arrow of time and the background initial conditions are

$t:-\infty\to\infty\implies\left(\epsilon,\,\eta\right):\left(\approx
0,\,\approx 0\right)\to\left(\sim 1,\,\sim 1\right)\,.$ (52)

Which means the slow-roll parameters are very small and positive during
inflation and inflation ends when the slow-roll parameters evolve to the order
of unity, which happens at some time $t_{\rm end}\gg 1$. The inflationary
quantum fluctuations are produced in the phase when
$\left(\epsilon,\,\eta\right)\ll 1$. As we discussed earlier, “time” in
quantum theory has totally different role and one must be very careful in
quantizing inflationary fluctuations. We emphasize that the quantum
fluctuations do not necessarily follow arrow of time prescribed by the initial
conditions in classical theory.

Unlike dS spacetime the quasi-dS metric does not have
$\mathcal{P}\mathcal{T}$-symmetry, therefore naturally one would expect the
$\mathcal{P}\mathcal{T}$-symmetry to be spontaneously broken at the quantum
level. To see this, we apply the scheme of quantization similar to what we
have done in the context of exact dS. Let us first focus on inflationary
scalar fluctuations which are described by the action (9). After the field
redefinition $\hat{v}$ is the canonical variable which we need to quantize
(12). As we postulated before quantum fluctuations always come as pair from
two different vacua related by $\mathcal{P}\mathcal{T}$ transformations. This
means that unlike the standard quantization of inflationary fluctuations,
where only one kind of quantum states are considered which only evolve forward
in time, here we describe a pair of quantum fluctuations that evolve both
forward and backward in time. Similar to the exact dS case we write the
canonical field operator as a direct sum of two quantum fields

$\hat{v}\left(\tau,\,\textbf{x}\right)=\frac{1}{\sqrt{2}}\hat{v}_{\rm
I}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{v}_{\rm
II}\left(-\tau,\,-\textbf{x}\right)\,.$ (53)

which emerges from two different vacua. The total vacuum is the direct sum of
these two which can be written as

$|0\rangle_{\rm qdS}=|0\rangle_{\rm qdS_{I}}\oplus|0\rangle_{\rm qdS_{II}}\,.$
(54)

The quantum field operators acting on the vacua $\hat{v}_{\rm
I}\left(\tau,\,\textbf{x}\right)|0\rangle_{\rm qdS_{II}}$ and $\hat{v}_{\rm
I}\left(-\tau,\,-\textbf{x}\right)|0\rangle_{\rm qdS_{II}}$ leads to the
creation of the pair of quantum fluctuations at x and at $-\textbf{x}$ which
propagate forward in time and back ward in time respectively. We will make it
clear shortly the meaning of quantum fluctuations that evolve forward and
backward in time in the context of quasi-dS.

Going into the details we expand the fields in terms of creation and
annhilation operators as explained below.

$\displaystyle\hat{v}_{\rm I}\left(\tau,\,\textbf{x}\right)$ (55)
$\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}}\int d\tau
d^{3}k\Bigg{[}c_{\textbf{k}}{v}_{\rm
I,\,k}\left(\tau\right)e^{i\textbf{k}\cdot\textbf{x}}+c_{\textbf{k}}^{\dagger}{v}_{\rm
I,\,k}^{\ast}\left(\tau\right)e^{-i\textbf{k}\cdot\textbf{x}}\Bigg{]}$

with $c_{\textbf{k}},\,c_{\textbf{k}}^{\dagger}$ being the creation and
annihilation operators of quasi-dS vacuum-I defined by

$c_{\textbf{k}}|0\rangle_{\rm qdS_{I}}=0\,,$ (56)

and ${v}_{\rm I,\,k}$ is the mode function obtained by solving MS-equation for
$v_{\rm I}\left(\tau\right)$

$v_{\rm
I,\,k}^{\prime\prime}+\left(k^{2}-\frac{{\nu}_{s}^{2}-\frac{1}{4}}{\tau^{2}}\right)v_{\rm
I,\,k}^{2}=0\,.$ (57)

Here $\tau:-\infty\to 0$ and the quasi-dS vacuum $|0\rangle_{\rm qdS_{I}}$
corresponds to the creation of positive frequency modes in the limit
$\tau\to-\infty$. Notice that unlike the dS case the MS-equation (18) is not
symmetric under time reversal because there is an additional time dependence
enters through the nearly constant variable $\nu_{s}$ which contains the slow-
roll parameters $\left(\epsilon,\,\eta\right)$ (4). Assuming
$0<\left(\epsilon,\,\eta\right)\ll 1$ and constant we obtain

$\displaystyle{v}_{\rm I,\,k}$ (58)
$\displaystyle=\frac{\sqrt{\pi}}{2}\left(-\tau\right)^{1/2}e^{\left(i\nu_{s}+1\right)}\Bigg{[}C_{k}H^{(1)}_{\nu_{s}}\left(-k\tau\right)+D_{k}H^{(2)}_{\nu_{s}}\left(-k\tau\right)\Bigg{]}\,.$

The above mode function corresponds to the creation of positive frequency
modes in the limit $\tau\to-\infty$ for the case
$\left(\mathcal{C}_{k},\,D_{k}\right)=\left(1,\,0\right)$ which corresponds to
the standard BD state that satisfies the Wronskian
$v_{I,k}v_{I,k}^{\prime\ast}-v_{I,k}^{\ast}v_{I,k}^{\prime}=i(\implies|C_{k}|^{2}-|D_{k}|^{2}=1)$.
We can expand the mode function for the case of
$\left(\mathcal{C}_{k},\,D_{k}\right)=\left(1,\,0\right)$ in the order of
slow-roll as

$\displaystyle{v}_{\rm I,\,k}$
$\displaystyle\approx\frac{\sqrt{\pi}}{2}\left(-\tau\right)^{1/2}e^{\left(i\nu_{s}+1\right)}H^{(1)}_{\nu_{s}}\left(-k\tau\right)\,$
(59)
$\displaystyle\approx\sqrt{\frac{1}{2k}}e^{-ik\tau}\left(1-\frac{i}{k\tau}\right)$
$\displaystyle+\left(\epsilon+\frac{\eta}{2}\right)\frac{\sqrt{\pi}}{2\sqrt{k}}\sqrt{\frac{k}{aH}}\frac{\partial
H^{(1)}_{\nu_{s}}\left(\frac{k}{aH}\right)}{\partial\nu_{s}}\Big{|}_{\nu_{s}=3/2}$

The second field $\hat{v}_{II\,,\textbf{k}}\left(-\tau,\,-\textbf{x}\right)$
is expanded as

$\displaystyle\hat{v}_{\rm II}\left(-\tau,\,-\textbf{x}\right)$ (60)
$\displaystyle=\frac{1}{\left(2\pi\right)^{3/2}}\int d\tau
d^{3}k\Bigg{[}d_{\textbf{k}}{v}_{\rm
II,\,k}\left(-\tau\right)e^{-i\textbf{k}\cdot\textbf{x}}+d_{\textbf{k}}^{\dagger}{v}_{\rm
I,\,k}^{\ast}\left(-\tau\right)e^{i\textbf{k}\cdot\textbf{x}}\Bigg{]}$

where $d_{\textbf{k}},\ ,d_{\textbf{k}}^{\dagger}$ are the creation and
annihilation operators that satisfy the canonical commutation relations and
the second vacuum is defined by444We note that
$\big{[}c_{\textbf{k}},\,d_{\textbf{k}^{\prime}}\big{]}=0,\quad\big{[}c^{\dagger}_{\textbf{k}},\,d^{\dagger}_{\textbf{k}^{\prime}}\big{]}=0$
similar to the dS case which means the pair of quantum fluctuations are not
causally connected.

$d_{\textbf{k}}|0\rangle_{\rm qdS_{II}}=0\,.$ (61)

The evolution of the mode function $v_{\rm II,\,k}$ is determined by solving
the following time reversed MS-equation

$v_{\rm
II,\,k}^{\prime\prime}+\left(k^{2}-\frac{\bar{\nu}_{s}^{2}-\frac{1}{4}}{\tau^{2}}\right)v_{\rm
II,\,k}^{2}=0\,.$ (62)

Here $\tau:+\infty\to 0$ and

$\bar{\nu}_{s}\approx\frac{3}{2}-\epsilon-\frac{\eta}{2}$ (63)

Here the time reversal transformations are

$t\to-t\implies H\to-H,\quad\epsilon\to-\epsilon,\quad\eta\to-\eta\,.$ (64)

which can be understood in the following sense. Our goal here is to define how
the quantum fluctuations propagate backward in time in an expanding Universe.
To achieve this goal we need to formulate the meaning of time reversal
operation. Logically, if the fluctuation propagates forward in time in a slow-
roll background, the fluctuation that goes backward in time experience
spacetime as a ”slow-climb”555In the standard slow-roll we have scalar field
rolling down the potential whose time reversal can be understood as a phantom
field that is slowly climbing the potential Piao and Zhang (2004).  which is
given by reversing the signs of the parameters $\left(\epsilon,\,\eta\right)$
as stated in (64). We impose this time reversal operation in a completely
quantum mechanical sense and this has no classical meaning i.e., our
background (classical) dynamics are completely determined by Friedmann
equations (5) and we do not apply at all time reversal to the classical
background. Since we treat time differently at quantum level we restrain
ourselves from any intuition from classical physics.666Notion of time is a
very non-trivial concept in physics and its meaning varies in different
contexts. We suggest the reader Rovelli (2004) for an extended physical
discussion. Our statement, that quantum states evolving backward in time has
no classical analog, is deeply rooted in the quantum gravity concept time as
it is presented in p. 184 of Rovelli (2004). Since dynamics of quantum fields
emerge from MS-equation (18) the functions $\left(\epsilon,\,\eta\right)$ are
now treated along with time in as parameters to specify the nature of quantum
state. This would encode a subtle difference between quantum fluctuations
propagating forward and backward in time.

$\displaystyle v_{\rm II,\,k}\left(-\tau\right)$ (65)
$\displaystyle=\dfrac{\sqrt{\pi}}{2}\left(\tau\right)^{1/2}e^{i\left(2\bar{\nu}_{s}+1\right)\pi}\Bigg{[}\bar{C}_{k}H^{(1)}_{\bar{\nu}_{s}}\left(k\tau\right)+\bar{D}_{k}H^{(2)}_{\bar{\nu}_{s}}\left(k\tau\right)\Bigg{]}.$

Now imposing the Wronskian condition $v_{\rm II,\,k}v_{\rm
II,\,k}^{\prime\ast}-v_{\rm II,\,k}^{\ast}v_{\rm II,\,k}^{\prime}=-i$ which
corresponds to the canonical commutation relation for a reversed arrow of time
Donoghue and Menezes (2019)

$\Big{[}\hat{v}_{\rm II}\left(-\tau,\,-\textbf{x}\right),\,\hat{\Pi}_{\rm
II}\left(-\tau,\,-\textbf{x}^{\prime}\right)\Big{]}=-i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right)\,.$
(66)

we obtain $|\bar{C}_{k}|^{2}-|\bar{D}_{k}|^{2}=1$. We consider
$\left(\bar{C},\,\bar{D}_{k}\right)=0$ which corresponds postive energy states
in the limit $\tau\to\infty$. Expanding (65) up to the leading order for
$\left(\bar{C},\,\bar{D}_{k}\right)=0$ we get

$\displaystyle{v}_{\rm II,\,k}$
$\displaystyle\approx\frac{\sqrt{\pi}}{2}\left(\tau\right)^{1/2}e^{\left(i\nu_{s}+1\right)}H^{(1)}_{\nu_{s}}\left(k\tau\right)\,$
(67)
$\displaystyle\approx\sqrt{\frac{1}{2k}}e^{ik\tau}\left(1+\frac{i}{k\tau}\right)$
$\displaystyle-\left(\epsilon+\frac{\eta}{2}\right)\frac{\sqrt{\pi}}{2\sqrt{k}}\sqrt{k\tau}\frac{\partial
H^{(1)}_{\nu_{s}}\left(k\tau\right)}{\partial\nu_{s}}\Big{|}_{\nu_{s}=3/2}$

Comparing (59) and (67) we can deduce that the two mode functions get
different slow-roll (quantum) corrections. This is because the parameter
$\nu_{s}$ that enters in the MS-equation (18) contains a tiny time dependence.
In the standard framework of quantization this is treated it to be nearly
constant there is only one type of quantum fluctuation (see Sec. II). But the
fact that MS-equation has time dependence in terms of $\epsilon,\,\eta$ is
crucial to probe the nature of quantum fluctuations. These quantities change
sign under discrete spacetime transformation and this has to be understood in
a completely quantum mechanical sense. Our scheme of quantization implies a
pair of quantum fluctuations: those that propagate forward in time
($\tau:-\infty\to 0$) at spatial position x which behave differently from
those fluctuations that propagate backward in time ($\tau:+\infty\to 0$) at
the spatial position $-\textbf{x}$. These pairs of fluctuations exit the
horizon in the two opposite directions and now we observe the modes as they
re-enter.

So far we applied our quantization method for the scalar mode
$v\left(\tau,\,\textbf{x}\right)$. We can apply the similar method of
quantization for the tensor modes (14) by writing the tensor fluctuation as
direct some fluctuations that are related by $\mathcal{P}\mathcal{T}$
transformations

$\hat{u}_{ij}\left(\tau,\,\textbf{x}\right)=\frac{1}{\sqrt{2}}\hat{u}^{\rm
I}_{ij}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{u}^{\rm
II}_{ij}\left(-\tau,\,-\textbf{x}\right)$ (68)

The above fields can be written in terms of creation and annihilation
operators similar to what we have done for (55) and (60). The corresponding
mode functions tensor modes can be straight forwardly derived as

$\displaystyle{u}^{\rm I}_{ij,\,k}$ $\displaystyle\approx
e_{ij}\sqrt{\frac{1}{2k}}e^{-ik\tau}\left(1-\frac{i}{k\tau}\right)$ (69)
$\displaystyle+e_{ij}\epsilon\frac{\sqrt{\pi}}{2\sqrt{k}}\sqrt{-k\tau}\frac{\partial
H^{(1)}_{\nu_{t}}\left(-k\tau\right)}{\partial\nu_{t}}\Big{|}_{\nu_{t}=3/2}$

and

$\displaystyle{u}^{\rm II}_{ij,\,k}$ $\displaystyle\approx
e_{ij}\sqrt{\frac{1}{2k}}e^{ik\tau}\left(1+\frac{i}{k\tau}\right)$ (70)
$\displaystyle-
e_{ij}\epsilon\frac{\sqrt{\pi}}{2\sqrt{k}}\sqrt{k\tau}\frac{\partial
H^{(1)}_{\nu_{t}}\left(k\tau\right)}{\partial\nu_{t}}\Big{|}_{\nu_{t}=3/2}$

where $e_{ij}$ denotes the polarization tensor. In the next section, we will
compute the inflationary power spectra for scalar and tensor modes.

##  VI Inflationary power spectra and predictions for hemispherical asymmetry

As we learned in the previous section, inflationary quantum fluctuations now
come in pair which exit the horizon on two opposite sides. The two point
correlations of these fluctuations can be computed as

$\displaystyle{}_{\rm qdS_{I}}\langle 0|\hat{v}_{\rm
I}\left(\textbf{x},\,\tau\right)\hat{v}_{\rm
I}\left(\textbf{x}+\boldsymbol{\xi},\,\tau\right)|0\rangle_{\rm qdS_{I}}=$
(71)
$\displaystyle\frac{4\pi}{\left(2\pi\right)^{3}}\int\frac{dk}{k}\frac{\sin
k\xi}{k\xi}k^{3}|v_{\rm I,\,k}|^{2}\,$ $\displaystyle{}_{\rm qdS_{II}}\langle
0|\hat{v}_{\rm II}\left(-\textbf{x},\,-\tau\right)\hat{v}_{\rm
II}\left(-\textbf{x}-\boldsymbol{\xi},\,-\tau\right)|0\rangle_{\rm qdS_{II}}=$
$\displaystyle\frac{4\pi}{\left(2\pi\right)^{3}}\int\frac{dk}{k}\frac{\sin
k\xi}{k\xi}k^{3}|v_{\rm II,\,k}|^{2}\,.$

Substituting the expressions (59) and (70) in the above expressions we learn
that the above two correlations are not equal. In the case of exact dS we
obtained the correlations to be equal (see (51)) but in the case of quasi-dS
we obtained an asymmetry because the background spacetime is not
$\mathcal{P}\mathcal{T}$ symmetric. As we know that the curvature perturbation
is frozen on super-horizon scales. To calculate the two point correlations of
curvature perturbation on super-horizon scales we re scale the canonical
fields with the classical background quantities (11). This implies

$\displaystyle{}_{\rm qdS}\langle
0|\zeta_{\textbf{k}}\zeta_{\textbf{k}^{\prime}}|0\rangle_{\rm qdS}$
$\displaystyle=\left(\frac{1}{2a^{2}\epsilon}\right)\Bigg{|}_{\rm
classical}\Big{[}\frac{1}{2}{}_{\rm qdS_{I}}\langle 0|\hat{v}_{\rm
I\,\textbf{k}}\hat{v}_{\rm I\,\textbf{k}^{\prime}}|0\rangle_{\rm qdS_{I}}$
(72) $\displaystyle\quad+\frac{1}{2}{}_{\rm qdS_{II}}\langle
0|\hat{v}_{\rm{II}\,\textbf{k}}\hat{v}_{\rm{II}\,\textbf{k}^{\prime}}|0\rangle_{\rm
qdS_{II}}\Big{]}$
$\displaystyle=\frac{2\pi^{2}}{k^{3}}\left(P_{\zeta_{1}}+P_{\zeta_{2}}\right)\delta\left(\textbf{k}+\textbf{k}^{\prime}\right)\,,$

where $\zeta_{1},\,\zeta_{2}$ are the curvature perturbations that exit the
horizon on two opposite directions and they become frozen on super-horizon
scales which follows from (71). With appropriate normalization we evaluate the
power spectrum of these curvature perturbations at the moment of horizon exit.
Computing the two power spectra at the horizon exit we obtain

$\displaystyle P_{\zeta 1}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{2a^{2}\epsilon}|v_{\rm
I,\,k}|^{2}\Bigg{|}_{\tau=-\frac{1}{aH}}$ (73)
$\displaystyle\approx\frac{H^{2}}{8\pi^{2}\epsilon}\left(1+\left(\frac{k}{aH}\right)^{2}\right)$
$\displaystyle\quad+\frac{H^{2}}{4\pi\epsilon}\left(\epsilon+\frac{\eta}{2}\right)\left(\frac{k}{aH}\right)^{3}H_{3/2}^{(1)}\left(\frac{k}{aH}\right)\frac{\partial
H^{(1)}_{\nu_{s}}\left(\frac{k}{aH}\right)}{\partial\nu_{s}}\Big{|}_{\nu_{s}=3/2}\,$
$\displaystyle P_{\zeta 2}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{2a^{2}\epsilon}|v_{\rm
II,\,k}|^{2}\Bigg{|}_{\tau_{\ast}=\frac{1}{aH}}$
$\displaystyle\approx\frac{H^{2}}{8\pi^{2}\epsilon}\left(1+\left(\frac{k}{aH}\right)^{2}\right)$
$\displaystyle-\frac{H^{2}}{4\pi\epsilon}\left(\epsilon+\frac{\eta}{2}\right)\left(\frac{k}{aH}\right)^{3}H_{3/2}^{(1)}\left(\frac{k}{aH}\right)\frac{\partial
H^{(1)}_{\nu_{s}}\left(\frac{k}{aH}\right)}{\partial\nu_{s}}\Big{|}_{\nu_{s}=3/2}\,.$

If we combine the above two expressions we obtain

$P_{\zeta}=\frac{1}{2}P_{\zeta 1}+\frac{1}{2}P_{\zeta
2}=\frac{H^{2}}{8\pi^{2}\epsilon}\Bigg{|}_{k=aH}$ (74)

which is the expression in the context of standard inflation (see Sec. II).
From (71) we can deduce that the power spectrum $P_{\zeta 1}$ can be mapped to
the two-point correlations in the direction $\hat{\textbf{n}}$ and $P_{\zeta
2}$ can be mapped to the two-point correlations in the opposite direction
$-\hat{\textbf{n}}$ of the CMB sky. The difference between the two power
spectra gives us the non-zero scale dependent contribution. Our formalism of
producing inflationary correlations in pair naturally explains the
hemispherical asymmetry and this can also be interpreted as an indication of
spontaneous breaking of $\mathcal{P}\mathcal{T}$ symmetry in the inflationary
background. In our context we can define the the amplitude of dipolar
modulation as (according to Eq. (2))

$A(k)=\frac{P_{\zeta 1}-P_{\zeta 2}}{4P_{\zeta}}$ (75)

From (73) we can notice that the two power spectra differ only by a small
scale dependent correction of the order of slow-roll parameters. In addition,
through CMB we can only probe very limited range of $k$ corresponding to
initial $7-8$ e-foldings centered around the pivot scale Martin (2005). At the
leading order the first terms in the two power spectra dominate which gives us
the tilt of the two power spectra nearly the same in the leading order in
slow-roll approximation

$\frac{d\ln P_{\zeta 1}}{d\ln k}\approx\frac{d\ln P_{\zeta 2}}{d\ln k}\approx
n_{s}-1\approx-2\epsilon-\eta\,.$ (76)

The above result is in-line with the data from the Planck satellite Mukherjee
_et al._ (2016); Axelsson  _et al._ (2013).

In Fig. 1 we depict the power asymmetry amplitude of the scalar power spectra
for $n_{s}=0.963$. This plot remains the same for any single-field slow-roll
inflation.

Figure 1: In the above plot we depict the amplitude of CMB dipolar modulation
or hemispherical asymmetry of scalar power spectra obtained from (75). In the
plot we choose the pivot scale $k_{\ast}=a_{\ast}H_{\ast}=0.05{\rm Mpc}^{-1}$
and fix $n_{s}=0.963$. The blue dot with error bar in the plot corresponds to
the observational constraint $|A|=0.066\pm 0.021$ on HPA at large angular
scales or at low-$\ell-2-64$ or $k\lesssim 10^{-1}k_{\ast}$. We can notice
from the plot that the HPA ceases to exist for the small angular scales or
high-$\ell$ which is compatible with the Planck satellite observations Ade
_et al._ (2014); Akrami  _et al._ (2014); Aiola  _et al._ (2015).

Similar to the scalar perturbations we propose the tensor perturbations to be
direct sum of two kinds of fluctuations related by our definition of
$\mathcal{P}\mathcal{T}$ transformation.

$\hat{u}_{ij}\left(\tau,\,\textbf{x}\right)=\frac{1}{\sqrt{2}}\hat{u}_{\rm
I,\,ij}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{u}_{\rm
II,\,ij}\left(-\tau,\,-\textbf{x}\right)\,.$ (77)

Similar to the scalar power spectra we obtain two tensor power spectra which
describe two point tensor correlations in the direction $\hat{\textbf{n}}$ and
$-\hat{\textbf{n}}$ respectively. The two power spectra of tensor correlations
are computed as

$\displaystyle P_{h1}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{4}{a^{2}}|u_{\rm
I\,k}|^{2}\Bigg{|}_{\tau=-\frac{1}{aH}}$ (78)
$\displaystyle\approx\frac{2H^{2}}{\pi^{2}}\left(1+\left(\frac{k}{aH}\right)^{2}\right)$
$\displaystyle\quad+\frac{H^{2}}{\pi}\epsilon\left(\frac{k}{aH}\right)^{3}H_{3/2}^{(1)}\left(\frac{k}{aH}\right)\frac{\partial
H^{(1)}_{\nu_{t}}\left(\frac{k}{aH}\right)}{\partial\nu_{t}}\Big{|}_{\nu_{t}=3/2}\,$
$\displaystyle P_{h2}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{4}{a^{2}}|u_{\rm
II,\,k}|^{2}\Bigg{|}_{\tau_{\ast}=\frac{1}{aH}}$
$\displaystyle\approx\frac{2H^{2}}{\pi^{2}}\left(1+\left(\frac{k}{aH}\right)^{2}\right)$
$\displaystyle\quad-\frac{H^{2}}{\pi}\epsilon\left(\frac{k}{aH}\right)^{3}H_{3/2}^{(1)}\left(\frac{k}{aH}\right)\frac{\partial
H^{(1)}_{\nu_{t}}\left(\frac{k}{aH}\right)}{\partial\nu_{t}}\Big{|}_{\nu_{t}=3/2}\,.$

The hemispherical power asymmetry of the tensor-power spectrum can be defined
as

$T(k)=\frac{P_{h1}-P_{h2}}{4P_{h}}$ (79)

To quantify the above amplitude $T(k)$ we choose Starobinsky or Higgs
inflationary potential in Einstein frame

$V(\phi)=\frac{\lambda}{4}\left(1-e^{-\sqrt{\frac{2}{3}}\phi}\right)^{2}\,.$
(80)

which gives the slow-roll parameters in terms of e-folds as Kehagias  _et al._
(2014)

$\epsilon=\frac{3}{4N^{2}},\quad\eta=\frac{2}{N}$ (81)

In Fig. 2 we depict HPA amplitude of the tensor power spectra for the case of
Starobinsky potential (80).

Figure 2: In the above plot we depict the amplitude of dipolar modulation for
primordial gravitational waves (PGWs) or HPA of tensor power spectra obtained
from (79). The above plot is for the case of Starobinsky or Higgs inflation
(80) for $N=55$ number of e-folds corresponding to the pivot scale
$k_{\ast}=0.05{\rm Mpc}^{-1}$.

Similar the case of the tilt of scalar power spectra (76) we obtain the tilt
of the tensor power spectra almost the same in the two opposite directions of
the sky

$\frac{d\ln P_{h1}}{d\ln k}\approx\frac{d\ln P_{h2}}{d\ln k}\approx
n_{t}\approx-2\epsilon\,.$ (82)

From Fig. 1 and Fig. 2 we can notice that the HPA decreases as we increase the
wave-number. This is a physically expected behavior because the short
wavelength modes do not see the curvature of spacetime therefore the asymmetry
should decrease with increasing $k$. In other words we also see it as how
$(\mathcal{C})\mathcal{P}\mathcal{T}$ symmetry is broken in a curved spacetime
as a function of length scales.

##  VII Conclusions

The success of inflationary cosmology is its mapping of quantum fluctuations
to the temperature fluctuations in the CMB. The HPA seems to pose a
significant challenge to the standard understanding of inflationary quantum
fluctuations because there is an asymmetry in the temperature correlations in
the northern and southern hemispheres of the CMB sky. In this paper, we
investigated whether the power asymmetry in the CMB can be originated from re-
formulating our understanding of the quantization of inflationary
fluctuations. Inspired from several open questions in the context of QFT in
curved spacetime and quantum cosmology we proposed that the inflationary
quantum fluctuations are generated in pairs which are related by discrete
transformations of spacetime. Our approach is inspired from the fact that QFT
in Minkowski spacetime is built on discrete symmetries such as
$\mathcal{C}\mathcal{P}\mathcal{T}$. In the case of curved spacetime i.e.,
when it involves gravity one must be careful in understanding spacetime
reflection symmetries because gravity introduces dynamics. In our framework,
we divide the notions of time into classical and quantum mechanical aspects.
In the context of inflationary cosmology the standard procedure is to find a
background geometry and quantize gravitational and matter degrees of freedom
respecting the classical arrow of time Martin (2005). However, we develop a
new framework of quantization in canonical single field inflation where at the
quantum level we allow the existence of quantum states that propagate both
forward and backward in time based on the $\mathcal{P}\mathcal{T}$
transformations (since we only deal with chargeless fields we omit
$\mathcal{C}$ transformation) of the spacetime. Both of these quantum states
in our framework co-exist within the co-moving horizon and they exit the
horizon in the two opposite directions. We explicitly showed that these two
type of quantum states even though created non-locally still do not violate
any causality and locality principles in our formulation of QFT. We showed
that these pairs of quantum fluctuations behave exactly the same in the case
of exact dS spacetime, but they are slightly different in the case of
inflationary spacetime, which can be interpreted as the spontaneous breaking
of $(\mathcal{C})\mathcal{P}\mathcal{T}$ symmetry. This slight deviation seems
to produce the power asymmetry hinted in the CMB observations. Our
quantization scheme also applied to the case of inflationary tensor power
spectrum and we predict a power asymmetry there as well. Notably, the HPA for
scalar and tensor power spectrum is significant only for large angular scales
or small wave numbers and it decreases for small angular scales or large wave
numbers. This is expected since the spontaneous breaking of
$(\mathcal{C})\mathcal{P}\mathcal{T}$ symmetry is significant for small $k$,
which are more affected by the background curved spacetime, compared to the
large $k$. We quantified all our predictions which serves as an observational
test of our formalism. If the future observations Li  _et al._ (2019); Zhai
and Blanton (2017) confirm these results we will learn significantly about the
nature of inflationary quantum fluctuations as well as QFT in curved
spacetime.

###### Acknowledgements.

KSK acknowledges the support from JSPS and KAKENHI Grant-in-Aid for Scientific
Research No. JP20F20320, and thank Mainz U. for hospitality where part of the
work has been carried out. J. Marto is supported by the grant
UIDB/MAT/00212/2020. We would like to thank Chris Ripken for the useful
discussions and suggestions on quantization procedure and for the initial
collaboration on the project. We thank Gerard ’t Hooft for his inspiring talks
and discussions related to QFT in curved spacetime. We thank Yashar Akrami,
Norma G. Sanchez, Masahide Yamaguchi, Alexei A. Starobinsky, Luca Buoninfante,
Francesco Di Fillippo, Yasha Neiman, Paolo Gondolo, Martin Reuter, Eiichiro
Komatsu, Paulo V. Moniz, Dhiraz Kumar Hazra and L. Sriramkumar for very useful
discussions.

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