[a,b]K. Sravan Kumar

# A robust explanation of CMB anomalies with a new formulation of inflationary
quantum fluctuations

João Marto

###### Abstract

The presence of CMB Hemispherical Asymmetry (HPA) challenges the current
understanding of inflationary cosmology which does not generically predict the
parity violation in the primordial correlations. In this paper, we shall
review the recently proposed resolution to this based on a new formulation of
quantizing inflationary fluctuations by focusing on the discrete spacetime
transformations in a gravitational context. The predictive power of this
formulation is that one can generate a scale dependent HPA in the context of
single field inflation for all the primordial modes including scalar and
tensor fluctuations without introducing any additional parameters. This result
can be seen as an indication of spontaneous breaking of
$\mathcal{C}\mathcal{P}\mathcal{T}$ symmetry in an expanding Universe, if
confirmed by future observations it would be a great leap in the subject of
quantum field theory in curved spacetime.

##  1 Introduction

Inflationary cosmology by definition quasi-de Sitter (qdS) expansion phase in
the early Universe and it has been the most successful paradigm in the early
Universe cosmology The predictions of scalar quantum fluctuations generated
during inflation have been amazingly consistent with the latest Planck
observations [1, 2, 3]. However, the latest observations from the Planck data
along with its predecessor NASA’s Wilkinson Microwave Anisotropy Probe (WMAP)
strongly indicate the presence of hemispherical power asymmetry (HPA) in the
CMB sky [4] which is an anomaly that strongly questions the nature of
inflationary quantum fluctuations [5, 6]. The significance of HPA CMB is
standing now at $3.3\sigma$ [7]. Being more precise, HPA means slightly more
two-point temperature correlations in the southern ecliptic hemisphere in
comparison with the northern ecliptic hemisphere [8]. This anomaly has been
found out to be significant at low-multipoles $\ell=2-64$ [9] ans less
significant but still present up to the multipoles $\ell\sim 600$ according to
the Planck 2013, Hansen _et al._ and Aiola _et al._ [10, 11, 12]. The position
dependent primordial power spectrum parametrization of the following,
phenomenologically defines HPA as [13, 14]

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

where $\mathcal{P}_{\mathcal{R}\>\>iso}(k)$ is the statistically homogeneous
and isotropic power spectrum, $A(k)$ is the amplitude of the observed HPA
which is constrained to be $|A|=0.066\pm 0.021$ ($3.3\sigma$) at the large
angular scales $\ell<64$ or at the wave numbers $k\lesssim 0.0045\textrm{
Mpc}^{-1}$. Here $\hat{\boldsymbol{p}}$ is the direction of maximal symmetry.
$\hat{\textbf{n}}=\frac{\boldsymbol{x}}{x_{\text{ls}}}$ is the line of sight
from earth and $x_{\text{ls}}=14,000\textrm{Mpc}^{-1}$ is the comoving
distance to the surface of last scattering. From (1) we can easily deduce [13]

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

Quantum fluctuations during inflation are somewhat special because we need to
quantize them in the qdS background. This means we would have to deal with
quantum field theory in curved spacetime which is a subject that has not yet
been fully understood. HPA strikingly show us the possible violation of parity
which is a discrete transformation. As we know from the success of quantum
field theory in Minkowski spacetime, the discrete symmetries play a very
important role, right from the $\mathcal{C}\mathcal{P}\mathcal{T}$ (charge
conjugation, parity and time reversal) invariance of scattering amplitudes to
the notion of anti-particle which is a particle that propagates backward in
time. In quantum theory time is a parameter whereas in classical General
Relativity (GR) time is a coordinate and when the spacetime it self is
dynamical and when one supposed to formulate a quantum theory in dynamical
spacetime several conceptual conundrums are inevitable [15]. The way forward
would be a very careful observation of the meaning of discrete spacetime
transformations in the gravitational context before defining quantum fields in
a given spacetime.

In this proceedings paper, we are going to review a new proposal for
quantizing fields in an expanding Universe [16] which explicitly defines the
meaning of $\mathcal{C}\mathcal{P}\mathcal{T}$ invariance and its spontaneous
symmetry breaking. In the later case one can precisely correlate it with the
HPA of primordial power spectra. The crux of this investigation lies in
distinguishing classical arrow of time (CAT) with the quantum mechanical arrow
of time (QAT). The CAT is associated with for example the expanding Universe.
CAT is also often called as thermodynamic arrow of time. The QAT on the other
hand is intrinsic property of quantum theory where the initial and final
conditions play a huge role. Just because Universe is found to be expanding,
one cannot assume the quantum fields in the expanding Universe evolve in the
same direction of time. In fact the expansion of Universe is only associated
with growing scale factor rather than the notion of time itself. It is this
subtle concept of time allow us to rethink on quantum fields in curved
spacetime especially in the context of inflationary background.

The paper is organized as follows. In Sec. 2 we discuss the de Sitter (dS)
spacetime in the flat Friedmann-Lemaître-Robertson-Walker (FLRW) coordinates
and the define the time reversal operation in the context of expanding
Universe. We discuss the direct-sum quantum field theory (DQFT) formalism in
dS and present analysis for the possible validity of
$\mathcal{C}\mathcal{P}\mathcal{T}$ invariance in a language of two-point
correlations. In Sec. 3 we present the definition of time reversal operation
in qdS expansion with the expectation that $\mathcal{C}\mathcal{P}\mathcal{T}$
must be spontaneously breaking in qdS by the so-called slow-roll parameters.
We then apply the DQFT to the inflationary flluctuations and compute the power
spectra of scalar and tensor modes. In Sec. 4 we explicitly witness the
spontaneous breaking of $\mathcal{C}\mathcal{P}\mathcal{T}$ symmetry
quantified by the amplitude of HPA which is of the order of slow-roll
parameters especially at low-$\ell$. We will also see how for the first time
we predict HPA for primordial gravitational wave spectra. In Sec. 5 we briefly
conclude with highlighting the significance of results discussed in this paper
and also comment on the future directions to explore.

##  2 Direct-sum quantum field theory (DQFT) in de Sitter spacetime

The dS manifold is maximally symmetric spacetime like Minkowski and it is a
solution of GR with a positive cosmological constant. Before we go to the
context of quantum fields during inflation, it is important to understand the
subject in dS. The dS spacetime is characterized the following relations
between curvature tensors and the metric as

$R^{\mu}_{\nu\rho\sigma}=\frac{R}{12}\left(\delta^{\mu}_{\rho}g_{\nu\sigma}-\delta^{\mu}_{\sigma}g_{\nu\rho}\right),\quad
R_{\nu\sigma}=\frac{R}{4}g_{\nu\sigma},\quad R=\mathrm{const}\,.$ (3)

The above definition is coordinate independent. In the context of cosmology
the relevant form of expressing dS metric is in terms of flat FLRW coordinates
which is given by

$ds^{2}=-dt^{2}+a(t)^{2}d\textbf{x}^{2}$ (4)

where the scale factor is

$a(t)=e^{Ht},\quad H^{2}=\left(\frac{1}{a}\frac{da}{dt}\right)^{2}>0\,,$ (5)

Here $H$ is called the Hubble parameter. In this dS space, one has the
comoving horizon defined by the radius

$r_{H}=\Big{|}\frac{1}{aH}\Big{|}$ (6)

which decreases during expansion since scale factor grows. For the purpose of
quantization it is useful to express (4) in terms of conformal time defined by

$d\tau=-\frac{dt}{a(t)}\,.$ (7)

Integrating this equation, we obtain

$\tau=\frac{1}{a(t)H}\,\text{,}$

Now the dS metric becomes conformal to Minkowski as

$ds^{2}=\frac{1}{H^{2}\tau^{2}}\left(-d\tau^{2}+d\textbf{x}^{2}\right)$ (8)

When the time coordinate $\tau\gtrless 0$ we get a coordinate singularity at
$\tau=0$ but it is just an artifact of the coordinate transformation. In
reality, There is no spacetime singularity because all the curvature
invariants in dS are constants. Just like Minkowski spacetime the dS metric
satisfies $\mathcal{P}\mathcal{T}$ symmetries as

$\mathcal{P}\mathcal{T}:\tau\to-\tau,\quad\textbf{x}\to-\textbf{x}\,.$ (9)

We can notice that the time ($\tau$) reversal operation actually leads to time
reversal in cosmic time, as well as the change of sign for the Hubble
parameter $H$ i.e.,

$\mathcal{T}:\tau\to-\tau\implies t\to-t,\,H\to-H\,.$ (10)

We can verify that $H\to-H$ operation does not effect at all the definition of
dS (3). For example, the Ricci scalar $R=12H^{2}$ in dS is completely
symmetric under $H\to-H$. Remember that time is a parameter in quantum theory
and here $\tau$ plays the role as a parameter for quantum field theory in dS.
Since the metric (8) is completely time symmteric (i.e., $\tau\to-\tau$)
quantum fields cannot see the difference between whether $\tau$ is positive or
negative. Indeed from (4), we can notice that

${\rm Expanding\,Universe:}\implies\begin{cases}t:-\infty\to+\infty,\quad
H>0&(\tau:+\infty\to 0)\\\ t:+\infty\to-\infty,\quad H<0&(\tau:-\infty\to
0)\end{cases}$ (11)

In (11) we see two arrows of cosmic time $t$ (together with sign of $H$), both
of which correspond to expanding Universe with the definition that the $r_{H}$
decreases as scale factor increases (6). Therefore, (11) exactly serves the
definition of time reversal in an expanding Universe and this is the cue for
our formulation of direct-sum quantum field theory (DQFT).

Let us start with direct-sum quantization of a massless scalar field whose
action for the metric (4) can be computed as

$S_{\phi}=-\frac{1}{2}\int d\tau
d^{3}xa^{2}\phi\left(\partial_{\tau}^{2}+2\mathcal{H}\partial_{\tau}+k^{2}\right)\phi\,,$
(12)

where $\mathcal{H}=\frac{1}{a}\frac{da}{d\tau}=aH$. The above action is
invariant under the $\mathcal{P}\mathcal{T}$ defined in (9). To quantize it is
convenient to rescale the field $\phi\to a\phi$, which bring us to a harmonic
oscillator form with time dependent mass

$S_{\phi}=\frac{1}{2}\int d\tau d^{3}x\Big{[}\phi^{\prime
2}-\left(\partial_{i}\phi\right)^{2}-\frac{2}{\tau^{2}}\phi^{2}\Big{]}\,.$
(13)

In the process of quantization, we promote the scalar field $\phi$ as an
operator $\hat{\phi}\left(\tau,\,\textbf{x}\right)$ which in DQFT formulation
expressed as direct-sum [17]

$\hat{\phi}\left(\tau,\,\textbf{x}\right)=\frac{1}{\sqrt{2}}\hat{\phi}_{I}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{\phi}_{II}\left(-\tau,\,-\textbf{x}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{\phi}_{I}\left(\tau,\,\textbf{x}\right)&0\\\
0&\hat{\phi}_{II}\left(-\tau,\,-\textbf{x}\right)\end{pmatrix}\,.$ (14)

where

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

The creation and annihilation operators here
$\hat{c}_{\pm\textbf{k}},\,\hat{c}^{\dagger}_{\pm\textbf{k}}$ satisfy the
following commutation relations

$\displaystyle=[\hat{c}_{II\,\textbf{k}},\,\hat{c}_{II\,\textbf{k}^{\prime}}^{\dagger}]=\delta^{(3)}\left(\textbf{k}-\textbf{k}^{\prime}\right)$
(16)
$\displaystyle[\hat{c}_{I\,\textbf{k}},\,\hat{c}_{II\,\textbf{k}^{\prime}}]$
$\displaystyle=[\hat{c}_{I\,\textbf{k}},\,\hat{c}_{II\,\textbf{k}^{\prime}}^{\dagger}]=[\hat{c}_{I\,\textbf{k}}^{\dagger},\,\hat{c}_{II\,\textbf{k}^{\prime}}^{\dagger}]=0\,.$

which leads to

$\displaystyle=i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right)$ (17)
$\displaystyle[\hat{\phi}_{II}\left(-t,-\textbf{x}\right),\,\pi_{II}\left(-t,\,-\textbf{x}^{\prime}\right)]$
$\displaystyle=-i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right)$
$\displaystyle\left[\hat{\phi}_{I}\left(\tau,\,\textbf{x}\right),\,\hat{\phi}_{II}\left(-\tau,\,-\textbf{x}\right)\right]$
$\displaystyle=0\,,$

where

$\pi_{I}\left(\tau,\,\textbf{x}\right)=\partial_{\tau}\phi_{I},\quad\pi_{II}\left(-\tau,\,-\textbf{x}\right)=-\partial_{\tau}\phi_{II}\,.$
(18)

The physical meaning of DQFT is the following. In the dS with flat FLRW we
demand the a quantum field within the comoving horizon evolves forward in time
at position x while it evolves backward in time at the position $-\textbf{x}$.
The third line of (17) is the most crucial one which ensures locality and
causality of our (D)QFT framework. We can notice here that the field operator
$\hat{\phi}_{II}$ canonical commutation relation in the second line of (17)
differs from the first line by the sign of $i\to-i$ which indicates the time
reversal $\tau\to-\tau$ because time reversal is an anti-Unitary operation in
quantum theory. In fact, in DQFT we divide the spatial region into two parts
by parity, lets call them as region I and II. A quantum field in region one
evolves forward in time and in the region II evolves backward in time in two
different Fock spaces called ${\rm dS}_{I}$ and ${\rm dS}_{II}$ whose direct-
sum forms the total Fock space of dS.

The mode functions $\phi_{I\,k},\,\phi_{II\,k}$ are

$\displaystyle\phi_{I,\,k}$
$\displaystyle=\alpha_{I\,k}\frac{e^{ik\tau}}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right)+\beta_{I\,k}\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)\,,$
(19) $\displaystyle\phi_{II,\,k}$
$\displaystyle=\alpha_{II\,k}\frac{e^{-ik\tau}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)+\beta_{II\,k}\frac{e^{ik\tau}}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right)\,,$

which satisfy $\phi_{I,\,k}\Big{|}_{\mathcal{T}:\,\tau\to-\tau}=\phi_{II,\,k}$
and are obtained by solving the Mukhanov-Sasaki equation [16]

$\displaystyle\phi_{m\,k}^{\prime\prime}+\left(k^{2}-\frac{2}{\tau^{2}}\right)\phi_{m\,k}=0\,.$
(20)

where $m=I,\,II$. In the limit $k^{2}\gg 2/\tau^{2}$ or $k^{2}\gg
2/r_{H}^{2}$, the mode functions $\phi_{m\,k}$ approach to the ones of
Minkowski spacetime which is expected because at short distance scales the
effects of curvature of spacetime should be negligible (See [15] for more
details). This fixes the Bogoliubov coefficients as

$\left(\alpha_{I\,k},\,\beta_{I\,k}\right)=\left(\alpha_{II\,k},\,\beta_{II\,k}\right)=\left(1,\,0\right)\,,$
(21)

which are compatible with the Wronskian conditions obtained from (17)

$\displaystyle\phi_{I,\,k}\phi^{\prime\ast}_{I,\,k}-\phi^{\ast}_{I,\,k}\varphi^{\prime}_{I,\,k}$
$\displaystyle=i$ (22)
$\displaystyle\phi_{II,\,k}\phi^{\prime\ast}_{II,\,k}-\phi^{\ast}_{II,\,k}\phi^{\prime}_{II,\,k}$
$\displaystyle=-i\,.$

The dS spacetime vacuum is now represented by the direct-sum

$|0\rangle_{dS}=\frac{1}{\sqrt{2}}\left(|0\rangle_{\rm
dS_{I}}\oplus|0\rangle_{\rm
dS_{II}}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}|0\rangle_{\rm dS_{I}}\\\
|0\rangle_{\rm dS_{II}}\end{pmatrix}\,.$ (23)

where $|0\rangle_{\rm dS_{I}},\,|0\rangle_{\rm dS_{II}}$ are vacuums defined
by

$c_{I\,\textbf{k}}|0\rangle_{\rm dS_{I}}=0,\quad
c_{II\,\textbf{k}}|0|0\rangle_{\rm dS_{II}}=0$ (24)

We can verify this, observing that

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

This (25) implies that the correlations of quantum fields related by
$\mathcal{P}\mathcal{T}$ transformations are identical in the case of dS
spacetime. Being more precise, quantum field correlations exiting the comoving
horizon at the angular coordinates $\left(\theta,\,\varphi\right)$ (of the
sphere of radius $r_{H}$ defined in (6) ) are exactly the same as those
exiting at the angular coordinates $\left(\pi-\theta,\,\pi+\varphi\right)$.
This indicates in dS spacetime DQFT formulation $\mathcal{P}\mathcal{T}$ or
$\mathcal{C}\mathcal{P}\mathcal{T}$ (if we include charged fields) symmetry
holds. In the case of qdS or inflationary spacetime we will see that the
equality of correlations in the spatial regions divided by parity does not
hold. We shall discuss this in the next section. We note here that in a soon
upcoming work more details of achieving Unitarity and observer complimentarity
in dS DQFT can be found [15].

##  3 DQFT in inflationary spacetime

In the previous section, we have discussed quantization in dS spacetime by
clear understanding of time reversal operation. In the context of inflationary
spacetime time reversal would be much more involved than dS. As we learned in
the previous section, the conformal time $\tau$ plays the role of parameter in
quantum theory and $\tau\to-\tau$ actually mean time reversal in an expanding
Universe for which we would have to change $t\to-t$ and $H\to-H$. This
implies, if we want to understand time reversal in the context of curved
spacetime or when there is gravity we would need additional parameters
associated with gravity or background curved spacetime. For example, the
Hubble parameter in the context of dS is the additional parameter along with
cosmic time (of course together we can just use $\tau$ as a parameter). In the
context of dynamical gravity like inflation, we would expect more parameters
to play the role in time reversal. This is a very subtle task but however,
with the generic expectation of $\mathcal{C}\mathcal{P}\mathcal{T}$ invariance
must be spontaneously broken in curved spacetime we define the time reversal
transformations in the inflationary context as

$t\to-t\implies H\to-H,\quad\epsilon\to-\epsilon,\quad\eta\to-\eta\,.$ (26)

where $\epsilon=-\dot{H}/H^{2},\eta=\dot{\epsilon}/(H\epsilon)$ are although
usual slow-roll parameters, here in the context of time reversal their role is
completely quantum mechanical and there is no classical analog (See [16] for
more details, also see [18] for more thought provoking discussions on time
reversal in quantum gravity).

Similar to the dS DQFT, we should split the field as direct-sum of the two
components as

$\hat{v}\left(\tau,\,\textbf{x}\right)=\frac{1}{\sqrt{2}}\hat{v}_{\rm+}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{v}_{\rm-}\left(-\tau,\,-\textbf{x}\right)=\frac{1}{\sqrt{2}}\begin{pmatrix}\hat{v}_{+}\left(\tau,\,\textbf{x}\right)&0\\\
0&\hat{v}_{-}\left(-\tau,\,-\textbf{x}\right)\end{pmatrix}\,.$ (27)

where $\hat{v}$ is the canonical Mukhanov-Sasaki variable corresponding to the
only propagating scalar in the context of single field inflation. Here
$\hat{v}_{+}\left(\tau,\,\textbf{x}\right)$ is the component when it acts on
the corresponding vacuum it gives the component of the quantum field which
evolves forward in time and similarly
$\hat{v}_{-}\left(-\tau,\,-\textbf{x}\right)$ when it acts on its
corresponding vacuum gives the component of the quantum field which evolves
backward in time. The total vacuum of inflationary spacetime is the direct-sum
of the two represented as

$|0\rangle_{\rm qdS}=|0\rangle_{\rm qdS_{I}}\oplus|0\rangle_{\rm
qdS_{II}}=\frac{1}{\sqrt{2}}\begin{pmatrix}|0\rangle_{\rm qdS_{I}}\\\
|0\rangle_{\rm qdS_{II}}\end{pmatrix}\,.$ (28)

The field operators $\hat{v}_{\pm}$ can be expanded as

$\displaystyle\hat{v}_{\pm}=\int\frac{d\tau
d^{3}k}{\left(2\pi\right)^{3/2}}\Bigg{[}c_{\left(\pm\right)\textbf{k}}{v}_{\pm,\,k}e^{\mp
i\textbf{k}\cdot\textbf{x}}+c_{\left(\pm\right)\textbf{k}}^{\dagger}{v}_{\pm,\,k}^{\ast}e^{\pm
i\textbf{k}\cdot\textbf{x}}\Bigg{]}$ (29)

with
$c_{\left(\pm\right)\textbf{k}},\,c_{\left(\pm\right)\textbf{k}}^{\dagger}$
being the creation and annihilation operators the define the vacua as

$c_{\left(+\right)\textbf{k}}|0\rangle_{\rm qdS_{I}}=0\quad
c_{\left(-\right)\textbf{k}}|0\rangle_{\rm qdS_{II}}=0$ (30)

The mode functions ${v}_{\pm,\,k}$ are obtained by solving Mukhanov-Sasaki
(MS) equations

$v_{\pm,\,k}^{\prime\prime}+\left(k^{2}-\frac{{\nu}_{s}^{\left(\pm\right)2}-\frac{1}{4}}{\tau^{2}}\right)v_{\pm,\,k}^{2}=0\,.$
(31)

where

$\nu_{s}^{\pm}\approx\frac{3}{2}\pm\epsilon\pm\frac{\eta}{2}$ (32)

We require that

$\displaystyle\big{[}c_{\left(+\right)\textbf{k}},\,c_{\left(-\right)\textbf{k}^{\prime}}\big{]}=0,\quad\big{[}c^{\dagger}_{\left(+\right)\textbf{k}},\,c^{\dagger}_{\left(-\right)\textbf{k}^{\prime}}\big{]}=0,\quad\big{[}c_{\left(+\right)\textbf{k}},\,c^{\dagger}_{\left(-\right)\textbf{k}^{\prime}}\big{]}=0\,.$
(33)

This implies

$\Big{[}\hat{v}_{+}\left(\tau,\,\textbf{x}\right),\,\hat{v}_{-}\left(-\tau,\,-\textbf{x}^{\prime}\right)\Big{]}=0$
(34)

$\displaystyle{v}_{\pm,\,k}=\frac{\sqrt{\pm\pi\tau}}{2}e^{\left(i\nu_{s}^{\pm}+1\right)}\Bigg{[}C_{k}^{\pm}H^{(1)}_{\nu_{s}^{\pm}}\left(\pm
k\tau\right)+D_{k}^{\pm}H^{(2)}_{\nu_{s}^{\pm}}\left(\pm
k\tau\right)\Bigg{]}.$ (35)

The above mode functions can define the creation of positive frequency modes
in the limit $\tau\to\pm\infty$ given we choose
$\left(C_{k}^{\pm},\,D_{k}^{\pm}\right)=\left(1,\,0\right)$ compatible with
the Wronskian

$v_{\pm,k}v_{\pm,k}^{\prime\ast}-v_{\pm,k}^{\ast}v_{\pm,k}^{\prime}=\pm\>i\>\>(\implies|C_{k}^{\pm}|^{2}-|D_{k}^{\pm}|^{2}=1).$
(36)

One can notice that the Wronskian condition equating to $-i$ corresponds to
the canonical commutation relation for a reversed arrow of time [19]

$\Big{[}\hat{v}_{-}\left(-\tau,\,-\textbf{x}\right),\,\hat{\Pi}_{-}\left(-\tau,\,-\textbf{x}^{\prime}\right)\Big{]}=-i\delta\left(\textbf{x}-\textbf{x}^{\prime}\right)\,.$
(37)

Now we can expand (35) up to the leading order in $\left(\epsilon,\eta\right)$
as

$\displaystyle{v}_{\pm,\,k}$ $\displaystyle\approx\sqrt{\frac{1}{2k}}e^{\pm
ik\tau}\left(1\pm\frac{i}{k\tau}\right)$ (38)
$\displaystyle\pm\left(\epsilon+\frac{\eta}{2}\right)\frac{\sqrt{\pi}}{2\sqrt{k}}\sqrt{\pm
k\tau}\frac{\partial H^{(1)}_{\nu_{s}^{\pm}}\left(\pm
k\tau\right)}{\partial\nu_{s}^{\pm}}\Big{|}_{\nu_{s}^{\pm}=3/2}$

We can clearly see from the above equation that $v_{\pm,\,k}$ are not
identical but rather differ by the order of slow-roll parameters and scale
dependence present in the form of Hankel functions. This exactly implies when
a mode $k\sim\frac{1}{r_{H}}$ exits the horizon, its value would be dictated
by $v_{+,\,k}$ at one end $\left(\theta,\,\varphi\right)$ and by $v_{-,\,k}$
on the other end $\left(\pi-\theta,\,\pi+\varphi\right)$. We can derive the
analogous result in the vase of inflationary tensor modes well [16]. When we
compute the equal time correlations of primordial modes on the two opposite
points (antipodal) of the three sphere of radius $r_{H}$ we would see a clear
deference between this. This would exactly generate HPA for primordial power
spectra.

##  4 HPA of primordial power spectra from DQFT of inflatonary quantum
fluctuations

Inflationary quantum fluctuations in the context of DQFT come with
deformations when they exit the comoving horizon on the two opposite sides. To
calculate the two point correlations of curvature perturbation on super-
horizon scales we need to rescale the canonical fields $\hat{v}$ using the
classical background quantities as

$\displaystyle{}_{\rm qdS}\langle
0|\zeta_{\textbf{k}}\zeta_{\textbf{k}^{\prime}}|0\rangle_{\rm qdS}$
$\displaystyle=\left(\frac{1}{2a^{2}\epsilon}\right)\Bigg{|}_{\rm
clas.}\frac{1}{2}\Big{[}{}_{\rm qdS_{I}}\langle
0|\hat{v}_{+\,\textbf{k}}\hat{v}_{+\,\textbf{k}^{\prime}}|0\rangle_{\rm
qdS_{I}}$ (39) $\displaystyle\quad+{}_{\rm qdS_{II}}\langle
0|\hat{v}_{-\,\textbf{k}}\hat{v}_{-\,\textbf{k}^{\prime}}|0\rangle_{\rm
qdS_{II}}\Big{]}$
$\displaystyle=\frac{2\pi^{2}}{k^{3}}\left(P_{\zeta_{+}}+P_{\zeta_{-}}\right)\delta\left(\textbf{k}+\textbf{k}^{\prime}\right)\,,$

where $\zeta_{+},\,\zeta_{-}$ are the power spectra of primordial scalar field
in the two hemispheres of the CMB sky. In deriving (39) we must use carefully
the direct-sum operation [17]

$\hat{v}|0\rangle_{\rm
qdS}=\frac{1}{2}(\hat{v}_{+}\oplus\hat{v}_{-})(|0\rangle_{\rm+}\oplus|0\rangle_{\rm-})=\frac{1}{2}\begin{pmatrix}\hat{v}_{+}|0\rangle_{+}\\\
\hat{v}_{-}|0\rangle_{-}\,.\end{pmatrix}$ (40)

The power spectra of curvature perturbation in the two hemispheres are

$\displaystyle P_{\zeta_{\pm}}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{1}{2a^{2}\epsilon}|v_{\pm,\,k}|^{2}\Bigg{|}_{\tau=\pm\frac{1}{aH}}$
(41)
$\displaystyle\approx\frac{H^{2}}{8\pi^{2}\epsilon}\left(1+\left(\frac{k}{aH}\right)^{2}\right)$
$\displaystyle\quad\pm\frac{H^{2}}{4\pi\epsilon}\left(\epsilon+\frac{\eta}{2}\right)\left(\frac{k}{aH}\right)^{3}H_{3/2}^{(1)}\>\>\frac{\partial
H^{(1)}_{\nu_{s}^{\pm}}}{\partial\nu_{s}^{\pm}}\Big{|}_{\nu_{s}^{\pm}=3/2}\,.$

Following the meaning of (2) the amplitude of HPA can be deduced as

$A(k)=\frac{P_{\zeta_{+}}-P_{\zeta_{-}}}{4P_{\zeta}}$ (42)

Computing the tilt of the power spectra in the two hemispheres of the CMB sky
we get that they are nearly equal.

$\frac{d\ln P_{\zeta_{+}}}{d\ln k}\approx\frac{d\ln P_{\zeta_{-}}}{d\ln
k}\approx n_{s}-1\approx-2\epsilon-\eta\,.$ (43)

This is because of the classical rescaling of the fields in (39) and there
would differences in the values of $n_{s}$ in the sub-leading order. This can
be verified easily by using the expressions of power spectrum (41). Our result
is consistent with the observations about $n_{s}$ in the hemispheres of the
CMB. [20, 21]. In Fig. 1 we depict the HPA amplitude of the scalar power
spectra for $n_{s}=0.963$. This is universal result and is valid for any
single-field inflation that is compatible with the constraints on $n_{s}$.

Figure 1: The amplitude of CMB HPA obtained from (42). In the plot we consider
the pivot scale $k_{\ast}=a_{\ast}H_{\ast}=0.05{\rm Mpc}^{-1}$ and
$n_{s}=0.963$. 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 $k\lesssim 10^{-1}k_{\ast}$.

Similarly, we can apply the DQFT for tensor modes by writing a tensor
fluctuation operator as a direct-sum of two components which describe the
tensor mode evolving forward in time at position x and the mode evolving
backward in time at position $-\textbf{x}$.

$\hat{u}_{ij}=\frac{1}{\sqrt{2}}\hat{u}^{+}_{ij}\left(\tau,\,\textbf{x}\right)\oplus\frac{1}{\sqrt{2}}\hat{u}^{-}_{ij}\left(-\tau,\,-\textbf{x}\right)$
(44)

Following extremely similar steps it is rather straight forward to compute the
tensor mode exiting the comoving horizon on the two opposite sides as

$\displaystyle{u}^{\pm}_{ij,\,k}$ $\displaystyle\approx
e_{ij}\sqrt{\frac{1}{2k}}e^{\pm ik\tau}\left(1\pm\frac{i}{k\tau}\right)$ (45)
$\displaystyle\pm e_{ij}\epsilon\frac{\sqrt{\pi}}{2\sqrt{k}}\sqrt{\pm
k\tau}\frac{\partial H^{(1)}_{\nu_{t}^{\pm}}\left(\pm
k\tau\right)}{\partial\nu_{t}^{\pm}}\Big{|}_{\nu_{t}^{\pm}=3/2}$

where $e_{ij}$ denotes the polarization tensor. As in the case of the scalar
power spectra, we also obtain here two tensor power spectra which describe two
point tensor correlations in the direction $\hat{\textbf{n}}$ and
$-\hat{\textbf{n}}$ respectively. Thus the tensor power spectra in the two
hemispheres of the primordial gravitational wave sky is give by

$\displaystyle P_{h_{\pm}}$
$\displaystyle=\frac{k^{3}}{2\pi^{2}}\frac{4}{a^{2}}|{u}^{\pm}_{ij,\,k}|^{2}\Bigg{|}_{\tau=\pm\frac{1}{aH}}$
(46)
$\displaystyle\approx\frac{2H^{2}}{\pi^{2}}\left(1+\left(\frac{k}{aH}\right)^{2}\right)$
$\displaystyle\quad\pm\frac{H^{2}}{\pi}\epsilon\left(\frac{k}{aH}\right)^{3}H_{3/2}^{(1)}\>\>\frac{\partial
H^{(1)}_{\nu_{t}^{\pm}}}{\partial\nu_{t}^{\pm}}\Big{|}_{\nu_{t}^{\pm}=3/2}\,.$

Very similar to (42) we can define the HPA of tensor power spectrum as

$T(k)=\frac{P_{h_{+}}-P_{h_{-}}}{4P_{h}}$ (47)

We can quantify the above amplitude $T(k)$ given a model of inflation. [22] In
Fig. 2 we depict HPA amplitude of the tensor power spectra for the case of
Starobinsky or Higgs inflation.

Figure 2: The amplitude of HPA for tensor power spectra obtained from (47). In
the above plot we consider Starobinsky or Higgs inflation for $N=55$ number of
e-folds corresponding to the pivot scale $k_{\ast}=0.05{\rm Mpc}^{-1}$.

Like the scalar power spectra tilt (43), we also obtain the same tilt of the
tensor power spectra in the two opposite directions of the sky i.e.,

$\frac{d\ln P_{h_{+}}}{d\ln k}\approx\frac{d\ln P_{h_{-}}}{d\ln k}\approx
n_{t}\approx-2\epsilon\,.$ (48)

From Fig. 1 and Fig. 2 we can notice that the HPA decreases as the wavenumber
increases. This is very much a reasonable result, short wavelength modes would
feel less the curvature of spacetime. Both Fig. 1 and Fig. 2 indicate a
spontaneous breaking of $\mathcal{C}\mathcal{P}\mathcal{T}$ if confirmed we
would transform our understanding of quantum fields in curved spacetime.

##  5 Conclusions

In this paper we discussed recent result of [16] hemispherical power asymmetry
of primordial power spectra which has emerged from fundamental questions of
discrete spacetime transformations and their role in quantizing fields in
curved spacetime. The result is based on a new scheme of quantizing fields in
curved spacetime especially in inflationary spacetime where field operators
and Fock space are proposed as direct-sum of two components defined by
$\mathcal{C}\mathcal{P}\mathcal{T}$ transformations in the gravitational
context. This method of quantization is called direct-sum quantum field theory
(DQFT) and it gives us a way to fully exploit the discrete spacetime
transformations which form the crux of quantum thoery. If the HPA of scalar
and tensor power spectra according to DQFT are confirmed in the future
observations of CMB and primordial gravitational waves, we would learn more
clearly about quantum origin of inflationary fluctuations. It is worth to
stress here that HPA of tensor power spectra is predicted for the first time
ever based on pure theoretical grounds. It is obviously a natural step to
explore further DQFT and also compute higher order correlations functions
within this program because as a matter of interest the recent Planck data [4]
appears to indicate presence of HPA even in the higher order correlations.

## Acknowledgments

KSK acknowledges the support from JSPS and KAKENHI Grant-in-Aid for Scientific
Research No. JP20F20320 and No. JP21H00069, and thank Mainz U. for hospitality
where part of the work has been carried out. KSK would also like to thank
Royal Society for the Newton International Fellowship. J. Marto is supported
by the grant UIDB/MAT/00212/2020. KSK would like to thank the organizers of
Tensions in cosmology 2022 for the kind invitation to present this talk.

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