Loop Corrections in Gravitational Wave Spectrum in Single Field Inflation

Hassan<EMAIL_ADDRESS>
School of Astronomy, Institute for Research in Fundamental Sciences (IPM)
P. O. Box 19395-5531, Tehran, Iran

###### Abstract

We study the one-loop corrections in power spectrum of long gravitational
waves induced from small scale modes in the models of single field inflation
undergoing a phase of ultra-slow-roll (USR). We show that the spectrum of long
tensor perturbations are largely unaffected by the loop corrections from the
short scalar modes. In particular, the spectrum of long tensor perturbations
is insensitive to the sharpness of the transition from the USR phase to the
final slow-roll phase. This is in contrast to the case of scalar power
spectrum in which the loop corrections can be large for a sharp transition
while it is slow-roll suppressed in a mild transition. We study the tensor-
scalar-scalar bispectrum in the squeezed limit and demonstrate that the
Maldacena consistency condition does hold.

##  1 Introduction

Recently the question of one-loop corrections in the power spectrum of large
CMB scale scalar perturbations from the small scale modes in the setup of
single field inflation undergoing a phase of ultra-slow-roll (USR) was debated
extensively [1, 2, 3, 4, 5, 6, 7, 8, 9, 10], for a related earlier work see
[11]. This is particularly an important question since the models of single
field inflation with an intermediate USR phase have been employed extensively
in recent years as a viable mechanism to generate primordial black holes
(PBHs) which may comprise all or parts of cold dark matter [12, 13, 14], for a
review see [15, 16]. More specifically, to have a successful mechanism of PBHs
formation, one requires the amplitude of curvature perturbations to be
enhanced by a factor of $10^{7}$ or so in the allowed small scales compared to
the large CMB scales. It turns out that an intermediate phase of USR inflation
can provide this enhancement naturally.

The USR setup is a phase of inflation in which the potential is very flat [17,
18, 19]. Consequently, the inflaton velocity falls off exponentially and the
curvature perturbations grow on superhorizon scales [20]. As the curvature
perturbations grows on superhorizon scales, it provides a non-trivial example
for the violation of the celebrated Maldacena consistency condition [21, 22]
for the non-Gaussianity of single field inflation [20, 23, 24, 25, 26, 27, 28,
29, 30]. More specifically, it was shown in [20] that the amplitude of local-
type non-Gaussianity in USR model is $f_{NL}=\frac{5}{2}$. This question was
further investigated in [31] in which it was demonstrated that the final
amplitude of $f_{NL}$ crucially depends on the sharpness of the transition
from the USR phase to the final slow-roll (SR) phase. In particular, in an
extreme sharp transition from the USR phase to the SR phase, as assumed in
[20], $f_{NL}$ reaches its maximum value $\frac{5}{2}$. However, if the
transition is mild, then the curvature perturbations evolve after the USR
phase until it reaches to is final attractor value. Correspondingly, much of
the amplitude of $f_{NL}$ is washed out and it ends up to a value at the order
of the slow-roll parameters though the Maldacena consistency condition is
still violated. The lesson is that the sharpness of the transition from the
USR phase to the final SR phase plays important roles to read off the
amplitude of cosmological observables at the end of inflation.

Originally, it was argued in [1], see also [2], that the one-loop corrections
from small USR modes can significantly affect the large CMB scale modes.
Therefore, it was argued that to keep these loop corrections under
perturbative control, the model loses its applicability to generate the
desired PBHs abundance. This conclusion was criticized in [3, 4] where it was
advocated that this conclusion is model-dependent and the dangerous one-loop
corrections can be harmless in a smooth transition. This question was further
investigated in [8] in a consistent manner where the effects of both cubic and
quartic Hamiltonians were taken into account. While the analysis in [8]
supported the conclusion of [1] for the setup with a sharp transition but it
was argued that the situation can be very different in a mild transition.
Finally, this question was further studied in [10] where, using $\delta N$
formalism, it was shown that for a mild transition the one-loop corrections
are suppressed by the slow-roll parameters and the setup can still be viable
for PBHs formation, in agreement with [3, 4]. The conclusion from these works,
as in the old story of $f_{NL}$ alluded to before, is that the amplitude of
one-loop corrections crucially depends on the sharpness of the transition from
the USR phase to the final SR phase. For a physical smooth transition, the
dangerous one-loop corrections are washed out during the subsequent evolutions
of the modes after the USR phase.

With the above discussions in mind, in this work we extend the motivation of
[1] and calculate the one-loop correction from small USR modes on large CMB
scale gravitational waves (GWs) perturbations 111For earlier works concerning
then loop corrections in tensor power spectrum during inflation see [32, 33,
34, 35, 36]. . On the physical ground, similar to the reasonings of [3, 4], it
is expected that the tensor perturbations to be less sensitive to the USR
phase transition. This is because the amplitude of GWs are determined by the
Hubble scale, $H$, during inflation. As the value of $H$ is not much modified
during the USR transition, then the background for GWs propagation is not much
modified either. Add to it the important effect that the tensor perturbations
are frozen on superhorizon scales at the linear level in perturbation theory
[37, 38, 39, 40]. However, the lesson of large loop corrections in a sharp
transition for the case of scalar power spectrum sets a non-trivial example to
examine more directly the validity of the above physical expectations for the
long GWs. This is the goal of this work.

##  2 The Setup

Here we briefly review our setup and present the formulas which will be
required for our subsequent analysis.

We consider a three-phase model of inflation in which a USR phase is
sandwiched between two phases of SR inflation ($SR\rightarrow USR\rightarrow
SR$). The early SR phase is when the large CMB scale mode leaves the horizon.
The USR phase is extended in the interval $t_{i}\leq t\leq t_{e}$ in which the
potential is flat $V(\phi)=V_{0}$. The background equations during the USR
phase are,

$\displaystyle\ddot{\phi}(t)+3H\dot{\phi}(t)=0\,,\quad\quad
3M_{P}^{2}H^{2}\simeq V_{0},$ (1)

where $M_{P}$ is the reduced Planck mass and $H$ is the Hubble expansion rate
during inflation. During the USR phase $H$ is very nearly constant while
$\dot{\phi}\propto\frac{1}{a^{3}}$.

The two slow-roll parameters related to $H$ are given as follows,

$\displaystyle\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}}{2M_{P}^{2}H^{2}}\,,\quad\quad\eta\equiv\frac{\dot{\epsilon}}{H\epsilon}\,.$
(2)

Since $\epsilon$ falls off like $a^{-6}$ during the USR setup, we see that
$\eta\simeq-6$ which is the hallmark of the USR inflation [17]. Going to
conformal time $d\tau=dt/a(t)$ with $aH\tau\simeq-1$, the evolution of
$\epsilon$ is given by

$\displaystyle\epsilon(\tau)=\epsilon_{i}\big{(}\frac{\tau}{\tau_{i}}\big{)}^{6}\,,$
(3)

in which $\epsilon_{i}$ is the value of $\epsilon$ at the start of USR phase.
Correspondingly, at the end of USR phase
$\epsilon_{e}=\epsilon_{i}\big{(}\frac{\tau_{e}}{\tau_{i}}\big{)}^{6}$. Using
the number of e-fold, $dN=Hdt$, the duration of the USR phase is denoted by
$\Delta N\equiv N(\tau_{e})-N(\tau_{i})$ so $\epsilon_{e}=e^{-6\Delta
N}\epsilon_{i}$.

As shown in [8], a crucial role is played by the sharpness of the transition
from the USR phase to the final SR phase. To take this into account, following
[31], we define the parameter associated to the sharpness of the transition,
$h$, as follows

$\displaystyle
h\equiv\frac{6\sqrt{2\epsilon_{V}}}{\dot{\phi}(t_{e})}=-6\sqrt{\frac{\epsilon_{V}}{\epsilon_{e}}}\,.$
(4)

Here, $\epsilon_{V}$ represents the slow-roll parameter at the final SR phase
when the system reaches its attractor regime. Since we assume (without lack of
generality) that $\phi$ is decreasing during USR phase, then $\dot{\phi}<0$ so
$h<0$.

As shown in [31] near the transition we can approximate $\eta$ as

$\displaystyle\eta=-6-h\theta(\tau-\tau_{e})\quad\quad\tau_{e}^{-}<\tau<\tau_{e}^{+}\,.$
(5)

In particular, for the derivative of $\eta$, we have

$\displaystyle\frac{d\eta}{d\tau}=-h\delta(\tau-\tau_{e})\,,\quad\quad\tau_{e}^{-}<\tau<\tau_{e}^{+}\,.$
(6)

In the following analysis we consider two cases of sharp transition: “natural”
sharp transition in which $\eta$ drops to zero immediately after transition
corresponding to $h=-6$. In this situation $\epsilon$ after the transition is
frozen to its value at the end of USR given by $\epsilon_{e}$. This limit was
studied in [1, 2]. The other case is “extreme” sharp transition where $|h|\gg
1$. In this situation, $\epsilon$ after the transition evolves towards the end
of inflation (or when the evolution in the final stage has reached its
attractor phase) so $\epsilon_{V}=\epsilon_{e}(\frac{h}{6})^{2}$.

As $\epsilon(\tau)$ falls off exponentially during the USR phase, the comoving
curvature perturbation ${\cal{R}}(\tau)$ grows exponentially during the USR
phase, ${\cal{R}}(\tau)\propto a(\tau)^{3}\propto\tau^{-3}$. After the USR
period, the curvature perturbation may evolve during the final USR phase until
it reaches its final attractor value to be measured at the end of inflation.
To read off the final value of ${\cal{R}}$, we have to track it from the first
phase of inflation towards the USR phase and then eventually into the final SR
phase. This is achieved by requiring that both ${\cal{R}}(\tau)$ and
${\cal{R}}^{\prime}(\tau)$ to be continuous across the transitions
$SR\rightarrow USR\rightarrow SR$.

Starting with a Bunch-Davies initial condition in the first SR phase, the mode
function in the Fourier space is given by

$\displaystyle{\cal{R}}^{(1)}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon_{i}k^{3}}}(1+ik\tau)e^{-ik\tau}\,,\quad\quad(\tau<\tau_{i})\,$
(7)

where $\epsilon_{i}$ is the value of the slow-roll parameter at the start of
inflation when the CMB scale mode leaves the horizon. The superscript (1)
indicates the first SR phase. During the USR phase, the mode function is given
formally by the superposition of the positive and negative frequency modes,

$\displaystyle{\cal{R}}^{(2)}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon_{i}k^{3}}}\big{(}\frac{\tau_{i}}{\tau}\big{)}^{3}\Big{[}\alpha^{(2)}_{k}(1+ik\tau)e^{-ik\tau}+\beta^{(2)}_{k}(1-ik\tau)e^{ik\tau}\Big{]}\,,$
(8)

with the coefficients $\alpha^{(2)}_{k}$ and $\beta^{(2)}_{k}$, after imposing
the matching condition at $\tau=\tau_{i}$, are obtained to be

$\displaystyle\alpha^{(2)}_{k}=1+\frac{3i}{2k^{3}\tau_{i}^{3}}(1+k^{2}\tau_{i}^{2})\,,\quad\quad\beta^{(2)}_{k}=-\frac{3i}{2k^{3}\tau_{i}^{3}}{(1+ik\tau_{i})^{2}}e^{-2ik\tau_{i}}\,.$
(9)

Finally, imposing the matching conditions at $\tau_{e}$, the mode function in
the final SR phase, denoted by the superscript (3), is obtained to be

$\displaystyle{\cal{R}}^{(3)}_{k}=\frac{H}{M_{P}\sqrt{4\epsilon(\tau)k^{3}}}\Big{[}\alpha^{(3)}_{k}(1+ik\tau)e^{-ik\tau}+\beta^{(3)}_{k}(1-ik\tau)e^{ik\tau}\Big{]}\,,$
(10)

with the coefficients $\alpha^{(3)}_{k}$ and $\beta^{(3)}_{k}$ given by,

$\displaystyle\alpha^{(3)}_{k}=\frac{1}{8k^{6}\tau_{i}^{3}\tau_{e}^{3}}\Big{[}3h(1-ik\tau_{e})^{2}(1+ik\tau_{i})^{2}e^{2ik(\tau_{e}-\tau_{i})}+(-2ik^{3}\tau_{i}^{3}+3k^{2}\tau_{i}^{2}+3)(4ik^{3}\tau_{e}^{3}-hk^{2}\tau_{e}^{2}-h)\Big{]}$

and

$\displaystyle\beta^{(3)}_{k}=\frac{-1}{8k^{6}\tau_{i}^{3}\tau_{e}^{3}}\Big{[}3(1+ik\tau_{i})^{2}(h+hk^{2}\tau_{e}^{2}+4ik^{3}\tau_{e}^{3})e^{-2ik\tau_{i}}-h(1+ik\tau_{e})^{2}(3+3k^{2}\tau_{i}^{2}-2ik^{3}\tau_{i}^{3})e^{-2ik\tau_{e}}\Big{]}$

Finally, the power spectrum of curvature perturbations at the end of inflation
$\tau=\tau_{0}\rightarrow 0$ for the mode in the interval $k_{i}<k<k_{e}$
which leaves the horizon during the USR phase is given by

$\displaystyle
P_{\cal{R}}(\tau_{0},k)=\Big{(}\frac{h-6}{h}\Big{)}^{2}\frac{H^{2}}{4M_{P}^{2}\epsilon_{e}k^{3}}=\Big{(}\frac{h-6}{h}\Big{)}^{2}P_{\cal{R}}(\tau_{e},k)\,,\quad\quad(k_{i}<k<k_{e})\,.$
(11)

Curiously, we see that the power spectrum is scaled with a factor
$\big{(}\frac{h-6}{h}\big{)}^{2}$ compared to its value at the end of USR
phase. In the limit of extreme sharp transition, $h\rightarrow-\infty$, we see
that $P_{\cal{R}}(\tau_{0},k)\simeq P_{\cal{R}}(\tau_{e},k)$. This is
expected, since in this limit the mode function is frozen immediately after
the USR phase and does not experience evolution after the USR phase. On the
other hand, for the case of natural sharp transition with $h=-6$, we see that
$P_{\cal{R}}(\tau_{0},k)\simeq 4P_{\cal{R}}(\tau_{e},k)$ so the power spectrum
actually becomes larger towards the end of inflation. This is because the mode
function is still evolving after the USR phase until it reaches to its final
attractor value. We comment that there are subleading correction of order
$O\big{(}\frac{k^{2}}{k_{e}^{2}}\big{)}$ in Eq. (11) which we have neglected.

On the other hand, the modes which leave the horizon during the first SR phase
are frozen during the intermediate USR phase. Correspondingly, for these modes
(at the tree level) we have

$\displaystyle
P_{\cal{R}}(\tau_{0},k)=\frac{H^{2}}{4M_{P}^{2}\epsilon_{i}k^{3}}\,,\quad\quad(k<k_{i})\,.$
(12)

##  3 Cubic and Quartic Hamiltonians

Our goal is to calculate the one-loop corrections in tensor power spectrum
induced by the scalar perturbations which experience a growth during the USR
phase. For this purpose, we need to calculate the cubic and quartic
interaction Hamiltonians. Schematically, the cubic Hamiltonian represents an
interaction of the type $\gamma{\cal{R}}^{2}$ while the quartic Hamiltonian is
in the form $\gamma^{2}{\cal{R}}^{2}$. A schematic view of the corresponding
one-loop diagrams associated to these interactions are presented in Fig. 1.
The left panel in Fig. 1 represents the contribution of the cubic Hamiltonian
involving a nested in-in integral while the right panel represents the
contribution of the quartic Hamiltonian involving a single in-in integral.

Figure 1:  The Feynman diagrams for the one-loop correction in tensor power
spectrum. The dotted line represents the tensor perturbations while the solid
line in the loop represents the scalar perturbations. The left and right panel
represent the contribution of the cubic and quartic Hamiltonians respectively.

We consider the tensor perturbations of the FLRW background as follows,

$\displaystyle ds^{2}=-dt^{2}+g_{ij}d{x}^{i}dx^{j}\,,\quad\quad g_{ij}\equiv
a(t)^{2}\hat{h}_{ij}\,,$ (13)

in which $\hat{h}_{ij}$ is expanded in terms of the tensor perturbations
$\gamma_{ij}$ as follows [21]

$\displaystyle\hat{h}_{ij}=\delta_{ij}+\gamma_{ij}+\frac{1}{2}\gamma_{i\ell}\gamma_{\ell
j}+\cdots\,.$ (14)

The tensor perturbations are transverse and traceless,
$\gamma^{i}_{i}=\partial_{i}\gamma_{ij}=0$ in which the indices are raised via
$\delta^{ij}$. With this construction, there is no contribution of
$\gamma_{ij}$ in $\sqrt{-g}$.

The total action is $S_{\rm total}=S_{\rm matter}+S_{\rm EH}$ in which $S_{\rm
matter}$ is the matter part of the action while $S_{\rm EH}$ represents the
usual Einstein-Hilbert action. To calculate the leading interaction
Hamiltonian, we use the effective field theory (EFT) of inflation [41, 42]. In
a near dS spacetime with a background inflaton field $\phi(t)$, the four-
dimensional diffeomorphism invariance is spontaneously broken to a three-
dimensional spatial diffeomorphism invariance. Starting with the unitary (or
comoving) gauge where the perturbations of inflaton are turned off, one is
allowed to write down all terms in the action which are consistent with the
remaining three-dimensional diffeomorphsim invariance. Upon doing so, the
background inflation dynamics is controlled via the known Hubble expansion
rate $H(t)$ and its derivative $\dot{H}(t)$. After writing the full action
consistent with the three dimensional diffeomorphsim invariance, one restores
the full four-dimensional diffeomorphsim invariance by introducing a scalar
field fluctuations, $\pi(x^{\mu})$, which is the Goldstone boson associated
with the breaking of the time diffeomorphsim invariance. One big advantage of
the EFT approach is when one works in the decoupling limit where the
gravitational back-reactions are neglected. In this limit one neglects the
slow-roll suppressed interactions in cubic and quartic actions while keeping
only the leading terms which can yield large non-Gaussianities. In our study
concerning the USR setup, these are the interactions which induce large
corrections in one-loop integrals. For earlier work employing EFT approach for
the bispectrum analysis in a general non-attractor setup (including the USR
setup) see [43]. The EFT approach was employed in [8] to study the one-loop
corrections in scalar power spectrum.

Assuming we have a canonical scalar field with a sound speed $c_{s}=1$, the
matter part of the action consistent with the FLRW inflationary background is
given by [41]

$\displaystyle S_{\rm
matter}=\int\\!d^{4}x\sqrt{-g}\Big{[}-M^{2}_{P}\dot{H}(t+\pi)\Big{(}\frac{1}{N^{2}}(1+\dot{\pi}-N^{i}\partial_{i}\pi)^{2}-g^{ij}\partial_{i}\pi\partial_{j}\pi\Big{)}$
$\displaystyle-M^{2}_{P}\left(3H^{2}(t+\pi)+\dot{H}(t+\pi)\right)\Big{]}\,,$
(15)

in which $N$ and $N^{i}$ are the lapse and shift function in the standard ADM
formalism. In the decoupling limit where the gravitational back-reactions are
neglected we set $N=1$, $N^{i}=0$ and $\sqrt{-g}=a^{3}$. Our goal is to read
off the interaction between $\pi$ and $\gamma_{ij}$. Since $\gamma_{ij}$ does
not contribute into $\sqrt{-g}$, the coupling between $\pi$ and $\gamma_{ij}$
to leading order comes via the interaction
$g^{ij}\partial_{i}\pi\partial_{j}\pi$. On the other hand, to quadratic order,
we have

$\displaystyle
g^{ij}=a^{-2}\big{(}\delta_{ij}-\gamma_{ij}+\frac{1}{2}\gamma_{i\ell}\gamma_{\ell
j}\big{)}\,,$ (16)

where in the right hand side above, we raise and lower the indices via
$\delta_{ij}$. Correspondingly, the interaction between $\pi$ and
$\gamma_{ij}$ to quartic order has the following terms

$\displaystyle
g^{ij}\partial_{i}\pi\partial_{j}\pi\rightarrow-\gamma_{ij}\partial_{i}\pi\partial_{j}\pi+\frac{1}{2}\gamma_{i\ell}\gamma_{\ell
j}\partial_{i}\pi\partial_{j}\pi\,.$ (17)

On the other hand, expanding $\dot{H}(t+\pi)$ to first order in $\pi$ we have

$\displaystyle\dot{H}(t+\pi)$ $\displaystyle=$
$\displaystyle\dot{H}+\ddot{H}\pi+\cdots,$ (18) $\displaystyle\simeq$
$\displaystyle-\epsilon H^{2}-\epsilon\eta H^{3}\pi\,.$

It is important to note that in the USR setup $\eta\simeq-6$, so we can not
discard the last term above.

Plugging Eqs. (18) and (17) in the action (3) the cubic action is obtained to
be [46]

$\displaystyle S_{\gamma\pi^{2}}=M_{P}^{2}H^{2}\int d\tau d^{3}x\,\epsilon
a^{2}\gamma_{ij}\partial_{i}\pi\partial_{j}\pi\,,$ (19)

while the quartic action is given by,

$\displaystyle S_{\gamma^{2}\pi^{2}}=M_{P}^{2}H^{2}\int d\tau d^{3}x\,\epsilon
a^{2}\left[-\frac{1}{2}\gamma_{i\ell}\gamma_{\ell
j}\partial_{i}\pi\partial_{j}\pi+\eta\pi\gamma_{ij}\partial_{i}\pi\partial_{j}\pi\right]\,.$
(20)

Correspondingly, the cubic and quartic interaction Hamiltonians are

$\displaystyle{\bf H_{3}}=-M_{P}^{2}H^{2}\int d^{3}x\,\epsilon
a^{2}\gamma_{ij}\partial_{i}\pi\partial_{j}\pi\,,$ (21)

and

$\displaystyle{\bf H_{4}}=M_{P}^{2}H^{2}\int d^{3}x\,\epsilon
a^{2}\left[\frac{1}{2}\gamma_{i\ell}\gamma_{\ell
j}\partial_{i}\pi\partial_{j}\pi-\eta\gamma_{ij}\pi\partial_{i}\pi\partial_{j}\pi\right]\,.$
(22)

As we see, the quartic Hamiltonian has two terms. One can easily check that
the second term above, containing
$\gamma_{ij}\pi\partial_{i}\pi\partial_{j}\pi$, does not contribute to
graviton power spectrum at one-loop level while it contributes to graviton
power spectrum at two-loop level. Therefore, in the following analysis where
we study the one-loop correction in graviton power spectrum, we neglect the
effects of the second term in ${\bf H_{4}}$.

From the above interaction Hamiltonians we see that both ${\bf H_{3}}$ and
${\bf H_{4}}$ contain spatial derivatives of the scalar perturbations. This is
required because the tensor perturbations carry the indices $i,j$ so they
should be contracted with the spatial derivatives of the scalar perturbations.
Consequently, one expects that the induced loop corrections in tensor power
spectrum to be suppressed compared to the case of scalar power spectrum.
However, the amplitude of one-loop corrections in tensor spectrum has yet to
be calculated.

Finally, note that curvature perturbations ${\cal{R}}$ is related to $\pi$ via
[8]

$\displaystyle{\cal{R}}=-H\pi+O(\pi^{2})\,,$ (23)

in which the higher order terms contain the derivatives of $\pi$ or $H$ [44,
45]. However, we calculate the two-point correlation functions at the end of
inflation $\tau=\tau_{0}\rightarrow 0$ where it is assumed that the system is
in the slow-roll regime and the perturbations are frozen on superhorizon
scales. In this case, the higher order corrections in Eq. (23) are suppressed
and we can simply use the linear relation between ${\cal{R}}$ and $\pi$ in the
following in-in integrals [8].

Going to Fourier space, the tensor perturbations are expended as follows:

$\displaystyle\gamma_{ij}(x)=\int\frac{d^{3}{\bf{k}}}{(2\pi)^{3}}\sum_{s=\pm}\epsilon_{ij}^{s}({\bf{k}})\gamma^{s}_{{\bf{k}}}e^{i{\bf{k}}\cdot{\bf{x}}}\,,$
(24)

in which $s=\pm$ are two polarizations of the tensor perturbation. The
polarization tensor is transverse and traceless,
$\epsilon_{ii}=k^{i}\epsilon_{ij}=0$ and satisfies

$\displaystyle\epsilon_{ij}^{s*}({\bf{k}})=\epsilon_{ij}^{s}(-{\bf{k}})\,,\quad\quad\epsilon_{ij}^{s}({\bf{k}})\epsilon_{ij}^{s^{\prime}*}({\bf{k}})=2\delta_{ss^{\prime}}\,.$
(25)

As an example of polarization tensor, taking $\widehat{\bf{k}}$ along the
third direction, we choose [37]

$\displaystyle\epsilon_{11}(\hat{z},\pm 2)=-\epsilon_{22}(\hat{z},\pm 2)=\mp
i\epsilon_{12}(\hat{z},\pm 2)=\mp i\epsilon_{21}(\hat{z},\pm
2)=\frac{1}{\sqrt{2}},\quad\epsilon_{i3}=\epsilon_{3i}=0\,.$ (26)

To quantize the free tensor perturbation, as usual we expand the Einstein-
Hilbert action to quadratic order in $\gamma_{ij}$ obtaining [38]

$\displaystyle S_{\gamma^{2}}=\frac{M_{P}^{2}}{8}\int d\tau
d^{3}x\,a^{2}\big{[}(\gamma_{ij}^{\prime})^{2}-(\nabla\gamma_{ij})^{2}\big{]}\,.$
(27)

Expanding the quantum operators in terms of the corresponding creation and
annihilation operators as,

$\displaystyle\gamma^{s}_{\bf{k}}=b^{s}_{\bf{k}}\gamma_{k}(\tau)+{b^{s\dagger}_{-{\bf{k}}}}\gamma_{k}(\tau)^{*},$
(28)

with the usual commutation relation
$[b^{s_{1}}_{\bf{k}},b^{s_{2}}_{{\bf{k}}^{\prime}}]=\delta^{s_{1}s_{2}}\delta^{3}({\bf{k}}-{\bf{k}}^{\prime})$,
the mode function is given by

$\displaystyle\gamma_{k}(\tau)=\frac{H\sqrt{2}}{M_{P}k^{\frac{3}{2}}}(1+ik\tau)e^{-ik\tau}.$
(29)

Correspondingly, the two-point correlation is given by

$\displaystyle\langle\gamma^{s}_{\bf{k}}\gamma^{s^{\prime}}_{{{\bf{k}}}^{\prime}}\rangle=\frac{\delta^{ss^{\prime}}}{2}P_{\gamma}(k)=\frac{2H^{2}}{k^{3}M_{P}^{2}}\delta^{ss^{\prime}}\,,$
(30)

with the dimensionless tensor power spectrum given by

$\displaystyle{\cal{P}}_{\gamma}=\frac{k^{3}}{2\pi^{2}}P_{\gamma}(k)=\frac{2H^{2}}{\pi^{2}M_{P}^{2}}\,.$
(31)

To calculate the loop corrections, we employ the standard in-in formalism [47]
in which the expectation value of the operator $\widehat{O}$ at the end of
inflation $\tau_{0}$ is given by the Dyson series,

$\displaystyle\big{\langle}\widehat{O}(\tau_{0})\big{\rangle}=\Big{\langle}\Big{[}\bar{\mathrm{T}}\exp\Big{(}i\int_{-\infty}^{\tau_{0}}d\tau^{\prime}H_{in}(\tau^{\prime})\Big{)}\Big{]}\,\widehat{O}(\tau_{0})\,\Big{[}\mathrm{T}\exp\Big{(}-i\int_{-\infty}^{\tau_{0}}d\tau^{\prime}H_{in}(\tau^{\prime})\Big{)}\Big{]}\Big{\rangle}\,,$
(32)

in which $\mathrm{T}$ and $\bar{\mathrm{T}}$ represent the time ordering and
anti-time ordering respectively while $H_{in}(t)$ collectively represents the
interaction Hamiltonian. In our case at hand $H_{in}(\tau)={\bf H}_{3}+{\bf
H}_{4}$.

##  4 Tensor-Scalar-Scalar Consistency Condition

While our main goal is to calculate the one-loop corrections in tensor power
spectrum, but as a prelude here we study the bispectrum of
$\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}{\cal{R}}_{{\bf{k}}_{2}}{\cal{R}}_{{\bf{k}}_{3}}\big{\rangle}$
in the squeezed limit $k_{1}\ll k_{2}\simeq k_{3}$. This is mainly to check
that our EFT approach with the interaction Hamiltonians given above are
trusted for the one-loop corrections in tensor power spectrum. While this
analysis in interesting and new (in the current $SR\rightarrow USR\rightarrow
SR$ setup), but the reader who is only interested in loop corrections can skip
directly to next section.

To calculate
$\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}{\cal{R}}_{{\bf{k}}_{2}}{\cal{R}}_{{\bf{k}}_{3}}\big{\rangle}$
in the squeezed limit we assume that the tensor perturbation has left the
horizon during the first SR phase while the scalar perturbations have left the
horizon during the intermediate USR phase. As such, the hierarchy
$k_{1}\rightarrow 0$ and $k_{2}\simeq k_{3}$ is assumed. On the physical
ground, as the tensor mode is frozen on superhorizon scale, we expect that a
consistency condition similar to that of Maldacena [21] for tensor-scalar-
scalar to hold.

To calculate
$\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}{\cal{R}}_{{\bf{k}}_{2}}{\cal{R}}_{{\bf{k}}_{3}}\big{\rangle}$
at the tree level, we only need the cubic interaction Hamiltonian ${\bf
H_{3}}$. Plugging ${\bf H_{3}}$ from Eq. (21) in the in-in integral (32), we
have

$\displaystyle\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{2}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{3}}(\tau_{0})\big{\rangle}=-2\mathrm{Im}\int_{-\infty}^{\tau_{0}}d\tau\big{\langle}{\bf
H_{3}(\tau)}\gamma^{\lambda}_{{\bf{k}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{2}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{3}}(\tau_{0})\big{\rangle}\,.$
(33)

Using the linear relation ${\cal{R}}=-H\pi$, and noting that
${\bf{k}}_{2}\simeq-{\bf{k}}_{3}$, we obtain

$\displaystyle\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{2}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{3}}(\tau_{0})\big{\rangle}^{\prime}=-4M_{P}^{2}\epsilon_{ij}^{\lambda}({\bf{k}}_{1})\widehat{\bf{k}}_{2i}\widehat{\bf{k}}_{2j}\,{\cal
I}\,,$ (34)

in which here and below a prime over $\langle...\rangle$ means we have pulled
out the overall factor
$(2\pi)^{3}\delta^{3}({\bf{k}}_{1}+{\bf{k}}_{2}+{\bf{k}}_{3})$. The factor
${\cal I}$ is calculated via the in-in integral as follows,

$\displaystyle{\cal I}\equiv
k_{2}^{2}\int_{-\infty}^{\tau_{0}}d\tau\epsilon(\tau)a^{2}\mathrm{Im}\Big{[}\gamma_{k_{1}}^{*}(\tau_{0}){\cal{R}}^{*}_{k_{2}}(\tau_{0})^{2}\gamma_{k_{1}}(\tau){\cal{R}}_{k_{2}}(\tau)^{2}\Big{]}\,.$
(35)

As the scalar modes leave the horizon during the USR period, there are two
contributions in the above integral, from the USR period
$\tau_{i}<\tau<\tau_{e}$ and after the USR period, $\tau_{e}<\tau<\tau_{0}$.
Performing the integral over the USR period and neglecting the contribution of
a rapidly oscillating term in the form of $\cos(2k_{2}\tau_{i})$, we obtain

$\displaystyle{\cal
I}(\tau_{i}<\tau<\tau_{e})=-\frac{3}{4}\big{(}\frac{h-6}{h}\big{)}^{2}\frac{H^{4}}{k_{1}^{3}k_{2}^{3}M_{P}^{4}\epsilon_{e}}+{\cal
O}\big{(}\frac{k_{2}^{2}}{k_{e}^{2}}\big{)}\,.$ (36)

On the other hand, calculating ${\cal I}$ for the period
$\tau_{e}<\tau<\tau_{0}$ we obtain

$\displaystyle{\cal
I}(\tau_{e}<\tau<\tau_{0})=-\frac{(6-h)(h-10)}{10h^{2}}\frac{H^{4}}{k_{1}^{3}k_{2}^{3}M_{P}^{4}\epsilon_{e}}\times\big{(}\frac{k_{2}^{2}}{k_{e}^{2}}\big{)}\,.$
(37)

For the modes which $k_{2}\ll k_{e}$, we may neglect the contribution ${\cal
I}(\tau_{e}<\tau<\tau_{0})$ and to leading order

$\displaystyle\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{2}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{3}}(\tau_{0})\big{\rangle}^{\prime}=\frac{3}{4}\epsilon_{ij}^{\lambda}({\bf{k}}_{1})\widehat{\bf{k}}_{2i}\widehat{\bf{k}}_{2j}P_{\cal{R}}(k_{2},\tau_{0})P_{\gamma}(k_{1},\tau_{0})\,,$
(38)

in which $P_{\cal{R}}(k_{2},\tau_{0})$ and $P_{\gamma}(k_{1},\tau_{0})$ are
the scalar and tensor power spectrum as given in Eqs. (11) and (30).

The above result is obtained employing a direct in-in calculation. However, as
the tensor mode is frozen on superhorizon scales and is not affected by the
USR phase, we expect a consistency condition similar to [21] to hold. Below we
demonstrate that this is indeed the case.

As $k_{1}\rightarrow 0$, one can assume that the long tensor mode only
modifies the background for the short scalar modes [21] in the form a
quadrupolar anisotropy by changing $k_{2}^{2}\rightarrow
k_{2}^{2}-\gamma_{ij}k_{2}^{i}k_{2}^{j}$. Following the logic of [21] we can
write

$\displaystyle\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}{\cal{R}}_{{\bf{k}}_{2}}{\cal{R}}_{{\bf{k}}_{3}}\big{\rangle}^{\prime}\simeq-\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}\gamma^{\lambda}_{{\bf{k}}_{1}}\big{\rangle}\,\epsilon_{ij}^{\lambda}({\bf{k}}_{1})\widehat{\bf{k}}_{2i}\widehat{\bf{k}}_{2j}\frac{\partial}{\partial
k_{2}^{2}}\langle{\cal{R}}_{{\bf{k}}_{2}}{\cal{R}}_{{\bf{k}}_{3}}\rangle\,.$
(39)

Using the specific form of the scalar power spectrum given in Eq. (11) we have

$\displaystyle\frac{\partial}{\partial
k_{2}^{2}}P_{\cal{R}}(k_{2})=-\frac{3}{2k_{2}^{2}}P_{\cal{R}}(k_{2})\,,$ (40)

and consequently, plugging this in Eq. (39), we obtain

$\displaystyle\big{\langle}\gamma^{\lambda}_{{\bf{k}}_{1}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{2}}(\tau_{0}){\cal{R}}_{{\bf{k}}_{3}}(\tau_{0})\big{\rangle}^{\prime}=\frac{3}{4}\epsilon_{ij}^{\lambda}({\bf{k}}_{1})\widehat{\bf{k}}_{2i}\widehat{\bf{k}}_{2j}P_{\cal{R}}(k_{2},\tau_{0})P_{\gamma}(k_{1},\tau_{0})\,,$
(41)

in exact agreement with Eq. (38).

As explained above, one expects that the above consistency condition to hold.
This is because the tensor perturbation has left the horizon during early SR
phase which is frozen afterwards and is largely unaffected by the USR phase.
Consequently, it can only modify the background for the short scalar modes,
which leave the horizon much later in USR phase, in a form of quadrupolar
anisotropy.

The above analysis confirms the applicability of our EFT approach. In
addition, as the consistency condition is unaffected, the above results imply
that the loop corrections from the short scalar perturbations to be minimal on
long tensor perturbations which have left the horizon much earlier. We study
this issue more directly in next section.

##  5 Loop Corrections in Tensor Power Spectrum

Now we study the one-loop corrections in long CMB scale gravitational power
spectrum
$\langle\gamma^{s_{1}}({\bf{p}}_{1})\gamma^{s_{2}}({\bf{p}}_{2})\rangle$
induced from the short scalar modes which leave the horizon during the
intermediate USR phase. In our convention the CMB scale tensor modes have
momentum ${\bf{p}}_{1}$ and ${\bf{p}}_{2}$ while that of short scalar
perturbations running in the loop is ${\bf{q}}$.

For a consistent one-loop corrections, we have to calculate the contributions
of both Feynman diagrams shown in Fig. 1. We start with the right panel which
is easier, containing a four vertex involving one in-in integral over the
quartic Hamiltonian ${\bf H_{4}}$.

###  5.1 Loop Corrections from Quartic Hamiltonian

With the quartic Hamiltonian given in Eq. (22) the one-loop correction from
the right panel of Fig. 1 is given by

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}_{{\bf
H_{4}}}=-2\mathrm{Im}\int_{-\infty}^{\tau_{0}}d\tau\big{\langle}\,{\bf
H_{4}(\tau)}\,\gamma_{s_{1}}({{\bf{p}}_{1}},\tau_{0})\gamma_{s_{2}}({{\bf{p}}_{1}},\tau_{0})\,\big{\rangle}\,,$
(42)

yielding

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{{\bf
H_{4}}}=-2M_{P}^{2}\,\mathrm{Im}\Big{[}\epsilon^{s_{1}}_{i\ell}(-{\bf{p}}_{1})\epsilon^{s_{2}}_{\ell
j}({\bf{p}}_{1})\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\,q_{i}q_{j}\,\,{\cal
I}_{4}(q)\Big{]}\,,$ (43)

in which the factor ${\cal I}_{4}(q)$ associated to the quartic Hamiltonian
in-in integral will be given shortly below.

Using the isotropy of the background, the integral
$\int{d^{3}{\bf{q}}}\,q_{i}q_{j}{\cal I}_{4}(q)$ is non-zero only $i=j$ so one
can replace this momentum integral by
$\frac{1}{3}\delta_{ij}\int{d^{3}{\bf{q}}}\,q^{2}{\cal I}_{4}(q)$. Now using
the properties of the polarization tensor given in Eq. (25) we obtain

$\displaystyle\epsilon^{s_{1}}_{i\ell}(-{\bf{p}}_{1})\epsilon^{s_{2}}_{\ell
j}({\bf{p}}_{1})\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\,q_{i}q_{j}{\cal
I}_{4}(q)=\frac{2}{3}\delta_{s_{1}s_{2}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\,q^{2}{\cal
I}_{4}(q)\,.$ (44)

Combining all together, we obtain

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{{\bf
H_{4}}}=-\frac{4\delta^{s_{1}s_{2}}}{3}M_{P}^{2}\,\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\,q^{2}\,\mathrm{Im}\,{\cal
I}_{4}(q)\,,$ (45)

in which the factor ${\cal I}_{4}(q)$ is given by

$\displaystyle{\cal
I}_{4}(q)\equiv\int_{-\infty}^{\tau_{0}}d\tau\epsilon(\tau)a^{2}\Big{[}\gamma^{*}(p_{1},\tau_{0})^{2}\gamma(p_{1},\tau)^{2}\Big{]}\big{|}{\cal{R}}(q,\tau)\big{|}^{2}\,.$
(46)

In performing the time integral above we should only consider the contribution
of the superhorizon modes, so the actual time interval in Eq. (46) should be
$-\frac{1}{q}<\tau<\tau_{0}$. This guarantees that we do not count the
contributions of the modes which are subhorizon (i.e. not yet classical) in
the time integral in Eq. (46). On the other hand, the modes which are
subhorizon during the USR phase are quantum mechanical in nature so their
contributions should be collected via a UV renormalization scheme. While
renormalization is an important issue to read off the final physical quantity
but here we are mainly interested in the effects of superhorizon modes to
obtain a rough estimate for the magnitude of the loop corrections. In
addition, as $q\tau\rightarrow 0$, the integral in Eq. (46) receives its
contribution from its lower end. In particular, the contribution from the
period after the USR phase $\tau_{e}<\tau<\tau_{0}$ is subleading.

In the limit that $p\rightarrow 0$, we have

$\displaystyle\mathrm{Im}\Big{[}\gamma^{*}(p_{1},\tau_{0})^{2}\gamma(p_{1},\tau)^{2}\Big{]}\simeq-\frac{8}{3}\frac{H^{4}\tau^{3}}{M_{P}^{4}p^{3}}\,.$
(47)

Furthermore, on the superhorizon in which $q\tau\rightarrow 0$, we have
$\epsilon(\tau)\big{|}{\cal{R}}(q,\tau)\big{|}^{2}\simeq\frac{H^{2}}{4q^{3}M_{P}^{2}}$,
yielding

$\displaystyle\mathrm{Im}\,{\cal
I}_{4}(q)\simeq-\frac{2H^{4}}{3M_{P}^{6}q^{3}p^{3}}\int_{-\frac{1}{q}}^{\tau_{0}}d\tau\tau\simeq\frac{H^{4}}{3M_{P}^{6}q^{5}p^{3}}\,.$
(48)

Plugging the above result in Eq. (45) and integrating over the USR modes
$q_{i}<q<q_{e}$, we obtain

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{{\bf
H_{4}}}\simeq-\frac{4\delta^{s_{1}s_{2}}}{9}\frac{H^{4}}{M_{P}^{4}p^{3}}\frac{\Delta
N}{2\pi^{2}}\,,$ (49)

in which $\Delta N=\ln\big{(}\frac{\tau_{i}}{\tau_{e}}\big{)}$ is the duration
of the USR phase.

It is convenient to express the loop correction in terms of the dimensionless
power spectrum ${\cal{P}}_{\gamma}$ defined in Eq. (31). Using the result from
Eq. (49), for the one-loop correction in tensor power spectrum from the
quartic Hamiltonian ${\bf H_{4}}$ we obtain

$\displaystyle{\cal{P}}_{\gamma}^{(\mathrm{loop})}\big{|}_{{\bf
H_{4}}}\simeq-\frac{\Delta N}{36}{\cal{P}}_{\gamma}^{2}\,.$ (50)

Since we calculate the loop corrections induced from the scalar perturbations
on tensor power spectrum, then one expects the loop correction to scale like
${\cal{P}}_{\gamma}{\cal{P}}_{\cal{R}}$. However, from Eq. (50) we see that
the loop correction actually scales like ${\cal{P}}_{\gamma}^{2}$. The reason
is that the interaction vertices in ${\bf H_{3}}$ and ${\bf H_{4}}$ contain
the factor $\epsilon$ so the combination $\epsilon{\cal{R}}^{2}$ appears
inside the in-in integral as in Eq. (46). Since
$\epsilon{\cal{R}}^{2}\sim{\cal{P}}_{\gamma}$, then the final result for loop
correction is given as ${\cal{P}}_{\gamma}^{2}$ instead of
${\cal{P}}_{\gamma}{\cal{P}}_{\cal{R}}$.

###  5.2 Loop Corrections from Cubic Hamiltonian

Now we calculate the loop corrections from the cubic Hamiltonian corresponding
to the left panel of Fig. 1. It involves a nested integral containing the
product of two three-vertices. More schematically, expanding the Dyson series
to second order in ${\bf H}_{3}$ we have

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}+\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}+\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(0,2)}$
(51)

in which

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}_{(2,0)}$
$\displaystyle=$
$\displaystyle-\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{\langle}{\bf
H}_{3}(\tau_{1}){\bf
H}_{3}(\tau_{2})\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}$
(52) $\displaystyle=$
$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\dagger}_{(0,2)}\,,$

and

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}=\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{0}}d\tau_{2}\,\big{\langle}{\bf
H}_{3}(\tau_{1})\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0}){\bf
H}_{3}(\tau_{2})\big{\rangle}\,.$ (53)

We leave the details of the in-in analysis into Appendix. After a long
calculation, one obtains

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\prime}_{{\bf
H}_{3}}=-8M_{P}^{4}\delta^{s_{1}s_{2}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\big{|}\epsilon^{s_{1}}_{ij}({\bf{p}})q_{i}q_{j}\big{|}^{2}\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\mathrm{Im}\,\big{[}X^{*}(\tau_{1})\delta(\tau_{2})\big{]}\,,$
(54)

in which

$\displaystyle X(\tau)\equiv\epsilon
a^{2}\gamma(p,\tau)\gamma^{*}(p,\tau_{0}){\cal{R}}(q,\tau)^{2}\,,$ (55)

and

$\displaystyle\delta(\tau)\equiv 2\epsilon
a^{2}{\cal{R}}(q,\tau)^{2}\mathrm{Im}\big{[}\gamma(p,\tau)\gamma^{*}(p,\tau_{0})\big{]}\,.$
(56)

Using the orthogonality properties of the polarization tensor one can show
that

$\displaystyle\int
d\Omega\big{|}\epsilon^{s_{1}}_{ij}({\bf{p}})\hat{q_{i}}\hat{q_{j}}\big{|}^{2}=\frac{16\pi}{15}\,.$
(57)

in which $d\Omega$ represents the angular parts of $d^{3}{\bf{q}}$.

Combining all contributions, we obtain (see Appendix for further details)

$\displaystyle{\cal{P}}_{\gamma}^{(\mathrm{loop})}\big{|}_{{\bf
H_{3}}}\simeq\big{(}c_{1}e^{\Delta N}+c_{2}\Delta
N\big{)}{\cal{P}}_{\gamma}^{2}\,.$ (58)

in which $c_{1}\simeq-0.003$ and $c_{2}\simeq 0.005$. As in quartic case the
loop correction scales like ${\cal{P}}_{\gamma}^{2}$ instead of
${\cal{P}}_{\cal{R}}{\cal{P}}_{\gamma}$.

Unlike the correction from the quartic case we see a mild dependence on
$e^{\Delta N}$. However, this does not cause much harm. Specifically, for the
typical USR setup employed for the PBHs formation, one has $\Delta N\sim{\cal
O}(1)$ so the contribution $c_{1}e^{\Delta N}<1$. For example, for $\Delta
N=5$, we obtain $c_{1}e^{\Delta N}\sim 0.4$. However, note that with $\Delta
N=5$ the loop corrections in scalar sector become already very large if the
transition is sharp [1, 8].

Now combining the results from the cubic and quartic interactions, Eqs. (58)
and (50), the total one-loop correction is obtained to be

$\displaystyle{\cal{P}}_{\gamma}^{(\mathrm{loop})}\simeq\big{(}c_{1}e^{\Delta
N}+c_{3}\Delta N\big{)}{\cal{P}}_{\gamma}^{2}$ (59)

in which $c_{3}\simeq-0.02$.

From the above result we see that the loop corrections in tensor power
spectrum induced from the USR modes are quite insensitive to the sharpness of
the transition from the USR phase to the SR phase. Indeed, we do not see any
explicit dependence to the sharpness parameter $h$ in Eq. (59). This is unlike
the loop corrections induced on long scalar perturbations in which the loop
corrections increase linearly with $h$ [8] for $|h|\gg 1$ in which
${\cal{P}}_{\cal{R}}^{(\mathrm{loop})}\sim
h{\cal{P}}_{\cal{R}}^{\mathrm{CMB}}\,{\cal{P}}_{\cal{R}}^{\mathrm{short}}\sim
h\big{(}{\cal{P}}_{\cal{R}}^{\mathrm{CMB}}\big{)}^{2}\,e^{6\Delta N}$. The
dependence to the duration of USR phase via the exponential factor $e^{6\Delta
N}$ is the hallmark of USR loop corrections in scalar power spectrum which can
invalidate the perturbative treatment.

In addition, we see that the induced loop corrections in GWs are quite small
in all practical setups. More specifically we obtain
$\frac{{\cal{P}}_{\gamma}^{(\mathrm{loop})}}{{\cal{P}}_{\gamma}}\sim
10^{-3}\,e^{\Delta N}{\cal{P}}_{\gamma}$. Assuming ${\cal{P}}_{\gamma}\lesssim
10^{-10}$ from the CMB observations, we need $\Delta N\sim 30$ in order for
the ratio $\frac{{\cal{P}}_{\gamma}^{(\mathrm{loop})}}{{\cal{P}}_{\gamma}}$ to
approach unity. However, this dos not happen, because by that time the scalar
power spectrum ${\cal{P}}_{\cal{R}}$ has increased by the gigantic amount
$e^{6\Delta N}\sim e^{180}$, invalidating the perturbative approach
completely. One may wonder if considering a mild transition can change the
above conclusion. More specifically, in a mild transition one expects that the
loop corrections in scalar power spectrum becomes suppressed (by slow-roll
parameters) so one may have more freedom in increasing $\Delta N$. However, we
note that the loop correction in scalar power spectrum scales like $e^{6\Delta
N}h$ [8]. So the sensitivity of $h$ is not strong enough to relax the value of
$\Delta N$ considerably. For example, changing $h$ by a factor $100$ from
$h=-6$ to a mild transition with $h=-0.06$ will increase $\Delta N$ by the
amount $\frac{1}{6}\ln(100)\simeq 0.8$ which can not change our conclusion
above that the loop correction in tensor spectrum is harmless.

The conclusion is that the long CMB scale gravitational waves are practically
unaffected by the short scalar perturbations which leave the horizon during
the USR phase. This conclusion is largely independent of the mechanism of the
transition from the USR phase to the final SR phase.

##  6 Summary and Discussions

In this work we have studied the one-loop correction in power spectrum of long
gravitational waves from small scale modes which leave the horizon during the
intermediate USR phase. This study is motivated by similar recent studies
performed for loop corrections in scalar power spectrum.

As one might have guessed, the results are quite different from what were
obtained for the case of scalar power spectrum. We have shown that the long
tensor power spectrum is largely unaffected by the loop corrections from small
USR modes. In particular, the one-loop corrections are quite insensitive to
the sharpness of the transition. This might have been expected from the
physical ground that the tensor perturbations only probe the Hubble expansion
rate of the corresponding inflationary background and are insensitive to slow-
roll parameters. Having said this, it is still a good cross check to verify
the validity of this physical expectation since a similar intuition,
suggesting that the scalar power spectrum should be unaffected by intermediate
short modes, proved to fail for the case of a sharp transition [1, 2, 8].
While our analysis was focused to the particular setup of $SR\rightarrow
USR\rightarrow SR$, but this conclusion may be general. As long as there is no
dramatic changes in the background Hubble expansion rate, then independent of
the nature of transitions in slow-roll parameters, the superhorizon tensor
modes are unaffected by the short scalar modes which may experience rapid
growth. It would be useful to verify this conjecture in its generality.

In addition we have shown that the Maldacena consistency condition for the
tensor-scalar-scalar bispectrum in the squeezed limit does hold. The fact that
the long tensor mode is frozen on superhorizon scale is the key reason for the
validity of this consistency condition. The long tensor perturbations only
induce small anisotropies on the background for the short modes yielding to
the expected tensor-scalar-scalar consistency condition.

We comment that the loop corrections on tensor power spectrum calculated here
should not be confused with the induced gravitational waves from second order
scalar perturbations which are actively investigated recently, for a review
see [48] and for works studying secondary GWs induced in models with non-
Gaussian feature or USR setup see [49, 50, 51, 52]. While these two questions
are related but the induced GWs from large second order scalar perturbations
are mostly concerned with small scale GWs, the modes near the peak of scalar
perturbations, which re-enter the horizon during the radiation dominated era.
Here, on the other hand, we look at the enhancement of GWs spectrum at the CMB
scales.

Acknowledgments: I am grateful to Mohammad Ali Gorji and Antonio Riotto for
useful discussions and for comments on the draft. I would like to thank Amin
Nassiri-Rad for checking the in-in analysis in Section 5. This research is
partially supported by the “Saramadan” Federation of Iran.

##  Appendix A In-In Analysis for Cubic Hamiltonian

In this Appendix we present the details of the in-in integral for the cubic
Hamiltonians ${\bf H_{3}}$.

As discussed before, the loop interaction from the cubic Hamiltonian is given
by

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{{\bf
H}_{3}}=\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(2,0)}+\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}+\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(0,2)}$
(60)

with

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}_{(2,0)}$
$\displaystyle=$
$\displaystyle-\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{\langle}{\bf
H}_{3}(\tau_{1}){\bf
H}_{3}(\tau_{2})\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}$
(61) $\displaystyle=$
$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\dagger}_{(0,2)}\,,$

and

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}=\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{0}}d\tau_{2}\,\big{\langle}{\bf
H}_{3}(\tau_{1})\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0}){\bf
H}_{3}(\tau_{2})\big{\rangle}\,.$ (62)

Let us start with
$\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle_{(1,1)}$.
Using the Hamiltonian (21), performing all contractions and employing the
properties of the polarization tensor given in Eq. (25) one obtains

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\prime}_{(1,1)}=4M_{P}^{4}\delta^{s_{1}s_{2}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\big{|}\epsilon^{s_{1}}_{ij}({\bf{p}})q_{i}q_{j}\big{|}^{2}\,\Big{|}\int_{-\infty}^{\tau_{0}}d\tau
X(\tau)\Big{|}^{2}\,,$ (63)

in which

$\displaystyle X(\tau)\equiv\epsilon
a^{2}\gamma(p,\tau)\gamma^{*}(p,\tau_{0}){\cal{R}}(q,\tau)^{2}\,.$ (64)

Similarly, for
$\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{(2,0)}$
we obtain

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{(2,0)}=-4M_{P}^{4}\delta^{s_{1}s_{2}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\big{|}\epsilon^{s_{1}}_{ij}({\bf{p}})q_{i}q_{j}\big{|}^{2}\,\int_{-\infty}^{\tau_{0}}d\tau_{1}X(\tau_{1})\int_{-\infty}^{\tau_{1}}d\tau_{2}Z(\tau_{2})\,,$
(65)

in which

$\displaystyle Z(\tau)\equiv\epsilon
a^{2}\gamma(p,\tau)\gamma^{*}(p,\tau_{0}){\cal{R}}^{*}(q,\tau)^{2}\,.$ (66)

Noting that
$\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}_{(2,0)}=\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\dagger}_{(0,2)}$,
we obtain

$\displaystyle\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{(2,0)}+\big{\langle}\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\big{\rangle}^{\prime}_{(0,2)}$
$\displaystyle=$
$\displaystyle-4M_{P}^{4}\delta^{s_{1}s_{2}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\big{|}\epsilon^{s_{1}}_{ij}({\bf{p}})q_{i}q_{j}\big{|}^{2}\,$
$\displaystyle\times$
$\displaystyle\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{[}X(\tau_{1})Z(\tau_{2})+X^{*}(\tau_{1})Z^{*}(\tau_{2})\big{]}\,.$

To proceed further, let us define

$\displaystyle Z(\tau)\equiv X^{*}(\tau)+i\delta(\tau)^{*}\,,$ (68)

in which the new variable $\delta$, from Eqs. (66) and (64), is obtained to be

$\displaystyle\delta(\tau)=2\epsilon
a^{2}{\cal{R}}(q,\tau)^{2}\mathrm{Im}\big{[}\gamma(p,\tau)\gamma^{*}(p,\tau_{0})\big{]}\,.$
(69)

With the above relation between $X(\tau)$ and $Z(\tau)$, one can show that the
nested time integrals in Eq. (A) is rearranged in the following form

$\displaystyle\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\big{[}X(\tau_{1})Z(\tau_{2})+X^{*}(\tau_{1})Z^{*}(\tau_{2})\big{]}$
$\displaystyle=$
$\displaystyle\int_{-\infty}^{\tau_{0}}d\tau\big{|}X(\tau)\big{|}^{2}$
$\displaystyle-2$
$\displaystyle\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\mathrm{Im}\big{[}X(\tau_{1})\delta^{*}(\tau_{2})\big{]}\,.$

We see that the first integral in Eq. (A) cancels the contribution of
$\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\prime}_{(1,1)}$
in Eq. (63) so at the end we are left with

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\prime}_{{\bf
H}_{3}}=-8M_{P}^{4}\delta^{s_{1}s_{2}}\int\frac{d^{3}{\bf{q}}}{(2\pi)^{3}}\big{|}\epsilon^{s_{1}}_{ij}({\bf{p}})q_{i}q_{j}\big{|}^{2}\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\mathrm{Im}\,\big{[}X^{*}(\tau_{1})\delta(\tau_{2})\big{]}\,.$
(71)

To go further, we need to calculate the contribution of the polarization
tensor in the above integral. With the specific form of the polarization
tensor given in Eq. (26), one can show that

$\displaystyle\epsilon^{\pm}_{ij}({\bf{p}})\widehat{q_{i}}\widehat{q_{j}}=\frac{1}{\sqrt{2}}\sin^{2}(\theta)e^{\pm
2i\phi}\,,$ (72)

in which the orientation of the unit vector $\widehat{q}$ in a coordinate
where $\widehat{\bf{p}}$ is along the third axis is specified by the angles
$(\phi,\theta)$ in which
$\widehat{q}=(\sin\theta\cos\phi,\sin\theta\sin\phi,\cos\theta)$.
Consequently, one can easily check that

$\displaystyle\int
d\Omega\big{(}\epsilon^{s_{1}}_{ij}({\bf{p}})\widehat{q_{i}}\widehat{q_{j}}\big{)}\big{(}\epsilon^{s_{2}*}_{mn}({\bf{p}})\widehat{q_{m}}\widehat{q_{n}}\big{)}=\frac{16\pi}{15}\delta^{s_{1}s_{2}}\,.$
(73)

Plugging the above result in Eq. (71) we obtain

$\displaystyle\langle\gamma^{s_{1}}_{{\bf{p}}_{1}}(\tau_{0})\gamma^{s_{2}}_{{\bf{p}}_{2}}(\tau_{0})\rangle^{\prime}_{{\bf
H}_{3}}=-\frac{16}{15\pi^{2}}M_{P}^{4}\delta^{s_{1}s_{2}}\int
dqq^{6}\int_{-\infty}^{\tau_{0}}d\tau_{1}\int_{-\infty}^{\tau_{1}}d\tau_{2}\mathrm{Im}\,\big{[}X^{*}(\tau_{1})\delta(\tau_{2})\big{]}\,.$
(74)

In performing the above nested integral, it is useful to note that

$\displaystyle\mathrm{Im}\big{[}\gamma(p,\tau)\gamma^{*}(p,\tau_{0})\big{]}=-\frac{2H^{2}}{3M_{P}^{2}}\tau^{3}\,,$
(75)

and

$\displaystyle\gamma(p,\tau)\gamma^{*}(p,\tau_{0})=\frac{2H^{2}}{M_{P}^{2}p^{3}}+{\cal
O}(p^{-1})\,.$ (76)

There is an important comment in order. We emphasis that we integrate over the
modes which become superhorizon during the USR phase, so the time integrals in
Eq. (74) are actually restricted to $-\frac{1}{q}<\tau_{2}<\tau_{1}<\tau_{e}$.
This it to make sure that we only count the modes which become classical
during the USR phase. The modes which are subhorizon during the USR phase are
not classical and their effects may be collected under a UV renormalization
scheme which is not our question of interest here. With the same logic, for
the integral over the momentum $q$ we integrate over the modes $q_{i}<q<q_{e}$
which become superhorizon during the USR phase.

Using the relations (75) and (76) for $\delta(\tau_{2})$ and $X(\tau_{1})$ in
the nested integral (74) we obtain Eq. (58) in the main text. We comment that
the main contribution in the time integral in Eq. (74) comes for the USR
period, $\tau_{i}<\tau_{1,2}<\tau_{e}$ while the contribution from the final
SR phase, $\tau_{e}<\tau_{1,2}<\tau_{0}$, is subleading. To perform the
analysis of the nested integral in Eq. (74) we use the Maple computational
software.

## References

  * [1] J. Kristiano and J. Yokoyama, [arXiv:2211.03395 [hep-th]].
  * [2] J. Kristiano and J. Yokoyama, [arXiv:2303.00341 [hep-th]].
  * [3] A. Riotto, [arXiv:2303.01727 [astro-ph.CO]].
  * [4] A. Riotto, [arXiv:2301.00599 [astro-ph.CO]].
  * [5] S. Choudhury, M. R. Gangopadhyay and M. Sami, [arXiv:2301.10000 [astro-ph.CO]].
  * [6] S. Choudhury, S. Panda and M. Sami, [arXiv:2302.05655 [astro-ph.CO]].
  * [7] S. Choudhury, S. Panda and M. Sami, [arXiv:2303.06066 [astro-ph.CO]].
  * [8] H. Firouzjahi, [arXiv:2303.12025 [astro-ph.CO]].
  * [9] H. Motohashi and Y. Tada, [arXiv:2303.16035 [astro-ph.CO]].
  * [10] H. Firouzjahi and A. Riotto, [arXiv:2304.07801 [astro-ph.CO]].
  * [11] S. L. Cheng, D. S. Lee and K. W. Ng, Phys. Lett. B 827, 136956 (2022).
  * [12] P. Ivanov, P. Naselsky and I. Novikov, Phys. Rev. D 50, 7173-7178 (1994).
  * [13] J. Garcia-Bellido and E. Ruiz Morales, Phys. Dark Univ. 18, 47-54 (2017).
  * [14] M. Biagetti, G. Franciolini, A. Kehagias and A. Riotto, JCAP 07, 032 (2018).
  * [15] O. Özsoy and G. Tasinato, [arXiv:2301.03600 [astro-ph.CO]].
  * [16] C. T. Byrnes and P. S. Cole, [arXiv:2112.05716 [astro-ph.CO]].
  * [17] W. H. Kinney, Phys. Rev. D 72, 023515 (2005) [gr-qc/0503017].
  * [18] M. J. P. Morse and W. H. Kinney, Phys. Rev. D 97, no.12, 123519 (2018).
  * [19] W. C. Lin, M. J. P. Morse and W. H. Kinney, JCAP 09, 063 (2019).
  * [20] M. H. Namjoo, H. Firouzjahi and M. Sasaki, Europhys. Lett. 101, 39001 (2013).
  * [21] J. M. Maldacena, JHEP 0305, 013 (2003) [astro-ph/0210603].
  * [22] P. Creminelli and M. Zaldarriaga, JCAP 10, 006 (2004).
  * [23] J. Martin, H. Motohashi and T. Suyama, Phys. Rev. D 87, no.2, 023514 (2013).
  * [24] X. Chen, H. Firouzjahi, M. H. Namjoo and M. Sasaki, Europhys. Lett. 102, 59001 (2013).
  * [25] X. Chen, H. Firouzjahi, E. Komatsu, M. H. Namjoo and M. Sasaki, JCAP 1312, 039 (2013).
  * [26] M. Akhshik, H. Firouzjahi and S. Jazayeri, JCAP 12, 027 (2015).
  * [27] S. Mooij and G. A. Palma, JCAP 11, 025 (2015).
  * [28] R. Bravo, S. Mooij, G. A. Palma and B. Pradenas, JCAP 05, 024 (2018).
  * [29] B. Finelli, G. Goon, E. Pajer and L. Santoni, Phys. Rev. D 97, no.6, 063531 (2018).
  * [30] S. Pi and M. Sasaki, Phys. Rev. Lett. 131, no.1, 011002 (2023).
  * [31] Y. F. Cai, X. Chen, M. H. Namjoo, M. Sasaki, D. G. Wang and Z. Wang, JCAP 05, 012 (2018).
  * [32] A. Ota, M. Sasaki and Y. Wang, [arXiv:2211.12766 [astro-ph.CO]].
  * [33] C. Chen, A. Ota, H. Y. Zhu and Y. Zhu, Phys. Rev. D 107, no.8, 083518 (2023).
  * [34] A. Ota, M. Sasaki and Y. Wang, [arXiv:2209.02272 [astro-ph.CO]].
  * [35] D. S. Meng, C. Yuan and Q. g. Huang, Phys. Rev. D 106, no.6, 063508 (2022).
  * [36] S. Brahma, A. Berera and J. Calderón-Figueroa, JHEP 08, 225 (2022).
  * [37] S. Weinberg, “Cosmology,” Oxford University Press, 2008.
  * [38] D. Baumann, “Cosmology,” Cambridge University Press, 2022,
  * [39] H. Kodama and M. Sasaki, Prog. Theor. Phys. Suppl. 78, 1-166 (1984).
  * [40] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Phys. Rept. 215, 203-333 (1992).
  * [41] C. Cheung, P. Creminelli, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JHEP 0803, 014 (2008).
  * [42] C. Cheung, A. L. Fitzpatrick, J. Kaplan and L. Senatore, JCAP 0802, 021 (2008).
  * [43] M. Akhshik, H. Firouzjahi and S. Jazayeri, JCAP 07, 048 (2015).
  * [44] P. R. Jarnhus and M. S. Sloth, JCAP 02, 013 (2008).
  * [45] F. Arroja and K. Koyama, Phys. Rev. D 77, 083517 (2008).
  * [46] T. Noumi and M. Yamaguchi, [arXiv:1403.6065 [hep-th]].
  * [47] S. Weinberg, Phys. Rev. D 72, 043514 (2005) .
  * [48] G. Domènech, Universe 7, no.11, 398 (2021) [arXiv:2109.01398 [gr-qc]].
  * [49] R. g. Cai, S. Pi and M. Sasaki, Phys. Rev. Lett. 122, no.20, 201101 (2019).
  * [50] J. Liu, Z. K. Guo and R. G. Cai, Phys. Rev. D 101, no.8, 083535 (2020).
  * [51] A. Talebian, S. A. Hosseini Mansoori and H. Firouzjahi, Astrophys. J. 948, no.1, 48 (2023).
  * [52] H. V. Ragavendra, Phys. Rev. D 105, no.6, 063533 (2022).


# Dual-State Capsule Networks for Text Classification

Piyumal Demotte, Surangika Ranathunga
Department of Computer Science and Engineering, University of Moratuwa
Katubedda 10400, Sri Lanka
<EMAIL_ADDRESS>
###### Abstract

Text classification systems based on contextual embeddings are not viable
options for many of the low resource languages. On the other hand, recently
introduced capsule networks have shown performance in par with these text
classification models. Thus, they could be considered as a viable alternative
for text classification for languages that do not have pre-trained contextual
embedding models. However, current capsule networks depend upon spatial
patterns without considering the sequential features of the text. They are
also sub-optimal in capturing the context-level information in longer
sequences. This paper presents a novel Dual-State Capsule (DS-Caps) network-
based technique for text classification, which is optimized to mitigate these
issues. Two varieties of states, namely sentence-level and word-level, are
integrated with capsule layers to capture deeper context-level information for
language modeling. The dynamic routing process among capsules was also
optimized using the context-level information obtained through sentence-level
states. The DS-Caps networks outperform the existing capsule network
architectures for multiple datasets, particularly for tasks with longer
sequences of text. We also demonstrate the superiority of DS-Caps in text
classification for a low resource language.

##  1 Introduction

Recent research has shown state-of-the-art results using BERT-like contextual
embedding models Devlin et al. (2018). However, these performances highly
depend on the comprehensiveness of the pre-trained contextual embedding
models. In other words, these models require a very large corpus to produce
optimal results in down-stream tasks such as text classification. However, for
many low resource languages, such large corpora are not available. Even if
available, building contextual embedding models for each and every language
has practical concerns with respect to resource requirements. As a solution,
multilingual models111https://github.com/google-
research/bert/blob/master/multilingual.md were introduced. However, still
there are many languages that are not included in these publicly available
multilingual models. It has also been shown that these multilingual models
give sub-optimal results compared to those trained on monolingual data
Dumitrescu et al. (2020). Thus, text classification for these languages still
has to rely on techniques that do not involve BERT-like models.

Another line of research has used language-specific linguistic features such
as Part of Speech (POS) as auxiliary input to neural models Qian et al.
(2016). Although these have also shown very impressive results, low resource
languages that do not have such resources cannot take benefit of this line of
research as well.

Among the Deep Learning models for text classification that do not rely on
contextual embeddings or auxiliary linguistic features, capsule networks are
in the fore-front Zhao et al. (2018); Kim et al. (2020). For example, for the
MR(2005) Pang and Lee (2005) results produced by Capsule Network models
closely trail behind linguistically enhanced and contextual embeddings-based
Deep Learning techniques Zhao et al. (2018).

This paper further improves the state-of-the-art capsule networks and presents
a novel capsule network architecture namely, Dual-State Capsule Networks (DS-
Caps). In particular, this is an enhancement to the existing capsule networks
model with dynamic routing, which has been employed for text classification
Zhao et al. (2018). DS-Caps is mainly inspired by the Sentence State LSTM
(S-LSTM) networks Zhang et al. (2018), which eliminates the sequential
dependencies of LSTMs to enable to capturing local n-grams as well as
sequential features of the text simultaneously. The proposed model could be
represented as a sophisticated language model that elevates the usage of
n-grams, sequential patterns, and capsules vector representation for language
modeling within a single architecture. DS-Caps further mitigate the
limitations with the vanilla capsule network of capturing contextual-level
data of the text while sequential processing.

We empirically show that our approach produces state-of-the-art performances
against the existing capsule networks, for many publicly available datasets
for text classification tasks. Furthermore, DS-Caps is evaluated against a
text classification task for a low resource language, for which there are no
pre-trained monolingual or multilingual embedding models. Even for this low
resource task, DS-Caps was able to outperform other capsule network-based
methods, as well as the recurrent models.

##  2 Related Work

Rather recent experiments suggest approaches based on combinations of
sequential models like RNNs, LSTMs, and CNNs to capture both contextual
information and n-gram features of text Wang et al. (2016). Yet, these
approaches still suffer from inherent limitations of sequential text
processing with RNNs based approaches and convolutions over sequences of text.
Zhang et al. (2018) argue that, in order to encode deep language
representations within a neural language model, both n-gram features, as well
as sequential information of text, should be utilized within a single language
model. Their language modeling procedure includes S-LSTM, a graph RNN. S-LSTM
encodes each word of a sentence as a separate state, as well as the sentence
as a global state in each recurrent step. For this, they utilize message
passing over graphs Scarselli et al. (2008) in order to capture the deep
neural representation of languages.

Furthermore, capsule networks that were initially employed in the image
processing tasks produced state-of-the-art results with the dynamic routing
procedure proposed by Sabour et al. (2017). The intention behind the capsule
strategy was to represent the features of objects within the data as vector
representation in order to identify the exact order or pose of the
information. The dynamic routing procedure between capsules mitigates the
issues of information loss of CNNs due to max-pooling and elevates the
advancement of the part-to-whole relationship between capsules for deeper
capsule representation for image classification tasks.

Inspired by the impressive performance of the capsule network architectures
for image classification, Wang et al. (2018) applied the same for sentiment
classifications with the combination of RNNs, which produced state-of-the-art
results at that time. Zhao et al. (2018) conducted an empirical experiment of
capsule networks with dynamic routing to validate utilization of capsule
networks for text classification. The implementation of capsule-A and
capsule-B, considering different n-gram variations produced the optimal
performances for text classification tasks. With even deeper analysis, Kim et
al. (2020) produced an approach based on static routing between capsules for
text classification tasks. This procedure alleviates the limitations provided
by the variations of text with background noise for capsules with dynamic
routing. However, capsules that are solely built upon convolutions do not
illustrate the ability to capture the context of the text when the length of
sequences increases.

##  3 Model Architecture

DS-Caps further improves the language representation capability of capsule
networks Zhao et al. (2018) with the use of sentence-level states and word-
level states, an idea borrowed from the S-LSTM architecture Zhang et al.
(2018). As illustrated in Figure 1, for the purpose of capturing n-grams
features with sequential dependencies of the text, sentence-level states, and
word-level states were employed on top of feature embeddings.

In particular, vanilla capsule models leverage pre-trained word-embeddings as
the input features. The idea behind the word-level states is to capture both
n-gram and sequential features of the text simultaneously and corporate with
vector representation of capsules to enhance language modeling. Therefore
instead of using word-embeddings as feature inputs for vanilla capsule,
feature maps based on concatenated word-level states were utilized.

Furthermore, the sentence-level states which carry global context-level
information were fed into the dynamic routing process between capsules. This
improves the context-sensitivity of capsules compared to vanilla capsule
networks considering the variability of background information of text.

###  3.1 Sentence-level and Word-level States

For the purpose of extracting n-gram features while preserving sequential
dependencies of text, (1.) global sentence-level states were utilized for
context-level feature extraction, and (2.) word-level states were employed for
each word to extract dependencies among words in the sequences of text as
n-gram features.

Generally, an encoded hidden state of a sentence using sentence-level states
($\Psi_{1}^{t}$ and $\Psi_{2}^{t}$) and word-level states ($h_{i}^{t}$) at a
given time step $t$ could be represented as Eq. 1.

$H^{t}={\langle}h_{1}^{t},h_{2}^{t},h_{3}^{t},......,h_{n}^{t},\Psi_{1}^{t},\Psi_{2}^{t}{\rangle}.$
(1)

Here the state of the neural representation of encoded hidden state $H^{t}$ of
S-LSTM layer consists of a sub-state $h_{i}^{t}$ for each word $w_{i}$. Since
two dynamic routing procedures exist between three capsule layers in a vanilla
capsule network, two sentence-level sub-states $\Psi_{1}^{t}$ and
$\Psi_{2}^{t}$ are introduced to utilize towards the optimizations of dynamic
routing procedure.

The suggested approach for the neural encoding of sentences utilizes recurrent
information exchange between word-level states and sentence-level states to
incrementally achieve a rich neural representation in each time step. The
initial states are set as $H^{0}$ = $h_{i}^{0}$ = $\Psi_{1}^{0}$ =
$\Psi_{1}^{0}$ = $h^{0}$ in the form of model parameters. The state transition
of a word state $h_{i}^{t}$ could be illustrated with the following equations.

$\begin{gathered}{\phi}_{i}^{t}=[h_{i-1}^{t-1},h_{i}^{t-1},h_{i+1}^{t-1}]\\\
\tilde{\Psi}=avg(\Psi_{1}^{t-1},\Psi_{2}^{t-1})\\\
\widehat{i}_{i}^{t}=\sigma(W_{i}\phi_{i}^{t}+U_{i}x_{i}+V_{i}\tilde{\Psi}+b_{i})\\\
\widehat{l}_{i}^{t}=\sigma(W_{l}\phi_{i}^{t}+U_{l}x_{i}+V_{l}\tilde{\Psi}+b_{l})\\\
\widehat{r_{i}^{t}}=\sigma(W_{r}\phi_{i}^{t}+U_{r}x_{i}+V_{r}\tilde{\Psi}+b_{r})\\\
\widehat{f}_{i}^{t}=\sigma(W_{f}\phi_{i}^{t}+U_{f}x_{i}+V_{f}\tilde{\Psi}+b_{f}\\\
\widehat{s}_{i1}^{t}=\sigma(W_{s1}\phi_{i}^{t}+U_{s1}x_{i}+V_{s1}\tilde{\Psi}+b_{s1})\\\
\widehat{s}_{i2}^{t}=\sigma(W_{s2}\phi_{i}^{t}+U_{s2}x_{i}+V_{s2}\tilde{\Psi}+b_{s2})\\\
{o_{i}^{t}}=\sigma(W_{o}\phi_{i}^{t}+U_{o}x_{i}+V_{o}\tilde{\Psi}+b_{o})\\\
{u_{i}^{t}}={tanh}(W_{u}\phi_{i}^{t}+U_{u}x_{i}+V_{u}\tilde{\Psi}+b_{u})\\\
i_{i}^{t},l_{i}^{t},r_{i}^{t},f_{i}^{t},s_{i}^{t}={softmax}(\widehat{i}_{i}^{t},\widehat{l}_{i}^{t},\widehat{r}_{i}^{t},\widehat{f}_{i}^{t},\widehat{s}_{i1}^{t},\widehat{s}_{i2}^{t})\\\
\begin{split}c_{i}^{t}=l_{i}^{t}{\odot}c_{i-1}^{t-1}+f_{i}^{t}{\odot}c_{i}^{t-1}+&r_{i}^{t}{\odot}c_{i+1}^{t-1}+s_{i1}^{t}{\odot}c_{\Psi_{1}}^{t-1}\\\
&+s_{i2}^{t}{\odot}c_{\Psi_{2}}^{t-1}+i_{i}^{t}{\odot}u_{i}^{t}\end{split}\\\
h_{i}^{t}=o_{i}^{t}{\odot}{tanh(c_{i}^{t})}\\\ \end{gathered}$

Here, $\phi_{i}^{t}$ is the representation of concatenated hidden vectors of
the context window. $\tilde{\Psi}$ represents the averaged state of two
sentence-level states $\Psi_{1}$ and $\Psi_{2}$. Following the S-LSTM network
Zhang et al. (2018) six gates were applied to control information flow from
cell state of previous time step, left cell state of previous time step, right
cell state of previous time step, two global sentence states of previous time
step, and input state of a given word to cell state of any given time step.
These gates are denoted respectively as $f_{i}^{t}$, $l_{i}^{t}$, $r_{i}^{t}$,
$s_{i1}^{t}$, $s_{i2}^{t}$ and $i_{i}^{t}$. $o_{i}^{t}$ represents the gate
that controls the information flow from the current cell state $c_{i}^{t}$ to
the current hidden state $h_{i}^{t}$. $W_{x}$, $U_{x}$, $V_{x}$ and $b_{x}$
are model parameters where x $\in$ {i, o, l, r, f, s1, s2, u}.

The following equations illustrate the state transitions for global sentence-
level states $\Psi_{1}$ and $\Psi_{2}$.

$\begin{gathered}\tilde{h}=avg(h_{1}^{t-1},h_{2}^{t-1},...,h_{n}^{t-1})\\\
\widehat{f}_{\Psi_{1}}^{t}=\sigma(W_{\Psi_{1}}\Psi_{1}^{t-1}+U_{\Psi_{1}}\tilde{h}+b_{\Psi_{1}})\\\
\widehat{f}_{\Psi_{2}}^{t}=\sigma(W_{\Psi_{2}}\Psi_{2}^{t-1}+U_{\Psi_{2}}\tilde{h}+b_{\Psi_{2}})\\\
\widehat{f}_{i}^{t}=\sigma(W_{f}\tilde{\Psi}+U_{fi}h_{i}^{t-1}+b_{fi})\\\
{o_{\Psi_{1}}^{t}}=\sigma(W_{o1}\Psi_{1}^{t-1}+U_{o1}\tilde{h}+b_{o1})\\\
{o_{\Psi_{2}}^{t}}=\sigma(W_{o2}\Psi_{2}^{t-1}+U_{o2}\tilde{h}+b_{o2})\\\
\begin{split}f_{1}^{t},f_{2}^{t},..,f_{n}^{t},{f_{\Psi_{2}}^{t}},{f_{\Psi_{2}}^{t}}={softmax}(\widehat{f}_{1}^{t},&\widehat{f}_{2}^{t},..,\widehat{f}_{n}^{t},\\\
\widehat{f}_{\Psi_{1}}^{t},\widehat{f}_{\Psi_{2}}^{t})\end{split}\\\
c_{\Psi_{1}}^{t}=f_{\Psi_{1}}^{t}{\odot}c_{\Psi_{1}}^{t-1}+\sum{f_{i1}^{t}{\odot}c_{i}^{t-1}}\\\
c_{\Psi_{2}}^{t}=f_{\Psi_{2}}^{t}{\odot}c_{\Psi_{2}}^{t-1}+\sum{f_{i}^{t}{\odot}c_{i}^{t-1}}\\\
\Psi_{1}^{t}=o_{1}^{t}{\odot}{tanh}(c_{\Psi_{1}}^{t})\\\
\Psi_{2}^{t}=o_{2}^{t}{\odot}{tanh}(c_{\Psi_{2}}^{t})\\\ \end{gathered}$

Here, $f_{1}^{t}$, $f_{2}^{t}$, …, $f_{n}^{t}$ and $f_{\Psi_{1}}^{t}$,
$f_{\Psi_{2}}^{t}$ are gates controlling information respectively from
$c_{1}^{t-1}$, $c_{2}^{t-1}$, ……, $c_{n}^{t-1}$ and $c_{\Psi_{1}}^{t-1}$,
$c_{\Psi_{2}}^{t-1}$ to $c_{\Psi_{1}}^{t}$ and $c_{\Psi_{2}}^{t}$.
$o_{\Psi_{1}}^{t}$ and $o_{\Psi_{2}}^{t}$ are gates that manipulate
information flow from $c_{\Psi_{1}}^{t}$ and $c_{\Psi_{2}}^{t}$ to
${\Psi_{1}}$ and ${\Psi_{2}}$. $W_{x}$, $U_{x}$ and $b_{x}$ are model
parameters where x $\in$ {${\Psi_{1}}$,${\Psi_{2}}$, f, o1, o2}.

###  3.2 Capsule Networks

As displayed in Figure 1, As a major enhancement to vanilla capsule network
consisting of 4 layers namely n-gram convolutional layer, primary capsules,
convolutional capsules and text capsules, concatenated word-level states,
${\langle}h_{1}^{t},h_{2}^{t},h_{3}^{t},......,h_{n}^{t}{\rangle}$ were fed as
input features. Furthermore, two global sentence-level states ($\Psi_{1}$ and
$\Psi_{2}$) that are consolidated with context-level-information, are
integrated with the dynamic routing process between primary capsules and
convolutional capsules, and between convolutional capsules and text capsules
Zhao et al. (2018); Kim et al. (2020). We empirically evaluate the usage of
the number of sentence-level states for optimal performance in Section 5.3.

####  3.2.1 N-gram Convolutional Layer

Figure 1: Dual-State Capsule Networks

For the purpose of extracting n-gram features from different positions of the
concatenated word-level states ($h_{i}^{t}$), a standard convolutional layer
was applied. Let $H\in{\rm I\\!R}^{d\times l}$ denotes the concatenated word-
level states with $l$ number of $d$ dimensional word-level states. The
convolution operation consists of filters of $C\in{\rm I\\!R}^{d\times f}$,
which are applied to a context window of size $f$ on top of the concatenated
word states $H^{\\{i\colon i+f\\}}$, where $i$ is the starting index of the
context window. This produces new features governed by the Eq. 2.

$x_{i}=\sigma(H^{\\{i\colon i+f\\}}\circ C+b_{0})$ (2)

Here, $x_{i}\in{\rm I\\!R}$ represents the generated feature, and $\circ$
denotes the element-wise multiplication between context window and filter
while $\sigma$ and $b_{0}$ are the non-linear activation function (ReLu or
tanh) and bias term respectively. This whole convolution process generates
$M_{i}\in{\rm I\\!R}^{d-{f}+1}$ as feature column consists of features
obtained through Eq. 2.With N number of filters, ultimately N number of
feature columns were generated according to the following Eq. 3.

$M=[M_{1},M_{2},...,M_{N}]\in{\rm I\\!R}^{(d-f+1)\times N}$ (3)

####  3.2.2 Primary Capsules

To convert the scalar output of the convolutional layer to the vector
representation of capsules, a capsule layer was applied with $v_{i}\in{\rm
I\\!R}^{u}$ as the instantiated parameters of the capsules with dimension $u$.
The capsules are generated based on the matrix multiplication that includes
matrix filters of $F\in{\rm I\\!R}^{N\times u}$. $(d-f+1)$ number of capsules
were generated with the matrix multiplication process governed by the
following Eq. 4.

$v_{i}=squash(F\otimes M_{i}+b_{1})$ (4)

Here, $v_{i}$ consists of a map of capsules that includes $(d-f+1)$ number of
capsules. $\otimes$ and $b_{1}$ denote the matrix multiplication operation and
bias term, respectively. The squash function refers to the non-linearity
displayed in Eq 10. With $D$ number of filters, $D$ number of maps were
created, which consist of $d$ dimensional $(d-f+1)$ number of capsules.
Therefore, the primary capsule layer could be represented as $V\in{\rm
I\\!R}^{(d-f+1)\times D\times u}$.

####  3.2.3 Convolutional Capsules

Inspired by the convolutional capsules introduced by Zhao et al. (2018), a
convolutional capsule layer was applied immediately after the primary capsule
layer, where a local region from the primary capsule layer is connected to the
capsules of the convolutional capsule layer. This procedure improves the model
performance due to the nature of the text, where certain objects of text
within the local regions of text could get activated when identifying the most
probable features towards the final outcome of the classification process.
Let the primary capsules in the region $l\times D$ to be mapped to the
convolutional capsules using the weight matrix $W^{c}\in{\rm I\\!R}^{E\times
u\times u}$, where $l\times D$ number of capsules are connected to each
convolutional capsule to incrementally learn child-to-parent relationship.
Here $E$ is the number of parent capsules in the convolutional layer. Given
each child capsule, the parent convolutional capsules are generated according
to the following Eq. 5.

${\widehat{u}_{j|i}}=W_{j}^{c}u_{i}+{\widehat{b}_{j|i}}$ (5)

Here ${\widehat{b}_{j|i}}$ is the bias term for the mapping. The child
capsules in the region $l\times D$ are denoted as $u_{i}$, which are mapped to
the parent convolutional capsules. $W^{c}_{j}$ is the $j^{th}$ weight matrix
for the learning by agreement procedure, which maps $u_{i}$ to the
${\widehat{u}_{j|i}}$. Thus, $(d-f-l+2)\times D$ total number of $u$
dimensional capsules were generated as convolutional capsules.

####  3.2.4 Text Capsules

The final capsule layer was designed to contain a number of capsules based on
the number of classes in the text classification tasks. The capsules in the
layer below were transformed based on the matrix multiplication process. This
generates text capsules while learning the child-to-parent relationship
process through the routing procedure.
The convolutional capsules were flattened to a list of capsules and were
transformed into text capsules by a transformation matrix $W^{d}\in{\rm
I\\!R}^{Z\times d\times d}$ considering the routing procedure to learn child-
to-parent relationship. The final capsules were produced as $x_{j}\in{\rm
I\\!R}^{d}$. Activation of the capsule denotes the class probability of a
given text category as $a_{j}\in{\rm I\\!R}$ for each class. Here, $Z$ denotes
the number of text capsules.

####  3.2.5 Dynamic Routing Algorithm

The routing by agreement procedure under the dynamic routing algorithm
incrementally ensures that appropriate child capsules are sent to parent
capsules by iteratively creating a non-linear mapping between capsules Sabour
et al. (2017). Under this experiment, two dynamic routing procedure exist in
the proposed architecture, between primary capsules and convolutional
capsules, and between convolutional capsules and text capsules.

The dynamic routing procedure is initiated by initializing the log prior
probabilities, which are the core components for the routing by agreement. As
the proposed enhancement, instead of initializing the log prior probabilities
to values in statistical distribution or zeros, the log prior probabilities
are assigned with values obtained through reshape operation on, global
sentence-level states. This strategy carries context-level information of the
text to validate the routing procedure which incrementally improves child-to-
parent relationship understanding background knowledge. The log prior
probabilities are initialized as in Eq. 6.

$b_{ij}\leftarrow reshape(\Psi)$ (6)

Here $b_{ij}$ represents the log prior probability between the child capsule
$i$ and the parent capsule $j$, while the reshape operation includes a matrix
multiplication that maps the dimensions of sentence-level state $\Psi$ to the
dimensions of $b_{ij}$ ( Both $\Psi_{1}$ and $\Psi_{2}$ sentence-level states
were considered under this operation.) The log-le prior probabilities between
each child capsule $i$ and each parent capsule $j$ were normalized using the
standard $softmax$ function during the iterative routing procedure, which sums
up coupling coefficients of each child capsule $i$ to all parent capsules $j$
in the layer above, to 1 as the Eq. 7.

$c_{ij}\leftarrow\frac{exp(b_{ij})}{\sum_{k}{exp(b_{ik})}}$ (7)

Here, $c_{ij}$ represents the coupling coefficient that is used when each
child capsule $i$ in the layer below maps to each parent capsule $j$ in the
layer above. These log prior probabilities $b_{ij}$ could be iteratively
learned at the same time as all other weights, where the log prior
probabilities only depend on the location of the child and parent capsules but
not on the given sequence of the input text. Then the log prior probabilities
between the child and parent capsules are updated in an iterative manner,
considering the agreement measurement that repeatedly measures the similarity
among predicted parent vectors by child capsules $\widehat{u}_{j|i}$ (
predicted parent capsule $j$ given child capsule $i$ ) and current capsule
$s_{j}$. The similarity measurement is a simple scalar product between the
predicted and current parent capsules.

$s_{j}=\sum_{i}{c_{ij}*{\widehat{u}_{j|i}}}$ (8)

$b_{ij}\leftarrow b_{ij}+\widehat{u}_{j|i}*s_{j}$ (9)

The current parent capsule is calculated based on the coupling coefficients
between the child and parent capsules ${c_{ij}}$, and the predicted parent
capsules by the child capsules denoted as $\widehat{u}_{j|i}$ according to the
Eq. 8. The log prior probabilities $b_{ij}$ are iteratively updated according
to the Eq. 9 considering the measurement of the agreement between current and
predicted parent capsules.

After the iterative routing procedure, the output capsules are normalized
using the non-linear squash function, which transforms the length of each
vector to represent the probability of the existence of the entity present
within a capsule. The squash function converts the length of long vectors near
to 1 while shrinking the length of short vectors to 0, which represents the
probability as the potential of an entity represented by each capsule. The
non-linear squash function applied for parent capsules $s_{j}$ that outputs
$v_{j}$ is defined as Eq. 10.

$v_{j}=\frac{\|s_{j}\|^{2}}{1+\|s_{j}\|^{2}}\frac{s_{j}}{\|s_{j}\|^{2}}$ (10)

Here $\|s_{j}\|$ denotes the standard norm for $s_{j}$, and as illustrated in
Algorithm 1, the non-linear squash function is applied outside the iterative
mapping procedure. This results in eliminating the degradation of log prior
probabilities in each routing iteration.

1

2procedure ROUTING(ûj|i,$\Psi$, $r$, $l$ )

3 forall  _capsules $i$ in $layer\;l$ and $j$ in $layer\;{l+1}$_ do

4 $b_{ij}\leftarrow reshape(\Psi)$

5

6for  _$r$ iterations_ do

7 forall  _capsules i in $layer\;l$_ do

8 $c_{i}\leftarrow$$softmax(b_{i})$

9 forall  _capsules j in $layer\;{l+1}$_ do

10 $s_{j}\leftarrow$$\sum_{i}c_{ij}*\widehat{u}_{ij}$

11 forall  _capsules $i$ in $layer\;l$ and $j$ in $layer\;{l+1}$_ do

12 $b_{ij}\leftarrow b_{ij}+u_{j}*s_{j}$

13

14

15forall  _capsules $j$ in $layer_{l+1}$ _ do

16 $v_{j}\leftarrow squash(s_{j})$

17

18return $v_{j}$

Algorithm 1 Dynamic Routing using Sentence State

####  3.2.6 Loss Function

For the text classification task, for each text capsule, we used a separate
margin loss Sabour et al. (2017) function to identify where a given text
category is present within a given capsule. For text capsule $s$, the margin
loss $L_{s}$ is given by;

$L_{s}=T_{s}{max(0,m^{+}-{\|v_{s}\|})^{2}}\\\
+\lambda{(1-T_{s})}{max(0,{\|v_{s}\|-m^{-}})^{2}}$ (11)

Here $T_{s}=1$ if the text category exists within the text capsule, otherwise
it is set to 0. $m^{+}$ and $m^{-}$ are set as 0.9 and 0.1 accordingly. The
down-weighting coefficient $\lambda$ is set to 0.25 with the optimal
performance.

##  4 Experiments

###  4.1 Data Sets

Experiments were conducted on six benchmark datasets covering multiple
classification tasks. These include movie review classification ( MR(2004)
Pang and Lee (2004), MR(2005), IMDB ), news article classification (
Reuters10, MPQA Wiebe et al. (2005) ), and question categorization ( TREC-QA
Li and Roth (2002) ). The details of each dataset are shown in table 1. For
measuring the performance of the model against low resource language
processing, a publicly available Sinhala dataset Senevirathne et al.  was
utilized. This includes 15059 news comments annotated with 4 sentiment
categories with an average comment length of 10.

###  4.2 Implementation

For the experimental analysis, we utilized the 300-dimensional
GloVe222https://nlp.stanford.edu/projects/glove/ word vectors, which consisted
of 840 billion words. Adam optimizer was used for the optimization process
with exponential learning rate decay. The models were trained on Google Colab
with Tensorflow as the implementation tool. The optimal hyperparameters for
models in each data set are indicated in Table 2.

For each experiment, the learning rate was set to $1e-3$, and the learning
rate decay was set to 0.95. The hidden word-level state dimension was set to
300, while sentence-level hidden state size was set to 600 dimensions. Max
sentence length was chosen as the sentence length, considering the variations
of the datasets. The n-gram convolutional layers were instantiated to extract
3-grams from each context window with 32 filters. Each capsule in the primary
capsule layer was instantiated with 8-dimensional vectors, while each
convolutional and text capsule was instantiated with 16-dimensional vectors.
The length of each capsule denotes the existence of an entity within a
capsule, which is further utilized with text capsules to identify the text
categories within a given sequence of text.

Data c Train Dev Test $l_{avg}$ $|V|$ MR(2004) 2 1620 180 200 779 40693
MR(2005) 2 8635 960 1067 22 18764 Reuters10 10 6472 720 2787 168 28482 TREC-QA
6 4843 539 500 9 8689 MPQA 2 8587 955 1067 3 6246 IMDB 2 22500 2500 25000 231
112540

Table 1:  Properties of datasets. c: Number of text categories, Train, Dev,
Test: Size of training, development, and test sets (respectively), $l_{avg}$:
Average sentence length, $|V|$: Size of the vocabulary

Data Batch Routing Steps CW Epochs MR(2004) 4 3 4 1 50 MR(2005) 8 3 8 2 20
Reuters10 8 2 4 2 20 TREC-QA 4 3 4 1 50 MPQA 4 4 4 1 20 IMDB 8 3 8 2 10

Table 2:  Optimal hyperparameters for each dataset. Batch: Batch size,
Routing: Number of iterations in the dynamic routing procedure, Steps: Number
of recurrent time steps used to encode the sequence of text utilizing word and
sentence states, CW: Number of word-states considered in both left and right
context for encoding word-states in a particular time step.

MR(2004) MR(2005) Reuters TREC-QA MPQA IMDB CNN-rand Kim (2014) - 76.1 - 91.2
- 83.4 CNN-static Kim (2014) - 81.0 - 92.8 - 89.6 CNN-non-static Kim et al.
(2020) 88.0 81.5 87.4 92.7 89.9 90.4 LSTM Zhao et al. (2018) - 75.9 - 86.8 - -
Bi-LSTM Zhao et al. (2018) - 79.3 - 89.6 - - Tree-LSTM Zhao et al. (2018) -
80.7 - 91.8 - - S-LSTM Zhang et al. (2018) 82.4 - - - - 87.2 Capsule-A Kim et
al. (2020) 85.0 79.4 87.7 90.8 88.3 89.3 Capsule-B Kim et al. (2020) 89.5 79.0
88.0 89.8 88.4 89.3 Capsule-static routing Kim et al. (2020) 89.6 81.0 87.5
94.8 90.6 89.7 Dual-state capsule networks (DS-Caps) 90.5 82.1 88.6 92.8 89.2
90.6

Table 3:  Text classification accuracies for benchmark datasets.

###  4.3 Baseline models

In this experiment, we empirically evaluated our models against several
baseline models such as LSTM, Bi-LSTM, CNN with randomly initialized vectors,
CNN with non-trainable embeddings (CNN-static), CNN with trainable embeddings
(CNN-non-static) Kim (2014) and and S-LSTM Zhang et al. (2018) as primary
baseline models. Furthermore, we evaluated the DS-Caps approach against
several capsule networks based techniques namely, Capsule-A, Capsule-B Zhao et
al. (2018), and capsule networks with static routing Kim et al. (2020). Some
evaluation records for datasets under certain baseline techniques are not
displayed in Table 3 as the results are not reported in the literature.

##  5 Evaluation

###  5.1 Evaluation on Benchmark Data Sets

Accuracy was chosen as the evaluation metric for our experiments, following
related research in the same domain Zhao et al. (2018). The experiment results
are summarized in the Table 3, against the six benchmark datasets.

Our DS-Caps network was able to achieve the best result for four out of six
benchmark results including MR(2004), MR(2005), Reuters and IMDB datasets,
outperforming the previous studies on capsule networks with static routing for
MR(2004) and MR(2005) datasets, Capsule-B for Reuters10 dataset, and CNN-non-
static for IMDB dataset. In particular, the DS-Caps network extensively and
consistently defeats sequential neural networks such as LSTM, BiLSTM, Tree
LSTM and S-LSTM networks, and spatial models such as CNN-rand, CNN-static,
CNN-non-static on all six datasets. The observation was expected due to the
language representation of capsules, where the vector representation could be
greatly associated with the exact order or pose of sequences of text. The
dynamic routing procedure introduced under the capsule networks further
eliminated the loss of information due to the pooling strategy used under
CNNs. This observation could be justified as the ability of the S-LSTM layer
to encode the neural representation of text more efficiently, which
subsequently enriches the dynamic routing procedure providing context-level
information through global sentence-level information.
Furthermore, the results obtained for datasets with higher sequence lengths
(MR(2004) and IMDB) illustrate greater performance compared to the baseline
models. This provides shreds of evidence for the capability of the DS-Caps
network to classify rather longer sequences of text utilizing sequential
features and n-gram features of the text, collaborating with the vector
representations of capsules.

###  5.2 Evaluation on Low Resource Languages

As mentioned in Section 1, to demonstrate that DS-Caps gives state-of-the-art
results on low resource languages, we selected Sinhala. Sinhala is a low
resource language, and the largest reported dataset is the CommonCrawl
dataset, which is just above 100M Lakmal et al. (2020). Furthermore, Sinhala
does not have a pre-trained BERT mode, nor is it included in multiBERT. Using
auxiliary features such as POS tags is also not an option for this language,
as the best-reported results for POS tagging are not optimal Fernando and
Ranathunga (2018).
The proposed DS-caps network was evaluated against a Sinhala multi-class
sentiment analysis task Senevirathne et al. . As the results displayed in
Table 4, DS-caps outperformed all previously tested deep learning approaches
for Sinhala sentiment analysis. The hyper-parameters used for the DS-Caps
networks were the same as the parameter used by Senevirathne et al.  for
Capsule-A and Capsule-B and fastText embeddings were used as primary features
for sentiment analysis task.

DS-caps produced promising results in the context of this low resourced
language, suggesting the possibility to be effective for any language as a
universal approach. Even though BERT models Devlin et al. (2018) indicated
superior performance in text classification tasks, they cannot be readily
applied for low resourced languages due to lack of pre-trained models and high
computation bottleneck. Also, the lack of linguistic resources for low
resource languages, makes it impossible to use highly resource-intensive
methodologies. Therefore DS-caps could be used as a proper substitution in
place of text classification models that are based on computationally
intensive BERT models.

Model Accuracy Precision Recall F1-score RNN 58.98 42.93 54.98 42.30 LSTM
62.88 70.95 51.93 54.50 GRU 62.78 60.93 62.78 54.83 Bi-LSTM 63.81 61.17 63.81
57.71 Stacked-BiLSTM 63.13 69.71 63.18 59.42 HAHNN 61.16 71.08 48.54 59.25
Capsule-A 61.89 56.12 61.89 53.55 Capsule-B 63.23 59.84 63.23 59.11 DS-Caps
64.03 61.68 64.03 61.33

Table 4:  10-fold cross-validated, weighted evaluation metrics for performance
on Sinhala multi-class sentiment analysis Senevirathne et al.

No. of Sentence states Loss function Context window Time steps Accuracy 0
Margin 2 5 80.6 0 Cross entropy 2 7 79.4 0 Spread 2 10 79.9 0 Margin + L2 3 4
80.3 1 Margin 1 3 80.5 1 Cross entropy 2 3 79.1 1 Spread 2 4 80.4 1 Margin +
L2 2 4 80.1 2 Margin 1 7 82.1 2 Cross entropy 2 7 79.9 2 Spread 2 3 79.7 2
Margin + L2 2 4 80.9

Table 5:  Performance of DS-Caps networks varying no. of sentence-level states
against lost function for MR(2005) dataset.

###  5.3 Ablation Study

For the purpose of analyzing the performance of the model varying different
components, an ablation study was conducted on MR(2005) dataset and reported
in Table 5. The performance was analyzed with respect to loss functions
proposed by Zhao et al. (2018) for capsule networks and further adding L2
regularization to the optimal loss function. Furthermore, the effect of the
number of sentence-level states was analyzed to find out the optimal number of
sentence-level states to be integrated with the dynamic routing process. Under
the experiments, the batch size was kept as 8 and, the number of epochs and
dynamic routing iterations were fixed as 20 and 3 respectively. Margin loss
with two sentence-level states produced the best performance for MR(2005)
dataset.

This result was expected due to the efficiency of margin loss for vanilla
capsule network Zhao et al. (2018). Further two distinct sentence-level states
facilitate separate learning procedures for dynamic routing algorithms to
learn log prior probabilities which elevate part-to-whole relationship
considering low-level and high-level-capsules.

##  6 Conclusion and Future Work

In this paper, we proposed a novel Dual-State Capsule (DS-Caps) network
architecture, which incrementally improves the language representation
considering local and global information of the text. As further enhancements,
attention mechanisms could be integrated with DS-Caps. It is also worthwhile
to integrate language resources such lexicons and contextual embeddings such
as BERT Devlin et al. (2018) with DS-Caps since most of the other deep
learning approaches have produced greater performances with such strategies
Saha et al. (2020).

## References

  * Devlin et al. (2018) Jacob Devlin, Ming-Wei Chang, Kenton Lee, and Kristina Toutanova. 2018.  Bert: Pre-training of deep bidirectional transformers for language understanding.  _arXiv preprint arXiv:1810.04805_.
  * Dumitrescu et al. (2020) Stefan Daniel Dumitrescu, Andrei-Marius Avram, and Sampo Pyysalo. 2020.  The birth of romanian bert.  _arXiv preprint arXiv:2009.08712_.
  * Fernando and Ranathunga (2018) Sandareka Fernando and Surangika Ranathunga. 2018.  Evaluation of different classifiers for sinhala pos tagging.  In _2018 Moratuwa Engineering Research Conference (MERCon)_ , pages 96–101. IEEE.
  * Kim et al. (2020) Jaeyoung Kim, Sion Jang, Eunjeong Park, and Sungchul Choi. 2020.  Text classification using capsules.  _Neurocomputing_ , 376:214–221.
  * Kim (2014) Yoon Kim. 2014.  Convolutional neural networks for sentence classification.  _arXiv preprint arXiv:1408.5882_.
  * Lakmal et al. (2020) Dimuthu Lakmal, Surangika Ranathunga, Saman Peramuna, and Indu Herath. 2020.  Word embedding evaluation for sinhala.  In _Proceedings of The 12th Language Resources and Evaluation Conference_ , pages 1874–1881.
  * Li and Roth (2002) Xin Li and Dan Roth. 2002.  Learning question classifiers.  In _COLING 2002: The 19th International Conference on Computational Linguistics_.
  * Pang and Lee (2004) Bo Pang and Lillian Lee. 2004.  A sentimental education: Sentiment analysis using subjectivity summarization based on minimum cuts.  _arXiv preprint cs/0409058_.
  * Pang and Lee (2005) Bo Pang and Lillian Lee. 2005.  Seeing stars: Exploiting class relationships for sentiment categorization with respect to rating scales.  _arXiv preprint cs/0506075_.
  * Qian et al. (2016) Qiao Qian, Minlie Huang, Jinhao Lei, and Xiaoyan Zhu. 2016.  Linguistically regularized lstms for sentiment classification.  _arXiv preprint arXiv:1611.03949_.
  * Sabour et al. (2017) Sara Sabour, Nicholas Frosst, and Geoffrey E Hinton. 2017.  Dynamic routing between capsules.  In _Advances in neural information processing systems_ , pages 3856–3866.
  * Saha et al. (2020) Tulika Saha, Srivatsa Ramesh Jayashree, Sriparna Saha, and Pushpak Bhattacharyya. 2020.  Bert-caps: A transformer-based capsule network for tweet act classification.  _IEEE Transactions on Computational Social Systems_.
  * Scarselli et al. (2008) Franco Scarselli, Marco Gori, Ah Chung Tsoi, Markus Hagenbuchner, and Gabriele Monfardini. 2008.  The graph neural network model.  _IEEE Transactions on Neural Networks_ , 20(1):61–80.
  * (14) Lahiru Senevirathne, Piyumal Demotte, Binod Karunanayake, Udyogi Munasinghe, and Surangika Ranathunga.  Sentiment analysis for sinhala using deep learning techniques.  _arXiv preprint arXiv:2011.07280v1_.
  * Wang et al. (2016) Xingyou Wang, Weijie Jiang, and Zhiyong Luo. 2016.  Combination of convolutional and recurrent neural network for sentiment analysis of short texts.  In _Proceedings of COLING 2016, the 26th international conference on computational linguistics: Technical papers_ , pages 2428–2437.
  * Wang et al. (2018) Yequan Wang, Aixin Sun, Jialong Han, Ying Liu, and Xiaoyan Zhu. 2018.  Sentiment analysis by capsules.  In _Proceedings of the 2018 world wide web conference_ , pages 1165–1174.
  * Wiebe et al. (2005) Janyce Wiebe, Theresa Wilson, and Claire Cardie. 2005.  Annotating expressions of opinions and emotions in language.  _Language resources and evaluation_ , 39(2-3):165–210.
  * Zhang et al. (2018) Yue Zhang, Qi Liu, and Linfeng Song. 2018.  Sentence-state LSTM for text representation.  _arXiv preprint arXiv:1805.02474_.
  * Zhao et al. (2018) Wei Zhao, Jianbo Ye, Min Yang, Zeyang Lei, Suofei Zhang, and Zhou Zhao. 2018.  Investigating capsule networks with dynamic routing for text classification.  _arXiv preprint arXiv:1804.00538_.