# One-loop tensor power spectrum from an excited scalar field during inflation

Atsuhisa Ota<EMAIL_ADDRESS>HKUST Jockey Club Institute for Advanced Study,
The Hong Kong University of Science and Technology, Clear Water Bay, Hong
Kong, P.R.China  Misao Sasaki  Kavli Institute for the Physics and Mathematics
of the Universe (WPI), University of Tokyo, Chiba, 277-8583, Japan  Center for
Gravitational Physics and Quantum Information, Yukawa Institute for
Theoretical Physics, Kyoto University, Kyoto, 606-8502, Japan  Leung Center
for Cosmology and Particle Astrophysics, National Taiwan University, Taipei
10617  Yi Wang  Department of Physics, The Hong Kong University of Science and
Technology, Clear Water Bay, Hong Kong, P.R.China  HKUST Jockey Club Institute
for Advanced Study, The Hong Kong University of Science and Technology, Clear
Water Bay, Hong Kong, P.R.China

###### Abstract

We present a consistent one-loop calculation for the inflationary tensor power
spectrum in the presence of an excited spectator scalar field using the in-in
formalism. We find that the super-horizon primordial power spectrum of the
tensor mode can be scale-invariantly enhanced or reduced by the loop effects
of a subhorizon scalar field. Our calculation also includes the scalar-induced
gravitational wave spectrum classically computed in the previous literature,
which is significant only near the scales where the scalar field is amplified.
The super-horizon enhancement is a higher-order effect of the interaction
Hamiltonian, which can be understood as a Bogoliubov transformation introduced
by nonlinear interactions. On the other hand, the scale-invariant reduction of
the tensor power spectrum may occur due to the fourth-order scalar-scalar-
tensor-tensor coupling. This phenomenon can be understood as the evolution of
an anisotropic Bianchi type-I background in the separate universe approach.
Our result suggests that large-scale measurements may indirectly test the
dramatic effects of small-scale cosmological perturbations through loop
corrections. This possibility opens a new ground in probing the small-scale
physics of the primordial Universe through gravitational wave detectors of
cosmological scales.

##  I Introduction

The one-loop corrections to the inflationary power spectra, and their UV and
IR divergences are important components in cosmological perturbation theory of
the primordial universe Weinberg:2005vy ; Urakawa:2008rb ; Senatore:2009cf ;
delRio:2018vrj ; Tan:2019czo ; Comelli:2022ikb ; Comelli:2022ikb ;
Dimastrogiovanni:2022afr . Most studies focus on tiny loop corrections
suppressed by the Hubble-to-Planck-mass ratio $H^{2}/M^{2}_{\rm pl}\sim
10^{-10}\epsilon$ with the first slow roll parameter
$\epsilon\equiv-\dot{H}/H$. Thus it seems the motivation of such loop analyses
is to understand what our theories entail, rather than to obtain
experimentally verifiable results.

Recent progress in cosmological observations may shed new light on the
situation. The Laser Interferometer Gravitational-Wave Observatory (LIGO) and
Virgo event in 2015 reported an unexpectedly massive black hole
LIGOScientific:2016aoc , which rejuvenated the idea of primordial black holes
(PBHs) Sasaki:2016jop ; Clesse:2016vqa ; Bird:2016dcv . PBHs are formed from
collapse of Hubble horizon size regions caused by large curvature
perturbations Hawking:1971ei ; Carr:1974nx ; Carr:1975qj ; therefore the
existence of PBHs suggests scale-dependence of scalar fluctuations and thus
drastic enhancement of quantum fields at some point during inflation
Alabidi:2012ex ; Drees:2011hb ; Drees:2011yz ; Garcia-Bellido:2017mdw ;
Ivanov:1994pa ; Ezquiaga:2017fvi ; Kannike:2017bxn ; Germani:2017bcs ;
Motohashi:2017kbs ; Yokoyama:1998pt ; Saito:2008em ; Zhou:2020kkf ;
Chen:2019zza ; Pi:2021dft . UV or IR divergences are no longer expected in
loop integrals if one focuses on contributions from excited states of a
quantum field. Instead, an excited quantum field may introduce sizable and
observable loop corrections. Thus, loop corrections motivated by PBHs may lead
to interesting phenomenology. The purpose of this paper is to present a
consistent one-loop calculation for such an excited state during inflation.

The inflationary loops arise from nonlinear interactions of cosmological
perturbations. In general, interactions of scalars are model-dependent and
thus complicated. In contrast, the coupling to the tensor fluctuations is
limited to the kinetic term regardless of the interaction among scalars for
minimally coupled scalar fields in general relativity. Hence, the possible
loop corrections to the tensor power spectrum are easier to deal with than
those in the scalar case. Moreover, ongoing and future gravitational wave (GW)
measurements are attracting growing attention from observational perspectives.
Therefore, we focus on the one-loop corrections to the tensor power spectrum
(see e.g., Refs. Kristiano:2022maq ; Inomata:2022yte  for recent works on
scalar inflationary loops from excited scalar fields.). The appropriate
formalism for the present purpose is a full quantum approach, known as the
“in-in formalism,” as we consider the quantum regime of inflationary
fluctuations.

One often considers a secondary GW production from cosmological perturbations
(see, e.g., Ref. Domenech:2021ztg  and references therein), where the
classical equation of motion for GWs with a source is integrated. The source
usually consists of a bilinear form of stochastic variables that represent the
scalar-type cosmological perturbation. The induced tensor fluctuations are
intensely investigated as their detection may be a smoking gun for the PBH
formation in the very early Universe. Conventionally the induced GW production
is understood from a viewpoint of classical nonlinear dynamics. However, the
same process may also be regarded as leading order quantum corrections to the
tensor modes from a viewpoint of quantum field theory. From this perspective,
we point out that the computation of the induced GWs corresponds to that in
the Born approximation of the leading order interaction Hamiltonian. Thus it
is important to understand effects beyond the Born approximation, which
include in particular the iterative (or higher-loop) corrections arising from
the leading order interaction Hamiltonian as well as those from higher-order
interaction Hamiltonians.

Our recent letter Ota:2022hvh  reported several surprising results when
including the iterative corrections as well as a higher order Hamiltonian in
the in-in formalism. We found that the super-horizon tensor power spectrum may
be scale-invariantly enhanced or reduced by the loop effect, which can be
understood as a Bogoliubov transformation introduced by the nonlinear
interactions. This paper provides detailed derivations of the result and an
analysis of a new model not discussed in Ref. Ota:2022hvh , including new
perspectives such as the separate universe approach.

We organize this paper as follows. In section II, we review the previous
approach to GW productions with stochastic variables and point out two
important missing components in it. In Section III, we provide a brief but
self-contained review of the in-in formalism for the one-loop calculation of
the tensor power spectrum. Section IV describes the cosmological setup in this
paper. We apply the in-in formulas to inflationary calculations. Then in
section V, we derive the loop spectra from an excited scalar field. In section
VI, we discuss physical implications of the results from a perspective of the
Bogoliubov transformation. In Section VII, we discuss the separate universe
approach to obtain a more intuitive understanding of our result for the scale
invariant Born approximation contribution. Conclusions are given in Section
IX, and we supplemented some mathematical details in the appendices.

##  II What is new?

In cosmology, one often computes gravitational waves in classical field
theory. The classical equation of motion for the cosmological tensor
perturbation $h_{ij}$ with source is obtained by perturbatively expanding the
Einstein equation as

$\displaystyle\frac{\partial^{2}h_{ij}}{\partial\tau^{2}}+2\mathcal{H}\frac{\partial
h_{ij}}{\partial\tau}-\nabla^{2}h_{ij}=\Pi_{ij}{}^{kl}S_{kl}\,,$ (1)

where $\tau$ is the conformal time, and $\mathcal{H}$ is the conformal Hubble
parameter, ${\mathcal{H}}=d\ln(a)/d\tau$. The source $S_{ij}$ is obtained by
expanding the Einstein tensor and the energy-momentum tensor, and
$\Pi_{ij}{}^{kl}$ is the projection operator onto the transverse-traceless
subspace. Given the Green’s function $G$ with an appropriate boundary
condition, Eq. (1) is formally integrated to give

$\displaystyle h_{ij}(x)$ $\displaystyle=h_{{\rm hom},ij}(x)+h_{{\rm
ind},ij}(x)\,;$ $\displaystyle h_{{\rm ind},ij}(x)$ $\displaystyle=\int
d^{4}x^{\prime}G(x;x^{\prime})\Sigma_{ij}(x^{\prime}),$ (2)

where $x$ and $x^{\prime}$ are 4-dimensional coordinates and
$\Sigma_{ij}\equiv\Pi_{ij}{}^{kl}S_{kl}$, where $h_{{\rm hom},ij}$ is a
homogeneous solution. With the retarded boundary condition on $G$, it is the
primordial tensor perturbation from inflation in the standard scenario of the
early universe. The inhomogeneous part represents gravitational waves
produced, e.g., by the nonlinear evolution of density fluctuations. The
inhomogeneous part $h_{{\rm ind},ij}$ in Eq. (2) may be regarded as a
classical field, but the source configuration is random, as the initial
condition of cosmological perturbations may originate from quantum
fluctuations during inflation. Hence $h_{{\rm ind},ij}$ should also be random,
whose statistical property is characterized by correlation functions. In
cosmology, we often consider the 2-point correlation function,

$\displaystyle\left\langle h_{ij}(x)h^{ij}(y)\right\rangle=\left\langle
h_{{\rm hom},ij}(x)h_{\rm hom}^{ij}(y)\right\rangle$ $\displaystyle+\int
d^{4}x^{\prime}d^{4}y^{\prime}G(x;x^{\prime})G(y;y^{\prime})\langle\Sigma_{ij}(x^{\prime})\Sigma^{ij}(y^{\prime})\rangle,$
(3)

where the first term due to the homogeneous part is usually assumed to be
negligible. Previous literature considered various origins of $\Sigma_{ij}$
and the associated secondary GWs. This is a standard prescription to compute
GWs from stochastic sources.

We mention that there are two additional important contributions that must be
included in the above computation (see also Ref. Chen:2022dah .). First, the
cross term of homogeneous and inhomogeneous solutions is missing in Eq. (3).
In addition to the auto-correlation of $h_{{\rm hom},ij}$, the following cross
term may be non-vanishing:

$\displaystyle\left\langle h_{ij}(x)h^{ij}(y)\right\rangle\supset$
$\displaystyle\int d^{4}y^{\prime}G(y;y^{\prime})\langle h_{{\rm
hom},ij}(x)\Sigma^{ij}(y^{\prime})\rangle.$ (4)

In fact, it is in general non-vanishing if we expand the source to third order
in perturbation. At leading order, the source is of the second order in
perturbation, and so is the induced part of $h_{ij}$. Therefore, the third-
order term is sub-leading in $h_{ij}$. However, the order of the cross-
correlation between the first- and third-order terms is equivalent to that of
the auto-correlation of the second-order term. Thus the cross-correlation may
also equally contribute to the correlation function. In the language of
quantum theory, ignoring the cross term implies an incomplete loop expansion.

The second missing contribution is from the iterative solution. The formal
solution (1) is well-defined in the Born approximation. However, once we take
nonlinear terms into account, the source term will contain the tensor
perturbation that appears on the left-hand side. Perturbatively, this means we
have to solve it iteratively. In particular, the iterative solution to the
next-to-leading order in the source needs to be included for consistency.
Namely, like the first issue, the first iterative correction is sub-leading at
the field level but may contribute to the correlation function at the same
order as the conventional induced GW part.

To summarize, we need to take into account all possible higher-order
interactions and iterative solutions to consistently calculate the nonlinear
corrections to cosmological correlation functions. In this paper, we perform a
consistent computation up to the one-loop order. We mention that similar
classical stochastic loops are discussed in the nonlinear matter power
spectrum of large-scale structure Bernardeau:2001qr . In that context, all
effects are included consistently.

The standard approach based on Eq. (1) applies to classical stochastic
systems. In cosmology, the method should explain the evolution of random
fields after inflation Chen:2022dah . We are interested in the quantum phase
during inflation in this paper, so we consider the in-in formalism instead. As
discussed in Section VI, the quantum evolution of the field operator $h_{ij}$
in the interaction picture is expanded into

$\displaystyle h_{ij}=h_{{\rm
I},ij}+i\lambda\int^{\tau}_{\tau_{0}}d\tau^{\prime}[H_{{\rm
int},I}(\tau^{\prime}),h_{{\rm I},ij}]$
$\displaystyle-\lambda^{2}\int^{\tau}_{\tau_{0}}d\tau^{\prime}\int^{\tau^{\prime}}_{\tau_{0}}d\tau^{\prime\prime}[H_{{\rm
int},I}(\tau^{\prime\prime}),[H_{{\rm int},I}(\tau^{\prime}),h_{{\rm I},ij}]]$
$\displaystyle+\mathcal{O}(\lambda^{3}),$ (5)

where $h_{{\rm I},ij}$ is the interaction picture field, i.e., the linear
tensor fluctuations, and $H_{{\rm int},I}$ is the interaction Hamiltonian in
the interaction picture. $\tau_{0}$ is the initial time, and the $\tau$
dependence is suppressed for notational simplicity. Order in $\lambda$ implies
the number of vertecies, and the standard induced GWs are the leading order
correction in the first line. The power of $H^{2}/M_{\rm pl}^{2}$ counts the
number of inflationary loops. The one-loop correction to the 2-point function
contains terms up to $\mathcal{O}(\lambda^{2})$, so there is no prior reason
to truncate the expansion at $\mathcal{O}(\lambda)$ in Eq. (5).

(In)equivalence of the classical stochastic approach and the quantum in-in
formalism during inflation is not obvious. A classical stochastic theory can
also describe a class of quantum systems, as Nelson discussed in Ref.
Nelson:1966sp . The comparison of the two approaches will be discussed in a
separate paper in future work. In the next section, we start with a lightning
review of the in-in formalism.

##  III In-in formalism: a review

We are interested in the time evolution of the vacuum expectation value (VEV)
of a field operator $O$ during inflation. In the Heisenberg picture, the VEV
should be written as

$\displaystyle\langle O\rangle=\langle\Omega|O|\Omega\rangle,$ (6)

where $O$ and $|\Omega\rangle$ are the operator and the vacuum in the
Heisenberg picture. The time evolution of $O$ in a general case is
complicated. In the interaction picture, we recast Eq. (6) into

$\displaystyle\langle\Omega|O|\Omega\rangle$ $\displaystyle\approx\langle
0|F^{\dagger}O_{I}F|0\rangle,$ (7)

where $|0\rangle$ is the free vacuum, and $F$ is a unitary operator
constructed from the free field operators. We evaluate the two-point function
up to the one-loop order by expanding $F$ to third-order in the free fields.
This section provides a brief but self-consistent derivation of Eq. (7) and
its perturbative expansions. One can also find reviews of the in-in formalism
in Refs. Maldacena:2002vr ; Weinberg:2005vy ; Chen:2010xka ; Wang:2013zva ,
and readers who are familiar with the in-in formalism may skip this section.

Quantization of classical fields is given by promoting Poisson (Dirac)
brackets in the (constraint) canonical formalism to the commutation relations.
Then the canonical equation of motion for a classical field $O$ with respect
to a Hamiltonian $H$ becomes the Heisenberg equation for the field operator.

$\displaystyle\dot{O}=i[H[\phi,\pi;\tau],O]+\frac{\partial O}{\partial\tau},$
(8)

where the over-dot is the ordinary derivative with respect to time, $\tau$.
The Hamiltonian operator $H$ is a functional of a canonical field variable
$\phi$ and conjugate momentum $\pi$. Using the Hamiltonian density,
$\mathcal{H}$, we have

$\displaystyle H[\phi,\pi;\tau]=\int d^{3}x\mathcal{H}(\phi,\pi;\tau).$ (9)

In cosmological perturbation theory, $H$ is explicitly dependent on $\tau$ as
we consider a time-dependent background. Extension to the multi-field
operators is straightforward, but here we only consider the single variable
for notational simplicity.

If $\partial O/\partial\tau=0$, the time evolution of $O$ is unitary. Using
the time evolution operator $U$ from $\tau_{0}$ to $\tau$, we get

$\displaystyle O=U^{-1}O_{0}U,$ (10)

where the subscript $0$ implies the operator evaluated at $\tau=\tau_{0}$.
Substituting Eqs. (10) into (8), we find

$\displaystyle\dot{O}=[-U^{-1}\dot{U},O].$ (11)

As Eq. (11) is satisfied for any operators with $\partial O/\partial\tau=0$,
Eqs. (8) and (11) yield

$\displaystyle\dot{U}=-iUH[\phi,\pi;\tau],$ (12)

which is different from the expression used in the literature, i.e., one often
assumes (see, e.g., Ref. Weinberg:2005vy .)

$\displaystyle\dot{U}=-iH[\phi,\pi;\tau]U.$ (13)

If $\partial H/\partial\tau=0$, we have $H=U^{-1}HU$, so we get Eq. (13) from
Eq. (12). However, this is not the case in cosmological perturbation theory.
Indeed, the correct operator ordering of Eq. (12) is used in the following
derivation.

If the Hamiltonian is written as $H=\bar{H}+H_{\rm int}$ with the free-field
Hamiltonian $\bar{H}$ and the interaction part $H_{\rm int}$, the perturbative
expansion with respect to the free-field operators is convenient. We introduce
the interaction picture field that evolves as

$\displaystyle\dot{O}_{I}=i[\bar{H}[\phi_{I},\pi_{I};\tau],O_{I}].$ (14)

The time evolution of $O_{I}$ is similarly written as

$\displaystyle O_{I}$ $\displaystyle=U_{I}^{-1}O_{I0}U_{I},$ (15)
$\displaystyle\dot{U}_{I}$
$\displaystyle=-iU_{I}\bar{H}[\phi_{I},\pi_{I};\tau].$ (16)

Assuming $O_{I0}=O_{0}$, the interaction picture field writes the Heisenberg
operator as

$\displaystyle O=F^{-1}O_{I}F,~{}F\equiv U_{I}^{-1}U.$ (17)

We find the equation of motion for $F$ by using Eqs. (12) and (16):

$\displaystyle\dot{F}$
$\displaystyle=-iFH[\phi,\pi;\tau]+i\bar{H}[\phi_{I},\pi_{I};\tau]F.$ (18)

Then,

$\displaystyle FH[\phi,\pi;\tau]F^{-1}$
$\displaystyle=U_{I}^{-1}H[\phi_{0},\pi_{0};\tau]U_{I}=H[\phi_{I},\pi_{I};\tau],$
(19)

is the Hamiltonian in the interaction picture, and Eq. (18) yields

$\displaystyle\dot{F}$
$\displaystyle=-i(H[\phi_{I},\pi_{I};\tau]-\bar{H}[\phi_{I},\pi_{I};\tau])F$
$\displaystyle=-iH_{\rm int}[\phi_{I},\pi_{I};\tau]F.$ (20)

Given the initial condition $F=1$ for $\tau=\tau_{0}$, the solution to this
equation is formally obtained as

$\displaystyle F=T\exp\left(-i\int^{\tau}_{\tau_{0}}d\tau^{\prime}H_{{\rm
int},I}(\tau^{\prime})\right),$ (21)

where $T$ is the time ordering operator. Hereafter the functional dependence
is suppressed for notational simplicity:

$\displaystyle H_{{\rm int},I}(\tau^{\prime})\equiv\int
d^{3}x\mathcal{H}_{{\rm
int}}\left(\phi_{I}(\tau^{\prime},\mathbf{x}),\pi_{I}(\tau^{\prime},\mathbf{x});\tau^{\prime}\right).$
(22)

$\tau$ dependence is also suppressed when it is obvious from the context.

We obtained the Heisenberg operator written by the interaction picture field
in Eq. (17). The next step is the calculation of the expectation value.
Following the prescription in the in-out formalism, one often shows

$\displaystyle\lim_{\tau_{0}\to-\infty}\langle\Omega|O|\Omega\rangle=\lim_{\tau_{0}\to-\infty(1-i\epsilon)}\langle
0|O|0\rangle,$ (23)

where $\tau_{0}$ is the initial time of inflation. We find the above equation
as follows. For simplicity, let us consider that $H$ is time-independent for
the $i\epsilon$ prescription and restore the time dependence of $H$ afterward.
Then we find

$\displaystyle\langle 0|O|0\rangle$ $\displaystyle=\langle
0|e^{i(\tau-\tau^{*}_{0})H}O_{0}e^{-i(\tau-\tau_{0})H}|0\rangle,$ (24)

where $O_{0}$ implies the operator at $\tau=\tau_{0}$. Inserting the identity
operator expanded by the Hamiltonian eigenstates,
$1=\sum_{n}|\Omega_{n}\rangle\langle\Omega_{n}|$ with
$H|\Omega_{n}\rangle=E_{\Omega_{n}}|\Omega_{n}\rangle$ and
$|\Omega_{0}\rangle=|\Omega\rangle$, one finds

$\displaystyle\langle 0|O|0\rangle$
$\displaystyle=\sum_{n,m}e^{-i(\tau-\Re[\tau_{0}])(E_{\Omega_{n}}-E_{\Omega_{m}})-\Im[\tau_{0}](E_{\Omega_{n}}+E_{\Omega_{m}})}$
$\displaystyle\times\langle
0|\Omega_{m}\rangle\langle\Omega_{m}|O_{0}|\Omega_{n}\rangle\langle\Omega_{n}|0\rangle.$
(25)

After the $i\epsilon$ prescription, we get

$\displaystyle\lim_{\tau_{0}\to-\infty(1-i\epsilon)}\langle
0|O|0\rangle=e^{-2\infty\epsilon E_{\Omega_{0}}}|\langle
0|\Omega\rangle|^{2}\langle\Omega|O_{0}|\Omega\rangle.$ (26)

Thus, $E_{\Omega_{n}}\neq E_{\Omega_{m}}$ configuration is oscillated away for
a $-i\epsilon$ rotated large time interval, and the diagonal components are
Boltzmann suppressed by $e^{-2\infty\epsilon E_{\Omega_{n}}}$. The
normalization factor is determined by considering $O=1$, and one finds
$e^{-2\infty\epsilon E_{\Omega_{0}}}|\langle 0|\Omega\rangle|^{2}=1$. Then
$\langle\Omega|O|\Omega\rangle$ is similarly expanded in the limit, and Eq.
(23) is shown. Substituting Eq. (21) into Eq. (17), and using Eq. (23), one
finds

$\displaystyle\langle O\rangle$
$\displaystyle=\lim_{\tau_{0}\to-\infty(1-i\epsilon)}\langle
0|\bar{T}\exp\left(i\lambda\int^{\tau}_{\tau_{0}^{*}}d\tau^{\prime}H_{{\rm
int},I}(\tau^{\prime})\right)$ $\displaystyle\times
O_{I}(\tau)T\exp\left(-i\lambda\int^{\tau}_{\tau_{0}}d\tau^{\prime\prime}H_{{\rm
int},I}(\tau^{\prime\prime})\right)|0\rangle,$ (27)

where we introduced an order counting parameter $\lambda$, and $\bar{T}$ is
the anti-time ordering operator. Eq. (27) is expanded into

$\displaystyle\langle O\rangle=\sum_{\lambda=0}^{\infty}\lambda^{n}\langle
O\rangle_{n},$ (28)

where we introduced

$\displaystyle\langle O\rangle_{0}$ $\displaystyle=\langle
0|O_{I}(\tau)|0\rangle,$ (29) $\displaystyle\langle O\rangle_{1}$
$\displaystyle=2\Im\int^{\tau}_{\tau_{0}}d\tau^{\prime}\langle
0|O_{I}(\tau)H_{{\rm int},I}(\tau^{\prime})|0\rangle,$ (30)
$\displaystyle\langle O\rangle_{2a}$
$\displaystyle=\int^{\tau}_{\tau_{0}^{*}}d\tau^{\prime}\int^{\tau}_{\tau_{0}}d\tau^{\prime\prime}$
$\displaystyle\times\langle 0|H_{{\rm int},I}(\tau^{\prime})O_{I}(\tau)H_{{\rm
int},I}(\tau^{\prime\prime})|0\rangle,$ (31) $\displaystyle\langle
O\rangle_{2b}$
$\displaystyle=-2\Re\int^{\tau}_{\tau_{0}}d\tau^{\prime}\int^{\tau^{\prime}}_{\tau_{0}}d\tau^{\prime\prime}$
$\displaystyle\times\langle 0|O(\tau)H_{{\rm int},I}(\tau^{\prime})H_{{\rm
int},I}(\tau^{\prime\prime})|0\rangle,$ (32)

with $\langle O\rangle_{2}\equiv\langle O\rangle_{2a}+\langle O\rangle_{2b}$.
In the end of calculation, we set $\tau=0$. Eqs. (30) to (32) are the basic
equations we use in this paper. Hereafter we set the order counting parameter
$\lambda=1$. $\mathcal{O}(\lambda^{2})$ and $\mathcal{O}(\lambda^{1})$
contributions are diagrammatically represented in Fig. 1.

Figure 1: The one-loop order Feynman diagrams considered here: $(a)$ and $(b)$
corresponds to $P_{h2}$ and $P_{h1}$, respectively, in our calculation.

##  IV Inflationary one-loop calculations

In the previous section, we reviewed the in-in formalism. We apply the method
to inflationary cosmology in this section. In cosmological perturbation
theory, we expand the full Hamiltonian around the inflationary background. The
second-order Hamiltonian is the free Hamiltonian in this setup, and
cosmological perturbations are the interaction picture fields. The interaction
Hamiltonian is the rest of the nonlinear corrections expanded by the linear
perturbations. In this section, we derive the interaction Hamiltonian from the
action and find the loop power spectrum.

###  IV.1 Cosmological setup

There exists gauge freedom in cosmological perturbation theory as identifying
a background spacetime with the physical spacetime is not unique. We need to
fix the gauge for quantization to properly count the dynamical degrees of
freedom and thus for the in-in formalism. In this paper, we consider the
uniform curvature gauge for the scalar perturbations where the scalar degrees
of freedom are those of matter sectors after solving the constraint equations
and the spatial curvature is zero. As we expand the action to second order in
the tensor perturbations, the gauge condition for tensor perturbations should
also be specified. Following Maldacena Maldacena:2002vr , the spatial
component of the metric tensor is written as

$\displaystyle g_{ij}=a^{2}e^{h_{ij}},$ (33)

where $a$ is the background scale factor, and we defined

$\displaystyle
e^{h_{ij}}\equiv\delta_{ij}+h_{ij}+\frac{1}{2}h_{i}{}^{k}h_{kj}+\mathcal{O}(h^{3}),$
(34)

and the transverse-traceless (TT) condition is given by

$\displaystyle\partial^{i}h_{ij}=\delta^{ij}h_{ij}=0.$ (35)

In this paper, the spatial index is raised and lowered by the background
spatial metric $\delta^{ij}$ and $\delta_{ij}$. In the previous literature,
one often used

$\displaystyle\bar{g}_{ij}=a^{2}(\delta_{ij}+\bar{h}_{ij}),$ (36)

instead of Eq. (34), which are equivalent at linear order. At nonlinear order,
the latter tensor gauge condition implies

$\displaystyle\ln\left(\sqrt{\det|\delta_{ij}+\bar{h}_{ij}|}\right)=\frac{1}{4}\bar{h}_{ij}\bar{h}^{ij}+\mathcal{O}(h^{3}).$
(37)

Thus, $\bar{h}_{ij}$ perturbs the volume element, which is the curvature
perturbation that should be zero in the uniform curvature gauge. Gravitational
waves or tensor fluctuations are distortions in spacetime without changing
volume. Hence, it should be introduced such that the volume element is not
perturbed by $h_{ij}$. Eq. (34) is an example of such a parameterization,
i.e., we have

$\displaystyle\det|e^{h_{ij}}|=1.$ (38)

Therefore, we use Eq. (33) instead of Eq. (36). The lapse and shift are slow-
roll suppressed compared to $\delta\chi$ in this gauge, so scalar fluctuations
are represented only by the matter action to the leading order in the slow-
roll parameter Maldacena:2002vr . This paper considers quantum corrections of
the tensor fluctuations due to a minimally coupled spectator scalar field
$\chi$. With Eq. (33), the possible interaction between $\chi$ and $h_{ij}$ is
model-independent and only appears in the kinetic term

$\displaystyle S_{\rm full}\supset-\frac{1}{2}\int
d^{4}x\sqrt{-g}g^{\mu\nu}\nabla_{\mu}\chi\nabla_{\nu}\chi,$ (39)

where $S_{\rm full}$ is the full inflationary action. We denote the spectator
scalar field fluctuation by $\delta\chi$. Then the relevant part of Eq. (39)
is written as

$\displaystyle\int d\tau a^{2}\int
d^{3}x\left(\frac{1}{2}h^{ij}-\frac{1}{4}h^{ik}h_{k}{}^{j}\right)\partial_{i}\delta\chi\partial_{j}\delta\chi.$
(40)

We read the interaction Lagrangian $L_{\rm int}$ from Eq. (40) and Legendre
transform gives $H_{\rm int}=-L_{\rm int}$ as the interaction do not involve
time derivative couplings. Then we find the interaction Hamiltonian

$\displaystyle H_{\rm int}\equiv a^{2}\int
d^{3}x\left(-\frac{1}{2}h^{ij}+\frac{1}{4}h^{ik}h_{k}{}^{j}\right)\partial_{i}\delta\chi\partial_{j}\delta\chi.$
(41)

As far as we consider minimally coupled scalar field, the interaction
Hamiltonian is written as Eq. (41) for a wide class of theories since
potential terms like $\sqrt{-g}V(\chi,\cdots)$ do not contain the tensor
fluctuation in the present gauge condition (33).

Fourier integrals of cosmological perturbations are written as

$\displaystyle\delta\chi(\tau,\mathbf{x})$
$\displaystyle=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\mathbf{q}\cdot\mathbf{x}}\delta\chi_{\mathbf{q}}(\tau),$
(42) $\displaystyle h_{ij}(\tau,\mathbf{x})$
$\displaystyle=\int\frac{d^{3}q}{(2\pi)^{3}}e^{i\mathbf{q}\cdot\mathbf{x}}\sum_{s=\pm
2}e_{ij}^{s}(\hat{q})h^{s}_{\mathbf{q}}(\tau),$ (43)

where $\hat{q}\equiv\mathbf{q}/|\mathbf{q}|$. The tensor fluctuations are
expanded by the polarization tensor $e_{ij}^{s}(\hat{q})$, which satisfies

$\displaystyle
e_{ij}^{s}(\hat{q})e^{ij,s^{\prime}}(\hat{q})^{*}=\delta^{ss^{\prime}},~{}e_{ij}^{s}(-\hat{q})=e_{ij}^{s}(\hat{q})^{*},$
(44)

where $\delta^{ss^{\prime}}$ is the Kronecker delta. Using Eqs. (42) and (43),
Eq. (41) is written as

$\displaystyle H_{\rm int}=H^{(3)}_{\rm int}+H^{(4)}_{\rm int},$ (45)

where Fourier integrals of the interaction Hamiltonians are written as

$\displaystyle H^{(3)}_{{\rm int}}$
$\displaystyle=\frac{1}{2}\prod_{A=1}^{3}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)(2\pi)^{3}\delta\left(\sum_{A=1}^{3}\mathbf{p}_{A}\right)\sum_{s}$
$\displaystyle\times$ $\displaystyle
a^{2}h^{s}_{\mathbf{p}_{1}}e^{ij,s}(\hat{p}_{1})p_{2i}p_{3j}\delta\chi_{\mathbf{p}_{2}}\delta\chi_{\mathbf{p}_{3}},$
(46) $\displaystyle H^{(4)}_{{\rm int}}$
$\displaystyle=-\frac{1}{4}\prod_{A=1}^{4}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)(2\pi)^{3}\delta\left(\sum_{A=1}^{4}\mathbf{p}_{A}\right)\sum_{s_{1},s_{2}}$
$\displaystyle\times a^{2}$ $\displaystyle
e^{ik,s_{1}}(\hat{p}_{1})e_{k}{}^{j,s_{2}}(\hat{p}_{2})p_{3i}p_{4j}h^{s_{1}}_{\mathbf{p}_{1}}h^{s_{2}}_{\mathbf{p}_{2}}\delta\chi_{\mathbf{p}_{3}}\delta\chi_{\mathbf{p}_{4}},$
(47)

where $\delta$ is the 3-dimensional delta function. As discussed in the
previous section, the second-order action describes the free theory in
cosmological perturbation theory, and the linear perturbations are the
interaction picture fields. Following the standard quantization procedures,
the Fourier transforms of cosmological perturbations are expanded into the
creation and annihilation operators as

$\displaystyle\delta\chi_{\mathbf{q}}(\tau)$
$\displaystyle=u_{q}(\tau)a_{\mathbf{q}}+u^{*}_{q}(\tau)a^{\dagger}_{-\mathbf{q}},$
(48) $\displaystyle h^{s}_{\mathbf{q}}(\tau)$
$\displaystyle=v_{q}(\tau)b^{s}_{\mathbf{q}}+v^{*}_{q}(\tau)b^{s\dagger}_{-\mathbf{q}},$
(49)

where $u_{q}$ and $v_{q}$ are the corresponding positive frequency mode
functions. For example, the ground state mode functions for free spectator
scalar and tensor fluctuations in general relativity are given as

$\displaystyle u^{\rm GS}_{q}(\tau)$
$\displaystyle=\frac{H}{\sqrt{2q^{3}}}(1+iq\tau)e^{-iq\tau},$ (50)
$\displaystyle v^{\rm GS}_{q}(\tau)$ $\displaystyle=\frac{2H}{M_{\rm
pl}\sqrt{2q^{3}}}(1+iq\tau)e^{-iq\tau}.$ (51)

The annihilation and creation operators of scalar and tensor fluctuations
satisfy

$\displaystyle a_{\mathbf{q}}|0\rangle$
$\displaystyle=b^{s}_{\mathbf{q}}|0\rangle=0,$ (52)

and the non-vanishing commutation relations are

$\displaystyle[a_{\mathbf{q}},a^{\dagger}_{-\bar{\mathbf{q}}}]$
$\displaystyle=(2\pi)^{3}\delta(\mathbf{q}+\bar{\mathbf{q}}),$ (53)
$\displaystyle[b^{s}_{\mathbf{q}},b^{\bar{s},\dagger}_{-\bar{\mathbf{q}}}]$
$\displaystyle=(2\pi)^{3}\delta^{s\bar{s}}\delta(\mathbf{q}+\bar{\mathbf{q}}).$
(54)

###  IV.2 Loop calculation

Given interaction Hamiltonians (46) and (47), we are ready to compute the VEV
of

$\displaystyle O_{I}(\tau)=\sum_{s=\pm
2}h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau),$ (55)

using Eq. (27). We compute the power spectrum $P_{h}$ defined as

$\displaystyle\left.\left\langle\sum_{s=\pm
2}h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)\right\rangle\right|_{\tau=0}$
$\displaystyle=(2\pi)^{3}\delta(\mathbf{q}+\bar{\mathbf{q}})P_{h}(q),$ (56)

order by order in $\lambda$ by using Eqs. (29) to (32). The total tensor power
spectrum, including one-loop correction, is

$\displaystyle P_{h}=P_{h0}+P_{h1}+P_{h2a}+P_{h2b},$ (57)

where the subscript implies the number of vertices in $\lambda$;
$P_{hn}=\mathcal{O}(\lambda^{n})$. Below we evaluate each term of Eq. (57).

### $\mathcal{O}(\lambda^{0})$

First, we derive the loop correction without any assumptions about the
background spacetime or the mode functions. Calculation of Eq. (29) is
straightforward. From Eq. (157), we find the tree-level power spectrum

$\displaystyle P_{h0}(q)=2|v_{q}(0)|^{2}.$ (58)

This is the power spectrum of the primordial gravitational waves in the
standard linear perturbation theory. For the ground state (51) at $\tau=0$, we
have

$\displaystyle P_{h0}(q)=\frac{4H^{2}}{M_{\rm
pl}^{2}q^{3}}=\frac{2\pi^{2}}{q^{3}}\cdot\frac{2H^{2}}{\pi^{2}M_{\rm
pl}^{2}}.$ (59)

### $\mathcal{O}(\lambda^{1})$

Next, we compute the diagram of $\mathcal{O}(\lambda^{1})$. Substituting Eqs.
(46) and (47) into Eq. (30), one finds

$\displaystyle\left\langle\sum_{s=\pm
2}h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)\right\rangle_{1}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\Im\int^{\tau}_{\tau_{0}}d\tau^{\prime}\prod_{A=1}^{4}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)(2\pi)^{3}\delta\left(\sum_{A=1}^{4}\mathbf{p}_{A}\right)$
$\displaystyle\times\sum_{s,s_{1},s_{2}}a(\tau^{\prime})^{2}e^{ik,s_{1}}(\hat{p}_{1})e_{k}{}^{j,s_{2}}(\hat{p}_{2})p_{3i}p_{4j}$
$\displaystyle\times\langle
0|h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)h^{s_{1}}_{\mathbf{p}_{1}}(\tau^{\prime})h^{s_{2}}_{\mathbf{p}_{2}}(\tau^{\prime})|0\rangle$
$\displaystyle\times\langle
0|\delta\chi_{\mathbf{p}_{3}}(\tau^{\prime})\delta\chi_{\mathbf{p}_{4}}(\tau^{\prime})|0\rangle.$
(60)

Note that the contribution from $H^{(3)}_{\rm int}$ vanishes since the linear
tensor fluctuation is Gaussian. Using Eqs. (156) and (159), we find

$\displaystyle\left\langle\sum_{s=\pm
2}h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)\right\rangle_{1}$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta\left(\mathbf{q}+\bar{\mathbf{q}}\right)\Im\int^{\tau}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}\int\frac{d^{3}p_{3}}{(2\pi)^{3}}\sum_{s}$
$\displaystyle\times
e^{ik,s}(\hat{q})^{*}e_{k}{}^{j,s}(\hat{q})p_{3i}p_{3j}v_{q}(\tau)^{2}v^{*}_{q}(\tau^{\prime})^{2}|u_{p_{3}}(\tau^{\prime})|^{2},$
(61)

where we dropped a (2-loop) bubble diagram eliminated from our consideration
after proper normalization. Indeed, those diagrams result in zero after the
polarization sum due to the angular momentum conservation. Using

$\displaystyle\int\frac{d\hat{p}_{3}}{4\pi}\hat{p}_{3i}\hat{p}_{3j}=\frac{\delta_{ij}}{3},$
(62)

and Eq. (44), we find

$\displaystyle\int\frac{d\hat{p}_{3}}{4\pi}\sum_{s}e^{ik,s}(\hat{q})^{*}e_{k}{}^{j,s}(\hat{q})\hat{p}_{3i}\hat{p}_{3j}=\frac{2}{3}$
(63)

Then we get

$\displaystyle P_{h1}=$ $\displaystyle
v_{q}(0)^{2}\Im\int^{0}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}v^{*}_{q}(\tau^{\prime})^{2}\int\frac{p^{4}_{3}dp_{3}}{3\pi^{2}}|u_{p_{3}}(\tau^{\prime})|^{2}.$
(64)

### $\mathcal{O}(\lambda^{2})$

Lastly, we consider $\mathcal{O}(\lambda^{2})$ contributions that we named
$P_{h2a}$ and $P_{h2b}$. Using Eqs. (31) and (46), one obtains

$\displaystyle\left\langle\sum_{s=\pm
2}h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)\right\rangle_{2a}$
$\displaystyle=$
$\displaystyle\frac{1}{4}\int^{\tau}_{\tau_{0}^{*}}d\tau^{\prime}a(\tau^{\prime})^{2}\int^{\tau}_{\tau_{0}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}\prod_{A=1}^{6}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)$
$\displaystyle\times(2\pi)^{3}\delta\left(\sum_{A=1}^{3}\mathbf{p}_{A}\right)(2\pi)^{3}\delta\left(\sum_{A=4}^{6}\mathbf{p}_{A}\right)$
$\displaystyle\times\sum_{s,s_{1},s_{4}}e^{ij,s_{1}}(\hat{p}_{1})p_{2i}p_{3j}e^{kl,s_{4}}(\hat{p}_{4})p_{5k}p_{6l}$
$\displaystyle\times\langle
0|h^{s_{1}}_{\mathbf{p}_{1}}(\tau^{\prime})h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)h^{s_{4}}_{\mathbf{p}_{4}}(\tau^{\prime\prime})|0\rangle$
$\displaystyle\times\langle
0|\delta\chi_{\mathbf{p}_{2}}(\tau^{\prime})\delta\chi_{\mathbf{p}_{3}}(\tau^{\prime})\delta\chi_{\mathbf{p}_{5}}(\tau^{\prime\prime})\delta\chi_{\mathbf{p}_{6}}(\tau^{\prime\prime})|0\rangle.$
(65)

Using Eqs. (158), (159), and the index symmetry for $\mathbf{p}_{2}$ and
$\mathbf{p}_{3}$, we find

$\displaystyle\left\langle\sum_{s=\pm
2}h^{s}_{\mathbf{q}}(\tau)h^{s}_{\bar{\mathbf{q}}}(\tau)\right\rangle_{2a}$
$\displaystyle=$
$\displaystyle(2\pi)^{3}\delta\left(\mathbf{q}+\bar{\mathbf{q}}\right)\int^{\tau}_{\tau_{0}^{*}}d\tau^{\prime}a(\tau^{\prime})^{2}\int^{\tau}_{\tau_{0}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}$
$\displaystyle\times\prod_{A=2}^{3}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)(2\pi)^{3}\delta\left(\sum_{A=2}^{3}\mathbf{p}_{A}-\mathbf{q}\right)$
$\displaystyle\times\sum_{s}e^{ij,s}(\hat{q})p_{2i}p_{3j}e^{kl,s}(\hat{q})^{*}p_{2k}p_{3l}v_{q}(\tau)v^{*}_{q}(\tau)$
$\displaystyle\times
v_{q}(\tau^{\prime})u_{p_{2}}(\tau^{\prime})u_{p_{3}}(\tau^{\prime})v^{*}_{q}(\tau^{\prime\prime})u^{*}_{p_{2}}(\tau^{\prime\prime})u^{*}_{p_{3}}(\tau^{\prime\prime}),$
(66)

where we dropped bubble graphs, and there is no contribution from tadpole
graphs. We can simplify the angular integrals in the above equation by using
Appendix B. Combining Eqs. (162), (163), and (164), one finds

$\displaystyle\prod_{A=2}^{3}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)(2\pi)^{3}\delta\left(\sum_{A=2}^{3}\mathbf{p}_{A}-\mathbf{q}\right)$
$\displaystyle\times\sum_{s=\pm
2}|e^{ij,s}(\hat{q})p_{2i}p_{3j}|^{2}f(p_{2},p_{3})$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}dp_{2}\int_{|p_{2}-q|}^{p_{2}+q}dp_{3}\bar{w}(q;p_{2},p_{3})f(p_{2},p_{3}),$
(67)

where we defined

$\displaystyle\bar{w}(q;p_{2},p_{3})$
$\displaystyle\equiv\frac{p_{2}p_{3}\left(p_{2}^{4}-2p_{2}^{2}\left(p_{3}^{2}+q^{2}\right)+\left(p_{3}^{2}-q^{2}\right)^{2}\right)^{2}}{128\pi^{2}q^{5}}.$
(68)

Using Eq. (67), Eq. (66) simplifies to

$\displaystyle P_{h2a}=$
$\displaystyle|v_{q}(0)|^{2}\int_{0}^{\infty}dp_{2}\int_{|p_{2}-q|}^{p_{2}+q}dp_{3}\bar{w}(q;p_{2},p_{3})$
$\displaystyle\times\left|\int^{0}_{\tau_{0}^{*}}d\tau^{\prime}a(\tau^{\prime})^{2}v_{q}(\tau^{\prime})u_{p_{2}}(\tau^{\prime})u_{p_{3}}(\tau^{\prime})\right|^{2}.$
(69)

$P_{h2b}$ can be calculated similarly. Using Eqs. (158) and (159) for Eqs.
(32) and (46), and dropping the bubble graphs, we find

$\displaystyle P_{h2b}$
$\displaystyle=-2v_{q}(0)^{2}\Re\int_{0}^{\infty}dp_{2}\int_{|p_{2}-q|}^{p_{2}+q}dp_{3}\bar{w}(q;p_{2},p_{3})$
$\displaystyle\times\int^{0}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}v^{*}_{q}(\tau^{\prime})u_{p_{2}}(\tau^{\prime})u_{p_{3}}(\tau^{\prime})$
$\displaystyle\times\int^{\tau^{\prime}}_{\tau_{0}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}v^{*}_{q}(\tau^{\prime\prime})u^{*}_{p_{2}}(\tau^{\prime\prime})u^{*}_{p_{3}}(\tau^{\prime\prime}).$
(70)

Note that $\Re$ and $\Im$ imply that we take the real and imaginary parts of
the whole equations, including the integral domain. This section derived
general one-loop formulas without specifying mode functions of scalar and
tensor perturbations. Eqs. (64), (69) and (70) will be useful for various
purposes.

##  V One-loop corrections from excited states

This section considers the loop corrections from an excited scalar field at
some specific momenta. Such a setup is interesting mainly for two reasons.
Firstly, the loop integral is simplified. We do not have to consider the UV or
IR divergence issue since the loop integral is evaluated for specific momenta
whose physics is completely determined by a given model. Secondly, such a
scale-dependent scalar field is phenomenologically considered in the context
of primordial black hole formation. The loop corrections from enhanced scalar
fields may be observationally testable. Let us derive the one-loop formulas
for monotonically excited scalar fields in this section.

###  V.1 One-loop formulas for sharp excited states

In the previous section, we saw that $\mathcal{O}(\lambda^{2})$ contributions
(69) and (70) take the following form

$\displaystyle
P_{h2}=\int_{0}^{\infty}dp_{2}\int_{|p_{2}-q|}^{p_{2}+q}dp_{3}\bar{w}(q;p_{2},p_{3})f(p_{2},p_{3}).$
(71)

The simplest situation is that we only have the scalar field excitation for
$p_{2}=p_{3}=p_{*}$ modes. In that case, the evaluation of Eq. (64) is
straightforward, and the integrand in Eq. (71) should be replaced as

$\displaystyle
f^{\delta}(p_{2},p_{3})=\delta\left(\ln(p_{2}/p_{*})\right)\delta\left(\ln(p_{3}/p_{*})\right)f(p_{*},p_{*}).$
(72)

Using Eq. (165), we obtain

$\displaystyle P_{h2}$
$\displaystyle=\frac{p_{*}^{4}(4p_{*}-q^{2})^{2}}{128\pi^{2}q}\Theta_{2-q/p_{*}}f(p_{*},p_{*}),$
(73)

where $\Theta_{2-q/p_{*}}$ is the Heaviside step function of $2-q/p_{*}$. The
Heaviside step function implies the momentum conservation upon generating
tensor fluctuations from the scalar fluctuations. On the other hand, the loop
integral in Eq. (64) is straightforward. Using Eq. (73) for Eqs. (69) and
(70), we obtain

$\displaystyle\frac{P^{\delta}_{h1}}{P_{h0}}=$
$\displaystyle\frac{p^{5}_{*}}{6\pi^{2}}\Im\int^{0}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}v^{*}_{q}(\tau^{\prime})^{2}|u_{p_{*}}(\tau^{\prime})|^{2},$
(74) $\displaystyle\frac{P^{\delta}_{h2a}}{P_{h0}}=$
$\displaystyle\frac{p_{*}^{4}(4p_{*}^{2}-q^{2})^{2}}{256\pi^{2}q}\Theta_{2-q/p_{*}}$
$\displaystyle\times\left|\int^{0}_{\tau_{0}^{*}}d\tau^{\prime}a(\tau^{\prime})^{2}v_{q}(\tau^{\prime})u_{p_{*}}(\tau^{\prime})^{2}\right|^{2},$
(75) $\displaystyle\frac{P^{\delta}_{h2b}}{P_{h0}}=$
$\displaystyle-\frac{p_{*}^{4}(4p_{*}^{2}-q^{2})^{2}\Theta_{2-q/p_{*}}}{128\pi^{2}q}$
$\displaystyle\times\Re\int^{0}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}v^{*}_{q}(\tau^{\prime})u_{p_{*}}(\tau^{\prime})^{2}$
$\displaystyle\times\int^{\tau^{\prime}}_{\tau_{0}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}v^{*}_{q}(\tau^{\prime\prime})u^{*}_{p_{*}}(\tau^{\prime\prime})^{2}.$
(76)

The delta functions in Eq. (72) are written as the superposition of plane
waves at all scales:

$\displaystyle\delta(\ln\tilde{q})=\frac{1}{2\pi}\int_{-\infty}^{\infty}e^{i(\ln\tilde{q})y}dy.$
(77)

From the causality perspective, a wave packet in real space should be
localized around a finite region. Hence, employing Eq. (77) implicitly
violates causality. To localize the real space field configuration, we
introduce a Gaussian window for the Fourier integral (77). Then we find

$\displaystyle\frac{1}{2\pi}\int_{-\infty}^{\infty}dye^{-\frac{1}{2}\Delta^{2}y^{2}}e^{i(\ln\tilde{q})y}=\frac{e^{-\frac{[\ln\tilde{q}]^{2}}{2\Delta^{2}}}}{\sqrt{2\pi}\Delta}.$
(78)

Thus, more realistic momentum distribution could be written by a lognormal
distribution with a finite width $\Delta$ around $p_{2}=p_{3}=p_{*}$. In this
case, we generalize Eq. (72) to

$\displaystyle f^{\rm
LN}(p_{2},p_{3})=\frac{e^{-\frac{[\ln(p_{2}/p_{*})]^{2}+[\ln(p_{3}/p_{*})]^{2}}{2\Delta^{2}}}}{2\pi\Delta^{2}}f(p_{*},p_{*}).$
(79)

In the $\Delta\to 0$ limit, the lognormal distribution recovers the delta
function. For the narrow lognormal peak, the momentum integral is approximated
by the peak value at $p_{2}=p_{3}=p_{*}$. Then Eq. (64) has the same form as
Eq. (74). For $P_{h2}$, we can integrate out one of the momenta trivially, and
the remaining integral gives a generalized step function

$\displaystyle\Theta^{\Delta}_{2-q/p_{*}}\equiv\frac{e^{2\Delta^{2}}}{2}\left[\text{erf}\left(\frac{2\Delta^{2}-\ln\left(|1-q/p_{*}|\right)}{\sqrt{2}\Delta}\right)\right.$
$\displaystyle\left.-\text{erf}\left(\frac{2\Delta^{2}-\ln\left(1+q/p_{*}\right)}{\sqrt{2}\Delta}\right)\right].$
(80)

Thus, the Heaviside step function in Eq. (73) is generalized to
$\Theta^{\Delta}$, as we have

$\displaystyle\lim_{\Delta\to
0}\Theta^{\Delta}_{2-q/p_{*}}=\Theta_{2-q/p_{*}}.$ (81)

For a small $q/p_{*}$, one finds

$\displaystyle\Theta^{\Delta}_{2p_{*}-\tilde{q}}=\sqrt{\frac{2}{\pi}}\frac{q}{p_{*}\Delta}+\mathcal{O}(q^{3}/p_{*}^{3}).$
(82)

Thus $\Theta^{\Delta}$ introduces additional $q/p_{*}$ factor for the log-
normal case. Hence, $P^{\rm LN}_{h2a/h2b}$ are obtained by replacing $\Theta$
with $\Theta^{\Delta}$ in $P^{\delta}_{h2a/h2b}$.

###  V.2 A model

We derived the one-loop formulas for the partly enhanced scalar fields in the
previous sections. This section considers a specific model of excited scalar
fields and evaluates the loop corrections more concretely. First, let us
consider an enhancement of spectator scalar fluctuation modeled by

$\displaystyle\delta\chi=e^{\mu H(t_{f}-t_{i})}\delta\chi^{\rm GS},$ (83)

where superscript “GS” implies the ground state, $\mu$ is a dimensionless time
constant, $t_{i}$ and $t_{f}$ are the physical time of the initial and final
time of the scalar enhancement. Note that the implementation of spectator
field enhancement is not unique. We also discuss the spectator scalar field
amplification via the Bogoliubov transformation in Section VI. In this paper,
we do not specify the enhancement process to amplify the scalar mode but
consider the consequences of Eq. (83). Using the conformal time, the
enhancement factor in Eq.(83) can be recast into

$\displaystyle e^{\mu
H(t_{f}-t_{i})}=\left(\frac{\tau_{i}}{\tau_{f}}\right)^{\mu},$ (84)

where $\tau_{i}$ and $\tau_{f}$ are the conformal time counterpart of $t_{i}$
and $t_{f}$. For simplicity, the scalar fluctuation is in the ground state
before the amplitude enhancement, and the enhancement factor is constant for
$\tau>\tau_{f}$. All details, including the time dependence of the enhancement
factor, are contained in

$\displaystyle\Xi(\tau)=\begin{cases}0,&(\tau<\tau_{i}),\\\
\left(\frac{\tau_{i}}{\tau}\right)^{\mu},&(\tau_{i}<\tau<\tau_{f}),\\\
\left(\frac{\tau_{i}}{\tau_{f}}\right)^{\mu},&(\tau_{f}<\tau),\end{cases}$
(85)

where $\Xi=0$ for $\tau<\tau_{i}$ implies that we subtracted the vacuum
contribution. We can change the time dependence of $\Xi$ for different models.
Time dependence at $\tau_{f}<\tau<0$ relies on a realization of how the scalar
amplitude is enhanced. More realistic enhancement factors could be oscillating
or decaying after $\tau=\tau_{f}$.

Let us consider that the enhancement happens only for $p=p_{*}$ modes. In this
case, Eq. (83) for $p=p_{*}$ is realized by simply multiplying $\Xi$ and the
ground state mode function as

$\displaystyle u_{p_{*}}=\Xi u^{\rm GS}_{p_{*}},$ (86)

and the tensor fluctuation is not changed: $v_{q}=v_{q}^{\rm GS}$. Eq. (86)
could be realized when, e.g., the sound speed of the spectator field is
exponentially suppressed. We may also consider such an exponential factor in
the parametric resonance.

We can straightforwardly use Eqs. (74) to (76) for Eq. (86), and we find

$\displaystyle\frac{P^{\delta}_{h1}}{P_{h0}}=$
$\displaystyle\frac{H^{2}}{M_{\rm
pl}^{2}}\Im\int^{0}_{x_{0}}dxX_{\tilde{q}}(x),$ (87)
$\displaystyle\frac{P^{\delta}_{h2a}}{P_{h0}}=$
$\displaystyle\frac{1}{2}\frac{H^{2}}{M_{\rm
pl}^{2}}\Theta^{\Delta}_{2-\tilde{q}}\left|\int^{0}_{x_{0}}Y_{\tilde{q}}(x)dx\right|^{2},$
(88) $\displaystyle\frac{P^{\delta}_{h2b}}{P_{h0}}=$
$\displaystyle-\frac{H^{2}}{M_{\rm
pl}^{2}}\Theta^{\Delta}_{2-\tilde{q}}\Re\int^{0}_{x_{0}}dx\int^{x}_{x_{0}}dx^{\prime}Y^{*}_{-\tilde{q}}(x)Y_{\tilde{q}}(x^{\prime}),$
(89)

where $x_{a}\equiv p_{*}\tau_{a}$ ($a=0,i,f$) and $\tilde{q}\equiv q/p_{*}$,
$a=-1/(H\tau)$ and we introduced

$\displaystyle X_{\tilde{q}}(x)$
$\displaystyle\equiv\frac{(1+x^{2})(1-i\tilde{q}x)^{2}}{6\pi^{2}\tilde{q}^{3}x^{2}}e^{2i\tilde{q}x}\Xi(x/p_{*}),$
(90) $\displaystyle Y_{\tilde{q}}(x)$
$\displaystyle\equiv\frac{(4-\tilde{q}^{2})(1-i\tilde{q}x)(1-ix)^{2}}{16\pi\tilde{q}^{2}x^{2}}e^{i(\tilde{q}+2)x}\Xi(x/p_{*}).$
(91)

The IR behavior of $P_{h1}$ can be found analytically. Expanding Eq. (87) with
respect to $\tilde{q}$,

$\displaystyle\frac{P_{h1}}{P_{h0}}$
$\displaystyle=-\frac{H^{2}}{18\pi^{2}M_{\rm
pl}^{2}}\left(\frac{x_{f}^{4}}{\mu-2}+\frac{x_{f}^{2}}{\mu-1}\right)\left(\frac{x_{i}}{x_{f}}\right)^{2\mu}$
$\displaystyle+\frac{H^{2}}{18\pi^{2}M_{\rm
pl}^{2}}\left(\frac{x_{i}^{4}}{\mu-2}+\frac{x_{i}^{2}}{\mu-1}\right)+\mathcal{O}(\tilde{q}).$
(92)

For $x_{i}<x_{f}<0$ and $\mu\geq 0$, we find

$\displaystyle\lim_{q/p_{*}\to
0}\frac{P_{h1}}{P_{h0}}=-\mathcal{O}(\Xi^{2})<0.$ (93)

Thus, $P_{h1}$ is scale-invariant and negative, i.e., the super-horizon
primordial gravitational waves will be reduced by this effect. The IR
contribution of $P^{\delta}_{h2}$ is given as

$\displaystyle\frac{P^{\delta}_{h2}}{P_{h0}}=\frac{H^{2}\Gamma[\Xi]}{24\pi^{2}M_{\rm
pl}^{2}\tilde{q}}+\mathcal{O}(\tilde{q}),$ (94)

where we defined a functional of $\Xi$ as

$\displaystyle\Gamma[\Xi]=\int^{0}_{x_{i}}dx^{\prime}\int^{x^{\prime}}_{x_{i}}dx^{\prime\prime}\frac{x^{\prime}}{x^{\prime\prime
2}}\Xi(x^{\prime}/p_{*})^{2}\Xi(x^{\prime\prime}/p_{*})^{2}$
$\displaystyle\times\left[\left(x^{\prime 2}+x^{\prime\prime
2}-4x^{\prime}x^{\prime\prime}-x^{\prime 2}x^{\prime\prime
2}-1\right)\sin(2x^{\prime}-2x^{\prime\prime})\right.$
$\displaystyle\left.+2\left(x^{\prime
2}x^{\prime\prime}-x^{\prime}x^{\prime\prime
2}+x^{\prime}-x^{\prime\prime}\right)\cos(2x^{\prime}-2x^{\prime\prime})\right],$
(95)

and we used the following relation for $P_{h2a}$:

$\displaystyle\int_{x_{0}}^{x}dx^{\prime}\int_{x_{0}}^{x}dx^{\prime\prime}A(x^{\prime},x^{\prime\prime})$
$\displaystyle=\int_{x_{0}}^{x}dx^{\prime}\int_{x_{0}}^{x^{\prime}}dx^{\prime\prime}\left[A(x^{\prime},x^{\prime\prime})+A(x^{\prime\prime},x^{\prime})\right].$
(96)

The red scale dependence of Eq. (94) comes from the fact that the monotonic
configuration in the Fourier space violates causality in real space. Properly
accounting for the causality by generalizing the delta-function-like
distribution, one finds an additional $\tilde{q}$ factor from Eq. (82), which
yields

$\displaystyle\frac{P^{\rm
LN}_{h2}}{P_{h0}}=\sqrt{\frac{2}{\pi}}\frac{H^{2}\Gamma[\Xi]}{24\pi^{2}M_{\rm
pl}^{2}\Delta}+\mathcal{O}(\tilde{q}^{2}).$ (97)

Thus, we obtain the scale-invariant correction of $\mathcal{O}(\Xi^{4})$.
$P_{h1}=\mathcal{O}(\Xi^{2})$, so the one-loop spectrum is dominated by
$P_{h2}$.

Figure 2: The one-loop corrections divided by the linear tensor power spectrum
in units of $r/0.01$ with the linear tensor-to-scalar ratio $r$ are presented.
We evaluate the spectra at $\tau=0$. The horizontal axis is the external
Fourier momentum $q$ divided by the scalar field peak momentum $p_{*}$, i.e.,
$\tilde{q}\equiv q/p_{*}$. The black dashed lines imply unity, and the spectra
above unity suggest that the loop contributions are dominant over the linear
tensor power spectrum. The solid blue, orange, green, red and purple curves
represent $P_{h1}$, $P^{\rm LN}_{2}$, $P^{\delta}_{2}$, $P^{\rm LN}_{\rm
ind.}$, and $P^{\delta}_{\rm ind.}$, respectively. The dotted curves mean the
negative part. $x_{i}\equiv p_{*}\tau_{i}$ and $x_{f}\equiv p_{*}\tau_{f}$ are
the dimensionless time at the initial and final time of enhancement, $\mu$ is
the time constant of the exponential amplification, and $\Delta$ is the width
of the lognormal distribution.

Further quantitative details are discussed numerically. $\Xi$ in Eq. (85) is
parameterized by 3 independent parameters: the dimensionless initial time
$x_{i}=p_{*}\tau_{i}$, and final time of amplification $x_{f}=p_{*}\tau_{f}$,
and the dimensionless time constant $\mu$. Fixing the final enhancement factor
as $(x_{i}/x_{f})^{\mu}=10^{3}$, we may consider the following physically
different situations:

  1. (A)

Near horizon contribution, $x_{f}=-1$, $x_{i}=-10$, $\mu=3$: the enhancement
of $\delta\chi$ happens and stops just before the horizon exit of $p=p_{*}$
modes.

  2. (B)

Subhorizon contribution, $x_{f}=-10$, $x_{i}=-100$, $\mu=3$: the enhancement
of $\delta\chi$ happens and stops well in advance of the horizon exit of
$p=p_{*}$ modes.

  3. (C)

Sub-to-near horizon contribution, $x_{f}=-1$, $x_{i}=-100$, $\mu=1.5$: the
enhancement of $\delta\chi$ happens well in advance and stops just before the
horizon exit of $p=p_{*}$ modes.

  4. (D)

Super-horizon contribution, $x_{f}=-0.1$, $x_{i}=-1$, $\mu=3$: the enhancement
of $\delta\chi$ happens at super-horizon scale.

Another unknown parameter is the inflationary Hubble parameter $H$, which
normalizes the linear tensor spectrum. We control the parameter by the linear
tensor-to-scalar ratio defined as

$\displaystyle r\equiv\frac{P_{h0}}{P_{\zeta 0}},$ (98)

where we used Eq. (59) and $P_{\zeta 0}$ is the linear power spectrum of the
adiabatic perturbation $\zeta$. From the CMB measurements, we have the almost
scale-invariant scalar spectrum. At the pivot scale $q_{p}=0.05{h/\rm Mpc}$,
we have $q_{p}^{3}P_{\zeta 0}(q_{p})/2\pi^{2}=2.196\times 10^{-9}$, which
yields

$\displaystyle\frac{H^{2}}{M_{\rm pl}^{2}}=\frac{\pi^{2}}{2}\cdot
r\cdot\frac{q^{3}P_{\zeta
0}}{2\pi^{2}}\approx\left(\frac{r}{0.01}\right)\times 10^{-10}.$ (99)

As shown in Eqs. (87) to (89), power of $H^{2}/M_{\rm pl}^{2}$ counts the
number of loops.

In Fig. 2, we show the results of numerical calculations for the above (A) to
(D). We present the one-loop corrections normalized by $P_{h0}$ in units of
$r/0.01$. Hence, the unity in the vertical axis implies that the loop
contribution is comparable to the linear tensor fluctuations for $r=0.01$. The
subhorizon spectator field enhances the super-horizon tensor fluctuations from
(A) to (C). On the other hand, the enhancement from the super-horizon scalar
field in (D) is relatively suppressed because of causality. The figure shows
that the enhancement of the scalar amplitude at a shorter scale introduces
larger one-loop corrections because $\delta\chi$ continuously enhances the
tensor fluctuation until $p\sim p_{*}$ modes exit the horizon. We see that the
IR behavior discussed analytically is reproduced in numerical calculation. The
size of $P_{h1}$ and $P_{h2}$ are loosely related as
$P_{h2}\approx\Xi^{2}P_{h1}$ near $\tilde{q}=1$ as the former involves two
additional scalar field operators. We also plot the induced tensor power
spectra discussed in the next section. As discussed later, the induced
components are included in the in-in calculation, which is dominant only for
subhorizon scales.

Figure 3: Time evolution of the one-loop spectrum in a cosmic time,
$\tau\partial P^{\text{1-loop}}/\partial\tau$ multiplied by $(P^{\rm
tree}_{h})^{-1}\cdot(r/0.01)^{-1}$ with the linear tensor-to-scalar ratio $r$.
The horizontal axis is the magnitude of time normalized by the wavenumber of
the tensor mode, and the time arrow is from right to left during inflation.
Curves are drawn for the fixed Fourier modes of $\tilde{q}\equiv q/p_{*}=1$,
and $0.1$, where $q$ is the wavenumber of the selected Fourier modes $p_{*}$
is that of the peak location. The scalar source and the tensor mode exit the
horizon simultaneously for $\tilde{q}=1$, whereas the tensor mode exit the
horizon prior to the scalar source for $\tilde{q}=0.1$. The orange (blue) and
red (green) curves represent $\tau\partial P_{h1}/\partial\tau$ and
$\tau\partial P^{\delta}_{2}/\partial\tau$ for $\tilde{q}=1~{}(0.1)$,
respectively. Each plot indicates that the evolution of the loop correction
lasts until the exponential amplification stops, regardless of the scale of
the tensor mode. Dotted curves represent the negative parts.

Several Fourier modes’ time evolution is presented in Fig. 3. The figure shows
that the nonlinear tensor fluctuations’ growth (decay) rate quickly converges
to zero after the scalar field amplification stops. After the exponential
enhancement, we keep the constant spectator scalar field, which introduces the
loop correction for $\tau>\tau_{f}$. The late time evolution could be the
dominant component of the final spectrum. At the same time, the IR scaling is
irrespective of the time dependence of the enhancement mechanism of the scalar
amplitude. In Fig. 4, we computed the loop correction when we set $\Xi=0$ for
$\tau>\tau_{f}$, that is, we subtracted the late time contribution. The early
time contribution is at most $\mathcal{O}(10)$% of the total correction for
the present parameters. Thus, the final spectrum is sensitive to the
implementation of the enhancement of scalar amplitude.

Figure 4: One-loop spectrum when we switch off the scalar field amplification
after the enhancement, that is, $\Xi=0$ for $\tau>\tau_{f}$. Definitions of
symbols are the same as Fig. 2.

##  VI Bogoliubov perspective

In the previous section, we saw the super-horizon reduction and enhancement of
the tensor power spectrum. One usually believes that the super-horizon tensor
fluctuation variation is prohibited from causality. Therefore, our results
look counterintuitive at first glance. Indeed, the GW production due to
physical processes must respect causality, and the induced GWs are produced
only on the subhorizon scale. This section explains that the super-horizon
correction is not considered as “induced GWs” because the tensor fluctuations
are not produced from “zero”.

###  VI.1 Super-horizon evolution as a Bogoliubov transformation

We may understand the physical implication of the effect more clearly by
inspecting the effect at the field level. We can see the field evolution by
directly solving the Heisenberg equation. The Heisenberg operator is written
by the interaction picture field as

$\displaystyle h^{s}_{\mathbf{q},{\rm H}}=$ $\displaystyle
F(\tau;\tau_{i})^{\dagger}h^{s}_{\mathbf{q}}F(\tau;\tau_{i}),$ (100)

where $h^{s}_{\mathbf{q}}$ is the interaction picture tensor fluctuation. The
time evolution operators are given as

$\displaystyle
F(\tau;\tau_{i})=1-i\lambda\int^{\tau}_{\tau_{i}}d\tau^{\prime}H_{{\rm
int},I}(\tau^{\prime})$
$\displaystyle-\lambda^{2}\int^{\tau}_{\tau_{i}}d\tau^{\prime}\int^{\tau^{\prime}}_{\tau_{i}}d\tau^{\prime\prime}H_{{\rm
int},I}(\tau^{\prime})H_{{\rm
int},I}(\tau^{\prime\prime})+\mathcal{O}(\lambda^{3}),$ (101)

where we temporarily restored the order counting parameter $\lambda$. As
$|\tau_{i}|\ll|\tau_{0}|$, we may consider $\tau_{i}=\tau_{i}^{*}$. In this
case, substituting Eqs. (101) into (100), we find Weinberg:2005vy

$\displaystyle h^{s}_{\mathbf{q},{\rm
H}}(\tau)=h^{s}_{\mathbf{q}}(\tau)+i\lambda\int^{\tau}_{\tau_{i}}d\tau^{\prime}[H_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]$
$\displaystyle-\lambda^{2}\int^{\tau}_{\tau_{i}}d\tau^{\prime}\int^{\tau^{\prime}}_{\tau_{i}}d\tau^{\prime\prime}[H_{{\rm
int},I}(\tau^{\prime\prime}),[H_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}]]+\mathcal{O}(\lambda^{3}).$ (102)

Eq. (102) is regarded as a perturbative solution to the quantum equation of
motion (8). Since we have $F^{\dagger}h^{2}F=(F^{\dagger}hF)(F^{\dagger}hF),$
Eq. (102) should reproduce the loop spectrum obtained in the in-in formalism.
The one-loop spectrum contributions from $\mathcal{O}(\lambda^{2})$ in Eq.
(102) can be understood as quantum correction for the first iterative
component. $\mathcal{O}(\lambda^{1})$ in Eq. (102) are understood as the Born
approximation effect.

$\mathcal{O}(\lambda^{1})$ corrections are composed of two terms. The first
term comes from the third-order interaction Hamiltonian:

$\displaystyle i\int^{\tau}_{\tau_{i}}d\tau^{\prime}[H^{(3)}_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]$ $\displaystyle=$
$\displaystyle\frac{i}{2}\int^{\tau}_{\tau_{i}}d\tau^{\prime}\prod_{A=1}^{3}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)(2\pi)^{3}\delta\left(\sum_{A=1}^{3}\mathbf{p}_{A}\right)\sum_{s_{1}}$
$\displaystyle\times$ $\displaystyle
a(\tau^{\prime})^{2}[h^{s_{1}}_{\mathbf{p}_{1}}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]e^{s_{1}}_{ij}(\hat{p}_{1})p_{2i}p_{3j}\delta\chi_{\mathbf{p}_{2}}(\tau^{\prime})\delta\chi_{\mathbf{p}_{3}}(\tau^{\prime}).$
(103)

Simplifying the commutator of the tensor fluctuations as

$\displaystyle[h^{s_{1}}_{\mathbf{p}_{1}}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]$
$\displaystyle=$
$\displaystyle(v_{q}(\tau^{\prime})v^{*}_{q}(\tau)-v^{*}_{q}(\tau^{\prime})v_{q}(\tau))(2\pi)^{3}\delta(\mathbf{q}+\mathbf{p}_{1})\delta^{ss_{1}},$
(104)

Eq. (103) yields

$\displaystyle i\int^{\tau}_{\tau_{i}}d\tau^{\prime}[H^{(3)}_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]=\int^{\tau}_{\tau_{i}}d\tau^{\prime}G_{q}(\tau;\tau^{\prime})S^{s}_{\mathbf{q}}[\delta\chi(\tau^{\prime})].$
(105)

where we defined

$\displaystyle G_{q}(\tau;\tau^{\prime})$
$\displaystyle=\frac{v_{q}(\tau^{\prime})v^{*}_{q}(\tau)-v^{*}_{q}(\tau^{\prime})v_{q}(\tau)}{W(\tau^{\prime})},$
(106) $\displaystyle W(\tau)$ $\displaystyle=\frac{i}{a(\tau)^{2}},$ (107)
$\displaystyle S^{s}_{\mathbf{q}}[\delta\chi]$
$\displaystyle=-\frac{1}{2}\prod_{A=2}^{3}\left(\int\frac{d^{3}p_{A}}{(2\pi)^{3}}\right)\delta\left(\mathbf{q}-\sum_{A=2}^{3}\mathbf{p}_{A}\right)$
$\displaystyle\times
e^{ij,s,*}(\hat{q})p_{2i}p_{3j}\delta\chi_{\mathbf{p}_{2}}\delta\chi_{\mathbf{p}_{3}}.$
(108)

Note that $W$ coincides with the Wronskian defined for the homogenous
solutions. Therefore, $G$ in Eq. (106) is the Green’s function constructed
from the Wronskian method when we solve the inhomogeneous classical equation
of motion, and $S$ in Eq. (108) corresponds to the second order source of the
scalar field fluctuations in the classical equation of motion. Therefore, we
may regard Eq. (108) as the induced GWs. Let us consider the IR behavior of
the induced GWs. For $q\ll p_{2},p_{3}$ and $|q\tau|,|q\tau^{\prime}|\ll 1$,
we find

$\displaystyle
S^{s}_{\mathbf{q}}[\delta\chi]=e^{ij,s,*}(\hat{q})\Sigma_{ij}[\Xi],$ (109)

where $\Sigma_{ij}$ is a $\mathbf{q}$ independent functional of $\Xi$, and the
Green’s function is expanded as

$\displaystyle G_{q}(0;\tau^{\prime})=-\frac{4H^{2}a^{\prime 2}\tau^{\prime
3}}{3M_{\rm pl}^{2}2}\left[1+\mathcal{O}(\tau^{\prime 2}q^{2})\right].$ (110)

Let us define the power spectrum of the induced GW as

$\displaystyle\left.\langle 0|\sum_{s}h^{s}_{\mathbf{q},{\rm
ind.}}(\tau)h^{s}_{\bar{\mathbf{q}},{\rm
ind.}}(\tau)|0\rangle\right|_{\tau=0}\equiv(2\pi)^{3}\delta(\mathbf{q}+\bar{\mathbf{q}})P_{\rm
ind}.$ (111)

The leading order $q$ independence of $G$ and $S$ implies

$\displaystyle\frac{P_{\rm ind}}{P_{h0}}=\mathcal{O}(\tilde{q}^{3}),$ (112)

which is the well-known IR behavior of the causally generated induced tensor
spectrum. Thus, $P_{\rm ind.}=\mathcal{O}(\lambda^{2})$ so it is included in
$P_{h2}$, but the induced GWs do not contribute to the super-horizon
enhancement of the one-loop tensor spectrum as they are causally produced. An
explicit form of the auto-power spectrum of the induced tensor mode at
$\tau=0$ is given as

$\displaystyle\frac{P_{\rm ind}}{P_{h0}}=$
$\displaystyle-\frac{1}{4}\int^{0}_{\tau_{i}}d\tau^{\prime}a(\tau^{\prime})^{2}\int^{\tau^{\prime}}_{\tau_{i}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}$
$\displaystyle\times(v_{q}(\tau^{\prime})-v^{*}_{q}(\tau^{\prime}))(v_{q}(\tau^{\prime\prime})-v^{*}_{q}(\tau^{\prime\prime}))$
$\displaystyle\times\int_{0}^{\infty}dp_{2}\int_{|p_{2}-q|}^{p_{2}+q}dp_{3}\bar{w}(q;p_{2},p_{3})$
$\displaystyle\times\left(u_{p_{2}}(\tau^{\prime})u^{*}_{p_{2}}(\tau^{\prime\prime})u_{p_{3}}(\tau^{\prime})u^{*}_{p_{3}}(\tau^{\prime\prime})\right.$
$\displaystyle\left.+u_{p_{2}}(\tau^{\prime\prime})u^{*}_{p_{2}}(\tau^{\prime})u_{p_{3}}(\tau^{\prime\prime})u^{*}_{p_{3}}(\tau^{\prime})\right).$
(113)

Note that $P_{\rm ind}$ is strictly non-negative as it is a squared quantity.
Then, for the delta-function-like scalar source, we find

$\displaystyle\frac{P^{\delta}_{\rm
ind}}{P_{h0}}=-\frac{p_{*}^{4}(4p_{*}-q^{2})^{2}}{512\pi^{2}q}\Theta_{2-q/p_{*}}$
$\displaystyle\times\int^{0}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}\int^{\tau^{\prime}}_{\tau_{0}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}$
$\displaystyle\times(v_{q}(\tau^{\prime})-v^{*}_{q}(\tau^{\prime}))(v_{q}(\tau^{\prime\prime})-v^{*}_{q}(\tau^{\prime\prime}))$
$\displaystyle\times(u^{2}_{p_{*}}(\tau^{\prime})(u^{*}_{p_{*}}(\tau^{\prime\prime}))^{2}+u^{2}_{p_{*}}(\tau^{\prime\prime})(u^{*}_{p_{*}}(\tau^{\prime}))^{2}),$
(114)

where the lognormal case is given by $\Theta\to\Theta^{\Delta}$. The induced
tensor modes are presented in Figs. 2 and 4.

Another $\mathcal{O}(\lambda^{1})$ correction originates from the fourth-order
interaction Hamiltonian:

$\displaystyle i\int^{\tau}_{\tau_{i}}d\tau^{\prime}[H^{(4)}_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]$
$\displaystyle=\frac{i}{2}\int^{\tau}_{\tau_{i}}d\tau^{\prime}a(\tau^{\prime})^{2}[v_{q}(\tau)v^{*}_{q}(\tau^{\prime})-v_{q}(\tau^{\prime})v^{*}_{q}(\tau)]$
$\displaystyle\times\prod_{i=2}^{4}\left(\int\frac{d^{3}p_{i}}{(2\pi)^{3}}\right)\delta\left(\sum_{i=2}^{4}\mathbf{p}_{i}-\mathbf{q}\right)\sum_{s_{2}}$
$\displaystyle\times
e^{s*}_{ik}(\hat{q})e^{s_{2}}_{kj}(\hat{p}_{2})p_{3i}p_{4j}\delta\chi_{\mathbf{p}_{3}}(\tau^{\prime})\delta\chi_{\mathbf{p}_{4}}(\tau^{\prime})h^{s_{2}}_{\mathbf{p}_{2}}(\tau^{\prime}).$
(115)

Taking the partial average over the scalar field, we can extract the component
parallel to $h^{s}_{\mathbf{q}}$ as

$\displaystyle i\int^{\tau}_{\tau_{i}}d\tau^{\prime}\langle[H^{(4)}_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]\rangle_{\delta\chi}$
$\displaystyle=$
$\displaystyle\frac{i}{6}\int^{\tau}_{\tau_{i}}d\tau^{\prime}a(\tau^{\prime})^{2}(v_{q}(\tau)v^{*}_{q}(\tau^{\prime})-v^{*}_{q}(\tau)v_{q}(\tau^{\prime}))$
$\displaystyle\times\int\frac{p_{3}^{4}dp_{3}}{2\pi^{2}}|u_{p_{3}}(\tau^{\prime})|^{2}h^{s}_{\mathbf{q}}(\tau^{\prime}),$
(116)

where the subscript $\delta\chi$ implies that we integrated out $\delta\chi$.
The cross-correlation of Eq. (116) and $h^{s}_{\mathbf{q}}$ reproduces the
result in the in-in formalism (64).

The variation of the super-horizon tensor fluctuation can be understood in a
quantum language as follows. Expanding Eq. (116) in terms of the tensor
annihilation and creation operators, we find

$\displaystyle
h^{s}_{\mathbf{q}}(\tau)+i\int^{\tau}_{\tau_{i}}d\tau^{\prime}\langle[H^{(4)}_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]\rangle_{\delta\chi}$
$\displaystyle=$ $\displaystyle
V_{q}(\tau)b^{s}_{\mathbf{q}}+V_{q}^{*}(\tau)b^{s\dagger}_{-\mathbf{q}},$
(117)

where we introduced the new positive frequency mode function

$\displaystyle
V_{q}(\tau)=\alpha(\tau)v_{q}(\tau)+\beta(\tau)v^{*}_{q}(\tau),$ (118)

with

$\displaystyle\alpha_{q}(\tau)$ $\displaystyle\equiv
1+\frac{i}{6}\int^{\tau}_{\tau_{i}}d\tau^{\prime}a(\tau^{\prime})^{2}\int\frac{p_{3}^{4}dp_{3}}{2\pi^{2}}|u_{p_{3}}(\tau^{\prime})|^{2}|v_{q}(\tau^{\prime})|^{2},$
(119) $\displaystyle\beta_{q}(\tau)$
$\displaystyle\equiv-\frac{i}{6}\int^{\tau}_{\tau_{i}}d\tau^{\prime}a(\tau^{\prime})^{2}\int\frac{p_{3}^{4}dp_{3}}{2\pi^{2}}|u_{p_{3}}(\tau^{\prime})|^{2}v^{2}_{q}(\tau^{\prime}).$
(120)

Then $\alpha$ and $\beta$ satisfy the following relation up to one-loop order

$\displaystyle|\alpha_{q}|^{2}-|\beta_{q}|^{2}=1.$ (121)

Therefore, the linear transformation from $(v,v^{*})$ to $(V,V^{*})$ in Eq.
(118) can be understood as a Bogoliubov transformation at least up to one-loop
order. The power spectrum is written as

$\displaystyle\frac{P_{h1}}{P_{h0}}=2|\beta_{q}(0)|.$ (122)

A Bogoliubov transformation also appears in $\mathcal{O}(\lambda^{2})$
correction. $\mathcal{O}(\lambda^{2})$ terms include mode mixing of tensor
fluctuations, but we can extract the component proportional to
$h^{s}_{\mathbf{q}}$ by taking the average over $\delta\chi$. After some
algebra, we find

$\displaystyle
h^{s}_{\mathbf{q}}(\tau)-\int^{\tau}_{\tau_{i}}d\tau^{\prime}\int^{\tau^{\prime}}_{\tau_{i}}d\tau^{\prime\prime}\langle[H^{(3)}_{{\rm
int},I}(\tau^{\prime\prime}),[H^{(3)}_{{\rm
int},I}(\tau^{\prime}),h^{s}_{\mathbf{q}}(\tau)]]\rangle_{\delta\chi}$
$\displaystyle=\tilde{V}_{q}(\tau)b^{s}_{\mathbf{q}}+\tilde{V}_{q}^{*}(\tau)b^{s\dagger}_{-\mathbf{q}},$
(123)

where defined

$\displaystyle\tilde{V}_{q}(\tau)=\tilde{\alpha}_{q}(\tau)v_{q}(\tau)+\tilde{\beta}_{q}(\tau)v^{*}_{q}(\tau),$
(124)

with

$\displaystyle\tilde{\alpha}_{q}$ $\displaystyle=1+\tilde{\beta}_{q}^{*},$
(125) $\displaystyle\tilde{\beta}_{q}$
$\displaystyle=\frac{1}{4}\int^{\tau}_{\tau_{0}}d\tau^{\prime}a(\tau^{\prime})^{2}\int^{\tau^{\prime}}_{\tau_{0}}d\tau^{\prime\prime}a(\tau^{\prime\prime})^{2}$
$\displaystyle\times\int^{\infty}_{0}dp_{2}\int^{p_{2}+q}_{|p_{2}-q|}dp_{3}\bar{w}(q;p_{2},p_{3})v_{q}^{*}(\tau^{\prime\prime})v_{q}(\tau^{\prime})$
$\displaystyle\times(u^{*}_{p_{3}}(\tau^{\prime\prime})u_{p_{3}}(\tau^{\prime})-u_{p_{3}}(\tau^{\prime\prime})u^{*}_{p_{3}}(\tau^{\prime}))$
$\displaystyle\times(u_{p_{2}}(\tau^{\prime\prime})u^{*}_{p_{2}}(\tau^{\prime})+u_{p_{2}}(\tau^{\prime})u^{*}_{p_{2}}(\tau^{\prime\prime})).$
(126)

Then, we find

$\displaystyle|\tilde{\alpha}_{q}|^{2}-|\tilde{\beta}_{q}|^{2}=1+\tilde{\beta}_{q}+\tilde{\beta}_{q}^{*}.$
(127)

The one-loop term in Eq. (127) does not straightforwardly vanish since, in
general, $v_{q}(\tau^{\prime\prime})\neq v_{q}(\tau^{\prime})$ unlike Eq.
(121). For the IR mode $|q\tau|\ll 1$, when
$|q\tau^{\prime}|,|q\tau^{\prime\prime}|\ll 1$, the time dependence of the
tensor mode functions is dropped and we get $v_{q}(\tau^{\prime\prime})\sim
v_{q}(\tau^{\prime})$. In this case, we can also think of the linear
transformation (124) as the Bogoliubov transformation up to one-loop order.

We concretely showed we have Bogoliubov transformations for several limited
circumstances, but Eq. (100) straightforwardly implies

$\displaystyle h^{s}_{\mathbf{q},{\rm H}}(\tau)$
$\displaystyle=v_{q}(\tau)B^{s}_{\mathbf{q}}(\tau)+v^{*}_{q}(\tau)B^{s,\dagger}_{-\mathbf{q}}(\tau),$
(128) $\displaystyle B^{s}_{\mathbf{q}}(\tau)$ $\displaystyle\equiv
F^{-1}(\tau)b^{s}_{\mathbf{q}}F(\tau),$ (129)

The unitarity of $F$ leads to

$\displaystyle[B^{s}_{\mathbf{q}},B^{\bar{s}}_{-\bar{\mathbf{q}}}]=[F^{-1}b^{s}_{\mathbf{q}}F,F^{-1}b^{s\dagger}_{-\mathbf{q}}F]=[b^{s}_{\mathbf{q}},b^{s\dagger}_{-\mathbf{q}}].$
(130)

Thus, the transformation $b\to B$ is a generalized Bogoliubov transformation.
The partial trace in terms of $\delta\chi$ might break the unitarity in
general, i.e., the Eq. (130) is not necessarily established once we take the
average of $\delta\chi$. At the same time, we confirmed that the unitarity is
kept at the one-loop order for the IR limit. $F^{-1}b^{s}_{\mathbf{q}}F$ is
not necessarily written by $b_{\mathbf{q}}$ and $b^{\dagger}_{\mathbf{q}}$
only as $F$ contains the mode coupling terms between different Fourier modes
and scalar fluctuations. Generally, the present Bogoliubov transformation
mixes the Fourier modes and $\delta\chi$, and the Bogoliubov coefficients are
operator-valued in the presence of $\delta\chi$, which satisfies the
generalized conditions for the multi-variable Bogoliubov transformation. We
leave the investigation for future work.

The Bogoliubov perspective also explains why the super-horizon scalar
fluctuations do not contribute to the loop correction, e.g., (D) in Fig. 2.
For super-horizon scalar fields, i.e., when
$p_{*}\tau^{\prime},p_{*}\tau^{\prime\prime}\ll 1$, we have
$u_{p_{*}}=u_{p_{*}}^{*}$. Then we find $\tilde{\alpha}_{q}=1$ and
$\tilde{\beta}_{q}=0$, which is why the super-horizon enhancement of the
spectator scalar field does not contribute to the loop effect. Ref.
Kanno:2018cuk  claimed that interactions with external classical fields
generate a coherent state as the unitary operator is written in a form of the
displacement operator. However, the claim does not apply to the present case
since the interacting scalar field is neither external nor classical.

###  VI.2 Bogoliubov transformation for scalar amplitude enhancement

So far, we have discussed that the loop correction can be understood as the
Bogoliubov transformation for tensor fluctuations. A Bogoliubov transformation
may also model a class of enhanced spectator scalar fields for $\delta\chi$
rather than a simple multiplicative enhancement of the scalar field (83).
Enhancement of $\delta\chi$ via a unitary evolution is categorized into this
class 111 Recent work Inomata:2022yte  considered inflationary scalar one-loop
calculation for the oscillatory features in the inflaton potential. They
numerically obtained the mode function by directly integrating the linear
equation of motion. Then the enhancement feature looks similar to the
enhancement via the Bogoliubov transformation. . In this case, the excited
state of the spectator field can be written as

$\displaystyle\delta\chi_{\mathbf{q}}=U_{q}a_{\mathbf{q}}+U^{*}_{q}a_{-\mathbf{q}}^{\dagger},$
(131)

where we consider

$\displaystyle
U_{q}=\frac{e^{i\phi_{1}}}{2}\left(\Xi+\frac{1}{\Xi}\right)u_{q}+\frac{e^{i\phi_{2}}}{2}\left(\Xi-\frac{1}{\Xi}\right)u_{q}^{*}.$
(132)

With the physical time we have

$\displaystyle\frac{\Xi+\Xi^{-1}}{2}$ $\displaystyle=\cosh[\mu
H(t-t_{i})],~{}$ (133) $\displaystyle\frac{\Xi-\Xi^{-1}}{2}$
$\displaystyle=\sinh[\mu H(t-t_{i})].$ (134)

In this parameterization, the power spectrum of the spectator scalar field is
amplified by $\Xi^{2}$, i.e., $|U_{q}|^{2}/|u_{q}|^{2}=\Xi^{2}$ for
$\phi_{1}=\phi_{2}$. Then, all equations above can be reused after $u_{q}\to
U_{q}$. Note that Eq. (132) is a most general parameterization of the
Bogoliubov transformation for a single field. $P_{h1}$ does not change much
for the new parameterization, while $P_{h2}$ changes drastically. Large $\Xi$
leads to $U_{q}\sim U_{q}^{*}$. We should note that the first iterative
component disappears, i.e., $\tilde{\alpha}_{q}=1$ and $\tilde{\beta}_{q}=0$
in Eqs. (125) and (126) if $U_{q}=U_{q}^{*}$. Therefore, the violation of
$U_{q}=U_{q}^{*}$ introduces non-vanishing scale invariant enhancement of the
loop. Hence, in Eq. (132), the appearance of $\Xi^{-1}$ terms is crucial. In
the new parameterization, we analytically find $\Gamma$ in Eq. (94) is
replaced as

$\displaystyle\tilde{\Gamma}[\Xi]=2\int^{0}_{x_{i}}dx^{\prime}\int^{x^{\prime}}_{x_{i}}dx^{\prime\prime}\frac{x^{\prime}}{x^{\prime\prime
2}}$ $\displaystyle\times(x^{\prime}\sin x^{\prime}+\cos
x^{\prime})(x^{\prime\prime}\sin x^{\prime\prime}+\cos x^{\prime\prime})$
$\displaystyle\times\left[\Xi^{\prime\prime 2}(x^{\prime}\cos x^{\prime}-\sin
x^{\prime})(x^{\prime\prime}\sin x^{\prime\prime}+\cos
x^{\prime\prime})\right.$ $\displaystyle\left.-\Xi^{\prime 2}(x^{\prime}\sin
x^{\prime}+\cos x^{\prime})(x^{\prime\prime}\cos x^{\prime\prime}-\sin
x^{\prime\prime})\right].$ (135)

Thus, the leading order contribution $\mathcal{O}(\Xi^{4})$ of the first
iterative correction is canceled, and the enhancement factor becomes
$\mathcal{O}(\Xi^{2})$. This cancellation is also seen from Eq. (126) as we
have

$\displaystyle
U^{*}_{p_{3}}(\tau^{\prime\prime})U_{p_{3}}(\tau^{\prime})-U_{p_{3}}(\tau^{\prime\prime})U^{*}_{p_{3}}(\tau^{\prime})$
$\displaystyle=$ $\displaystyle
u^{*}_{p_{3}}(\tau^{\prime\prime})u_{p_{3}}(\tau^{\prime})-u_{p_{3}}(\tau^{\prime\prime})u^{*}_{p_{3}}(\tau^{\prime}).$
(136)

In the present setup, the magnitude of $P_{h1}$ and $P_{h2}$ are comparable in
the IR region, and their signs can be opposite. The exact cancellation should
not happen as we compare the different interactions. Numerical calculation for
(A) to (D) in the new parameterization is presented in Fig. 5. The super-
horizon enhancement is relatively suppressed, as we discussed. (B) implies the
secondary spectrum is dominated by the induced tensor modes for subhorizon
scalar fields, while (A) and (D) suggest near horizon scalar fluctuation
hardly produces the induced tensor fluctuations in contrast to Figs. 2 and 4.

Figure 5: One-loop spectrum for the scalar field excited via the Bogoliubov
transformation (131). Definitions of symbols are the same as Fig. 2. The
magnitude of $P_{h1}$ and $P_{h2}$ are comparable as suggested by Eq. (135).

For multi-field inflation, Bogoliubov transformation for a scalar field is not
necessarily written as Eq. (132). Eq. (131) contains different operators, and
we cannot use the above results straightforwardly. We leave the detailed
analysis for the multi-spectator fields for future work.

##  VII Separate universe approach in Bianchi type-I cosmology

We considered several quantum loops during inflation, using the in-in
formalism. The new contributions change the super-horizon tensor spectrum,
which may sound odd from causality. We claimed that the time evolution could
be regarded as a Bogoliubov transformation. It just changes the definition of
the vacuum state via the nonlinear interaction so that the super-horizon
tensor fluctuations are seen differently without violation of causality.

For skeptical readers, we provide another look at our results in this section,
focusing on the classical counterpart of the effects. We may understand the
classical evolution of super-horizon perturbations from the separate Universe
approach. We can think of different realizations of local FLRW universes as
fluctuations of e-folding numbers, i.e., the super-horizon curvature
perturbations. Similarly, realizations of super-horizon tensor fluctuations
may be regarded as those of different homogenous but anisotropic universes,
i.e., Bianchi type-I spacetimes, whose metric may be written as

$\displaystyle
ds^{2}=a^{2}\left(-d\tau^{2}+e^{2b}dx^{2}+e^{-2b}dy^{2}+dz^{2}\right).$ (137)

When expanding Eq. (137) to linear order in $b$, one finds that $b$ coincides
with the realization of a plus mode propagating along the $z$ axis. The
transverse condition holds non-perturbatively as the components are in the
plane perpendicular to the propagating direction. On the other hand, the
traceless condition is not well-defined without a background spacetime metric
Ota:2021fdv . The physical implication of the traceless condition is that the
volume element is not affected by the perturbation. In this sense, $b$ is
properly defined such that the volume element is not perturbed to all orders
in Eq. (137). Hence we consider Eq. (137) as a nonperturbative extension of
the tensor perturbation in the separate Universe approach. Eq. (137) can also
be obtained by focusing on a single mode of $h_{ij}$ in Eq. (33) and
diagonalizing it. In the anisotropic spacetime, we consider a minimally
coupled classical scalar field $\chi$ whose action is written as

$\displaystyle S_{\chi}=-\frac{1}{2}\int
d^{4}x\sqrt{-g}g^{\mu\nu}\partial_{\mu}\chi\partial_{\nu}\chi+\cdots,$ (138)

where $\cdots$ implies the model-dependent potential terms, which determine
the details of the scalar dynamics.

Let us consider the effect of the scalar field on the spacetime anisotropy
$b$. First, the Einstein tensor is written as (without perturbative expansion
for $b$)

$\displaystyle G_{00}$ $\displaystyle=3\mathcal{H}^{2}-\dot{b}^{2},$ (139)
$\displaystyle G_{11}$
$\displaystyle=e^{2b}(-\mathcal{H}^{2}-2\dot{\mathcal{H}}+2\mathcal{H}\dot{b}+\ddot{b}-\dot{b}^{2}),$
(140) $\displaystyle G_{22}$
$\displaystyle=e^{-2b}(-\mathcal{H}^{2}-2\dot{\mathcal{H}}-2\mathcal{H}\dot{b}-\ddot{b}-\dot{b}^{2}),$
(141) $\displaystyle G_{33}$
$\displaystyle=-\mathcal{H}^{2}-2\dot{\mathcal{H}}-\dot{b}^{2},$ (142)

where the overdot is the derivative with respect to the conformal time, and we
introduced the conformal Hubble parameter $\mathcal{H}\equiv\dot{a}/a$, and
then the energy-momentum tensor is given as

$\displaystyle T_{00}$
$\displaystyle=\partial_{0}\chi\partial_{0}\chi+\frac{1}{2}a^{2}(X+\cdots),$
(143) $\displaystyle T_{11}$
$\displaystyle=\partial_{1}\chi\partial_{1}\chi-\frac{1}{2}a^{2}e^{2b}(X+\cdots),$
(144) $\displaystyle T_{22}$
$\displaystyle=\partial_{2}\chi\partial_{2}\chi-\frac{1}{2}a^{2}e^{-2b}(X+\cdots),$
(145) $\displaystyle T_{33}$
$\displaystyle=\partial_{3}\chi\partial_{3}\chi-\frac{1}{2}a^{2}(X+\cdots),$
(146)

and we defined

$\displaystyle X\equiv
g^{\rho\sigma}\partial_{\rho}\chi\partial_{\sigma}\chi.$ (147)

Then, the Einstein equation yields

$\displaystyle\ddot{b}+2\mathcal{H}\dot{b}=\frac{e^{-2b}\partial_{1}\chi\partial_{1}\chi-e^{2b}\partial_{2}\chi\partial_{2}\chi}{2M_{\rm
pl}^{2}}.$ (148)

Thus, the model-dependent terms do not contribute to the equation of motion
for $b$. We limit our calculation for the Born approximation for simplicity;
we only consider the classical counterpart of $P_{h1}$-like contribution. When
expanding $\chi$ with iterative solutions, additional terms appear in Eq.
(148) for consistent iteration.

Without the scalar field, we get

$\displaystyle
b=c_{1}\int^{\tau}\frac{d\tau^{\prime}}{a(\tau^{\prime})^{2}}+c_{2}.$ (149)

The constant solution $c_{2}$ corresponds to the super-horizon conserved
tensor mode. $c_{1}$ is the decaying mode.

For a non-vanishing scalar field, statistical isotropy of $\chi$ implies that
we have

$\displaystyle\langle\partial_{x}\chi\partial_{x}\chi\rangle=\langle\partial_{y}\chi\partial_{y}\chi\rangle=\frac{1}{3}\langle(\partial_{i}\chi)^{2}\rangle\geq
0.$ (150)

the leading order EoM for $b$ becomes

$\displaystyle b^{\prime\prime}+2\mathcal{H}b^{\prime}+m_{\rm
eff}^{2}b=\mathcal{O}(b^{2}),$ (151)

where we defined

$\displaystyle m_{\rm
eff}^{2}=\frac{2\langle(\partial_{i}\chi)^{2}\rangle}{3M_{\rm pl}^{2}}\geq 0.$
(152)

Thus, the back reaction of $\chi$ behaves like a mass term of the tensor mode
$b$. This is a harmonic oscillator in a friction force with an effective mass
$m_{\rm eff}$, which results in the decaying solution whose time scale is
given by $m_{\rm eff}^{-1}$. Suppose we have a scalar field enhancement at
$p=p_{*}$ mode during inflation. We have

$\displaystyle m_{\rm eff}^{2}=\frac{1}{3}\frac{H^{2}}{M_{\rm
pl}^{2}}(1+\tau^{2}p_{*}^{2})p_{*}^{2}\Xi^{2}.$ (153)

The effective mass squared is suppressed by a factor of $H^{2}/M_{\rm
pl}^{2}$, so the mass is usually negligibly small. $b$ is a realization of
super-horizon tensor fluctuations that exit the horizon in the early stage of
inflation, which is constant at the initial time. Once we get a large $\Xi$
when the enhanced scalar mode is inside the horizon $|p_{*}\tau|\gtrsim 1$, we
get $|m_{\rm eff}\tau|\gtrsim 1$, and $b$ oscillates and decays. Thus, we
showed that the background anisotropy could evolve due to the scalar field.
Then the super-horizon suppression of $P_{h1}$ is explained in the separate
Universe approach. A similar argument will be possible for $P_{h2}$, including
the first iterative solution in $\chi$. Solving Eq. (151) is a non-
perturbative way to find the super-horizon variation. For example, we can find
the solution for constant $m_{\rm eff}$, which is given by a linear
combination of the Bessel functions with the time constant $m_{\rm eff}$. The
Bessel functions account for the effects of $m^{2}_{\rm eff}\propto
H^{2}/M_{\rm pl}^{2}$ to all order non perturbatively.

##  VIII Observations

One normally considers that scalar fields during inflation decay in the end to
realize the radiation-dominant Universe. Hence $\delta\chi$ should decay at
some point, and the one-loop correction to the tensor fluctuation is fixed.
The decay of $\delta\chi$ is model-dependent, which is not considered in this
paper. Once $\delta\chi$ decays, the time evolution of the tensor fluctuations
afterward should be the same as the linear tensor modes. The tensor
fluctuations with the loop re-enter the horizon and propagate as gravitational
waves. If large curvature perturbations are produced by the end of inflation
from the large $\delta\chi$, a similar GW reduction or enhancement will happen
in the Universe after reheating Chen:2022dah .

We can apply the standard CMB polarization analysis for the corrected
gravitational waves since the time evolution of the GWs is the same as the
linear one in the late universe. However, the tensor-to-scalar-ratio
constrained in the observation should be related to

$\displaystyle r_{\rm NL}=\frac{P_{h0}+P_{h1}+P_{h2}+\cdots}{P_{\zeta
0}+P_{\zeta 1}+P_{\zeta 2}+\cdots},$ (154)

where we also expect the loop correction for the adiabatic perturbations for
the same reason. The loop correction for $\zeta$ is model-dependent. Also, the
coupling to $\zeta$ is more slow-roll suppressed, so that the correction may
be relatively small.

Calculation of $\Omega_{\rm GW}\equiv\rho_{\rm GW}/\rho_{\rm tot}$ with the GW
energy density $\rho_{\rm GW}$ and the total energy density $\rho_{\rm tot}$
can be done as if they are linear GWs; the energy density of GWs are computed
as Isaacson:1968zza ; Isaacson:1968hbi

$\displaystyle\frac{d\ln\rho_{\rm GW}}{d\ln q}$ $\displaystyle=\frac{M_{\rm
pl}^{2}}{4a^{2}}\left(\frac{dT_{h}(q\tau)}{d\tau}\right)^{2}\frac{q^{3}(P_{h0}+P_{h1}+P_{h2}+\cdots)}{2\pi^{2}},$
(155)

where $T_{h}$ is the linear transfer function of the tensor mode. Given an
enhancement model of the scalar amplitude, we can put some constraints on the
parameters from the current or future measurements of $r_{\rm NL}$ and
$\Omega_{\rm GW}$.

A remaining issue is the gauge dependence of the loop contribution. Tensor
fluctuations are gauge-independent at linear order but are not at nonlinear
order. The gauge issue has been discussed in the context of induced GWs in the
literature Inomata:2019yww ; DeLuca:2019ufz ; Domenech:2020xin ; Ota:2021fdv .
Only the propagating component of the tensor modes can be regarded as GWs,
which are gauge independent in the subhorizon limit, as far as we choose a
gauge condition where the metric perturbations do not diverge in the short
scale. We expect similar arguments for the nonlinear interaction. We showed
that the super-horizon scalar field hardly contributes to the loop correction.
Therefore, as far as the gauge-dependent component of $\delta\chi$ is small
before horizon exit, we expect the gauge independent loop. In the in-in
formalism, equations are expanded by the linear perturbations. Therefore,
gauge transformation can be truncated at linear order. We can always construct
the linear gauge invariant from the gauge-dependent perturbations. The gauge
invariants can express our Hamiltonian, and the loop correction should be
gauge invariant.

##  IX Conclusions

This paper presented a full-quantum calculation of the one-loop inflationary
tensor power spectrum in the presence of an excited spectator scalar field. We
found that the one-loop correction from an excited subhorizon scalar mode may
enhance or reduce the super-horizon primordial tensor power spectrum scale-
invariantly. The first impression of the results is odd, as we believe that
the super-horizon tensor fluctuations should be constant until horizon re-
entry from causality. Our setup differs from the previous literature about
induced GWs based on the classical equation of motion. We employed the in-in
formalism, and the full one-loop effect was considered. Our calculation
includes the causal production of GWs, limited to the subhorizon scale, as we
expected. The super horizon loop effects are considered as Bogoliubov
transformation. Nonlinear interaction among the tensor and scalar changes the
definition of the vacuum, and then the initial super horizon tensor
fluctuations are observed differently. We examined the separate Universe
approach with Bianchi type-I local universes to explain the super-horizon
evolution of tensor fluctuations. We find that the background anisotropy
reduces when a scalar field exists, a damping effect introduced by an
effective mass. This is the separate Universe counterpart of $P_{h1}$.

We considered several forms of enhanced scalar fields. For example, a simple
multiplication of the mode functions by $\Xi$ leads to the loop correction of
$\mathcal{O}(\Xi^{4})$. This amplification may happen when, e.g., we modify
the sound speed. In contrast, we found a leading order cancellation for a
scalar enhancement via the Bogoliubov transformation, which results in
$\mathcal{O}(\Xi^{2})$ in the super Hubble region. The size of the one-loop
correction is sensitive to the amplitude and the initial and final
amplification time. We saw several physically different situations (A) to (D)
in section V.2; depending on the situations, the final spectrum varies. Hence,
we need to specify a model and parameters for quantitative analysis. We found
that the loop correction may exceed the tree level power spectrum, which
indicates the breakdown of the perturbative expansion in the presence of a
highly enhanced scalar field.

Once the scalar peak disappears, the super-horizon correction is fixed, and we
cannot distinguish it from the original primordial GWs. Time evolution after
the horizon re-entry is linear, and observational constraints on the linear
tensor power spectrum straightforwardly apply to the corrected spectrum.

Before concluding this paper, let us discuss some interesting extensions of
this work. First, our calculation is based on several toy models of an
enhancement, and the relationship between $\delta\chi$ and the final $\zeta$
is unknown. Considering the loop effect for a concrete model with scalar
amplitude enhancement should be interesting since we can predict the
relationship between the PBH formation history and the one-loop inflationary
spectrum (see, e.g. recent works Kristiano:2022maq ; Inomata:2022yte .).
Computing a one-loop bispectrum or trispectrum should also be interesting as
the loop correction suggests large non-Gaussianity for the tensor
fluctuations. This property could be useful to distinguish the linear tensor
power spectrum from the one-loop. As we mentioned in Section II, the
(in)equivalence of our quantum approach and classical field theory with the
stochastic initial condition is not obvious. As far as we compute the induced
GWs in the quantum approach, the spectrum looks quite similar. Solving a
consistent iteration in the EoM may tell us the quantum nature during
inflation. The loop correction for the scalar power spectrum should be
technically complicated since it is fully model-dependent. However, we expect
the same momentum structure for the one-loop terms for the scalar power
spectrum. Our result suggests that large-scale measurements may indirectly
test the short-scale enhancement of cosmological perturbations, so combining
gravitational wave detectors at all scales is crucial for future surveys
Kogut:2011xw ; Andre:2013afa ; Matsumura:2013aja ; Desvignes:2016yex ;
Brazier:2019mmu ; Kerr:2020qdo ; LISA:2017pwj ; Kawamura:2011zz ; Ruan:2018tsw
; TianQin:2015yph .

###### Acknowledgements.

The authors would like to thank Keisuke Inomata for useful comments. AO would
like to thank Chen Chao, Yuhang Zhu and Huiyu Zhu for useful discussions. AO
and YW are supported in part by the National Key R&D Program of China
(2021YFC2203100), the NSFC Excellent Young Scientist Scheme (Hong Kong and
Macau) Grant No. 12022516, and the RGC of Hong Kong SAR, China (GRF 16303621).
MS is partly supported by the JSPS KAKENHI grant Nos. 19H01895, 20H04727, and
20H05853.

##  Appendix A Operator products

In this paper, we often use the following products of the scalar and tensor
fluctuations:

$\displaystyle\langle
0|\delta\chi_{\mathbf{p}_{1}}\delta\chi_{\mathbf{p}_{2}}|0\rangle$
$\displaystyle=(2\pi)^{3}\delta(\mathbf{p}_{1}+\mathbf{p}_{2})u_{p_{1}}u^{*}_{p_{2}},$
(156) $\displaystyle\langle
0|h^{s_{1}}_{\mathbf{p}_{1}}h^{s_{2}}_{\mathbf{p}_{2}}|0\rangle$
$\displaystyle=(2\pi)^{3}\delta(\mathbf{p}_{1}+\mathbf{p}_{2})\delta^{s_{1}s_{2}}v_{p_{1}}v^{*}_{p_{2}}.$
(157)

Also, the products of the four operators are given as

$\displaystyle\langle
0|\delta\chi_{\mathbf{p}_{1}}\delta\chi_{\mathbf{p}_{2}}\delta\chi_{\mathbf{p}_{3}}\delta\chi_{\mathbf{p}_{4}}|0\rangle$
$\displaystyle=$
$\displaystyle(2\pi)^{6}\delta(\mathbf{p}_{1}+\mathbf{p}_{2})\delta(\mathbf{p}_{3}+\mathbf{p}_{4})u_{p_{1}}u^{*}_{p_{2}}u_{p_{3}}u^{*}_{p_{4}}$
$\displaystyle+$
$\displaystyle(2\pi)^{6}\delta(\mathbf{p}_{1}+\mathbf{p}_{3})\delta(\mathbf{p}_{2}+\mathbf{p}_{4})u_{p_{1}}u^{*}_{p_{3}}u_{p_{2}}u^{*}_{p_{4}}$
$\displaystyle+$
$\displaystyle(2\pi)^{6}\delta(\mathbf{p}_{1}+\mathbf{p}_{4})\delta(\mathbf{p}_{2}+\mathbf{p}_{3})u_{p_{1}}u^{*}_{p_{4}}u_{p_{2}}u^{*}_{p_{3}},$
(158)

and

$\displaystyle\langle
0|h^{s_{1}}_{\mathbf{p}_{1}}h^{s_{2}}_{\mathbf{p}_{2}}h^{s_{3}}_{\mathbf{p}_{3}}h^{s_{4}}_{\mathbf{p}_{4}}|0\rangle$
$\displaystyle=$
$\displaystyle(2\pi)^{6}\delta(\mathbf{p}_{1}+\mathbf{p}_{2})\delta(\mathbf{p}_{3}+\mathbf{p}_{4})\delta^{s_{1}s_{2}}\delta^{s_{3}s_{4}}v_{p_{1}}v^{*}_{p_{2}}v_{p_{3}}v^{*}_{p_{4}}$
$\displaystyle+$
$\displaystyle(2\pi)^{6}\delta(\mathbf{p}_{1}+\mathbf{p}_{3})\delta(\mathbf{p}_{2}+\mathbf{p}_{4})\delta^{s_{1}s_{3}}\delta^{s_{2}s_{4}}v_{p_{1}}v^{*}_{p_{3}}v_{p_{2}}v^{*}_{p_{4}}$
$\displaystyle+$
$\displaystyle(2\pi)^{6}\delta(\mathbf{p}_{1}+\mathbf{p}_{4})\delta(\mathbf{p}_{2}+\mathbf{p}_{3})\delta^{s_{1}s_{4}}\delta^{s_{2}s_{3}}v_{p_{1}}v^{*}_{p_{4}}v_{p_{2}}v^{*}_{p_{3}}.$
(159)

##  Appendix B Angular integrals

In this appendix, we discuss the treatment for the polarization sum and the
angular dependence in the loop integrals. Eq. (66) can be computed in a
convenient coordinate system where $\hat{q}=(0,0,1)$. In this frame, the
polarization tensor can be written as (up to a phase factor)

$\displaystyle e_{ij}^{\pm 2}(\hat{q})\equiv\frac{1}{2}\begin{pmatrix}1&\pm
i&0\\\ \pm i&-1&0\\\ 0&0&0\end{pmatrix},$ (160)

and then the rest of the momentum vectors are

$\displaystyle\mathbf{p}_{A}=p_{A}\begin{pmatrix}\sin\theta_{A}\cos\phi_{A}&\sin\theta_{A}\sin\phi_{A}&\cos\theta_{A}\end{pmatrix}$
(161)

Using these specific representations, we find

$\displaystyle\sum_{s=\pm 2}|e^{ij,s}(\hat{q})p_{2i}p_{3j}|^{2}$
$\displaystyle=\frac{p_{2}^{2}p_{3}^{2}}{2}\sin^{2}\theta_{2}\sin^{2}\theta_{3},$
(162)

which can be written by $p_{2}$ and $p_{3}$ since we have

$\displaystyle\sin^{2}\theta_{2}\sin^{2}\theta_{3}$
$\displaystyle=\frac{1}{16p_{2}^{2}p_{3}^{2}q^{4}}(p_{2}+p_{3}+q)^{2}(p_{2}+p_{3}-q)^{2}$
$\displaystyle\times(p_{2}-p_{3}+q)^{2}(-p_{2}+p_{3}+q)^{2}.$ (163)

Thus, momentum integrals in Eq. (66) is only a function of $p_{2}$ and
$p_{3}$. When the integrand is independent of the angular coordinates, we have

$\displaystyle\int\frac{d^{3}p_{2}d^{3}p_{3}}{(2\pi)^{6}}(2\pi)^{3}\delta(\mathbf{q}-\mathbf{p}_{2}-\mathbf{p}_{3})$
$\displaystyle=$
$\displaystyle\frac{1}{(2\pi)^{2}q}\int_{0}^{\infty}dp_{2}\int_{|p_{2}-q|}^{p_{2}+q}dp_{3}p_{2}p_{3}.$
(164)

Combining Eqs. (162), (163), and (164), we integrate out the angular
coordinates. A derivation of Eq. (164) is given as

$\displaystyle\int\frac{d^{3}p_{2}d^{3}p_{3}}{(2\pi)^{6}}(2\pi)^{3}\delta(\mathbf{q}-\mathbf{p}_{2}-\mathbf{p}_{3})$
$\displaystyle=$ $\displaystyle\int
d^{3}xe^{i\mathbf{q}\cdot\mathbf{x}}\int\frac{d^{3}p_{2}d^{3}p_{3}}{(2\pi)^{6}}e^{i\mathbf{p}_{2}\cdot\mathbf{x}}e^{i\mathbf{p}_{3}\cdot\mathbf{x}}$
$\displaystyle=$
$\displaystyle(4\pi)^{2}\sum_{\ell\ell_{2}\ell_{3}mm_{2}m_{3}}i^{\ell_{2}+\ell_{3}}\int
d^{3}x\int\frac{d^{3}p_{2}d^{3}p_{3}}{(2\pi)^{6}}$ $\displaystyle\times
Y_{\ell m}(\hat{x})Y_{\ell_{2}m_{2}}(\hat{x})Y_{\ell_{3}m_{3}}(\hat{x})Y_{\ell
m}^{\star}(\hat{q})Y_{\ell_{2}m_{2}}(\hat{p}_{2})Y_{\ell_{3}m_{3}}(\hat{p}_{3})$
$\displaystyle\times j_{\ell}(qx)j_{\ell_{2}}(p_{2}x)j_{\ell_{3}}(p_{3}x)$
$\displaystyle=$ $\displaystyle
4\pi\int\frac{p_{2}^{2}dp_{2}}{2\pi^{2}}\int\frac{p_{3}^{2}dp_{3}}{2\pi^{2}}\int
x^{2}dx$ $\displaystyle\times j_{0}(qx)j_{0}(p_{2}x)j_{0}(p_{3}x)$
$\displaystyle=$ $\displaystyle
4\pi\int\frac{p_{2}^{2}dp_{2}}{2\pi^{2}}\int\frac{p_{3}^{2}dp_{3}}{2\pi^{2}}\frac{\pi}{8p_{2}p_{3}q}\left(\frac{1}{\text{sgn}(p_{2}-p_{3}+q)}\right.$
$\displaystyle\left.+\frac{1}{\text{sgn}(-p_{2}+p_{3}+q)}+\frac{1}{\text{sgn}(p_{2}+p_{3}-q)}-1\right),$
(165)

where we used the partial wave expansion for the plain waves in the third
line. The bracket in the last line reduces to a top hat filter for
$|q-p_{3}|\leq p_{3}\leq q+p_{3}$.

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