# Why Must Primordial Non-Gaussianity Be Very Small?

Jason Kristiano<EMAIL_ADDRESS>Research Center for the
Early Universe (RESCEU), Graduate School of Science, The University of Tokyo,
Tokyo 113-0033, Japan  Department of Physics, Graduate School of Science, The
University of Tokyo, Tokyo 113-0033, Japan  Jun’ichi Yokoyama
<EMAIL_ADDRESS>Research Center for the Early Universe
(RESCEU), Graduate School of Science, The University of Tokyo, Tokyo 113-0033,
Japan  Department of Physics, Graduate School of Science, The University of
Tokyo, Tokyo 113-0033, Japan  Kavli Institute for the Physics and Mathematics
of the Universe (Kavli IPMU), WPI, UTIAS, The University of Tokyo, Kashiwa,
Chiba 277-8568, Japan  Trans-scale Quantum Science Institute, The University
of Tokyo, Tokyo 113-0033, Japan

###### Abstract

One-loop correction to the power spectrum in generic single-field inflation is
calculated by using standard perturbation theory. Because of the enhancement
inversely proportional to the observed red tilt of the spectral index of
curvature perturbation, the correction turns out to be much larger than
previously anticipated. As a result, the primordial non-Gaussianity must be
much smaller than the current observational bound in order to warrant the
validity of cosmological perturbation theory.

inflation, cosmological perturbation, power spectrum, loop corrections, non-
Gaussianity

††preprint: RESCEU-4/21

Inflation in the early Universe [1, 2, 3] (see, e.g., [4] for a review) has
been a part of standard cosmology not only to solve the horizon, flatness, and
monopole problems, but also to account for the origin of large-scale
structures. Inflation is well described by a homogeneous scalar field dubbed
as inflaton $\phi$ in quasi-de Sitter (dS) space. Properties of the inflaton,
such as the forms of its kinetic and potential terms as well as its coupling
to gravity are yet to be clarified both from theoretical and observational
viewpoints. Quantum fluctuations of scalar field and gravitational field
generated during inflation [5, 6, 7, 8, 9] serve as probes of its physics that
can be tested by observations of cosmic microwave background (CMB) and large-
scale structures.

The simplest class of inflation models is the potential-driven canonical slow-
roll inflation where the kinetic term of the inflaton has the canonical form
and its potential energy drives inflation. In this model, fluctuations can be
expressed approximately as a massless free scalar field in dS space minimally
coupled to gravity. As a result, its linear perturbation calculation predicts
a nearly scale-invariant spectrum with highly Gaussian distribution (see,
e.g., [10, 11] for a review), in good agreement with observations [12, 13]. In
these canonical models, deviations from the scale-invariant Gaussian
distribution are controlled by the slow-roll parameters [14]. Observationally,
exact scale-invariant power spectrum has already been ruled out with more than
two-$\sigma$ confidence level with the red-tilted spectral index $n_{s}<1$
[13], but the primordial non-Gaussianity characterized by the bispectrum has
not been detected so far, and only constraints on the nonlinearity parameter
$f_{\mathrm{NL}}$ of various types have been obtained so far [15].

In this situation, a number of extended inflation models have been proposed so
far that can realize sizable non-Gaussianity while reproducing the observed
red-tilted spectrum. Theoretically, the simplest local-type non-Gaussianity
[16] may be produced by curvaton [17, 18] or modulated reheating [19, 20]
scenarios, both of which require another fluctuating field in addition to the
inflaton. Since there is no observational evidence requiring multiple
fluctuating fields during inflation [12, 13], we do not consider such models
but stick to the single-field inflation models, whose non-Gaussian signature
is mostly the equilateral one. To realize sizable non-Gaussianity in single-
field models, some models modify the kinetic terms as in $k$ or $G$ inflation
[21, 22, 23], ghost condensate [24, 25], and Dirac-Born Infeld inflation [26],
while other models extend the gravitational sector [5, 27, 28, 29, 30], or
both [31]. In these noncanonical models, enhanced non-Gaussianity may be
realized due to the smallness of sound speed during inflation [32, 33, 34, 35,
36] and still consistent with the observation [15]. Hence primordial non-
Gaussianity serves as a good probe of new physics on which these extended
models are based [37, 38].

In the language of quantum field theory (QFT), power spectrum of perturbation
corresponds to the vacuum expectation value (VEV) of the fluctuation two-point
functions. The lowest order of non-Gaussianity is the VEV of the fluctuation
three-point functions. Technically, the non-Gaussianity exists because of
higher-order interaction terms and it is evaluated by in-in perturbation
theory with two-point function as the input to the calculation. However, such
interaction terms also generate higher-order corrections to the two-point
function which are called loop corrections in QFT terminology. Such correction
must be analyzed carefully to ensure its smallness compared with the tree-
level amplitude, or the result of calculations based on the linear
perturbation theory, on which most cosmologists rely to set the initial
condition of the post-inflationary Universe, would lose its significance.

In this Letter, we calculate one-loop corrections to the power spectrum
arising from the three-body interactions of perturbations which also generate
primordial non-Gaussianity. As a result we find that in order to ensure that
such loop corrections are small enough to warrant the validity of the
conventional lowest-order calculation, the primordial non-Gaussianity must be
much smaller than the current observational bound for generic single-field
inflation models.

Loop corrections of cosmological perturbations have been of interest since it
was introduced in [39, 40]. A number of authors have calculated them for
various inflation models with the discussions on regularization and
renormalization schemes [41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52].
Infrared regularity and its relation to quantum states has been discussed in
[53, 54], while loop corrections due to external particles have been discussed
in [55, 56]. Loop corrections related to a specific vacuum phase transition
model was discussed in [57].

We discuss a single-field inflation model whose Lagrangian is a general
function of $X\equiv-\nabla_{\mu}\phi\nabla^{\mu}\phi/2$ and $\phi$, namely
$P(\phi,X)$, in Einstein gravity. It is the simplest inflation model that can
generate a large non-Gaussianity. The action of the inflaton is given by

$S=\frac{1}{2}\int\mathrm{d}^{4}x\sqrt{-g}\left[M_{\mathrm{pl}}^{2}R+2P(X,\phi)\right],$
(1)

where $g=\mathrm{det}(g_{\mu\nu})$, $g_{\mu\nu}$ is a metric tensor, and $R$
and $M_{\mathrm{pl}}$ are the Ricci scalar and the reduced Planck scale,
respectively. Small perturbation from the homogeneous part of the inflaton
$\bar{\phi}(t)$ and metric can be expressed as

$\displaystyle\phi(\mathbf{x},t)=\bar{\phi}(t)+\delta\phi(\mathbf{x},t)$
$\displaystyle
ds^{2}=-N^{2}\mathrm{d}t^{2}+\gamma_{ij}(\mathrm{d}x^{i}+N^{i}\mathrm{d}t)(\mathrm{d}x^{j}+N^{j}\mathrm{d}t)$
(2)

where $\gamma_{ij}$ is a three-dimensional metric on slices of constant $t$,
$N$ is a lapse function, and $N^{i}$ is a shift vector. We work in the
comoving gauge

$\delta\phi(\mathbf{x},t)=0,~{}\gamma_{ij}(\mathbf{x},t)=a^{2}(t)[1+2\zeta(\mathbf{x},t)]\delta_{ij},$
(3)

where $N$ and $N^{i}$ are solutions of Hamiltonian and momentum constraint
equations. Also, we do not consider tensor perturbation. After some algebra,
the second-order action of the curvature perturbation $\zeta(\mathbf{x},t)$
reads

$S^{(2)}=M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}\frac{\epsilon}{c_{s}^{2}}\left[\dot{\zeta}^{2}-\frac{c_{s}^{2}}{a^{2}}(\partial_{i}\zeta)^{2}\right],$
(4)

where a dot denotes differentiation with respect to time, and
$\epsilon\equiv-\dot{H}/{H^{2}}$ is a slow-roll parameter of the Hubble
parameter $H$, which has a very weak time dependence during inflation. $c_{s}$
is the sound speed expressed as

$c_{s}^{2}=\frac{P_{,X}}{P_{,X}+2XP_{,XX}},$ (5)

with a comma representing a partial derivative.

Before calculating the loop correction to the inflationary power spectrum, we
briefly review the standard quantization of cosmological perturbation by
introducing the Mukhanov-Sasaki (MS) variable [6, 58]

$v\equiv
M_{\mathrm{pl}}z\zeta,~{}~{}~{}~{}z\equiv\frac{a}{c_{s}}\sqrt{2\epsilon},$ (6)

with which the second-order action (4) becomes canonically normalized

$S^{(2)}=\frac{1}{2}\int\mathrm{d}\tau~{}\mathrm{d}^{3}x\left[(v^{\prime})^{2}-c_{s}^{2}(\partial_{i}v)^{2}+\frac{z^{\prime\prime}}{z}v^{2}\right],$
(7)

where $\tau$ is conformal time and the prime denotes derivative with respect
to $\tau$. During inflation we find, to the lowest order in the slow-roll
parameters,

$aH=-\frac{1}{(1-\epsilon)\tau},~{}~{}a(\tau)\propto\tau^{-(1+\epsilon)},~{}{\rm
and}~{}~{}\frac{z^{\prime\prime}}{z}=\frac{2+3\epsilon}{\tau^{2}}.$ (8)

Hence, the operator expansion of the MS variable reads

$\hat{v}(\mathbf{p},\tau)=M_{\mathrm{pl}}z\zeta(\mathbf{p},\tau)=v_{p}(\tau)\hat{a}_{\mathbf{p}}+v^{*}_{p}(\tau)\hat{a}_{-\mathbf{p}}^{\dagger},$
(9)

with the mode function

$v_{p}(\tau)=\left(-\frac{\pi\tau}{4}\right)^{1/2}H_{\nu}^{(1)}(-pc_{s}\tau),~{}~{}\nu=\frac{3}{2}+\epsilon,$
(10)

where $\hat{a}_{-\mathbf{p}}^{\dagger}$ and $\hat{a}_{\mathbf{p}}$ are
creation and annihilation operators with the commutation relation

$\left[\hat{a}_{\mathbf{p}},\hat{a}_{-\mathbf{q}}^{\dagger}\right]=(2\pi)^{3}\delta^{3}(\mathbf{p}+\mathbf{q}).$
(11)

The mode function corresponds to the Bunch-Davies vacuum $\ket{0}$ at early
time, which is defined as the state annihilated by $\hat{a}_{\mathbf{p}}$. To
simplify notation, let us express the two-point function of curvature
perturbation and power spectrum at a late time during inflation, $\tau_{0}$,
as

$\displaystyle\left\langle\zeta(\mathbf{p})\zeta(\mathbf{q})\right\rangle=(2\pi)^{3}\delta^{3}(\mathbf{p}+\mathbf{q})\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle,$
(12)
$\displaystyle\Delta^{2}_{s}(p)\equiv\frac{p^{3}}{2\pi^{2}}\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle,$
(13)

and take a limit $\tau_{0}\rightarrow 0$, when all the relevant modes are in
the superhorizon regime and the observed power spectrum is evaluated. Here the
bracket denotes the VEV, $\langle\cdots\rangle=\bra{0}\cdots\ket{0}$, and
$\Delta^{2}_{s}(p)$ is the power spectrum multiplied by the phase space
density.

Since the mode function is well described as

$\zeta_{p}(\tau)=\frac{v_{p}(\tau)}{zM_{\mathrm{pl}}}=\left(\frac{H^{2}}{4M_{\mathrm{pl}}^{2}\epsilon
c_{s}}\right)_{H}^{\frac{1}{2}}\frac{e^{-ic_{s}p\tau}}{p^{3/2}}(1+ic_{s}p\tau),$
(14)

the power spectrum is given by

$\Delta^{2}_{s(0)}(p)=\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}c_{s}\epsilon}\right)_{H}=\Delta^{2}_{s(0)}(p_{*})\left(\frac{p}{p_{*}}\right)^{n_{s}-1},$
(15)

to the lowest order in slow-roll parameters. Here subscripts $(0)$ and $H$
denote tree-level contribution and the quantity evaluated at the sound horizon
crossing $c_{s}p=aH$, respectively. $p_{*}$ is an arbitrary pivot momentum.
The power spectrum is almost scale invariant with the deviation parametrized
by the spectral index $n_{s}-1=O(\epsilon)$. Indeed, there is a weak momentum
dependence because of time dependence of the quantities in the parenthesis at
the horizon crossing of each mode.

Beyond second-order action, we can expand more to third-order action. In the
limit of $c_{s}\ll 1$, the third-order action is [33, 34]

$S_{\mathrm{int}}=\int\mathrm{d}t~{}\mathrm{d}^{3}x\left[-\frac{2\lambda}{H^{3}}a^{3}\dot{\zeta}^{3}+\frac{\epsilon
M_{\mathrm{pl}}^{2}}{Hc_{s}^{2}}a\dot{\zeta}(\partial_{i}\zeta)^{2}\right],$
(16)

where $\lambda=X^{2}P_{,XX}+(2/3)X^{3}P_{,XXX}$. It is also predicted by the
effective field theory of inflation [59], where the two interaction terms are
independent so observable quantities generated by one term will not affect the
other. Such third-order action is equivalent to cubic self-interaction term of
the curvature perturbation.

In this Letter, we focus on calculating one-loop correction generated by the
second term in (16), which generates primordial non-Gaussianity of the form

$f_{\mathrm{NL}}=\frac{\langle\zeta\zeta\zeta\rangle}{\langle\zeta\zeta\rangle^{2}}\propto\frac{1}{c_{s}^{2}},$
(17)

resulting in large non-Gaussianity for smaller sound speed.

Loop correction can be computed by using standard in-in perturbation theory as

$\langle\mathcal{O(\tau)}\rangle=\left\langle\left[\bar{\mathrm{T}}\exp\left(i\int_{-\infty}^{\tau}\mathrm{d}\tau^{\prime}H_{\mathrm{int}}(\tau^{\prime})\right)\right]\mathcal{\hat{O}}(\tau)\left[\mathrm{T}\exp\left(-i\int_{-\infty}^{\tau}\mathrm{d\tau^{\prime}}H_{\mathrm{int}}(\tau^{\prime})\right)\right]\right\rangle,$
(18)

where $\mathcal{\hat{O}}(\tau)$ is an operator at a fixed time $\tau$,
$\mathrm{T}$ and $\bar{\mathrm{T}}$ denote time and antitime ordering.
Although the interactions (16) contain the time derivative of the field, the
relation $H_{\mathrm{int}}=-\int\mathrm{d}^{3}x~{}\mathcal{L}_{\mathrm{int}}$
still holds with $\mathcal{L}_{\mathrm{int}}$ defined by the integrand of (16)
[60, 48]. In our case, the operator is $\zeta(\mathbf{p})\zeta(-\mathbf{p})$
evaluated at $\tau=\tau_{0}~{}(\rightarrow 0)$. First-order expansion
vanishes, yielding an odd-point correlation function. Second-order expansion
of the perturbation theory is

$\displaystyle\langle\mathcal{O(\tau)}\rangle=\langle\mathcal{O(\tau)}\rangle_{(0,2)}^{\dagger}+\langle\mathcal{O(\tau)}\rangle_{(1,1)}+\langle\mathcal{O(\tau)}\rangle_{(0,2)},$
$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(1,1)}=\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\int_{-\infty}^{\tau}\mathrm{d}\tau_{2}\left\langle
H_{\mathrm{int}}(\tau_{1})\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{2})\right\rangle,$
(19)
$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(0,2)}=-\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}\left\langle\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{1})H_{\mathrm{int}}(\tau_{2})\right\rangle.$
(20)

We start the calculation for the $(1,1)$ term by substituting the operator
expansion

$\displaystyle\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle_{(1,1)}=$
$\displaystyle~{}M_{\mathrm{pl}}^{4}\int_{-\infty}^{0}\mathrm{d}\tau_{1}\frac{a^{2}(\tau_{1})\epsilon(\tau_{1})}{H(\tau_{1})c_{s}^{2}}\int_{-\infty}^{0}\mathrm{d}\tau_{2}\frac{a^{2}(\tau_{2})\epsilon(\tau_{2})}{H(\tau_{2})c_{s}^{2}}\int\prod_{a=1}^{6}\left[\frac{\mathrm{d}^{3}k_{a}}{(2\pi)^{3}}\right]\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3})~{}\delta^{3}(\mathbf{k}_{4}+\mathbf{k}_{5}+\mathbf{k}_{6})$
$\displaystyle(\mathbf{k}_{2}\cdot\mathbf{k}_{3})(\mathbf{k}_{5}\cdot\mathbf{k}_{6})\left\langle\dot{\zeta}(\mathbf{k}_{1},\tau_{1})\zeta(\mathbf{k}_{2},\tau_{1})\zeta(\mathbf{k}_{3},\tau_{1})\zeta(\mathbf{p})\zeta(-\mathbf{p})\dot{\zeta}(\mathbf{k}_{4},\tau_{2})\zeta(\mathbf{k}_{5},\tau_{2})\zeta(\mathbf{k}_{6},\tau_{2})\right\rangle.$
(21)

For small loop momentum $k\ll p$, performing Wick contraction and substituting
mode function yields

$\displaystyle\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(1,1)}=2p^{4}\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0)}$
$\displaystyle\int_{-\infty}^{0}\mathrm{d}\tau_{1}\int_{-\infty}^{0}\mathrm{d}\tau_{2}(1+ic_{s}p\tau_{1})^{2}(1-ic_{s}p\tau_{2})^{2}e^{-2ic_{s}p(\tau_{1}-\tau_{2})}$
$\displaystyle\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\frac{k}{4c_{s}^{2}p^{6}}\left(\frac{H^{2}}{4M_{\mathrm{pl}}^{2}\epsilon
c_{s}}\right)_{H}.$ (22)

We find that contribution from small loop momentum in the limit $k\rightarrow
0$ is zero, consistent with Maldacena’s theorem [14]. In the same limit, the
$(0,2)$ term will generate the same momentum dependence.

Hence, the loop correction comes from other domain, namely, $k>p$. In this
domain, the divergent integral comes from the $(0,2)$ term, while the $(1,1)$
term converges in a similar way to [47]. The $(0,2)$ term is given by

$\displaystyle\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle_{(0,2)}=$
$\displaystyle-
M_{\mathrm{pl}}^{4}\int_{-\infty}^{0}\mathrm{d}\tau_{1}\frac{a^{2}(\tau_{1})\epsilon(\tau_{1})}{H(\tau_{1})c_{s}^{2}}\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}\frac{a^{2}(\tau_{2})\epsilon(\tau_{2})}{H(\tau_{2})c_{s}^{2}}\int\prod_{a=1}^{6}\left[\frac{\mathrm{d}^{3}k_{a}}{(2\pi)^{3}}\right]\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3})~{}\delta^{3}(\mathbf{k}_{4}+\mathbf{k}_{5}+\mathbf{k}_{6})$
$\displaystyle(\mathbf{k}_{2}\cdot\mathbf{k}_{3})(\mathbf{k}_{5}\cdot\mathbf{k}_{6})\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\dot{\zeta}(\mathbf{k}_{1},\tau_{1})\zeta(\mathbf{k}_{2},\tau_{1})\zeta(\mathbf{k}_{3},\tau_{1})\dot{\zeta}(\mathbf{k}_{4},\tau_{2})\zeta(\mathbf{k}_{5},\tau_{2})\zeta(\mathbf{k}_{6},\tau_{2})\right\rangle.$
(23)

Performing Wick contraction, it becomes

$\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0,2)}=-M_{\mathrm{pl}}^{4}\int_{-\infty}^{0}\mathrm{d}\tau_{1}\frac{a^{2}(\tau_{1})\epsilon(\tau_{1})}{H(\tau_{1})c_{s}^{2}}\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}\frac{a^{2}(\tau_{2})\epsilon(\tau_{2})}{H(\tau_{2})c_{s}^{2}}\zeta_{p}(\tau_{0})\zeta_{p}(\tau_{0})f(p,\tau_{1},\tau_{2}),$
(24) $\displaystyle f(p,\tau_{1},\tau_{2})$
$\displaystyle=\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\left[8(\mathbf{p}\cdot\mathbf{q})^{2}\dot{\zeta}_{k}(\tau_{1})\dot{\zeta}^{*}_{k}(\tau_{2})\zeta_{p}^{*}(\tau_{1})\zeta_{p}^{*}(\tau_{2})\zeta_{q}(\tau_{1})\zeta_{q}^{*}(\tau_{2})\right.$
(25)
$\displaystyle\left.-8(\mathbf{p}\cdot\mathbf{k})(\mathbf{p}\cdot\mathbf{q})\dot{\zeta}_{k}(\tau_{1})\dot{\zeta}^{*}_{q}(\tau_{2})\zeta_{p}^{*}(\tau_{1})\zeta_{p}^{*}(\tau_{2})\zeta_{q}(\tau_{1})\zeta_{k}^{*}(\tau_{2})-8(\mathbf{p}\cdot\mathbf{q})(\mathbf{k}\cdot\mathbf{q})\dot{\zeta}_{k}(\tau_{1})\dot{\zeta}^{*}_{p}(\tau_{2})\zeta_{p}^{*}(\tau_{1})\zeta_{k}^{*}(\tau_{2})\zeta_{q}(\tau_{1})\zeta_{q}^{*}(\tau_{2})\right.$
$\displaystyle\left.-8(\mathbf{p}\cdot\mathbf{q})(\mathbf{k}\cdot\mathbf{q})\dot{\zeta}^{*}_{p}(\tau_{1})\dot{\zeta}^{*}_{k}(\tau_{2})\zeta_{k}(\tau_{1})\zeta_{p}^{*}(\tau_{2})\zeta_{q}(\tau_{1})\zeta_{q}^{*}(\tau_{2})+4(\mathbf{k}\cdot\mathbf{q})^{2}\dot{\zeta}^{*}_{p}(\tau_{1})\dot{\zeta}^{*}_{p}(\tau_{2})\zeta_{k}(\tau_{1})\zeta_{k}^{*}(\tau_{2})\zeta_{q}(\tau_{1})\zeta_{q}^{*}(\tau_{2})\right],$

where $\mathbf{q}=\mathbf{k}-\mathbf{p}$. Then, substituting mode function and
performing time integration, we find ultraviolet (UV) divergent terms with
different power of $k$. The terms that diverge with a positive power of $k$
are to be regularized and renormalized in the same way as QFT in flat
spacetime, and are irrelevant in cosmological consideration. The most
important term here is the term with its $k$-integrand proportional to
$k^{-3}$, which would yield a logarithmic divergence in pure dS background
with a scale-invariant spectrum [47, 48]. Fortunately, however, in a realistic
situation with a red-tilted power spectrum, such a term is multiplied by a
function with a weak $k$ dependence suppressed by a slow-roll parameter, which
automatically removes the logarithmic divergence as we see now.

Extracting terms with the above mentioned $k$ dependence from (24) yields

$\displaystyle\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0,2)}$
$\displaystyle\cong\frac{51}{80c_{s}^{4}}\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0)}\int_{p}^{\infty}\mathrm{d}k~{}k^{2}\frac{1}{k^{3}}\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon
c_{s}}\right)_{H}$
$\displaystyle=\frac{51}{80c_{s}^{4}}\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0)}\int_{p}^{\infty}\mathrm{d}k~{}k^{2}\frac{1}{k^{3}}\Delta^{2}_{s(0)}(k_{*})\left(\frac{k}{k_{*}}\right)^{n_{s}-1}.$
(26)

For $n_{s}<1$, which is the case in our Universe, the momentum integration
converges as

$\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0,2)}=\frac{51}{80c_{s}^{4}}\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0)}\frac{\Delta^{2}_{s(0)}(p)}{1-n_{s}}.$
(27)

Hence, adding the contribution of the $(2,0)$ term, the total one-loop
correction to the power spectrum is

$\Delta^{2}_{s(1)}(p)=\frac{51[\Delta^{2}_{s(0)}(p)]^{2}}{40c_{s}^{4}(1-n_{s})}.$
(28)

Note that preceding analyses [47, 48], which studied the same type of loop
correction, did not take the spectral tilt into account, but stuck to the
scale-invariant spectrum. As a consequence, they encountered a logarithmic
divergence and introduced dimensional regularization

$\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}k^{3}}\rightarrow\int\frac{\mathrm{d}^{3+\delta}k}{(2\pi)^{3}k^{3}},$
(29)

where $\delta$ is a very small arbitrary parameter. Then, they got a
$1/\delta$ pole and renormalize it to make the observable free of an arbitrary
parameter. In our observed Universe, however, the spectrum has a red tilt,
which makes the momentum integral convergent. Therefore, we do not need
regularization and renormalization introducing an extra parameter $\delta$,
which should not be confused with a physical observable $1-n_{s}$.

In order for the perturbative calculation to be credible, one-loop correction
must be suppressed compared with the tree-level contribution, otherwise the
power spectrum would become infinitely large if we calculate higher-order loop
correction. Thus we require

$\frac{\Delta_{s(0)}^{2}}{c_{s}^{4}(1-n_{s})}\ll 1.$ (30)

We now estimate the left-hand side of the above inequality from observational
results [13]. Although $\Delta_{s(0)}^{2}$ corresponds to the ”bare” quantity
in QFT language, let us put the observed amplitude of the scalar power
spectrum at the pivot scale $k_{\ast}=0.05~{}\mathrm{Mpc}^{-1}$,
$\Delta_{s(0)}^{2}\equiv A_{2}=2.1\times 10^{-9}$ [13] according to the common
practice of cosmology which we are about to refine. From the observed value of
the spectral index, $n_{s}=0.9649\pm 0.0042$ based on $TT$, $TE$, $EE$, low
$E$, and lens of Planck 2018 [13], let us take a conservative value
$1-n_{s}=0.0393$ from its one-$\sigma$ lower bound. Then the inequality (30)
imposes a constraint on the sound speed as $c_{s}\gg 0.02$.

Interestingly, this inequality practically coincides with the observational
bound on the sound speed imposed by the equilateral non-Gaussianity [15],
namely, $c_{s}>0.021$ at 95% confidence level. Since our inequality is a
strong one with $\gg$, this means that the amplitude of the actual non-
Gaussianity must be much smaller than the current observational bound;
otherwise the conventional cosmological perturbation theory, which has been
successfully describing the evolution of the Universe in agreement with
observations, would lose the predictability 111Here one may recall Einstein’s
saying that the eternal mystery of the world is its comprehensibility.,
running into the strong coupling regime [62].

If the lower limit of sound speed is increased to $c_{s}\sim 0.1$, it will be
safe from a dangerous loop correction as it is suppressed to $O(0.001)$, but
it is still as large as slow-roll parameter corrections to the power spectrum
[63].

We may compare our result to the constraint presented in [48], where they
considered an exact scale-invariant power spectrum so they did not get
observationally relevant inverse of the $1-n_{s}$ factor. A similar constraint
was also discussed in [50], where they only analyzed the coupling constant of
the interaction action. As we have shown here, however, proper incorporation
of background dynamics is essentially important to get the enhancement factor.
Therefore, our constraint is more severe and places primordial non-Gaussianity
in a stricter limit. This is important as non-Gaussianity has been proposed as
the probe of new fundamental physics.

So far we have assumed a single power-law spectrum in the entire wave number
range. If we relax this assumption and consider a wider possibility of the
shape of the spectrum on smaller scales that cannot be probed by CMB
observations, we may find various interesting implications to cosmology of the
early Universe. For example, if small-scale perturbations with wave number
$p^{\prime}$ have a different amplitude and red spectral index
$n^{\prime}_{s}$, then the condition (30) may rather be interpreted as a
constraint on the amplitude of fluctuations

$\Delta_{s(0)}^{2}(p^{\prime})\ll c_{s}^{4}(1-n^{\prime}_{s})\approx
f_{\mathrm{NL}}^{-2}(1-n^{\prime}_{s}).$ (31)

For $c_{s}=0.1$ and $1-n^{\prime}_{s}=0.1$, for example, we find
$\Delta_{s(0)}^{2}(p^{\prime})\ll 10^{-5}$, which is already more stringent
than some known constraints [64, 65].

More interesting is the case in which the power spectrum has some peaky shape,
which may induce formation of primordial black holes (PBHs) [66, 67, 68, 69]
and large stochastic gravitational wave background [70, 71, 72, 73]. Since the
shape of the power spectrum as well as its generation mechanism vary model to
model, let us consider a simple toy model where $\zeta_{p}(t)$ has the same
functional form as (14), but is enhanced by a constant factor $B\gg 1$ for a
finite wave number range from $k_{p}$ to $k_{p}e^{N_{p}}$ where $N_{p}={\cal
O}(1)$. Repeating the same calculations from (23) to (26), we find that the
wave number integral in (25) is multiplied by a factor $B^{4}$ for
$k_{p}<k<k_{p}e^{N_{p}}$, assuming $k_{p}\gg p$, so that (28) acquires an
extra component,

$\Delta^{2}_{s(1)}(p)=\frac{51[\Delta^{2}_{s(0)}(p)]^{2}}{40c_{s}^{4}(1-n_{s})}+{\cal
O}\left(\frac{N_{p}}{c_{s}^{4}}[\Delta^{2}_{s(0)}(k_{p})]^{2}\right).$ (32)

Then the perturbativity condition reads

$\Delta^{2}_{s(0)}(k_{p})\ll\left(\frac{c_{s}^{4}}{N_{p}}\Delta^{2}_{s(0)}(p)\right)^{\frac{1}{2}}\sim(5\times
10^{-5})c_{s}^{2}.$ (33)

If we take this constraint at face value, formation of an appreciable amount
of PBHs nor observable amplitude of stochastic gravitational wave background
are by no means possible unless deci-hertz interferometer gravitational wave
observatory (DECIGO) or big bang observer (BBO) are realized for the latter
[71].

In fact, however, this result should be interpreted with caveats, as it has
been derived under a number of assumptions such as the form of interaction
(16) taking $c_{s}$ small and an significantly simplified mode function.
Nevertheless, this simple analysis certainly motivates us to study the
constraint in more realistic single-field inflation models predicting PBH
formation (e.g., [74, 75, 76]) or models inducing large-amplitude stochastic
gravitational wave background [77] to which our analysis does not directly
apply because these authors assumed a different type of non-Gaussianity than
we consider here.

Consistency with Maldacena’s theorem implies that contribution from small loop
momentum to the one-loop correction (26) is zero, which we have confirmed by
direct manipulations. However, there are some known single-field examples that
violate Maldacena’s consistency condition, such as the ultra slow-roll (USR)
inflation [78, 79, 80]. One may wonder if such models suffer from infrared
divergence, as they are not protected by Maldacena’s theorem. Since such
models are realized as a nonattractor solution, which does not last long,
modes relevant to the USR regime span only a finite range of wave number [81],
and hence they are free from the infrared problem. These modes, however, may
contribute significantly to the one-loop corrections, as the would-be decaying
mode of perturbation may grow during such regime in a similar way to [75]. In
such a case, the mode function has a different form from (14), so a separate
analysis is required. Note that the non-Gaussianity generated in this model
has a different shape, too [82, 83].

In conclusion, we calculated the one-loop correction to the inflationary power
spectrum by using standard in-in perturbation theory. The red tilt of the
spectral index makes the loop’s momentum integration convergent. Because of
it, the loop correction is enhanced by an inverse factor of $1-n_{s}$, which
is a small positive number according to the latest observation [13]. As a
result, in order for the loop correction to be small enough to warrant the
validity of the standard perturbation theory, the amplitude of equilateral
non-Gaussianity must be much smaller than the current observational bound for
the case where the primordial spectrum has a simple power-law spectrum. This
consideration opens up the possibility of further interesting consequences for
a more general spectrum, which will be reported elsewhere.

We thank Enrico Pajer for a very helpful discussion about Maldacena’s
consistency condition. We also thank Yusuke Yamada and Kohei Kamada for the
discussion in the early stage of writing this Letter. J. K. acknowledges the
support from Monbukagakusho (MEXT) scholarship and Global Science Graduate
Course (GSGC) program of The University of Tokyo. J. Y. is supported by JSPS
KAKENHI Grant No. 20H05639 and Grant-in-Aid for Scientific Research on
Innovative Areas 20H05248.

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