# Ruling Out Primordial Black Hole Formation From Single-Field Inflation

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

The most widely studied formation mechanism of a primordial black hole (PBH)
is collapse of large-amplitude perturbation on small scales generated in
single-field inflation. In this Letter, we calculate one-loop correction to
the large-scale power spectrum in such a model. We find models producing
appreciable amount of PBHs induce nonperturbative coupling on large scale
probed by cosmic microwave background radiation. We therefore conclude that
PBH formation from cosmological perturbation theory in single-field inflation
is ruled out.

inflation, cosmological perturbation, power spectrum, loop corrections,
primordial black holes

††preprint: RESCEU-20/22

Primordial black holes (PBHs) have been a research interest for more than 50
years [1, 2, 3], although there has been no observational evidence for them.
They could be light enough for Hawking radiation to be important [4], they are
a potential dark matter candidate [5, 6, 7, 8, 9, 10, 11, 12, 13, 14]
(reviewed in [15, 16]), and they can explain LIGO-Virgo gravitational wave
events [17, 18, 19, 20, 21, 22].

A number of formation mechanisms of PBHs in the early Universe has been
proposed. The most well-studied one makes use of quantum fluctuations [23, 24,
25, 26] generated in cosmic inflation [27, 28, 29]. Observations of CMB
anisotropy [30, 31, 32] tightly constrain these fluctuations on large scales.
Their power spectrum is almost scale invariant with the amplitude $2.1\times
10^{-9}$. On smaller scales that cannot be probed by CMB observations,
observational constraints are loose enough [33, 34, 35, 36, 37, 38, 39].
Therefore, it is possible to have a theory which produce large fluctuations
with the amplitude of power spectrum $\mathcal{O}(0.01)$ that satisfy
observational constraints and many models have been proposed to realize such a
feature [7, 40, 8, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55,
56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74,
75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93].
Peaks of such fluctuations may collapse into PBHs with appreciable abundance
[94, 95] after entering the horizon, which also produce large stochastic
gravitational wave background [96, 97, 98, 99, 100] that can be probed by
future gravitational wave observations as well as pulsar timing array
experiments [101].

The simplest inflation model that is consistent with current observational
data [31, 32] is canonical slow-roll (SR) inflation as reviewed in [102]. It
is described by a scalar field $\phi$, called inflaton, with a canonical
kinetic term and potential $V(\phi)$ in quasi-de Sitter space. The standard SR
inflation generates nearly scale-invariant adiabatic curvature perturbation
that behaves classically as the decaying mode decreases exponentially during
inflation, so that the perturbation variable and its conjugate momentum
practically commute with each other [103]. In order to be consistent with CMB
observations [31, 32], the shape of the potential is tightly constrained for a
finite range of $\phi$.

If the inflaton passes through an extremely flat region of the potential with
$\mathrm{d}V/\mathrm{d}\phi\approx 0$ after the comoving scales probed by CMB
have left the horizon, it may produce large-amplitude fluctuations on small
scales. In this region, slow-roll condition fails, and the inflation goes into
a temporary ultraslow-roll (USR) period [104, 105, 106, 107, 108]. During this
regime the non-constant mode of fluctuations, which would decay exponentially
in SR inflation, actually grows, as observed in other models [109, 81],
resulting in enhanced power spectrum on specific scales. This may also imply
the importance of quantum effects as we will see below.

Many inflation models with a flat region or inflection point of the potential
have been proposed inspired by high energy theories such as supergravity [42,
43, 44, 45, 46, 47, 48], axion monodromy [49, 50], scalaron in $R^{2}$-gravity
[51], $\alpha$-attractor [52, 53, 54], and string theory [55, 56, 57], as well
as in Higgs inflation which does not require theories beyond the standard
model [58, 59, 60, 61, 62]. As an extension of USR period, constant-roll
inflation can also produce large amplitudes [110, 75].

Theoretically, the power spectrum is described by the vacuum expectation value
(VEV) of the fluctuation two-point functions in quantum field theory, to which
only wavevectors with equal magnitude and opposite direction contribute. As we
expand the theory to higher-order in fluctuations, we will get higher-order
interaction terms, which generate primordial non-Gaussianity or VEV of the
higher-point functions which are calculated by in-in perturbation theory [111,
112, 113]. At the same time, such interactions also generate back reaction to
the two-point function which is called loop correction [114, 115, 116, 117,
118, 119, 120, 121, 122]. These corrections behave non-linearly, where
fluctuations with different wavenumber magnitude can contribute. Therefore,
small-scale fluctuations can contribute to the loop corrections of the CMB-
scale fluctuation two-point functions.

As mentioned above, in order to realize PBH formation appropriately, we need
an inflation model producing enhanced power spectrum with the amplitude
$\mathcal{O}(0.01)$ on a certain small scale while keeping the amplitude at
$2.1\times 10^{-9}$ on CMB scale. However, such a requirement is only a tree-
level statement. So far, understanding of inflation models accommodating PBH
formation is very limited beyond tree-level, although it has been discussed in
[123, 124, 125, 126, 127, 128, 129] for some specific models. It is important
to ensure that one-loop correction is suppressed compared to the tree-level
contribution, so that we can still trust the perturbation theory.

In our previous papers [121, 122], we showed that one-loop perturbativity
bound can strongly constrain single-field inflation models. We have also
qualitatively pointed out a possible problem in PBH formation mechanism. In
this Letter, we use one-loop perturbativity requirement to examine the
possibility of PBH formation models in single-field inflation. We calculate
contribution of the peak of power spectrum on small scale to one-loop
correction of the CMB-scale power spectrum. Requiring one-loop correction to
be much smaller than tree-level contribution, we obtain an upper-bound on the
power spectrum on small scale.

We specifically consider a PBH formation from an extremely flat region in the
potential that leads to a temporary USR motion of the inflaton. At the end, we
will explain that our result can be generalized to other PBH formation models
in single-field inflation.

The action of canonical inflation is given by

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

where $M_{\mathrm{pl}}$ is reduced Planck scale,
$g=\mathrm{det}~{}g_{\mu\nu}$, $g_{\mu\nu}$ and $R$ are metric tensor and its
Ricci scalar. Consider a spatially flat, homogeneous and isotropic background

$\mathrm{d}s^{2}=-\mathrm{d}t^{2}+a^{2}(t)\mathrm{d}\mathbf{x}^{2}=a^{2}(\tau)(-\mathrm{d}\tau^{2}+\mathrm{d}\mathbf{x}^{2}),$
(2)

where $\tau$ is conformal time. Equations of motion for the scale factor
$a(t)$ and the homogeneous part of the inflaton $\phi(t)$ are the Friedmann
equations

$H^{2}=\frac{1}{3M_{\mathrm{pl}}^{2}}\left(\frac{1}{2}\dot{\phi}^{2}+V(\phi)\right),~{}\dot{H}=-\frac{\dot{\phi}^{2}}{2M_{\mathrm{pl}}^{2}},$
(3)

with $H=\dot{a}/{a}$ being the Hubble parameter, and the Klein-Gordon equation

$\ddot{\phi}+3H\dot{\phi}+\frac{\mathrm{d}V}{\mathrm{d}\phi}=0.$ (4)

When CMB-scale fluctuations leave the horizon at around $\phi_{\mathrm{CMB}}$
(see Fig. 1), the potential is slightly tilted to realize slow-roll inflation,
satisfying

$\absolutevalue{\frac{\ddot{\phi}}{\dot{\phi}H}}\ll
1,~{}\epsilon\equiv-\frac{\dot{H}}{H^{2}}=\frac{\dot{\phi}^{2}}{2M_{\mathrm{pl}}^{2}H^{2}}\ll
1,$ (5)

where $\epsilon$ is a SR parameter. In the SR period, $\epsilon$ is
approximately constant. Then the inflaton goes through an extremely flat
region of the potential, between time $t_{s}$ to $t_{e}$, experiencing an USR
period. When inflaton enters this region with
$\mathrm{d}V/\mathrm{d}\phi\approx 0$, Eq. (4) becomes
$\ddot{\phi}\approx-3H\dot{\phi}$, so $\dot{\phi}\propto a^{-3}$, which breaks
SR approximation [107]. This makes $\epsilon$ strongly time-dependent and
extremely small as

$\epsilon=\frac{\dot{\phi}^{2}}{2M_{\mathrm{pl}}^{2}H^{2}}\propto a^{-6}.$ (6)

We also define the second SR parameter

$\eta\equiv\frac{\dot{\epsilon}}{\epsilon
H}=2\epsilon+2\frac{\ddot{\phi}}{\dot{\phi}H},$ (7)

which is approximately constant and very small in SR period $\eta\ll 1$, but
large in USR period $\eta\approx-6$. The latter regime satisfies the condition
of the growth of the non-constant mode of perturbation found in [81], namely,
$3-\epsilon+\eta<0$, so that enhanced spectrum is obtained then.

After the USR period, the inflaton enters SR period again until the end of
inflation. In both SR and USR period, because $\epsilon$ is very small, the
scale factor can be approximated as $a=-1/H\tau\propto e^{Ht}$.

Figure 1:  Schematic picture of the inflaton potential realizing PBH
formation. When the inflaton is around $\phi_{\mathrm{CMB}}$, scales probed by
CMB observations leave horizon and it is in the SR regime. It enters an
extremely flat region at $t=t_{s}$ undergoing an USR period. It enters SR
period again at $t=t_{e}$ until $\phi_{\mathrm{end}}$, the end of inflation.

Small perturbation from the homogeneous part, $\phi(t)$, of the inflaton
$\phi(\mathbf{x},t)$ and metric can be expressed as

$\displaystyle\phi(\mathbf{x},t)=\phi(t)+\delta\phi(\mathbf{x},t),$
$\displaystyle\mathrm{d}s^{2}=-N^{2}\mathrm{d}t^{2}+\gamma_{ij}(\mathrm{d}x^{i}+N^{i}\mathrm{d}t)(\mathrm{d}x^{j}+N^{j}\mathrm{d}t),$
(8)

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

$\delta\phi(\mathbf{x},t)=0,~{}\gamma_{ij}(\mathbf{x},t)=a^{2}(t)e^{2\zeta(\mathbf{x},t)}\delta_{ij},$
(9)

where $\zeta(\mathbf{x},t)$ is curvature perturbation. Here, tensor
perturbation is not relevant. Also, $N$ and $N^{i}$ are obtained by solving
constraint equations.

Expanding the action (1) up to the second-order of the curvature perturbation
yields

$S^{(2)}=M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}\epsilon\left[\dot{\zeta}^{2}-\frac{1}{a^{2}}(\partial_{i}\zeta)^{2}\right].$
(10)

In terms of Mukhanov-Sasaki variable $v=zM_{\mathrm{pl}}\zeta$, where
$z=a\sqrt{2\epsilon}$, the action becomes canonically normalized

$S^{(2)}=\frac{1}{2}\int\mathrm{d}\tau~{}\mathrm{d}^{3}x\left[(v^{\prime})^{2}-(\partial_{i}v)^{2}+\frac{z^{\prime\prime}}{z}v^{2}\right],$
(11)

where a prime denotes derivative with respect to $\tau$ [23, 130]. In momentum
space, quantization is performed by promoting the Mukhanov-Sasaki variable as
an operator

$\hat{v}(\mathbf{k},\tau)=M_{\mathrm{pl}}z\zeta(\mathbf{k},\tau)=v_{k}(\tau)\hat{a}_{\mathbf{k}}+v^{*}_{k}(\tau)\hat{a}_{-\mathbf{k}}^{\dagger},$
(12)

where mode function $v_{k}(\tau)$ approximately satisfies

$v_{k}^{\prime\prime}+\left(k^{2}-\frac{2}{\tau^{2}}\right)v_{k}=0,$ (13)

in both SR and USR regimes, and the operators satisfy the commutation relation
$[\hat{a}_{\mathbf{k}},\hat{a}_{-\mathbf{k^{\prime}}}^{\dagger}]=(2\pi)^{3}\delta^{3}(\mathbf{k}+\mathbf{k^{\prime}})$
under the normalization condition

$v_{k}^{\prime*}v_{k}-v_{k}^{\prime}v_{k}^{*}=i.$ (14)

The general solution of mode function $v_{k}(\tau)$ is

$v_{k}(\tau)=\frac{\mathcal{A}_{k}}{\sqrt{2k}}\left(1-\frac{i}{k\tau}\right)e^{-ik\tau}+\frac{\mathcal{B}_{k}}{\sqrt{2k}}\left(1+\frac{i}{k\tau}\right)e^{ik\tau},$
(15)

where $\mathcal{A}_{k}$ and $\mathcal{B}_{k}$ are determined by boundary
conditions.

At an early time, $t\lesssim t_{s}$, the inflaton was in SR period with Bunch-
Davies initial vacuum, a state $\ket{0}$ defined by
$\hat{a}_{\mathbf{k}}\ket{0}=0$ with $\mathcal{A}_{k}=1$ and
$\mathcal{B}_{k}=0$. Mode function of the curvature perturbation
$\zeta_{k}=v_{k}/zM_{\mathrm{pl}}$ is

$\zeta_{k}(\tau)=\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{\epsilon_{\mathrm{SR}}}}\right)_{\star}\frac{e^{-ik\tau}}{k^{3/2}}(1+ik\tau),$
(16)

where $\epsilon_{\mathrm{SR}}$ is $\epsilon$ in SR period and subscript
$\star$ denotes the value at the horizon crossing epoch $\tau=-1/k$.

At $t_{s}\lesssim t\lesssim t_{e}$, the inflaton is in USR period. We define
$\tau_{s}$ and $\tau_{e}$ as conformal time corresponding to $t_{s}$ and
$t_{e}$, respectively. The SR parameter $\epsilon$ can be written as
$\epsilon(\tau)=\epsilon_{\mathrm{SR}}(\tau/\tau_{s})^{6}$ based on
proportionality in (6). Therefore, the curvature perturbation becomes

$\displaystyle\zeta_{k}(\tau)=$
$\displaystyle~{}\left(\frac{iH}{2M_{\mathrm{pl}}\sqrt{\epsilon_{\mathrm{SR}}}}\right)_{\star}\left(\frac{\tau_{s}}{\tau}\right)^{3}\frac{1}{k^{3/2}}$
$\displaystyle\times\left[\mathcal{A}_{k}e^{-ik\tau}(1+ik\tau)-\mathcal{B}_{k}e^{ik\tau}(1-ik\tau)\right],$
(17)

where coefficients $\mathcal{A}_{k}$ and $\mathcal{B}_{k}$ are determined by
matching to the SR solution (16) at the boundary. We consider instantaneous
transition from SR to USR, because it is a good approximation to numerical
solutions [131]. Solutions of the coefficients by requiring continuity of
$\zeta_{k}(\tau)$ and $\zeta_{k}^{\prime}(\tau)$ at transition $\tau=\tau_{s}$
are [132, 133, 134, 135, 136, 131, 137]

$\mathcal{A}_{k}=1-\frac{3(1+k^{2}\tau_{s}^{2})}{2ik^{3}\tau_{s}^{3}},~{}\mathcal{B}_{k}=-\frac{3(1+ik\tau_{s})^{2}}{2ik^{3}\tau_{s}^{3}}e^{-2ik\tau_{s}}.$
(18)

At late time, $t\gtrsim t_{e}$, the inflaton goes back to SR dynamics. We
define $k_{s}$ and $k_{e}$ as wavenumbers which cross the horizon at
$\tau_{s}$ and $\tau_{e}$, respectively. For perturbation with wavenumber
$k\lesssim k_{e}$, the mode function approaches constant as
$\zeta_{k}(\tau)\approx\zeta_{k}(\tau_{e})$. For perturbation with $k\gtrsim
k_{e}$, the mode function can be approximated as (16).

The two-point functions of curvature perturbation and power spectrum at the
end of inflation, $\tau_{0}~{}(\rightarrow 0)$, can be written as

$\displaystyle\left\langle\zeta(\mathbf{k})\zeta(\mathbf{k^{\prime}})\right\rangle\equiv(2\pi)^{3}\delta^{3}(\mathbf{k}+\mathbf{k^{\prime}})\left\langle\\!\left\langle\zeta(\mathbf{k})\zeta(-\mathbf{k})\right\rangle\\!\right\rangle,$
(19)
$\displaystyle\Delta^{2}_{s}(k)\equiv\frac{k^{3}}{2\pi^{2}}\left\langle\\!\left\langle\zeta(\mathbf{k})\zeta(-\mathbf{k})\right\rangle\\!\right\rangle,$
(20)

the bracket $\langle\cdots\rangle=\bra{0}\cdots\ket{0}$ denotes VEV, and
$\Delta^{2}_{s}(k)$ is the power spectrum multiplied by the phase space
density. For $k\lesssim k_{e}$, because
$\zeta_{k}(\tau_{0})\approx\zeta_{k}(\tau_{e})$, the power spectrum is

$\Delta^{2}_{s(0)}(k)=\frac{k^{3}}{2\pi^{2}}\absolutevalue{\zeta_{k}(\tau_{0})}^{2}\simeq\frac{k^{3}}{2\pi^{2}}\absolutevalue{\zeta_{k}(\tau_{e})}^{2},$
(21)

where $\zeta_{k}(\tau_{e})$ is given by (17) with coefficients in (18), and
the subscript $(0)$ denotes tree-level contribution.

On large scale, the power spectrum approaches an almost scale-invariant limit

$\Delta_{s(\mathrm{SR})}^{2}(k)=\Delta^{2}_{s(0)}(k\ll
k_{s})=\left(\frac{H^{2}}{8\pi^{2}M_{\mathrm{pl}}^{2}\epsilon_{\mathrm{SR}}}\right)_{\star},$
(22)

with a small wavenumber dependence due to the horizon crossing condition
manifested in the spectral tilt

$n_{s}-1=\frac{\mathrm{d}\log\Delta_{s(\mathrm{SR})}^{2}}{\mathrm{d}\log
k}=-2\epsilon_{\mathrm{SR}}-\eta_{\mathrm{SR}},$ (23)

where $\eta_{\mathrm{SR}}$ is $\eta$ in SR period. This large-scale limit must
be consistent with CMB observation.

On small scale with larger wavenumber, $k_{s}\lesssim k\lesssim k_{e}$, the
power spectrum is oscillating. Compared to the large-scale power spectrum, it
is enhanced to

$\Delta_{s(\mathrm{PBH})}^{2}\approx\Delta_{s(\mathrm{SR})}^{2}(k_{s})\left(\frac{k_{e}}{k_{s}}\right)^{6},$
(24)

whose high density peak may collapse into PBHs. Plot of the typical power
spectrum is shown in Fig. 2.

Figure 2:  Power spectrum of the curvature perturbation. At CMB scale, $k\ll
k_{s}$, the power spectrum is almost scale invariant.
$p_{*}=0.05~{}\mathrm{Mpc}^{-1}$ is the pivot scale with amplitude
$\Delta_{s(\mathrm{SR})}^{2}(p_{*})=2.1\times 10^{-9}$, based on observational
result [31]. At small scale, between $k_{s}$ and $k_{e}$, the power spectrum
is amplified to typically $\Delta_{s(\mathrm{PBH})}^{2}\sim\mathcal{O}(0.01)$
to form appreciable amount of PBHs.

So far, we have explained the tree-level contribution of the power spectrum.
If we expand the action (1) until higher-order in curvature perturbation, we
can calculate loop corrections to the power spectrum. Expanding (1) to third-
order of $\zeta$ yields the interaction action [111]

$\displaystyle
S_{\mathrm{int}}=M_{\mathrm{pl}}^{2}\int\mathrm{d}t~{}\mathrm{d}^{3}x~{}a^{3}\left[\epsilon^{2}\dot{\zeta}^{2}\zeta+\frac{1}{a^{2}}\epsilon^{2}(\partial_{i}\zeta)^{2}\zeta\right.$
$\displaystyle\left.-2\epsilon\dot{\zeta}\partial_{i}\zeta\partial_{i}\chi-\frac{1}{2}\epsilon^{3}\dot{\zeta}^{2}\zeta+\frac{1}{2}\epsilon\zeta(\partial_{i}\partial_{j}\chi)^{2}+\frac{1}{2}\epsilon\dot{\eta}\dot{\zeta}\zeta^{2}\right],$
(25)

where $\partial^{2}\chi=\epsilon\dot{\zeta}$. In standard SR inflation, the
first three terms and the last three terms in (25) have coupling
$\mathcal{O}(\epsilon^{2})$ and $\mathcal{O}(\epsilon^{3})$, respectively. The
same situation happens in the context of inflation with PBH formation
scenario, except the last term in (25), which has a coupling
$\mathcal{O}(\epsilon)$ because $\eta$ can have $\mathcal{O}(1)$ transition
[138, 139, 140, 141, 142], approximately from $0$ to $-6$.

We now calculate one-loop correction generated by cubic self-interaction (25)
in the context of PBH formation using the standard in-in perturbation theory

$\displaystyle\langle\mathcal{O(\tau)}\rangle=\left\langle\left[\bar{\mathrm{T}}\exp\left(i\int_{-\infty}^{\tau}\mathrm{d}\tau^{\prime}H_{\mathrm{int}}(\tau^{\prime})\right)\right]\mathcal{\hat{O}}(\tau)\right.$
$\displaystyle\left.\left[\mathrm{T}\exp\left(-i\int_{-\infty}^{\tau}\mathrm{d\tau^{\prime}}H_{\mathrm{int}}(\tau^{\prime})\right)\right]\right\rangle,$
(26)

where $\mathcal{\hat{O}}(\tau)$ is an operator at a fixed time $\tau$, and
$\mathrm{T}$ and $\bar{\mathrm{T}}$ denote time and antitime ordering. Also,
$H_{\mathrm{int}}=-\int\mathrm{d}^{3}x~{}\mathcal{L}_{\mathrm{int}}$ is
Hamiltonian corresponding to the Lagrangian $\mathcal{L}_{\mathrm{int}}$,
defined by the integrand of (25). In our case, the operator is
$\zeta(\mathbf{p})\zeta(-\mathbf{p})$, where $\mathbf{p}$ is CMB scale
wavevector, 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

$\displaystyle\langle\mathcal{O(\tau)}\rangle=\langle\mathcal{O(\tau)}\rangle_{(0,2)}^{\dagger}+\langle\mathcal{O(\tau)}\rangle_{(1,1)}+\langle\mathcal{O(\tau)}\rangle_{(0,2)},$
(27)
$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(1,1)}=\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\int_{-\infty}^{\tau}\mathrm{d}\tau_{2}\langle
H_{\mathrm{int}}(\tau_{1})\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{2})\rangle,$
$\displaystyle\langle\mathcal{O(\tau)}\rangle_{(0,2)}=-\int_{-\infty}^{\tau}\mathrm{d}\tau_{1}\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}\langle\mathcal{\hat{O}}(\tau)H_{\mathrm{int}}(\tau_{1})H_{\mathrm{int}}(\tau_{2})\rangle.$

The leading cubic self-interaction is the last term in (25) with interaction
Hamiltonian 111Strictly speaking, this interaction Hamiltonian is a function
of redefined field $\bm{\zeta}$, which is first introduced by Maldacena [111].
The relation between $\zeta$ and $\bm{\zeta}$ is
$\zeta=\bm{\zeta}+\frac{1}{4}\eta\bm{\zeta}^{2}+\frac{1}{H}\dot{\bm{\zeta}}\bm{\zeta}+\dots$,
where dots represent SR or superhorizon suppressed term. It is shown that
field redefinition method is equivalent to considering boundary interaction
[149]. Moreover, quartic self-interaction with first-order perturbation theory
might generate the same order of one-loop correction to cubic self-interaction
with second-order perturbation theory. Based on [150], such contribution from
quartic self-interaction generates much smaller one-loop correction than those
induced by the cubic self-interaction.

$H_{\mathrm{int}}(\tau)=-\frac{1}{2}M_{\mathrm{pl}}^{2}\int\mathrm{d}^{3}x~{}\epsilon\eta^{\prime}a^{2}\zeta^{\prime}\zeta^{2}.$
(28)

After substituting the interaction Hamiltonian to the perturbation theory, we
find

$\displaystyle\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\rangle_{(1,1)}=$
$\displaystyle~{}\frac{1}{4}M_{\mathrm{pl}}^{4}\int_{-\infty}^{0}\mathrm{d}\tau_{1}~{}a^{2}(\tau_{1})\epsilon(\tau_{1})\eta^{\prime}(\tau_{1})\int_{-\infty}^{0}\mathrm{d}\tau_{2}~{}a^{2}(\tau_{2})\epsilon(\tau_{2})\eta^{\prime}(\tau_{2})\int\prod_{a=1}^{6}\left[\frac{\mathrm{d}^{3}k_{a}}{(2\pi)^{3}}\right]\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3})~{}$
$\displaystyle\times\delta^{3}(\mathbf{k}_{4}+\mathbf{k}_{5}+\mathbf{k}_{6})\left\langle\zeta^{\prime}(\mathbf{k}_{1},\tau_{1})\zeta(\mathbf{k}_{2},\tau_{1})\zeta(\mathbf{k}_{3},\tau_{1})\zeta(\mathbf{p})\zeta(-\mathbf{p})\zeta^{\prime}(\mathbf{k}_{4},\tau_{2})\zeta(\mathbf{k}_{5},\tau_{2})\zeta(\mathbf{k}_{6},\tau_{2})\right\rangle,$
(29) $\displaystyle\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\rangle_{(0,2)}=$
$\displaystyle~{}-\frac{1}{4}M_{\mathrm{pl}}^{4}\int_{-\infty}^{0}\mathrm{d}\tau_{1}~{}a^{2}(\tau_{1})\epsilon(\tau_{1})\eta^{\prime}(\tau_{1})\int_{-\infty}^{\tau_{1}}\mathrm{d}\tau_{2}~{}a^{2}(\tau_{2})\epsilon(\tau_{2})\eta^{\prime}(\tau_{2})\int\prod_{a=1}^{6}\left[\frac{\mathrm{d}^{3}k_{a}}{(2\pi)^{3}}\right]\delta^{3}(\mathbf{k}_{1}+\mathbf{k}_{2}+\mathbf{k}_{3})~{}$
$\displaystyle\times\delta^{3}(\mathbf{k}_{4}+\mathbf{k}_{5}+\mathbf{k}_{6})\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\zeta^{\prime}(\mathbf{k}_{1},\tau_{1})\zeta(\mathbf{k}_{2},\tau_{1})\zeta(\mathbf{k}_{3},\tau_{1})\zeta^{\prime}(\mathbf{k}_{4},\tau_{2})\zeta(\mathbf{k}_{5},\tau_{2})\zeta(\mathbf{k}_{6},\tau_{2})\right\rangle.$
(30)

To evaluate the time integral, we note that $\eta$ is almost constant in both
SR and USR periods, so $\eta^{\prime}(\tau)\approx 0$ except for sharp
transitions around $\tau=\tau_{s}$ and $\tau=\tau_{e}$. Therefore, the time
integral can be evaluated as

$\int_{-\infty}^{0}\mathrm{d}\tau~{}\eta^{\prime}(\tau)f(\tau)=\Delta\eta~{}f(\tau_{e}),$
(31)

where $f(\tau)$ is a general continuous function. We neglect contribution from
$\tau=\tau_{s}$ because it is much smaller than that from $\tau=\tau_{e}$.
Nevertheless, contribution from $\tau=\tau_{s}$ is essential to recover
Maldacena’s theorem for the bispectrum, as we show in our companion paper
[144]. Performing operator expansion and Wick contraction, we obtain total
one-loop correction

$\displaystyle\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(1)}=\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(1,1)}+2\mathrm{Re}\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(0,2)}=\frac{1}{4}M_{\mathrm{pl}}^{4}\epsilon^{2}(\tau_{e})a^{4}(\tau_{e})(\Delta\eta)^{2}$
(32)
$\displaystyle\times\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\left\\{\zeta_{p}(\tau_{0})\zeta_{p}^{*}(\tau_{0})\left(4\zeta_{p}^{\prime}\zeta_{p}^{\prime*}\zeta_{q}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime*}\zeta_{p}\zeta_{q}^{\prime}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime}\zeta_{p}^{*}\zeta_{q}^{\prime*}\zeta_{q}\zeta_{k}\zeta_{k}^{*}+16\zeta_{p}\zeta_{p}^{*}\zeta_{q}^{\prime}\zeta_{q}^{\prime*}\zeta_{k}\zeta_{k}^{*}\right)_{\tau=\tau_{e}}\right.$
$\displaystyle\left.-\mathrm{Re}\left[\zeta_{p}(\tau_{0})\zeta_{p}(\tau_{0})\left(4\zeta_{p}^{\prime*}\zeta_{p}^{\prime*}\zeta_{q}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime*}\zeta_{p}^{*}\zeta_{q}^{\prime}\zeta_{q}^{*}\zeta_{k}\zeta_{k}^{*}+8\zeta_{p}^{\prime*}\zeta_{p}^{*}\zeta_{q}^{\prime*}\zeta_{q}\zeta_{k}\zeta_{k}^{*}+16\zeta_{p}^{*}\zeta_{p}^{*}\zeta_{q}^{\prime}\zeta_{q}^{\prime*}\zeta_{k}\zeta_{k}^{*}\right)_{\tau=\tau_{e}}\right]\right\\},$

where $\mathbf{q}=\mathbf{k}-\mathbf{p}$. After some algebra, for $p\ll k$,
the leading term is simplified to

$\displaystyle\left\langle\\!\left\langle\zeta(\mathbf{p})\zeta(-\mathbf{p})\right\rangle\\!\right\rangle_{(1)}=\frac{1}{4}M_{\mathrm{pl}}^{4}\epsilon^{2}(\tau_{e})a^{4}(\tau_{e})(\Delta\eta)^{2}\absolutevalue{\zeta_{p}(\tau_{0})}^{2}$
$\displaystyle\times
16\int\frac{\mathrm{d}^{3}k}{(2\pi)^{3}}\left[\absolutevalue{\zeta_{k}}^{2}~{}\mathrm{Im}(\zeta_{p}\zeta_{p}^{\prime*})~{}\mathrm{Im}(\zeta_{q}\zeta_{q}^{\prime*})\right]_{\tau=\tau_{e}}.$
(33)

Expressing (14) in terms of curvature perturbation during USR period, we
obtain

$[\mathrm{Im}(\zeta_{q}\zeta_{q}^{\prime*})]_{\tau=\tau_{e}}=\frac{1}{4M_{\mathrm{pl}}^{2}\epsilon(\tau_{e})a^{2}(\tau_{e})},$
(34)

which is independent of wavenumber $q$. It leads to commutation relation
$[\zeta,\dot{\zeta}]\propto a^{3}$, which is in line with large quantum loop
correction that we will find shortly. Then, substituting it to (33) yields
one-loop correction to the power spectrum

$\Delta_{s(1)}^{2}(p)=\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(0)}^{2}(p)\int_{k_{\mathrm{IR}}}^{k_{\mathrm{UV}}}\frac{\mathrm{d}k}{k}\Delta^{2}_{s(0)}(k).$
(35)

In principle, the wavenumber integral should be extended from well below CMB
pivot scale $k_{\mathrm{IR}}\ll p_{*}=0.05~{}\mathrm{Mpc}^{-1}$ to ultraviolet
(UV) cutoff scale $k_{\mathrm{UV}}\rightarrow\infty$. Contributions from large
length scale including the CMB scale can be neglected because they are much
smaller than those from amplified perturbation on small scale. On the other
hand, contribution from UV scale diverges. Because we are interested in the
finite effect of the amplified perturbation on specific small scale due to the
USR period to one-loop correction, we conservatively restrict the wavenumber
integration domain from $k_{\mathrm{IR}}=k_{s}$ to $k_{\mathrm{UV}}=k_{e}$.
The UV cutoff issue will be discussed later. For
$k_{\mathrm{UV}}/k_{\mathrm{IR}}=k_{e}/k_{s}\gg 1$, (35) reads

$\Delta_{s(1)}^{2}(p)=\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(0)}^{2}(p)\Delta_{s(\mathrm{PBH})}^{2}\left(1.1+\log\frac{k_{e}}{k_{s}}\right).$
(36)

In order for the standard cosmological perturbation theory to be trustable,
one-loop correction must be much smaller than the tree-level contribution,
namely $\Delta_{s(1)}^{2}\ll\Delta_{s(0)}^{2}$. It leads to a strong
inequality

$\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\left(1.1+\log\frac{k_{e}}{k_{s}}\right)\ll
1.$ (37)

We can obtain a bound on $k_{e}/k_{s}$ by substituting numerical values
$\Delta_{s(\mathrm{SR})}^{2}(p_{*})=2.1\times 10^{-9}$ and $n_{s}=0.9649\pm
0.0042$ at pivot scale based on observational result [31]. Also,
$\Delta\eta=6$ as transition from USR to SR period. Solving inequality (37)
numerically leads to an upper bound $k_{e}/k_{s}\ll 15$, which is equivalent
to

$\Delta_{s(\mathrm{PBH})}^{2}\ll\frac{1}{(\Delta\eta)^{2}}\simeq 0.03.$ (38)

So far, we have considered finite contribution of superhorizon perturbation at
$\tau=\tau_{e}$ to the one-loop correction. In addition, there is contribution
from perturbations which are still inside the horizon at $\tau=\tau_{e}$. To
identify such contributions, we introduce physical UV cutoff $\Lambda$ to the
upper bound of wavenumber integral such as 222This can be derived by
introducing physical wavenumber cutoff $\Lambda a(\tau)$ to the upper bound of
wavenumber integral in (29) and (30). After performing time integral (31), the
cutoff is evaluated at $\tau=\tau_{e}$, which leads to (39)

$\int_{k_{\mathrm{IR}}}^{k_{\mathrm{UV}}}\frac{\mathrm{d}k}{k}\Delta^{2}_{s(0)}(k)=\left(\int_{k_{s}}^{k_{e}}+\int_{k_{e}}^{\Lambda
a(\tau_{e})}\right)\frac{\mathrm{d}k}{k}\Delta^{2}_{s(0)}(k).$ (39)

Including the divergence, the one-loop correction is

$\displaystyle\Delta_{s(1)}^{2}(p)=\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(0)}^{2}(p)\Delta_{s(\mathrm{PBH})}^{2}$
$\displaystyle\times\left(1.1+\log\frac{k_{e}}{k_{s}}+\log\frac{\Lambda}{H}+\frac{\Lambda^{2}-H^{2}}{2H^{2}}\right).$
(40)

The bare power spectrum at one-loop order is given by

$\displaystyle\Delta_{s}^{2}(p)=\Delta_{s(0)}^{2}(p_{*})\left(\frac{p}{p_{*}}\right)^{n_{s}-1}\left\\{1+\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\right.$
$\displaystyle\left.\times\left(1.1+\log\frac{k_{e}}{k_{s}}+\log\tilde{\Lambda}+\frac{\tilde{\Lambda}^{2}-1}{2}\right)\right.$
$\displaystyle\left.+\mathcal{O}\left[\left((\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\right)^{2}\right]\right\\},$
(41)

where $\tilde{\Lambda}=\Lambda/H$ is defined. Such power spectrum diverges as
we take $\Lambda\rightarrow\infty$ limit 333Dimensional regularization leads
to a similar expression, except $\log\tilde{\Lambda}$ is changed to
$1/\delta$, where $\delta$ is a small correction to the spatial dimension.
Another difference is dimensional regularization cannot capture the quadratic
divergence $\tilde{\Lambda}^{2}$, as expected. Compared to one-loop
computation for case considered in [118, 119], the time integral in our case
is dominated by $\tau=\tau_{e}$, not CMB horizon crossing time $\tau=-1/p$..
To make it finite, we need to renormalize order by order the coefficient
$\Delta_{s(0)}^{2}(p_{*})$ so it depends on $\tilde{\Lambda}$. This procedure
is equivalent to introducing a counterterm $\delta Z$, defined in
$\Delta_{s(0)}^{2}(p_{*})\equiv(1+\delta
Z)\Delta_{s(0)}^{2}(p_{*},\tilde{\mu})$. At one-loop order, after substituting
(24), $\delta Z$ can be read off from the renormalized coefficient

$\displaystyle\Delta_{s(0)}^{2}(p_{*})\equiv\Delta_{s(0)}^{2}(p_{*},\tilde{\mu})\left\\{1+\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(0)}^{2}(p_{*},\tilde{\mu})\right.$
$\displaystyle\left.\times\left(\frac{k_{e}}{k_{s}}\right)^{6}\left(-1.1-\log\frac{k_{e}}{k_{s}}+\log\frac{\tilde{\mu}}{\tilde{\Lambda}}+\frac{\tilde{\mu}^{2}-\tilde{\Lambda}^{2}}{2}\right)\right.$
$\displaystyle\left.+\mathcal{O}\left[\left((\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\right)^{2}\right]\right\\},$
(42)

where $\tilde{\mu}=\mu/H$ with $\mu$ being a renormalization scale. Therefore,
we obtained the renormalized power spectrum

$\displaystyle\Delta_{s}^{2}(p)=\Delta_{s(0)}^{2}(p_{*},\tilde{\mu})\left(\frac{p}{p_{*}}\right)^{n_{s}-1}$
$\displaystyle\times\left\\{1+\frac{1}{4}(\Delta\eta)^{2}\Delta_{s(0)}^{2}(p_{*},\tilde{\mu})\left(\frac{k_{e}}{k_{s}}\right)^{6}\left(\log\tilde{\mu}+\frac{\tilde{\mu}^{2}-1}{2}\right)\right.$
$\displaystyle\left.+\mathcal{O}\left[\left((\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\right)^{2}\right]\right\\}.$
(43)

Taking renormalization scale $\mu=H$, the renormalized power spectrum reads

$\displaystyle\Delta_{s}^{2}(p)=$
$\displaystyle~{}\Delta_{s(0)}^{2}(p_{*},\mu=H)\left(\frac{p}{p_{*}}\right)^{n_{s}-1}$
$\displaystyle\times\left\\{1+\mathcal{O}\left[\left((\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\right)^{2}\right]\right\\},$
(44)

where $\Delta_{s(0)}^{2}(p_{*},\mu=H)$ is fixed to the observed power spectrum
$2.1\times 10^{-9}$.

Such order by order renormalization is justified if higher-order loop
correction is suppressed compared to the lower-order one. In this context,
this is possible only if $(\Delta\eta)^{2}\Delta_{s(\mathrm{PBH})}^{2}\ll 1$,
which is the same condition as (38). Therefore, condition (38) is a necessary
condition for the theory to be perturbative, or requirement to avoid strong
coupling. Indeed, this perturbative coupling requirement is needed to justify
the use of in-in perturbation theory (27). Although we focus on loop
correction to large-scale power spectrum, such coupling also affects small-
scale power spectrum and the same constraint is needed to avoid strong
coupling.

In single-field inflation, PBH formation models can be classified into two
categories [131]. The first category is models with features in the
inflationary potential or non-minimally coupled inflaton with potential
defined in the Einstein frame. Models with an extremely flat feature fall in
this category including those with an inflection point in the potential [7,
42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 55, 56, 57, 58, 59, 60, 61, 62, 52,
53, 54, 63, 64]. Other examples of feature are a tiny bump or dip [65, 66, 67,
68, 69, 70], an upward or downward step [71, 72, 74, 75, 73, 76], polynomial
shape [77, 78, 79, 80] and Coleman-Weinberg potential [41, 81, 82]. In these
examples, modification of potential makes a sharp $\eta$ transition on the
inflaton dynamics.

The second category is models with modified gravity or beyond non-minimally
coupled inflaton. For example, models based on $k$ [147] or $G$ [148]
inflation [83, 84, 85], the effective field theory of inflation [86, 87, 88],
$f(R)$ gravity [89], a non-minimal derivative coupling [90, 91], Gauss-Bonnet
inflation [92], and bumpy axion inflation [93]. In these examples,
amplification of small-scale perturbation can be realized by a sharp
transition of $\eta$ and/or other parameters. In [86, 87], amplification of
small-scale perturbation is caused by a sharp transition of the sound speed,
$c_{s}$, a quantity that parametrizes deviation from canonical kinetic term.
In this case, coupling $\eta^{\prime}$ in cubic self-interaction (28) is
modified to $(\eta/c_{s}^{2})^{\prime}$ [112, 113], so such theory might also
have large one-loop correction. Therefore, constraint
$\Delta_{s(\mathrm{PBH})}^{2}\ll\mathcal{O}(0.01)$ can be imposed to almost
every PBH formation model in single-field inflation that has been studied,
although for a few number of models we have to examine it more carefully.

In conclusion, we have calculated the one-loop correction of the inflationary
power spectrum in single-field inflation realizing PBH formation. We have
shown that models realizing appreciable amount of PBH formation with the
enhanced small-scale spectrum by USR inflaton dynamics inevitably induces
nonperturbative coupling to the power spectrum on CMB scale. We therefore
conclude that PBH formation from cosmological perturbation theory single-field
inflation with an USR dynamics is ruled out.

J. K. thanks Hayato Motohashi for a fruitful discussion. J. K. acknowledges
the support from JSPS KAKENHI Grant No. 22J20289 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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