# Large Field Polynomial Inflation:
Parameter Space, Predictions and (Double) Eternal Nature

Manuel Drees  and Yong Xu

###### Abstract

Simple monomial inflationary scenarios have been ruled out by recent
observations. In this work we revisit the next simplest scenario, a
single–field model where the scalar potential is a polynomial of degree four
which features a concave “almost” saddle point. We focus on trans–Planckian
field values. We reparametrize the potential, which greatly simplifies the
procedure for finding acceptbale model parameters. This allows for the first
comprehensive scan of parameter space consistent with recent Planck and
BICEP/Keck 2018 measurements. Even for trans–Planckian field values the
tensor–to–scalar ratio $r$ can be as small as $\mathcal{O}(10^{-8})$, but the
model can also saturate the current upper bound. In contrast to the
small–field version of this model, radiative stability does not lead to strong
constraints on the parameters of the inflaton potential. For very large field
values the potential can be approximated by the quartic term; as well known,
this allows eternal inflation even for field energy well below the reduced
Planck mass $M_{\rm Pl}$, with Hubble parameter $H\sim 10^{-2}M_{\rm Pl}$.
More interestingly, we find a region of parameter space that even supports two
phases of eternal inflation. The second epoch only occurs if the slope at the
would–be saddle point is very small, and has $H\sim 10^{-5}M_{\rm Pl}$; it can
only be realized if $r\sim 10^{-2}$, within the sensitivity range of
next–generation CMB observations.

December 2022

##  1 Introduction and Motivation

Inflation, as invented in the 1980s [1, 2, 3, 4], is an elegant paradigm of
early universe physics. It not only solves the horizon, flatness and monopole
problems of standard cosmology, but also generates initial seeds for structure
formation arising from vacuum fluctuations [5]. In the simplest, “slow–roll”
implementation of this idea, a spin$-0$ inflaton field is introduced which
slowly rolls down a sufficiently flat potential; see Ref. [6] for a
comprehensive review of models of inflation.

The simplest model assumes a monomial $\phi^{p}$ potential; in renormalizable
models whose potential is bounded from below $p$ is either $2$ or $4$.
However, such potentials are sufficiently flat, i.e. the first and second
derivatives of the potential are sufficiently small compared to the potential
itself, only at large field values. This leads to the overproduction of tensor
modes, i.e. the tensor–to–scalar ratio $r$ is predicted beyond the upper bound
established by recent measurements of Cosmic Microwave Background (CMB)
anisotropies [7, 8].111One may also consider monomials with fractional power,
e.g. the monodromy inflationary model where $V(\phi)\propto\phi^{2/3}$ [9,
10]. The BICEP/Keck 2018 results [8] then require $p<0.53$ at $95\%$ c.l. if
CMB scales experienced no more than 60 e–folds of inflation, in strong tension
with monodromy inflation.

In this paper, we still assume that the inflaton is a real scalar field
$\phi$, but we allow a general, renormalizable polynomial potential. This next
to simplest scenario has been analyzed many times since 1990 [11, 12, 13, 14,
15, 16, 17, 18, 19, 20]. Such polynomial inflation can also be realized in
string theory [21]. All these analyses have been performed before the release
of the 2018 Planck and BICEP/Keck results [7, 8]. Unlike previous
investigations, we aim to work out the full parameter space that agrees with
the latest measurements, and derive the allowed range of $r$. This is timely
since future precise observations, for example by CORE [22], AliCPT [23],
LiteBIRD [24], and CMB-S4 [25], should greatly extend the sensitivity, down to
$r\sim\mathcal{O}(10^{-3})$.

As noted above, the potential should be flatter than a monomial at not too
large field values. Here we achieve this by canceling several contributions,
with different powers of the field, around a (near) saddle point, where both
the first and second derivative of the potential become small. This is similar
to the inflection point inflationary scenario [26, 27, 28, 29, 30, 31, 32];
however, we employ a purely renormalizable potential, i.e. only allow terms up
to $\phi^{4}$. Just below the would–be saddle point the potential has a
concave shape, as favored by the Planck 2018 data [7]. We rewrite the
potential in terms of the location $\phi_{0}$ of the would–be saddle point, a
quantity $\beta$ which governs the slope of the potential at $\phi_{0}$, and a
multiplicative factor which only affects the overall normalization of the CMB
anisotropies. Based on this reparametrization, we work out, for the first
time, the full parameter space with predictions (power spectrum, spectral
index and its running) consistent with Planck and BICEP/Keck 2018 measurements
[7]. We find that the current upper bound on $r$ can be saturated, which means
that part of the parameter space should be testable in the near future.

Another aim of this work is to investigate at which scale(s) eternal
inflation222See e.g. Ref. [33] for a review. might have occurred. In slow–roll
inflation the classical change of the inflaton field during one Hubble time
dominates over its quantum fluctuation; the inflaton field thus moves
essentially deterministically downhill towards its minimum, until the end of
inflation. However, in the opposite situation, where the quantum fluctuations
dominate over the classical evolution, the inflaton field can move uphill
rather than downhill. The Hubble patches where this happens inflate longer; in
fact, in this case some such patches will inflate forever, i.e. inflation
becomes eternal, although in our patch inflation obviously must have ended.
The possibility that inflation can be eternal was first discussed in Ref.
[34]. Later it was shown [35] that eternal inflation is in fact inevitable if
the potential is a monomial with positive power, assuming only that the
initial field value is sufficiently large; this occurs at energy scales well
below the reduced Planck mass, $M_{\rm Pl}\simeq 2.4\cdot 10^{18}$ GeV. One
thus may not need to worry about quantum gravity effects when describing
eternal inflation [36]. “Hilltop” models can also lead to eternal inflation
[37, 38].

During the eternal expansion, infinitely many independent “mini–universes” (or
“pocket universes”) with different de Sitter vacua are generated via a self-
reproducing process [39]. It has been speculated that this process can
“populate” (or probe) the landscape of string theory [40, 41, 42]. This
mechanism also naturally provides a scientific justification for the (weak)
anthropic principle. From the perspective of eternal inflation, nearly
everything is possible, provided only that the overall energy density is
dominated by the potential energy of the inflaton field. For example,
independent mini–universes may feature different types of compactification
leading to different fundamental physical laws and/or different values of
physical “constants” (which are field–dependent in superstring theory). Some
of these laws and constants support life of our type, and clearly we (as
living beings) can only observe those mini–universes where this is indeed the
case [39, 43]. Finally, eternal inflation may help to relax the initial
conditions problem. By this we mean the probability that some initial
configuration of the inflaton field, and of the other dynamical degrees of
freedom, gives rise to sufficiently long exponential expansion of the
universe.333For a review see e.g. Refs. [44, 45]. As argued in [46], any
initial configuration that leads to eternal inflation will produce an infinite
spacetime volume, making the probability of this initial configuration less
significant.

Since for large field values our potential is dominated by the $\phi^{4}$
term, it is not surprising that it leads to eternal inflation at sufficiently
large values of $\phi$; the minimal required inflaton field energy turns out
to be slightly lower than in pure $\phi^{4}$ inflation. More intriguingly, for
sufficiently small values of $\beta$, i.e. a sufficiently flat potential near
the would–be saddle point, a second period of eternal inflation can occur, at
a value of the Hubble parameter smaller by more than three orders of magnitude
than that during the first epoch of eternal inflation. However, this can only
be realized if
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>15M_{\rm
Pl}$, with
$r\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>0.01$.
These scenarios can therefore be probed in the near future.

The remainder of this paper is organized as follows. In sec. 2 we describe the
general setup, with emphasis on the reparametrization of the inflaton
potential. In sec. 3 methods to scan the allowed parameter space are shown,
and the corresponding predictions are given. The radiative stability of the
potential is checked and the maximal reheating temperature is determined in
sec. 4. In sec. 5 we investigate the possibility of realizing eternal
inflation in our scenario, with focus on the calculation of the corresponding
energy scale(s). Finally, sec. 6 summarizes this work. In this paper, we use
Planckian units, i.e. we set the reduced Planck mass $M_{\rm
Pl}=\sqrt{\frac{1}{8\pi G}}\simeq 2.4\cdot 10^{18}$ GeV to unity.

##  2 The Setup

In this section we first describe the inflaton potential and the resulting
expressions for the parameters of inflation; in the second subsection we
present simplified analytical results in some limits.

###  2.1 General Analysis

The action for the inflaton field in the Einstein frame is given by:

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}g_{\mu\nu}\partial^{\mu}\phi\partial^{\nu}\phi-V(\phi)\right]\,,$
(2.1)

where $g$ is the determinant of the metric; we assume it to be of the
Friedmann–Robertson–Walker (FRW) type, i.e. $g_{\mu\nu}={\rm
diag}(+1,-a^{2},-a^{2},-a^{2})$ with $a$ denoting the scale factor. The
corresponding Euler–Lagrange equation of motion for the classical background
field is

$\displaystyle\ddot{\phi}+3H\dot{\phi}-\frac{1}{a^{2}}\nabla^{2}\phi+V^{\prime}(\phi)=0\,.$
(2.2)

Here $\vec{\nabla}$ denotes derivatives with respect to comoving spatial
coordinates, and $H\equiv\frac{\dot{a}}{a}$ is the Hubble parameter, which is
determined by the Friedmann equation:

$\displaystyle
H^{2}=\frac{1}{3}\rho(\phi)=\frac{1}{3}\left[\frac{1}{2}(\dot{\phi})^{2}+V(\phi)+\frac{1}{2a^{2}}(\nabla\phi)^{2}\right]\,.$
(2.3)

We make the usual assumption that the classical background $\phi$ is
homogeneous, i.e. $\phi\equiv\phi(t)$ is a function of the cosmic time $t$
only, so that all gradient terms for the background field vanish. The
potential we are considering is the general renormalizable one444A linear term
can be removed through a shift of $\phi$. We neglect the tiny cosmological
constant term, which would be generated by a constant term in the potential.:

$\displaystyle V(\phi)=b\,\phi^{2}+c\,\phi^{3}+d\,\phi^{4}\,.$ (2.4)

We need $d>0$ for the potential to be bound from below, and we consider $b>0$
so that the minimum of the potential is at $\phi=\phi_{\rm min}=0$, with
$V(\phi_{\rm min})=0$. Since the potential is symmetric under the
transformation $c\to-c$, $\phi\to-\phi$, we set $c\leq 0$ without loss of
generality, so that inflation occurs at positive field values.555The simpler
case with $c=0$, so that the potential has only two terms, has been
investigated in Ref. [47]; our analysis shows that this scenario is no longer
viable. In Ref. [48], the two-term scenario with radiative corrections is
investigated. The first and second derivatives of the potential are given by:

$\displaystyle V^{\prime}(\phi)=2b\,\phi+3c\,\phi^{2}+4d\,\phi^{3}\,;\ \ \
V^{\prime\prime}(\phi)=2b+6c\,\phi+12d\,\phi^{2}\,.$ (2.5)

We need the potential to be very flat over some range of field values. Suppose
first that the potential features an exact saddle point at $\phi=\phi_{0}$,
i.e. $V^{\prime}(\phi_{0})=V^{\prime\prime}(\phi_{0})=0$, which requires

$\displaystyle\phi_{0}=-\frac{3c}{8d}\,;b=\frac{9c^{2}}{32d}\,,$ (2.6)

from which we learn that the ratio $c/d$ determines the position of the saddle
point. Allowing for a finite slope even at $\phi_{0}$, one can reparametrize
the potential as

$\displaystyle V(\phi)$
$\displaystyle=d\left[\phi^{4}+\frac{c}{d}\left(1-\beta\right)\phi^{3}+\frac{9}{32}\left(\frac{c}{d}\right)^{2}\phi^{2}\right]$
$\displaystyle=d\left[\phi^{4}+A\left(1-\beta\right)\phi^{3}+\frac{9}{32}A^{2}\phi^{2}\right]\,.$
(2.7)

Here $A\equiv\frac{c}{d}\equiv-\frac{8}{3}\phi_{0}$ determines the location of
the flat region of the potential. Note that the potential (2.1) still contains
three free parameters, $d,\,A$ and $\beta$, i.e. it is a genuine
reparametrization of the general ansatz (2.4). However, the form (2.1) is far
more convenient, since the overall multiplicative factor $d$ only affects the
overall normalization of the density perturbations, while $\beta$ directly
controls the slope near $\phi_{0}$. For $\beta<0$ the potential has a second
minimum at $\phi>\phi_{0}$ where the inflaton field may get stuck, in which
case there would be no hot Big Bang. We therefore require $\beta\geq 0$;
recall that for $\beta=0$ the potential has an exact saddle point at
$\phi_{0}$.

In this paper, we focus on the large field inflation scenario where
$\phi_{0}\geq 1$; a detailed analysis of the small field case can be found in
Ref. [49].

The traditional potential slow–roll (SR) parameters [50] are:

$\displaystyle\epsilon_{V}\equiv\frac{1}{2}\left(\frac{V^{\prime}}{V}\right)^{2}=\frac{2}{\phi^{2}}\left[\frac{9A^{2}-48A(\beta-1)\phi+64\phi^{2}}{9A^{2}-32A(\beta-1)\phi+32\phi^{2}}\right]^{2}\,;$
$\displaystyle\eta_{V}\equiv\frac{V^{\prime\prime}}{V}=\frac{6}{\phi^{2}}\left[\frac{3A^{2}-32A(\beta-1)\phi+64\phi^{2}}{9A^{2}-32A(\beta-1)\phi+32\phi^{2}}\right]\,;$
(2.8)
$\displaystyle\xi_{V}^{2}\equiv\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}}=-\frac{384\left[A(\beta-1)-4\phi\right]}{\phi^{3}}\frac{\left(9A^{2}-48A(\beta-1)\phi+64\phi^{2}\right)}{\left(9A^{2}-32A(\beta-1)\phi+32\phi^{2}\right)^{2}}\,.$

These quantities do not depend on $d$. During SR inflation, all these
parameters must be small, $\epsilon_{V}$, $|\eta_{V}|$ and $|\xi_{V}^{2}|\ll
1$.

Inflation ends at a field value $\phi_{\text{end}}$ where
$\epsilon_{V}(\phi_{\text{end}})=1$. Since for $\phi\ll\phi_{0}$ the term
$9A^{2}=64\phi_{0}^{2}$ dominates over other terms in eq. (2.1),
$\epsilon_{V}\approx 2/\phi^{2}\approx\eta_{V}$ and hence
$\phi_{\text{end}}\approx 1.41$ if $\phi_{0}\gg 1$. For smaller $\phi_{0}$,
$\phi_{\text{end}}$ is closer to $\phi_{0}$ so that the cubic and quartic
terms need to be included in its determination.

Another important quantity is the total number $N_{\text{CMB}}$ of e–folds of
inflation that occurred after the CMB pivot scale
$k_{\star}=0.05\rm{Mpc}^{-1}$ first crossed out the horizon; in our model it
can be computed analytically (within the SR approximation). The full result is
given in the Appendix; for $\beta\ll 1$ it reduces to

$\displaystyle N_{\text{CMB}}$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{1}{\sqrt{2\epsilon_{V}}}d\phi$
$\displaystyle\simeq\frac{1}{24}\left\\{3\phi^{2}-4\phi\phi_{0}+15\phi_{0}^{2}-\phi_{0}^{2}\sqrt{\frac{2}{\beta}}\arctan\left(\frac{\phi_{0}-\phi}{\sqrt{2\beta}\phi_{0}}\right)-\phi_{0}^{2}\ln\left[(\phi_{0}-\phi)^{2}\right]\right\\}\Bigg{|}^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\,.$
(2.9)

Here $\phi_{\text{CMB}}$ denotes value of the field when $k_{\star}$ crossed
out of the horizon. In order to solve the flatness and horizon problems,
$\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>50$
e–folds of inflation are needed.

During SR inflation with a quasi de Sitter spacetime, Gaussian curvature
perturbations are generated, with power [50]

$\mathcal{P}_{\zeta}=\frac{V}{24\pi^{2}\epsilon_{V}}\,.$ (2.10)

The spectral index $n_{s}$ and its running $\alpha$ are given by

$n_{s}=1-6\epsilon_{V}+2\eta_{V}\,;\ \ \
\alpha=16\epsilon_{V}\eta_{V}-24\epsilon_{V}^{2}-2\xi_{V}^{2}\,;$ (2.11)

measurements of these quantities can be used to constrain the model parameters
$\beta$ and $A$.

The final observable of interest is the tensor–to–scalar ratio $r$, which is
given by [50]

$r=16\epsilon_{V}=\frac{32}{\phi^{2}}\left[\frac{9A^{2}-48A(\beta-1)\phi+64\phi^{2}}{9A^{2}-32A(\beta-1)\phi+32\phi^{2}}\right]^{2}\,.$
(2.12)

The final Planck 2018 measurements at the pivot scale
$k_{\star}=0.05\rm{Mpc}^{-1}$, including their own measurements plus results
on baryonic acoustic oscillations (BAO), give in a 7 parameter cosmological
model (baseline $\Lambda$CDM plus running $n_{s}$) [7]:

$\mathcal{P}_{\zeta}=(2.1\pm 0.1)\cdot 10^{-9}\,;\ n_{s}=0.9659\pm 0.0040\,;\
\alpha=-0.0041\pm 0.0067\,.$ (2.13)

So far no evidence for a non–vanishing tensor–to–scalar ratio $r$ has been
found. The most recent upper bound, from BICEP/Keck 2018 results [8], is

$r_{0.05}<0.035\,$ (2.14)

at 95% C.L., after extrapolation to our pivot scale.666Note that in the
experimental literature (e.g. the $r-n_{s}$ plots of Planck [7] or BICEP/Keck
[8]) the bound on $r$ is usually quoted at scale $k=0.002\rm{Mpc}^{-1}$,
denoted by $r_{0.002}$. For our choice of the pivot scale
$k_{\star}=0.05\rm{Mpc}^{-1}$, one has $r_{0.05}\simeq
r_{0.002}\left(\frac{0.05}{0.002}\right)^{n_{T}}\simeq
r_{0.002}\left(\frac{0.05}{0.002}\right)^{-r_{0.002}/8}$, where the (small)
running of the tensor spectral index $n_{T}\simeq-r/8$ has been neglected.

The combined constraint on $r$ and $n_{s}$ (adapted from Ref. [8]) is shown in
Fig. 1. We also present predictions for three sets of free parameters of the
potential (2.1), chosen such that $\mathcal{O}(10^{-3})\lesssim
r\lesssim\mathcal{O}(10^{-2})$, which should be testable in the near future
[22, 24, 23, 25].

Figure 1: The blue shaded region is the currently allowed part of the
$r-n_{s}$ plane; it has been adapted from the recent BICEP/Keck 2018 results
[8]. The red lines show predictions for our model, defined in eq.(2.1), where
the parameters have been chosen such that $\mathcal{O}(10^{-3})\lesssim
r\lesssim\mathcal{O}(10^{-2})$. The small and big red dots correspond to
$N_{\text{CMB}}=50~{}\text{and}~{}60$, respectively.

###  2.2 Approximations

Eqs.(2.1) to (2.12) allow a fully analytical calculation of all SR parameters
in terms of the free parameters of the potential plus the value of
$\phi_{\text{CMB}}$, which is also a free parameter. However, these equations
are too complicated to be solved analytically for the free parameters, for
given $n_{S}$, $\alpha$ and $N_{\text{CMB}}$. In this section, we therefore
present simplified analytical expressions, which work well in some limits.

  1. 1.

$\phi\approx\phi_{0}$:

In this regime, analytical results for the inflationary predictions can be
obtained by rewriting the field as [49]

$\phi=\phi_{0}(1-\delta)\,.$ (2.15)

Decreasing $\phi$ corresponds to increasing $\delta$. Note that fluctuations
at scales probed by observations of the CMB must have been created at
$\phi<\phi_{0}$, where the inflaton potential is concave, i.e. $\eta_{V}<0$,
so that $n_{s}<1$ can be reproduced. Since both $\delta$ and $\beta$ are
rather small (as we will see,
$\delta\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>{\cal
O}(\sqrt{\beta})$ is needed, so that $\beta\ll\delta\ll 1$), it will be
sufficient to keep only terms linear $\beta$ and up to quadratic in $\delta$
in the analysis.

The SR parameters defined in (2.1) can then be approximated as [49]:

$\epsilon_{V}\simeq\frac{72\left(2\beta+\delta^{2}\right)^{2}}{\phi_{0}^{2}};\,\eta_{V}\simeq\frac{24\left(2\beta-\delta\right)}{\phi_{0}^{2}};\,\xi_{V}^{2}\simeq\frac{288(2\beta+\delta^{2})}{\phi_{0}^{4}}\,.$
(2.16)

Using the simplified result for $\epsilon_{V}$, the number of e–folds becomes
[49]

$\displaystyle N_{\rm CMB}$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{1}{\sqrt{2\epsilon_{V}}}d\phi$
$\displaystyle=-\frac{\phi_{0}^{2}}{12}\int^{\delta_{\rm CMB}}_{\delta_{\rm
end}}\frac{d\delta}{\left(2\beta+\delta^{2}\right)}$
$\displaystyle\simeq\frac{\phi_{0}^{2}}{12\sqrt{2\beta}}\left[\frac{\pi}{2}-\arctan\left(\frac{\delta_{\rm
CMB}}{\sqrt{2\beta}}\right)\right]\,.$ (2.17)

The normalization of the power spectrum, its spectral index and the running of
the spectral index defined in (2.11), simplify to [49]

$\mathcal{P}_{\zeta}\simeq\frac{d\phi_{0}^{6}}{5184\pi^{2}(\delta^{2}+2\beta)^{2}}\,;$
(2.18) $n_{s}\simeq 1-\frac{48\delta}{\phi_{0}^{2}}\,;$ (2.19)
$\alpha\simeq-\frac{576(2\beta+\delta^{2})}{\phi_{0}^{4}}\,.$ (2.20)

Finally, the tensor–to–scalar ratio $r$ defined in eq.(2.12) reduces to [49]

$r\simeq\frac{1152\left(2\beta+\delta^{2}\right)^{2}}{\phi_{0}^{2}}\,.$ (2.21)

Evidently eq.(2.19) immediately determines $\delta$, i.e. $\phi_{\text{CMB}}$.
$\beta$ can then be fixed using eq.(1): the argument of the $\arctan$ needs to
be ${\cal O}(1)$, giving $\beta\sim{\cal O}(\delta^{2})$, as claimed above.
Moreover, $|\eta_{V}|\gg\epsilon_{V}$ in this scenario. The SR conditions thus
cease to be satisfied at
$\phi_{\text{end}}\simeq\phi_{0}\left(1-\phi_{0}^{2}/24\right)$, where
$\eta_{V}$ crosses $-1$ [49]. However, exponential expansion continues until
$\epsilon_{V}\simeq 1$, which is satisfied for
$\phi\simeq\phi_{0}\left(1-\sqrt{\phi_{0}/(6\sqrt{2})}\right)$.

In ref.[49] it was shown that this approximation works very well for
$\phi_{0}\leq 1$. Here we find that it still works quite well even for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$.

  2. 2.

$\phi\gg\phi_{0}$:

Here the potential is dominated by the quartic term:

$V\simeq d\phi^{4}\,.$ (2.22)

The corresponding SR parameters are:

$\epsilon_{V}=\frac{8}{\phi^{2}}\,;\ \ \ \eta_{V}=\frac{12}{\phi^{2}}\,,$
(2.23)

while the spectral index is given by

$n_{s}=1-6\epsilon_{V}+2\eta_{V}=1-\frac{24}{\phi^{2}}\,.$ (2.24)

The Planck central value $n_{s}=0.9659$ (cf. eq.(2.13)) thus requires
$\phi_{\text{CMB}}\simeq 27$, which implies $r=\frac{128}{\phi^{2}}\simeq
0.18$. This is well above the current upper bound on $r$, see fig. 1.
Moreover, $\phi_{\text{CMB}}=27$ with a $\phi^{4}$ potential yields
$N_{\text{CMB}}\simeq 90$, which is also well above the allowed range. This
re–derives the by now quite well–known result that simple $\phi^{4}$ inflation
is excluded; as we noted in the Introduction, this holds for any monomial
inflaton potential, $V(\phi)\propto\phi^{p}$ with $p\geq 2$.

  3. 3.

$\phi\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>\phi_{0}$:

If $\phi$ is very close to $\phi_{0}$ the first approximation can again be
used, but now with $\delta<0$. Eq.(2.19) shows that then $n_{s}>1$ if
$\beta\ll 1$, in conflict with observation. Solutions with
$\phi_{\text{CMB}}>\phi_{0}$ require larger values of $\beta$ than those with
$\phi_{\text{CMB}}<\phi_{0}$ in order to keep $N_{\text{CMB}}$ within the
acceptable range. In fact, for $\beta>0.1$, $n_{s}<1$ even for
$\phi_{\text{CMB}}=\phi_{0}$, since $6\epsilon_{V}>2\eta_{V}$ then. However,
for $\phi_{\text{CMB}}=\phi_{0}$ and $\beta=0.1$ one has
$r=128/(9\phi_{0}^{2})$. $r<0.035$ then requires $\phi_{0}>20$, which in turn
yields777For such large values of $\beta$ eq.(2.1) is no longer valid, and the
full expression given in the Appendix should be used. $N_{\text{CMB}}>150$. In
order to obtain a value of $n_{s}$ close to its upper bound (2.13) at
$\phi_{\text{CMB}}=\phi_{0}$ one even needs
$\beta\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>0.2$;
$r<0.035$ then requires $\phi_{0}>28$ and $N_{\text{CMB}}>245$, about a factor
of $4$ above the desired range.

In the regime with $\phi>\phi_{0}$ but neither $\phi\sim\phi_{0}$ nor
$\phi\gg\phi_{0}$, the potential remains convex, i.e. $\eta_{V}>0$. As for the
case $\phi_{\text{CMB}}\simeq\phi_{0}$ one can still get $n_{s}<1$, since for
sufficiently large $\beta$ and/or $\phi_{\text{CMB}}$ $\epsilon_{V}$ becomes
larger than $\eta_{V}/3$. However, once again this leads to too large values
for $N_{\text{CMB}}$ or $r$ (or both). Therefore no viable solution with
$\phi_{\text{CMB}}\geq\phi_{0}$ exists.

  4. 4.

$\phi\ll\phi_{0}$:

If $\phi\ll\phi_{0}$, the quadratic term in the potential (2.1) dominates:

$\displaystyle V(\phi)\simeq b\,\phi^{2}\,.$ (2.25)

The corresponding SR parameters are

$\displaystyle\epsilon_{V}=\frac{2}{\phi^{2}}=\eta_{V}\,,$ (2.26)

giving a spectral index

$\displaystyle n_{s}=1-6\epsilon_{V}+2\eta_{V}=1-\frac{8}{\phi^{2}}\,.$ (2.27)

The central value (2.13) of $n_{s}$ measured by Planck 2018 is reproduced for
$\phi_{\text{CMB}}\simeq 15$, which in turn leads to $r\simeq 0.14$, well
above its upper bound. ($N_{\text{CMB}}\simeq 56$ comes out correctly in this
case.)

  5. 5.

$\phi<\phi_{0}$:

In the regime with $\phi<\phi_{0}$ but with $\phi_{\text{CMB}}$ neither close
to $\phi_{0}$ nor $\phi_{\text{CMB}}\ll\phi_{0}$, the potential can maintain a
small and negative curvature due to the negative contribution from the cubic
term slightly overcompensating the positive contributions from the quadratic
and quartic terms. In fact, for $\beta\ll 1$ the potential remains concave for
$\phi_{0}(1-2\beta)>\phi>\phi_{0}(1+2\beta)/3$. If
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>1$,
$|\eta_{V}|$ begins to exceed $1$, signaling the end of SR inflation, already
at a value of $\phi$ close to $\phi_{0}$, see eq.(2.16). However, for
$\phi_{0}>1$ SR inflation can extend to field values well below $\phi_{0}$. In
fact for very large $\phi_{0}$, the potential is effectively approaching a
quadratic one again, since the last
$\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>65$
e–folds of inflation happen at $\phi\ll\phi_{0}$; this leads back to the case
discussed in the previous paragraph, which is excluded by the upper bound on
$r$. This argument shows that there must be an upper bound on $\phi_{0}$ in
our model. In the remainder of this paper we will explore the parameter space
with $\phi_{0}>1$ and $\phi_{\text{CMB}}<\phi_{0}$ in detail.

##  3 Model Parameters and Predictions

In this section, we first describe our methods to search for acceptable model
parameters and then scan over the full parameter space that is consistent with
the latest CMB observations (2.13) and (2.14) at the $2\sigma$ level.

###  3.1 Method to Find Model Parameters and Examples

We have learned that the location of the plateau, i.e. $\phi_{0}$, is
determined by the parameter $A$ of the rewritten potential (2.1); we treat it
as a free parameter. The slope of the plateau is determined by $\beta$, i.e.
for given field value (not too far from $\phi_{0}$) the SR parameter
$\epsilon_{V}$ will become larger when $\beta$ is increased. Of course, the SR
parameters, as well as $N_{\text{CMB}}$ and $r$, also depend on
$\phi_{\text{CMB}}$. On the other hand, the overall coupling $d$ in eq.(2.1)
only affects the normalization of the power spectrum, see eq.(2.10).

In practice we first fix $\phi_{0}$. The parameters $\phi_{\text{CMB}}$ and
$\beta$ should then be chosen such that $n_{s}$ and $N_{\text{CMB}}$ have the
desired values. As argued in the previous subsection, viable solutions only
exist for $\phi_{\text{CMB}}<\phi_{0}$. Reducing $\phi_{\text{CMB}}$ for given
$\beta$ means that one is moving away from the flattest part of the potential
(apart from the region near the minimum, which cannot lead to inflation); this
increases $\epsilon_{V}$ and reduces $\eta_{V}$ (often making it more
negative). This means that $1-n_{s}$ and $r$ both increase when
$\phi_{\text{CMB}}$ is reduced, but $N_{\text{CMB}}$ becomes smaller. Reducing
$\beta$ for fixed $\phi_{\text{CMB}}$ has the opposite effect: the potential
becomes flatter, which increases $N_{\text{CMB}}$ but decreases $r$ and
usually also $1-n_{s}$.

For
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$
the approximation described by eqs.(2.15)–(2.21) still works fairly well. As
already noted in the corresponding discussion, in this case one can use the
spectral slope $n_{s}$ to determine $\phi_{\text{CMB}}$, and then chose
$\beta$ such that $N_{\text{CMB}}$ is reproduced. For this range of parameters
$r$ is still very small, well below the present bound.

This procedure yields $1-\phi_{\text{CMB}}/\phi_{0}\propto\phi_{0}^{2}$, so
the approximation $\phi_{0}-\phi_{\text{CMB}}\ll\phi_{0}$ begins to break down
for $\phi_{0}\geq 5$. The free parameters $\phi_{\text{CMB}}$ and $\beta$ then
need to be determined together. We find that this can still be done
iteratively. One starts with a guess for $\beta$, e.g. the small$-\phi_{0}$
value $\simeq 10^{-6}\phi_{0}^{4}$. For this value of $\beta$,
$\phi_{\text{CMB}}$ is selected such that the spectral index $n_{s}$ comes out
as desired. One then fixes $\phi_{\text{CMB}}$ and varies $\beta$ until
$N_{\text{CMB}}$ takes the desired value. With this new value of $\beta$, a
new value of $\phi_{\text{CMB}}$ can be computed using $n_{s}$, and so on.
This iteration usually converges fairly quickly. At the end, the overall
coupling strength $d$ is determined using eq.(2.10) with ${\cal
P}_{\zeta}=2.1\cdot 10^{-9}$.

Of course, one should also check that $r$ and the running of the spectral
index $\alpha$ have acceptable values. We find that $\alpha$ is always
negative, and lies within the currently allowed range given in (2.13). On the
other hand, for large $\phi_{0}$ the tensor–to–scalar ratio $r$ may come out
too large. Moreover, while for sufficiently small $\phi_{0}$ the desired
values of $n_{s}$ and $N_{\text{CMB}}$ can always be attained, this is not
necessarily true for larger values of $\phi_{0}$.

Figure 2: The values of $\phi_{\text{CMB}}$ (black, divided by $100$) and
$\beta$ (red) that lead to $n_{s}=0.9659$ and $N_{\text{CMB}}=60$, as function
of $\phi_{0}$. The blue curve shows the resulting prediction for the
tensor–to–scalar ratio $r$.

This procedure is illustrated in Fig. 2, for $\phi_{0}\geq 5$ where the
deviation from the small$-\phi_{0}$ solution begins to be sizable. Here we
have chosen $N_{\text{CMB}}=60$ and $n_{s}=0.9659$, the current central value.
We see that for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10$,
$\phi_{\text{CMB}}$ (shown in black) has to remain quite close to $\phi_{0}$.
For fixed ratio $\phi_{\text{CMB}}/\phi_{0}$ an increase of $\phi_{0}$ reduces
the SR parameters due to the overall $1/\phi^{2}$ factors in eqs.(2.1), moving
$n_{s}$ closer to 1. This has to be compensated by decreasing
$\phi_{\text{CMB}}/\phi_{0}$. On the other hand, $N_{\text{CMB}}$ depends not
only on the SR parameter $\epsilon_{V}$, but also on the range of field values
over which the integral in eq.(2.1) has to be evaluated. An increase of
$\phi_{0}$ thus has to be compensated by an increase in $\beta$, shown by the
red curve, in order to leave $N_{\text{CMB}}$ unchanged. This leads to a rapid
increase of $\epsilon_{V}$, and hence of $r$ (shown in blue); however, in this
region of parameter space we still have $\epsilon_{V}\ll|\eta_{V}|$, i.e. the
spectral index $n_{s}$ is essentially determined by $\eta_{V}$.

For $\phi_{0}>10$ the curve for $\phi_{\text{CMB}}$ flattens out. Recall that
even for a purely quadratic potential $\phi_{\text{CMB}}\simeq 15$, see
eq.(2.27), and our potential is significantly flatter, hence requiring smaller
$\phi_{\text{CMB}}$ in order to give the correct $N_{\text{CMB}}$. At the same
time $\epsilon_{V}$ keeps increasing, so that its contribution to $n_{s}$
becomes significant. This flattens the increase of $\beta$, which reaches a
maximum at $\phi_{0}\simeq 17$. For yet larger values of $\phi_{0}$,
$\phi_{\text{CMB}}$ becomes almost independent of $\phi_{0}$, i.e. inflation
now occurs further and further away from the saddle point. Keeping the
potential sufficiently flat then requires a reduction of $\beta$. For the
given choice of $n_{s}$ and $N_{\text{CMB}}$ no solution can be found for
$\phi_{0}\geq 25.4$; the red curve drops very steeply at the end, since the
potential at $\phi_{\text{CMB}}\sim 0.55\phi_{0}$ depends only weakly on the
slope at $\phi_{0}$.

Note that $r$ keeps increasing even when $\beta$ is decreasing; for the chosen
parameters, it exceeds the bound of $0.035$ for $\phi_{0}\geq 17.5$. However,
for other choices of $N_{\text{CMB}}\in[50,65]$ and $n_{s}\in[0.9579,0.9739]$
(the $2\sigma$ range) the solution terminates before the bound on $r$ is
saturated.888As long as only field values below $\phi_{0}$ are considered, one
could find additional solutions with larger $\phi_{0}$ and negative $\beta$.
Recall, however, that in this case the potential features a second minimum
above $\phi_{0}$, in which the inflaton would get stuck if it came from yet
larger field values; in particular, in scenarios featuring a phase of eternal
inflation, as discussed below. This is why we restrict ourselves to $\beta\geq
0$. Notice also that $\beta$ remains quite small throughout, i.e. the
potential indeed needs to feature a near–inflection point.

$\phi_{0}$ | $d/10^{-14}$ | $\beta/10^{-3}$ | $\phi_{\text{CMB}}$ | $n_{s}$ | $r/10^{-3}$ | $\alpha/10^{-3}$ | $N_{\rm{CMB}}$
---|---|---|---|---|---|---|---
$1$ | $0.00636$ | $0.0009$ | $0.999203$ | $0.9619$ | $6.81\cdot 10^{-6}$ | $-1.40$ | $64.6$
$1$ | $0.164$ | $0.0017$ | $0.999285$ | $0.9659$ | $1.76\cdot 10^{-5}$ | $-2.24$ | $54.6$
$1$ | $0.268$ | $0.0023$ | $0.999368$ | $0.9699$ | $2.87\cdot 10^{-5}$ | $-2.87$ | $50.2$
$2$ | $0.27$ | $0.015$ | $1.99356$ | $0.9619$ | $4.64\cdot 10^{-4}$ | $-1.43$ | $64.3$
$2$ | $0.65$ | $0.027$ | $1.99418$ | $0.9659$ | $1.11\cdot 10^{-3}$ | $-2.22$ | $55.2$
$2$ | $1.03$ | $0.036$ | $1.99482$ | $0.9699$ | $1.77\cdot 10^{-3}$ | $-2.80$ | $51.1$
$3$ | $0.66$ | $0.08$ | $2.97783$ | $0.9619$ | $5.74\cdot 10^{-3}$ | $-1.47$ | $63.8$
$3$ | $1.52$ | $0.14$ | $2.97976$ | $0.9659$ | $1.32\cdot 10^{-2}$ | $-2.24$ | $55.2$
$3$ | $2.49$ | $0.19$ | $2.98175$ | $0.9699$ | $2.17\cdot 10^{-2}$ | $-2.87$ | $50.7$
$4$ | $1.16$ | $0.25$ | $3.9461$ | $0.9619$ | $3.18\cdot 10^{-2}$ | $-1.43$ | $64.4$
$4$ | $2.77$ | $0.45$ | $3.9499$ | $0.9659$ | $7.62\cdot 10^{-2}$ | $-2.22$ | $55.2$
$4$ | $4.39$ | $0.60$ | $3.9542$ | $0.9699$ | $1.21\cdot 10^{-1}$ | $-2.81$ | $51.0$
$5$ | $2.23$ | $0.7$ | $4.8899$ | $0.9619$ | $0.15$ | $-1.55$ | $61.9$
$5$ | $4.29$ | $1.1$ | $4.8968$ | $0.9659$ | $0.29$ | $-2.16$ | $55.2$
$5$ | $7.02$ | $1.5$ | $4.9037$ | $0.9699$ | $0.48$ | $-2.77$ | $50.6$
$8$ | $4.54$ | $4.0$ | $7.4815$ | $0.9619$ | $2.04$ | $-1.23$ | $62.3$
$8$ | $8.76$ | $6.7$ | $7.4879$ | $0.9659$ | $4.01$ | $-1.72$ | $55.0$
$8$ | $13.05$ | $9.0$ | $7.5048$ | $0.9699$ | $6.07$ | $-2.12$ | $51.2$
$11$ | $5.19$ | $10$ | $9.4906$ | $0.9619$ | $8.5$ | $-0.84$ | $62.1$
$11$ | $9.43$ | $19$ | $9.4772$ | $0.9659$ | $16.1$ | $-1.18$ | $54.7$
$11$ | $11.99$ | $25$ | $9.5432$ | $0.9699$ | $21.1$ | $-1.35$ | $52.7$
$14$ | $6.93$ | $26$ | $10.6090$ | $0.9619$ | $29.8$ | $-0.82$ | $53.7$
$14$ | $7.05$ | $33$ | $10.8235$ | $0.9659$ | $31.5$ | $-0.83$ | $55.3$
$14$ | $7.36$ | $41$ | $11.0439$ | $0.9699$ | $34.2$ | $-0.86$ | $56.6$
$17$ | $3.28$ | $8$ | $11.6109$ | $0.9619$ | $27.1$ | $-0.49$ | $61.1$
$17$ | $3.43$ | $22$ | $11.8913$ | $0.9659$ | $30.0$ | $-0.51$ | $62.6$
$17$ | $3.73$ | $39$ | $12.1859$ | $0.9699$ | $34.9$ | $-0.55$ | $63.6$
$20$ | $2.25$ | $3$ | $12.4500$ | $0.9651$ | $32.9$ | $-0.42$ | $64.3$
$20$ | $2.28$ | $8$ | $12.5183$ | $0.9659$ | $33.9$ | $-0.42$ | $64.6$
$20$ | $2.30$ | $13$ | $12.6000$ | $0.9668$ | $34.7$ | $-0.43$ | $65.0$
Table 1: Examples of model parameters and corresponding predictions. The
overall coupling strength $d$ has been chosen to reproduce the central value
of power spectrum, i.e. $\mathcal{P}_{\zeta}\simeq 2.1\cdot 10^{-9}$. The
predictions for $n_{s}$ and $\alpha$ are consistent with Planck 2018 results
(2.13) at the $1\sigma$ level. Predictions for the tensor–to–scalar ratio $r$
satisfy the current bound $r<0.035$ (from the recent BICEP/Keck 2018 [8]) and
range from $\mathcal{O}(10^{-8})$ to $\mathcal{O}(10^{-2})$.

In Table 1 we explore a wider range999Generically $N_{\rm CMB}<65$ unless
there is an exotic reheating phase following the end of inflation [51, 52].
For our case, inflation ends with a usual quadratic potential, hence we adopt
$50\leq N_{\text{CMB}}\leq 65$. of $N_{\text{CMB}}$ and $n_{s}$, for
$\phi_{0}$ between $1$ and $20$. The overall trends are as in Fig. 2:
$\phi_{\text{CMB}}$ remains close to $\phi_{0}$ for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10$,
but increases only slowly once $\phi_{0}>15$; and $\beta$ at first increases
quickly, but reaches a maximum at $\phi_{0}$ around $15$ and then quickly
diminishes again. We also see that the model can saturate the upper bound on
$r$ for $\phi_{0}\geq 14$.

Moreover, the running $\alpha$ of the spectral index is always negative, and
well within the currently allowed range given in (2.13). It is essentially
independent of $\phi_{0}$ for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$,
but becomes smaller in magnitude for $\phi_{0}\geq 8$. In fact, $\xi_{V}^{2}$
changes sign at $\phi_{\text{CMB}}=2\phi_{0}(1-\beta)/3$; for the parameters
used in Fig. 2 this happens at $\phi_{0}\simeq 18$. However, by then
$\epsilon_{V}$ has become so large that the first two terms in the expression
for $\alpha$ in eq.(2.11) dominate; these terms are negative since
$\eta_{V}<0$.

Table 1 also lists the values of $d$ needed to reproduce the observed
normalization of the spectrum of CMB anisotropies, computed from eq.(2.10). At
small $\phi_{0}$ the coupling scales like $d\propto\phi_{0}^{2}$, since
$V(\phi_{\text{CMB}})\propto d\phi_{0}^{4}$ and
$\epsilon_{V}\propto\phi_{0}^{6}$. However, for $\phi_{0}>5$ the growth of
$\epsilon_{V}$ with increasing $\phi_{0}$ slows down. As a result, $d$ reaches
a maximum value near $10^{-13}$ at $\phi_{0}\sim 10$, and decreases again for
yet larger values of $\phi_{0}$. As a result, the physical inflaton mass
$m_{\phi}=2\sqrt{d}\phi_{0}$ increases $\propto\phi_{0}^{2}$ for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$,
but depends only weakly on $\phi_{0}$ for $\phi_{0}>10$.

###  3.2 Complete Scan of Parameter Space

Figure 3: The numerical scan over the free parameters
$\phi_{0},\,\phi_{\text{CMB}}$ and $\beta$, requiring $50\leq
N_{\text{CMB}}\leq 65$, fills the entire allowed region of the $(n_{s},\,r)$
plane.

In order to explore the full parameter space, we scan over the parameters
$\phi_{0},\,\phi_{\text{CMB}}$ and $\beta$; $d$ has again been fixed such that
$\mathcal{P}_{\zeta}\simeq 2.1\cdot 10^{-9}$. We accept all combinations of
parameters that yield $50\leq N_{\rm CMB}\leq 65$ and satisfy the constraints
on $n_{s}$ and $r$ shown in Fig. 1. We see from Fig. 3 that this scan in fact
fills the entire presently allowed region of the $(n_{s},\,r)$ plane.

Figure 4: Blue dots represent allowed sets of model parameters (left frames)
and the corresponding predictions (right frames) using the full expressions.
The red lines depict the analytical approximation valid for
$\phi_{0}-\phi_{\text{CMB}}\ll\phi_{0}$, for fixed $n_{s}=0.9659$ and
$N_{\text{CMB}}=55$, as described in the text.

The allowed ranges of the model parameters and the resulting predictions for
$r,\,n_{s}$ and $N_{\text{CMB}}$ are further shown as function of $\phi_{0}$
in Fig. 4.

The upper left panel shows $\phi_{\text{CMB}}$. For
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$,
$\phi_{\text{CMB}}\simeq\phi_{0}$ can be analytically obtained by [49]

$\displaystyle\phi_{\text{CMB}}=\phi_{0}(1-\delta_{\rm
CMB})\simeq\phi_{0}\left(1-7.10\cdot 10^{-4}\phi_{0}^{2}\right)\,;$ (3.1)

here we have used $\delta_{\rm CMB}\simeq 1-\frac{\phi_{0}^{2}}{48}(1-n_{s})$
(cf. Eq.(2.19)) with $n_{s}=0.9659$. This analytical approximation is shown by
the red line, which describes the numerical results very well for small
$\phi_{0}$. As we already saw in Fig. 2, this approximation breaks down for
$\phi_{0}>5$, where the difference between $\phi_{0}$ and $\phi_{\text{CMB}}$
increases quickly. Eventually $\phi_{\text{CMB}}$ becomes nearly independent
of $\phi_{0}$, taking values around $12$; larger $\phi_{\text{CMB}}$ would
require larger $\epsilon_{V}$ in order to keep $N_{\text{CMB}}$ within the
acceptable range, in conflict with the upper bound on $r$.

The middle left panel of Fig. 4 gives $\beta$ as function of $\phi_{0}$. Here
the analytical small $\phi_{0}$ approximation gives [49]

$\displaystyle\beta\simeq 1.65\cdot 10^{-6}\phi_{0}^{4}\,,$ (3.2)

for our choices $N_{\rm CMB}=55$ and $n_{s}=0.9659$; again, this reproduces
the numerical results for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$.
On the other hand, for $\phi_{0}\geq 15$, a relatively wide range of values of
$\beta$ is allowed, since for $\beta\ll 1$ the potential at
$\phi\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>\phi_{\text{CMB}}$
only weakly depends on $\beta$; recall from the top–left frame that
$\phi_{\text{CMB}}\leq 0.7\phi_{0}$ lies well below the near–inflection point
for these large values of $\phi_{0}$.

The lower left panel of Fig. 4 shows $d$ as function of $\phi_{0}$. In this
case the small $\phi_{0}$ approximation yields [49]

$\displaystyle d\simeq 1.55\cdot 10^{-15}\phi_{0}^{2}\,,$ (3.3)

for $\mathcal{P}_{\zeta}=2.1\cdot 10^{-9}$ and our default values of $n_{s}$
and $N_{\text{CMB}}$.101010The larger scatter of the blue points around the
red curve in this frame, compared to the top left and middle left frames, is
largely a plotting artifact. The $y-$axis in the latter spans 6 orders of
magnitude compared to “only” 4 orders of magnitude in the lower left frame,
making the blue “bars” appear correspondingly shorter. Moreover, for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$
the dynamics is more usefully described by the scaled difference $\delta$
introduced in eq.(2.15); the scatter in $\delta_{\rm CMB}$ for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$
is similar to that in $\beta$ and $d$. On the other hand, we already saw in
the discussion of Table 1 that for $\phi_{0}>10$, a smaller coupling $d$ is
required in order to obtain the correct power spectrum.

The upper right panel of Fig. 4 depicts $r$ as function of $\phi_{0}$. For
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$
this is again quite well described by the analytical approximation [49]

$\displaystyle r\simeq 1.66\cdot 10^{-8}\phi_{0}^{6}\,.$ (3.4)

For $\phi_{0}>5$ $r$ increases slightly less quickly with increasing
$\phi_{0}$; nevertheless for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>12$
solutions can be found that saturate the current upper bound on $r$. In fact,
for $\phi_{0}>22$ all solutions that give $N_{\text{CMB}}\leq 65$ predict too
large a value of $r$.

The middle right panel of Fig. 4 gives $n_{s}$ as function of $\phi_{0}$. For
$\phi_{0}\leq 13$ essentially the entire currently allowed range can be
covered by our model. For larger values of $\phi_{0}$ the parameter space
begins to get squeezed by the conflicting constraints $N_{\text{CMB}}\leq 65$
and $r\leq 0.035$. Probably coincidentally the prediction of $n_{s}$ for the
largest allowed $\phi_{0}$ is quite close to the present central value.

Finally, the lower right panel of Fig. 4 gives results for $N_{\rm CMB}$.
Again for $\phi_{0}\leq 13$ all values between our chosen limits of $50$ and
$65$ can be reproduced. For larger $\phi_{0}$ the upper bound on $r$ leads to
a lower bound on $N_{\text{CMB}}$, since a flatter potential reduces the
former but increases the latter. As we saw above, the two constraints become
incompatible for $\phi_{0}>22$.

Let us end this section by summarizing the parameter space for the polynomial
inflation model (2.1):

$\displaystyle 6\cdot 10^{-16}\lesssim d\lesssim 2\cdot 10^{-13}\,;\ \ \
0<\beta\lesssim 4\cdot 10^{-2}\,;\ \ \ \phi_{0}\lesssim 21.5\,.$ (3.5)

The upper bound on $\phi_{0}$ immediately yields a lower bound on the cubic
potential parameter, $A\gtrsim-57$. Recall that these limits have been derived
from the observed value of $n_{s}$, the constraint on $N_{\text{CMB}}$, and
the upper bound on $r$. It might be worth mentioning that in our model the
constraint on $n_{s}$ suffices to derive a theoretical upper bound on the
tensor–to–scalar ratio, $r\lesssim 0.06$, if
$N_{\text{CMB}}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>60$;
however, for
$N_{\text{CMB}}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>55$
significantly larger values of $r$ could have been reproduced. The recent
bound on $r$ therefore significantly reduces the allowed parameter space of
our model. The above discussion also shows that a tighter upper bound on $r$
would further limit the parameter space; in particular, the upper bound on
$\phi_{0}$ would become smaller if next–generation experiments fail to detect
CMB tensor modes.

##  4 Radiative Stability and Reheating

So far our analysis has been based on the tree–level potential (2.1). This is
only justified if loop corrections to the potential are small. In order to
check this one can compute the $1-$loop Coleman–Weinberg (CW) corrections to
the potential [53] and require that they are subdominant compared to the
tree–level potential. Here we follow the procedure outlined in Ref. [49]. In
particular, we focus on the potential at $\phi_{0}$. Here the first and second
derivatives of the tree–level potential are suppressed by the small parameter
$\beta$, which increases the relative importance of loop corrections.

The bound on the inflaton self–coupling can be computed by requiring the
second derivative of the $1-$loop potential at $\phi=\phi_{0}$ to be
comparable to the tree–level value. This gives

$\displaystyle\Bigl{\lvert}\frac{d^{2}\ln(16d\beta)}{\pi^{2}}\Bigr{\rvert}<8d\beta\,.$
(4.1)

Since for $\phi_{0}\geq 1$ we typically have
$\beta\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>10^{-6}$
while
$d\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10^{-13}$,
this inequality is nearly always satisfied. As shown in Fig. 4, for
$\phi_{0}\gtrsim 14.5$, $\beta$ can in principle become arbitrarily small.
However, the inflationary parameters basically do not change when $\beta$ is
increased from, say, $10^{-15}$ to a loop corrected value of $10^{-13}$; our
predictions therefore remain stable in this part of parameter space even if
the inequality (4.1) is violated. This therefore does not lead to an
additional theoretical constraint on the parameter space, in contrast to the
small–field version of this model [49].

###  4.1 Reheating

In a complete model of very early universe cosmology the inflaton field has to
couple to external particles for reheating [50, 54]. Here we assume $\phi$
couples to daughter particles with trilinear couplings in order to fully drain
the inflaton energy so that a radiation dominated epoch is reproduced after
reheating. For a scalar field $\phi^{\prime}$, e.g. the standard model Higgs
field, we introduce a term $g\phi|\phi^{\prime}|^{2}$ in the Lagrangian; for a
fermionic field $\chi$, e.g. right–handed neutrino, the corresponding term is
$y\phi\bar{\chi}\chi$.111111The non–perturbative preheating effect in our
model is negligible. In the bosonic case, i.e. preheating with Higgs
production, the trilinear coupling induce tachyonic instabilities [55], which
tends to make preheating efficient. However the Higgs self–coupling gives rise
to a positive effective mass
$\propto\lambda_{\phi^{\prime}}\langle\phi^{\prime 2}\rangle$ (where
$\lambda_{\phi^{\prime}}\sim\mathcal{O}(0.1)$ is the Higgs self-coupling and
$\langle\phi^{\prime 2}\rangle$ denotes the variance of the produced Higgs
field), which quickly dominates over the (possibly negative) contribution
$\propto g\phi$ (where the small trilinear coupling $g$ is bounded by Eq.
(4.5)), thereby blocking further non–perturbative
$\phi\rightarrow\phi^{\prime}$ energy transfer [49, 56]. In the fermionic
case, Pauli blocking implies that only a small fraction of the energy stored
in the inflaton field can be non–perturbatively transferred to $\chi$
particles, unless the daughter particles decay very fast [57]. The radiative
stability conditions [49] then read:

$\displaystyle\Bigl{\lvert}\frac{y^{4}-3y^{4}\ln(y^{2})}{4\pi^{2}}\Bigr{\rvert}<16d\beta\,;$
(4.2)
$\displaystyle\frac{1}{8\pi^{2}}\left(\frac{g}{\phi_{0}}\right)^{2}\Bigl{\lvert}\ln\left(\frac{g}{\phi_{0}}\right)-1\Bigr{\lvert}<8d\beta\,.$
(4.3)

For typical value of $d\sim 10^{-14}$ and
$\beta\lesssim\beta_{\text{max}}\simeq 4\cdot 10^{-2}$ (cf. Eq. (3.5)), one
has

$\displaystyle y$ $\displaystyle\lesssim y_{\text{max}}\simeq 2.7\cdot
10^{-4}\,;$ (4.4) $\displaystyle\left(\frac{g}{\phi_{0}}\right)$
$\displaystyle\lesssim\left(\frac{g}{\phi_{0}}\right)_{\text{max}}\simeq
1.2\cdot 10^{-7}\,.$ (4.5)

These upper bounds on the inflaton couplings immediately lead to upper bounds
on the corresponding partial widths for
$\phi\rightarrow\phi^{\prime}\phi^{\prime}$ and
$\phi\rightarrow\bar{\chi}\chi$ decays, which in turn imply upper bounds on
the post–inflationary reheating temperature. In the instantaneous decay
approximation the latter is given by [54]

$\displaystyle T_{\text{rh}}\simeq 1.41g_{\star}^{-1/4}\Gamma_{\phi}^{1/2}\,.$
(4.6)

For the fermionic reheating channel, this becomes

$\displaystyle T_{\text{rh}}^{\chi}\simeq
1.41g_{\star}^{-1/4}\left(2\phi_{0}\frac{y^{2}}{8\pi}\sqrt{d}\right)^{1/2}\lesssim
1.1\cdot 10^{11}~{}\text{GeV}\,,$ (4.7)

where we have considered
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>20$,
$g_{\star}=106.75$ and used Eq. (4.4) for the upper bound on $y$. For bosonic
reheating, the analogous calculation allows even higher reheat temperatures,

$\displaystyle T_{\text{rh}}^{\phi^{\prime}}\simeq
1.41g_{\star}^{-1/4}\left(\frac{g^{2}}{8\pi\,2\phi_{0}\sqrt{d}}\right)^{1/2}\,\lesssim
2.5\cdot 10^{14}~{}\text{GeV}\,,$ (4.8)

where the maximum value $g_{\text{max}}$ reported in Eq. (4.5) has been
utilized.

Both bounds are saturated at the largest allowed value of $\phi_{0}$, where
the physical mass of the inflaton $m_{\phi}=2\sqrt{d}\phi_{0}\simeq 10^{13}$
GeV. This is comfortably above the bound (4.7), which can thus be saturated
using simple perturbative $\phi\rightarrow\chi\bar{\chi}$ decays. On the other
hand, $m_{\phi}$ is an order of magnitude below the bound (4.8). This bound
can therefore only be saturated if one can turn an ensemble of $\phi^{\prime}$
particles with energy $\simeq m_{\phi}/2\sim 5\cdot 10^{12}$ GeV into a
thermal bath with much higher temperature, which can only happen via
scattering reactions that reduce the number of particles, in particular
$3\rightarrow 2$ scattering reactions. It is not clear whether the rate of
such reactions is sufficiently high that the bound (4.8) can be saturated; in
the absence of such reactions it would have to be replaced by
$T_{\text{rh}}^{\phi^{\prime}}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>m_{\phi}/2$.
The same remark holds for the maximal temperature of the radiation bath, which
is typically attained well before reheating is completed and can be
significantly larger than $T_{\text{rh}}$ [54]: in the absence of fast
$3\rightarrow 2$ reactions it is also bounded by $m_{\phi}/2$.

##  5 Eternal Polynomial Inflation

In the previous sections, we have worked out the parameter space consistent
with Planck 2018 (2.13) and BICEP/Keck 2018 (2.14) and investigated the
radiative stability of the inflaton potential as well as perturbative
reheating for the polynomial inflation model. All this happened at field
values (well) below $\phi_{0}$. In this section we analyze the situation at
$\phi\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>\phi_{0}$.
In particular, we are interested in the energy scale(s) at which “eternal”
inflation could have occurred within the allowed parameter space.

###  5.1 Eternal Phase I

During the SR phase, one can neglect the acceleration term in eq. (2.2), so
that the classical inflaton field evolves as

$\displaystyle\dot{\phi}\approx-\frac{V^{\prime}(\phi)}{3H}\,.$ (5.1)

This predicts a classical field excursion per Hubble time

$\displaystyle\Delta\phi_{\rm
cl}=\frac{|\dot{\phi}|}{H}\approx\frac{|V^{\prime}(\phi)|}{3H^{2}}\approx\frac{|V^{\prime}(\phi)|}{V}=\sqrt{2\epsilon_{V}}\,.$
(5.2)

On the other hand, since the inflaton field is very weakly coupled its quantum
fluctuations follow a Gaussian probability distribution. In the quasi de
Sitter background during SR inflation, the typical size of these quantum
fluctuation over one Hubble time is given by $\delta\phi_{\rm
qu}=\frac{H}{2\pi}$ [58, 59].

Assume that the quantum fluctuation dominates over the classical field
excursion, i.e. $\delta\phi_{\rm qu}>\Delta\phi_{\rm cl}$. In this case the
inflaton field in a given Hubble volume is almost as likely to move “uphill”,
towards larger values, as it is to move towards the minimum of the potential.
Since after a small number of e–folds of inflation the inflaton field fills a
great many Hubble volumes which henceforth evolve independently, it is
virtually guaranteed that in some of these volumes the field does indeed move
uphill. Since these regions subsequently will expand faster (owing to the
larger Hubble parameter), in this picture “most” of space will continue
inflating forever, even though inflation clearly must have ended in our own
Hubble patch. This is known as eternal inflation.

From eq.(5.2) and $\delta\phi_{\rm qu}=\frac{H}{2\pi}$, the condition for
eternal inflation is

$\displaystyle\frac{H}{2\pi}>\sqrt{2\epsilon_{V}}\Leftrightarrow\frac{H^{2}}{8\pi^{2}\epsilon_{V}}>1\,.$
(5.3)

Hence the condition for eternal inflation is satisfied if the leading order
prediction of the amplitude of curvature perturbations $\mathcal{P}_{\zeta}$
exceeds unity [38].

As already emphasized in sec. 2.2, for $\phi\gg\phi_{0}$ effectively our model
behaves like quartic inflation, i.e. the inflaton potential can be simplified
to $V=d\,\phi^{4}$, so that

$\displaystyle H=\sqrt{\frac{d}{3}}\phi^{2}\,,$ (5.4)

and

$\displaystyle\frac{H^{2}}{8\pi^{2}\epsilon_{V}}=\frac{d\phi^{6}}{192\pi^{2}}\,.$
(5.5)

From condition (5.3) eternal inflation then requires

$\displaystyle\phi^{2}>\left(\frac{192\pi^{2}}{d}\right)^{1/3}\,.$ (5.6)

This in turn leads to a lower bound on the Hubble parameter:

$\displaystyle H>H^{\rm c}_{\rm
EI}=4\pi^{2/3}\left(\frac{d}{3}\right)^{1/6}\,,$ (5.7)

which only depends on $d$. Once $H>H^{\rm c}_{\rm EI}$, an eternal
inflationary phase can occur.

The usual monomial chaotic $\lambda\phi^{4}$ inflation model requires
$\lambda\sim 10^{-12}$ [33] in order to match the normalization of the power
spectrum; eq. (5.7) then yields $H^{\rm c\prime}_{\rm EI}\approx 0.07$. In the
last section we saw that in our polynomial scenario the quartic coupling $d$
needs to be somewhat smaller, $6\cdot 10^{-16}\lesssim d\lesssim 2\cdot
10^{-13}$. The corresponding threshold value of the inflaton field is (in
Planckian units):

$\displaystyle 460\lesssim\phi_{\rm c}\lesssim 1211\,,$ (5.8)

which evidently is indeed well above $\phi_{0}$. One can further work out the
threshold of the corresponding inflationary scale (again in Planckian units):

$\displaystyle 0.02\lesssim H^{\rm c}_{\rm EI}\lesssim 0.05\,.$ (5.9)

Evidently the corresponding energy scale is well below the Planck scale (and
also somewhat below the scale required for eternal inflation in pure quartic
inflation); hence our semi–classical treatment, which ignores “quantum
gravity” effects, may be valid.

We thus conclude that our model does allow for an epoch of “eternal” while
reproducing all present measurements of inflationary parameters.

###  5.2 Eternal Phase II

The discussion in the previous section shows that eternal inflation should
occur in the polynomial model if the inflaton field and the Hubble parameter
ever exceeded the critical values (5.8) and (5.9), respectively. Here we show
that, at least in part of the allowed parameter space, a second, later epoch
of eternal inflation will occur.

We have seen in Fig. 4 that the parameter $\beta$ can be arbitrarily small if
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>15$.
Recall that a very small $\beta$ implies that the potential at $\phi_{0}$ is
very flat. Hence it is expected that a (second) eternal phase can occur. Since
this eternal phase appears when the inflaton is near the saddle point
$\phi_{0}$, which is much smaller than $\phi_{\rm c}$ analyzed above, one can
expect that the corresponding Hubble scale should be much lower than $H^{\rm
c}_{\rm EI}$ of eq.(5.9).121212This has some similarity to eternal hilltop
inflation investigated in [38]. However, in our case the inflaton first rolls
down to a plateau around the saddle point; in hilltop inflation, one has to
impose as initial condition that the inflaton starts near a (local) maximum of
the potential, which can also be very flat.

In order to obtain the maximum value of $\beta$ that allows a second phase of
eternal inflation, we again use condition Eq. (5.3), identical to the leading
order prediction of the power spectrum being larger than unity as mentioned
earlier. Since the potential is flattest at $\phi=\phi_{0}$, we can use
eq.(2.18) with $\delta=0$ to derive the condition for the existence of a
second epoch of eternal inflation:

$\beta<\frac{\sqrt{d}\phi_{0}^{3}}{144\pi}\simeq 10^{-6}\,;$ (5.10)

For the numerical value we have used $d\sim 10^{-14}$ and $\phi_{0}\sim 20$.

The potential at $\phi_{0}$ is given by $d\phi_{0}^{4}/3$, up to corrections
of relative order $\beta$ which are evidently completely negligible here. This
epoch of eternal inflation would thus have a Hubble parameter

$\displaystyle H_{\rm EI}=\frac{\sqrt{d}}{3}\phi_{0}^{2}\sim 10^{-5}\,;$
(5.11)

this is at least three orders of magnitude smaller than the one given in
eq.(5.9).

The width of this second region in field space that allows eternal inflation
is also important. Again from eq.(2.18) we see that the evolution of the
inflaton field will be dominated by quantum fluctuations as long as

$\delta<\delta_{\rm c}=\frac{d^{1/4}\phi_{0}^{3/2}}{6\sqrt{2\pi}}\,.$ (5.12)

In units of the size $H_{\rm EI}/(2\pi)$ of random walk steps the half width
of the region in field space allowing eternal inflation is thus

$\frac{\delta_{\rm c}\phi_{0}}{H_{\rm
EI}/(2\pi)}=\sqrt{\frac{\pi}{2}}\frac{\phi_{0}^{1/2}}{d^{1/4}}\simeq 1.8\cdot
10^{4}\,.$ (5.13)

Given that the average excursion in a random walk is proportional to the
square of the number of steps taken times the (typical) step size, starting
from $\phi=\phi_{0}$ the field would thus typically need more than $10^{8}$
steps to leave the region where quantum fluctuations dominate the dynamics.
Moreover, half the time the random walk would end at
$\phi>\phi_{0}\left(1+\delta^{\rm c}\right)$, in which case the classical
field evolution would bring the field back into the range where quantum
effects dominate. These arguments indicate that in our model indeed “most of”
space would inflate eternally if $\phi$ ever reached the region very close to
$\phi_{0}$ and $\beta<10^{-6}$.

Figure 5: Evolution of the Hubble parameter as function of the classical
prediction for the number of e–folds $N_{\rm cla}$ (depending on $\phi$ via
Eq. (2.1)), for $\phi_{0}=20,~{}\beta=10^{-7}$ and $d\simeq 2\cdot 10^{-14}$.
Inflation can be eternal in the gray shaded region where $H>H^{\rm c}_{\rm
EI}\simeq 0.04~{}M_{\rm Pl}$. For $H<H^{\rm c}_{\rm EI}$ a period of the usual
SR inflation follows. For the given choice of a very small $\beta$ a second
epoch of eternal inflation occurs (between the dashed vertical lines), with
Hubble parameter as low as $\mathcal{O}(10^{-5})~{}M_{\rm Pl}$.

A typical inflationary trajectory starting at very large field values
$\phi>\phi_{\rm c}$ of eq.(5.8) is shown in Fig. 5, which plots the Hubble
parameter as function of the classical prediction $N_{\rm cla}$ of the number
of e–folds that occur after the inflaton field had a certain value, given by
eq.(2.1); larger $N_{\rm cla}$ correspond to larger $\phi$. Of course, for the
field ranges allowing eternal inflation, i.e. for $\phi>\phi_{\rm c}$ and for
$\phi\in[\phi_{0}(1-\delta_{\rm c}),\phi_{0}(1+\delta_{c})]$, the actual
number of e–folds by which our Hubble patch expanded was likely very much
larger than the classical prediction.

For field values below $\phi_{\rm c}$ the Hubble parameter seems to drop very
steeply. However, if
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>4$
the SR conditions are satisfied for all
$\phi\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>\phi_{0}$;
essentially deterministic inflation therefore lasts from the end of the first,
high–energy stage of eternal inflation to the onset of the second epoch of
eternal inflation where $\phi\simeq\phi_{0}$. This is in contrast to the
small–field version of this model, where the SR conditions are violated for
some range of field values above $\phi_{0}$, and inflation around $\phi_{0}$
is always deterministic [49]. Finally, once $\phi<\phi_{0}\left(1-\delta_{\rm
c}\right)$ inflation is deterministic again, including the last $\sim 65$
e–folds of inflation with $\phi\leq\phi_{\text{CMB}}$.

Recalling the results of Fig. 4, we conclude that model parameters (a subset
of (3.5)) with:

$\displaystyle 15\lesssim\phi_{0}\lesssim 21.5\,;\ \ \
0<\beta\lesssim\mathcal{O}(10^{-6})\,;\ \ \ 2\cdot 10^{-14}\lesssim d\lesssim
6\cdot 10^{-14}\,,$ (5.14)

satisfy all observational constraints and allow a second eternal inflationary
phase. Note that the second eternal phase can only occur if $\phi_{0}$ is
rather large, which implies $r\sim\mathcal{O}(10^{-2})$. The part of parameter
space allowing the (unusual) low scale second eternal phase can thus be tested
by the next generation CMB experiments, e.g. CORE [22], AliCPT [23], LiteBIRD
[24] and CMB-S4 [25], with expected sensitivity down to
$r\sim\mathcal{O}(10^{-3})$.

##  6 Summary and Conclusions

In this paper we revisited large field inflation with a single inflaton. We
investigated a model where the inflaton potential is a polynomial of degree
four. Current observations then require that the potential features a near
saddle point at $\phi_{0}$, making the potential concave at
$\phi\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>\phi_{\text{CMB}}<\phi_{0}$
as required by the Planck 2018 data.

The model was described in Sec. 2. The potential contains three free
parameters: an overall (quartic) coupling strength $d$; the location
$\phi_{0}$ of the almost saddle point; and $\beta\ll 1$ to determine the
deviation from a true saddle point, with smaller $\beta$ making the potential
flatter for $\phi\sim\phi_{0}$. The parameters $\phi_{0}$ and $\beta$ thus
determine the shape of the potential, while the overall normalization, given
by $d$, can be fixed from the normalization of the power spectrum of curvature
perturbation $\mathcal{P}_{\zeta}$. The value $\phi_{\text{CMB}}$ of the
inflaton field when the cosmic microwave background (CMB) scales first left
the horizon is another important free parameter. It allows us to choose
$\phi_{0}$ within a rather wide range, and then use the measured spectral
index $n_{s}$ as well as the number $N_{\text{CMB}}$ of e–folds of inflation
generated after CMB scales first left the horizon to determine, or constrain,
$\phi_{\text{CMB}}$ and $\beta$. We also discuss various analytical
approximations. In particular, for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>5$
(in Planckian units, which we use throughout)
$\phi_{0}-\phi_{\text{CMB}}\ll\phi_{0}$ and the fully analytical treatment
developed for the small field version of this model [49] still holds to good
approximation.

In Sec. 3 a full scan of the parameter space consistent with most resent
Planck and BICEP/Keck 2018 observations at the $2\sigma$ level (cf. (2.13) and
(2.14)) is described. The final result is summarized in (3.5). The predictions
for $r$ range from unobservably small, $\mathcal{O}(10^{-8})$, to the current
upper bound. In fact, the current bound $r\leq 0.035$ together with the
constraint $N_{\text{CMB}}\leq 65$ leads to the upper bound
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>21.5$.
Moreover, we predict negative running of the spectral index,
$\alpha\sim-\mathcal{O}(10^{-3})$, which might be testable in the near future
[60]. A large set of examples are listed in Table 1. To our knowledge, this is
the first such comprehensive scan of parameter space of polynomial inflation
taking into account the most recent CMB data. Of course, the currently allowed
region described by eqs.(3.5), in particular the upper bound on $\phi_{0}$,
should be further constrained once more precise CMB experiments are performed,
such as CORE [22], AliCPT [23], LiteBIRD [24] and CMB-S4 [25], which could
probe all
$r\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>\mathcal{O}(10^{-3})$.

In Sec. 4 we showed that radiative stability of the inflaton potential near
the inflection-point $\phi_{0}$ leads to relatively mild constraints for our
large field model, in sharp contrast to its small field version [49]. In
particular, the one–loop Coleman–Weinberg (CW) corrections to the potential
due to the self–interactions of the inflaton are always harmless. Moreover,
large reheat temperatures, up to $10^{11}$ $(2\cdot 10^{14})$ GeV are in
principle possible for perturbative inflaton decay into fermionic (bosonic)
final states. However, we remind the reader that temperatures above half the
inflaton mass $m_{\phi}$ can only be reached if the rate for reactions that
reduce the number of particles (e.g. $3\rightarrow 2$) is sufficiently high;
note that $m_{\phi}\leq 10^{13}$ GeV in this model.

In Sec. 5 the possibility of eternal inflation is discussed. This is generally
expected to occur in models of polynomial inflation, assuming the field value
or, equivalently, the inflationary Hubble parameter reached sufficiently high
values. In our case the critical Hubble parameter only depends on the size of
the quartic coupling $d$, and lies in the range $0.02\lesssim H^{\rm c}_{\rm
EI}\lesssim 0.05$; this is somewhat below the value of $0.07$ needed in the
monomial $\lambda\phi^{4}$ model (which is in any case excluded by the upper
bound on $r$).

More interestingly, we find that there exists another possibility to realize
eternal inflation in our scenario. This occurs when the potential is very flat
around $\phi_{0}$: for
$\beta\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10^{-6}$
quantum fluctuations can dominate over the classical evolution already at
$\phi\simeq\phi_{0}$, compared to
$\phi\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>{\cal
O}(10^{3})$ in the first epoch of eternal inflation. This second epoch of
eternal inflation features a much smaller Hubble parameter,
$H\sim\mathcal{O}(10^{-5})$, which is of the same order of magnitude as that
when the CMB pivot scale $k_{\star}=0.05\ \rm{Mpc}^{-1}$ first crossed out of
the horizon. Nevertheless there will be many e–folds of deterministic
inflation between the end of the second epoch of eternal inflation and the era
when the CMB scales first crossed out of the horizon. We are therefore not
aware of any immediate observational consequences of this “late” epoch of
eternal inflation. However, since
$\beta\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle<}}{{\sim}}$}\>10^{-6}$
is possible only for
$\phi_{0}\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>15$,
which in turn implies
$r\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>0.01$,
at least in our model this possibility can be tested by the next round of CMB
experiments.

This novel scenario featuring two epochs of eternal inflation, as depicted in
fig. 5, might also be conceptually interesting. Eternal inflation is the only
known mechanism that might be able to populate the “landscape” of superstring
theory [40, 41, 42]. During eternal inflation not only the inflaton field
undergoes a random walk in field space, but so does every field whose mass is
below the inflationary Hubble parameter. This might include many of the scalar
fields (from the four–dimensional perspective) that determine the sizes of
physical couplings in string theory. During the first, high scale epoch of
eternal inflation these fields will be sampled with a typical step size
$H/(2\pi)\>\raisebox{-2.15277pt}{$\stackrel{{\scriptstyle\textstyle>}}{{\sim}}$}\>5\cdot
10^{-3}$. Since the second epoch of eternal inflation has a thousand times
smaller Hubble parameter, the step size of the random walk in field space is
also thousand times smaller during this second epoch. This might allow to much
more efficiently “home in” on relatively small features of the landscape.

In summary, we have presented a successful large field polynomial model,
worked out the complete allowed parameter space (3.5) and offered the
corresponding inflationary predictions. Combined with the earlier analysis of
the small field version of this model [49] this offers the most complete
analysis of the polynomial inflation model after Planck and BICEP/Keck 2018.
We also pointed out for the first time that in our model the early history of
the universe might feature two epochs of eternal inflation, at quite different
energy scales.

##  Appendix A General Expression for the Number of e-Folds

$\displaystyle N_{\rm CMB}$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{1}{\sqrt{2\epsilon_{V}}}d\phi$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{\phi}{2}\left[\frac{32\phi^{2}-32A\left(\beta-1\right)\phi+9A^{2}}{64\phi^{2}-48A\left(\beta-1\right)\phi+9A^{2}}\right]d\phi\,$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{\phi}{2}\left[1+\frac{-32\phi^{2}+16A\left(\beta-1\right)\phi}{64\phi^{2}-48A\left(\beta-1\right)\phi+9A^{2}}\right]d\phi\,$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{\phi}{2}\left\\{1+\frac{-32\phi^{2}+16A\left(\beta-1\right)\phi}{\left[8\phi-3A(\beta-1)\right]^{2}-9A^{2}(\beta^{2}-2\beta)}\right\\}d\phi\,$
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{\phi}{2}\left\\{1+\frac{-32\phi^{2}+16A\left(\beta-1\right)\phi}{64\left[\phi-3/8A(\beta-1)\right]^{2}-9A^{2}(\beta^{2}-2\beta)}\right\\}d\phi\,$
(A.1)
$\displaystyle=\int^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\frac{\phi}{2}\left\\{1-\frac{1}{2}\frac{\phi^{2}}{\left[\phi+\phi_{0}(\beta-1)\right]^{2}-\phi_{0}^{2}(\beta^{2}-2\beta)}-\frac{2}{3}\frac{\phi_{0}\left(\beta-1\right)\phi}{\left[\phi+\phi_{0}(\beta-1)\right]^{2}-\phi_{0}^{2}(\beta^{2}-2\beta)}\right\\}d\phi\,.$

The three integrals can be evaluated analytically:

$\int d\phi\frac{\phi}{2}=\frac{\phi^{2}}{4}\,;$ (A.2)
$\displaystyle\int-\frac{1}{4}\frac{\phi^{3}}{\left[\phi-m\right]^{2}+n^{2}}\,d\phi=\frac{1}{8}$
$\displaystyle\Bigg{\\{}5m^{2}-4m\,\phi-\phi^{2}+\frac{2(m^{3}-3m\,n^{2})\,}{n}\arctan\left[\frac{m-\phi}{n}\right]$
$\displaystyle-(3m^{2}-n^{2})\log\left[n^{2}+(m-\phi)^{2}\right]\Bigg{\\}}\,;$
(A.3)

and

$\displaystyle\int$
$\displaystyle\frac{1}{3}\frac{d\phi\,m\,\phi^{2}}{\left[\phi-m\right]^{2}+n^{2}}=\frac{m}{3}\left\\{\phi-\frac{(m^{2}-n^{2})}{n}\arctan\left[\frac{m-\phi}{n}\right]+m\log\left[n^{2}+(m-\phi)^{2}\right]\right\\}\,,$
(A.4)

with $m=\phi_{0}(1-\beta)$ and $n^{2}=\phi_{0}^{2}(2\beta-\beta^{2})$.
Combining these results, we obtain

$\displaystyle N_{\text{CMB}}$
$\displaystyle=\left\\{-\frac{m^{3}+5mn^{2}}{12n}\arctan\left(\frac{m-\phi}{n}\right)+\frac{5m^{2}}{8}-\frac{m\phi}{6}+\frac{\phi^{2}}{8}\right.$
$\displaystyle\left.\hskip
19.91692pt-\frac{(m^{2}-3n^{2})}{24}\ln\left[n^{2}+(m-\phi)^{2}\right]\right\\}\Bigg{|}^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}$
(A.5)
$\displaystyle\simeq\frac{1}{24}\left\\{3\phi^{2}-4\phi\phi_{0}+15\phi_{0}^{2}-\phi_{0}^{2}\sqrt{\frac{2}{\beta}}\arctan\left(\frac{\phi_{0}-\phi}{\sqrt{2\beta}\phi_{0}}\right)-\phi_{0}^{2}\ln\left[(\phi_{0}-\phi)^{2}\right]\right\\}\Bigg{|}^{\phi_{\text{CMB}}}_{\phi_{\text{end}}}\,.$

In the last step we have assumed $\beta\ll 1$, which is true in the allowed
parameter space, but cannot be assumed a priori.

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