# Regularization of Single Field Inflation Models

Josh Hoffmann Corresponding author<EMAIL_ADDRESS>Department of
Physics, Lancaster University, Lancaster, United Kingdom LA1 4YB  David Sloan
Department of Physics, Lancaster University, Lancaster, United Kingdom LA1 4YB

###### Abstract

There are many single field inflationary models that are consistent with the
recent Planck 2018 measurements of the spectral index $n_{s}$ and tensor-to-
scalar ratio $r$. Despite good agreement with observational data some of these
models suffer from having unregularized potentials which would produce a
collapsing universe shortly after the end of inflation. In this paper we show
that how one chooses to correct the behaviour potential towards the end of
inflation can have a significant effect on the inflationary predictions of the
model, specifically in the case of quartic hilltop and radiatively corrected
Higgs inflation.

##  1 Introduction

Cosmological inflation represents one of the most important aspects of modern
cosmology and is the focus of a great amount of theoretical and experimental
physics [1, 2, 3, 4, 5, 6]. Inflation is a period of exponential expansion of
the early universe some time between $10^{-36}$ and $10^{-32}$ seconds after
the big bang and is essential for understanding the structure of the CMB, a
cornerstone of observational cosmology [4, 5, 6, 7]. The theory of
cosmological inflation was originally posited as a means to explain several
observations about our universe, namely the horizon, flatness and monopole
problems [8]. However its true success is in providing a mechanism for
structure formation in the early universe [9, 10, 11, 12, 13, 14, 15, 16, 17,
18, 19, 20, 21, 22]. Inflation allows for quantum mechanical density
perturbations in the otherwise homogeneous matter content of the early
universe to become amplified to large scales, eventually leading to the
formation of structures we are more familiar with today such as stars,
galaxies and the CMB. As such, it is important to provide an explanation as to
how inflation occurs and to corroborate this with experimental observations.

It has been particularly fruitful to study the inflationary paradigm through
the introduction of a homogeneous field $\phi(t)$ (typically a scalar)
minimally coupled to gravity and governed by a potential $V(\phi)$ which
essentially defines the model. The Planck 2018 [23] survey has made it
possible to precisely test a huge variety of inflationary models. Two
important and model dependant cosmological observables are the scalar tensor
ratio $r$ and spectral index $n_{s}$. These parameters and their compatibility
with the parameter space determined by the Planck 2018 results are studied for
many models in [24].

The Planck 2018 survey estimates $n_{s}=0.9649\pm 0.0042$ at $68\%$ CL &
$r\lesssim 0.056$ at $95\%$ CL. However a more recent BICEP/Keck array
measurement [25] further constrained this to $r\lesssim 0.036$ at $95\%$ CL.

The spectral index $n_{s}$ and tensor fraction are directly related to the
inflationary potential through the slow-roll parameters $\varepsilon(\phi)$
and $\eta(\phi)$

$n_{s}=1-6\varepsilon+2\eta$ (1) $r=16\varepsilon$ (2)

where

$\varepsilon(\phi)=\frac{1}{2}m_{pl}^{2}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2}$
(3) $\eta(\phi)=m_{pl}^{2}\frac{V^{\prime\prime}(\phi)}{V(\phi)}$ (4)

and thus allow us to directly test our inflationary models by specifying a
form of the potential $V(\phi)$.

A number of the single scalar field models analysed in [24], including some of
those most favoured by the Planck survey, involve potentials which are
unbounded from below after the end of inflation. Such models may be able to
provide satisfactory inflationary dynamics, however if they are taken to be
valid beyond inflation the potentials quickly become negative and due to their
unboundedness, lead to universes that collapse on a time scale $\tau\sim
H^{-1}$. Clearly this is not the cosmology we observe today, so such behaviour
must be corrected by regularizing the potentials to introduce a local minima
shortly after inflation ends. Naturally one may ask if modifying the behaviour
after the end of inflation significantly changes the models predictions of
inflationary observables such as the tensor fraction and spectral index, even
50-60 efolds before the end of inflation.

In this paper we explore the effect of generic correction terms on a
collection of inflationary models. The correction terms are designed only to
be the simplest possible options that prevent collapse of the universe after
the end of inflation. Such terms which stabilise the inflaton VEV may, and do,
appear in in more physically motivated in other physically motivated
models.Although in this work we do not attempt to explain the physical origin
of such terms. We show that such corrections will have an effect on the
inflationary predictions of the model. Thus, regardless of how corrections of
these forms may appear, they do indeed need to be accounted for before the
inflationary predictions of the model can be trusted.

We show that the addition of correction terms designed to stabilise the
inflaton VEV affect the reheating temperature $T_{re}$ predicted by the model
as a function of the spectral index $n_{s}$ at a given number of efolds before
the end of inflation $N_{k}$. Therefore by requiring that the predicted
reheating temperature stays within the loose bounds set by the big bang
nucleosynthesis scale and the energy scale of inflation, $0.01\text{GeV}\leq
T_{re}\leq 10^{16}\text{GeV}$, the model under consideration may fall into a
further restricted region of $r-n_{s}$ space that is acceptable under the
Planck 2018 results. Thus we arrive at a key result of our analysis: In
considering models which require regularization, it is insufficient to examine
their predictions without taking such regularization into account. We have
chosen to remain agnostic on the form that such regularization should take,
motivating the correction terms solely by their role in ensuring that the
potential remain positive with zero minimum. This allows us to demonstrate
that generic regularizations should be taken into account regardless of their
physical origins. In specific cases there are good physical reasons which
inform the precise nature of the regularization, such as the contribution of
higher loop corrections to the radiatively corrected Higgs inflation from the
Jordan frame. However our goal is not simply to test specific models and
regularizations but to make the broader point that since regularizations do
make an impact on physical observables for such models, a wide range of
potentials requiring regularization should only have their observational
consequences examined with such regularizations in place.

In [26] we addressed for the case of quartic hilltop inflation, a model well
favoured by Planck, corrected by one particular modification scheme which
involves squaring the potential. It is shown that modifying the quartic
hilltop potential in this way does have a significant effect on the final
$(r,n_{s})$ parameter space that is consistent with the Planck 2018 results.
In this paper, we explore several different modification types for quartic
hilltop, radiatively corrected Higgs inflation and Exponential SUSY inflation.
These are all single field inflation models featured in [24] which all suffer
from the same issue of unboundedness. We consider generic corrections to these
models, motivated only by preventing collapse of the universe after the end of
inflation, but we show that should such correction terms arise from physical
motivations, they may have significant effects on the inflationary predictions
of the models. Therefore one must carefully consider exactly how a candidate
potential exits inflation. As we will show in this paper, it can have a
significant effect on the inflationary predictions. Models that produce such a
collapsing universe, must be corrected before calculations of $r$ and $n_{s}$
are compared to data. In all three models we explore the effects of simply
squaring the potential to form a potential that is bounded below (for quartic
hilltop this is investigated in detail in our previous paper [26]). We then
explore the simplest possible correction terms that may be used to extend the
models, these are polynomial and inverse polynomial terms for QH and RCHI
respectively.

The rest of this paper is structured as follows. In section 2 we will recap
the quartic hilltop model (QH), a promising candidate potential. We summarise
the current comparisons of the hilltop model predictions to the Planck 2018
survey. We also review recent analytical investigations of the model in [27,
26]. Polynomial corrections are then added to the hilltop model and their
effects on inflationary observables are investigated. In section 3 we move on
to the second model identified in Encyclopaedia Inflationaris [24],
radiatively corrected Higgs inflation (RCHI), which suffers from the same
unboundedness issue as quartic hilltop. We explore a squared version of the
potential as well as adding inverse polynomial correction terms to modify the
behaviour of the potential towards the end of inflation, and their effects are
investigated. It is certainly worth noting that the RCHI model derives from
calculating quantum corrections to the Higgs inflation model in the Jordan
frame and only then transforming to the Einstein frame potential. However
there exists another method of consistently correcting the RCHI model, in
which the quantum corrections are calculated in the Einstein frame [28, 29],
this will be discussed in more detail in section 3. In section 4 we discuss
the final model, exponential SUSY inflation (ESI). ESI also suffers from the
same unboundedness as QH and RCHI but is one of the most promising in terms of
it’s compatibility with Planck 2018. We find that using the approximation
techniques in this paper, it is not possible to give the ESI potential a local
mininma near where the field exits inflation at $\phi=0$ by adding inverse
polynomial terms. However, a squared version of this potential is
investigated.

##  2 Hilltop Models

###  2.1 Recap of Hilltop Inflation and Its Current State

After the Planck 2018 survey, one of the most popular inflationary models that
was consistent with the observational data was the Quartic Hilltop model. This
model belongs to the wider class of hilltop models, those defined by a
potential of the form

$V(\phi)=V_{0}\left(1-\lambda\frac{\phi^{n}}{m_{pl}^{n}}+...\right)$ (5)

The characteristic features of hilltop models make them very attractive for
the slow-roll approximation. Namely that, in hilltop models contain a broad,
flat plateau (or “hilltop”) near $\phi=0$. The two most popular cases are
quadratic and quartic hilltop models, which have $n=2$ and $n=4$ respectively.
Early analytical investigations of hilltop models [30] found that for $n>2$,
provided the higher order terms are heavily suppressed during inflation, the
spectral index when the cosmological scale exits the horizon should be

$n_{s}\simeq 1-2\left(\frac{n-1}{n-2}\right)\frac{1}{N}$ (6)

where $N$ is the remaining number of efolds of inflation after the scale
$k_{*}$ exits the horizon. For the $n=4$ case of quartic hilltop inflation

$V(\phi)=V_{0}\left[1-\lambda\left(\frac{\phi}{m_{pl}}\right)^{4}\right]$ (7)

the spectral index at $N=50$ and $N=60$ is $n_{s}=0.94$ and $n_{s}=0.95$
respectively. This is far too low considering the 2$\sigma$ Planck 2018 bound
of $n_{s}\gtrsim 0.9607$

Figure 1: The quartic hilltop potential at $\lambda=10$. This value if
$\lambda$ is chosen only to clearly illustrate the shape of the potential,
much smaller values of $\lambda$ are required to produce a spectral index and
tensor fraction in agreement with the Planck 2018 measurements.

The $n=2$ case is simply that of Higgs inflation for $\phi\ll m_{pl}$ [31, 32]
for which

$n_{s}=1-\frac{2}{N},\quad r=\frac{8}{N}$ (8)

The spectral index at $N=50$ and $N=60$ are within an acceptable range,
however the tensor fraction is far larger than that allowed by the Planck 2018
bound $r\lesssim 0.1$.

This initial analytical treatment of the hilltop potential was reliant on two
key assumptions. Firstly that during inflation, the value of the scalar field
is much smaller than it’s vacuum expectation value
$\phi_{0}=m_{pl}\lambda^{-\frac{1}{4}}$. Secondly, in order to calculate the
spectral index and tensor fraction as functions of $N$, one must first
calculate

$N(\phi)=\frac{1}{m_{pl}}\int^{\phi}_{\phi_{\textrm{end}}}\frac{V(\phi)}{V^{\prime}(\phi)}d\phi$
(9)

In the initial calculations in [30], it’s assumed that the contribution from
the value of the inflaton field at the end of inflation is negligible. This is
essentially a statement that the model can be assumed to exit inflation on the
plateau where $V\simeq V_{0}$.

The numerical analysis of these models carried out in [24] however revealed
that such models can be compatible with observational data from the Planck
2013 survey if one considers the correct parameter range for $\lambda$. Namely
$\lambda\ll 1$ such that the VEV is super Planckian $\langle\phi\rangle\gg
m_{pl}$. In the Planck 2018 measurements of the spectral index and tensor
fraction, the quartic hilltop (QH) model was featured as one of the more
favourable models. [23]. The results from the extensive numerical
investigations of the quartic hilltop model in [24] were still found to be
compatible with the tighter constraints put on $n_{s}$ and $r$ from the 2018
measurements. However numerical simulations alone do not explain why the model
behaves in this way, only the results that we should expect in such a
parameter range. The behaviour of the QH model was made fully transparent by a
recent analytical calculation of the spectral index and tensor fraction [27],
in which the two previous assumptions are relaxed. When performing these
calculations one finds specifically that the contribution of the inflation
field at the end of inflation $\phi_{\text{end}}$ is

$N_{\textrm{end}}=\frac{1}{4\sqrt{\lambda}}$ (10)

which is clearly large for sufficiently small $\lambda$. This analytical
treatment of the quartic hilltop model made it possible to derive a closed
form expression for the tensor fraction $r$ as a function of the spectral
index $n_{s}$ and remaining number of efolds of inflation (after the scale
exits the horizon) $N$.

$r(n_{s},N)=\frac{8}{3}(1-n_{s})\left[1-\frac{\sqrt{3[2(1-n_{s})N-3]}}{(1-n_{s})N}\right]$
(11)

Equation 11 explains exactly the behaviour of the QH model in numerical
simulations [24, 33]. The relaxing of the assumptions of previous analytical
investigations allows the value of the field during inflation to come closer
to its VEV which is super-planckian in the small $\lambda$ regime, where the
potential’s plateau is broader and flatter allowing for more efolds of
inflation to take place. We are also provided with an explanation for why the
$\phi_{\text{end}}$ contribution cannot be disregarded in equation 9, since
this contribution is proportional to $\lambda^{-\frac{1}{2}}$. It’s clear that
in order for the QH model to be reconcile with observational data, we must
take care in considering how the model exits inflation since $\phi(N)$ is
allowed to come close to its VEV. Furthermore, it is well established that the
potential 7 approximates a linear potential near the VEV, which is already
known to be incompatible with the recent Planck data. Thus to accurately
calculate the predictions of hilltop inflation, one has to account for the
contribution of the vacuum stabilising terms in the potential which are
suppressed during early inflation when $\phi(N)$ is small but become
significant around the VEV.

The importance of these terms is discussed extensively in [34]. The authors
consider general hilltop potentials of the form 5 (reproduced here for
brevity)

$V(\phi)=V_{0}\left(1-\lambda\frac{\phi^{n}}{m_{pl}^{n}}+...\right)$ (12)

stabilised by corrections, denoted “…”, which generically give the potential a
local minima at $\phi_{0}=\lambda^{-1/n}m_{pl}$. For such models, in the limit
$\lambda\gg 1$, inflation ends before $\phi$ is near the VEV, and since
$\phi\ll\phi_{0}$ for the duration of inflation, the ambiguity in the
stabilising terms does not affect the predictions of $n_{s}$ and $r$, so it is
suitable to disregard them in this limit. Of course, as we have outlined, the
predictions of hilltop models in this limit are known to be poorly compatible
with recent measurements, and the model is only a suitable fit in the
$\lambda\ll 1$ regime where the behaviour of the potential around the VEV
becomes significant. However the behaviour of the quartic hilltop model is
vastly unphysical in this region. Since the QH potential is unbounded from
below, after inflation ends, the potential soon becomes negative and the
inflaton field has access to arbitrarily low energy states, and the universe
would begin to collapse on a time scale $t\sim H^{-1}$. As the authors in [34]
point out, this is a generic result of potentials $V(\phi/\mu)$ which are
unbounded from below, and such models should be excluded from the space of
candidate inflationary potentials as they do not produce any sensible
inflationary cosmology. Only those models with regularised potentials can be
considered suitable candidates. Considering the QH model specifically, there
are many ways of ensuring that the potential has a stable vacuum. The simplest
options would just be to add terms which are higher order in
$(\phi/m_{pl})^{n}$. One could also capture the effect of infinitely many such
terms by simply taking the square of the QH potential as suggested in [34]

$V(\phi)=V_{0}\left[1-\lambda\left(\frac{\phi}{m_{pl}}\right)^{4}\right]^{2}$
(13)

Figure 2: The quartic hilltop squared potential with $\lambda=1$ for
illustrative purposes.

For small $\phi$ this potential is $V(\phi)\simeq
V_{0}(1-2\lambda\phi^{4}/m_{pl}^{4})$, thus the behaviour is only modified
near the VEV. This “Quartic Hilltop-Squared” (QHS) potential has been
investigated numerically in [34], and like it’s quartic hilltop cousin, was
found to be in good agreement with the Planck 2018 constraints on $r$ and
$n_{s}$, but with the added benefit of producing sensible, and physically
viable inflationary cosmology due to the vacuum stabilising terms contained in
it’s series expansion.

Accounting for these terms also makes the potential suitable for analysing
reheating after inflation. Which is ultimately necessary as the inflationary
predictions of the model must be compatible with it’s predictions of the
temperature at the end of reheating $T_{re}$ and the number of efolds of
reheating $N_{re}$ which constrain $N$.

###  2.2 Corrected Hilltop Models

While the predictions of the tensor fraction and spectral index for the QHS
model are consistent with the Planck 2018 bounds, the numerical analysis that
these results are derived from still acts in an opaque manner. It tells us
only the results that we should expect for $r(n_{s})$ and not precisely why
the model behaves in a certain way. Further more, this analysis is incomplete
with regards to reheating after inflation. Since the QHS potential is UV
regulated, there exists a local minima of the potential just after inflation
ends around which the inflaton oscillate. The damped oscillations cause the
inflaton field to dump it’s energy into the thermal bath of the universe,
leading eventually the production of ordinary matter. The temperature of the
universe at the end of reheating $T_{re}$ and the number of reheating efolds
$N_{re}$ depend explicitely on the number of efolds of inflation after the
scale exits $N_{k}$.

$T_{re}=V_{0}\left[\left(\frac{43}{g_{re}}\right)^{\frac{1}{3}}\left(\frac{a_{0}T_{0}}{k}\right)H_{k}e^{-N_{k}}\left(\frac{45}{\pi^{2}g_{re}}\right)^{-\frac{1}{3(1+w_{re})}}\right]^{\frac{3(1+w_{re})}{3w_{re}-1}}$
(14)
$N_{re}=\frac{4}{1-3w_{re}}\left(\frac{1}{4}\ln\frac{\pi^{2}g_{re}}{45}+\frac{1}{3}\ln\frac{11}{43g_{re}}+\ln\frac{a_{0}T_{0}}{k}-\ln\frac{V_{end}^{\frac{1}{4}}}{H_{k}}-N_{k}\right)$
(15)

where $g_{re}$ is the effective number of relativistic species at the end of
reheating, $T_{0}$ is the present day CMB temperature, $V_{end}$ is the value
of the inflaton potential at the end of inflation, $H_{k}$ is the value of the
Hubble parameter when the scale exits and $w_{re}$ is the effective equation
of state parameter of the inflaton during reheating. Throughout this paper we
will assume a mean reheating parameter of $w_{re}=1$ as this providers the
most conservative estimates on the predictions of reheating [35].

For a fixed value of $N_{k}$, we see that the $(r,n_{s})$ predictions of the
QH model differ quite drastically from the QHS model, this behaviour can be
better understood by also preforming an analytical investigation of the QHS
model. The analytical methods developed in [27] to investigate the quartic
hilltop model have also been used in [26] analyse the inflationary predictions
of QHS. In that analysis we derive a closed form expression for $r(n_{s})$

$r=\begin{cases}\frac{1}{4N}\left(\frac{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15+\sqrt{8N\tilde{n}_{s}-15}}{4N\tilde{n}_{s}-N^{2}\tilde{n}_{s}^{2}+\sqrt{8N\tilde{n}_{s}-15}}\right)\left[1+g_{+}(\tilde{n}_{s},N)\right]\left[g_{+}(\tilde{n}_{s},N)-7\right],\quad\lambda\geq\lambda_{c}\\\
\\\
\frac{1}{4N}\left(\frac{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15-\sqrt{8N\tilde{n}_{s}-15}}{4N\tilde{n}_{s}-N^{2}\tilde{n}_{s}^{2}-\sqrt{8N\tilde{n}_{s}-15}}\right)\left[1+g_{-}(\tilde{n}_{s},N)\right]\left[g_{-}(\tilde{n}_{s},N)-7\right],\quad\lambda<\lambda_{c}\end{cases}$
(16)

where

$g_{\pm}(\tilde{n}_{s},N)=\sqrt{\frac{12N\tilde{n}_{s}+62N^{2}\tilde{n}_{s}^{2}-16N^{3}\tilde{n}_{s}^{3}-15\pm(1+16N\tilde{n}_{s})\sqrt{8N\tilde{n}_{s}-15}}{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15\pm\sqrt{8N\tilde{n}_{s}-15}}}$
(17)

and the critical parameter value is $\lambda_{c}=(60N)^{-\frac{1}{2}}$. As
noted previously, in the QH model, there is a contribution
$N_{end}^{QH}=1/4\sqrt{\lambda}$ to the total efolds of inflation coming from
the value of the field at the end of inflation $\phi_{end}$. In the QHS model
this contribution is found to be $N_{end}^{QHS}=1/8\sqrt{\lambda}$ (when
$\lambda\ll 1$) the contribution is half as small as in the QHS model, for
small $\lambda$ the higher order terms begin to contribute significantly, and
push the field out of slow roll inflation sooner than in QH. For all of the
corrected quartic hilltop models in this paper, we work in the small $\lambda$
regime, where the symmetry breaking scale $\mu=m_{pl}(4\lambda)^{-1/4}$ is
large [36], thus avoiding the region of parameter space that is not compatible
with the Planck results.

Furthermore, the authors complete the analysis of QHS model by computing it’s
reheating temperature and efolds. Computing the reheating parameters allows us
to further constrain the parameter space for QHS by demanding that the
reheating temperature be bounded below by the Big Bang Nucleosynthesis scale
$T_{BBN}\lesssim 0.01\text{GeV}\lesssim T_{re}$ and bounded above by the
energy scale of inflation $T_{re}\lesssim 10^{16}\text{GeV}$. In fact, the
size of the acceptable region in $(r,n_{s})$ space of the QHS model reduces
significantly after reheating consistency is taken into account. Before
reheating, whilst only bounded by the Planck 2018 data, the QHS model was
constrained to the region defined by

$\displaystyle 0.9607\lesssim$ $\displaystyle n_{s}\lesssim 0.9691$ (18)
$\displaystyle 55\lesssim$ $\displaystyle N_{k}$

whereas after taking into account reheating consistency, this is reduced to

$\displaystyle 0.9607\lesssim$ $\displaystyle n_{s}\lesssim 0.9691$ (19)
$\displaystyle 63\lesssim$ $\displaystyle N_{k}\lesssim 68$

The fact that the inflationary potential must be UV regularised and that the
models predictions of the tensor fraction and spectral index must be
consistent with reheating bounds significantly constrains the models parameter
space which is in agreement with the Planck 2018 bounds.

Choosing to regularise the potential by squaring it, is simply one possibility
out of a great many. In this work, we show that how one chooses the regularize
the shape of the potential near the end of inflation can have an important and
quantifiable effect on the models predictions of $n_{s}$ and $r$. The QHS
potential essentially amounts to adding a $(\phi/m_{pl})^{8}$ correction term
to the original $QH$ potential, which is seen simply by expanding out the
potential and rescaling the coupling parameter $\lambda$.

$V_{\textrm{QH}}(\phi)=V_{0}\left[1-\lambda\left(\frac{\phi}{m_{pl}}\right)^{4}\right]^{2}$
(20) $\displaystyle V_{\textrm{QHS}}(\phi)$
$\displaystyle=V_{0}\left[1-2\lambda\left(\frac{\phi}{m_{pl}}\right)^{4}+\lambda^{2}\left(\frac{\phi}{m_{pl}}\right)^{8}\right]$
(21)
$\displaystyle=V_{0}\left[1-\tilde{\lambda}\left(\frac{\phi}{m_{pl}}\right)^{4}+\frac{1}{4}\tilde{\lambda}^{2}\left(\frac{\phi}{m_{pl}}\right)^{8}\right]$

As well as squaring the potential, one may also consider what the effect of
adding single polynomial terms $\phi^{p}$ to the potential are. We only wish
to correct the behaviour of the potential for large $\phi$, so we look at
positive values of $p$. Furthermore, the coefficient of the $\phi^{p}$ term
must be fine-tuned to ensure that the VEV remains at $V(\phi_{0})=0$ so that
inflation ends in a finite time. Therefore consider potentials of the form

$V(\phi)=V_{0}\left[1-\lambda\left(\frac{\phi}{m_{pl}}\right)^{4}+\alpha_{p}\left(\frac{\phi}{m_{pl}}\right)^{p}\right],\quad
p>4$ (22)

The potential has stationary points at

$\phi^{3}\left(p\alpha_{p}\phi^{p-4}-4\lambda\right)=0$ (23)

and thus the VEV is located at

$\phi_{0}=\left(\frac{4\lambda}{p\alpha_{p}}\right)^{\frac{1}{p-4}}$ (24)

Requiring that $V(\phi_{0})=0$ allows us to solve for the fine-tuning of the
coefficient $\alpha_{p}$

$\alpha_{p}=4\lambda^{\frac{p}{4}}p^{-\frac{p}{4}}\left(p-4\right)^{\frac{p-4}{4}}$
(25)

So to add polynomial corrections to the Quartic Hilltop model, we work with
potentials of the form

$V(\phi)=V_{0}\left[1-\lambda\left(\frac{\phi}{m_{pl}}\right)^{4}+4\lambda^{\frac{p}{4}}p^{-\frac{p}{4}}\left(p-4\right)^{\frac{p-4}{4}}\left(\frac{\phi}{m_{pl}}\right)^{p}\right]$
(26)

Henceforth we will refer to such models as $\text{QH}_{p}$, with the quartic
hilltop squared model being equivalent to $\text{QH}_{8}$ as per equation 21.

The lowest order term that we can add to modify the small-field behaviour of
the potential but retain its plateau and large-field shape is $p=5$ since the
QH model already contains a $\phi^{4}$ term. As we shall see from the results
later in this section, there is no need to investigate larger than $p=10$, so
we consider only terms in this range. Starting from $p=5$ through to $p=10$ we
have calculated the tensor ratio $r$ as a function of the spectral index
$n_{s}$ and the temperature at the end of reheating $T_{re}$ as a function of
the spectral index between $50\leq N_{k}\leq N_{k}=70$. We then determine
which of these curves are reheating-consistent by demanding that they lie
within the rectangle on the $T_{re}-n_{s}$ plots. The horizontal bounds on the
reheating temperature plots are $0.01\text{GeV}\lesssim T_{re}\lesssim
10^{16}\text{GeV}$. The reheating temperature is bounded below by the energy
scale of Big Bang Nucleosynthesis (BBN) [37, 38] $T_{BBN}\sim 10\text{MeV}$.
Measurements of the CMB anisotropies constraining the tensor fraction $r$ are
equivalent to upper bounds on the energy scale of inflation [23, 38, 39] since

$V_{*}^{\frac{1}{4}}=\left(\frac{3\pi^{2}A_{s}}{2}rm_{pl}\right)^{\frac{1}{4}}\lesssim
10^{16}\text{GeV}$ (27)

where $V_{*}$ is the energy scale of inflation and $A_{s}$ is the amplitude of
scalar perturbations. This bounds $T_{re}$ from above.

In all of the following $r-n_{s}$ plots, we produce results from numerical
solutions of the tensor fraction and spectral index and the corresponding
temperature at the end of reheating for $\text{QH}_{p}$ models with $p=5,10$.
Figures for $p=6,7,9$ are contained in Appendix B (omitting
$\text{QH}_{8}\equiv\text{QHS}$). The solid and dashed black curves are the
Planck 2018 $1\sigma$ and $2\sigma$ bounds respectively. The horizontal black
line is the BICEP/Keck Array bound on the tensor fraction $r<0.036$ [39].
Similarly, in all following $T_{re}-n_{s}$ plots, the upper horizontal line is
the energy scale of inflation bound $T_{re}\lesssim 10^{16}\text{GeV}$ and the
lower horizontal line is the BBN bound $0.01\text{GeV}\lesssim T_{re}$. The
vertical solid lines represent the Planck 2018 bounds $0.9607\lesssim
n_{s}\lesssim 0.9691$.

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{QH}_{5}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{QH}_{5}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 3:

Starting with the $\text{QH}_{5}$ model, only those curves with $58\lesssim
N_{k}\lesssim 68$ are within the acceptable region of reheating temperate and
spectral index in figure 3(b). However when one also takes into account the
$1\sigma$ region in figure 3(a) this is further reduced to $60\lesssim
N_{k}\lesssim 68$. Requiring that we consider both the Planck-consistent and
reheating-consistent curves thus drastically reduces the acceptable region of
$r-n_{s}$ parameter space for the corrected quartic hilltop model. This is
significant as the corrections themselves are required for such models to even
be taken seriously as candidates for single-field slow-roll inflation as
without regularizing the potentials they produce a cosmology entirely
incompatible with a universe that does not immediately collapse after
inflation ends.

For $\text{QH}_{5}$, the resulting acceptable region on the $r-n_{s}$ plot
lies in the mid to upper-left corner of the $1\sigma$ bound. If further, more
precise measurements were to constrain the tensor fraction and spectral index
away from this region, $\text{QH}_{5}$ would quickly become untenable.

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{QH}_{10}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{QH}_{10}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 4:

Considering the $\text{QH}_{10}$ in figure 4(a), we see that increasing $p$
causes more curves to enter the $1\sigma$ region of the $r-n_{s}$ plot. If one
were to not consider reheating consistency this would place a lower bound of
$56\lesssim N_{k}$ on the number of efolds of inflation after the scale exits.
However as we increase the power of the correction term, the curves in the
(log) reheating temperature plot move very slowly and by $p=10$ no new curves
have entered or left the acceptable region in figure 4(b). Reheating
consistency of $\text{QH}_{10}$ thus still requires $60\lesssim N_{k}\lesssim
68$ just as with the lowest order correction $\text{QH}_{5}$. In contrast to
$\text{QH}_{5}$ we see that now the curves for which $54\lesssim N_{k}\lesssim
58$ have now entered the $1\sigma$ region in figure 4(a) however must still
reject these as they remain outside the acceptable region of the reheating
plot.

(a) The $\text{QH}_{5}$ potential at $\lambda=1$

(b) The $\text{QH}_{10}$ potential at $\lambda=1$

Figure 5:

In figures 5(a) and 5(b) we plot the potentials of two of the corrected
hilltop models $\text{QH}_{5}$ and $\text{QH}_{10}$ for $\lambda=1$. The
potentials now have a stable vacuum about which the field can oscillate during
the reheating phase.

Overall, the region of reheating-consistent parameter space in the $r-n_{s}$
plot actually increases with increasing $p$, as the distance between the
curves remains relatively constant and they are only shifted along laterally.
Initially it may then seem that one could simply add arbitrarily high powers
to the quartic hillop model in order to increase the region of validity.

##  3 Radiatively Corrected Higgs Inflation

The case of the quartic hilltop model demonstrates clearly that we must only
consider those potentials that are regularised and will not collapse the
universe immediately after exiting inflation. How one regularises the
potential is important if the model is to be compared to measurements. A huge
number of candidate single field inflation models are analysed in [24]. Out of
these models there are two which suffer from the same vacuum stabilisation as
quartic hilltop. The first of these is the Radiatively Corrected Higgs
Inflation (RCHI) model. This model derives from taking into account 1-loop
corrections to the Higgs inflation model, in which the inflaton is a Higgs
particle [31, 40, 41, 42, 43]. If one considers the standard model with the
Higgs non-minimally coupled to classical gravity, the simplest such action in
the Jordan frame is

$S_{T}=\int d^{4}x\sqrt{-g}\left[\frac{1}{2}m_{pl}^{2}R+\xi
H^{\dagger}HR+\mathcal{L}_{SM}\right]$ (28)

where $H^{\dagger}$ is the Higgs doublet and $\xi>0$ is a dimensionless
coupling parameter.

If we consider only the gravi-scalar sector of the theory and make use of the
unitary gauge for which $H=(0,h/\sqrt{2})$ where $h$ is a real scalar field,
then the action of interest is

$S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}F(h)R-\frac{1}{2}\partial_{\mu}h\partial^{\mu}h-W(h)\right]$
(29)

the functions $F(h)$ and $W(h)$ which we define now for later convenience are

$F_{1}(h)=(m_{pl}^{2}+\xi h^{2}),\quad W(h)=\frac{\lambda}{4}(h^{2}-\nu^{2})$
(30)

where $\nu$ is the elctroweak symmetry breaking scale. After a conformal
transformation of the metric

$g_{\mu\nu}\rightarrow\theta(x)g_{\mu\nu},\quad\theta(x)=\frac{m_{pl}^{2}}{m_{pl}^{2}+\xi
h^{2}}$ (31)

we obtain the Einstein frame action

$S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}m_{pl}^{2}R-\frac{1}{2}m_{pl}^{2}K(\theta)\partial_{\mu}\theta\partial^{\mu}\theta-V(\theta)\right]$
(32)

in which the scalar field $\theta$ has a non-trivial kinetic term

$K(\theta)=\frac{1}{4|a|\theta^{2}}\left(\frac{1-6|a|\theta}{1-\theta}\right),\quad
a=-\frac{\xi}{1+6\xi}$ (33)

After rescaling to a canonically normalised field the action becomes

$S=\int
d^{4}x\sqrt{-g}\left[\frac{1}{2}m_{pl}^{2}R-\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-V(\phi)\right]$
(34)

where the field $\phi$ satisfied the differential equation

$\frac{1}{M_{pl}^{2}}\left(\frac{d\phi}{d\theta}\right)^{2}=K(\theta)$ (35)

The exact solution, which we omit here for brevity, is given in [31]. At tree
level, where the coupling in the Jordan frame $\xi$ is large and $a\simeq 1/6$
the Einstein frame potential is approximately

$V(\phi)=\frac{M_{pl}^{4}\lambda}{4\xi^{2}}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{\phi}{m_{pl}}}\right)^{2}$
(36)

We see from equation 36 that the Higgs self coupling $\lambda$ and coupling to
gravity $\xi$ only enter the potential through it’s overall normalisation so
the ratio $\sqrt{\lambda}/\xi$ is completely determined from the CMB
normalisation. However, the 1-loop radiative corrections to the effective
action 29 in the Jordan frame contribute more significantly at large values of
$\xi$, so it is not sufficient to consider only the tree level approximation
[44]. The radiative corrections modify the functions $F(h)$ and $W(h)$. To
first order these corrections are

$F(h)=m_{pl}^{2}+\xi
h^{2}+\frac{C}{16\pi^{2}}h^{2}\ln\left(\frac{m_{pl}^{2}h^{2}}{\mu^{2}}\right)$
(37)

$W(h)=\frac{1}{4}\lambda\left(h^{2}-\nu^{2}\right)^{2}+\frac{\lambda
A}{128\pi^{2}}h^{4}\ln(\frac{h^{2}}{\mu^{2}})$ (38)

The modifications to these functions change the resulting Einstein frame
potential from which we compute the inflationary observable $n_{s}$ and $r$.
The Radiatively Corrected Higgs Inflation (RCHI) potential is thus

$V(\phi)\simeq
V_{0}\left(1-2e^{-\frac{2\phi}{\sqrt{6}m_{pl}}}+\frac{A_{I}}{16\pi^{2}}\frac{\phi}{\sqrt{6}m_{pl}}\right)$
(39)

where $A_{I}$ is a free parameter as given in eq 4.12 of [24].

Before moving on to calculating corrections terms to the RCHI model we take
note that there exists in the literature a modification to the Higgs inflation
model that produces a regularized potential (as opposed to equation 39) which
is unbounded from below. Firstly, in the literature there has been debate over
the validity of the Higgs inflation model due to it’s UV cutoff being very
close to the energy scale of inflation $H\sim\sqrt{\lambda}m_{pl}/\xi$ [28,
29, 45, 46, 47] The cutoff scale, at which unitarity is broken for tree-level
amplitudes, is calculated in [29] as $\Lambda\sim m_{pl}/\xi$ in the Einstein
frame. For non-minimal coupling $\xi$ and $\lambda\sim O(1)$ the effective
field theory may not be a valid description of inflationary dynamics. In [28]
the authors point out that these calculations of the cutoff scale are
performed in the small field approximation $\phi\approx h$. However the
authors note that the inflationary regime has $\phi\gg 1$, in which the
Einstein frame potential is

$\displaystyle V(\phi)$
$\displaystyle\approx\frac{\lambda}{4\xi^{2}}\left(1+e^{-\frac{2\phi}{\sqrt{6}}}\right)^{2}$
(40)
$\displaystyle\approx\frac{\lambda}{4\xi^{2}}\left(1+2e^{-\frac{2\phi}{\sqrt{6}}}\right)$

If one considers the series expansion of the potential in powers of $\phi$,
the result is quite different to what would have been obtained in the small
field approximation

$V(\phi)\approx\frac{3\lambda}{4\xi^{2}}-\frac{\lambda}{\xi^{2}\sqrt{6}}\phi+\frac{\lambda}{6\xi^{2}}\phi^{2}-\frac{\lambda}{9\sqrt{6}\xi^{2}}\phi^{3}+...$
(41)

thus the one-loop quantum corrections will contribute a cutoff

$\Lambda\sim\frac{\xi^{2}}{\lambda}m_{pl}$ (42)

This unitarity bound is well above the energy scale of inflation and so the
effective field theory remains a safe description of inflationary physics with
a regularized potential given by equation 40. In this paper however we focus
only on the RCHI model featured in [24] as a means of demonstrating the effect
of correction terms required by such unregularized potentials.

The Radiatively Corrected Higgs Inflation model shares similarities to Quartic
Hilltop in that it also fits within the $1\sigma$ region of the Planck 2018
data over a given range of it’s free parameter $A_{I}$, but does not produce a
cosmology consistent with a universe that does not collapse

(a) Numerical solutions for tensor fraction $r$ and spectral index $n_{s}$ of
the RCHI model over the range $1\leq A_{I}\leq 40$ for $50\leq N_{k}\leq 70$.
The solid and dashed curves represent the Planck 2018 $1\sigma$ and $2\sigma$
bounds respectively. The horizontal black line is the BICEP tensor fraction
bound $r\lesssim 0.032$.

(b) Numerical solutions for reheating temperature $T_{re}$ and spectral index
$n_{s}$ of the RCHI model over the range $1\leq A_{I}\leq 40$ for $50\leq
N_{k}\leq 70$. The vertical solid lines represent the Planck 2018 spectral
index bounds $0.9607\lesssim n_{s}\lesssim 0.9691$ and the horizontal solid
lines represent the reheating temperature bounds $0.01\text{GeV}\lesssim
T_{re}\lesssim 10^{16}\text{GeV}$.

Figure 6:

In figure 6(a) that there are indeed a few curves that fit well within the
$1\sigma$ region provided by the Planck 2018 data, specifically those with
$50\lesssim N_{k}\lesssim 60$. When we take into account reheating consistency
over the same parameter range in figure 6(b) we see that lie within the
acceptable region formed by the rectangular bounds, essentially ruling out
this model as a possibility. Like the quartic hilltop model, this potential
also fails to produce a cosmology compatible with what we observe today. In
the RCHI model, we exit inflation at small $\phi$ and shortly after the
potential becomes negative. This behaviour of the potential needs to be
corrected. There are a variety of ways one may correct the behaviour of this
potential, as we did to form the QHS model. As such we could first consider
simply squaring the potential to produce one which is regularised. However for
RCHI, the potential only needs to be corrected at small field values as we
would like to retain the flatness of the potential during slow-roll. Squaring
the potential does not achieve this very well as it makes the potential very
steep at large $\phi$.

(a) Numerical solutions for tensor fraction $r$ and spectral index $n_{s}$ of
the RCHI-Squared model over the range $1\leq A_{I}\leq 40$ for $50\leq
N_{k}\leq 70$. The solid and dashed curves represent the Planck 2018 $1\sigma$
and $2\sigma$ bounds respectively. The horizontal black line is the BICEP
tensor fraction bound $r\lesssim 0.032$

(b) Numerical solutions for reheating temperature $T_{re}$ and spectral index
$n_{s}$ of the RCHI model over the range $1\leq A_{I}\leq 40$ for $50\leq
N_{k}\leq 70$. The vertical solid lines represent the Planck 2018 spectral
index bounds $0.9607\lesssim n_{s}\lesssim 0.9691$ and the horizontal solid
lines represent the reheating temperature bounds $0.01\text{GeV}\lesssim
T_{re}\lesssim 10^{16}\text{GeV}$.

Figure 7:

Compared to the standard RCHI model, the $r-n_{s}$ curves in figure 7(a) are
elongated and far more dramatically curved outside of the $2\sigma$ region.
Whilst the curves with $50\lesssim N_{k}\lesssim 58$ lie within the $1\sigma$
region, it still remains the case that none of the curves are able to be made
reheating-consistent. Furthermore, it is also worth noting that one needs at
least 60 efolds of inflation to solve the horizon problem, making a squared
version of the RCHI potential even less favourable.

(a) The RCHI potential at $A_{I}=1$

(b) The RCHI-Squared potential at $A_{I}=1$.

Figure 8:

The simplest such terms that one could add the only modify the potential at
small $\phi$ are inverse powers of the form $\phi^{-p}$ where $p>0$.
Henceforth we refer to such models as $\text{RCHI}_{p}$. Just as we did when
adding polynomial correction to the quartic hilltop model, we must ensure that
the coefficient of the correction term keeps the scalar fields VEV at zero in
order for inflation to end in a finite time. That is, consider a potential of
the form

$V(\phi)=1-2e^{-\frac{2\phi}{\sqrt{6}m_{pl}}}+\frac{A_{I}}{16\pi^{2}}\frac{\phi}{\sqrt{6}m_{pl}}+\alpha_{p}6^{\frac{p}{2}}\phi^{-p}$
(43)

The coefficient $\alpha_{p}$ must be such that $V(\phi_{0})=0$, where
$\phi_{0}$ is the field value at the minimum of the potential and therefore
satisfies $V^{\prime}(\phi_{0})=0$

$\frac{4\phi_{0}}{\sqrt{6}m_{pl}}e^{-\frac{2\phi_{0}}{\sqrt{6}m_{pl}}}+\frac{A_{I}}{16\pi^{2}\sqrt{6}m_{pl}}-p\alpha_{p}6^{\frac{p}{2}}\phi_{0}^{-(p+1)}=0$
(44)

Equation 44 cannot be solved analytically for $\phi_{0}$ but we may make use
of an approximation. The minima of the potential is at small field values
$\phi_{0}/m_{pl}\lesssim 1$ and thus

$e^{-\frac{2\phi_{0}}{\sqrt{6}m_{pl}}}\approx
1-\frac{2\phi_{0}}{\sqrt{6}m_{pl}}$ (45)

The $\phi_{0}^{-(p+1)}$ term will dominate the expression and thus we may
solve for $\phi_{0}(\alpha_{p})$

$\phi_{0}\simeq\sqrt{6}\left(\frac{16\pi^{2}p\alpha_{p}m_{pl}}{A_{I}+64\pi^{2}}\right)^{\frac{1}{p+1}}$
(46)

Now substituting approximate solution 46 into $V(\phi_{0})=0$ allows us to
solve for the coefficient

$\alpha_{p}=\left(\frac{16\pi^{2}m_{pl}}{A_{I}+64\pi^{2}}\right)^{p}p^{p}(1+p)^{-(p+1)}$
(47)

and so we work with an approximate potential

$V(\phi)=1-2e^{-\frac{2\phi}{\sqrt{6}m_{pl}}}+\frac{A_{I}}{16\pi^{2}}\frac{\phi}{\sqrt{6}m_{pl}}+\left(\frac{16\pi^{2}m_{pl}}{A_{I}+64\pi^{2}}\right)^{p}p^{p}(1+p)^{-(p+1)}6^{\frac{p}{2}}\phi^{-p}$
(48)

In the following figures we display the $r-n_{s}$ and $T_{re}-n_{s}$ plots for
$\text{RCHI}_{p}$ with $1\leq p\leq 10$. The bounds on the parameter spaces
are the same as in all previous figures.

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{1}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{1}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 9:

Starting with the $\text{RCHI}_{1}$ model which contains a $\phi^{-1}$
correction term, it can be seen in figure 9(a), that the $1\sigma$ region
contains those curves with $50\lesssim N_{k}\lesssim 62$. Taking into account
reheating consistency in figure 9(b), we see that the reheating-consistent
curves are those with $58\lesssim N_{k}\lesssim 62$. However, these curves
only slightly fit into the acceptable region of the $T_{re}-n_{s}$ plot over
the parameter range $1\leq A_{I}\leq 40$, so the likelihood that they would
survive any further constraining by more precise measurements of $n_{s}$ is
low.

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{10}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{10}$ model against
the spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 10:

As we increase the power of the correction term through to $p=10$, the effect
of this term in the potential is of course much stronger at small $\phi$ where
the model exits inflation, and much weaker at large $\phi$ where it starts.
Figures for $2\leq p\leq 9$ are given in Appendix C. The effect on the
$r-n_{s}$ plot compared to that of $\text{RCHI}_{1}$ is to push out of a few
of the curves, leaving those with $50\lesssim N_{k}\lesssim 60$ in the
$1\sigma$ region. This is a relatively small change in the $r-n_{s}$ parameter
space. Conversely, in figure 10(b) we see that all of the curves have been
pushed out of the acceptable in the reheating plot, leaving none that are
reheating consistent over this parameter range.

Clearly the models become less viable as one increases the strength of the
correction term. We also see from figures 11(a) and 11(b) that as the the
power of the correction term increases the approximation for the coefficient
$\alpha_{p}$ given in equation 47 becomes less accurate and the local minima
is displaced from $V(\phi_{0})\simeq 0$. Given that the $\text{RCHI}_{1}$
model only just survives reheating-consistency, it’s likely that RCHI and
corrections thereof will not fair well under any future, more precise
measurements.

(a) The $\text{RCHI}_{1}$ potential at $A_{I}=1$

(b) The $\text{RCHI}_{10}$ potential at $A_{I}=1$.

Figure 11:

##  4 Exponential SUSY Inflation

The final model we will investigate is that of Exponential SUSY Inflation
(ESI). ESI models are governed by potentials of the form

$V(\phi)-V_{0}\left(1-e^{-q\frac{\phi}{m_{pl}}}\right)$ (49)

where $q$ is the free parameter of order 1.

Potentials of this form appear in a broad range of literature [48, 49, 50, 51,
52], so we will not discuss the precise nature in which they occur in detail
and only focus on the inflationary predictions of the potential 49 and
modifications thereof. Exponential SUSY inflation is a large field inflation
model, one can calculate exactly the field value when inflation ends [24]

$\frac{\phi_{\textrm{end}}}{m_{pl}}=\frac{1}{q}\ln\left(1+\frac{q}{\sqrt{2}}\right)$
(50)

For $q\sim\mathcal{O}(1)$, inflation ends at small field values
$\phi_{end}/m_{pl}\lesssim 1$, after which the potential becomes negative. The
ESI model is another of the single field inflation models featured in [24]
which fits well within the Planck 2018 bounds. However like RCHI and QH, the
potential remains unregularized and thus the model cannot lead to any
physically realised cosmology.

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the ESI model with $0<q<10$ over $50\leq N_{k}\leq 70$. The solid
and dashed curves represent the Planck 2018 $1\sigma$ and $2\sigma$ bounds
respectively.

(b) The (log) reheating temperature of the ESI model with against the spectral
index $n_{s}$ with $0<q<10$ over $50\leq N_{k}\leq 70$. The vertical solid
lines represent the Planck 2018 spectral index bounds $0.9607\lesssim
n_{s}\lesssim 0.9691$ and the horizontal solid lines represent the reheating
temperature bounds $0.01\text{GeV}\lesssim T_{re}\lesssim 10^{16}\text{GeV}$

Figure 12:

If one considers only the $r-n_{s}$ plot in figure 12(a), ESI is among the
most promising of the three models investigated in this paper, with all curves
between $52\lesssim N_{k}\lesssim 68$ lying within the $1\sigma$ region. In
figure 12(b) however, only the curves for $56\lesssim N_{k}\lesssim 64$ lie
withing the acceptable region, and much like for the $\text{RCHI}_{p}$ models,
only a small proportion of the curves fit within the Planck 2018 $n_{s}$
bounds, particularly for larger $N_{k}$, making it difficult for this model to
remain viable if the parameter space were to be further constrained towards
the centre of the rectangular bound. Of course we argue that these results are
of little significance due to the unregularized behaviour of the potential.
Just as with QH and RCHI this can be corrected by taking the square of the
potential and calculating the resulting spectral index, tensor fraction and
temperature at the end of reheating.

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the ESI-Squared model with $0<q<10$ over $50\leq N_{k}\leq 70$.
The solid and dashed curves represent the Planck 2018 $1\sigma$ and $2\sigma$
bounds respectively.

(b) The (log) reheating temperature of the ESI-Squared model with against the
spectral index $n_{s}$ with $0<q<10$ over $50\leq N_{k}\leq 70$. The vertical
solid lines represent the Planck 2018 spectral index bounds $0.9607\lesssim
n_{s}\lesssim 0.9691$ and the horizontal solid lines represent the reheating
temperature bounds $0.01\text{GeV}\lesssim T_{re}\lesssim 10^{16}\text{GeV}$

Figure 13:

Just as with quartic hilltop and RCHI, one may try simply squaring the ESI
potential to obtain a new potential that is bounded below. The squaring of the
potential is motivated simply as a means of obtaining a new potential which
behaves similarly to the original potential during inflation but has modified
behaviour as the inflaton exits inflation. Namely that the potential is
bounded from below and has a stable vacuum. From both figures 13(a) & 13(b) we
see that squaring the potential does little to change the inflationary
predictions of the model but gives allows the potential to produce a cosmology
compatible with a universe that does not immediately collapse after inflation.
The reheating-consistent curves remain those with $56\lesssim N_{k}\lesssim
64$, although they still cover only a small fraction of the parameter space in
figure 13(b).

(a) The ESI potential at $q=1$

(b) The ESI-Squared potential at $q=1$.

Figure 14:

Unlike QH, we cannot add polynomial terms as an option to correct the ESI
model at the end of inflation, since the exponential function will dominate
over any $\phi^{p}$ term we can add towards the end of inflation. We could
however, consider adding an inverse power correction of the form $\phi^{-p}$
as we did with RCHI to form the family of $\text{RCHI}_{p}$ models.

Consider a potential of the form

$V(\phi)=1-e^{-q\frac{\phi}{m_{pl}}}+\alpha_{p}\phi^{-p}$ (51)

Just as with the $\text{QH}_{p}$ and $\text{RCHI}_{p}$ models, we must adjust
the coefficient $\alpha_{p}$ to ensure that the potentials at the VEV remains
at $V(\phi_{0})\simeq 0$ to first order so that inflation ends in a finite
time. The VEV $\phi_{0}$ satisfies $V^{\prime}(\phi)=0$ and thus

$\frac{q}{m_{pl}}e^{-q\frac{\phi_{0}}{m_{pl}}}-p\alpha_{p}\phi_{0}^{-(p+1)}=0$
(52)

For sufficiently small $\phi_{0}$ equation 52 is dominated by the
$\phi_{0}^{-(p+1)}$ term and thus to first order

$\phi_{0}^{p+1}\simeq\frac{p\alpha_{p}m_{pl}}{q}$ (53)

Demanding that the potential at the VEV is zero, one arrives at the equation

$\phi_{0}\left(1+\frac{1}{p}\right)=0$ (54)

for generic $p$ this would require $\alpha_{p}=0$,thus to a first
approximation in the methods used here, there are no such correction terms
that we can find for ESI. There may be other regularization methods outside
the scope of this analysis which could perform this role.

##  5 Discussion

In this paper we have investigated three promising candidate single field
inflation models. These models all belong to the subset of candidate inflation
models that are capable of producing acceptable an acceptable spectral index
and tensor fraction in light of the Planck 2018 data, but do not predict any
kind of sensible cosmology that is consistent with what we are simply around
to observe the universe today. The issue lies in the unstabilized nature of
the potentials after inflation ends. The quartic hilltop, radiatively
corrected higgs and exponential SUSY potentials do not posses local minima and
so there is no stable vacuum about which the inflaton may oscillate and reheat
the universe after expansion. In fact, since the potentials are unbounded from
below, if the energy density of the universe continues to be dominated by the
inflaton it will collapse on a time scale $t=H^{-1}$ after the end of
inflation. This is a generic feature of any such potential that is not
stabilized after inflation. The question remains as to whether regularizing
the behaviour of these potentials has a significant effect on the inflationary
predictions of the models, which indeed it does.

The corrections that are made to the inflationary potentials in this model are
not motivated by a particular physical principle, however such potentials do
indeed have to be corrected, otherwise the resulting cosmology is simply not
compatible with the cosmology we observe today. We show that any such
correction terms are significant in terms of their effect on the inflationary
predictions of the model. There are of course examples of more physically
motivated corrections to inflationary models that previously suffered from the
same issue of unboundedness. Brane inflation models featured in [24] can be
consistently modified in a way that does not introduce new fine tuning
parameters, to fit the Planck 2018 data in KKTLI inflation [53]. Likewise, the
Higgs inflation model featured also has a physically motivated modification
scheme that is consistent with Planck [28, 29]. In this approach, the
corrections to the functions $F(h)$ and $W(h)$ in equations 37 & 38 are
calculated in the Einstein frame, giving an inflationary potential 40 that is
everywhere positive with a local minima.

Potentials with regularized behaviour may undergo the reheating period after
inflation without the universe collapsing in the process. The reheating
temperature must at least as large as the BBN energy scale, and no greater
than the energy scale of inflation, and so is loosely bounded by
$0.01\text{GeV}\lesssim T_{re}\lesssim 10^{16}\text{GeV}$. Since $T_{re}$
depends explicitly on the amount of inflation that has occurred, through
$N_{k}$ and $H_{k}$. Demanding reheating-consistency further constrains the
acceptable region of the $r-n_{s}$ parameter space when combined with the
Planck 2018 and BICEP/Keck Array measurements. Furthermore, at least
approximately 60 efolds of inflation are required to solve the horizon problem
[8], this consideration also allows us to pinpoint the regions of $r-n_{s}$
and $T_{re}-n_{s}$ inhabited by these regularized models.

For the quartic hilltop model investigated in [26], before taking into account
reheating consistency, one may fit all curves with $53\lesssim N_{k}$ into the
acceptable region of $r-n_{s}$ parameter space, however after regularizing to
form the QHS model, reheating consistency reduces this to only the curves with
$63\lesssim N_{k}\lesssim 68$ corresponding to reheating temperatures
$1.8\times 10^{-2}\text{GeV}\lesssim T_{re}\lesssim 6.0\times
10^{4}\text{GeV}$. In this paper we investigate further options for correcting
the behaviour of QH by adding polynomial terms $\phi^{p}$ forming the class of
$QH_{p}$ models for $5\leq p\leq 10$. The lowest order corrected model
$QH_{5}$ has reheating consistent curves $60\lesssim N_{k}\lesssim 68$
corresponding to reheating temperatures $1\text{GeV}\lesssim T_{re}\lesssim
7.2\times 10^{10}\text{GeV}$. As we increase the strength of the correction
term through to $QH_{10}$, the reheating-consistent curves in the $r-n_{s}$
parameter space still remain only those with $60\lesssim N_{k}\lesssim 68$ as
the spectral index and tensor fraction $N_{k}$ efolds before the end of
inflation are not particularly sensitive to the strength of the correction
term which kicks in towards the end of inflation. However the reheating
temperature is sensitive to this change and for $QH_{10}$ these curves
correspond to reheating temperatures of $8.2\times 10^{-2}\text{GeV}\lesssim
T_{re}\lesssim 1.1\times 10^{13}\text{GeV}$

The next model in our discussion is that of Radiatively Corrected Higgs
Inflation. Out of the three models considered in this paper, RCHI appears to
be the least favourable, having already started in a precarious position even
before regularization. Un-regularized RCHI potential allows only those curves
with $50\lesssim N_{k}\lesssim 60$ in the acceptable region of $r-n_{s}$
parameter space, considering that we require at least 60 efolds for the
horizon problem that leaves only a singular curve $N_{k}\simeq 60$ as a
potential candidate. However, none of the curves enter the region of
reheating-consistency in the $T_{re}-n_{s}$ space. As we did with quartic
hilltop, we may attempt to regularize this model by simply squaring the
potential. The RCHI-Squared potential however only admits curves with
$50\lesssim N_{k}\lesssim 58$ in the acceptable region of $r-n_{s}$ space, so
it does not address the horizon problem, which is an important corner stone of
scalar field inflation models. Squaring the RCHI potential significantly
alters the shape of the potential during inflation, so we may look for options
that only affect the potential towards the end of inflation when the inflation
field $\phi$ is small. The simplest such choices would be inverse power
corrections of the form $\phi^{-p}$, forming a family of $\text{RCHI}_{p}$
models. We investigate these for $1\leq p\leq 10$. $\text{RCHI}_{1}$ admits
curves with $58\lesssim N_{k}\lesssim 62$ in the acceptable region of
$r-n_{s}$ space, further constrained to $60\lesssim N_{k}\lesssim 62$ to be
consistent with the horizon problem, corresponding to reheating temperatures
$2.4\times 10^{7}\text{GeV}\lesssim T_{re}\lesssim 3.6\times
10^{9}\text{GeV}$. This simple correction makes the model very tightly bound
and ideal for further analysis under more precise measurements of the spectral
index and tensor fraction. As we increase the power of the correction term,
there are less reheating-consistent curves available in the $r-n_{s}$
parameter space. The highest order corrected model we look at,
$\text{RCHI}_{10}$ contains no reheating-consistent curves at all.

The final model discussed in this paper is that of exponential SUSY inflation.
This potential in its un-regulairzed form already inhabits promising regions
of the $r-n_{s}$ and $T_{re}-n_{s}$ parameter spaces. When squared to form a
regularized ESI-squared model, the reheating consistent curves are those with
$56\lesssim N_{k}\lesssim 64$. Taking into account the horizon problem this is
further reduced to $60\lesssim N_{k}\lesssim 64$, a corresponding bound on the
reheating temperature of $1.5\times 10^{2}\text{GeV}\lesssim T_{re}\lesssim
4.9\times 10^{8}\text{GeV}$. Unlike QH, adding polynomial corrections to ESI
does not stabilise the potential as the exponential function dominates any
such term that we could add. We show also that is it not possible to add
inverse power terms that can be investigated analytically to first order.

Single field inflation models provide a versatile landscape of models to
approach the problem of explaining cosmological inflation, but one must take
great care that any model under consideration produces a cosmology after
inflation that is consistent with what we observe today. In particular, it is
important that the inflationary models we consider in this paper have stable
vacuums which prevent the universe from collapsing shortly after inflation
ends. This is an important consideration for as we have shown in the cases of
quartic hilltop and radiatively corrected Higgs inflation, how one decides to
regularize the potential can have significant effects on the observable that
we measure today and may determine the viability of a given model when
compared to observational data. In this work we make use of polynomial and
inverse power correction terms, it is certainly worth noting that for a scalar
field theory in $d=4$ dimensions, only $\phi^{4}$ interaction terms are
renormalizable [54, 55]. So even models such as these with corrected
potentials cannot be quantized to form self-consistent QFT’s, they can only be
considered as classical effective field theories for computing inflationary
observables.

##  Appendix A Numerical Simulations

Throughout this work we make frequent use of numerical simulations to
calculate the spectral index, tensor fraction and temperate at the end of
reheating. These calculations are performed using a bespoke Mathematica
notebook available at https://github.com/JoshHoffmann/SFSRIA. This code allows
for the analysis of generic single field slow-roll inflation models by
numerical integrating the exact equations of motion LABEL:EoM at a fixed
number of remaining efolds of inflation $N_{k}$, over parameter ranges and
suitable initial conditions set by the user. Integrating the equations of
motion up to the end of inflation allows one to extract the spectral index
$n_{s}$ and tensor fraction $r$ as well as the energy density of the scalar
field at the end of inflation and Hubble parameter when the picot scale exits
the horizon $H_{k}$. Thus we are also able to calculate the reheating
temperature and number of reheating efolds given by equations 14 & 15
respectively.

##  Appendix B $\text{QH}_{p}$ Correction figures

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{QH}_{6}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{QH}_{6}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 15:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{QH}_{7}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{QH}_{7}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 16:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{QH}_{9}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{QH}_{9}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 17:

##  Appendix C $\text{RCHI}_{p}$ Correction Figures

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{2}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{2}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 18:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{3}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{3}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 19:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{4}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{4}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 20:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{5}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{5}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 21:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{6}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{6}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 22:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{7}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{7}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 23:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{8}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{8}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 24:

(a) Numerical solutions of the tensor-scalar ratio $r$ and spectral index
$n_{s}$ for the $\text{RCHI}_{9}$ model over the $50\leq N_{k}\leq 70$.

(b) The (log) reheating temperature of the $\text{RCHI}_{9}$ model against the
spectral index $n_{s}$ over $50\leq N_{k}\leq 70$.

Figure 25:

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