# Squared Quartic Hilltop Inflation

Josh Hoffmann Corresponding author<EMAIL_ADDRESS>Department of
Physics, Lancaster University, Lancaster UK  David Sloan  Department of
Physics, Lancaster University, Lancaster UK

###### Abstract

A correction to the Quartic Hilltop inflationary model is proposed to account
for stabilising terms in the potential. We derive analytical predictions for
the spectral index $n_{s}$ and tensor-scalar ratio $r$ which lie within the
Planck 2018 survey bounds. The reheating predictions of the corrected model
are investigated and by considering the reheating temperature we further
constrain the $(n_{s},r)$ parameter space. It is shown that the correction
terms are physically important during inflation and generally need to be
accounted for when considering Hilltop models as a candidate for inflation.

##  1 Quartic Hilltop Inflation

With the release of the Planck 2018 data it has been possible to accurately
test a variety of candidate inflationary models [1]. Among the models that
best fit the data is the Quartic Hilltop (QH) model. The QH model belongs to
the wider class of Hilltop models which are defined by potentials of the form
[2]

$V(\phi)=\Lambda\left[1-\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{q}+...\right]$ (1)

where … indicates higher order terms. Hilltop models are characterised by
inflation occurring near the maxima of the potential on a broad, flat plateau
or “hilltop”. This feature makes Hilltop models attractive for slow roll
inflation as the conditions are easy for such a model to fulfil. Furthermore,
simple potentials such as these are easier to understand from a particle
physics and QFT perspective and do occur in symmetry breaking theories [3],
supergravity [4, 5, 6, 7, 8, 9, 10, 11, 12], supersymmetry [13, 14] and
superstring theory models [15]. Broadly speaking, inflation models can be
divided into two categories [16]. These are large field models, where
inflation starts with the field at a large value $\phi>m_{\rm pl}$, and small
field models for which the inflation starts with the field at a small value
$\phi<m_{\rm pl}$.

Hilltop potentials are small field models. These are a widely studied class of
models that are easily viewed from the perspective of an effective field
theory in particle physics, particularly because the field varies on a scale
less than $m_{\rm pl}$, which should be the cutoff of such a theory. They are
also good at accommodating slow-roll inflation [16, 17, 18, 19], which
provides for excellent agreement with observations. In slow roll inflation,
the potential energy of the inflaton field $V(\phi)$ dominates over its
kinetic energy and the field evolves slowly. To provide a sufficient amount of
slow-roll inflation, potentials which are close to being flat are
advantageous. This is well accommodated by small and large field models, whose
potentials can be constructed to feature long, flat plateaus for the field to
evolve through whilst the universe expands greatly.

The most common matter sources for inflationary theories are constructed from
a matter Lagrangian describing usually one (but possibly many) scalar field(s)
minimally coupled to gravity [20]

$S=\int
d^{4}x\sqrt{-g}\left(\frac{1}{2}R+\mathcal{L}_{m}(\phi,\partial_{\mu}\phi)\right)$
(2)

in units where $8\pi Gc^{-4}=1$, $g_{\mu\nu}$ is the FRLW metric with scale
factor $a(t)$ and the Lagrangian for the scalar field is

$\mathcal{L}_{m}(\phi,\partial_{\mu}\phi)=\frac{1}{2}\dot{\phi}^{2}-V(\phi)$
(3)

One need only then specify a potential $V(\phi)$ to determine cosmological
quantities relevant to inflation. Such a scalar field has a simple equation of
motion that includes a friction term from the Hubble parameter $H=\dot{a}/a$
and a force term from the potential $V^{\prime}$

$\ddot{\phi}+3H\dot{\phi}+V^{\prime}(\phi)=0$ (4)

with the energy density and pressure given by

$\rho=V(\phi)+\frac{1}{2}\dot{\phi}^{2}$ (5)
$p=-V(\phi)+\frac{1}{2}\dot{\phi}^{2}$ (6)

Inflation requires that the acceleration of the scale factor $\ddot{a}(t)$,
given by

$\frac{\ddot{a}}{a}=-\frac{4\pi G}{3}(\rho+3p)$ (7)

is positive. This in turn requires the energy density and pressure to satisfy
$\rho+3p<0$. Given equations (5) & (6) this is possible if
$\dot{\phi}^{2}<2V(\phi)$, hence the field changes slowly over the course of
inflation. In slow roll inflation, the acceleration of the field is also
negligibly small $\ddot{\phi}\simeq 0$. The equation of motion then gives the
attractor solution

$\dot{\phi}\simeq-\frac{1}{3H}V^{\prime}(\phi)$ (8)

By defining the dimensionless slow roll parameters

$\varepsilon=\frac{1}{2}m_{\rm pl}^{2}\left(\frac{V^{\prime}}{V}\right)^{2}$
(9) $\eta=m_{\rm pl}^{2}\frac{V^{\prime\prime}}{V}$ (10)

where $m_{\rm pl}$ is the reduced Planck mass, the conditions for slow roll
inflation can be summarised as requiring $\varepsilon<1$, $|\eta|<1$ and that
the field tends to the attractor solution, equation (8). One of the greatest
successes of inflation is that it provides a mechanism by which small density
perturbations in the early universe can later give rise to structure formation
[21, 22, 23, 24, 25, 26, 27, 28]. The main cosmological observables of
interest in models such as those described here derive from fluctuations in
the scalar field $\phi$ and tensor field $g_{\mu\nu}$, these are the spectral
index $n_{s}(k)$ and the tensor-scalar ratio $r$ [29, 30]. The spectral index
is defined through the power spectrum of scalar fluctuations
$\mathcal{P}_{\phi}$

$n_{s}(k)-1=\frac{d\ln{\mathcal{P}_{\phi}}}{d\ln{k}}$ (11)

such that if the spectral index does not vary much with the scale $k$, the
spectrum follows a power law $\mathcal{P}_{\phi}\propto k^{n_{s}-1}$

Hilltop models thus have many attractive features for an inflationary model.
In Hilltop models, when the parameter is large enough, $\lambda\gtrsim 1$
inflation occurs at a small value of the field $\phi/m_{\rm pl}\ll\lambda$ and
thus the higher order terms in the potential are heavily suppressed and do not
affect inflationary predictions of the model. For this reason, Hilltop models
containing only the lowest order terms have usually been considered in
previous studies.

The Quartic Hilltop (QH) model invokes a simple potential consisting of a
constant term and a quartic term in the field $\phi$.

$V(\phi)=\Lambda\left[1-\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{4}\right]$ (12)

The QH model has recently received more attention in light of the Planck 2018
data, to which it showed a satisfactory fit under a numerical analysis [31],
whereas earlier attempts at an analytical treatment had ruled QH out on the
basis that the predictions for the spectral index $n_{s}$ as a function of the
number of remaining e-folds $N$ of inflation after the scale $k_{*}$ exits the
horizon [32]

$n_{s}=1-\frac{3}{N}$ (13)

was too small at $N=50$ & $60$ to fit within the bounds of the Planck 2018
data, as this places $n_{s}$ between 0.94 and 0.95. To align with the Planck
data we require $n_{s}\gtrsim 0.96$.

Figure 1: The Quartic Hilltop potential (12) of the inflation field $\phi$ at
various values of $\lambda$. The numerical values of $\lambda$ chosen for this
figure best illustrate the shape of the potential, but the physically
interesting values will be for $\lambda\ll 1$.

Numerical treatments indicate that the QH model does achieve this, so there is
an apparent discrepancy between the results of numerical and analytic
investigations. The reason is that in the earlier attempts at solving the
model analytically, it was assumed that the physically relevant parameter
space has $\lambda\sim 1$. In [33], it is shown that this is in fact not the
case and that the parameter $\lambda$ is actually very small. This allows for
a more detailed analytic investigation of the QH model, by revising two key
assumptions used in deriving the result (13).

  1. 1.

The value of the inflation field $\phi(N)$ at a given $N$ is much smaller than
its vacuum expectation value (VEV) i.e. $\phi(N)\ll\langle\phi\rangle\sim
m_{\rm pl}/\lambda^{1/4}$.

  2. 2.

The contribution of the value of the inflation field at the end of inflation
$\phi_{\rm end}$ to the calculation of $N(\phi)$ is negligible.

By relaxing both of these assumptions it is possible to derive a theoretical
prediction for the $r-n_{s}$ curve which is in close agreement with numerical
analysis and thus better fits the Planck 2018 data. In the small $\lambda$
regime, the higher order terms become of greater importance and as we will
show in this work, accounting for such terms leads to distinct results when
compared to the QH model in the $(n_{s},r)$ parameter space favoured by the
Planck 2018 survey. Ultimately what is derived in [33] is a relationship
between the scalar tensor ratio $r$ and the spectral index $n_{s}$ at a given
number of e-folds before the end of inflation $N$.

$r(n_{s},N)=\frac{8}{3}(1-n_{s})\left[1-\frac{\sqrt{3[2(1-n_{s})N-3]}}{(1-n_{s})N}\right]$
(14)

Whilst this treatment of the QH models yields a result that is much more
satisfactory due to being a better fit to the Planck data, the model is not
without its issues as discussed in [34]. Namely, that the QH potential turns
negative shortly after inflation ends and is unbounded from below [34]. This
is clearly unsatisfactory as such a theory, when quantized would posses
arbitrarily low energy states. The potential needs to be stabilised by the
higher order terms that have been neglected in (12).

We will take the investigation of the QH model in [33] as a guide for
investigating a model which accounts for the higher order terms. That is, we
will follow steps analogous to those for investigating the QH model

  * •

Compute the number of e-folds, $N(\phi)$, of inflation after the cosmological
scale leaves the horizon

  * •

Derive the contribution to $N$ from the inflaton at the end of slow-roll in
terms of $\lambda$, $N_{\rm end}(\lambda)$ by solving the conditions
$\varepsilon(\phi_{\rm end})=1$ & $|\eta(\phi_{\rm end})|=1$

  * •

Express the spectral index $n_{s}$ in terms of $N$ and $\lambda$ through our
solution for $N_{\rm end}(\lambda)$

  * •

Invert the expression for $n_{s}(\lambda,N)$ to find $\lambda(n_{s},N)$ and
use this to derive the relationship between $r$ and $n_{s}$ for $N=50$ and
$N=60$

Although the general method for the QH model and the model containing higher
order terms will be the same, a major difference comes in the final step when
deriving the relationship between the spectral index and tensor-scalar ratio
in which we find that, unlike in the QH case, we find a one-to-many mapping of
$n_{s}$ to $r$ for given values of $N$.

Whilst we will derive analytical results for the relationship between the
tensor-scalar ratio and spectral index at a given number of remaining efolds
of inflation $N$ which allows accurate comparison of the predictions of the
two models. However, this does not necessarily allow us to evaluate how well
the Quartic Hilltop model or its corrected version explain the data collected
by Planck and other such surveys of the CMB. The $r-n_{s}$ relationship is
only really one half of the picture since, given some model defined by a
potential $V(\phi)$ as we are considering here, one may fix any arbitrary
$(n_{s},r)$ by simply choosing an appropriate time at which the scale $k$
exits the horizon, which translates into an amount of remaining efolds $N$.
There are of course some a priori restrictions on $N$ as it is well
established that approximately 50 efolds of inflation are required to solve
the horizon problem. However as we show in section 4, by considering the
reheating dynamics of the corrected model, which one cannot do for the UV-
incomplete QH model, it is possible to place further restrictions on the
possible values of $N$ so that the resulting temperature of the universe at
the end of reheating is consistent with standard cosmology, which may not be
the case if one simply chooses $N$ to get a desired $(n_{s},r)$ pair.

##  2 The Quartic Hilltop Squared Model

As noted in [33], the Quartic Hilltop (QH) model111The parameter in the QH
model is taken to be $2\lambda$ to keep the form consistent with the series
expansion of (16).

$V(\phi)=\Lambda\left[1-2\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{4}\right]$ (15)

approximates a linear potential near $V(\phi)=0$ which will become negative
for $\phi>m_{\rm pl}$ shortly before the end of inflation, and needs to be
stabilised by higher order terms. This can be achieved by working with the
Quartic Hilltop Squared Model (QHS) as suggested in [34] for which
$V(\phi)\geq 0$.

$V(\phi)=\Lambda\left[1-\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{4}\right]^{2}$ (16)

The approach of squaring the Hilltop potential to investigate the effects of
accounting for stabilising terms has previously been investigated in the case
of the Quadratic Hilltop potential[31] (its corrected form known as Double
Well Inflation) and in [35] where the authors discuss different aspects of the
same corrected Quartic Hilltop potential. As such, it is known that such
models, whilst agreeing closely with the uncorrected versions at small field
values as one would expect from Taylor expanding the potential, still produce
substantially different predictions for the tensor-scalar ratio and spectral
index when the VEV is in the super-Planckian regime. Hilltop models generally
require a super-Planckian VEV for their predictions of $r$ and $n_{s}$ to be
in lie with current measurements and this is shown for both the Quadratic [31]
and Quartic Hilltop models [33].

Figure 2: The Quartic Hilltop-Squared potential (16) at various values of
$\lambda$. The values of $\lambda$ are chosen to illustrate the shape of the
potential, which unlike the QH potential, is bounded from below and gains a
local minima from the higher order terms but retains a flat plateau needed for
slow-roll inflation.

We will calculate analytically the spectral index $n_{s}$ and tensor-scalar
ratio $r$. The analytic calculations can be compared to the detailed data
collected by the Planck satellite. Just as the QH model is investigated
analytically in [33], analogous steps can be performed for the QHS model. The
spectral index and tensor-scalar ratio can be written in terms of the slow
roll parameters $\varepsilon$ & $\eta$.

$n_{s}=1-6\varepsilon+2\eta$ (17) $r=16\varepsilon$ (18)

where the slow roll parameters are

$\displaystyle\varepsilon(\phi)$ $\displaystyle=\frac{1}{2}m_{\rm
pl}^{2}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2}$ (19)
$\displaystyle=32\lambda^{2}\left(\frac{\phi}{m_{\rm
pl}}\right)^{6}\frac{1}{\left[1-\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{4}\right]^{2}}$ $\displaystyle\eta(\phi)$ $\displaystyle=m_{\rm
pl}^{2}\frac{V^{\prime\prime}(\phi)}{V(\phi)}$ (20)
$\displaystyle=\varepsilon(\phi)-24\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{2}\frac{1}{\left[1-\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{4}\right]}$

It is easiest to work with the spectral index and tensor-scalar ratio in terms
of a parameter $Z$ which will be defined shortly and $N$, the remaining number
of e-folds until the end of inflation after the cosmological scale has exited
the horizon. This is calculated as

$\displaystyle N$ $\displaystyle=\frac{1}{m_{\rm pl}^{2}}\int_{\phi_{\rm
end}}^{\phi(N)}\frac{V}{V^{\prime}}d\phi$ (21)
$\displaystyle=\frac{1}{16}\left(\frac{\phi}{m_{\rm
pl}}\right)^{2}+\frac{1}{16\lambda}\left(\frac{\phi}{m_{\rm
pl}}\right)^{-2}-N_{\rm end}$

where the value of the scalar field at the end of inflation is $\phi_{\rm
end}$ and

$N_{\rm end}=\frac{1}{16}\left(\frac{\phi_{\rm end}}{m_{\rm
pl}}\right)^{2}+\frac{1}{16\lambda}\left(\frac{\phi_{\rm end}}{m_{\rm
pl}}\right)^{-2}$ (22)

By defining

$\bar{N}\equiv N+N_{\rm end}=\frac{1}{16}\left(\frac{\phi}{m_{\rm
pl}}\right)^{2}+\frac{1}{16\lambda}\left(\frac{\phi}{m_{\rm pl}}\right)^{-2}$
(23)

and $Z=64\lambda\bar{N}^{2}$, equation (21) may be solved for
$\left(\frac{\phi}{m_{\rm pl}}\right)^{2}$ as

$\left(\frac{\phi}{m_{\rm pl}}\right)^{2}=8\bar{N}[Z]$ (24)

The function $[Z]$ is defined as

$[Z]=1-\sqrt{1-\frac{1}{Z}}$ (25)

A property of this function that we will make use of later is that it
satisfies $[Z]^{2}=2[Z]-1/Z$. The appearance of this function $[Z]$ sets a
constraint $Z>1$. Since the $Z$ parameter is defined as
$Z=64\lambda\bar{N}^{2}$, this constraint is saying that for a given value of
$\lambda$, which is a free parameter in the model, there will always be a
minimum amount of inflation given by

$\bar{N}>\frac{1}{8\sqrt{\lambda}}$ (26)

In the QH model there is an analogous bound $\bar{N}_{\rm
QH}>1/4\sqrt{\lambda}$, in fact it is shown that in the small $\lambda$ regime
$\bar{N}_{\rm QH}\simeq N+1/4\sqrt{\lambda}$. This suggests that for the QHS
model we should expect $\bar{N}\simeq N+1/8\sqrt{\lambda}$, and in fact we
will later show that this is indeed the case.

Using equation (24), the slow roll parameters can be written as

$\varepsilon=\frac{4}{\bar{N}}\frac{Z^{2}[Z]^{3}}{\left(1-Z[Z]^{2}\right)^{2}}$
(27) $\eta=\varepsilon-\frac{3}{\bar{N}}\frac{Z[Z]}{1-Z[Z]^{2}}$ (28)

which allows a calculation of the spectral index and tensor-scalar ratio as

$n_{s}=1-\frac{6}{\bar{N}}\frac{Z[Z]}{1-Z[Z]^{2}}-\frac{16}{\bar{N}}\frac{Z^{2}[Z]^{3}}{\left(1-Z[Z]^{2}\right)^{2}}$
(29) $r=\frac{64}{\bar{N}}\frac{Z^{2}[Z]^{3}}{(1-Z[Z]^{2})^{2}}$ (30)

Equation (29) can be written as a quadratic in $F(Z)=\sqrt{Z/(Z-1)}$ by virtue
of the purely algebraic relations

$1-Z[Z]^{2}=2\left(1-Z[Z]\right)$ (31)
$\left(\frac{Z[Z]}{1-Z[Z]}\right)^{2}=\frac{Z}{Z-1}$ (32)

Thus we have

$n_{s}=1+\frac{1}{\bar{N}}\sqrt{\frac{Z}{Z-1}}-\frac{4}{\bar{N}}\left(\frac{Z}{Z-1}\right)$
(33)

with the solution

$\sqrt{\frac{Z}{Z-1}}=\frac{1+\sqrt{1+16\bar{N}\tilde{n}_{s}}}{8}$ (34)

where $\tilde{n}_{s}=1-n_{s}$.

The goal of this investigation is to produce an analytic relationship between
the tensor-scalar ratio $r$ and spectral index $n_{s}$ at a given number of
e-folds before the end of inflation $N$ for $\lambda\ll 1$. Recalling the
definitions $Z=64\lambda\bar{N}^{2}$ and $\bar{N}=N+N_{\rm end}$, to produce
such a function $r(n_{s},N)$, we are required to eliminate $\lambda$ &
$\bar{N}$ from the equations.

$N_{\rm end}$ is the contribution of the value of the inflation field at the
end of inflation to the total number of e-folds, which as noted in [33] can be
large for the QH model at small $\lambda$. Investigation of this $N_{\rm end}$
term results in the approximation $\bar{N}_{\rm QH}\simeq
N+1/4\sqrt{\lambda}$, so we are motivated to find an analogous approximation
for the QHS model.

Thus far we have extracted both the spectral index and tensor-scalar ratio in
terms of two newly defined parameters $Z$ and $\bar{N}$. The end goal of this
analysis is to produce a closed form expression for the tensor-scalar ratio in
terms of only the spectral index $n_{s}$ and remaining number of e-folds $N$.
A strategy for proceeding can the be devised as follows: An explicit
expression for $N_{\rm end}(\lambda)$ will be derived in the small $\lambda$
limit. This allows for the total number of e-folds $\bar{N}$ to be written
only in terms of $N$ and $\lambda$ and hence also the $Z$ parameter and
spectral index. The resulting expression for $n_{s}(N,\lambda)$ is invertible
for $\lambda(N,n_{s})$. The expressions for $\bar{N}(N,\lambda)$ and
$\lambda(N,n_{s})$ can then be substituted into the definition of $Z$ which is
used in equation (30) to find the target closed form relationship between $r$
and $n_{s}$ for a given $N$.

##  3 Small $\lambda$ behaviour of $N_{\rm end}$

Guided by the QH model and Planck data, we are mainly interested in the region
of parameter space where $\lambda$ is very small, roughly $\lambda\lesssim
10^{-4}$. The small $\lambda$ behaviour of $N_{\rm end}$ can be investigated
by considering separately the cases where inflation is ended by the slow roll
parameters $\varepsilon$ and $\eta$.

###  3.1 $\varepsilon(\phi_{\rm end})=1$

Consider first the case where slow roll inflation is ended by $\varepsilon$
reaching unity before $\eta$. In this case equation (19) becomes

$\frac{1}{\lambda}=\left(\frac{\phi_{\rm end}}{m_{\rm
pl}}\right)^{3}\left(\frac{\phi_{\rm end}}{m_{\rm pl}}+4\sqrt{2}\right)$ (35)

For $\lambda\ll 1$, we have $\phi_{\rm end}\gg m_{\rm pl}$ and thus one has
approximately

$\left(\frac{\phi_{\rm end}}{m_{\rm
pl}}\right)^{2}\simeq\frac{1}{\sqrt{\lambda}}$ (36)

This result may be substituted into equation (22) to get an approximate
expression for $N_{\rm end}$ in the $\lambda\ll 1$ limit.

$N_{\rm end}\simeq\frac{1}{8\sqrt{\lambda}}$ (37)

###  3.2 $|\eta(\phi_{\rm end})|=1$

Consider now the second case where the end of slow roll is characterized by
$|\eta(\phi_{\rm end})|=1$, which of course has two sub cases $\eta(\phi_{\rm
end})=1$ & $\eta(\phi_{\rm end})=-1$. starting with the $\eta(\phi_{\rm
end})=1$ case we have

$\left(\frac{\phi_{\rm end}}{m_{\rm pl}}\right)^{8}-56\left(\frac{\phi_{\rm
end}}{m_{\rm pl}}\right)^{6}-\frac{2}{\lambda}\left(\frac{\phi_{\rm
end}}{m_{\rm pl}}\right)^{4}+\frac{24}{\lambda}\left(\frac{\phi_{\rm
end}}{m_{\rm pl}}\right)^{2}+\frac{1}{\lambda^{2}}=0$ (38)

Again, for $\lambda\ll 1$, $\left(\phi_{\rm end}/m_{\rm pl}\right)\gg 1$ and
(38) is approximately

$\left(\frac{\phi_{\rm end}}{m_{\rm
pl}}\right)^{8}+\frac{1}{\lambda^{2}}\simeq 0$ (39)

which has no real solutions.

In second sub case $\eta(\phi_{\rm end})=-1$ we come to the same conclusion.
For $\lambda\ll 1$, there are no real, positive solutions. From these cases we
conclude that, to a first approximation, for small $\lambda$ the end of slow-
roll inflation is marked only by $\varepsilon(\phi)\geq 1$ .

###  3.3 Small $\lambda$ Relationship Between $r$ and $n_{s}$

Given the results of section 2, for small $\lambda$ we can use the
approximation

$\bar{N}\simeq N+\frac{1}{8\sqrt{\lambda}}$ (40)

which should come as no surprise given the earlier bound derived on $\bar{N}$
in equation (26). We see that just as in the case of the QH model, the term
contributed by $\phi_{\rm end}$ is proportional to $\lambda^{-\frac{1}{2}}$ so
this term certainly cannot be neglected when $\lambda\ll 1$. With this
expression we can now write equation (34) in terms of $\lambda,n_{s}$ and $N$.

$\sqrt{1+\frac{1}{16N\sqrt{\lambda}+64N^{2}\lambda}}=\frac{1}{8}\left[1+\sqrt{1+16\tilde{n}_{s}\left(N+\frac{1}{8\sqrt{\lambda}}\right)}\right]$
(41)

Which can be solved as a quadratic in terms of $\sqrt{\lambda}$, and as such
there will be two solutions, both of which are valid. This is of course
because $\lambda$ is a free parameter in the model, a given $r$ may map to
multiple values of $\lambda$.

$\left(\sqrt{\lambda}\right)_{\pm}=\frac{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15\pm\sqrt{8N\tilde{n}_{s}-15}}{8N\left(15-8N\tilde{n}_{s}+N^{2}\tilde{n}_{s}^{2}\right)}$
(42)

The crossover point between the two branches where
$\left(\sqrt{\lambda}\right)_{+}=\left(\sqrt{\lambda}\right)_{-}=\sqrt{\lambda_{c}}$
is at $\sqrt{8N\tilde{n}_{s}-15}=0\rightarrow N\tilde{n}_{s}=15/8$, this also
defines the maximum possible value of $n_{s}$ at a given $N$

$n_{s}^{\rm max}=1-\frac{15}{8N}$ (43)

The calculated spectral index bounds $n_{s}^{\rm max}(N=60)=0.96875$ and
$n_{s}^{\rm max}(N=50)=0.9625$ are consistent with the observational data
provided by Planck 2018 [1]

Substituting $N\tilde{n}_{s}=\frac{15}{8N}$ into (42) one finds
$\sqrt{\lambda}_{c}=\frac{1}{60N}$, resulting in the critical values of
$\lambda_{c}\simeq 7.72\times 10^{-8}$ and $\lambda_{c}\simeq 1.11\times
10^{-7}$ for $N=60$ & $N=50$ respectively.

The tensor-scalar ratio $r$ which we aim to calculate, can be expressed in
terms of the slow-roll parameter $\varepsilon$ as

$\displaystyle r$ $\displaystyle=16\varepsilon$ (44)
$\displaystyle=\frac{16}{\bar{N}}\left(\frac{Z}{Z-1}\right)\left(1-\sqrt{\frac{Z-1}{Z}}\right)$

Using equation (34) and the solutions for $\sqrt{\lambda}$ (42), we can
calculate the quantities $\bar{N}$ and $Z/(Z-1)$ in terms of $n_{s}$ and $N$
as desired.

$r=\begin{cases}\frac{1}{4N}\left(\frac{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15+\sqrt{8N\tilde{n}_{s}-15}}{4N\tilde{n}_{s}-N^{2}\tilde{n}_{s}^{2}+\sqrt{8N\tilde{n}_{s}-15}}\right)\left[1+g_{+}(\tilde{n}_{s},N)\right]\left[g_{+}(\tilde{n}_{s},N)-7\right],\quad\lambda\geq\lambda_{c}\\\
\\\
\frac{1}{4N}\left(\frac{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15-\sqrt{8N\tilde{n}_{s}-15}}{4N\tilde{n}_{s}-N^{2}\tilde{n}_{s}^{2}-\sqrt{8N\tilde{n}_{s}-15}}\right)\left[1+g_{-}(\tilde{n}_{s},N)\right]\left[g_{-}(\tilde{n}_{s},N)-7\right],\quad\lambda<\lambda_{c}\end{cases}$
(45)

Where we define the function

$g_{\pm}(\tilde{n}_{s},N)=\sqrt{\frac{12N\tilde{n}_{s}+62N^{2}\tilde{n}_{s}^{2}-16N^{3}\tilde{n}_{s}^{3}-15\pm(1+16N\tilde{n}_{s})\sqrt{8N\tilde{n}_{s}-15}}{12N\tilde{n}_{s}-2N^{2}\tilde{n}_{s}^{2}-15\pm\sqrt{8N\tilde{n}_{s}-15}}}$
(46)

Figure 3: Analytical solutions (bold) for the spectral index $n_{s}$ and
tensor-scalar ratio $r$ of the QHS model equation (45), compared with their
numerical solutions (dotted) for $10^{-8}\leq\lambda\leq 1$ at $N=50$ and
$N=60$ e-folds before the end of inflation.

By comparison to numerical solutions for a broad range of $\lambda$ from
$\lambda\ll 1$ to $\lambda\approx 1$ in figure (3) we can see that this
approximation is quite accurate, even in the regime when $\lambda$ is large
($\lambda\sim 1$). We can further validate the analytical solution by
comparing the $N=60$ and $N=50$ solutions to numerical solutions calculated up
to the predicted critical $\lambda_{c}$ values. In figures (4(a)) and (4(b))
it can be seen that if the system is solved numerically up to the predicated
$\lambda_{c}$ values, the solution branches do indeed stop at the crossover
point between the $r_{+}$ and $r_{-}$ branches of the analytical solution.

(a) QHS $N=60$ numerical solution calculated up to the
crossover point between $r_{\pm}$ branches at $\lambda_{c}$ compared to
analytical solution

(b) QHS $N=50$ numerical solution calculated up
to the crossover point between $r_{\pm}$ branches at $\lambda_{c}$
compared to analytical solution

Figure 5: Comparisons of the analytic $r(n_{s})$ solutions for Quartic
Hilltop-Squared (45) calculated in this work and Quartic Hilltop (14)
calculated in [33] at $N=50$ & $N=60$ e-folds before the end of inflation.

Comparing the solution for the QH model (14) and QHS model (45), we see that
the effect of including the higher order stabilising terms causes the
predictions of the QHS model to diverge quite dramatically from those of the
QH model. The models begin to diverge around $n_{s}\simeq 0.95$ which is well
outside of the region expected from the Planck 2018 data making them quite
distinct within the acceptable regions. This is an important result as it
makes clear that different methods of stabilising the potential must be
considered as distinct if we are to take Hilltop potentials as serious
candidates for inflation. It is not sufficient to ignore these stabilising
terms in the potential since in the small $\lambda$ regime they have a
dramatic effect on inflationary predictions of the theory.

Figure 6: Analytical QHS solutions (bold) given by equation (45) at $N=50$ &
$N=60$ e-folds before the end of inflation, compared to the Planck 2018 data
bounds represented by the filled regions ($2\sigma$ in lighter, $1\sigma$ in
darker).

The result of including the stabilising terms, as shown in figure (6), is that
the N = 50 branch has shifted out of the $1\sigma$ region whilst remaining
just within the $2\sigma$ bound. In the QH model, both branches remains within
the $1\sigma$ region, however this does not spell such bad news for the QHS
model since $N\simeq 60$ is generally favoured by reheating constraints.

Another feature that is seen in the QHS model but not the QH model is that the
solution for $r(n_{s})$ is not a function, given $n_{s}$ may correspond to
multiples values of $r$. As such, the curves in figure 3 representing the pass
vertical and. The points at which the curves reach their maximal values of the
spectral index are indeed the bounds calculated earlier $n_{s}^{\rm
max}(N=60)=0.96875$ and $n_{s}^{\rm max}(N=50)=0.9625$. This arises due to the
stabilisation, and is a general feature of stabilised Hilltop models. This can
be seen by considering the spectral index $n_{s}$ and tensor ratio as
functions of the slow roll parameters

$\displaystyle n_{s}$ $\displaystyle=1-6\varepsilon(\phi)+2\eta(\phi)$ (47)
$\displaystyle r$ $\displaystyle=16\varepsilon(\phi)$

In the QHS case, the slow roll parameters are

$\eta(\phi)=\varepsilon(\phi)-24\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{2}\frac{1}{1-\lambda\left(\frac{\phi}{m_{\rm pl}}\right)^{4}}$
(48)

Defining

$\Gamma(\phi,\lambda)=24\lambda\left(\frac{\phi}{m_{\rm
pl}}\right)^{2}\frac{1}{1-\lambda\left(\phi/m_{\rm pl}\right)^{4}}$ (49)

the two equations in (48) can be combined to give

$r=8\left(1-n_{s}\right)-32\Gamma(\phi,\lambda)$ (50)

From the analytical results in this work, specifically equation (42), for a
given $n_{s}<n_{s}^{\rm max}$ there will be one or two corresponding values of
lambda $\lambda_{\pm}$ (one in the case $N\tilde{n}_{s}=15/8$ and two
otherwise). And thus for a given $n_{s}$ there will be one or two values of
$\Gamma(\phi,\lambda)$ and hence one or two corresponding values of $r$. The
QH model does not exhibit this behaviour because in this particular case
$\eta(\phi)$ and $\varepsilon(\phi)$ are linearly independent. From a physical
standpoint, we expect this to be a generic feature of all Hilltop models that
take higher order correction terms into account. Consider the spectral index
$n_{s}$ written in terms of the Hilltop potential $V(\phi)$.

$n_{s}=1-3m_{\rm
pl}^{2}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2}+2m_{\rm
pl}^{2}\frac{V^{\prime\prime}(\phi)}{V(\phi)}$ (51)

The dependence of $n_{s}$ on the ratios of the derivatives of the potential to
the potential itself suggests that we should write this in terms of
$W(\phi)=\ln V(\phi)$ for which

$n_{s}=1-m_{\rm pl}^{2}W^{\prime}(\phi)+2m_{\rm pl}^{2}W^{\prime\prime}(\phi)$
(52)

If one includes the stabilising terms in $V(\phi)$ then the potential will
have a local minima and thus the second derivative will also have a local
minima $\phi_{min}<\phi_{\rm end}$ for sufficiently small $\lambda$. Because
the curvature of $V(\phi)$ has a turning point, $W^{\prime\prime}(\phi)$ will
have a form such that if we fix a particular value
$W^{\prime\prime}_{*}=W(\phi_{0},\lambda_{0})$, we can find another set
$(\phi_{1},\lambda_{1})$ for which
$W^{\prime\prime}_{*}=W(\phi_{1},\lambda_{1})$. Since the spectral index is
just a linear combination of the first and second derivatives it will also
inherit this property. So for a fixed value of the spectral index $n_{s}^{*}$
at $(\phi_{0},\lambda_{0})$, we may also find another set
$(\phi_{1},\lambda_{1})$ which corresponds to the same value of the spectral
index. However the gradient term $W^{\prime}(\phi)$ is not necessarily the
same at $(\phi_{0},\lambda_{0})$ and $(\phi_{1},\lambda_{1})$ and since the
tensor-scalar ratio is proportional to only this term, $n_{s}^{*}$ will
necessarily correspond to multiple values of $r$.

##  4 Reheating in QHS

###  4.1 Generic Reheating Analysis and Subsequent Bound on $w_{re}$

Whilst it is useful to be able to calculate $(n_{s},r)$ pairs for the Quartic
Hilltop Squared model, and verify that for $50\lesssim N\lesssim 60$ the
predictions of the model lie within the parameter space determined by Planck
2018 data, this does not fully validate the model as a viable candidate for
inflation. As discussed in section D of [36], given a generic potential
$V(\phi)$ one may calculate and desired $(n_{s},r)$ pair by through equations
17 & 18 by choosing an appropriate time at which the scale $k$ exits the
horizon. This will correspond to a number of efolds of inflation $N_{k}$
remaining when the scale exits. This however, may not result in acceptable
reheating predictions of the model defined by $V(\phi)$. The number of
reheating efolds and the temperature at the end of reheating $T_{re}$ depends
explicitly on $N_{k}$[36, 37]. This dependence is derived by considering the
evolution of the horizon scale and the energy density throughout the evolution
of the universe.

Figure 7: A schematic outlining the evolution of the horizon scale $(aH)^{-1}$
as a function of the (log) scale factor $a(t)$ and dominant matter source at
each time from inflation to the present day at $a_{0}(t)$.

Let the comoving scale at the time it exits the horizon be $k=a_{k}H_{k}$,
this can be compared to the horizon scale today

$\frac{k}{a_{0}H_{0}}=\frac{a_{k}}{a_{end}}\frac{a_{end}}{a_{re}}\frac{a_{re}}{a_{eq}}\frac{a_{eq}}{a_{0}}\frac{H_{eq}}{H_{0}}\frac{H_{k}}{H_{eq}}$
(53)

We the write $N_{k}=\ln{a_{end}/a_{k}}$ for the number of remaining inflation
efolds after the scale exits the horizon, $N_{re}=\ln{a_{re}/a_{end}}$ for the
number of reheating efolds, $N_{RD}=\ln{a_{eq}/a_{re}}$ for the number of
efolds during radiation domination.

$\ln{\frac{k}{a_{0}H_{0}}}=-N_{k}-N_{re}-N_{RD}-\ln{\frac{a_{eq}}{a_{0}}}+\ln{\frac{H_{eq}}{H_{0}}}+\ln{\frac{H_{k}}{H_{eq}}}$
(54)

Assuming a constant equation of state during reheating $w_{re}$, we may track
the evolution of the energy density from the end of inflation to the end of
reheating

$\frac{\rho_{end}}{\rho_{re}}=\left(\frac{a_{end}}{a_{re}}\right)^{-3(1+w_{re})}$
(55)

In terms of the reheating efolds this is

$N_{re}=\frac{1}{3(1+w_{re})}\ln{\frac{\rho_{end}}{\rho_{re}}}$ (56)

where the energy densities are given by

$\rho_{re}=\frac{\pi^{2}}{30}g_{re}T_{re}^{4}$ (57)
$\rho_{end}=\frac{3}{2}V_{end}$ (58)

where $V_{end}$ is the value of the potential at the end of inflation (i.e.
when $\varepsilon\simeq 1$) , $g_{re}$ is the effective number of relativistic
species at the end of reheating, and the reheating temperature $T_{re}$ can be
written in terms of the CMB temperature today $T_{0}$

$\frac{T_{re}}{T_{0}}=\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\frac{a_{0}}{a_{eq}}\frac{a_{eq}}{a_{re}}$
(59)

By taking the logarithm of Eq. 59 and combining with Eq. 54 we obtain the
reheating efolds and temperature (Equivalently one may also consider the
reheating energy density 58 rather than the temperature as is done in some
literature [36]).

$N_{re}=\frac{4}{1-3w_{re}}\left(\frac{1}{4}\ln{\frac{\pi^{2}g_{re}}{45}}+\frac{1}{3}\ln{\frac{11}{43g_{re}}}+\ln{\frac{a_{0}T_{0}}{k}}-\ln{\frac{V_{end}^{\frac{1}{4}}}{H_{k}}}-N_{k}\right)$
(60)
$\left[\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\left(\frac{a_{0}T_{0}}{k}\right)H_{k}e^{-N_{k}}\left[\frac{45V_{end}}{\pi^{2}g_{re}}\right]^{-\frac{1}{3(1+w_{re})}}\right]^{\frac{3(1+w_{re})}{3w_{re}-1}}$
(61)

For the model to at least be considered as an acceptable candidate one needs
both that the predicted $(n_{s},r)$ are within the bounds of current
measurements and that the corresponding remaining number of inflation efolds
$N_{k}$ gives an acceptable reheating temperature $T_{re}$. We may calculate
analytically the model dependant terms in equations 60 & 61. Firstly the value
of the potential at the end of inflation is simply $V(\phi)$ evaluated on the
solution to equation 35. However using just the first-order approximate
solution 36 would result in $V_{end}=0$. This may be sufficient for
calculating $N_{re}$ since it only depends logarithmically on $V_{end}$, but
the reheating temperature $T_{re}$ is sensitive to $V_{end}$ so one needs at
least the second order term in the power series solution to 35 to calculate
the reheating temperature.

$\left(\frac{\phi_{end}}{m_{pl}}\right)\simeq\lambda^{-\frac{1}{4}}+3\lambda^{-\frac{7}{4}}\qquad\lambda\ll
1$ (62)

and thus we have the value of the potential at the end of inflation

$V_{end}\simeq 144\Lambda\lambda^{-3}$ (63)

The value of the Hubble parameter $H_{k}$ at the pivot scale is

$H_{k}\simeq\pi m_{pl}\sqrt{\frac{A_{s}r}{2}}$ (64)

Finally, given $N_{end}\simeq 1/8\sqrt{\lambda}$ and equation 21, the number
of remaining efolds of inflation $N_{k}$ may be written in terms of the field
value at the pivot scale $\phi_{k}$.

$N_{k}=\frac{1}{16}\left(\frac{\phi_{k}}{m_{pl}}\right)^{2}+\frac{1}{16\lambda}\left(\frac{\phi_{k}}{m_{pl}}\right)^{-2}-\frac{1}{8\sqrt{\lambda}}$
(65)

And thus the reheating efolds and temperature are expressed as

$\displaystyle N_{re}=\frac{4}{1-3w_{re}}$
$\displaystyle\left[\frac{1}{4}\ln{\frac{\pi^{2}g_{re}}{45}}+\frac{1}{3}\ln{\frac{11}{43g_{re}}}+\ln{\frac{a_{0}T_{0}}{k}}-\ln\left[\frac{\sqrt{2}(144\Lambda)^{\frac{1}{4}}}{\lambda^{\frac{3}{4}}\pi
m_{pl}\sqrt{A_{s}r}}\right]\right.$
$\displaystyle\left.+\frac{1}{8\sqrt{\lambda}}-\frac{1}{16}\left(\frac{\phi_{k}}{m_{pl}}\right)^{2}-\frac{1}{16\lambda}\left(\frac{\phi_{k}}{m_{pl}}\right)^{-2}\right]$
$\displaystyle
T_{re}=\left[\left(\frac{43}{11g_{re}}\right)^{\frac{1}{3}}\left(\frac{a_{0}T_{0}}{k}\right)\pi
m_{pl}\sqrt{\frac{A_{s}r}{2}}e^{-N_{k}}\left[\frac{6480\Lambda}{\lambda^{3}}\pi^{2}g_{re}\right]^{-\frac{1}{3(1+w_{re})}}\right]^{\frac{3(1+w_{re})}{3w_{re}-1}}$

To investigate the reheating predictions of the model we consider $r-n_{s}$
curves for $50\leq N_{k}\leq 65$ and their corresponding reheating efolds and
temperatures.

Figure 8: QHS numerical solutions of the tensor-scalar ratio $r$ and spectral
index $n_{s}$ for $50\leq N_{k}\leq 65$ over the parameter range
$10^{-3}\lesssim\lambda\lesssim 10^{-7}$. The solid black lines represent the
Planck 2018 bounds just as in figure 6

(a) Plots of the reheating efolds $N_{re}$ corresponding to the numerical
$(n_{s},r)$ solutions for the QHS model in figure 8 with $w_{re}=0$. The solid
vertical black lines mark the Planck 2018 $1\sigma$ bound on the spectral
index $n_{s}$. For $N_{k}>55$ all of the constant $N_{k}$ curves lie below the
$N_{re}=0$ line, thus in this case the model does not produce any reheating if
there are more than 55 efolds of inflation.

(b) Plots of the (log) reheating temperature $T_{re}$ corresponding to the
numerical $(n_{s},r)$ solutions for the QHS model in figure 8 with $w_{re}=0$.
The solid vertical black lines mark the Planck 2018 $1\sigma$ bound on the
spectral index $n_{s}$.

Figure 9:

From the plots of the reheating efolds in figure 9(a), one sees that if there
are more than approximately 55 remaining efolds of inflation when the pivot
scale exits the horizon, then it it not possible to get $N_{re}>0$. However
for $N_{k}\leq 55$, all of the $(n_{s},r)$ predictions of the model lie
outside of the $1\sigma$ bound in figure 8 and thus an equation of state
parameter of $w_{re}=0$ leaves the model with no acceptable predictions
compared to the Planck 2018 data.

(a) Plots of the reheating efolds $N_{re}$ corresponding to the numerical
$(n_{s},r)$ solutions for the QHS model in figure 8 with $w_{re}=-1/3$. The
solid vertical black lines mark the Planck 2018 $1\sigma$ bound on the
spectral index $n_{s}$. Just as in figure 9(a) The model cannot produce any
reheating efolds if there are more than 55 efolds of inflation.

(b) Plots of the (log) reheating temperature $T_{re}$ corresponding to the
numerical $(n_{s},r)$ solutions for the QHS model in figure 8 with
$w_{re}=-1/3$. The solid vertical black lines mark the Planck 2018 $1\sigma$
bound on the spectral index $n_{s}$.

Figure 10:

(a) Plots of the reheating efolds $N_{re}$ corresponding to the numerical
$(n_{s},r)$ solutions for the QHS model in figure 8 with $w_{re}=2/3$. The
solid vertical black lines mark the Planck 2018 $1\sigma$ bound on the
spectral index $n_{s}$. In this case the behaviour of the reheating efolds is
inverted from figures 9(a) & 10(a), the model only produces non-zero reheating
efolds $N_{re}\geq 0$ for $N_{k}\geq 55$. However, also restricting to the
region of the spectral index’s $1\sigma$ bound makes this inequality
noninclusive $N_{k}>55$.

(b) Plots of the (log) reheating temperature $T_{re}$ corresponding to the
numerical $(n_{s},r)$ solutions for the QHS model in figure 8 with
$w_{re}=2/3$. The solid vertical black lines mark the Planck 2018 $1\sigma$
bound on the spectral index $n_{s}$.

Figure 11:

(a) Plots of the reheating efolds $N_{re}$ corresponding to the numerical
$(n_{s},r)$ solutions for the QHS model in figure 8 with $w_{re}=1$. The solid
vertical black lines mark the Planck 2018 $1\sigma$ bound on the spectral
index $n_{s}$. As in figure 11(a), the constant $N_{k}$ curves lie above the
$N_{re}=0$ line in the $1\sigma$ $n_{s}$ region for $N_{k}>55$.

(b) Plots of the (log) reheating temperature $T_{re}$ corresponding to the
numerical $(n_{s},r)$ solutions for the QHS model in figure 8 with $w_{re}=1$.
The solid vertical black lines mark the Planck 2018 $1\sigma$ bound on the
spectral index $n_{s}$.

Figure 12:

Taking a negative equation of state parameter $w_{re}=-1/3$ as in figures
10(a) and 10(b) serves to push the solutions slightly tighter together
compared to the $w_{re}=0$ case, as one would expect given that the dependence
of the reheating efolds $N_{re}$ on $w_{re}$ follows $(1-3w_{re})^{-1}$. In
this case there are still no acceptable solutions for $N_{k}>55$. We may
however also try positive equation of state parameters.

Using a positive equation of state parameter in figures 11 and 12 inverts the
behaviour of $N_{re}$ and $T_{re}$ with respect to $N_{k}$ so that the number
of reheating efolds increases with $N_{k}$ and the reheating temperature
decreases for with $N_{k}$. In this case there are now solutions which lie in
the $1\sigma$ region of figure 8 (i.e. $N_{k}>55$) and have $N_{re}>0$.

###  4.2 Reheating Temperature Restrictions On The $r-n_{s}$ Parameter Space.

In section 4.1 we have established that in order to get a non-zero amount of
reheating efolds from the QHS model consistent with Planck measurements of the
spectral index $n_{s}$, one is restricted to a positive equation of state
parameter $0<w_{re}\leq 1$. The temperature of the universe at the end of
reheating is loosely bounded from below in order to be consistent with
standard cosmology. If reheating is to occur before Big Bang Nucleosynethsis
(BBN) then one requires $T_{re}>T_{BBN}$. BBN occurs on temperature scales of
roughly $T_{BBN}\lesssim 10\text{MeV}$ [38] so the reheating temperature is
constrained to be at least larger than $0.01\text{GeV}$. The reheating
temperature is also loosely bounded from above by the fact that it should not
exceed the energy scale of inflation $T_{re}\lesssim 10^{16}\text{GeV}$ [39],
but this bound may be further restricted if one accepts supersymmetry and
considers the effects of gravitino production during inflation on BBN [40, 41,
42, 43]. This restricts the reheating temperature to much smaller range
$0.01\text{GeV}\lesssim T_{re}\lesssim 10^{8}\text{GeV}$. Considering the
extremal case where the equation of state parameter $w_{re}=1$, the reheating
temperature bounds can be used to restrict the QHS model to a much tighter
region of the $r-n_{s}$ parameter space, whose reheating efolds are non-zero
and reheating temperature is compatible with standard cosmology.

Figure 13: We plot the (log) reheating temperature of the QHS model over the
range $60\leq N_{k}\leq 70$ where as in previous figures the vertical black
lines mark the $1\sigma$ Planck 2018 bounds on the spectral index $n_{s}$. The
horizontal black lines mark the reheating temperature bounds
$0.01\text{GeV}\lesssim T_{re}\lesssim 10^{8}\text{GeV}$. The range of the
parameter $\lambda$ is the same as in the figure 8.

If we consider the fixed $N_{k}$ curves in figure 13 who lie within the
reheating temperature bounds for all values of $\lambda$ within the $1\sigma$
region, one is restricted to the range $63\leq N_{k}\leq 68$ of remaining
efolds of inflation. In figure 14 we plot the $r-n_{s}$ curves for $N_{k}=63$
and $N_{k}=68$. These curves bound a region of parameter space that is both
compatible with the Planck 2018 measurements and such that the model produces
reheating temperatures within the range $0.01\text{GeV}\lesssim T_{re}\lesssim
10^{8}\text{GeV}$.

Figure 14: The reheating temperature limits restrict us to the area bounded by
the $N_{k}=63$ and $N_{k}=68$ curves in the $r-n_{s}$ parameter space which is
considerably smaller than just the Planck 2018 bounds.

##  5 Discussion

We have derived analytical predictions for the cosmological parameters $n_{s}$
and $r$ in a Quartic Hilltop Squared model. This model accounts for a higher
order correction term which stabilises the potential in [33], such that it is
nowhere negative. The work of [33] makes it clear that when working with
Hilltop models in the small $\lambda$ regime (the one in which they best fit
observational data), one cannot assume that when calculating the remaining
number of e-folds of inflation, the contribution of the inflaton field at the
end of inflation will be small enough to be negligible. In fact one find that
the contribution is $N_{\rm end}\propto\lambda^{-1/2}$. Clearly this will be
large for small $\lambda$. In the case of the Quartic Hilltop model the
relevant range of values of $\lambda$ are $\lambda\leq 10^{-4}$, yet one only
needs $\lambda=4\times 10^{-4}$ for the $\lambda^{-1/2}$ contribution to be as
large as 50 e-folds. In this paper we find that the same relationship exists
when accounting for the higher order terms in the Quartic Hilltop Squared
model. The only difference is that the constant of proportionality is smaller.
For the QH model $N_{\rm end}^{\rm QH}/\lambda^{-1/2}=1/4$ and for the QHS
model $N_{\rm end}^{\rm QHS}/\lambda^{-1/2}=1/8$. Because $N_{\rm end}$ and
$\lambda$ follow the same relationship in the QHS model studied in this paper
we also conclude that one cannot assume this contribution to be small in the
calculation of the remaining number of e-folds and that it must be accounted
for in order to explain why the model fits well to observational data.

When the analytical solution derived in this work is compared with that of the
QH model in [33] and the Planck 2018 data there are two important observations
to be made. Firstly, both Hilltop models provide very good fits to the
observational data, being within the $1\sigma$ region with the exception only
of the QHS $N=50$ branch. The QH model was already known to be a good
candidate inflationary model and we show here that whilst accounting for the
higher order terms does shift the positions of the $N$-branches quite
dramatically, the model still remains within a region of the parameter space
that makes it an attractive inflationary model. In addition to both models
being favourable with respect to the Planck 2018 data, it is clear from
comparing the two solutions, such as in figure (5), that the two solutions
deviate quite dramatically long before either model enters the region of
parameter space favoured by the Planck 2018 data. This is because the higher
order terms actually become large during inflation and are therefore relevant
to the calculation of observables and dramatically change the inflationary
predictions of the model by shifting the $r-n_{s}$ spectrum far to the left.
Clearly the QH and QHS models need to be treated distinctly in the context of
inflation, and that one cannot simply take the Quartic Hilltop model as an
accurate approximation of Quartic Hilltop Squared.

Throughout this work frequent references and comparisons are made to numerical
solutions for the Hilltop models in question. The numerical solutions help
evaluate the validity of the results and form part of ongoing work by the
author to develop numerical solutions for a wide range of inflationary models.
The ability to solve models numerically will aid in investigating the generic
features of Hilltop models. For example whether or not the $N_{\rm
end}\propto\lambda^{-1/2}$ relationship holds for other Hilltop models or if
there is some other power law. Furthermore, the numerical solutions also make
it possible to investigate how dramatically the $r-n_{s}$ changes in other
Hilltop models, say for example quintic models as opposed to quartic and
whether or not those models require us to account for the higher order
correction terms.

At a first glance, the fact that both the QH and QHS model predictions for the
tensor-scalar ratio and spectral index fit nicely into the Planck 2018 bounds
seems encouraging, however as we note, and has already been discussed at
length in the literature, this does not necessarily mean the model goes a long
way in helping explain the data that has been presented since we may generate
any $(n_{s},r)$ pair by appropriate choice of $N_{k}$, that may or may not
correspond to reheating predictions that are consistent with standard
cosmology.

Correcting the models UV behaviour allows us to explore the reheating dynamics
of the QHS model through the numerical solutions and further constrain its
$r-n_{s}$ parameter space to the region bounded by $63\leq N_{k}\leq 68$, by
considering Big Bang Nucleosynthesis limits on the reheating temperature.
Whilst this does not drastically reduce the range of allowed values of $n_{s}$
and $r$, the area of $(n_{s},r)$ pairs is significantly smaller than if one
were to just consider the bounds derived by the Planck 2018 survey.

## Acknowledgements

The authors are indebted to the anonymous referee whose input has greatly
improved this paper.

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