# Gravitational Waves and Gravitino Mass in No-Scale Supergravity Inflation
with Polonyi Term

Miguel Crispim Romão Email<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Southampton. Southampton SO17 1BJ, United Kingdom  Laboratório
de Instrumentação e Física Experimental de Partículas, Escola de Ciências,
Departamento de Física, Universidade do Minho, 4701-057 Braga, Portugal
Stephen F. King Email<EMAIL_ADDRESS>School of Physics and Astronomy,
University of Southampton. Southampton SO17 1BJ, United Kingdom

###### Abstract

We study a No-Scale supergravity inflation model which has a non-minimal
deformation of the Kähler potential and a Wess-Zumino superpotential extended
by the inclusion of a Polonyi mass term. The non-minimal structure of the
Kähler potential is responsible for an inflexion point that can lead to the
production of gravitational waves at late stages of inflation, while the
Polonyi term breaks supersymmetry at the end of inflation, generating a non-
vanishing gravitino mass. After a thorough parameter space scan, we identify
promising points for gravitational wave production. We then study the
resulting gravitational wave energy density for this set of points, and we
observe that the gravitational waves should be observable in the next
generation of both space-based and ground-based interferometers. Finally, we
show how the presence of the Polonyi term can be used to further boost the
gravitational wave energy density, which is correlated with the gravitino
mass. The code used for the scan and the numerical analysis is provided at
https://gitlab.com/miguel.romao/gw-and-m32-no-scale-inflation-polonyi.

##  1 Introduction

With the first observation of Gravitational Waves (GW) by LIGO and Virgo
collaborations [1, 2, 3], we have witnessed a new era of Cosmology. Since
then, multiple signals have been observed from astrophysical phenomena and,
more recently, the NANOGrav collaboration has observed evidence for the
existence of a stochastic GW background [4, 5]. With multiple future GW
experiments lined up, including space-based interferometers like LISA, BBO,
and DECIGO [6, 7, 8], as well as next-generation ground-based interferometers
such as the Cosmic Explorer and the Einstein Telescope [9, 10], sensitivity to
GW signals will soon include a multitude of phenomena beyond astrophysical
events.

A particularly interesting source of GW that is not from astrophysical
phenomena is the one associated with an enhancement of the scalar power
spectrum (henceforth in this work refereed solely as power spectrum) during
inflation. In this scenario, the first order scalar perturbations will drive
second order perturbations that are themselves responsible for the production
of GW [11, 12, 13, 14]. In order to achieve this, the effective potential of
the inflaton needs to exhibit a non-trivial feature where the inflaton
considerably slows down, driving an enhancement of the power spectrum. Such a
feature could be an inflexion point at a later stage of inflationary dynamics,
where the inflaton slows down while traversing it due to the Hubble attrition.
Such a scenario has been extensively studied recently within the context of
Primordial Black-Holes (PBH) production [15, 16], and more recently as a
mechanism to generate GW in a No-Scale Supergravity [17], which we will follow
closely throughout this work.

Inflation [18, 19, 20, 21, 22] is a highly motivated framework that provides a
solution for some of the problems unexplained by the Cosmological Standard
Model [23], such as the flatness and horizon problems, the absence of
cosmological relics, and the origin of cosmological fluctuations. Inflation is
characterised by a period of sustained accelerated expansion due to a negative
pressure. This can be obtained if all physical quantities are slowly varying.
In particular, it can be described by the so-called slow-roll inflationary
[24, 25] paradigm, where one (or many) scalar field(s) slowly evolve along a
flat direction of the effective scalar potential. Currently, there is no model
independent way of testing inflation, but observational data from the Planck
satellite [26] can constrain inflationary dynamics and rule out many candidate
models of inflation. Many models of inflation have been proposed over the past
years [27], but most have been ruled out by the Cosmic Microwave Background
(CMB) data from the Planck satellite. The remaining viable models fall inside
a few classes, such as non-minimal gravity models, such as the $R^{2}$
Starobinsky inflation [28, 29], Higgs and related inflation [30, 31, 32], and
hybrid low scale inflation [33, 34]. While these classes of models survive
Planck data, they are very sensitive to their possible Ultra-Violet (UV)
completions, where generic corrections to the potential can lead to large
corrections to the slow-roll parameters, spoiling inflation in what is known
as the $\eta$-problem.

In the context of inflation, supersymmetry (SUSY) presents itself as a
promising framework for models since in such models flat directions of the
scalar potential are common, with UV corrections being naturally controlled by
SUSY. For these reasons, SUSY has been a highly motivated ingredient for
inflation [35, 36, 37]. Given that inflationary dynamics occurs so early in
our Universe’s history, and given the expected role that UV contributions
might have, it is natural to study SUSY in its local form of Supergravity
(SUGRA). In SUGRA, the F-term scalar potential receives contributions from two
functions: the superpotential and the Kähler potential. It has been known that
quadratic terms in the Kähler potential, which appear naturally if the Kähler
potential is given by a minimal expansion in the (super)fields, will lead to
the $\eta$-problem in SUGRA inflationary models. A solution to this problem is
to consider the so-called No-Scale SUGRA, where the Kähler potential is given
by a logarithmic form of the (super)fields, which are common in string theory
compactifications, an idea that has gathered considerable interest lately
[38].

In [39, 40], Ellis, Nanopoulos, and Olive (ENO) have motivated a No-Scale
Kähler potential arising from a non-compact $SU(2,1)/U(2)\times(1)$ coset
compactification parametrised by two fields, a modulus $T$ and a chiral
superfield $\Phi$. Using this Kähler potential in a Wess-Zumino framework for
the superpotential, they showed that (for a given point in the parameter
space) this SUGRA model is conformally equivalent to an $R+R^{2}$ Starobinsky
inflation. Shortly after, we (CRK) have proposed a minimal extension of the
Wess-Zumino superpotential by considering the existence of a Polonyi mass term
[41]. In our work, we showed that the Polonyi mass term leads to SUSY
breaking, and that the viable region of the parameter space for inflation also
set a bound on the gravitino mass, in the range of current and near future
colliders. The phenomenology of SUSY breaking from our model has also been
explored [42] (for a general discussion of the status of no-scale SUGRA
phenomenology see [43]). Recently, Nanopoulos, Spanos, and Stamou (NSS) [15]
suggested that a deformation of the compact space considered by ENO would lead
to an enhancement of the power spectrum that could be exploited to produce
PBH. This new Kähler potential has been further been understood in more
generic terms as breaking the non-compact coset symmetry of the
compactification [16], and it was also shown that it can lead to the
production of GW [17].

In this work we revisit the No-Scale CRK model [41], which extended the one
discussed by ENO by considering the presence of a Polonyi term in the
superpotential, and we consider the impact of the more general Kähler
potential with the deformation proposed by NSS. Our goal is to assess whether
the phenomenology of the gravitino mass, a cornerstone feature of our previous
model, is impacted by the presence of the NSS Kähler potential deformation,
and, conversely, whether the phenomenology of the GW, a central feature of the
NSS model, is impacted by the presence of the Polonyi term. In order to study
this question, we perform a thorough analysis of the parameter space,
constraining it by the Planck satellite data and viable inflation
requirements, followed by a careful numerical analysis of the power spectrum
and resulting GW energy density spectrum for a selection of promising points.
We shall find that the gravitino mass in the CRK model is indeed correlated
with the energy density of the GW signal.

This work is organised as follows. In section 2 we introduce the No-Scale
Supergravity framework for inflationary models, where we set the notation,
review main results, and present our model and its respective inflaton scalar
potential in section 2.4. Next, in section 3, we perform a thorough scan of
the parameter space in order to identify regions in the parameter space that
are in agreement with the Planck satellite constraints whilst producing viable
inflation, and that show promising features that can lead to an enhancement of
the power spectrum, which is needed for GW production. After identifying these
promising points, in section 4 we start by computing the full numerical
solution for the power spectrum in section 4.1, which we finally use to derive
the present day energy density of the GW spectrum in section 4.2. In section 5
we draw our conclusions and discuss future directions of work.

##  2 The No-Scale Supergravity Inflation Framework

We review the No-Scale SUGRA inflation framework to set notation, present our
model, and derive all relevant formulae for later analysis.

###  2.1 $\mathcal{N}=1$ Supergravity Preamble

The main quantity in $\mathcal{N}=1$ SUGRA [44, 45] is the dimensionless real
Kähler function, $G$, given by

$G=K(\Phi,\Phi^{*})+\log|W(\Phi)|^{2}$ (1)

where $\Phi$ represents all the superfields in the theory, $K$ is the Kähler
potential, itself a real function of the superfields, and $W$ is the
superpotential, which is a holomorphic function of the superfields. We also
notice that we are following the customary SUGRA notations where we use the
same symbol to refer to both the superfield and its scalar component, and the
reduce Planck mass, $m_{Pl}$, is set to unity, i.e. $m_{Pl}=1$, and therefore
all dimensionful quantities in this work are considered in units of $m_{Pl}$
unless explicitly given in GeV during the discussion.

The scalar F-term potential of the whole theory, $V_{F}$, is given by

$\displaystyle V_{F}$ $\displaystyle=e^{G}(K^{ij^{*}}G_{i}G_{j^{*}}-3)$ (2)
$\displaystyle=K_{i^{*}j}F^{i^{*}}F^{j}-3e^{K}|W|^{2}\ ,$ (3)

where the derivatives are taken with respect to the superfields with
$\partial_{i}=\partial/\partial\Phi^{i}$,
$\partial_{i^{*}}=\partial/\partial\Phi^{*i}$, and
$\partial_{i^{*},j}=\partial^{2}/(\partial\Phi^{*i}\partial\Phi^{j})$, while
the expression is evaluated on the scalar field component of the superfield,
and

$\displaystyle K_{i^{*}j}$ $\displaystyle=\partial_{i^{*},j}K$ (4)
$\displaystyle F^{i}$
$\displaystyle=-e^{K/2}K^{ij^{*}}(\partial_{j^{*}}W^{*}+W^{*}\partial_{j^{*}}K)\
,$ (5)

where $K_{i^{*}j}$ is the Kähler metric, $K^{ij^{*}}$ its inverse (i.e.,
$K^{ij^{*}}K_{j^{*}k}=\delta^{i}_{j}$), and $F^{i}$ is the SUGRA F-term. For
gauge singlets, which will be focus of our work, the total scalar potential is
given by $V=V_{F}$ as there is no D-term scalar potential contributions.

A very important observable in SUGRA is gravitino mass. In the case for
vanishing cosmological constant, it can be written as

$m_{3/2}^{2}=e^{K}|W|^{2}=\frac{1}{3}K_{i^{*}j}F^{i^{*}}F^{j}\ ,$ (6)

where the last equality is obtained from eq. 3 by imposing $V=V_{F}=0$. Since
is is non-vanishing only when $\langle F^{i}\rangle\neq 0$, it can be taken to
the order parameter of SUSY breaking.

Next, we introduce the forms of the Kähler potential and superpotential that
we will be studying for the rest of the paper, and how the shape of the
potential is affected by different values of the its parameters.

###  2.2 The No-Scale Kähler Potential

The general form of the No-Scale Kähler potential that we will study is

$K=-3\log\left(T+T^{*}+f(\Phi,\Phi^{*})\right)\ ,$ (7)

where $T$ is a modulus field, to be stabilised by some other mechanism defined
by the UV completion of the theory, and $\Phi$ corresponds to a hidden sector
matter superfield, whose scalar part will drive inflation, while $f$ is a real
function.

In the $(\Phi,T)$ basis, eq. 7 leads to the Kähler metric

$K_{i^{*}j}=\frac{3}{(T+T^{*}+f)^{2}}\begin{pmatrix}|\partial_{\Phi}f|^{2}-(T+T^{*}+f)\partial_{\Phi^{*},\Phi}f&\partial_{\Phi^{*}}f\\\
\partial_{\Phi}f&1\end{pmatrix}\ ,$ (8)

and we note that an important entry is the $(1,1)$ element

$K_{\Phi^{*}\Phi}=\frac{3}{\left(T+T^{*}+f\right)^{2}}\left(|\partial_{\Phi}f|^{2}-(T+T^{*}+f)\partial_{\Phi^{*},\Phi}f\right)\
,$ (9)

which, once $T$ is stabilised, needs to accounted for when studying the
canonically normalised field evolution.

Two forms of the Kähler potential have been thoroughly studied. In [39, 40]
ENO, proposed the simple form for a space defined by the non-compact
$SU(2,1)/SU(2)\times U(1)$ coset, parametrised by a modulus $T$ and a
superfield $\Phi$ as

$K_{ENO}=-3\log\left(T+T^{*}-\frac{\Phi^{*}\Phi}{3}\right)\ ,$ (10)

and they showed that leads to Starobinsky inflation when paired with the a
Wess-Zumino superpotential, which we will discuss in the next section. More
recently, NSS [15] (and [16]) extended the discussion to deformation of the
space described above, given by a Kähler potential of the form

$K_{NSS}=-3\log\left(T+T^{*}-\frac{\Phi^{*}\Phi}{3}+ae^{-b(\Phi+\Phi^{*})^{2}}(\Phi+\Phi^{*})^{4}\right)\
,$ (11)

where $a$ and $b$ are free parameters, and it was shown that the new
exponential factor inside the logarithm will produce a feature in the
potential which can take the form of an inflexion point from which primordial
black holes and gravitational waves can be produced [15, 16, 17]. For both
Kähler potentials, the modulus $T$ is assumed stabilised and real, for which
we take

$\langle T\rangle=\langle T^{*}\rangle=\frac{c}{2}\ ,$ (12)

with vanishing derivatives.

####  2.2.1 Kinetic Term and Canonically Normalised Field

An immediate consequence of the Kähler metric, eq. 8, not being diagonal is
that the field $\Phi$ is not canonically normalised. Taking the modulus $T$ to
be stabilised as eq. 12, the kinetic term of $\Phi$ has a factor given by the
$(1,1)$ entry of the Kähler matrix, eq. 9. In order to proceed, we need to
identify a canonically normalised field which will play the role of the
inflaton. Consider the complex field

$\chi=\frac{1}{\sqrt{2}}(x+iy)$ (13)

as the canonically normalised field associated with $\Phi$,

$\mathcal{L}_{Kin}\supset
K_{\Phi^{*}\Phi}\partial_{\mu}\Phi^{*}\partial^{\mu}\Phi=\partial_{\mu}\chi^{*}\partial^{\mu}\chi=\frac{1}{2}\partial_{\mu}x\partial^{\mu}x+\frac{1}{2}\partial_{\mu}y\partial^{\mu}y\
,$ (14)

and assume now that the imaginary part has a vanishing vacuum expectation
value (VEV) and it is not dynamic, so that we can focus on $\phi=\Re(\Phi)$.
From the previous equation, we see that the field $x$, which we will identify
as the inflaton, is now the canonically normalised version of $\phi$, with

$\partial_{\mu}x=\frac{\partial x}{\partial\phi}\partial_{\mu}\phi\ ,$ (15)

where the map between the two, $x=x(\phi)$, is given by integrating the
exchange of variables coefficient given by the Kähler metric entry

$\frac{\partial x}{\partial\phi}=\sqrt{2K_{\Phi^{*}\Phi}\big{|}_{\Phi=\phi}}\
.$ (16)

For the case of the ENO Kähler potential, eq. 10, it is possible to perform
the integration eq. 16 and obtain a closed-form expression

$\phi_{ENO}=\sqrt{3c}\tanh\left(\frac{x}{\sqrt{6}}\right)\ .$ (17)

where $c$ is the modulus VEV in eq. 12 However, for more complicated Kähler
potentials, such as the NSS eq. 11, eq. 16 can only be integrated numerically.

###  2.3 Superpotential

The second ingredient required to define the scalar potential is the
superpotential. In [39, 40], ENO assumed the minimal Wess-Zumino
superpotential

$W_{ENO}=\frac{\hat{\mu}}{2}\Phi^{2}-\frac{\lambda}{3}\Phi^{3}\ ,$ (18)

which, with eq. 10, produces a scalar potential where the inflaton is
identified with $x$, as in the previous section, producing a Starobinsky-like
inflation. In fact, this superpotential and eq. 10 have an exact Starobinsky
limit for $\hat{\mu}=\mu/\sqrt{c/3}$ and $\lambda=\mu/3$. This superpotential
was also considered by NSS paired with their proposed Kähler potential eq. 11.

Later, we have proposed extending the Wess-Zumino superpotential with the
addition of the a Polonyi mass term [41]

$W_{CRK}=M^{2}\Phi+W_{ENO}\ .$ (19)

The addition of the extra Polonyi mass term shifts the VEV of the inflaton at
the end of inflation to a non-vanishing configuration, $\langle x\rangle\neq
0$, producing a non-vanishing gravitino mass, which is bounded from above for
the regions of the parameter space that produce good inflation. We will
revisit this result in the next section.

###  2.4 Scalar F-Term Potential

Regardless of the choice of superpotential, the generic No-Scale Kähler
potential from eq. 7, leads to a scalar F-term potential for $\phi$ of the
form

$V=\frac{1}{-3\partial_{\Phi^{*},\Phi}f}\frac{1}{\left(f+T+T^{*}\right)^{2}}|\partial_{\Phi}W|^{2}\
\Big{|}_{\Phi=\Phi^{*}=\phi},$ (20)

and we have organised the expression as the product of three terms, where the
last two are always positive semi-definite, but the first can take negative
values for a generic real function $f=f(\Phi,\Phi^{*})$. Note, however, that
for the ENO Kähler potential, eq. 10, one has $f=-|\Phi|^{2}/3$, and therefore
the potential is always positive semi-definite. The same cannot be said for
the NSS Kähler potential, eq. 11.

We now turn to case of the NSS Kähler potential, eq. 11, with the CRK
superpotential, eq. 19, which will be the focus of our work. To simplify
notation, we also use the following parameter redefinitions for reasons that
will become clear shortly

$\displaystyle\hat{\mu}$ $\displaystyle\equiv\mu\sqrt{c/3}$ (21)
$\displaystyle M$ $\displaystyle\equiv\sqrt{\mu d}$ (22)
$\displaystyle\lambda$ $\displaystyle\equiv\mu l\ ,$ (23)

where the replacement of $\hat{\mu}$, $M$ and $\lambda$ by three new
parameters $\mu$, $d$ and $l$ does not change the total parametric freedom,
which also includes the Kähler parameters $a,b$ in eq. 11 and the modulus VEV
$c$. This corresponds to six real parameters in total, although $\mu^{2}$ can
be factored out of the scalar potential which depends on the five real
parameters $a,b,c,d,l$, as we now discuss.

With these, we can obtain the final scalar F-term potential for the non-
canonically normalised inflaton field, $\phi=\Re(\Phi)$,

$V=\frac{1}{1-24a\phi^{2}e^{-4b\phi^{2}}\left(4b\phi^{2}\left(8b\phi^{2}-9\right)+6\right)}\frac{1}{\left(c-\frac{\phi^{2}}{3}+16ae^{-4b\phi^{2}}\phi^{4}\right)^{2}}\mu^{2}(d+\sqrt{c/3}\phi-l\phi^{2})^{2}\
,$ (24)

which can be shown to reproduce our previous results for $a=0$ (with the ENO
results obtained by further considering $d=0$), as well as exhibiting a
Starobinsky limit for $l=1/3$111The expression eq. 24 differs to the one in
[15]. After contacting the authors, we were able to identify the source of the
mismatch from their side.. Furthermore, we can factorise $\mu^{2}$ as a result
of the parametric redefinitions listed above.

With the expression for the scalar potential of the non-canonically normalised
field $\phi$, eq. 24, and the change of variables from $\phi$ to the
canonically normalised inflaton field, $x$, given by eq. 16, we can plot the
potential for different values of the parameters to assess their impact in the
potential shape. In fig. 1 we present the potential where we take each
parameter to vary in a certain range while keeping the remainder constant.

(a) Varying $a\in[-2,0]$.

(b) Varying $b\in[10,30]$.

(c) Varying $c\in[0.03,0.1]$.

(d) Varying $d\in[10^{-9},10^{-4}]$.

(e) Varying $l\in\frac{1}{3}\times[0.999,1.001]$.

Figure 1: The inflaton potential displays a kink structure which depends
sensitivity on the different parameters. When fixed, the parameters take the
values $a=-1$, $b=22$, $c=0.065$, $d=10^{-4}$, $l=1/3$.

In the first row, figs. 1(a), 1(b) and 1(c), we observe how the shape of the
potential varies as we vary the three parameters that arise from the Kähler
potential $a$, $b$, and $c$, respectively. We observe that they all impact on
the shape of the kink-like feature, although this always happens around $x\sim
1$. In fig. 1(a) we see that increasing the magnitude of $a$ will make a more
pronounced feature in the potential, where the kink can become a wall. This is
easily understandable, as the larger the magnitude of $a$ the bigger will be
the contribution of the exponential part of $f$. Similarly, in fig. 1(b) we
see that a smaller $b$ will lead to a more pronounced perturbation of the
potential. This happens for the same reason as $a$, as $b$ tends to $0$ the
exponential becomes leading inside logarithm in the Kähler potential. However,
unlike $a$, the values of $b$ can impact the potential at around $x\sim 5$,
where the potential plateaus and where inflation is expected to start,
suggesting that too small values of $b$ will spoil the desired inflationary
characteristics of the potential. With regards to the last Kähler potential
parameter, $c$, in fig. 1(c) we see that for very small values we witness the
kink becoming again more pronounced. This is understandable as for small $c$
the remaining terms of the Kähler potential become leading, increasing their
impact in the shape and position of the kink. Like $b$, for extreme values of
$c$ the potential is also considerably changed around $x\sim 5$, spoiling good
inflationary dynamics. Finally, we notice that $b$ and $c$ appear to change
the eight and the width of the kink, whereas $a$ can only turn the kink into a
wall, but does not change its position.

In the second row, figs. 1(d) and 1(e), we observe how the shape of the
potential is affected by the ranges of values for the superpotential
parameters $d$, $l$, respectively. The first observation is that neither of
these parameters affect the kink. This is unsurprising, as the kink is a
feature arising from the new contributions to the Kähler potential proposed by
NSS, eq. 11. In fig. 1(d) we see how the magnitude $d$, which is proportional
to the Polonyi term c.f. eq. 22, has limited impact on the shape of the
potential unless it takes large values. This is in agreement with the findings
in [41], where $M$ was found to have minimal impact on Starobinsky-like
inflation unless the gravitino mass (which is mainly driven by $M$, as we will
see below) is large. Finally, in fig. 1(e) we see the impact of $l$, c.f. eq.
23 in the shape of potential. Especially, we note how the plateau where
inflation is expected to start is highly sensitive to a deviation of $l\neq
1/3$, with $l=1/3$ producing a Starobinsky-like inflationary model.

####  2.4.1 Potential Positive Semi-Definiteness and Global Minimum

As it can be seen from eq. 20, the scalar potential for the inflaton is not
always positive semi-definite. In particular, this means that we cannot
identify the global minimum of the potential to be the field configuration
where $V=0$ as it was done in [41]. To understand this better, we notice that
the potential is found to be positive semi-definite if the first factor in eq.
24 is positive, i.e.

$1-24a\phi^{2}e^{-4b\phi^{2}}\left(4b\phi^{2}\left(8b\phi^{2}-9\right)+6\right)>0\
.$ (25)

Figure 2: Positivity condition as a function of the parameter $b$ and the
field $\phi$ value for different values of $a$. Red (green) indicates a
negative (positive) value of the scalar potential.

In fig. 2 we study when eq. 25 holds for different values of $a$, $b$, and
$\phi$. We find that as long as $a\gtrsim-1.5$ and $b\gtrsim 15$, that the
potential is positive semi-definite for all values of $\phi$ during inflation.
In such case, the minimum of the potential is given by $V=0$, or conversely,
c.f. eq. 20, by

$\displaystyle\partial_{\Phi}W\Big{|}_{\Phi=\langle\Phi\rangle}=0\ ,$ (26)

which happens to be a simple quadratic equation in $\Phi$, with solution

$\langle\Phi\rangle=\frac{\hat{\mu}\pm\sqrt{\hat{\mu}^{2}+4M\lambda}}{2\lambda}\
,$ (27)

or, alternatively using the parameter redefinitions and focusing on $\phi$ we
have

$\langle\phi\rangle=\frac{1}{2l}\left(\sqrt{\frac{c}{3}}\pm\sqrt{\frac{c}{3}+4ld}\right)\
.$ (28)

Therefore, for $a\gtrsim-1.5$ and $b\gtrsim 15$ the inflaton, which is
identified as the canonically normalised version of $\phi$, will achieve the
global minimum at $V=0$ and be stabilised at a non-vanishing VEV, breaking
SUSY and producing a non-vanishing gravitino mass

$\langle\phi\rangle\neq 0\Rightarrow\langle F^{\Phi}\rangle\neq 0\Rightarrow
m_{3/2}\neq 0.$ (29)

##  3 Parameter Space Scan: Planck Data and Inflation

As it is observable in fig. 1, the non-trivial contribution in $K_{NSS}$, eq.
11, can lead to a kink-like feature in the potential that can become wall-
like, rather than an inflexion point or a plateau. As a consequence, the
canonically normalised inflaton field, $x$, which is the real part of $\chi$
in eq. 13, can get stuck in the kink at later e-fold times instead of rolling
down to the global minimum. Therefore, in order to study the parameter space
of our model, we need to dynamically evolve the inflaton from the top of the
potential to the global minimum. To do this, we need to solve its equation of
motion, which in e-fold time, $N$, is given by [17]

$\displaystyle\frac{d^{2}x}{dN^{2}}+\left(3-\frac{1}{2}\left(\frac{dx}{dN}\right)^{2}\right)\frac{dx}{dN}+\left(3-\frac{1}{2}\left(\frac{dx}{dN}\right)^{2}\right)\partial_{x}\ln
V=0\ .$ (30)

The closed-form for the potential appearing in the last term is given by eq.
24 is only known in terms of $\phi$, not of $x$. However, given the field
redefinition transformation, eq. 16, we can use the expression of the
potential in terms of $\phi$ as long as we take into account the field
redefinition transformation in the derivative with regards to $x$, i.e. by
replacing

$\displaystyle\partial_{x}\to\frac{1}{\sqrt{2K_{\Phi^{*}\Phi}\Big{|}_{\Phi=\phi}}}\partial_{\phi}\
.$ (31)

With this transformation, eq. 30 has now an explicit dependence on $\phi$,
therefore we need to evaluate it jointly with the e-fold time evolution of
$\phi$, which is given by

$\displaystyle\frac{d\phi}{dN}=\frac{1}{\sqrt{2K_{\Phi^{*}\Phi}\Big{|}_{\Phi=\phi}}}\frac{dx}{dN}\
.$ (32)

To integrate the equations above we need to set the initial conditions. We
integrate from the beginning inflation, which we set to be $N_{i}=0$, until
$N=70$ so that the integration covers the desired number of e-folds, $\Delta
N$, which should take values $\Delta N=N_{f}-N_{i}=\int_{a_{i}}^{a_{e}}d\ln
a\simeq 50-60$. The beginning of inflation happens at the high plateau of the
potential. This is identified with the pivot scale, $k^{*}=0.05\text{
Mpc}^{-1}$, i.e. $N^{*}\simeq N_{i}=0$, which is the scale at which the CMB
crosses the decreasing comoving Hubble radius. Therefore, the initial
conditions for $x^{*}=x(N^{*})$ are set such that it is given asymptotically
by

$\displaystyle
x^{*}=\sqrt{6}\operatorname{arctanh}\left(\frac{\phi^{*}}{\sqrt{3c}}\right)\
,$ (33)

which is justified by the fact that for large values of $\phi$ we have
$K_{NSS}\to K_{ENO}$ and so we recover the field redefinition eq. 17. In turn,
this sets an upper bound on $\phi^{*}\lesssim\sqrt{3c}$. As for the derivative
initial condition, we set it to the so-called slow-roll attractor, with the
choice of sign such that the inflaton rolls to decreasing values of $x$

$\displaystyle\left(\frac{dx}{dN}\right)^{*}\simeq-\left|\frac{1}{V}\partial_{x}V\right|_{x=x^{*}}\
.$ (34)

With the above integration of the equations of motion we will be able to
assess whether a point of the parameter space produces the desired inflation.
In addition to this, at the pivot scale we will compare the predictions for
the scalar to tensor ratio, $r$, and the scalar tilt, $n_{s}$, which were
measured by the Planck satellite to be (at 3 sigma) [26]

$\displaystyle r$ $\displaystyle<0.055$ (35) $\displaystyle n_{s}$
$\displaystyle\in[0.9536,0.9782]\ .$ (36)

These can be computed for a choice of parameters at top of the potential using
the potential slow-roll parameters, $\epsilon_{V}$ and $\eta_{V}$,

$\displaystyle r$ $\displaystyle\simeq 16\epsilon_{V}$ (37) $\displaystyle
n_{s}$ $\displaystyle\simeq 1-6\epsilon_{V}+2\eta_{V}\ ,$ (38)

with

$\displaystyle\epsilon_{V}$
$\displaystyle=\frac{1}{2}\left(\frac{\partial_{x}V}{V}\right)^{2}$ (39)
$\displaystyle\eta_{V}$ $\displaystyle=\frac{\partial_{x,x}V}{V}\ ,$ (40)

where both quantities can be computed using the expressions in terms of $\phi$
when taking eq. 31, and are evaluated at $x=x^{*}$, i.e. at $\phi=\phi^{*}$.

Both $n_{s}$ and $r$ are independent of $\mu$, as it can be factored out from
$V$ and it cancels in the expressions above. However, the value for $\mu$ can
be fixed by another Planck observable, which is the amplitude of the power
spectrum, $P_{R}$, measure at the pivot scale. At the pivot scale, we can use
the slow-roll approximation for $P_{R}$, which reads

$\displaystyle P_{R,sr}\simeq\frac{1}{8\pi^{2}}\frac{H^{2}}{\epsilon_{H}}\ ,$
(41)

where we add the label $s.r.$ to indicate it as a slow-roll approximation, $H$
is the Hubble radius, and $\epsilon_{H}$ the Hubble $\epsilon$ parameter,
given by

$\displaystyle H^{2}$ $\displaystyle=\frac{V}{3-\epsilon_{H}}$ (42)
$\displaystyle\epsilon_{H}$
$\displaystyle=\frac{1}{2}\left(\frac{dx}{dN}\right)^{2}\ ,$ (43)

where $dx/dN$ is obtained from the numerical integration222Alternatively, at
the pivot scale the full slow-roll approximation can be taken
$\epsilon_{H}\simeq\epsilon_{V}$ and $\epsilon_{V}\ll 1$ s.t.
$P_{R,s.r.}\simeq(1/8\pi^{2})V^{2}/(3\epsilon_{V})$.. The power spectrum is
measured by Planck to be $P_{R}\big{|}_{k=k^{*}}\simeq 2.1\times 10^{-9}$.

The last observable that we will study is the gravitino mass at the end of
inflation, i.e.

$\displaystyle m^{2}_{3/2}=e^{K}|W|^{2}\Big{|}_{\Phi=\langle\phi\rangle}\ ,$
(44)

where we choose the minus sign in eq. 28 as it corresponds to a global minimum
to the left of the maximum of the potential, i.e. in the direction where
$\phi$ is rolling to as set by eq. 34.

###  3.1 Random Scan

For the range of the parameters, we follow the discussion in section 2.4 and
in [41, 17] to focus the scan on a promising region of the parameter space:

$\displaystyle a$ $\displaystyle\in[-1.5,-0.5]$ (45) $\displaystyle b$
$\displaystyle\in[15,30]$ (46) $\displaystyle c$ $\displaystyle\in
0.065\times[0.95,1.05]$ (47) $\displaystyle d$ $\displaystyle\in 10^{[-8,-2]}$
(48) $\displaystyle l$ $\displaystyle\in[0.33327,1/3]$ (49)
$\displaystyle\phi^{*}$ $\displaystyle\in\sqrt{3c}\times[0.95,1.00]\ .$ (50)

We randomly sample $10^{6}$ points from this parameter space. For each point,
we collect the values of the observables listed above, the final value of
$N_{f}$ at which the numerical integration converges, and the value of the
inflaton field, $x$, at the end of inflation, $\langle x\rangle_{end}$. This
scan was performed in Mathematica, with the system of differential equations
eqs. 30 and 32 solved jointly using Mathematica’s NDSolve.

The first noticeable feature observed in the scan, is that not all parameter
points lead to the inflaton rolling down to the global minimum, as anticipated
by the exploratory analysis in the previous section. In fig. 3 we see how the
final field configuration for $x$, $\langle x\rangle_{end}$, is affected by
different values, and render each point either green or red depending on
whether the predicted slow-roll observables agree with Planck or not,
respectively. We see that there are two main clusters of points: those where
the inflaton gets stuck at the kink, $\langle x\rangle_{end}\gtrsim 1$, and
those where the inflaton rolls down to the global minimum, $\langle
x\rangle_{end}\lesssim 0.1$.

Figure 3: Inflaton field configuration, $\langle x\rangle_{end}$ at the end of
the numerical integration versus the parameters of the scalar potential. Green
(Red) points agree (disagree) with Planck constraints eq. 36.

For the points that agree with Planck cosmological constraints, we can further
see how the parameters $a$ and $b$ impact the value of $\langle
x\rangle_{end}$. More precisely, we see how for $a\to 0$ or $b\to\infty$ it is
easier for the inflaton to roll down to the global minimum, as in these
limiting cases the kink disappears completely. Conversely, for $a\to-\infty$
or $b\to 0$, the impact on the scalar potential from the non-trivial
contribution in eq. 11 becomes more pronounced and the kink goes from a
plateau to a wall, preventing $x$ from rolling down to the global minimum.

Since we are interested in the cases where the inflaton reaches the global
minimum, we now restrict to the case the points where $\langle
x\rangle_{end}<0.1$. For this subset of points, we show in fig. 4 the number
of e-folds, $\Delta N$, against the same parameters of the scalar potential.
As expected, there are fewer points for smaller values of $a$ and $b$, as in
this region the kink becomes as wall.

Figure 4: Number of e-folds, $\Delta N$ at the end of the numerical
integration versus the parameters of the scalar potential. Green (Red) points
agree (disagree) with Planck constraints eq. 36.

Focusing on the points that agree with the Planck satellite experiment, we see
that there is a clear preference for $l\to 1/3$, which represents the
Starobinsky limit of the ENO inflationary model. Additionally, we observe that
$\phi^{*}$ needs to be fairly close to its upper bound for $\Delta N>50$, this
is because for lower values of $\phi^{*}$ the inflaton starts very close to
the inflexion of the potential providing a very short inflationary period.
Finally, see how there is an upper bound on the parameter $d$ that sets the
value of the Polonyi term. Since the Polonyi term is responsible for the non-
gravitino mass, this in turn means that there is an upper bound on the values
of the gravitino mass. In fig. 5 we show the possible values of the gravitino
mass against $d$, and we observe that there is an upper bound of
$\mathcal{O}(10^{3})$ TeV on the gravitino mass for points with successful
inflation. This is in agreement with our previous results from [41].

Figure 5: Gravitino mass (in GeV) as a function of $d$ (in units of $m_{Pl}$).
Green points are in agreement with Planck constraints, and produce the
required inflationary dynamics with the inflaton rolling down to the global
minimum. Black points represent the whole sample. Horizontal lines are given
for gravitino mass values of 100 GeV, 10 TeV, 1000 TeV.

In fig. 6 we explore the sensitivity of the gravitino mass with regards the
rest of the parameters. In the top row, we show the gravitino mass versus the
Kähler potential parameters $a$, $b$, and $c$, and we observe that $m_{3/2}$
is uncorrelated to these. In the bottom row, we show $m_{3/2}$ versus the
other two superpotential parameters $\mu$ and $l$. We observe that the
gravitino mass is unrelated to $l$, but has a non-trivial dependence on $\mu$.
However, in the region with valid points in green, we can see that the
gravitino mass is uncorrelated with $\mu$, with $\mu\gtrsim 10^{-5}$ for all
valid points.

Figure 6: Gravitino mass (in GeV) versus (Top) Kähler potential and (Bottom)
superpotential parameters. Green points agree with Planck constraints eq. 36.

###  3.2 Exploratory Analysis of the Power Spectrum Peak

So far we have only studied the parameter space points under Planck
constraints and the requirements for the desirable inflationary dynamics.
However, a crucial part of this work is to study the production of GW by an
enhancement of the power spectrum originating by the presence of an inflexion
point in the inflaton potential. As it has been discussed before [15, 16], to
fully study the peak of the power spectrum we need to go beyond the slow-roll
approximation. However, this process is too computationally intensive to
systematically study the parameter space. Instead, in this section we perform
an initial analysis of the power spectrum using the slow-roll approximation,
eq. 41, to identify potentially interesting points. Once a set of interesting
points has been identified, we will compute the full numerical solution for
the power spectrum in section 4.2.

We first look at the points that lead to good inflationary dynamics and
observables, i.e,

  * •

Agree with Planck constraints

  * •

$50\leq\Delta N\leq 60$

  * •

$\langle x\rangle_{end}<0.1$ .

In addition, we further restrict to points where the (slow-roll) power
spectrum, $P_{R,s.r.}$, has a maximum when the inflaton rolls through the
inflexion point. Defining $\tilde{N}$ as the e-fold time at which the (slow-
roll) has a maximum, we then restrict it to be around
$30\lesssim\tilde{N}\lesssim 45$. Out of $10^{6}$ points, only $46$ satisfy
these conditions. In fig. 7 we show the power spectrum, $P_{R,s.r.}$, the
inflaton field value, $x$, in terms of the e-fold time, $N$, and the potential
for these $46$ points.

Figure 7: For the $46$ points passing all constraints we present, from left to
right: The power spectrum in terms of the number of e-folds, the inflaton
field value in terms of e-folds, and the inflaton potential in terms of $x$.
However the enhancement of the power spectrum in these examples is not
sufficient to produce observable GW.

Although all the $46$ points pass the constraints listed above, they are
actually not good candidates for GW production. The reason being that
$P_{R,s.r.}$ is not enhanced enough, as we would need $P_{R,s.r.}$ to reach
around $10^{-4}$ to produce observable gravitational waves. In principle, this
could be obtained by tweaking the value of the parameter $b$, which has been
observed to produced the desired enhancement with significant level of fine-
tuning [16]. The tuning, however, does not work for these points for the
following reason. Enhancing the $P_{R,s.r.}$ peak requires $x$ to slow down as
it passes the plateau even further. Looking at the middle and right plots in
fig. 7, this requires $x$ to spend _more time_ rolling through the lower
plateau. This can be achieved by slightly decreasing the value of $b$, as this
enhances kink. However, when tuning $b$ for these points we find that $\Delta
N$ can easily go beyond $60$, producing _too much inflation_ , while the
$P_{R,s.r.}$ peak never raises significantly. We show this explicit for one of
the four points where $\Delta N\simeq 50$, given by the parameters

$\displaystyle\\{a,b,c,d,l,\phi^{*}\\}=\\{-0.827,19.0177,0.0666,5.35\times
10^{-6},0.3333,0.432\\}\ ,$ (51)

and where we keep more significant digits for $b$ and $l$ parameters to which
the potential and the power spectrum are especially sensitive to. We now tune
the $b$ parameter by reducing it by a small amount parametrised by $\delta b$
such that

$\displaystyle b\to b\times(1-\delta b)\ ,$ (52)

where we scan the for the possible values $\delta b\in
4\times[10^{-5},10^{-4}]$.

In fig. 8 we show the changes as we tune the value of $b$. In the leftmost
plot, we see that even though $P_{R,s.r.}$ can be enhanced by an order of
magnitude, it fails to be significantly increased. In the middle plot, we
observe how the number of e-folds increase beyond $60$ as we decrease $b$.
This is easily understood as by decreasing $b$ we are slowing down the
inflaton field at the plateau, prolonging inflation. In the limit that we
significantly decrease $b$, we can see how the inflaton can get stuck at the
plateau, as $x$ does not roll down to the bottom of the potential. Finally, on
the rightmost plot we see that changes to $b$ of the order of the tuning,
$\mathcal{O}(10^{-4})$, has no visible effect on the shape of the potential,
even thought it has a significant impact on the $P_{R,s.r.}$ and the number of
e-folds. Similar conclusions can be drawn for the remaining 45 points that
pass all of the constraints.

Figure 8: For the point in eq. 51 we present, from left to right and for
various values of $b$: The power spectrum in terms of the number of e-folds,
the inflaton field value in terms of e-folds, the inflaton potential. As
before the enhancement of the power spectrum in these examples is not
sufficient to produce observable GW.

The previous discussions opens up for the possibility that there may be points
that have not passed all the required constraints, but might be tunable into
promising points. We now focus on points that have a good agreement with
Planck data, but that the inflaton rolls down to the global minimum earlier,
also producing an earlier $P_{R,s.r.}$ peak, i.e.

$\displaystyle 45\leq\Delta N\leq 50$ (53) $\displaystyle 30<\tilde{N}<40\ ,$
(54)

as we expect to be able to _delay_ both quantities by tuning $b$ accordingly.
Out of the million sampled points, $38$ points satisfy these requirements. In
figs. 9 and 10 we present the power spectrum, the inflaton field values, and
the potential for these points, and where the colour of the plots is given by
the value of the parameters $a$ and $b$, respectively. In accordance to the
discussion in section 2.4, we observe that lower values of $b$ lead to the
plateau (arising from the inflexion point) happens _earlier_ , i.e. for
smaller values of both the e-fold time, $N$, and the value of the inflaton
field, $x$.

Figure 9: Promising points that agree with Planck, but with shorter
inflationary period of $45\leq\Delta N\leq 50$ and earlier $P_{R}$ peak
($30<\tilde{N}<40$). Gradient represents the value of $a$ of each point. From
left to right: The power spectrum in terms of the number of e-folds, the
inflaton field value in terms of e-folds, the inflaton potential. However the
enhancement of the power spectrum in these examples is still insufficient to
produce observable GW. Figure 10: Promising points that agree with Planck, but
with shorter inflationary period of $45\leq\Delta N\leq 50$ and earlier
$P_{R}$ peak ($30<\tilde{N}<40$). Gradient represents the value of $b$ of each
point. From left to right: The power spectrum in terms of the number of
e-folds, the inflaton field value in terms of e-folds, the inflaton potential.
Again the enhancement of the power spectrum in these examples is not
sufficient to produce observable GW.

Perhaps more surprisingly, we also see the same trend for $a$, which is a
phenomenon not supported by the earlier discussion in section 2.4. In fact, in
fig. 11 we show that for these points333In fact, the same is observed for the
points that pass all the constraints shown in fig. 7, but this is not shown to
declutter the discussion., $a$ and $b$ are highly anti-correlated. In light of
the discussion in section 2.4, this can be understood as follows. Decreasing
$b$ will produce an earlier plateau around the inflexion point, but it might
also turn the plateau into a wall. On the other hand, decreasing $a$ can
mitigate the appearance of the wall after the inflexion point, and to an even
shorten the plateau444We have also performed an equivalent analysis for the
parameter $c$, but there was no correlation..

Figure 11: $(a,b)$ scatter plot for the points that agree with Planck, but
with shorter inflationary period of $45\leq\Delta N\leq 50$ and earlier
$P_{R}$ peak ($30<\tilde{N}<40$).

Another feature that is noticeable from figs. 9 and 10 is that for larger
values of $a$ the power spectrum has a wider and rounder peak, similar to what
can be seen in fig. 8 and for most of the points in fig. 7. As it was studied
in fig. 8, these are unlikely to produced a high enough power spectrum
enhancement. We therefore focus on the points where the power spectrum has a
sharp enhancement, which happens for $a\leq-0.9$. This cut produces only three
points, which we will denote Early $\tilde{N}$ points. We then proceed to tune
$b$ in order to maximise $P_{R,s.r.}$ while keeping $\Delta N\lesssim 60$ and
avoiding $x$ getting stuck at the plateau. For completeness of our analysis,
we will also include a point with a slightly later $\tilde{N}$, i.e. a point
that initially (prior to tuning) respects eq. 53 but not eq. 54. The four
points identified by the random scan as showing promising features for
gravitational waves production to be studied in more detail in section 4.2 are
presented in table 1 and in fig. 12.

$a$ $b$ $c$ $d$ $l$ $\phi^{*}$ $\delta b$ $b\times(1-\delta b)$ Early
$\tilde{N}$ (1) $-0.932$ $20.967463$ $0.0671$ $8.56\times 10^{-6}$ $0.33331$
$0.434$ $0.000056$ $20.966291$ Early $\tilde{N}$ (2) $-0.914$ $20.642496$
$0.0671$ $9.26\times 10^{-7}$ $0.33330$ $0.433$ $0.000206$ $20.638251$ Early
$\tilde{N}$ (3) $-0.901$ $20.686070$ $0.0618$ $9.64\times 10^{-6}$ $0.33328$
$0.415$ $0.000475$ $20.676242$ Late $\tilde{N}$ $-1.180$ $25.427327$ $0.0645$
$2.02\times 10^{-6}$ $0.33333$ $0.426$ $0.000007$ $25.427144$

Table 1: Promising points with $b$ tuned in order to maximise $P_{R,s.r.}$ and
to achieve $\Delta N\gtrsim 50$. Given the level of tuning associated with $b$
and the sensitivity of the shape of potential to $l$, we express these
parameters with more significant digits than $a$, $c$, $d$, and $\phi^{*}$.
Figure 12: Promising points from table 1 with $b$ tuned. In this case the
enhancement of the power spectrum in these examples is sufficient to produce
observable GW.

In table 2 we present the observables $n_{s}$, $r$, and $m_{3/2}$ for the
tuned points listed in table 1, where we highlight they all agree with Planck
satellite constraints from eq. 36. On the other hand, the gravitino mass
values are within what we expected from the discussion earlier in this
section. It is worth noticing that the observables in this table changed at
most $\mathcal{O}(10^{-5})$ as we tuned $b$, which is far below the number of
significant digits presented. This reinforces the idea that tuning $b$ will
have a considerable impact on the power spectrum peak and number of e-folds,
but not on the remainder of the observables, including the gravitino mass.

| $n_{s}$ | $r$ |  $m_{3/2}$ (GeV)
---|---|---|---
Early $\tilde{N}$ (1) | $0.9677$ | $0.011$ | $7.24\times 10^{5}$
Early $\tilde{N}$ (2) | $0.9576$ | $0.009$ | $7.74\times 10^{3}$
Early $\tilde{N}$ (3) | $0.9674$ | $0.017$ | $1.31\times 10^{6}$
Late $\tilde{N}$ | $0.9586$ | $0.007$ | $3.35\times 10^{4}$
Table 2: Observables for the tuned points from table 1.

##  4 Full Numerical Solution and Gravitational Waves Production

In the previous section we have performed a thorough parameter space scan
where we identified points that produce the desired inflationary observables,
and that could be tuned such that the slow-roll approximated power spectrum
was enhanced, while ensuring that the inflaton rolls down to the global
minimum after around $50$ e-folds. However, it has been pointed out [15, 16,
46, 17] that the slow-roll approximated power spectrum in eq. 41 fails to
capture the true power spectrum, where the peak can be both higher and be
shifted. This is easily understood as the dynamics around the plateau break
the slow-roll approximation, since the inflaton field arrives at the plateau
with high velocity, invalidating $\epsilon_{V}\ll 1$, and is met with an
increase in deceleration, invalidating $|\eta_{V}|\ll 1$. Once the full
numerical solution for the power spectrum is found, we can then compute the
present day gravitational wave density spectrum.

###  4.1 Full Numerical Solution for Power Spectrum

We now focus on accurately computing the power spectrum. The procedure
presented here is detailed in [27], and it has been applied in [46, 17]. For a
given mode of comoving wavelength $k$, the associated power spectrum is given
by

$\displaystyle P_{R}$ $\displaystyle=\frac{k^{3}}{2\pi^{2}}|R_{k}|^{2}$ (55)
$\displaystyle R_{k}$ $\displaystyle=\Psi+\frac{\delta x}{dx/dN}\ ,$ (56)

where $\Psi$ is the Bardeen potential and $\delta x$ is the inflaton
fluctuations field555Strictly speaking, the $\delta x$ refers to the mode $k$,
i.e. $\delta x_{k}$, as we are studying the contribution of each mode to the
power spectrum. Here we follow the slightly simplified notation also found
elsewhere in the literature.. The fields $\Psi$ and $\delta x$ are obtained by
evolving their respective equation of motion

$\displaystyle\frac{d^{2}\delta x}{dN^{2}}$
$\displaystyle=-\left(3-\frac{1}{2}\left(\frac{dx}{dN}\right)^{2}\right)\frac{d\delta
x}{dN}-\frac{1}{H^{2}}\partial_{x,x}V\delta x-\frac{k^{2}}{a^{2}H^{2}}\delta
x+4\frac{d\Psi}{dN}\frac{dx}{dN}-\frac{2\Psi}{H^{2}}\partial_{x}V$ (57)
$\displaystyle\frac{d^{2}\Psi}{dN^{2}}$
$\displaystyle=-\left(7-\frac{1}{2}\left(\frac{dx}{dN}\right)^{2}\right)\frac{d\Psi}{dN}-\left(2\frac{V}{H^{2}}+\frac{k^{2}}{a^{2}H^{2}}\right)\Psi-\frac{1}{H^{2}}\partial_{x,x}V\delta
x\ .$ (58)

The initial conditions are set at subhorizon scales, $k\gg aH$, such that we
have the so-called Bunch-Davies vacuum. For both $\delta x$ and $\Psi$, as
well as their derivatives, the initial conditions can be written solely as
dependent on the the comoving wavelength, $k$, and quantities that can be
derived from the background solution for $x$:

$\displaystyle(\delta x)_{i.c.}$
$\displaystyle=\frac{1}{\sqrt{2k}}\frac{1}{a_{i.c.}}$ (59)
$\displaystyle\left(\frac{d\delta x}{dN}\right)_{i.c.}$
$\displaystyle=-\frac{1}{a_{i.c.}\sqrt{2k}}\left(1+i\frac{k}{a_{i.c.}H_{i.c.}}\right)$
(60) $\displaystyle(\Psi)_{i.c.}$
$\displaystyle=\frac{1}{2\left(\epsilon_{H,i.c.}-\frac{k^{2}}{a^{2}_{i.c.}H^{2}_{i.c.}}\right)}\left[\left(\frac{dx}{dN}\right)_{i.c.}\left(\frac{d\delta
x}{dN}\right)_{i.c.}+(\delta
x)_{i.c.}\left(3\left(\frac{dx}{dN}\right)_{i.c.}+\frac{1}{H_{i.c.}}(\partial_{x}V)_{i.c.}\right)\right]$
(61) $\displaystyle\left(\frac{d\Psi}{dN}\right)_{i.c.}$
$\displaystyle=\frac{1}{2}\left(\frac{dx}{dN}\right)_{i.c.}(\delta
x)_{i.c.}-\Psi_{i.c.}\ ,$ (62)

where every quantity with subscript $i.c.$ is computed at an initial
conditions e-fold time, $N_{i.c.}$.

The process to obtain the power spectrum as a function of the comoving
wavelength of the modes, $P_{R}=P_{R}(k)$, is highlighted in [27] and we
summarise it now.

  1. 1.

The background solution for $x$, eq. 30 is integrated from $N^{*}=0$ until
$N_{end}=\Delta N$, from which we obtain $x(N)$ and $dx/dN(N)$.

  2. 2.

For each comoving wavelength $k^{\prime}$, we can find the e-fold time at
which it crosses the Hubble radius as $k^{\prime}=a(N^{\prime})H(N^{\prime})$.

  3. 3.

The initial conditions for mode $k^{\prime}$ are set at a time $N_{i.c.}$
earlier than $N^{\prime}$, ascribed by the relation [27]

$k^{\prime}=C_{q}a(N_{i.c.})H(N_{i.c.})\ ,$ (63)

where we set $C_{q}=100$. The value of $N_{i.c.}$ is obtained by producing an
interpolation map $N(k)$ obtained by inverting the function
$k(N)=a(N)H(N)$666There are two subtle details to refer in this step. The
first one, is that using the background solution, $x$, and
$k^{*}=a(N^{*})H(N^{*})$ we can obtain $a(N^{*})=a(0)$, from which we can find
$a(N)$ for any other e-fold time $a(N)=a(0)e^{N}$. The second detail is more
nuanced: for the pivot scale measured by the CMB, $k^{*}=0.05\ Mpc^{-1}$, we
have $N^{\prime}=N*=0>N_{i.c.}$ and therefore the initial conditions have to
be set for a negative $N_{i.c.}$. To workaround this, the background solution
for $x$ is in fact evolved from $N=-3$ until $\Delta N$. Effectively this adds
a few more e-folds to inflation prior to the CMB crossing the Hubble radius,
while keeping the pivot scale well defined and identified with the Planck
experiment.. All quantities in the initial conditions, eqs. 59, 60, 61 and 62,
can be computed using the background solution for $x$ and its derivative.

  4. 4.

The equations of motion eqs. 30, 57 and 58 are then integrated jointly from
$N_{i.c.}$ until $N_{end}=\Delta N$ in order to provide to obtain the present
day power spectrum associated with the mode of wavelength $k$, and eq. 55 is
finally evaluated.

  5. 5.

The process is repeated for values of $k^{\prime}\in[k^{*},k(\Delta N)]$, in
logarithm steps. This produces a list of tuples $(k,P_{R})$ from which an
interpolating function $P(k)$ can be obtained.

Using the steps above, we were able to reproduce the power spectra in [15, 16,
17]777Our results slightly disagree some of theirs, but this is due to a
difference in the final form of the scalar potential already discussed in
section 2.4 and this was resolved after contacting the authors.

Figure 13: Comparison between the slow-roll approximation power spectrum, c.f.
eq. 41, against the full numerical solution power spectrum, c.f. eq. 55. The
points shown are those identified in section 3 and are listed in table 1.

In fig. 13 we compare the power spectrum obtained by computing the full
numerical solution of the equations of motion against the one provided by the
slow-roll approximation for the points in table 1. A few things are worth
noticing at this stage. Firstly, both methodologies agree for early stages of
inflation, from $N\simeq N^{*}$ ($k\simeq k^{*}=0.05$ Mpc-1) up until the cusp
of the high plateau at $N\simeq 30$, which is the regime when the slow-roll
approximation is intended to be used. Secondly, we notice that for points
where the power spectrum is sharply enhanced by the plateau, it gets further
amplified by at least a few orders of magnitude, whereas for the remainder
points there is no discernable amplification. Thirdly, the power spectrum peak
is shifted to the left, i.e. to happen earlier. Lastly, given that, in human
units,

$\displaystyle 1\ \text{Mpc}^{-1}=0.97154\times 10^{-14}\ \text{Hz},$ (64)

we expect, to first approximation, that for earlier peaks the produced
gravitational waves to be produced for frequencies $\mathcal{O}(0.1-1)$ Hz,
while for later peaks the frequencies will be around $\mathcal{O}(100-1000)$
Hz.

###  4.2 Gravitational Waves Production

With the power spectrum obtained from the previous section, we have now all
the ingredients to compute the present day gravitational wave density.
Following [14, 17], the present day density gravitational wave with wave
number $k$ is given by888For numerical purposes, as the integration in $v$
spans several orders of magnitude over the argument of $P_{R}$, we performed
the integration over $\tilde{v}=\log v$ with the appropriate jacobian.

$\displaystyle
h^{2}\Omega_{GW}(k)=\frac{\Omega_{r}}{36}\int_{0}^{1/\sqrt{3}}du\int_{1/\sqrt{3}}^{\infty}dv\left[\frac{(u^{2}-1/3)(v^{2}-1/3)}{u^{2}-v^{2}}\right]^{2}P_{R}(kX)P_{R}(kY)(I^{c}_{2}+I^{2}_{s})\
,$ (65)

where $\Omega_{r}\simeq 5.4\times 10^{-5}$ is the present day radiation
density,

$\displaystyle X$ $\displaystyle=\frac{\sqrt{3}}{2}(u+v)$ (66) $\displaystyle
Y$ $\displaystyle=\frac{\sqrt{3}}{2}(-u+v)\ ,$ (67)

and we use the analytical approximation for the functions $I_{c}$ and $I_{s}$
(see [14] for more details)

$\displaystyle I_{c}$
$\displaystyle=-36\pi\frac{(u^{2}+v^{2}-2)^{2}}{(-u^{2}+v^{2})^{3}}\Theta(v-u)$
(69) $\displaystyle I_{s}$
$\displaystyle=-36\frac{(u^{2}+v^{2}-2)^{2}}{(u^{2}-v^{2})^{2}}\left[\frac{(u^{2}+v^{2}-2)}{(-u^{2}+v^{2})}\log\left|\frac{u^{2}-1}{v^{2}-1}\right|+2\right]\
.$ (70)

For each point in table 1, we compute eq. 65 for $k$ in the range
$[k(30),k_{max}]$ where $k_{max}=\min(k(\Delta N),10^{19}\ \text{Mpc}^{-1})$,
and this way we cover the range where the power spectrum is amplified, c.f.
fig. 13. The present day gravitational spectrum, as a function of the
wavelength, is presented in fig. 14 for the points presented in table 1.

Figure 14: In red: gravitational wave spectrum for the points in table 1. The
sensitivity curves are from [47].

We observe that, for earlier peaks with enough power spectrum enhancement, the
produced gravitational waves are within the detectable range of space-based
interferometers, more concretely BBO and DECIGO. On the other hand, for later
peaks, the gravitational wave signal is also in the range of ground-based
interferometers, such as the Cosmic Explorer and the Einstein Telescope,
whilst also being detectable by the mentioned space-based interferometers.
Finally, for points with a wide power spectrum enhancement, identified as
those with a long plateau appearing earlier during inflation, the
gravitational wave signal is expected to evade any possible detection on
current and near-future interferometers.

###  4.3 The Effect of the Gravitino Mass on the Gravitational Waves Spectrum

In section 3 we studied how the different parameters of the potential impact
the Planck constrained observables and the gravitino mass. It was shown that
the gravitino mass is largely insensitive to the parameters arising from the
Kähler potential. On the other hand, these are the parameters that govern the
shape and the size of the kink which ultimately drive the enhancement of the
power spectrum that source the production of the GW. Whilst the discussion so
far has concerned itself on how the presence of the kink might affect the
value of the gravitino mass, we have not yet discussed the converse statement:
how the presence of a non-vanishing gravitino mass impacts the phenomenology
of the GW.

To explore this, we select one of the points from table 1 studied above. More
precisely, we choose the second point with early power spectrum enhancement,
which we have labelled as Early $\tilde{N}(2)$. This point has the lowest
value of $d$, at $d=9.25\times 10^{-7}$, and consequently the lowest gravitino
mass from the set, c.f. table 2. We will now change the value of $d$ to take
the following values999We cannot take $d$ to be much greater than $10^{-6}$
for this point as it leads to unsatisfactory inflation.

$\displaystyle d\in\\{0,\ 1.0\times 10^{-7},\ 2.5.0\times 10^{-7},\ 5.0\times
10^{-7},\ 7.5\times 10^{-7},\ 1.0\times 10^{-6}\\}\ ,$ (71)

and for each case we compute the resulting power spectrum and the GW energy
density spectrum.

Figure 15: Gravitational wave spectrum for the second point from table 1, with $d$ taking values from eq. 71. The sensitivity curves are from [47]. $d$ | $n_{s}$ | $r$ |  $m_{3/2}$ (GeV) | $\Delta N$
---|---|---|---|---
$0$ | $0.9566$ | $0.00902$ | $0.0$ | $52.4$
$1.0\times 10^{-7}$ | $0.9567$ | $0.00906$ | $8.9\times 10^{1}$ | $52.5$
$2.5\times 10^{-7}$ | $0.9568$ | $0.00912$ | $5.6\times 10^{2}$ | $52.7$
$5.0\times 10^{-7}$ | $0.9571$ | $0.00921$ | $2.2\times 10^{3}$ | $53.1$
$7.5\times 10^{-7}$ | $0.9574$ | $0.00930$ | $5.1\times 10^{3}$ | $53.6$
$1.0\times 10^{-6}$ | $0.9576$ | $0.00940$ | $9.0\times 10^{3}$ | $54.2$
Table 3: Impact on the inflation observables, gravitino mass, and number of
e-folds by changing the parameter the value of the parameter $d$ for the
second point in table 2 over eq. 71.

For each value of $d$ in the range eq. 71, we show the resulting GW energy
density spectrum in fig. 15, and the inflationary observables, the gravitino
mass, and the number of e-folds in table 3. The results show a somehow
unexpected phenomenon: as $d$ increases the GW energy density increases. Since
$d$ drives the gravitino mass, one can then conclude that for a given
combination of parameters, a higher gravitino mass will be associated with a
higher GW energy density. Notice that GW would still be detectable in the
limit of a massless gravitino, i.e. for $d=0$. While the existence of the
kink, and consequently of observable GW, is governed by the Kähler potential
parameters $a$, $b$, and $c$, the total energy density of the GW spectrum can
be boosted by the Polonyi term, governed by $d$, which also drives the value
of the gravitino mass. From table 3 we can see that increasing $d$ has a
similar effect as decreasing $b$ through tuning: the inflaton _spends more
time_ traversing the plateau, leading to a slightly higher number of e-folds,
but in a far less dramatic way than when tuning $b$. Also from the same table,
we see how the Planck constrained observables are minimally altered, as
expected from the discussion above and from our previous work. Therefore,
while a GW signal is still dependent on fine tuning the parameter $b$, as it
was observed and discussed in [16], our model provides a new source of GW
energy density enhancement via the Polonyi term with reduced fine tuning.

##  5 Conclusions

In this work we studied the production of GW in No-Scale SUGRA inflation. In
section 2 we presented the main No-Scale SUGRA inflation framework. We
revisited our previous model, where the Wess-Zumino superpotential is extended
to include a Polonyi term, and further extended it to include a non-minimal
Kähler superpotential, which describes a deformation of the original space
described by a non-compact $SU(2,1)/SU(2)\times U(1)$ coset. This deformation
leads to a kink feature in the effective scalar potential of the inflaton,
which, for certain values of the parameters, exhibits an inflexion point where
the inflaton slows down. This slowing down enhances the power spectrum, which
in turn can source the production of GW at later stages of inflation.

After discussing how the shape of the inflaton potential changes for different
values of the parameters in section 2.4, we proceeded to study the resulting
inflationary dynamics across the parameter space by performing a thorough scan
in section 3. We observed that there are regions of the parameter space, for
$a\ll 0$ (with large $|a|$) and small $b$, where the inflaton can get stuck at
the inflexion point where the kink feature becomes effectively a wall. This
reinforces the bounds on the Kähler potential parameters $a\gtrsim-1.5$ and
$b\gtrsim 15$, which were previously identified to ensure that the potential
is positive semi-definite with a global minimum at $V=0$. Focusing on the
points that allow for the inflaton to roll down to the global minimum, we
obtained an upper bound on the parameter $d$ that sets scale of the Polonyi
term for the points that pass the Planck satellite constraints. This parameter
is responsible for SUSY breaking at the end of inflation, producing a
gravitino mass that is bounded from above to be at most $\mathcal{O}(1000)$
TeV for the region of the parameter space that produces the desirable
inflationary dynamics, in agreement with our previous work. As in our previous
work, this means that the SUSY scale cannot be exceed the PeV scale, and
therefore that SUSY should eventually be discovered.

Having identified the parameter space points that produce good inflationary
dynamics and observables, we then turned to the study of the power spectrum.
We first performed an extensive analysis of the power spectrum in section 3.2
using the slow-roll approximation. We found that in general the points that
exhibit a power spectrum peak around the time that the inflaton traverses the
inflexion point fall short in enhancing the power spectrum for the production
of observable GW. We then investigated whether this generic expectation of
unobservable GW is a robust conclusion, or whether it is indeed possible for
the model to produce an observable GW signal is certain corners of parameter
space. Since it is known from the literature that the parameter $b$ can be
tuned to augment the power spectrum peak around the inflexion point by slowing
down the inflaton, we then changed our strategy and focused on points that
pass the Planck satellite constraints, but end inflation earlier than desired,
$\Delta N<50$. From this set of points, we further identified three promising
points as those for which the power spectrum is narrowly peaked, corresponding
to $a\lesssim-0.9$, i.e., $|a|\simeq 1$. We were able to tune the parameter
$b$ of these four points, obtaining a pronounced power spectrum peak
enhancement, while producing $50\leq\Delta N\leq 60$. These three points also
produce a relatively earlier peak of the power spectrum. To complement these
points, we also identified a fourth point with a later power spectrum.

We then proceeded to compute the full numerical solution for the power
spectrum and the resulting GW energy density spectrum for these four points in
section 4. In section 4.1 we confirmed the result that a crucial
characteristic to have an enhancement of the power spectrum is for its peak to
be narrow, with a sharp increase. For the three points where the power
spectrum exhibits this characteristic, the full numerical solution produces a
peak at least an order of magnitude higher than the one suggested by the slow-
roll approximation. We then proceeded to study the resulting GW spectrum in
section 4.2, where we found that for the points with sufficiently enhanced
power spectrum peak, that a GW signal can be detectable by near future
interferometers. More precisely, for the points with an earlier peak,
$\tilde{N}\lesssim 40$ the spectrum falls within the detection range of space-
based interferometers BBO and DECIGO, while for the point with a later peak,
$\tilde{N}\simeq 45$, it would also be possible to detect the signal in
ground-based interferometers such as the Cosmic Explorer and the Einstein
Telescope.

In section 4.3 we performed an analysis of how the presence of the Polonyi
term would affect the GW energy density spectrum. Choosing one of the four
points identified before, we varied the value of the parameter $d$ governing
the magnitude of the Polonyi term, and found that the GW energy density is
augmented by the presence of a non-vanishing Polonyi term. While our result
does not cure the high level of tuning associated with the parameter $b$, it
does provide a further mechanism to increase the GW signal in near future
interferometers. Notwithstanding that the main parameter governing the power
spectrum peak and the resulting GW spectrum is a highly tuned $b$, our result
still suggests an interplay between the detection of GW and the discovery of
SUSY at current and near future colliders.

Although our analysis focuses on No-Scale SUGRA inflationary models with an
inflexion point arising from a non-minimal Kähler potential, it would be an
interesting avenue of research to assess whether more general inflationary
potentials with similar features lead to equivalent findings. Furthermore, as
an enhanced power spectrum peak is also known to lead to the production of
PBH, which can contribute to both the NANOGrav observed GW cosmic background,
but also to the Dark Matter relic density, it would be interesting to study
whether there is an interplay between the gravitino mass and the Dark Matter
phenomenology. We leave these open questions for future work.

## Acknowledgements

We are immensely grateful to Dr. Ioanna Stamou for her patience in helping us
to validate our methodology and to reproduce the results in the literature.
SFK acknowledges the STFC Consolidated Grant ST/L000296/1 and the European
Union’s Horizon 2020 Research and Innovation programme under Marie Sklodowska-
Curie grant agreement HIDDeN European ITN project (H2020-MSCA-
ITN-2019//860881-HIDDeN).

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