# Reentrant supersolidity

Leo Radzihovsky  Department of Physics and Center for Theory of Quantum Matter
University of Colorado, Boulder, CO 80309<EMAIL_ADDRESS>
(December 2015)

###### Abstract

A “supersolid” – a crystal that exhibits an off-diagonal long-range order and
a superflow – has been a subject of much research since its first
proposalAndreevLifshitz69 , but has not been realized as a ground state of
short-range interacting bosons in a continuum. In this note I point out a
simple and generic mechanism for a thermally-driven reentrant supersolidity,
and discuss challenges of experimental realization of this idea. In the limit
of bosons in a periodic potential, this mechanism reduces to a reentrant low-
temperature normal-superfluid transition, that should be accessible to
simulations and in current experiments on bosonic atoms in an optical periodic
potential.

Introduction and background: A “supersolid” – a crystal that can superflow is
a putative state of matter that spontaneously “breaks” global translational
and $U(1)$ phase rotational symmetries, associated with momentum and particle
number conservation, respectively.AndreevLifshitz69 ; Chester70  This novel
“incommensurate crystal” 111In an incommensurate crystal the number of lattice
sites is decoupled from the number of atoms. ground state is driven by quantum
fluctuations that can lower the kinetic energy due delocalization of a
fraction of atoms from their lattice sites below the interaction energy cost
of vacancies and interstitials, i.e., a vanishing of the Mott gap while
retaining crystalline order. Such putative state has been somewhat of a holy
grail in experimentsKimChanNature04 ; KimChanScience04 ; RittnerReppy06 ;
KetterleNature17 ; Fraass89  and simulationsMeiselReviewSS92 ; Ceperley04 ;
ProkofevSvistunov05 , as a ground state of short-range interacting bosons in a
continuum.222A generalized supersolid with its definition expanded to include
bosons with dipolar and/or spin-orbit interactions, vortex lattices of rotated
condensates, and lattice bosons in a periodic potentials with a fractional
filling have indeed been realized, and many others, like the Fulde-Ferrel-
Larkin-Ovchinikov and Pair-Density Wave superconductors as promising
candidates. However, despite these promising but controversial
experimentsKimDyChanPRB14 , that rekindled this seasoned subject, to my
knowledge for a continuous translational symmetry no conclusive observation
exist to date.

Despite this currently discouraging experimental status, in this note I point
out a simple generic mechanism – with its own challenges, that I will discuss
– for a temperature-tuned reentrant supersolid state, that should be of
interest even if a zero-temperature supersolid is not realized. One way to
characterize the supersolid state is as a superfluid of vacancies and/or
interstitials that are driven by quantum fluctuations to proliferate even in
the ground state ($T=0$) of a quantum crystal of bosonic atoms. Such putative
state will then generically be stable at low temperatures, but will transition
into a conventional crystal when at high temperature vacancies and
interstitials undergo a phase transition into a normal liquid or a thermal
Boltzmann gas, as any gas inevitably does.

BEC of vacancies/interstitials - a reentrant supersolid: As in a conventional
Bose-condensate (BEC), a phase transition to a BEC (SF) state of vacancies and
interstitials is controlled by the dimensionless phase-space density
$n\lambda_{T}^{3}$ exceeding a constant of order $1$. This corresponds to the
condition that the three-dimensional (3D) thermal de Broglie wavelength
$\lambda_{T}=\sqrt{2\pi\hbar^{2}/mT}$ (using units with $k_{B}=1$) exceeds
inter-particle spacing $n^{-1/3}$. Equivalently, it is useful to state the
criterion for a BEC transition in terms of a critical boson density,

$n_{c}(T)=\left(2mT/\hbar^{2}\right)^{3/2},$ (1)

with BEC emerging for $n>n_{c}(T)$ (up to a constant of order 1).

Somewhat counter-intuitively, here I consider a crystal ground state, which at
zero temperature is a conventional “commensurate” (nonsuper-) solid, and focus
on its fate upon warming it to a nonzero temperature. I then note that there
are two competing effects of increasing temperature of such a state. On one
hand, as usual, increasing temperature decoheres a quantum state, reducing
particles’ thermal de Broglie wavelength. On the other hand, since vacancies
and interstitials in a crystal are point defects with a finite core energy
$E_{c}$, a finite density of them, $n(T)$ is induced at any nonzero
temperature through thermal activation,

$n(T)=n_{0}e^{-E_{c}/T},$ (2)

where $n_{0}\sim 1/a^{3}$ is the total density of bosonic atoms (with $a$ the
crystal’s lattice constant), or equivalently the density of the bosonic fluid
were the crystal to be melted. With the vacancies and interstitials’ core
energy $E_{c}$ set by short-scale elastic energies I expect it to be
proportional to the bulk modulus $B$ and roughly given by

$E_{c}\sim a^{3}B.$ (3)

As illustrated in Fig.1, because $n(T)$ exhibits an essential singularity at
zero temperature, plateauing to $n_{0}$ at high $T$, while $n_{c}(T)$ is a
power-law as a function of temperature, generically (without fine-tuning) the
two functions are guaranteed to either not cross for $E_{c}\gg T_{BEC}(n_{0})$
or to cross twice for $E_{c}\ll T_{BEC}(n_{0})$. In the former case (Fig.1(b))
vacancies and interstitials remain too dilute to Bose-condense, with a
conventional crystal appearing at all temperatures below the melting
temperature $T_{m}$. In the latter case (Fig.1(a)) $n(T)$ and $n_{c}(T)$ cross
at two temperatures $T_{c1}$ and $T_{c2}$ set by $n(T)=n_{c}(T)$,

$n_{0}e^{-E_{c}/T}=\left(2mT/\hbar^{2}\right)^{3/2},$ (4)

as illustrated in Fig.1(a).

Figure 1: Phase diagrams for a quantum crystal under two scenarios. (a)
$E_{c}\ll T_{BEC}(n_{0})$, exhibiting reentrant supersolidity, where a
conventional crystal (Solid) undergoes a phase transition at $T_{c1}$ into a
Supersolid as thermally-proliferated vacancies/interstitials Bose-condense
with increasing $T$, followed by a reentrant transition at $T_{c2}$ back into
a conventional crystal (with a thermal gas of vacancies/interstitials). The
solid then subsequently melts at $T_{m}$. (b) $E_{c}\gg T_{BEC}(n_{0})$,
exhibiting a conventional crystal (Solid) that melts at $T_{m}$ directly into
a conventional fluid (Liquid) phases.

As long as the crystal does not melt this corresponds to a reentrant
supersolidity as a function of temperature,

$\displaystyle 0<T<T_{c1},\ \text{conventional crystal S},$ (5) $\displaystyle
T_{c1}<T<T_{c2},\ \text{supersolid SS},\;\;\;\;\;\;$ (6) $\displaystyle
T_{c2}<T,\ \text{conventional crystal S}.$ (7)

It is straightforward to see that these phase boundaries are set by,

$\displaystyle T_{c1}$ $\displaystyle\sim$ $\displaystyle E_{c},$ (8)
$\displaystyle T_{c2}$ $\displaystyle\sim$ $\displaystyle
T_{BEC}=\hbar^{2}n_{0}^{2/3}/2m,$ (9)

Thus, for above scenario of $E_{c}\ll T_{BEC}(n_{0})$, a supersolidity emerges
as a result of increasing temperature, for $T>T_{c1}$. For $T<T_{c1}$ the
density of vacancies and interstitials is too low to reach the necessary
thermal density $n_{c}(T)$, and so the gas is thermal and the state is a
conventional solid. In the higher temperature range of $T_{c1}<T<T_{c2}$,
$n(T)$ increases beyond $n_{c}(T)$ and the gas of vacancies and interstitials
Bose-condenses, and thereby leads to a supersolid state, despite a concomitant
decrease in the thermal deBroglie length. Then, for $T>T_{c2}$, the thermal
deBroglie length gets sufficiently short that $n(T)$ again drops below
$n_{c}(T)$ leading to a conventional solid with a thermal gas of vacancies and
interstitials, which then can melt at $T_{m}$ into a normal fluid. Another
scenario (not illustrated in Fig. 1) is that the melting temperature falls in
the range $T_{c1}<T_{m}<T_{c2}$, that corresponds to a supersolid melting into
a superfluid, which then transitions into a normal fluid at $T_{c2}$.

Challenges to the realization: I now turn to deliberations about different
physical scenarios and challenges to their realization. To this end, I first
note that if the core energy $E_{c}$ is determined by the Coulomb energy at
the lattice scale, (i.e., on the scale of a Rydberg (Ry)), it is highly
unfavorable for reentrant supersolidity, since then $T_{BEC}(n_{0})$ is
estimated to be three orders of magnitude smaller than $E_{c}$ by the ratio of
an electron to proton mass.

A more favorable condition for the reentrant supersolid is expected if $E_{c}$
is controlled by the energetics of the van der Waals interaction in neutral
atoms, orders of magnitude lower than a Ry. This then allows for the scenario
$E_{c}\ll T_{BEC}(n_{0})$, a necessary but not sufficient condition for the
realization of reentrant supersolidity. Another necessary condition is on the
melting temperature, $T_{m}\gg E_{c}$. If satisfied, indeed a reentrant
supersolid will appear for $T_{c1}<T<T_{c2}$, as illustrated in Fig.1(a). On
the other hand, generically one may expect that $E_{c}$, controlled by short-
scale solid’s cohesion energies is comparable to the melting temperature,
$T_{m}$. If so then instead of a rentrant supersolid, for $T_{c1}\sim
T_{m}<T<T_{c2}$ a low-temperature conventional crystal will simply melt
directly into a superfluid (BEC), which will then become a thermal fluid
(Boltzmann gas) for $T>T_{c2}$.

Currently, in solid Helium-4 there is no experimental evidence for a
thermally-reentrant supersolid in many hundreds of torsional oscillator
experiments typically performed between $30$ mK and $1.5$ KMeiselReviewSS92 ;
KimChanNature04 ; KimChanScience04 ; RittnerReppy06 . This is consistent with
X-ray measurements by Simmons, et al.Fraass89  and with Quantum Monte Carlo
simulationsCeperley04 ; ProkofevSvistunovTroyer06 , that estimate $E_{c}$ on
the order of $10-15$K, thus much larger than $k_{B}T_{BEC}(n_{0})\sim 1$K.

I note that there is no fundamental obstruction for elastically-controlled
core energy $E_{c}\sim T_{c1}$ to be lower than the mass- and density-
dependent $T_{BEC}(n_{0})\sim T_{c2}$. In contrast, as emphasized above, in a
single-component crystal there is a natural elastic link between $E_{c}$ and
$T_{m}$. Thus to realize $E_{c}\ll T_{BEC}(n_{0})\ll T_{m}$ requires a system
in which vacancy/interstitial core energy $E_{c}$ is decoupled from its
melting temperature $T_{m}$. One way to realize this is via e.g., a two-
component crystal, where high $T_{m}$ crystallization is controlled by a
strongly interacting atom and vacancy/interstitial core energy $E_{c}$ by a
weakly interacting smaller atom. One can view such system in a limit of a
positionally well-ordered single-component crystal, determining $T_{m}$,
acting as a periodic potential for a second weakly-interacting atom, whose
vacancy/interstitial core energies $E_{c}\ll T_{m}$. This may be realizable in
current highly tunable ultracold-atom experiments.

Reentrant superfluid: In fact, one can consider an extreme
$T_{m}\rightarrow\infty$ limit with a single atomic component in a
commensurate periodic potential, that is routinely realized in ultra-cold
atoms via an optical potential of interfering laser beams. Because of the
imposed periodic potential atoms are by construction in a crystal state, with
the atomic structure function exhibiting Bragg peaks at all temperatures. With
melting obstruction out of the way, the $T_{c1}\ll T_{c2}$ requirement may be
satisfied by working at low, commensurate filling fraction and short-range
interactions, natural for neutral ultracold atoms.

However, one may then rightly object that the remaining problem has nothing to
do with supersolidity, since the translational symmetry is explicitly broken
by the imposed optical periodic potential. Indeed this is correct, and the
problem in this limit instead reduces to a putative thermally-induced
reentrant superfluidity of interacting lattice bosons, as I discuss below and
illustrate in Fig. 2.

At zero temperature and commensurate filling, such lattice bosons exhibit an
extensively studied Superfluid to Mott-insulator (SF-MI) quantum phase
transitionDoniach81 ; FisherMISF89  as a function of the ratio $U/t\equiv g$
of the onsite interaction strength $U$ to the hopping amplitude $t$,
realizable in ultracold atomsJaksch98 ; Greiner02 . Focussing on the Mott-
insulator side of the quantum SF-MI critical point, I consider a nonzero
temperature at which a nonzero fraction of bosons $n(T)=n_{0}e^{-E_{c}/T}$ is
thermally activated above the Mott gap $E_{c}$, whose scale is set by $U$.
This gas of bosons at nonzero temperature can then be in a superfluid or a
thermal state, depending on the ratio of its density $n(T)$ and the deBroglie
condensation density $n_{c}(T)$, latter controlled by the hopping amplitude,
$t$. Thus, in this context, the scenario discussed above is then simply a
prediction of a possible (depending on microscopic details) temperature-driven
reentrant Normal-to-SF-to-Normal phase transition of finite-temperature bosons
on a lattice, as illustrated in Fig.2. To realize such a scenario I note that
on one hand, to be in the zero-temperature MI state the system must have
$t<U$, but on the other hand, to have reentrance, $T_{c1}\sim U<T_{c2}\sim t$,
i.e., must have $U<t$. These considerations give a crude estimate of the phase
boundaries

$\displaystyle e^{-E_{c}/T}$ $\displaystyle\sim$ $\displaystyle T/t\;,$ (10)
$\displaystyle e^{-1/\hat{T}}$ $\displaystyle\approx$ $\displaystyle
cg\hat{T}\;,$ (11)

(akin to that in the continuum, summarized by Eq. (4)), where $\hat{T}\equiv
T/E_{c}$ is the temperature in units of MI gap $E_{c}$, that I expect to be
proportional to $U$, $E_{c}/t\approx cU/t=cg$, with $c$ the proportionality
constant. Thus from (11) we see that the existence of the superfluid
reentrance requires $c<g^{-1}<1$, and is then a matter of microscopic details
that determine constants of $O(1)$, inaccessible to above generic arguments
and qualitative estimates. With the above motivation, a demonstration of the
reentrant superfluidity requires a detailed quantitative analysis of the
nonzero-temperature phase boundary that must include next- and next-to-nearest
neighbors interactions and hopping, and faithfully treated thermal
fluctuations. With all the tunability afforded by AMO systems, I expect this
reentrant-superfluid scenario to be realizable in a degenerate atomic Bose gas
in an optical periodic potential.Greiner02

Summary: In this brief note, I presented generic arguments for a possible
realization of a thermally-driven reentrant supersolid and superfluid. The key
to the mechanism is a strong low-temperature dependence of the density of
bosonic excitations, activated over a gap of the insulating state such as a
non-supersolid crystal and a Mott insulator. I demonstrated that although such
putative reentrance is in principle possible it relies on favorable
microscopic details, that lie beyond present analysis. A role of interactions,
and quantum and thermal fluctuations will quantitatively modify above crude
estimates of the phase boundary. Thus, a detailed first-principles numerical
analysis for a specific system is necessary to assess the feasibility of the
thermal reentrance scenario outlined here. Despite this uncertainty, the
possibility of realizing a temperature-driven reentrant supersolidity and
superfluidity is quite intriguing, and I hope will stimulate numerical and
experimental searches of this phenomenology.

Figure 2: A schematic putative finite temperature MI-SF phase diagram showing
a reentrant superfluid phase above a zero-temperature MI state.

_Acknowledgments_. I thank Moses Chan, Anton Kapustin, Jorge Kurchan, Alan
Dorsey, Liang Fu, Victor Gurarie, Boris Svistunov and Nikolay Prokofév for
feedback and stimulating discussions. I acknowledge support by the Simons
Investigator Award from the James Simons Foundation. I thank The Kavli
Institute for Theoretical Physics for its hospitality during workshops while
this manuscript was in preparation, supported by the National Science
Foundation under Grant No. NSF PHY-1748958 and PHY-2309135.

## References

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UoA

August 27, 2024

Gravitational Waves From No-Scale Supergravity

Vassilis C. Spanos and Ioanna D. Stamou

National and Kapodistrian University of Athens, Department of Physics,
Section of Nuclear and Particle Physics, GR–15784 Athens, Greece

In this paper we study four concrete models, based on no-scale supergravity
with SU(2,1)/SU(2)$\times$ U(1) symmetry. We modify either the Kähler
potential or the superpotential, which are related to the no-scale theory with
this symmetry.  In this scenario, the induced Gravitational Waves, are
calculated to be detectable by the future space-based observations such as
LISA, BBO and DECIGO. The models under study are interrelated, as they all
yield the Starobinsky effective-like scalar potential in the unmodified case.
We evaluate numerically the scalar power spectrum and the stochastic
background of the Gravitational Waves, satisfying the observational Planck
cosmological constraints for inflation.

###### Contents

  1. 1 Introduction
  2. 2 Basic Considerations
    1. 2.1 General Aspect Of Supergravity
    2. 2.2 Curvature Perturbations
  3. 3 Gravitational Waves from the No-Scale models
    1. 3.1 Modifying the superpotential
    2. 3.2 Modifying the Kähler potential
  4. 4 Producing Gravitational Waves
    1. 4.1 Modifying the superpotential
    2. 4.2 Modifying the Kähler potential
  5. 5 Conclusions
  6. A The forms of the scalar potential
    1. A.1 Model I
    2. A.2 Model II
    3. A.3 Model III
    4. A.4 Model IV
  7. B Fine-tuning analysis and a comparison between the models

##  1 Introduction

The detection of the Gravitational Waves (GWs) is undeniably regarded as a
milestone in cosmology. The GWs emitted by a binary black hole merges,
reported by the LIGO and Virgo collaborations [1, 2, 3, 4, 5], draw attention
to both experimental and theoretical researches for exploring the stochastic
background of GWs. The signal of the GWs is expected to be detectable by
future space-based GW interferometers such as LISA, BBO and DECIGO[6, 7].

The detection of these events by the LIGO/Virgo collaborations has been
rekindled the old study of Primordial Black Holes (PBHs)[8, 9, 10, 11].
Specifically, it has been claimed that the PBHs, which are formed during
radiation dominated epoch, can be considered as reliable candidates for a
large, or the whole amount of the Dark Matter (DM) of the Universe[12, 13, 14,
15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27]. In this case, there is a
strong possibility to detect induced GWs’ signals, which are generated during
the radiation phase and will be available in the future from the
aforementioned observations. The contribution of the first order scalar
perturbations, to the generation of second order perturbations associated to
the GWs, has already been studied in detail [28, 29, 30, 31, 32, 33, 34].

Several mechanisms for producing enhancement in the scalar power spectrum have
been proposed in the literature. In the case of the single field inflation,
the mechanism associated to the presence of an inflection point has been
already studied extensively [13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 26,
27]. At the inflection point, both first and second derivatives of the
effective scalar potential become approximately zero. This feature in the
effective scalar potential, significantly decreases the velocity of the
inflaton. This decrease is imprinted in a peak in power spectrum. The slow-
roll approximation fails to give the correct results and one has to solve
numerically the exact equation of the field perturbations, because the slow-
roll parameter $\eta$ growths significantly. In this work we will consider
this mechanism for producing such amplifications. However, alternative
mechanisms have been proposed regarding single field inflation. One of them is
models with a feature like steps in the potential[35, 36, 37, 38]. Moreover,
models with ultra slow-roll inflation have been studied [39, 40]. Two field
inflationary model have also been proposed [41, 24, 42, 43, 44].

The relation between the first and second order perturbations, that is the
scalar and the tensor power spectrum, gives the insight to study the GWs by
analysing and measuring only the scalar power spectrum [28, 29, 30, 31, 32,
33, 45, 46, 47]. There are numerous studies based on this scenario [24, 25,
48, 43, 49, 50, 51, 52, 53, 54, 55, 37, 56] and we briefly discuss some of
them. In Refs. [24, 25] the authors study two scalar field models with a non-
canonical kinetic term. Other two-field models, in order to generate GWs have
been studied in Refs. [24, 25, 42, 43, 48]. The production of GWs within a
single field inflation has been considered in Refs. [56, 55, 50, 37].

Supergravity (SUGRA) models can provide us with inflationary potentials
compatible with observational data. To this end, a significant enhancement in
the scalar power spectrum as described above, can occur [21, 22, 18, 17, 26,
27]. In particular, it has been claimed that no-scale theory can be regarded
as a theory worthy of consideration, in order to study enhancement in power
spectrum [21, 22]. This theory treats many problematic issues such as the
fine-tuning in order to obtain a vanishing cosmological constant and it gives
a natural solution to the $\eta$ problem [57, 58, 59, 60, 61].

In [22] a mechanism to produce PBHs was proposed, based on the breaking of the
non-compact SU(2,1)/SU(2)$\times$ U(1) symmetry. It is proved that, the
effective scalar potentials related to inflation, not only can explain the
production of PBHs, but also they can conserve the transformation laws,
related to the coset SU(2,1)/SU(2)$\times$ U(1). In this work we will consider
these potentials, in order to explain the generation of GWs and we will
numerically evaluate the energy density. Moreover, we generate GWs by
modifying the kinetic term in the Lagrangian and keeping the superpotentials,
which lead to the Starobinsky-effective potential such as the Wess-Zumino or
the Cecotti, unchanged. All the models present in this work are in consistence
with the Planck constraints of the spectral index $n_{s}$ and the tensor-to-
scalar ratio $r$ [62, 63].

The layout of the paper is as follows: In section 2 we briefly review some
basic aspects for the SUGRA theory and the perturbation theory, which are used
in our analysis. In section 3 we explain how the enhancement of power spectrum
can be produced in the framework of no-scale theory. In section 4 we present
the GWs for our analysis. Finally, in section 5 we give our conclusions and
perspectives.

##  2 Basic Considerations

In this section we discuss two basic considerations used in our analysis. The
first is an introduction to SUGRA theory, which is relevant to the inflation
models we will employ below. The second is about the general tools used for
studying the field perturbations during inflation.

###  2.1 General Aspect Of Supergravity

In this subsection we present some basic aspects of SUGRA and particularly the
no-scale theory. The SUGRA Lagrangian coupled to matter is given in the
following expression

$\mathcal{L}=\frac{1}{2}R-K^{\Phi\bar{\Phi}}\partial_{\mu}\Phi\partial^{\mu}\bar{\Phi}-V(\Phi,\bar{\Phi})\,,$
(2.1)

where $K$ denotes the Kähler potential, $W$ is the superpotential and the
indices $\Phi$ and $\bar{\Phi}$ denote the derivatives in respect to the
corresponding fields. We remark that throughout this study we work in reduced
Planck units ($M_{P}=1$).

The F-term of scalar potential is given as follows

$V=e^{K}(D_{\Phi}WK^{\bar{\Phi}\Phi}D_{\bar{\Phi}}\bar{W}-3|W|^{2})\,,$ (2.2)

where

$D_{\Phi}W=\frac{\partial W}{\partial\Phi}+\frac{\partial K}{\partial\Phi}W$
(2.3)

is the Kähler convariant derivative.

The minimal no-scale model is given by the Kähler potential in the present of
a single chiral field $\Phi$ [57, 58, 59, 60, 61]

$K=-3\ln(\Phi+\bar{\Phi}).$ (2.4)

By considering this Kähler potential, one can notice that the term $-3|W|^{2}$
in Eq.(2.2) is vanishing because of the following expression [57]

$K^{\Phi\bar{\Phi}}K_{\Phi}K_{\bar{\Phi}}=3.$ (2.5)

In the case of non-compact symmetry SU(2,1)/SU(2)$\times$ U(1), a no-scale
supergravity model is described by the following two equivalent Kähler
potentials [64]

$K=-3\ln\left(T+\bar{T}-\frac{|\varphi|^{2}}{3}\right)$ (2.6)

and

$K=-3\ln\left(1-\frac{|y_{1}|^{2}}{3}-\frac{|y_{2}|^{2}}{3}\right)$ (2.7)

where $y_{1}$, $y_{2}$, $T$ and $\varphi$ are chiral fields. The Kähler
potential in $(y_{1},y_{2})$ basis is related to this in the $(T,\varphi)$
basis, as analyzed in [65]. The relevant transformation is

$y_{1}=\Big{(}\frac{2\varphi}{1+2T}\Big{)},\quad
y_{2}=\sqrt{3}\Big{(}\frac{1-2T}{1+2T}\Big{)}\,,$ (2.8)

while its inverse is

$T=\frac{1}{2}\Big{(}\frac{1-y_{2}/\sqrt{3}}{1+y_{2}/\sqrt{3}}\Big{)},\quad\varphi=\Big{(}\frac{y_{1}}{1+y_{2}/\sqrt{3}}\Big{)}\,.$
(2.9)

For a given superpotential, in the $(y_{1},y_{2})$ basis, we can derive the
corresponding in $(T,\varphi)$ by using the following relation [65]

$W(T,\varphi)\rightarrow\bar{W}(y_{1},y_{2})=\left(1+y_{2}/\sqrt{3}\right)^{3}W\,.$
(2.10)

In order to present inflationary models which are in agreement with the
current observational data on CMB anisotropies [62, 63], we consider the
Starobinsky-type potential. We note that in [62, 63] the $\mathcal{R}^{2}$
inflationary models are in accordance with the current data. Interestingly,
the no-scale theory described above, provides us with potentials of this kind
(Starobinsky-like potential) [66, 67]. Furthermore, there are other
supergravity models, which can lead to Starobinsky-like potential as well. We
briefly mention the $\alpha$-attractor models, which are described in [68, 69,
70]. In these models the Eq. (2.7) is multiplied by a term $\alpha$, which can
be different to one:

$K=-3\alpha\ln\left(1-\frac{|y_{1}|^{2}}{3}-\frac{|y_{2}|^{2}}{3}\right).$
(2.11)

As a result the condition (2.5) is not conserved. However, these models have
other interesting features, e.g. they are invariant under the conformal
transformations [68, 69, 70].

In our analysis, the Starobinsky-like effective scalar potential can be
derived with two ways from no-scale theory, as it is described below. One is
this of the Wess-Zumino model [67]

$W=\frac{\hat{\mu}}{2}\varphi^{2}-\frac{\lambda}{3}\varphi^{3}\,,$ (2.12)

where $\hat{\mu}$ is a mass term and $\lambda$ a trilinear coupling in the
Lagrangian. This is written in the $(T,\varphi)$ basis. In order to get the
Starobinsky-like effective scalar potential, we use the Eq.(2.2) and we assume
that the inflationary direction is ${\textrm{Im}}\varphi={\textrm{Im}}T=0$ and
${\textrm{Re}}T=c$. We define $\hat{\mu}=\mu\sqrt{c/3}$.  In addition, in
order to evaluate the effective potential we need to fix the non-canonical
kinetic term. To do this we use the following redefinition of the field

$\varphi=\sqrt{3}\tanh\left(\frac{\phi}{\sqrt{6}}\right)$ (2.13)

where $\phi$ is the field with fixed the non-canonical kinetic term.  From
Eq.(2.2) we get

$V=\mu^{2}e^{-\sqrt{2/3}x}\sinh^{2}\left(\frac{\phi}{\sqrt{6}}\right),$ (2.14)

where we have assumed $\lambda/\mu=1/3$.

The superpotential in Eq.(2.12) can be written in $(y_{1},y_{2})$ basis, using
the Eqs. (2.9) and (2.10) [65]. Hence, we get

$W=\frac{\hat{\mu}}{2}\Big{(}y_{1}^{2}+\frac{y_{1}^{2}y_{2}}{\sqrt{3}}\Big{)}-\lambda\frac{y_{1}^{3}}{3}$
(2.15)

and the kinetic term is fixed using the redefinition

$y_{1}=\sqrt{3}\tanh\left(\frac{\phi}{\sqrt{6}}\right)\,,$ (2.16)

where the Starobinsky effective scalar potential appears assuming that the
$y_{1}$ is the inflaton field and $y_{2}$ is the modulo.  The resulting
effective scalar potential is the same with this derived in the $(T,\varphi)$
basis.

Alternatively, the Starobinsky-like potential can be derived by the Cecotti
superpotential as given in $(T,\varphi)$ basis as [66]

$W=\sqrt{3}\,m\,\varphi\,\left(T-\frac{1}{2}\right)\,,$ (2.17)

where $m$ is a mass term. The Starobinsky effective scalar potential arises
thought the direction $\varphi=\text{Im}T=0$. In order to have canonical
kinetic terms, one has to use the field redefinition

$T=\frac{1}{2}e^{\sqrt{\frac{2}{3}}\phi}\,.$ (2.18)

In general, the kinetic term can be fixed by the transformation

$\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi=K_{\chi\chi}\partial_{\mu}x\partial^{\mu}\chi\Rightarrow\frac{d\phi}{d\chi}=\sqrt{2K_{\chi\chi}}\,.$
(2.19)

Although the relation (2.19) in the case of the Kähler potentials (2.6) and
(2.7) yields analytical solution, it is also generally useful even in the
cases that there is no analytical solution. In the following, we will consider
both analytical and numerical solutions of Eq. (2.19).

The superpotential in Eq. (2.17) can be written in the $(y_{1},y_{2})$ basis
as

$W=m\Big{(}-y_{1}y_{2}+\frac{y_{2}y_{1}^{2}}{\sqrt{3}}\Big{)}$ (2.20)

and as before the redefinition for the canonical kinetic term is

$y_{1}=\sqrt{3}\tanh\left(\frac{\phi}{\sqrt{6}}\right)\,,$ (2.21)

with $y_{1}$ being the inflaton field  and $y_{2}$ the modulo. If we evaluate
the effective potential, we can obtain the same Starobinsky-like potential in
both bases $(T,\varphi)$ and $(y_{1},y_{2})$.

Therefore, there is an important equivalence between the models of Wess-Zumino
and Cecotti, as they can provide us with Starobinsky-like potential, when they
are embedded in no-scale theory [64]. Moreover, when we study the forms
written in $(y_{1},y_{2})$ basis in Eq.(2.7) and the corresponding
superpotential given in (2.15) and (2.20) we can obtain that these models obey
the symmetry identified by the matrix

$\displaystyle U=\begin{bmatrix}\alpha&\beta&0\\\ -\beta^{*}&\alpha^{*}&0\\\
0&0&1\end{bmatrix}\,,$ (2.22)

where $\alpha,\beta\in\mathbb{C}$ and $|\alpha|^{2}+|\beta|^{2}=1$. By this
matrix and the analysis shown in Ref. [64] we can obtain the following
transformation laws for the fields

$y_{1}\rightarrow\alpha y_{1}+\beta y_{2},\quad
y_{2}\rightarrow-\beta^{*}y_{1}+\alpha^{*}y_{2}.$ (2.23)

This symmetry is analyzed in [64]. In the next section we will present models
that satisfies this symmetry.

###  2.2 Curvature Perturbations

The equation of motion of the inflaton field is

$\phi^{\prime\prime}+3\phi^{\prime}-\frac{1}{2}\phi^{\prime
3}+\left(3-\frac{1}{2}\phi^{\prime 2}\right)\frac{d\ln V(\phi)}{d\phi}=0\,,$
(2.24)

where primes denote the derivative in efold time.

The general Friedmann-Robertson-Walker (FRW) metric written in the conformal
Newtonian gauge is [71, 72]

$ds^{2}=-a^{2}\left(1+2\Phi\right)d\tau^{2}+a\left((1+2\Psi)\delta_{ij}+\frac{1}{2}h_{ij}\right)dx^{i}dx^{j}\,,$
(2.25)

where $\tau$ is the conformal time, $a$ is the scale factor, $h_{ij}$ are
tensor perturbations, $\Phi$ and $\Psi$ are the Bardeen potentials, which are
equal in the absence of anisotropy in the stress energy tensor. Assuming that
the perturbation of the field is $\phi+\delta\phi$, the equation of the
curvature perturbation takes the form [73]

${\delta\phi^{\prime\prime}=-\left(3-\frac{1}{2}\phi^{\prime
2}\right)\delta\phi^{\prime}-\frac{1}{H^{2}}\frac{d^{2}V}{d\phi^{2}}\delta\phi-\frac{k^{2}}{a^{2}H^{2}}\delta\phi+4\Psi^{\prime}\phi^{\prime}-\frac{2\Psi}{H^{2}}\frac{dV}{d\phi}}\,,$
(2.26)

where the Bardeen potential $\Psi$ results from the equation

${\Psi^{\prime\prime}=-\left(7-\frac{1}{2}\phi^{\prime
2}\right)\Psi^{\prime}-\left(2\frac{V}{H^{2}}+\frac{k^{2}}{a^{2}H^{2}}\right)\Psi-\frac{1}{H^{2}}\frac{dV}{d\phi}\delta\phi\,.}$
(2.27)

With $H$ we denote the Hubble parameter, which is

${H^{2}=\frac{V}{(3-\frac{1}{2}{\phi^{\prime}}^{2})}\,.}$ (2.28)

The initial condition for the perturbation, assuming Bunch-Davies vacuum, is

$aQ\rightarrow\frac{e^{-ik\tau}}{\sqrt{2k}}\,,$ (2.29)

where $Q$ is the Mukhanov-Sassaki variable. The complete expressions for the
initial conditions for the perturbation, as well as the initial conditions for
the Bardeen potential are

$\begin{split}{\delta\phi_{ic}=\frac{1}{\sqrt{2k}}\frac{1}{a_{ic}},\quad\left(\delta\phi^{\prime}\right)_{ic}=-\frac{1}{a_{ic}\sqrt{2k}}\left(1+i\frac{k}{a_{ic}H_{ic}}\right)}\end{split}$
(2.30)

and

$\begin{split}\Psi_{ic}=&\frac{1}{2\left(\varepsilon_{H,ic}-\frac{k^{2}}{a_{ic}^{2}H_{ic}^{2}}\right)}\left(\phi^{\prime}_{ic}\delta\phi^{\prime}_{ic}+\delta\phi_{ic}\left[3\left(\phi^{\prime}\right)_{ic}+\frac{1}{H_{ic}^{2}}\left(\frac{dV}{d\phi}\right)_{ic}\right]\right)\\\
\Psi^{\prime}_{ic}=&\frac{1}{2}\left(\phi^{\prime}\right)_{ic}\delta\phi_{ic}-\Psi_{ic}\,.\end{split}$
(2.31)

With the indices $ic$ we denote the initial conditions and with the
$\varepsilon_{H}$ we denote the slow roll parameter

$\varepsilon_{H}=\frac{1}{2}\phi^{\prime 2}.$

We note that these equations are valid for a Lagrangian with a canonical
kinetic term. Otherwise one should take into account the field transformation
in order to fix the canonical kinetic term, as we have mentioned in the
previous subsection in Eq.(2.19).

Using Eqs. (2.24) -(2.28) we can evaluate the power spectrum as

$P_{R}=\frac{k^{3}}{2\pi^{2}}\left|R_{k}\right|^{2},$ (2.32)

where $k$ is the comoving wavenumber of the Fourier mode and

$R_{k}=\Psi+\frac{\delta\phi}{\phi^{\prime}}\,.$ (2.33)

The numerical procedure, we follow is descibed in Refs. [21, 22].

There are many works where the analytical evaluation of enhanced power
spectrum has been studied, such as [36, 37, 40]. The power spectrum in the
vicinity of the peak can be approximated analytically by the power-law
spectrum which is given as [74, 75]:

$P_{R}=\begin{cases}0&\text{if $k<k_{peak}$}\\\
P_{0}\left(\frac{k}{k_{peak}}\right)^{-n}&\text{if
$k>k_{peak}$}\,\,,\end{cases}$ (2.34)

where $k_{peak}$ is the comoving wavenumber of the peak. $P_{0}$ is related to
the peak amplitude and $n$ is related to the spectral index. In the following
we present both numerical and analytical results for the sake of comparison.

The models we study in this work are based on $\mathcal{R}^{2}$ inflation and
are consistent with the current observational data of inflation. To this end,
we use the slow-roll parameter $\eta_{H}$, defined as

$\eta_{H}=\varepsilon_{H}-\frac{1}{2}\frac{d\ln\varepsilon_{H}}{dN}\,.$ (2.35)

The slow-roll parameters $\varepsilon_{V}$ and $\eta_{V}$ are defined also in
terms of the inflationary potential as

$\varepsilon_{V}=\frac{1}{2}\left(\frac{V^{\prime}(\phi)}{V(\phi)}\right)^{2},\quad\eta_{V}=\frac{V^{\prime\prime}(\phi)}{V(\phi)}\,,$
(2.36)

where primes denote the derivatives in respect of the field $\phi$. The
spectral index is defined as

$n_{s}-1=\frac{d\ln P_{R}}{d\ln k}\,.$ (2.37)

In terms of slow roll conditions for the prediction of the observable $n_{s}$,
we get

$n_{s}\simeq 1+2\eta_{V}-6\varepsilon_{V}\,.$ (2.38)

The prediction for the ratio tensor-to-scalar $r$ are

$r\simeq 16\varepsilon_{V}.$ (2.39)

The corresponding values for these quantities, which are evaluated at the
pivot scale of $k_{*}=0.05Mpc^{-1}$ are [63]

$n_{s}=\begin{cases}0.9659\pm 0.0041&\text{Planck TT,TE,EE+lowEB+lensing}\\\
0.9651\pm 0.0041&\text{Planck TT,TE,EE+lowEB+lensing+BK15}\\\ 0.9668\pm
0.0037&\text{Planck TT,TE,EE+lowEB+lensing+BK15+BAO}\end{cases}$ (2.40)

and

$r<\begin{cases}0.11&\text{Planck TT,TE,EE+lowEB+lensing}\\\
0.061&\text{Planck TT,TE,EE+lowEB+lensing+BK15}\\\ 0.063&\text{Planck
TT,TE,EE+lowEB+lensing+BK15+BAO}\,.\end{cases}$ (2.41)

In the following we present the predictions for the observables of the
proposed models and we will check their consistency against the Planck data.

##  3 Gravitational Waves from the No-Scale models

Below we will study how modifications in the superpotential or in the Kähler
potential can be used in order to explain the generation of GWs. We present
two different ways to produce such peaks in power spectrum, which as we will
see it is related to the spectrum of GWs. The first is by modifying the Wess-
Zumino and Cecotti superpotential, given in Eqs. (2.12), (2.15), (2.13) and
(2.17). The second is by modifying the Kähler potential, meaning the kinetic
term on the Lagrangian, taking into account that both Wess-Zumino
superpotential and Cecotti superpotential are unchanged. Therefore, we present
two different ways in order to get significant peaks in the power spectrum by
conserving (modifying superpotentials) and breaking (modifying Kähler
potential) the SU(2,1)/SU(2)$\times$ U(1) symmetry. All models give the
Starobinsky effective scalar potential in the unmodified case.

###  3.1 Modifying the superpotential

It was shown that the non-compact SU(2,1)/SU(2)$\times$ U(1) symmetry leads to
the same Starobinsky-like effective scalar potential, demonstrating an
important equivalence between the different models [64]. Moreover, it can be
shown that this symmetry can be preserved [22] under proper modification of
the superpotential, in order to generate a peak in power spectrum and produce
significant amount of the PBHs in the Universe. Therefore, we consider the no-
scale supergravity coset SU(2,1)/SU(2)$\times$ U(1). This coset can be
identified by two equivalent forms of the Kähler potential

$\begin{split}K=-3\ln(1-\frac{|y_{1}|^{2}}{3}-\frac{|y_{2}|^{2}}{3})\\\
K=-3\ln(T+\bar{T}-\frac{|\varphi|^{2}}{3})\,,\end{split}$ (3.1)

which can be written in two basis, the $(T,\varphi)$ and $(y_{1},y_{2})$.

First, by a proper modification of the Wess-Zumino model, we can derive the
same effective scalar potential in both $(T,\varphi)$ and $(y_{1},y_{2})$
basis. The superpotential adopted in Ref. [22], for modified the Wess-Zumino
model in $(y_{1},y_{2})$ basis, is

$W=\left[\frac{\hat{\mu}}{2}\left(y_{1}^{2}+\frac{y_{2}^{2}y_{1}}{\sqrt{3}}\right)-\lambda\frac{y_{1}^{3}}{3}\right]\left[1+e^{-b_{1}y_{1}^{2}}\left(c_{1}{y_{1}}^{2}+c_{2}{y_{1}}^{4}\right)\right]\,,$
(3.2)

where $\hat{\mu}$, $\lambda$, $b_{1}$, $c_{1}$ and $c_{2}$ are free
parameters, fixed by observations. The effective scalar potential can be
calculated by using Eq. (2.2) and the redefinition of the field, which is
given in Eq. (2.16). Using this modification the resulting effective scalar
potential is not affected by the transformations $y_{1}\rightarrow-y_{1}$ and
$y_{2}\rightarrow-y_{2}$. Alternatively, using $y_{1}\rightarrow-y_{2}$ and
$y_{2}\rightarrow y_{1}$, one can derive the same effective scalar potential,
in the inflationary direction  $y_{2}={\textrm{Im}}\,y_{1}=0$, by fixing the
non-canonical kinetic term as

$y_{1}=-\sqrt{3}\tanh\left(\frac{\phi}{\sqrt{6}}\right)\,.$ (3.3)

This analysis is presented in [64].

Figure 1: The effective potential (solid line) for the superpotential in Eq.
(3.2) with the choice of parameters $\lambda/\mu=1/3$, $c_{1}=1.9$,
$c_{2}=-0.73$ and $b_{1}=3.9615$. The dashed like corresponds to the
Starobinsky potential.

The superpotential in Eq. (3.2) can be written in the $(T,\varphi)$ basis, if
one considers the Eq. (2.10). Hence

$W=\left(\frac{\hat{\mu}}{2}\varphi^{2}-\frac{\lambda}{3}\varphi^{3}\right)\left(1+e^{-b_{1}\left(\frac{2\varphi}{1+2T}\right)^{2}}\left[c_{1}\left(\frac{2\varphi}{1+2T}\right)^{2}+c_{2}\left(\frac{2\varphi}{1+2T}\right)^{4}\right]\right)\,,$
(3.4)

where $\varphi$ is the inflaton field and $T$ is the modulo. This yields the
same effective scalar potential as in Eq. (3.2) using the redefinition of the
field (2.13), in order to get canonical kinetic terms.  In Fig. 1 we present
the effective scalar potential given for the superpotential (3.2) (or
equivalently Eq.(3.4)). The details for this calculation are described in
Appendix A. In order to derive this potential we redefine the field in order
to have canonical kinetic terms. We notice that there is a region around
$\phi\approx 1.4$, where there is a near inflection point. We expect that this
feature can be imprinted in the scalar power spectrum. On the other hand,
there are many mechanisms in order to obtain an enhancement in scalar power
spectrum, such as two field models, models with bumpy features[76, 77]. In
Appendix B there is a comparison between some models. In Fig. 1 we also depict
the case of $c_{1}=c_{2}=0$ which correspond to the Starobinsky potential.

In addition to that, by a proper modification in Cecotti superpotential, given
in (2.17), we can produce significant peak in the power spectrum in the
$(y_{1},y_{2})$ basis. In Ref.[22] the following superpotential is considered

$W=m\left(-y_{1}y_{2}+\frac{y_{2}y_{1}^{2}}{l\sqrt{3}}\right)\left(1+c_{3}e^{-b_{2}{y_{1}}^{2}}{y_{1}}^{2}\right)\,,$
(3.5)

where $m$, $l$, $c_{3}$, $b_{1}$ are the parameters of the model. This choice
of modification is made for keeping the resulting effective scalar potential
unchanged under the transformations $y_{1}\rightarrow-y_{1}$,
$y_{2}\rightarrow-y_{2}$. Also, if one considers the case of exchanging the
modulo and the inflaton field by changing $y_{1}\rightarrow-y_{2}$ and
$y_{2}\rightarrow y_{1}$, the effective scalar potential remains the same. For
more detail see Refs. [22].

In the $(T,\varphi)$ basis, the superpotential (3.5) takes the following form
using Eq. (2.10)

$W=\frac{\sqrt{3}}{2}m\varphi\Big{(}\frac{1}{2}-T\Big{)}\left(-1-2T-\frac{1-2T}{l}\right)\left[1+3c_{3}e^{-3b_{2}\left(\frac{1-2T}{1+2T}\right)^{2}}\left(\frac{1-2T}{1+2T}\right)^{2}\right]\,.$
(3.6)

This gives the same effective scalar potential as in Eq. (3.5), if we consider
the redefinition of the field of Eq. (2.18).

The superpotentials (3.2) and (3.5) not only preserve the same resulting
effective scalar potential in both $(y_{1},y_{2})$ and $(T,\varphi)$ basis, by
the transformation given in Eq. (2.10), but also they can produce significant
enhancement in the power spectra, explaining the production of PBHs, as shown
in [22]. In the following we will show that these peaks can also produce GWs
spectra sizeable enough to be detected in the corresponding experiments. These
peaks are related to the inflection points in effective scalar potential,
which one can find by a proper choice of parameters, as shown in [22].

To sum up, the new terms that are present in Eqs.(3.2) and (3.5) not only
preserve the transformation laws of the symmetry SU(2,1)/SU(2)$\times$ U(1) as
given in Eq. (2.23), but also result to a significant enhancement in scalar
power spectrum. Hence, these modifications have the following interesting
properties. First, they conserve the SU(2,1)/SU(2)$\times$ U(1) symmetry.
Second, they give the same form of the potential in both bases $(y_{1},y_{2})$
and $(T,\varphi)$. Third, they satisfy the no-scale condition (2.5), in order
to ensure the vanishing of the cosmological constant. Last but not least, they
can result to an inflection point under proper choice of parameters and
finally, therefore can yield to a significant amount of PBHs and GWs.

###  3.2 Modifying the Kähler potential

In the previous section we study the generation of significant peaks in power
spectrum by modifying well-known superpotentials, which yield to the
Starobinsky-like effective scalar potential. These significant peaks can be
also presented by modified only the Kähler potential and keeping the
superpotential unchanged. Hence, this amplification can be found via breaking
the SU(2,1)/SU(2)$\times$ U(1) symmetry by adding extra terms in the kinetic
part of the Lagrangian. In this section we present two models in order to
produce such peaks, one by keeping the Wess-Zumino superpotential (2.12)
unchanged and the other by keeping the Cecotti superpotential unchanged
(2.17).

In a previous work it was shown that sizable amplification in the power
spectrum can be produced through the scheme[21]

$K=-3\ln(T+\bar{T}-\frac{\varphi\bar{\varphi}}{3}+c_{3}e^{-b_{3}(\varphi+\bar{\varphi})^{2}}(\varphi+\bar{\varphi})^{4})\,,$
(3.7)

$W=\frac{\hat{\mu}}{2}\varphi^{2}-\frac{\lambda}{3}\varphi^{3}\,,$ (3.8)

where $c_{3}$ and $b_{3}$ are free parameters of the model. This model has to
deal with just an extra parameter $b_{3}$, which gives us both peaks in power
spectrum and the correct prediction of spectral index $n_{s}$ and the tensor-
to-scalar ratio $r$. The advantage of this model is that it leads to a reduced
level of fine-tuning, as we will see later.

Also it is considered that the superpotential is given by the Cecotti form,
Eq. (2.17). Specifically, we have

$K=-3\ln(T+\bar{T}-\frac{\varphi\bar{\varphi}}{3}+F(T+\bar{T},\varphi+\bar{\varphi}))$
(3.9)

$W=\sqrt{3}m\varphi\left(T-\frac{1}{2}\right)$ (3.10)

where $F(T+\bar{T},\varphi+\bar{\varphi})$ is a function of both chiral
fields. Choosing

$F(T+\bar{T},\varphi+\bar{\varphi})=e^{-d(T+\bar{T})}\left(c_{4}e^{-b_{4}(T+\bar{T})}(T+\bar{T})^{2}+\bar{\lambda}(\varphi+\bar{\varphi}+\bar{\mu})\right)\,,$
(3.11)

where $b_{4}$, $c_{4}$, $\bar{\lambda}$, $\bar{\mu}$ and $d$ are free
parameters. Although, this scheme is more complicated than the previous one,
one can derive effective scalar potential which is flat in the large values of
field.

To summarize, we have presented two different mechanisms, in order to explain
the production of GWs. The first of them is by modifying the superpotential
and preserve the transformation laws of the symmetry SU(2,1)/SU(2)$\times$
U(1), which provide us with an important equivalent between the models [64].
The other is by modifying the Kähler potential and keeping the superpotential
unchanged. In all models presented in this work we remark that we are in
complete consistence with the Planck constraints.

##  4 Producing Gravitational Waves

Based on the previous analysis we can evaluate the amount of GWs produced
during the radiation dominated epoch. The GWs are calculated by the tensor
perturbations (2.25). The equation of motion of GWs reads as

$\frac{d^{2}h_{k}}{d\tau}+2aH\frac{dh_{k}}{d\tau}+k^{2}=\hat{\mathcal{S}}_{k}\,,$
(4.1)

where $h_{k}$ corresponds to the tensor metric perturbation written in Fourier
space from Eq. (2.25) and the $\hat{\mathcal{S}_{k}}$ is the source term in
Fourier space. In the radiation dominated era the solution of Eq.(4.1) is
obtained through the method of Green function as

$h_{k}(\tau)=\frac{1}{a(\tau)}\int
d\tilde{\tau}\,G_{k}(\tau,\tilde{\tau})\,a(\tau)\,\hat{\mathcal{S}}_{k}\,,$
(4.2)

where the Green function is given as follows

$G_{k}(\tau,\tilde{\tau})=\frac{\sin\left(k(\tau-\tilde{\tau})\right)}{k}\,\Theta(\tau-\tilde{\tau})\,.$
(4.3)

The power spectrum of GWs is related to the scalar power spectrum, as the
$h_{k}$ can be expressed in curvature perturbation $R$ as

$h_{k}(\tau)=\frac{4}{9}\int\frac{d^{3}p}{(2\pi)^{3}}\frac{1}{k^{3}\tau}e(\textbf{k},\textbf{p})R(\textbf{p})R(\textbf{k}-\textbf{p})\,\left[I_{c}(\tilde{x},\tilde{y})\cos(k\tau)+I_{s}(\tilde{x},\tilde{y})\sin(k\tau)\right]\,,$
(4.4)

where $\tilde{x}=p/k$, $\tilde{y}=|\textbf{k}-\textbf{p}|/k$ and $e$ is the
polarization tensor of the graviton. The functions $I_{c}$ and $I_{s}$ are
given below. Hence, the spectrum of second order GWs can be evaluated through
the first order scalar perturbations [30, 31, 32, 33].

The general expression, in order to generate the present-day GWs density
function ${\Omega}_{GW}$, is given by [78]

${\Omega_{GW}(k)}=\frac{1}{24}\left(\frac{k}{aH}\right)^{2}\overline{P_{h}(\tau,k)}\,,$
(4.5)

where $P_{h}$ is the tensor power spectrum and the overline denotes the
average over time. The $P_{h}$ is directly related to the scalar power
spectrum. Moreover, the energy density of the GWs in terms of scalar power
spectrum is [33, 74, 79]

${\Omega_{GW}(k)}=\frac{\Omega_{r}}{36}\int^{\frac{1}{\sqrt{3}}}_{0}\mathrm{d}d\int^{\infty}_{\frac{1}{\sqrt{3}}}\mathrm{d}s\left[\frac{(s^{2}-1/3)(d^{2}-1/3)}{s^{2}+d^{2}}\right]^{2}\,P_{R}(kx)\,P_{R}(ky)\,(I_{c}^{2}+I_{s}^{2}).$
(4.6)

The radiation density $\Omega_{r}$ has the present day value
$\Omega_{r}\approx 5.4\times 10^{-5}$. The variables $x$ and $y$ are

$x=\frac{\sqrt{3}}{2}(s+d),\quad y=\frac{\sqrt{3}}{2}(s-d).$ (4.7)

Finally, the functions $I_{c}$ and $I_{s}$ are given by the expressions

$\displaystyle
I_{c}=-36\pi\frac{(s^{2}+d^{2}-2)^{2}}{(s^{2}-d^{2})^{3}}\Theta(s-1)$ (4.8)
$\displaystyle
I_{s}=-36\frac{(s^{2}+d^{2}-2)^{2}}{(s^{2}-d^{2})^{2}}\left[\frac{(s^{2}+d^{2}-2)}{(s^{2}-d^{2})}\log\left|\frac{d^{2}-1}{s^{2}-1}\right|+2\right].$
(4.9)

Using that

$1\,{\rm Mpc}^{-1}=0.97154\times\,10^{-14}\,{\rm s}^{-1}$ and $k=2\pi f$

we can evaluate the energy density of power spectrum as a function of the
frequency $f$. We notice that we must consider the possible modification of
the relativistic degrees of freedom, as described in [79]. Although a detailed
discussion on this, goes beyond the scope of this work, we have checked
numerically that our results have a quite weak dependence on this
modification.

An analytical estimation of the energy density of the GWs has been shown in
[74]. The power spectrum can be approximated by the power-law spectrum as in
Eq.(2.34). In Ref. [74] the following approximation is considered

${\Omega_{GW}(k)}=Q(n)\left(P_{0}\left(\frac{k}{k_{peak}}\right)^{-n}\right)^{2},$
(4.10)

where the coefficients $Q(n)$ are given in [74] for specific values of $n$.
This equation is valid in ultraviolet (UV) region, if $k\gg k_{peak}$[80, 81].
There are two assumption for the value of $n$

$n=\begin{cases}n_{UV}&,\quad 0<n_{UV}<4\\\ 2+n_{UV}/2&,\quad
n_{UV}>4\,,\end{cases}$ (4.11)

where $n_{UV}$ is a positive constant and it is referred as the spectral index
in UV region. For $k\ll k_{peak}$ in the infrared (IR) regions we have the
following approximation

${\Omega_{GW}(k)}=Q(n)P_{0}^{2}\left(\frac{k}{k_{peak}}\right)^{3}\,.$ (4.12)

In [82, 83] is shown that there is a general behaviour of the power spectrum
of GWs in the IR regions. In particular, in the IR scales the energy density
of GWs has been shown that follows the power law [82]:

$\Omega_{GW}\propto f^{3-2/\ln(f_{c}/f)},$ (4.13)

where $f_{c}$ is roughly the peak of the frequency. In the region of the peak
this approximation becomes

$\Omega_{GW}\propto f^{2-2/\ln(f_{c}/f)}$,

and it is valid if the power spectrum of scalar curvature perturbation is
narrow [82].

###  4.1 Modifying the superpotential

Now we turn to the results of the power spectrum and the stochastic spectrum
of GWs in the case of superpotentials (3.2) and (3.5), where the Kähler
potential gets the form in $(y_{1},y_{2})$ basis given in Eq. (2.7). By
evaluating the effective scalar potential as shown in Eq.(2.1), we can get
inflection points, which can yield significant enhancement in the power
spectrum, as calculated from Eq. (2.32). We remark that using the Kähler
potential in the $(T,\varphi)$ basis given from Eq. (3.1) and the
corresponding superpotential from Eqs. (2.23) and (3.10), we derive the same
effective scalar potential and hence the same results. In the following, we
will use the basis $(y_{1},y_{2})$, in spite the fact that we derive the same
effective potential in the basis ($T,\varphi$) if we consider the
transformation laws given in Eq. (2.8) and analyzed in [64, 22].

Having evaluated the effective scalar potential using the superpotentials in
(3.2) and (3.5), as well as the Kähler potential in (2.7), we proceed to the
numerical integration of the background dynamics, as given in Eq.(2.24), and
the perturbation of the field as given in the Eq. (2.26) and Eq. (2.27). The
numerical procedure, which we follow, is shown in Refs. [21, 22]. For initial
conditions of the field we use those which that predict $n_{s}$ and $r$
compatible to Planck measurements [62, 63] and result to a peak in power
spectrum in the appropriate place, in order to explain the generation of GWs.
The values of the initial condition, as well as the prediction for $n_{s}$ and
$r$ for the cases (3.2) and (3.5) are given in Table 1 respectively. With
$\phi$ we denote the field after the redefinition given in Eq. (2.16) for
obtaining canonical kinetic terms.

| $\phi_{ic}$ | $n_{s}$ | $r$
---|---|---|---
1 | $4.98$ | $0.9691$ | $0.0103$
2 | $4.84$ | $0.9639$ | $0.0153$
Table 1: Initial conditions for the effective scalar potential in Eqs. (3.2)
and (3.5) and the predictions for $n_{s}$ and $r$.

Afterwards, we evaluate numerically the energy density of GWs, which is given
in Eq. (4.6), as we described before. In Figs. 2 and 4 we present the power
spectrum and the density abundance of GWs for the cases (3.2) and (3.5)
respectively. One can remark that the enhancement of scalar spectrum can be
imprinted in the density of GWs. We present both the exact power spectrum
given from (2.33) with solid black lines and the spectrum from the power-law
formula (2.34) with dashed red line. The choice of parameters, in order to
achieve the inflection point in the effective scalar potential is given in the
caption. We notice that it is important the position of the peak, to be within
$10^{12}-10^{14}Mpc^{-1}$, in order to lie in the observational range of the
future GWs experiments.

In order to compare with the numerical evaluation of the GWs abundance, we
show in Fig. 3 the corresponding example of Fig. 2. We evaluate the IR slope
given from Eq.(4.13) with red dashed line and the UV slope given from
Eq.(4.10) with blue dashed line. We use $n_{UV}=1.8$, that lies in the region
$n_{UV}<4$.

Figure 2: The power spectrum (left) with solid line (numerical) and with
dashed (analytical from Eq. 2.34). The density of stochastic GWs (right). Both
plots are for the case of modifying Wess-Zumino superpotential. We choose
$\lambda/\mu=1/3$, $c_{1}=1.9$, $c_{2}=-0.73$ and $b_{1}=3.9615$. Figure 3:
The energy density of GWs for the case of modifying Wess-Zumino
superpotential. Dashed lines corresponds to analytical approximation of the IR
slope (red) and UV slope (blue).

Figure 4: The power spectrum (left) with solid line (numerical) and with
dashed (analytical from Eq. 2.34). The density of stochastic GW (right). Both
plots are for the case of modifying Cecotti superpotential. We choose
$c_{3}=13.2$, $l=1.00028$ and $b_{2}=8.575$.

###  4.2 Modifying the Kähler potential

In this section we present the results of the power spectrum and the
abundances of GWs in the case of the modified Kähler potential, given in Eqs.
(3.9) and (3.11). This case is more labored than the previous one, because
there is analytic expression for transformation of the field, in order to have
canonical kinetic terms. Hence, the Eq. (2.19) should be used. We proceed with
the numerical integration of background and curvature perturbations of the
field, assuming that we are in Bunch-Davies vacuum, given in Eqs. (2.24),
(2.26) and (2.27), as we have discussed before.

In Figs. 5 and 6 we present the resulting power spectrum and the density
abundance of GWs for the Eqs. (3.9) and (3.11), respectively. The choice of
initial conditions and the prediction for $n_{s}$ and $r$ are given in Table
2. In this table we present the initial condition for the canonical normalized
field, which we denote with $\phi$. In this Table we show that the prediction
of $n_{s}$ and $r$ are consistent with the Planck constraints[62, 63], as in
the previous cases.

| $\phi_{ic}$ | $n_{s}$ | $r$
---|---|---|---
1 | $4.899$ | $0.9612$ | $0.0121$
2 | $4.097$ | $0.9601$ | $0.0092$
Table 2: Initial conditions for the field for the Eqs. (3.7) and (3.9) and the
prediction of $n_{s}$ and $r$.

Figure 5: The power spectrum (left) with solid line (numerical) and with
dashed (analytical from Eq. 2.34). The density of stochastic GWs (right) for
the case of modifying Wess-Zumino. We choose for the parameters $b_{3}=87.38$,
$c=0.065$, $c_{3}=-4$ and $\lambda/\mu=0.33327$

Figure 6: The power spectrum (left) with solid line (numerical) and with
dashed (analytical from Eq. 2.34). The density of stochastic GWs (right) for
the case of modifying Cecotti. The parameters are $d=-0.054$, $b_{4}=7.51511$,
$c_{4}=8.8$, $\bar{\lambda}=-0.5$ and $\bar{\mu}=1.0$

In left panels in the Figs. 5 and 6 we also show the numerical and analytical
results of power spectrum from Eqs. (2.33) and (2.34). The value of $n$ in Eq.
(2.34) ranges from $0.15$ to $2.7$ [75]. In both models, either by modifying
the Kähler potential or by modifying the superpotential), $n$ ranges from 0.8
to 1.8. Hence, in order to estimate the effect of the non-gaussianities the
analysis of [75] can be used. In particular, this related to the peaks of
power spectrum $P_{peak}=10^{-3}-10^{-2}$ as $f_{NL}\lessapprox 10^{-2}$.
Generally, the amplitude $f_{NL}$ is defined as

$f_{NL}=\frac{5}{6}\frac{\mathcal{B}(k_{1},k_{2},k_{3})}{P_{R}(k_{1})P_{R}(k_{2})+P_{R}(k_{1})P_{R}(k_{3})+P_{R}(k_{2})P_{R}(k_{3})}\,,$
(4.14)

where $\mathcal{B}$ is the bispectrum. According to the study [75] the non-
gausianities of models with an enhanced power spectrum can be large enough not
to satisfy the constraints on the PBHs abundance. We note that in this paper
we focus in the production of GWs, as the production of the PBHs for these
models has already presented elsewhere [21, 22].

The production of PBHs using similar modifications in the SUGRA model has been
studied in [22]. There, it was concluded that the maximum value of the peak of
the power spectrum should be around $3-5\times 10^{-2}$. In this analysis
here, that refers to the production of GWs this restriction does not apply.

That is, the power spectra of GWs, will be detected by future experiments,
such as LISA or DECIGO even if the high of the peak of the power spectrum is
smaller than $10^{-2}$. On the other hand, the abundance of the PBHs is
exponentially sensitive to the mass fraction of the Universe that collapse to
PBHs [84, 85, 86, 87, 88]. Consequently, it is expected that the amount of the
fine-tuning in the parameters of the extra terms, will be reduced.  We discuss
this point in Appendix B.

##  5 Conclusions

In this paper we introduce four models in order to propose a possible scenario
for the production of GWs, that can be detected by future space-based
experiments of GWs.  A well-known way to get amplified production of the GWs
is an enhancement in the scalar power spectrum, since the first and second
order spectra are related. There are many mechanisms proposed in the
literature, in order to get such an enhancement. In our study we employed this
of the near inflection point, in the context of single field inflation. We
embedded this scenario in models derived by no-scale SUGRA theory and
especially we consider models that conserve and break the
SU(2,1)/SU(2)$\times$ U(1) symmetry.

First, we modified the Wess-Zumino and the Cecotti superpotentials. These
modifications leads to the effective scalar potential, which is equivalent in
both bases $(T,\varphi)$ and $(y_{1},y_{2})$. Both of these models can
conserve the SU(2,1)/SU(2)$\times$ U(1) symmetry, as we can properly interplay
between the fields $y_{1}$ and $y_{2}$ and we can derive the same effective
scalar potential. The same potential can be found, if we transform the fields
and use the equivalent forms in the $(T,\varphi)$ basis. The equivalence
between the models has been studied for the unmodified case. However, in our
study we have the additional feature of an inflection point that leads to a
significant enhancement of the scalar power spectrum.

Secondly, we keep these two superpotentials, Wess-Zumino and Cecotti,
unchanged and we modify the Kähler potential in $(T,\varphi)$ basis. The
resulting effective scalar potential can only be given numerically, as there
is not an analytical solution with canonical kinetic term. Hence, we derive
four inflationary potentials with a near inflection point.

This enhancement of scalar power spectrum is expected to be imprinted in the
energy density of GWs. In order to evaluate this energy density, we calculate
numerically the perturbations of the field for all the proposed potentials. We
evaluate the scalar power spectrum and then the corresponding energy densities
of GWs. We show that these models give sizable GWs spectra, which can be
detected by the future experiments. Last but not least, all the models
presented in this work are consistent with the observational constraints on
inflation.

##  Appendix A The forms of the scalar potential

In general the scalar potential can be evaluated in the framework of
supergravity, if we use the following expression:

$V=e^{K}(D_{\Phi}WK^{\bar{\Phi}\Phi}D_{\bar{\Phi}}\bar{W}-3|W|^{2})\,,$ (A.1)

where

$D_{\Phi}W=\frac{\partial W}{\partial\Phi}+\frac{\partial K}{\partial\Phi}W.$
(A.2)

In this Appendix we sum up the four models proposed in the text and we present
the effective scalar potentials for the two first models.

###  A.1 Model I

For the choice of Kähler potential

$K=-3\ln(1-\frac{|y_{1}|^{2}}{3}-\frac{|y_{2}|^{2}}{3})$ (A.3)

we modify the superpotential given in Eq. (2.12) as:

$W=\left[\frac{\hat{\mu}}{2}\left(y_{1}^{2}+\frac{y_{2}^{2}y_{1}}{\sqrt{3}}\right)-\lambda\frac{y_{1}^{3}}{3}\right]\left[1+e^{-b_{1}y_{1}^{2}}\left(c_{1}{y_{1}}^{2}+c_{2}{y_{1}}^{4}\right)\right]\,$
(A.4)

where $\hat{\mu}$, $\lambda$, $b_{1}$, $c_{1}$ and $c_{2}$ are free
parameters. If we use Eq. (A.1), the effective scalar potential becomes:

$\begin{split}V/\mu^{2}&=\sinh^{2}\left(\frac{\phi}{\sqrt{6}}\right)\Big{[}9\tanh^{4}\left(\frac{\phi}{\sqrt{6}}\right)e^{-6b_{1}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)}\left(\mu-2\lambda\tanh\left(\frac{\phi}{\sqrt{6}}\right)\right)^{2}\times\\\
&\times\left(\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)(6{c_{2}}-3b_{1}c_{1})-9b_{1}{c_{2}}\tanh^{4}\left(\frac{\phi}{\sqrt{6}}\right)+c_{1}\right)^{2}-\\\
&-3\sinh^{2}\left(\frac{\phi}{\sqrt{6}}\right)e^{-6b_{1}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)}\left(\mu-2\lambda\tanh\left(\frac{\phi}{\sqrt{6}}\right)\right)\times\\\
&\times\left(2\mu-6\lambda\tanh\left(\frac{\phi}{\sqrt{6}}\right)+\mu\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)\right)\cosh^{2}\left(\frac{\phi}{\sqrt{6}}\right)\left(\mu-3\lambda\tanh\left(\frac{\phi}{\sqrt{6}}\right)\right)^{2}\\\
&\times\left(3\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)e^{-3b_{1}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)}\left(c_{1}+3{c_{2}}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)\right)+1\right)^{2}+\\\
&+\left(e^{3b_{1}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)}+3c_{1}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)+9{c_{2}}\tanh^{4}\left(\frac{\phi}{\sqrt{6}}\right)\right)\times\\\
&\times\left(3\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)(b_{1}c_{1}-2{c_{2}})+9b_{1}{c_{2}}\tanh^{4}\left(\frac{\phi}{\sqrt{6}}\right)-c_{1}\right)\Big{]}\end{split}$
(A.5)

where $\hat{\mu}=\mu\sqrt{c/3}$. We have considered that the inflationary
direction for this example is given as:

$y_{1}=\varphi,\quad y_{2}=0.$ (A.6)

Moreover, in order to have fix the non- canonical kinetic term we consider the
redefinition of the field:

$y_{1}=-\sqrt{3}\tanh\left(\frac{\phi}{\sqrt{6}}\right)\,.$ (A.7)

We remark that the same effective scalar potential can be derived if we have
the following transformations: $y_{1}\rightarrow-y_{1}$ and $y_{2}\rightarrow-
y_{2}$ [22, 64].

The same form of the potential can be derived, if we use the basis
$(T,\varphi)$ with the tranformation described in the text from Eqs.(2.8),
(2.9) and (2.10). Hence, we have the following forms:

$K=-3\ln(T+\bar{T}-\frac{|\varphi|^{2}}{3})$ (A.8)

$W=\left(\frac{\hat{\mu}}{2}\varphi^{2}-\frac{\lambda}{3}\varphi^{3}\right)\left(1+e^{-b_{1}\left(\frac{2\varphi}{1+2T}\right)^{2}}\left[c_{1}\left(\frac{2\varphi}{1+2T}\right)^{2}+c_{2}\left(\frac{2\varphi}{1+2T}\right)^{4}\right]\right).$
(A.9)

We choose the following direction $T=Im\varphi=0$ and $Re\varphi=\phi$ and we
apply Eq. (A.1).

###  A.2 Model II

For the choice of Kähler potential (A.3) we modify the superpotential given in
Eq. (2.17) as:

$W=m\left(-y_{1}y_{2}+\frac{y_{2}y_{1}^{2}}{l\sqrt{3}}\right)\left(1+c_{3}e^{-b_{2}{y_{1}}^{2}}{y_{1}}^{2}\right)\,.$
(A.10)

From Eq. (A.1) we derive the form of effective scalar potential, which is
given as follows:

$\begin{split}V/m^{2}&=\frac{3}{l^{2}}\sinh^{2}\left(\frac{\phi}{\sqrt{6}}\right)\cosh^{2}\left(\frac{\phi}{\sqrt{6}}\right)e^{-6b\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)}\left(l-\tanh\left(\frac{\phi}{\sqrt{6}}\right)\right)^{2}\\\
&\times\left(e^{3b\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)}+3c_{3}\tanh^{2}\left(\frac{\phi}{\sqrt{6}}\right)\right)^{2}.\end{split}$
(A.11)

As in model 1, we choose the inflationary direction (A.6) and we fix the non-
canonical kinetic terms. This potential has the following free parameters:
$l$, $b_{2}$ and $c_{3}$.

We can use the other form of the Kähler potential (A.8) and with proper
tranformations given from Eqs.(2.8), (2.9) and (2.10) we can take the
equivalent form of the superpotential in the basis $(T,\varphi)$. The
superpotential is given as follows:

$W=\frac{\sqrt{3}}{2}m\varphi\Big{(}\frac{1}{2}-T\Big{)}\left(-1-2T-\frac{1-2T}{l}\right)\left[1+3c_{3}e^{-3b_{2}\left(\frac{1-2T}{1+2T}\right)^{2}}\left(\frac{1-2T}{1+2T}\right)^{2}\right]\,.$
(A.12)

###  A.3 Model III

In this model we do not change the superpotential but the Kähler potential in
$(T,\varphi)$ basis. For superpotential we choose this of Wess Zumino model
described in Eq. (2.12).

$K=-3\ln(T+\bar{T}-\frac{\varphi\bar{\varphi}}{3}+c_{4}e^{-b_{3}(\varphi+\bar{\varphi})^{2}}(\varphi+\bar{\varphi})^{4})\,,$
(A.13) $W=\frac{\hat{\mu}}{2}\varphi^{2}-\frac{\lambda}{3}\varphi^{3}.$ (A.14)

We can evaluate the scalar potential from (A.1). In this case the non-
canonical kinetic term can be fixed only numerically. The direction of
inflation is $T=c/2$ and $\varphi=\phi$. The free parameters for this case are
$c$, $\lambda$, $\mu$, $c_{3}$, $b_{3}$ and $c$.

###  A.4 Model IV

In this model we modify the described in Eq.(A.8) and we keep unchanged the
Cecotti superpotential given in (2.17). Hence we have:

$K=-3\ln(T+\bar{T}-\frac{\varphi\bar{\varphi}}{3}+F(T+\bar{T},\varphi+\bar{\varphi}))$
(A.15) $W=\sqrt{3}m\varphi\left(T-\frac{1}{2}\right)$ (A.16)

where $F(T+\bar{T},\varphi+\bar{\varphi})$ is a function of both chiral
fields. We choose the following form

$F(T+\bar{T},\varphi+\bar{\varphi})=e^{-d(T+\bar{T})}\left(c_{4}e^{-b_{4}(T+\bar{T})}(T+\bar{T})^{2}+\bar{\lambda}(\varphi+\bar{\varphi}+\bar{\mu})\right)\,,$
(A.17)

and we assume that the inflationary direction is

$T=ReT=\phi,\quad ImT=\varphi=0.$ (A.18)

The scalar power spectrum can be evaluated numerically by using Eq. (A.1). We
need to consider the transformation of the field in order to have canonical
kinetic term. This redefinition can be evaluated only numerically as in the
previous case. The free parameters are $m$, $d$, $b_{4}$, $c_{4}$,
$\bar{\lambda}$ and $\bar{\mu}$.

##  Appendix B Fine-tuning analysis and a comparison between the models

It is well-known that the enhancement of scalar power spectrum, which occurs
due to the inflection point in the effective scalar potential requires a lot
of fine-tuning [19]. The value of this enhancement can be smaller in the case
of studying the production of GWs than in the case of studying the amount of
DM from the PBHs. In Ref. [22] there is a discussion explaining how the
parameters of the potential presented in this work arise. In this Appendix we
analyse the level of fine-tuning by considering the parameter $b_{i}$, which
is presented throughout this work and is the parameter, which depends on the
power spectrum’s peak and demands more fine-tuning.

The role of the parameter $b_{i}$ is to trigger an enhancement in the power
spectrum. In order to analyze the level of fine-tuning, we calculate the
parameter $\Delta_{b}$, which is shown in Refs. [89, 90] and it is given as
the max value of the follow quantity

$\Delta_{b}=\left|\frac{\partial\ln(P_{R}^{peak})}{\partial\ln(b_{i})}\right|\,.$
(B.1)

In the following, we study the fine-tuning of the function
$P_{R}^{peak}(b_{i})$. Large value of the maximum of the quantity $\Delta_{b}$
means that a high level of fine-tuning is required. As we mentioned before, we
expect that the fine-tuning of the parameters can be decreased in the study of
generation of GWs, due to its wider range of peak’s height in comparison with
the production of PBHs.

Figure 7: The level of fine-tuning for the case (3.7).

In Fig.7 we show the quantity $\Delta_{b}$ as a function of the peak of the
power spectrum.  In particular we show how the quantity $\Delta_{b}$ varies in
respect to the values of the spectrum’s peak in the case of study PBHs (blue
line) and in the case of GWs (red line). The case for this plot is the model
described from the Eqs.(3.7) and( 3.8) (model III of Appendix A). As one can
notice we need more fine-tuning when we study PBHs. This occurs not only due
to smaller peaks, which are needed in the study of GWs, but also due to the
specific value of the threshold presented in the evaluation of the production
of PBHs. Specifically, in order to obtain a significant amount of DM from PBHs
and to use the proposed values for threshold, as analysed in [84, 85, 86, 87,
88], we need to properly adjust the extra parameters of the model. This
increases the fine-tuning in the study of PBHs.

We remark that in the previous study of Refs [22], where the fine-tuning for
the PBHs production is analysed, the maximum value of $(\Delta_{b})$ was at
the order of magnitude of $10^{6}$. In a relevant study of Ref. [19] the
amount of fine-tuning for PBHs was at level of $10^{8}$. It is possible that
the fine-tuning can be decreased further. In Table 3, we show the value of the
maximum of $\Delta_{b}$, assuming that we have the minimal enhancement of peak
of the power spectrum, in order to predict the future space based experiment
DECIGO. In other words, we calculate the max( $\Delta_{b}$) by considering
that the peak of power spectrum is around $k=10^{-5}$. In Table 3 we present,
for comparison, the maximum value of $\Delta_{b}$ for PBHs, as adopted in
Refs. [21, 22]. One can notice that there is sizable difference with respect
to the case of the production of the GWs. Specifically, the fine-tuning in our
current analysis is smaller at least two orders of magnitude.

case | max$(\Delta_{b})_{PBHs}$ | max$(\Delta_{b})_{GWs}$
---|---|---
1 |  $7.9$ $\times 10^{5}$ |  $3.5$ $\times 10^{3}$
2 |  $9.8$ $\times 10^{5}$ |  $3.6$ $\times 10^{3}$
3 |  $1.6$ $\times 10^{5}$ | $8.5\times 10^{2}$
4 |  $4.2$ $\times 10^{6}$ |  $5.6$ $\times 10^{4}$
Table 3: The maximum value for $\Delta_{b}$ for the cases in Eqs. (3.2),
(3.5),(3.7) and (3.9) respectively. The abundances of PBHs are adopted by [21,
22]. Details given are in main text.

Figure 8: The level of fine-tuning for our model (blue) with comparison with
this amount from other models.

Finally, in order to have a comparison with the previous mechanisms with an
enhancement in scalar power spectrum analysed in the literature, we depict the
quantity $\Delta_{b}$ for different models. In particular, in Fig.8 we show
the power spectrum for the case given Eqs.(3.7) and( 3.8) with blue lines.
Orange lines corresponds to the model presented in [77] where the enhancement
of power spectrum comes from a bulky potential. Green lines to the two- model
shown in [24]. In this work a two field model with a non-canonical kinetic
term has been proposed. In both these models the underlying parameter b needs
fine-tuning in order to obtain the proper enhancement. The left panel of the
Fig.8 shows the power spectrum of these models, reproduced by the methodology
of subsection 2.2 and the right panel shows the quantity $\Delta_{b}$. We
notice that models with an inflection point have more fine-tuning than the
other ones. Models with a bumb feature in the potential have significantly
less fine-tuning than the models with an inflection point. Finally, models
with two fields experience the less fine-tuning of all cases.

## Acknowledgments

This research work was supported by the Hellenic Foundation for Research and
Innovation (H.F.R.I.) under the “First Call for H.F.R.I. Research Projects to
support Faculty members and Researchers and the procurement of high-cost
research equipment grant” (Project Number: 824).

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