# Inflection-Point Sgoldstino Inflation in no-Scale Supergravity

Constantinos Pallis
Laboratory of Physics, Faculty of Engineering, Aristotle University of
Thessaloniki, GR-541 24 Thessaloniki, GREECE
e-mail address<EMAIL_ADDRESS>
###### Abstract

Abstract: We propose a modification of no-scale supergravity models which
incorporates sgoldstino stabilization and supersymmetry (SUSY) breaking with a
tunable cosmological constant by introducing a Kähler potential which yields a
kinetic pole of order one. The resulting scalar potential may develop an
inflection point, close to which an inflationary period can be realized for
subplanckian field values consistently with the observational data. For
central value of the spectral index $n_{\rm s}$, the necessary tuning is of
the order of $10^{-6}$, the tensor-to-scalar ratio is tiny whereas the running
of $n_{\rm s}$ is around $-3\cdot 10^{-3}$. Our proposal is compatible with
high-scale SUSY and the results of LHC on the Higgs boson mass.

PACs numbers: 12.60.Jv, 04.65.+e

##  I Introduction

One of the most tantalizingly evasive problems in Particle Physics is the
explanation of the large hierarchies existing in the fundamental scales of the
modern theories. Two of these scales are related to the cosmological problems
of inflation and _Dark Energy_ (DE) whereas a third one is related to the
scale of _Supersymmetry_ (SUSY) breaking. The first of the scales above, is
expected to be less than $10^{4}~{}{\mbox{\rm EeV}}$, the second one is
related to the present acceleration of the Universe and is really very tiny
($\Lambda\sim 1~{}{\rm meV}$) whereas the scale of the SUSY partners,
$\widetilde{m}$, is continuously pushed more and more beyond the TeV region
due to the lack of any positive signal at LHC until now.

In a set of recent papers ns89 ; de , two of the aforementioned scales
($\widetilde{m}$ and $\Lambda$) are systematically interconnected within the
framework of no-scale _Supergravity_ (SUGRA) old . Adopting simple Kähler
potentials parameterizing flat or curved (compact or non-compact) Kähler
manifolds, one can initially derive superpotentials which yield SUSY breaking
along Minkowski flat directions. Combining two types of these superpotentials,
_de Sitter_ (dS) vacua are achieved which may, in addition, explain the
notorious DE problem finely tuning a single superpotential parameter. In other
words, these models offer a _technically convenient_ resolution to the DE
problem. The construction can be easily extended to multi-modular settings of
mixed geometry. Mild deformations of the adopted moduli geometry can cure
possible instabilities and/or massless excitations. _Soft SUSY-Breaking_ (SSB)
parameters of the order of the gravitino mass can be also derived. It is worth
mentioning that the method above does not require any external mechanism for
vacuum uplifting – see, e.g., Ref. kallosh ; antst .

It would be certainly beneficial if we could incorporate in the aforementioned
framework observationally acceptable inflation as done in Ref. nsinfl ;
nsreview . There the SUSY breaking sector is supplemented with the
inflationary one which assures Starobinsky-like inflation consistently with
data – for similar attempts see Ref. king ; lhclinde ; ketov ; kai ; scroest .
In contrast to those models, though, we here adopt the most economical
possible interface of both sectors postulating that the scalar component of
the goldstino superfield, – which is responsible for the SUSY breaking – plays
the role of the inflaton – cf. Ref. postma ; ipisugra ; ant1 ; aattr ; froest
. Such a construction predicts directly the scale $\widetilde{m}$, since the
fundamental mass parameter, $m$, entering the superpotential of goldstino is
relied on the normalization plcp  of the power spectrum of the curvature
perturbation – cf. Ref. ant1 ; aattr ; ipisugra . As a consequence, the
gravitino mass and $\widetilde{m}$ lie at the EeV mass scale. Therefore, the
set-up of high-scale SUSY hall ; strumia  naturally arises, as can be
demonstrated by coupling the goldstino sector to the _Minimal SUSY Standard
Model_ (MSSM) martin  and deriving the relevant SSB parameters soft . It is
worth mentioning that these $\widetilde{m}$ values are compatible with the
Higgs boson mass discovered at LHC lhc  stabilizing, thereby, the electroweak
vacuum.

To implement the inflationary scenario above we adopt a Kähler potential which
generates a kinetic pole of order one epole  in the SUGRA lagrangian. The
presence of this pole essentially restricts the dynamics of both ingredients
of our set-up (inflation and SUSY breaking) to field values below the Planck
scale $m_{\rm P}$. For specific values of the parameters the SUGRA potential
develops an inflection point ipisusy ; ipilinde ; aterm ; ipirad ; drees ;
ipisugra  which supports inflationary solutions at the cost of a tuning of the
order of $10^{-6}$ in the selection of a relevant parameter. The realized
_Inflection-Point inflation_ (IPI) is a typical inflationary model – see,
e.g., Ref. review  – which was first introduced in the context of MSSM ipisusy
; aterm  and then employed in various frameworks ipilinde ; ipirad ; drees ;
ipisugra . To our knowledge, none of them does not combine IPI with the
aforementioned merits of no-scale SUGRA. IPI discussed here occurs for
subplanckian field values and is consistent with data plin ; bk15  predicting
negligible production of primordial gravitational waves and testable asdrees
running of $n_{\rm s}$. After its end, the universe is reheated up to a
temperature rh  of PeV level via the perturbative decay of the heavy
sgoldstino into gravitinos and/or MSSM (s)particles baerh ; antrh ; nsrh ;
full  via SUGRA-based interactions.

We start our presentation implementing the transition from Minkowski to dS
vacua in the framework of our model in Sec. II. Then, in Sec. III, we verify
the generation of the DE potential energy at the stable dS vacuum and, in Sec.
IV, we extract the SSB terms. In Sec. V we focus on the exploration of the
inflationary stage and, finally, we summarize our results in Sec. VI. Unless
otherwise stated, we use units where the reduced Planck scale $m_{\rm
P}=2.4\cdot 10^{18}~{}{\mbox{\rm GeV}}$ is taken to be unity, a subscript of
type $,\chi$ denotes derivation _with respect to_ (w.r.t.) the field $\chi$
and charge conjugation is denoted by a star. We also recall that
$1~{}{\mbox{\rm PeV}}=10^{6}~{}{\mbox{\rm GeV}}$ and $1~{}{\mbox{\rm
EeV}}=10^{9}~{}{\mbox{\rm GeV}}$.

##  II Model Set-up

We work in the context of SUGRA employing just one chiral superfield $Z$ – cf.
Ref. scroest ; froest ; ipisugra . The F–term SUGRA potential is given by

$V=e^{G}\left(G^{ZZ^{*}}G_{Z}G_{Z^{*}}-3\right),$ (1)

where $G$ is the Kähler -invariant function defined in terms of the Kähler
potential $K=K(Z,Z^{*})$ and the superpotential $W=W(Z)$ as follows

$G:=K+\ln|W|^{2}~{}~{}\mbox{with}~{}~{}G_{ZZ^{*}}=K_{ZZ^{*}}=K_{,ZZ^{*}}$ (2)

denoting the Kähler metric and $G^{ZZ^{*}}$ is its inverse. We concentrate on
the following $K$

$K=-N\ln\Omega~{}~{}\mbox{with}~{}~{}\Omega=1-(Z+Z^{*})/2+k^{2}Z_{\rm v}^{4},$
(3a) where $N>0$ and we include in the argument of $\ln$ the stabilization
term $Z_{\rm v}=Z+Z^{*}-2{\rm v}.$ (3b)

Here $k$ and ${\rm v}$ are two real free parameters. The $Z$ space generated
by $K$ for $k=0$ is hyperbolic and invariant under a set of transformations
related to the group $U(1,1)$ – see Ref. epole . Small $k$’s are completely
natural, according to the ’t Hooft argument symm , thanks to the enhanced
symmetry above.

Following our strategy in Ref. ns89 ; de  we first substitute $K$ in Eq. (3a)
with $k=0$ together with an unspecified $W$ into Eq. (1) taking the limit
$Z={Z^{*}}$. We obtain

$V=\omega^{-N}W^{2}\left(\frac{4\omega^{2}}{N}\left(\frac{N}{2\omega}+\frac{dW}{dZW}\right)^{2}-3\right),$
(4)

where we introduce the shorthand notation

$\omega=\Omega(Z=Z^{*},k=0)=1-Z.$ (5)

A $Z$-flat direction with Minkowski vacua can be assured if we seek $W=W_{0}$
such that $V=0$ for any $Z$. $W_{0}$ can be determined solving the resulting
ordinary differential equation

$\frac{dW_{0}}{W_{0}}=n_{\pm}\frac{dZ}{\omega}~{}~{}\mbox{with}~{}~{}n_{\pm}=\frac{1}{2}\left(N\pm\sqrt{3N}\right).$
(6)

For $N=3$ we obtain $n_{\pm}=0$ and we reveal the archetypal no-scale model
old  which exclusively yields Minkowski vacua. For $N\neq 3$, we obtain two
possible forms of $W_{0}$,

$W_{0}^{(\pm)}(Z)=m\omega^{n_{\pm}},$ (7)

where $m$ is an arbitrary mass parameter. The exponents $n_{\pm}\neq 0$ may,
in principle, acquire any real value, if we consider $W_{0}$ as an effective
superpotential including perturbative and non-perturbative contributions from
string theory ns89 . However, when $N/3$ is a perfect square, integer
$n_{\pm}$ values may arise too. E.g., for $N=12,27$ and $48$ we obtain
$(n_{-},n_{+})=(3,9),(9,18)$ and $(18,30)$ respectively.

The solutions in Eq. (7) can be combined as follows

$W_{\Lambda}=W_{0}^{(+)}-C_{\Lambda}W_{0}^{(-)}=m\omega^{n_{+}}C_{\omega}^{-},$
(8)

where we normalize somehow the relevant coefficients – cf. Ref. de  – setting
that multiplying $W_{0}^{(+)}$ equal to unity. Also,

$C_{\omega}^{\pm}:=1\pm C_{\Lambda}\omega^{-\sqrt{3N}}$ (9)

and reduces to unity for tiny $C_{\Lambda}$ – see below. The resulting
potential, $V=V_{\Lambda}$, obtained after replacing Eqs. (3a) and (8) into
Eq. (1), is

$V_{\Lambda}=m^{2}\Omega^{-N}\omega^{2n_{+}}\left(|U/2\omega|^{2}-3|C_{\omega}^{-}|^{2}\right),$
(10)

where we define the quantity

$U=\frac{\sqrt{2N}}{J\Omega}\left(\left(\sqrt{3}C_{\omega}^{+}+\sqrt{N}C_{\omega}^{-}\right)\Omega-2\sqrt{N}C_{\omega}^{-}\Omega_{,Z}\omega\right).$
(11)

Here $J$ is related to the canonical normalization of the complex scalar field
$Z=ze^{i{\theta}}$ according to which

${d\widehat{z}/dz}=\sqrt{2K_{ZZ^{*}}}=J~{}~{}\mbox{and}~{}~{}\widehat{{\theta}}=J{\theta}z,$
(12)

where $J$ can be expressed in terms of $\Omega$ as follows

$J=\sqrt{2N}\left(\frac{\Omega_{,Z}^{2}}{\Omega^{2}}-\frac{\Omega_{,ZZ^{*}}}{\Omega}\right).$
(13)

The involved derivatives of $\Omega$ are found to be

$\Omega_{,Z}=-1/2+4\kappa^{2}Z_{\rm
v}^{3}~{}~{}\mbox{and}~{}~{}\Omega_{,ZZ^{*}}=12\kappa^{2}Z_{\rm v}^{2}.$ (14)

The structure of $V_{\Lambda}$ in Eq. (10) is shown on Fig. 1 where the
dimensionless quantity $10^{2}V_{\rm I}/m^{2}m_{\rm P}^{2}$ is plotted as a
function of $z$ and ${\theta}$ for the $N,{\rm v}$, $k$, $m$ and $C_{\Lambda}$
values shown in Table 1– we select the lower $N$ which yields integer
$n_{\pm}>0$. We see that $V_{\Lambda}$ develops along the stable direction
${\theta}=0$ (i) A dS vacuum for $z={\rm v}=0.5$ which interprets DE – see
Sec. III – and breaks SUSY – see Sec. IV –, and (ii) an inflection point for
$z=z_{0}\simeq 0.71$, suitable for driving IPI – see Sec. V. The last finding
seems to be a consequence of the adopted $\Omega$ in Eq. (3a), since replacing
its $k$-independent part with $Z+Z^{*}$ or $1-|Z|^{2}$ and repeating the
procedure above, the resulting $V_{\Lambda}$ does not posses inflection point
but it acquires a shape resembling that presented in Ref. ant1 ; aattr .

##  III Dark Energy

Fig. 1: The (dimensionless) SUGRA potential $10^{2}V_{\Lambda}/m^{2}m_{\rm
P}^{2}$ in Eq. (10) as a function of $z$ and ${\theta}$ defined above Eq. (12)
for the inputs shown in Table 1. The location of the dS vacuum at
$(\left\langle{z}\right\rangle,{\theta})=(0.5,0)$ and the inflection point at
$(z_{0},{\theta})\simeq(0.71,0)$ is also depicted by two thick points.

The stability of $V_{\Lambda}$ at its dS – for $C_{\Lambda}>0$ – vacuum
noticed by Fig. 1 can be analytically verified in general. Namely, we can show
that the vacuum

$\left\langle{z}\right\rangle={\rm
v}~{}~{}\mbox{and}~{}~{}\left\langle{{\theta}}\right\rangle=0~{}~{}\mbox{with}~{}~{}\left\langle{V_{\Lambda}}\right\rangle=12C_{\Lambda}m^{2}$
(15)

is stable against fluctuations of the various excitations for $N>3$. In fact,
for these $N$ values the resulting scalar masses squared are positive since we
find

$\displaystyle\widehat{m}_{z}$ $\displaystyle=$ $\displaystyle
48m_{3/2}kN^{-1/2}\left\langle{\omega^{3/2}{C_{\omega}^{+}}/{C_{\omega}^{-}}}\right\rangle;$
(16a) $\displaystyle\widehat{m}_{{\theta}}$ $\displaystyle=$ $\displaystyle
2m_{3/2}\left(1-(3/N)\left\langle{C_{\omega}^{+}/C_{\omega}^{-}}\right\rangle^{2}\right)^{1/2}.$
(16b)

The restriction on $N$ stems from the requirement
$\widehat{m}_{{\theta}}^{2}>0$. The expressions above contain the
$\widetilde{G}$ mass determined as follows

$m_{3/2}=\left\langle{e^{G}}\right\rangle=\left\langle{m\omega^{\sqrt{3N}}C_{\omega}^{-}}\right\rangle.$
(17)

Since $m$ is included in the last formula, all the masses above share
approximately the same size, as can be confirmed from the sample values
accumulated in the last arrow of Table 1.

An interpretation of DE can be achieved by demanding

$\left\langle{V_{\Lambda}}\right\rangle=\Omega_{\Lambda}\rho_{\rm c0}=7.2\cdot
10^{-121}m_{\rm P}^{4},$ (18)

where $\Omega_{\Lambda}=0.6889$ and $\rho_{\rm c0}=2.4\cdot
10^{-120}h^{2}m_{\rm P}^{4}$ with $h=0.6732$ plcp  is the density parameter of
DE and the current critical energy density of the universe. As shown in Table
1, the required value of $C_{\Lambda}$ signals a serious fine tuning whose the
origin remains elusive within our proposal. However, this value does not
influence the remaining sectors of the model and can be selected in the
definition of $W_{\Lambda}$.

Table 1: Sample-Point Parameters With Units Reinstalled
(Values in curly brackets are obtained by the analytic expressions) Model
Parameters
---
$m/m_{\rm P}$ | $C_{\Lambda}/10^{-108}$ | $N$ | $k/0.1$ | ${\rm v}/m_{\rm P}$
$5.6\cdot 10^{-7}$ | $2.3$ | $12$ | $4.0167291$ | $0.5$
$n_{+}$ | $n_{-}$ | $|A|/{\mbox{\rm EeV}}$ | $|B|/{\mbox{\rm EeV}}$ | $\widehat{\mu}/{\mbox{\rm EeV}}$
$9$ | $3$ | $1000$ | $1200$ | $81$
Inflection Point Localization
$k_{0}/0.1$ | $z_{0}/0.1m_{\rm P}$ | $\delta k/10^{-6}$ | $\delta z_{\star}/10^{-4}m_{\rm P}$
$4.0166971$ | $7.07433$ | $3.20232$ |  $-1.5$ {$-1.1$}
Expansion Parameters
$v_{0}/(mm_{\rm P})^{2}$ | $v_{1}/(mm_{\rm P})^{2}$ | $v_{2}/(mm_{\rm P})^{2}$ | $v_{3}/(mm_{\rm P})^{2}$ | $J_{0}$
$3.9\cdot 10^{-3}$ | $1.5\cdot 10^{-6}$ | $-2.1\cdot 10^{-6}$ | $2.2$ | $5.4$
Inflation Results
$n_{\rm s}$ | $r/10^{-8}$ | $-a_{\rm s}/10^{-3}$ | $10^{5}A_{\rm s}$ | ${N_{\star}}$
$0.966$ {$0.97$} |  $4.8$ {$3.9$} |  $3.3$ {$3.2$} |  $4.59$ {$4.27$} |  $46.5$ {$45$}
$V_{\rm I\star}^{1/4}/{\mbox{\rm EeV}}$ | $H_{\rm I\star}/{\mbox{\rm EeV}}$ | $\delta z_{\rm f}/10^{-2}m_{\rm P}$ | $m_{{\theta}\rm I\star}/H_{\rm I\star}$
$4.6\cdot 10^{5}$ | $49.5$ |  $-1.16$ {$-0.87$} | $5.1$
Post-Inflation Results
$m_{z}/{\mbox{\rm EeV}}$ | $m_{\theta}/{\mbox{\rm EeV}}$ | $m_{3/2}/{\mbox{\rm EeV}}$ | ${\Gamma}_{3/2}/{\rm keV}$ | $T_{\rm rh}/~{}{\mbox{\rm PeV}}$
$319$ | $281$ | $162$ | $85$ | $4.9$

##  IV SUSY Breaking

The SUSY breaking occurred at the vacuum in Eq. (15) can be transmitted to the
visible world if we specify a reference model, such as MSSM which includes the
chiral superfields

$\displaystyle\Phi_{\alpha}={Q},{L},{d}^{c},{u}^{c},{e}^{c},{H_{d}}~{}~{}\mbox{and}~{}~{}{H_{u}},$

where we suppress the generation indices. We assume the following structure –
cf. Ref. soft  – for the total superpotential, $W_{\rm\Lambda O}$, of the
theory

$W_{\rm\Lambda O}=W_{\Lambda}(Z)+W_{\rm MSSM}\left(\Phi_{\alpha}\right),$ (19)

where $W_{\rm MSSM}$ has the well-known form written in short as

$W_{\rm
MSSM}=h_{{\alpha}{\beta}\gamma}\Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}/6+\mu{H_{u}}{H_{d}}.$
(20)

Here we denote the three non-vanishing Yukawa coupling constants as
$h_{{\alpha}{\beta}\gamma}=h_{D},h_{U}$ and $h_{E}$ for
$({\alpha},{\beta},\gamma)=(Q,{H_{d}},d^{c}),(Q,{H_{u}},u^{c})$ and
$(L,{H_{d}},e^{c})$ respectively. Also, $\mu$ is a free parameter which is
expected to be close to $m_{3/2}$. On the other hand, we consider two simple
variants of the total $K$ of the theory, $K_{\rm\Lambda O}$, ensuring SSB
parameters for $\Phi_{\alpha}$:

$\displaystyle K_{1\rm\Lambda O}$
$\displaystyle=K(Z)+\mbox{$\sum_{\alpha}$}|\Phi_{\alpha}|^{2};~{}~{}~{}$ (21a)
$\displaystyle K_{2\rm\Lambda O}$ $\displaystyle=K(Z)+N_{\rm
O}\ln\left(1+\mbox{$\sum_{\alpha}$}|\Phi_{\alpha}|^{2}/N_{\rm O}\right),$
(21b)

where $N_{\rm O}$ may remain unspecified. Note that if we expand $K_{\rm
2\Lambda O}$ for low $\Phi_{\alpha}$ values, the result coincidences with
$K_{\rm 1\Lambda O}$.

Adapting the general formulae of Ref. soft , we find universal (i.e.,
$\widetilde{m}_{\alpha}=\widetilde{m}$ and $A_{{\alpha}{\beta}\gamma}=A$) SSB
terms in the effective low energy potential which can be written as

$V_{\rm
SSB}=\widetilde{m}^{2}|\Phi_{\alpha}|^{2}+\left(\frac{1}{6}A\widehat{h}_{{\alpha}{\beta}\gamma}\Phi_{\alpha}\Phi_{\beta}\Phi_{\gamma}+B\widehat{\mu}H_{u}H_{d}+{\rm
h.c.}\right)$ (22)

where the normalized (hatted) parameters are defined as

$(\widehat{h}_{{\alpha}{\beta}\gamma},\widehat{\mu})=\left\langle{\omega}\right\rangle^{-N/2}(h_{{\alpha}{\beta}\gamma},\mu),$
(23)

whereas the SSB parameters are found to be

$\widetilde{m}=m_{3/2},~{}~{}|A|=\sqrt{3N}m_{3/2}~{}~{}\mbox{and}~{}~{}|B|=(1+\sqrt{3N})m_{3/2}.$
(24)

Values for these parameters are displayed in Table 1 for
$\widehat{\mu}=m_{3/2}/2$. We see that $|A|>m_{3/2}$ and $|B|>m_{3/2}$ due to
large $N$ adopted there. However, these parameters have very suppressed impact
on the SUSY mass spectra.

Similar values for the gauginos of MSSM are also expected. E.g., we may select
the gauge-kinetic function soft  as

$f_{\rm a}=\lambda_{\rm a}Z\,$ (25)

where $\lambda_{\rm a}$ is a free parameter and ${\rm a}=1,2,3$ runs over the
factors of the gauge group of MSSM, $U(1)_{Y}$, $SU(2)_{\rm L}$ and
$SU(3)_{\rm c}$ with gauge coupling constants $g_{\rm a}$ respectively. In a
such case, we find soft

$|M_{\rm a}|=\sqrt{3/N}\left\langle{\omega/z}\right\rangle m_{3/2},$ (26)

which is obviously of the order of $m_{3/2}$.

Scenarios with large $\widetilde{m}$, although not directly accessible at the
LHC, can be probed via the measured value of the Higgs boson mass. Indeed,
updated analysis requires lhc ; strumia

$3\lesssim\widetilde{m}/{\mbox{\rm EeV}}\lesssim 300,$ (27)

for degenerate sparticle spectrum, low $\tan\beta$ values and minimal stop
mixing. From Eq. (24) and the values in Table 1 we conclude that our setting
is comfortably compatible with the requirement above.

##  V Inflection-Point Inflation

In this section we analyze the inflationary sector of our model. In Sec. A we
outline our method for the determination of the inflection point of the
potential and in Sec. B we describe our semi-analytic approach to inflationary
dynamics. Then, in Sec. C, we discuss the reheat process and in Sec. D the
parameters of our model are confronted with observations.

Fig. 2: Dimensionless inflationary potential $V_{\rm I}/m^{2}m_{\rm P}^{2}$ as
a function of $z$ for $k=0.40167291$ (black line) or $k=0.2$ (dot-dashed line)
or $k=0.6$ (dashed line) and the remaining inputs in Table 1. The dotted line
is obtained by Eq. (32). The values of $z_{\star}$, $z_{\rm f}$ and $z_{0}$
(for the first case) are also indicated.

###  A Inflection-Point Conditions

The inflationary potential $V_{\rm I}=V_{\rm I}(z)$ is obtained from
$V_{\Lambda}(Z)$ in Eq. (10), replacing ${\theta}=0$ and $C_{\Lambda}\simeq
0$. The resulting one-dimensional $V_{\rm I}(z)$ is plotted by a solid line as
a function of $z$ in Fig. 2 for the model parameters shown in Table 1. Thanks
to the interplay of the two opposing contributions in the parenthesis in Eq.
(10), which (almost) cancel each other out, a step is generated for $z<1$ –
note that for $z\simeq 1$, $V_{\rm I}$ approaches a singularity due to the
presence of $\Omega^{-N}$ in Eq. (10). The emergence of the inflationary
plateau, though, crucially depends on $k$. Indeed, for $k=0.2$ and $k=0.6$ and
keeping the residual inputs in Table 1 we obtain the dot-dashed and dashed
lines respectively in Fig. 2 where $V_{\rm I}$ does not develop inflection
point.

Fig. 3: Location of the inflection point in the $z_{0}-k_{0}$ plane for
various $N$’s indicated in the plot. Shown is also the variation of ${\rm v}$
in grey along the lines.

To localize the position of the inflection point, we impose

$V_{\rm I}^{\prime}(z)=V_{\rm I}^{\prime\prime}(z)=0,$ (28)

where prime stands for derivation w.r.t $z$. These conditions can be
translated as follows

$\displaystyle\sqrt{3N}\Omega+N(\Omega+\omega\Omega^{\prime})=2\Omega
U\frac{U+\omega U^{\prime}}{U^{2}-12\omega^{2}};$ (29a) $\displaystyle
N\Omega\omega\left(4\Omega^{\prime}(n_{+}-1)U^{2}+\omega\left(12\omega\left(\sqrt{3N}+N^{2}\right)+2UU^{\prime}\right)\right)$
$\displaystyle+\Omega^{2}\Big{(}U^{2}N_{1}+\omega\big{(}4(1-n_{+})U^{2\prime}-\omega\left(12N_{2}-(U^{2})^{\prime\prime})\right)\big{)}\Big{)}$
$\displaystyle=(12\omega^{2}-U^{2})\left(\Omega^{\prime\prime}\omega+N(1+N)\Omega^{\prime}\omega^{2}\right),$
(29b)

where we define the quantities

$\displaystyle N_{1}$ $\displaystyle=6-5\sqrt{3N}-2N+\sqrt{3N^{3}}+N^{2};$
(30) $\displaystyle N_{2}$ $\displaystyle=2N-\sqrt{3N}+2\sqrt{3N^{3}}+N^{2}.$

Note that the conditions above are independent from the parameters $m$ and
$C_{\Lambda}$ of $W_{\Lambda}$. Solving the conditions above w.r.t $k$ and
$z$, for every selected ${\rm v}$ and $N$, we can specify $k=k_{0}$, for which
we obtain an inflection point, and its position at $z=z_{0}$. The output of
this procedure is given in Fig. 3 where we plot $(k_{0},z_{0})$ for $N=4,10$
and $30$ (dashed, solid and dot-dashed line respectively). Along each line we
show the variation of ${\rm v}$ in grey. We observe that the maximal allowed
${\rm v}$ decrease as $N$ increases whereas for ${\rm v}<0.01$ the
$(k_{0},z_{0})$ values remain almost constant. We also remark that $z_{0}>{\rm
v}$ and it remains subplanckian whereas the required $k_{0}<1$ is quite
natural, although it has to be determined using a precision up to six digits
to be reliable.

###  B Inflation Analysis

Once we determine the $(k_{0},z_{0})$ values, we can investigate the
realization of IPI which is delimited by the condition review

${\footnotesize\sf max}\\{\epsilon(\widehat{z}),|\eta(\widehat{z})|\\}\leq 1,$
(31a) where the slow-roll parameters $\epsilon$ and $\eta$ read
$\epsilon=\left({V_{{\rm I},\widehat{z}}/\sqrt{2}V_{{\rm
I}}}\right)^{2}~{}~{}\mbox{and}~{}~{}\eta={V_{{\rm
I},\widehat{z}\widehat{z}}/V_{\rm I}}$ (31b)

and can be derived by employing $V_{\rm I}$ in Eq. (10) for ${\theta}=0$ and
$J$ in Eq. (13), without express explicitly $V_{\rm I}$ in terms of
$\widehat{z}$. Due to the complicate form of $V_{\rm I}$, the analytic
approach to the inflationary dynamics is not doable. Some progress can be
made, if we use as input for our analytic treatment the numerical expansions
of $V_{\rm I}$ and $J$ about $z=z_{0}$,

$V_{\rm I}\simeq v_{0}+v_{1}\delta z+v_{2}\delta z^{2}+v_{3}\delta
z^{3}~{}~{}\mbox{and}~{}~{}J\simeq J_{0},$ (32)

where $\delta z=z-z_{0}$, $v_{0}=V_{\rm I}(z_{0})$ and $J_{0}=J(z_{0})$. Since
the observationally relevant part of IPI takes place quite close to $z_{0}$,
the approximation above is quite accurate. This is confirmed in Fig. 2 where
the dotted line, obtained by Eq. (32), is plotted against the exact result of
$V_{\rm I}$ for the inputs in Table 1. The relevant expansion coefficients are
listed there. Thanks to the conditions in Eq. (28), $v_{1}=V_{\rm
I}^{\prime}(z_{0})$ and $v_{2}=V_{\rm I}^{\prime\prime}(z_{0})/2$ are quite
suppressed compared to $v_{0}$ and $v_{3}$ and so we may neglect terms with
$v_{1}^{2},v_{2}^{2}$ and $v_{1}v_{2}$ below. Namely, inserting Eq. (32) into
Eq. (31b) we arrive at the following results

$\sqrt{\epsilon}\simeq\frac{v_{1}+\delta z(2v_{2}+3\delta
zv_{3})}{\sqrt{2}J_{0}v_{0}}~{}~{}\mbox{and}~{}~{}\eta\simeq\frac{2(v_{2}+3\delta
zv_{3})}{J_{0}^{2}v_{0}},$ (33)

from which we can verify that Eq. (31a) is saturated for $\delta z=\delta
z_{\rm f}$, found from the condition

$\eta\left(\delta z_{\rm f}\right)\simeq 1~{}~{}\Rightarrow~{}~{}\delta z_{\rm
f}\simeq-(J_{0}^{2}v_{0}+2v_{2})/6v_{3}.$ (34)

Given that $J_{0}^{2}v_{0}\gg v_{2}$, we expect $\delta z_{\rm f}<0$ or
$z_{\rm f}<z_{\star}$.

The number of e-foldings ${N_{\star}}$ that the scale $k_{\star}=0.05/{\rm
Mpc}$ experiences during IPI and the amplitude $A_{\rm s}$ of the power
spectrum of the curvature perturbations generated by $z$ can be computed using
the standard formulae

${N_{\star}}=\int_{\widehat{z}_{\rm
f}}^{\widehat{z}_{\star}}d\widehat{z}\frac{V_{\rm I}}{V_{{\rm
I},\widehat{z}}}~{}~{}~{}\mbox{and}~{}~{}~{}A_{\rm
s}=\left.\frac{1}{12\pi^{2}}\frac{V_{\rm I}^{3}}{V^{2}_{{\rm
I},\widehat{z}}}\right|_{\widehat{z}=\widehat{z}_{\star}},$ (35)

where $z_{\star}~{}[\widehat{z}_{\star}]$ is the value of $z~{}[\widehat{z}]$
when $k_{\star}$ crosses the inflationary horizon. From the leftmost relation
we find

${N_{\star}}=(f_{N\star}-f_{N{\rm
f}})/p_{N}~{}~{}\mbox{where}~{}~{}p_{N}={\sqrt{3v_{1}v_{3}}}/{J_{0}^{2}v_{0}}$
(36)

and $f_{N\star}=f_{N}(\delta z_{\star})$ and $f_{N{\rm f}}=f_{N}(\delta z_{\rm
f})$ with

$f_{N}(z)=\arctan\left({v_{2}+3zv_{3}}/{\sqrt{3v_{1}v_{3}}}\right).$ (37)

Since $\delta z_{\rm f}$ turns out to be close to $\delta z_{\star}$ – as
depicted in Fig. 2 – both contributions in Eq. (36) are important. Solving it
w.r.t $\delta z_{\star}$ we obtain

$\delta
z_{\star}\simeq-\frac{v_{2}}{3v_{3}}+\sqrt{\frac{v_{1}}{3v_{3}}}\tan\left(\frac{\sqrt{3}{N_{\star}}}{J_{0}^{2}v_{0}}+f_{N{\rm
f}}\right),$ (38)

from which we can deduce that $z_{\star}<z_{0}$ – see inset of Fig. 2.
Plugging it into the rightmost equation in Eq. (35), we obtain

$A_{\rm s}\simeq\frac{J_{0}v_{0}^{3/2}}{2\sqrt{3}\pi
v_{1}}\cos^{2}\left(p_{N}{N_{\star}}+f_{N{\rm f}}\right)$ (39)

The remaining inflationary observables are found from

$\displaystyle n_{\rm s}=\>1-6\epsilon_{\star}\ +\
2\eta_{\star},~{}~{}r=16\epsilon_{\star},$ (40a) $\displaystyle a_{\rm
s}=\>2\left(4\eta_{\star}^{2}-(n_{\rm s}-1)^{2}\right)/3-2\xi_{\star}.$ (40b)

Here and hereafter the variables with subscript $\star$ are evaluated at
$z=z_{\star}$ and $\xi={V_{{\rm I},\widehat{}z}V_{{\rm
I},\widehat{}z\widehat{}z\widehat{}z}/V^{2}_{\rm I}}$. Inserting $\delta
z_{\star}$ from Eq. (38) into Eq. (31b) and then into equations above we
obtain

$\displaystyle n_{\rm s}$ $\displaystyle\simeq$ $\displaystyle
1+4p_{N}\tan\left(p_{N}{N_{\star}}+f_{N{\rm f}}\right),$ (41a) $\displaystyle
r$ $\displaystyle\simeq$ $\displaystyle
8v_{1}^{2}\cos^{-4}\left(p_{N}{N_{\star}}+f_{N{\rm
f}}\right)/J_{0}^{2}v_{0}^{2},$ (41b) $\displaystyle a_{\rm s}$
$\displaystyle\simeq$
$\displaystyle-4p_{N}\cos^{-2}\left(p_{N}{N_{\star}}+f_{N{\rm f}}\right).$
(41c)

For the inputs of Table 1, the results of our semianalytic approach are
displayed in curly brackets and compared with those obtained using the pure
numerical program. The proximity of both results is certainly impressive.

From Table 1 we may also infer that the semiclassical approximation, used in
our analysis, is perfectly valid since $V_{\rm I\star}^{1/4}\ll m_{\rm P}$.
Moreover, the ${\theta}=0$ direction is well stabilized and does not
contribute to the curvature perturbation, since for the relevant effective
mass $m_{{\theta}\rm I}$ we find $m^{2}_{{\theta}\rm I}>0$ for $N>3$ and
$m_{{\theta}\rm I\star}/H_{\rm I\star}>1$ where $H_{\rm I}=(V_{\rm
I}/3)^{1/2}$. We also checked that the one-loop radiative corrections,
$\Delta\widehat{V}_{\rm HI}$, to $V_{\rm I}$ induced by $m_{{\theta}\rm I}$
let intact our inflationary outputs, provided that we take for the
renormalization-group mass scale $Q=m_{{\theta}\rm I\star}$. Since $Q$ is
close to $V_{\rm I\star}^{1/4}$ we do not expect sizable running of the
quantities measured at $Q$.

###  C Sgoldstino Decay

Soon after the end of IPI, the (canonically normalized) sgoldstino

$\widehat{\updelta
z}=\left\langle{J}\right\rangle{\updelta}z~{}~{}\mbox{with}~{}~{}{\updelta}z=z-{\rm
v}~{}~{}\mbox{and}~{}~{}\left\langle{J}\right\rangle=\sqrt{\frac{N}{2}}\frac{1}{\left\langle{\omega}\right\rangle}$
(42)

settles into a phase of damped oscillations abound the minimum in Eq. (15)
reheating the universe at a temperature rh

$T_{\rm rh}=\left({72/5\pi^{2}g_{\rm
rh*}}\right)^{1/4}{\Gamma}_{{\updelta}z}^{1/2}m_{\rm P}^{1/2},$ (43a) where
$g_{\rm rh*}=106.75$ counts the effective number of the relativistic degrees
of freedom at $T_{\rm rh}$. The total decay width, ${\Gamma}_{{\updelta}z}$,
of $\widehat{\updelta z}$ predominantly includes the contributions
${\Gamma}_{{\updelta}z}\simeq{\Gamma}_{3/2}+{\Gamma}_{{\theta}}+{\Gamma}_{\tilde{h}},$
(43b)

where the individual decay widths – which stem from SUGRA-induced interactions
full ; baerh ; antrh ; nsrh  – are found to be

$({\Gamma}_{3/2},{\Gamma}_{{\theta}},{\Gamma}_{\tilde{h}})\simeq\left(\frac{\left\langle{\omega}\right\rangle^{-\sqrt{3N}}m_{z}^{5}}{96\pi
m^{2}m_{\rm P}^{2}},~{}~{}\frac{m_{z}^{3}}{16N\pi{\rm v}m_{\rm
P}},\frac{N\widehat{\mu}^{2}m_{z}}{16\pi m_{\rm P}^{2}}\right).$ (44)

They express decay of $\widehat{\updelta z}$ into gravitinos, pseudo-
goldstinos and higgsinos via the $\mu$ term respectively. Thanks to the
appearance of $N$ in ${\Gamma}_{\tilde{h}}$, it is rather enhanced for large
$N$’s.

###  D Parameter Space

Fig. 4: Allowed curves in the $(-\delta z_{\star})-\delta k$ plane for $n_{\rm
s}\simeq 0.966$ and various $N$’s indicated in the plot. Shown is also the
variation of ${\rm v}$ in grey along the lines.

In order to delineate the available parameter space of the model, we confront
the quantities in Eq. (35) with the observational requirements plcp

${N_{\star}}\simeq 61+\ln\left(\pi v_{0}T_{\rm
rh}^{2}\right)^{1/6}~{}~{}\mbox{and}~{}~{}A_{\rm s}\simeq 2.105\cdot 10^{-9}$
(45)

where we assume that IPI is followed in turn by a oscillatory phase, with mean
equation-of-state parameter $w_{\rm rh}\simeq 0$, radiation and matter
domination. The remaining observables must be in agreement with the fitting of
the Planck TT, TE, EE+lowE+lensing, BK15 bk15  and BAO data plin  with the
$\Lambda$CDM$+r+a_{\rm s}$ model which requires

$n_{\rm s}=0.9658\pm 0.008~{}~{}\mbox{and}~{}~{}a_{\rm s}=-0.0066\pm 0.014$
(46)

at 95$\%$ confidence level. Recall that the upper bound on $r$ is irrelevant
in our case since $r$ is negligible – see Sec. B.

In our numerical program for any selected $N$ and ${\rm v}$ – see Eq. (10) –
we compute $(k_{0},z_{0})$ solving numerically Eq. (28). Then enforcing Eq.
(45) we can restrict $\delta k=k-k_{0}$ and $m$ whereas the leftmost
observable in Eq. (46) determines $\delta z_{\star}$. From Eq. (40b) the
model’s predictions regarding $a_{\rm s}$ and $r$ can be computed. Increasing
$\delta k$ allows us to increase the slope of the plateau around $z_{0}$
decreasing, thereby, ${N_{\star}}$. Note that the $\delta k$ does not appear
in the formulae in Sec. B, since the relevant information is encoded in Eq.
(32).

The outputs of our numerical investigation can be presented in the $(-\delta
z_{\star})-\delta k$ plane as in Figs. 4 and 5. In the first one we fix $N$ to
three representative values $4,10$ and $30$ and display the allowed curves
(dot-dashed, solid and dashed lines respectively) taking the central $n_{\rm
s}$ value in Eq. (46). The variation of ${\rm v}$ along each line is displayed
in gray. On the other hand, in Fig. 5 we set ${\rm v}=0.5$ and identify the
allowed (shaded) region allowing $n_{\rm s}$ to vary in the margin of Eq.
(46). The variation of $N$ is shown along each line. The allowed region is
bounded by (i) the solid black line, which corresponds to the upper bound in
Eq. (27), (ii) the dashed black line which originates from the lower bound on
$N$ derived in Sec. III and (iii) the dot-dashed and dashed gray lines along
which the lower and upper bounds on $n_{\rm s}$ in Eq. (46) are saturated
respectively. We remark that increasing $|\delta z_{\star}|$, decreases
$n_{\rm s}$ with fixed $\delta k$. Fixing $n_{\rm s}\simeq 0.966$, we obtain
the gray solid line, along which we obtain

$2.3\lesssim\frac{m_{z}}{100~{}{\mbox{\rm EeV}}}\lesssim
4.4,~{}~{}\mbox{and}~{}~{}8.9\lesssim\frac{m_{\theta}}{10~{}{\mbox{\rm
EeV}}}\lesssim 59,$ (47)

where the lower bound on $m_{\theta}$ coincides with that on $m_{3/2}$. For
$N<10$, ${\Gamma}_{{\updelta}z}\simeq{\Gamma}_{3/2}+{\Gamma}_{\tilde{h}}$
whereas for larger $N$’s ${\Gamma}_{{\updelta}z}\simeq{\Gamma}_{\tilde{h}}$
and so Eq. (43a) yields $T_{\rm rh}\simeq(4-20)~{}{\mbox{\rm PeV}}$ resulting
to ${N_{\star}}\simeq(45.5-46.7)$. The obtained $a_{\rm
s}\simeq-(3.1-3.2)\cdot 10^{-3}$ might be detectable in future asdrees . From
both figures we deduce that the achievement of observationally acceptable IPI
requires a tuning of the order of $10^{-6}$ which is somehow ameliorated as
$N$ increases. This tuning though is milder than that needed within the
conventional MSSM aterm .

Fig. 5: Allowed (shaded) region in the $(-\delta z_{\star})-\delta k$ plane
for ${\rm v}=0.5$. The constraint fulfilled along each line and the variation
of $N$ are also shown in black and grey respectively.

##  VI Conclusions

Taking advantage from the self-stabilized no-scale models of SUSY breaking,
established in Ref. ns89 ; de , we proposed a variant which incorporates IPI –
i.e., inflection-point inflation – realized by the sgoldstino. The
inflationary model results to an adjustable $n_{\rm s}$, a small $r$ and a
sizable $a_{\rm s}$ of the order of $-10^{-3}$. Linking our model to MSSM we
showed that SUSY may be broken at a dS vacuum, providing the correct DE
density parameter – at the cost of a fine tuned parameter – and a SUSY mass
scale $\widetilde{m}\sim 100~{}{\mbox{\rm EeV}}$ which is consistent with the
Higgs boson mass measured at LHC. Needless to say, the stability of the
electroweak vacuum up to the Planck scale is automatically assured within our
framework.

It would be interesting to investigate if the intermediate-scale lightest
neutralino with mass $M_{1}$ in the interval $T_{\rm rh}<M_{1}<T_{\rm max}$ is
a good cold dark matter candidate. Here $T_{\rm max}$ is the maximal
temperature during reheating and may be as high as $10^{3}T_{\rm rh}$ kolb .
Also baryogenesis via non-thermal leptogensis ntlepto  may be activated even
without direct coupling of the sgoldstino to right-handed neutrinos. Our
model, most probably, does not belong to the string swampland vafa  but it is
amenable to modifications sevilla  which may render it more friendly with the
string ultraviolet completions.

#### Acknowledgment

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: 2251).

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