# PeV-Scale SUSY and Cosmic Strings
from F-term Hybrid Inflation

C. Pallis
School of Civil Engineering
Faculty of Engineering
Aristotle University of Thessaloniki
GR-541 24 Thessaloniki  GREECE
<EMAIL_ADDRESS>
###### Abstract

We consider F-term hybrid inflation (FHI) and SUSY breaking in the context of
a $B-L$ extension of MSSM which largely respects a global $U(1)$ $R$ symmetry.
The hidden sector Kaehler manifold enjoys an enhanced $SU(1,1)/U(1)$ symmetry
with the scalar curvature determined by the achievement of a SUSY-breaking de
Sitter vacuum without ugly tuning. FHI turns out to be consistent with data,
provided that the magnitude of the emergent soft tadpole term is confined in
the range $(1.2-100)$ TeV, and it is accompanied with the production of $B-L$
cosmic strings. If these are metastable they interpret the present
observations from PTA experiments on the stochastic background of
gravitational waves with dimensionless tension ${G\mu_{\rm
cs}}\simeq(1-9.2)\cdot 10^{-8}$. The $\mu$ parameter of MSSM arises by
appropriately adapting the Giudice-Masiero mechanism and facilitates the out-
of-equilibrium decay of the $R$ saxion at a reheat temperature lower than
about $71$ GeV. Due to the prolonged matter dominated era the gravitational
wave signal is suppressed at high frequencies. The SUSY mass scale turns out
to lie in the PeV region.

In memory of Prof. G. Lazarides

Keywords: Cosmology, Inflation, Supersymmetric Models
PACS codes: 98.80.Cq, 12.60.Jv, 95.30.Cq, 95.30.Sf

##  1 Introduction

_Supersymmetric_ (SUSY) hybrid inflation [1] based on F terms, called for
short henceforth _F-term hybrid inflation_ (FHI) [2] is undoubtedly a well-
motivated inflationary model – for reviews see Ref. [3]. The most notable
reasons which support our statement above are the following:

  * •

FHI is tied to a renormalizable superpotential uniquely determined by a gauge
and a global $U(1)$ $R$ symmetries;

  * •

FHI does not require fine-tuned parameters and transplanckian inflaton values;

  * •

FHI can be reconciled with the Planck data [4] – fitted to the standard power-
law cosmological model with _Cold Dark Matter_ (CDM) and a cosmological
constant ($\Lambda$CDM) – if we take properly into account not only _radiative
corrections_ (RCs) but also corrections originated by _supergravity_ (SUGRA)
[5, 6, 8, 7] as well as soft SUSY-breaking terms [10, 11, 9, 13, 12].

  * •

FHI can be naturally followed by a _Grand Unified Theory_ (GUT) phase
transition which may lead to the production of cosmological defects, if these
are predicted by the symmetry breaking scheme. In the large majority of GUT
breaking chains the formation [14] of _cosmic strings_ (CSs) cannot be
avoided.

Although the last feature above is often used to criticize the powerfulness of
the embedding of FHI in several GUTs – see e.g. Ref. [15] –, it recently
appears as an interesting ingredient in the cosmological model building. This
is, because the announced data from several _pulsar timing array_ (PTA)
experiments [16] – most notably the _NANOGrav 15-years results_ (NANOGrav15)
[17] – provide strong support for the discovery of a _gravitational wave_ (GW)
background around the nanohertz frequencies.

Given that the interpretation of this signal in terms of the merging of
supermassive black hole binaries is somewhat disfavored [18], its attribution
to gravitational radiation emitted by topologically unstable superheavy CSs –
which may arise after the end of FHI – attracts a fair amount of attention
[22, 21, 20, 23, 19, 24]. In particular, the observations can be interpreted
if the CSs are meta- [25] or quasi-stable [26]. Both types of topologically
unstable CSs arise from the symmetry breaking
$\mathbb{G}\rightarrow\mathbb{H}\times U(1)$, which produces monopoles. The
subsequent $U(1)$ breaking yields CSs which connect monopoles with
antimonopoles. We here assume that these last ones are inflated away, but can
appear on metastable CSs via quantum pair creation. Therefore, metastable CSs
can be easily achieved if a $U(1)$ symmetry is embedded in a gauge group with
higher rank such as the Pati-Salam [22], the flipped $SU(5)$ [21] or $SO(10)$
[20]. For this reason, in this work we focus on FHI realized in a $B-L$
extension of _Minimal SUSY Standard Model_ (MSSM) – cf. Ref. [27, 22] – which
dilutes possibly preexisting monopoles and is naturally accompanied by the
production of a network of CSs. On the other hand, we do not specify the
mechanism of the monopole production and the metastability of CSs, as done,
e.g., in Ref. [19, 24].

We adopt the minimal possible framework [10, 11] which supports
observationally acceptable FHI. It employs minimal Kähler potential for the
inflaton field, RCs and soft SUSY-breaking terms. From the last ones, the
tadpole term plays a crucial role in establishing compatibility with data. Its
magnitude can be motivated by intertwining [28] the _inflationary sector_ (IS)
with a _hidden sector_ (HS) as that introduced in Ref. [29] which exhibits a
mildly violated $R$ symmetry similar to that used for FHI [1] – for earlier
attempts see Refs. [31, 30, 33, 32, 34]. The consequences of the
interconnection of the two sectors above are the following:

  * •

The $R$ charge $2/\nu$ of the goldstino superfield – which is related to the
geometry of the HS [29] – is constrained to values with $0<\nu<1$.

  * •

The SUSY breaking is achieved not only in a Minkowski vacuum but also in a _de
Sitter_ (dS) one which allows us to control the notorious _Dark Energy_ (DE)
problem by mildly tuning a single superpotential parameter to a value of order
$10^{-12}$.

  * •

The sgoldstino is stabilized [31, 33, 32, 34] to low values during FHI. This
fact together with the selection of a minimal Kähler potential for the
inflaton assists us to resolve the $\eta$-problem.

  * •

The solution to the $\mu$ problem [35] of MSSM is achieved by suitably
applying [29] the Giudice-Masiero mechanism [36, 37]. Contrary to similar
attempts [39, 38, 30] the $\mu$ term here plays no role during FHI.

  * •

The domination of the energy density of the universe by the sgoldstino (or $R$
saxion) condensate which decays [41, 40, 44, 43, 42] before the onset of the
_Big Bang Nucleosynthesis_ (BBN) at cosmic temperature $(2-4)~{}{\mbox{\rm
MeV}}$ [45] thanks the aforementioned $\mu$ term. As a byproduct, we obtain a
reduction [46, 47] of the spectrum of GWs at high frequencies ($f>0.1~{}{\rm
Hz}$) which assists us to avoid any conflict with the third run advanced
_LIGO-VIRGO-KAGRA_ (LVK) data [48].

  * •

The SUSY mass scale $\widetilde{m}$ is predicted to be close to the PeV scale
[49]. It fits well with the Higgs boson mass, discovered at LHC, as it is
estimated [50] within high-scale SUSY if we assume a relatively low
$\tan\beta$ and stop mixing. Note that the connection of inflation with SUSY
breaking has been extensively discussed in literature [51] the last years.

In this feature paper we review further the model introduced in Ref. [28]
focusing exclusively on its implementation within a version of MSSM endowed by
a ${B-L}$ Higgs sector. We present the complete particle content of the model,
paying special attention on the computation of the GWs emerging from the CS
decay, under the assumption that those are metastable. We also explain the
generation of neutrino masses taking into account SUGRA contributions from
Ref. [52].

The remaining manuscript is built up as follows: In Sec. 2 we introduce our
framework. Following, we revise the salient features of our model as regards
its vacuum in Sec. 3 and the inflationary era in Sec. 4. Then, we study the
reheating process in Sec. 5 and the production of the GWs from the CSs in Sec.
6. Our predictions for the SUSY-mass scale are exposed in Sec. 7. Our
conclusions are summarized in Sec. 8. Details for the derivation of neutrino
masses are given in Appendix 8.

##  2 Model Set-up

We focus on an extension of MSSM invariant under the gauge group
$\mathbb{G}_{B-L}=\mathbb{G}_{\rm SM}\times U(1)_{B-L}$, where
$\mathbb{G}_{\rm SM}$ is the Standard Model gauge group. The charge
assignments under these symmetries of the various matter and Higgs superfields
are listed in Table 1. Namely, the $i$th generation $SU(2)_{\rm L}$ doublet
left-handed quark and lepton superfields are denoted by $Q_{i}$ and $L_{i}$
respectively, whereas the $SU(2)_{\rm L}$ singlet antiquark [antilepton]
superfields by $u^{c}_{i}$ and ${d_{i}}^{c}$ [$e^{c}_{i}$ and $N^{c}_{i}$]
respectively. The electroweak Higgs superfields which couple to the up [down]
quark superfields are denoted by ${H_{u}}$ [${H_{d}}$]. Besides the MSSM
particle content, the model is augmented by seven superfields: a gauge singlet
$S$, three $N^{c}_{i}$’s, a pair of Higgs superfields $\Phi$ and $\bar{\Phi}$
which break $U(1)_{B-L}$ and the goldstino superfield $Z$. In addition to the
local symmetry, the model possesses also the baryon and lepton number
symmetries and an $R$ symmetry $U(1)_{R}$. The latter plays a crucial role to
the construction of the superpotential (see Sec. 2.1) and the Kähler potential
(see Sec. 2.2).

Superfields | Representations | Global Symmetries
---|---|---
| under $\mathbb{G}_{B-L}$ |  $R$ |  $B$ | $L$
Matter Superfields
$e^{c}_{i}$ | $({\bf 1,1},1,1)$ | $0$ | $0$ | $-1$
$N^{c}_{i}$ | $({\bf 1,1},0,1)$ | $0$ | $0$ | $-1$
$L_{i}$ | $({\bf 1,2},-1/2,-1)$ | $0$ | $0$ | $1$
$u^{c}_{i}$ | $({\bf 3,1},-2/3,-1/3)$ | $0$ | $-1/3$ | $0$
$d^{c}_{i}$ | $({\bf 3,1},1/3,-1/3)$ | $0$ | $-1/3$ | $0$
$Q_{i}$ | $({\bf\bar{3},2},1/6,1/3)$ | $0$ | $1/3$ | $0$
Higgs Superfields
${H_{d}}$ | $({\bf 1,2},-1/2,0)$ | $2$ | $0$ | $0$
${H_{u}}$ | $({\bf 1,2},1/2,0)$ | $2$ | $0$ | $0$
$S$ | $({\bf 1,1},0,0)$ | $2$ | $0$ | $0$
$\Phi$ | $({\bf 1,1},0,2)$ | $0$ | $0$ | $-2$
$\bar{\Phi}$ | $({\bf 1,1},0,-2)$ | $0$ | $0$ | $2$
Goldstino Superfield
$Z$ | $({\bf 1,1},0,0)$ | $2/\nu$ | $0$ | $0$
Table 1: The representations under $\mathbb{G}_{B-L}$ and the extra global
charges of the superfields of our model.

###  2.1 Superpotential

The superpotential of our model naturally splits into five parts:

$W=W_{\rm I}+W_{\rm H}+W_{\rm GH}+W_{\rm MSSM}+W_{\rm MD},$ (1)

where the subscripts “I” and “H” stand for the IS and HS respectively and the
content of each term is specified as follows:

##### (a)

$W_{\rm I}$ is the IS part of $W$ which reads [1]

$W_{\rm I}=\kappa S\left(\bar{\Phi}\Phi-M^{2}\right),$ (2a) where $\kappa$ and
$M$ are free parameters which may be made positive by field redefinitions.

##### (b)

$W_{\rm H}$ is the HS part of $W$ written as [29]

$W_{\rm H}=mm_{\rm P}^{2}(Z/m_{\rm P})^{\nu}.$ (2b)

Here $m_{\rm P}=2.4~{}{\mbox{\rm ReV}}$ is the reduced Planck mass – with
${\mbox{\rm ReV}}=10^{18}~{}{\mbox{\rm GeV}}$ –, $m$ is a positive free
parameter with mass dimensions, and $\nu$ is an exponent which may, in
principle, acquire any real value if $W_{\rm H}$ is considered as an effective
superpotential valid close to the non-zero _vacuum expectation value_ (v.e.v)
of $Z$, $\langle{Z}\rangle$. We assume though that the effective
superpotential is such that only positive powers of $Z$ appear. If we also
assume that $W$ is holomorphic in $S$ then mixed terms of the form
$S^{\nu_{s}}Z^{\nu_{z}}$ can be forbidden in $W$ since the exponent of a such
term has to obey the relation

$\displaystyle\nu_{s}+\nu_{z}/\nu=1~{}~{}\Rightarrow~{}~{}\nu_{z}=(1-\nu_{s})\nu,$

leading to negative values of $\nu_{z}$. This conclusion contradicts with our
assumptions above.

##### (c)

$W_{\rm GH}$ is a term which mixes the HS and the $B-L$ gauge fields of the
IS. It has the form

$W_{\rm GH}=-\lambda m_{\rm P}(Z/m_{\rm P})^{\nu}\bar{\Phi}\Phi$ (2c)

with $\lambda$ a real coupling constant. The magnitude of $\lambda$ can be
restricted by the DE requirement as we see below.

##### (d)

$W_{\rm MSSM}$ is the part of $W$ which contains the usual trilinear terms of
MSSM, i.e.,

$W_{\rm
MSSM}=h_{ijD}{d}^{c}_{i}{Q}_{j}{H_{d}}+h_{ijU}{u}^{c}_{i}{Q}_{j}{H_{u}}+h_{ijE}{e}^{c}_{i}{L}_{j}{H_{d}}.$
(2d)

The selected $R$ assignments in Table 1 prohibit the presence in $W_{\rm
MSSM}$ of the bilinear $\mu{H_{u}}{H_{d}}$ term of MSSM and other mixing terms
– e.g. $\lambda_{\mu}S{H_{u}}{H_{d}}$ [39] which is frequently employed to
generate this $\mu$ term. This term is generated here via $K_{\mu}$ – see Sec.
2.2 below.

##### (e)

$W_{\rm MD}$ is the part of $W$ which is provides masses to neutrinos

$W_{\rm MD}=h_{ijN}N^{c}_{i}L_{j}{H_{u}}+\lambda_{[N^{c}}i]\left(S+(Z/m_{\rm
P})^{\nu}\right)\bar{\Phi}N^{c2}_{i}.$ (2e)

The first term in the right-hand side of Eq. (2e) is responsible for Dirac
neutrino masses whereas for the Majorana masses – cf. Refs. [10, 15]. The
scale of the latter masses is intermediate since
$\langle{\bar{\Phi}}\rangle\sim 1~{}{\mbox{\rm YeV}}$ and
$\langle{Z}\rangle\sim m_{\rm P}$. The cooperation of both terms lead to the
light neutrino masses via the well-known (type I) seesaw mechanism – see also
Appendix 8.

###  2.2 Kähler Potential

The Kähler potential has the following contributions

$K=K_{\rm I}+K_{\rm H}+K_{\mu}+K_{\rm D}+|Y_{\alpha}|^{2},$ (3)

which left-handed chiral superfields of MSSM denoted by $Y_{\alpha}$ with
${\alpha}=1,...,7$, i.e.,

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

where the generation indices are suppressed.

##### (a)

$K_{\rm I}$ is the part of $K$ which depends on the fields involved in FHI –
cf. Eq. (2a). We adopt the form

$K_{\rm I}=|S|^{2}+|\Phi|^{2}+|\bar{\Phi}|^{2},\\\ $ (4a) which represents the
simplest possible choice – cf. Ref. [10, 2].

##### (b)

$K_{\rm H}$ is the part of $K$ devoted to the HS. We adopt the form introduced
in Ref. [29] where

$K_{\rm H}=Nm_{\rm P}^{2}\ln\left(1+\frac{|Z|^{2}-k^{2}Z_{-}^{4}/m_{\rm
P}^{2}}{Nm_{\rm P}^{2}}\right)~{}~{}\mbox{with}~{}~{}Z_{\pm}=Z\pm Z^{*}.$ (4b)

Here, $k>0$ mildly violates $R$ symmetry endowing $R$ axion with
phenomenologically acceptable mass. The selected $K_{\rm H}$ is not motivated
by the string theory but it can be considered as an interesting
phenomenological option for two reasons: It largely respects the $R$ symmetry,
which is a crucial ingredient for FHI, and it ensures – as we see in Sec. 3 –
a dS vacuum of the whole field system with tunable cosmological constant for

$N=\frac{4\nu^{2}}{3-4\nu}~{}~{}\mbox{with}~{}~{}\frac{3}{4}<\nu<\frac{3}{2}~{}~{}\mbox{for}~{}~{}N<0.$
(5)

Our favored $\nu$ range finally is $3/4<\nu<1$. Since $N<0$, $K_{\rm H}$
parameterizes the $SU(1,1)/U(1)$ hyperbolic Kähler manifold for $k\sim 0$.

##### (c)

$K_{\mu}$ includes higher order terms which generate the needed mixing term
between ${H_{u}}$ and ${H_{d}}$ in the lagrangian of MSSM – cf. Ref. [36, 29]
– and has the form

$K_{\mu}=\lambda_{\mu}\left({{Z^{*}}^{2\nu}}/{m_{\rm
P}^{2\nu}}\right){H_{u}}{H_{d}}\ +\ {\rm h.c.},$ (6a) where the dimensionless
constant $\lambda_{\mu}$ is taken real for simplicity.

##### (d)

$K_{\rm D}$ an unavoidable term which mixes the observable sector with the HS
as follows

$K_{\rm D}=\lambda_{ij\rm D}\left(Z^{*\nu}/m_{\rm
P}^{\nu+1}\right)N^{c}_{i}L_{j}{H_{u}}+{\rm h.c}.$ (6b)

It provides (subdominant) Dirac masses for $\nu_{i}$ [52] as shown in Appendix
8.

#####

The total $K$ in Eq. (3) enjoys an enhanced symmetry for the $S,Y^{\alpha}$
and $Z$ fields, namely

$\prod_{\alpha}U(1)_{Y^{\alpha}}\times
U(1)_{S}\times\left(SU(1,1)/U(1)\right)_{Z},$ (7)

where the indices indicate the moduli which parameterize the corresponding
manifolds. Thanks to this symmetry, mixing terms of the form
$S^{\tilde{\nu}_{s}}Z^{*\tilde{\nu}_{z}}$ can be ignored although they may be
allowed by the $R$ symmetry for $\tilde{\nu}_{z}=\nu\tilde{\nu}_{s}$.

##  3 SUSY and $\mathbb{G}_{B-L}$ Breaking – Dark Energy

The vacuum of our model is determined by minimizing the F–term (tree level)
SUGRA scalar potential $V_{\rm F}$ derived [28] from $W$ in Eq. (1) and $K$ in
Eq. (3). Note that D–term contributions to the total SUGRA scalar potential
vanish if we confine ourselves during FHI and at the vacuum along the D-flat
direction

$|\bar{\Phi}|=|\Phi|~{}~{}\mbox{which assures}~{}~{}V_{\rm
D}=\frac{g^{2}}{2}\left(|\Phi|^{2}-|\bar{\Phi}|^{2}\right)^{2}=0.$ (8)

Here $g$ is the unique (considered unified) gauge coupling constant of
$\mathbb{G}_{B-L}$and we employ the same symbol for the various superfields
$X^{\alpha}=S,Z,\Phi,\bar{\Phi}$ and their complex scalar components.

As we can verify numerically, $V_{\rm F}$ is minimized at the
$\mathbb{G}_{B-L}$-breaking vacuum

$\left|\langle{\Phi}\rangle\right|=\left|\langle{\bar{\Phi}}\rangle\right|=M.$
(9)

It has also a stable valley along $\langle{{\theta}}\rangle=0$ and
$\langle{{\theta_{S}}/m_{\rm P}}\rangle=\pi$, where these fields are defined
by

$Z=(z+i{\theta})/\sqrt{2}~{}~{}\mbox{and}~{}~{}S=\sigma\
e^{i{\theta_{S}}/m_{\rm P}}/\sqrt{2}.$ (10)

Substituting Eq. (10) in $V_{\rm F}$ and minimizing it _with respect to_
(w.r.t.) the various directions we arrive at the results

$\sigma=-2^{(1-\nu)/2}\left(\lambda(M^{2}+m_{\rm P}^{2})-mm_{\rm P}\right)\
z^{\nu}/m_{\rm
P}^{(\nu+1)}~{}~{}\mbox{and}~{}~{}\langle{z}\rangle=2\sqrt{2/3}|\nu|m_{\rm
P},$ (11)

which yield the constant potential energy density

$\langle{V_{\rm
F}}\rangle=\left(\frac{16\nu^{4}}{9}\right)^{\nu}\left(\frac{\lambda
M^{2}-mm_{\rm P}}{\kappa m_{\rm
P}^{2}}\right)^{2}\omega^{N}\left(\lambda(M^{2}+m_{\rm P}^{2})-mm_{\rm
P}\right)^{2}~{}~{}\mbox{with}~{}~{}\omega=e^{\frac{\langle{K_{\rm
H}}\rangle}{Nm_{\rm P}^{2}}}\simeq\frac{2}{3}(3-2\nu).$ (12)

Tuning $\lambda$ to a value $\lambda\sim m/m_{\rm P}\simeq 10^{-12}$ we may
wish identify $\langle{V_{\rm F}}\rangle$ with the DE energy density, i.e.,

$\langle{V_{\rm F}}\rangle=\Omega_{\Lambda}\rho_{\rm c}=7.3\cdot
10^{-121}m_{\rm P}^{4},$ (13)

where the density parameter of DE and the current critical energy density of
the universe are respectively given by [58]

$\Omega_{\Lambda}=0.6889~{}~{}\mbox{and}~{}~{}\rho_{\rm c}=2.31\cdot
10^{-120}h^{2}m_{\rm P}^{4}~{}~{}\mbox{with}~{}~{}h=0.6766.$ (14)

Therefore, we obtain a post-inflationary dS vacuum which explains the
notorious DE problem. Moreover, Eq. (11) yields $\langle{\sigma}\rangle\simeq
0$.

The particle spectrum of the theory at the vacuum in Eqs. (9) and (11)
includes the gravitino ($\widetilde{G}$) which acquires mass [29]

$m_{3/2}=\langle{e^{{K_{\rm H}}/{2m_{\rm P}^{2}}}W_{\rm H}/m_{\rm
P}^{2}}\rangle\simeq 2^{\nu}3^{-\nu/2}|\nu|^{\nu}m\omega^{N/2}.$ (15a)
Diagonalizing the the mass-squared matrix of the field system
$S-\Phi-\bar{\Phi}-Z$ we also find out that the IS acquire a common mass
$m_{\rm I}=e^{{K_{\rm H}}/{2m_{\rm
P}^{2}}}\sqrt{2}\left(\kappa^{2}M^{2}+(4\nu^{2}/3)^{\nu}(1+4M^{2}/m_{\rm
P}^{2})m^{2}\right)^{1/2},$ (15b) where the second term arises due to the
coexistence of the IS with the HS – cf. Ref. [10]. Similar mixing does not
appear in the mass spectrum of HS which contains the (canonically normalized)
sgoldstino (or $R$ saxion) and the pseudo-sgoldstino (or $R$ axion) with
respective masses
$m_{z}\simeq\frac{3\omega}{2\nu}m_{3/2}~{}~{}\mbox{and}~{}~{}m_{\theta}\simeq
12k\omega^{3/2}m_{3/2}.$ (15c)

Applying finally the relevant formulas of Refs. [37, 29], we find that
$K_{\mu}$ induces a non-vanishing $\mu$ term in the superpotential of MSSM
whereas $W_{\rm MSSM}$ and $K_{\mu}+|Y^{\alpha}|^{2}$ in Eqs. (2d) and (3)
lead to a common soft SUSY-breaking mass parameter $\widetilde{m}$, at the
vacuum of Eq. (11), which indicatively represents the mass level of the SUSY
partners. Namely, we obtain

$W\ni\mu
H_{u}H_{d}~{}~{}\mbox{with}~{}~{}|\mu|=\lambda_{\mu}\left(\frac{4\nu^{2}}{3}\right)^{\nu}(5-4\nu)m_{3/2}~{}~{}\mbox{and}~{}~{}\widetilde{m}=m_{3/2}.$
(16)

Variant form of the terms $|Y^{\alpha}|$ in $K$ – see Eq. (3) – do not alter
essentially our results [29, 28].

##  4 Inflation Analysis

It is well known [1, 3] that in global SUSY FHI takes place for sufficiently
large $|S|$ values along a F- and D- flat direction of the SUSY potential

$\bar{\Phi}={\Phi}=0,~{}~{}\mbox{where}~{}~{}V_{\rm
SUSY}\left({\Phi}=0\right)\equiv V_{\rm
I0}=\kappa^{2}M^{4}~{}~{}\mbox{are}~{}~{}H_{\rm I}=\sqrt{V_{\rm I0}/3m_{\rm
P}^{2}}$ (17)

are the constant potential energy density and correspoding Hubble parameter
which drive FHI – the subscript $0$ means that this is the tree level value.
In a SUGRA context, though, we first check – in Sec. 4.1 – the conditions
under which such a scheme can be achieved and then in Sec. 4.2 we give the
final form of the inflationary potential. Lastly, we present our results in
Sec. 4.4 imposing a number of contraints listed in Sec. 4.3.

###  4.1 Hidden Sector’s Stabilization

The attainment of FHI is possible if $Z$ is well stabilized during it. The
relative mechanism is pretty well known [34]. $Z$ is transported due to
$V_{\rm I0}$ form its value in Eq. (9) to a value well below $m_{\rm P}$. To
determine this, we construct the complete expression for $V_{\rm F}$ along the
inflationary trajectory in Eq. (17) and then expand the resulting expression
for low $S/m_{\rm P}$ values, assuming that the ${\theta}=0$ direction is
stable as in Eq. (9). Under these conditions $V_{\rm F}$ is minimized for the
value

$\langle{z}\rangle_{\rm I}\simeq\left(\sqrt{3}\cdot 2^{\nu/2-1}H_{\rm
I}/m\nu\sqrt{1-\nu}\right)^{1/(\nu-2)}m_{\rm P}.$ (18)

This result is in good agreement with its precise value derived numerically.
Note that $\nu<1$ assures a real value of $\langle{z}\rangle_{\rm I}$ with
$\langle{z}\rangle_{\rm I}\ll m_{\rm P}$ since $H_{\rm I}/m\ll 1$.

The (canonically normalized) components of sgoldstino, acquire masses squared,
respectively,

$m_{{\rm I}z}^{2}\simeq 6(2-\nu)H_{\rm I}^{2}~{}~{}\mbox{and}~{}~{}m_{\rm
I\theta}^{2}\simeq 3H_{\rm I}^{2}-m^{2}\left(8\nu^{2}m_{\rm
P}^{2}-3\langle{z}\rangle_{\rm I}^{2}\right)\frac{4\nu(1-\nu)m_{\rm
P}^{2}+(1-96k^{2}\nu)\langle{z}\rangle_{\rm I}^{2}}{2^{3+\nu}\nu m_{\rm
P}^{2\nu}\langle{z}\rangle_{\rm I}^{2(2-\nu)}},$ (19a) whereas the mass of
$\widetilde{G}$ turns out to be $m_{\rm
I3/2}\simeq\left(\nu(1-\nu)^{1/2}m^{2/\nu}/\sqrt{3}H_{\rm
I}\right)^{\nu/(2-\nu)}.$ (19b)

It is clear from the results above that $m_{{\rm I}z}\gg H_{\rm I}$ and
therefore it is well stabilized during FHI whereas $m_{\rm I{\theta}}\simeq
H_{\rm I}$ and gets slightly increased as $k$ increases. However, the
isocurvature perturbation is expected to be quite suppressed since it becomes
observationally dangerous only for $m_{\rm I\theta}\ll H_{\rm I}$.

###  4.2 Inflationary Potential

Expanding $V_{\rm F}$ for low $S$ values, introducing the canonically
normalized inflaton $\sigma=\sqrt{2}|S|$ and taking into account the RCs [1,
3] we derive [28] the inflationary potential $V_{\rm I}$ which can be cast in
the form

$V_{\rm I}\simeq V_{\rm I0}\left(1+C_{\rm RC}+C_{\rm SSB}+C_{\rm
SUGRA}\right).$ (20)

The individual contributions are specified as follows:

##### (a)

$C_{\rm RC}$ represents the RCs to $V_{\rm I}/V_{\rm I0}$ which may be written
as [3, 2]

$C_{\rm RC}={\kappa^{2}\over 128\pi^{2}}\left(8\ln{\kappa^{2}M^{2}\over
Q^{2}}+8x^{2}\tanh^{-1}\left(\frac{2}{x^{2}}\right)-4(\ln 4-x^{4}\ln
x)+(4+x^{4})\ln(x^{4}-4)\right),$ (21a) with
$x=(\sigma-\sqrt{2}\lambda\langle{Z}\rangle_{\rm I}^{\nu}m_{\rm
P}^{1-\nu}/\kappa)/M>\sqrt{2}$, beyond which the expression above ceases to be
valid. Here we take into account that the multiplicity of the waterfall fields
is ${\sf N}_{\rm G}=1$ since these are $U(1)_{B-L}$ non-singlets.

##### (b)

$C_{\rm SSB}$ is the contribution to $V_{\rm I}/V_{\rm I0}$ from the soft
SUSY-breaking effects [9] parameterized as follows:

$C_{\rm SSB}=m_{\rm I3/2}^{2}\sigma^{2}/2V_{\rm I0}-{\rm
a}_{S}\,\sigma/\sqrt{2V_{\rm I0}},$ (21b)

where the tadpole parameter reads

${\rm a}_{S}=2^{1-\nu/2}m\frac{\langle{z}\rangle_{\rm I}^{\nu}}{m_{\rm
P}^{\nu}}\left(1+\frac{\langle{z}\rangle_{\rm I}^{2}}{2Nm_{\rm
P}^{2}}\right)\left(2-\nu-\frac{3\langle{z}\rangle_{\rm I}^{2}}{8\nu m_{\rm
P}^{2}}\right).$ (21c)

The minus sign results from the stabilization of ${\theta}$ – see Eq. (10) –
at zero and the minimization of the factor
$(S+S^{*})=\sqrt{2}\sigma\cos(\theta_{S}/m_{\rm P})$ which occurs for
$\theta_{S}/m_{\rm P}=\pi~{}({\sf mod}~{}2\pi)$ – the decomposition of $S$ is
shown in Eq. (10). We further assume that $\theta_{S}$ remains constant during
FHI so that the simple one-field slow-roll approximation is valid. Possible
variation of $\theta_{S}$ is investigated in Ref. [11] where they found that
acceptable solutions with $\theta_{S}/m_{\rm P}\neq\pi$ require a significant
amount of tuning. The first term in Eq. (21b) does not play any essential role
in our set-up due to low enough $m_{3/2}$’s – cf. Ref. [10].

##### (c)

$C_{\rm SUGRA}$ is the SUGRA correction to $V_{\rm I}/V_{\rm I0}$, after
subtracting the one in $C_{\rm SSB}$. It reads

$C_{\rm SUGRA}=c_{2\nu}\frac{\sigma^{2}}{2m_{\rm
P}^{2}}+c_{4\nu}\frac{\sigma^{4}}{4m_{\rm
P}^{4}}~{}~{}\mbox{with}~{}~{}c_{2\nu}=\frac{\langle{z}\rangle_{\rm
I}^{2}}{2m_{\rm
P}^{2}}~{}~{}\mbox{and}~{}~{}c_{4\nu}=\frac{1}{2}\left(1+{\langle{z}\rangle_{\rm
I}^{2}}{m_{\rm P}^{2}}\right).$ (21d)

Thanks to the minimality of $K_{\rm I}$ in Eq. (4a) and the smallness of
$\langle{z}\rangle_{\rm I}$ the coefficients above are low enough and allow
for FHI to be established.

###  4.3 Observational Requirements

The analysis of FHI can be performed in the slow-roll approximation if we
calculate the slow-roll parameters [4]

$\epsilon={m^{2}_{\rm P}}\left(\frac{V^{\prime}_{\rm I}}{\sqrt{2}V_{\rm
I}}\right)^{2}\simeq\frac{m_{\rm P}^{2}}{2}\left(C^{\prime}_{\rm
RC}+C^{\prime}_{\rm SSB}\right)^{2}~{}~{}\mbox{and}~{}~{}\eta=m^{2}_{\rm
P}~{}\frac{V^{\prime\prime}_{\rm I}}{V_{\rm I}}\simeq m_{\rm
P}^{2}C^{\prime\prime}_{\rm RC},$ (22)

where the derivatives of the various contributions read

$\displaystyle C^{\prime}_{\rm SSB}$ $\displaystyle\simeq-{\rm
a}_{S}/\sqrt{2V_{\rm I0}},$ (23a) $\displaystyle C^{\prime}_{\rm RC}$
$\displaystyle\simeq\frac{kp^{2}x}{32M\pi^{2}}\left(4\tanh^{-1}\left(\frac{2}{x^{2}}\right)+x^{2}\ln\left(1-\frac{4}{x^{4}}\right)\right),$
(23b) $\displaystyle C^{\prime\prime}_{\rm RC}$
$\displaystyle\simeq\frac{\kappa^{2}}{32M^{2}\pi^{2}}\left(4\tanh^{-1}\left(\frac{2}{x^{2}}\right)+3x^{2}\ln\left(1-\frac{4}{x^{4}}\right)\right).$
(23c)

The required behavior of $V_{\rm I}$ in Eq. (20) can be obtained thanks to the
relation $C^{\prime}_{\rm RC}\simeq-C^{\prime}_{\rm SSB}$ which is established
for carefully selecting $\kappa$ (or $M$) and ${\rm a}_{S}$. Apparently, we
have $C^{\prime}_{\rm SSB}<0$ and $C^{\prime}_{\rm RC}>0$ for
$\sigma_{\star}<\sigma_{\rm max}$ since
$|4\tanh^{-1}\left({2}/{x^{2}}\right)|>|x^{2}\ln(1-4/x^{4})|$. On the
contrary, $C^{\prime\prime}_{\rm RC}<0$, since the negative contribution
$3x^{2}\ln(1-4/x^{4})$ dominates over the first positive one, and so we obtain
$\eta<0$ giving rise to acceptably low $n_{\rm s}$ values.

Our model of FHI can be qualified if we test it against a number of
observational requirements. Namely:

##### (a)

The number of e-foldings elapsed between the horizon crossing of the pivot
scale $k_{\star}=0.05/{\rm Mpc}$ and the end of FHI has to be adequately large
for the resolution of the horizon and flatness problems of standard Big Bang
cosmology. Taking into account – see Sec. 5 –, that FHI is followed in turn,
by a matter and a radiation dominated era, the relevant condition takes the
form [2, 4]:

${N_{\rm I\star}}=\int_{\sigma_{\rm
f}}^{\sigma_{\star}}\frac{d\sigma}{m^{2}_{\rm P}}\>\frac{V_{\rm
I}}{V^{\prime}_{\rm I}}\simeq 19.4+{2\over 3}\ln{V^{1/4}_{\rm
I0}\over{1~{}{\rm GeV}}}+{1\over 3}\ln{T_{\rm rh}\over{1~{}{\rm GeV}}},$ (24)

where the prime denotes derivation w.r.t. $\sigma$, $\sigma_{\star}$ is the
value of $\sigma$ when $k_{\star}$ crosses outside the horizon of FHI and
$\sigma_{\rm f}$ is the value of $\sigma$ at the end of FHI. This is normally
obtained by the critical point $\sigma_{\rm c}=\sqrt{2}|S_{\rm c}|$ , i.e.,
the end of inflation coincides with the onset of the $B-L$ phase transition.
Note that $\langle{\sigma}\rangle\simeq 0$, as mentioned below Eq. (13), and
so it does not disturb the inflationary dynamics which governs the $\sigma$
evolution for $\sigma\geq\sigma_{\rm c}$.

##### (b)

The amplitude $A_{\rm s}$ of the power spectrum of the curvature perturbation
generated by $\sigma$ during FHI must be consistent with the data [58] on
_cosmic microwave background_ (CMB), i.e.,

$A_{\rm s}=\frac{1}{12\,\pi^{2}m^{6}_{\rm P}}\;\left.\frac{V_{\rm
I}^{3}(\sigma_{\star})}{|V^{\prime}_{\rm
I}(\sigma_{\star})|^{2}}\right.\simeq\>2.105\cdot 10^{-9}.$ (25)

The observed curvature perturbation is generated wholly by $\sigma$ since the
other scalars are massive enough during FHI – see Sec. 4.1.

##### (c)

The remaining observables – the scalar spectral index $n_{\rm s}$, its running
$\alpha_{\rm s}$, and the scalar-to-tensor ratio $r$ – which are calculated by
the following standard formulas

$n_{\rm s}=1-6\epsilon_{\star}\ +\ 2\eta_{\star},\alpha_{\rm
s}={2}\left(4\eta_{\star}^{2}-(n_{\rm
s}-1)^{2}\right)/3-2\xi_{\star}~{}~{}\mbox{and}~{}~{}r=16\epsilon_{\star},$
(26)

(where $\xi\simeq m_{\rm P}^{4}~{}V^{\prime}_{\rm
I}V^{\prime\prime\prime}_{\rm I}/V^{2}_{\rm I}$ and all the variables with the
subscript $\star$ are evaluated at $\sigma=\sigma_{\star}$) must be in
agreement with data. We take into account the latest _Planck release 4_ (PR4)
– including TT,TE,EE+lowE power spectra [58] –, _Baryon Acoustic Oscillations_
(BAO), CMB-lensing and BICEP/Keck (BK18) data [59]. Fitting it with
$\Lambda$CDM$+r$ we obtain approximately

$n_{\rm s}=0.965\pm 0.009~{}~{}\mbox{and}~{}~{}r\lesssim 0.032,$ (27)

at 95$\%$ _confidence level_ (c.l.) with negligible $|\alpha_{\rm s}|\ll
0.01$.

###  4.4 Results

As deduced from Sec. 4.1 – 4.3, our model depends on the parameters

$\displaystyle\kappa,~{}M,~{}m,~{}\lambda,~{}\nu,~{}k~{}~{}\mbox{and}~{}~{}\lambda_{\mu}.$

Recall that $N$ is related to $\nu$ via Eq. (5). Enforcing Eq. (13) fixes
$\lambda$ at a rather low value which does not influence our remaining
results. Moreover, $k$ affects exclusively $m_{\theta}$ and $m_{\rm I\theta}$
via Eqs. (15c) and (19a). We select throughout the value $k=0.1$ which assures
the avoidance of massless modes. We here present the ${\rm a}_{S}$ values as a
function of $\kappa$ or $M$ which assist the confrontation of FHI with data –
cf. Ref. [10] – and postpone their derivation from $m$ and $\nu$ via Eq. (21c)
in Sec. 7. As regards $T_{\rm rh}$, which influences Eq. (24), we adopt a
value close to those met in our set-up, $T_{\rm rh}\simeq 1~{}{\mbox{\rm
GeV}}$.

Benchmark |  |  | Benchmark |  |
---|---|---|---|---|---
Point: | A | B | Point: | A | B
Inputs
$M~{}({\mbox{\rm YeV}})$ | $1.4$ | $2.1$ | $\kappa$ | $0.0005$ | $0.001$
$m~{}({\mbox{\rm PeV}})$ | $0.5$ | $3.5$ | $\lambda~{}(10^{-12})$ | $0.2$ | $1.4$
Inflationary Parameters
${\rm a}_{S}~{}({\mbox{\rm TeV}})$ | $2.63$ | $25.3$ | $\Delta_{\rm c\star}~{}(\%)$ | $2.6$ | $8.2$
$H_{\rm I}~{}({\mbox{\rm EeV}})$ | $0.23$ | $1.05$ | $\sigma_{\rm max}/M$ | $1.49$ | $1.66$
$\sigma_{\star}/M$ | $1.45$ | $1.53$ | $\Delta_{\rm max*}~{}(\%)$ | $2.7$ | $7.7$
${N_{\rm I\star}}$ | $40.5$ | $40.8$ | $\sigma_{\rm min}/M$ | $35.5$ | $30.8$
Observables
$n_{\rm s}$ | $0.965$ | $-\alpha_{\rm s}~{}(10^{-4})$ | $2.4$ | $3.1$
$r$ | $9\cdot 10^{-13}$ | $1.8\cdot 10^{-11}$ | ${G\mu_{\rm cs}}~{}(10^{-7})$ | $2.3$ | $6$
$z$ v.e.v and Particle Spectrum
During FHI | at the Vacuum $\langle{z}\rangle=1.4m_{\rm P}$
$\langle{z}\rangle_{\rm I}~{}(10^{-3}m_{\rm P})$ | $1.3$ | $2$ | $m_{\rm 3/2}~{}({\mbox{\rm PeV}})$ | $0.9$ | $6.2$
$m_{\rm I3/2}~{}({\mbox{\rm TeV}})$ | $1.2$ | $11.2$ | $m_{z}~{}({\mbox{\rm PeV}})$ | $1.2$ | $8.8$
$m_{{\rm I}z}~{}({\mbox{\rm EeV}})$ | 0.6 | $2.7$ | $m_{\theta}~{}({\mbox{\rm PeV}})$ | $0.9$ | $5.6$
$m_{\rm I\theta}~{}({\mbox{\rm EeV}})$ | $0.08$ | $0.5$ | $m_{\rm I}~{}({\mbox{\rm ZeV}})$ | $1.7$ | $5.2$
Reheating Process
$\mu/\widetilde{m}$ | $T_{\rm max}~{}({\mbox{\rm PeV}})$ | $\mu/\widetilde{m}$ | $-{\uptau_{\rm max}}$
$3$ | $0.3$ | $2.2$ | $3$ | $66.6$ | $67$
$1/3$ | $0.1$ | $0.7$ | $1/3$ | $67.4$ | $67.6$
$\mu/\widetilde{m}$ | $T_{\rm rh}({\mbox{\rm GeV}})$ | $\mu/\widetilde{m}$ | $-{\uptau_{\rm rh}}$
$3$ | $0.21$ | $3.5$ | $3$ | $28.3$ | $31.3$
$1/3$ | $0.04$ | $0.43$ | $1/3$ | $26.3$ | $29.1$
Table 2: A benchmark table of our scenario. We fix $\nu=7/8$ (resulting to
$N=-49/8$) and $k=0.1$. Recall that $1~{}{\mbox{\rm PeV}}=10^{6}~{}{\mbox{\rm
GeV}}$, $1~{}{\mbox{\rm EeV}}=10^{9}~{}{\mbox{\rm GeV}}$, $1~{}{\mbox{\rm
ZeV}}=10^{12}~{}{\mbox{\rm GeV}}$ and $1~{}{\mbox{\rm
YeV}}=10^{15}~{}{\mbox{\rm GeV}}$.

Enforcing Eqs. (24) and (25) we can restrict $M$ and $\sigma_{\star}$ as
functions of our free parameters $\kappa$ and ${\rm a}_{S}$. It is apparent
that the allowed ranges of parameters is similar to those explored in Ref.
[10], where the HS is not specified. Some deviations are only due to the
improvements on the determination of the $n_{\rm s}$ values in Eq. (27). The
correct values of that quantity are attained if FHI becomes of hilltop type
[10, 11] . I.e., $V_{\rm I}$ is non-monotonic and develops a maximum at
$\sigma_{\rm max}$ and a minimum at $\sigma_{\rm min}\gg\sigma_{\rm max}$. For
$\sigma>\sigma_{\rm max}$ $V_{\rm I}$ becomes a monotonically increasing
function of $\sigma$ and so its boundedness is assured. FHI takes place for
$\sigma<\sigma_{\rm max}$. The position of $\sigma_{\rm max}$ is predominantly
dominated by $C^{\prime}_{\rm RC}$ and $C^{\prime}_{\rm SSB}$. On the other
hand, $\sigma_{\rm min}$ can be roughly found by the interplay of
$C^{\prime}_{\rm SUGRA}$ and $C^{\prime}_{\rm SSB}$. Namely, we obtain [28]

$\sigma_{\rm max}\simeq\frac{\kappa^{3}M^{2}}{4\sqrt{2}\pi^{2}{\rm
a}_{S}}~{}~{}\mbox{and}~{}~{}\sigma_{\rm min}\simeq\left(\frac{{\rm
a}_{S}m_{\rm P}^{4}}{\sqrt{2}c_{4\nu}\kappa M^{2}}\right)^{1/3}\cdot$ (28)

Note that $\sigma_{\rm max}$ is independent from $C_{\rm SUGRA}$ and
$\sigma_{\rm min}$ from $C_{\rm RC}$. The attainment of successful hilltop FHI
requires the establishment of the hierarchy $\sigma_{\rm
c}<\sigma_{\star}<\sigma_{\rm max}$. From our numerical computation we observe
that, for constant $\kappa$ and ${\rm a}_{S}$, $n_{\rm s}$ decreases as
$\sigma_{\star}$ approaches $\sigma_{\rm max}$. To quantify somehow the amount
of these tunings, we define the quantities

$\Delta_{\rm c\star}={\sigma_{\star}/\sigma_{\rm
c}}-1~{}~{}\mbox{and}~{}~{}\Delta_{\rm max*}={1-\sigma_{\star}/\sigma_{\rm
max}}\,.$ (29)

The naturalness of the hilltop FHI increases with $\Delta_{\rm c\star}$ and
$\Delta_{\rm max*}$.

To get an impression of the required values of the parameters of the model, we
construct first the benchmark table Table 2. We display there the results of
our analysis for two _benchmark points_ (BPs) with $\kappa=0.0005$ (BPA) and
$\kappa=0.001$ (BPB) and $n_{\rm s}$ fixed to its central value in Eq. (27).
We also employ some representative $\nu$ and $k$ values. In all cases, we
obtain $N_{\rm I\star}\sim 40$ from Eq. (24). From the observables listed in
Table 2 we infer that $|\alpha_{\rm s}|$ turns out to be of order $10^{-4}$,
whereas $r$ is extremely tiny, of order $10^{-11}$, and therefore far outside
the reach of the forthcoming experiments devoted to detect primordial GWs.
From the entries of Table 2 related to $\Delta_{\rm c\star}$ and $\Delta_{\rm
max*}$, we notice that $\Delta_{\rm max*}>\Delta_{\rm c\star}$, their values
may be up to $10\%$, and increase as ${\rm a}_{S}$ or $\kappa$ (and $M$)
increases. From the mass spectra arranged in Table 2 – see Sec. 4.1 – we see
that $m_{\rm I3/2}$ is similar to ${\rm a}_{S}\sim{\mbox{\rm TeV}}$ whereas
the other masses are of order EeV whereas at the vacuum $m_{\rm I}$ turns out
to be of order $1~{}{\mbox{\rm ZeV}}=10^{12}~{}{\mbox{\rm GeV}}$ whereas
$m_{3/2},m_{z}$ and $m_{\theta}$ lie in the PeV range – see Sec. 3. In the
same Table we find it convenient to accumulate the values of some parameters
related to the reheating process, specified in Sec. 5 below, and the formed
CSs – see Sec. 6.

Figure 1: Allowed (shaded) regions as determined by Eqs. (24), (25), and (27)
in the $\kappa-{\rm a}_{S}$ (a) and $M-{\rm a}_{S}$ (b) plane. The conventions
adopted for the various lines are also shown.

To explore further the parameter space of our model allowed by the
inflationary requirements in Eqs. (24), (25) and (27) we present the gray
shaded regions in the $\kappa-{\rm a}_{S}$ and $M-{\rm a}_{S}$ plane – see
Fig. 1-(a) and Fig. 1-(b). The boundaries of the allowed areas in Fig. 1 are
determined by the dashed [dot-dashed] lines corresponding to the upper [lower]
bound on $n_{\rm s}$ in Eq. (27) – note that in Fig. 1-(a) we obtain a very
narrow strip from the $n_{\rm s}$ variation, We also display by solid lines
the allowed contours for $n_{\rm s}=0.965$. The maximal $r$’s are encountered
in the upper right end of the dashed lines – corresponding to $n_{\rm
s}=0.974$. On the other hand, the maximal $|\alpha_{\rm s}|$’s are achieved
along the dot-dashed lines and the minimal value is $\alpha_{\rm s}=-3.2\cdot
10^{-4}$. Summarizing our findings from Fig. 1 for central $n_{\rm s}$ in Eq.
(27) we end up with the following ranges:

$0.07\lesssim{M/{\rm YeV}}\lesssim 2.6~{}~{}\mbox{and}~{}~{}0.1\lesssim{{\rm
a}_{S}/{\mbox{\rm TeV}}}\lesssim 100.$ (30)

Within the margins above, $\Delta_{\rm c\star}$ ranges between $0.5\%$ and
$9.5\%$ and $\Delta_{\rm max*}$ between $0.4\%$ and $8.2\%$ with the relevant
values increasing with $M$ and ${\rm a}_{S}$, as deduced from Table 2 too. The
lower bounds of the inequalities above are expected to be displaced to
slightly upper values due to the post-inflationary requirements – see Eq. (37)
below – which are not considered here for the shake of generality.

##  5 Reheating Process

Soon after FHI, the IS and $z$ enter into an oscillatory phase about their
minima in Eqs. (9) and (11) and eventually decay via their coupling to lighter
degrees of freedom. Note that ${\theta}$ and ${\theta_{S}}$ remain well
stabilized at their values shown below Eq. (9) during and after FHI and so
they do not participate in the phase of dumped oscillations. The commencement
of $z$-dominated phase occurs for values of Hubble rate $H_{z\rm I}\sim
m_{z}$. Given that $\langle{z}\rangle\sim m_{\rm P}$ – see Eq. (11) –, the
initial energy density of its oscillations $\rho_{z\rm I}$ is comparable with
the energy density of the universe $\rho_{z\rm It}$ at the onset of these
oscillations since

$\rho_{z\rm I}\sim
m_{z}^{2}\langle{z}\rangle^{2}~{}~{}\mbox{and}~{}~{}\rho_{z\rm It}=3m_{\rm
P}^{2}H_{z\rm I}^{2}\simeq 3m_{\rm P}^{2}m_{z}^{2}.$ (31)

Therefore, we expect that $z$ will dominate the energy density of the universe
until completing its decay through its weak gravitational interactions. Due to
weakness of these interactions we expect that the reheating temperature
$T_{\rm rh}$ will be rather low. This is the notorious cosmic moduli problem
[41, 40] which plagues the vast majority of the SUGRA settings.

In our model though $T_{\rm rh}$ adequately increases thanks to the large
enough $m_{z}$ and $\mu$ originated by $K_{\mu}$ in Eq. (6a). To show this
fact we estimate $T_{\rm rh}$ by [55]

$T_{\rm rh}=\left({72/5\pi^{2}g_{\rm
rh*}}\right)^{1/4}{\Gamma}_{{\delta}z}^{1/2}m_{\rm P}^{1/2},$ (32)

where $g_{\rm rh*}\simeq 10.75-100$ counts for the effective number of the
relativistic degrees of freedom at $T_{\rm rh}$. This is achieved for

$\tilde{\uptau}_{\rm rh}\simeq-\frac{2}{3}\ln\frac{2}{5}\sqrt{3}m_{\rm
P}{\Gamma}_{{\delta}z}\rho_{z\rm I}^{-1/2},$ (33)

where $\tilde{\uptau}=\ln\left(R/R_{\rm I}\right)$ with $R$ the scale factor
of the universe and $R_{\rm I}$ its value at the onset of the $z$
oscillations. Also ${\Gamma}_{{\delta}z}$ is the total decay width of the
(canonically normalized) sgoldstino

$\widehat{\delta
z}=\langle{K^{1/2}_{ZZ^{*}}}\rangle{\delta}z~{}~{}\mbox{with}~{}~{}{\delta}z=z-\langle{z}\rangle~{}~{}\mbox{and}~{}~{}\langle{K_{ZZ^{*}}}\rangle=\langle{\omega}\rangle^{-2}$
(34)

which predominantly includes the contributions from its decay into pseudo-
sgoldstinos, ${\theta}$ and electroweak higgs, ${H_{u}}$ and ${H_{d}}$ via the
kinetic terms $K_{XX^{*}}\partial_{\mu}X\partial^{\mu}X^{*}$ with
$X=Z,{H_{u}}$ and ${H_{d}}$ [42, 41, 44, 43] of the Lagrangian. In particular,
we have

${\Gamma}_{{\delta}z}\simeq{\Gamma}_{{\theta}}+{\Gamma}_{\tilde{h}},$ (35)

where the individual decay widths are found to be

${\Gamma}_{{\theta}}\simeq\frac{\lambda_{\theta}^{2}m_{z}^{3}}{32\pi m_{\rm
P}^{2}}\sqrt{1-\frac{4m_{\theta}^{2}}{m_{3/2}^{2}}}~{}~{}\mbox{with}~{}~{}\lambda_{\theta}=\frac{\langle{z}\rangle}{N}=\frac{4\nu-3}{\sqrt{6}\nu}~{}~{}\mbox{and}~{}~{}{\Gamma}_{\tilde{h}}=\frac{2^{4\nu-1}}{3^{2\nu-1}}\lambda_{\mu}^{2}\frac{\omega^{2}}{4\pi}\frac{m_{z}^{3}}{m_{\rm
P}^{2}}\nu^{4\nu}\,.$ (36)

From the expressions above we readily recognize that ${\Gamma}_{{\delta}z}$ is
roughly proportional to $m_{z}^{3}/m_{\rm P}^{2}$ as expected for any typical
modulus [41, 40]. We explicitly checked that $z$-decay channels into gauge
bosons through anomalies and three-body MSSM (s)particles are subdominant. On
the other hand, we kinematically block the decay of $\widehat{\delta z}$ into
$\widetilde{G}$’s selecting $\nu>3/4$ which ensures $m_{z}<2m_{3/2}$ – see Eq.
(15c). We do so in order to protect our setting from the so-called [53]
moduli-induced $\widetilde{G}$ problem, i.e., the possible late decay of the
produced $\widetilde{G}$ and the problems with the abundance of the
subsequently produced lightest SUSY particles – cf. Ref. [41].

Figure 2: Allowed strip in the $\kappa-T_{\rm rh}$ plane compatible with the
inflationary requirements in Sec. 4.3 for $n_{\rm s}=0.965$. We take $\nu=7/8$
and $\mu=\widetilde{m}$ (solid line), $\mu=\widetilde{m}/3$ (dot-dashed line)
or $\mu=3\widetilde{m}$ (dashed line). The BBN lower bounds on $T_{\rm rh}$
for hadronic branching ratios ${\sf B}_{\rm h}=1$ and $0.001$ are also
depicted by two thin lines.

If we enforce the compatibility between theoretical and observational values
of light element abundances predicted by BBN we achieve Ref. [45] a lower
bound on $T_{\rm rh}$ which, for large $m_{z}\sim 0.1~{}{\mbox{\rm PeV}}$,
entails

$T_{\rm rh}\geq 4.1~{}{\mbox{\rm MeV}}~{}~{}\mbox{for}~{}~{}{\sf B}_{\rm
h}=1~{}~{}\mbox{and}~{}~{}T_{\rm rh}\geq 2.1~{}{\mbox{\rm
MeV}}~{}~{}\mbox{for}~{}~{}{\sf B}_{\rm h}=10^{-3},$ (37)

where ${\sf B}_{\rm h}$ is the hadronic branching ratio. The bound above is a
little softened for larger $m_{z}$ values.

Taking $\kappa$ and $m_{z}$ values allowed by the inflationary part of our
model in Sec. 4.4, we evaluate $T_{\rm rh}$ as a function of $\kappa$ and
delineate the regions allowed by the BBN constraints in Eq. (37) – see Sec.
4.3 below. The results of a such computation are displayed in Fig. 2, where we
design allowed contours in $\kappa-T_{\rm rh}$ plane for $\nu=7/8$. This is an
intermediate value in the selected here margin $(3/4-1)$. The boundary curves
of the allowed region correspond to $\mu=\widetilde{m}/3$ or
$\lambda_{\mu}=0.22$ (dot-dashed line) and $\mu=3\widetilde{m}$ or
$\lambda_{\mu}=1.96$ (dashed line) whereas the solid line is obtained for
$\mu=\widetilde{m}$ or $\lambda_{\mu}=0.65$. Note that the relation between
$\lambda_{\mu}$ and $\mu/\widetilde{m}$ is given in Eq. (16). We see that
there is an ample parameter space consistent with the BBN bounds in Eq. (37)
depicted with two horizontal lines. Since the inflationary requirements
increase the scale $m$ with $\kappa$ and $m$ heavily influences $m_{z}$ and
also $T_{\rm rh}$ – see Eq. (32) – $T_{\rm rh}$ increases with $\kappa$. The
maximal value of $T_{\rm rh}$ for the selected $\nu$ is obtained for
$\mu=3\widetilde{m}$ and is estimated to be

$T_{\rm rh}^{\max}\simeq 19~{}{\mbox{\rm GeV}}$ (38)

Obviously, decreasing $\mu$ below $\widetilde{m}/3$, $\lambda_{\mu}$,
${\Gamma}_{{\delta}z}$, and so $T_{\rm rh}$ decrease too and the slice cut
from the BBN bound increases.

It is worth emphasizing that the reheating stage is not instantaneous. In
particular, the maximal temperature $T_{\rm max}$ during this period is much
larger than $T_{\rm rh}$, which can be better considered as the largest
temperature of the radiation domination [54]. To be quantitative, $T_{\rm
max}$ can be calculated as [55]

$T_{\rm max}=\left(3/8\right)^{3/20}12^{1/4}3^{1/8}m_{\rm
P}^{1/4}{\Gamma}_{{\delta}z}^{1/4}\rho_{z\rm I}^{1/8}/g_{\rm
rh*}^{1/4}\pi^{1/2}.$ (39)

It is achieved [55] for $\tilde{\uptau}=\tilde{\uptau}_{\rm
max}=0.39\ll\tilde{\uptau}_{\rm rh}$. The large hierarchy between $T_{\rm
max}$ and $T_{\rm rh}$ can be appreciated by their numerical values displayed
in Table 2 for BPA and BPB. We see that $T_{\rm max}\sim 1~{}{\mbox{\rm PeV}}$
whereas $T_{\rm rh}\sim 1~{}{\mbox{\rm GeV}}$. As a consequence, the
electroweak sphalerons are still operative and so baryogenesis via
leptogenesis is in principle feasible.

Figure 3: Evolution of $\log\rho_{i}$ with $i=z$ – dashed line for $T_{\rm
rh}=1.5~{}{\mbox{\rm GeV}}$ and dot-dashed line for $T_{\rm
rh}=0.5~{}{\mbox{\rm GeV}}$ – and $i=$ r, m (solid lines) as a function of
${\uptau}=-\ln(1+{\text{z}})$. Shown are also ${\uptau_{\rm max}}$,
${\uptau_{\rm rh}}$, ${\uptau_{\rm eq}}$and ${\uptau_{\rm dc}}$for
$\kappa=0.001$ and ${\rm a}_{S}=25.3~{}{\mbox{\rm TeV}}$.

To obtain a more complete picture of our post-inflationary cosmological
setting we introduce the logarithmic time ${\uptau}$ which is defined as a
function of the redshift z (not to be confused with saxion field $z$) or the
scale factor $R$ as [57]

${\uptau}=-\ln(1+{\text{z}})=\ln\left(R/R_{0}\right),$ (40)

where the subscript $0$ is refereed hereafter to present-day values. $T_{\rm
max}$ is achieved at ${\uptau_{\rm max}}$ which can be found from

${\uptau_{\rm max}}=\tilde{\uptau}_{\rm max}-\tilde{\uptau}_{\rm
rh}+{\uptau_{\rm rh}}~{}~{}\mbox{since}~{}~{}R_{\rm max}/R_{0}=\left(R_{\rm
max}/R_{I}\right)\cdot\left(R_{I}/R_{\rm rh}\right)\cdot\left(R_{\rm
rh}/R_{0}\right)$ (41)

taking into account Eqs. (33) and (39). For ${\uptau}\geq{\uptau_{\rm rh}}$,
${\uptau}$ can be found using the entropy conservation through the relation

${\uptau}=-\ln\left(g_{s*}/g_{0s*}\right)^{1/3}\left(T/T_{0}\right)$ (42)

where $g_{s*}$ is the entropy effective number of degrees of freedom at
temperature $T$ and $T_{0}=2.35\cdot 10^{-13}~{}{\mbox{\rm GeV}}$. Its precise
numerical values are evaluated by using the tables included in public packages
[56] and assuming the particle spectrum of the MSSM. E.g, at BBN
($T=2~{}{\mbox{\rm MeV}}$) ${\uptau}_{\rm BBN}\simeq-23$ and at the time of
radiation-matter equidensity ${\uptau_{\rm eq}}\simeq-8.1$ corresponding to
${\text{z}}_{\rm eq}=3387$. The values of ${\uptau_{\rm rh}}$ and
${\uptau_{\rm max}}$ are accumulated in Table 2 for some sample $T_{\rm rh}$
and ratios $\mu/\widetilde{m}$. We remark that ${\uptau_{\rm max}}$ values are
similar for both $T_{\rm rh}$ but ${\uptau_{\rm rh}}$ is smaller for larger
$T_{\rm rh}$.

The energy density of the oscillating $z$ condensate $\rho_{z}$, radiation
$\rho_{\rm r}$ and matter $\rho_{\rm m}$ may be evaluated as follows[57, 55]

$\rho_{z}=\rho_{\rm rh}e^{-3({\uptau}-{\uptau_{\rm rh}})},~{}\rho_{\rm
r}=\frac{\pi^{2}}{30}g_{*}T^{4}~{}~{}\mbox{and}~{}~{}\rho_{\rm m}=\Omega_{{\rm
m0}}\rho_{c}e^{-3{\uptau}},$ (43)

where $\Omega_{\rm m0}=0.311$ and the energy density at reheating is given by
$\rho_{\rm rh}=\rho_{\rm r}(T=T_{\rm rh})$. Taking advantage from the formulae
above we illustrate in Fig. 3 our cosmological scenario. Namely, we present
the evolution of the various $\log\rho_{i}$ – with $\rho_{i}$ normalized to
$\rho_{\rm c}$ given in Eq. (14) – as functions of ${\uptau}$ keeping, for
simplicity, only the dominant component of the universal energy density for
each ${\uptau}$. As regards $\log\rho_{i}$ with $i=z$ we employ the parameters
of BPB in Table 2 besides $\mu$ which is $\mu=5\widetilde{m}/4$ for $T_{\rm
rh}=1.5~{}{\mbox{\rm GeV}}$ (dashed line) or $\mu=2\widetilde{m}/5$ for
$T_{\rm rh}=0.5~{}{\mbox{\rm GeV}}$ (dot-dashed line). We also draw with solid
lines $\log\rho_{i}$ with $i=$ r, m. From the plot it is apparent that our
scenario is distinguishable from the standard scenario according to which the
radiation domination commences after a high $T_{\rm rh}\sim 1~{}{\mbox{\rm
EeV}}$ corresponding to ${\uptau_{\rm rh}}\sim-50$. We observe that
$\log\rho_{z}$ and $\log\rho_{\rm m}$ have the same slop since both are
proportional to $-3{\uptau}$ whereas $\log\rho_{\rm r}$ is more stiff since it
is proportional to $-4{\uptau}$. In the plot we also localize the position of
${\uptau_{\rm max}}$, which is the same for both $T_{\rm rh}$’s, the two
${\uptau_{\rm rh}}$ values and the equidensity point ${\uptau_{\rm eq}}$.
Shown is also the ${\uptau}$ value which corresponds to the CS decay
${\uptau_{\rm dc}}$ for ${r_{\rm ms}^{1/2}}=8$ – see Sec. 6.2. It is located
within the radiation dominated era.

##  6 Metastable CSs And An Early Matter Domination

The $U(1)_{B-L}$ breaking which occurs for $\sigma\simeq\sigma_{\rm c}$
produces a network of CSs which may be stable meta- or quasi- stable. This
network has the potential to undergo decay via the Schwinger production of
monopole-antimonopole pairs leading thereby to the generation of a stochastic
GW background. We below compute the tension of these CSs in Sec. 6.1 and their
GW spectrum in Sec. 6.2 under the assumption that these are metastable.

###  6.1 CS Tension

The dimensionless tension ${G\mu_{\rm cs}}$ of the $B-L$ CSs produced at the
end of FHI can be estimated by [10, 27]

${G\mu_{\rm cs}}\simeq\frac{1}{2}\left(\frac{M}{m_{\rm
P}}\right)^{2}{\epsilon_{\rm cs}}({r_{\rm
cs}})~{}~{}\mbox{with}~{}~{}{\epsilon_{\rm cs}}({r_{\rm
cs}})=\frac{2.4}{\ln(2/{r_{\rm cs}})}~{}~{}\mbox{and}~{}~{}{r_{\rm
cs}}=\kappa^{2}/8g^{2}\leq 10^{-2},$ (44)

where we take into account that $(B-L)(\Phi)=2$ – cf. Ref. [28]. Here
$G=1/8\pi m_{\rm P}^{2}$ is the Newton gravitational constant and $g\simeq
0.7$ is the gauge coupling constant at a scale close to $M$. For the
parameters in Eq. (30) we find

$0.59\lesssim{G\mu_{\rm cs}}/10^{-8}\lesssim 9.2.$ (45)

To qualify the result above, we single out the cases:

##### (a)

If the CSs are stable, the range in Eq. (45) is acceptable by the level of the
CS contribution to the observed anisotropies of CMB which is confined by
Planck [60] in the range

${G\mu_{\rm cs}}\lesssim 2.4\cdot 10^{-7}~{}~{}\mbox{at 95$\%$ c.l.}$ (46)

On the other hand, the results of Eq. (45) are completely excluded by the
recent PTA bound which requires [18]

${G\mu_{\rm cs}}\lesssim 2\cdot 10^{-10}~{}~{}\mbox{at 95$\%$ c.l.}$ (47)

##### (b)

If the CSs are metastable, the explanation [25, 18] of the recent NANOGrav15
data [17, 16] on stochastic GWs is possible for

$M\gtrsim 0.9~{}{\mbox{\rm YeV}}~{}~{}\mbox{and}~{}~{}\kappa\gtrsim 0.0003.$
(48)

This is, because the obtained ${G\mu_{\rm cs}}$ values, through Eq. (44), from
the ranges above is confined in the range dictated by the interpretation of
the recent data [18]

$10^{-8}\lesssim{G\mu_{\rm cs}}\lesssim 2.4\cdot
10^{-4}~{}~{}\mbox{for}~{}~{}8.2\gtrsim\sqrt{{r_{\rm ms}}}\gtrsim
7.5~{}~{}\mbox{at 95$\%$ c.l.}$ (49)

where the metastability factor ${r_{\rm ms}}$ is the ratio of the monopole
mass squared to $\mu_{\rm cs}$. Since we do not investigate the monopole
formation in our work, the last restriction does not impact on our parameters.
Moreover, the GWs obtained from CSs have to be consistent with the upper bound
on their abundance $\Omega_{\rm GW}h^{2}$ originating from the advanced LVK
third observing run [48]

$\Omega_{\rm GW}h^{2}\lesssim 6.96\cdot
10^{-9}~{}~{}\mbox{for}~{}~{}f=32~{}{\rm Hz},$ (50)

where $f$ is the frequency of the observation. At last, although not relevant
for our computation, let us mention for completeness that $\Omega_{\rm
GW}h^{2}$ should be smaller than the limit on dark radiation which is encoded
in an upper limit on $\Delta N_{\rm eff}$ from BBN and CMB observations [61]

$\Omega_{\rm
GW}h^{2}\lesssim\frac{7}{8}\left(\frac{4}{11}\right)^{4/3}\Omega_{\gamma
0}h^{2}=5.6\cdot 10^{-6}\Delta N_{\rm eff}~{}~{}\mbox{with}~{}~{}\Delta N_{\rm
eff}\lesssim 0.28~{}~{}\mbox{\rm at $95\%$ c.l.},$ (51)

taking into account TT,TE,EE+lowE+lensing+BAO data sets [58]. Here,
$\Omega_{\gamma 0}h^{2}\simeq 2.5\cdot 10^{-5}$ is the photon relic abundance
at present and we assume that $\Omega_{\rm GW}h^{2}$ develops a flat shape in
a broad range of $f$ values so as the $f$ dependence in Eq. (51) to be very
weak.

###  6.2 GWs from Metastable CSs with Low Reheating

We focus here on case (b) of Sec. 6.1 and compute the spectrum of the GW
produced by the CSs. The presence of the long-lasting matter domination
obtained in our set up due to the $z$ oscillations after the end of FHI – see
Sec. 4 – has important ramifications in the shape of the spectrum of GW. This
is, because a decaying-particle-dominated era can be approximated by a matter
domination which leads to a spectral suppression at relatively large
frequencies [46, 47].

To verify this fact in the case of our model we compute the emitted GW
background at frequency $f$ following the standard formula of Ref. [47]

$\Omega_{\text{GW}}h^{2}=\frac{8\pi h^{2}}{3H_{0}^{2}}f({G\mu_{\rm
cs}})^{2}\sum_{{\text{k}}=1}^{\infty}C_{\text{k}}(f)P_{\text{k}}~{}~{}\mbox{with}~{}~{}H_{0}=\sqrt{\rho_{\rm
c}/{3m_{\rm P}^{2}}}.$ (52)

Here $\rho_{\rm c}$ is given in Eq. (14) and $P_{\text{k}}$ is the power
spectrum of GWs emitted by the ${\text{k}}^{\text{th}}$ harmonic of a CS loop.
Assuming cusps as the main source of GWs, $P_{\text{k}}$ is given by [47]

$P_{\text{k}}\simeq{\Gamma}/{\zeta({4}/{3})}{\text{k}}^{-4/3}~{}~{}\mbox{with}~{}~{}\Gamma=50~{}~{}\mbox{and}~{}~{}\zeta({4}/{3})=3.6.$
(53)

In Eq. (52) $\Omega_{\rm GW}h^{2}$ is expressed as a sum of the contributions
from an infinite number of normal modes. On technical ground, however, we take
a sum up to ${\text{k}}_{\rm max}\sim 10^{5}$ to achieve a sufficiently
accurate result.

Figure 4: GW spectra from $B-L$ metastable CSs for the inputs of BPB and (a)
${r_{\rm ms}^{1/2}}=8$ and $\mu=\widetilde{m}/3$ (dotted line) or
$\mu=3\widetilde{m}$ (solid line) (b) $\mu=\widetilde{m}$ and various
metastability factors ${r_{\rm ms}^{1/2}}$ indicated in the plot. The shaded
areas in the background indicate the sensitivities of the current – i.e.
NANOGrav [17] and LVK [48] – and future – SKA [62], THEIA [63], $\mu$Ares
[64], LISA [65], Taiji [66], TianQin [67], BBO [68], DECIGO [69], ET [70] and
CE [71] – experiments.

Following our cosmological scenario, the number of loops emitting GWs,
observed at a given frequency $f$, can be found from

$\displaystyle C_{\text{k}}(f)$ $\displaystyle=$
$\displaystyle\frac{2{\text{k}}}{f^{2}}\left(\int_{{\uptau_{\rm
max}}}^{{\uptau_{\rm rh}}}d{\uptau}e^{5{\uptau}}H_{z}n_{\rm
m}\right.+\int_{{\uptau_{\rm rh}}}^{{\rm min}\\{{\uptau_{\rm eq}},{\uptau_{\rm
dc}}\\}}d{\uptau}e^{5{\uptau}}H_{\rm st}n_{\rm r}$ (54) $\displaystyle+$
$\displaystyle\int_{{\uptau_{\rm eq}}}^{{\uptau_{\rm
dc}}}d{\uptau}e^{5{\uptau}}H_{\rm st}n_{\rm rm}\Theta({\uptau_{\rm
dc}}-{\uptau_{\rm eq}})+\left.\int_{{\uptau_{\rm eq}}}^{{\uptau_{\rm
dc}}}d{\uptau}e^{5{\uptau}}H_{\rm st}n_{\rm m}\Theta({\uptau_{\rm
dc}}-{\uptau_{\rm eq}})\right),$

where we use as integration variable ${\uptau}$ – cf. Ref. [47] – taking into
account Eq. (40) and $d{\text{z}}=-e^{-{\uptau}}d{\uptau}$. Also ${\uptau_{\rm
max}}$, ${\uptau_{\rm rh}}$ and ${\uptau_{\rm eq}}$ are given in Sec. 5 and
${\uptau_{\rm dc}}$ corresponds to the decay ${\uptau}$ of CS network, which
is estimated from

${\uptau_{\rm
dc}}=-\ln\Bigg{(}\left({70}/{H_{0}}\right)^{1/2}\left(\Gamma\Gamma_{\rm
dc}{G\mu_{\rm cs}}\right)^{1/4}+1\Bigg{)}~{}~{}\mbox{with}~{}~{}\Gamma_{\rm
dc}=4{G\mu_{\rm cs}}m_{\rm P}^{2}e^{-\pi{r_{\rm ms}}}$ (55)

the rate per unit length of the metastable CSs [31]. For the loop number
density per unit length $n(\ell,t)$ – with mass dimension $4$ – we adopt the
expressions [47]

$\displaystyle n_{\rm r}(\ell,t)$ $\displaystyle=$
$\displaystyle\frac{0.18}{t^{3/2}{\ell_{\Gamma}}^{5/2}}\Theta(0.1t-\ell),$
(56a) $\displaystyle n_{\rm m}(\ell,t)$ $\displaystyle=$
$\displaystyle\frac{0.27-0.45(\ell/t)^{0.31}}{t^{2}\ell_{\Gamma}^{2}}\Theta(0.18t-\ell),$
(56b) $\displaystyle n_{\rm rm}(\ell,t)$ $\displaystyle=$
$\displaystyle\frac{0.18t_{\rm
eq}^{1/2}}{t^{2}{\ell_{\Gamma}}^{5/2}}\Theta(0.09t_{\text{eq}}-{\ell_{\Gamma}}-\ell),$
(56c)

where the subscripts r and m are refereed to CSs produced and radiating in a
radiation- and a matter-dominated era respectively whereas rm means that the
loops produced during the radiation domination, but radiating during the
matter domination. Also ${\ell_{\Gamma}}$ is the initial length of a loop
which has length $\ell$ at a cosmic time $t$ and is given by

${\ell_{\Gamma}}=\ell+\Gamma{G\mu_{\rm
cs}}t~{}~{}\mbox{with}~{}~{}\Gamma\simeq
50~{}~{}\mbox{and}~{}~{}\ell={2{\text{k}}}e^{\uptau}/f.$ (57)

Here $\Gamma$ is related to the energy emission from CS [46] and introduces
some uncertainty in the computation. The Hubble parameter during the standard
cosmological evolution, $H_{\rm st}$, and during the $z$ oscillation, $H_{z}$,
can be found from the formulas [57, 54, 55]

$H_{\rm st}={1\over\sqrt{3}m_{\rm P}}\left(\Omega_{\Lambda}\rho_{\rm
c0}+\rho_{\rm m}+\rho_{{\rm
r}}\right)^{1/2}~{}~{}\mbox{and}~{}~{}H_{z}=\rho^{1/2}_{z}/\sqrt{3}m_{\rm P},$
(58)

where the various $\rho_{i}$ are given in Eq. (43). Lastly, the cosmic time as
a function of ${\uptau}$ is written as

$\displaystyle t({\uptau})$ $\displaystyle=$ $\displaystyle\int_{{\uptau_{\rm
max}}}^{{\uptau}}\frac{d{\uptau}^{\prime}}{H_{z}}\Theta({\uptau}-{\uptau_{\rm
max}})\Theta({\uptau_{\rm rh}}-{\uptau})+\int_{{\uptau_{\rm
max}}}^{{\uptau}}\frac{d{\uptau}^{\prime}}{H_{\rm
st}}\Theta({\uptau}-{\uptau_{\rm rh}})$ (59) $\displaystyle\simeq$
$\displaystyle\begin{cases}2/3H_{z}&\mbox{for}\>\>\>{\uptau_{\rm
max}}\leq{\uptau}<{\uptau_{\rm rh}},\\\ 2/3(1+w)H_{\rm
st}&\mbox{for}\>\>\>{\uptau}>{\uptau_{\rm rh}}.\end{cases}$

In particular, we have $w=1/3$ or $w=0$ for ${\uptau_{\rm
rh}}<{\uptau}\leq{\uptau_{\rm eq}}$ or ${\uptau}>{\uptau_{\rm eq}}$
respectively.

Armed with the formulae above we compute $\Omega_{\rm GW}h^{2}$ for the GWs
produced from the CS formatted in our setting under the assumption that these
are metastable. Employing the inputs of BPB in Table 2 – which yield
${G\mu_{\rm cs}}=6\cdot 10^{-8}$ – we obtain the outputs displayed in Fig. 4.
Namely, in Fig. 4-(a) we show $\Omega_{\rm GW}h^{2}$ as a function of $f$ for
${r_{\rm ms}^{1/2}}=8$ and $\mu=\widetilde{m}/3$ which yields $T_{\rm rh}=0.4$
(dotted line) or $\mu=3\widetilde{m}$ which results to $T_{\rm rh}=3.5$ (solid
line). On the other hand, for the GW spectra depicted in Fig. 4-(b) we employ
$\mu=\widetilde{m}$ resulting to $T_{\rm rh}=1.2~{}{\mbox{\rm GeV}}$ and fix
${r_{\rm ms}^{1/2}}$ to $7.9$ (dotted line) $8$ (solid line) and $8.2$ (dashed
line) – see Eq. (49). In both panels of Fig. 4 we see that the derived GW
spectra can explain NANOGrav15 data shown with gray almost vertical lines. We
see though that, as ${r_{\rm ms}^{1/2}}$ increases, the increase of
$\Omega_{\rm GW}h^{2}$ becomes sharper and provide better fit to the
observations. Also, in both panels the shape of GW signal suffers a
diminishment above a turning frequency $f_{\rm rh}\sim 0.03~{}{\mbox{\rm Hz}}$
which enables us to satisfy Eq. (50) more comfortably than in the case with
high reheating – cf. Ref. [22, 24]. As $T_{\rm rh}$ decreases, the reduction
of $\Omega_{\rm GW}h^{2}$ becomes more drastic in accordance with the findings
of Ref. [46, 47]. The plots also show examples of sensitivities of possible
future observatories [62, 63, 64, 65, 66, 67, 68, 69, 70, 71] which can test
the signals at various $f$ values. Needless to say, the bound in Eq. (51) is
not depicted since it is lies above the $\Omega_{\rm GW}h^{2}$ values
displayed in the plots.

It would be preferable to obtain larger $M$, and so ${G\mu_{\rm cs}}$, values
(e.g., $M\simeq 10~{}{\mbox{\rm YeV}}$ yields ${G\mu_{\rm cs}}\simeq 10^{-6}$)
such that the resulting $\Omega_{\rm GW}h^{2}$ enters the dense part of the
NANOGrav15 favorable region. This can be achieved – see e.g. Ref. [22, 12] –
if we use ${\theta}=\pi$ and a next-to-minimal version [2] for $K_{\rm I}$.
However, the generation of ${\rm a}_{S}$ from $Z$ with ${\theta}=0$ in Eq.
(21c) fixes the sign of the second term in Eq. (21b) and does not allow for
the alternative arrangement mentioned above. On the other hand, next-to-next-
to-minimal $K_{\rm I}$ may accommodate [2, 8, 12] such an augmentation of the
$M$ value without disturbing the $z$ stabilization during FHI – see Sec. 4.1.
In that case we expect that ${\rm a}_{S}$ would not be constrained by the
requirements of successful FHI and it could be selected so that $T_{\rm rh}$
lies at the convenient level which allows for the comfortable evasion of the
exclusion limits from the LVK [48] experiment. This setting opens up an
interesting interconnection between the GW experiments and $\widetilde{m}$.

##  7 Predictions for the SUSY-Mass Scale

The ${\rm a}_{S}$ values displayed in Fig. 1 and Eq. (30) – which result from
the observational constraints to FHI – give us the opportunity to gain
information about the mass scale of SUSY particles through the determination
of $\widetilde{m}=m_{3/2}$ – see Eq. (16). This aim can be achieved by solving
Eq. (21c) w.r.t. $m$ as follows

$m\simeq\left(\frac{{\rm
a}_{S}}{2^{1+\nu}(2-\nu)}\right)^{(2-\nu)/2}\left(\frac{3H_{\rm
I}^{2}}{(1-\nu)\nu^{2}}\right)^{\nu/4},$ (60)

where we take into account Eq. (18) and the fact that $\langle{z}\rangle_{\rm
I}/m_{\rm P}\sim 10^{-3}$. As a consequence, the analytic result above
approximates well the numerical one which is obtained by extracting
consistently $m$ as a function of $\kappa$ and $M$ determined by the
conditions in Eqs. (24) and (25). Note that an iterative process has to be
realized introducing a trial $m$ value which allows us to use as input the
form of $V_{\rm I}$ in Eq. (20). The aforementioned smallness of
$\langle{z}\rangle_{\rm I}$ causes a diminishment of $m$ compared to ${\rm
a}_{S}$ rendering therefore $\widetilde{m}$ via Eqs. (16) and (15a) of the
order of PeV scale. Indeed, for fixed $\nu$, Eq. (60) yields $m$ and then
$m_{3/2}$ , $m_{z}$ and $m_{\theta}$ can be easily estimated by Eqs. (15a) and
(15c) whereas $\widetilde{m}$ and $T_{\rm rh}$ by Eqs. (16) and (32). Their
numerical values are presented in the BPs A and B in Table 2. The magnitude of
the derived $\widetilde{m}$ values and the necessity for
$\mu\sim\widetilde{m}$, established in Sec. 5 hints towards the high-scale
MSSM.

Figure 5: Allowed by Eqs. (24) and (25) region in the $\kappa-\widetilde{m}$
plane for $\widetilde{m}/3\leq\mu\leq 3\widetilde{m}$, $n_{\rm s}=0.965$ and
$3/4<\nu<1$. The allowed contour for $\nu=7/8$ are also depicted. Hatched is
the region excluded by BBN for ${\sf B}_{\rm h}=0.001$.

To highlight numerically our expectations above, we display the allowed (gray
shaded) region in the $\kappa-\widetilde{m}$ plane fixing $n_{\rm s}=0.965$
and varying $\nu$ and $\mu$ within their possible respective margins
$(0.75-1)$ and $(1/3-3)\widetilde{m}$ – see Fig. 5. Along the solid line we
set $\nu=7/8$. From Eq. (60) we can convince ourselves that the lower boundary
curve of the displayed region is obtained for $\nu\simeq 0.751$, whereas the
upper one corresponds to $\nu\simeq 0.99$. Assuming also $\mu=\widetilde{m}$
we can determine the slice of the area which can be excluded due to the BBN
bound in Eq. (37). In all, we find that $\widetilde{m}$ turns out to be
confined in the range

$1.2\lesssim{\rm a}_{S}/{\mbox{\rm TeV}}\lesssim
100~{}~{}\mbox{and}~{}~{}0.34\lesssim\widetilde{m}/{\mbox{\rm PeV}}\lesssim
13.6,$ (61)

whereas $T_{\rm rh}^{\max}\simeq 71~{}{\mbox{\rm GeV}}$ attained for
$\mu=3\widetilde{m}$ and $\nu\simeq 0.99$. The derived allowed margin of
$\widetilde{m}$ and the employed $\mu$ values render our proposal compatible
with the mass of the Higgs boson discovered in LHC, if we adopt as a low
energy effective theory the high-scale version of MSSM [50]. Indeed, within
high-scale SUSY, updated analysis requires [50]

$3~{}{\mbox{\rm TeV}}\lesssim\widetilde{m}\lesssim 0.3~{}{\mbox{\rm ZeV}},$
(62)

for degenerate sparticle spectrum, $\widetilde{m}/3\leq\mu\leq
3\widetilde{m}$, $1\leq\tan\beta\leq 50$ and varying the stop mixing. On the
other hand, our setting does not fit well with natural [41] or split [50] SUSY
which assume $\mu\ll\widetilde{m}$.

##  8 Conclusions

We analyzed the implementation of FHI together with the SUSY breaking and the
CS formation in a B-L extension of MSSM. We adopted the super- and Kähler
potentials in Eqs. (1) and (3) applying an approximate $R$ symmetry. The model
offers the following interesting achievements:

  * •

Observationally acceptable FHI of hilltop-type adjusting the tadpole parameter
${\rm a}_{S}$ and the $B-L$ breaking scale $M$ in the ranges of Eq. (30).

  * •

A prediction for the SUSY-mass scale $\widetilde{m}$ which turns out to be of
the order of PeV.

  * •

An interpretation of the DE problem without extensive tuning. We obtain
$\lambda\sim 10^{-12}$ in Eq. (2c).

  * •

Explantation of the $\mu$ term of MSSM with $|\mu|\sim m_{3/2}$ – see Eq. (6a)
– by invoking a appropriate modification of the Giudice-Masiero mechanism.

  * •

Reheating generated due to the domination of the sgoldstino condensate after
the end of FHI; since $\mu$ is of the order PeV, the resulting $T_{\rm rh}$ is
higher than its lower bound from BBN.

  * •

$B-L$ CSs which, if metastable, explain rather well the recent observations on
GWs. A characteristic of the obtained spectra is their suppression at
relatively large frequencies ($f>0.1~{}{\rm Hz}$) which is due to the long-
lasting matter domination caused by the sgolstino oscillations.

  * •

Generation of neutrino masses via the type I see-saw mechanism supported by
$W_{\rm MD}$ and $K_{\rm D}$ in Eqs. (2e) and (6b).

A potential shortcoming of our proposal is that baryogenesis is made difficult
due to the low reheat temperature. Non-thermal leptogenesis is not operative
since the sgoldstino (with mass of order $1~{}{\mbox{\rm PeV}}$) is lighter
than $N_{i}^{c}$ which acquire mass of order $1~{}{\mbox{\rm ZeV}}$ – cf. Ref.
[10, 15] – and so the direct decay of $z$ into $N_{i}^{c}$ is forbidden.
However, there are extensions of MSSM [72] where the late decay of the
sgoldstino may generate non-thermally the baryon asymmetry of the universe.
Alternatively, this problem may be overcome applying improved attempts [73]
based on the idea of cold electroweak baryogenesis [74]. As regards the CDM,
the candidacy of lightest neutralino has to be investigated thoroughly solving
precisely the relevant Bolzmann equations, as in Ref. [55, 75]. In case that
the abundance is low we can open up slightly the decay channel of $z$ into
$\widetilde{G}$ so that we obtain a controllable production of neutralinos.
Another aspect is the fate of $R$ axion which remains stable throughout our
setting [76].

As regards naturalness, it is a puzzle why the SUSY should appear at such
scale, which is higher than the electroweak scale making the fine-tuning
severe, while it is much smaller than the fundamental energy scale such as the
GUT or Planck scales. Our scenario provides a possible solution to this issue:
this may be due to the inflationary selection. Namely, the apparent fine-
tuning could be a result of combination of the FHI and a bias toward high-
scale SUSY in the landscape.

##### Acknowledgments:

I would like to thank R. Kolb and R. Maji for useful discussions. 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).

##### Dedication:

It is with deep sadness that I dedicate this work to the memory of my Ph. D
advisor, Prof. George Lazarides. He has been a giving mentor and an invaluable
collaborator who has constantly supported me over the past thirty years.

Dirac and Majorana Masses to Neutrinos

####

We check here if the presence of $K_{\rm D}$ in Eq. (3) – which is absent in
other similar settings [10, 15] – has some impact on the derivation of the
neutrino masses. This is because $K_{\rm D}$ contributes to Dirac neutrino
masses according to the formulae of Ref. [52]. In our case the non-vanishing
contributions are

$\displaystyle m_{{\alpha}{\beta}}$
$\displaystyle=\frac{1}{2}\left(\langle{\partial^{2}_{{\alpha}{\beta}}W_{\rm
MD}}\rangle-\langle{K_{\rm
H}^{ZZ^{*}}\partial^{3}_{{\alpha}{\beta}Z^{*}}K_{\rm HO}\partial_{Z}W_{\rm
H}}\rangle\right)$ $\displaystyle-\frac{m_{3/2}}{2}\left(\langle{K_{\rm
H}^{ZZ^{*}}\partial^{3}_{{\alpha}{\beta}Z^{*}}K_{\rm HO}\partial_{Z}K_{\rm
H}}\rangle-\langle{\partial^{2}_{{\alpha}{\beta}}K_{\rm HO}}\rangle\right),$
(63)

where $\partial_{\alpha}:=\partial/\partial{Y^{\alpha}}$,
$\partial_{Z}:=\partial/\partial{Z}$ and we define

$K_{\rm HO}=K_{\rm H}+K_{\mu}+K_{\rm
D}+|Y^{\alpha}|^{2}~{}~{}\mbox{with}~{}~{}Y^{\alpha}={H_{u}},{H_{d}},\Phi,\bar{\Phi},L~{}~{}\mbox{and}~{}~{}N^{c}.$
(64)

If we confine ourselves to the case of the third generation we obtain the
following mass matrix for the neutrino masses

$\mbox{ $\left\lgroup\begin{matrix}0&m_{[\rm D}K]+m_{[\rm D}W]\cr m_{[\rm
D}K]+m_{[\rm D}W]&m_{\rm M}\end{matrix}\right\rgroup$},$ (65)

where the various contributions read

$\displaystyle m_{[\rm D}W]=h_{N}\langle{{H_{u}}}\rangle/2~{}~{}\&~{}~{}m_{\rm
M}=(2\nu)^{\nu}M/3^{\nu/2}$ (66) $\displaystyle m_{[\rm
D}K]=\frac{4^{n}m}{3^{n}m_{\rm P}}\lambda_{\rm
D}\nu^{2\nu}\langle{{H_{u}}}\rangle\omega\left(\omega^{N/2-2}\left(\frac{1}{2}-\nu+\frac{2}{3}\nu^{2}\right)-\frac{3}{8}\right),$
(67)

from which those in the first line come from the terms of Eq. (2e) whereas
this in the second line originates from Eq. (6b). Note that $\omega$ is given
in Eq. (12). For the inputs of BPB in Table 2 we obtain from the well-known
(type I) see-saw formula – see e.g. Ref. [15] – for the mass of the third
generation neutrino, $\nu_{3}$, and $N^{c}$, $N^{c}_{3}$, respectively

$m_{3\nu}\simeq m_{[\rm D}W]^{2}/M\simeq 0.05~{}{\mbox{\rm
eV}}~{}~{}\mbox{and}~{}~{}M_{3N^{c}}\simeq 63~{}{\mbox{\rm
ZeV}}~{}~{}\mbox{if}~{}~{}h_{3N}=0.5~{}~{}\mbox{and}~{}~{}\lambda_{\rm
3N^{c}}=0.03.$ (68)

The obtained $m_{3\nu}$ is phenomenologically acceptable if we assume that the
three active $\nu_{i}$’s have the normal mass hierarchy so that $m_{3\nu}$ is
equal to the mass induced by atmospheric $\nu_{i}$ experiments. For the sample
values above we remark that $m_{[\rm D}W]\sim 40~{}{\mbox{\rm GeV}}$ dominates
over $m_{[\rm D}K]\sim 0.01~{}{\mbox{\rm eV}}$. However, if the terms in
$W_{\rm DM}$ were prohibited then $m_{[\rm D}K]$ could naturally account for
the neutrino masses thanks to the high $m_{3/2}\sim{\mbox{\rm PeV}}$ [49].
Indeed, the resulting $m_{[\rm D}K]$ takes the correct value for $\lambda_{\rm
D}\simeq 2.05$.

In conclusion, our model assures acceptable neutrino masses without sizable
contribution from the terms emerging in $K$.

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