# Large curvature fluctuations from no-scale supergravity
with a spectator field

Ioanna D. Stamou  Service de Physique Théorique, Université Libre de Bruxelles
(ULB), Boulevard du Triomphe, CP225, B-1050 Brussels, Belgium

###### Abstract

We investigate the large curvature perturbations which can lead to the
formation of primordial black holes (PBHs) in the context of no-scale
supergravity. Our study does not depend on any exotic scenario, such as scalar
potentials with inflection points or bulks, and aims to avoid the fine-tuning
of model parameters to achieve the formation of PBHs. This formation relies on
the quantum fluctuations of a light spectator stochastic field after the
inflationary period. Our analysis is based on the $\mathrm{SU(2,1)/SU(2)\times
U(1)}$ symmetry, considering both the inflaton and the spectator field.
Specifically, we examine existing no-scale models with Starobinsky-like scalar
potentials that are consistent with observable constraints on inflation from
measurements of the cosmic microwave background (CMB). These models involve
two chiral fields: the inflaton and the modulus field. We propose a novel role
for the modulus field as a spectator field, responsible for generating PBHs.
Our hypothesis suggests that while the inflaton field satisfies the CMB
constraints of inflation, it is the modulus field acting as the spectator that
leads to large curvature perturbations, capable to explain the production of
PBHs. Additionally, we prioritize retaining the inflationary constraints from
the CMB through the consideration of spectator fluctuations. Therefore, by
exploring the relationship between these fields within the framework of the
$\mathrm{SU(2,1)/SU(2)\times U(1)}$ symmetry, our aim is to unveil their
implications for the formation of PBHs.

##  I Introduction

The enigma surrounding the origin of dark matter remains one of the most
pressing challenges in modern cosmology. A hypothesis gaining interest is that
primordial black holes (PBHs) contribute significantly to the dark matter
content. This idea has been supported from various observations, including the
groundbreaking detection of gravitational waves emitted by a binary black hole
merger by Ligo/VIRGO/Kagra (LVK) collaboration [1, 2, 3, 4, 5]. Numerous other
studies have also provided evidence, such as investigations into the size and
mass-to-light ratio of ultra-faint dwarf galaxies, identification of several
microlensing candidates, analysis of spatial correlations in source-subtracted
cosmic infrared and X-ray backgrounds, and the observation of supermassive
black holes at high redshifts [6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17].
Hence, these studies have triggered the theoretical investigations exploring
the existence of PBHs and their intriguing link to dark matter [18, 19, 20,
21].

Inflationary models within the framework of supersymmetry have gained
prominence due to their ability to address the energy scale requirements of
inflation, which are typically smaller than the Planck scale [22, 23]. They
offer advantages by circumventing problematic features encountered in other
models. For instance, they address the challenge of achieving a vanishing
cosmological constant at the classical level without resorting to excessive
fine-tuning [24]. Additionally, no-scale supergravity tackles the $\eta$
problem, which arises in attempts to explain the smallness of the inflaton
mass compared to the Planck scale [24, 25, 26, 27, 28]. Hence, the combination
of supergravity, and specifically, the merits of no-scale supergravity,
provides a compelling framework for studying inflationary models that
effectively address the energy scale requirements and other challenges
encountered in cosmology.

Supergravity models have been proposed in the literature in order to explain
the production of PBHs [29, 30, 31, 32, 33, 34, 35, 36]. These models often
involve modified versions of supergravity models that incorporate extra
features such as inflection points or bulks. Additionally, no-scale models
have been previously studied in order to explain the production and fractional
abundance of PBHs as candidates for dark matter [37, 38, 39, 40]. These models
provide significant results in explaining a substantial fraction of dark
matter in the Universe while respecting the constraints of inflation reported
by the Planck collaboration [41]. However, they typically rely on additional
features in the effective scalar potential, which require substantial fine-
tuning of the underlying parameters.

In this study, we propose an intriguing alternative scenario within the
framework of no-scale supergravity, where the desired features can be achieved
without the requirement of fine-tuning. Our approach is guided by the
symmetries proposed in Refs. [42, 23], which are based on a non-compact coset
SU(2,1)/SU(2)$\times$U(1). In particular, no-scale models have previously been
studied and they can be in accordance with constraints of inflation from
cosmic microwave background (CMB) anisotropies measurements [43]. In order to
reach these observables, there is the need of introducing one additional
chiral field. Hence, one of these fields acts as the inflaton and the other
one is considered as the modulus field. We propose that the modulus field can
play the role of a spectator field and despite the fact that it does not
contribute to the inflation, it is able to play an important role after
inflation and lead to the formation of PBHs. Therefore, within this framework
of no scale, we consider the presence of two chiral fields, with one field
acting as the inflaton and the other as a spectator field.

The central concept behind the generation of PBHs through a spectator field
lies in the quantum stochastic fluctuations experienced by the field. These
fluctuations cause the field to acquire distinct mean values within various
Hubble patches. Given the immense number of these patches, it is inevitable
that some of them will exhibit the necessary conditions for the spectator
field to possess the precise value required for subsequent quantum
fluctuations to engender significant curvature fluctuations across horizon-
sized regions during PBHs formation. In essence, the existence of these
specific patches triggers an additional expansion of the Universe, resulting
in the amplification of fluctuations and ultimately leading to the formation
of PBHs. The analysis of this mechanism has been previously discussed in
Ref[44].

In our analysis we can derive an extra expansion of the Universe using the
$\delta\rm N$ formalism for the aforementioned no-scale supergravity model.
The outcomes of our study reveal the existence of non-Gaussian tails in the
distribution of induced curvature perturbations. These non-Gaussian tails lead
to the production of PBHs. Furthermore, these results can have implications
for the GW background reported by NANOGrav. Finally, our findings align with
the constraints on inflation obtained from measurements of CMB.

The structure of this paper is organized as follows. In Section II we provide
an overview of the fundamental principles of the no-scale theory that are
needed in our analysis. In Section III we show how a spectator in the
framework of no-scale supergravity can lead to large fluctuations at small
scales. In Section IV we present the production of PBHs for the aforementioned
fluctuations. In Section V we discuss the fine-tuning of this model. In
Section VI we briefly propose an alternative supergravity model motivated from
$\alpha$-attractors models and other no-scale supergravity models. Lastly, in
Section VII, we present our concluding remarks.

##  II No-scale framework

In this section, we describe the key characteristics of no-scale supergravity
essential for our analysis. Firstly, we consider the $\rm{N=1}$ supergravity
action as follows:

$\mathcal{S}=\int
d^{4}x\sqrt{-g}\left(K_{i\bar{j}}\partial_{\mu}\Phi^{i}\partial^{\mu}\bar{\Phi}^{\bar{j}}-V(\Phi)\right),$
(1)

where $\mathrm{\Phi^{i}}$ denotes the chiral fields and
$\mathrm{\bar{\Phi}^{\bar{j}}}$ the conjugate fields. $\mathrm{K}$ is the
Kähler potential, which is a real function and $\mathrm{K_{i\bar{j}}}$ is
given by:

$K_{i\bar{j}}(\Phi,\bar{\Phi})=\frac{\partial^{2}K}{\partial\Phi^{i}\partial\bar{\Phi}^{\bar{j}}}.$

We work in Planck units ($\mathrm{M_{P}=1}$) with $\mathrm{M_{P}}$ is the
reduced mass Planck.

The scalar potential $\mathrm{V}$ can be expressed in terms of the Kähler
potential $\mathrm{K}$ and the superpotential $\mathrm{W}$ as:

$V=e^{K}\Big{(}K^{i\bar{j}}\mathcal{D}iW\mathcal{D}{\bar{j}}\bar{W}-3|W|^{2}\Big{)}+\frac{\tilde{g}^{2}}{2}(K^{i}{\mathcal{T}}^{a}\Phi_{i})^{2}.$
(2)

In this equation, $\mathrm{K^{i\bar{j}}}$ is the inverse of the Kähler metric
and $\mathcal{D}$ represents the covariant derivative, which are defined as:

$\begin{split}&\mathcal{D}_{i}W\equiv\partial_{i}W+K_{i}W,\\\
&\mathcal{D}^{i}W\equiv\partial^{i}W-K^{i}W\,.\end{split}$ (3)

The term of Eq.(2) proportional to $\mathrm{e^{K}}$ is the $\mathrm{F}$-term
of the potential. The last term of this equation proportional to
$\mathrm{\tilde{g}^{2}}$ corresponds to the $\mathrm{D}$-term potential, which
is zero in this case since the chiral fields are gauge singlets. By this
expression, the scalar potential $\mathrm{V}$ is derived from Eq.(2) for a
given Kähler potential and superpotential.

A basic example of no-scale model is represented by the following equation:

$K=-3\ln(T+\bar{T})$ (4)

where $\rm{T}$ is a single chiral field. From Eq. (2) the term $-3|W|^{2}$
vanishes due to the identity [24]:

$K^{T\bar{T}}K_{T}K_{\bar{T}}=3.$ (5)

Therefore, in the framework of no-scale theory, the scalar potential is
identically zero. The Kähler potential in the no-scale theory given in Eq. (4)
can be identified with the coset SU(1,1)/U(1), which is invariant under the
SU(1,1) group of isometric transformation [26]:

$T\rightarrow\frac{\alpha T+i\beta}{i\gamma
T+\delta},\quad\alpha,\beta,\gamma,\delta\in\rm\mathbb{R}\quad\text{with}\quad\alpha\delta+\beta\gamma=1.$
(6)

This identification allows for a number of attractive features in the theory,
including the prediction of a naturally vanishing cosmological constant, a
mechanism for generating a large hierarchy between the electroweak and Planck
scales, and a natural explanation for the smallness of neutrino masses[24, 45,
46, 47, 48, 49, 50].

The SU(1,1)/U(1) symmetry becomes more apparent when we introduce the complex
field $y$ as [27, 23]:

$T=\frac{1}{2}\left(\frac{1-y/\sqrt{3}}{1+y\sqrt{3}}\right).$ (7)

Then the Kähler potential takes the form:

$K=-3\ln\left(1-\frac{y^{2}}{3}\right).$ (8)

The SU(1,1)/U(1) coset space can be parameterized by the $2\times 2$ complex
matrix ${U^{\dagger}}JU$ with:

$U=\begin{bmatrix}A&B\\\ \bar{B}&\bar{A}\end{bmatrix}\quad\text{where}\quad
J=\begin{bmatrix}1&0\\\ 0&-1\end{bmatrix}\text{and}\quad A,B\in\rm\mathbb{C}.$
(9)

In this new basis the Lagrangian is invariant under the transformation law:

$y\rightarrow\frac{Ay+B}{\bar{B}y+\bar{A}}\quad\text{with}\quad|A|^{2}-|B|^{2}=1.$

The minimal no-scale SU(1,1)/U(1) model lacks the ability to generate the
Starobinsky effective scalar potential due to the absence of superpotentials
that support it [23]. To overcome this limitation, an extension of the
SU(1,1)/U(1) coset to the SU(2,1)/SU(2)$\times$U(1) coset is considered. In
this new coset the Kähler potential is given from the equation [27]:

$K=-3\ln\left(T+\bar{T}-\frac{\phi\bar{\phi}}{3}\right)$ (10)

where $\mathrm{T}$ and $\mathrm{\phi}$ are two chiral superfields. The
equivalent form of the Kähler potential in the more revealing basis of two
chiral fields $\mathrm{(y_{1},y_{2})}$, as discussed in Ref. [42] and it is
given as follows:

$K=-3\ln\left(1-\frac{|y_{1}|^{2}}{3}-\frac{|y_{2}|^{2}}{3}\right).$ (11)

These complex fields $(y_{1},y_{2})$ are related to $(T,\varphi)$ by the
following expressions:

$y_{1}=\Big{(}\frac{2\varphi}{1+2T}\Big{)},\quad
y_{2}=\sqrt{3}\Big{(}\frac{1-2T}{1+2T}\Big{)}$ (12)

and the inverse relations by:

$T=\frac{1}{2}\Big{(}\frac{1-y_{2}/\sqrt{3}}{1+y_{2}/\sqrt{3}}\Big{)},\quad\phi=\Big{(}\frac{y_{1}}{1+y_{2}/\sqrt{3}}\Big{)}.$
(13)

The superpotential transforms as:

$W(T,\varphi)\rightarrow\bar{W}(y_{1},y_{2})=(1+y_{2}/\sqrt{3})^{3}W.$ (14)

The Kähler potential presented in Eq. (11) is embedded within the
SU(2,1)/SU(2)$\times$U(1) coset manifold. This coset can be parametrized by
the following matrix [42][23]:

$\displaystyle U=\begin{bmatrix}\alpha&\beta&0\\\ -\beta^{*}&\alpha^{*}&0\\\
0&0&1\end{bmatrix}$ (15)

where $\alpha,\beta\in\mathbb{C}$ and $|\alpha|^{2}+|\beta|^{2}=1$. By this
matrix and the analysis shown in Ref. [42] we can obtain the following
transformation laws for the fields:

$y_{1}\rightarrow\alpha y_{1}+\beta y_{2},\quad
y_{2}\rightarrow-\beta^{*}y_{1}+\alpha^{*}y_{2}.$ (16)

By applying the transformation described in Eq. (16) to the Kähler potential
given in Eq. (11), we observe that it remains invariant. However, the
superpotential undergoes non-trivial modifications. Nevertheless, by carefully
choosing the parameters $\alpha$ and $\beta$, it is indeed possible to obtain
an invariant form of the superpotential. This intriguing feature allows us to
explore the symmetries and transformations within the theory, revealing deeper
insights into its underlying structure.

A carefully chosen form of the superpotential can yield an effective scalar
potential reminiscent of the Starobinsky model. One notable example is the
Cecotti superpotential, presented as follows:

$W=\sqrt{3}m\phi\left(T-\frac{1}{2}\right)$ (17)

where the field $\rm{T}$ plays the role of inflaton and $\rm{\phi}$ is the
modulus field. The Eq.(17) can be written in the equivalent form in the basis
$\rm{y_{1},y_{2}}$ as [23]:

$W=m\Big{(}-y_{1}y_{2}+\frac{y_{2}y_{1}^{2}}{\sqrt{3}}\Big{)}.$ (18)

where $\rm{y_{1}}$ is the inflaton and $\rm{y_{2}}$ is the modulus field. The
Starobinsky-like potential can be derived from Eq.(2) in the direction
$\rm{y_{2}=Imy_{1}=0}$. One should also take into account to fix the non
canonical kinetic term in the Lagrangian. Hence we have the field redefinition
as:

$\varphi\rightarrow\sqrt{3}\tanh\left(\frac{\varphi}{\sqrt{6}}\right)$ (19)

where $\rm{\varphi=Re(y_{1})}$.

An alternative superpotential that leads to the Starobinsky scalar potential
is found in the simplest globally symmetric model known as the Wess-Zumino
model [43]. This model involves a single chiral superfield, denoted as $\phi$,
and is characterized by a mass term, denoted as $\hat{\mu}$, and a trilinear
coupling, denoted as $\lambda$. The superpotential takes the form:

$W=\frac{\hat{\mu}}{2}\phi^{2}-\frac{\lambda}{3}\phi^{3}.$ (20)

By considering the field $\mathrm{\phi}$ as the inflaton field and the field
$\mathrm{T}$ as the modulus field, and setting $\mathrm{\lambda/\mu=1/3}$ and
$\mathrm{\hat{\mu}=\mu\sqrt{1/3}}$, one can identify a specific direction in
field space where $\mathrm{T=\bar{T}=c}$ and $\rm{Im(\phi)=0}$. In this
direction, the scalar potential can be reduced to the Starobinsky potential.
The equivalent form of this superpotential in the basis $\rm{(y_{1},y_{2})}$
is given as [23]:

$W=\frac{\hat{\mu}}{2}\Big{(}y_{1}^{2}+\frac{y_{1}^{2}y_{2}}{\sqrt{3}}\Big{)}-\lambda\frac{y_{1}^{3}}{3}.$
(21)

Here $\rm{y_{1}}$ represents the inflaton and $\rm{y_{2}}$ is the modulus
field, and in order to derive the Starobinsky potential we need to consider
the field redefinition from Eq(19).

Finally, additional superpotentials capable of yielding Starobinsky-like
potentials include the following:

$W=my_{2}(1-e^{-y_{1}})(3-y_{1}^{2}).$ (22)

and

$W=my_{2}y_{1}(3-y_{1}^{2}),$ (23)

where this class of models encompasses the asymmetric T-model
$\alpha$-attractor and the assymetric E-model respectively [51, 52]. We assume
the inflationary direction $\rm{y_{2}=Im(y_{1})=0}$ and
$\rm{\varphi=Re(y_{1})}$. As previously mentioned, it is necessary to address
the non-canonical kinetic term in the Lagrangian, as outlined in Eq. (19). We
need to remark here that this direction is not always stabilized. However an
elegant way to stabilize it can be given by adding an extra term in the Kähler
potential [53, 54, 55], which is given as:

$K=-3\ln\Big{(}1-\frac{|y_{1}|^{2}+|y_{2}|^{2}}{3}+\frac{g|y_{2}|^{4}}{3-|y_{1}|^{2}}\Big{)}.$
(24)

The extra g term does not affect the resulting scalar potential in the
inflationary direction.

Therefore, in the literature on no-scale models that give rise to Starobinsky-
like scalar potentials, two chiral fields have been commonly used: the
inflaton and the modulus field. In this study, we retain the properties of the
inflaton field that are favoured by the constraints of inflation.
Additionally, we propose that the modulus field can play the role of spectator
field, responsible for the generation of large fluctuations. Therefore, we
hypothesize that while the inflaton field satisfies the CMB constraints of
inflation, it is the modulus field, which is regarded as the spectator, that
facilitates the formation of PBHs.

##  III No-scale with a spectator field

The key idea behind generating PBHs with a spectator field is the quantum
stochastic fluctuations that lead to different mean values in various patches.
With a large number of patches, some will meet the conditions for the
curvature fluctuations during PBHs formation[56, 44, 57]. In our study, we
incorporate this mechanism into a no-scale supergravity model.

In the framwork of no-scale $\mathrm{SU(2,1)/SU(2)\times U(1)}$ theory we
consider the Kähler potential given in Eq.(11).

$K=-3\ln\left(1-\frac{|\Phi|^{2}}{3}-\frac{|S|^{2}}{3}\right),$ (25)

where we assume that the field $\mathrm{\Phi}$ is the inflaton and the
$\mathrm{S}$ is the spectator. However, as we will notice later it is possible
to exchange the role of these two field and obtain the same results. The
superpotential is given by

$W=W_{\text{inf}}+W_{\text{spect}}$ (26)

or more specifically,

$\begin{split}W=&M_{\text{inf}}S(3-\Phi^{2})\left(1-e^{b_{\text{inf}}\Phi}\right)^{2}+\\\
&M_{\text{S}}\Phi(3-S^{2})(1-e^{b_{\text{S}}S})^{2}\end{split}$ (27)

Here, $M_{\text{inf}}$ and $M_{\text{S}}$ represent mass scales associated
with the inflationary and spectator sectors, respectively, while
$b_{\text{inf}}$ and $b_{\text{S}}$ are parameters modulating the field decay
in each sector. We assume that these two terms operate at different scales of
the Universe’s expansion, as indicated by:

$M_{\text{inf}}\gg M_{\text{S}}.$ (28)

The stabilization of this system is studied in Appendix A.

Our proposed model respects the transformation laws of the coset
SU(2,1)/SU(2)$\times$U(1) described in Eq.(16). Specifically if we introduce
the

$\Phi\rightarrow-\Phi$ and $S\rightarrow-S$,

we derive the identical scalar potential from Eq.2 . To achieve this, it is
imperative to properly address the non-canonical kinetic terms in the
Lagrangian. For example, if we assume the direction $\rm{Im\Phi=S=0}$ and
$\rm{\varphi=Re\Phi}$ , we introduce a redefinition for the canonical kinetic
term as follows:

$\varphi\rightarrow-\sqrt{3}\tanh\left(\frac{\varphi}{\sqrt{6}}\right).$ (29)

Moreover, if we implement the fields’ transformation in our case described in
Eqs. (25) and (27) as

$\Phi\rightarrow S$ and $S\rightarrow\Phi$,

we can notice that effectively it changes the roles of the inflaton and
spectator fields. Concurrently, we adjust the values of $\rm{b_{i}}$ and
$\rm{M_{i}}$, where i={inf,S}, to reflect the changed dynamics and to meet the
requirements for the inflationary and post-inflationary evolution of the
Universe. As the Universe cools down from the hot, dense state, it undergoes
phase transitions. Such transitions could naturally lead to changes in
$\rm{b_{i}}$ and $\rm{M_{i}}$, reflecting the symmetry properties of different
phases[58, 48, 59].

To sum up, our analysis underscores the significance of the coset
SU(2,1)/SU(2)×U(1) symmetry in shaping the foundational aspects of our model.
In the following, we will explore the dynamics of this no-scale model,
focusing on its behavior during inflation, where the inflaton plays a dominant
role, and post-inflation, where the spectator field takes precedence.

###  III.1 No-scale model during inflation

One of the intriguing consequences of the no-scale theory is its ability to
naturally generate the Starobinsky potential. We assume the general Kähler
potential given in Eq.(25) and the superpotential given in Eq. (27) where, as
we mentioned, that the $\rm{\Phi}$ plays the role of inflaton. Hence we have
the following inflationary direction

$S=Im\Phi=0,\quad Re\Phi=\varphi.$ (30)

From Eq. (2) we find that the effective scalar potential for the inflaton is
given as

$V_{\text{inf}}=M_{1}^{2}\left(1-\exp\left[\sqrt{3}b_{\text{inf}}\tanh\left(\frac{\varphi}{\sqrt{6}}\right)\right]\right)^{4}$
(31)

where $M_{1}=3M_{\text{inf}}$. In order to evaluate the scalar potential given
in (31), it is essential to apply the appropriate field redefinition for the
canonical kinetic term in the Lagrangian, as specified in Eq. (19).

Figure 1:  The effective scalar potential for the inflaton field, Eq.(31),
assuming $\rm{b_{\text{inf}}=-M_{P}}$.

In Fig 1 we depict the potential for Eq. (31).

The evolution of the inflaton field $\rm{\varphi}$ in terms of cosmic time is
given as

$\begin{split}\ddot{\varphi}&+3H\dot{\varphi}+\frac{\partial
V_{inf}}{\partial\varphi}=0.\\\ \end{split}$ (32)

where dots denote derivatives in cosmic time. $\mathrm{H}$ is the Hubble
parameter which is defined as follows

$\begin{split}H=\dot{N}&=\sqrt{\frac{\rho}{3}}.\end{split}$ (33)

and $\rho$ denotes the energy density.

The slow roll parameters $\mathrm{\epsilon_{1}}$ and $\mathrm{\epsilon_{2}}$
defined as

$\epsilon_{1}\equiv-\frac{d\ln
H}{dN},\quad\epsilon_{2}\equiv\frac{d\ln\epsilon_{1}}{dN}.$ (34)

The prediction of $\mathrm{n_{s}}$ and $\mathrm{r}$ can be approximated from:

$n_{\rm s}\simeq 1-2\epsilon_{1*}-\epsilon_{2*},\quad r=16\epsilon_{1*}.$ (35)

Here, the asterisk (*) denotes the values evaluated at the pivot scale
$k_{*}=0.05{\rm Mpc}^{-1}$ .

If we assume as initial condition $\mathrm{\varphi_{ic}=6}$ $\rm{M_{P}}$ and
we solve the Eq.(32) we can obtain the following prediction of spectral index
$\mathrm{n_{s}}$ and tensor-to-scalar ratio $\mathrm{r}$:

$n_{s}=0.96907,\quad r=0.002716,$ (36)

and the corresponding slow-roll parameters are given as:

$\epsilon_{1*}=0.00017,\quad\epsilon_{2*}=0.03058.$ (37)

The power spectrum at the pivot scale is given from:

$A_{\rm s}=2.1\times 10^{-9}\simeq\frac{H^{2}_{*}}{8\pi^{2}\epsilon_{1}M_{\rm
P}^{2}}$ (38)

If we consider Eqs. (37) and (38), the value of the Hubble parameter is given
by:

$H_{*}=5.36\times 10^{-6}M_{P}.$ (39)

Therefore, in the inflationary direction Eq.(30) we have a model which is in
complete consistence with the observable constraints of inflation by Planck
collaboration [41]. The remarkable agreement between the predictions of the
no-scale Starobinsky model and observational data not only support to the
theory but also highlights the importance of study no-scale supergravity
models.

###  III.2 No-scale model after inflation

Following the end of inflation, the dynamics of the spectator field come into
play, leading to its eventual dominance in the energy density of the Universe.
This field triggers an additional phase of expansion, during which the
dynamics of the spectator field play a crucial role in shaping the subsequent
evolution of the Universe.

We consider that the field $\mathrm{S}$ in Eqs. (25) and (27) plays the role
of the spectator. On scales larger than the Hubble radius, the spectator field
remains constant. However, at smaller scales, quantum fluctuations cause the
field to undergo fluctuations around its mean value. These mean values can
vary across different patches that extend beyond the Hubble radius. If the
mean value of the field exhibits sufficient flatness, it will assume a
position of dominance in the Universe’s density and initiate a brief secondary
inflationary period. This secondary phase gives rise to the generation of
significant fluctuations in the spectator field. In the following, we will
delve into the formalism employed to characterize the origin of these
fluctuations in detail.

We examine the Kähler potential and superpotential as specified in Eqs. (25)
and (27). For the spectator field, we adopt the following direction:

$\Phi=ImS=0,\quad ReS=\chi.$ (40)

Using Eq. (2) to calculate the scalar potentialwe find that

$V_{S}=M_{2}^{2}\left(1-\exp\left[-\sqrt{3}b_{S}\tanh\left({\chi}/{\sqrt{6}}\right)\right]\right)^{4}$
(41)

where $M_{2}=3M_{s}$ and we have fixed the non-canonical kinetic term as
before

$\chi\rightarrow\sqrt{3}\tanh\left(\frac{\chi}{\sqrt{6}}\right).$ (42)

The parameter $\rm{M_{S}}$ signifies the scale of the post-inflationary era
and does not influence the result of the field’s evolution. The parameter
$\rm{b_{S}}$ is approximately defined as $\rm{b_{S}\approx{-20}/{H_{*}}}$,
with a detailed examination of its value presented in Ref. [56]. In Fig. 2, we
illustrate the potential, highlighting the chosen parameter value in the
figure’s label. It is evident that our analysis operates within the regime of
small field values $\rm{\chi\ll 1}$, ensuring the applicability of Eq. (41).

Figure 2:  The effective scalar potential for the spectator field, Eq. (41),
assuming $\rm{\mathrm{b_{S}=-4\times 10^{6}}M_{P}}$.

The dynamical behavior of the spectator field $\rm{\chi}$ in cosmic time is
governed by the equation:

$\ddot{\chi}+3H\dot{\chi}+\frac{\partial V_{S}}{\partial\chi}=0,$ (43)

For the energy density $\rm{\rho}$, we consider the following equation:

$\rho=\rho_{\text{rad}}e^{-4N}+V(\chi),$ (44)

under the premise that the PBHs are formed during a radiation-dominated epoch.
We specify $\rm{\rho_{\text{rad}}=10V(\chi_{\text{ic}})}$, where
$\rm{\chi_{\text{ic}}}$ signifies the initial conditions of the field.

Figure 3:  The number of e-fold when the spectator field is dominant.
$\rm{\chi_{ic}}$ represents the initial conditions for the evolution of the
Eq.(43). Figure 4:  The evolution of the spectator field (purple line) after
inflation and its derivative (orange line). Blue region corresponds to the
acceptable range of the mean value of the field.

The results of the numerical evaluation of Eq.(2) using the $\rm{\delta N}$
formalism are depicted in Fig.3, where we observe the additional expansion of
the universe that we require. In Fig. 4, the purple line represents the
evolution of the field, and the orange line corresponds to its derivative. It
is essential to highlight that the amplitude of the spectator field in this
model is consistent with the constraints from the power spectrum at CMB
scales. Specifically, by computing the power spectrum amplitude with the
equation:

$P_{R(s)}=\frac{H^{2}_{CMB}}{4\pi^{2}}\left(\frac{d\chi}{dN}\right)^{-2}\Big{|}_{\chi=\langle\chi\rangle},$
(45)

where $\langle\chi\rangle$ is the mean value of the field, we demand to obtain
values $\mathrm{P_{R(s)}\ll 2.1\times 10^{-9}}$. In Fig. 3, the region where
the derivative of the field complies with the power spectrum at CMB scales is
highlighted in blue. Requiring $\rm{P_{R}<2.1\times 10^{-10}}$ as an upper
limit and referring to Eq.(39), we deduce that the mean value of the field
must be less than $\rm{2\times\mathrm{10^{-6}}M_{P}}$. Dash-dotted lines on
the graph indicate the maximum permissible value to avoid violating the CMB
power spectrum constraints. Thus, this model not only potentially contributes
to a significant abundance of PBHs, capable of accounting for the dark matter
in the Universe, as discussed later but also adheres to the observational
constraints at CMB scales.

##  IV Curvature perturbations and PBHs production

The behavior of the spectator field involves natural randomness, which can be
described using a Fokker-Planck equation [60]. This equation describes the
evolution of the probability density distribution associated with the field’s
fluctuations over time. By solving the Fokker-Planck equation, we can obtain
the statistical behavior of the field, allowing us to analyze its properties
and study the formation of structures, such as PBHs.

In our study, we employ the $\rm{\delta N}$ formalism to describe the
curvature perturbation $\rm{\zeta}$, which is expressed as:

$\zeta(x)=N(\delta\chi_{\rm in}+\delta\chi_{\rm
out}+\langle\chi\rangle)-\langle N\rangle,$ (46)

where $\rm{\langle N\rangle}$ represents the average number of e-folds. The
probability distribution is evaluated from the following integral [56, 57]

$P(\zeta_{\rm in}-\zeta_{\rm out})=\int{\rm d}\delta\chi_{\rm
out}P(\delta\chi_{\rm in})P(\delta\chi_{\rm out})\left.\frac{{\rm d}\chi}{{\rm
d}N}\right|_{\chi_{\rm out}}~{},$ (47)

where $P(\delta\chi_{\rm out})$ and $P(\delta\chi_{\rm in})$ are Gaussian
distributions of the fluctuations of the spectator field and their
corresponding variances come from the inflationary epoch. The subscripts ”in”
and ”out” correspond to fluctuations of the field within each patch and around
the patches, respectively. The evolution of the variance fluctuations
$\langle\delta{\chi{\text{out}}}(N_{\text{inf}})\rangle$ and
$\langle\delta{\chi_{\text{in}}}(N_{\text{inf}})\rangle$ during inflation can
be approximated by the following relations.[56]:

$\langle\delta\chi_{\text{out}}^{2}(N_{\text{inf}})\rangle\simeq\frac{{H_{{*}}^{2}}}{{8\pi^{2}\epsilon_{1*}}}\left(1-e^{-2\epsilon_{1*}N_{\text{inf}}}\right)$
(48)

$\langle\delta\chi_{\text{in}}^{2}(N_{\text{inf}})\rangle\simeq\frac{{H_{*}^{2}}}{{4\pi^{2}}}\exp\left(-2\frac{\epsilon_{1*}}{\epsilon_{2*}}[e^{\epsilon_{2*}N_{\text{inf}}}-1]\right)$
(49)

The values of the slow-roll parameters are given in Eqs. (37) and (39). For
more details about the methodology see Refs.[56].

Figure 5:  The probability distribution by Eq.(47) assuming
$\langle\chi\rangle=10^{-6}M_{P}$.  Figure 6:  The fractional abundance of
PBHs with $\langle\chi\rangle=10^{-6}M_{P}$.

The formation of PBHs occurs when a cosmological perturbation collapses to a
black hole if its amplitude exceeds a certain threshold value, denoted as
$\rm{\zeta_{cr}}$. The threshold value depends on the specific expansion
mechanism and properties of the collapsing object, such as its mass, size and
density profile [61, 62, 63, 64, 65, 66]. In Fig. 5, the probability
distribution is depicted as a function of $\rm{\zeta}$, with the gray vertical
lines representing the critical threshold $\rm{\zeta_{cr}}$. The blue line
represents the corresponding Gaussian distribution with the same inflationary
slow -roll parameters of the model given previously. As the e-folding time
increases, we note that the non-Gaussian tails of the distribution widen,
surpassing the $\rm{\zeta_{cr}}$ threshold and hence they lead to the
formation of PBHs.

The probability, which is related to the mass fraction of PBHs, is evaluated
by the integration of the probability distribution described in (47). This
probability, which depends of the $\mathrm{N_{inf}}$, is connected to the mass
fraction of the Universe collapse to PBHs as

$\beta(M_{\rm PBH})=\int_{\zeta_{\rm cr}}^{\infty}P(\zeta){\rm d}\zeta\,.$
(50)

The fraction abundance of PBH is given from

$f_{\rm PBH}(M_{\rm PBH})\approx 2.4\beta(M_{\rm PBH})\left(\frac{2.8\times
10^{17}M_{\odot}}{M_{\rm PBH}}\right)^{1/2}~{},$ (51)

where $\mathrm{M_{PBH}}$ is the mass fraction of PBHs. More details about the
evaluation are shown in Ref.[44, 56, 57]. The fraction abundance of PBHs
resulting from our proposed model is illustrated in Fig. 6. It is notable that
our model can account for a significant fraction of dark matter, offering a
compelling explanation. Finally, our model can leave an imprint in the GW
background, specifically in relation to the observations reported by the
NANOGrav collaboration[67, 68].

##  V Fine-tuning Discussion

Inflationary models that facilitate the formation of PBHs often face criticism
for the extensive fine-tuning required to amplify the scalar power spectrum
adequately, as discussed in [38, 69]. This section explores the fine-tuning
aspects of our model, offering a comparative analysis with other models
referenced in the literature.

To address the fine-tuning concern linked with a given parameter of a model
$\rm{p}$, we evaluate the quantity $\Delta_{p}$, as introduced in Refs. [70,
71]. This quantity is calculated as the maximum value of the logarithmic
derivative:

$\Delta_{p}=\text{Max}\left|\frac{\partial\ln(f)}{\partial\ln(x)}\right|=\text{Max}\left|\frac{\partial
f}{\partial x}\cdot\frac{x}{f}\right|,$ (52)

where $f(x)$ denotes a general function. High $\Delta_{p}$ values mean the
model needs a lot of fine-tuning, showing it’s very sensitive to changes in
its parameters.

One can numerically compute Eq. (52) by assuming the function $\rm{f}$
represents the fractional abundances of PBHs, $\rm{f_{PBH}(p)}$. In models
featuring an inflection point, recent work has indicated that this quantity
ranges from $\rm{10^{4}}$ to $\rm{10^{9}}$ [69]. The specific range varies
depending on the model. However, the significant level of fine-tuning required
across these scenarios raises concerns regarding the naturalness of such
models. Hybrid models have been proposed as a more natural mechanism for
generating PBHs. The degree of fine-tuning in these models has been analyzed
in [72, 73]demonstrating a significant reduction in the parameter described by
Eq. (52), with values approximately to $\rm{10^{2}}$.

In our analysis, the only additional parameter is the $b_{S}$ of the
potential. This parameter dependents on the inflationary model we have and
especially on the value of the Hubble parameter, $\mathrm{H_{*}}$. Its value
is crucial in order to obtain correct values for the power spectrum at CMB
scales and it does not affect the fractional abundance. Other values of
$b_{S}$, which are smaller than the one provided here (over than one order of
magnitude), lead to obtaining significant fraction of PBHs but they affect the
in Eq.(45). A detailed discussion about the fine-tuning is provided in
Ref.[56]. By evaluating the quantity Eq. (52) we find that this takes values
of $\rm{\mathcal{O}(1)}$. Therefore, this model can indeed avoid the need of
fine tuning of the parameters in order to obtain a significant fraction of
PBHs.

As we mentioned before, PBHs from no-scale have previously been studied in
Refs.[37, 40]. The amount of fine-tuning has been computed to be at the range
of $\mathrm{10^{6}}$ [74, 38]. Moreover, these models have been criticised
from the evaluation of one-loop corrections to the large-scale power spectrum
according to the study of Refs. [75, 76, 77, 78, 79, 80, 81, 82]. An other
issue, which have these models to deal with, is the prediction of extremely
light PBHs, lighter than the range of the future detection and, hence, their
results cannot be tested in the near future. Finally, in these models the
condition for obtaining a vanishing cosmological constant given in Eq.(5) is
not conserved as well as the properties of the SU(2,1)/SU(2)$\times$U(1) coset
manifold.

In this proposed model, on the other hand, it is important to remark that
slightly changes of the parameter $b_{S}$ does not affect the results. Hence
this model does not suffer for the unnatural fine-tuning of the inflection-
point models. In addition to that, in our analysis we use the $\mathrm{\delta
N}$ formalism, which evades the problem with the one-loop correction [83].
Moreover, the prediction of masses are in the range of
$\mathrm{[10^{0}-10^{2}]}$ $\mathrm{M_{\odot}}$ which are supported by
subsolar mass black hole secondary component, which is considered as GW signal
[84]. Finally, the properties of the group SU(2,1)/SU(2)$\times$U(1) are
conserved, as it is analysed previously.

##  VI Alternative scenario in context of supergravity models

###  VI.1  An $\alpha$-attractors supergravity model

Several models can apply the proposed mechanism of spectator field analyzed in
Ref. [56] in order to elucidate PBHs production. This section presents an
alternative model for generating substantial curvature perturbations necessary
for the formation of PBHs. This alternative scenario is based on
superconformal $\alpha$-attractors supergravity models.

In the framework of superconformal field theory, the landscape is enriched by
symmetries that extend beyond local supersymmetry [85]. These include Special
Conformal Symmetry, Special Supersymmetry, and Weyl Symmetry.

The alternative model is based on the following choice of Kähler potential and
superpotential, extensively explored in the literature [53, 86, 54, 55, 51,
85, 87]

$K_{2}=-3\alpha\ln\Big{(}1-\frac{|S|^{2}+|\Phi|^{2}}{3}+\frac{g|\Phi|^{4}}{3-|S|^{2}}\Big{)}$
(53)

and

$W_{2}=\Phi M\frac{S}{\sqrt{3}}\left(3-S^{2}\right)^{(3\alpha-1)/2}.$ (54)

The $\alpha$ term in $\alpha$-attractor models represents a parameter that
controls the curvature of the potential in the inflationary landscape,
influencing the inflationary trajectory and the resulting predictions for
cosmological observables like the spectral index and the tensor-to-scalar
ratio. In our scenario, this term is applied to the spectator field and does
not directly impact the inflationary observable constraints.

The potential in the direction of the spectator field $\rm{\Phi=ImS=0}$ and
$\rm{ReS=\chi}$ is simply given as

$V_{2}(\chi)={M}^{2}\tanh^{2}\left(\frac{\chi}{\sqrt{6\alpha}}\right).$ (55)

where we have fixed the non-canonical kinetic term. Regarding the inflaton
field, we consider a model that yields slow-roll parameters and observable
constraints predictions akin to those in the previously examined no-scale
model.

Figure 7:  The extra expansion of the Universe assuming $\alpha=10^{-13}$
during the phase when spectator field is dominant.  Figure 8:  The probability
distribution for the case of $\alpha-$attractors.

We use the $\rm{\delta}$N formalism to analyze the dynamics of the spectator
field, as in the previous case of no-scale model. The resulting additional
expansion of the Universe is illustrated in Fig. 7. Thus, this model
facilitates the extra expansion necessary for the formation of PBHs.

In conclusion, we determine the probability distribution using Eq. (47) and we
depict our results in Fig. 8. The non-Gaussian tails widen as the number of
e-folds increases. During certain inflationary e-folds, these tails may
surpass the critical threshold, inducing substantial curvature fluctuations
sufficient to produce PBHs.

###  VI.2  Alternative examples based on no-scale

In this section, we present alternative superpotential choices that can lead
to additional expansion of the Universe, and consequently, a significant
fraction of PBHs within the framework of no-scale supergravity.

First, we adopt the Kähler potential from Eq. (25) and consider the following
superpotential for the spectator field in Eq. (26):

$W_{S}=M_{S}\left(-S\Phi+\frac{\Phi
S^{2}}{\sqrt{3}}\right)\left(1+c_{2}e^{-b_{2}S^{2}}S^{2}\right),$ (56)

where S represents the spectator field and $\rm{\Phi}$ is the inflaton and
$\rm{c_{2}}$, $\rm{b_{2}}$ are parameters. This choice, explored in [38], can
lead to an inflection point. With an appropriate choice of parameters, this
point can be positioned very close to the potential’s origin, thereby allowing
fluctuations around this point to generate large curvature perturbations, as
proposed in Ref. [57]. Positioning the inflection point near small field
values ensures accordance with the power spectrum constraints at CMB scales.

Additionally, we explore the different superpotentials, such as:

$W=M_{\text{inf}}\left(-\Phi S+\frac{S\Phi^{2}}{\sqrt{3}}\right)+M_{S}\Phi
S^{b_{3}}(3-S^{2}),$ (57)

or

$W=M_{\text{inf}}\left(-\Phi
S+\frac{S\Phi^{2}}{\sqrt{3}}\right)+M_{\text{S}}\Phi(1-e^{b_{\text{S}}S})(3-S^{2}),$
(58)

where $\rm{S}$ is the spectator field, $\rm{\Phi}$ is the inflaton (as
before), and $\rm{b_{3}}$ is a parameter, with the Kähler potential again
given by Eq. (25). We can derive a potential which exhibits a plateau at small
field values and it can lead to similar extra expansion of the Universe
capable to facilitate the PBHs production.

Therefore, within the framework of supergravity models, there are multiple
avenues to achieve an extra expansion of the Universe, similar to the results
of this work.

##  VII Conclusion

This study delves into the realm of PBHs within the no-scale supergravity
framework, offering an alternative avenue avoiding the need of inflection
points. Our analysis focuses on the $\mathrm{SU(2,1)/SU(2)\times U(1)}$
symmetry and considers both the inflaton and spectator fields. Specifically,
we examine existing no-scale models that generate scalar potentials resembling
the Starobinsky model, which typically involve two chiral fields: the inflaton
and the modulus field. By interplaying the proposed symmetries from previous
works, we introduce these two chiral fields, with the inflaton field driving
inflation and the modulus field acting as a spectator throughout the process.

We have developed a mechanism in which a small expansion of the spectator
field, approximately one e-fold, plays a crucial role in the formation of PBHs
during the radiation epoch. By carefully exploring this concept within the
framework of an inflationary model, we have ensured that our predictions
adhere to the rigorous observational constraints set by the amplitude of CMB
fluctuations on large scales. We evaluate the fractional abundance of PBHs and
we find that this model can predict a significant fraction of dark matter in
the Universe with large values of mass of PBHs.

As a next step in our research, we plan to delve into the exploration of the
mass distribution of PBHs within the framework of this model. By analyzing the
mass spectrum, we aim to understand better the role that PBHs play as
potential dark matter candidates. Additionally, we intend to examine the
compatibility of our model’s predictions with existing observational
constraints.

Overall, our study contributes to the growing body of research on PBHs and
their connection to dark matter and the evolution of the universe. By offering
an alternative mechanism within the no-scale supergravity framework and
providing insights into the interplay between chiral fields, we deepen our
understanding of the intricate dynamics that give rise to PBHs. This research
opens up new avenues for exploring the nature of dark matter and the
mechanisms that shape the Universe’s structure.

##  Appendix A Study the stabilization

In this appendix, we examine the stabilization of the fields presented in the
Kähler potential and superpotential in Eqs. (25) and (27).

First, we calculate the squared mass matrix using the method outlined in Ref.
[88] for our proposed model.

$m^{2}_{s}=\begin{pmatrix}\alpha_{\Phi\bar{\Phi}}&\alpha_{\Phi\bar{S}}\\\
\alpha_{S\bar{\Phi}}&\alpha_{S\bar{S}}\end{pmatrix}.$ (59)

We calculate the elements of the matrix and we set $\Phi=\bar{\Phi}$ and
$S=\bar{S}$ (see Eq.(18) in [88]). The elements are:

$\alpha_{\Phi\bar{\Phi}}=(K^{-1})^{i}_{k}D^{k}\partial_{j}V$ (60)

$\alpha_{\Phi\bar{S}}=(K^{-1})^{k}_{i}D_{k}\partial_{j}V$ (61)

$\alpha_{S\bar{\Phi}}=(K^{-1})^{i}_{k}D^{k}\partial^{j}V$ (62)

$\alpha_{S\bar{S}}=(K^{-1})^{k}_{i}D_{k}\partial^{j}V$ (63)

where $\rm{i,j,k}$ denote the fields and the covariant derivatives are given
in Eq.(3). The scalar potential is described in Eq.(2) and it is depicted in
Fig.9. The inflaton slowly rolls through the valley at the direction of
$\rm{S=Im\Phi=0}$ until the end of inflation. Then at values very close to the
origin of the potential the spectator field $\rm{S}$ starts to dominate the
Universe. In Fig.10 we depict various slices of the potential along the real
and imaginary part of each field. Our results are similar to those in Ref.[89]

Figure 9:  The scalar potential assuming $\rm{\Phi=\bar{\Phi}=Re\Phi}$ and
$\rm{\bar{S}=S=ReS}$ using (25) and (27).

Figure 10:  Slices through the effective potential for the model from Eqs.(25)
and (27) along the direction $(\rm{Re\Phi,Im\Phi})$ (left plot) and
$(\rm{ReS,ImS})$ (right plot).

After identifying the matrix elements in Eq.(59), we set the expectation
values $\langle S\rangle=0$ and $\langle\Phi\rangle=0$. In order to find this
values, it becomes necessary to numerically solve the following system
$\partial{V_{j}}\big{|}_{j=\langle j\rangle}=0$ where j represents the fields.
Given our selected parameters and within the chosen field region, we find that
the mass matrix exhibits positive eigenvalues. This calculation is repeated
for the imaginary direction, confirming that, given our parametric selection,
the eigenvalues of the matrix have positive solutions. However, considering a
more extensive range of parameters one need to incorporate the introduction of
additional stabilizing terms to the Kähler potential. Such modifications align
with the suggestions in Refs. [53, 90, 91], indicating that stabilization is
achievable. Incorporating these stabilizer terms is particularly model in the
context of a two-field system, suggesting that a more comprehensive analysis
is imperative for a generalized scenario.

## Acknowledgement

We would like to express our gratitude to S. Clesse, P. Fayet, M.A. Garcia
Garcia & M. Vanvlasselaer for their fruitful discussions.

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