aainstitutetext: Center for Cosmology and Particle Physics, New York
University, 726 Broadway, New York, NY 10003, USAbbinstitutetext: INAF/OAS
Bologna, via Gobetti 101, I-40129 Bologna, Italyccinstitutetext: Stanford
Institute for Theoretical Physics and Department of Physics,
Stanford University, Stanford, CA 94305, USAddinstitutetext: INFN, Sezione di
Bologna, via Irnerio 46, I-40126 Bologna, Italy

#  Hybrid $\alpha$-attractors, primordial black holes and gravitational wave
backgrounds

Matteo Braglia c Andrei Linde c Renata Kallosh b,d Fabio Finelli
<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We investigate the two-stage inflation regime in the theory of hybrid
cosmological $\alpha$-attractors. The spectrum of inflationary perturbations
is compatible with the latest Planck/BICEP/Keck results, thanks to the
attractor properties of the model. However, at smaller scales, it may have a
very high peak of controllable width and position, leading to a copious
production of primordial black holes (PBH) and generation of a stochastic
background of gravitational waves (SGWB).

##  1 Introduction

There has recently been a growing interest in the possibility to generate
large peaks in the amplitude of perturbations at the late stages of inflation.
Such peaks may lead to the formation of primordial black holes (PBHs) and to
generate a stochastic gravitational wave background (SGWB) which can be
explored with future gravitational wave interferometers.

One of the first fully developed models to generate such peaks in the
primordial power spectrum (PPS) of the perturbations and consequently seed PBH
formation was proposed in Garcia-Bellido:1996mdl  in the context of the hybrid
inflation scenario Linde:1991km ; Linde:1993cn . There, it was argued that
under certain conditions such PBHs may provide a significant contribution to
dark matter. Many other interesting mechanisms were proposed to produce a bump
in the PPS from inflation. Among them is the PBH production due to large
isocurvature perturbations Dolgov:1992pu , amplification of perturbations
during rapid turns of inflationary trajectory Fumagalli:2020adf ;
Palma:2020ejf ; Braglia:2020eai ; Braglia:2020taf ; Iacconi:2021ltm ;
Kallosh:2022vha , a sudden slowing down of the inflaton evolution often
referred to as Ultra Slow Roll Ivanov:1994pa ; Garcia-Bellido:2017mdw ;
Motohashi:2017kbs ; Germani:2017bcs ; Ballesteros:2017fsr ; Dalianis:2018frf ;
Ketov:2021fww ; Cai:2021zsp ; Geller:2022nkr  or other types of features in
the inflaton potential Inomata:2021uqj ; Inomata:2021tpx ; Dalianis:2021dbs ;
Boutivas:2022qtl ; Inomata:2022yte  and creation of black holes and wormholes
by colliding walls Garriga:2015fdk ; Deng:2016vzb . For a review of these and
other mechanisms of PBH formation and their cosmological implications see e.g.
Carr:2021bzv .

In this paper, we will revisit the mechanism of Garcia-Bellido:1996mdl  in the
recently proposed $\alpha$-attractor generalization Kallosh:2022ggf  of the
hybrid inflation scenario Linde:1991km ; Linde:1993cn  and consider its
consequences on PBH and SGWB production. While, due to the large uncertainties
in the critical process of PBH formation, we will just touch upon it, we will
instead provide a full characterization of the SGWB produced by the large
perturbations at horizon re-entry during the radiation dominated era
Carbone:2004iv ; Ananda:2006af ; Baumann:2007zm , which can be used in the
future to test this model. We will pay particular attention to the consistency
with large scale measurements of the Cosmic microwave Background (CMB)
anisotropies from the latest Planck/BICEP/Keck release Planck:2018jri ;
BICEPKeck:2021gln ; Paoletti:2022anb , which is not a trivial task since
models producing a peak in the small scale power spectrum often tend to
predict a spectral index which is slightly redder than the Planck best fit.

Let us start with a quick reminder of the basic hybrid inflation model
Linde:1991km ; Linde:1993cn . The effective potential of this model is given
by

$V(\chi,\phi)={1\over 4\lambda}(M^{2}-\lambda\chi^{2})^{2}+{m^{2}\over
2}\phi^{2}+{g^{2}\over 2}\phi^{2}\chi^{2}\ .$ (1)

For $\phi>\phi_{c}=M/g$ the only minimum of the effective potential
$V(\chi,\phi)$ with respect to $\chi$, sometimes referred to as hybrid field,
is at $\chi=0$. The mass squared of the field $\chi$ at $\chi=0$ is equal to
$V_{\chi,\chi}(\chi=0)=-M^{2}+g^{2}\phi^{2}.$ At large $\phi$, the curvature
of the effective potential in the $\chi$-direction is much larger than in the
$\phi$-direction. Thus, during the first stages of expansion of the universe,
the field $\chi$ rolls down to $\chi=0$, whereas the field $\phi$ could remain
large and drive inflation for a much longer time. At that time, the potential
of the inflaton field $\phi$ is given by $V(\phi)={m^{2}\over
2}\phi^{2}+V_{\rm uplift}$, where the uplift potential is given by
${M^{2}\over 4\lambda}$.

At the moment when the inflaton field $\phi$ becomes smaller than
$\phi_{c}=M/g$, the effective mass squared of the field $\chi$ at $\chi=0$
becomes negative (tachyonic), quantum fluctuations of this field begin to
grow, and a so-called waterfall phase transition with symmetry breaking
occurs.

In the simplest versions of the hybrid inflation scenario, the absolute value
of the tachyonic mass of the field $\chi$ becomes much greater than the Hubble
constant $H$ shortly after the field $\phi$ becomes smaller than $\phi_{c}$.
In that case, the tachyonic instability leads to an abrupt end of inflation at
$\phi\approx\phi_{c}$ Linde:1991km ; Linde:1993cn . However, in some models
the tachyonic mass of the field $\chi$ at $\phi\lesssim\phi_{c}$ may remain
much smaller than the Hubble constant $H$. In the Higgs-type potential
$V(\chi)\sim(\chi^{2}-\chi_{0}^{2})^{2}$ used in Linde:1991km ; Linde:1993cn ;
Garcia-Bellido:1996mdl , this regime occurs for $\chi_{0}\gg 1$ (in Planck
mass units $M_{\rm pl}=1$). In such cases, inflation continues while the field
$\chi$ slowly rolls down towards the minimum of $V(\chi)$. As we will explain
later, the amplitude of perturbations produced at the second stage of
inflation in hybrid inflation models with such properties can be extremely
large.

In fact, these perturbations can be so large that in addition to producing PBH
they may also trigger the process of eternal inflation inside some regions of
the observable part of the universe Garcia-Bellido:1996mdl . One way to
regularize the amplitude of the perturbations, which we explore in this paper,
is by a minor modifications of the theory, i.e. by adding a tiny linear term
$\sim\chi$ to the potential (1). This allows to control the strength of the
effect and the amplitude of the peak in the PPS.

Whereas the basic mechanism of the PBH production described above is quite
general and can be implemented in a broad class of hybrid inflation models,
the simplest versions of this scenario Linde:1991km ; Linde:1993cn  resulted
in the spectral index $n_{s}\approx 1+2m^{2}/V_{\rm uplift}>1$, which did not
match the Planck data Planck:2013jfk .

An attempt to overcome this problem was made in Clesse:2015wea . The authors
changed the sign of $m^{2}$ making the potential tachyonic, $m^{2}<0$, which
allowed to have $n_{s}<1$. This paper, together with the detection of
gravitational waves from a binary black hole merger by the LIGO-Virgo
collaboration Abbott:2016blz , attracted new attention to the possibility that
PBH may constitute a substantial portion of the dark matter Bird:2016dcv ;
Sasaki:2016jop ; Clesse:2016vqa . However, the tachyonic potential $V(\phi)$
used in the toy model discussed in Clesse:2015wea  is unbounded from below,
and the field $\phi\gtrsim\phi_{c}$ instead of moving towards $\phi=0$ tends
to run to $\phi\to\infty$. This makes it difficult to solve the problem of
initial conditions for inflation in this scenario unless the potential is
further modified at $\phi>\phi_{c}$.

Fortunately, these problems can be addressed in the $\alpha$-attractor
versions Kallosh:2022ggf  of the original hybrid inflation scenario
Linde:1991km ; Linde:1993cn . For a broad choice of parameters, these models
have the standard universal $\alpha$-attractor predictions Kallosh:2013hoa ;
Ferrara:2013rsa ; Kallosh:2013yoa ; Kallosh:2021mnu ; Kallosh:2022feu
matching the Planck/BICEP/Keck data Planck:2018jri ; BICEPKeck:2021gln . In
particular, $n_{s}=1-{2\over N}\sim 0.96-0.97$ for the exponential attractors,
where $N$ is the number of $e$-foldings Kallosh:2013yoa . However, in the
large uplift limit $V_{\rm uplift}\gg{m^{2}\over 2}\phi^{2}$ these models have
another attractor regime, with $n_{s}=1$. By changing the parameters, one can
continuously interpolate between the two attractor predictions,
$n_{s}=1-{2\over N}$ and $n_{s}=1$ Kallosh:2022ggf . Because of the double
attractor structure of such models, their predictions are compatible with all
presently available cosmological data. In the models where the tachyonic mass
of the field $\chi$ at $\phi\lesssim\phi_{c}$ is smaller than the Hubble
constant $H$, which we will study in this paper, the range of possible values
of $n_{s}$ is even broader.

In this paper, we will study the simplest versions of such models, explain how
one can solve the problem of initial conditions there, and evaluate the
spectrum of perturbations in these models. We will demonstrate that one can
keep $n_{s}$ and $r$ consistent with the latest CMB data, while simultaneously
controlling the shape, the width and the height of the peak of the spectrum of
perturbations which determine the features of the PBHs and the stochastic SGWB
background produced in this scenario.

This paper is organized as follows. We start with a discussion of hybrid
inflation in the context of $\alpha$-attractors in Section 2. After reviewing
the basic results of Kallosh:2022ggf , we discuss the problem of initial
conditions in our model in Section 3. In Section 4, we will make a step back
and study the perturbations in the $\chi$-sector of hybrid inflation,
temporarily ignoring the field $\varphi$. As we will see, this will help to
understand the origin of the high peak of perturbations produced in various
versions of the hybrid inflation scenario. In Section 5, we explain in details
how the model parameters control distinct properties of the bump in the PPS
and discuss the constraints from Planck/BICEP/Keck data. Then, in Section 6,
we describe the gravitational wave phenomenology of our model providing clear
targets for gravitational wave interferometers and discussing how to interpret
a future detection of a SGWB in terms of our model parameters. In Section 7 we
will generalize our results by considering models based on polynomial
attractors or KKLTI potentials Kallosh:2022feu . We conclude in Section 8. In
the Appendix A, we provide an implementation of our model in SUGRA
constructions. We set $M_{\rm pl}=1$ throughout our analysis. The numerical
results in this paper are obtained using a multifield extension of the BINGO
code Hazra:2012yn , presented in Braglia:2020fms , which is however not yet
public.

##  2 Hybrid $\alpha$-attractors

Before describing the hybrid attractors Kallosh:2022ggf , we will represent
the original model (1), together with a small additional linear term
$\mu^{3}\chi$, in a slightly different form, which is more convenient for our
investigation111One should also add a tiny constant term $\Delta
V\approx-\mu^{3}\chi_{0}$ to the potential (2) to ensure nearly vanishing
value of the cosmological constant, but it can be ignored in the discussion of
inflation.:

$V(\chi,\phi)=M^{2}\left[{({\chi^{2}-\chi_{0}^{2}})^{2}\over
4\chi_{0}^{2}}+{{\tilde{m}}^{2}\over 2}\phi^{2}+{{\tilde{g}}^{2}\over
2}\phi^{2}\chi^{2}+d\,\chi\right].$ (2)

Here $\tilde{m}=m/M$, $\tilde{g}=g/M$, and $d={\mu^{3}\over M^{2}}$. In this
paper we will consider models with $\chi_{0}=O(1)$ and $M\sim m\sim\mu\sim
g\sim 10^{{-5}}$, with masses given in the Planck mass units $M_{\rm pl}=1$.
In that case $\tilde{m}\sim\tilde{g}=O(1)$, whereas the linear term can be
very small, with $d\sim 10^{{-5}}$.

In the original version of the hybrid inflation Linde:1991km ; Linde:1993cn ,
both $\phi$ and $\chi$ are canonically normalized. To present the simplest
$\alpha$-attractor generalization of the original model, it is sufficient to
modify the kinetic term of the field $\phi$:

${{\cal L}\over\sqrt{-g}}={R\over 2}-{(\partial_{\mu}\phi)^{2}\over
2\bigl{(}1-{\phi^{2}\over 6\alpha}\bigr{)}^{2}}-{(\partial_{\mu}\chi)^{2}\over
2}-V(\chi,\phi)\,.$ (3)

One may also make a similar generalization of the kinetic term of the field
$\chi$ Kallosh:2022ggf , but it is not required in the context of our
investigation. The geometric interpretation of this modification can be found
in Kallosh:2015zsa  and its supergravity implementation for hybrid attractors
is given in Kallosh:2022ggf  and in the Appendix A of our paper.

Upon a transformation to the canonical variable $\varphi$, the hybrid
inflation potential becomes

$\displaystyle V(\chi,\varphi)=M^{2}\left[{({\chi^{2}-\chi_{0}^{2}})^{2}\over
4\chi_{0}^{2}}+3\alpha(\tilde{m}^{2}+\tilde{g}^{2}\chi^{2})\tanh^{2}{\varphi\over\sqrt{6\alpha}}+d\,\chi\right].$
(4)

The potential at $\chi=0$ is given by

$V(\varphi)=M^{2}\left({\chi_{0}^{2}\over
4}+3{\tilde{m}}^{2}\alpha\tanh^{2}{\varphi\over\sqrt{6\alpha}}\right)\ .$ (5)

This is the simplest $\alpha$-attractor potential Kallosh:2013hoa ;
Ferrara:2013rsa ; Kallosh:2013yoa ; Kallosh:2021mnu ; Kallosh:2022feu ,
uplifted by the term $V_{\rm uplift}={M^{2}\chi_{0}^{2}\over 4}$. Another
important parameter is the mass squared of the field $\chi$ at $\chi=0$,

$M^{2}_{\chi}=V_{\chi,\chi}(\chi=0)=M^{2}\left(-1+6\alpha\tilde{g}^{2}\tanh^{2}{\varphi\over\sqrt{6\alpha}}\right).$
(6)

The last term in this equation stabilizes the inflationary trajectory $\chi=0$
at large $\varphi$.

In this paper, we will consider the case $6\alpha\tilde{g}^{2}>1$. In that
case $M^{2}_{\chi}>0$ at sufficiently large $\varphi>\varphi_{c}$, where

$\tanh^{2}{\varphi_{c}\over\sqrt{6\alpha}}={1\over 6\alpha\tilde{g}^{2}}.$ (7)

During inflation, when the field $\varphi$ decreases below $\varphi_{c}$, the
mass squared of the field $\chi$ becomes negative and the tachyonic
instability with generation of the scalar field $\chi$ develops. At
$\varphi=0$ this mass has its largest absolute value,
$M^{2}_{\chi}(\varphi=0)=-M^{2}$.

The potential (4) for some particular values of parameters is shown in Fig. 1.

Figure 1: Hybrid inflation potential for the model (4) with
$\tilde{m}=1/3,\chi_{0}=2,\tilde{g}=1,\alpha=1,\,d=-5\times 10^{-6}$. The blue
dot corresponds to $\varphi_{c}$ (7), below which the mass of the field $\chi$
becomes tachyonic. Note that the effects of the linear term are almost
invisible by eye.

It is instructive to compare $M^{2}_{\chi}(\varphi=0)=-M^{2}$ with the Hubble
constant at $\varphi=0$,

$H_{0}=\sqrt{V_{\rm uplift}\over 3}={M\chi_{0}\over 2\sqrt{3}}\ .$ (8)

The main regime explored in the original formulation of the hybrid inflation
scenario was $\chi_{0}\ll 1$ Linde:1991km ; Linde:1993cn , in which case the
absolute value of the tachyonic mass $M_{\chi}$ becomes much larger than the
Hubble constant as soon as $\phi$ becomes smaller than $\phi_{c}$. This leads
to an abrupt termination of inflation at $\phi\approx\phi_{c}$ Linde:1991km ;
Linde:1993cn ; Kallosh:2022ggf .

In this paper, following Garcia-Bellido:1996mdl , we will be interested in the
opposite regime $\chi_{0}\gtrsim 2\sqrt{3}$. In this regime
$|M_{\chi}(\chi=0)|\lesssim H$ for all $\varphi<\chi\lesssim\varphi_{c}$. This
mass vanishes when the field $\varphi$ is close to $\varphi_{c}$. As a result,
the tachyonic instability is very slow to develop. Importantly, at any nonzero
value of $\varphi$, its contribution to the equation of motion of the field
$\chi$ is only slowing it down, and at $\varphi\ll\varphi_{c}$ the
contribution of the field $\varphi$ to the equation of motion of the field
$\chi$ becomes negligible.

That is why one can learn quite a lot about the waterfall regime in such
models by studying the evolution of the field $\chi$ from the top of the
single-field inflationary potential
$V(\chi)=({\chi^{2}-\chi_{0}^{2}})^{2}\,M^{2}\,/\,4\chi_{0}^{2}$, ignoring the
field $\varphi$. We will discuss it in Section 4. This will explain in an
intuitive way why it is natural to expect a very high peak for the
inflationary perturbations in such models, and how one can control its height.
In subsequent sections, we will return to the full scenario describing
classical and quantum evolution of both fields.

But before doing it, we will discuss the problem of initial conditions in this
scenario. And the first question is: How did the scalar fields found their way
into the narrow infinitely long valley with $\chi=0$?

##  3 Initial conditions for inflation

As we can see from equation (2), the potential $V(\chi,\varphi)$ at
$\varphi\gg\sqrt{6\alpha}$ practically does not depend on $\varphi$ because in
this regime the function $\tanh{\varphi_{c}\over\sqrt{6\alpha}}$ becomes
exponentially close to $1$:

$\displaystyle V(\chi,\varphi)\approx
M^{2}\left[{({\chi^{2}-\chi_{0}^{2}})^{2}\over
4\chi_{0}^{2}}+3\alpha(\tilde{m}^{2}+\tilde{g}^{2}\chi^{2})+d\,\chi\right].$
(9)

Thus at large $\varphi$ one can apply to this theory the standard arguments of
the single field chaotic inflation scenario Linde:1983gd ; Linde:1985ub ;
Linde:2017pwt . One can consider a tiny Planck size domain of the universe
with the Planckian value of the potential energy of the field $\chi$, and let
it fall. If the kinetic and gradient energy of the fields initially are
smaller that its potential energy, each such tiny domain becomes exponentially
expanding.

The potential $V(\chi,\varphi)$ approaches the Planck boundary at
$\chi^{2}\sim 2\chi_{0}/M\gg 1$ for any value of $\varphi$ in the infinitely
large range $-\infty<\varphi<+\infty$. Therefore in this context it is natural
to consider initial conditions where both fields are extremely large, but
$\varphi\gg\chi$. For most of such initial conditions the field $\chi$
gradually falls down to the valley, oscillates with a rapidly decreasing
amplitude, and relaxes at the bottom of the valley, and, only after that, the
field $\varphi$ begins to move towards $\varphi=0$.

Figure 2: [Left] Initial conditions for hybrid attractors. The parameters used
are those in Section 5, Eq. (22). The potential is shown in units of $M^{2}$.
The fields begin at large values of $\varphi$ and $\chi$. Then $\chi$ rapidly
falls down, and, after that, inflation driven by the field $\varphi$ begins.
Eventually, after $\varphi$ crosses $\varphi_{c}$ the field $\chi$ falls to
one of the two minima of the potential $\chi\approx\pm\chi_{0}$ depending on
the sign of $d$. The initial conditions are shown on the right panel. [Right]
Evolution of the field $\chi$ for different initial conditions compared to the
attractor solution in Eq. (11). The agreement of the attractor solution with
the numerical (exact) solution of $\chi$ is remarkable. As expected, it starts
to deviate from the position of the minimum of the potential (11) close to
when $\varphi\sim\varphi_{c}$. The power spectrum is clearly unaffected by the
initial conditions.

Note that, because of the tiny linear term $d\,\chi$, the field $\chi$ does
not relax at $\chi=0$, but it is slightly displaced in the direction depending
on the sign of the coefficient $d$. At $\chi\ll\chi_{0}$ and $d\not=0$ the
valley is at the minimum of the potential

$V(\chi,\varphi)\approx{M^{2}}\left[{1\over
2}\left(-1+6\alpha\tilde{g}^{2}\tanh^{2}{\varphi\over\sqrt{6\alpha}}\right)\chi^{2}+d\,\chi\right]\
.$ (10)

This potential admits an attractor for $\chi$ defined by its minimum at

$\chi_{\rm attr}=-{d\over
6\alpha\tilde{g}^{2}\tanh^{2}{\varphi\over\sqrt{6\alpha}}-1}\ .$ (11)

As mentioned in the previous section, we study models with
$6\alpha\tilde{g}^{2}>1$. At $\varphi\gg\sqrt{6\alpha}$, we have
$\tanh^{2}\varphi/\sqrt{6\alpha}\simeq 1$ and the minimum with respect to
$\chi$ is at a constant $\varphi$-independent distance from $\chi=0$:

$\chi=-{d\over 6\alpha\tilde{g}^{2}-1}\ .$ (12)

Because of the rapid decrease of the kinetic energy of the field $\chi$ during
inflation, the amplitude of its oscillations about this minimum decreases
exponentially fast. Then, the field $\varphi$ starts slowly moving towards its
smaller values, see Fig. 2. At $\sqrt{6\alpha}\gtrsim\varphi>\varphi_{c}$, the
minimum of this potential moves further away from zero. Close to
$\varphi=\varphi_{c}$, the part of the potential $\sim\chi^{2}$ vanishes, so
it can no longer protect the field $\chi$ from growing due to the linear term
$d\,\chi$.

Thus, even though this term $d\,\chi$ is very small, it pushes the field
$\chi$ away from the $\chi=0$. Eventually, the fields falls down to one of the
two minima at $|\chi|\approx\chi_{0}$ at $\varphi=0$. The choice of the
minimum is completely independent on the initial conditions on the two fields,
and rather depends on the sign of the coefficient $d$ of the tiny linear term.

A more important statement is that if the initial values of the field
$\varphi$ are sufficiently large, the original oscillations of the field
$\chi$ becomes exponentially damped, and the final results of our calculations
of the spectrum of perturbations do not depend on initial conditions.

Simplicity and robustness of this process is one of the advantages of the
original hybrid inflation scenario, as well as of its $\alpha$-attractor
generalization. Its most important part is the existence of a long valley to
which the field $\chi$ falls, and the positive slope of the potential with
respect to the field $\varphi$ which pushes it along the valley towards
$\varphi=0$ Linde:1991km ; Felder:1999pv ; Clesse:2008pf .

Figure 3: Initial conditions problem for the model (1) of Ref. Clesse:2015wea
. If evolution begins at smaller values of $\phi$, as shown by the black line,
the field evolution is chaotic, and hybrid inflation regime typically does not
emerge. If evolution begins at greater values of $\phi$, as shown by the blue
line, the field $\phi$ turns, runs to infinity, its tachyonic potential
becomes negative, and the universe collapses. In this example, we use the
parameters of Fig. 3 in Ref. Clesse:2015wea  with $\phi_{c}=0.1$ and
$(\phi_{i},\,\chi_{i})\,=(3,\,3),\,(4,\,-2)$ for the black and blue lines
respectively.

The situation can be very different if one attempts to make the parameter
$m^{2}$ negative, as in the toy model described in Clesse:2015wea . In that
case, if one considers small initial values of the field $\phi$, the field
moves to $\phi=0$, but without a significant fine-tuning of initial conditions
it is difficult to achieve a long hybrid inflation regime starting with large
$\chi$, see Fig. 3. On the other hand, if one considers large initial values
of $\phi$, the field $\phi$ initially moves to smaller $\phi$, but then it
turns and rolls down along the valley towards very large $\phi$ with
$V(\phi)<0$, and the universe collapses. It is possible to solve this problem
by a proper modification of the potential used in Clesse:2015wea  at
$\phi>\phi_{c}$. Fortunately, this issue does not appear in the original
version of the hybrid inflation scenario Linde:1991km ; Linde:1993cn , and in
the hybrid attractor models described in our paper.

##  4 Evolution of the field $\chi$ in the single-field approximation

As we have argued in section 2, one can learn quite a lot about the waterfall
stage of inflation in hybrid inflation with $\chi_{0}\gtrsim 2\sqrt{3}$ by
ignoring the field $\varphi$ and investigating the single field model with

$V(\chi)={M^{2}\over 4\chi_{0}^{2}}({\chi^{2}-\chi_{0}^{2}})^{2}\ .$ (13)

Many of our results will be valid for any single field inflationary potential;
they will be based on the general theory of eternal inflation
Steinhardt:1982kg ; Linde:1982ur ; Vilenkin:1983xq ; Linde:1986fd  which we
will remind here following the general theory of eternal inflation developed
in Linde:1986fd .

In the slow-roll approximation, ignoring quantum fluctuations, during each
$e$-folding the scalar field decreases by

$\Delta\chi={V_{\chi}\over V}\ .$ (14)

where $V_{\chi}={\partial V\over\partial\chi}$. During that time the volume of
each inflationary domain of the horizon size $\mathcal{O}(H^{-1})$ increases
$e^{3}\sim 20$ times, so we get $\sim 20$ independent horizon-size
inflationary domains where the average value of the field $\chi$ decreases by
$\Delta\chi={V_{\chi}/V}$.

However, during the same time, inflationary quantum fluctuations may increase
the value of $\chi$ by

$\delta\chi\sim{H\over 2\pi}={\sqrt{V}\over 2\sqrt{3}\,\pi}\ .$ (15)

If $\delta\chi>|\Delta\chi|$, these quantum jumps may bring half of the 20
horizon-size domain uphill, back to where we started. This leads to the regime
of eternal inflation Linde:1986fd . Remarkably, as one can easily see, the
condition $\delta\chi<|\Delta\chi|$ required for the absence of eternal
inflation is equivalent to the condition that the amplitude of perturbations
is smaller than $\mathcal{O}(1)$:

$A_{s}={V^{3}\over 12\pi^{2}V_{\chi}^{2}}\lesssim 1.$ (16)

This is not a coincidence, since eternal inflation would imply that inflation
continues in some parts of the universe, whereas in many other parts of the
universe inflation is over and the energy density rapidly becomes small.

In the context of the model (13), the criterion (16) is always violated at the
top of the potential where $V_{\chi}$. However, quantum fluctuations in each
horizon size domain during one $e$-folding of inflation shift the field $\chi$
from $\chi=0$ by $\delta\chi\sim{H\over 2\pi}$. Thus one can use this estimate
for the initial beginning of the inflationary trajectory. Expanding
$V(\chi)=V+V_{\chi,\chi}\chi^{2}/2$ we find a simple criterion for the
existence of eternal inflation at the top of the potential:
$|V_{\chi,\chi}|\lesssim V$.

A similar and perhaps a slightly more accurate estimate can be obtained by
investigation of eternally inflating domain walls separating parts of the
universe with $\chi>0$ from the parts with $\chi<0$ Linde:1994hy ;
Vilenkin:1994pv ; Linde:1994wt . According to Sakai:1995nh , this happens for

${|V_{\chi,\chi}|\over V}<{9\over 2\pi}\ .$ (17)

This means that if this slow-roll condition is satisfied at the top of the
potential, inflation there is eternal.

In application to the model (13), eternally inflating topological domain walls
appear if

$M\lesssim 2H\ ,$ (18)

or, equivalently, if Sakai:1995nh

$\chi_{0}\gtrsim 1.7\ .$ (19)

What relation does it have to the two-field evolution in the hybrid inflation
scenario? The answer is that if the field $\varphi$ initially was large,
during the long process of its rolling along the valley of the potential at
$\varphi>\varphi_{c}$ any original oscillations of the field $\chi$ disappear,
and it becomes stabilized at $\chi=0$. Then when the field $\varphi$ passes
the critical point $\varphi_{c}$, the field $\chi$ may start falling down.
Since $V_{\chi}=0$ at $\chi=0$, the field $\chi$ can fall down from $\chi=0$
only due to generation of inflationary perturbations.

Note that while the field $\varphi$ rolls to $\varphi=0$, the Hubble constant
decreases, whereas the absolute value of the tachyonic mass squared
$|M^{2}_{\chi}|$ increases until it reaches $M^{2}$. Therefore if the
condition (17), (18) is satisfied at $\varphi=0$, it is satisfied everywhere
along the ridge $\chi=0$, $\varphi<\varphi_{c}$. In that case we enter the
eternal inflation regime, the process which generates enormous density
perturbations on the scale well within the observable part of the universe.
There perturbations are sufficient for producing not only PBH, but also
eternally inflating domain walls separating parts of the universe with
$\chi>0$ from the parts with $\chi<0$.

Thus the first conclusion which follows from this analysis is that in hybrid
inflation with $M\lesssim H$ it is very easy to produce PBH. The only problem
here is how to avoid overproducing PBH and inflating topological defects.

The simplest way to avoid eternal inflation and to decrease the amplitude of
perturbations below $O(1)$ is to consider models with $\chi_{0}\ll 1$. It is
still possible to have a second stage of fast roll inflation in this scenario
producing large perturbations Garcia-Bellido:1996mdl ; Linde:2001ae . However,
this does not address the problem of superheavy topological defects.

To solve both problems simultaneously, we added the small term $\mu^{3}\,\chi$
to the potential (2). In that case one has
$V_{\chi}(\chi=0,\varphi=0)=\mu^{3}$, and the condition $A_{s}<1$ reads

$\mu^{3}\gtrsim{V^{3/2}\over 2\sqrt{3}\pi}\ .$ (20)

To give a particular example, for $V\sim 10^{{-10}}$ this condition implies
that $\mu^{3}\gtrsim 10^{-16}$. This means that by adding a tiny term
$10^{-16}\chi$ to the potential one can avoid the problems discussed above.
The corresponding constraint on $d=\mu^{3}/M^{2}$ is

$d\gtrsim{M\chi_{0}^{3}\over 2^{4}\,\sqrt{3}\pi\,}\ .$ (21)

As we mentioned in section 3, the term $d\,\chi$ leads to an additional
modification of this scenario. In its absence, the field $\chi$ is rolling
along the minimum of the potential at $\chi=0$, which is why in our previous
estimates of the amplitude of the perturbations we used $\chi=0$ as a starting
point of the inflationary waterfall. However, the term $d\,\chi$ pushes the
field $\chi$ slightly away from $\chi=0$. This happens well before $\varphi$
becomes smaller than $\varphi_{c}$, see (12), (11). Thus, even though the
linear term $d\,\chi$ is very small, it pushes the field $\chi$ away from the
ridge of the potential and from its initial equilibrium state (11). This
additionally decreases the height of the peak of the perturbations. On the
other hand, during the several e-foldings near $\varphi_{c}$ the average
amplitude of the perturbations may grow slightly above $H\over 2\pi$, by a
factor $O(1)$. Therefore it would be interesting to perform a more detailed
investigation of stochastic effects during inflation, following
Starobinsky:1986fx ; Goncharov:1987ir ; Linde:1993xx ; Garcia-Bellido:1993fsr
; Felder:2000hj ; Felder:2001kt ; Finelli:2008zg ; Finelli:2010sh ;
Demozzi:2010aj ; Kawasaki:2015ppx ; Assadullahi:2016gkk ; Pinol:2020cdp .
However, we believe that the simple estimates (20), (21) give a good estimate
of the range of validity of the perturbative analysis to be used in this
paper.

Thus we see that by adding the term $d\chi$ to the potential, one can control
the height of the peak of the perturbations. A detailed analysis of the
perturbations produced in the full two-field scenario is rather sophisticated
and will be given in the next section. Interestingly, we will find that the
spectrum of the perturbations produced at $\varphi$ sufficiently far below the
critical point $\varphi_{c}$ is well described by the theory of perturbations
in the single-field model described in this section, while around the critical
point $\varphi_{c}$ multifield effects play a non-negligible role.

##  5 Hybrid exponential attractors and the spectrum of perturbations

In this section we will perform a full numerical investigation of
perturbations in hybrid attractors. For illustration, we will consider the
exponential attractor model in Eq. (2) and consider the following combination
of parameters as a benchmark

$M=1.47\times
10^{-5}\,,\,\,\,\,\,\alpha=1\,,\,\,\,\,\,\tilde{g}=0.8\,,\,\,\,\,\,\tilde{m}=0.3\,,\,\,\,\,\,\chi_{0}=2.5\,,\,\,\,d=-5\times
10^{-6}\,,$ (22)

which we denote as the baseline parameters and explore how changing each
parameter affects the shape and amplitude of the bump in the primordial power
spectrum that is produced at small scales. Note that we always set the field
$\chi$ initially at rest at $\chi_{i}=0$ and we adopt the initial condition
$\varphi_{i}=3.4$, which corresponds to $83$ $e$-folds of inflation. The
trajectory is shown in the left panel of Fig. 4. Our baseline parameters in
Eq. (22) are chosen so that

$\ln\,10^{10}\,A_{s}=3.043\,,\,\,\,\,\,\,\,n_{s}=0.9618\,,\,\,\,\,\,\,\,\alpha_{s}=3.1\times
10^{-6}\,,\,\,\,\,\,\,\,r_{0.002\,{\rm Mpc}^{-1}}=0.01\,,$ (23)

well consistent with Planck/BICEP/Keck latest constraints at large scales.
Note that the amplitude $\ln\,10^{10}\,A_{s}$, the tilt $n_{s}$ and the
running of the tilt $\alpha_{s}$ are computed at the pivot scale
$k_{*}=0.05\,{\rm Mpc}^{-1}$ assuming that it crosses the Hubble radius
$N_{*}=55$ $e$-folds before the end of inflation222$N_{*}$ is affected by
uncertainties that comes from our incomplete understanding of reheating.
Clearly a larger/smaller $N_{*}$ would shift the peak in the power spectrum
towards smaller/larger scales and give a bluer/redder $n_{s}$ compared to what
we report in the main text.. Changing some of the parameters in (22) also
slightly affects predictions at CMB scales. This effect can always be
counterbalanced by tweaking more parameters at the same time.

We note that recent efforts to solve the so-called $H_{0}$ tension require a
reinterpretation of available data, which, taken at face value, would imply
higher values of $n_{s}$, all the way up to $n_{s}=1$, i.e. the Harrison-
Zeldovich value Braglia:2020bym ; Jiang:2022uyg ; Cruz:2022oqk ; Giare:2022rvg
. While it is not our goal to discuss such claims, we show at the end of this
Section that we can easily obtain larger values of $n_{s}$ in our model.

We will also consider another example described by the parameters:

$M=9.48\times
10^{-6}\,,\,\,\,\,\,\alpha=1\,,\,\,\,\,\,\tilde{g}=1\,,\,\,\,\,\,\tilde{m}=1\,,\,\,\,\,\,\chi_{0}=2.58\,,\,\,\,d=-10^{-5}\,,$
(24)

Its trajectory is shown in the right panel of Fig. 4. As can be seen, compared
to the baseline model, the hybrid field starts to move only when $\varphi$ has
approached $\varphi\simeq 0$. While this example has a spectral index which is
too red to fit Planck data, the waterfall phase here is closer to a single-
field stage of inflation, so it is useful for illustrative purposes and to be
compared to the intuitive arguments presented in the previous Section.

Figure 4: Trajectory in the exponential attractor model for the baseline
parameters in Eq. (22) (left) and for the parameters in Eq. (24) (right). To
better appreciate the differences, we set the same range for the y-axis.

Let us briefly summarize here the mechanism that leads to a peak in
$\mathcal{P}(k)$, also anticipated in the previous Sections, from the
perspective of the full multifield dynamics. The piece of the trajectory
producing the perturbations at the CMB scales consists in a simple slow-roll
phase along the $\varphi$ direction. During this stage, the field $\chi$
slowly increases due to the linear term. Inflation is dominated by the slowly-
rolling inflaton $\varphi$ and curvature perturbations do not experience
amplification. Eventually, when $\varphi$ crosses the critical value
$\varphi_{c}$, the effective mass squared of the field $\chi$ becomes
negative, it develops an instability and isocurvature perturbations start to
grow exponentially.

Even during this period, however, curvature perturbations experience the usual
evolution and freeze to a constant value after Hubble crossing. The trajectory
turns and the field $\chi$ rolls towards the minimum of the potential at
$\chi_{0}$, taking a few $e$-folds to terminate inflation. At the moment of
the turn, the coupling between curvature and isocurvature perturbations is
switched on and power in the latter is quickly transferred to the former,
leading to a large peak in the PPS. We emphasize that the sourcing entirely
occurs on super-Hubble scales, as opposed to recently proposed models where
the turn sources amplification around Hubble crossing Fumagalli:2020adf ;
Palma:2020ejf ; Braglia:2020eai ; Braglia:2020taf ; Iacconi:2021ltm ;
Kallosh:2022vha . It is also important to understand that, while isocurvature
perturbations do grow during inflation, almost all of the power is transferred
to the curvature ones during the turn, leaving the spectrum of primordial
perturbations after inflation adiabatic, consistently with tight bounds on
primordial isocurvature from CMB data Planck:2018jri .

To understand this process, it is useful to investigate the behavior of two
important quantities, i.e. the effective mass squared of isocurvature modes on
superhorizon scales:333We note that sometimes the quantity $m_{\rm
iso}^{2}/H^{2}=m_{\rm eff,\,iso}^{2}/H^{2}-4\eta^{2}_{\perp}$ is also
discussed in the literature. The two quantities are conceptually different as
$m_{\rm eff,\,iso}^{2}/H^{2}$ is the effective mass squared of isocurvature
perturbations obtained integrating out the evolution of curvature
perturbations on super-horizon scales, while $m_{\rm iso}^{2}/H^{2}$ really
corresponds to the mass of isocurvature perturbations. However, in hybrid
inflation, most of the growth of isocurvature perturbations occurs before the
turn, when $\eta_{\perp}^{2}\simeq 0$ and the two quantities are numerically
equivalent.

$\frac{m_{\rm
eff,\,iso}^{2}}{H^{2}}=\frac{1}{\left(\dot{\varphi}^{2}+\dot{\chi}^{2}\right)\,H^{2}}\left[V_{\varphi\varphi}\,\dot{\chi}^{2}-2V_{\chi\varphi}\,\dot{\varphi}\dot{\chi}+V_{\chi\chi}\,\dot{\varphi}^{2}\right]+3\eta^{2}_{\perp},$
(25)

which describes the exponential growth of isocurvature modes, and the turn
rate

$\eta_{\perp}^{2}=\frac{1}{\left(\dot{\varphi}^{2}+\dot{\chi}^{2}\right)\,H^{2}}\left[V_{\varphi}\,\dot{\chi}-V_{\chi}\,\dot{\varphi}\right]^{2},$
(26)

which controls the coupling between curvature and isocurvature modes and the
resulting amplification of the former.

Figure 5:  [Left] Time evolution of the dimensionless spectra of curvature and
isocurvature perturbations in the baseline case. We show the evolution for the
peak scale $k_{\rm peak}\simeq 5.3\times 10^{9}\,{\rm Mpc}^{-1}$ and, for a
comparison, the pivot scale $k_{*}=0.05\,{\rm Mpc}^{-1}$. Green and red areas
correspond to the regions where $\eta^{2}_{\perp}/H^{2}>10^{-2}$ and $m_{\rm
eff,\,iso}^{2}/H^{2}<0$ respectively, as shown in the lower part of the plot.
[Right] PPS at the end of inflation. The yellow-shaded area is the region
$\mathcal{P}_{\mathcal{R}}(k)\gtrsim 10^{-3}$ where we expect a sizable
production of PBHs. The dashed black line is $\mathcal{P}_{\mathcal{R}}=0.01$,
which is the value commonly used as criterium for PBH formation. The
parameters used to produce this figure are the baseline parameters in (22).

These quantities are plotted in the left panel of Fig. 5, together with the
evolution of the dimensionless spectra of curvature $\mathcal{R}_{k}$ and
isocurvature perturbations $\mathcal{S}_{k}$ for some illustrative wavenumbers
$k$. As can be seen, as soon as $\varphi$ crosses $\varphi_{c}$ and $m_{\rm
eff,\,iso}^{2}/H^{2}$ becomes negative, isocurvature perturbations start to
grow. This amplification is experienced by all the modes, however, since
$m_{\rm eff,\,iso}^{2}/H^{2}>0$ for $\varphi>\varphi_{c}$, isocurvature modes
which cross the Hubble radius long before the instability decay before have
sufficiently decayed so that their amplitude is too small to source curvature
perturbations, even if they get amplified (see dashed orange line in the left
panel of Fig. 5).

The net effect of such amplification is to produce a broad bump in the PPS. If
the amplitude of the bump is large enough, the perturbations can collapse into
PBHs when they re-enter the Hubble radius during radiation dominated era
Garcia-Bellido:1996mdl . The criterium for PBHs to form is very sensitive to
many assumptions, and it is not the purpose of this paper to study it in
details. Nevertheless, it generically requires
$\mathcal{P}_{\mathcal{R}}\gtrsim 10^{-3}-10^{-2}$ (see e.g. Refs.
Sasaki:2018dmp ; Carr:2021bzv ; Escriva:2022duf ) as indicated by the yellow
shaded area in the right panel of Fig. 5. This can easily be satisfied by our
model.

The next issue to consider in the masses of PBH formed after inflation. This
is a rather complicated story, we will limit ourselves to simple estimates.
Consider perturbations produced at the peak, $\Delta N=N_{\rm end}-N_{\rm
peak}$ e-foldings from the end of inflation. The mass of PBH can be estimated
by the total energy inside the horizon when these perturbations re-enter the
horizon. If the evolution until that time was matter dominated (which happens
if reheating is very inefficient), then one can roughly estimate $M_{\rm
PBH}\sim O(10)\,H^{{-1}}e^{{3\Delta N}}$, in Planck mass units, where $H$ is
the Hubble constant at the end of inflation. For a radiation dominated
universe (efficient reheating) the masses are expected to be much smaller,
$M_{\rm PBH}\sim O(10)\,H^{{-1}}e^{{2\Delta N}}$ Garcia-Bellido:1996mdl ;
Linde:2012bt . The reason for the smaller mass in the second case is the
redshift of radiation, which occurs prior to the horizon re-enter. In more
realistic situations, where the PBH formation occurs in a universe with
$p=\rho/3$ after a long stage of matter domination, the redshift is less
efficient, and one may expect $M_{\rm PBH}$ between the two limits discussed
above Dalianis:2018frf ; Iacconi:2021ltm .

Figure 6:  Variation of model parameters and their imprints on the PPS. For
each case, we also plot the evolution of $m_{\rm eff,\,iso}^{2}/H^{2}$ and
$\eta^{2}_{\perp}/H^{2}$. [Top-left] Variation of $d$, which controls the
height of the peak. [Top-right] Variation of $\chi_{0}$, which controls the
location of the peak. [Bottom-left] Variation of $\tilde{g}$, which controls
the width of the peak. [Bottom-right] Variation of $\tilde{g}$ for the
parameters in (24). To highlight that the last case is not compatible with
Planck observations, we plot it on top of a red background ($n_{s}$ is redder
in that case). We also note that the x-scale axes on the bottom $N_{\rm
end}-N$ correspond to the time at which the wavenumbers on the top axes cross
the horizon i.e. $k=a(N_{\rm end}-N)H(N_{\rm end}-N)$.

To give a particular example, consider the universe which was radiation
dominated soon after the end of inflation. In this case one may use the
expression that relates the scale of a perturbation to the mass of the formed
PBH Wang:2019kaf :

$\frac{M(k)}{M_{\odot}}\simeq
10^{-16}\left(\frac{\gamma}{0.2}\right)\left(\frac{g(T_{k})}{106.75}\right)^{-1/6}\left(\frac{k}{10^{14}\,{\rm
Mpc}^{-1}}\right)^{-2},$ (27)

where $\gamma$ is a factor that encodes the efficiency of the collapse to PBHs
and $g(T_{k})$ is the effective number of relativistic degrees of freedom at
the time (or, equivalently, temperature) of formation. For example, assuming
$\gamma=0.2$ and $g(T_{k})=106.75$, the baseline power spectrum would produce
a PBH mass function peaked around $M\sim 10^{-8}\,M_{\odot}$. While this is
excluded by microlensing experiments Escriva:2022duf , we will shortly see
that the parameters can be tuned to shift the location of the peak and give
rise to much heavier or much lighter PBHs. For example, the green line in the
top-right panel of Fig. 6 shows a peak at $k_{\rm peak}=1.3\times
10^{12}\,{\rm Mpc}^{-1}$, corresponding to PBHs of masses $M_{\rm PBH}\sim
6\times 10^{-13}\,M_{\odot}$, that can constitute the totality of the Cold
Dark Matter in our Universe Escriva:2022duf . Also note that the abundance of
PBHs depends exponentially on ratio of the primordial power spectrum to the
critical value of the density contrast $\delta_{c}$, therefore, depending on
the specific value adopted, the power spectrum shown in Fig. 5 could result in
no PBHs to an overproduction of PBHs.

The bump-like shape of the power spectrum at small scales suggests that we can
control properties such as its amplitude, location and width by choosing
different combinations of the model parameters. Phenomenologically, this does
not strongly impact the formation of PBHs, which, being a critical process, is
mainly sensitive to the peak amplitude Cole:2022xqc , so the relevant features
are just its amplitude and location. However, the frequency profile has
implications on the SGWB (see next Section), which is sensitive to a wider
portion of the bump. Therefore, we now analyze the effects of varying the
model parameters and show our results in Fig. 6.

Figure 7:  We compare the single-field slow-roll approximations for the
primordial power spectrum to the exact numerical results for the baseline
example in Eq. (22) (left) and for the example in Eq. (24) (right). Red and
green regions have the same meaning as in Fig. 5.

The parameter controlling the amplitude of the peak is the amplitude $d$ of
the linear term in the potential. Intuitively, this is very simple to
understand. As we have shown in the previous section, in the absence of the
linear term in the models with $\chi_{0}\gtrsim 1.7$ one has eternal inflation
at the ridge of the potential at $\chi=0$, which results in $\mathcal{O}(1)$
perturbations. The term $d\,\chi$ moves us away from the eternal inflation
regime, decreasing the amplitude in a controllable way. When the field
$\varphi$ moves sufficiently far away of the critical point $\varphi_{c}$, its
equations of motion depend less and less on the field $\varphi$. Following the
argument in the previous Section, we would like to approximate the resulting
amplitude of perturbations produced at this stage by the simple single-field
equation (16). We illustrate it in Fig. 7.

As can be seen, the approximation is quite bad for the amplitude of the peak
in the baseline example though it becomes reasonable for the spectrum away
from the peak. This approximation becomes much better in the second example,
shown in the right sides of Figures 4, 7. The reason is that in the second
example the turn is sharper and the waterfall stage is entirely driven by
$\chi$, i.e. the trajectory is in the $\chi$ direction, with the field
$\varphi$ being approximately constant $\varphi\simeq 0$. Therefore, during
the last $\sim 10$ $e$-folds after the turn with $\eta_{\perp}^{2}/H^{2}\sim
0$, the numerical power spectrum matches well the slow-roll prediction
$\mathcal{P}_{\mathcal{R}}\simeq V^{3}/12\pi^{2}V_{\chi}^{2}\simeq
H^{2}/8\pi^{2}\epsilon$. On the other hand, in the baseline case the
trajectory is still slowly turning close to the end of inflation, and the
approximation to the single-field inflation driven by $\chi$ is rough.

We note that, though $\epsilon$ only becomes large close to the end of
inflation, it does grow reasonably fast during the waterfall stage, so its
logarithmic derivative, i.e. the second slow-roll parameter $\eta$ is not
negligible. Therefore the simple expressions $\mathcal{P}_{\mathcal{R}}\simeq
V^{3}/12\pi^{2}V_{\chi\,(\varphi)}^{2}\simeq H^{2}/8\pi^{2}\epsilon$, which
are derived at first order in slow-roll receive higher order corrections. In
particular, the expression for the spectrum at horizon crossing $k\sim aH$
becomes Schwarz:2001vv :

$\mathcal{P}_{\mathcal{R}}\simeq\frac{H^{2}}{8\pi^{2}\epsilon}\left[\frac{1}{\pi}\left(2-2\epsilon\right)^{2\nu-1}\,\Gamma^{2}(\nu)\right],$
(28)

where $\nu=1/2\,+\,1/(1-\epsilon)\,+\,\eta/2$ and $\Gamma$ is the Gamma
function. Expressions for this quantity during the stages of inflation driven
by $\varphi$ or $\chi$ can be obtained by using
$\epsilon\simeq\epsilon_{V}=\left(V_{\varphi\,(\chi)}/V\right)^{2}\,/\,2$ and
$\eta\simeq 4\epsilon_{V}-2\eta_{V}$, where
$\eta_{V}=V_{\varphi\varphi\,(\chi\chi)}/V$.

As can be seen from the dashed lines in Fig. 7, the agreement on the spectrum
produced after the peak is now much better.

Another interesting feature that can be understood from Fig. 7 is that the
peak in the power spectrum is very close to the scale that crosses the horizon
when $\varphi=\varphi_{c}$, i.e. the one for which isocurvature perturbations
experience the largest growth without having decayed on superhorizon scales
prior to $\varphi=\varphi_{c}$. While, in the second example, we see from the
plot that the amplitude of the spectrum at $k_{\rm peak}$ can be approximated
quite well by $\mathcal{P}_{\mathcal{R}}\simeq V^{3}/12\pi^{2}V_{\chi}^{2}$,
we stress again, that this approximation is rigorous only after the turn, when
inflation is effectively single-field, and that we must compute the full
multifield dynamics to get exact results.

The location of the peak is controlled by the duration of the waterfall phase,
which essentially depends on the $\chi_{0}$ (see top-right panel of Fig. 6).
Smaller values of $\chi_{0}$ imply that the minimum in the $\chi$ direction is
closer to $\chi\simeq 0$ and the waterfall stage is thus shorter. For this
reason, the amplified modes cross the Hubble radius closer to the end of
inflation and the peak in the power spectrum occurs at smaller scales (larger
wavenumbers). Therefore, tuning $\chi_{0}$, we can easily control the mass of
the produced PBHs, which scales exponentially with $N$, i.e. $M\sim k^{-2}\sim
e^{2(N_{\rm end}-N_{\rm peak})}$, see Eq. (27). Clearly, shifting $\chi_{0}$,
we would need to adjust other parameters to maintain the consistency with
large scale data, as explained below.

Finally, the coupling between the two fields $\chi$ and $\varphi$ is
responsible for the width of the peak. While the coupling does not change the
stage with $m_{\rm eff,\,iso}^{2}/H^{2}<0$, it significantly affects the
evolution of $m_{\rm eff,\,iso}^{2}/H^{2}$ at earlier times. In particular, a
larger value of $g$ makes the effective squared mass of isocurvature
perturbations much larger for $\varphi>\varphi_{c}$. Therefore, isocurvature
modes that are super-horizon before the tachyonic instability decay very fast
and their subsequent amplification is not enough to impact the evolution of
curvature perturbations. On the other hand, modes that cross the Hubble radius
after the tachyonic instability has developed are not affected by this. As
such, the shape of the bump is only modified at scales $k<k_{\rm peak}$ (see
bottom-left panel of Fig. 6). This effect is particularly pronounced in the
bottom-right panel, where we show the variation with $\tilde{g}$ for a
different combination of parameters.

Armed with our understanding of the model parameters and their effects on the
power spectrum, we would like to ask a final question. What is the relation of
our results to the simple $\alpha$-attractor prediction444We only consider the
spectral index here for simplicity. $n_{s}=1-\frac{2}{N}$? Or, in other words,
how much are we taking advantage of the attractor nature of the model?

As we will shortly discuss, these questions are partly related to the last
parameter of the model, whose effects we have not explored so far, i.e. the
parameter $\tilde{m}$, which controls the amplitude of the plateau at large
values of $\varphi$ relative to the uplift. We note that the following
discussion can be straightforwardly generalized to polynomial attractors model
that we construct in Section 7, which have different attractor predictions,
see e.g. Eq. (39).

Let us first clarify the meaning of $N$ in $n_{s}(N)=1-\frac{2}{N}$. In
single-field $\alpha$ attractors models, $N$ represents the number of
$e$-folds from the moment at which the pivot scale crosses the Hubble radius
to the end of inflation. Therefore, to be as clear as possible, the quantity
we are interested is $n_{s}(N_{*})$. As stressed above, we adopt $N_{*}=55$ in
our paper. More specifically, the attractor expression is derived by
analytically solving the slow-roll equation for the inflaton $\varphi(N)$ as a
function of $N$ and plugging it into the slow-roll approximation for the
spectral index $n_{s}\simeq 1-6\epsilon_{V}+2\eta_{V}$.

However, in our model, where we have two stages of inflation, the equation for
$n_{s}$ needs to be modified as Iacconi:2021ltm ; Kallosh:2022ggf :

$n_{s}=1-\frac{2}{N_{*}-\Delta N}.$ (29)

This is very easy to understand, as the second stage shifts the end of
inflation by $\Delta N$. In this way, we relate the value of $n_{s}$ to the
region of the plateau that is producing the perturbations at $N_{*}$. However,
how do we define $\Delta N$? An intuitive way to do so would be to define it
as the distance from the time where $\eta_{\perp}^{2}/H^{2}$ peaks, which
roughly separates the region two stages of inflation driven $\varphi$ and
$\chi$, to the end of inflation. However, in our model, there are situations
where this is not the best definition for $\Delta N$. Take the second example
in Fig. 4. There, $\varphi$ approaches $0$ some $e$-folds after
$\epsilon_{\varphi}=(V^{\prime}/V)^{2}/2$ reaches its maximum, i.e. the point
of maximum velocity of $\varphi$, and significantly slows down afterwards.
However, since $\chi$ is moving very slowly and its kinetic energy is much
smaller than that of $\varphi$, it takes roughly $\sim 10$ $e$-folds before
$\eta_{\perp}^{2}/H^{2}$ peaks and the trajectory turns into the $\chi$
direction. This is different from what would happen in the single-field case,
where inflation would end shortly after $\epsilon_{\varphi}=$ reaches its
maximum value, which in that context is $\epsilon_{\varphi}\,=\,1$.

A more sensible definition, which resembles more the one single-field
$\alpha$-attractor model, is in fact to choose the time at which
$\epsilon_{\varphi}$ peaks, so that $\Delta N\equiv
N_{\rm\epsilon_{\varphi}\,=\,{\rm maximum}}-N_{\rm end}$555We stress that this
is $\Delta N$ is not the duration of the waterfall stage, which instead begins
when $\varphi=\varphi_{c}$ and $\chi$ develops an instability. . In this way,
regardless of what happens after $N_{\rm\epsilon_{\varphi}\,=\,{\rm
maximum}}$, Eq. (29) should match quite well the prediction, as long as the
first stage of inflation is independent from the first one. This, in turn
depends mainly on the combination of parameters $\tilde{m}\,\sqrt{\alpha}$.

Also, note that, even after the modification $N_{*}\to N_{*}-\Delta N$ the
simple Eq. (29) would be adequate only if there was no uplifting potential. In
that case, the equation needs to be further modified as:

$n_{s}-1=1-\frac{2V_{0}\gamma^{2}}{V_{0}\gamma^{2}\left(N_{*}-\Delta
N\right)+\left(V_{\rm up}+V_{0}\right)e^{\gamma s_{c}}},$ (30)

where $\gamma=\sqrt{2/3\alpha}$, $s_{c}=\varphi_{c}-\frac{1}{\gamma}\ln 4$,
$V_{0}=3\alpha\tilde{m}^{2}M^{2}$ and $V_{\rm up}=M^{2}\chi_{0}^{2}/4$, see
Eq. (4.10) of Ref. Kallosh:2022ggf . Note that Eq. (30) reduces to Eq. (29) if
$V_{\rm up}+V_{0}\ll V_{0}\gamma^{2}(N_{*}-\Delta N)e^{-\gamma s_{0}}$. On the
other hand, in the large uplift limit $V_{\rm up}+V_{0}\gg
V_{0}\gamma^{2}(N_{*}-\Delta N)e^{-\gamma s_{0}}$ the value of $n_{s}$ moves
towards $n_{s}=1$ Kallosh:2022ggf .666Note that with an increase of $V_{\rm
up}$ the attractor $n_{s}=1$ may be reached twice. Indeed, when $V_{\rm up}$
increases, the evolution of the field $\varphi$ slows down, so the last $55$
e-foldings of inflation may happen at $\varphi<\sqrt{6\alpha}$, when the
potential $\sim\tanh^{2}{\varphi\over\sqrt{6\alpha}}$ is reduced to the
original quadratic potential. In this case, with the growth of $V_{\rm up}$
the value of $n_{s}$ crosses $n_{s}=1$, continues to grow for a while, and
then decreases and approaches its final attractor value $n_{s}=1$. And with
the account taken of the full two-field evolution, the situation becomes even
more complicated, which is why we needed a detailed numerical analysis
described in this section.

Figure 8:  Varying $\tilde{m}$ and its effect on the field evolution (left)
and the primordial power spectrum (right).

In Fig. 8, we show the sensitivity of our results to the parameter
$\tilde{m}$. We plot the evolution of the field $\varphi$ together with the
corresponding primordial power spectra. The change in $\tilde{m}\sqrt{\alpha}$
is accompanied by a change in the initial condition $\varphi_{i}$ to get the
same number of $e$-folds in all the cases and by a change in the $d=-10^{-5}$
in the the case with $\tilde{m}\sqrt{\alpha}=1$ and $\tilde{g}=2.5$ in the
case with $\tilde{m}\sqrt{\alpha}=5$, so that all the spectra show a peak of a
comparable amplitude. $\alpha$ and $\chi_{0}$ are kept fixed to the baseline
values. For a comparison, we show in red the evolution of $\varphi$ in the
single-field $\alpha$-attractor model.

As can be seen, the evolution for $\varphi$ gets more and more similar to the
corresponding single-field evolution as $\tilde{m}\sqrt{\alpha}$ increases,
recovering the attractor predictions for large values of $V_{0}/V_{\rm up}$.
This is confirmed by the values of $n_{s}$ reported in Fig. 8, which agree
with Eq. (30). Therefore, the conclusion is that, if we want to take advantage
of the attractor predictions, we need a large $V_{0}/V_{\rm up}$. In that
case, however, for typical values of $\Delta N$ required to have interesting
phenomenological effects at small scales, $n_{s}\sim 1-2/(N_{*}-\Delta N)$ is
too red to fit Planck data.

Let us stress that this does not mean that our model is excluded by
observations. In fact, as discussed above, the baseline example perfectly
agrees with Planck measurements. However, in that example, we must go slightly
off the attractor regime, and compensate for the decrease of $n_{s}$ by an
increase of $n_{s}$ due to the uplift by $V_{\rm up}=M^{2}\chi_{0}^{2}/4$
Kallosh:2022ggf ).

Things are different for polynomial hybrid attractors that we introduce in the
next Section, where one does not need to rely on uplifting for consistency
with the Planck data. But for the exponential attractors discussed in this
Section, the proximity of the attractor regime is also very helpful. For
example, if $\Delta N=10$ and $N_{*}\sim 55$, the attractor prediction
ignoring the uplifting would be $n_{s}\sim 0.956$. This would inform us that
with an account taken of uplifting the value of $n_{s}$ does not go below
$0.956$, and we only need an uplift increasing $n_{s}$ by less than $0.01$ to
make it consistent with the Planck results. It is always possible to do it; in
the large uplift limit one can bring $n_{s}$ all the way up to $n_{s}=1$
Kallosh:2022ggf . This healing power of uplift may play a significant role in
the development of advanced inflationary models compatible with the changing
observational landscape, where the attempts to resolve the $H_{0}$ problem may
push us towards considering inflationary models with large $n_{s}$
Braglia:2020bym ; Jiang:2022uyg ; Cruz:2022oqk ; Giare:2022rvg .

Figure 9:  Evolution of perturbations for the cases shown in the top left
panel of Fig. 6 with $d=-5\times 10^{-6}$ (left) and $d=-10^{-6}$ (right). We
plot the evolution for the mode corresponding to the peak in the power
spectrum and we denote the horizon crossing with a small blue circle. The red
and green shaded regions have the same meaning as in Fig. 5.

Growth of perturbations. Before concluding, we note that since the
perturbations in our model grow very large, care should be taken for them not
to backreact on the background evolution. As a naive criterium, which should
anyway give a good estimate of when backreaction is negligible, we compare the
background values of $\varphi$ and $\chi$ to the evolution of the
corresponding perturbations $\sim\sqrt{\mathcal{P}_{Q_{\varphi}}}$ and
$\sim\sqrt{\mathcal{P}_{Q_{\chi}}}$. Here, we denote the gauge-invariant
perturbations to the two fields by $Q_{\varphi}$ and $Q_{\chi}$. We show this
in Fig. 9, where we plot the evolution of the $k_{\rm peak}$ modes, which is
the one that takes larger values.

In the left panel, corresponding to the baseline case, where
$\mathcal{P}_{\mathcal{R}}(k_{\rm peak})\sim 0.007$, we see that the
perturbations are always subdominant with respect to the background fields. In
particular, after the perturbation to $\chi$ crosses the horizon, where it
takes the typical value $H/2\pi$, it starts growing because of the tachyonic
instability, but we see that the background value for $\chi$ is always at
least $\sim 15$ times larger. We therefore conclude that in this case the
perturbations should not affect the background evolution and our results are
robust. As another example, we show the evolution for $k_{\rm peak}$ in the
case of a smaller $d=-10^{-6}$, also shown in Fig. 6. We see that also in this
case perturbations remain small compared to the background value, although now
$\sqrt{\mathcal{P}_{Q_{\chi}}}$ is only a factor of $\sim 3$ smaller, so the
two quantities are of the same order of magnitude. However, this case has a
very large peak amplitude $\mathcal{P}_{\mathcal{R}}(k_{\rm peak})\sim 0.17$
so that it is excluded by the over-production of PBHs and also by current
Pulsar Timing Array (PTA) limits on the SGWB induced by the scalar
perturbations at second order (see next Section).

#### A larger $n_{s}$.

Although the example above is perfectly consistent with CMB observations, it
is interesting to explore whether larger values of $n_{s}$ can be attained in
our model. We show two examples in Fig. 10, where $n_{s}$ falls both on the
right of the sweet spot of the Planck/BICEP/Keck. Note that also a non-
negligible amount of running is produced, which is nevertheless consistent
with the constraints from Planck Planck:2018jri . As seen in the left panel,
the main difference with respect to the baseline evolution is that, since $g$
is larger and $\varphi_{c}$ smaller, the hybrid field $\chi$ starts to move
later during inflation, with a resulting shorter waterfall stage. The CMB
scales therefore test the region where the plateau is flatter and $n_{s}$
bluer. However, also in this case, the uplifting potential is quite large,
i.e. the ratio $V_{0}/V_{\rm up}$ is small, and we cannot rely on the simple
attractor relation $n_{s}=1-2/(N_{*}-\Delta N)$, as discussed above.

We also note another interesting feature of the power spectrum in Fig. 10. The
power spectra in the yellow-shaded region rises with a spectral index of
roughly $10.9$ and $11.2$ for Case A and B respectively. This growth is much
steeper than $n_{s}\sim 5$ (plus a weak running) which is the limit in the
case of canonical single field inflation Byrnes:2018txb ; Carrilho:2019oqg ,
confirming that multiple field effects can easily overcome such limit
Palma:2020ejf ; Fumagalli:2020adf ; Braglia:2020taf .

Figure 10:  Examples with a larger $n_{s}$. We use the following parameters.
Case A:
$\alpha=0.9\,,\,\tilde{m}=0.35\,,\,\chi_{0}=2.5\,,\,\,\,\tilde{g}=3.7\,,\,\,\,d=5\times
10^{-6}\,$. Case B:
$\alpha=0.6\,,\,\tilde{m}=0.35\,,\,\chi_{0}=2.5\,,\,\,\,\tilde{g}=5\,,\,\,\,d=5\times
10^{-6}\,$. The parameter $M$ is fixed to match the COBE normalization and
$\varphi_{i}$ to achieve $~{}70$ $e$-folds of inflation. The potential is
shown in units $M^{2}$. The trajectory in the left panel is shown for Case A,
but is representative also of Case B, which shares the same features.

##  6 Stochastic Gravitational Wave background generated at horizon re-entry

It is well known that scalar and tensor perturbations are decoupled at linear
order. However, they are mixed at higher orders in perturbation theory. In
particular, scalar perturbations act as a source of gravitational waves at
second order in perturbation theory when they re-enter the Hubble radius
during the radiation dominated era777In principle, since the perturbations
produced during inflation are quite large, in addition to the gravitational
waves produced during the radiation era, we could also expect a sizable
background to be sourced already during inflation Biagetti:2013kwa . However,
as shown in Ref. Fumagalli:2021mpc , its amplitude is suppressed, relative to
that produced after inflation, by powers of $\epsilon$ and $\eta_{\perp}/H$.
Since both factors are much smaller than $1$ in our model, we ignore this
contribution. Carbone:2004iv ; Ananda:2006af ; Baumann:2007zm . If their
amplitude is large enough, the resulting SGWB can fall into the sensitivity of
future gravitational wave interferometers, that can therefore be used to test
the physics of Hybrid inflation. It is therefore crucial to make accurate
predictions on the spectral shape of the SGWB.

Figure 11:  SGWB sourced by the large amplitude perturbations. We consider the
same parameters as in Fig. 6. Magenta, turquoise-green and grey shaded regions
correspond to the sensitivities of SKA Zhao:2013bba , LISA
LISACosmologyWorkingGroup:2022jok  and BBO/DECIGO Yagi:2011wg .

The energy density of the SGWB at present time is given by Kohri:2018awv ;
Espinosa:2018eve :

$\Omega_{\rm ind}(k)=0.387\,\Omega_{\rm
R}\left(\frac{g_{*,s}^{4}g_{*}^{-3}}{106.75}\right)^{-\frac{1}{3}}\frac{1}{6}\int_{-1}^{1}\mathrm{d}x\int_{1}^{\infty}\mathrm{d}y\,\mathcal{P}_{\mathcal{R}}\left(\frac{y-x}{2}k\right)\mathcal{P}_{\mathcal{R}}\left(\frac{x+y}{2}k\right)F(x,y)\,,$
(31)

where $\Omega_{\rm R}=5.38\times 10^{-5}$ is the radiation abundance
Planck:2018vyg , the effective numbers of degrees of freedom, $g_{*,s}$ and
$g_{*}$, are evaluated at the moment when the constant abundance is reached,
roughly coinciding with the horizon crossing time, and

$\displaystyle F(x,y)=$
$\displaystyle\frac{288(x^{2}+y^{2}-6)^{2}(x^{2}-1)^{2}(y^{2}-1)^{2}}{(x-y)^{8}(x+y)^{8}}$
(32) $\displaystyle\hskip
28.45274pt\times\left[\left(x^{2}-y^{2}+\frac{x^{2}+y^{2}-6}{2}\log\left|\frac{y^{2}-3}{x^{2}-3}\right|\right)^{2}+\frac{\pi^{2}}{4}(x^{2}+y^{2}-6)^{2}\theta(y-\sqrt{3})\right]\,.$
(33)

We show in Fig. 11 the SGWB spectra corresponding to the scalar power spectra
shown in Fig. 6, compared to the sensitivity of future experiments. We also
plot in cyan the recent claim of a detection from the NANOGrav collaboration
NANOGrav:2020bcs . However, since there is no evidence for the so-called
Hellings and Downs curve, which would be the smoking gun for a gravitational
wave background detection, we interpret the data point as an upper limit.

We note that $\Omega_{\rm GW}$ depends on the model parameters qualitatively
in the same way as the curvature power spectrum. This is obvious because the
integral in Eq. (31) suggests that $\Omega_{\rm
GW}(f)\propto\mathcal{P}_{\mathcal{R}}^{2}(f)$. We see that the background
produced in Hybrid inflationary model typically falls in the sensitivity
region of PTA experiments (magenta shaded area) and space-based gravitational
waves interferometers such as LISA (turquoise-green shaded area) and more
futuristic ones like BBO and DECIGO (grey shaded area). Some values of the
parameter space, in particular those featuring a too-large amplification of
the spectrum are already excluded by current PTA limits from the NANOGrav
collaboration. Interestingly, the models excluded are those for which our
analysis is less reliable and we should start to take into account stochastic
effects the backreaction on the background dynamics during inflation, see a
discussion in Section 4, and in the discussion of Fig. 9.

Reconstructing the spectral shape of the SGWB is a target of future
gravitational wave experiments Caprini:2019pxz ; Flauger:2020qyi , it is thus
important to provide simple templates for $\Omega_{\rm GW}(f)$ that can be
readily used in such searches. Since the computation of the primordial
curvature power spectrum, and therefore of the induced SGWB, has to be
performed numerically, such template is intended to be purely phenomenological
and a one-to-one connection with the model parameters is not straightforward.
Nevertheless, having a template that closely matches the exact result has the
clear advantage of simplicity. In case of a detection of the phenomenological
template, the exact signal can be compared numerically and constraints on the
true model parameters can be set.

Focusing for simplicity on LISA, we see that, depending on where the peak is
located, we may need a more complex template to capture the signal. For
example, if the peak falls outside the sensitivity, the SGWB can be just
described by a simple power law, as for the blue line in the top-left panel of
Fig. 11. If, on the other hand, the peak is exactly in the sensitivity range
of LISA, a simple power-law would not be precise enough to describe the richer
structure of $\Omega_{\rm GW}(f)$ and some more advanced functions to describe
the peak will be needed. This is the case of the green line in the top-right
panel of Fig. 11.

As we can see, a simple lognormal shape for $\Omega_{\rm GW}$, i.e. a Gaussian
in log-space, captures very well the region close to the peak, but it fails to
describe the power-law behaviors at frequencies smaller and larger than the
peak frequency $f_{\rm peak}$. While this accuracy is probably acceptable for
LISA, whose sensitivity band is narrow enough not to see this deviation, it
would be nice to have an even more accurate template, suitable for the next-
generation of space based experiments. Also, this template is important to
describe situations, such as the green line in the bottom-left panel of Fig.
11, where the background falls in the sensitivity range of multiple
experiments.

We see that a simple broken power law is not able to reproduce our signal
either. A solution is to smooth the transition between the two power-laws at
the pivot frequency $f_{\rm peak}$ and adopt the following smoothed broken
power-law template:

$h^{2}\Omega_{\rm GW}(f)=A\,\frac{\mathcal{F}(f,\,f_{\rm
peak},\,n_{1},\,n_{2},\,\Delta)}{\mathcal{F}(f_{\rm peak},\,f_{\rm
peak},\,n_{1},\,n_{2},\,\Delta)}\ ,$ (34)

where

$\mathcal{F}(f,\,f_{\rm
peak},\,n_{1},\,n_{2},\,\Delta)=\frac{\left[\frac{f}{f_{\rm
peak}}\left(-\frac{n_{1}}{n_{2}}\right)^{\Delta}\right]^{n_{1}}}{\left\\{\frac{1}{2}\left[1+\left(\frac{f}{f_{\rm
peak}}\left(-\frac{n_{1}}{n_{2}}\right)^{\Delta}\right)^{1/\Delta}\right]\right\\}^{\Delta(n_{1}-n_{2})}}\
.$ (35)

As can be seen, with a proper choice of parameters, the template accurately
matches the exact result and can therefore used in phenomenological searches
for the SGWB produced in hybrid attractors models.

##  7 Hybrid polynomial attractors

The hybrid attractor scenario described in this paper was mainly devoted to
the so-called exponential attractors, such as
$V_{0}\tanh^{2}\big{(}{\varphi\over\sqrt{6\alpha}}\big{)}$, where the
potential of the field $\varphi$ approaches the plateau exponentially,

$V(\varphi)=V_{0}(1-e^{-\varphi/\mu}+...)\ .$ (36)

But there is yet another broad class of cosmological attractors, where the
potential approaches the plateau more slowly, not exponentially but as inverse
powers of the inflaton field,

$V\sim V_{0}(1-{\mu^{k}\over\varphi^{k}}+...)\ ,$ (37)

where $k$ can be any (integer or not) positive constant. The simplest examples
of such potentials are given by

$V=V_{0}\,{\varphi^{k}\over\varphi^{k}+\alpha^{k}}\,,$ (38)

They were called polynomial attractors Kallosh:2022feu . Such potentials may
appear in several different contexts, such as KKLTI inflation Kachru:2003sx ,
pole inflation Kallosh:2019hzo , and also as a special version of
$\alpha$-attractors Kallosh:2022feu .

In all of these cases, at $\alpha\ll 1$ in the large $N$ limit these
potentials have universal attractor predictions, independently of other
details of the potential indicated by the ellipsis $...$ in Eq. (37). In
particular, the spectral index $n_{s}$ depends only on $k$ Kallosh:2018zsi :

$n_{s}=1-{2\over N}{k+1\over k+2}\ .$ (39)

Here we will consider the hybrid inflation scenario where instead of the
potential ${m^{2}\over 2}\phi^{2}$ we will use a potential $V={m^{2}\over
2}{\alpha^{2}\varphi^{2}\over\varphi^{2}+\alpha^{2}}$. At
$\varphi^{2}\ll\alpha^{2}$ this potential is given by the familiar expression
${m^{2}\over 2}\varphi^{2}$, but at $\varphi^{2}\gg\alpha^{2}$ it approaches a
plateau $V_{0}={m^{2}\alpha^{2}\over 2}$.

Ignoring, for simplicity, the hybrid inflation uplifting, these models have
universal predictions Kallosh:2018zsi

$n_{s,{\rm KKLTI}}=1-{3\over 2N}\ ,\qquad r={\sqrt{2}\alpha\over N^{3/2}}\ .$
(40)

The corresponding hybrid inflation potential based on this particular version
of the polynomial attractor is given by

$V_{\rm{\rm
poly}}(\chi,\varphi)=M^{2}\left[{({\chi^{2}-\chi_{0}^{2}})^{2}\over
4\chi_{0}^{2}}+{\tilde{m}^{2}\over
2}{\alpha^{2}\varphi^{2}\over\varphi^{2}+\alpha^{2}}+\frac{\tilde{g}^{2}}{2}\varphi^{2}\chi^{2}+d\chi\right].$
(41)

Figure 12:  [Left] Evolution of the field $\varphi$ in the hybrid polynomial
attractors model. [Right] Primordial power spectra. The parameters for each
example are reported in the Table in the main text.

In Fig. 12, we provide some examples of the evolution of the field $\varphi$
in this model, together with the associated power spectra. We fix the
following parameters $\alpha=0.05$, $\lambda=1/\chi_{0}^{2}=0.3$ and $M$ is
chosen to match the COBE normalization, as usual. The small value of $\alpha$
is chosen here to take advantage of the attractor regime of polynomial
attractors Kallosh:2019hzo . The other parameters are varied according the
following table. As discussed in the previous Section, the number of $e$-folds
that is used in Eq. (40) is $N_{\rm KKLTI}=N_{*}-\Delta N=55-\Delta N$, where
we assume $N_{*}=55$.

| $d$ | $\tilde{m}\alpha$ | $\tilde{g}$ | $\Delta N$ | $n_{s,{\rm KKLTI}}$ | $n_{s,\,{\rm num}}$
---|---|---|---|---|---|---
Example 1 | $-2\times 10^{-7}$ | $0.7$ | $2$ | $9.43$ | $0.9671$ | $0.9670$
Example 2 | $-10^{-7}$ | $0.3$ | $6$ | $11.00$ | $0.9659$ | $0.9652$
Example 3 | $-6\times 10^{-8}$ | $0.05$ | $8$ | $8.91$ | $0.9674$ | $0.9662$

We can see that the main features of the peak in the power spectrum are
essentially the same as in the exponential hybrid attractor models. The main
difference is that the predictions at large scales follow a different
attractor. Like in the previous Section, we observe that increasing
$\tilde{m}\alpha$ brings the analytical predictions into closer agreement with
the numerical results for $n_{s}$. However, unlike in the case of exponential
attractors, here this allows us to take full advantage of the attractor nature
of the model. Indeed, since now $n_{s}=1-3/2(N_{*}-\Delta N)$, the attractor
predictions are consistent with Planck data even for the value of $\Delta
N\sim 10$ needed to get a phenomenologically interesting peak at small scales
in this model. Therefore in this context we do not need to rely on uplifting
to shift the attractor prediction upwards to larger values of $n_{s}$.
However, we can use this mechanism if we want to consider models with larger
values of $\Delta N$, or if we want to increase $n_{s}$ even further.

##  8 Discussion

Models of inflation producing a large amplitude peak at small scales are the
subject of a very active field of research, primarily because they potentially
lead to the production of PBHs and a SGWB testable with future gravitational
wave facilities. Historically, hybrid inflation Linde:1991km ; Linde:1993cn
was one of the first models where such an amplification of power was
considered Garcia-Bellido:1996mdl , see also Randall:1995dj ; Kawasaki:1997ju
; Yokoyama:1998pt ; Frampton:2010sw ; Kawasaki:2016pql ; Choi:2021yxz ;
Spanos:2021hpk ; Kawai:2022emp . In this paper, we investigated the
possibility to generate perturbations with a very high peak in the spectrum in
its recently developed $\alpha$-attractor generalization Kallosh:2022ggf  of
the hybrid inflation scenario and discussed its consequences for PBH and SGWB
production.

This scenario describes an inflaton field $\varphi$ interacting with a Higgs-
type field $\chi$ in such a way that the latter becomes tachyonic when
$\varphi$ becomes smaller than some critical value $\varphi_{c}$. In the
simplest cases, the absolute value of the tachyonic mass of the field $\chi$
becomes much greater than the Hubble constant $H$ at $\varphi<\varphi_{c}$. Is
such cases, the field $\chi$ falls down to the minimum of the potential at
$\chi=\chi_{0}$ and inflation abruptly ends at $\varphi\approx\varphi_{c}$
Linde:1991km ; Linde:1993cn .

However, for $\chi_{0}\gg 1$ the tachyonic mass of the field $\chi$ at
$\varphi\lesssim\varphi_{c}$ remains much smaller than $H$. In such cases,
inflation continues while the field $\chi$ slowly rolls down towards the
minimum of $V(\chi)$. As we explained in this paper, in this case, at
$\varphi<\varphi_{c}$, the universe enters the eternal inflation regime: In
some parts of the universe inflation eternally continues at $\chi=0$, whereas
in many other parts inflation ends within a relatively small number of
$e$-foldings $\Delta N$. This leads to enormous perturbations of density on
the scale proportional to $e^{\Delta N}$, which results in copious production
of PBHs and eternally inflating parts of the universe, including eternally
inflating topological defects.

This is a very general property of this class of models, which does not
require any fine-tuning, so it is very easy to produce a very high peak of
perturbations and PBH in this scenario. The only problem is how to keep the
situation under control, to avoid overproduction of both PBHs and eternally
inflating topological defects inside the observable part of the universe.

One way to do it is to reduce $\chi_{0}$, which stops eternal inflation and
suppress the amplitude of perturbations Garcia-Bellido:1996mdl . However, this
is not sufficient to solve the problem of supermassive topological defects.
Another problem is how to make a large bump at some intermediate range of
$e$-foldings and simultaneously have perturbation with $n_{s}\sim 0.96-0.97$
compatible with the Planck data.

In this paper, we showed that it is possible to solve all of these problems in
the $\alpha$-attractor generalization of hybrid inflation Kallosh:2022ggf  if
we add a tiny linear term $\mu^{3}\chi$ to the potential. This term pushes the
inflationary trajectory slightly off the ridge of the potential. This solves
the problem of topological defects and simultaneously decreases the amplitude
of the perturbation in a controllable way. The position and the width of the
peak can be controlled by other parameters of the hybrid inflation potential.

Phenomenologically, key features of this model are the agreement with CMB
measurements from Planck and the production of PBHs and SGWB at smaller
scales, that offer nice prospects for testing it in the future. Indeed, many
realizations of hybrid inflation were excluded by the earliest Planck release
Planck:2013jfk . Thanks to the $\alpha$-attractor modification of the model,
one can find a broad range of parameters where $n_{s}$ and $r$ are consistent
with the latest Planck/BICEP/Keck results. On smaller scales, the model is
flexible enough to produce bumps at different locations and with different
shapes, making it possible to produce very light to heavy PBHs, possibly
constituting the totality of the Cold Dark Matter in our universe if the peak
is located at the right position. While the formation of PBHs is mainly
sensitive to the overall amplitude of the peak, future observations of the
induced SGWB have the potential to unveil its spectral shape over orders of
magnitude in frequency. Our model thus constitute a primary target of future
searches for stochastic backgrounds of gravitational waves. To this purpose,
we have also derived a very simple template that perfectly capture the
frequency profile of the gravitational wave energy spectrum, that can be
readily used by the gravitational wave community to test our model.

We note that the hybrid attractor model proposed in Kallosh:2022ggf , and
further developed in this paper, represents a simple generalization of the
basic hybrid inflation model Linde:1991km ; Linde:1993cn . In the Appendix A
it is shown how this scenario can be implemented in supergravity. One can also
generalize this two-field model to include many other fields. In particular,
instead of the inflaton potential ${m^{2}\over 2}\varphi^{2}$ one can consider
O(2) invariant potential ${m^{2}\over 2}(\varphi_{1}^{2}+\varphi_{2}^{2})$. In
such model instead of domain walls we would have global strings. If we
consider a potential ${m^{2}\over
2}(\varphi_{1}^{2}+\varphi_{2}^{2}+\varphi_{3}^{2})$ we would have global
monopoles. By considering more general potentials for the field $\chi$,
including hilltop inflationary potentials such as
$(\chi^{4}-\chi_{0}^{4})^{2}$, or the Coleman-Weinberg potential used in new
inflation, one may have a second inflationary stage with an extremely small
value of $\chi_{0}$. In such models one may find a different way to control
the amplitude of the spectrum and avoid problems with topological defects. In
fact, as shown in Vilenkin:2018zol ; Matsuda:2005ey , in certain cases such
defects may independently contribute to the PBH production.

## Acknowledgement

R.K. and A.L. are supported by SITP and by the US National Science Foundation
Grant PHY-2014215. A.L. is grateful to Juan Garcia-Bellido for many useful
discussions and collaboration on related projects. F.F. acknowledges financial
support by the agreement n. 2020-9-HH.0 ASI-UniRM2 “Partecipazione italiana
alla fase A della missione LiteBIRD”.

##  Appendix A Supergravity implementation of hybrid attractors

###  A.1 Hybrid $\alpha$-attractors with a canonical waterfall field, adding
axions

In the context of the PBH production hybrid inflation was studied in
supersymmetric F- and D-term inflation Clesse:2015wea . All fields in these
models have canonical kinetic terms, therefore the choice of parameters
consistent with CMB as well as leading to PBH production is not easy to make.
For example in D-term supergravity model the Planck-like FI term and a
superpotential with large coupling are required.

Our models have plateau potentials with non-canonical geometric inflaton at
the first stage of inflation, which facilitates their agreement with the CMB
data.

The $\alpha$-attractor versions of hybrid inflation models were proposed and
studied in Kallosh:2022ggf . The first stage of inflation was defined by a
plateau potential for the field $\varphi$ originating from a geometric
hyperbolic disk geometry with the Kähler curvature ${\cal R}=-{2\over
3\alpha}$. The waterfall phase of hybrid inflation is due to a second field
$\chi$ which was also defined in Kallosh:2022ggf  by a geometric hyperbolic
disk geometry with the Kähler curvature ${\cal R}=-{2\over 3\beta}$.

In the models we study here we take the waterfall $\chi$ to be a canonical
field, $\beta\to\infty$. For our purpose to study high peaks in inflationary
perturbations any value of $\beta>1$ is suitable, not much depends on it.
Small values of $\beta$ are more interesting if we design the second stage of
inflation as a quintessence cosmology, which is not the purpose of this paper.
Therefore, in our case a canonical waterfall field is the simplest
possibility.

Here we will present the supergravity version of the cosmological models for a
geometric inflaton $\varphi$ and a canonical waterfall field $\chi$ studied in
this paper. These models can also be obtained from supergravity versions of
general hybrid inflation in Kallosh:2022ggf  in the limit $\beta\to\infty$.

We start as follows

${{\cal L}\over\sqrt{-g}}={R\over 2}-{(\partial_{\mu}\phi)^{2}\over
2\bigl{(}1-{\phi^{2}\over 6\alpha}\bigr{)}^{2}}-{(\partial_{\mu}\chi)^{2}\over
2}-V(\chi,\phi)\,,$ (42)

where

$V(\phi,\chi)=M^{2}\Big{(}V_{\rm hybrid}(\chi,\phi)+V_{\rm
lin}(\chi)\Big{)}\,,$ (43)

Here

$V_{\rm hybrid}(\phi,\chi)={m^{2}\over 2}\phi^{2}+{g^{2}\over
2}\phi^{2}\chi^{2}+{({\chi^{2}-\chi_{0}^{2}})^{2}\over 4\chi_{0}^{2}}\ ,$ (44)
$V_{\rm lin}(\chi)=d\chi\ .$ (45)

To be careful, one should also subtract a tiny constant $\approx d\,\chi_{0}$
to ensure smallness of the cosmological constant at the minimum, but this term
can be ignored in the investigation of inflation.

Upon transformation to canonical variables
$\phi=\sqrt{6\alpha}\,\tanh{\varphi\over\sqrt{6\alpha}}$ we find

${{\cal L}\over\sqrt{-g}}={R\over 2}-{1\over
2}(\partial_{\mu}\varphi)^{2}-{1\over
2}(\partial_{\mu}\chi)^{2}-V(\varphi,\chi)\,.$ (46)

The hybrid potential (43) as a function of $\varphi$ and $\chi$ becomes (4)

$\displaystyle
V(\varphi,\chi)=M^{2}\Big{[}3\alpha\,\tanh^{2}{\varphi\over\sqrt{6\alpha}}(\tilde{m}^{2}+\tilde{g}^{2}\,\chi^{2})+{({\chi^{2}-\chi_{0}^{2}})^{2}\over
4\chi_{0}^{2}}+d\chi\Big{]}\ .$ (47)

The model given in eqs. (46), (47) is a model with 2 bosonic fields. In the
main part of the paper we studied the cosmological properties of this model.

We would like to embed this bosonic model into a supergravity model. Such an
embedding is not unique, we choose here a relatively simple one. The first
step is to relate each of the scalars $\varphi$ and $\chi$ to complex scalars
$Z_{1},Z_{2}$. This means also that two scalars of the original bosonic models
are supplemented by two axions.

$Z_{1}=ze^{i\theta_{1}}=\tanh{\varphi\over\sqrt{6\alpha}}e^{i\theta_{1}}\
,\qquad Z_{2}={1\over\sqrt{2}}\chi e^{i\theta_{2}}\ .$ (48)

Now we need to make sure that there is a minimum in $\theta_{i}$ directions at
$\theta_{i}=0$, so that both axions are stabilized at their vanishing value.

Many cosmological supergravity models associated with string theory and
uplifting anti-D3 branes which were proposed and studied during the last few
years involve an additional nilpotent chiral superfield $X$ such that
$X^{2}=0$. It represents non-linearly realized supersymmetry of the Volkov-
Akulov type.

The reason why the embedding proposed in (48) is relatively simple is the
fact, established earlier in Carrasco:2015uma ; Carrasco:2015pla ;
Kallosh:2017wnt , that there is a universal geometric mechanism of
stabilization of the inflaton partner during inflation (or in general, of the
imaginary part of the complex scalar), based on the bisectional curvature. In
our case, this is a statement that the fields ${\rm Im}\,Z_{i}$ can be
stabilized at ${\rm Im}\,Z_{i}=0$, i. e. at $\theta_{1}=\theta_{2}=0$, which
reduces our supergravity model to the bosonic cosmological model we study in
this paper. In presence of the nilpotent superfield one can add to the Kähler
potential a bisectional curvature term of the form

$K_{\rm bisec}=X\bar{X}A(Z,\bar{Z}){\rm Im}^{2}Z$ (49)

for each of the axions ${\rm Im}\,Z$ which we want to stabilize. These terms
with appropriate choice of the function $A(Z,\bar{Z})$ add a positive mass
terms to each of the axions at all positions in inflationary trajectory.

In fact, very often our earlier models in Kallosh:2017wnt  were stabilized at
the vanishing values of the axions even without the need for bisectional
curvature terms. This feature is provided by special properties of these
models which we also used in Kallosh:2021vcf  and will use here.

###  A.2 Supergravity model with stabilized axions

There are 3 chiral superfields, $Z_{1},Z_{2}$ and a nilpotent one $X$. The
Kähler potential and the superpotential are:

$K=K_{X\bar{X}}X\bar{X}-3\alpha\log\left(1-{Z_{1}}\overline{{Z_{1}}}\right)+{Z_{2}}\overline{{Z_{2}}}\,,\qquad
K_{X\bar{X}}=\frac{F_{X}^{2}}{{F_{X}}^{2}+{\cal
V}\left({Z_{1}},\overline{{Z_{1}}},{Z_{2}},\overline{{Z_{2}}}\right)}\ ,$ (50)
$W=M(F_{X}X+W_{0})\left(1-Z_{1}^{2}\right)^{\frac{3\alpha}{2}}e^{-\frac{{Z_{2}}^{2}}{2}}\
.$ (51)

Here ${\cal
V}\left({Z_{1}},\overline{{Z_{1}}},{Z_{2}},\overline{{Z_{2}}}\right)$ is an
arbitrary function. To match the potential of the hybrid $\alpha$-attractors
(47) we will take

${\cal
V}=M^{2}\left(3\alpha{Z_{1}}\overline{Z_{1}}\Big{(}\tilde{m}^{2}+\tilde{g}^{2}{Z_{2}}\overline{Z_{2}}\Big{)}+{1\over
4\chi_{0}^{2}}\left(\left(\frac{{Z_{2}}+\overline{{Z_{2}}}}{\sqrt{2}}\right)^{2}-\chi_{0}^{2}\right)^{2}+d\frac{{Z_{2}}+\overline{{Z_{2}}}}{\sqrt{2}}\right)\
.$ (52)

The total potential

$V^{\rm super}=e^{K}(|DW|^{2}-3|W|^{2})$ (53)

as a function of complex fields $Z_{1},Z_{2}$ at $X=0$ is complicated, but at
real $Z_{1},Z_{2}$ it is reduced to (47) where in addition there is a
cosmological constant term $\Lambda=F_{X}^{2}-3W^{2}$

$V^{\rm super}|_{Z_{i}=\overline{Z}_{i},X=0}=\Lambda+V(\varphi,\chi)\ .$ (54)

Using Kähler invariance of the potential (53) we can present an alternative
form of $K$ and $W$ which helps to explain the reason why the axions are easy
to stabilize in this class of models. The first term in $K$ is the same as in
(50) but the second and the third, as well as $W$ are different due to a
Kähler transformation preserving the potential

$K=K_{X\bar{X}}X\bar{X}-{3\alpha\over
2}\log{\left(1-{Z_{1}}\overline{{Z_{1}}}\right)\over(1-Z_{1}^{2})(1-\overline{{Z_{1}}}^{2})}-{1\over
2}({Z_{2}}-\overline{{Z_{2}}})^{2}\ ,$ (55) $W=M\,(F_{X}X+W_{0})\ .$ (56)

The Kähler potential $-{3\alpha\over
2}\log{\left(1-{Z_{1}}\overline{{Z_{1}}}\right)\over(1-Z_{1}^{2})(1-\overline{{Z_{1}}}^{2})}-{1\over
2}({Z_{2}}-\overline{{Z_{2}}})^{2}$ vanishes at $Z_{i}=\overline{Z}_{i}$ by
design, and it has terms proportional to axions
$-(Z_{i}-\overline{Z}_{i})^{2}$. The shift symmetry for axions $Z\to Z+ic$ is
therefore broken already by $K$ and in most cases axions are stabilized even
without additional terms with bisectional curvature (49), as shown in
Kallosh:2017wnt .

It was also important here that one can make a choice of the function ${\cal
V}\left({Z_{1}},\overline{{Z_{1}}},{Z_{2}},\overline{{Z_{2}}}\right)$ in eq.
(50) in the Kähler potential of the nilpotent field $K_{X\bar{X}}$ supporting
the stabilization of the axions initiated by the choice of the Kähler
potential of the chiral superfields $Z_{i}$.

In particular, for the parameters used in this paper for the model in (47) we
have checked that axions are stabilized at $Z_{i}=\overline{Z}_{i}$ and they
have large positive masses even without bisectional curvature terms (49). For
more general parameters, in any case the bisectional curvature terms (49) can
always be added and therefore we have promoted our bosonic models to
supergravity models consistently.

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