††institutetext: Stanford Institute for Theoretical Physics and Department of
Physics,
Stanford University, Stanford, CA 94305, USA

#  Hybrid cosmological attractors

Renata Kallosh and  Andrei Linde<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

We construct $\alpha$-attractor versions of hybrid inflation models. In these
models, the potential of the inflaton field $\varphi$ is uplifted by the
potential of the second field $\chi$. This uplifting ends due to a tachyonic
instability with respect to the field $\chi$, which appears when $\varphi$
becomes smaller than some critical value $\varphi_{c}$. In the large $N$
limit, these models have the standard universal $\alpha$-attractor
predictions. In particular, $n_{s}=1-{2\over N}$ for the exponential
attractors. However, in some special cases the large $N$ limit is reached only
beyond the horizon, for $N\gtrsim 60$. This may change predictions for the
cosmological observations. For any fixed $N$, in the limit of large uplift
$V_{\rm up}$, or in the limit of large $\varphi_{c}$, we find another
attractor prediction, $n_{s}=1$. By changing the parameters $V_{\rm up}$ and
$\varphi_{c}$ one can continuously interpolate between the two attractor
predictions $n_{s}=1-{2\over N}$ and $n_{s}=1$.

##  1 Introduction

In this paper we will study two-field cosmological attractors, using the
$\alpha$-attractor generalization of the original version of hybrid inflation
as an example Linde:1991km ; Linde:1993cn .

In cosmological $\alpha$-attractors of a single inflaton field $\varphi$, the
predictions for $n_{s},r$ are very stable with respect to significant
modifications of the theory Kallosh:2013hoa ; Ferrara:2013rsa ;
Kallosh:2013yoa ; Galante:2014ifa ; Kallosh:2015zsa ; Kallosh:2019eeu ;
Kallosh:2019hzo . In exponential $\alpha$-attractors Kallosh:2013yoa , where
the field approaches the plateau exponentially fast, in the large $N$ limit,
where $N$ is the number of e-foldings, one has

$V=V_{0}(1-e^{-\varphi/\mu}+\dots)\,,\qquad\mu=\sqrt{3\alpha\over 2}\,,\qquad
n_{s}\approx 1-{2\over N}\,,\qquad r\approx{12\alpha\over N^{2}}\ .$ (1)

for $\mu\lesssim O(1)$. For example, for $N=55$

$n_{s}\approx 1-{2\over 55}\approx 0.963\ .$ (2)

Predictions of the simplest models of this class can completely cover the left
part of the $n_{s}-r$ area favored by the latest Planck/BICEP/Keck data.

For the family of polynomial $\alpha$-attractors Kallosh:2022feu , where the
potential approaches a plateau as inverse powers of the inflaton field, one
has

$V\sim V_{0}\large(1-{\mu^{k}\over\varphi^{k}}+\dots\large)\,,\qquad
n_{s}=1-{2\over N}{k+1\over k+2}\,,\qquad r\approx{8k^{2}\mu^{2k\over
k+2}\over[(k(k+2))N]^{2k+2\over k+2}}\ .$ (3)

Here $k$ can take any positive value. For example, in $k=2$ case

$V\sim V_{0}\large(1-{\mu^{2}\over\varphi^{2}}+\dots\large)\,,\qquad
n_{s}\approx 1-{3\over 2N}\,,\qquad r\approx{\sqrt{2}\,\mu\over N^{3/2}}$ (4)

For $N=N_{\rm total}=55$ we have

$n_{s}\approx 1-{3\over 110}\approx 0.973\ .$ (5)

By taking smaller $k$, one can increase the value of $n_{s}$ in this scenario
from $1-{2\over N}$ to $1-{1\over N}$. As a result, predictions of the
simplest models of exponential and polynomial attractors completely cover the
$n_{s}-r$ area favored by the latest Planck/BICEP/Keck data, see Fig. 3 of
Kallosh:2022feu .

Thus it would seem that a rather simple set of models with deep roots in
supergravity can describe any set of data which any future observations may
bring. However, there are still some issues which one may try to address.

1) One may wonder whether it is possible to increase $n_{s}$ to cover the
right part of the $n_{s}-r$ area favored by the latest Planck/BICEP/Keck data
within the more familiar class of exponential $\alpha$-attractors (1).

2) There are ongoing efforts to solve the $H_{0}$ and $S_{8}$ problems by
modifying the standard $\Lambda$CDM model Riess:2021jrx ; Abdalla:2022yfr .
Some of these efforts require a significant re-interpretation of the available
data, resulting in much higher values of $n_{s}$, all the way up to the
Harris-Zeldovich value $n_{s}=1$, see Jiang:2022uyg ; Smith:2022hwi  and
references therein. Thus one may wonder whether one may find some versions of
$\alpha$-attractors which would be compatible with such values of $n_{s}$.

3) In models of $\alpha$-attractors inspired by string theory and M-theory,
one may encounter many interacting scalar fields, each of which may have
inflaton potentials with different values of $\alpha$ Ferrara:2016fwe ;
Kallosh:2017ced ; Kallosh:2017wnt ; Achucarro:2017ing ; Linde:2018hmx ;
Gunaydin:2020ric ; Kallosh:2021fvz ; Kallosh:2021vcf . Therefore it is
important to explore multi-field $\alpha$-attractors. In the simplest cases,
one may have several different stages of inflation, but in many models the
last $N\sim 50$ \- 60 e-foldings of inflation are described by a single stage
of inflaton, with the predictions described above.

However, this is not always the case. For example, suppose that there is a
short secondary stage of inflation describing $\Delta N$ e-foldings after the
$\alpha$-attractor stage. In this case, we must carefully distinguish between
the total number of e-foldings $N_{e}\sim 50$ \- 60 responsible for the
observable structure of the universe, and its part $N$ related to inflation in
the $\alpha$-attractor regime:

$N=N_{e}-\Delta N\ .$ (6)

The observational predictions of $\alpha$-attractors are still described by
(1), (4), but the value of $N=N_{e}-\Delta N$ becomes smaller than $N_{e}\sim
50$ \- 60 Christodoulidis:2018qdw ; Linde:2018hmx . This may significantly
decrease the value of $n_{s}$, which may contradict the observational data
unless the second stage of inflation is very short.

This issue is less important for polynomial attractors (3) because they
predict higher values of $n_{s}$. That is why some of the popular models of
large PBH formation Braglia:2020eai  can be formulated in the context of the
KKLTI polynomial $\alpha$-attractors Kallosh:2022vha , whereas similar models
based on exponential $\alpha$-attractors tend to predict very small PBHs
Iacconi:2021ltm . It would be interesting to see whether one may overcome
these limitations and find a way to increase $n_{s}$, if required.

In this paper we will show how one can significantly increase $n_{s}$ in two-
field inflationary models. As an example, we will study the original version
of the hybrid inflation scenario Linde:1991km ; Linde:1993cn , and then
explore its $\alpha$-attractor implementation. In these models, the potential
of the inflaton field $\varphi$ is uplifted by the potential of the second
field $\chi$, but this uplifting ends due to a tachyonic instability with
respect to the field $\chi$, which happen when the field $\varphi$ becomes
smaller than its critical value $\varphi_{c}$. This instability typically
leads to a nearly instant end of inflation and rapid reheating, but it may
also occur slowly, in a secondary inflationary stage.

We will confirm that the main attractor predictions (1), (4) remain true in
these models in the large $N$ limit. However, we will show that in some models
the large $N$ limit is achieved only for $N>60$, and for $N\lesssim 60$ one
may have an intermediate asymptotic regime with $n_{s}$ that can be greater
than the attractor values (1), (4). In particular, for any fixed $N$ (e.g. for
$N\sim 50$), in the large uplift limit, or in the limit of large value of
$\varphi_{c}$, we find another attractor prediction, the Harrison-Zeldovich
spectrum with $n_{s}\rightarrow 1$.

##  2 Single field $\alpha$-attractors

We will begin with describing single field $\alpha$-attractors. The simplest
example is given by the theory

${{\cal L}\over\sqrt{-g}}={R\over 2}-{(\partial_{\mu}\phi)^{2}\over
2\bigl{(}1-{\phi^{2}\over 6\alpha}\bigr{)}^{2}}-V(\phi)\,.$ (7)

Here $\phi(x)$ is the scalar field, the inflaton. In the limit
$\alpha\to\infty$ the kinetic term becomes the standard canonical term
$-{(\partial_{\mu}\phi)^{2}\over 2}$. The new kinetic term has a singularity
at $|\phi|=\sqrt{6\alpha}$. However, one can get rid of the singularity and
recover the canonical normalization by solving the equation
${\partial\phi\over 1-{\phi^{2}\over 6\alpha}}=\partial\varphi$, which yields
$\phi=\sqrt{6\alpha}\,\tanh{\varphi\over\sqrt{6\alpha}}$. The full theory, in
terms of the canonical variables, becomes a theory with a plateau potential

${{\cal L}\over\sqrt{-g}}={R\over 2}-{(\partial_{\mu}\varphi)^{2}\over
2}-V\big{(}\sqrt{6\alpha}\,\tanh{\varphi\over\sqrt{6\alpha}}\big{)}\,.$ (8)

We called such models T-models due to their dependence on the
$\tanh{\varphi\over\sqrt{6\alpha}}$. Asymptotic behavior of the potential at
large $\varphi>0$ is given by

$V(\varphi)=V_{0}-2\sqrt{6\alpha}\,V^{\prime}_{0}\ e^{-\sqrt{2\over
3\alpha}\varphi}\ .$ (9)

Here $V_{0}=V(\phi)|_{\phi=\sqrt{6\alpha}}$ is the height of the plateau
potential, and $V^{\prime}_{0}=\partial_{\phi}V|_{\phi=\sqrt{6\alpha}}$. The
coefficient $2\sqrt{6\alpha}\,V^{\prime}_{0}$ in front of the exponent can be
absorbed into a redefinition (shift) of the field $\varphi$. Therefore
inflationary predictions of this theory in the regime with $e^{-\sqrt{2\over
3\alpha}\varphi}\ll 1$ are determined only by two parameters, $V_{0}$ and
$\alpha$, i.e. they do not depend on many other features of the potential
$V(\phi)$. That is why they are called attractors.

At large $N$, predictions of these models for $A_{s}$, $n_{s}$ and $r$
coincide in the small $\alpha$ limit, nearly independently of the detailed
choice of the potential $V(\phi)$:

$A_{s}={V_{0}\,N^{2}\over 18\pi^{2}\alpha}\ ,\qquad n_{s}=1-{2\over N}\
,\qquad r={12\alpha\over N^{2}_{e}}\ .$ (10)

These models are compatible with the presently available observational data
for sufficiently small $\alpha$.

Importantly, these results depend on the height of the inflationary plateau,
which is given by $V_{0}=V(\phi)|_{\phi=\sqrt{6\alpha}}$, but they do not
depend on many other details of behavior of the potential $V(\phi)$ in (7).
This explains, in particular, stability of the predictions of these models
with respect to quantum corrections Kallosh:2016gqp .

The amplitude of inflationary perturbations in these models matches the Planck
normalization $A_{s}\approx 2.01\times 10^{{-9}}$ for ${V_{0}\over\alpha}\sim
10^{{-10}}$, $N=60$, or for ${V_{0}\over\alpha}\sim 1.5\times 10^{{-10}}$,
$N=50$. For the simplest model $V={m^{2}\over 2}\phi^{2}$ one finds

$V=3m^{2}\alpha\tanh^{2}{\varphi\over\sqrt{6\alpha}}\ .$ (11)

This simplest model is shown by the prominent vertical yellow band in Fig. 8
of the paper on inflation in the Planck2018 data release Planck:2018jri . In
this model, the condition ${V_{0}\over\alpha}=3m^{2}=\sim 10^{{-10}}$ reads
$m\sim 0.6\times 10^{{-5}}$. The small magnitude of this parameter accounts
for the small amplitude of perturbations $A_{s}\approx 2.01\times 10^{{-9}}$.
No other parameters are required to describe all presently available
inflation-related data in this model. If the inflationary gravitational waves
are discovered, their amplitude can be accounted for by the choice of the
parameter $\alpha$ in (10).

The results described above are valid under assumptions that the potential
$V(\phi)$ and its derivatives are non-singular at the boundary
$|\phi|=\sqrt{6\alpha}$. If one keeps the requirement that the potential
$V(\phi)$ is non-singular, but allows its derivatives to be singular, the
potential $V(\varphi)$ remains a plateau potential in canonical variables, but
it may become a polynomial attractor, with properties and predictions
described in (3), (4) Kallosh:2022feu .

One should note also, that these results rely on a hidden assumption that
inflation occurs in the single field regime with a potential (1) or (3), and
ends when the slow-roll conditions are no longer satisfied. This assumption is
natural indeed, but one can find, or engineer, some models where it may be
violated.

As we already mentioned in the previous section, the simplest possibility to
do it is to arrange for a second stage of inflation with duration $\Delta N$.
This modification decreases $n_{s}$. For exponential $\alpha$-attractors (1)
this decrease is not particularly desirable.

However, there is yet another possibility, which may allow many interesting
variations of the main theme. One may consider multi-field models, where the
single-field inflation regime ends prematurely because of the instability of
the inflationary trajectory, or because of its sharp turn.

The simplest well-known example is provided by hybrid inflation Linde:1991km ;
Linde:1993cn . In this scenario, inflation driven by the field $\phi$ is
terminated because of the tachyonic waterfall instability with spontaneous
generation of the second field $\sigma$. This mechanism involves two
ingredients, each of which allow to control (increase) $n_{s}$. First of all,
this scenario involves uplift of an inflationary potential by some potential
depending on $\sigma$. This uplift disappears after the waterfall instability,
but during inflation with $\phi>\phi_{c}$ the uplift increases $V$ while
keeping $V^{\prime}$ intact. This decreases slow-roll parameters and increases
$n_{s}$ for $\phi>\phi_{c}$. Secondly, one can control the value of $\phi_{c}$
by a proper choice of parameters. As a result, one can also control the value
of the field $\phi_{N}$ corresponding to $N$ e-foldings prior to termination
of inflation. This provides an additional tool to control $n_{s}$.

In this paper we will consider hybrid models of $\alpha$-attractors and
explain how both of these mechanisms affect inflationary predictions for
$n_{s}$ and $r$. To avoid misunderstandings, we should emphasize that hybrid
$\alpha$-attractors are more complicated than the single-field
$\alpha$-attractors. However, realistic inflationary models often involve more
than one scalar field. As we will see, investigation of their
$\alpha$-attractor versions can be quite instructive.

##  3 Hybrid inflation

###  3.1 Original hybrid inflation model

Let us first consider the simplest hybrid inflation model Linde:1991km ;
Linde:1993cn . The effective potential of this model is given by

$V(\sigma,\phi)={1\over 4\lambda}(M^{2}-\lambda\sigma^{2})^{2}+{m^{2}\over
2}\phi^{2}+{g^{2}\over 2}\phi^{2}\sigma^{2}\ .$ (12)

To illustrate the main features of this potential, we show it in Fig. 1.

Figure 1: Hybrid inflation potential (12) for $m=0.2,M=1,\lambda=0.5,g=0.8$.

The effective mass squared of the field $\sigma$ at $\sigma=0$ is equal to

$V_{\sigma,\sigma}(\sigma=0)=-M^{2}+g^{2}\phi^{2}\ .$ (13)

For $\phi>\phi_{c}=M/g$ the only minimum of the effective potential
$V(\sigma,\phi)$ with respect to $\sigma$ is at $\sigma=0$. The curvature of
the effective potential in the $\sigma$-direction is much greater than in the
$\phi$-direction. Thus we expect that at the first stages of expansion of the
Universe the field $\sigma$ rolled down to $\sigma=0$, whereas the field
$\phi$ could remain large for a much longer time.

The potential at $\sigma=0$ can be written as

$V(\sigma=0,\phi)=V_{\rm up}+{m^{2}\over 2}\phi^{2}\ ,$ (14)

where the uplifting potential is

$V_{\rm up}={M^{4}\over 4\lambda}\ .$ (15)

At the moment when the inflaton field $\phi$ becomes smaller than
$\phi_{c}=M/g$, the phase transition with the symmetry breaking occurs. For a
proper choice of parameters, this phase transition occurs very fast, and
inflation abruptly ends Linde:1991km ; Linde:1993cn . However, there are some
situations where inflation may continue for a while in the process of
spontaneous symmetry breaking, which may lead to production of primordial
black holes (PBHs) Garcia-Bellido:1996mdl .

Unfortunately, these models are disfavored by the data in most of its
parameter space: at ${m^{2}\over 2}\phi^{2}\gtrsim V_{\rm up}$ the tensor-to-
scalar ratio is too high, whereas at ${m^{2}\over 2}\phi^{2}\ll V_{\rm up}$
the spectral index $n_{s}$ is too high: $n_{s}>1$ Planck:2013jfk .

Once we switch to $\alpha$-attractor version of hybrid inflation, the first of
these problems disappears. As we will show later, the second problem may also
disappear: in the large $N$ limit these models lead to the standard
$\alpha$-attractor predictions (1), (3). The issue we need to carefully
examine is whether $N\sim 60$ is large enough to be described by the large $N$
limit.

###  3.2 Hybrid $\alpha$-attractors

Here we will explore what may happen if we generalize the hybrid inflation
model (12) by embedding it in the context of exponential $\alpha$-attractors
Kallosh:2013hoa .111We will discuss polynomial attractors Kallosh:2022feu  in
section 9.

${{\cal L}\over\sqrt{-g}}={R\over 2}-{(\partial_{\mu}\phi)^{2}\over
2\bigl{(}1-{\phi^{2}\over
6\alpha}\bigr{)}^{2}}-{(\partial_{\mu}\sigma)^{2}\over
2\bigl{(}1-{\sigma^{2}\over 6\beta}\bigr{)}^{2}}-V(\sigma,\phi)\,.$ (16)

Upon a transformation to canonical variables $\varphi$ and $\chi$, the hybrid
inflation potential becomes

$\displaystyle V(\chi,\varphi)$ $\displaystyle=$ $\displaystyle{1\over
4\lambda}\large(M^{2}-6\beta\lambda\tanh^{2}{\chi\over\sqrt{6\beta}}\large)^{2}+3m^{2}\alpha\tanh^{2}{\varphi\over\sqrt{6\alpha}}$
(17) $\displaystyle+$ $\displaystyle
18g^{2}\alpha\beta\,\tanh^{2}{\varphi\over\sqrt{6\alpha}}\,\tanh^{2}{\chi\over\sqrt{6\beta}}.$

The shape of this potential for some particular values of parameters is shown
in Fig. 2.

Figure 2: Hybrid inflation potential for the model (17) with
$m=0.2,M=1,\lambda=0.5,g=0.8,\alpha=1,\beta=1$. It looks very similar to the
original potential shown in Fig. 1, but the potential along the valley
$\chi=0$ is much more flat.

The curvature of the potential in the $\chi$ direction at $\chi=0$ coincides
with the curvature with respect to $\sigma$ at $\sigma=0$:

$V_{\chi,\chi}(\chi=0)=V_{\sigma,\sigma}(\sigma=0)=-M^{2}+g^{2}\phi^{2}=-M^{2}+6\alpha\,g^{2}\tanh^{2}{\varphi\over\sqrt{6\alpha}}.$
(18)

For $\phi>\phi_{c}=M/g$, this curvature is positive, and the inflationary
trajectory with $\chi=0$ remains stable until field $\phi$ rolls below the
critical point

$\phi_{c}=\sqrt{6\alpha}\tanh{\varphi_{c}\over\sqrt{6\alpha}}=M/g\ .$ (19)

If the last 60 e-foldings of inflation occur when $|\phi|\ll\sqrt{6\alpha}$,
$|\sigma|\ll\sqrt{6\beta}$, then most cosmological consequences of this model
will coincide with those of the original version of hybrid inflation
Linde:1991km ; Linde:1993cn . In particular, inflation ends at
$\phi_{c}\approx\varphi_{c}=M/g$. In this paper we will be interested in the
opposite possibility, when the last 60 e-foldings occur in the
$\alpha$-attractor regime where $\varphi_{c}\gg\sqrt{6\alpha}$.

One should note also that the standard scenario with the waterfall phase
transition shown in Fig. 2 occurs only if $\phi_{c}=M/g<\sqrt{6\alpha}$. In
the opposite case $\phi_{c}=M/g>\sqrt{6\alpha}$ the field $\chi$ does not
vanish at any values of $\varphi$, because all values of $\varphi$ correspond
to $\phi<\sqrt{6\alpha}$. The amplitude of spontaneous symmetry breaking grows
during inflation starting from $\chi^{2}={M^{2}-6\alpha\over\lambda}$ at
$\varphi\to\infty$, and gradually approaching its maximal value
$\chi^{2}={M^{2}\over\lambda}$ at $\phi=0$. This scenario does not suffer from
the problem with topological defects which may appear in the scenario shown in
Figs. 1, 2.

To illustrate what happens for $M/g>\sqrt{6\alpha}$, we plot in the left panel
of Fig. 3 the potential (17) for the same values of parameters as in Fig. 2.
The only parameter we change is $g$, which we take smaller, $g=0.35$.

Figure 3: Left panel shows potential (17) for
$m=0.2,M=1,\lambda=0.5,g=0.35,\alpha=1,\beta=1$. Right panel shows potential
(17) for $m=0.2,M=1,\lambda=0.5,g=0.8,\alpha=1,\beta=1/4$.

This is not the last of the surprises which may await us after introducing
hybrid $\alpha$-attractors, see the right panel in Fig. 3, where we plot the
same potential for the same parameters as in Fig. 2, but for a smaller value
of $\beta$. As we see, in this case the position of the minimum of the
potential with respect to $\chi$ disappears, and we end up with the potential
describing the $\alpha$-attractor generalization Dimopoulos:2017zvq ;
Akrami:2017cir ; Braglia:2020bym  of the quintessential inflation
Peebles:1998qn ; Felder:1999pv . This happens because for sufficiently small
$\beta$ the position of the minimum of the potential with respect to $\sigma$
moves outside the boundary of the moduli space at $\sigma=\sqrt{6\beta}$.

It is not our goal to describe all of these interesting possibilities in this
paper. In what follows we will study the more traditional regime described by
Fig. 2. In this regime, the initial stages of inflation occur at $\chi=0$,
until the field reaches a critical point $\varphi_{c}$. After that, the
tachyonic instability with respect to the field $\chi$ terminates the stage of
inflation at $\chi=0$. Depending on the parameters of the model, this may lead
either to an abrupt end of inflation, or to a beginning of a short additional
period of inflation. We will concentrate on the first of these two possible
outcomes, and calculate inflationary parameters $A_{s}$, $n_{s}$ and $r$
assuming that inflation ends at the moment when the field $\varphi$ reaches
$\varphi_{c}$ (19).

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

$V(\varphi)={M^{4}\over
4\lambda}+3m^{2}\alpha\tanh^{2}{\varphi\over\sqrt{6\alpha}}\ .$ (20)

Using equation (9), one can represent this potential during inflation at
$\varphi\gg\sqrt{6\alpha}$ in this model as

$V=V_{\rm up}+V_{0}\,(1-4\ e^{-\sqrt{2\over 3\alpha}\varphi}+...)\ ,$ (21)

where $V_{\rm up}={M^{4}\over 4\lambda}$ is the value of the uplifting
potential ${1\over 4\lambda}(M^{2}-\lambda\sigma^{2})^{2}$ at $\sigma=0$, and
$V_{0}=3m^{2}\alpha$ is the value of the $\alpha$-attractor potential
$3m^{2}\alpha\tanh^{2}{\varphi\over\sqrt{6\alpha}}$ at its plateau.

Let us first consider the regime $V_{up}\gg V_{0}$, i.e

$M^{4}\gg 12\alpha\lambda m^{2}\ .$ (22)

The Hubble constant in this case is

$H^{2}={M^{4}\over 12\lambda}\ .$ (23)

Thus $M^{2}\gg H^{2}$ for

$M^{2}\ll 12\lambda\ .$ (24)

If $M^{2}\ll 12\lambda$, then shortly after the field $\phi$ moves below the
critical value $\phi_{c}=M/g$, the effective mass squared of the field $\chi$
becomes negative. Once its absolute value becomes greater than $H^{2}$, the
tachyonic instability of the field $\chi$ develops, which leads to an abrupt
termination of inflation at $\phi\approx\phi_{c}$, as in the standard version
of the hybrid inflation scenario Linde:1991km ; Linde:1993cn .

##  4 Supergravity version of hybrid $\alpha$-attractors

There are several popular versions of the hybrid inflation models in
supergravity which are known as F-term and D-term inflation Copeland:1994vg ;
Dvali:1994ms ; Binetruy:1996xj ; Halyo:1996pp . However, the models, just as
the original hybrid inflation model Linde:1991km ; Linde:1993cn , require
modifications to become compatible with observations.

Here we would like to present a supergravity version of the $\alpha$-attractor
generalization (17) of the original hybrid inflation model (12). This can be
done by introducing two chiral multiplets $Z_{1}$ and $Z_{2}$, both described
by some hyperbolic geometries with non-canonical kinetic terms,

$Z_{i}=z_{i}e^{i\theta_{i}}=\tanh{\varphi_{i}\over\sqrt{6\alpha_{i}}}e^{i\theta_{i}}\
,$ (25)

and one nilpotent multiplet $X$.

The supergravity version is designed to have the angular fields $\theta_{1}$
and $\theta_{2}$ stabilized at their minimum $\theta_{1}=\theta_{2}=0$
Kallosh:2017ced ; Kallosh:2017wnt ; Gunaydin:2020ric ; Kallosh:2021vcf ;
Kallosh:2021fvz . The class of models describe in equations (16), (17) can be
presented by the following Kähler potential and superpotential (here we call
$\varphi=\varphi_{1},\chi=\varphi_{2}$ and $\alpha=\alpha_{1}$ and
$\beta=\alpha_{2}$).

$K=-3\sum_{i=1,2}\alpha_{i}\log(1-Z_{i}\overline{Z}_{i})+{F_{X}^{2}\over
F_{X}^{2}+V_{\rm infl}(Z_{i},\overline{Z}_{i})}X\overline{X},$ (26)

and superpotential

$W=(W_{0}+F_{X}X)\prod_{i=1,2}(1-Z_{i}^{2})^{3\alpha_{i}/2}\ ,$ (27)

which yields

$V_{\rm total}(z_{i})=\Lambda+V_{\rm
infl}(Z_{i},\overline{Z}_{i})_{|_{Z_{i}=\overline{Z}_{i}=z_{i}}}\ ,$ (28)

where $\Lambda=F_{X}^{2}-3W_{0}^{2}$. For
$z_{i}=\tanh{\varphi_{i}\over\sqrt{6\alpha_{i}}}$ this provides a supergravity
embedding of the models like (16), (17).

##  5 Inflationary predictions of hybrid $\alpha$-attractors

In our investigation of perturbations in the hybrid inflation, we will try to
be as model-independent as possible. The results to be obtained in this
section will be applicable not only to hybrid inflation, but to any
$\alpha$-attractor potentials uplifted by an additional term similar to the
first term in (12). We will also assume that the single-field regime may end
not because of the violation of the slow-roll conditions, but because some
kind of instability terminating the original stage of inflation in a vicinity
of a critical field $\phi_{c}$, as in the hybrid inflation scenario.

The general $\alpha$-attractor potential (9) at large $\varphi$ can be
represented as

$V(s)=V_{0}\large(1-e^{-\sqrt{2\over 3\alpha}s}+...\large)\ ,$ (29)

where $s$ is given by

$s=\varphi-\sqrt{3\alpha\over
2}\ln\Bigl{(}2\sqrt{6\alpha}\,{V^{\prime}_{0}\over V_{0}}\Bigr{)}\ ,$ (30)

and $V^{\prime}_{0}=\partial_{\phi}V|_{\phi=\sqrt{6\alpha}}$ at the boundary
$\phi=\sqrt{6\alpha}$, as in (9). To give a particular example, in the
simplest T-model (11) one has

$s=\varphi-\sqrt{3\alpha\over 2}\ln 4\approx\varphi-1.7\sqrt{\alpha}\ .$ (31)

Thus for $\alpha\lesssim 1$ one has $\varphi=s+O(1)$.

Now we will uplift this potential by adding to it a constant $V_{up}$. In the
hybrid inflation model (12) one has $V_{\rm up}={M^{4}\over 4\lambda}$. The
full potential becomes

$V(s)=V_{\rm up}+V_{0}(1-e^{-\gamma s})\ ,$ (32)

where $\gamma$ is related to the Kähler curvature

$\gamma=\sqrt{2\over 3\alpha}\,,\qquad\gamma^{2}={2\over 3\alpha}\ .$ (33)

This form correctly describes the potential for

$e^{-\gamma s}\ll 1\ .$ (34)

We consider a stage of $N\gg 1$ e-foldings of inflation which begins at
$s_{N}$ and ends at $s_{c}$. Inflation may continue when the field reaches
$s_{c}$, or it may end abruptly if the inflationary trajectory changes at
$s_{c}$ because of the waterfall instability at in hybrid inflation.

Equation describing evolution of $s$ in the slow-roll regime is

${ds\over dN}={{dV\over ds}\over V(s)}={V_{0}\gamma\ e^{-\gamma\,s}\over
V_{\rm up}+V_{0}(1-\ e^{-\gamma\,s})}\ .$ (35)

We are interested in the regime $e^{-\gamma\,s}\ll 1$. In that case one can
ignore the exponent in the denominator and find a solution of this equation:

$e^{\gamma\,s_{N}}={\gamma^{2}V_{0}N\over V_{\rm up}+V_{0}}+e^{\gamma\,s_{c}}\
.$ (36)

where $s_{N}$ is the value of the field $s$ at $N$ e-foldings before the end
of this stage of inflation before it reaches $s_{c}$, i. e. $s_{N}=s_{c}$ at
$N=0$.

The standard expression for $n_{s}$ is

$n_{s}=1-3\left({V^{\prime}\over V}\right)^{2}+2{V^{\prime\prime}\over
V}\approx 1-{3V_{0}^{2}\gamma^{2}\ e^{-2\gamma\,s_{N}}\over(V_{\rm
up}+V_{0})^{2}}-{2V_{0}\gamma^{2}e^{-\gamma\,s_{N}}\over V_{\rm
up}+V_{0}}\approx 1-{2V_{0}\gamma^{2}e^{-\gamma\,s_{N}}\over V_{\rm
up}+V_{0}}\ .$ (37)

Here the derivatives are taken with respect to $s$. Using equation (36), we
find

$n_{s}=1-\frac{2V_{0}\gamma^{2}}{V_{0}\,\gamma^{2}N+(V_{\rm
up}+V_{0})e^{\gamma s_{c}}}\ .$ (38)

In the large $N$ limit we always have the standard universal
$\alpha$-attractor prediction, independently of all other parameters of the
model,

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

However, the range accessible to observations is limited, $N\lesssim 50-60$.
For

$e^{\gamma s_{c}}\gg{\gamma^{2}V_{0}N\over V_{\rm up}+V_{0}}\ ,$ (40)

one has, in accordance with (36),

$e^{\gamma s_{N}}\approx e^{\gamma s_{c}},$ (41)

and instead of the large $N$ limit, one has a different limiting case,

$1-n_{s}={2V_{0}\gamma^{2}e^{-\gamma\,s_{N}}\over V_{\rm
up}+V_{0}}\approx{2V_{0}\gamma^{2}e^{-\gamma\,s_{c}}\over V_{\rm
up}+V_{0}}\ll{2\over N}\ ,$ (42)

where the last inequality follows from (40). Thus in the large $V_{\rm up}$
limit (for large ratio $V_{\rm up}/V_{0}$), or in the large $s_{c}$ limit (for
$\gamma s_{c}\gg 1$), when inequality (40) is satisfied, we have $n_{s}\to 1$,
i.e. the Harrison-Zeldovich spectrum.

Interpolating between these two limiting cases by changing $V_{\rm up}/V_{0}$,
or by changing $s_{c}$, one can find any value of $n_{s}$ in the range

$1-{2\over N}\ \lesssim\ n_{s}\ \lesssim\ 1.$ (43)

In particular, for

$V_{\rm up}+V_{0}=V_{0}\,\gamma^{2}Ne^{-\gamma s_{c}}$ (44)

we have

$n_{s}=1-{1\over N}\ .$ (45)

Let us consider the implications for the amplitude of perturbations $A_{s}$
and for $r$.

$A_{s}={V^{3}\over 12\pi^{2}(V^{\prime})^{2}}\approx{(V_{\rm up}+V_{0})^{3}\
e^{2\gamma\,s_{N}}\over 12\pi^{2}V_{0}^{2}\gamma^{2}}\ .$ (46)

In the large $N$ limit one finds

$A_{s}={(V_{\rm up}+V_{0})N^{2}\over 18\alpha\pi^{2}\ }\ ,$ (47)

Meanwhile for $V_{\rm up}+V_{0}\gg V_{0}\,\gamma^{2}Ne^{-\gamma s_{c}}$ one
has

$A_{s}={\alpha(V_{\rm up}+V_{0})^{3}\ e^{2\sqrt{2\over 3\alpha}s_{c}}\over
8\pi^{2}V_{0}^{2}}\ .$ (48)

and for $V_{\rm up}+V_{0}=V_{0}\,\gamma^{2}Ne^{-\gamma s_{c}}$ one has

$A_{s}={2(V_{\rm up}+V_{0})N^{2}\over 9\alpha\pi^{2}\ }\ .$ (49)

Finally, let us calculate the tensor to scalar ration $r$:

$r=8\left({V^{\prime}\over
V}\right)^{2}={8V_{0}^{2}\gamma^{2}e^{-2\gamma\,s_{N}}\over(V_{\rm
up}+V_{0})^{2}}\\\ \ .$ (50)

In the large $N$ limit one has the standard $\alpha$-attractor result

$r={12\alpha\over N^{2}}\ ,$ (51)

Meanwhile for $V_{\rm up}+V_{0}\gg V_{0}\,\gamma^{2}Ne^{-\gamma s_{c}}$ the
value of $r$ is smaller,

$r={8V_{0}^{2}\gamma^{2}e^{-2\gamma\,s_{c}}\over(V_{\rm
up}+V_{0})^{2}}\ll{12\alpha\over N^{2}}\ ,$ (52)

and for $V_{\rm up}+V_{0}=V_{0}\,\gamma^{2}Ne^{-\gamma s_{c}}$ one has

$r={3\alpha\over N^{2}}\ .$ (53)

What is the meaning of these results? First of all, we confirmed that in the
large N limit

$N\gg{3\alpha\over 2}e^{\sqrt{2\over 3\alpha}s_{c}}\left({V_{\rm up}\over
V_{0}}+1\right)\ ,$ (54)

the predictions of $\alpha$-attractors are universal, as shown in equation
(10). To be more precise, the amplitude of the perturbations $A_{s}$ in (47)
now depends not on $V_{0}$, but on the total height of the plateau $V_{\rm
up}+V_{0}$.

Meanwhile, for smaller values of $N$ (smaller wavelengths), such that

$N\ll{3\alpha\over 2}e^{\sqrt{2\over 3\alpha}s_{c}}\left({V_{\rm up}\over
V_{0}}+1\right)\ ,$ (55)

which may still exceed $N\sim 50-60$ for sufficiently large $V_{\rm up}$ and
$\sqrt{2\over 3\alpha}s_{c}$, the predictions approach the flat Harrison-
Zeldovich spectrum:

$A_{s}\approx{\alpha(V_{\rm up}+V_{0})^{3}\ e^{2\sqrt{2\over
3\alpha}s_{c}}\over 8\pi^{2}V_{0}^{2}},\quad
1-n_{s}=\frac{4V_{0}e^{-\sqrt{2\over 3\alpha}s_{c}}}{3\alpha(V_{\rm
up}+V_{0})}\ll{2\over N},\quad r\approx{16V_{0}^{2}e^{-2\sqrt{2\over
3\alpha}s_{c}}\over 3\alpha(V_{\rm up}+V_{0})^{2}}\ll{12\alpha\over N^{2}}\ .$
(56)

Note that these predictions are also universal. They do depend on constants
$V_{\rm up}$, $V_{0}$, $\alpha$ and $s_{c}$, but not on the detailed choice of
the original $\alpha$-attractor potential.

All results obtained above are formulated in terms of the field $s$ related to
the field $\varphi$ by the equation (30). As we already noted, in the simplest
T-model (11) one has $s=\varphi-\sqrt{3\alpha\over 2}\ln
4\approx\varphi-1.7\sqrt{\alpha}$. Thus for $\alpha\lesssim 1$ one has
$\varphi=s+O(1)$. In many cases this difference can be ignored, but if an
exact relation is needed, one can always return back from $s$ to $\varphi$ in
the final results using (30).

In particular, for the simplest hybrid inflation model (12) one has

$N\approx{3\alpha(V_{\rm up}+V_{0})\over 8V_{0}}\left(e^{\sqrt{2\over
3\alpha}\varphi_{N}}-e^{\sqrt{2\over 3\alpha}\varphi_{c}}\right)\ .$ (57)

We have also derived this formula in Appendix A directly for the model (12).

In the limit of large $V_{\rm up}$ and/or large $\varphi_{c}$ one has

$A_{s}\approx{\alpha(V_{\rm up}+V_{0})^{3}\ e^{2\sqrt{2\over
3\alpha}\varphi_{c}}\over 128\pi^{2}V_{0}^{2}}\ ,\quad
1-n_{s}\approx\frac{16V_{0}e^{-\sqrt{2\over
3\alpha}\varphi_{c}}}{3\alpha(V_{\rm up}+V_{0})}\ll{2\over N}\ ,\quad
r\approx{256V_{0}^{2}e^{-2\sqrt{2\over 3\alpha}\varphi_{c}}\over
3\alpha(V_{\rm up}+V_{0})^{2}}\ll{12\alpha\over N^{2}}\ .$ (58)

where $V_{\rm up}={M^{4}\over 4\lambda}$ and $V_{0}=3m^{2}\alpha$.

##  6 Interpretation and some examples

Since the hybrid inflation models considered in the previous section belong to
the general class of $\alpha$-attractors, some of the formal results obtained
above may seem rather unexpected, especially the existence of the Harrison-
Zeldovich attractor with $n_{s}=1$. In this section we will provide a simple
interpretation of our results.

The standard approach to evaluation of $n_{s}(N)$ consists of two steps. First
of all, we find the point where the slow-roll approximation breaks down and
inflation ends. Then we solve equations of motion to find the values of the
fields driving inflation $N$ e-foldings back in the cosmological evolution,
and find $n_{s}(\varphi)$ at that time.

In hybrid inflation, the approach is somewhat different. We find the position
of the inflaton field $\varphi_{c}$ (or $s_{c}$) where the slow-roll
conditions with respect to the field $\varphi$ may still be satisfied, but
inflation ends because of the tachyonic instability with respect to the field
$\chi$. The value of the field $\varphi_{c}$ depends on parameters $M$ and
$g$, so by taking proper values of these parameters one can dial almost any
desirable value of the field $\varphi_{c}$. After that one finds $\varphi_{N}$
(or, equivalently, $s_{N}$), see equation (36).

We found that in the limit of large uplift and/or large $s_{c}$ (or
$\varphi_{c}$) one has $\varphi_{N}\approx\varphi_{c}$ (41). And once
$\varphi_{N}$ is known, one can further increase $V_{up}$ without changing
$V^{\prime\prime}$. One may also exponentially decrease
$V^{\prime\prime}(\varphi_{N})$ by increasing $\varphi_{c}$. In both cases,
the slow roll parameters decrease, and $n_{s}$ asymptotically increases up to
the Harrison-Zeldovich value $n_{s}=1$.

To explain potential implications of these results, we will consider some
simple numerical examples illustrating these ideas A fully developed example
of a hybrid inflation model will be considered in the next section.

1) Let us take $\gamma=1$, $V_{\rm up}=V_{0}$. Suppose first that we want to
achieve $N=50$ e-foldings of inflation, and then trigger the waterfall
transition along the lines of the hybrid inflation scenario at $s_{c}=1$. Then
$n_{s}$ will be given by equation (39), $n_{s}=0.96$ for $N=50$. The value of
$s_{50}$ will be determined by equation (36) with $\gamma=1$,

$e^{s_{N}}={N\over 2}\ .$ (59)

Here we ignored $e^{s_{0}}\approx 2.7$ as compared to ${N\over 2}=25$ (large
$N$ approximation). This gives $s_{50}\approx\ln{25}=3.2$.

2) Now let us change our game. Let us trigger the end of inflation not at
$s_{c}=1$ but at $s^{*}_{c}=3.2$. We put here a star to emphasize that this is
a different regime, where inflation ends at the point $s^{*}_{c}=s_{50}\approx
3.2$. In that case (for $\gamma=1$, $V_{\rm up}=V_{0}$) the point from which
inflation goes for $N=50$ e-foldings until it reaches $s^{*}_{c}=s_{50}=3.2$
will be given by

$e^{s^{*}_{50}}={50\over 2}+s^{*}_{c}=50\ .$ (60)

Equation for $n_{s}$ for $N=50$ will read

$n_{s}=1-{1\over N}=0.98\ .$ (61)

That is a significant modification of $n_{s}$ achieved by changing the point
at which $N$ e-foldings of inflation end. This is achieved because if not for
the waterfall, inflation from the point $s^{*}_{N}$ would last $2N=100$
e-foldings. We just interrupted it midway, but the calculation of $n_{s}$ for
the perturbations prior to the waterfall goes the same way as if it began at
the beginning of inflation of duration $N=100$. That is why instead of
$n_{s}=1-2/50$ we have $n_{s}=1-2/100=0.98$.

3) Let us change the game once more. Suppose that after (or during) the
waterfall phase transition at $s^{*}_{c}=s_{50}\approx 3.2$ inflation does not
end, but continues in the waterfall regime for additional $\Delta N=20$
e-foldings. Then the inflationary perturbations that we are going to see at
the horizon are the ones generated in the $\alpha$-attractor regime during
$N=50-\Delta N=30$ e-foldings prior to the waterfall. This corresponds to the
point from which (if not for the waterfall), the field would roll during
$N=80$ e-foldings. This yields

$n_{s}=1-{2\over 80}=0.975\ .$ (62)

4) Finally, suppose that the waterfall occurs at $s_{c}=1$. Naively, in that
case one would not expect any major changes in $n_{s}$. However, this is not
the case if the uplift $V_{\rm up}={M^{4}\over 4\lambda}$ is much greater than
$V_{0}=3m^{2}\alpha$. This condition is very similar to the standard
assumption $H^{2}={M^{4}\over 12\lambda}\gg m^{2}$ made in the original hybrid
inflation scenario Linde:1991km ; Linde:1993cn . In particular, from (45) one
may conclude that for $\alpha=1$, $s_{c}=1$, $N=50$ and $V_{up}\approx
11V_{0}$ one would have

$n_{s}=1-{1\over 50}=0.98\ .$ (63)

These examples show that a large uplifting, or a premature ending of the
$\alpha$-attractor stage of inflation at $\gamma s_{c}\gg 1$, may lead to a
significant increase of $n_{s}$ in the $\alpha$-attractor versions of the
hybrid inflation models.

##  7 A fully developed example

In this section we will give a fully developed example including all
parameters of the hybrid inflation model (12). In all estimates we will
assume, for definiteness, that $\alpha=1$ (i.e. $\gamma=\sqrt{2/3}$), the
number of e-foldings is $N=50$ and the critical value of the field is given by
$s_{c}=2$. This corresponds to $\varphi_{c}\approx s_{c}+1.7=3.7$. In terms of
the original geometric field $\phi$, the critical point is at $\phi_{c}=2.22$.

To evaluate the importance of the effects considered in the previous sections,
we study here the intermediate regime (44), where

$n_{s}=1-{1\over N}=0.98\ ,$ (64)

see (45). For $\alpha=1$ one can use (53) to find

$r={3\over N^{2}}=0.0012\ .$ (65)

For $\alpha=1$, $s_{c}=2$ the condition (44) reads

$V_{\rm up}+V_{0}=V_{0}\,{100\over 3}e^{-2\sqrt{2/3}}=6.5\,V_{0}\ .$ (66)

Using (49) and Planck normalization $A_{s}=2.1\times 10^{{-9}}$ for $\alpha=1$
and $V_{0}=3m^{2}$, we find

$m=1.95\times 10^{{-6}}\ ,$ (67)

and

$V_{\rm up}=6.3\times 10^{{-11}}\ .$ (68)

Then using (66), we find

$M=0.004\,\lambda^{1/4}\ .$ (69)

To have the critical point at $\phi_{c}=2.22$ one should take
$g=M/\phi_{c}=0.0018\lambda^{1/4}$.

To understand the dynamics of the waterfall instability in this model is
important to compare the tachyonic mass $-M^{2}$ at $\chi=\varphi=0$ with the
square of the Hubble constant at that point:

$H^{2}(0)={V_{\rm up}\over 3}=2.1\times 10^{{-11}}\ .$ (70)

The Hubble constant at the critical point $\phi_{c}$ is very similar.
Meanwhile

$M^{2}=1.5\times 10^{-5}\,\sqrt{\lambda}\ .$ (71)

Thus $M^{2}\gg H^{2}$ unless $\lambda\lesssim 10^{{-12}}$. This means that
unless $\lambda$ is extremely small, the absolute value of the tachyonic mass
$-M^{2}+g^{2}\phi_{c}^{2}$ of the field $\chi$ becomes much greater than
$H^{2}$ almost instantly after the inflaton field $\varphi$ becomes smaller
than its critical value $\varphi_{c}$, and inflation ends, just as in the
original version of the hybrid inflation scenario Linde:1991km ; Linde:1993cn
.

Thus we gave here a particular example of the $\alpha$-attractor version of
hybrid inflation, where $n_{s}=1-1/N=0.98$ instead of the standard result
$n_{s}=1-2/N=0.96$ (for $N=50$). This shows that by changing $V_{\rm up}$ and
$\varphi_{c}$ one can change $n_{s}$ anywhere in the range from $n_{s}=1-2/N$
to $n_{s}=1$.

This does not mean that the theory of $\alpha$-attractors is not predictive.
In order to modify the standard prediction $n_{s}=1-2/N$ we needed to consider
two-field models with very special properties, such as uplifting $V_{\rm up}$
and a premature end of the $\alpha$-attractor stage of inflation.
Nevertheless, it is important to know that such models do exist, and can be
easily constructed in the familiar framework of hybrid inflation. Other
mechanisms which may lead to a premature end of inflation were reviewed for
example in Renaux-Petel:2021yxh .

Finally, let us try to understand what is so special about the exceptional
regime $\lambda\lesssim 10^{{-12}}$. The amplitude of spontaneous symmetry
breaking in the Higgs potential ${1\over
4\lambda}(M^{2}-\lambda\sigma^{2})^{2}$ for $\lambda\lesssim 10^{{-12}}$ is
given by

$\sigma=M\lambda^{-1/2}\gtrsim 4\ .$ (72)

In this case, the Higgs potential ${1\over
4\lambda}(M^{2}-\lambda\sigma^{2})^{2}$ becomes an inflationary potential,
because the length of the slope from $\sigma=0$ to $\sigma=M\lambda^{-1/2}$ is
super-Planckian. This length is even greater in terms of the canonically
normalized field $\chi$. This means that inflation may not end at the critical
point, but may continue during the process of spontaneous symmetry breaking in
this model.

A detailed theory of this second stage of inflation in the context of the
hybrid inflation scenario is described in Garcia-Bellido:1996mdl . The second
stage of inflation may last long, or it can be short, the duration $\Delta N$
being controlled by $\lambda$. The amplitude of perturbations produced at the
onset of the second stage of inflation can be very large, all the way up to
$O(1)$, leading to copious formation of black holes, with masses depending
exponentially on the number of e-foldings $\Delta N$ at the second stage of
inflation. As proposed in Garcia-Bellido:1996mdl ; Clesse:2015wea , primordial
black holes produced in such models may be sufficiently abundant to play the
role of dark matter.

The existence of the second stage of inflation means that the number of
e-foldings at the $\alpha$-attractor stage is $N_{e}-\Delta N$. For example,
for $N_{e}=50$ and $\Delta N=20$, it leaves only $N=30$ e-foldings for
$\alpha$-attractors. Then the standard expression $n_{s}=1-2/N$ would lead to
$n_{s}\sim 0.933$, which is ruled out by Planck2018 Planck:2018jri . However,
in the regime studied above one has $n_{s}=1-1/N\approx 0.967$, which is in a
very good agreement with the Planck data.

##  8 The second $\alpha$-attractor regime in the same hybrid inflation model

It could seem that we already fully explored the basic hybrid inflation model
(17) shown in Fig. 2. But even this simple model has some other interesting
features, which are not apparent in Fig. 2. To reveal them, we show the
potential of this model in Fig. 4, with the same parameters as in Fig. 2, but
in a larger range of values of $\varphi$ and $\chi$.

As one can see, this potential has not one, but two flat directions,
corresponding to each of the inflaton fields $\varphi$ and $\chi$. Until now
we studied only the scenario where the field $\varphi$ rolls down along the
yellow valley at $\chi=0$, see Figs. 2 and 4, and then the inflationary
trajectory turns towards one of the two red minima of the potential at
$\chi\not=0$. All results obtained until now are describing this possibility.

The second possibility is that initially the field $\varphi$ was small,
whereas the field $\chi$ was large, and it was playing the role of the
inflaton field, rolling down along the blue valley towards one of the two
minima of its potential shown as red areas in Fig. 4.

Figure 4: The view from the top at the hybrid inflation potential for the
model (17) with $m=0.2$, $M=1$, $\lambda=0.5$, $g=0.8$, $\alpha=1$, $\beta=1$.
This is the same potential as the one shown in Fig. 2, with the same
parameters, but now we show it for a much larger range of values of $\varphi$
and $\chi$.

Fortunately, investigation of this second scenario is fairly simple. The
potential of the field $\chi$ along the valley $\varphi=0$ is not uplifted by
the potential of the field $\varphi$, inflation ends in the standard way at
the end of the slow-roll regime, so all observational consequences are
described by the standard $\alpha$-attractor predictions (1).

This means that there are two sets of cosmological predictions for the hybrid
inflation model (17), depending on initial conditions for inflation. The first
set corresponds to the hybrid inflation regime starting at $\chi=0$ and large
$\varphi$. These predictions are described in the previous sections. The
second set of predictions corresponds to the usual single-field
$\alpha$-attractor regime, which begins and ends at $\varphi=0$, with the
predictions given in (1).

##  9 Hybrid polynomial attractors

Similar results can be obtained for other types of plateau inflation models.
Let us consider, as an example, KKLTI models with potentials approaching the
plateau as inverse powers of the canonically normalized inflaton field
$\varphi$:

$V\sim V_{0}(1-{\mu^{k}\over\varphi^{k}}+...)\ ,$ (73)

where $k$ can be any (integer or not) positive constant. Such models, which
were invented in the context of D-brane inflation Dvali:1998pa ; Dvali:2001fw
; Burgess:2001fx ; Kachru:2003sx ; Lorenz:2007ze ; Martin:2013tda ;
Kallosh:2018zsi  and pole inflation scenario Galante:2014ifa ; Terada:2016nqg
; Karamitsos:2019vor ; Kallosh:2019hzo , were recently incorporated in the
general $\alpha$-attractor framework Kallosh:2022feu .

As before, we uplift this potential by adding to it $V_{\rm up}$, which is
going to disappear after an instability at $\varphi=\varphi_{c}$. We will only
consider here the spectral index $n_{s}$. Before the uplift, the spectral
index in the large $N$ approximation is given by

$n_{s}=1-{2\over N}{1+k\over 2+k}\ .$ (74)

After the uplift, we have

$n_{s}=1-{2V_{0}k(1+k)\mu^{k}\over V_{0}k(2+k)\mu^{k}N+(V_{\rm
up}+V_{0})\,\varphi_{c}^{{2+k}}}\ .$ (75)

In the large $N$ limit one has the original result (74). In the large uplift
limit (or large $\varphi_{c}$ limit) one finds

$n_{s}=1-{2V_{0}k(1+k)\mu^{k}\over(V_{\rm up}+V_{0})\,\varphi_{c}^{{2+k}}}\ .$
(76)

In the small $k$ limit, one has the Harrison-Zeldovich result $n_{s}=1$,
whereas in the intermediate case with $(V_{\rm
up}+V_{0})\,\varphi_{c}^{{2+k}}=V_{0}k(2+k)\mu^{k}N$ one has

$n_{s}=1-{1\over N}{1+k\over 2+k}\ .$ (77)

As in the case of exponential attractors, depending on initial conditions,
there is also the standard single-field $\alpha$-attractor regime, similar to
the one described in the previous section. In that case, the predictions are
given by (3).

##  10 Discussion

In this paper we constructed $\alpha$-attractor versions of the simplest two-
field hybrid inflation models. We found that the standard inflationary
predictions of $\alpha$ attractors, such as $n_{s}=1-{2\over N}$, remain valid
in the limit of large number of e-foldings $N$. However, in some special cases
the large $N$ limit is reached only beyond the horizon, for $N\gtrsim 60$,
which changes predictions for the cosmological observations at $N\lesssim 60$.

This happens because the end of inflation in the hybrid inflation scenario is
not related to breaking of the slow-roll condition for the inflaton field
$\varphi$, but is due to the waterfall instability with respect to the field
$\chi$. Prior to the instability, which happens at $\varphi<\varphi_{c}$, the
potential of the field $\chi$ contributes to the inflaton potential, but after
the instability this contribution disappears, and inflation either ends, or
continues in a very different regime.

The critical value $\varphi_{c}$ is controlled by a combination of different
parameters of the model. We studied the situations where $\varphi_{c}$ belongs
to the $\alpha$-attractor plateau of the potential (1) or (3), and the
universe experienced $N$ e-foldings of inflation before the field $\varphi$
rolled down from $\varphi_{N}$ to $\varphi_{c}$. We confirmed the validity of
the standard predictions of $\alpha$-attractors in the large $N$ limit. But we
also found that for any particular value of $N$ there is another attractor
point: In the limit of large uplift, or of large value of $\varphi_{c}$, the
position of the point $\varphi_{N}$ moves very close to $\varphi_{c}$, all
slow roll parameters become very small, and the spectral index approaches the
Harrison-Zeldovich attractor point $n_{s}=1$.

This also implies that by changing the uplifting contribution $V_{\rm up}$ of
the field $\chi$, or the position of the critical point $\varphi_{c}$, one can
dial any desirable value of $n_{s}$ in the broad range $1-2/N\leq n_{s}\leq
1$. This does not take anything away from the universality of the standard
single-field $\alpha$-attractor predictions (1) or (3), because this
flexibility comes at a price of introducing a very specific two-field model
(12), (16) with many free parameters. However, there are many situations where
such flexibility can be desirable.

In this paper we only briefly outlined some other aspects of this flexibility.
In particular, now we can have a second stage of inflation during the
waterfall instability without violating the observational constraints on
$n_{s}$. Under some conditions (or with slight modifications of the original
hybrid inflation model), this instability may lead to production of PBHs,
which may be abundant enough to play the role of dark matter Garcia-
Bellido:1996mdl ; Clesse:2015wea .

In the models with $M/g>\sqrt{6\alpha}$ the original inflationary trajectory
shifts away from $\sigma=0$, as shown in the left part of Fig. 3. This allows
to avoid production of topological defects, while preserving most of the
results obtained in this paper.

Finally, there is a large spectrum of possibilities related to the potential
shown in the right part of Fig. 3. It shows the potential for which the
position of the minimum at $\sigma=M/\sqrt{\lambda}$ is beyond the boundary of
the moduli space $\sigma=\sqrt{6\beta}$. In terms of the canonical variable
$\chi$, this would mean that instead of having a minimum at $\chi\not=0$, we
have an infinitely long plateau describing quintessence/dark energy, similar
to quintessential inflation in single-field or two-field $\alpha$-attractor
models studied in Dimopoulos:2017zvq ; Akrami:2017cir .

Depending on the parameters $M$ and $\lambda$, this dark energy stage may be
preceded by a short waterfall stage and reheating, or a secondary inflation
stage during the waterfall. For extremely small $V_{\rm up}$, one may also
have a primary stage of dark energy domination during the waterfall, followed
by the secondary dark energy regime during the rolling along the exponentially
flat quintessential potential. Taking into account that this rolling may end
up in the universe with vacuum energy that can be either positive, negative,
or zero, and there can be various phase transitions along the way, we have
lots of interesting possibilities to be explored.

We should also mention that whereas in this paper we described hybrid
inflation, some of our qualitative results may apply to other multi-field
models as well, such as cascade inflation, which may occur in some string
theory motivated inflationary models Kallosh:2017ced ; Kallosh:2017wnt ;
Gunaydin:2020ric ; Kallosh:2021vcf ; Kallosh:2021fvz .

## Acknowledgement

We are grateful to Y. Yamada for useful comments on this work and for the
collaboration on closely related projects which inspired this paper. This work
is supported by SITP and by the US National Science Foundation Grant
PHY-2014215.

##  Appendix A Inflationary evolution in models $V_{\rm
up}+V_{0}\tanh^{2}\frac{\varphi}{\sqrt{6a}}$

In section 5 we analyzed inflationary evolution in general $\alpha$-attractor
models with potentials of the type

$V(s)=V_{0}\large(1-e^{-\sqrt{2\over 3\alpha}s}+...\large)\ ,$ (78)

where $s$ is given by

$s=\varphi-\sqrt{3\alpha\over
2}\ln\Bigl{(}2\sqrt{6\alpha}\,{V^{\prime}_{0}\over V_{0}}\Bigr{)}\ ,$ (79)

and $V^{\prime}_{0}=\partial_{\phi}V|_{\phi=\sqrt{6\alpha}}$ at the boundary
$\phi=\sqrt{6\alpha}$. Here we will do it directly in terms of the field
$\varphi$, for the simplest model

$V=V_{\rm up}+V_{0}\tanh^{2}\frac{\varphi}{\sqrt{6a}}\ ,$ (80)

which is a part of the hybrid inflation model (17).

The number of e-foldings $N$ for inflation beginning at the point
$\varphi_{N}$ and proceeding via slow-roll up to the point $\varphi_{c}$ is
given by

$N\simeq\int^{\varphi_{N}}_{\varphi_{c}}d\varphi\,{V\over V_{\varphi}}\ .$
(81)

Here

$V_{\varphi}=\sqrt{\frac{2}{3\alpha}}V_{0}\tanh\frac{\varphi}{\sqrt{6\alpha}}\text{sech}^{2}\frac{\varphi}{\sqrt{6\alpha}}\
,$ (82)

and

$\int d\varphi\,{V\over V_{\varphi}}={3\alpha\over
4V_{0}}\left((V_{\varphi}+V_{0})\cosh\sqrt{{2\over 3\alpha}\varphi}+V_{\rm
up}\left(4\log\Bigl{(}\sinh\frac{\varphi}{\sqrt{6\alpha}}\Bigr{)}-1\right)\right)\
.$ (83)

Thus in the slow roll approximation

$\displaystyle N\simeq\int^{\varphi_{N}}_{\varphi_{c}}d\varphi\,{V\over
V_{\varphi}}={3\alpha\over 4V_{0}}\left((V_{\rm up}+V_{0})\cosh\sqrt{{2\over
3\alpha}\varphi_{N}}+4V_{\rm
up}\log\Bigl{(}\sinh\frac{\varphi_{N}}{\sqrt{6\alpha}}\Bigr{)}\right)$ (84)
$\displaystyle-$ $\displaystyle{3\alpha\over 4V_{0}}\left((V_{\rm
up}+V_{0})\cosh\sqrt{{2\over 3\alpha}\varphi_{c}}+4V_{\rm
up}\log\Bigl{(}\sinh\frac{\varphi_{c}}{\sqrt{6\alpha}}\Bigr{)}\right)\ .$ (85)

In the $\alpha$-attractor regime with $\varphi_{N}>\varphi_{c}$ and ${2\over
3\alpha}\varphi_{c}\gg 1$ this equation reads

$N\simeq{3\alpha(V_{\rm up}+V_{0})\over 8V_{0}}\left(e^{\sqrt{2\over
3\alpha}\varphi_{N}}-e^{\sqrt{2\over 3\alpha}\varphi_{c}}\right)\ .$ (86)

Using equation (30), one can show that is equivalent to equation (36), which
was obtained for generic $\alpha$-attractors.

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