# Hybrid inflation and gravitational waves
from accidentally light scalars

Felix Brümmer  _Laboratoire Univers et Particules de Montpellier (LUPM),
University of Montpellier and CNRS, Montpellier, France_ Giacomo Ferrante
_Laboratoire Univers et Particules de Montpellier (LUPM), University of
Montpellier and CNRS, Montpellier, France_ Michele Frigerio  _Laboratoire
Charles Coulomb (L2C), University of Montpellier and CNRS, Montpellier,
France_

###### Abstract

We construct a hybrid-inflation model where the inflaton potential is
generated radiatively, as gauge symmetries guarantee it to be accidentally
flat at tree level. The model can be regarded as a small-field version of
Natural Inflation, with inflation ending when the mass of a second scalar, the
waterfall field, turns tachyonic. This provides a minimal, robust realisation
of hybrid inflation, which predicts specific correlations among CMB
observables. Tachyonic preheating leads to the production of gravitational
waves which, for a low inflationary scale, might be detected by upcoming
experiments. Simple variations of the model can give rise to topological
defects, such as unstable domain walls. Their dynamics produces a stochastic
gravitational-wave background, which can be compatible with the recent
detection by pulsar timing arrays.

##  1 Introduction

Field-theoretical models of cosmic inflation can be roughly divided into two
classes. In large-field models, the field excursion of the inflaton is
transplanckian, while in small-field models it remains below the Planck scale.
When allowing for large field excursions, very simple shapes of the potential
can give rise to inflation (although many are now excluded by increasingly
accurate CMB measurements [1, 2]). However, it is generally dubious if large-
field models can be trustable effective field theories, without being embedded
in a complete quantum gravity framework. Small-field models, on the other
hand, generally require more involved field-theoretic constructions. In
particular, some mechanism should be included to protect the inflaton
potential from radiative corrections. Once this is achieved, effects from
transplanckian physics can be argued to be under control in the effective
field theory. The only necessary fine-tuning left is that of the cosmological
constant after inflation, and potentially of the initial conditions leading to
an inflationary phase.

In small-field models, the flatness of the inflaton potential could be
ensured, for example, by (spontaneously broken) supersymmetry, or by an
(approximate) shift symmetry if the inflaton is a pseudo Nambu-Goldstone boson
(pNGB). In the present work, we propose to employ a novel mechanism to this
end: the inflaton is an accidentally massless scalar field, i.e. a scalar
whose potential is constrained by symmetry to be flat up to loop corrections
and higher-dimensional operators [3]. Similarly to the pNGB case, a small
slope is generated radiatively.

At first sight, our inflaton potential is very similar to that of Natural
Inflation (NI), where the inflaton is realized as a $\mathrm{U}(1)$ pNGB [4].
Minimal NI, however, is a large-field model, and moreover is excluded by
Planck data [2]. Small-field NI needs additional fields responsible for ending
inflation, and such models of “hybrid NI” turn out to be difficult to
construct, since the couplings between the additional fields and the inflaton
should not spoil the flatness of the potential (although this can be achieved
with suitable discrete symmetries [5, 6]).

Our model, by contrast, is simple and economical; in the minimal case the
symmetry is simply a gauged $\mathrm{SO}(3)$ times a $\mathbb{Z}_{2}$ parity.
Indeed, models with accidentally light scalars [3] lend themselves
particularly well to hybrid inflation model-building. This is because they
feature an almost-flat inflaton direction by construction, yet the other
scalar fields couple to the inflaton at the tree level, which allows to
naturally end inflation by a second field becoming tachyonic.

We will compute the Cosmic Microwave Background (CMB) observables as a
function of the model parameters, and study their correlations. We will also
present naturalness arguments to identify the preferred region of parameters,
and derive the corresponding predictions.

While naturalness would prefer an inflationary scale not too far from the
Planck scale, lower-scale scenarios are more interesting for the potential
observation of gravitational wave (GW) signals. In hybrid models such as ours,
inflation ends because a steep tachyonic “waterfall” direction appears in
field space [7], which leads to a transition from the inflationary slow-roll
regime to a phase of tachyonic preheating [8, 9]. In this phase, GWs can be
produced, and may be within the reach of future detectors, if the inflationary
scale is low. In addition, spontaneous symmetry breaking (SSB) in the
inflaton-waterfall potential may lead to the production of topological
defects, which also source GWs. We will present slightly extended versions of
the minimal model, which produce cosmic strings or domain walls after the end
of inflation. These scenarios are phenomenologically interesting only if the
inflationary scale is sufficiently low. Then, the annihilations of topological
defects produce GWs which can be within experimental reach.

We will compute these various GW signals, study their correlations with the
CMB observables, and compare them with present constraints from LIGO-Virgo-
KAGRA [10] and NANOGrav [11], as well as with the expected sensitivity of
future GW detectors.

Both large-field and small-field models are the subject of several swampland
conjectures, which aim to characterize the set of effective field theories
that are supposedly incompatible with general principles of quantum gravity;
see [12] for a review. Specifically, models with transplanckian field
excursions are concerned by the Distance Conjecture [13]. This conjecture
states roughly that, at large distances in field space, an infinite tower of
fields becomes light, invalidating the effective field theory. On the other
hand, slow-roll potentials with subplanckian field excursions and positive
vacuum energy are claimed to be ruled out by various (arguably more
speculative) de Sitter conjectures [14, 15, 16]. While these conjectures are
certainly interesting, none of them is firmly established. For this work we
therefore assume that single-field slow-roll inflation is not per se in the
swampland, and that small-field models, in particular, can be viable effective
field theories of inflation.

##  2 Hybrid inflation with an accidentally light scalar

Models with accidentally massless scalar fields were recently constructed in
Ref. [3], where the focus was on models with a single scalar in an irreducible
representation and no ad-hoc discrete symmetries. To construct a model of
hybrid inflation, however, we need scalar fields in at least two different
multiplets, one of which will contain the inflaton and the other the waterfall
field. While our model will share many features with the ones presented in
Ref. [3], the details of its field content and symmetry structure differ.

The model is defined by a $G=\mathrm{SO}(3)$ symmetry with a real scalar
$\phi$ transforming in the ${\bf 5}$ of $\mathrm{SO}(3)$, and a real scalar
$\chi$ transforming in the ${\bf 3}$. We also impose a $\mathbb{Z}_{2}$
symmetry under which $\phi$ is odd. The most general renormalizable potential
is

$V=-\frac{1}{2}\mu_{\phi}^{2}\phi^{2}-\frac{1}{2}\mu_{\chi}^{2}\chi^{2}+\frac{\lambda_{\phi}}{4}(\phi^{2})^{2}+\frac{\lambda_{\chi}}{4}(\chi^{2})^{2}+\frac{\varepsilon}{4}\phi^{2}\chi^{2}+\frac{\zeta}{4}\,T^{a}_{AC}T^{b}_{CB}\phi_{A}\phi_{B}\chi^{a}\chi^{b}\,.$
(1)

Here $T^{a}_{BC}$ are the $\mathrm{SO}(3)$ generators in the ${\bf 5}$-plet
representation ($a=1,2,3$ and $A,B,C=1\ldots 5$). The continuous symmetry of
this potential is strictly $G=\mathrm{SO}(3)$: there is no larger custodial-
type symmetry present. More precisely, the symmetry of the kinetic term is
$\mathrm{SO}(8)$, which is explicitly broken to
$\mathrm{SO}(5)_{\phi}\times\mathrm{SO}(3)_{\chi}$ by the mass terms and the
$\lambda_{\phi,\chi}$ and $\epsilon$ quartics. The $\zeta$ coupling then
breaks the symmetry explicitly to $\mathrm{SO}(3)$.

It is instructive to consider first the case $\mu_{\phi}^{2}>0$ and
$\mu_{\chi}^{2}<0$ with all quartic couplings positive. This parameter choice
does not lead to hybrid inflation, but it is useful in order to understand how
a slow-roll direction emerges. The potential is minimized at
$\langle\chi\rangle=0$ and $|\langle\phi\rangle|=v$, where
$v^{2}=\mu_{\phi}^{2}/\lambda_{\phi}$. Therefore, the vacuum manifold is a
four-sphere, with $G$ completely broken at generic points. Three of the four
flat directions are Nambu-Goldstone directions corresponding to the broken $G$
generators. If $G$ is gauged, the associated NGBs are absorbed by the gauge
bosons. The fourth flat direction corresponds to an accidentally (tree level-)
massless scalar field. It is not associated to any symmetry of the scalar
potential and as such is neither an NGB nor, strictly speaking, a pNGB. The
flat direction will be lifted by loop corrections through both the scalar
quartic coupling $\zeta$ and the gauge coupling $g$. In the limit of small
$\zeta$, it can be understood as a genuine pNGB direction, as the scalar
potential acquires a larger symmetry for $\zeta\to 0$.

At a special point along the flat direction, corresponding to a null
eigenvector of one of the $G$ generators, an
O$(2)=\mathrm{SO}(2)\times\mathbb{Z}_{2}$ subgroup of $G$ is preserved. At
this point there are only two NGBs, while not one but two scalars remain
accidentally massless at the tree level; the second one is the scalar which is
absorbed by the $\mathrm{SO}(2)$ gauge boson outside the special point. This
point turns out to be a minimum of the one-loop effective potential.

For concreteness, consider an explicit parametrization of the tree-level flat
direction by writing the isospin-2 field $\phi$ as a $3\times 3$ traceless
symmetric matrix $\Phi=\phi_{A}\lambda_{A}$. Here $\lambda_{A}$ are the the
symmetric Gell-Mann matrices $\lambda_{1,3,4,6,8}$. By an $\mathrm{SO}(3)$
transformation, $\Phi$ can be diagonalized, such that its VEV becomes

$\langle\Phi\rangle=v\left(\lambda_{3}\sin\alpha+\lambda_{8}\cos\alpha\right)\,,\qquad
v^{2}=\mu_{\phi}^{2}/\lambda_{\phi}\,.$ (2)

The mass of the radial mode in $\Phi$ is

$m_{\rho}^{2}=2\lambda_{\phi}v^{2}$ (3)

and the $\chi$ masses are

$m_{\chi^{1,2}}^{2}=-\tilde{\mu}_{\chi}^{2}+2\,v^{2}\zeta\,\sin^{2}\left(\alpha\pm\frac{\pi}{3}\right)\,,\quad
m_{\chi^{3}}^{2}=-\tilde{\mu}_{\chi}^{2}+2\,\zeta v^{2}\sin^{2}\alpha$ (4)

where we have defined

$\tilde{\mu}_{\chi}^{2}=\mu_{\chi}^{2}-\frac{\varepsilon}{2}v^{2}\,.$ (5)

Since, for the time being, we chose $\mu_{\chi}^{2}<0$ and $\epsilon$ and
$\zeta$ positive, we have $\tilde{\mu}_{\chi}^{2}<0$ and so all masses are
positive. The $\alpha$-dependence of the masses is a crucial feature of this
model which will eventually allow us to realize hybrid inflation. As $G$ is
gauged, the gauge bosons are also massive at the tree level. The vacuum
manifold of physically inequivalent points is parametrized by
$\alpha\in[0,\,\pi/6]$, since points related by $\alpha\,\rightarrow\,-\alpha$
and $\alpha\,\rightarrow\,\alpha+\pi/3$ are identified under $G$. At
$\alpha=0$, there is a symmetry-enhanced point with a massless
$\mathrm{SO}(2)$ gauge field and one additional massless scalar field (which
is otherwise absorbed by the third massive gauge boson).

When including one-loop corrections to the potential, the vacuum degeneracy is
lifted, the physical vacuum is at the point $\alpha=0$, and the two tree-
level-massless scalars become massive. The only massless state is the gauge
boson of the unbroken $\mathrm{SO}(2)$. However, there remains a one-loop mass
hierarchy between the six heavy scalar modes having obtained their masses at
the tree level, and the two accidentally light ones with one-loop masses. We
emphasize that the latter are calculable and finite as functions of
$\mu_{\phi}^{2}$ and $\mu_{\chi}^{2}$, and therefore they are quadratically
sensitive to quantum corrections at the cutoff scale only at two loops (since
contributions to the mass parameters themselves are quadratically divergent at
one loop), much as in Little Higgs models (for a review of the latter see e.g.
Ref. [17]).

Focusing on the one-loop effective potential along the tree-level flat
direction, one finds that it is given by 111To calculate the effective
potential, we are using the renormalization prescription detailed in Ref. [3].
As a periodic and even function, $V_{\rm eff}(a)$ can be expanded into its
Fourier cosine series $\sum_{n}c_{n}\cos(na/f)$.

$V_{\rm eff}(a)=V_{0}-M^{4}\cos\frac{a}{f}+(\text{higher harmonics})~{},$ (6)

where $a=\alpha v$ is the canonically normalized “accident” field
parametrizing the tree-level flat direction, $f=v/6$, and we have arbitrarily
renormalized the vacuum energy to be $V_{0}-M^{4}$. Moreover, we find

$M^{4}=\frac{1}{160\,\pi^{2}}\left[9\,g^{4}\,v^{4}+2\,\frac{\tilde{\mu}_{\chi}^{10}}{\zeta^{3}\,v^{6}}\left(F\left(\frac{\zeta
v^{2}}{\tilde{\mu}_{\chi}^{2}}\right)-T_{F,5}\left(\frac{\zeta
v^{2}}{\tilde{\mu}_{\chi}^{2}}\right)\right)\right]~{},$ (7)

where $g$ is the $\mathrm{SO}(3)$ gauge coupling, $F(x)=(1-2\,x)^{5/2}$, and
$T_{F,5}(x)=1-5\,x+\frac{15}{2}x^{2}-\frac{5}{2}x^{3}-\frac{5}{8}x^{4}-\frac{3}{8}x^{5}$
is the Taylor polynomial of $F(x)$ of degree 5. Higher harmonics in the
effective potential are suppressed relative to the first harmonic by small
numerical factors.222For example, the coefficient of the second harmonic
$\cos(2a/f)$ induced by $\zeta$ can be shown to always be smaller than its
counterpart of Eq. (7) by a factor $\gtrsim 10$ in the region of field space
of interest. Likewise, the second harmonic induced by $g$ is always suppressed
with respect to the first. They will be neglected in the following.

The shape of the one-loop potential, therefore, closely resembles what one
would obtain if $a$ were the pNGB of an approximate global $\mathrm{U}(1)$
symmetry. Identifying $a$ with the inflaton and setting $V_{0}=M^{4}$, one
obtains a realization of the famous Natural Inflation (NI) model [4]. However,
vanilla NI is excluded by Planck data [2]. Simple modifications of the minimal
NI model, such as a non-minimal coupling to gravity [18, 19, 20] or allowing
for some exotic phase of reheating [21] can improve the fit, but in the
preferred parameter region one finds $f\gtrsim M_{\mathrm{P}}$.333We designate
by $M_{\mathrm{P}}$ the reduced Planck mass, $M_{\mathrm{P}}\equiv(8\pi
G_{\mathrm{\scriptscriptstyle{N}}})^{-1/2}=2.4\times 10^{18}\,{\rm GeV}$. As
already discussed, transplanckian field excursions, apart from being
constrained by the Swampland Distance Conjecture, imply that the effects of
higher-dimensional operators are not under control (although models have been
constructed which address this issue [22, 23]). To overcome these problems, it
is desirable to identify a variant of the model where the field excursion is
sub-Planckian, i.e. a small-field model of inflation.

In a small-field version of NI, the inflationary potential should be dominated
by the constant term $V_{0}$ in the vacuum energy. This implies that slow-roll
is possible for $f<M_{\mathrm{P}}$, but it also implies that slow-roll never
ends on its own because the slow-roll parameters never grow to become ${\cal
O}(1)$, and that the vacuum energy density $V_{0}$ somehow needs to be
converted into radiation eventually. To this end, one may extend the model
into one of hybrid inflation [7]: add a second scalar field, the waterfall
field, which is heavy during inflation but becomes tachyonic for some value of
the inflaton, triggering the end of inflation and the start of preheating.

Models of hybrid inflation with pNGBs as inflatons were constructed, for
example, in Refs. [24, 23]. In these models, the approximate symmetry
protecting the inflaton is explicitly broken by the couplings to the waterfall
field. These must be chosen ad hoc such as to preserve the flatness of the
potential. Related models were subsequently built [5, 6], where it was shown
that the required structure of the potential can be protected by non-abelian
discrete symmetries. By contrast, we will now show that our model with an
accidentally light scalar can furnish a natural model of hybrid inflation with
the simple potential of Eq. (1).

So far the role of the field $\chi$ was solely to provide a source for loop
corrections to the effective potential which lift the flatness of the inflaton
potential. We will now identify $\chi$ with the waterfall field by choosing
the model parameters such that the true vacuum of the model is no longer at
$\langle\chi\rangle=0$ and $\langle\phi\rangle\neq 0$ but rather at
$\langle\phi\rangle=0$ and $\langle\chi\rangle\neq 0$. Our previous discussion
of the vacuum structure applies as long as the effective $\chi$ masses, given
in Eq. (4), are all positive, regardless of the sign of the mass parameter
$\tilde{\mu}_{\chi}^{2}$ of Eq. (5). Even for $\tilde{\mu}_{\chi}^{2}>0$, the
effective potential is still given by Eqs. (6) and (7) upon replacing $M^{4}$
by its real part. It is only when a tree-level mass becomes tachyonic that our
computation of $V_{\rm eff}$ can no longer be trusted.

Now choosing parameters such that $\tilde{\mu}_{\chi}^{2}$ satisfies

$0<\tilde{\mu}_{\chi}^{2}<\frac{\zeta}{2}v^{2}\,,$ (8)

one observes that $m^{2}_{\chi^{3}}$ is positive for $\alpha=\pi/6$, negative
for $\alpha=0$, and crosses zero at

$\alpha_{c}=\arcsin\sqrt{\frac{\tilde{\mu}_{\chi}^{2}}{2\zeta\,v^{2}}}\,,$ (9)

while all other masses remain strictly positive for $\alpha\geq\alpha_{c}$.
This can be achieved by choosing $\mu_{\chi}^{2}>0$ and suitable (positive)
values for the quartics.

In this regime, the inflaton $a$ starts rolling close to $\alpha=\pi/6$ or
equivalently $a=\pi f$, where all scalars except $a$ have positive tree-level
masses. In particular, $\chi$ is stabilized at $\chi=0$. For
$\alpha>\alpha_{c}$ the one-loop effective potential Eq. (6) can be trusted,
and provides a small slope which the inflaton rolls down. As the inflaton
crosses the point $\alpha_{c}$, the tree-level vacuum will no longer be given
by $\chi=0$ and Eq. (2), since a tree-level tachyon develops in the $\chi^{3}$
direction. Slow-roll ends as the tachyonic $\chi$ mass grows large, and $\chi$
modes are being excited. Eventually, the potential energy of the inflaton is
converted into energy in the $\chi$ sector. In Fig. 1 we provide a sketch of
the potential along the $a$ (inflaton) and $\chi^{3}$ (waterfall) directions,
for the sake of illustration.

Figure 1: Pictorial representation of the inflationary potential, normalised
to the inflationary scale $V_{0}$. Close to $a=\pi f$, where inflation begins,
the potential along the $\chi^{3}$ direction (purple lines) has positive and
large curvature, stabilizing the inflationary trajectory (red line). At the
$a=a_{*}$ the pivot scale crosses the Hubble horizon, $RH=k_{*}$ (with $R$ the
scale factor). As the inflaton slow-rolls down its potential,
$m_{\chi^{3}}^{2}$ decreases until it becomes $0$ at the critical point
$a_{c}$, where inflation ends. For $a<a_{c}$, the effective potential cannot
be trusted, as the accidentally flat direction is no longer a minimum of the
tree-level potential. A value of $a_{*}/f$ ($a_{c}/f$) in the green (blue)
region would correspond to a tuning $\gtrsim 10\%$ on the initial (final)
point of observable inflation, and therefore it is theoretically disfavoured.

Reheating should then proceed via some coupling, such as a Higgs portal
coupling, between $\chi$ and the Standard Model; in this work we will not
explore reheating in detail. It is easily checked that, for suitable values of
the couplings, the true vacuum of the model is located at

$\chi^{3}=\sqrt{\frac{\mu^{2}_{\chi}}{\lambda_{\chi}}}\equiv
v_{\chi}\,,\qquad\text{all other fields}=0$ (10)

up to $G$-transformations. The residual symmetry is $\mathrm{SO}(2)\subset G$,
and all fields except for the unbroken gauge boson pick up tree-level masses.
Since the true vacuum should be Minkowski up to a tiny cosmological constant,
we can identify

$V_{0}\simeq\frac{\mu_{\chi}^{4}}{4\,\lambda_{\chi}}-\frac{\mu_{\phi}^{4}}{4\,\lambda_{\phi}}\,,$
(11)

which is the (tree-level) potential energy difference between the inflationary
field configuration and the true vacuum. In the following we will assume that
the second term in Eq. (11) is not only subdominant but even negligible.

##  3 Inflation parameters and CMB observables

As inflation proceeds along the tree-level flat direction, the waterfall field
$\chi$ is stabilized at $\chi=0$ by its effective mass, hence the model
qualifies as a single-field model. The evolution of scalar and tensor
perturbations is then dictated by the potential in Eq. (6), with the slow-roll
parameters defined by

$\epsilon\equiv\frac{M_{\mathrm{P}}^{2}}{2}\left(\frac{V_{\rm
eff}^{\prime}}{V_{\rm eff}}\right)^{2}\,,\qquad\eta\equiv
M_{\mathrm{P}}^{2}\frac{V_{\rm eff}^{\prime\prime}}{V_{\rm
eff}}\,,\qquad\xi\equiv M_{\mathrm{P}}^{4}\frac{V_{\rm eff}^{\prime}V_{\rm
eff}^{\prime\prime\prime}}{V_{\rm eff}^{2}}\,,\qquad\sigma\equiv
M_{\mathrm{P}}^{6}\frac{\left(V^{\prime}_{\rm
eff}\right)^{2}V^{\prime\prime\prime\prime}_{\rm eff}}{V_{\rm eff}^{3}}\,,$
(12)

where primes denote derivatives with respect to the inflaton field $a$.

At the leading order in the slow-roll parameters, the tensor-to-scalar ratio
$r$, the amplitude of the curvature power spectrum $A_{s}$, the spectral index
of the curvature power spectrum $n_{s}$, the running of the spectral index
$n_{r}$ and the running of the running $n_{rr}$ can be computed as

$\displaystyle r=16\epsilon_{*}\,,\qquad
A_{s}=\frac{H_{*}^{2}}{8\pi^{2}M_{\mathrm{P}}^{2}}\frac{1}{\epsilon_{*}}\,,\qquad
n_{s}=1+2\eta_{*}-6\epsilon_{*}\,,$ (13) $\displaystyle
n_{r}=16\epsilon_{*}\eta_{*}-24\epsilon_{*}^{2}-2\xi_{*}\,,\qquad
n_{rr}=-192\epsilon_{*}^{3}+192\epsilon_{*}^{2}\eta_{*}-32\epsilon_{*}\eta_{*}^{2}-24\epsilon_{*}\xi_{*}+2\eta_{*}\xi_{*}+2\sigma_{*}\,,$

where the asterisk means that all the quantities are evaluated at the moment
when the pivot scale, $k_{*}=0.05\;\rm{Mpc^{-1}}$, exits the Hubble horizon.
These quantities are constrained by CMB observations. The most recent
constraints on the scalar power spectrum are given by the Planck collaboration
[1]: $\log\left(10^{10}A_{s}\right)=3.047\,\pm\,0.014$,
$n_{s}=0.9647\,\pm\,0.0043$, $n_{r}=0.0011\,\pm\,0.0099$,
$n_{rr}=0.009\,\pm\,0.012$. The tensor-to-scalar ratio is constrained to be
$r<0.036$ by the BICEP/Keck collaboration [25].

The independent model parameters entering the calculation of the CMB power
spectrum can be taken as $V_{0}$, $M$, $f$ and $a_{*}$. Considering that
$V_{0}\gg M^{4}$, the observables are

$\displaystyle r\simeq
8\,\left(\frac{M_{\mathrm{P}}}{f}\frac{M^{4}}{V_{0}}\right)^{2}\sin^{2}\frac{a_{*}}{f}\,,\qquad$
$\displaystyle
A_{s}\simeq\frac{1}{12\pi^{2}}\frac{f^{2}\,V_{0}^{3}}{M_{\mathrm{P}}^{6}\,M^{8}}\sin^{-2}\frac{a_{*}}{f}\,,\qquad
n_{s}\simeq
1+2\,\frac{M_{\mathrm{P}}^{2}}{f^{2}}\frac{M^{4}}{V_{0}}\cos\frac{a_{*}}{f}\,,$
(14) $\displaystyle n_{r}\simeq
2\,\left(\frac{M_{\mathrm{P}}^{2}}{f^{2}}\frac{M^{4}}{V_{0}}\right)^{2}\sin^{2}\frac{a_{*}}{f}\,,\qquad$
$\displaystyle
n_{rr}\simeq-4\,\left(\frac{M_{\mathrm{P}}}{f}\right)^{6}\left(\frac{M^{4}}{V_{0}}\right)^{3}\sin^{2}\frac{a_{*}}{f}\cos\frac{a_{*}}{f}\,,$

while the number of $e$-folds of observable inflation is given by

$N\simeq\frac{1}{M_{\mathrm{P}}^{2}}\int_{a_{\rm
end}}^{a_{*}}\frac{V_{\rm{eff}}}{V^{\prime}_{\rm{eff}}}da\simeq\frac{f^{2}}{M_{\mathrm{P}}^{2}}\frac{V_{0}}{M^{4}}\left(\log\tan\frac{a_{*}}{2f}-\log\tan\frac{a_{\rm
end}}{2f}\right)\,.$ (15)

We identify the field value $a_{\rm end}$ where inflation ends with the
critical point $a_{c}=6f\alpha_{c}$, where the waterfall field becomes
tachyonic, see Eq. (9). We will comment on the assumptions underlying this
identification at the end of this section. As the end of inflation is
controlled by the coupling between the inflaton and the waterfall field,
$a_{\rm end}$ is an independent parameter of our model, which can be freely
adjusted: by suitably choosing $a_{\rm end}$, we are always able to
accommodate enough $e$-folds of inflation. For concreteness, in what follows
we fix $N=60$ as a reference value.

A characteristic feature of the model is to predict positive $n_{r}$ and
$n_{rr}$. A second feature concerns the running of the running of the spectral
index, which for this model is not independent from $n_{s}$ and $n_{r}$,
$n_{rr}\simeq(1-n_{s})n_{r}$, leading to the constraint $n_{rr}<4\times
10^{-4}$. This is more stringent than the direct experimental bound on
$n_{rr}$. The upcoming satellite mission SPHEREx [26] will measure the running
of the spectral index with a target sensitivity $\Delta n_{r}\sim 10^{-3}$,
and therefore it could rule out the model in the near future.

The four parameters of the model are not all determined by the CMB
observables. Requiring that the model parameters do not suffer from fine-
tuning problems will lead to further relations between them, see section 3.1.
To better understand how observation constrains them, we invert the relations
given in Eq. (LABEL:eq:CMBObs) and obtain

$\displaystyle V_{0}=\frac{3\pi^{2}}{2}A_{s}rM_{\mathrm{P}}^{4}\,,\qquad$
$\displaystyle
M^{4}=\frac{3\pi^{2}}{16}\frac{A_{s}}{n_{r}}\sqrt{2n_{r}+\left(n_{s}-1\right)^{2}}r^{2}M_{\mathrm{P}}^{4}\,,$
(16) $\displaystyle
f=\frac{1}{2}\sqrt{\frac{r}{n_{r}}}M_{\mathrm{P}}\,,\qquad$
$\displaystyle\frac{a_{*}}{f}=\arccos\left[\frac{n_{s}-1}{\sqrt{2n_{r}+(n_{s}-1)^{2}}}\right]\,.$

By imposing $A_{s}=2.105\times 10^{-9}$, $n_{s}=0.965$, $n_{r}<0.011$ and
$r<0.036$, we can derive the inequalities

$V_{0}^{1/4}<6\times 10^{-3}M_{\mathrm{P}}\,,\qquad\frac{M^{4}}{f^{4}}\simeq
3\sqrt{2}\pi^{2}A_{s}n_{r}^{3/2}<10^{-10}\,.$ (17)

In the left-hand panel of Fig. 2 we chart the region allowed by observations
in the $(n_{r},r)$ plane, once $A_{s}$ and $n_{s}$ are fixed. All the above
relations have been derived under the assumption that $V_{0}$ drives
inflation, i.e. $V_{0}\gg M^{4}$. Below we quantify what separation is needed
between the two scales, in order not to end inflation before the critical
point $a=a_{c}$. In particular, we require $\epsilon<1$ all along the
inflationary trajectory.444For slow-roll dynamics to persists we also have to
check that $\lvert\eta\rvert<1$ for any value of $a$, but this leads to a
constraint which is very similar to Eq. (19).

$\epsilon=\frac{M_{\mathrm{P}}^{2}}{2}\left(\frac{\frac{M^{4}}{f}\sin{\frac{a}{f}}}{V_{0}-M^{4}\cos{\frac{a}{f}}}\right)^{2}<1\,,\quad\forall\;a\,.$
(18)

This is equivalent to requiring

$\frac{M^{4}}{V_{0}}<\sqrt{\frac{2f^{2}}{2f^{2}+M_{\mathrm{P}}^{2}}}\iff
r<n_{r}\left(-1+\sqrt{1+\frac{64}{2n_{r}+(n_{s}-1)^{2}}}\right)\,.$ (19)

The region where this condition is not satisfied is shaded in dark grey in the
left-hand panel of Fig. 2.

Figure 2: Left-hand panel. The values of $M$ and $f$ in units of
$M_{\mathrm{P}}$, as a function of $n_{r}$ and $r$, while fixing
$A_{s}=2.105\times 10^{-9}$ and $n_{s}=0.965$. The largest displayed values of
$n_{r}$ and $r$ correspond to the present experimental upper bounds. Solid
(dashed) lines are contours of constant $M$ ($f$). The dark-grey region
violates the slow-roll condition before the critical point. Right-hand panel.
The white region is the one which complies with naturalness criteria,
according to the discussion in section 3.1. In the orange regions the value of
$f$ is unnaturally small, while in the red region $f>M_{\mathrm{P}}$. The
parameter space covered by the blue and green regions requires a significant
fine-tuning of the inflation initial and final conditions, respectively.

In Fig. 3 we show the predictions of the model in the $(n_{s},r)$ plane,
together with the region allowed by the latest CMB measurements [2, 25]. Our
model is in agreement with observations for natural values of $f$, i.e. for
$f$ smaller than, but not too far from $M_{\mathrm{P}}$.

Figure 3: The purple areas show the 95% CL and 68 % CL preferred regions from
Planck 2018 + BAO + lensing + BICEP/Keck 2018 data [25]. We fixed the
amplitude of the scalar power spectrum at $A_{s}=2.105\times 10^{-9}$. In the
left-hand panel we set the running of the spectral index at $n_{r}=10^{-3}$,
while in the right-hand panel we took $n_{r}=10^{-4}$. The horizontal, green
lines are obtained by fixing $f=0.2\,M_{\mathrm{P}}$ and varying $M$, while
along the vertical, blue lines $M/f=3\times 10^{-3}$ is fixed and $f$ varies.
The orange and the red regions are disfavoured by naturalness (see section
3.1): low values of the tensor-to-scalar ratio $r$ require unnaturally small
$f$ while a large $r$ is associated to transplanckian $f$. The relation
between $f$ and $r$ depends on the value assumed for $n_{r}$, according to Eq.
(16).

We require the end of inflation to occur within one Hubble time after the
critical point has been reached by the accident. This allows us to identify
$a_{c}=6f\alpha_{c}$ with the point $a_{\rm end}$ in field space in which
inflation ends. For this requirement to hold, the Hubble rate at the critical
point, $H_{c}$, must be much lower than the value of the waterfall mass, after
$\Delta t\sim H_{c}^{-1}$ has elapsed since $a=a_{c}$

$\lvert m_{\chi^{3}}^{2}\left(a_{c}+\Delta a\right)\rvert\gg H_{c}^{2}\,,$
(20)

where $\Delta a$ is the accident excursion during $\Delta t$. This condition
is also known as the “waterfall condition” [7]. Given that the energy density
during inflation is dominated by the false vacuum, the Hubble rate can be
approximated to $H_{c}\simeq V_{0}/3M_{\mathrm{P}}^{2}$. Moreover, the
evolution of $a$ is governed, in the slow-roll approximation, by
$3\,H_{c}\,\dot{a}\simeq\partial V_{\rm eff}/\partial a$. With these
considerations in mind the above waterfall condition translates into a lower
bound on the inflaton-waterfall quartic coupling,

$\zeta\gg\frac{V_{0}^{2}}{18\,M^{4}M_{\mathrm{P}}^{4}}\,\frac{1}{\sin\left(a_{c}/f\right)\sin\left(a_{c}/3f\right)}\,.$
(21)

By making use of the relations in Eqs. (16) and (15), we find that the right-
hand side of the above condition is a monotonic, increasing function of
$n_{r}$, once $A_{s}$, $n_{s}$ and $N$ are fixed. For $n_{r}$ taking its
largest value allowed by observations, Eq. (21) becomes $\zeta\gg 10^{-6}$,
meaning that we can always choose a perturbative value of $\zeta$ for which
the waterfall condition holds. We note that the requirement for inflation to
end as soon the critical point is reached is not imposed by observations. In
principle, one could also allow inflation to proceed along the waterfall
direction for some 20 to 40 $e$-folds as realised in the so called “mild
hybrid inflation” scenario [27, 28, 29].

###  3.1 Naturalness considerations

The aim of this section is to asses which portion of the parameter space
allowed by observations complies with naturalness. An important motivation for
considering this model in the first place is that the inflaton potential is
nearly flat for generic values of the couplings, without requiring fine-
tuning. To what extent does the model remain “natural”, or minimally fine-
tuned, once we impose that it should give rise to a quantitatively realistic
phenomenology?

It turns out that there is a preferred range of natural values for the running
of the spectral index $n_{r}$. A lower bound on $n_{r}$ is obtained as
follows: from Eq. (LABEL:eq:CMBObs) we have

$n_{r}=\frac{1}{2}\left(n_{s}-1\right)^{2}\tan^{2}\frac{a_{*}}{f}\,.$ (22)

Hence, fixing $n_{s}$ at its value preferred by CMB measurements, $n_{r}$
could be arbitrarily small, but only at the cost of tuning the start of
observable inflation $a_{*}$ arbitrarily close to $a_{*}=\pi f$. However, when
remaining agnostic about any pre-inflationary physics, the inflaton could
start rolling anywhere in field space. Taking, for instance, 10% as an
acceptable degree of fine-tuning of the initial condition,

$\frac{a_{*}}{\pi f}\lesssim 0.9\,,$ (23)

one obtains the naturalness bound $n_{r}\gtrsim 6.5\times 10^{-5}$. Requiring
the fine-tuning to be less than 1% gives, instead, $n_{r}\gtrsim 6\times
10^{-7}$. Such lower bounds are shown as blue-shaded regions in the right-hand
panel of Fig. 2.

An upper bound on $n_{r}$ is similarly given by the following argument: fixing
the number $N$ of observable e-folds to $N=60$, the parameter combination
$f^{2}V_{0}/(M_{\mathrm{P}}^{2}M^{4})$ could in principle be made arbitrarily
small by tuning the ending point of inflation $a_{\rm end}$ close to zero, see
Eq. (15) (and by simultaneously tuning $a_{*}$ close to $a_{*}=\frac{\pi}{2}f$
to obtain the observed value of $n_{s}$ according to Eqs. (LABEL:eq:CMBObs)).
This would imply, again by Eqs. (LABEL:eq:CMBObs), that $n_{r}$ could be
arbitrarily large. But $a_{\rm end}$ is given by a combination of independent
model parameters which have no reason to conspire to produce $a_{\rm
end}\simeq 0$. Imposing for concreteness

$\frac{a_{\rm end}}{\pi f}\gtrsim 0.1\,,$ (24)

this yields $n_{r}\lesssim 2\times 10^{-3}$, shown by green shading in the
right-hand panel of Fig. 2.

Moreover, naturalness requires $f$ to be not too far from the cutoff scale
$\Lambda_{\rm UV}$ of the model. This is nothing but the well-known hierarchy
problem of models with elementary scalar fields: quadratically divergent loop
corrections to the mass parameters $\mu_{\phi}^{2}$ and $\mu_{\chi}^{2}$
signal that they depend sensitively on the ultraviolet dynamics at the scale
$\Lambda_{\rm UV}$. At this scale, the model might be embedded in a
supersymmetric field theory, or in a strongly coupled field theory where
$\phi$ and $\chi$ emerge as bound states. However, the most minimal assumption
is that there is no field-theoretic embedding which cures the UV sensitivity,
in which case we should identify $\Lambda_{\rm UV}$ with the quantum gravity
scale.

Quantum gravity corrections are, of course, greatly model-dependent.
Identifying $\Lambda_{\rm UV}=M_{\mathrm{P}}$ for definiteness, and assuming
that UV states have $\lesssim{\cal O}(1)$ couplings to $\phi$ and $\chi$, we
can derive a tentative naturalness bound on the tensor-to-scalar ratio $r$.
Cutting off the one-loop corrections to the $\mu_{\phi}^{2}$ parameter at the
scale $M_{\mathrm{P}}$ and requiring that they do not exceed the tree-level
value leads to

$\mu_{\phi}^{2}\gtrsim\left(7\,\lambda_{\phi}+\frac{3}{2}\,\varepsilon+3\,\zeta\right)\frac{M_{\mathrm{P}}^{2}}{16\pi^{2}}\,.$
(25)

Since $f^{2}=\mu_{\phi}^{2}/6\lambda_{\phi}$, this implies $f\gtrsim
0.1\,M_{\mathrm{P}}$, under the technically natural assumption that
$\varepsilon$ and $\zeta$ are at most of the order of $\lambda_{\phi}$.
Therefore, the orange-shaded region in the right-hand panel of Fig. 2 is
disfavoured. From Eqs. (LABEL:eq:CMBObs) it follows that
$r=4\,n_{r}\,(f/M_{\mathrm{P}})^{2}$ and thus

$r\gtrsim(4\times 10^{-2})\,n_{r}\,,$ (26)

shown in orange shading in Fig. 3. Depending on how much fine-tuning one is
willing to accept, this bound can of course shift by several orders of
magnitude, as also indicated in the right-hand panel of Fig. 2.

On the other hand, values of $f$ too close to $\Lambda_{\rm UV}$ imply that
higher-dimensional operators, which may spoil the tree-level flatness of the
inflaton potential, can no longer be neglected. In that sense, the “least
fine-tuned” region of parameter space is also the one with the least
theoretical control. In Figs. 2 and 3 we shaded in red the regions with
$f\gtrsim M_{\mathrm{P}}$, or equivalently $r\gtrsim 4n_{r}$.

Such naturalness constraints translate into some bounds on the model
parameters. For instance, the inflationary scale is given by
$V_{0}\simeq\lambda_{\chi}v_{\chi}^{4}/4$, and the above bounds on $r$ and
$n_{r}$ can be translated into bounds on $V_{0}$ once $A_{s}$ is fixed,
according to Eq. (16). This, together with $0.1\,M_{\mathrm{P}}\lesssim
v_{\chi}\lesssim M_{\mathrm{P}}$ (we require $v_{\chi}$ to be subplanckian,
analogously to $f$) gives a natural range of values for the $\chi$ quartic
coupling,

$10^{-13}\lesssim\lambda_{\chi}\lesssim 10^{-5}\,.$ (27)

##  4 Gravitational waves from preheating

Hybrid inflation is followed by a stage of tachyonic preheating [8, 9]. The
tachyonic instability of the waterfall field exponentially enhances field
inhomogeneities. Such inhomogeneous, bubble-like structures behave as scalar
waves and scatter off each other until the Universe is homogenized and driven
to local thermal equilibrium. A stochastic gravitational wave background
(SGWB) is produced in the process [30, 31, 32, 33]. GWs quickly decouple from
the background dynamics and propagate (almost) undisturbed throughout cosmic
history, hence carrying information about the process that sourced them. Given
the energy scale of inflation, which is typically large, the frequency at
which GWs are produced during preheating lies well beyond the reach of current
or even planned interferometers. However, it has been shown, e.g. in Ref.
[33], that by sufficiently lowering the scale of inflation, the signal might
be brought within the projected sensitivity of the Einstein Telescope (ET)
[34]. It is also worth mentioning that there have been proposals of using
optical atomic clocks to detect high-frequency GWs, see e.g. Ref. [35] and
references therein.

The spectrum of GWs produced during preheating presents a peaked shape, with
the frequency and amplitude at the peak given by [33]

$f_{\star}\sim 4\times 10^{10}\,{\rm
Hz}\frac{k_{\star}}{{\rho_{c}}^{1/4}}\,,\qquad h^{2}\Omega_{\star}\sim
10^{-6}\left(\frac{H_{c}}{k_{\star}}\right)^{2}\,,$ (28)

where $k_{\star}$ is the typical momentum enhanced by the tachyonic
instability, related to the characteristic size of inhomogeneities as
$R_{\star}\sim 1/k_{\star}$.555Here $k_{\star}$ is unrelated to the pivot
scale of CMB observables $k_{*}$ of Section 3. Here, $\rho_{c}\simeq V_{0}$
and $H_{c}^{2}\simeq V_{0}/(3M_{\mathrm{P}}^{2})$ are the energy density and
the Hubble rate at the end of inflation or, analogously, at the onset of the
tachyonic instability. The value of $k_{\star}$ depends on the inflationary
dynamics close to the waterfall transition.

From Eq. (28) one may already deduce that for a signal of GWs from preheating
to be within the reach of the ET, the inflation scale must lie in the range

$3\times 10^{5}\,{\rm GeV}\lesssim V_{0}^{1/4}\lesssim 5\times 10^{11}\,{\rm
GeV}\,,$ (29)

which is associated to an unobservably small tensor-to-scalar ratio, according
to Eq. (16). Should $r$ be measured in the future, along with a SGWB at the ET
which can be traced back to tachyonic preheating, this would therefore rule
out our model (and other models of hybrid inflation, where tachyonic
preheating is an unavoidable feature).

In principle, two different regimes for GW production from preheating can be
identified [33], depending on the velocity $\dot{a}_{c}$ of the inflaton at
the critical point $a=a_{c}$ where the waterfall field becomes tachyonic. In
the first regime, $a$ is fast enough for the tachyonic instability to be
driven by its classical dynamics. Another possibility is that $\dot{a}_{c}$ is
sufficiently small, so that preheating is triggered by quantum fluctuations.
In the present work, we take model parameters such that inflation ends as soon
as the inflaton crosses the critical point, see Eq. (21). In that case
classical rolling always dominates.

We are interested in an order-of-magnitude estimate of the amplitude and
frequency of the GWs signal at the peak. Due to the fact that tachyonic
preheating is a non-linear process, a more precise computation of the GW
spectrum would require lattice simulations which is beyond the scope of this
work. In the slow-roll approximation, $\ddot{a}\ll 3H\dot{a}$, the inflaton
velocity at the critical point reads

$\dot{a}_{c}\simeq\frac{M_{\mathrm{P}}}{\sqrt{3}f}\frac{M^{4}}{V_{0}^{1/2}}\sin\frac{a_{c}}{f}\,.$
(30)

If its velocity is large enough, the inflaton will keep rolling classically
past the critical point. At $a=a_{c}$, the mass-squared of the waterfall field
vanishes and then starts growing negative so that after an interval $\Delta t$
it becomes $m_{\chi^{3}}^{2}\simeq 6\,\zeta\,f\,\dot{a}_{c}\,\Delta
t\,\sin\left(a_{c}/3f\right)$. The exponential growth of quantum fluctuations
becomes efficient when the tachyonic mass of the waterfall field has grown
large enough such that $\sqrt{|m_{\chi^{3}}^{2}|}\gtrsim\Delta t^{-1}$. This
leads to the enhancement of the mode $k_{\star}\sim 1/\Delta
t\lesssim\sqrt{|m_{\chi^{3}}^{2}|}$, so that

$k_{\star,{\rm cl}}^{3}\simeq
2\sqrt{3}\,\zeta\,\frac{M^{4}M_{\mathrm{P}}}{V_{0}^{1/2}}\sin\frac{a_{c}}{f}\sin\frac{a_{c}}{3f}\,.$
(31)

Fig. 4 shows the values of $r$ (or, analogously, of the inflationary scale
$V_{0}$) and $n_{r}$ which correspond to a GW signal within the ET projected
sensitivity for $\tilde{\mu}_{\chi}^{2}/f^{2}=1$ (left-hand panel) and
$\tilde{\mu}_{\chi}^{2}/f^{2}=10^{-6}$ (right-hand panel), where the ratio
$\tilde{\mu}_{\chi}^{2}/f^{2}$ measures the steepness of the waterfall
direction.

Figure 4: Sensitivity region (yellow) of the ET [34] for the GW spectrum
produced during tachyonic preheating, as a function of $n_{r}$ and $r$ (or
equivalently $V_{0}$). In the left-hand panel we take
$\tilde{\mu}_{\chi}^{2}/f^{2}=1$, while in the right-hand one we fix
$\tilde{\mu}_{\chi}^{2}/f^{2}=10^{-6}$. The projected sensitivity for the ET
is the one derived in Ref. [11] and it is shown in Fig. 5. To guide the eye we
also plot lines of constant peak frequency and amplitude for the preheating GW
spectrum. The brown region shows the integrated sensitivity for the third
observing run of the LIGO-Virgo-KAGRA network [10].

We conclude this section by mentioning that, in principle, tachyonic
instability could also lead to production of primordial black holes
(PBHs).666As a side note, we touch on the possibility of producing PBHs at the
end of inflation thanks to an enhancement of curvature perturbations driven by
a large running of the spectral index. It has been shown in Ref. [36] that
this eventuality bounds $n_{r}\lesssim 10^{-2}$ in single-field, slow-roll
inflation. However, larger values for $n_{r}$ are nowadays excluded by CMB
observations. In our model, however, inflation ends almost immediately after
reaching the critical point. This leads to enhancement of perturbations at
very small scales and to a subsequent production of extremely light PBHs which
have completely evaporated by now. It is possible to envision a scenario in
which inflation goes on for a certain number of e-folds along the waterfall
direction, hence producing PBHs which can be of phenomenological relevance
[37, 38].

##  5 Topological defects

Models with spontaneously broken symmetries may give rise to topological
defects such as magnetic monopoles, cosmic strings (CSs), and domain walls
(DWs).

Magnetic monopoles can appear if the second homotopy group of the vacuum
manifold is nonzero. When $G=\mathrm{SO}(3)$ is broken to $\mathrm{SO}(2)$ by
a triplet VEV, as is the case in our model, this gives rise to the celebrated
’t Hooft-Polyakov monopole [39, 40] with
$\pi_{2}(\mathrm{SO}(3)/\mathrm{SO}(2))=\pi_{2}[S^{2}]=\mathbb{Z}$. However,
in our models the relevant phase transition takes place before inflation, and
therefore no monopoles are likely to remain in our Hubble patch. To be
precise, symmetry breaking in our model proceeds in two steps: first, at the
scale $v$, the symmetry group $G$ is broken completely by the 5-plet VEV,
while the triplet VEV is zero in this phase. The vacuum manifold (or rather
the scalar field manifold which minimizes the potential for all fields except
the slowly-rolling inflaton) is $G$, and in different causally connected
regions of the universe, the scalar fields will take VEVs in different domains
of $G$. However, the subsequent phase of inflation stretches out these
inhomogeneities, such that, at all scales relevant to us, the $5$-plet VEV
becomes effectively a single point in field space. Nontrivial field
configurations which might evolve into monopoles are “inflated away”. The
second step of symmetry breaking happens at the waterfall transition at the
end of inflation, when the $5$-plet VEV relaxes to zero and a triplet VEV is
generated, thereby restoring an $\mathrm{SO}(2)$ symmetry. The vacuum manifold
becomes $\mathrm{SO}(3)/\mathrm{SO}(2)\simeq S^{2}$; however, no new monopoles
can be created, since this would require energies sufficiently high to restore
the entire $\mathrm{SO}(3)$ symmetry. We conclude that, for better or worse,
magnetic monopoles will not affect the cosmology of our model.

Cosmic strings can appear if the scalar vacuum manifold is not simply
connected. This is not the case in the minimal model, but it is interesting to
consider a non-minimal extension to study the phenomenological consequences.
Consider a version of the model in which the triplet $\chi$ is a complex
scalar field transforming under $G=\mathrm{SU}(2)\times\mathrm{U}(1)_{\chi}$,
with $\mathrm{U}(1)_{\chi}$ acting as a phase rotation. All or part of $G$ may
be thought of as gauged. We can further extend the model by promoting also
$\phi$ to a complex field and gauging $\mathrm{U}(1)_{\phi}$, which is
appealing because then the model no longer relies on any ad-hoc discrete
symmetries, the role of the $\mathbb{Z}_{2}$ in the original model now being
played by $\mathrm{U}(1)_{\phi}$. The technical details of these model
versions are worked out in Appendix A. These models do allow for additional
quartic scalar couplings, but the qualitative features of the inflaton
potential, as well as CMB predictions, do not change: crucially, there is
still an accidentally light scalar field which can play the role of the
inflaton, with a cosinusoidal loop-induced potential analogous to Eq. (6).
Likewise, the effective mass parameter of the waterfall field $\chi$ depends
on the value of the inflaton, and parameters can be chosen such that it
becomes tachyonic at the end of inflation. The waterfall field is zero along
the inflationary trajectory, so $\mathrm{U}(1)_{\chi}$ remains unbroken during
inflation while $\mathrm{SU}(2)\times\mathrm{U}(1)_{\phi}$ is completely
broken. The true vacuum is at $\langle\phi\rangle=0$ and
$\langle|\chi|\rangle=\langle|\chi^{3}|\rangle=v_{\chi}\neq 0$, so the vacuum
manifold is $S^{2}\times\mathrm{U}(1)_{\chi}$.

As $\pi_{1}[U(1)]=\mathbb{Z}$, the SSB phase transition results in the
production of a network of stable, local CSs with a typical tension [41]

$G_{\mathrm{\scriptscriptstyle{N}}}\mu\simeq
2\pi\,G_{\mathrm{\scriptscriptstyle{N}}}\,v_{\chi}^{2}\,,$ (32)

where $v_{\chi}$ is the breaking scale of $\mathrm{U}(1)_{\chi}$. It is well
known that this network quickly reaches a scaling regime, where the fraction
of energy density stored in the CSs redshifts in the same way as the energy
density of the background [42, 41]. Strings can intercommute and produce
loops, which oscillate under their own tension and radiate energy in the form
of GWs. Such a signal of GWs appears to us as a SGWB with a characteristic
scale-invariant spectrum (if standard cosmology is assumed [43]). The peak and
the amplitude of the GWs signal for stable strings are both given by the
string tension, which, in order to match the signal recently reported by the
NANOGrav collaboration, has to be
$G_{\mathrm{\scriptscriptstyle{N}}}\,\mu\simeq 10^{-10}$, even though stable
CSs seem to be disfavoured with respect to other cosmological sources [11].
The reason is that, for stable CSs, both the amplitude and the shape of the GW
signal are controlled by $G_{\mathrm{\scriptscriptstyle{N}}}\mu$; once we fix
the string tension to match the amplitude of the reported signal, the CS
spectrum turns out to be flat in the nHz region, with data preferring a blue-
tilted spectrum. $G_{\mathrm{\scriptscriptstyle{N}}}\mu\simeq 10^{-10}$
represents a trade-off between a signal which is not too weak and one which is
not too flat. However, as the computation of the expected signal of GWs from a
CS network is subject to large uncertainties, we can take as upper bound on
the string tension the one given by the CMB constraint on the impact of CSs on
the CMB power spectrum [44], $G_{\mathrm{\scriptscriptstyle{N}}}\mu\lesssim
10^{-7}$, which corresponds to an inflationary scale
$V_{0}^{1/4}\lesssim\left(\lambda_{\chi}+\lambda_{\chi}^{\prime}\right)\left(3\times
10^{14}\right)\,{\rm GeV}$.

Finally, domain walls can appear if the scalar vacuum manifold is not
connected. This, again, is not the case in our minimal model. In the following
we exploit a non-minimal model with disconnected minima, which is described in
detail in Appendix B. In this case the PTA signal can be fit with GWs produced
from annihilating DWs, as explained in the next section. The waterfall field
$\chi$ is taken to be a complex scalar triplet, subject to a discrete symmetry
$\mathbb{Z}_{4\chi}:\chi\,\rightarrow\,i\chi$. This leads to disconnected
minima which are degenerate, corresponding to stable DWs. However, the vacuum
degeneracy can be easily lifted by softly breaking the $Z_{4\chi}$ discrete
symmetry, allowing for the DWs to annihilate. Also in this model the
inflationary predictions are the same as those for the minimal model.

###  5.1 Gravitational waves from unstable domain walls

In the model described in Appendix B, after the end of inflation the $\chi$
scalar potential has two disconnected, degenerate minima, so that DWs are
formed with a surface energy density [41]

$\sigma=\frac{2\sqrt{2}}{3}\lambda_{\chi}^{1/2}v_{\chi}^{3}\,,$ (33)

where $\lambda_{\chi}$ is the $\chi$ quartic coupling and $v_{\chi}$ its VEV.
The motion of DWs is initially damped by their interaction with the
surrounding plasma. Numerical simulations show that, after some time, friction
becomes negligible and the DW network reaches the “scaling regime” [45] in
which their energy density scales with time as $\rho_{\rm
DW}=\sigma\mathcal{A}/t$, with $\mathcal{A}\simeq 0.8$ [46]. If the vacuum
degeneracy were exact, the DWs would be stable and they would quickly
overclose the Universe, as, in the scaling regime, $\rho_{\rm DW}$ redshifts
more slowly than the energy density of the background. DWs start dominating
when $3H^{2}M_{\mathrm{P}}^{2}=\rho_{\rm DW}$, which for a previously
radiation-dominated universe corresponds to the time

$t_{\rm dom}^{\rm DW}=\frac{3\,M_{\mathrm{P}}^{2}}{4\,\mathcal{A}\,\sigma}\,.$
(34)

However, the discrete symmetry ensuring the vacuum degeneracy may be softly
broken, which lifts the degeneracy and introduces a potential bias $\Delta V$,
see Eq. (B6). Provided that $\Delta V<0.795\,V_{0}$ [45], where $V_{0}$ is the
potential energy at $\chi=0$, DWs are formed, but they are unstable as long as
$\Delta V\neq 0$. The different domains annihilate when the bias pressure
$\Delta V$ dominates over the DW surface tension at

$t_{\rm ann}\simeq C_{d}\mathcal{A}\frac{\sigma}{\Delta V}\,,$ (35)

with $C_{d}\simeq 3$ [47], until all the observable Universe lies in the true
vacuum. We require the DW network to disappear before the bias energy density
of the false-vacuum patches dominates the Universe,

$t_{\rm dom}^{\Delta V}\simeq\frac{\sqrt{3}M_{\mathrm{P}}}{2\sqrt{\Delta
V}}\,.$ (36)

The requirement $t_{\rm ann}<t_{\rm dom}^{\Delta V}$ leads to the constraint
$\Delta V\gtrsim 8\,\sigma^{2}/M_{\mathrm{P}}^{2}$.

In an early phase, as far as we know, the universe might well have been
dominated by DWs, provided that they annihilate before big bang
nucleosynthesis (BBN). However, the dynamics of a DW-dominated Universe is not
well understood, and its treatment requires dedicated numerical studies [45],
so we restrict ourselves to a radiation-dominated scenario. By combining this
requirement together with that of having a not too large $\Delta V$, in order
to have DWs appear in the first place, we obtain $v_{\chi}\lesssim
0.3\,M_{\mathrm{P}}$, where we have made use of Eq. (33) and
$V_{0}=\lambda_{\chi}v_{\chi}^{4}$.

As the DW network evolves, the DWs are driven to relativistic speed and
radiate GWs. The signal of GWs, appearing to us as a SGWB, has a peak
frequency given by the Hubble parameter at the moment of annihilation, which
after taking into account redshift becomes [45]

$f_{p}\simeq\frac{R\left(t_{\rm ann}\right)}{R\left(t_{0}\right)}H_{\rm
ann}\simeq 1.6\times 10^{-7}\;{\rm Hz}\left(\frac{g_{*}(T_{\rm
ann})}{100}\right)^{1/6}\frac{T_{\rm ann}}{{\rm GeV}}\,.$ (37)

Here $R\left(t_{\rm ann}\right)$ and $R\left(t_{0}\right)$ are the scale
factors at the time of DW annihilation and today, respectively, $g_{*}$ is the
number of effective degrees of freedom in relativistic energy, and $T_{\rm
ann}$ is the temperature of the Universe at the moment of DW annihilation,

$T_{\rm ann}=\left[\frac{45}{2\pi^{2}g_{*}(T_{\rm
ann})}\;\frac{M_{\mathrm{P}}^{2}}{t_{\rm ann}^{2}}\right]^{1/4}\,.$ (38)

The present energy density of radiated GWs is [45]

$h^{2}\Omega_{\rm GW}\left(f\right)\simeq 1.6\times
10^{-5}\left(\frac{100}{g_{*}(T_{\rm
ann})}\right)^{1/3}\,\frac{3}{32\pi}\,\tilde{\epsilon}\,\alpha_{\rm
ann}^{2}\,\mathcal{S}(f/f_{p}),$ (39)

where $\tilde{\epsilon}=0.7$ is an efficiency parameter taken from simulations
[46], $\alpha_{\rm ann}$ is the DW network energy density fraction at the time
of annihilation,

$\alpha_{\rm ann}\equiv\frac{\rho_{\rm DW}(t_{\rm ann})}{\rho_{r}(t_{\rm
ann})}\simeq\frac{4}{3}\,C_{d}\,\mathcal{A}^{2}\frac{\sigma^{2}}{M_{\mathrm{P}}^{2}\,\Delta
V}\,,$ (40)

and the spectral function $\mathcal{S}(x)$ can be modeled as [11]

$\mathcal{S}(x)=\left(\frac{a+b}{b\,x^{-a/c}+a\,x^{b/c}}\right)^{c}\,.$ (41)

Numerical simulations have shown that $b,c\simeq 1$ for $\mathbb{Z}_{2}$-type
DWs [46], and our model falls in this category, while causality fixes $a=3$
[48].777In principle, the causality tail of the GW spectrum is affected by the
QCD phase transition and the subsequent change in the number of relativistic
degrees of freedom. This leads to a deviation from the pure power-law
behaviour with $a=3$. Including the effect of the QCD phase transition is
beyond the scope of this work and we refer to Ref. [49] for a detailed
discussion on the topic. By plugging Eq. (35) and Eq. (38) into Eq. (37), we
observe that the GW peak frequency scales with the DW network parameters as
$f_{p}\sim\sqrt{\Delta V/\sigma}$. Similarly, by plugging Eq. (40) into Eq.
(39), we find that $h^{2}\Omega_{\rm GW}(f_{p})\sim(\sigma^{2}/\Delta V)^{2}$.

By taking the quartic couplings to be of order one, the DW tension is related
to the inflationary scale as $V_{0}\simeq\sigma^{4/3}$. This allows us to
relate the GW signal generated by the DW network to inflation. It has been
shown [11, 47, 50] that a network of unstable DWs is a good candidate to
explain the signal of a SGWB recently detected by the NANOGrav collaboration
[11]. We find that a model with $V_{0}^{1/4}\simeq 2\times 10^{5}\,{\rm GeV}$
and a suitable bias $\Delta V$ leads to a GW signal which fits the NANOGrav
detection. The peak of the GW signal is within the reach of the upcoming
space-borne interferometer LISA [51] for $3\times 10^{7}\,{\rm GeV}\lesssim
V_{0}^{1/4}\lesssim 2\times 10^{10}\,{\rm GeV}$, while for $3\times
10^{10}\,{\rm GeV}\lesssim V_{0}^{1/4}\lesssim 7\times 10^{12}\,{\rm GeV}$ the
SGWB might be detected by the ET in the future. Fig. 5 shows several different
benchmark GW spectra together with the sensitivity curves of current and
planned experiments.

The energy density released by DW annihilation can potentially alter the
expansion rate of the Universe. In order not to spoil BBN, the products of DW
annihilation have to decay before the BBN epoque, either into dark radiation
(DR) or into SM particles [52]. Indeed, $\chi$ can decay into the
$\mathrm{SO}(2)$ dark photon, or into SM degrees of freedom via e.g. a Higgs-
portal coupling. The dominant decay channel depends on the relative magnitude
of the couplings and is related to the reheating mechanism, which we do not
explore in this paper. The usual assumption, which also leads to the weakest
constraints, is that such decays are already active at $T_{\rm ann}$ [52].
Assuming the DW energy density is transferred entirely into DR, the extra
number of neutrinos species receives a correction [53]

$\Delta N_{\rm
eff}=\frac{8}{7}\left(\frac{11}{4}\right)^{4/3}\frac{g_{*}(T_{\rm
BBN})}{2}\alpha_{\rm ann}\,,$ (42)

where the effective number of SM relativistic degrees of freedom at the time
of BBN is $g_{*}(T_{\rm BBN})\simeq 3.36$. This quantity is constrained by
BBN, $\Delta N_{\rm eff}\leq 0.3$ [54], leading to an upper bound on the
amplitude of GWs generated by a network of DWs decaying into DR. The bound is
shown in pink in Fig. 5. On the other hand, if DWs mainly decay into SM
degrees of freedom, such an upper bound is relevant only for networks decaying
after the neutrino decoupling temperature, $T_{\rm ann}\lesssim 1\,{\rm MeV}$
[55, 56]. Indeed, in this scenario, neutrinos produced by the DW decay would
not be in thermal equilibrium by the time of BBN. Such a low annihilation
temperature, however, is associated to much lower frequencies of the GW
signal.

Figure 5: GW spectra produced by unstable DWs for different benchmark points.
The red curve is obtained with DW tension $\sigma^{1/3}=3.2\times 10^{5}\,{\rm
GeV}$ and bias $\Delta V^{1/4}=0.23\,{\rm GeV}$. The blue solid curve
corresponds to $\sigma^{1/3}=3.2\times 10^{8}\,{\rm GeV}$, and the purple
solid one to $\sigma^{1/3}=1.2\times 10^{11}\,{\rm GeV}$ ($\Delta V$ is varied
accordingly to keep the amplitude of the signal fixed). The dashed spectra are
obtained from the solid ones by decreasing $\sigma$ while keeping $f_{p}$
fixed: the dashed blue curve corresponds to $\sigma^{1/3}=9.3\times
10^{7}\,{\rm GeV}$, while the dashed purple one to $\sigma^{1/3}=3.5\times
10^{10}\,{\rm GeV}$. The gray violins show the SGWB reported by the NANOGrav
15 yrs experiment [11]. The green and yellow shaded regions show the projected
sensitivities of LISA [51] and ET [34], respectively, as derived in Ref. [11],
assuming a signal-to-noise-ratio equal to unity and an observing time of one
year. The integrated sensitivity of the LIGO-Virgo-KAGRA network during the
third observing run [10] is shown in brown. The region where the bias energy
density dominates the Universe, leading to local departures from radiation
domination, is shown in orange, while the black dashed line shows the bound on
the amplitude of primordial GWs given by BBN [51]. If the DW network decays
into dark radiation, the pink region is excluded by the BBN bound on the
effective number of neutrino species.

##  6 Conclusions

Models with accidentally light scalars [3] are particularly well suited for
realising hybrid inflation. They feature tree-level flat directions, lifted by
loop corrections, along which slow-roll inflation takes place. They also
feature tree-level couplings of the inflaton to other scalar fields, which can
become tachyonic along the inflationary direction, thereby ending inflation by
the waterfall mechanism.

In this paper we have presented a simple yet realistic example. It is defined
by a symmetry group $G=\mathrm{SO}(3)$ and two scalar multiplets $\phi$ and
$\chi$, transforming in the representation $\boldsymbol{5}$ and
$\boldsymbol{3}$ respectively. The two fields may also carry an abelian charge
(discrete or continuous), depending on the variant of the model. Gauging
$\mathrm{SO}(3)$ and allowing for all renormalizable potential terms
compatible with the symmetry, one degree of freedom $a$ in $\phi$ remains
massless at tree level: this is what we call an accidentally light field,
which we identify with the inflaton. The inflationary potential arises at the
one-loop level, induced by the gauge coupling and a quartic coupling between
$\phi$ and $\chi$. It has a form similar to that of Natural Inflation [4]. One
component of $\chi$ plays the role of the waterfall field: its mass-squared
starts out positive and decreases during inflation until, at some critical
point along the inflationary direction, it turns tachyonic. This destabilizes
the waterfall field which terminates inflation by fast-rolling down its steep
potential.

The flatness of the inflationary potential is naturally protected: it is
guaranteed by symmetry at the tree level and it gets lifted by one-loop
radiative corrections. Unlike pseudo-Goldstone inflatons, here the inflaton
has tree-level non-derivative couplings to other scalars, which are necessary
for an efficient waterfall. These couplings are typically a concern in earlier
realisations of hybrid natural inflation, as they spoil the flatness, unless
they are forbidden by some complicated additional symmetries or fine-tuned.

The predictions of the model are in agreement with the latest CMB measurements
[1, 2] for a wide region of the parameter space. We identify a portion of
parameter space for which the parameters are natural (assuming the UV cutoff
is identified with $M_{\mathrm{P}}$), and fine-tuning of the initial and final
points of inflation is less than 10%. Inside this region, the predicted
tensor-to-scalar ratio is $\left(4\times 10^{-2}\right)n_{r}\lesssim r\lesssim
4\,n_{r}$, with $n_{r}$ natural in the range $6.5\times 10^{-5}\lesssim
n_{r}\lesssim 2\times 10^{-3}$.

Inflation ends as the waterfall field becomes tachyonic, and the Universe is
reheated with a reheating temperature of the order of the inflationary scale,
$T_{\rm RH}\sim{\cal O}(V_{0}^{1/4})$. A negative curvature of the potential
triggers tachyonic instability of the waterfall field quantum fluctuations,
which are exponentially enhanced. This leads to production of GWs, which reach
us as a SGWB signal. The peak frequency of the GW spectrum is typically as
large as $\mathcal{O}\left(10^{10}\right)\,{\rm Hz}$ but it can be lowered by
sufficiently decreasing the inflationary scale. We find that the signal of GWs
produced during tachyonic instability is within the reach of the Einstein
Telescope for an inflationary scale in the range $3\times 10^{5}\,{\rm
GeV}\lesssim V_{0}^{1/4}\lesssim 5\times 10^{11}{\rm GeV}$. Such a low
inflationary scale, despite pointing to some UV completion well below
$M_{\mathrm{P}}$, is perfectly viable and is associated to an unobservably
small tensor-to-scalar ratio. This confirms that GWs from a tachyonic
instability can act as a possible probe of low-scale inflationary scenarios.

The inflationary phase may be followed by a production of topological defects,
depending on the specific symmetries which undergo SSB. In a model variation
with $\phi$ and $\chi$ charged under an additional $U(1)$ gauge symmetry, the
vacuum manifold is such that CSs are produced during the waterfall transition.
The tension of the string network is bounded by CMB, and this constrains the
inflationary scale to be $V_{0}^{1/4}\lesssim$ a few $\times 10^{14}\,{\rm
GeV}$. Another model variation features a softly-broken $\mathbb{Z}_{4}$
symmetry acting on $\phi$ and $\chi$. In this case the vacuum manifold is
disconnected, and unstable DWs are generated at the end of inflation. They
produce a SGWB, which can fit the signal recently detected by the NANOGrav
collaboration [11], albeit for an extremely low inflationary scale,
$V_{0}^{1/4}\simeq 2\times 10^{5}\,{\rm GeV}$, assuming order one quartic
couplings. The DW annihilation signal can also be relevant for future
detection by LISA (ET) for $3\times 10^{7}\,{\rm GeV}\lesssim
V_{0}^{1/4}\lesssim 2\times 10^{10}\,{\rm GeV}$ ($3\times 10^{10}\,{\rm
GeV}\lesssim V_{0}^{1/4}\lesssim 7\times 10^{12}\,{\rm GeV}$).

Upcoming experiments will soon be able to further constrain or rule out our
models, for example, by measuring the sign of $n_{r}$, or by a discovery of a
SGWB by the LIGO-Virgo-KAGRA network. Concerning future directions, it would
be interesting to couple our models to the Standard Model in order to study
reheating in detail. Finally, the accident mechanism could be applied to other
realisations of inflation.

## Acknowledgements

We thank Pierre Auclair, Yann Gouttenoire, Thomas Hambye, Antonio Junior
Iovino, Lucas Pinol, Cristophe Ringeval, Simon Schreyer and Alfredo Urbano for
useful discussions. MF has received support from the European Union Horizon
2020 research and innovation program under the Marie Skłodowska-Curie grant
agreements No 860881-HIDDeN and No 101086085–ASYMMETRY.

##  Appendix A A model with complex scalars, giving rise to cosmic strings

In Section 2 we presented a minimal model with two real scalar
$\mathrm{SU}(2)$ multiplets, a five-plet $\phi$ and a triplet $\chi$. Here we
take these fields to be complex, charged under additional $\mathrm{U}(1)$
symmetries.

Let $\phi$ be a complex scalar transforming in the 5 of $\mathrm{SU}(2)$ with
unit $\mathrm{U}(1)_{\phi}$ charge. The most general renormalizable potential
is

$V_{\phi}=-\mu_{\phi}^{2}\,\phi^{\dagger}\phi+\lambda\,(\phi^{\dagger}\phi)^{2}+\kappa\left[(\phi^{\dagger}\phi)^{2}-|\phi^{T}\phi|^{2}\right]+\delta\,(\phi^{\dagger}T^{a}\phi)^{2}\,,$
(A1)

with the generators $T^{a}$ defined below Eq. (1). A detailed study of this
potential can be found in ref. [3].888The quartic couplings
$\lambda,\kappa,\delta$ here are normalised to be a factor 2 smaller than in
ref. [3]. The quartic couplings $\lambda$, $\kappa$ and $\delta$ and the mass
parameter $\mu_{\phi}^{2}$ are chosen positive. Notice that the $\kappa$ and
$\delta$ terms vanish when $\phi$ is real. Cubic terms are forbidden by
$\mathrm{U}(1)_{\phi}$, with no need for ad-hoc discrete symmetries. The
minimum of the potential is at $\langle\phi^{\dagger}\phi\rangle=v^{2}/2$,
where $v^{2}=\mu_{\phi}^{2}/\lambda$. When
$\mathrm{SU}(2)\times\mathrm{U}(1)_{\phi}$ is gauged, all NGBs are absorbed by
the gauge bosons. However, there remains a continuous family of vacua at the
tree level, parameterized by an angle $\alpha\in[0,\pi/6]$ as in Eq. (2). Thus
$\alpha$ is an accidentally flat direction. For generic $\alpha$, all of the
gauge symmetry is broken, but at the special point $\alpha=0$, a subgroup
$\mathrm{U}(1)^{\prime}\subset\mathrm{SU}(2)$ remains unbroken. At this point
there is an additional massless scalar in the spectrum, besides the one
corresponding to the flat direction.

In the limit $\delta\rightarrow 0$, the potential recovers an enlarged
$\mathrm{SO}(5)\times\mathrm{U}(1)_{\phi}$ symmetry, spontaneously broken to
$\mathrm{SO}(4)$. Switching on $\delta$ explicitly breaks the symmetry of the
potential to $\mathrm{SU}(2)\times\mathrm{U}(1)_{\phi}$, so that the
continuous symmetry of $V$ is strictly no larger than the gauge symmetry:
therefore, the accidentally light mode is not a pNGB in the sense of [57].

Loop corrections lift the accidentally flat direction, analogously to the
minimal model of Section 2, with $\alpha=0$ being selected as the true vacuum.
The one-loop effective potential along the tree-level flat direction takes the
form $V_{\rm eff}\simeq
V_{0}^{\prime}-\left(M^{\prime}\right)^{4}\cos\left(a/f\right)$ (up to higher
harmonics whose coefficients are numerically small for all values of the
couplings of interest). The amplitude of the potential is given by

$\left(M^{\prime}\right)^{4}=\frac{v^{4}}{160\,\pi^{2}}\left[9\,g^{4}-\frac{\kappa^{5}}{\delta^{3}}\left(F\left(-\frac{2\delta}{\kappa}\right)-T_{F,5}\left(-\frac{2\delta}{\kappa}\right)\right)\right]\,,$
(A2)

where $g_{2}$ is the $\mathrm{SU}(2)$ gauge coupling, and $F(x)$ and $T_{F,5}$
are defined below Eq. (7). The tree-level massless scalar picks up a mass
which is one-loop suppressed with respect to the masses of the other fields.

Now add to the model a complex scalar $\chi$ transforming in the ${\bf 3}$ of
$\mathrm{SU}(2)$ with unit $\mathrm{U}(1)_{\chi}$ charge, which will play the
role of the waterfall field. The most general renormalizable scalar potential
now also includes the terms

$\begin{split}V_{\chi}=&\;-\mu^{2}_{\chi}\chi^{\dagger}\chi+\lambda_{\chi}\,(\chi^{\dagger}\chi)^{2}+\lambda^{\prime}_{\chi}\,|\chi^{T}\chi|^{2}\,,\\\
V_{\phi\chi}=&\;\varepsilon\,\phi^{\dagger}\phi\chi^{\dagger}\chi+\zeta\,T^{a}_{AB}T^{b}_{BC}\phi^{*}_{A}\phi_{C}\chi^{*a}\chi^{b}+\vartheta\,T^{a}_{AB}(i\varepsilon^{abc})\phi^{*}_{A}\phi_{B}\chi^{*b}\chi^{c}\,.\end{split}$
(A3)

Notice that one can replace the phase rotations
$\mathrm{U}(1)_{\phi}\times\mathrm{U}(1)_{\chi}$ by a single gauged
$\mathrm{U}(1)$, with charges $q_{\phi}$ and $q_{\chi}$ chosen to forbid
additional mixed couplings, beside $\epsilon,\zeta$ and $\vartheta$. In this
case, by construction, the scalar potential has an accidental global
$\mathrm{U}(1)_{\phi}\times\mathrm{U}(1)_{\chi}$ symmetry. As a consequence,
when $\langle\chi\rangle\neq 0$ is generated after inflation, one produces
cosmic strings with observable consequences, as discussed in section 5.

For $\mu^{2}_{\chi}<0$ and positive quartic couplings in Eq. (A3), the tree-
level potential is easily seen to be minimized at $\chi=0$, with the tree-
level vacuum structure for $\phi$ unchanged. In fact, when $\phi$ is set to
its tree-level minimum of Eq. (2), the terms $\mu^{2}_{\chi}$ and
$\varepsilon$ give rise to a universal, $\alpha$-independent mass term for the
three components of $\chi$, while the $\zeta$ term splits their masses in an
$\alpha$-dependent fashion (and the $\vartheta$ term does not contribute to
the mass matrix). The mass eigenvalues for the $\chi$ fields are the same as
in Eq. (4). To obtain a model of hybrid inflation, we choose
$\tilde{\mu}_{\chi}^{2}\equiv\mu_{\chi}^{2}-\frac{\varepsilon}{2}v^{2}>0$, and
$\zeta v^{2}>2\tilde{\mu}_{\chi}^{2}$, such that the overall $\chi^{3}$ mass-
squared is positive for $\alpha=\pi/6$ but negative for $\alpha=0$. At the
intermediate value $\alpha=\alpha_{c}$ (given in Eq. (9)) the effective
$\chi^{3}$ mass crosses zero.

For suitable values of the couplings, the true vacuum of the model is located
at

$\langle\lvert\chi^{3}\rvert\rangle=\sqrt{\frac{\mu^{2}_{\chi}}{2\left(\lambda_{\chi}+\lambda_{\chi}^{\prime}\right)}}\,,\qquad\text{all
other fields}=0$ (A4)

up to gauge transformations. The residual gauge symmetry is
$\mathrm{U}(1)^{\prime}\subset\mathrm{SU}(2)$, and all fields except the
unbroken gauge boson pick up tree-level masses. This holds for
$\lambda_{\chi}^{\prime}>0$, while for
$-\lambda_{\chi}<\lambda_{\chi}^{\prime}<0$, the VEV of $\chi$ is aligned
differently, and it breaks $\mathrm{SU}(2)$ entirely.

Loops of $\chi$ contribute to the accident effective potential so that Eq.
(A2) becomes

$\begin{split}\left(M^{\prime}\right)^{4}=\frac{v^{4}}{160\,\pi^{2}}\Bigg{[}&9\,g_{2}^{4}-\frac{\kappa^{5}}{\delta^{3}}\left(F\left(-\frac{2\delta}{\kappa}\right)-T_{F,5}\left(-\frac{2\delta}{\kappa}\right)\right)\\\
&+4\,\frac{\tilde{\mu}_{\chi}^{10}}{\zeta^{3}\,v^{10}}\left({\rm
Re}\;F\left(\frac{\zeta
v^{2}}{\tilde{\mu}_{\chi}^{2}}\right)-T_{F,5}\left(\frac{\zeta
v^{2}}{\tilde{\mu}_{\chi}^{2}}\right)\right)\Bigg{]}\,.\end{split}$ (A5)

Note that the ${\rm Re}\,F$ term in the second line vanishes for the parameter
region of physical interest $\tilde{\mu}_{\chi}^{2}>0$.

The calculation of CMB observables in this model is done in the same way as
outlined in Section 3, upon replacing $M\,\rightarrow\,M^{\prime}$ and
$V_{0}\,\rightarrow\,V_{0}^{\prime}$.

##  Appendix B A model with disconnected minima, giving rise to unstable
domain walls

In this variant of the model, the potential has two physically distinct minima
which can be chosen to be near-degenerate. Let $\phi$ be a real five-plet of
$\mathrm{SO}(3)$ which is odd under $\mathbb{Z}_{2\phi}:\phi\to-\phi$, and add
to the model a pair of triplet waterfall fields $\chi_{R}$ and $\chi_{I}$; we
represent them as a single complex field $\chi=(\chi_{R}+i\chi_{I})/\sqrt{2}$,
although no $\mathrm{U}(1)$ symmetry is implied. We do however impose a
$\mathbb{Z}_{4\chi}$ symmetry acting as $\chi\rightarrow i\chi$. With these
fields and symmetries, the most general renormalisable potential reads

$\displaystyle V=$
$\displaystyle-\frac{1}{2}\mu_{\phi}^{2}\phi^{2}+\frac{\lambda_{\phi}}{4}(\phi^{2})^{2}-\mu_{\chi}^{2}\chi^{*}\chi+\lambda_{\chi}\left(\chi^{*}\chi\right)^{2}+\delta\,{\chi^{*}}^{2}\chi^{2}+\dfrac{1}{2}\left(\kappa\,\chi^{2}\chi^{2}+{\rm
h.c.}\right)$ (B1)
$\displaystyle+\frac{\varepsilon}{2}\phi^{2}\left(\chi^{*}\chi\right)+\frac{\zeta}{2}\,T^{a}_{AC}T^{b}_{CB}\phi_{A}\phi_{B}{\chi^{a}}^{*}\chi^{b}\,,$

where $T^{a}$ are the $\mathrm{SO}(3)$ five-plet generators, chosen as
$5\times 5$ imaginary and anti-symmetric matrices. For simplicity, we assume
$\kappa$ real, which corresponds to a CP invariance
$\chi\leftrightarrow\chi^{*}$, and all the quartic couplings to be positive.

Let us first consider the case $\mu_{\phi}^{2}>0$ and $\mu_{\chi}^{2}<0$. The
minimum is at
$\langle\phi^{2}\rangle=v^{2}\equiv\mu_{\phi}^{2}/\lambda_{\phi}$,
$\langle\chi\rangle=0$, where the mass spectrum resembles the one of the
minimal model of section 2. After the spontaneous breaking of
$\mathrm{SO}(3)$, one observes the emergence of an accidentally flat direction
parametrized by Eq. (2). The masses of the complex $\chi$ components are given
by Eq. (4). One-loop corrections give rise to the usual potential along the
accidentally flat direction $V_{\rm
eff}=V_{0}^{\prime\prime}-\left(M^{\prime\prime}\right)^{4}\cos\left(a/f\right)$,
with $M^{\prime\prime}$ given by

$\left(M^{\prime\prime}\right)^{4}=\frac{1}{160\,\pi^{2}}\left[9\,g^{4}\,v^{4}+4\,\frac{\tilde{\mu}_{\chi}^{10}}{\zeta^{3}\,v^{6}}\left({\rm
Re}\;F\left(\frac{\zeta
v^{2}}{\tilde{\mu}_{\chi}^{2}}\right)-T_{F,5}\left(\frac{\zeta
v^{2}}{\tilde{\mu}_{\chi}^{2}}\right)\right)\right]\,,$ (B2)

and $F$ and $T_{F,5}$ defined below Eq. (7).

We now switch to the case interesting for inflation: $\mu_{\phi}^{2}>0$ and
$\mu_{\chi}^{2}>0$. The $\chi_{3}$ mass is positive at the beginning of
inflation, close to $\alpha=\pi/6$, and it crosses $0$ at $\alpha=\alpha_{c}$,
given by Eq. (9). With this choice of couplings, the potential has a minimum
at $\langle\phi\rangle=0$ and $\langle\chi\rangle\neq 0$. The six equations to
solve in order to find the extrema of the potential are

$\displaystyle\chi_{R}^{a}\left[-\mu_{\chi}^{2}+\lambda_{\chi}\left(\chi_{R}^{2}+\chi_{I}^{2}\right)+(\kappa+\delta)(\chi_{R}^{2}-\chi_{I}^{2})\right]-2\chi_{I}^{a}\left(\chi_{R}\cdot\chi_{I}\right)\left(\kappa-\delta\right)$
$\displaystyle=0\,,$ (B3)
$\displaystyle\chi_{I}^{a}\left[-\mu_{\chi}^{2}+\lambda_{\chi}\left(\chi_{R}^{2}+\chi_{I}^{2}\right)-(\kappa+\delta)(\chi_{R}^{2}-\chi_{I}^{2})\right]-2\chi_{R}^{a}\left(\chi_{R}\cdot\chi_{I}\right)\left(\kappa-\delta\right)$
$\displaystyle=0\,,$

for $a=1,2,3$. It is convenient to employ $\mathrm{SO}(3)$ invariance to set
$\chi^{1}_{R}=\chi^{2}_{R}=0$ as well as $\chi^{1}_{I}=0$. Then, the system
reduces to

$\displaystyle\chi^{2}_{I}\chi^{3}_{I}\chi^{3}_{R}=0\,,$ (B4)
$\displaystyle\chi_{I}^{2}\left[-\mu_{\chi}^{2}+\lambda_{\chi}[(\chi_{R}^{3})^{2}+(\chi_{I}^{2})^{2}+(\chi_{I}^{3})^{2}]-(\kappa+\delta)[(\chi_{R}^{3})^{2}-(\chi_{I}^{2})^{2}-(\chi_{I}^{3})^{2}]\right]=0\,,$
$\displaystyle\chi_{R}^{3}\left[-\mu_{\chi}^{2}+\lambda_{\chi}[(\chi_{R}^{3})^{2}+(\chi_{I}^{2})^{2}+(\chi_{I}^{3})^{2}]+(\kappa+\delta)[(\chi_{R}^{3})^{2}-(\chi_{I}^{2})^{2}]-(3\kappa-\delta)(\chi_{I}^{3})^{2}\right]=0\,,$
$\displaystyle\chi_{I}^{3}\left[-\mu_{\chi}^{2}+\lambda_{\chi}[(\chi_{R}^{3})^{2}+(\chi_{I}^{2})^{2}+(\chi_{I}^{3})^{2}]+(\kappa+\delta)[(\chi_{I}^{2})^{2}+(\chi_{I}^{3})^{2}]-(3\kappa-\delta)(\chi_{R}^{3})^{2}\right]=0\,.$

For a certain hierarchy among the quartic couplings
($0<\delta<\kappa<\lambda_{\chi}+\delta<\varepsilon/2+\kappa$), the potential
is minimized at

$(\chi_{R}^{3})^{2}=(\chi_{I}^{3})^{2}=w^{2}\equiv\frac{\mu_{\chi}^{2}}{2\left(\lambda_{\chi}+\delta-\kappa\right)}\,,\quad\chi_{I}^{2}=0\,.$
(B5)

In these minima the symmetry is spontaneously broken as
$\mathrm{SO}(3)\times\mathbb{Z}_{4\chi}\to\mathrm{SO}(2)\times\mathbb{Z}_{2\chi}$.
There are four solutions to Eq. (B5) but only two physically distinct vacua:
one $(+)$ where the $\chi_{R}$ and $\chi_{I}$ VEVs are parallel,
$\chi_{R}^{3}=\chi_{I}^{3}=\pm w$, and the other $(-)$ where they are
antiparallel, $\chi_{R}^{3}=-\chi_{I}^{3}=\pm w$. Thus, the vacuum manifold
has two disconnected components, and DWs can be produced during the SSB phase
transition, as discussed in section 5.

For a viable phenomenology, the DW network needs to be unstable, as discussed
in section 5.1. We therefore assume that $\mathbb{Z}_{4\chi}$ is softly broken
by a term $V_{\rm soft}=im_{\chi}^{2}\chi\chi+{\rm h.c.}$, with $m_{\chi}^{2}$
real. This term preserves $\mathbb{Z}_{2\chi}:\chi\to-\chi$, while it breaks
the CP symmetry $\chi\to\chi^{*}$. Consequently, it breaks the degeneracy
between the two disconnected vacua ($+$) and ($-$), introducing a bias in the
potential. We assume that soft $\mathbb{Z}_{4\chi}$ breaking is a small
effect, $m_{\chi}^{2}\ll\mu_{\chi}^{2}$. Then the potential bias is, to
leading order in $m_{\chi}^{2}$,

$\Delta V\equiv
V_{(+)}-V_{(-)}=-\frac{2\,m_{\chi}^{2}\mu_{\chi}^{2}}{\lambda_{\chi}+\delta-\kappa}+\mathcal{O}\left(m_{\chi}^{4}\right)\,,$
(B6)

where $V_{(+)}$ and $V_{(-)}$ are the potential energies evaluated at the
minimum configuration in which $\chi_{R}$ and $\chi_{I}$ are parallel and
anti-parallel, respectively.

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