# VISH$\nu$: a unified solution to five SM shortcomings with a protected
electroweak scale

Alexei H. Sopov<EMAIL_ADDRESS>Raymond R. Volkas
<EMAIL_ADDRESS>ARC Centre of Excellence for Dark Matter Particle
Physics, School of Physics,
The University of Melbourne, Victoria 3010, Australia

###### Abstract

We propose a Standard Model extension, coined VISH$\nu$ (Variant-axIon Seesaw
Higgs $\nu$-trino), that is an $N_{\text{DW}}=1$ variation of its predecessor,
the $\nu$DFSZ model. In accounting for the origin of neutrino masses, dark
matter and the baryon asymmetry of the universe, VISH$\nu$ inherits the
explanatory power of $\nu$DFSZ while, of course, resolving the strong $CP$
problem. In both models, the electroweak scale is naturally protected from a
high seesaw scale that is identified with the Peccei-Quinn (PQ) spontaneous
symmetry breaking scale. Through a flavour variant coupling structure,
VISH$\nu$ evades a domain wall problem, extending the cosmological reach of
the $\nu$DFSZ to include a viable period of inflation. The primary focus of
this paper is on the inflationary dynamics of VISH$\nu$ and their naturalness
in the sense of radiative stability. We find that non-minimal gravitational
couplings, generically developed by the VISH$\nu$ scalar fields, naturally
support a viable inflaton field which typically has both a PQ scalar and Higgs
component. An axion mass window [$40\mu\text{eV},\sim 2\text{meV}$] accessible
to forthcoming searches, results for the case where PQ symmetry is restored
during the (p)reheating phase.

††preprint: APS/123-QED

##  I Introduction

The Standard Model (SM) features both remarkable experimental agreement and
considerable phenomenological deficit, predicting neither neutrino masses nor
a viable dark matter candidate. Given that sub-Planckian energies were
explored in the early universe, cosmological phenomena also lie within its
estimation, and while the SM predicts an inflaton field if one admits non-
minimal coupling of the Higgs doublet to gravity [1], it cannot generate an
acceptable baryon asymmetry of the universe (BAU).

It may be that these shortcomings point to additional physics that does not
exceed the electroweak scale. In the $\nu$MSM [2, 3], three additional right-
handed neutrino singlets with sub-electroweak masses can already support
type-I seesaw [4, 5, 6, 7] masses for SM neutrinos, a keV-range dark matter
candidate (the lightest neutrino singlet) and a version of ARS leptogenesis
[8], without spoiling Higgs-driven inflation. However, the SM _also_ has a
strong $CP$ problem (why is $\bar{\theta}\ll 1$?), for which viable axion [9]
solutions require the presence of a new fundamental scale that exceeds the
electroweak scale by several orders of magnitude.

Whenever there is such a measurable hierarchy of scales, there is a
naturalness issue concerning the stability of the lower mass-scale under large
and physically meaningful radiative corrections induced by the higher mass-
scale. While its origin, a “hierarchy problem”, is perhaps answerable only in
the far UV, we can avoid introducing new naturalness problems through cautious
model-building, provided the larger scale is sub-Planckian. Hidden-sector
extensions, discussed in Ref. [10], are free of naturalness problems arising
from any new non-gravitational interactions. This is because the high-scale
physics is assumed to interact with the low-energy SM sector only through a
weakly-coupled interaction Lagrangian whose small couplings
($\lambda_{\text{int},i}$) are protected by an enhanced Poincaré symmetry
manifest in the decoupling limit ($\lambda_{\text{int},i}\rightarrow 0$). The
electroweak scale is therefore stabilised under the induced corrections by
parameters which remain appropriately small under running.111 Of course, to
completely address this electroweak naturalness problem, one should also
understand the effect of gravitational corrections arising from a Planckian
scale. (This is the case for the SM.) Our focus, however, is ensuring that any
new high-scale physics we introduce beyond the SM does not create naturalness
problems of its own: a necessary (but not sufficient) condition for solving
the electroweak naturalness problem posed in a given SM extension. This is
exactly the restricted sense in which we refer to a “protected” electroweak
scale.

An exemplar, the $\nu$DFSZ model [11, 12], is a minimal hidden sector
extension with a radiatively stable hierarchy of scales built around the DFSZ
axion [13, 14] and three right-handed neutrinos. The new fundamental scale,
$f_{PQ}\sim 10^{10}$–$10^{11}\,\text{GeV}$, is associated with a spontaneously
broken Peccei-Quinn symmetry $U(1)_{\text{PQ}}$ to realise an invisible axion,
and also serves as the seesaw neutrino mass scale [15, 16]. In Ref. [12] a
critical density of axionic dark matter and the BAU were shown also to be
viable outcomes without requiring unnatural inter-sector couplings. The BAU is
generated by the $CP$-violating decays of the lightest right-handed neutrino
out-of-equilibrium [17], as in standard thermal leptogenesis [18].
Subsequently, the SMASH model [19, 20, 21, 22, 23] was proposed, built around
the KSVZ axion [24, 25], with similar explanatory power to the $\nu$DFSZ
(neutrino masses, axion dark matter, BAU, $\bar{\theta}\rightarrow 0$) but
supplemented by an analysis of an early cosmological history (inflation
followed by a period of reheating [20, 21, 22, 23]). However, unlike the SMASH
model [21], the $\nu$DFSZ model can preserve the high-scale validity of its
Higgs sector, without introducing a destabilising contribution to the Higgs
mass parameters [26], and does not require an exotic quark.

A primary objective of this paper is to extend the original analysis of the
$\nu$DFSZ model to also include an early period of inflation [27, 28], without
reintroducing the naturalness concerns it serves to resolve. This is
accomplished by non-minimal couplings to gravity of the scalar sector which,
for a general range of initial conditions [29, 30], gives rise to effectively
single-field inflation along valleys of the conformally stretched large-field
potential in the Einstein frame, producing an excellent fit to CMB data [31].
We identify the possible inflaton configurations for a general non-minimally
coupled three-scalar model, before specialising to the hidden-sector parameter
space where the inflaton field generally has both a singlet scalar and Higgs
component. When non-minimal couplings are much larger than unity, inflationary
predictions replicate Higgs inflation, while smaller couplings, unscathed by
unitarity concerns, are also viable and compatible with both naturalness and
stability requirements.

We also remove a well-known cosmological axion domain wall problem [32], which
afflicts the $\nu$DFSZ model if PQ symmetry is restored after the inflation,
using a variant axion. Drawing inspiration from original proposals in Refs.
[33, 34], we recast the model with a flavour-dependent $U(1)_{PQ}$ charge
assignment that realises a unit domain wall number which is cosmologically
benign [35, 36, 37, 38] and leads to generation non-universal quark Yukawa
couplings. For the sake of definiteness, we focus on a top-specific Yukawa
structure, in which one of two Higgs doublets is responsible for giving a
heavier mass to the top quark and no other fermion, gaining a natural
explanation for the top mass as a remarkable bonus. This structure leads to
interesting phenomenology [39, 40] which we briefly review alongside its
discovery prospects.

We outline our $\nu$DFSZ-variant, top-specific “VISH$\nu$” model222Though in
this paper we single out the top-specific model for definiteness and
phenomenological simplicity, we have in mind a family of models, dubbed
“avatars” of VISH$\nu$, encompassing all unit axion domain wall number
$\nu$DFSZ-like models with viable flavour-variant structures. in Sec. II. We
then arrive at inflationary predictions in Sec. III, discuss implications for
dark matter predictions in Sec. IV and establish a parameter space which is
consistent with reviewed naturalness desiderata, as well as high-scale
validity and possible unitarity concerns, in Sec. V. We conclude in Sec. VI
with a summary of our findings.

##  II The model and its collider phenomenology

###  II.1 The model

Preserving the SM gauge group, our model extends the SM fermion content by
three right-handed neutrinos singlets $\nu_{R}\sim(1,1,0)$ and upgrades its
scalars to two Higgs electroweak doublets $\Phi_{1,2}\sim(1,2,1)$ with an
additional gauge singlet $S\sim(1,1,0)$. To realise an invisible axion
solution to the strong $CP$ problem, an anomalous chiral global
$U(1)_{\text{PQ}}$ symmetry constrains possible renormalisable terms in the
potential and Yukawa sectors.

Gauge invariance and renormalisability forces two independent choices for non-
trivially $U(1)_{\text{PQ}}$ invariant terms in the potential. So that, if we
denote as $X_{1},X_{2}$ the PQ charges of $\Phi_{1},\Phi_{2}$, and fix the
charge of $S$ to unity:

$\begin{split}V&=M_{11}^{2}\Phi^{\dagger}_{1}\Phi_{1}+M_{22}^{2}\Phi^{\dagger}_{2}\Phi_{2}+M_{SS}^{2}S^{*}S\\\
&\quad+\frac{\lambda_{1}}{2}(\Phi^{\dagger}_{1}\Phi_{1})^{2}+\frac{\lambda_{2}}{2}(\Phi^{\dagger}_{2}\Phi_{2})^{2}+\frac{\lambda_{S}}{2}(S^{*}S)^{2}\\\
&\quad+\lambda_{3}(\Phi^{\dagger}_{1}\Phi_{1})(\Phi^{\dagger}_{2}\Phi_{2})+\lambda_{4}(\Phi^{\dagger}_{1}\Phi_{2})(\Phi^{\dagger}_{2}\Phi_{1})\\\
&\quad+\lambda_{1S}(\Phi^{\dagger}_{1}\Phi_{1})(S^{*}S)+\lambda_{2S}(\Phi^{\dagger}_{2}\Phi_{2})(S^{*}S)\\\
&+\begin{cases}\kappa\Phi_{1}^{\dagger}\Phi_{2}S\ +\text{h.c.}\ \ \text{if}\
X_{2}-X_{1}=1\ [\text{VISH}\nu]\\\
\epsilon\Phi_{1}^{\dagger}\Phi_{2}S^{2}+\text{h.c.}\ \ \text{if}\
X_{2}-X_{1}=2\ [\nu\text{DFSZ}].\end{cases}\end{split}$ (1)

While the second choice of final term ($X_{2}-X_{1}=2$) was considered in both
the original DFSZ model and the $\nu$DFSZ papers [11, 12], we adopt the
_first_ throughout. Our Yukawa sector is determined by the $U(1)_{\text{PQ}}$
charge assignments given in Table 1. Implying summation over the generation
indices $a=1,2$ and $j,k=1,2,3$:

$\begin{split}-\mathcal{L}_{Y}&=\overline{q_{L}}^{j}y_{u1}^{j3}\widetilde{\Phi}_{1}u_{R}^{3}+\overline{q_{L}}^{j}y_{u2}^{ja}\widetilde{\Phi}_{2}u_{R}^{a}+\overline{q_{L}}^{j}y_{d}^{jk}\Phi_{2}d_{R}^{k}\\\
&\quad+\overline{l_{L}}^{j}y_{e}^{jk}\Phi_{2}e_{R}^{k}+\overline{l_{L}}^{j}y_{\nu}^{jk}\widetilde{\Phi}_{2}\nu_{R}^{k}\\\
&\quad+\frac{1}{2}\overline{(\nu_{R})^{c}}^{j}y_{N}^{jk}S\nu_{R}^{k}+\text{h.c.},\end{split}$
(2)

where, of the right-handed fermions, $\Phi_{1}$ Yukawa-couples exclusively to
$u^{3}_{R}$ and we work hereafter in a basis where $y_{N}$ is real and
diagonal. The potential (1) is minimised when the scalar fields develop the
real-valued vacuum expectation values (vevs), for $i=1,2$:

$\langle
S\rangle=\frac{v_{S}}{\sqrt{2}},\quad\langle\Phi_{i}\rangle=\frac{1}{\sqrt{2}}\begin{pmatrix}0\\\
v_{i}\end{pmatrix},$ (3)

where $v=\sqrt{v_{1}^{2}+v_{2}^{2}}\simeq 246$ GeV and, since we require
$v_{S}\gg v_{1},v_{2}$ for an “invisible” axion,
$v_{S}\simeq\sqrt{-2M_{SS}^{2}/\lambda_{S}}\sim 10^{10}$–$10^{11}\
\text{GeV}$. Integrating out $S$ leads to the following quadratic terms in the
Higgs potential:

$V\supset
m_{11}^{2}\Phi^{\dagger}_{1}\Phi_{1}+m_{22}^{2}\Phi^{\dagger}_{2}\Phi_{2}-m^{2}_{12}(\Phi^{\dagger}_{1}\Phi_{2}+\Phi_{1}\Phi_{2}^{\dagger}),$
(4)

where $m^{2}_{ii}=M^{2}_{ii}+\lambda_{iS}\langle S\rangle^{2}$ and
$m^{2}_{12}=\kappa\langle S\rangle$. Together with a weakly-coupled axion
($f_{A}\sim v_{S}$), the low energy model is the $\nu$2HDM [17] up to
$O(v/v_{S})$ corrections and top-specific Yukawa couplings.

Table 1:  $U(1)_{\text{PQ}}$ charge assignments for the VISH$\nu$ model. In order to decouple the $U(1)_{\text{PQ}}$ current from the Goldstone boson eaten by the $Z$ boson, it is sufficient to impose $X_{1}=(-\cot^{2}\beta)X_{2}$ [13], where $X_{1},X_{2}$ denote the PQ charges of $\Phi_{1},\Phi_{2}$. Field: | $q_{L}$ | $u^{a}_{R}$ | $u^{3}_{R}$ | $d_{R}$ | $l_{L}$ | $e_{R}$ | $\nu_{R}$ | $\Phi_{1}$ | $\Phi_{2}$ | $S$
---|---|---|---|---|---|---|---|---|---|---
Charge: | 0 | $-\sin^{2}\beta$ | $\cos^{2}\beta$ | $\sin^{2}\beta$ | $\sin^{2}\beta-\frac{1}{2}$ | $2\sin^{2}\beta-\frac{1}{2}$ | $-\frac{1}{2}$ | $\cos^{2}\beta$ | $-\sin^{2}\beta$ | 1

Values for the mass parameters, $m_{ij}$, impact two important predictions of
the model. These are: the viability of thermal hierarchical leptogenesis, and
the protection of the electroweak scale from physically meaningful corrections
induced by the PQ scale. For instance, we see that _tree-level_ naturalness
already requires that $\lambda_{iS}\langle S\rangle^{2}$ to not exceed
$M_{ii}^{2}$. In Ref. [17] it was shown that both predictions can be
additionally met if one of the Higgs vevs, namely $v_{2}$, is slightly smaller
than the other ($\sim$ few GeV). This is due to the convergence of a Vissani-
like naturalness bound, ensuring $\nu_{R}$-induced corrections to the
electroweak Higgs masses are controlled, and a Davidson-Ibarra bound, ensuring
sufficient $CP$ asymmetry during leptogenesis, under the assumption of
perturbative validity.333We direct the reader to Fig. 5 of Ref. [17] for a
more detailed representation of the bounds on $v_{2}$, and their dependence on
the lightest neutrino mass. To realise the observed value $v\simeq 246$ GeV,
this forces $v_{1}$ to a comparatively larger value ($\sim 10^{2}$ GeV) than
$v_{2}$. As $\Phi_{1}$ is responsible only for giving mass to the top quark,
an explanation for its large mass in comparison to other SM fermions is an
immediate consequence.444This remarkable bonus hints suggestively at an inroad
to the flavour puzzle, since richer VISH$\nu$ flavour-structures are also
possible. We require that $|m^{2}_{12}|/m^{2}_{22}\ll 1$ to protect the
resulting small hierarchy in Higgs vevs from radiative corrections. As
discussed in Ref. [12], the ensuing values of the vevs, $v_{i}$, mean that it
is typical for the mass parameters to take the values $m^{2}_{11}\simeq-(88\
\text{GeV})^{2}$ and $m_{22}\sim 10^{3}$ GeV together with
$m^{2}_{22}/\tan^{2}\beta\ll|m^{2}_{11}|$, where $\tan\beta\equiv
v_{2}/v_{1}$. Taken together, we find that the consistency of VISH$\nu$
predictions with our naturalness criteria require that $|\lambda_{1S}|\lesssim
10^{-18}$, $|\lambda_{2S}|\lesssim 10^{-16}$ and $|\kappa|\ll O(0.1)$ keV,
where the final condition replaces its analogue $|\epsilon|\ll 10^{-18}$ in
the $\nu$DFSZ model [12].

Observe that the model manifests two decoupling limits which render weak
inter-sector interactions technically natural in the Poincaré-protection
sense, as will be fully discussed in Sec. V. The heavy sector
$\\{S,\nu_{R}\\}$ is decoupled from the Higgs (and SM) sector in the limit
$\lambda_{1S},\lambda_{2S},\kappa,y_{\nu}\rightarrow 0$, while $\\{S\\}$ can
itself be decoupled by $\lambda_{1S},\lambda_{2S},\kappa,y_{N}\rightarrow 0$.
Additionally, there is an enhanced $U(1)$ symmetry in the limits
$y_{N},y_{\nu},\kappa\rightarrow 0$ which individually protect small values of
$y_{N},y_{\nu}$ and $\kappa$. Necessary and sufficient bounded-from-below
conditions for the potential (1) were derived in Ref. [41]. For
$\widetilde{\lambda}=\max\\{\lambda_{3},\lambda_{3}+\lambda_{4}\\}$, they
describe the parameter space: $\\{\lambda_{S},\lambda_{i}>0,\
\lambda_{iS}>-\sqrt{\lambda_{i}\lambda_{S}},\
\widetilde{\lambda}>-\sqrt{\lambda_{1}\lambda_{2}},\
\lambda_{1S}\geq-\lambda_{2S}\sqrt{\lambda_{1}/\lambda_{2}}\\}\cup\\{\lambda_{S},\lambda_{i}>0,\
\sqrt{\lambda_{2}\lambda_{S}}\geq\lambda_{2S}>-\sqrt{\lambda_{2}\lambda_{S}},\
-\lambda_{2S}\sqrt{\lambda_{1}/\lambda_{2}}\geq\lambda_{1S}>-\sqrt{\lambda_{1}\lambda_{S}},\
\lambda_{S}\widetilde{\lambda}>\lambda_{1S}\lambda_{2S}-\sqrt{(\lambda^{2}_{1S}-\lambda_{1}\lambda_{S})(\lambda^{2}_{2S}-\lambda_{2}\lambda_{S})}\\}$.
For the very weak interactions coupling the Higgs and $S$ sectors, the
conditions diminish to those of a general two Higgs doublet model [42, 41, 43,
17] with couplings limited by $U(1)_{\text{PQ}}$:

$\lambda_{1}>0,\ \lambda_{2}>0,\
\widetilde{\lambda}\gtrsim-\sqrt{\lambda_{1}\lambda_{2}},$ (5)

together with the requirement that $\lambda_{S}>0$, up to small corrections.
To ensure that (3) is the global minimum of the potential, we also require
that $\lambda_{1}\lambda_{2}-(\lambda_{3}+\lambda_{4})^{2}>0$, which
strengthens the final condition of (5).

###  II.2 A brief summary of collider phenomenology

We now mention some interesting phenomenological differences between VISH$\nu$
and the $\nu$DFSZ model born from the top-specific flavour structure,
referring the reader to Refs. [12, 17] for the natural explanations of
neutrino masses and the BAU, which remain the same, and Sec. IV for an updated
discussion of axionic dark matter production.

Unlike the standard DFSZ model, flavour-variant models generically feature
neutral Higgs-induced flavour-changing processes at tree level. We now explain
how VISH$\nu$ produces a SM-like mass eigenstate $h$ in a natural alignment
limit, and how a top-specific structure abides by current experimental
constraints. Since $\Phi_{1}$ couples only to the right-handed top-quark while
$\Phi_{2}$ couples to right-handed up- and charm-quarks, there are tree-level
FCNC decays of the top quark via mixing of the $CP$-even neutral Higgs states.
Hence, the collider signature of our flavour structure is the decay
$t\rightarrow ch$ (or $t\rightarrow uh$), where $h$ is the observed Higgs [39,
40, 44].

Upper limits on the observed (expected) branching ratios of each decay have
recently been made more stringent by the CMS collaboration [45]:

$\begin{split}&\mathcal{B}(t\rightarrow hc)<0.073\ (0.051)\ \%,\\\
&\mathcal{B}(t\rightarrow hu)<0.019\ (0.031)\ \%,\end{split}$ (6)

with $\sqrt{s}=$ 13 TeV and 137 fb-1 data, representing no significant excess
above the predicted background. For the case where $t\rightarrow hc$ is
dominant (which we expect anyway from mass-mixing [44]), this corresponds to
the constraint on model parameters [40]:

$a^{2}\sin^{2}\rho<0.023,$ (7)

where $a=(\tan\beta+\cot\beta)\cos(\beta-\alpha)$, $\alpha$ is the rotation
angle which diagonalises the neutral Higgs scalars and $\rho$ is a $t-c$
mixing angle defined in Ref. [40].

In addition to the usual decoupling suppression ($v_{1}/m^{2}_{22}\ll 1$)
[17], the quantity $\cos(\beta-\alpha)$ is also suppressed555See, for example,
Eq. 15 of Ref. [17]. in VISH$\nu$ by the approximate $U(1)$ symmetry realised
for $m^{2}_{12}/m^{2}_{22}\ll 1$. Note that, in the full theory, this
corresponds to the natural limit $\kappa\rightarrow 0$ discussed above. As a
consequence, the light $CP$-even neutral scalar _naturally_ resembles the SM
Higgs. Since the product (7) is suppressed in the alignment limit,
$\cos(\beta-\alpha)\rightarrow 0$, $t-c$ mixing does not have to be
unnaturally small to remain compatible with exclusion limits. In fact, while
permitting in-principle tree-level FCNC decays associated to the _SM-like_
Higgs, it follows that flavour-variants of the $\nu$DFSZ may automatically
arrange their suppression.

An additional way to access the FCNC top-netural-Higgs interaction, without
alignment suppression, is in the top-associated production of the heavy scalar
bosons $H,A$ and $H^{+}$ [44] via the channels: $cg\rightarrow tH$ or $tA$
[46] as well as $cg\rightarrow bH^{+}$ [47], which can be searched for at the
LHC and its high-luminosity upgrade [44].

##  III Inflation

A scalar inflaton field which is non-minimally coupled to gravity remains
among the best fits to CMB data [31], and may be identified with the SM Higgs
field [1]. However, there also exist realistic models, such as invisible axion
or multi-Higgs doublet models, which employ an extended scalar sector.
Compellingly, non-minimal gravitational couplings of a multi-scalar sector
generically give rise to one or more attractors for inflation trajectories
which replicate the successful inflationary predictions of the single-field
model [29], with some exceptional cases [48]. This behaviour has, for example,
motivated analyses of an inflation phase in non-minimally coupled versions of
the two Higgs doublet [49, 50, 51] and KSVZ models [20, 21, 22].

In this section, we extend each of these analyses to include a third scalar
non-minimally coupled to gravity, with couplings limited by our
$U(1)_{\text{PQ}}$ symmetry. Since this simply restricts to a $Z_{2}$ symmetry
on the quartic potential relevant in the inflation regime, our analysis is
quite general.666In particular, Table 2 may be useful for inflation model-
builders. In Sec. III.3, we discuss implications of weak inter-sector
couplings and present fits to CMB observations in Sec. III.4 and Sec. III.5.
Of course, these findings are independent of our flavour structure, and so
constitute a general analysis of both non-minimally coupled VISH$\nu$ and
cubic DFSZ models for inflation, the results of which are summarised in Figure
1, Figure 2 and Table 3.

###  III.1 VISH$\nu$ in the Einstein frame

If they respect the $U(1)_{\text{PQ}}$ symmetry, mass-dimension four non-
minimal couplings of the VISH$\nu$ scalar fields to gravity will generically
arise in curved spacetime, since they are generated radiatively, and are
required for renormalisation [52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63].
These modify the Einstein-Hilbert action as follows:

$\frac{\mathcal{L}^{\mathcal{J}}}{\sqrt{-g^{\mathcal{J}}}}\supset\left(\frac{M_{P}^{2}}{2}+\xi_{1}\Phi_{1}^{\dagger}\Phi_{1}+\xi_{2}\Phi_{2}^{\dagger}\Phi_{2}+\xi_{S}S^{\dagger}S\right)R^{\mathcal{J}},$
(8)

where $\xi_{1},\xi_{2},\xi_{S}$ are real dimensionless couplings which we will
assume to be non-negative, $M_{P}$ is the reduced Planck mass and
$R^{\mathcal{J}}$ is the Ricci scalar in the Jordan frame. Since the non-
minimal couplings are diagonal, we parameterise the neutral scalars as a
modulus which, at large values, plays a role in driving inflation, and a
phase:

$\Phi^{0}_{1}=\frac{\rho_{1}}{\sqrt{2}}e^{i\vartheta_{1}/v_{1}},\
\Phi^{0}_{2}=\frac{\rho_{2}}{\sqrt{2}}e^{i\vartheta_{2}/v_{2}},\
S=\frac{\sigma}{\sqrt{2}}e^{i\vartheta_{S}/v_{S}}$ (9)

(we set the charged Higgs components to zero for the inflation analysis [49,
50, 64]). We will assume the inflation phase begins with a displacement of one
or more of the modulus fields from their minimum which satisfies a _large-
field_ regime ($\xi_{1}\rho_{1}^{2}+\xi_{2}\rho_{2}^{2}+\xi_{S}\sigma^{2}\gg
M_{P}^{2}$).

To analyse the predicted inflation, it is standard to work in the Einstein
frame ($\mathcal{E}$), which is related to Jordan frame ($\mathcal{J}$) of (8)
by a local rescaling of the spacetime metric:

$\begin{split}&g^{\mathcal{J}}_{\mu\nu}\rightarrow
g^{\mathcal{E}}_{\mu\nu}=\Omega^{2}(\rho_{1},\rho_{2},\sigma)g^{\mathcal{J}}_{\mu\nu},\\\
&\text{where}\quad\Omega^{2}(\rho_{1},\rho_{2},\sigma)\equiv
1+\frac{\xi_{1}\rho_{1}^{2}+\xi_{2}\rho_{2}^{2}+\xi_{S}\sigma^{2}}{M_{P}^{2}}.\end{split}$
(10)

The effect of the Weyl transformation is both to restore a minimal coupling to
gravity and flatten out the potential for large values of the modulus fields,
but this necessarily comes at the expense of non-canonical kinetic terms [65],
so that the gravi-scalar action becomes:

$\frac{\mathcal{L}^{\mathcal{E}}}{\sqrt{-g^{\mathcal{E}}}}\supset\frac{M_{P}^{2}}{2}R^{\mathcal{E}}-\frac{1}{2}\sum_{I,J}\mathcal{G}^{\mathcal{E}}_{IJ}\partial_{\mu}\varphi^{I}\partial^{\mu}\varphi^{J}-V^{\mathcal{E}}(\varphi^{I}),$
(11)

where $\varphi^{I}=(\rho_{1},\rho_{2},\sigma)$ and
$\mathcal{G}^{\mathcal{E}}_{IJ}$ is the non-trivial metric induced by (10) on
the scalar field space.777We discuss implications of terms with non-
renormalisable mass dimension in Sec. V. It is:

$\mathcal{G}^{\mathcal{E}}_{IJ}=\frac{\delta_{IJ}}{\Omega^{2}}+\frac{3M_{P}^{2}}{2}\frac{\partial\log\Omega^{2}}{\partial\varphi^{I}}\frac{\partial\log\Omega^{2}}{\partial\varphi^{J}}.$
(12)

In the large-field regime ($\Omega^{2}\gg 1$) the Einstein frame scalar
potential is:

$\begin{split}V^{\mathcal{E}}(\varphi^{I})&=\Omega^{-4}(\varphi^{I})V^{\mathcal{J}}(\varphi^{I})\\\
&=\frac{M_{P}^{4}}{8}\frac{\lambda_{i}\rho_{i}^{4}+2\lambda_{34}\rho_{1}^{2}\rho_{2}^{2}+2\lambda_{iS}\rho_{i}^{2}\sigma^{2}+\lambda_{S}\sigma^{4}}{(\xi_{i}\rho_{i}^{2}+\xi_{S}\sigma^{2})^{2}}\\\
&\quad\times\left[1-\mathcal{O}\left(\frac{M_{P}^{2}}{\xi_{i}\rho_{i}^{2}+\xi_{S}\sigma^{2}}\right)\right]^{2},\end{split}$
(13)

where $\lambda_{34}=\lambda_{3}+\lambda_{4}$ and we will compress sums over
$i=1,2$ when what we mean is obvious.

###  III.2 Inflation model with three non-minimally coupled scalars

We will now outline the inflation scenarios for a model of three real scalar
fields, $\chi^{I}=(\chi_{1},\chi_{2},\chi_{3})$, with non-minimal
gravitational couplings, $\xi_{k}=(\xi_{1},\xi_{2},\xi_{3})$, and which
respects independent $\mathbb{Z}_{2}$ transformations of the components of
$\chi^{I}$. In other words, we will analyse the scalar potential which, in the
large-field regime, is:

$V(\chi^{I})\simeq\frac{M^{4}_{P}}{8}\frac{\sum_{ij}\lambda_{ij}(\chi_{i}\chi_{j})^{2}}{\left[\sum_{k}\xi_{k}(\chi_{k})^{2}\right]^{2}}\left[1-\mathcal{O}\left(\frac{M_{P}^{2}}{\sum_{k}\xi_{k}(\chi_{k})^{2}}\right)\right]^{2},$
(14)

and which is obtained in the Einstein frame by a Weyl transformation that
induces a field-space metric in direct analogy to (10) and (12).888Of course,
(14) is merely a generalisation of (13). We do this so that the calculations
of this subsection apply now to an arbitrary parameter space,
$\\{\xi_{I},\lambda_{IJ}\\}$, with the minimal requirements: $\xi_{I}\geq 0$,
and $\sum_{I,J}\lambda_{IJ}(\chi^{I}\chi^{J})^{2}\geq 0$. We reserve Sec.
III.3 for the hidden-sector parameter space of VISH$\nu$.

Unless there is an internal $O(3)$ symmetry of the field space, which forces
both the equality of all quartic couplings $\lambda_{ij}$ and all non-minimal
couplings $\xi_{k}$, (14) describes a manifold with higher-dimensional
analogues of ridges and valleys [66]. At large field-values, and with respect
to a radial cross-section, we will denote as “hyper-valleys” features which
are local minima, reserving “ridge” for features which either resemble a local
maximum or are saddle-shaped. For general large-field initial displacements
$\chi_{0}^{I}$, and a broad range of initial velocities $\dot{\chi}_{0}^{I}$,
the hyper-valleys correspond to effectively single-field attractors for
inflaton trajectories due to Hubble friction [29]. After settling into a
hyper-valley, the field trajectory: satisfies the single field slow-roll
approximation, exhibits a negligible turn rate during the inflation, and
orthogonal directions can acquire an effective mass which generally exceeds
the inflationary Hubble scale, so that both non-Gaussianities and isocurvature
pertubations are negligible [29, 66, 48, 30].999As explained in Ref. [48],
there are exceptions to this general situation which can arise when the
curvature near a ridge is sufficiently weak. When this is the case, severely
finely-tuned initial field trajectories can either drive a period of inflation
along a ridge which is incompatible with observations, or persist there for a
time, before falling into a valley at a later stage. This leads, for example,
to the generation of sizeable isocurvature modes [48], which are generally
suppressed for the effectively single-field trajectories and initial
conditions we are interested in.

To determine the possible inflation scenarios, we thus identify the hyper-
valleys of the manifold (14) compatible with a given parameter space. We do
this by characterising the curvature near the extrema of the surface at large
field displacement (17). While the existence of ridges and hyper-valleys will
be independent of the parametrisation we use for the underlying field space,
we can simplify the problem using spherical coordinates:

$\chi_{1}=r\cos\phi\sin\theta\quad\chi_{2}=r\sin\phi\sin\theta\quad\chi_{3}=r\cos\theta$
(15)

where $r^{2}=\sum_{k}\chi_{k}^{2}$. The azimuthal and polar angles have the
definitions when $\chi_{1}\neq 0$:

$\tan\phi=\frac{\chi_{2}}{\chi_{1}},\quad\cos\theta=\frac{\chi_{3}}{r},$ (16)

and we focus on the first octant without loss of generality:
$\phi\in[0,\pi/2]$, $\theta\in[0,\pi/2]$ and $r>0$. In the large-field regime,
the leading order part of the potential (14) is therefore independent of the
radial variable (with $c_{\psi},s_{\psi}\equiv\cos\psi,\sin\psi$ and
$\lambda_{ij}=\lambda_{ji}$):

$\begin{split}V\simeq\Lambda[\phi,\theta]=\
&\frac{M_{P}^{4}}{8}[s_{\theta}^{2}(\xi_{1}c_{\phi}^{2}+\xi_{2}s_{\phi}^{2})+\xi_{3}c^{2}_{\theta}]^{-2}\\\
&\times\left\\{(\lambda_{1}c_{\phi}^{4}+2\lambda_{12}c_{\phi}^{2}s_{\phi}^{2}+\lambda_{2}s_{\phi}^{4})s^{4}_{\theta}\right.\\\
&\quad\quad\left.+2[(\lambda_{23}-\lambda_{13})c^{2}_{\phi}+\lambda_{13}]c^{2}_{\theta}s^{2}_{\theta}\right.\\\
&\quad\quad\left.+\lambda_{3}c_{\theta}^{4}\right\\}\end{split}$ (17)

and, when extremised, represents a false vacuum energy density
($\Lambda_{\text{inf}}$) responsible for a de Sitter expansion, to leading
order.

Table 2:  We supply the locations $(\overline{\phi},\overline{\theta})$ of the critical points of the false vacuum energy density which is the leading order contribution of the inflation potential (17). The parameters $\kappa_{ij}$ and $A_{ij}$ are defined in (18), and we have indicated which components in the $\chi_{k}$ basis take non-trivial values at each location. For the existence of each critical point, we use “$\checkmark$” for when this is guaranteed in general and derive the non-trivial conditions using restrictions on the codomain of the cosine function, namely that it is real-valued and does not exceed unity. For the conditions which must be satisfied to ensure a hyper-valley (a valley of the 3-manifold) at the point $(\overline{\phi},\overline{\theta})$, we require the positivity of both principal curvatures which, for a critical point, constitute the eigenvalues of the Hessian matrix of double derivatives. For brevity, we omit the existence conditions for the case where a principal curvature vanishes. The $\overline{\phi}$ value for the critical point along the $\chi_{3}$ axis is degenerate. $\cos^{2}\overline{\phi}$ | $\cos^{2}\overline{\theta}$ | Non-zero components | Existence of critical point | Hyper-valley conditions
---|---|---|---|---
$1$ | $0$ | $\chi_{1}$ | $\checkmark$ | $\kappa_{12},\kappa_{13}>0$
0 | 0 | $\chi_{2}$ | $\checkmark$ | $\kappa_{21},\kappa_{23}>0$
deg. | 1 | $\chi_{3}$ | $\checkmark$ | $\kappa_{31},\kappa_{32}>0$
$\dfrac{\kappa_{21}}{\kappa_{12}+\kappa_{21}}$ | $0$ | $\chi_{1},\chi_{2}$ | $\kappa_{12}\kappa_{21}>0$ | $\kappa_{12},\kappa_{21},A_{12}<0$
$1$ | $\dfrac{\kappa_{13}}{\kappa_{13}+\kappa_{31}}$ | $\chi_{1},\chi_{3}$ | $\kappa_{13}\kappa_{31}>0$ | $\kappa_{13},\kappa_{31},A_{13}<0$
$0$ | $\dfrac{\kappa_{23}}{\kappa_{23}+\kappa_{32}}$ | $\chi_{2},\chi_{3}$ | $\kappa_{23}\kappa_{32}>0$ | $\kappa_{23},\kappa_{32},A_{23}<0$
$\dfrac{A_{23}}{A_{23}+A_{13}}$ | $\dfrac{A_{12}}{A_{12}+A_{23}+A_{13}}$ | $\chi_{1},\chi_{2},\chi_{3}$ | $A_{13}A_{23},\ A_{12}(A_{23}+A_{13})>0$ | $A_{12},A_{23},A_{13}>0$

In a general parameter space, we identify seven possible critical points of
the surface $(\phi,\theta,\Lambda[\phi,\theta])$. Despite a non-trivial field-
space metric, they merely occur when both $\partial_{\phi}\Lambda=0$ and
$\partial_{\theta}\Lambda=0$, as the function $\Lambda$ is a field-space
scalar. Their locations are given in Table 2. To compress expressions, we
introduce the parameters, for $i\neq j\neq k$ and no implied sums:

$\begin{split}&\kappa_{ij}\equiv\lambda_{ij}\xi_{i}-\lambda_{i}\xi_{j},\quad\gamma_{ij}=\gamma_{ji}\equiv\lambda_{i}\lambda_{j}-\lambda^{2}_{ij}>0,\\\
&A_{ij}=A_{ji}\equiv\xi_{k}\gamma_{ij}+\lambda_{ik}\kappa_{ji}+\lambda_{jk}\kappa_{ij}\end{split}$
(18)

which have geometric roles. The parameter $\kappa_{ij}$ is proportional to one
of the pair of principal curvatures at the critical point in the
$\chi_{i}$-direction, namely that which is associated to the
$\chi_{j}$-direction, while $\gamma_{ij}>0$ forbids unwanted minima of the
Jordan frame potential in the $\chi_{i}\chi_{j}$-plane. The $A_{ij}$ arise in
the products $\kappa_{31}A_{13}$, $\kappa_{32}A_{23}$ and
$(\kappa_{12}+\kappa_{21})A_{12}$ which respectively fix the sign of the
principal curvature associated to the $\chi_{1}$, $\chi_{2}$ and $\chi_{3}$
directions at the critical point misaligned in the $\chi_{1}\chi_{3}$-,
$\chi_{2}\chi_{3}$\- and $\chi_{1}\chi_{2}$-planes. In fact, the relevant
curvature data for all seven critical point factorises into the $\kappa_{ij}$
and $A_{ij}$, and so their signs are sufficient to determine when each
realises a local minimum (or maximum/saddle point) of the leading order
potential, and hence a hyper-valley (or ridge) of the full potential. We
supply this information in the final column of Table 2 and, hence, exhaust the
possible attractor orientations for a general parameter space
$\\{\lambda_{ij},\xi_{k}\\}$.

In Sec. III.3, we show how the results of this subsection can be used identify
the hyper-valleys of (13) by restricting the parameter space to realise the
hidden-sector structure, as this is the parameter space which is relevant to
the VISH$\nu$ model, and the remainder of our discussion. Having established
the possible inflation scenarios, it remains to analyse the effectively
single-field inflationary dynamics within each hyper-valley and demonstrate
their reproduction of observed features of the CMB power spectrum. Although
our discussion will focus on the VISH$\nu$ model, these results are generally
independent of the parameter space, with the exception of the scalar power
spectrum amplitude constraint ($A_{s}$), and are complementary to previous
studies. In Sec. III.4, we assume that $\xi_{k}\gg 1$ whenever a hyper-valley
has a component in the $\chi_{k}$ direction (we call this the “large
$\xi_{k}$” regime) and these inflationary predictions are degenerate with the
Higgs inflation model [1]. In Sec. III.5, we use the parametric freedom of our
extended scalar sector to explore representative limiting cases that continue
the parameter space outside the large $\xi_{k}$ regime, i.e. “small
$\xi_{k}$”, and this regime can avoid unitarity concerns. As we soon
demonstrate, a simplification achieved in either context is the suppression of
the kinetic mixing of the field components induced by the field-space metric
in a convenient basis. In Appendix A we compare the masses of canonically
normalised fluctuations orthogonal to the hyper-valleys with the inflationary
Hubble scale.

###  III.3 Inflation scenarios in VISH$\nu$

Let us now discuss the possible orientations of hyper-valleys specific to the
case of the large-field VISH$\nu$ potential (13), for which we require
$\lambda_{iS}\ll\lambda_{i},\lambda_{S},\lambda_{34}$ to realise the hidden
sector structure (see Sec. V.1). To do this, we merely identify the components
of $\varphi^{I}=(\rho_{1},\rho_{2},\sigma)$ with the components of $\chi^{I}$
in the preceding section. Correspondingly, we will substitute “$S$” for the
“$3$” index of the results we have obtained, and identify the coefficients of
(14) with (13) in the obvious way, e.g. $\lambda_{34}=\lambda_{12}$.

In general, the effect that decoupling $S$ from the Higgs sector has on the
curvature of the inflation potential is encoded by the set of parameters
$\\{\kappa_{1S}$, $\kappa_{S1}$, $\kappa_{2S}$, $\kappa_{S2}\\}$ which all
become negative, and $A_{12}$, now positive. When this is the case, initial
trajectories will quickly develop a $\sigma$ component, but, as the $\sigma$
axis forms a ridge ($\kappa_{S1},\kappa_{S2}<0$), there are at most three
possible hyper-valleys where the trajectories can stabilise. We will denote
the corresponding inflation scenarios by the fields which give rise to a non-
zero component in the inflaton field parameterising the radial time-evolution.
Hence, for the general situation where are all non-minimal couplings are
sizeable, there are three candidate scenarios to consider:
$\Phi_{1}S$-Inflation can occur if $A_{1S}<0$ ($\kappa_{12}>0$),
$\Phi_{2}S$-Inflation can occur if $A_{2S}<0$ ($\kappa_{21}>0$), while
$\Phi_{1}\Phi_{2}S$-Inflation can occur if both $A_{1S}>0$ ($\kappa_{12}<0$)
and $A_{2S}>0$ ($\kappa_{21}<0$).101010 $\Phi_{1}\Phi_{2}$-Inflation is
possible for the special case that $\gamma_{12}\simeq 0$ and $\lambda_{iS}>0$,
but any approximate $SO(2)$ symmetry of the Higgs potential realised when
$\gamma_{12}$ is set to zero along with the inter-sector couplings is
explicitly broken by the Yukawa terms. This means that $\gamma_{12}\simeq 0$
is not radiatively stable in the sense we define in Sec. V and so, unless
$\xi_{S}\simeq 0$, requires a fine-tuning of the quartic parameters to become
a valley. The corresponding condition in brackets holds when the sectors are
fully decoupled. As $\kappa_{12}$ and $\kappa_{21}$ cannot both be positive if
$\xi_{1},\xi_{2}>0$, there is exactly one candidate inflaton field for each of
the three disjoint regions of $\\{\kappa_{12},\kappa_{21}\\}$ parameter space
we identify.111111We begin with
$\sqrt{\lambda_{1}\lambda_{2}}-\lambda_{34}>0$, which follows from
$\gamma_{12}>0$ and boundedness-from-below, and we assume that
$\xi_{1},\xi_{2}>0$. It follows that
$(\sqrt{\lambda_{1}}\xi_{1}-\sqrt{\lambda_{1}}\xi_{2})^{2}+2\xi_{1}\xi_{2}(\sqrt{\lambda_{1}\lambda_{2}}-\lambda_{34})>0$,
because we have multiplied by, and then added, positive real numbers.
Rearrangement shows the left-hand quantity to be
$-[\xi_{1}\kappa_{21}+\xi_{2}\kappa_{12}]$ which, accordingly, must be
positive. This forbids the case where both $\kappa_{12}>0$ and
$\kappa_{21}>0$. Generalising our results, it is clear that a hidden-sector
extension cannot constitute an inflaton sector if a hidden scalar develops a
non-minimal gravitational coupling in addition to the SM Higgs. Symmetrically,
scalars which are very weakly coupled to the Higgs boson inevitably play an
inflationary role when both are non-minimally coupled to gravity. This is
because directions aligned with the field axes constitute ridges, while hyper-
valleys have a non-trivial orientation. For each of the three scenarios
generically compatible with the VISH$\nu$ parameter space
($\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{i}S$-Inflation), masses associated
to the orthogonal modulus field directions exceed the Hubble scale of the
inflation by several orders of magnitude (this is demonstrated in Appendix A),
which, with the exception of fluctuations in the axion field discussed in Sec.
IV, suppresses isocurvature modes ($\beta_{\text{iso}}\sim 0$) even in the
limit of sector decoupling.

However, there are special exceptions to these findings. To support either
single component hyper-valleys (aligned with a field axis), or those which are
misaligned in the Higgs plane but orthogonal to the $\sigma$ axis, we can
specialise to a parameter space where only some fields develop sizeable non-
minimal couplings, hence reversing the sign of the curvatures $\kappa_{ij}$
which otherwise forbid them, for the case where $\lambda_{iS}>0$.
Specifically, if $\xi_{S}\simeq 0$, then $\kappa_{iS},A_{12}>0$ and so
$\Phi_{1}\Phi_{2}$-Inflation, $\Phi_{1}$-Inflation and $\Phi_{2}$-Inflation
become possible — the Higgs sector can be an inflaton sector. While if
$\xi_{1},\xi_{2}\simeq 0$ then $\kappa_{Si}>0$, so $S$-Inflation becomes
possible — the hidden sector can be an inflaton sector.121212Of course, the
naturalness of this parameter space must be defended. Given the $\xi_{k}$ are
coupled under renormalisation group evolution, their hierarchical organisation
may not be radiatively stable. However, in the limit of sector decoupling, the
running of $\xi_{S}$ should likewise decouple from $\xi_{i}$, so that
$\xi_{S}\gg\xi_{1},\xi_{2}$ and $\xi_{S}\ll\xi_{1},\xi_{2}$ may not provoke a
naturalness concern, while $\xi_{S},\xi_{1}\gg\xi_{2}$ may be considered
unnatural. This decoupling is evinced in a two-scalar context by RG equations
in Ref. [67]. For these special deformations of the potential, one finds that
$\kappa_{ij}\simeq 0$, due to $|\lambda_{iS}|\ll 1$, approximating a regime of
zero or extremely weak curvature considered in Refs. [68, 29].131313It is
worth noting that the resulting inflaton potential is, nonetheless, very
different to the $O(3)$-symmetric case. It approaches flatness at large field
values only along a single radial direction, or in the Higgs plane, but
quartically diverges in a general direction. To study this scenario, we will
conservatively assume the inflaton field is initialised sufficiently close to
the hyper-valley, with negligible velocities in the orthogonal direction, to
ensure the trajectory stabilises to the attractor before observationally
relevant times. These initial data are more finely-tuned than the general case
due to mass suppression of the orthogonal directions by $\lambda_{iS}\ll 1$
and may not be compulsory. Within these limiting hyper-valleys, and preserving
the hidden sector structure, the mass associated to at least one orthogonal
direction in the modulus field-space can be very light compared to the Hubble
scale, and the corresponding modulus field develops non-negligible
isocurvature fluctuations during inflation. In general, these will be erased
during reheating if PQ symmetry is restored (and since $\lambda_{iS}\ll
1$).141414If non-restored, inflationary fluctuations in the axion field can
lead to a significant isocurvature fraction (see Sec. IV). A worst-case
estimate for the preserved isocurvature fraction due only to modulus
fluctuations is $\beta_{\text{iso}}\sim 10^{-5}$ (inferred from Ref. [29]).

###  III.4 Inflation model at large non-minimal coupling

The large $\xi_{k}$ inflation model is obtained in the limit that $\xi_{k}\gg
1$ whenever a hyper-valley has a non-trivial component in the $\varphi_{k}$
direction. We find that this regime, like Higgs inflation [1] and the
Starobinsky model [27], generally reproduces the potential (23), which remains
among the best fits to CMB data [31], despite a multi-field scalar sector.

####  III.4.1 Analytical approximation

We have classified the orientations and existence conditions for the hyper-
valleys using spherical coordinates. However, to analyse the inflationary
dynamics in the large $\xi_{k}$ regime, it is most convenient to modify the
spherical coordinates (15) to an ellipsoidal form (with
$\xi_{1},\xi_{2},\xi_{S}>0$ and $c_{\psi},s_{\psi}\equiv\cos\psi,\sin\psi$):

$\begin{split}\rho_{1}&=\frac{M_{P}}{\sqrt{\xi_{1}}}\
e^{\frac{\chi_{r}}{M_{P}\sqrt{6}}}c_{\phi^{\prime}}s_{\theta^{\prime}}\simeq\frac{M_{P}}{\sqrt{\xi_{1}}}\
\Omega\ c_{\phi^{\prime}}s_{\theta^{\prime}},\\\
\rho_{2}&=\frac{M_{P}}{\sqrt{\xi_{2}}}\
e^{\frac{\chi_{r}}{M_{P}\sqrt{6}}}s_{\phi^{\prime}}s_{\theta^{\prime}}\simeq\frac{M_{P}}{\sqrt{\xi_{2}}}\
\Omega\ s_{\phi^{\prime}}s_{\theta^{\prime}},\\\
\sigma&=\frac{M_{P}}{\sqrt{\xi_{S}}}\
e^{\frac{\chi_{r}}{M_{P}\sqrt{6}}}c_{\theta^{\prime}}\simeq\frac{M_{P}}{\sqrt{\xi_{S}}}\
\Omega\ c_{\theta^{\prime}};\end{split}$ (19)

where $\chi_{r}\equiv\sqrt{\frac{3}{2}}M_{P}\log(\Omega^{2}-1)$ and $\Omega>0$
parametrises large radial displacement:

$\Omega^{2}-1\simeq\frac{\xi_{1}}{M_{P}^{2}}\rho_{1}^{2}+\frac{\xi_{2}}{M_{P}^{2}}\rho_{2}^{2}+\frac{\xi_{S}}{M_{P}^{2}}\sigma^{2}.$
(20)

This is because, up to $\mathcal{O}(\xi^{-1}_{k})$ corrections, the kinetic
mixing vanishes, $\chi_{r}$ is canonically normalised and the background
angular fields are static (see Appendix A). In this parametrisation, the
large-field Einstein frame VISH$\nu$ potential (13) becomes:

$V^{\mathcal{E}}\simeq\Lambda[\phi^{\prime},\theta^{\prime}]\left[1-e^{-\frac{2}{\sqrt{6}}\frac{\chi_{r}}{M_{P}}}\right]^{2}\
;$ (21)

where, to leading order in the inter-sector couplings, the energy density:

$\begin{split}&\Lambda[\phi^{\prime},\theta^{\prime}]\simeq\\\ &\
\frac{M_{P}^{4}}{4}\left(c^{4}_{\theta^{\prime}}\frac{\lambda_{S}}{\xi_{S}^{2}}+s^{4}_{\theta^{\prime}}\left[\frac{\lambda_{1}}{\xi_{1}^{2}}c_{\phi^{\prime}}^{4}+\frac{2\lambda_{34}}{\xi_{1}\xi_{2}}c_{\phi^{\prime}}^{2}s_{\phi^{\prime}}^{2}+\frac{\lambda_{2}}{\xi_{2}^{2}}s_{\phi^{\prime}}^{4}\right]\right),\end{split}$
(22)

is minimised along a hyper-valley. While a reparameterisation of the field-
space does not alter the topography of the potential, so that the existence
conditions in Table 2 as well as $\Lambda_{\text{min}}$ are invariant, the
hyper-valley co-ordinates will transform, as
$(\phi^{\prime},\theta^{\prime})\neq(\phi,\theta)$ in general.

Recall that during the early stages of inflation, the trajectory rapidly
settles into the hyper-valley before observationally relevant times and the
corresponding background angular field values induce an effectively single-
field potential for the inflaton field $\chi_{r}$:

$V^{\mathcal{E}}\simeq\frac{M_{P}^{4}}{8}\frac{\lambda_{\text{eff}}}{\xi^{2}_{\text{eff}}}\left[1-e^{-\frac{2}{\sqrt{6}}\frac{\chi_{r}}{M_{P}}}\right]^{2},$
(23)

which describes the slow-roll inflationary dynamics along the adiabatic
direction. In $\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{i}S$-Inflation this is
a general outcome and we respectively compute (defining
$L\equiv\lambda_{1}\lambda_{2}-\lambda_{34}^{2}$):

$\begin{split}\frac{\lambda_{\text{eff}}}{\xi^{2}_{\text{eff}}}\simeq\
&\frac{\lambda_{S}L}{\lambda_{S}(\lambda_{2}\xi_{1}^{2}-2\lambda_{34}\xi_{1}\xi_{2}+\lambda_{1}\xi_{2}^{2})+\xi_{S}^{2}L}\\\
&\text{and}\quad\frac{\lambda_{S}\lambda_{i}}{\lambda_{S}\xi_{i}^{2}+\lambda_{i}\xi_{S}^{2}}.\end{split}$
(24)

As explained in Sec. III.3, for $S$-Inflation ($\xi_{1},\xi_{2}\simeq 0$),
$\Phi_{1}\Phi_{2}$-Inflation and $\Phi_{i}$-Inflation ($\xi_{S}\simeq 0$) the
trajectory stabilises to the hyper-valley for more specific initial conditions
due to sub-Hubble masses of the orthogonal directions. Respectively, we find
that:

$\frac{\lambda_{\text{eff}}}{\xi^{2}_{\text{eff}}}\simeq\frac{\lambda_{S}}{\xi_{S}^{2}},\
\
\frac{L}{\lambda_{2}\xi_{1}^{2}-2\lambda_{34}\xi_{1}\xi_{2}+\lambda_{1}\xi_{2}^{2}},\
\ \text{and}\ \ \frac{\lambda_{i}}{\xi_{i}^{2}}.$ (25)

This information is summarised in Table 3. Observe that
$\lambda_{\text{eff}}/\xi^{2}_{\text{eff}}>0$ ensures positivity of the
inflationary energy density in each scenario.

Table 3:  We summarise some results from Sec. III.3, Sec. III.4 and Sec. III.5. The required parameter space is for the existence of a hyper-valley permitting each inflation scenario. Recall that $\kappa_{12}=\lambda_{34}\xi_{1}-\lambda_{1}\xi_{2}$, $\kappa_{21}=\lambda_{34}\xi_{2}-\lambda_{2}\xi_{1}$ and $L=\lambda_{1}\lambda_{2}-\lambda_{34}^{2}$ (which is positive). The condition $\lambda_{2}\xi_{1}^{2}-2\lambda_{34}\xi_{1}\xi_{2}+\lambda_{1}\xi_{2}^{2}>0$ ensures $V\sim M_{P}^{4}\lambda_{\text{eff}}/8\xi^{2}_{\text{eff}}>0$ for $\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{1}\Phi_{2}$-Inflation. Statements about initial conditions concern the neutral modulus field-space: we make more conservative assumptions in the final four rows due to the presence of at least one light direction, these assumptions are absent in the first three scenarios which we therefore designate as “generic”. (Although we still assume, for example, a large displacement of the fields from the minimum of the potential at the onset of the inflation phase.) We use a $\checkmark$ in the final column to indicate when the small $\xi_{k}$ regime is viable and preserves naturalness requirements (we forbid small Higgs-sector parameters). Inflation scenario | Hyper-valley conditions | $\lambda_{\text{eff}}/\xi^{2}_{\text{eff}}$ | Initial conditions | Small $\xi_{k}$ model
---|---|---|---|---
$\Phi_{1}\Phi_{2}S$-Inflation | $\kappa_{12},\kappa_{21}<0$ | $\dfrac{\lambda_{S}L}{\lambda_{S}(\lambda_{2}\xi_{1}^{2}-2\lambda_{34}\xi_{1}\xi_{2}+\lambda_{1}\xi_{2}^{2})+L\xi_{S}^{2}}$ | Generic | ✓
$\Phi_{1}S$-Inflation | $\kappa_{12}>0$ | $\dfrac{\lambda_{S}\lambda_{1}}{\lambda_{S}\xi_{1}^{2}+\lambda_{1}\xi_{S}^{2}}$ | Generic | ✓
$\Phi_{2}S$-Inflation | $\kappa_{21}>0$ | $\dfrac{\lambda_{S}\lambda_{2}}{\lambda_{S}\xi_{2}^{2}+\lambda_{2}\xi_{S}^{2}}$ | Generic | ✓
$S$-Inflation | $\xi_{i}\simeq 0,\ \lambda_{iS}>0$ | $\dfrac{\lambda_{S}}{\xi_{S}^{2}}$ | $\rho_{i},\dot{\rho}_{i}\sim 0$ | ✓
$\Phi_{1}\Phi_{2}$-Inflation | $\xi_{S}\simeq 0,\ \lambda_{iS}>0$ | $\dfrac{L}{\lambda_{2}\xi_{1}^{2}-2\lambda_{34}\xi_{1}\xi_{2}+\lambda_{1}\xi_{2}^{2}}$ | $\sigma,\dot{\sigma}\sim 0$ | Unnatural
$\Phi_{1}$-Inflation | $\xi_{S}\simeq 0,\ \lambda_{iS},\kappa_{12}>0$ | $\dfrac{\lambda_{1}}{\xi_{1}^{2}}$ | $\sigma,\dot{\sigma}\sim 0$ | Unnatural
$\Phi_{2}$-Inflation | $\xi_{S}\simeq 0,\ \lambda_{iS},\kappa_{21}>0$ | $\dfrac{\lambda_{2}}{\xi_{2}^{2}}$ | $\sigma,\dot{\sigma}\sim 0$ | Unnatural

####  III.4.2 Fit to CMB data

During the inflation, the inflaton field $\chi_{r}$ develops quantum
fluctuations which source almost scale-invariant spectra of primordial scalar
and tensor perturbations. Respectively, these give rise to large-scale
structure formation and relic gravitational waves which, through observations
of the CMB, constitute the observational basis by which the inflation model is
constrained. One expresses the primordial spectra as a power law about a pivot
scale $k_{*}$. For the case of the scalar power spectrum, we define:

$\Delta^{2}_{s}(k)=A_{s}(k_{*})\left(k/k_{*}\right)^{-1+n_{s}(k_{*})+\frac{1}{2}\alpha_{s}(k_{*})\log(k/k_{*})},$
(26)

where $A_{s}$ is the amplitude, $n_{s}$ is the scalar spectral index and
$\alpha_{s}$ describes first-order running in the spectral index. Analogous
definitions follow for the tensor power spectrum, and we define the tensor-to-
scalar ratio: $r\equiv A_{t}/A_{s}$.

The Planck 2018 release [31] of CMB temperature and polarisation data
confirmed a small deviation from scale invariance in the scalar power
spectrum, with the scalar spectral index measured as:

$\begin{split}n_{s}=\ &0.9649\pm 0.0042\\\ &\text{(68 \% CL, Planck
TT,TE,EE+lowE+lensing)}.\end{split}$ (27)

If running in the scalar spectral index is also considered, it is constrained
to be [31]:

$\begin{split}\alpha_{s}\equiv&\frac{\mathrm{d}n_{s}}{\mathrm{d}\log k}=\
-0.0045\pm 0.0067\\\ &\text{(68 \% CL, Planck
TT,TE,EE+lowE+lensing)}.\end{split}$ (28)

The upper bound on the tensor-to-scalar ratio has recently been made more
stringent by BICEP/Keck Array 2018 (BK18) data [69], given to 95% confidence
level as:

$r<0.036.$ (29)

These constraints (and $A_{s}$ below) are evaluated at the CMB pivot scale
$k_{*}=0.05$ Mpc-1.

Our predictions for $r,\ n_{s}$ and $\alpha_{s}$ may be computed directly from
the Einstein frame model as follows. In the slow-roll approximation, we have
that

$r\simeq 16\tilde{\epsilon},\quad n_{s}\simeq
1-6\tilde{\epsilon}+2\eta,\quad\alpha_{s}\simeq-24\tilde{\epsilon}^{2}+6\tilde{\epsilon}\eta-2\zeta,$
(30)

where the first151515Let us distinguish $\epsilon$, the $\nu$DFSZ model
parameter, from $\tilde{\epsilon}$, the first slow-roll parameter, using a
tilde., second and third slow-roll parameters are

$\tilde{\epsilon}\equiv\frac{M_{P}^{2}}{2}\left(\frac{V^{\prime}}{V}\right)^{2},\quad\eta\equiv
M_{P}^{2}\frac{V^{\prime\prime}}{V},\quad\zeta\equiv
M_{P}^{4}\frac{V^{\prime}V^{\prime\prime\prime}}{V^{2}},$ (31)

Here $V\equiv V^{\mathcal{E}}$ is the inflaton potential (23) and ′ denotes a
derivative with respect to $\chi_{r}$. Hence, defining
$x\equiv\exp[2\chi_{r}/(\sqrt{6}M_{P})]$, we can re-express (30) as [21]:

$r\simeq\frac{64}{3(x-1)^{2}},\ n_{s}\simeq 1-\frac{8(x+1)}{3(x-1)^{2}},\
\alpha_{s}\simeq\frac{-32x(x+3)}{9(x-1)^{4}}.$ (32)

Let us define $N_{*}$ as the interval in e-folds between the end of the
inflation phase and the earlier horizon exit of the CMB pivot scale $k_{*}$:

$N_{*}\simeq\frac{1}{M_{P}}\int^{\chi_{*}}_{\chi_{f}}\frac{\mathrm{d}\chi}{\sqrt{2\tilde{\epsilon}}}\simeq\frac{3}{4}\left[x_{*}-x_{f}-\frac{2(\chi_{*}-\chi_{f})}{\sqrt{6}M_{P}}\right],$
(33)

where $\chi_{f}$ corresponds to the field value at the end of the inflation
phase, where $\tilde{\epsilon}(\chi_{f})\simeq 1$ and the slow-roll
approximation breaks down. Then, the central value of $(n_{s})_{*}$ is
obtained for $N_{*}\simeq 55$ e-folds of inflation, corresponding to
$\chi_{*}\simeq 5.4M_{P}$. This means that (at least one) of the modulus
fields takes a value $\sim\mathcal{O}(10)M_{P}/\sqrt{\xi}$ and we find that

$(n_{s})_{*}\simeq 0.965,\ r_{*}\simeq 0.00351,\
(\alpha_{s})_{*}\simeq-0.000623,$ (34)

which are all in perfect agreement with available data. We plot the dependence
of our $r$ and $n_{s}$ predictions on the e-folds of inflation in Figure 1,
which show a range of acceptable values in the large $\xi_{k}$ regime. In
fact, at leading order, we can express (32) in terms of $N_{*}$:

$r_{*}\simeq\frac{12}{N_{*}^{2}}\quad(n_{s})_{*}\simeq 1-\frac{2}{N_{*}},$
(35)

if running in the scalar spectral index is ignored.

Figure 1:  Plots of the tensor to scalar ratio $r_{*}$ and spectral index
$(n_{s})_{*}$ at horizon-exit of the CMB pivot scale, exhibiting dependence on
e-folds of inflation ($N_{*}$) in the large $\xi_{k}$ limit. The green regions
correspond to those favoured by CMB data, leading to an acceptable range for
$N_{*}$ which is ultimately determined by the reheating dynamics.

Finally, we must also match the measured scalar power spectrum amplitude at
horizon exit, $(A_{s})_{*}\sim 2.1\times 10^{-9}$ [70]. Since in the slow-roll
regime this is

$(A_{s})_{*}\simeq\frac{1}{24\pi^{2}\epsilon_{*}}\frac{V_{*}}{M_{P}^{4}}\simeq\frac{N_{*}}{72\pi^{2}}\frac{\lambda_{\text{eff}}}{\xi_{\text{eff}}^{2}},$
(36)

we obtain a relation between the effective quartic and non-minimal couplings
(for acceptable values of $N_{*}$):

$\frac{\lambda_{\text{eff}}}{2\xi^{2}_{\text{eff}}}\sim 4.4\times 10^{-10},$
(37)

which serves finally to constrain the parameters of the inflaton potential —
compare (32), which depended only on $\chi_{*}$ (or $N_{*}$). In particular,
(37) determines the height of the inflation plateau and, hence, the Hubble
scale of the expansion, $H_{\text{inf}}=V/3M_{P}^{2}\sim\text{few}\times
10^{13}$ GeV. The large relative size of the effective couplings implied by
(37), poses a _prima facie_ challenge to the naturalness of the model, which
we address in Sec. V.

###  III.5 Inflation model at small non-minimal coupling

In Sec. III.4, we analysed the case where all modulus fields which take a
large value during the inflation phase have a large non-minimal coupling to
gravity, so that $\max_{k}\\{\xi_{k}\\}\gg\mathcal{O}(1)$. In addition to this
regime, we now show that a range of values $\max_{k}\\{\xi_{I}\\}\lesssim
O(1)$ are also viable. As explained in Sec. V, this regime avoids possible
issues associated to a low unitarity cutoff and can only be naturally realised
for inflation scenarios where $\xi_{S}\sigma^{2}>M^{2}_{P}$. This condition is
satisfied in $\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{i}S$-Inflation, which
are compatible with a generic parameter space and initial data, as well as
$S$-Inflation, which can result for $\xi_{1},\xi_{2}\simeq 0$.

While, outside of the large $\xi_{k}$ regime, the kinetic mixing of the
modulus fields is non-vanishing for non-trivial hyper-valley orientations,
this is not the case for hyper-valleys which are aligned with the field axis.
To illustrate this, let us consider the relevant case of $S$-Inflation
($\overline{\rho_{i}}=0$), for which the induced field metric has the non-zero
components, where $\Omega^{2}=1+\xi_{S}\sigma^{2}/M_{P}^{2}$:

$\overline{\mathcal{G}}_{11}=\overline{\mathcal{G}}_{22}=\frac{1}{\Omega^{2}},\quad\overline{\mathcal{G}}_{SS}=\frac{1}{\Omega^{4}}\left[1+\xi_{S}(1+6\xi_{S})\frac{\sigma^{2}}{M_{P}^{2}}\right].$
(38)

In fact, since we require $\xi_{i}/\xi_{S}<\lambda_{iS}/\lambda_{S}\lesssim
10^{-8}$ for the $\sigma$ axis to form a valley, $\Phi_{1}\Phi_{2}S$-Inflation
and $\Phi_{i}S$-Inflation essentially degenerate to $S$-Inflation for this
parameter space. The field metric components for these scenarios are (38) up
to $\mathcal{O}(\xi_{i}/\xi_{S})$ corrections and the trajectories are,
likewise, close to the $\sigma$ axis. Hence, (38) and the inflaton potential:

$V(\sigma)\simeq\frac{1}{8}\frac{\lambda_{S}\sigma^{4}}{(1+\xi_{S}\sigma^{2}/M_{P}^{2})^{2}},$
(39)

describe the effectively single field dynamics appropriate to each hyper-
valley, up to small $\mathcal{O}(\xi_{i}/\xi_{S})$ and
$\mathcal{O}(\lambda_{iS})$ corrections. In all four cases, we define the
canonically normalised scalar inflaton field $\chi$:

$\frac{\mathrm{d}\chi}{\mathrm{d}\sigma}\simeq\sqrt{\mathcal{G}_{SS}}\simeq\frac{\sqrt{1+\xi_{S}(1+6\xi_{S})(\sigma/M_{P})^{2}}}{{1+\xi_{S}(\sigma/M_{P})^{2}}},$
(40)

where the equation is exact for the special case of $S$-Inflation and can be
solved analytically [71]. The first and second slow-roll parameters (31) then
become [72]:

$\tilde{\epsilon}=\frac{M_{P}^{2}}{2}\left(\frac{V^{\prime}}{V\chi^{\prime}}\right)^{2},\
\eta=M_{P}^{2}\left(\frac{V^{\prime\prime}}{V[\chi^{\prime}]^{2}}-\frac{V^{\prime}\chi^{\prime\prime}}{V[\chi^{\prime}]^{3}}\right),$
(41)

where ′ denotes a derivative with respect to $\sigma$. We summarise the
resulting inflationary predictions in Figure 2, where we have used $A_{s}$ to
fix the relation between $\lambda_{S}$ and $\xi_{S}$, exhibiting the
$\xi_{S}$-dependence of $r$ and $n_{s}$ (ignoring running in the latter),
which promptly asymptote to values discussed above in the large $\xi_{k}$
regime. Observe that values of $\xi_{S}>\text{few}\times 10^{-2}$ abide by
confidence limits on $n_{s}$ and the exclusion limit on $r$ for a range of
standard values for $N_{*}$.

We now consider the general case, for which the $\sigma$ axis now forms a
ridge and there are three possible inflation scenarios:
$\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{i}S$-Inflation. Although in the
$\\{\rho_{1},\rho_{2},\sigma\\}$ field-basis there is non-trivial kinetic
mixing, it is possible to make a gauge transformation of the field-space
coordinates which diagonalises the kinetic terms during the inflation phase,
once the trajectory settles into a hyper-valley. For concreteness, consider
$\Phi_{1}S$-Inflation ($\kappa_{12}>0$) and the limiting case that
$\xi_{1}=\xi_{S}$. A transformation which accomplishes diagonal kinetic terms
is merely a rotation of the field basis about the $\rho_{2}$ axis by the non-
zero background angle which defines the hyper-valley orientation,
$\overline{\theta}$:

$\rho^{\prime}_{1}=\cos\overline{\theta}\ \rho_{1}-\sin\overline{\theta}\
\sigma,\ \ \rho^{\prime}_{2}=\rho_{2},\ \sigma^{\prime}=\sin\overline{\theta}\
\rho_{1}+\cos\overline{\theta}\ \sigma.$ (42)

The outcome of this transformation is that the $\sigma^{\prime}$ direction is
aligned with the hyper-valley (for which $\rho_{1}=\tan\overline{\theta}\
\sigma$). We obtain analogous transformations for the case of
$\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{2}S$-Inflation. The reason we
consider the limiting case $\xi_{i}=\xi_{S}$ is that this suppresses to zero
any residual kinetic mixing of the primed fields after the transformation
(42). An analysis of the limiting case $\xi_{i}=\xi_{S}$ is, of course, the
leading-order analysis of the more general situation where the sizes of non-
minimal couplings are comparable ($\xi_{1}\sim\xi_{S}$). Eqns. (38), (39) and
(40) are reproduced in the rotated field-basis for the substitution
$\sigma\rightarrow\sigma^{\prime}$ and with:

$\lambda_{S}\rightarrow\lambda_{\text{eff}}=\lambda_{S}\left[1-\mathcal{O}\left(\frac{\lambda_{S}}{\lambda_{1}}\right)\right]\left[1+\mathcal{O}\left(\frac{|\xi_{S}-\xi_{1}|}{\xi_{S}}\right)\right],$
(43)

provided corrections due to relative sizes of the $\xi_{k}$ are kept small
(setting inter-sector couplings to zero). As a consequence of the assumed
degeneracy in $\xi_{k}$, we plot the dependence of the inflationary
observables on a single non-minimal coupling, $\xi_{S}$, and we re-obtain the
plots in Figure 2 (assuming
$\lambda_{1},\lambda_{2},\lambda_{34}\sim\mathcal{O}(1)$ for the plot of
required $\lambda_{S}$). For the case of $\Phi_{2}S$-Inflation and
$\Phi_{1}\Phi_{2}S$-Inflation, we also obtain expressions for
$\lambda_{\text{eff}}$ which reproduce (24) and (25), and the resulting plots
are degenerate with those in 2, where $\lambda_{S}$ is responsible for the
suppression of the scalar amplitude.

Finally, we remark that the small values of $\lambda_{S}\lesssim 10^{-8}$
required for successful inflation in a regime of small $\xi_{k}$ results in a
suppression of the Higgs component in the inflaton fields for
$\Phi_{1}\Phi_{2}S$-Inflation and $\Phi_{i}S$-Inflation (which is to say, the
orientation of the hyper-valley aligns more closely with the $\sigma$-axis).
This is in contrast to the large non-minimal coupling regime, where the
magnitude of the Higgs component can be up to a comparable size to the $S$
component. This is because, in $\Phi_{i}S$-Inflation,
$\cos^{2}\overline{\theta}\sim 1-\mathcal{O}(\lambda_{S}/\lambda_{i})$, while
in $\Phi_{1}\Phi_{2}S$-Inflation, $\cos^{2}\overline{\theta}\sim
1-\mathcal{O}(\lambda_{S}(\kappa_{12}+\kappa_{21})/\xi_{S}L)$, using the
results of Sec. III.2. So, it is also sensible to use the approximation
$\overline{\rho}_{i}\simeq 0$, in these scenarios, if
$\mathcal{O}(\lambda_{S}/\lambda_{i})$ corrections can be safely ignored.

Beyond the two representative regimes for the $\xi_{k}$ (hierarchical and
degenerate), the analysis is more involved due to a priori large kinetic
mixing but no important differences are anticipated: there will be an
effective non-minimal coupling which, when large, reproduces in the findings
of Sec. III.4, and, when small, remains perfectly compatible with confidence
limits on the $r$-$n_{s}$ plane, up to lower bound $\sim\mathcal{O}(0.01)$.

Figure 2:  Plots exhibiting the $\xi_{S}$-dependence (common horizontal axis)
of the tensor to scalar ratio $r_{*}$ and spectral index $(n_{s})_{*}$ at the
horizon exit of the CMB pivot scale in the parameter regimes
$\xi_{i}\simeq\xi_{S}$ and $\xi_{i}\ll\xi_{S}$, as well as the quartic-
coupling $\lambda_{S}$ at the inflation scale and the corresponding
$H_{\text{inf}}$. The solid contours correspond to inflation scenarios with
$N_{*}=60$, while $N_{*}=50$ is represented by the dashed contours and delimit
the favoured values for $N_{*}$ [31]. The green regions correspond to (27) and
(29), and are favoured by CMB data, placing a lower limit on $\xi_{S}$.

##  IV Dark matter

The solution offered by VISH$\nu$ to the origin and nature of the observed
cold dark matter abundance depends on the existence of a range of $f_{A}$ for
which the variant DFSZ axion, which arises as part of a solution to the strong
$CP$ problem, also becomes a viable dark matter candidate. Hence, the
predicted axion abundance becomes sensitive to the particular cosmological
history and the possibility that the PQ symmetry was restored during the
inflation or reheating phases.

###  IV.1 Dependence on the reheating phase

Importantly, in the inflation scenarios we have analysed, the PQ symmetry is
generally _broken_ during the inflation phase by the large instantaneous
values of the modulus fields $\varphi^{I}$ driving the expansion, since both
$S$ and $\Phi_{1,2}$ carry PQ charges, corresponding to a large displacement
from the minimum of the VISH$\nu$ potential161616 The exception to the general
situation that PQ is broken obtains when only _one_ of $\rho_{1}$ and
$\rho_{2}$ takes large values, while $\sigma$ does not, since in those cases a
linear combination of weak hypercharge and the original PQ charge remains
unbroken, which is a redefined PQ charge. In each of those cases the
associated redefined PQ symmetry can be restored during inflation because
$f_{A}$ is less than $T_{\text{GH}}=H_{\text{inf}}/2\pi$, the inflationary
Gibbons-Hawking temperature [73, 74], realising a post-inflationary axion.
(1). This can gives rise to an effective value for the axion decay constant
[75] ($f_{A,\text{inf}}$) — either $\sigma_{*}$ [72] or induced by large Higgs
vevs [51] depending on the inflaton field — and establishes the axion field
during the inflation phase. Unless the axion field develops a large
inflationary mass (see Ref. [51]), it will develop unsuppressed
fluctuations.171717Note that, because we can neglect the tri-linear term in
the inflation, the Higgs pseudoscalar is also massless and develops
fluctuations. As in Ref. [51], we assume that these do not contribute to
observable isocurvature perturbations. If these axion fluctuations are not
erased by a restoration of the PQ symmetry after inflation, these give rise to
non-negligible (CDM) isocurvature fluctuations in the CMB temperature power
spectrum which are severely constrained [31], imposing a bound on the
inflationary Hubble scale which is prima facie incompatible with the inflation
model analysed in Sec. III.

It is possible for $U(1)_{PQ}$ to be non-thermally restored during the early
stages of reheating in inflation scenarios where the scalar partner of the
axion is embedded into the inflaton, which is the general case. Once inflation
comes to an end (i.e. when $\tilde{\epsilon}(\chi_{f})\simeq 1$), the identity
of the inflaton field and the effectively single-field background field
evolution are generically preserved by the non-minimal gravitational couplings
[30], and the inflaton condensate begins to oscillate about the minimum of the
potential with a large initial amplitude. Non-perturbative effects will be
responsible for the resonant amplification of scalar field fluctuations in a
phase of pre-heating [76, 77], and the efficiency of this effect grows with
non-minimal coupling size [78]. In Ref. [79], it is shown that fluctuations in
the axion field ($\sim\vartheta_{S}$) will grow to effectively restore the
$U(1)_{\text{PQ}}$ symmetry for $f_{A}\lesssim 10^{16}$ GeV, and this result
is replicated by more recent simulations in the context of the SMASH model
[21, 80] (see also [81, 82] for related discussions), for which the inflaton
has a small Higgs component. This finding is based on three assumptions. The
first is that dimensionful parameters in the potential can be ignored (this is
what applies the upper-bound to $f_{A}$), the second is that non-minimal
coupling sizes are at most $\sim 1$ and the third is that the Higgs fields do
not affect the growth of the $S$ fluctuations [21]. The first and second
conditions can be replicated in VISH$\nu$ models, as the tri-linear coupling
and Higgs mass parameters can be ignored, while $m^{2}_{SS}$ induces the same
bound, and Sec. III.5 outlines viable predictions for $\xi_{k}\sim 1$.
VISH$\nu$ can also naturally realise the third condition by the effective
decoupling of the inflaton field, which is mostly aligned with $\sigma$ for
small $\xi_{k}$, from the Higgs fields. Hence, we can expect that $U(1)_{PQ}$
is non-thermally restored, in general, for the small $\xi_{k}$ inflation
model. However, this conclusion should be supported by dedicated simulations,
such as Ref. [80], which we do not perform here.181818Ref. [80] does not scan
the hidden-sector parameter space, but uses inputs which are standard for the
SMASH model, which requires significantly larger inter-sector couplings to
successfully preserve the stability of the Higgs potential (see Sec. V). We
speculate that a regime of larger $\xi_{k}$ will increase the efficiency of
the non-thermal restoration, rather than hampering it, but (to our knowledge)
this has not been examined before. It has also been suggested that Higgs-
sector oscillations following, for example, $\Phi_{1}\Phi_{2}$-Inflation and
$\Phi_{i}$-Inflation, can induce fluctuations in $S$ that restore the symmetry
[51], and this was demonstrated in toy model simulations by Ref. [80]. We do
not expect that this conclusion holds in VISH$\nu$, since the hidden-sector
parameter space will suppress the effect in general.

An additional possibility is that $U(1)_{PQ}$ is thermally restored during the
later stages of reheating, if the maximum temperature after inflation (which
may be larger than the reheating temperature $T_{\text{reh}}$ [83]) exceeds
the critical temperature for the PQ phase transition (which may be lower than
$f_{A}$ [21]). Non-minimally coupled models can predict a high reheating
temperature when large amplitude oscillations of the background inflaton
condensate lead to efficient SM gauge boson production. In VISH$\nu$, such a
phase of reheating will occur through the Higgs fields, which either form a
sizeable inflaton component ($\xi_{k}\gtrsim 1$), are effectively coupled to
the inflaton ($\xi_{k}\ll 1$) or effectively decoupled ($\xi_{1},\xi_{2}\simeq
0$). In the large $\xi_{k}$ regime, Higgs inflation has been shown to result
in $T_{\text{reh}}\sim 10^{13}$ GeV [84, 71, 1], while reheating through a
tree-level coupling of the inflaton to the Higgs can vary in
efficiency.191919The variation is due primarily to the size of the portal
coupling(s) compared with the effective inflaton self-coupling. However,
comparable reheating temperatures are achievable [85]. A comprehensive study
of the small $\xi_{k}$ regime has been completed for the SMASH model [21],
where reheating is less efficient ($T_{\text{reh}}\sim 10^{10}$ GeV) due to
the largeness of the Higgs portal coupling compared to the effective inflaton
self-coupling. As this is the opposite regime to VISH$\nu$, our results will
differ and we can expect greater efficiency (like SMASH, the production of
$\nu_{R}$ will not affect this result due to suppressed decays). Each of these
temperatures are sufficient to realise a viable post-inflationary scenario,
even if non-thermal restoration of the symmetry fails, and should result in
axion thermalisation with the SM sector [86], avoiding restrictive dark
radiation constraints discussed in Ref. [21].202020A high reheating
temperature is also consistent with our realisation of thermal leptogenesis.
Some efforts to achieve viable reheating without a tree-level inflaton
coupling to the Higgs, for example Ref. [87], cannot be successfully realised
in the $S$-Inflation scenario due to light neutral Higgs modulus masses during
the inflation.212121We do not expect efficient reheating for the $S$-Inflation
scenario in the absence of such a mechanism, due to $\lambda_{iS}\ll 1$.

Let us summarise how dark matter predictions come to depend on reheating
outcomes. For a given inflation scenario, and non-minimal coupling regime,
there exists a particular value for the axion decay constant in VISH$\nu$
($f^{res}_{A}$) below which the PQ symmetry is restored during reheating. For
the region of parameter space $f_{A}<f^{res}_{A}$, there is only one value
($f^{*}_{A}$) which entirely reproduces the observed CDM density, provided
$f^{*}_{A}<f^{res}_{A}$. We can expect VISH$\nu$ to realise the final
condition, in general, due to thermal and/or non-thermal processes once
inflation ends.222222The generality of this statement applies to the non-
minimal coupling sizes which, with the exception of limiting cases, lead to
$\Phi_{i}S$-Inflation or $\Phi_{1}\Phi_{2}S$-Inflation, so guaranteeing both a
singlet component in the inflaton (typically responsible for the efficient
growth of fluctuations in the axion field during a phase of preheating) and a
Higgs component (typically responsible for a high reheating temperature).
However, a complete analysis of the VISH$\nu$ reheating phase is beyond the
scope of this paper. In the opposite regime ($f_{A}>f^{res}_{A}$), the PQ
symmetry remains broken and there can be a range of $f_{A}$ values which
respect the CDM constraint, up to a maximal value ($f^{iso}_{A}$) where
isocurvature bounds are violated, provided $f^{iso}_{A}>f^{res}_{A}$. This
requires a mechanism to suppress axion fluctuations, which can be fulfilled by
a generic subset of the inflation scenarios, assuming the symmetry is not
restored. We now consider both scenarios (restored and non-restored) in the
context of VISH$\nu$ predictions for dark matter.

###  IV.2 Restored $U(1)_{PQ}$ scenario

In the scenario where the PQ symmetry is restored and PQ breaking occurs for
the last time after inflation, the predicted axion abundance depends only on a
calculable axion mass [88] which should not overproduce the measured CDM
density and ideally reproduces it. In combination with astrophysical results,
namely stellar cooling fits which constrain axion interactions, we can
determine a representative range, see (46) below, of viable VISH$\nu$ _post-
inflationary_ axion masses.

One contribution to axion-production arises from the “vacuum misalignment”
mechanism [89, 90, 91]. The random selection of initial “misalignment angle”
field-values in many distinct domains populating the universe leads to an
averaged axion CDM contribution [92]:

$\Omega^{\text{VR}}_{A}h^{2}\simeq(3.8\pm 0.6)\times
10^{-3}\left(\frac{f_{A}}{10^{10}\ \text{GeV}}\right)^{1.165}.$ (44)

Naïvely, this gives rise to the totality of dark matter when $m_{A}\sim 28(2)\
\mu$eV [93], or equivalently $f_{A}\sim 2\times 10^{11}$ GeV. However, there
are additional sources of axion density, even besides thermal processes, which
owe to the decay of one and two-dimensional topological defects that are not
inflated away. The spontaneous breakdown of $U(1)_{\text{PQ}}$ results in the
formation of cosmic strings which radiate axions as they evolve [94]. While
this can furnish a second contribution to the CDM abundance, it is afflicted
by large theoretical uncertainties which lead to a range of lower limits for
$m_{A}$ below. Later, for temperatures around the QCD phase transition,
instanton effects not only give the axion a mass, which at zero-temperature is
[95]:

$m_{A}\simeq 5.691(51)\left(\frac{10^{12}\ \text{GeV}}{f_{A}}\right)\
\mu\text{eV},$ (45)

but also subsequently induce the formation of a network of domain walls
bounded by axionic strings [96]. As the VISH$\nu$ model has _unit_ domain wall
number, the network is unstable [97] — thus avoiding a domain wall problem —
and its ensuing decay into axions also generically contributes to the CDM
density [98].

Assembling the three contributions, the _full_ observed CDM density is
explained by the abundance of VISH$\nu$ axions of mass232323The upper value is
favoured by Refs. [99, 88] and the lower value encloses a range favoured in
Ref. [100]. $m_{A}\sim(40$–$500)\,\mu\text{eV}$ [99, 88, 100], corresponding
to values for the axion decay constant $f_{A}\sim(1.1\times
10^{10}$–$1.4\times 10^{11})\,\text{GeV}$. To ensure that CDM is not
overproduced beyond observational limits, we should understand this as the
range of upper (lower) bounds for $f_{A}$ ($m_{A}$), allowing for theoretical
uncertainties. Note that both bounds assume $N_{\text{DW}}=1$, and the lower
bound on the mass of the VISH$\nu$ axion is the same as that for minimal KSVZ
models.

Since interacting axions can accelerate energy transport out of stars,
observed stellar cooling rates and lifetimes serve to constrain axion
couplings to photons, electrons and nucleons. In the top-specific VISH$\nu$
model, the axion-photon coupling is the same as in the DFSZ-I model (see
Appendix C), but may be enhanced in alternative flavour structures [101, 102].
However, the coefficient of the axion-electron coupling ($C_{ae}$) is a factor
of three larger [101], leading to rescaled lower bounds on $f_{A}$ from the
white-dwarf luminosity function [103], which becomes $f_{A}>(2.4\times 10^{9}\
\text{GeV})\sin^{2}\beta$ and the tip of the red-giant branch constraint [104,
105], which becomes $f_{A}>(\sim 3\times 10^{9}\ \text{GeV})\sin^{2}\beta$,
both slightly more stringent in the top-specific VISH$\nu$ model than for
ordinary DFSZ models.

The collection of anomalous stellar cooling hints discussed in Ref. [106] may
also be interpreted by a VISH$\nu$ model. An analysis of variant axion models
in Ref. [101], suggests that the tree-level top-specific structure both
accommodates WD, RGB and HB hints, and abides by the SN 1987A bound [106] for
the $2\sigma$ range of axion masses $m_{A}\in(0.46,29)\,\text{meV}$ compatible
with perturbativity.242424We expect that the $R_{2}$ bounds discussed in Ref.
[107] may further constrain this parameter space. Interestingly, there is an
overlap around $0.5\,\textrm{meV}$ with the mass range, identified above, for
which top-specific VISH$\nu$ axions explain the totality of dark matter. In
comparison, the hints are not well-interpreted by KSVZ models, but may be in
extensions by right-handed neutrinos due to loop-induced contributions to the
axion couplings at the cost of a tension with perturbativity [106, 101].

Taken together, cosmological and astrophysical constraints point to the
following representative mass range for the post-inflationary VISH$\nu$ axion:

$m_{A}\ \in\ (40\ \mu\text{eV},\ \sim 2\ \text{meV}),$ (46)

where, for the upper limit, we take $\tan\beta\gtrsim 8$, required by the
natural explanation for leptogenesis [17, 12], in the red giant bound
mentioned above. Note that the lower bound252525We recall that the lower bound
is naïvely as low as $28(2)\mu$eV if topological defects do not significantly
contribute to axion CDM. is comparable to SMASH [22] (when updated using
recent simulations of the topological defect contributions) while the
generally $\beta$-dependent upper bound is DFSZ-like.

Of the direct detection experiments which probe the $\mu$eV ballpark for the
axion mass in the QCD band, only ADMX has reached the sensitivity of the DFSZ
axion [108, 109, 110], but for a mass range lighter than the VISH$\nu$
prediction. In the years to come, our representative mass window (46) may
eventually be accessible to searches including, in addition to ADMX, MADMAX
[111], CAPP [112], HAYSTAC [113], ALPHA [114], ORGAN [115] and others.
Particularly if the Vissani-like bound on $\tan\beta$ can be somewhat relaxed,
as suggested in Ref. [17], a larger VISH$\nu$ axion mass may also be in the
projected reach of the helioscopes IAXO [116] and IAXO+.

###  IV.3 Non-restored $U(1)_{PQ}$ scenario

We now consider the scenario where the PQ symmetry, broken during inflation,
fails to be restored and discuss how the pre-inflationary VISH$\nu$ (and
$\nu$DFSZ) axion may still abide by CMB isocurvature constraints while
reproducing the measured CDM density.

In order for the non-restored case to be compatible with the high-scale
inflation model we have analysed in Sec. III, the axion fluctuations during
inflation must be suppressed and there are two ways this can be achieved in
the model parameter space, distinguished by the composition of the inflaton
field. The first case, which is more natural, consists in scenarios where
there is a Planckian displacement of $\sigma$ during the inflation (as in
$\Phi_{1}\Phi_{2}S$-Inflation, $\Phi_{i}S$-Inflation and $S$-Inflation). This
induces $f_{A,\text{inf}}\gg f_{A}$ and there is a signficant suppression of
axion isocurvature absent in the usual case [72, 21, 117]. Such a scenario
respects constraints for $f_{A}\lesssim 10^{14}$ GeV [21], compatible with
cosmological and astrophysical bounds. However, the growth of axion
fluctuations during pre-heating should also be properly accounted for and this
tends to restore the PQ symmetry unless $f_{A}$ is considerably larger [80].

The second case to consider is $\Phi_{1}\Phi_{2}$-Inflation (recall that
$\Phi_{i}$-Inflation can satisfy the PQ-restored scenario). The required
suppression of axion fluctuations can occur if the axion obtains a large
inflationary mass which exceeds the Hubble scale. Remarkably, Ref. [51] have
shown that this situation can arise in the DFSZ model through the
$\epsilon$-coupled term, when the inflaton field is supplied only by the Higgs
sector. Then, during the inflation, the axion mass exceeds the Hubble scale to
suppress fluctuations, while the Higgs pseudoscalar is almost massless but
develops harmless fluctuations. This mass relationship is reversed after
inflation, because the Higgs are charged under $U(1)_{\text{PQ}}$, and the
background Higgs field values subside. However, this explanation requires that
inter-sector couplings exceed the tiny values, required to suppress tree-level
corrections to the electro-weak scale, by many orders of magnitude, and so
fails the strict naturalness criteria we adopt in Sec. V. This is not very
serious for our model, as $\Phi_{1}\Phi_{2}$-Inflation only occurs for a small
subset of the parameter space and non-thermal restoration is difficult to
avoid [80].

As axion contributions from the density of axion strings, formed when PQ
symmetry is broken for the last time, are diluted away during the inflation,
the only contribution to the axionic CDM abundance we need to consider is from
the vacuum misalignment mechanism. In particular, there is no longer a domain
wall problem to be overcome, and the pre-inflationary scenario is compatible
with the ($N_{DW}\neq 1$) DFSZ axion of the $\nu$DFSZ. The explanation,
however, differs to the post-inflationary case considered above, since the
initial randomly-selected misalignment angle $\theta_{i}\in[-\pi,\pi]$ is now
the same in our Hubble volume (and not averaged over, as other values are
inflated away). Hence, the resulting CDM axion abundance [93, 118] depends
also on $\theta_{i}$ (and a factor $F$ defined in Ref. [118]):

$\Omega^{\text{VR}}_{A}h^{2}\simeq 0.12\left(\frac{f_{A}}{9\times 10^{11}\
\text{GeV}}\right)^{1.165}F\theta_{i}^{2}.$ (47)

It follows that, for $F\theta_{i}^{2}\sim 1$, the pre-inflationary axion can
account for the totality of the measured CDM density for $f_{A}\sim 10^{12}$
GeV. As discussed in Ref. [12], re-interpreting this as an upper bound
preventing CDM over-production, smaller values of $f_{A}$ are permitted in so
far as they are astrophysically viable — including values which are relevant
to stellar evolution considered above. Note that larger $f_{A}$ are possible
under the assumption of smaller initial values $\theta^{2}_{i}$, provided they
respect $f^{\text{res}}_{A}<f_{A}<f^{\text{iso}}_{A}$.

##  V Naturalness

In this section, we first review our naturalness philosophy and how our model
obviates any new naturalness problems posed by the non-gravitational
interactions of its hidden sector, and remains fully consistent with vacuum
stability. We also address the particular naturalness challenges posed by the
inflation, which arise for general values of non-minimal couplings, $\xi_{k}$.
This is a non-trivial consideration, since some of the inflation scenarios we
have considered could provoke a “unitarity problem” in the large $\xi_{k}$
limit of VISH$\nu$.

###  V.1 Our naturalness philosophy

A famous naturalness criterion, originally due to ’t Hooft [119], is that of
_technical naturalness_. It says that a parameter $\lambda$, _or_ a relation
among model parameters $f(\lambda_{I})$, can take a value262626For parameters,
$\mu_{I}$, of general mass dimension $d$, we define:
$\mu_{I}\equiv\lambda_{I}\Lambda^{d}$, where $\Lambda$ is the fundamental
scale and $\lambda_{I}$ are the dimensionless arguments of a (possibly
trivial) relation $f$. $|\lambda|\ll 1$ provided that $\lambda\rightarrow 0$
results in a new symmetry of the quantum theory. The upshot of such a
postulate is that a small value for $\lambda$ is _radiatively stable_ , that
is, quantum corrections do not drive $|\lambda|$ to orders-of-magnitude larger
values. This is because when $\lambda=0$ is set at tree level, radiative
corrections cannot generate a nonzero value for it, because those corrections
must obey the symmetry of the quantum theory. So, corrections to $|\lambda|\ll
1$ will only ever be _proportional_ to that small symmetry-violating parameter
relation and thus themselves small.

The smallness of $|\lambda|$, which may be required for phenomenological
reasons at a particular scale, thus becomes a consistent feature of the model
across the entire regime of validity, as it is stable under renormalisation
group evolution. We imagine that the model is replaced in the far UV by a more
fundamental one — as it surely must be to include gravity — which ascribes a
dynamical origin to the hierarchy of parameter sizes. Nevertheless, the
particular origin of $|\lambda|\ll 1$ need not concern us in the IR of such a
model, since the enhanced symmetry structure already furnishes _a priori_
justification.

However, this is merely a sufficient condition for naturalness in the sense of
radiative stability. We now introduce an extension of this principle, which we
call _second-order technical naturalness_. We argue that a parameter
$\lambda$, or a relation among model parameters $f(\lambda_{I})$, can take a
natural value $|\lambda|\ll 1$ provided that (i) $|g(\lambda_{I})|\ll 1$
guarantees the radiative stability of $\lambda$ and (ii)
$g(\lambda_{I})\rightarrow 0$ results in a new symmetry of the quantum theory,
where $g$ is another relation. Or, what is the same, $|g(\lambda_{I})|\ll 1$
is technically natural and this is inherited by $\lambda$.

As hidden-sector extensions of the SM, the VISH$\nu$ and $\nu$DFSZ models come
with a set of technically natural, “inter-sector” (or “sector-mixing”)
parameters: $y_{\nu}$, $\lambda_{1S}$, $\lambda_{2S}$ and either $\kappa$ or
$\epsilon$. This is because, in the limit of sector decoupling, in which every
inter-sector coupling is set to zero, there is an enhanced Poincaré
symmetry,272727This is only true in the limit that gravity is neglected (see
footnote 1), because this trivially forbids the possibility of any surviving
sector-mixing gravitational interaction, which must be treated separately. We
note that a variation of this remaining issue already afflicts the SM. which
is manifest at the level of the action [10]:

$S=S_{\text{SM}}+S_{\text{HS}}=\int\mathrm{d}^{4}x\
\mathcal{L}_{\text{SM}}(x)+\int\mathrm{d}^{4}x^{\prime}\
\mathcal{L}_{\text{HS}}(x^{\prime}),$ (48)

where $S_{\text{SM}}$ is the electroweak scale action, $S_{\text{HS}}$ is the
$f_{A}\sim 10^{11}$ GeV hidden sector ($\\{S,\nu_{R}\\}$) action and the
inter-sector Lagrangian has disappeared. Observe that we can now make two
independent Poincaré transformations of the spacetime which leave the sector
actions, $S_{\text{SM}}$ and $S_{\text{HS}}$, separately conserved [10]. So,
there is a custodial symmetry which automatically protects weak inter-sector
interactions. Recall that there is also a custodial $U(1)$ in the limits
$y_{\nu},y_{N},\kappa\rightarrow 0$, which separately protects small,
technically natural values of each parameter [12].

In the plain SM, excluding potentially dangerous gravity effects, there is no
naturalness problem. This is because there is now no other high physical scale
which induces destabilising contributions to the Higgs mass parameter,
$m_{h}$. Although $m_{h}\rightarrow 0$ is not a symmetry of the quantum
theory, it does realise _classical_ scale invariance, resulting in only
harmless logarithmic running of $m_{h}$, with the $\beta$-function vanishing
in the $m_{h}\to 0$ limit. On the other hand, the introduction of a new heavy
fundamental scale to solve the strong $CP$ problem and furnish axionic dark
matter _does_ generically provoke a concern over naturalness, beyond existing
concerns over gravity. In this situation, $m_{h}\rightarrow 0$ no longer
realises classical scale invariance, owing to the presence of the new scale.
The failure of the Higgs mass to be technically natural means some other way
must be found of suppressing the accumulation of large physical corrections
induced by inter-sector couplings of the SM to the new high-scale physics,
which could serve to drive $m_{h}$ unnaturally large values, well above the
electroweak scale ($\delta m^{2}_{h}\gg m_{h}^{2}$).

In VISH$\nu$, and the $\nu$DFSZ model, $m_{ii}^{2}$ is _prima facie_
vulnerable to large $\sim v_{S}^{2}$ corrections from $\Phi_{i}$ couplings to
the heavy sector. However, since all inter-sector couplings take values which
ensure that $\delta m^{2}_{ii}\lesssim m_{ii}^{2}$, the Higgs mass parameters
_are_ radiatively stable. If these custodial parameter values were assumed by
fiat at a particular scale, but took general values elsewhere, this would
ordinarily require fine-tuning. However we have seen that precisely this set
of values is technically natural in the $\nu$DFSZ and VISH$\nu$, owing to a
custodial Poincaré symmetry.

Hence, while the Higgs mass parameters are not technically natural at first-
order, our solution _in the presence_ of high-scale physics, is that they are
technically natural at second-order: inheriting the technical naturalness of
the inter-sector parameters. We may therefore call the electroweak sector
“Poincaré protected” [10]. Note the importance of $S$ being a gauge-singlet,
since additional tree-level couplings of the hidden sector to the SM can
reintroduce large inter-sector couplings at higher loop order. That is, the
decoupling limit must _fully_ decouple the heavy sector, as it does in both
VISH$\nu$ and the $\nu$DFSZ. Note too that provided there is no intrinsic
naturalness-violating hierarchy in the decoupled sector actions, the mass
parameters of each are again subject to a custodial classical scale invariance
that results in harmless logarithmic running about each fundamental scale.

Ref. [12] gives a detailed analysis and demonstration of the naturalness of
the $\nu$DFSZ model under RG evolution, in the sense just described, which we
will not reproduce here for VISH$\nu$ as there are no important differences.
On the other hand, the radiative stability of some fundamental parameters in
theory must be reconsidered in light of parameter choices which permit viable
or predictive inflation; we do this below.

###  V.2 Naturalness of the inflation model

Since our results in Sec. III did not depend on the inter-sector couplings at
leading order, the radiative stability of the electroweak scale is a
consistent feature of the model supplemented by a viable period of inflation,
as laid out in Sec. III.

For each of the seven inflation scenarios, (37) furnishes hierarchical
constraints on the quartic parameters $\lambda_{J}$ of each sector Lagrangian
as compared to the sizes of the non-minimal couplings $\xi_{I}$. One way the
model can become unnatural is if, in some viable region of parameter space,
the $\lambda_{J}$, or some relation between them (e.g.
$L=\lambda_{1}\lambda_{2}-\lambda^{2}_{34}$), must take a very small value
without _a priori_ justification. If this is true, our naturalness
expectations mean that such a scenario is ruled out in VISH$\nu$. Since large
values of the non-minimal couplings, $\max_{I}\\{\xi_{I}\\}\in[10^{3},10^{4}]$
satisfy (37) for naturally-sized $|\lambda_{J}|\in(0.01,1)$, this represents a
regime where there is no tree-level conflict between our inflation predictions
and our expectations for naturalness.

####  V.2.1 Small non-minimal couplings

As we have demonstrated in Sec. III.5 for $\xi_{i}\ll\xi_{S}$ and
$\xi_{i}\simeq\xi_{S}$, smaller values of $\xi_{k}$ additionally abide by
constraints on primordial fluctuations. However, the case where
$\xi_{S}\ll\xi_{i}$ is unnatural: the viability of Higgs-sector
$\Phi_{i}$-Inflation and $\Phi_{1}\Phi_{2}$-Inflation scenarios requires that
$\lambda_{i},L\lesssim 10^{-8}$. Since the limits $\lambda_{i},L\rightarrow 0$
do not correspond separately to enhanced symmetries of the theory, they are
not technically natural limits. As neither can be made technically natural at
second-order, the radiative stability of their small values cannot be
guaranteed in the model. Hence, these three inflation scenarios meet our
naturalness expectations only for $\xi_{i}\gg 1$.

The remaining four possibilities — $S$-Inflation, $\Phi_{i}S$-Inflation
$\Phi_{1}\Phi_{2}S$-Inflation — can all meet our naturalness expectations,
even in a limited regime of small $\xi_{k}$. This is because, in the limit
$y_{N}\rightarrow 0$, small values of $\lambda_{S}$ become natural at second
order, because they inherit the technical naturalness of the inter-sector
couplings in the decoupling limit. Hence, a window of radiative stability is
opened up for small values of $\lambda_{S}$, the range of which depends on the
values of $y_{N}$, which can take small values protected by a custodial $U(1)$
symmetry. So, while small $\lambda_{S}$ meets the criteria for second-order
technical naturalness _a priori_ , we must also note that the right-handed
neutrino Yukawas must still be sufficiently large to permit leptogenesis.

This can be seen through the RGE for $\lambda_{S}$, which to 1-loop reads:

$\frac{\mathrm{d}\lambda_{S}}{\mathrm{d}\log\mu}\simeq\frac{1}{16\pi^{2}}\left(10\lambda_{S}^{2}+2\sum_{i}\left[\lambda_{S}(y^{i}_{N})^{2}-(y^{i}_{N})^{4}\right]\right),$
(49)

and we suppress the contributions $\sim\lambda_{iS}^{2}$ which are at most
$O(10^{-32})$. To ensure the radiative stability of a parameter $X$, namely
that it is not driven to very different values under running from, say, the
electroweak scale to the Planck scale, it suffices that:

$\left|\frac{\mathrm{d}\log_{10}X}{\mathrm{d}\log_{10}\mu}\right|=\left|\frac{1}{X}\frac{\mathrm{d}X}{\mathrm{d}\log\mu}\right|<\mathcal{O}(0.1),$
(50)

since $[\log_{10}(M_{P}/m_{Z})]^{-1}\sim\mathcal{O}(0.1)$. As much, for
example, is guaranteed for technically natural parameters, since the loop
factor is $\mathcal{O}(0.1)$ and corrections are only ever _proportional_ to
the symmetry-violating parameters (a glance at the RGEs for the inter-sector
parameters in Appendix B makes this clear). Hence, an estimate for the right-
handed neutrino Yukawa sizes permitted by the radiative stablity of
$\lambda_{S}>0$ is $\sum_{i}(y_{N}^{i})^{4}<\lambda_{S}$. To discover a range
of naturally small $\lambda_{S}$, we should therefore identify the range of
$y^{i}_{N}$ deemed viable in the natural solutions to the origin of neutrino
masses and baryogenesis found in Ref. [12].

Much like $v_{2}$, the lightest right-handed neutrino mass, $M_{1}$, comes to
be constrained by its role in leptogenesis. It has a Vissani-like upper bound
and a Davidson-Ibarra lower bound, the latter being adapted for the two-Higgs
doublet scenario [12], yielding $10^{5}$ GeV $\lesssim M_{1}\lesssim 10^{7}$
GeV. Our reproduction of the BAU employs hierarchical right-handed neutrino
masses $M_{1}\ll M_{2},M_{3}$, setting aside the quasi-degenerate case [120]
which usually requires some resonant enhancement [121]. However, a Vissani-
like upper bound still applies to the second and third generation right-handed
neutrino masses [17], thus $M_{2},M_{3}~{}\lesssim~{}10^{7}$ GeV. So, since
$y^{2,3}_{N}\sim M_{2,3}/f_{A}\sim 10^{-4}$–$10^{-3}$, we expect that values
$\lambda_{S}\gtrsim 10^{-13}$ are radiatively stable using our estimated
neutrino Yukawa bound. This was confirmed under RG running for standard values
of input parameters. The most stringent naturalness bound on $\lambda_{S}$,
therefore, still permits the full range of viable values for $\xi_{S}$.

####  V.2.2 Stability of the singlet direction

As a direct application of the inherited radiative stability of $\lambda_{S}$
we show that the bounded-from-below condition ($\lambda_{S}>0$) in the
$S$-direction is also preserved under radiative corrections away from the
minimum of the potential, to accommodate small positive $\lambda_{S}$ at large
inflationary field values. This can be seen using the effective Coleman-
Weinberg [122] quartic potential in the $S$ direction, for which the largest
radiative contribution to the effective quartic coupling, denoted
$\hat{\lambda}_{S}$, owes to the largest right-handed neutrino Yukawa,
$y^{\text{max}}_{N}$, at first loop order. Hence, with $k=1/16\pi^{2}$,

$\hat{\lambda}_{S}(\sigma_{*})\sim\lambda_{S}(\mu)-\mathcal{O}(k)\left(y^{\text{max}}_{N}\right)^{4}\log\left(\sigma_{*}/\mu\right),$
(51)

where, as in Ref. [21], we can set $\mu\sim m_{\sigma}$, the mass of the PQ
scalar modulus (a renormalisation point relevant to the potential minimum),
and $\sigma_{*}$ is the large inflationary field-value. Additionally, we use
the fact that the $y_{N}$ evolve slowly, which is guaranteed by our custodial
$U(1)$ symmetry, so that our neutrino mass bounds are not greatly modified
under running. Even with a background field value
$\sigma_{*}\sim\mathcal{O}(10)M_{P}$, $y^{\text{max}}_{N}<10^{-3}$ is
sufficient to keep $\hat{\lambda}_{S}>0$, and the correction smaller in
magnitude than $\lambda_{S}\gtrsim 10^{-11}$ relevant to viable small
$\xi_{k}$ inflation. Unlike Ref. [21], we are therefore able to achieve the
ensuing stability in the $S$-direction, along with a natural explanation of
leptogenesis, without admitting degeneracy in the masses of the right-handed
neutrinos.

####  V.2.3 Stability of the Higgs sector

The extended scalar sector of VISH$\nu$ evades other stability concerns
addressed, for example, in Refs. [20, 21, 22], without compromising the
hidden-sector structure. In particular, having demonstrated the stability of
the S direction due to the second-order technical naturalness of
$\lambda_{S}$, we must also consider the vacuum stability of the Higgs sector.

For current measured values of the Higgs and top masses, it is highly probable
that the electroweak vacuum of the SM Higgs potential is metastable [123].
This can pose a threat to the validity of the Higgs inflation models, since it
is possible that the instability scale, at which the Higgs quartic coupling
runs negative, may not be sufficiently large to accommodate the high-scale
inflation model with Planckian field values.

Let us understand how this concern may be ameliorated by the addition of an
extra scalar state, as in Ref. [21]. The scalar potential of the ($N_{DW}=1$)
minimal KSVZ model [24, 25] can be written as:

$\begin{split}V=\
&\lambda_{H}\left(H^{\dagger}H-v^{2}/2\right)^{2}+\lambda_{S}\left(S^{*}S-v_{S}^{2}/2\right)^{2}\\\
&+2\lambda_{HS}\left(H^{\dagger}H-v^{2}/2\right)\left(S^{*}S-v_{S}^{2}/2\right),\end{split}$
(52)

where $H$ is the Higgs field and $S$ is the gauge singlet complex scalar whose
phase is the axion, which obtain the vevs $v\simeq 246$ GeV and $v_{S}=f_{A}$,
giving rise to the Higgs mass parameter
$M_{H}^{2}=-\lambda_{H}v^{2}-\lambda_{HS}v_{S}^{2}$. Furthermore, we will
assume that $f_{A}\sim 10^{10}$–$10^{11}$ GeV, as in Ref. [21]. Once $|S|$ is
integrated out to realise the SM Higgs potential, one requires [118]:

$m_{H}^{2}=M^{2}_{H}+\lambda_{HS}v_{S}^{2}\simeq-(88\ \text{GeV})^{2},$ (53)

and for tree-level naturalness, we argue that the inter-sector coupling
satisfies $|\lambda_{HS}|~{}\lesssim~{}10^{-18}$ to prevent a tuning.

In order to forbid unstable regions of the potential during the cosmological
evolution, one can require that the bounded-from-below conditions are valid at
high scales, and (52) is positive away from the absolute minimum. The point of
this criterion is to eliminate the possibility that unsuppressed fluctuations
developed during inflation, or thermally-sourced for very high reheating
temperatures [124], will harmfully probe or induce tunnelling into the
unstable regions of the potential, even if the inflation naïvely proceeds
along a “stable” direction to a long-lived vacuum [21]. However, the inter-
sector coupling $\lambda_{HS}$ — which leads to positive contributions to the
$\beta$ function of $\lambda_{H}$ that could help to counteract the
destabilising effect of the top Yukawa — can often be quite small, making the
loop-level stabilisation difficult to achieve (and certainly unnatural). A
remaining avenue to preserve $\lambda_{H}>0$ is so-called threshold
stabilisation [125, 126] (see also Ref. [21]) which can outsize the previous
effect by leveraging the inter-sector tree-level contribution to induce a
matching condition for $\lambda_{H}$, below the instability scale, that keeps
it positive at high-scales. In Ref. [21], it is found that this typically
requires $|\lambda_{HS}|\gtrsim 10^{-7}$ at best, for the smallest values of
$\lambda_{S}$ consistent with viable inflation in the $S$ direction. In
comparison to the tree-level naturalness bound, this is sufficient to spoil
the electroweak scale protection _a priori_ achievable in the model if an SM-
like vacuum stability is accepted.

Let us turn our attention now to the top-specific VISH$\nu$ model. Setting
aside the possibility of threshold stabilisation by the heavy sector, the need
for additional positive contributions from the electroweak sector to the
running of the Higgs quartic couplings $\lambda_{1,2}$ can be met by (and
motivates) the extension of the Higgs sector to include an additional Higgs
doublet. In general, this ensures the $\beta$-functions for $\lambda_{1,2}$
more easily become positive at high scales, due to the radiative accumulation
of additional scalar couplings, whose sizes must nevertheless respect
perturbativity bounds. In particular, Ref. [26] demonstrates that (i) the
absence of Landau poles, (ii) vacuum stability and (iii) the high-scale
validity of the bounded-from-below conditions, can be met for a Poincaré-
protected DFSZ Higgs sector282828Since, as remarked in Sec. II, the decoupled
Higgs sector is essentially the Type II 2HDM model with $U(1)_{PQ}$-limited
charges, Refs. [127, 128, 129, 130, 131] are additionally relevant. up to the
Planck scale. Although with rather restrictive implications for the parameter
space, as explained in Ref. [26], these predictions can still, quite
remarkably, remain compatible with results of collider searches and —
importantly — be met naturally [17] by $|\cos(\beta-\alpha)|\ll 1$, with
$\approx m_{22}$ masses of the heavy scalar states [12] and for a range of
$\tan\beta$ permitting the natural explanation of leptogenesis discussed in
Refs. [12, 17], for which the Vissani-like bound protects the Higgs sector
from sizeable right-handed neutrino Yukawa contributions. Additionally, the
top-specific Yukawa structure we consider here should not give rise to
important differences; nor the replacement of the $\epsilon$-coupled quartic
term with the cubic $\kappa$-coupled tri-linear term required by a
($N_{DW}=1$) VISH$\nu$ model, since both induce, and therefore realise, the
same bound on $m^{2}_{12}$ by construction. Hence, our analysis of an early
period of inflation driven by the Poincaré-protected top-specific VISH$\nu$
model is well-motivated.

####  V.2.4 Unitarity

The small $\xi_{k}$ regime of VISH$\nu$ (Sec. III.5) not only leads to
successful inflationary predictions but is, in principle, a valid description
of physics up to a Planckian cutoff. However, large non-minimal couplings
($\max_{I}\\{\xi_{I}\\}\gg 1$) could introduce new sub-Planckian unitarity-
violating scale, $\Lambda_{UV}$, and this should be interpreted as a cutoff
for the validity of the model. This was pointed out in the context of SM Higgs
inflation, e.g. by Refs. [132, 133], for which the measured Higgs quartic
coupling $\lambda\sim\mathcal{O}(0.1)$ requires $\xi\sim\mathcal{O}(10^{4})$,
though the running of $\lambda$ to small (positive) values may tolerate
smaller $\xi$. We recall that this is not the case for VISH$\nu$, where the
quartic couplings are not as tightly constrained and we expect smaller values
of the $\xi_{I}$ to be likewise viable ($\lambda_{S}\lesssim 10^{-8}$ realise
the Planckian cutoff for small $\xi_{k}$). Nevertheless, given that our model
additionally provides viable predictions throughout a regime of large
$\xi_{k}$, we find it necessary to understand what consequences to
predictivity, if any, follow from a new scale $\Lambda_{UV}$ in VISH$\nu$, but
will not add anything new to the debate over unitarity problems for non-
minimally coupled inflation models.

There are additional operators in the Einstein frame Lagrangian of the Higgs
inflation model which induce the cutoff
$\Lambda^{\text{Higgs}}_{UV}=M_{P}/\xi$ for $\xi\gg 1$, in the present
vacuum.292929The same result can be obtained in the Jordan frame by expanding
the metric around the flat background [134]. In VISH$\nu$, the corresponding
cutoff is $\Lambda_{UV}=\min_{IJ}\\{M_{P}/\sqrt{\xi_{I}\xi_{J}}\\}$. This is
comparable to $H_{\text{inf}}$, and below both the energy density scale
$V^{1/4}$ and the non-negligible background modulus field values during the
inflation. This could lead to legitimate concern over the predictivity of the
large $\xi_{k}$ inflation model, following the arguments of Ref. [132, 133],
but does not afflict $\max_{k}\\{\xi_{k}\\}\lesssim 1$ as the cutoff breaches
the Planck scale.

An alternative approach takes into account the classical inflationary
background, which renders the cutoff field-dependent. In this interpretation,
the cutoff in the Einstein frame has a larger value during the inflation which
is $\sim M_{P}$ [135, 134] (and this falls to $\sim M_{P}/\sqrt{\xi}$ during
reheating). This is above the inflationary Hubble scale, and greater than (or
comparable to) the inflationary modulus field-values and energy density scale.
It is commonly claimed, then, that unitarity concerns over the tree-level
_inflation_ are alleviated in non-minimally coupled models, such as the one we
have analysed in Sec. III, although this is not a universal view.

Unitarity violation may also occur during the production of longitudinal gauge
bosons in the early stages of reheating [136, 137, 138], but the scope of this
problem has been limited by Ref. [139]. The subset of large $\xi_{k}$ regime
of VISH$\nu$ inflationary scenarios involving a large-valued neutral Higgs
component may also confront this issue. Introducing $R^{2}$ gravitational
corrections to the gravi-scalar action can avoid this problem and reinstate
the Planck-scale cutoff [140], which may be a well-motivated UV completion to
the large $\xi_{k}$ model if the unitarity issue is shown to be
serious.303030Ref. [64] has analysed inflation with an $R^{2}$-corrected 2HDM
and we expect analogous tree-level results for a mixed $R^{2}$ and VISH$\nu$
$\Phi_{1}\Phi_{2}$-Inflation scenario, as $U(1)_{\text{PQ}}$ merely limits
possible terms. We also note that Palatini Higgs inflation avoids this problem
[141] (we work in the metric formulation). The problem is avoided without
further modification in the small $\xi_{k}$ model.

##  VI Summary and Conclusion

The top-specific VISH$\nu$ model, and its flavour-variants, serve as an
existence proof that weakly-coupled high-scale physics, which does not
introduce new destabilising contributions to an electroweak-scale Higgs mass,
can nonetheless address important phenomenological and theoretical problems of
the Standard Model (SM) in a unified and minimal way. This is achieved without
eschewing technical naturalness of non-gravitational interactions and with an
experimentally accessible dark matter candidate.

The model inherits the explanatory framework of the $\nu$DFSZ, predicting:
small active neutrino masses, BAU through thermal hierarchial leptogenesis,
axionic dark matter and $CP$ invariant strong interactions; while removing a
domain wall problem through an $N_{DW}=1$ flavoured Yukawa structure. The
present work extends (and refines) these predictions in the following ways.

A period of cosmic inflation is invoked by the standard cosmological paradigm,
but lacks a unique particle physics explanation. A simple model is the slow-
roll evolution of a scalar inflaton field initially displaced from a potential
minimum and this is successfully realised in the Einstein frame of a theory
with direct coupling of a single scalar field, such as the SM Higgs, to Ricci
scalar curvature. Realistic extensions of the SM, such as invisible axion
models, necessitate a multi-scalar explanation, which can be fulfilled by non-
minimal gravitational couplings ($\xi_{k}$) of the additional scalars.

While the two-scalar case has been extensively analysed elsewhere, we classify
the possible inflaton configurations for a generic non-minimally coupled
three-scalar case for the first time.313131This analysis is made possible by
arguments in Refs. [29, 30] that the attractors of generic inflaton
trajectories in this class of models can often be identified with valleys of
the Einstein frame potential, which realise effectively single-field
inflation. Our findings are more general than a three-scalar discussion in
Ref. [142]. We find that restriction to the VISH$\nu$ parameter space places a
meaningful limitation on the seven inflationary scenarios, enforcing that a
large-field valley of the conformally stretched Einstein frame potential is
only realised in general when there is both an electroweak sector ($\Phi_{i}$)
and hidden-sector ($S$) component. The three possibilities are
$\Phi_{1}S$-Inflation, $\Phi_{2}S$-Inflation and
$\Phi_{1}\Phi_{2}S$-Inflation, corresponding to the disjoint regions of
parameter space: $\kappa_{12}>0$, $\kappa_{21}>0$ and
$\kappa_{12},\kappa_{21}<0$, respectively.323232These are parameters we
introduce in Sec. III and have the definitions:
$\kappa_{12}=\lambda_{34}\xi_{1}-\lambda_{1}\xi_{2}$ and
$\kappa_{21}=\lambda_{34}\xi_{2}-\lambda_{2}\xi_{1}$. It is also possible for
$S$ ($\xi_{1},\xi_{2}\simeq 0$) or a Higgs direction ($\xi_{S}\simeq 0$) to
play the role of the inflaton at the expense of one or more light neutral
modulus field directions (absent in the general case for $\xi_{S}\gtrsim
0.2$).

We show that, in addition to a Higgs sector which can be valid at high-scales
[26], the model non-trivially preserves the stability of the singlet direction
for $\lambda_{S}>10^{-13}$. This fact means that the non-minimal coupling
sizes can be naturally extended to smaller magnitudes for
$\Phi_{1}\Phi_{2}S$-Inflation, $\Phi_{i}S$-Inflation and $S$-Inflation, while
matching the amplitude of CMB temperature anisotropies. Accordingly, we
distinguish two regimes for the non-minimal couplings (large and small),
obtaining predictions of the tensor-to-scalar ratio, scalar spectral index and
its running, which are in excellent agreement with present exclusion limits
from CMB observations in both regimes (and for intermediate values). In
particular, large $\xi_{k}\gg 1$ reproduce the inflationary potential of the
Starobinsky and Higgs inflation models. While for small
$\xi_{S}\lesssim\mathcal{O}(1)$, we cure the non-minimal inflation model of
(actively debated) unitarity concerns and remain within excellent agreement
with CMB data for $\xi_{S}>0.01$. The suppression of $\lambda_{S}$ means the
magnitude of the $S$ inflaton component will generally dominate the Higgs
component, while in the large $\xi_{k}$ regime the ratio is democratic. Our
analysis of the small $\xi_{k}$ regime is performed for two limiting cases
that suppress kinetic mixing of the inflaton and orthogonal directions:
$\xi_{i}\ll\xi_{S}$, which results in $S$-Inflation only, as well as
$\xi_{i}\simeq\xi_{S}$, resulting in $\Phi_{i}S$-Inflation or
$\Phi_{1}\Phi_{2}S$-Inflation.

Finally, as a remarkable bonus, we find that the top-specific structure unites
natural explanations for a large top quark mass and hierarchical thermal
leptogenesis due to a small protected hierarchy of the Higgs vevs. We also
show that the mixing of the PQ and L symmetries does not alter the variant
axion-photon and axion-electron couplings beyond top-specific DFSZ
predictions.

As the PQ symmetry is generally broken during the high-scale inflation, there
are significant implications for dark matter predictions. While the full
details of the post-inflation reheating phase remain a subject for further
research, a high reheating temperature and restoration of the PQ symmetry are
reasonable expectations following $\Phi_{i}S$-Inflation and
$\Phi_{1}\Phi_{2}S$-Inflation, due to the generic admixture of PQ scalar and
Higgs-sector components preserved in the inflaton background. This results in
a post-inflationary axion scenario with a tightly constrained axion mass range
$(40\mu\text{eV},\sim 2\text{meV})$ within the projected reach of many
searches. The lower limit, subject to theoretical uncertainties, is KSVZ-like
and forbids CDM over-production, while the upper limit is DFSZ-like and is
imposed by astrophysical constraints. The mass-range also permits an
interpretation of stellar cooling hints. As the inflaton generally carries an
$S$ component, the axion isocurvature fluctuations during the inflation will
be sufficiently suppressed to rule-in the pre-inflationary axion for any
values $f_{A}\lesssim 10^{14}$ GeV [21] which do not lead to restoration. This
scenario leads to a viable mass-range which is less predictively constrained.

The DFSZ-based VISH$\nu$ model, like its KSVZ cousin SMASH, provides a
remarkably simple and economical way to solve the currently most important
phenomenological problems of particle physics within a cosmology that
implements successful inflation using exclusively the scalar fields required
to meet particle physics objectives.

###### Acknowledgements.

This work was supported in part by the Australian Research Council through the
ARC Centre of Excellence for Dark Matter Particle Physics, CE200100008. AHS is
supported by the Australian Government Research Training Program Scholarship
initiative. We thank Jackson Clarke for helpful correspondence.

##  Appendix A Discussion of the inflationary kinetic-mixing

In this appendix, we study the relationship between the masses of fluctuations
in directions orthogonal to the inflaton direction in the neutral modulus
field-space, and the inflationary Hubble scale. This will require the
suppression of kinetic-mixing between the fields parameterising each
direction, so that they may be canonically normalised. Our ultimate objective
is to identify which scenarios have only one light modulus direction (for
which slow-roll is induced generically) and distinguish these from scenarios
where there is an additional light direction (which are made possible by the
approximate decoupling of the Higgs and hidden sectors). We recall that light
modes have sub-Hubble masses. Our analysis of the two-scalar case follows
discussions in Refs. [49, 50].

Let us first consider the large $\xi_{k}$ regime and use an ellipsoidal
parametrisation (with $\xi_{1},\xi_{2},\xi_{3}>0$):

$\begin{split}\chi_{1}=\frac{M_{P}}{\sqrt{\xi_{1}}}\
e^{\frac{\chi_{r}}{M_{P}\sqrt{6}}}\cos\phi^{\prime}\sin\theta^{\prime},\quad\chi_{2}=\frac{M_{P}}{\sqrt{\xi_{2}}}\
e^{\frac{\chi_{r}}{M_{P}\sqrt{6}}}\sin\phi^{\prime}\sin\theta^{\prime},\quad\chi_{3}=\frac{M_{P}}{\sqrt{\xi_{3}}}\
e^{\frac{\chi_{r}}{M_{P}\sqrt{6}}}\cos\theta^{\prime};\end{split}$ (54)

where
$\chi_{r}\equiv\sqrt{\frac{3}{2}}M_{P}\log(\frac{\xi_{1}}{M_{P}^{2}}\chi_{1}^{2}+\frac{\xi_{2}}{M_{P}^{2}}\chi_{2}^{2}+\frac{\xi_{3}}{M_{P}^{2}})$.

This brings the kinetic terms, $G^{\prime}_{IJ}\partial\chi^{\prime
I}\partial\chi^{\prime J}$ where $\chi^{\prime
I}=(\chi_{r},\phi^{\prime},\theta^{\prime})$, into an appealing form:

$G^{\prime}_{IJ}=\begin{pmatrix}1+\frac{1}{6}\left[\frac{1}{\xi_{1}}c^{2}_{\phi^{\prime}}+\frac{1}{\xi_{2}}s^{2}_{\phi^{\prime}}\right]s^{2}_{\theta^{\prime}}+\frac{1}{6\xi_{3}}c^{2}_{\theta^{\prime}}\
&\frac{1}{\sqrt{6}}\left[\frac{1}{\xi_{2}}-\frac{1}{\xi_{1}}\right]c_{\phi^{\prime}}s_{\phi^{\prime}}s^{2}_{\theta^{\prime}}&\frac{1}{\sqrt{6}}\left[\frac{1}{\xi_{1}}c^{2}_{\phi^{\prime}}+\frac{1}{\xi_{2}}s^{2}_{\phi^{\prime}}-\frac{1}{\xi_{3}}\right]c_{\theta^{\prime}}s_{\theta^{\prime}}\\\\[10.00002pt]
\frac{1}{\sqrt{6}}\left[\frac{1}{\xi_{2}}-\frac{1}{\xi_{1}}\right]c_{\phi^{\prime}}s_{\phi^{\prime}}s^{2}_{\theta^{\prime}}&\left[\frac{1}{\xi_{2}}c^{2}_{\phi^{\prime}}+\frac{1}{\xi_{1}}s^{2}_{\phi^{\prime}}\right]s^{2}_{\theta^{\prime}}&\left[\frac{1}{\xi_{2}}-\frac{1}{\xi_{1}}\right]c_{\phi^{\prime}}s_{\phi^{\prime}}c_{\theta^{\prime}}s_{\theta^{\prime}}\\\\[10.00002pt]
\frac{1}{\sqrt{6}}\left[\frac{1}{\xi_{1}}c^{2}_{\phi^{\prime}}+\frac{1}{\xi_{2}}s^{2}_{\phi^{\prime}}-\frac{1}{\xi_{3}}\right]c_{\theta^{\prime}}s_{\theta^{\prime}}&\left[\frac{1}{\xi_{2}}-\frac{1}{\xi_{1}}\right]c_{\phi^{\prime}}s_{\phi^{\prime}}c_{\theta^{\prime}}s_{\theta^{\prime}}&\left[\frac{1}{\xi_{1}}c^{2}_{\phi^{\prime}}+\frac{1}{\xi_{2}}s^{2}_{\phi^{\prime}}\right]c^{2}_{\theta^{\prime}}+\frac{1}{\xi_{3}}s^{2}_{\theta^{\prime}}\end{pmatrix}.$
(55)

If we consider the limit $\xi_{k}\rightarrow\infty$, the radial field
($\chi_{r}$) is canonically normalised and all other components are exactly
zero. Retaining $\mathcal{O}(\xi_{k}^{-1})$ terms, a field transformation must
be made to bring $G^{\prime}_{IJ}$ into a canonical form. As we are interested
in the effective mass of perturbations about the false vacua of the leading
order potential (17), which are orthogonal to the radial time-evolution
parametrised by $\chi_{r}$, we define the shifted angular fields
$\phi^{\prime}=\overline{\phi}^{\prime}+\delta\phi^{\prime}/M_{P}$ and
$\theta^{\prime}=\overline{\theta}^{\prime}+\delta\theta^{\prime}/M_{P}$. We
denote by $(\overline{\phi}^{\prime},\overline{\theta}^{\prime})$, the
background field values attained at a given critical point of (17), and by
$(\delta\phi^{\prime},\delta\theta^{\prime})$, the massive first-order
perturbations.

Let us consider the simplest case where a hyper-valley is aligned with an axis
in the $\\{\chi_{k}\\}$ basis, so the candidate inflaton field is proportional
to the single modulus field parametrising the large-field radial displacement
along the axis. We choose the $\chi_{1}$ axis without loss of generality, as
results generalise under a permutation of indices. The background angular
field-values are thus $\phi^{\prime}=0$ and $\theta^{\prime}=\pi/2$, bringing
the kinetic terms into a diagonal form:

$\mathcal{L}\supset\frac{1}{2}(\partial{\chi_{r}})^{2}+\frac{1}{2\xi_{2}}[\partial(\delta\phi^{\prime})]^{2}+\frac{1}{2\xi_{3}}[\partial(\delta\theta^{\prime})]^{2}$
(56)

where we retain only the leading order term for $G_{\chi_{r}\chi_{r}}$ (we
only need to assume $\xi_{1}\gg 1$) and we rescale the angular perturbations
($\delta\phi^{\prime\prime}=\delta\phi^{\prime}/\sqrt{\xi_{2}}$,
$\delta\theta^{\prime\prime}=\delta\theta^{\prime}/\sqrt{\xi_{3}}$) to achieve
the desired canonical normalisation. The effective mass terms for the
perturbations arise from the quadratic part of the perturbative expansion of
the leading-order potential around the background angular fields:

$\Lambda^{(2)}[\delta\phi^{\prime\prime},\delta\theta^{\prime\prime}]\supset\frac{\kappa_{12}}{2\xi_{1}^{2}}M_{P}^{2}(\delta\phi^{\prime\prime})^{2}+\frac{\kappa_{13}}{2\xi_{1}^{2}}M_{P}^{2}(\delta\theta^{\prime\prime})^{2}$
(57)

As the inflationary Hubble scale predicted for this regime is
$H_{\text{inf}}\sim 10^{13}\ \text{GeV}$, the masses of the rescaled,
orthogonal modes exceed $H_{\text{inf}}$ when both
$\kappa_{12},\kappa_{13}\gtrsim(\xi_{1}/10^{5})^{2}$.

In the context of the VISH$\nu$ model, $\Phi_{i}$-Inflation ($\xi_{S}\simeq
0$) it is possible to find a generic region of parameter space where the
orthogonal Higgs direction is heavy, however the orthogonal direction
associated to the modulus of the PQ scalar is generically light (and positive
for $\lambda_{iS}>0$), since $\lambda_{iS}\ll 10^{-10}\xi_{i}$. In
$S$-Inflation, both the orthogonal Higgs directions have sub-Hubble masses. To
ensure the motion is stabilised in the hyper-valley it suffices to assume that
initial field velocities in the orthogonal directions are negligible, and the
fields are initialised close to the attractor, which is a more finely-tuned
scenario than the general case, where these requirements are absent. For these
cases, the dynamics in the adabiatic direction follows the discussion in Sec.
III.4 and Sec. III.5, while the isocurvature direction experiences non-
negligible fluctuations which must be separately analysed (we do not expect
these to survive reheating).

Consider, now, a hyper-valley misaligned in the $\chi_{1}\chi_{2}$-plane
(requiring $\kappa_{12},\kappa_{21},A_{12}<0$), the results for the
trajectories in the $\chi_{1}\chi_{3}$\- and $\chi_{2}\chi_{3}$-planes
ultimately being identical under the interchanges of indices:
$2\leftrightarrow 3$ and $1\leftrightarrow 2$, respectively. The kinetic terms
read, to leading order in $\xi_{1}^{-1}$ and $\xi_{2}^{-1}$:

$\begin{split}&\overline{G}^{\prime}_{IJ}\simeq\begin{pmatrix}1&\alpha&0\\\
\alpha&\beta&0\\\ 0&0&\xi_{3}^{-1}\end{pmatrix}\quad\text{where:}\\\
&\alpha=\left[\frac{1}{\xi_{2}}-\frac{1}{\xi_{1}}\right]c_{\overline{\phi}^{\prime}}s_{\overline{\phi}^{\prime}},\
\beta=\frac{1}{\xi_{2}}c^{2}_{\overline{\phi}^{\prime}}+\frac{1}{\xi_{1}}s^{2}_{\overline{\phi}^{\prime}}.\end{split}$
(58)

To analyse the upper-left submatrix we may closely follow the discussions for
the two-scalar case in Refs. [49, 50]. We make the transformation:
$\chi_{r}^{\prime}=\sqrt{\lambda_{+}}(c_{\psi}\chi_{r}+s_{\psi}\delta\phi^{\prime}$),
$\delta\phi^{\prime\prime}=\sqrt{\lambda_{-}}(-s_{\psi}\chi_{r}+c_{\psi}\delta\phi^{\prime})$
and $\delta\theta^{\prime\prime}=\sqrt{\lambda_{2}}\delta\theta^{\prime}$,
where to diagonalise the kinetic terms:

$\tan\psi=\frac{2\alpha}{1-\beta+\sqrt{(1-\beta)^{2}+4\alpha^{2}}},$ (59)

and we have rescaled by the eigenvalues:
$\lambda_{\pm}=\frac{1}{2}(1+\beta\pm\sqrt{(1-\beta)^{2}+4\alpha^{2}})$ and
$\lambda_{2}=\xi^{-1}_{3}$ to achieve a canonical normalisation. Hence, the
effective mass terms give rise to the mass eigenvalues:

$\begin{split}&m_{-}^{2}=0,\quad
m_{+}^{2}=\frac{2M_{P}^{2}}{\xi_{1}\xi_{2}}\frac{-\kappa_{12}\kappa_{21}}{\kappa_{21}\xi_{1}+\kappa_{12}\xi_{2}}\left[\frac{s_{\psi}^{2}}{\lambda_{+}}+\frac{c^{2}_{\psi}}{\lambda_{-}}\right],\\\
&m^{2}_{2}=M_{P}^{2}\cdot\frac{A_{12}}{\kappa_{21}\xi_{1}+\kappa_{12}\xi_{2}}.\end{split}$
(60)

Because we assume $\xi_{1},\xi_{2}\gg 1$, we know that $\alpha,\beta\ll 1$,
from which it follows that $\lambda_{+}\sim 1$,
$\theta\sim\mathcal{O}(\xi^{-1}_{k})$ and $\lambda_{-}\sim\beta$. So, the mode
which is approximately massless is $\simeq\chi_{r}$, and the orthogonal motion
is generically stabilised for:

$\frac{-\kappa_{12}\kappa_{21}}{\kappa_{21}\xi_{1}^{2}+\kappa_{12}\xi_{2}^{2}},\
\frac{A_{12}}{\kappa_{21}\xi_{1}+\kappa_{12}\xi_{2}}\gtrsim 10^{-10}.$ (61)

Let us now consider the generic VISH$\nu$ scenario of $\Phi_{i}S$-Inflation.
Setting $\lambda_{iS}\simeq 0$ in the analogue of (61), assuming
$\lambda_{i},\lambda_{34}\sim\mathcal{O}(0.1)$ and $\xi_{i}\sim\xi_{S}$, the
orthogonal direction in the $\rho_{i}\sigma$-plane is heavy since
$-\kappa_{iS}\kappa_{Si}/(\kappa_{Si}\xi_{i}^{2}+\kappa_{iS}\xi_{S}^{2})\sim\lambda_{S}/\xi_{S}^{2}\sim\text{few}\times
10^{-10}$ (fitting to $A_{s}$), while the Higgs modulus direction is heavy due
to
$A_{iS}/(\kappa_{Si}\xi_{i}+\kappa_{iS}\xi_{S})\sim\lambda_{S}/\xi_{S}\gtrsim
10^{-10}$. Hence, we can integrate out the orthogonal degrees of freedom [143,
144, 145, 146] to obtain the effective field theory in general. In
$\Phi_{1}\Phi_{2}$-Inflation, we find that while the orthogonal direction
aligned with the Higgs plane is generically heavy, the $\sigma$ direction is
always light due to suppressed inter-sector couplings (but positive for
$\lambda_{iS}>0$).

The final situation to analyse concerns the hyper-valley misaligned with
respect to all three axes. We make statements only for the special case of the
hidden-sector parameter space where the results are more simplified and
pertinent. The more general calculation is, of course, the direct
generalisation of the two-scalar cases discussed above. We set
$\lambda_{13},\lambda_{23}\simeq 0$ to simplify the mass matrix, and assume
$\xi_{1}\simeq\xi_{2}$, which suppresses kinetic mixing between the angular
directions to replicate the block-diagonal form just discussed, and we also
set $\lambda_{i},\lambda_{34}\sim\mathcal{O}(0.1)$. Hence, we find that the
masses of the orthogonal directions are $\sim(\lambda_{S}/\xi_{S})M_{P}^{2}$,
which are heavy when compared with $H_{\text{inf}}$. Taken together, this
demonstrates that a tilt of the hyper-valley towards both the $S$ direction
and a Higgs direction is sufficient to steepen the orthogonal direction of the
valley in spite of an approximate decoupling of the Higgs and singlet sectors,
in the large $\xi_{k}$ regime.

The above conclusions hold for $\max_{k}\\{\xi_{k}\\}\gtrsim\mathcal{O}(10)$,
but can be suitably extended into a the
$\max_{k}\\{\xi_{k}\\}\lesssim\mathcal{O}(1)$ regime as follows. The
parametrisation used above no longer canonically normalises the kinetic terms
and we resort to the regimes discussed in Sec. III.5. Let us assume that
$\xi_{S}\gg\xi_{i}$ which diagonalises the kinetic terms and enforces that
$\rho_{i}\simeq 0$. As remarked in Sec. III.5, $\Phi_{1}\Phi_{2}S$-Inflation,
$\Phi_{i}S$-Inflation degenerate to $S$-Inflation for this extremal parameter
space. This leads to the mass eigenvalues for canonically normalised
orthogonal modes:
$m^{2}_{\rho^{\prime}_{i}}\simeq\lambda_{iS}\sigma^{2}\Omega^{-2}(\sigma)$.
The non-negligible dependence of the masses on the inflaton field-value is a
generic consequence of the small $\xi_{k}$ regime absent from the large
$\xi_{k}$ discussion. In particular, this allows for the masses of the
orthogonal directions to become light as the inflaton field evolves. To
prevent this, we consider when the masses remain heavy up until the inflaton
field-value ($\sigma_{f}$) at the end of the inflation,
$\widetilde{\epsilon}(\sigma_{f})\simeq 1$. For the case that $\xi\simeq
0.01$, we find that $\sigma_{f}\simeq 2.7M_{P}$ while for $\xi\simeq 1$, we
find that $\sigma\simeq M_{P}$. In either case (and for the range of $\xi_{k}$
so delimited) we find that the orthogonal directions are light, but positive
for $\lambda_{iS}>0$, during the entire inflation and are aligned with the
Higgs modulus directions. For $\xi_{S}\simeq\xi_{i}$, the orthogonal
directions in $\Phi_{i}S$-Inflation and $\Phi_{1}\Phi_{2}S$-Inflation are
heavy throughout the entire phase for $\xi_{S}\gtrsim 0.2$, but become
slightly sub-Hubble as the inflation phase ends for smaller values of the non-
minimal coupling. This is because, for
$\lambda_{i},\lambda_{34}\sim\mathcal{O}(0.1)$, the masses of the orthogonal
directions are $\sim\lambda_{S}M_{P}^{2}$, and $\lambda_{S}$ can be expressed
as a function of $\xi_{S}$ through the fit to $A_{s}$. Hence, for $\xi\gtrsim
0.2$, the large $\xi_{k}$ findings are replicated: the inflation model is
compatible with generic initial conditions and negligible isocurvature
fluctuations develop in the neutral modulus directions. For $\xi_{S}<0.2$, the
initial conditions which ensure the motion is stabilised in the hyper-valley
are potentially less general (the orthogonal masses will be
$\sim\mathcal{O}(0.1)H_{\text{inf}}$), and this could be investigated by
numerical simulations.

To summarise: for $\max\\{\xi_{k}\\}\gtrsim 0.2$, the orthogonal directions in
the neutral modulus field-space for the generic $\Phi_{i}S$-Inflation and
$\Phi_{1}\Phi_{2}S$-Inflation scenarios obtain an effective mass which remains
heavy in comparison to the Hubble scale throughout the inflation phase, while
in all other inflation scenarios, and for smaller values of the non-minimal
couplings, there is at least one additional light neutral modulus direction
(with sub-Hubble but positive mass for $\lambda_{iS}>0$).

##  Appendix B $\beta$-functions

To illustrate features of the running in both the top-specific VISH$\nu$ and
$\nu$DFSZ models, we give the $\beta$ functions to 1-loop order, which we have
calculated using PyR@TE [147]. In what follows, we adopt the convention
$\mu\frac{dX}{d\mu}\equiv\frac{1}{\left(4\pi\right)^{2}}\beta^{(1)}[X]$.

###  B.1 Top-specific VISH$\nu$ model

The $\beta$ functions for the fundamental top-specific VISH$\nu$ couplings
are: