KUNS-3009

Two-field inflation from one complex scalar
with symmetry breaking

Yoshihiko<EMAIL_ADDRESS>Toshimasa
<EMAIL_ADDRESS>and Koichi
<EMAIL_ADDRESS>
1Department of Physics, University of Wisconsin-Madison, Madison, WI 53706,
USA
2Department of Physics, Kyoto University, Kyoto 606-8502, Japan

###### Contents

  1. 1 Introduction
  2. 2 Motions of inflaton with symmetry breaking
    1. 2.1 Complex scalar with non-minimal coupling
    2. 2.2 Slow-roll and slow-turn approximation
      1. 2.2.1 Equations of motion and inflation parameters
      2. 2.2.2 Behavior of slow-roll parameters
    3. 2.3 Slow roll versus slow turn
    4. 2.4 Classification of inflation trajectory
  3. 3 Constraints from cosmological observations
    1. 3.1 Transfer functions and observables
    2. 3.2 Spectral index and running spectrum
    3. 3.3 Constraints on potential parameters
      1. 3.3.1 $\phi_{*}>v_{\phi}$ and $v_{\phi}\ll 1$
      2. 3.3.2 $\phi_{*}>v_{\phi}$ and $v_{\phi}=\mathcal{O}(10)$
      3. 3.3.3 $\phi_{*}<v_{\phi}$ and $v_{\phi}=\mathcal{O}(10)$
      4. 3.3.4 $\phi_{*}<v_{\phi}$ and $v_{\phi}=\mathcal{O}(1)$
  4. 4 Cosmic reheating
    1. 4.1 Coupling to RH neutrinos
    2. 4.2 Reheating and leptogenesis
  5. 5 Conclusions
  6. A Scalar field dynamics
    1. A.1 EOMs
    2. A.2 Geometry of scalar fields
  7. B Slow-roll and slow-turn parameters
  8. C Formulae of cosmological observables
    1. C.1 Transfer functions
    2. C.2 Cosmological observables

##  1 Introduction

Inflation in the early universe gives a natural solution to the horizon and
the flatness problems of the big-bang theory, and generates the primordial
perturbations [1, 2, 3, 4, 5, 6]. The inflationary expansion driven by a
single scalar field has been extensively studied. Further, multiple fields
inflation models have also been studied with field theoretical [7, 8, 9, 10,
11, 12, 13, 14, 15, 16] and other motivations such as supergravity and string
theory [17, 18].

We consider a two-field inflation model from a single complex scalar field
with a non-minimal coupling to gravity, which is assumed to develop a non-
vanishing vacuum expectation value (VEV). If we consider only one of the
radial and phase (Nambu-Goldstone) components to play the role of the
inflaton, the inflationary expansion is realized by the similar dynamics to
the new inflation [3], the chaotic inflation [6], the natural inflation [19,
20, 21, 22, 23, 24], the Higgs inflation [25, 26, 27, 28, 29], etc. The VEV
and the soft-breaking mass parameter of the phase mode determine the preferred
inflation scenario.

In this paper, we study the details of two-field inflation from a single
complex scalar with symmetry breaking. The inflation trajectories are
classified according to the field component that drives inflation, with the
VEV and soft-breaking mass parameter being varied. Subsequently, we show the
parameter space of our inflation model that is consistent with the current
cosmological observations. The classification of trajectories enables us to
comprehend the typical inflation features and predictions. It is found that
the Higgs-like inflation driven by the radial mode is favored in the wide
region, while various other types can realize the successful inflation.
Furthermore, we investigate the possibility of reheating the universe through
the decay to right-handed (RH) neutrinos from a complex inflaton field with
the non-minimal coupling.

This paper is organized as follows. In Section 2, we introduce our model and
classify the inflation trajectories into typical three categories. Section 3
shows the parameter space consistent with the experimental constraints from
the current cosmological observations. In Section 4, we discuss the reheating
via the inflaton decay to RH neutrinos and evaluate the reheating temperature
consistent with the thermal leptogenesis. Section 5 is devoted to our
conclusions. The details of the notation and definition of parameters are
summarized in Appendices.

##  2 Motions of inflaton with symmetry breaking

We first introduce our two-field inflation model and analyze the inflation
dynamics by solving the equations of motion (EOMs). In this paper, we set the
reduced Planck scale $M_{P}\approx 2.4\times 10^{18}~{}\mathrm{GeV}$ to be
unity otherwise stated. We use the Minkowski metric convention
$\eta_{\mu\nu}=\mathop{\mathrm{diag}}(+1,-1,-1,-1)$.

###  2.1 Complex scalar with non-minimal coupling

We consider a complex scalar field $\Phi$ with a non-minimal coupling to
gravity. In this paper, we start our discussion from the following action:

$\displaystyle S=\int
d^{4}x\sqrt{-g_{J}}\left[-\frac{1}{2}\Omega^{2}(\Phi)R_{J}+g^{\mu\nu}_{J}\partial_{\mu}\Phi\partial_{\nu}\Phi^{*}-U(\Phi,\Phi^{*})\right],$
(2.1)

where we introduce the non-minimal coupling $\xi$,

$\displaystyle\Omega^{2}(\Phi)=1+2\xi|\Phi|^{2}.$ (2.2)

$R_{J}$ is the Ricci scalar obtained from the Jordan frame metric
$g_{J\mu\nu}$. Let the potential $U(\Phi,\Phi^{*})$ be given by

$\displaystyle
U(\Phi,\Phi^{*})=\frac{\lambda}{2}|\Phi|^{4}-\frac{\mu_{\Phi}^{2}}{2}|\Phi|^{2}-\frac{m_{\chi}^{2}}{4}(\Phi^{2}+\Phi^{*2})+U_{0},$
(2.3)

as considered in Ref. [30] in the context of dark matter. The third term
explicitly breaks the $\mathrm{U}(1)$ phase rotation symmetry of $\Phi$ to
$\mathbb{Z}_{2}$ and generates the pseudo Nambu-Goldstone boson (pNGB) mass.
The parameter $m_{\chi}(\geq 0)$ is called as the soft-breaking mass. For a
transformation $\Phi\mapsto e^{i\pi/2}\Phi$, the action is invariant under a
sign flip of $m_{\chi}^{2}$, $m_{\chi}^{2}\mapsto-m_{\chi}^{2}$. That implies
we can impose $m_{\chi}^{2}\geq 0$ consistently in the model. The ultraviolet
completeness of this pNGB mass term is not specified in this paper, but its
origin is discussed in several works [31, 32, 33, 34, 35, 36].

We suppose that the scalar field $\Phi$ develops a non-vanishing VEV
$v_{\phi}$. The stationary condition allows us to derive the relations among
$\mu_{\Phi}^{2}$, $U_{0}$, and $v_{\phi}$, demanding that $U=0$ at the vacuum.
If we parameterize $\Phi$ in the non-linear representation as

$\displaystyle\Phi=\frac{\phi}{\sqrt{2}}e^{i\chi},$ (2.4)

the scalar potential for the fields $\phi$ and $\chi$ is given by

$\displaystyle
U(\phi,\chi)=\frac{\lambda}{8}(\phi^{2}-v_{\phi}^{2})^{2}+\frac{m_{\chi}^{2}}{4}\phi^{2}(1-\cos
2\chi).$ (2.5)

We note that $\chi$ is a dimensionless variable.

In order to move to the Einstein frame from the Jordan frame [37, 38, 25, 39,
40, 28, 29], we consider the field-dependent Weyl rescaling

$\displaystyle g_{\mu\nu}=\Omega^{2}g_{J\mu\nu}=(1+\xi\phi^{2})g_{J\mu\nu},$
(2.6)

where $g_{\mu\nu}$ denotes the metric in the Einstein frame. This redefinition
results in the following action

$\displaystyle S=\int
d^{4}x\sqrt{-g}\biggl{[}-\frac{R}{2}+\frac{1+\xi\phi^{2}+6\xi^{2}\phi^{2}}{2(1+\xi\phi^{2})^{2}}\partial_{\mu}\phi\partial^{\mu}\phi+\frac{\phi^{2}}{2(1+\xi\phi^{2})}\partial_{\mu}\chi\partial^{\mu}\chi-V(\phi,\chi)\biggr{]},$
(2.7)

where $R$ represents the Ricci scalar of $g_{\mu\nu}$, and the inflaton
potential is given by

$\displaystyle
V(\phi,\chi)\coloneqq\frac{U(\phi,\chi)}{(1+\xi\phi^{2})^{2}}=\frac{\lambda(\phi^{2}-v_{\phi}^{2})^{2}+2m_{\chi}^{2}\phi^{2}(1-\cos
2\chi)}{8(1+\xi\phi^{2})^{2}}.$ (2.8)

In the following, we refer to the first term in (2.8) as the Higgs potential
and the second term as the pNGB potential, respectively. Note that the pNGB
potential differs from the usual natural inflation potential in Refs. [19, 20,
21, 23, 22, 24] in that the overall coefficient of the potential depends on
$\phi$, which naturally arises from the pNGB soft-breaking term in our case.

The field space metric $K_{ab}(\varphi)$ for real scalar fields $\bm{\varphi}$
is generally defined by the kinetic term as

$\displaystyle\bm{L}_{\text{kin.}}=\frac{1}{2}K_{ab}(\varphi)\partial_{\mu}\varphi^{a}\partial_{\nu}\varphi^{b}g^{\mu\nu},$
(2.9)

where $\bm{\varphi}$ is the multiplet of scalar fields and indices $a$ and $b$
run all components. In the present model, $\varphi^{1}=\phi$ and
$\varphi^{2}=\chi$, and their metric reads from (2.7) as

$\displaystyle
K_{ab}=\mathop{\mathrm{diag}}\left(\frac{1+\xi\phi^{2}+6\xi^{2}\phi^{2}}{(1+\xi\phi^{2})^{2}},\frac{\phi^{2}}{1+\xi\phi^{2}}\right).$
(2.10)

###  2.2 Slow-roll and slow-turn approximation

####  2.2.1 Equations of motion and inflation parameters

The equations of motion for the background fields are given by

$\displaystyle\frac{d^{2}\phi}{dN^{2}}+\gamma^{1}_{11}\left(\frac{d\phi}{dN}\right)^{2}+\gamma^{1}_{22}\left(\frac{d\chi}{dN}\right)^{2}+(3-\varepsilon)\left(\frac{d\phi}{dN}+K^{11}\frac{\partial}{\partial\phi}\ln
V\right)$ $\displaystyle=0,$ (2.11)
$\displaystyle\frac{d^{2}\chi}{dN^{2}}+2\gamma^{2}_{12}\frac{d\phi}{dN}\frac{d\chi}{dN}+(3-\varepsilon)\left(\frac{d\chi}{dN}+K^{22}\frac{\partial}{\partial\chi}\ln
V\right)$ $\displaystyle=0,$ (2.12)

where the e-folding $N$ is defined by $dN=Hdt$ with $H$ being the Hubble
parameter, and $\gamma^{a}_{bc}$ is the Levi-Civita connection derived from
the field space metric $K_{ab}$. The slow-roll (SR) parameter $\varepsilon$ is
defined by

$\displaystyle\varepsilon\coloneqq-\frac{\dot{H}}{H^{2}}=\frac{1}{2}K_{11}\left(\frac{d\phi}{dN}\right)^{2}+\frac{1}{2}K_{22}\left(\frac{d\chi}{dN}\right)^{2}.$
(2.13)

For the detail, see Appendix A. In order to realize the SR inflation, the
first SR condition

$\displaystyle\varepsilon\ll 1,$ (2.14)

is imposed [41, 42, 43, 44, 45, 46, 47, 48]. This condition also appears in
the case of single-field inflation.

The acceleration of the inflaton is characterized by the parameters

$\displaystyle\eta^{1}$
$\displaystyle=\frac{d^{2}\phi}{dN^{2}}+\gamma^{1}_{11}\left(\frac{d\phi}{dN}\right)^{2}+\gamma^{1}_{22}\left(\frac{d\chi}{dN}\right)^{2},$
(2.15) $\displaystyle\eta^{2}$
$\displaystyle=\frac{d^{2}\chi}{dN^{2}}+2\gamma^{2}_{12}\frac{d\phi}{dN}\frac{d\chi}{dN},$
(2.16)

in the $\phi$\- and $\chi$-directions, respectively. As there are two fields,
it is beneficial to decompose them into the direction of the inflation
trajectory and the direction of rotation [49, 42, 43]. The former is called as
the parallel direction and its unit vector is $\hat{\bm{e}}_{\parallel}$. The
latter is the vertical direction to $\hat{\bm{e}}_{\parallel}$ and its unit
vector is $\hat{\bm{e}}_{\perp}$. Using these unit vectors, the conditions of
the acceleration for successful inflation are given by

$\displaystyle\Bigl{|}\frac{\eta_{\parallel}}{v}\Bigr{|}$
$\displaystyle\coloneqq\biggl{|}\frac{K_{ab}\hat{e}^{a}_{\parallel}\eta^{b}}{v}\biggr{|}\ll
1,$ (2.17) $\displaystyle\frac{\eta_{\perp}}{v}$
$\displaystyle\coloneqq\frac{K_{ab}\hat{e}^{a}_{\perp}\eta^{b}}{v}\ll 1,$
(2.18)

where $v=\sqrt{2\varepsilon}$ is the speed of field vector [41, 42, 43, 46].
Eq. (2.17) is the second SR condition and Eq. (2.18) is called the slow-turn
(ST) condition. The ST condition is one of the indication of small vertical
acceleration. Once the SR conditions (2.14), (2.17) and the ST condition
(2.18) are satisfied, the SR and ST parameters are well approximately
expressed with the inflaton potential $V$ [46],

$\displaystyle\varepsilon$
$\displaystyle\approx\frac{1}{2}K^{ab}\partial_{a}\ln V\partial_{b}\ln V,$
(2.19) $\displaystyle\frac{\eta_{\parallel}}{v}$
$\displaystyle\approx-\hat{e}_{\parallel}^{a}(\partial_{a}\partial_{b}\ln
V-\gamma^{c}_{ab}\partial_{c}\ln V)\hat{e}^{b}_{\parallel},$ (2.20)
$\displaystyle\frac{\eta_{\perp}}{v}$
$\displaystyle\approx-\hat{e}_{\parallel}^{a}(\partial_{a}\partial_{b}\ln
V-\gamma^{c}_{ab}\partial_{c}\ln V)\hat{e}^{b}_{\perp}.$ (2.21)

The unit vector in the parallel and vertical directions are expressed as

$\displaystyle\hat{e}_{\parallel}^{a}$
$\displaystyle=\frac{1}{v}\,\frac{d\varphi^{a}}{dN}\approx\frac{-K^{ab}\partial_{b}\ln
V}{\sqrt{K^{cd}\partial_{c}\ln V\partial_{d}\ln V}},$ (2.22)
$\displaystyle\hat{e}_{\perp}^{a}$
$\displaystyle\approx\frac{\pm(K^{ab}-\hat{e}_{\parallel}^{a}\hat{e}_{\parallel}^{b})(\partial_{b}\partial_{c}\ln
V-\gamma^{d}_{bc}\partial_{d}\ln
V)\hat{e}_{\parallel}^{c}}{\sqrt{(K^{ab}-\hat{e}_{\parallel}^{a}\hat{e}_{\parallel}^{b})(\partial_{a}\partial_{c}\ln
V-\gamma^{e}_{ac}\partial_{e}\ln V)(\partial_{b}\partial_{d}\ln
V-\gamma^{f}_{bd}\partial_{f}\ln
V)\hat{e}_{\parallel}^{c}\hat{e}_{\parallel}^{d}}}.$ (2.23)

The direction of $\hat{e}_{\perp}$ is chosen so that $\eta_{\perp}\geq 0$. We
can use the EOMs to show that the above expression of the vertical unit vector
is equivalent to $K^{ab}J_{bc}\hat{e}_{\parallel}^{c}$ with some asymmetric
$J_{ab}$, which is another expression according to the definition of
perpendicular to $\hat{e}_{\parallel}$. In the case of two-dimensional field
space, a simple solution is $J_{ab}=\sqrt{\det K}/v\epsilon_{ab}$, where
$\epsilon_{ab}$ is the totally asymmetric tensor. Note that all of the above
quantities are written in terms of fields and their derivatives through the
potential $V$ and the field space metric $K_{ab}$, which values are obtained
by solving the equations of motion (2.11) and (2.12).

####  2.2.2 Behavior of slow-roll parameters

Figure 1:  The $m_{\chi}$ and $\phi$ dependence of the slow-roll parameters.
In the left panels, we set $\lambda=1$, $v_{\phi}=10^{-4}$, $\phi=1$, and
$\chi=\frac{\pi}{3}$. In the right panels, $\lambda=1$, $m_{\chi}=10^{-4}$,
$\chi=\pi/3$ and $\xi=10^{5}$ ($\xi=10^{-2}$) are used for $v_{\phi}=10^{-4}$
($v_{\phi}=20$). The dashed lines indicate that the SR conditions are not
satisfied.

We start our discussion from the parameter dependence of the SR parameters
$\varepsilon$ and $|\eta_{\parallel}/v|$. Figure 1 shows the dependence of the
soft-breaking mass $m_{\chi}$ and the field value $\phi$. In the left panels,
the $m_{\chi}$ dependence is given with $\lambda=1$, $v_{\phi}=10^{-4}$,
$\phi=1$, and $\chi=\frac{\pi}{3}$. The blue, orange, and green lines
represent the results obtained for $\xi=10,\ 10^{3}$, and $10^{5}$,
respectively. The value of VEV $v_{\phi}$ is motivated from the energy scale
of grand unification or lepton number violation. When $m_{\chi}$ is
sufficiently smaller than the Planck scale, the inflation is primarily driven
by $\phi$ and effectively reduces to the single-field inflation, analogous to
the Higgs inflation. In this case, the SR parameters are approximately given
by

$\displaystyle\varepsilon$ $\displaystyle\approx\frac{8(1+\xi
v_{\phi}^{2})^{2}\phi^{2}}{(\phi^{2}-v_{\phi}^{2})^{2}(1+\xi\phi^{2}+6\xi^{2}\phi^{2})},$
(2.24) $\displaystyle\frac{\eta_{\parallel}}{v}$
$\displaystyle\approx\frac{4(1+\xi
v_{\phi}^{2})(1+\xi\phi^{2})\bigl{[}v_{\phi}^{2}+\phi^{2}(1+2\xi\phi^{2}+12\xi^{2}\phi^{2})\bigr{]}}{(\phi^{2}-v_{\phi}^{2})^{2}(1+\xi\phi^{2}+6\xi^{2}\phi^{2})^{2}}.$
(2.25)

These expressions do not involve the soft-breaking mass. When $m_{\chi}$ is
sufficiently large, the pNGB potential tends to dominate the inflation.
Consequently, the SR parameters do not involve $m_{\chi}$ again since
$m_{\chi}$ becomes an overall parameter of the inflaton potential. We note
that a larger $\xi$ makes the inflaton potential flatter, which results in
smaller SR parameters. For the detail expressions of the SR parameters, see
Appendix B.

The right panels of Figure 1 show the $\phi$ dependence of SR parameters with
$\lambda=1$, $m_{\chi}=10^{-4}$, $\chi=\pi/3$ and $\xi=10^{5}$ ($\xi=10^{-2}$)
for $v_{\phi}=10^{-4}$ ($v_{\phi}=20$). This large VEV is motivated by the
fact that the natural inflation requires VEV beyond the Planck scale [21]. A
large value of $\phi$ ($>v_{\phi}$) results in a reduction of the SR
parameters due to the proportionality $\varepsilon\propto\phi^{-4}$ and
$\eta_{\parallel}/v\propto\phi^{-2}$, as derived from the approximate formulae
presented in Appendix B. These behaviors can be attributed to the dominance of
the Higgs potential in the large $\phi$ region. In contrast, for a small value
of $\phi$ ($<v_{\phi}$), the SR parameters are expressed as

$\displaystyle\varepsilon$
$\displaystyle\approx\frac{8\phi^{2}}{\lambda^{2}v_{\phi}^{8}}\big{(}A^{2}+m_{\chi}^{4}\sin^{2}2\chi\big{)},$
(2.26) $\displaystyle\frac{\eta_{\parallel}}{v}$
$\displaystyle\approx\frac{4}{\lambda
v_{\phi}^{4}}\frac{A^{3}+3Am_{\chi}^{4}\sin^{2}2\chi-2m_{\chi}^{6}\cos
2\chi\sin^{2}2\chi}{A^{2}+m_{\chi}^{4}\sin^{2}2\chi},$ (2.27)

where $A=\lambda v_{\phi}^{2}(1+\xi v_{\phi}^{2})-2m_{\chi}^{2}\sin^{2}\chi$.
It is found that $\varepsilon$ becomes small and $|\eta_{\parallel}/v|$ is
constant for sufficiently small $\phi$ due to the flatness of the inflaton
potential. When $m_{\chi}\ll\sqrt{\lambda}v_{\phi}$ and $\xi v_{\phi}^{2}\ll
1$, we have

$\displaystyle\left|\frac{\eta_{\parallel}}{v}\right|\approx\frac{4}{v_{\phi}^{2}},$
(2.28)

which implies that the VEV must exceed the Planck scale in order to achieve a
SR inflation with the Higgs potential in the region of small field value
$\phi$.

###  2.3 Slow roll versus slow turn

Let us consider the condition characterizing the two-field nature of the
inflationary trajectory according to Eqs. (2.11) and (2.12). Because the turn
in the field space can be dominant if $|\eta_{\parallel}/v|\ll\eta_{\perp}/v$
[46], let us define

$\displaystyle\mu:=\max\biggl{(}\frac{\eta_{\perp}/v}{|\eta_{\parallel}/v|}\biggr{)},$
(2.29)

which is evaluated on the inflationary trajectory. This $\mu$ parameter
represents how much the two-field nature originating from the movements of
$\phi$ and $\chi$ is strong.

(a) $\chi_{*}=23\pi/48$

(b) $\chi_{*}=\pi/3$

(c) $\chi_{*}=\pi/100$

Figure 2:  The contours of the $\mu$ parameter (2.29) in the $(m_{\chi},\xi)$
plane. We set $\lambda=1$ and $\phi_{*}>v_{\phi}=10^{-4}$. Each horizontal
purple line means $\phi_{*}$ takes a fixed value explicitly stated along each
line. These plots are evaluated with $N$ being fixed at $55$. In the right-
upper white region, either SR or ST condition is violated until reaching
$N=55$.

We use $\phi_{*}$ and $\chi_{*}$ for the horizon-exit values of these fields.
As typical values of $\chi_{*}$, we choose $\chi_{*}=23\pi/48,\ \pi/3$, and
$\pi/100$, which respectively correspond to the following three cases:

  1. (a)

Near the top of the pNGB potential

  2. (b)

Middle of the pNGB potential

  3. (c)

Near the bottom of the pNGB potential

Figure 2 shows the $\mu$ parameter in the $(m_{\chi},\xi)$ plane for the
aforementioned values of $\chi_{*}$. We set $\lambda=1$ and
$\phi_{*}>v_{\phi}=10^{-4}$ as in Section 2.2.2, and evaluate $\mu$ at the
e-folding $N=55$.

For large $m_{\chi}$ or $\xi$, it is not possible to satisfy the SR and/or ST
conditions for a fixed e-folding, as shown by the white region in Figure 2. In
the figure, $\phi$ tends to drive the inflation with the field-turning effect
becoming dominant when the pNGB potential is effective as $m_{\chi}$
increases. This implies the SR/ST violating (white) region for large
$m_{\chi}$ with a fixed $\xi$. Conversely, a larger $\xi$ makes $\phi_{*}$
smaller in the $\phi_{*}>v_{\phi}$ region due to the flattering effect of the
inflaton potential. For too large $\xi$, the inflaton potential for $\phi$
becomes too flat, which may result in a too-slow acceleration. That implies a
large $\xi$ results in large $\mu$ for a fixed $m_{\chi}$ (a fixed size of the
pNGB potential).

If $m_{\chi}$ or $\xi$ is relatively large, the inflaton tends to move in the
$\chi$-direction. It can be seen from the ratio of the gradient of the
inflaton potential

$\displaystyle\frac{K^{22}\partial_{2}\ln V}{K^{11}\partial_{1}\ln
V}=\frac{m_{\chi}^{2}(1+\xi\phi^{2}+6\xi^{2}\phi^{2})\sin
2\chi}{\phi[\lambda\phi^{2}+2m_{\chi}^{2}(1-\xi\phi^{2})\sin^{2}\chi]},$
(2.30)

which is proportional to $m_{\chi}^{2}$ and $\xi^{2}$ when the dynamics of
$\phi$ is dominated by the Higgs potential. We also find a simple approximated
formula for the $\mu$ parameter for small $v_{\phi}$ and $m_{\chi}$

$\displaystyle\mu\approx\frac{m_{\chi}^{2}\xi^{\frac{5}{2}}v_{\phi}^{2}\sin(2\chi_{*})}{8\lambda}.$
(2.31)

It is found from this expression that $\mu$ is vanishing for $m_{\chi}=0$, and
is positively proportional to $\xi$ as $\mu\propto m_{\chi}^{2}\xi^{5/2}$.
From these behaviors, we find that a large $\mu$ can be realized in the large
$m_{\chi}$ and large $\xi$ region, which implies the strong nature of two-
field inflation.

It is noted that the inflaton potential (2.8) is rewritten as

$\displaystyle
V(\phi,\chi)=\lambda\frac{(\phi^{2}-v_{\phi}^{2})^{2}+2\big{(}\frac{m_{\chi}}{\sqrt{\lambda}}\big{)}^{2}\phi^{2}(1-\cos
2\chi)}{8(1+\xi\phi^{2})^{2}}.$ (2.32)

While we have shown Figure 2 with a fixed value $\lambda=1$, the $\mu$
parameter for other values of $\lambda$ can be obtained by rescaling
$m_{\chi}$ as $m_{\chi}/\sqrt{\lambda}$, since $\mu$ is roughly given by the
ratio of the potential and the overall $\lambda$ dependence is cancelled, as
seen from the above potential.

###  2.4 Classification of inflation trajectory

(a) $m_{\chi}=10^{-10}$

(b) $m_{\chi}=10^{-4}$

(c) $m_{\chi}=10^{-2}$

(d) $m_{\chi}=10^{-10}$

(e) $m_{\chi}=10^{-4}$

(f) $m_{\chi}=10^{-2}$

Figure 3:  Typical inflation trajectories in the present two-field model. The
red and blue lines represent the trajectories with $N=50$ and $N=60$,
respectively. The red and blue circle represent the initial value of the
inflaton with $N=50$ and $N=60$, respectively. The arrows indicate the
potential gradients $K^{ab}\partial_{b}\ln V$, which corresponds to the field
velocity in the slow-roll and slow-turn limits. The SR and/or ST condition is
not satisfied in the shaded region. The parameters are chosen as follows:
(Upper panels) $\lambda=10^{-10}$, $v_{\phi}=10^{-4}$, $\xi=10^{-2}$,
$\chi_{*}=34\pi/75$ ($N=50$), and $\chi_{*}=47\pi/100$ ($N=60$). (Lower
panels) $\lambda=1$, $v_{\phi}=20$, $\xi=10^{-2.6}$, $\chi_{*}=5\pi/17$
($N=50$), and $\chi_{*}=20\pi/63$ ($N=60$).

The parameter $\mu$ defined in Eq. (2.29) is an appropriate quantity for
characterizing the multi-field nature of the inflaton motion. In this section,
we classify the inflation trajectory with the help of $\mu$. This
classification enables us to comprehend the multi-field nature of inflation
without having to examine in great detail the complex motions of inflatons due
to their interactions with each other.

Figure 3 shows typical inflation trajectories in the present two-field model.
The vectors in the figures indicate the potential gradients appearing in the
EOMs, $K^{ab}\partial_{b}\ln V$ with $a,b=1,2$. The red and blue solid lines
represent the trajectories with $N=50$ and $N=60$, respectively. In the shaded
region, the SR and/or ST condition is not satisfied. We here adopt the values
of $v_{\phi}=10^{-4}$ and $v_{\phi}=20$ as in the previous case. In the upper
(lower) three panels, the VEV of $\Phi$ is chosen as $v_{\phi}=10^{-4}$
($v_{\phi}=20$), which is the large (small) VEV case under the situation
$\phi_{*}>v_{\phi}$ ($\phi_{*}<v_{\phi}$).

When $m_{\chi}$ is small and the pNGB potential contribution is irrelevant
(the left panels of Figure 3), the inflation is mainly driven by $\phi$. This
is similar to the single-field inflation scenario such as the Higgs inflation
and chaotic inflation. In this case, $\mu\ll 1$ as a result of little motion
in the $\chi$-direction during the inflation. As $m_{\chi}$ increases, $\chi$
also plays a part of the inflaton as well as $\phi$, which is shown in the
middle panels of Figure 3. As illustrated in the figures, the multi-field
effect becomes pronounced, frequently resulting in $\mu\gtrsim 1$. In the
right panels, where $m_{\chi}$ is large, the inflaton trajectory is mainly
along the pNGB $\chi$-direction, similar to the natural inflation. In this
case, the inflation process is also driven by a single field, which results in
the situation of $\mu\ll 1$.

Let us classify the inflaton trajectories into the following three categories
in the present two-field inflation model:

  * •

Higgs inflation type driven by $\phi$ (shown by (a) and (d))

  * •

Mixed type with both $\phi$ and $\chi$ as inflaton (shown by (b) and (e))

  * •

Natural inflation type mainly driven by $\chi$ (shown by (c) and (f))

In the following section, we examine the relation between inflationary
trajectories and the cosmological observables. The symbols H(iggs), M(ixed),
and N(atural) will be used to indicate which of these types the inflation
trajectory we are referring to.

##  3 Constraints from cosmological observations

This section presents an analysis of the inflation observables and the
parameter constraints for the current two-field inflation model.

###  3.1 Transfer functions and observables

In the context of single-field inflation, the observables are derived from the
fluctuations of the inflaton field and the curvature perturbation, denoted by
$\mathcal{R}$, in the co-moving gauge on hypersurfaces at a constant time [50,
51, 52, 53]. In the case of multi-field inflation, this scalar fluctuation is
defined as a projection of the multi-field fluctuations onto the tangent
direction of the inflation trajectory:

$\displaystyle\mathcal{R}=\frac{K_{ab}\hat{e}_{\parallel}^{a}\delta\varphi_{f}^{b}}{v}.$
(3.1)

The perturbation $\delta\bm{\varphi}_{f}$ is the multi-field version of the
Mukhanov-Sasaki variable [53, 54, 46]

$\displaystyle\delta\bm{\varphi}_{f}=\delta\bm{\varphi}+\psi\frac{d\bm{\varphi}}{dN},$
(3.2)

where $\delta\bm{\varphi}$ is the scalar field fluctuation and $\psi$ is a
part of the diagonal metric perturbation [50, 51, 54]. Furthermore, it is
necessary to consider the fluctuations in the vertical direction with respect
to the inflation trajectory, which give rise to the isocurvature perturbation
[55, 46]

$\displaystyle\mathcal{S}=\frac{K_{ab}\hat{e}_{\perp}^{a}\delta\varphi_{f}^{b}}{v}.$
(3.3)

These two perturbations $\mathcal{R}$ and $\mathcal{S}$ have the following
relation [56, 55, 46, 57]

$\displaystyle\begin{pmatrix}\mathcal{R}\\\
\mathcal{S}\end{pmatrix}=\begin{pmatrix}1&T_{\mathcal{RS}}\\\
0&T_{\mathcal{SS}}\end{pmatrix}\begin{pmatrix}\mathcal{R}_{*}\\\
\mathcal{S}_{*}\end{pmatrix}.$ (3.4)

The values at the horizon exit are denoted with $\ast$ such as
$\mathcal{R}_{*}$ and $\mathcal{S}_{*}$. This relation indicates that the
curvature perturbation $\mathcal{R}$ is not frozen after the horizon exit and
the isocurvature perturbation $\mathcal{S}$ contributes to it through the
transfer function $T_{\mathcal{RS}}$ [52, 42, 43, 58, 57, 27].

In contrast to the single-field inflation scenario, $T_{\mathcal{RS}}$ can
contribute to the inflationary observables in the multi-field case. The power
spectrum $\mathcal{P_{R}}$ and tensor-to-scalar ratio $r$ are given by [41,
46]

$\displaystyle\mathcal{P_{R}}=\frac{V_{*}}{24\pi^{2}\varepsilon_{*}}(1+T_{\mathcal{RS}}^{2}),\qquad
r=\frac{16\varepsilon_{*}}{1+T_{\mathcal{RS}}^{2}}.$ (3.5)

In addition, we consider the spectral index $n_{s}$ and the running spectrum
$\alpha$ given by

$\displaystyle n_{s}=1+\frac{d\ln\mathcal{P_{R}}}{d\ln
k},\qquad\alpha=\frac{dn_{s}}{d\ln k},$ (3.6)

and the isocurvature fraction [59, 60, 57, 27]

$\displaystyle\beta_{\mathrm{iso}}=\frac{T_{\mathcal{SS}}^{2}}{1+T_{\mathcal{RS}}^{2}+T_{\mathcal{SS}}^{2}}.$
(3.7)

These quantities also depend on the transfer functions [61, 46, 27, 57], and
all of these can be expressed by the parameters and field values in the
present inflation model. See Appendix C for the detail.

These inflationary observables are experimentally constrained by the Planck
Collaboration [62]

$\displaystyle\mathcal{P_{R}}=(2.1\pm 0.1)\times 10^{-9},\qquad r<0.07,\qquad
n_{s}=0.9649\pm 0.0042,$ $\displaystyle\alpha=-0.0045\pm
0.0067,\qquad\beta_{\mathrm{iso}}<0.038.$ (3.8)

###  3.2 Spectral index and running spectrum

Figure 4:  Typical predictions for the spectral index and the tensor-to-scalar
ratio for a fixed initial value $\chi_{*}=\pi/3$. The red and blue lines
correspond to $N=50$ and $N=60$. The dashed, solid and dotted lines in the
left panel correspond to $\xi=1$, $10^{-2}$, and $10^{-3}$, respectively. The
dashed (solid) lines in the right panel are for $\xi=10^{-3}$ ($\xi=10^{-6}$).
The soft-breaking mass takes $0\leq m_{\chi}\leq 1.4\times 10^{-5}$
($\xi=10^{-3}$), $0\leq m_{\chi}\leq 2.3\times 10^{-5}$ ($\xi=10^{-2}$) and
$0\leq m_{\chi}\leq 1.1\times 10^{-4}$ ($\xi=1$) in the left panel, and $0\leq
m_{\chi}\leq 7.0\times 10^{-6}$ in the right panel. The arrows on the lines
indicate the direction $m_{\chi}$ increases. The marks mean the fixed
$m_{\chi}$ behaviors: $m_{\chi}=0$ (circle), $6\times 10^{-6}$ (square),
$7.6\times 10^{-6}$ (diamond), $5.9\times 10^{-6}$ (blue star), $7.0\times
10^{-6}$ (red star), $6.2\times 10^{-6}$ (blue triangle), and $7.0\times
10^{-6}$ (red triangle). The contours are the experimental bounds with
$1\sigma$ and $2\sigma$ confidence levels by the Planck observation.

Figure 4 shows typical predictions in the $(n_{s},r)$ plane in the present
two-field model. The power spectrum is normalized by $\lambda$ as
$\mathcal{P_{R}}=2.1\times 10^{-9}$ such that it is consistent with the Planck
observation. The parameters $v_{\phi}$ and $\chi_{*}$ are fixed at specific
values. In the left panel, we explicitly use $v_{\phi}=10^{-4}$ and
$\chi_{*}=\pi/3$, on the other hand $v_{\phi}=20$, $\chi_{*}=\pi/3$ in the
right panel. In addition to these, we explicitly use $0\leq m_{\chi}\leq
1.4\times 10^{-5}$ ($\xi=10^{-3}$), $0\leq m_{\chi}\leq 2.3\times 10^{-5}$
($\xi=10^{-2}$) and $0\leq m_{\chi}\leq 1.1\times 10^{-4}$ ($\xi=1$) in the
left panel and $0\leq m_{\chi}\leq 7.0\times 10^{-6}$ ($\xi=10^{-6}$ and
$\xi=10^{-3}$) in the right panel.

We first focus on the left panel of Figure 4. When $m_{\chi}$ is sufficiently
small, the Higgs potential dominates the inflaton potential. Consequently, the
prediction of the observables is similar to the Higgs inflation case. In this
cases, $n_{s}$ ($r$) becomes large (small) as $N$ increases. This is due to
the fact that the initial value $\phi_{*}$ must be sufficiently large in order
to flatten the inflaton potential during the inflation, which results in
larger $n_{s}$ and small $r$. The change in $n_{s}$ occurs more quickly when
$N$ is larger, which makes $\phi_{*}$ larger, since $m_{\chi}$ appears in the
Lagrangian in the combination of $m_{\chi}^{2}\phi^{2}$. As $m_{\chi}$
increases, the inflaton motion in the $\chi$-direction also contributes to the
inflation. Consequently, $\phi_{*}$ becomes more closely at the bottom of the
Higgs potential in order to achieve a specific fixed e-folding. This results
in a smaller inflaton potential, which in turn implies smaller
$\varepsilon_{*}$ due to the Planck normalization. From the single-field
inflation analogy, this gives rise to an increase in the value of $n_{s}$.
From this consideration, we can argue that the value of $n_{s}$ increases when
the pNGB mode also contributes to the inflation, provided that
$\phi_{*}>v_{\phi}$. Conversely, sufficiently large $m_{\chi}$ can drive the
natural-like inflation. The left panel of Figure 4 shows that larger
$m_{\chi}$ results in a reduction in both $n_{s}$ and $r$. This feature is
also seen in the context of natural inflation. For small $\xi$ and large
$m_{\chi}$, the pNGB part of the action (2.7) during the inflation is
approximated as

$\displaystyle S\approx\int
d^{4}x\sqrt{-g}\Bigg{[}\frac{\phi_{*}^{2}}{2}\partial_{\mu}\chi\partial^{\mu}\chi-\frac{m_{\chi}^{2}\phi_{*}^{2}}{4}(1-\cos
2\chi)\Bigg{]}.$ (3.9)

This action shows that $\phi_{*}$ being larger than Planck scale facilitates
the flatter pNGB potential for the natural-like inflation. Consequently, we
can conclude that the natural-like inflation is feasible despite $v_{\phi}$
being smaller than the Planck scale. We note that a larger $\xi$ flattens the
inflaton potential enough and tends to result in a smaller value of $r$.

Let us move to the right panel of Figure 4. When $m_{\chi}$ is sufficiently
small, we can say that an increase of $N$ results in an increase of $n_{s}$
and a decrease of $r$. This is due to the fact that, in order to ensure the
flatness of the inflaton potential during inflation, it is necessary for a
larger value of $N$ to initiate the inflation from the part of flatter
potential. In this region, the inflation driven by almost only the radial
component $\phi$. On the other hand, if the Higgs potential is no longer
dominant, the natural inflation occurs with a particular $m_{\chi}$, which
leads to a large $r$. This behavior can be seen in the figure, and we can say
that natural-like inflation also occurs for a large VEV as the single-field
natural inflation. Finally, we discuss the relation between $\xi$ and the
observables. When $m_{\chi}$ is sufficiently small, the Higgs potential is
dominant, so a larger $\xi$ tilts the inflaton potential in the region of
$\phi<v_{\phi}$. This results in smaller $n_{s}$ and larger $r$. Conversely,
if the pNGB potential is the dominant one, a larger $\xi$ leads to a flatter
inflaton potential, which in turn results in a smaller $r$. These can be read
from the right panel of Figure 4.

###  3.3 Constraints on potential parameters

We show the parameter space in the $(m_{\chi},\xi)$ plane which is consistent
with the cosmological observations by considering three typical initial values
$\chi_{*}$, as discussed in Section 2.3:

$\displaystyle\chi_{*}=\frac{23\pi}{48},\ \frac{\pi}{3},\ \frac{\pi}{100}.$
(3.10)

The discussion is mainly divided into the cases of $\phi_{*}>v_{\phi}$ and
$\phi_{*}<v_{\phi}$. These cases typically correspond to the large and small
field inflation scenarios along with $\phi$, respectively. We investigate the
inflationary features using $v_{\phi}=10^{-4}$ as the case of small VEV ($\ll
1$) and $v_{\phi}=20$ as the case of large VEV ($\sim\mathcal{O}(10)$). In
addition to them, we also investigate the third case with $v_{\phi}=5$ as a
typical intermediate VEV scale ($\sim\mathcal{O}(1)$). In the following, we
consider the four patterns of VEV and the $\phi$ direction:

  1. (i)

$\phi_{*}>v_{\phi}=10^{-4}\ll 1$

  2. (ii)

$\phi_{*}>v_{\phi}=20=\mathcal{O}(10)$

  3. (iii)

$\phi_{*}<v_{\phi}=20=\mathcal{O}(10)$

  4. (iv)

$\phi_{*}<v_{\phi}=5=\mathcal{O}(1)$

In each pattern, we analyze the inflation with various values of the soft-
breaking mass $m_{\chi}$.

The inflaton trajectories are classified into three types, as discussed in
Section 2.4 and referred to by the symbols H, M and N. In all figures of
parameter constraints given below, the Planck normalization
$\mathcal{P_{R}}=2.1\times 10^{-9}$ is imposed to determine the coupling
constant $\lambda$.

####  3.3.1 $\phi_{*}>v_{\phi}$ and $v_{\phi}\ll 1$

(a) $\chi_{*}=23\pi/48$

(b) $\chi_{*}=\pi/3$

(c) $\chi_{*}=\pi/100$

Figure 5:  The allowed parameter region in the $(m_{\chi},\xi)$ plane for
$\phi_{*}>v_{\phi}=10^{-4}$. The pink region is consistent with the
observations and $N=55$. The gray, blue, and red colored regions are excluded
by the constraint from the observations on $n_{s}$, $r$, and $\alpha$,
respectively. $\phi_{*}=1$ and $\lambda=1$ hold on the black and brown dashed
lines, respectively.

First, we examine the case where the VEV is small and $\phi$ is rolling down
from a large initial value. Figure 5 shows the parameter space which is
consistent with the observations, where we set $v_{\phi}=10^{-4}$,
$\phi_{*}>v_{\phi}$, and $N=55$. In the white region, the condition $N=55$ is
not satisfied, which is found in the large $m_{\chi}$ region. This is due to
the fact that the Planck normalization provides an upper limit for
$m_{\chi}\phi_{*}$, and the SR and ST conditions exclude small $\phi_{*}$ for
enough e-folding. The pink region is consistent with the observations and
$N=55$. The gray, blue, red colored regions are excluded by the constraints
from the observations on $n_{s}$, $r$, and $\alpha$, respectively. The black
(brown) dashed line represents $\phi_{*}=1$ ($\lambda=1$). The Planck
normalization and the perturbativity of the quartic coupling $\lambda$
provides the upper bound on the non-minimal coupling, which is given by
$\xi\lesssim 10^{5}$, as discussed in Refs. [25, 26, 29]. The soft-breaking
mass parameter $m_{\chi}$ determines which contribution of potential is
dominant, the Higgs potential or the pNGB potential. In the following, the
parameter space is analyzed with paying attention to the value of $m_{\chi}$.

We find some small allowed region labeled by N around
$(m_{\chi},\xi)\sim(10^{-5},10^{-3})$ in Figure 5(a) where the inflation is
mainly driven by $\chi$, and $\phi$ oscillates at the end of inflation. It is
noticed that the inflationary expansion in this region is similar to the one
discussed in Refs. [12, 63].

#####  Small $m_{\chi}$ region:

In the small $m_{\chi}$ region, the inflation is dominantly driven by $\phi$,
which region is labelled by H in the figure. There the spectral index and the
tensor-to-scalar ratio are approximately given by

$\displaystyle n_{s}$ $\displaystyle\approx
1-\frac{8(3+5\xi\phi_{*}^{2}+24\xi^{2}\phi_{*}^{2}+2\xi^{2}\phi_{*}^{4}+12\xi^{3}\phi_{*}^{4})}{\phi_{*}^{2}(1+\xi\phi_{*}^{2}+6\xi^{2}\phi_{*}^{2})^{2}}+\mathcal{O}(m_{\chi}^{2}),$
(3.11) $\displaystyle r$
$\displaystyle\approx\frac{128}{\phi_{*}^{2}(1+\xi\phi_{*}^{2}+6\xi^{2}\phi_{*}^{2})}+\mathcal{O}(m_{\chi}^{2}),$
(3.12)

where the e-folding is expressed as [64]

$\displaystyle N$
$\displaystyle\approx\frac{1-\sqrt{1+32\xi+192\xi^{2}}+2(1+6\xi)\xi\phi_{*}^{2}+12\xi\ln\frac{1+12\xi+\sqrt{1+32\xi+192\xi^{2}}}{2(1+6\xi)(1+\xi\phi_{*}^{2})}}{16\xi}.$
(3.13)

When $\xi$ is also small, these expressions reduce to

$\displaystyle n_{s}$ $\displaystyle\approx
1-\frac{3}{N}+\mathcal{O}(m_{\chi}^{2}),$ (3.14) $\displaystyle r$
$\displaystyle\approx\frac{16}{N}+\mathcal{O}(m_{\chi}^{2}),$ (3.15)

and are excluded by the observations as seen in Figure 5. This is due to the
fact that a smaller $\xi$ results in a more tilted inflaton potential, which
makes the spacetime away from the de Sitter space and $r$ becomes larger than
the observed value. The above discussion and the approximate formulae lead to
the conclusion that there is a lower bound for the non-minimal coupling $\xi$.
This can be read from Figure 5 as $\xi\gtrsim 10^{-2}$, which is consistent
with the result in Ref. [64].

Next, let us discuss the running spectrum. In the small $\xi$ region, it is
approximately given by

$\displaystyle\alpha\approx-\frac{3}{N^{2}}+\mathcal{O}(m_{\chi}^{2}),$ (3.16)

which implies that $\alpha$ is consistent with the observation for
sufficiently large e-folding $N$ under the current situation. This result is
to be expected since the magnitudes of $|1-n_{s}|$ and $|\alpha|$ are
generally hierarchical in single-field inflation [65]. On the other hand, for
large $\xi$, we obtain the approximate formula for $\alpha$,

$\displaystyle\alpha$ $\displaystyle\approx-\frac{32}{9\xi^{2}\phi_{*}^{4}}.$
(3.17)

A large value of $\xi$ ensures that $\alpha$ remains sufficiently small and
consistent with the experimental data. That means a hierarchy between
$|1-n_{s}|$ and $|\alpha|$, and the inflation is mainly achieved by the Higgs
potential part and reduces to the single-field inflation scenario effectively.

We also discuss the isocurvature fraction for small $m_{\chi}$. The general
form is given by (3.7) and the transfer function $T_{\mathcal{SS}}$ is written
as

$\displaystyle
T_{\mathcal{SS}}=\exp\int_{N_{*}}^{N}\tilde{\beta}\,dN^{\prime}.$ (3.18)

The explicit form of the function $\tilde{\beta}$ is found in Appendix C,
which can be determined by the model parameters and field values obtained by
solving the EOMs. In the present case, $\tilde{\beta}$ is approximately given
by

$\displaystyle\tilde{\beta}\approx$
$\displaystyle-\frac{8}{\phi^{2}}\quad(\xi\ll 1),$ (3.19)
$\displaystyle\tilde{\beta}\approx$
$\displaystyle-\frac{4v_{\phi}^{2}}{3\phi^{2}}\quad(\xi\gg 1).$ (3.20)

A negative, moderate $\tilde{\beta}$ leads to an exponentially small
$T_{\mathcal{SS}}$, and thus $\beta_{\mathrm{iso}}$ is sufficiently small to
be consistent with the observational data. Note here that the condition
$\xi\gg 1$ implies $\phi_{*}\ll 1$.

#####  Large $m_{\chi}$ region:

If we have a large $m_{\chi}$ and the $\chi$ contribution, both $\phi$ and
$\chi$ can play the roll of the inflaton when $\xi$ is small, as shown in the
left and middle panels of Figure 5. For the spectral index, we find that
$n_{s}$ becomes larger with increasing of $m_{\chi}$ under $\xi\ll 1$ as
discussed in Figure 4, which leads to a contradiction to the observational
data. Conversely, the parameter regions where the inflation is primarily
driven by $\chi$ can be allowed as shown in the left panel. In particular, the
spectral index is approximated as

$\displaystyle n_{s}\approx
1-\frac{4(2+T_{\mathcal{RS}}^{2})}{(1+T_{\mathcal{RS}}^{2})\phi_{*}^{2}\sin^{2}\chi_{*}},$
(3.21)

when $m_{\chi}\gg\sqrt{\lambda}v_{\phi}$ and $\xi\ll 1$. The spectral index on
this inflation trajectory can match to the observations. That is driven by the
pNGB mode with a help of the dynamical radial component. Furthermore, the
inflation may also be allowed in the case of $\xi\gg 1$, where the pNGB mode
contributes sufficiently, as shown in the left and middle panels of Figure 5.
This is due to the fact that a large $\xi$ not only flattens the inflaton
potential but also facilitates the movement in the $\chi$-direction as
discussed in Section 2.3. On the other hand, if the initial value $\chi_{*}$
is sufficiently small as in the right panel, the inflaton moves almost no
further in the $\chi$-direction, independently of $v_{\phi}$.

Finally, we give a comment on the isocurvature bound $\beta_{\mathrm{iso}}$.
The function $\tilde{\beta}$ appearing in Eq. (3.18) is approximated as

$\displaystyle\tilde{\beta}$
$\displaystyle\approx-\frac{2}{\phi^{2}\sin^{2}\chi}\quad(\xi\ll 1),$ (3.22)
$\displaystyle\tilde{\beta}$
$\displaystyle\approx-\frac{2\xi}{\sin^{2}\chi}\quad(\xi\gg 1),$ (3.23)

where we assume $m_{\chi}\gg\sqrt{\lambda}v_{\phi}$ and $v_{\phi}\ll 1$. Since
$\tilde{\beta}$ is negative in the both cases, we can say that
$\beta_{\mathrm{iso}}$ is sufficiently small in this case.

####  3.3.2 $\phi_{*}>v_{\phi}$ and $v_{\phi}=\mathcal{O}(10)$

(a) $\chi_{*}=23\pi/48$

(b) $\chi_{*}=\pi/3$

(c) $\chi_{*}=\pi/100$

Figure 6:  The allowed parameter region in the $(m_{\chi},\xi)$ plane for
$\phi_{*}>v_{\phi}=20$. For the meanings of the colored regions, see the
caption of Figure 5.

We consider $\phi_{*}>v_{\phi}$ with the VEV being a relatively large value of
$v_{\phi}=20$. Figure 6 shows the allowed parameter region for the inflation
under this VEV and field value. The aforementioned colors are used to show the
experimentally-excluded regions. In the white region, the e-folding $N=55$
cannot be satisfied with a large $m_{\chi}$, similar to Figure 5.

#####  Small $m_{\chi}$ region:

In the small $m_{\chi}$ region with $\phi_{*}>v_{\phi}=20$ and $\xi\gg 1$, the
spectral index, the tensor-to-scalar ratio, and the running spectral are
approximately given by

$\displaystyle n_{s}$ $\displaystyle\approx
1-\frac{8v_{\phi}^{2}}{3\phi_{*}^{2}},$ (3.24) $\displaystyle r$
$\displaystyle\approx\frac{64v_{\phi}^{4}}{3\phi_{*}^{4}},$ (3.25)
$\displaystyle\alpha$
$\displaystyle\approx-\frac{32v_{\phi}^{4}}{9\phi_{*}^{4}}.$ (3.26)

The relation between $\phi_{*}$ and the e-folding reads

$\displaystyle
N=\frac{(1+6\xi)(\phi_{*}^{2}-\phi_{e}^{2})-v_{\phi}^{2}\ln\frac{\phi_{*}^{2}}{\phi_{e}^{2}}-6(1+\xi
v_{\phi}^{2})\ln\frac{1+\xi\phi_{*}^{2}}{1+\xi\phi_{e}^{2}}}{8(1+\xi
v_{\phi}^{2})},$ (3.27)

where we introduce $\phi_{e}$ for the final value of $\phi$, which can be
derived from the SR and ST conditions. From these approximated formulae, we
find that the inflation in this case is consistent with the observational
data. This is mainly due to the fact that, as in the previous case, a large
$\xi$ results in the flattening of the inflaton potential. In the case of
small $\xi$, on the other hand, the gradient of the inflaton potential
increases, resulting in the inconsistency between theoretical predictions and
the observational data, as seen in Figure 6. Since $\tilde{\beta}$ is
approximately given by (3.19) and (3.20) also in this case, we find
$\beta_{\mathrm{iso}}$ sufficiently small. The inflaton behavior discussed
here is similar to the previous one with the small VEV $v_{\phi}\ll 1$, since
both inflation motions are mainly driven by the radial mode only.

#####  Large $m_{\chi}$ region:

In this case, the inflaton trajectories largely depend on the initial value
$\chi_{*}$. For large $m_{\chi}$, we have a significant contribution from the
pNGB potential, which in turn allows the inflaton to move in the
$\chi$-direction. In particular, the natural inflation can be achieved if the
initial field values and parameters are tuned, as seen in the middle panel of
Figure 6. If both scalar fields drive the inflation, a larger $\xi$ can lead
to the inflation consistent with the observations, as seen in the left and
middle panels. A small value of $\xi$ makes the inflation the chaotic or the
natural one, which in any case is excluded by the observational data.

In the large VEV case, we have the upper bound on $m_{\chi}$. For $\xi\gg 1$,
the power spectrum is approximated as

$\displaystyle\mathcal{P_{R}}\approx\frac{(\lambda\phi_{*}^{2}+4m_{\chi}^{2}\sin^{2}\chi_{*})^{3}(1+T_{\mathcal{RS}}^{2})}{1536\pi^{2}\xi^{3}\phi_{*}^{2}m_{\chi}^{4}\sin^{2}2\chi_{*}}.$
(3.28)

A larger $v_{\phi}$ requires a larger $\phi_{*}$, which in turn results in a
larger $m_{\chi}$ due to the Planck normalization $\mathcal{P_{R}}$.
Accordingly, we find that the experimentally allowed parameter value of
$m_{\chi}$ becomes larger in the present case than that of $v_{\phi}\ll 1$
with $\xi\gg 1$. On the other hand, for $\xi\ll 1$, the power spectrum is
approximated as

$\displaystyle\mathcal{P_{R}}\approx\frac{\phi_{*}^{4}(\lambda\phi_{*}^{2}+4m_{\chi}^{2}\sin^{2}\chi_{*})^{3}(1+T_{\mathcal{RS}}^{2})}{1536\pi^{2}[\lambda^{2}\phi_{*}^{4}+4m_{\chi}^{2}(\lambda\phi_{*}^{2}+m_{\chi}^{2})\sin^{2}\chi_{*}]}.$
(3.29)

When $m_{\chi}$ is large, namely the pNGB potential is dominant, we have
$\mathcal{P_{R}}\propto m_{\chi}^{2}$ since the soft-breaking mass parameter
is the overall coefficient of the inflaton potential. Therefore, we find that
a large $v_{\phi}$ leading a large $\phi_{*}$ requires a smaller $m_{\chi}$
for $\xi\ll 1$. This behavior is opposite to the case of $\xi\gg 1$.

As for the isocurvature bound, the function $\tilde{\beta}$ in this case
becomes

$\displaystyle\tilde{\beta}$
$\displaystyle\approx-\frac{2}{\phi^{2}\sin^{2}\chi}\quad(\xi\ll 1),$ (3.30)
$\displaystyle\tilde{\beta}$
$\displaystyle\approx-\frac{2\xi}{\sin^{2}\chi}\quad(\xi\gg 1),$ (3.31)

for $m_{\chi}\gg\sqrt{\lambda}v_{\phi}$ and $v_{\phi}\gg 1$. We find that
$\beta_{\mathrm{iso}}$ tends to be exponentially small in this type of
inflation, similar to the case of $v_{\phi}\ll 1$.

####  3.3.3 $\phi_{*}<v_{\phi}$ and $v_{\phi}=\mathcal{O}(10)$

(a) $\chi_{*}=23\pi/48$

(b) $\chi_{*}=\pi/3$

(c) $\chi_{*}=\pi/100$

Figure 7:  The allowed parameter region in the $(m_{\chi},\xi)$ plane for
$\phi_{*}<v_{\phi}=20$. The green shaded region is excluded by the
isocurvature bound $\beta_{\mathrm{iso}}$. The meanings of the other colored
regions are same as in Figure 5.

Let us move on to discussing the different type of inflation with
$\phi_{*}<v_{\phi}$, namely, $\phi$ rolls down to the vacuum from some smaller
field value. As mentioned before, we choose two typical values for the VEV as
$v_{\phi}=5$ and $20$. We note that the SR inflation is not possible for a
small value of VEV, as shown in Eq. (2.28).

We first consider the large VEV case, namely, $v_{\phi}=20$. Figure 7 shows
the allowed parameter region in the $(m_{\chi},\xi)$ plane for
$\phi_{*}<v_{\phi}$ and $N=55$. It is noted that a rather small value of $\xi$
is not excluded because the inflaton potential is sufficiently flat when
$\phi_{*}<v_{\phi}$. On the other hand, the large $\xi$ region is excluded
because the inflaton potential becomes tilted when $\phi_{*}<v_{\phi}$. These
behaviors are in contrast to the previous case with $\phi_{*}>v_{\phi}$.

#####  Small $m_{\chi}$ region:

When $m_{\chi}$ is small, the observables are explicitly written as

$\displaystyle n_{s}$ $\displaystyle\approx 1-\frac{8(1+\xi
v_{\phi}^{2})}{v_{\phi}^{2}}-\frac{8(5+6\xi
v_{\phi}^{2}+\xi^{2}v_{\phi}^{4})\phi_{*}^{2}}{v_{\phi}^{4}},$ (3.32)
$\displaystyle r$ $\displaystyle\approx\frac{128(1+\xi
v_{\phi}^{2})^{2}\phi_{*}^{2}}{v_{\phi}^{4}(1+\xi\phi_{*}^{2})},$ (3.33)
$\displaystyle\alpha$ $\displaystyle\approx-\frac{64(1+\xi
v_{\phi}^{2})^{2}(5+\xi
v_{\phi}^{2}-12\xi^{2}v_{\phi}^{2})\phi_{*}^{2}}{v_{\phi}^{6}},\vskip 12.0pt
plus 4.0pt minus 4.0pt$ (3.34)

in the region where the SR and ST conditions are satisfied.

If $\xi\ll 1$, the observables $r$ and $\alpha$ become sufficiently small to
be consistent with the observations due to the suppression by a large value of
$v_{\phi}$. Furthermore, $n_{s}$ is in the experimentally allowed range for
$v_{\phi}\sim 20$, as shown in Figure 7. We can say again this behavior is due
to the flat inflaton potential in the region $\phi_{*}<v_{\phi}$.

On the other hand, $\xi\lesssim\mathcal{O}(1)$ leads to a more tilted inflaton
potential in the region $\phi_{*}<v_{\phi}$. That makes $n_{s}$ smaller than
the observed value. We find this result from the approximate formula of
$n_{s}$ in this case,

$\displaystyle n_{s}\approx 1-\frac{8}{v_{\phi}^{2}}-8\xi.$ (3.35)

The severe constraint can be read from this formula and also found in Figure
7. A smaller $\xi$ is required for the successful inflation under the current
conditions.

Finally, we discuss the isocurvature fraction $\beta_{\mathrm{iso}}$ in this
case. The green shaded region in Figure 7 is experimentally excluded by the
isocurvature bound. In the region of $m_{\chi}\ll v_{\phi}$, the function
$\tilde{\beta}$ in (3.18) is approximated as

$\displaystyle\tilde{\beta}\approx-\frac{8\phi^{2}}{3v_{\phi}^{4}}(1+\xi
v_{\phi}^{2})(3+4\xi+12\xi^{2}-5\xi^{2}v_{\phi}^{2}+12\xi^{3}v_{\phi}^{2}).$
(3.36)

For sufficiently small or large $\xi$, $\tilde{\beta}$ is always negative and
thus $\beta_{\mathrm{iso}}$ is found to be small enough. When
$\xi\sim\mathcal{O}(1)$, on the other hand, $\tilde{\beta}$ turns out to be
positive, which can lead to a large $\beta_{\mathrm{iso}}$ and is
experimentally disfavored. This behavior is actually seen in Figure 7.

#####  Large $m_{\chi}$ region:

In this region, we have three different patterns of inflation, depending on
the initial value $\chi_{*}$. In the case where $\chi_{*}$ is small, the
inflation is mostly driven by $\phi$. If $\chi_{*}$ is large, both $\phi$ and
$\chi$ contribute to the inflation, as in the left and middle panels of Figure
7. These two patterns are denoted by the symbols H and M in the figure. As
previously discussed, large values of $\xi$ and $m_{\chi}$ lead to a small
$n_{s}$. This is due to the fact that the inflaton potential of the radial
component is dominated by the quadratic term. Therefore the inflation in which
both components join is challenging to achieve in the case of
$\phi_{*}<v_{\phi}$ as opposed to the scenario of $\phi_{*}>v_{\phi}$.

We also have a characteristic behavior in the region with a large $m_{\chi}$
and a small $\xi$ in the middle panel of Figure 7. In this region, the
inflation similar to the natural inflation occurs with a very specific value
of $m_{\chi}$, as previously discussed in Figure 4. That appears under the
conditions that $\chi_{*}$ is tuned to realize a fixed value of e-folding $N$
almost only by the $\chi$ motion. Consequently, $r$ becomes larger than the
observed value, which is a characteristic feature found from the approximated
formula [26]

$\displaystyle r\approx\frac{32}{\phi_{*}^{2}\sin^{2}\chi_{*}}.$ (3.37)

Since this natural-like inflation is almost driven by a single field $\chi$,
the hierarchy between $|1-n_{s}|$ and $|\alpha|$ is hold. We note that in the
region where natural-like inflation occurs, $\tilde{\beta}$ is approximated as

$\displaystyle\tilde{\beta}\approx-\frac{2}{\phi^{2}\sin^{2}\chi},$ (3.38)

which also makes $\beta_{\mathrm{iso}}$ exponentially small due to its
negative value.

####  3.3.4 $\phi_{*}<v_{\phi}$ and $v_{\phi}=\mathcal{O}(1)$

(a) $\chi_{*}=23\pi/48$

(b) $\chi_{*}=\pi/3$

(c) $\chi_{*}=\pi/100$

Figure 8:  The parameter region in the $(m_{\chi},\xi)$ plane for
$\phi_{*}<v_{\phi}=5$. The meanings of the colored regions are same as in
Figure 7. No experimentally-allowed region is found.

As the final pattern of inflation in the present two-field model, consider the
intermediate value of VEV $\sim\mathcal{O}(1)$. Figure 8 shows the
experimentally allowed parameter region for $\phi_{*}<v_{\phi}=5$. One can see
that a large portion of area is more severely constrained by the condition
$N=55$, compared to the large VEV case of $\phi_{*}<v_{\phi}=20$. A large
$\xi$ is not allowed when $v_{\phi}$ is small in order to realize a flat
inflaton potential. Besides, compared to the large VEV case, the region with a
large $m_{\chi}$ is also excluded. This is because the inflaton potential is
flatter for a larger $v_{\phi}$ and a smaller $m_{\chi}$. For large $m_{\chi}$
and $\phi_{*}<v_{\phi}$, the power spectrum is approximated as

$\displaystyle\mathcal{P_{R}}\approx\frac{\lambda^{3}v_{\phi}^{12}(1+T_{\mathcal{RS}}^{2})}{1536\pi^{2}\phi_{*}^{2}[\lambda^{2}v_{\phi}^{4}+4(\lambda
v_{\phi}^{2}+m_{\chi}^{2})m_{\chi}^{2}\sin^{2}\chi_{*}]},$ (3.39)

and it is found that a small $v_{\phi}$ only allows a small value of
$m_{\chi}$ because of the Planck normalization.

The parameter region shown in Figure 8, especially in the middle panel, is
limited also by the inconsistency with the observed value of $n_{s}$. We find
an approximate formula for it as

$\displaystyle n_{s}\approx 1-\frac{8(1+\xi
v_{\phi}^{2})}{v_{\phi}^{2}}+\mathcal{O}(m_{\chi}^{2}),$ (3.40)

which shows that $n_{s}$ gives smaller values than the observation in this
case [66]. This is due to the large acceleration of inflaton in this region.
The tensor-to-scalar ratio is also approximated as

$\displaystyle r\approx\frac{128\phi_{*}^{2}(1+\xi
v_{\phi}^{2})^{2}}{v_{\phi}^{4}}+\mathcal{O}(m_{\chi}^{2}),$ (3.41)

which is still smaller in the entire region where the inflaton potential is
flat. The approximated formula for $\alpha$ is expressed as Eq. (3.34). The
current case leads to the conclusion $\phi_{*}\ll 1$, which in turn implies
that $\alpha$ is consistent with the observations provided that $m_{\chi}$ is
sufficiently small.

Finally, we discuss the isocurvature fraction $\beta_{\mathrm{iso}}$. This is
found to be large in most entire region in this case. The approximation of
$\tilde{\beta}$ is given by Eq. (3.36). In contrast to the large VEV case, we
find $\tilde{\beta}\approx 0$ at the beginning of inflation. The inflaton is
almost insensitive to the $\phi$-direction initially, which results in the
approximation $\tilde{\beta}\approx 0$ being realized for a prolonged period
during inflation. This means that a relatively small VEV under the condition
$\phi_{*}<v_{\phi}$ leads to a large $\beta_{\mathrm{iso}}$, which is
inconsistent with the observation.

##  4 Cosmic reheating

In this section, we introduce RH neutrinos with Yukawa coupling to the complex
scalar field $\Phi$, and consider the reheating [67, 68] so that the inflaton
heats the thermal bath by the decay to the neutrino sector. In the vacuum
after the inflation epoch, the VEV $v_{\phi}$ generates the RH neutrino masses
and also the pNGB associated with the lepton number $\mathrm{U}(1)_{L}$
symmetry, called the majoron [69, 70, 71, 72]. The majoron has been widely
studied in the context of dark matter physics [73, 74, 75, 76, 77, 78, 79, 80,
81, 82, 83] and the lepton asymmetry generation [84, 85, 86, 87, 88]. In this
section, we consider this type of RH neutrino couplings can also be available
to the reheating process for non-minimally coupled inflaton.

###  4.1 Coupling to RH neutrinos

We consider the following action for the RH neutrinos $N_{i}$ in the Einstein
frame,

$\displaystyle S_{N}=\int
d^{4}x\sqrt{-g}\biggl{[}\frac{1}{2(1+2\xi|\Phi|^{2})^{\frac{3}{2}}}\overline{N_{i}}i\not{D}N_{i}-\frac{1}{2(1+2\xi|\Phi|^{2})^{2}}(f_{i}\Phi\overline{N_{i}}P_{R}N_{i}+\text{h.c.})\biggr{]},$
(4.1)

where the generation-diagonal Yukawa couplings are introduced between the
inflaton and the RH neutrinos, and $f_{i}$ are generally complex-valued. The
covariant derivative $\not{D}$ includes the spin connection and $P_{R}$ is the
chiral projection operator. This form of the action, in particular the $\Phi$
dependence of the coefficients, can be derived from the original canonical
action for $N_{i}$ in the Jordan frame and Weyl rescaled with the inflaton
field, similarly to the inflaton action in Section 2.1. The action has the
lepton number symmetry under which $N_{i}$ and $\Phi$ have the charges $+1$
and $-2$, respectively. This symmetry may correspond to the U(1) rotation of
inflaton field (the origin of pNGB and its soft-breaking mass), which implies
a natural support of two-field inflation from a complex scalar. In the
following, we focus on the case that $\phi$ field oscillates at the end of the
inflation and decay to the thermal bath through RH neutrinos. We also assume
$\chi$ stays at a constant value $\bar{\chi}$ during the reheating, and
perform the chiral rotation of RH neutrino fields

$\displaystyle N_{i}\to e^{-i\gamma^{5}(\bar{\chi}+\arg f_{i})/2}N_{i},$ (4.2)

such that the Yukawa couplings (then the RH neutrino masses) become real
valued. During the reheating era, the inflaton oscillates at the bottom of the
potential which is determined by the VEV $u_{\phi}$,

$\displaystyle u_{\phi}=\mathop{\mathrm{Re}}\sqrt{\frac{\lambda
v_{\phi}^{2}\Omega^{2}(v_{\phi})-m_{\chi}^{2}(1-\cos
2\bar{\chi})}{\lambda\Omega^{2}(v_{\phi})-\xi m_{\chi}^{2}(1-\cos
2\bar{\chi})}}.$ (4.3)

Note that $u_{\phi}$ can be vanished if the pNGB potential has a large
contribution during the reheating. We expand the scalar field around this VEV
so that the scalar field has a canonically normalized kinetic term as

$\displaystyle\phi=u_{\phi}+\frac{\Omega^{2}(u_{\phi})}{\sqrt{\Omega^{2}(u_{\phi})+6\xi^{2}u_{\phi}^{2}}}\rho,$
(4.4)

and we obtain the action for the scalar fluctuation and the (normalized) RH
neutrinos

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\Biggl{[}\frac{1}{2}\partial_{\mu}\rho\partial^{\mu}\rho-V(\rho)+\frac{1}{2}\overline{N_{i}}(i\not{D}-m_{N_{i}})N_{i}-\frac{1}{2}y_{i}\rho\overline{N_{i}}N_{i}\Biggr{]},$
(4.5)

The inflaton coupling to the RH neutrinos $y_{i}$ and their mass eigenvalues
$m_{N_{i}}$ and $m_{\rho}$ are given by

$\displaystyle y_{i}$
$\displaystyle=\frac{\Omega(u_{\phi})}{\sqrt{\Omega^{2}(u_{\phi})+6\xi^{2}u_{\phi}^{2}}}\frac{f_{i}}{\sqrt{2}},$
(4.6) $\displaystyle m_{N_{i}}$
$\displaystyle=\frac{1}{\Omega(u_{\phi})}\frac{f_{i}}{\sqrt{2}}u_{\phi},$
(4.7) $\displaystyle m_{\rho}$
$\displaystyle=\frac{\sqrt{2\lambda\Omega^{2}(v_{\phi})\big{[}u_{\phi}^{2}(1+2\Omega^{2}(v_{\phi}))-v_{\phi}^{2}\big{]}+m_{\chi}^{2}(1-5\xi
u_{\phi}^{2})\sin^{2}\bar{\chi}}}{2\Omega^{2}(u_{\phi})\sqrt{\Omega^{2}(u_{\phi})+6\xi^{2}u_{\phi}^{2}}}.$
(4.8)

###  4.2 Reheating and leptogenesis

We examine whether the inflaton coupling to RH neutrinos can be consistent
with the reheating of the universe after inflation. A simplified analysis is
performed here by taking into account of rough estimations of seesaw-induced
neutrino masses and the condition for leptogenesis.

The inflaton decays to the RH neutrino pair, $\rho\to N_{i}N_{i}$, during the
oscillation. The width is determined by the Yukawa coupling $y_{i}$,

$\displaystyle\Gamma_{\rho\to
N_{i}N_{i}}=\frac{y_{i}^{2}}{16\pi}m_{\rho}\bigg{(}1-\frac{4m_{N_{i}}^{2}}{m_{\rho}^{2}}\bigg{)}^{3/2}.$
(4.9)

Once the energy is converted to the $N_{i}$ sector, the Standard Model (SM)
fields are thermalized by the decay and scattering of $N_{i}$ with
sufficiently large neutrino Yukawa couplings $y_{\nu_{i}}$. The decay process
$N_{i}\to L_{j}H$, where $L_{j}$ and $H$ are the leptons and Higgs fields, is
effective when $N_{i}$ is heavier than the electroweak scale and the decay
width is larger than the Hubble parameter at the temperature $T$. This
condition is roughly given by the present model parameters as

$\displaystyle
T<21\,g_{*}^{-1/4}y_{i}v_{\phi}\Big{(}\frac{m_{\nu_{i}}}{10^{-1}\,\text{eV}}\Big{)}^{1/2}\eqqcolon
T_{D},$ (4.10)

where $g_{*}$ is the effective degrees of freedom in the radiation with which
the temperature is defined. The seesaw-induce light neutrino mass is given by
$m_{\nu_{i}}=(y_{\nu_{i}}v_{h})^{2}/m_{N_{i}}$ with $v_{h}=175$ GeV (the
generation structure is omitted for simplicity of discussion). The scattering
of $N_{i}$ to the SM sector is also effective if the scattering rate is
sufficiently large. This process may be utilized to make the SM sector
thermalized radiation component, when the decay process is unavailable in any
sense. We roughly evaluate its effectiveness condition by considering the
collision term from the scattering, which can be expressed in terms of the
present model parameters as

$\displaystyle
T<0.17\,g_{*}^{-3/2}y_{i}v_{\phi}\Big{(}\frac{m_{\nu_{i}}}{10^{-1}\,\text{eV}}\Big{)}\eqqcolon
T_{S}.$ (4.11)

We have approximated the effective degrees of freedom for the entropy density
as equal to that for the energy density. The SM sector is decoupled from the
scattering process when its temperature is above $T_{S}$.

As a typical example of this type of reheating, we focus on the parameter
region which we have shown, in the middle panel of Figure 5, consistent with
the inflationary observable. Figure 9 shows the classification of the
reheating and leptogenesis on the allowed region of $(m_{\chi},\xi)$ plane.
The uncolored region is excluded by the inflation phenomena only.

Figure 9: Typical reheating temperature $T_{R}$ from the inflaton decay to RH
neutrinos with the coupling $y_{i}=1,\,10^{-2},\,10^{-5}$. We show $T_{R}$ on
the parameter space consistent with the inflationary observables (as an
example, Figure 5(b) where the parameters are set to $v_{\phi}=10^{-4}$,
$\phi_{*}>v_{\phi}$ and $\chi_{*}=\pi/3$). In the purple (brown) region, the
inflaton can decay while oscillating around the origin (the non-trivial
vacuum). The decay channel is not open in pink area. We also show the hatched
regions where the thermal leptogenesis by RH neutrino decay can work after the
reheating is completed ($T_{R}>m_{N_{i}}$).

The purple regions in Figure 9 correspond to $u_{\phi}=0$, namely, the
inflaton oscillates around the origin. In this case, the decay process may be
ineffective if the RH neutrino mass is regarded as small. According to (4.11),
the SM sector is thermalized up to the temperature $T_{S}$ with $g_{*}=112$,
at which the SM sector decouples from the thermal bath. Since the width (4.9)
is large enough, the RH neutrino sector is subsequently heated and reaches the
temperature $T_{N}$,

$\displaystyle T_{N}=0.17\,g_{*}^{-1/4}y_{i}m_{\rho}^{1/2},$ (4.12)

with $g_{*}=5.25$. Here $T_{N}$ is defined by the Hubble time being equal to
the width (4.9). (If $T_{N}<T_{S}$, the $N_{i}$ reheating era does not exist
and the reheating temperature is given by $T_{N}$.) As soon as this reheating
is over, $\rho$ is assumed to immediately roll down to the non-trivial vacuum,
where the RH neutrinos are massive and the decay process becomes valid for
heating the SM sector. If $T_{N}<T_{D}$, the SM sector is quickly thermalized,
and otherwise, the SM thermalization occurs after the universe cools down to
$T_{D}$. We then find the final reheating temperature $T_{R}$ for the SM
sector is given by

$\displaystyle T_{R}=\begin{cases}T_{N}&(T_{N}<T_{S})\\\
\bigl{(}(1-\omega)T_{S}^{3}+\omega
T_{N}^{3}\bigr{)}^{1/3}&(T_{S}<T_{N}<T_{D})\\\
\bigl{(}(1-\omega)T_{S}^{3}+\omega
T_{D}^{3}\bigr{)}^{1/3}&(T_{S}<T_{D}<T_{N})\end{cases},$ (4.13)

where $\omega\coloneqq g_{*}^{3N}/(g_{*}^{\text{SM}}+g_{*}^{3N})$. In the
region of interest shown in Figure 9, the relation $T_{S}<T_{N}<T_{D}$ is
found to be numerically satisfied everywhere, and the reheating temperature of
the radiation then becomes $T_{R}\sim 0.36\,T_{N}\sim
0.027y_{i}m_{\chi}^{1/2}$. These parameter dependence on $y_{i}$ and
$m_{\chi}$ can be seen in the purple regions of Figure 9.

We comment on the leptogenesis, the lepton number generation by the out-of-
equilibrium decay of RH neutrinos [89]. The mechanism can work if at least
$T_{R}>m_{N_{i}}$ is satisfied. This condition now reads
$0.027m_{\chi}^{1/2}>v_{\phi}$, which implies $m_{\chi}>1.3\times 10^{-5}$ for
the parameters used in Figure 9. Note that this bound on $m_{\chi}$ is
independent of the Yukawa couplings and common to three panels in Figure 9.
One can see from this bound that only a small portion of the purple region is
consistent with successful reheating and leptogenesis in the present inflation
scenario.

Another case is that the VEV $u_{\phi}$ is non-vanishing, namely, the inflaton
oscillates around the non-trivial vacuum. In Figure 9, it corresponds to the
colored regions other than purple, where it is numerically checked that the
VEV $u_{\phi}$ is roughly similar to $v_{\phi}$. In this case, the RH
neutrinos $N_{i}$ are massive and the reheating by the inflaton decay to
$N_{i}$ is possible if $m_{\rho}>2m_{N_{i}}$. This minimum requirement for
reheating imposes the following condition on the model parameters:

$\displaystyle\Omega^{2}(v_{\phi})\lambda^{1/2}>2(\Omega^{2}(v_{\phi})+6\xi^{2}v_{\phi}^{2})y_{i},$
(4.14)

where we have neglected small $\bar{\chi}$ contributions. Furthermore, the RH
neutrino decay to the SM sector can be sufficiently effective, unlike the
$u_{\phi}=0$ case. The reheating by the scattering is subdominant as can be
seen by comparing the conditions (4.10) and (4.11), and is not included in the
analysis below. The reheating temperature is then given by (4.12) with
$g_{*}=112$, if this temperature is smaller than $T_{D}$ and hence the decay
process is effective until the universe is heated up to (4.12). That is
converted to the constraint on the model parameters as

$\displaystyle\lambda^{1/2}<1.5\times
10^{4}\,v_{\phi}\sqrt{\Omega^{2}(v_{\phi})+6\xi^{2}v_{\phi}^{2}}\,\Big{(}\frac{m_{\nu_{i}}}{10^{-1}\,\text{eV}}\Big{)}^{2}.$
(4.15)

If not satisfied, the reheating temperature for the SM sector is determined by
$T_{D}$, as discussed above. We find the inequality (4.15) generally holds in
the present inflation scenario unless $v_{\phi}$ is quite small, and the
reheating temperature $T_{R}$ is given by (4.12) with $g_{*}=112$:

$\displaystyle
T_{R}=0.052\frac{\lambda^{1/4}y_{i}v_{\phi}^{1/2}}{(\Omega^{2}(v_{\phi})+6\xi^{2}v_{\phi}^{2})^{1/4}}\bigg{(}1-\frac{4(\Omega^{2}(v_{\phi})+6\xi^{2}v_{\phi}^{2})^{2}y_{i}^{2}}{\Omega^{4}(v_{\phi})\lambda}\bigg{)}^{3/4},$
(4.16)

where the phase space factor is explicitly included, though it is not so
important for evaluating $T_{R}$. As seen from Figure 9, the reheating
temperature is not continuously connected to the $u_{\phi}=0$ case. For small
$\xi$, it scales as $T_{R}\propto\xi^{1/2}$ if a rough estimation
$\lambda\propto\xi^{2}$ from the power spectrum is supposed. On the other
hand, for larger $\xi$ ($\xi v_{\phi}\gg 1$), $T_{R}$ is almost constant and
depends linearly on the inflaton coupling $y_{i}$ only. The minimum
requirement for the reheating (4.14) also leads to a non-trivial upper bound
on $y_{i}$. In Figure 9, the pink regions are excluded by this reheating
condition. A smaller $y_{i}$ is favored that can be read off from the fact
that the brown region in Figure 9 becomes wider as $y_{i}$ decreases. For a
fixed $y_{i}$, (4.14) turns out to give a lower (upper) bound on $\xi$ for
small (large) value of $\xi$, if a rough estimation $\lambda\propto\xi^{2}$ is
taken into account. That indeed corresponds to the horizontal brown bands in
the $(m_{\chi},\xi)$ plane.

The thermal leptogenesis is also possible for the $u_{\phi}\neq 0$ case. The
reheating temperature needs to be at least higher than the (lightest) RH
neutrino mass scale, and that implies the parameter bound from (4.7) and
(4.16),

$\displaystyle
4y_{i}^{2}<\frac{\lambda\Omega^{4}(v_{\phi})}{(\Omega^{2}(v_{\phi})+6\xi^{2}v_{\phi}^{2})^{2}}-\frac{50(\Omega^{2}(v_{\phi})\lambda
v_{\phi})^{2/3}}{\Omega^{2}(v_{\phi})+6\xi^{2}v_{\phi}^{2}}.$ (4.17)

This bound is satisfied in the brown hatched regions in Figure 9 where the
leptogenesis is available. Assuming $\lambda$ and $\xi$ are determined by the
inflation phenomena and $v_{\phi}$ is a fixed parameter in our analysis,
(4.17) gives a upper bound on the inflaton Yukawa coupling $y_{i}$. This is
the phase space bound of the inflaton decay and does not strongly affect the
model parameter constraints once the decay channel is open. These behaviors
can be seen from the $y_{i}$ difference in Figure 9. Note that (4.17) (the
brown hatched region) is necessarily stronger than (4.14) (the brown region)
since the latter is the vanishing limit of the phase space. For a fixed
$y_{i}$, (4.17) gives the bound on $\xi$, almost independently of $m_{\chi}$.
Taking a rough relation $\lambda\propto\xi^{2}$ obtained from the power
spectrum in mind, a lower (upper) bound on $\xi$ for small (large) value of
$\xi$ can be derived from (4.17), which corresponds to the horizontal brown-
hatched bands in the $(m_{\chi},\xi)$ plane.

We finally comment on several other possibilities of reheating in the present
inflation model. In the pink region of Figure 9, the inflaton decay to heavy
$N_{i}$ is forbidden and the main decay mode to the SM sector would be a top
quark pair. The reheating only with this mode gives a roughly estimated
temperature $T_{R}\sim
10^{-14}\lambda^{1/4}v_{\phi}^{1/2}(m_{\nu_{i}}/\text{eV})^{2}$, which is
suppressed by large $N_{i}$ masses and becomes around
$\mathcal{O}(\text{MeV})$ to $\mathcal{O}(\text{GeV})$ for the parameters in
the pink regions. This temperature is just before the big bang
nucleosynthesis, but would need to be improved, e.g., for the leptogenesis
being successful. As another possibility, the inflaton can also interact with
the scalar sector through the Higgs portal coupling, inflaton potential terms,
etc. While we do not consider this type of reheating process in this paper, it
would be interesting to examine it in connection with the inflaton shift to
the non-trivial vacuum, and will be left for future work.

##  5 Conclusions

In this paper, we study the two-field inflation originated from one complex
scalar field with a non-vanishing VEV and a non-minimal coupling to gravity.
We introduce the $\mu$ parameter to characterize the multi-field nature and
classify the inflaton trajectories in the field space to three typical types,
depending on which field components contribute to the inflationary expansion.
According to this classification, there are several cases that only the radial
component contributes to the inflation, only the pNGB mode contributes, and
both do. The parameter $\mu$ tends to be large when both modes contribute to
the inflation.

Furthermore, we make the detailed classifications on the VEV, the soft-
breaking mass $m_{\chi}$, and the initial value of the inflaton field
$\phi_{*}$ and $\chi_{*}$, as the classification covers the above three
typical patterns. For each classification, we show the parameter space
consistent with the current cosmological observations. It is found that the
experimental results favor the inflation mainly driven by the radial component
of the scalar field, but we also find other parameter space consistent with
the observations for these three types of inflationary expansion.

While the natural inflation is known to generally require that the VEV is
larger than the Planck scale, we show that VEVs smaller than the Planck scale
allow the successful inflation with a sufficient contribution from pNGB modes
under the condition $\phi_{*}>v_{\phi}$ and $\xi\lesssim 1$. Moreover, we find
that even when VEV is rather small, the inflation is also successful for large
$m_{\chi}$ and $\chi_{*}$ where the contribution from the pNGB mode is
comparable to the radial mode. For a VEV larger than the Planck scale, the
pNGB-contributed inflation is possible without fine tuning. On the other hand,
the small field inflation with $\phi_{*}<v_{\phi}$ is shown to generally
contradict with the observations unless sufficient contribution of the radial
component exists. It is also found that a tiny value of VEV is not favored for
$\phi_{*}<v_{\phi}$ even when the pNGB mode contributes to some extent.

We also examine whether the reheating by the inflaton decay to the RH
neutrinos is possible where the complex scalar field generating the Majorana
mass term plays the roll of inflaton. The VEV at the end of the inflation can
be away from the scalar potential vacuum due to non-trivial field value which
is a result of the inflation. Taking this into the account and evaluating the
reheating temperature, we show that the parameter region with large $m_{\chi}$
is allowed in addition to the region with moderately large $\xi$. This is
because the inflaton oscillates around the origin in the region where
$m_{\chi}$ is sufficiently large. We would need detailed numerical analysis to
examine the consistency with successful leptogenesis, but that is beyond the
scope of this paper. Some mechanism to move the background $\chi$ field is
also a left future problem so that the non-trivial potential vacuum is
realized after the inflation dynamics and reheating.

## Acknowledgments

The work of Y.A. is supported by JSPS Overseas Research Fellowships. The work
of K.Y. is supported by JSPS KAKENHI Grant No. JP20K03949.

##  Appendix A Scalar field dynamics

In this appendix, we summarize the details of the scalar sector.

###  A.1 EOMs

The general form of the action for the scalar fields $\varphi^{a}$ is

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\biggl{[}-\frac{R}{2}+\frac{1}{2}K_{ab}(\varphi)\partial_{\mu}\varphi^{a}\partial^{\mu}\varphi^{b}-V(\varphi)\biggr{]}.$
(A.1)

From this, the general EOM of the scalar field reads

$\displaystyle\nabla^{2}\varphi^{a}+\gamma^{a}_{bc}(\varphi)\partial_{\mu}\varphi^{b}\partial^{\mu}\varphi^{c}+K^{ab}(\varphi)\partial_{b}V(\varphi)=0,$
(A.2)

where $\nabla$ is the covariant derivative evaluated by the Einstein frame
metric $g_{\mu\nu}$. The Levi-Civita connection in the scalar field geometry
is given by

$\displaystyle\gamma^{a}_{bc}=\frac{1}{2}K^{ad}(\partial_{b}K_{dc}+\partial_{c}K_{bd}-\partial_{d}K_{bc}).$
(A.3)

We assume

$\displaystyle ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=dt^{2}-a(t)^{2}d\vec{x}^{2},$
(A.4)

and $\varphi^{a}$ depends only on the time. Then the EOM reduces to

$\displaystyle\ddot{\varphi}^{a}+3H\dot{\varphi}^{a}+\gamma^{a}_{bc}\dot{\varphi}^{b}\dot{\varphi}^{c}+K^{ab}\partial_{b}V=0,$
(A.5)

where the dot is the derivative with respect to $t$, and the Hubble parameter
is defined as $H\coloneqq\dot{a}/a$. When we introduce the e-folding $N$ by
$dN=Hdt$, the EOMs (A.5) become

$\displaystyle\frac{d^{2}\varphi^{a}}{dN^{2}}+\gamma^{a}_{bc}\frac{d\varphi^{b}}{dN}\frac{d\varphi^{c}}{dN}+(3-\varepsilon)\biggl{(}\frac{d\varphi^{a}}{dN}+K^{ab}\partial_{b}\ln
V\biggr{)}=0.$ (A.6)

In this derivation, we have used the Friedmann equation

$\displaystyle
H^{2}=\frac{1}{3}\biggl{(}\frac{1}{2}K_{ab}\dot{\varphi}^{a}\dot{\varphi}^{b}+V(\varphi)\biggr{)},\qquad\dot{H}=-\frac{1}{2}K_{ab}\dot{\varphi}^{a}\dot{\varphi}^{b},$
(A.7)

and introduced the slow-roll parameter $\varepsilon$ given by

$\displaystyle\varepsilon\coloneqq-\frac{\dot{H}}{H^{2}}=\frac{1}{2}K_{ab}\frac{d\varphi^{a}}{dN}\frac{d\varphi^{b}}{dN}.$
(A.8)

###  A.2 Geometry of scalar fields

We consider the complex scalar $\Phi=\phi e^{i\chi}/\sqrt{2}$ with the non-
minimal coupling $\xi$. The metric of the scalar field space is found to be

$\displaystyle
K_{ab}=\mathop{\mathrm{diag}}\left(\frac{1+\xi\phi^{2}+6\xi^{2}\phi^{2}}{(1+\xi\phi^{2})^{2}},\frac{\phi^{2}}{1+\xi\phi^{2}}\right).$
(A.9)

The non-vanishing Levi-Civita connection in this space is given by

$\displaystyle\gamma^{1}_{11}$
$\displaystyle=-\frac{\xi\phi(1-6\xi+\xi\phi^{2}+6\xi^{2}\phi^{2})}{(1+\xi\phi^{2})(1+\xi\phi^{2}+6\xi^{2}\phi^{2})},$
(A.10) $\displaystyle\gamma^{1}_{22}$
$\displaystyle=-\frac{\phi}{1+\xi\phi^{2}+6\xi^{2}\phi^{2}},$ (A.11)
$\displaystyle\gamma^{2}_{12}$
$\displaystyle=\gamma^{2}_{21}=\frac{1}{\phi(1+\xi\phi^{2})}.$ (A.12)

Here the index $1$ and $2$ correspond to $\phi$ and $\chi$, respectively. The
Ricci scalar of this geometry becomes

$\displaystyle\mathscr{R}=\frac{4\xi(1+3\xi+\xi\phi^{2}+6\xi^{2}\phi^{2})}{(1+\xi\phi^{2}+6\xi^{2}\phi^{2})^{2}}.$
(A.13)

We note $\mathscr{R}\to 0$ as the non-minimal coupling $\xi\to 0$.

##  Appendix B Slow-roll and slow-turn parameters

In the space of field $\varphi^{a}$, the velocity is given by its derivative
with respect to the e-folding, i.e., $d\varphi^{a}/dN$, and the (covariant)
acceleration of the field vector is
$\eta^{a}=d^{2}\varphi^{a}/dN^{2}+\gamma^{a}_{bc}(d\varphi^{c}/dN)(d\varphi^{b}/dN)$.
The slow-roll and slow-turn parameters are defined by the (half) size of the
velocity vector and the parallel and vertical components of the acceleration
vector as [46]:

$\displaystyle\varepsilon$
$\displaystyle=\frac{1}{2}K_{ab}\frac{d\varphi^{a}}{dN}\frac{d\varphi^{b}}{dN},$
(B.1) $\displaystyle\frac{\eta_{\parallel}^{a}}{v}$
$\displaystyle=\frac{1}{v}\,K_{ab}\hat{e}_{\parallel}^{a}\Big{(}\frac{d^{2}\varphi^{b}}{dN^{2}}+\gamma^{b}_{dc}\frac{d\varphi^{c}}{dN}\frac{d\varphi^{d}}{dN}\Big{)},$
(B.2) $\displaystyle\frac{\eta_{\perp}^{a}}{v}$
$\displaystyle=\frac{1}{v}\,K_{ab}\hat{e}_{\perp}^{a}\Big{(}\frac{d^{2}\varphi^{b}}{dN^{2}}+\gamma^{b}_{dc}\frac{d\varphi^{c}}{dN}\frac{d\varphi^{d}}{dN}\Big{)}.$
(B.3)

For the acceleration vector, dividing by the speed $v$ implies to express the
rate of the change of the field velocity. When the slow-roll and slow-turn
approximation are satisfied,
$\varepsilon,\,|\eta_{\parallel}|/v,\,\eta_{\perp}/v\ll 1$, these parameters
can be explicitly written by the inflaton potential and the field space
metric, as given in (2.19)–(2.21).

We here present the slow-roll and slow-turn parameters in our two-field
inflation model, especially in the approximation used in the text.

When the soft-breaking mass $m_{\chi}$ is sufficiently large, the inflation is
mainly driven by the $\chi$ potential, and the slow-roll parameters are given
by

$\displaystyle\varepsilon$
$\displaystyle\approx\frac{2}{\phi^{2}}\bigg{[}\frac{(1-\xi\phi^{2})^{2}}{1+\xi\phi^{2}+6\xi^{2}\phi^{2}}+(1+\xi\phi^{2})\cot^{2}\chi\bigg{]},$
(B.4) $\displaystyle\frac{\eta_{\parallel}}{v}$
$\displaystyle\approx\frac{1+\xi\phi^{2}}{\phi^{2}}\Bigl{[}\,\frac{2(1-\xi\phi^{2})^{2}(1+3\xi\phi^{2}+12\xi^{2}\phi^{2})\tan^{2}\chi}{(1+\xi\phi^{2}+6\xi^{2}\phi^{2})^{2}}$
$\displaystyle\quad+\csc^{2}\chi\bigl{[}1+5\xi\phi^{2}+12\xi^{2}\phi^{2}+2\xi^{2}\phi^{4}+12\xi^{3}\phi^{4}+(1-\xi\phi^{2})\cos
2\chi\bigr{]}$
$\displaystyle\qquad+4(1-\xi\phi^{2})\Big{]}\Big{/}\Big{[}(1-\xi\phi^{2})^{2}\tan^{2}\chi+(1+\xi\phi^{2})(1+\xi\phi^{2}+6\xi^{2}\phi^{2})\Big{]}.$
(B.5)

Both parameters do not depend on $m_{\chi}$ since the inflaton $\chi$
potential is proportional to it.

When $\phi$ takes a larger value than $v_{\phi}$ during the inflation, the
slow-roll parameters are expressed as

$\displaystyle\varepsilon$
$\displaystyle\approx\frac{8}{\lambda^{2}\xi(1+6\xi)\phi^{4}}\Big{[}\big{[}\lambda(1+\xi
v_{\phi}^{2})-2\xi
m_{\chi}^{2}\sin^{2}\chi\big{]}^{2}+m_{\chi}^{4}\xi^{2}(1+6\xi)\sin^{2}2\chi\Big{]},$
(B.6) $\displaystyle\frac{\eta_{\parallel}}{v}$
$\displaystyle\approx\frac{8\xi}{\lambda\phi^{2}}\biggl{[}\frac{\big{[}\lambda(1+\xi
v_{\phi}^{2})-2\xi
m_{\chi}^{2}\sin^{2}\chi\big{]}^{3}}{\xi^{2}(1+6\xi)^{2}}+\frac{2\big{[}\lambda(1+\xi
v_{\phi}^{2})-2\xi
m_{\chi}^{2}\sin^{2}\chi\big{]}m_{\chi}^{4}\sin^{2}2\chi}{1+6\xi}$
$\displaystyle\qquad-\xi m_{\chi}^{6}\cos
2\chi\sin^{2}2\chi\biggr{]}\bigg{/}\biggl{[}\frac{\big{[}\lambda(1+\xi
v_{\phi}^{2})-2\xi m_{\chi}^{2}\sin^{2}\chi\big{]}^{2}}{\xi(1+6\xi)}+\xi
m_{\chi}^{4}\sin^{2}2\chi\biggr{]},$ (B.7)

On the other hand, in the small field region with $\phi<v_{\phi}$, the slow-
roll parameters become

$\displaystyle\varepsilon$
$\displaystyle\approx\frac{8\phi^{2}}{\lambda^{2}v_{\phi}^{8}}\Big{[}\big{[}\lambda
v_{\phi}^{2}(1+\xi
v_{\phi}^{2})-2m_{\chi}^{2}\sin^{2}\chi\big{]}^{2}+m_{\chi}^{4}\sin^{2}2\chi\Big{]},$
(B.8) $\displaystyle\frac{\eta_{\parallel}}{v}$
$\displaystyle\approx\frac{4}{\lambda v_{\phi}^{4}}\Big{[}\big{[}\lambda
v_{\phi}^{2}(1+\xi
v_{\phi}^{2})-2m_{\chi}^{2}\sin^{2}\chi\big{]}^{3}+3\big{[}\lambda
v_{\phi}^{2}(1+\xi
v_{\phi}^{2})-2m_{\chi}^{2}\sin^{2}\chi\big{]}m_{\chi}^{4}\sin^{2}2\chi$
$\displaystyle\qquad-2m_{\chi}^{6}\cos
2\chi\sin^{2}2\chi\Big{]}\Big{/}\Big{[}\big{[}\lambda v_{\phi}^{2}(1+\xi
v_{\phi}^{2})-2m_{\chi}^{2}\sin^{2}\chi\big{]}^{2}+m_{\chi}^{4}\sin^{2}2\chi\Big{]}.$
(B.9)

##  Appendix C Formulae of cosmological observables

###  C.1 Transfer functions

In the multi-field inflation, the isocurvature mode turn into the curvature
fluctuations, thus, it is important to represent the expression for the
transport functions describing the super-horizon evolution of the
perturbation, according to Ref. [46]. The effective mass matrix is defined as

$\displaystyle M^{a}{}_{b}\coloneqq K^{ac}(\partial_{c}\partial_{b}\ln
V-\gamma^{d}_{cb}\partial_{d}\ln
V)+\frac{1}{3}\varepsilon\mathscr{R}\hat{e}_{\perp}^{a}K_{bc}\hat{e}_{\perp}^{c}.$
(C.1)

With the expression of the basis vector (2.23) and the Ricci scalar in the
field space (A.13) at hand, the matrix $M$ is fully determined by the
couplings and background field values in the present inflation model. With
this matrix, the transfer functions are defined as

$\displaystyle T_{\mathcal{SS}}$
$\displaystyle=\exp\bigg{[}\int_{N_{*}}^{N}dN^{\prime}(K_{ac}\hat{e}_{\parallel}^{c}M^{a}{}_{b}\hat{e}_{\parallel}^{b}-K_{ac}\hat{e}_{\perp}^{c}M^{a}{}_{b}\hat{e}_{\perp}^{b})\bigg{]},$
(C.2) $\displaystyle T_{\mathcal{RS}}$
$\displaystyle=\int_{N_{*}}^{N}dN^{\prime}\,\frac{\eta_{\perp}}{v}\exp\bigg{[}\int_{N_{*}}^{N^{\prime}}dN^{\prime\prime}(K_{ac}\hat{e}_{\parallel}^{c}M^{a}{}_{b}\hat{e}_{\parallel}^{b}-K_{ac}\hat{e}_{\perp}^{c}M^{a}{}_{b}\hat{e}_{\perp}^{b})\bigg{]}.$
(C.3)

These are also the functions of model parameters and field values through the
expressions of the basis vectors and the slow-roll parameter (2.21). Note that
the transfer functions (C.2) and (C.3) always take positive values.

###  C.2 Cosmological observables

Here we give the exact form of the cosmological observables. First, the power
spectrum and the tensor-to-scalar ratio are given as the following explicit
forms in our model,

$\displaystyle\mathcal{P_{R}}$
$\displaystyle=\frac{(1+\xi\phi_{*}^{2})^{4}(1+T_{\mathcal{RS}}^{2})V(\phi_{*},\chi_{*})^{3}}{3\pi^{2}\phi_{*}^{2}}$
$\displaystyle\qquad\bigg{/}\bigg{[}\frac{[\lambda(1+\xi
v_{\phi}^{2})(\phi_{*}^{2}-v_{\phi}^{2})+2m_{\chi}^{2}(1-\xi\phi_{*}^{2})\sin^{2}\chi_{*}]^{2}}{1+\xi\phi_{*}^{2}+6\xi^{2}\phi_{*}^{2}}+m_{\chi}^{4}(1+\xi\phi_{*}^{2})\sin^{2}2\chi_{*}\bigg{]},$
(C.4) $\displaystyle r$
$\displaystyle=\frac{2\phi_{*}^{2}}{(1+\xi\phi_{*}^{2})^{4}(1+T_{\mathcal{RS}}^{2})V(\phi_{*},\chi_{*})^{2}}$
$\displaystyle\qquad\times\bigg{[}\frac{[\lambda(1+\xi
v_{\phi}^{2})(\phi_{*}^{2}-v_{\phi}^{2})+2m_{\chi}^{2}(1-\xi\phi_{*}^{2})\sin^{2}\chi_{*}]^{2}}{1+\xi\phi_{*}^{2}+6\xi^{2}\phi_{*}^{2}}+m_{\chi}^{4}(1+\xi\phi_{*}^{2})\sin^{2}2\chi_{*}\bigg{]}.$
(C.5)

The power spectrum consisting of the correlation function of scalar
fluctuations depends on the positive power of $T_{\mathcal{RS}}$. On the other
hand, $r$, which consists of the ratio of tensor fluctuations to scalar ones,
depends on the inverse power of $T_{\mathcal{RS}}$. The spectral index $n_{s}$
and its running $\alpha$ are also written by the field values in the present
model under the slow-roll and slow-turn approximation [46],

$\displaystyle n_{s}$
$\displaystyle=1-2\varepsilon_{*}+2K_{ab*}\hat{e}_{N*}^{a}M^{b}{}_{c*}\hat{e}_{N*}^{c},$
(C.6) $\displaystyle\alpha$ $\displaystyle=[-K^{ab}\partial_{a}\ln
V\partial_{b}(-2\varepsilon+2K_{cd}\hat{e}_{N}^{c}M^{d}{}_{e}\hat{e}_{N}^{e})]_{*},$
(C.7)

where

$\displaystyle\hat{\bm{e}}_{N}=\frac{1}{\sqrt{1+T_{\mathcal{RS}}^{2}}}\hat{\bm{e}}_{\parallel}+\frac{T_{\mathcal{RS}}}{\sqrt{1+T_{\mathcal{RS}}^{2}}}\hat{\bm{e}}_{\perp}.$
(C.8)

The isocurvature fraction is determined by the transfer functions as

$\displaystyle\beta_{\mathrm{iso}}=\frac{T_{\mathcal{SS}}^{2}}{1+T_{\mathcal{RS}}^{2}+T_{\mathcal{SS}}^{2}}.$
(C.9)

Using the explicit expressions of the basis vectors (2.22), (2.23) and the
transfer functions (C.2), (C.3) defined by using the matrix (C.1), all these
cosmological observables can also be expressed by the parameters and field
values in the present inflation model. In the text, various approximations of
these formulae are presented, depending on the magnitude of coupling constants
and field values.

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