# Dynamics of inflation with mutually orthogonal vector fields in a closed
universe

Tomoaki Murata<EMAIL_ADDRESS>Department of Physics, Rikkyo University,
Toshima, Tokyo 171-8501, Japan  Tsutomu Kobayashi<EMAIL_ADDRESS>Department of Physics, Rikkyo University, Toshima, Tokyo 171-8501, Japan

###### Abstract

We study the dynamics of a homogeneous, isotropic, and positively curved
universe in the presence of a SU(2) gauge field or a triplet of mutually
orthogonal vector fields. In the SU(2) case we use the previously known ansatz
for the gauge-field configuration, but the case without non-abelian symmetries
is more nontrivial and we develop a new ansatz. We in particular consider
axion-SU(2) inflation and inflation with vector fields having
U(1)$\times$U(1)$\times$U(1) symmetry, and analyze their dynamics in detail
numerically. Novel effects of the spatial curvature come into play through
vector fields, which causes unconventional pre-inflationary dynamics. It is
found that the closed universe with vector fields is slightly more stable
against collapse than that filled solely with an inflaton field.

††preprint: RUP-21-12

##  I Introduction

Inflation [1, 2, 3] is an accelerated expansion that occurred in the early
Universe. It can explain homogeneity, isotropy, and spatial flatness of the
Universe in a natural way. Furthermore, it can give rise to primordial
fluctuations that are consistent with observations of the CMB and large-scale
structure of the Universe. Inflation was thus introduced to resolve the
problems on the initial conditions in the standard Big-Bang model. This does
not, however, necessarily mean that the Universe is homogeneous, isotropic,
and spatially flat at around the beginning of inflation. Therefore, it is
important to assess to what extent inflation is likely to occur and thus to
erase inhomogeneity, anisotropy, and spatial curvature starting from generic
initial conditions.

In spite of its importance, not so many papers (as compared to a huge number
of papers on inflation) have been devoted to the problem of the initial
conditions for inflation. For example, Ref. [4] showed that homogeneous and
anisotropic Bianchi models except Bianchi type-IX universes always evolve
toward an isotropic attractor in the presence of a positive cosmological
constant. The exceptional case corresponds to the universe with positive
spatial curvature; the universe would collapse if the spatial curvature is as
large as the cosmological constant. As shown in Ref. [5], the positive spatial
curvature reduces (but not significantly) the fraction of the initial
conditions that lead to successful inflation in the case of a massive scalar
inflaton field (see also Ref. [6]). The effect of the initial inhomogeneity on
the onset of inflation has been addressed in Refs. [7, 8, 9, 10, 11, 12, 13,
14, 15, 16]. More recently, the problem of the initial inhomogeneity was
investigated in the context of multi-field inflation, with somewhat nontrivial
results [17].

While most of the inflationary models are based on one or more scalar fields,
models with a triplet of vector fields have been proposed recently. Vector
fields are apparently incompatible with isotropic cosmology, but by assuming a
triplet of mutually orthogonal vector fields one can achieve an isotropic
configuration. An earlier model of inflation driven by such vector fields
nonminimally coupled to gravity is given in Ref. [18]. An interesting example
in this class of models motivated by particle physics is chromo-natural (or
axion-SU(2)) inflation [19] (see Ref. [20] for a review). Cosmological models
with multiple generalized Proca fields [21, 22] and three copies of U(1)
vector fields [23] have also been considered in the literature. See also Refs.
[24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38] for models of
similar kinds.

The problem of initial conditions in the above inflationary models with vector
fields is more subtle and less studied than that in usual inflation where only
a scalar field participates in the dynamics. In the context of the axion-SU(2)
model, Bianchi type-I anisotropic cosmology has been discussed [39, 40], with
the conclusion that the initial anisotropies always dilute away immediately
[41]. However, the dynamics of the other Bianchi types of the axion-SU(2)
system is still unclear (see however Ref. [42] for the dynamics of Bianchi
universes in the presence of an SU(2) gauge field coupled to a particular
scalar field theory). The problem of initial conditions in the other vector
field models is also awaited to be explored.

To take a step forward, in this paper, we explore the spatially curved
generalization of homogeneous and isotropic cosmologies in the axion-SU(2)
model and in (certain variants of) multiple generalized Proca theories, and
study the dynamics of the pre-inflationary universe. It is not trivial to
introduce a triplet of dynamical vector fields in such a way that their
configuration is consistent with homogeneity and isotropy in a spatially
curved universe. In fact, the consistent ansatz for the SU(2) gauge field
configuration in a curved universe has long been known [43, 44, 45]. However,
the ansatz introduced in Ref. [43, 44, 45] relies on the non-abelian-specific
structure, and hence it cannot be extended straightforwardly to the case, for
example, of the vector field model with the U(1)$\times$U(1)$\times$U(1)
symmetry [23]. This point is also addressed in the present paper.

The paper is organized as follows. In Sec. II, we add the spatial curvature to
the axion-SU(2) model, and then discuss its consequences on the cosmological
dynamics on the basis of the analytic argument and the results of numerical
calculations. In Sec. III, we consider spatially curved cosmological models in
the presence of a triplet of (generalized) Proca fields, with a focus on two
particular examples in the literature. Finally, we draw our conclusions in
Sec. IV, with a comment on the extension of the present work to the Bianchi
type-IX geometry.

##  II Axion-SU(2) in a Curved Universe

###  II.1 Basic equations

The axion-SU(2) inflation model [19] is described by the Lagrangian

$\displaystyle{\cal L}_{\rm CN}$ $\displaystyle=\frac{M_{\rm
Pl}^{2}}{2}R-\frac{1}{4}F_{\mu\nu}^{a}F_{a}^{\mu\nu}-\frac{1}{2}(\partial\chi)^{2}-V(\chi)$
$\displaystyle\quad-\frac{\lambda}{4f}\chi\widetilde{F}_{\mu\nu}^{a}F^{\mu\nu}_{a},$
(1)

where $\chi$ is the axion field,

$\displaystyle V(\chi)$
$\displaystyle=\mu^{4}\left(1+\cos\frac{\chi}{f}\right),$ (2)

is its potential, and $F_{\mu\nu}^{a}$ is the field strength defined as

$\displaystyle
F_{\mu\nu}^{a}=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}+g_{A}\epsilon^{a}_{\;bc}A_{\mu}^{b}A_{\nu}^{c}.$
(3)

The last term in Eq. (1) is written more explicitly as

$\displaystyle\widetilde{F}_{\mu\nu}^{a}F^{\mu\nu}_{a}$
$\displaystyle=\frac{1}{2}\varepsilon^{\mu\nu\rho\lambda}F_{\mu\nu}^{a}F_{\rho\lambda}^{a},$
(4)

with $\varepsilon^{\mu\nu\rho\lambda}=\epsilon^{\mu\nu\rho\lambda}/\sqrt{-g}$
being the Levi-Cività tensor and $\epsilon^{0123}=1$.

A flat FLRW universe is compatible with the SU(2) gauge field configuration

$\displaystyle A_{0}^{a}=0,\quad A_{i}^{a}=a(t)\psi(t)\delta_{i}^{a}.$ (5)

In contrast, it is not so straightforward to see whether a curved universe is
also compatible with the vector fields having nonvanishing spatial components.
However, actually it has long been known that there is a homogeneous and
isotropic SU(2) gauge field configuration for a curved universe (see, e.g.,
Refs. [43, 44, 45]). Following these previous works, here we derive the basic
equations governing the axion and SU(2) gauge field dynamics in a spatially
curved universe.

The metric for a spatially curved FLRW universe is given by

$\displaystyle\mathrm{d}s^{2}=-N^{2}(t)\mathrm{d}t^{2}+a^{2}(t)\left[\mathrm{d}r^{2}+S^{2}(r)\left(\mathrm{d}\theta^{2}+\sin^{2}\theta\mathrm{d}\varphi^{2}\right)\right],$
(6)

where

$\displaystyle
S(r)=\frac{\sin(\sqrt{\mathcal{K}}r)}{\sqrt{\mathcal{K}}},\quad\frac{\sinh(\sqrt{-\mathcal{K}}r)}{\sqrt{-\mathcal{K}}}$
(7)

for the closed ($\mathcal{K}>0$) and open ($\mathcal{K}<0$) cases,
respectively. In our notation, ${\cal K}$ has the dimension of (length)-2. We
may always set $N=1$.

The gauge field configuration compatible with a homogeneous, isotropic, and
spatially curved FLRW universe is given as follows. First, we fix the gauge
freedom so that the time component vanishes. Then, following [44], we consider
the ansatz

$\displaystyle g_{A}A_{0}^{a}=0,$ (8) $\displaystyle g_{A}A_{1}^{a}=a\psi
L_{1}^{a},$ (9) $\displaystyle g_{A}A_{2}^{a}=a\psi
SL_{2}^{a}-\left(1-\sqrt{1-\mathcal{K}S^{2}}\right)L_{3}^{a},$ (10)
$\displaystyle
g_{A}A_{3}^{a}=\left[\left(1-\sqrt{1-\mathcal{K}S^{2}}\right)L_{2}^{a}+a\psi
SL_{3}^{a}\right]\sin\theta,$ (11)

where $\psi=\psi(t)$ is a function of $t$ only and $L_{i}^{a}$ is defined as

$\displaystyle
L_{1}^{a}=(\sin\theta\cos\varphi,\sin\theta\sin\varphi,\cos\theta),$ (12)
$\displaystyle
L_{2}^{a}=(\cos\theta\cos\varphi,\cos\theta\sin\varphi,-\sin\theta),$ (13)
$\displaystyle L_{3}^{a}=(-\sin\varphi,\cos\varphi,0).$ (14)

(See also [46] for a detailed derivation.) For this gauge field configuration,
we see that the following terms in the Lagrangian depend only on $t$:

$\displaystyle
F_{\mu\nu}^{a}F_{a}^{\mu\nu}=-\frac{6}{g_{A}^{2}N^{2}}\left[\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)^{2}-N^{2}\left(\psi^{2}-\frac{\mathcal{K}}{a^{2}}\right)^{2}\right],$
(15)
$\displaystyle\widetilde{F}_{\mu\nu}^{a}F_{a}^{\mu\nu}=\frac{12}{g_{A}^{2}N}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)\left(\psi^{2}-\frac{\mathcal{K}}{a^{2}}\right),$
(16)

showing that the above ansatz is indeed consistent with a homogeneous,
isotropic, and spatially curved FLRW universe. Here a dot denotes
differentiation with respect to $t$. Note that ${\cal K}$ appears directly in
Eqs. (15) and (16), which implies that the axion-SU(2) dynamics depends
nontrivially on the spatial curvature. In other words, the effective potential
for $\psi$ depends explicitly on ${\cal K}$. The previous spatially flat
result can be reproduced by rescaling $\psi$ as $\psi=g_{A}\psi_{\rm
previous}$ and setting ${\cal K}=0$. In the spatially flat case, the rescaling
of $\psi$ allows us to consider the $g_{A}\to 0$ limit. However, in the case
of ${\cal K}\neq 0$, one cannot take such a smooth $g_{A}\to 0$ limit. In this
sense, the fact that the vector field is the SU(2) gauge field is crucial for
the above ansatz.

From the Lagrangian (1) and our ansatz for the gauge field, we obtain the
field equations as

$\displaystyle\ddot{\chi}+3H\dot{\chi}-\frac{\mu^{4}}{f}\sin\frac{\chi}{f}=-\frac{3\lambda}{fg_{A}^{2}}\left(\dot{\psi}+H\psi\right)\left(\psi^{2}-\frac{\mathcal{K}}{a^{2}}\right),$
(17)
$\displaystyle\ddot{\psi}+3H\dot{\psi}+\left(\dot{H}+2H^{2}\right)\psi=\left(\frac{\lambda}{f}\dot{\chi}-2\psi\right)\left(\psi^{2}-\frac{\mathcal{K}}{a^{2}}\right),$
(18)

where $H:=\dot{a}/a$ is the Hubble parameter and we set $N=1$. The Einstein
equations read

$\displaystyle 3M_{\rm Pl}^{2}\left(H^{2}+\frac{\mathcal{K}}{a^{2}}\right)$
$\displaystyle=\rho_{\chi}+\rho_{\psi},$ (19) $\displaystyle-2M_{\rm
Pl}^{2}\left(\dot{H}-\frac{\mathcal{K}}{a^{2}}\right)$
$\displaystyle=\rho_{\chi}+\rho_{\psi}+p_{\chi}+p_{\psi},$ (20)

where

$\displaystyle\rho_{\chi}$
$\displaystyle=\frac{\dot{\chi}^{2}}{2}+\mu^{4}\left(1+\cos\frac{\chi}{f}\right),$
(21) $\displaystyle p_{\chi}$
$\displaystyle=\frac{\dot{\chi}^{2}}{2}-\mu^{4}\left(1+\cos\frac{\chi}{f}\right),$
(22) $\displaystyle\rho_{\psi}$
$\displaystyle=\frac{3}{2g_{A}^{2}}\left[\left(\dot{\psi}+H\psi\right)^{2}+\left(\psi^{2}-\frac{\mathcal{K}}{a^{2}}\right)^{2}\right],$
(23) $\displaystyle p_{\psi}$ $\displaystyle=\frac{\rho_{\psi}}{3}.$ (24)

We will solve these equations numerically to see the nontrivial dynamics
brought by the spatial curvature.

###  II.2 Numerical Results

For a given set of parameters we impose the initial conditions
$(\chi_{0},\psi_{0},\dot{\chi}_{0},\dot{\psi}_{0},H_{0})$ at $a=a_{0}=1$ so
that they satisfy the Friedmann equation (19). We then solve numerically the
dynamical equations (17), (18), and (20). We confirm that the Friedmann
equation is satisfied at each time step. The parameters used in our numerical
calculations (except for the ones in which $g_{A}$ is varied) are listed in
Table 1. (In the actual numerical calculations we adopt the units $M_{\rm
Pl}=1$.) Here we introduce for convenience the scale of the (would-be)
inflationary Hubble parameter, $H_{*}$, defined as $H_{*}:=\mu^{2}/M_{\rm
Pl}$. We set $H_{*}$ to be the GUT scale.

Table 1: Parameters $f$ | $\mu$ | $\lambda$ | $g_{A}$ | $H_{*}$
---|---|---|---|---
$10^{-2}M_{\rm Pl}$ | $10^{-3}M_{\rm Pl}$ | $2\times 10^{2}$ | $2\times 10^{-6}$ | $10^{-6}M_{\rm Pl}$

We are interested in particular in the case of a closed universe. To clarify
the role of the gauge field in the inflationary dynamics with ${\cal K}>0$,
let us first consider the case without the gauge field. In this case, the
curvature term acts as effective negative energy density. Suppose that at some
initial moment the curvature term is as large as $M_{\rm Pl}^{2}{\cal
K}/a^{2}\sim V\sim\mu^{4}$. Such a universe would typically collapse within
one e-fold. If the kinetic energy of the axion field is sufficiently large
(i.e., $M_{\rm Pl}^{2}H^{2}\sim\dot{\chi}^{2}\gg M_{\rm Pl}^{2}{\cal
K}/a^{2}\sim V$), the universe could avoid to collapse, but the axion would
then roll down to the potential minimum very rapidly, preventing prolonged
inflation.

Let us next switch on the gauge field and investigate the axion-SU(2) dynamics
in a flat universe [19]. It is assumed that $\dot{H}$, $\ddot{\chi}$, and
$\ddot{\psi}$ can be ignored in Eqs. (17) and (18) (with ${\cal K}=0$). We can
then rearrange these equations so that $\dot{\chi}$ and $\dot{\psi}$ are
expressed in terms of $\chi$ and $\psi$, and find

$\displaystyle 3H\dot{\psi}\simeq-V_{{\rm eff},\psi},$ (25)

with

$\displaystyle V_{{\rm
eff},\psi}=3H^{2}\psi-\frac{g_{A}^{2}\mu^{4}H\sin(\chi/f)}{\lambda\psi^{2}},$
(26)

where we assumed that

$\displaystyle fg_{A}H\ll\lambda\psi^{2}.$ (27)

Although this effective potential is obtained by making some approximations
and assumptions, it is useful for understanding roughly the evolution of the
system. The minimum of the effective potential is given by

$\displaystyle\psi_{\rm min}=\left(\frac{g_{A}^{2}\mu^{4}}{3\lambda
H}\sin\frac{\chi}{f}\right)^{1/3}.$ (28)

One usually assumes that $\psi$ takes this value from the beginning,
$\psi\simeq\psi_{\rm min}$, and study the dynamics of the axion $\chi$.
However, the actual dynamics of the axion depends on the initial condition for
$\psi$. If $\psi\gtrsim\psi_{\rm min}$ initially, it rolls down to the minimum
of the effective potential within a few e-folds, leading to successful
$\psi$-assisted inflation as proposed originally in [19]. If $\psi$ is
displaced from $\psi_{\rm min}$ to the left to some extent initially, then the
axion field quickly falls down into the minimum of its potential without
sufficient inflation. The reason is as follows. For sufficiently small $\psi$,
the right-hand side of Eq. (17) (with ${\cal K}=0$), i.e., the interaction
term between the axion and the gauge field, is much smaller than the bare
potential of $\chi$. Thus, $\chi$ quickly rolls down to the minimum before
$\psi$ becomes sufficiently large to assist slow-roll inflation. This occurs
when the condition (27) is violated.

Figure 1 shows the e-folding number,

$\displaystyle{\cal N}:=\int_{t_{\rm i}}^{t_{\rm f}}H{\rm d}t,$ (29)

calculated for different initial conditions for the gauge field. Here, $t_{\rm
i}$ is the initial time and $t_{\rm f}$ is defined as the time at which the
energy density of $\chi$ becomes as small as $\rho_{\chi}=10^{-3}M_{\rm
Pl}^{2}H_{*}^{2}$. If this does not occur and inflation lasts for 60 e-folds,
we stop the calculation and just set ${\cal N}=60$. The initial condition for
the axion is fixed as

$\displaystyle\left[\begin{array}[]{c}\chi_{0}\\\
\dot{\chi}_{0}\end{array}\right]=\left[\begin{array}[]{c}\pi f\times
10^{-2}\\\ 0\end{array}\right].$ (34)

For our choice of the parameters, we have $\psi_{\rm min}/g_{A}\simeq 0.07$,
and we can see from Fig. 1 the aforementioned behavior, though the dynamics
depends also on the initial value of $\dot{\psi}$.

Figure 1: Initial condition dependence of the e-folding number in the flat
model. The parameters are given in Table 1.

We now move to the discussion on the case with ${\cal K}>0$. The evolution of
the system would be more complicated due to the various ${\cal K}$-dependent
terms in the basic equations. Now positive spatial curvature does not simply
act as effective negative energy density, and hence large positive ${\cal K}$
does not necessarily make the universe collapse. Spatial curvature also
modifies the shapes of the effective potentials for $\chi$ and $\psi$, as seen
from Eqs. (17) and (18), and thus affects directly the evolution of these
fields.

Let us estimate the effect of spatial curvature in Eq. (19). Assuming that the
curvature is large and the kinetic energy of the fields is small enough, Eq.
(19) reduces to

$\displaystyle 3M_{\rm Pl}^{2}H^{2}$
$\displaystyle\simeq\frac{3}{2g_{A}^{2}}\frac{{\cal K}^{2}}{a^{4}}-3M_{\rm
Pl}^{2}\frac{{\cal K}}{a^{2}}+V,$ (35)

and $V\simeq 2\mu^{4}=2M_{\rm Pl}^{2}H_{*}^{2}$. This can be recast into a
dimensionless form as

$\displaystyle 1\simeq\frac{1}{2}\left(\frac{H}{g_{A}M_{\rm
Pl}}\right)^{2}\left(\frac{{\cal K}}{a^{2}H^{2}}\right)^{2}-\frac{{\cal
K}}{a^{2}H^{2}}+\frac{2}{3}\left(\frac{H_{*}}{H}\right)^{2}.$ (36)

We consider the case where $H\sim H_{*}$ and ${\cal K}/a^{2}H_{*}^{2}\sim 1$.
It can then be seen that if $g_{A}\sim H_{*}/M_{\rm Pl}$, the first term in
the right-hand side can cancel the second term, supporting the universe
against collapse. For larger $g_{A}$, the first term becomes less important.
Note that $g_{A}$ cannot be too small in order for the Friedmann equation to
be satisfied in the universe with large spatial curvature.

The above argument can be verified by the numerical result shown in Fig. 2. We
run the numerical code for different values of ${\cal K}$ and $g_{A}$. The
initial conditions for the axion and the Hubble parameter are given
respectively by Eq. (34) and $H_{0}=H_{*}$, while the initial value of the
gauge field is given by $\psi_{0}=0.05\times g_{A}$. The initial velocity of
the gauge field, $\dot{\psi}_{0}$, is then determined from the Friedmann
equation (19), which is different for different $(g_{A},{\cal K})$. For ${\cal
K}$ above a certain value the universe immediately collapses. In Fig. 2 this
critical value of ${\cal K}$ is shown as a function of $g_{A}$. We see that
the universe with smaller $g_{A}$ is more stable against the inclusion of
positive spatial curvature.

Figure 2:  The critical values of ${\cal K}$ are shown as black dots, above
which the universe immdediately collapses.

The initial condition dependence of the e-folding number is also altered by
the presence of spatial curvature, as can be seen by comparing Fig. 1 with
Fig. 3. We find that, as long as the initial velocity of $\psi$ is small, we
have a sufficient duration of the inflationary phase for a wide range of
initial conditions $\psi_{0}$. Note here that the initial conditions plotted
in Fig. 3 all satisfy $\psi_{0}^{2}\ll{\cal K}/H_{*}^{2}$. The typical
dynamics of the axion and gauge fields in this case is shown in Fig. 4, which
is different from the conventional one in the early stage, but leads in the
end to prolonged inflation. In the early stage, the interaction term in the
right-hand side of Eq. (17) is maintained even for tiny $\psi_{0}$ for ${\cal
O}(1)$ e-folds, because we have $\psi^{2}-{\cal K}/a^{2}\simeq-{\cal
K}/a^{2}$. Then, both fields change their values rapidly at around
$H_{*}t\simeq 3$. This corresponds to the moment at which the right-hand sides
of Eqs. (17) and (18) vanishes, $\psi^{2}\simeq{\cal K}/a^{2}$. After that,
$\psi$-assisted slow-roll inflation occurs at the different side of the
potential from the one where the axion field is initially placed. We thus have
the unconventional pre-inflationary dynamics leading to successful inflation.

Figure 3: Initial condition dependence of the e-folding number in the closed
model with $\mathcal{K}=0.5H_{*}^{2}$. The initial condition for the axion
field is given by Eq. (34).

Figure 4: The evolution of the axion and gauge fields with the curvature
$\mathcal{K}=0.5H_{*}^{2}$ for the initial condition given by Eq. (34),
$\psi_{0}=10^{-7}M_{\rm Pl}$, and $\dot{\psi}_{0}=-10^{-12}M_{\rm Pl}^{2}$
(for which $H_{0}=0.555H_{*}$).

So far we have considered the case where the axion is the inflaton field. Let
us briefly comment on the case where there is another scalar field that plays
the role of the inflaton and the axion is a spectator field [47]. Suppose in
this spectator axion-SU(2) model that the inflaton’s potential energy is given
by $V_{\rm inf}\sim 2\alpha\mu^{4}$ ($\alpha>1$). This amounts to rescaling
$H_{*}$ as $H_{*}\to(1+\alpha)^{1/2}H_{*}$ in Eq. (35). Therefore, for larger
inflaton’s contribution (i.e., for larger $\alpha$), $g_{A}/H_{*}$ becomes
effectively smaller, rendering the universe more stable against the inclusion
of positive spatial curvature.

##  III Multiple vector fields without the non-abelian-specific structure

Let us move to the case of multiple generalized Proca fields [21, 22]. In this
case, the vector fields do not have the non-abelian-specific structure and
hence we cannot use the same ansatz in a spatially curved universe as in the
case of the SU(2) gauge field. In this section, we therefore start from
reconsidering the metric and vector field ansatz.

The Lagrangian for the model considered in Ref. [22] contains three vector
fields $A_{\mu}^{a}$ ($a=1,2,3$) and is given by

$\displaystyle{\cal L}=\mathcal{L}_{2}+\mathcal{L}_{4},$ (37)

with

$\displaystyle\mathcal{L}_{2}$
$\displaystyle=G_{2}\left(X,Y,Z,W_{1},W_{2},W_{3}\right),$ (38)
$\displaystyle\mathcal{L}_{4}$
$\displaystyle=G_{4}(X)R+G_{4,X}\left[\nabla_{\mu}A^{\mu}_{a}\nabla^{\nu}A_{\nu}^{a}-\nabla_{\mu}A_{\nu}^{a}\nabla^{\nu}A^{\mu}_{a}\right],$
(39)

where

$\displaystyle X:=-\frac{1}{2}A_{\mu}^{a}A^{\mu}_{a},\quad
Y:=-\frac{1}{4}F_{\mu\nu}^{a}F^{\mu\nu}_{a},$ $\displaystyle
Z:=-\frac{1}{4}F_{\mu\nu}^{a}\widetilde{F}^{\mu\nu}_{a},\quad
W_{1}:=A_{\mu}^{a}A^{\nu}_{a}F^{\mu\rho}_{b}F_{\nu\rho}^{b},$ $\displaystyle
W_{2}:=A_{\mu}^{a}A^{\nu}_{b}F^{\mu\rho}_{a}F_{\nu\rho}^{b},\quad
W_{3}:=A_{\mu}^{a}A^{\nu}_{b}F^{\mu\rho}_{b}F_{\nu\rho}^{a},$ (40)

and note that in the present case

$\displaystyle
F_{\mu\nu}^{a}:=\partial_{\mu}A_{\nu}^{a}-\partial_{\nu}A_{\mu}^{a}.$ (41)

To explore the configuration of the vector fields, we first note that the
metric of a closed universe can be written using the left-invariant 1-forms as

$\displaystyle\mathrm{d}s^{2}=-N^{2}(t)\mathrm{d}t^{2}+\frac{a^{2}(t)}{4}\delta_{ab}\omega^{a}\omega^{b},$
(42)

where

$\displaystyle\omega^{1}$
$\displaystyle=-\sin\left(\sqrt{\mathcal{K}}x^{3}\right)\mathrm{d}x^{1}+\sin\left(\sqrt{\mathcal{K}}x^{1}\right)\cos\left(\sqrt{\mathcal{K}}x^{3}\right)\mathrm{d}x^{2},$
(43) $\displaystyle\omega^{2}$
$\displaystyle=\cos\left(\sqrt{\mathcal{K}}x^{3}\right)\mathrm{d}x^{1}+\sin\left(\sqrt{\mathcal{K}}x^{1}\right)\sin\left(\sqrt{\mathcal{K}}x^{3}\right)\mathrm{d}x^{2},$
(44) $\displaystyle\omega^{3}$
$\displaystyle=\cos\left(\sqrt{\mathcal{K}}x^{1}\right)\mathrm{d}x^{2}+\mathrm{d}x^{3}.$
(45)

Notice that this expression can be used only for a closed universe. We then
assume that the vector fields take the form111In Appendix A, we show that the
same ansatz can also be used for the SU(2) gauge field, reproducing the same
result as in Sec. II up to field redefinition.

$\displaystyle A_{0}^{a}=0,\quad
A_{i}^{a}\mathrm{d}x^{i}=\frac{a}{2}\psi(t)\omega^{a},$ (46)

where we set $A_{0}^{a}=0$ because it is indeed a trivial solution. Under the
above ansatz we have

$\displaystyle X$ $\displaystyle=-\frac{3}{2}\psi^{2},$ (47) $\displaystyle Y$
$\displaystyle=\frac{3}{2}\left[\frac{1}{N^{2}}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)^{2}-\frac{4\mathcal{K}}{a^{2}}\psi^{2}\right],$
(48) $\displaystyle Z$
$\displaystyle=\frac{6}{N}\psi\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)\frac{\sqrt{\mathcal{K}}}{a},$
(49) $\displaystyle W_{1}$
$\displaystyle=-3\psi^{2}\left[\frac{1}{N^{2}}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)^{2}-\frac{8\mathcal{K}}{a^{2}}\psi^{2}\right],$
(50) $\displaystyle W_{2}$
$\displaystyle=-\frac{9}{N^{2}}\psi^{2}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)^{2},$
(51) $\displaystyle W_{3}$
$\displaystyle=-3\psi^{2}\left[\frac{1}{N^{2}}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)^{2}+\frac{8\mathcal{K}}{a^{2}}\psi^{2}\right],$
(52)

and

$\displaystyle\nabla_{\mu}A^{\mu}_{a}\nabla^{\nu}A_{\nu}^{a}-\nabla_{\mu}A_{\nu}^{a}\nabla^{\nu}A^{\mu}_{a}=-\frac{6}{N^{2}}\frac{\dot{a}}{a}\dot{\psi}\psi+\frac{6\mathcal{K}}{a^{2}}\psi^{2}.$
(53)

We see that each piece in the Lagrangian depends only on $t$, and therefore
the above assumed configuration of the vector fields is consistent with a
closed universe. In contrast to the spatially flat case [22], $Z$ is no longer
vanishing in a positively curved universe.

Let us present two applications of the our result. The first example is the
vector inflation model proposed in Ref. [18], whose Lagrangian is given by

$\displaystyle{\cal L}=\left(\frac{M_{\rm Pl}^{2}}{2}-\xi X\right)R+Y+m^{2}X,$
(54)

where $\xi$ and $m$ are constants, and below we will set $\xi=1/6$. Strictly
speaking, this does not belong to the multiple generalized Proca theory, as
the Lagrangian (54) is obtained by omitting $G_{4,X}$ by hand [22].
Nevertheless, one can use the ansatz (46) to analyze the dynamic of the vector
inflation model of [18] with ${\cal K}>0$, since, as seen from Eqs. (47) and
(48), both $X$ and $Y$ are dependent only on $t$. Our field equations are
given by

$\displaystyle\ddot{\psi}+3H\dot{\psi}+\left(m^{2}+\frac{3\mathcal{K}}{a^{2}}\right)\psi=0,$
(55) $\displaystyle 3M_{\rm
Pl}^{2}\left(H^{2}+\frac{\mathcal{K}}{a^{2}}\right)=\frac{3}{2}\left(\dot{\psi}^{2}+m^{2}\psi^{2}\right)+\frac{9\mathcal{K}}{2a^{2}}\psi^{2},$
(56) $\displaystyle 2M_{\rm
Pl}^{2}\left(\dot{H}-\frac{\mathcal{K}}{a^{2}}\right)=-3\dot{\psi}^{2}-\frac{3\mathcal{K}}{a^{2}}\psi^{2}.$
(57)

Open and closed FLRW models in the theory (54) have been studied earlier in
Ref. [48]. Our result disagrees with that of Ref. [48]; it seems that the
consistent ansatz for the vector fields has not been employed in [48]. The
sign of the vector-field-induced curvature terms in Eqs. (55)–(57) is opposite
compared to the result of [48], and accordingly the dynamics should be
qualitatively different as compared to [48].

The second example is the model with a triplet of vector fields coupled to a
scalar field. Obviously, the ansatz (46) can be used in such a case as well.
For example, the model studied in Ref. [23] is described by the Lagrangian

$\displaystyle\mathcal{L}=\frac{M_{\rm
Pl}^{2}}{2}R-\frac{1}{2}(\partial\phi)^{2}-V(\phi)+f^{2}(\phi)\left(Y+\theta
Z\right),$ (58)

where $\theta$ is a constant. In this case, the vector fields have the
U(1)$\times$U(1)$\times$U(1) symmetry. (One may thus use this symmetry to
eliminate $A_{0}^{a}$.) Let us investigate the cosmological background
dynamics of this model in more detail. The field equations are given by

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V^{\prime}-3ff^{\prime}(\dot{\psi}+H\psi)^{2}$
$\displaystyle+12ff^{\prime}\psi\left[\frac{{\cal
K}}{a^{2}}\psi+\theta\frac{\sqrt{\cal K}}{a}(\dot{\psi}+H\psi)\right]=0,$ (59)
$\displaystyle\ddot{\psi}+3H\dot{\psi}+\left(\dot{H}+2H^{2}\right)\psi+\frac{2f^{\prime}}{f}\dot{\phi}\left(\dot{\psi}+H\psi\right)$
$\displaystyle+4\psi\left(\frac{\mathcal{K}}{a^{2}}-\theta\frac{\sqrt{\mathcal{K}}}{a}\frac{f^{\prime}}{f}\dot{\phi}\right)=0,$
(60) $\displaystyle 3M_{\rm
Pl}^{2}\left(H^{2}+\frac{\mathcal{K}}{a^{2}}\right)=\frac{\dot{\phi}^{2}}{2}+V+\frac{6\mathcal{K}}{a^{2}}f^{2}\psi^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\quad+\frac{3}{2}f^{2}\left(\dot{\psi}+H\psi\right)^{2},$
(61) $\displaystyle M_{\rm Pl}^{2}\left(\dot{H}-\frac{\cal
K}{a^{2}}\right)=-\frac{\dot{\phi}^{2}}{2}-\frac{4\mathcal{K}}{a^{2}}f^{2}\psi^{2}-f^{2}\left(\dot{\psi}+H\psi\right)^{2},$
(62)

where a prime stands for differentiation with respect to $\phi$. In the
following, we will consider the particular form of the potential and the
coupling function given by

$\displaystyle V=\frac{1}{2}m^{2}\phi^{2},\quad f=f_{*}\exp\left[\frac{2M_{\rm
Pl}^{-2}}{1-I}\int^{\phi}_{\phi_{*}}\frac{V(\widetilde{\phi})}{V^{\prime}(\widetilde{\phi})}{\rm
d}\widetilde{\phi}\right],$ (63)

where $m$, $I$, $f_{*}$, and $\phi_{*}$ are constants. This particular case
was also studied in Ref. [23] and shown to admit an inflationary attractor
solution with nonvanishing vector fields [49].

For a fixed value of ${\cal K}/a_{0}^{2}H_{0}^{2}$, we numerically solve the
field equations to see whether the universe continues to expand toward the
inflationary attractor or stops expanding at a certain moment and eventually
collapses. Let us first switch off the vector fields by setting $f_{*}=0$ and
consider the case where ${\cal K}/a_{0}^{2}H_{0}^{2}=7.5$ at the initial
moment. We take $m=10^{-2}M_{\rm Pl}$. It is easy to see that for the initial
condition $(\phi_{0},\dot{\phi}_{0})=(10M_{\rm Pl},-0.1M_{\rm Pl}^{2})$ the
universe stops expanding within one e-fold and then collapses. We then include
the vector fields with $f(\phi_{0})=0.1$, $I=0.1$ and $\theta=0$. For the
initial condition $(\phi_{0},\dot{\phi}_{0},\psi_{0},\dot{\psi}_{0})=(10M_{\rm
Pl},-0.1M_{\rm Pl}^{2},M_{\rm Pl},0)$, we find that the universe expands
toward the inflationary attractor rather than collapses even in the presence
of such a large spatial curvature at the initial moment. This implies that the
triplet of vector fields can save the universe from collapsing.

Figure 5 shows the numerical results for different initial conditions
satisfying $(1/2)\dot{\phi}_{0}^{2}+V(\phi_{0})=10^{-2}M_{\rm Pl}^{4}$, with
$(\psi_{0},\dot{\psi}_{0})=(M_{\rm Pl},0)$. The parameters are the same as
above: ${\cal K}/a_{0}^{2}H_{0}^{2}=7.5$, $m=10^{-2}M_{\rm Pl}$, $I=0.1$, and
$\theta=0$. For each calculation we introduce $f_{*}$ and $\phi_{*}$ so that
$f(\phi_{0})=0.1$, We see that the fraction of initial conditions leading to
successful inflation indeed increases with the help of the vector fields,
though the effect is not so significant.

Figure 5: The circles represent the initial conditions that result in
inflation both with and without the vector fields, while for the initial
conditions represented by the crosses the universe immediately collapses in
any case. For the initial conditions represented by the triangles, the
universe without the vector fields collapses, but with the help of the vector
fields it avoids to do so, leading to successful inflation. No initial
conditions are found for which the vector fields hinder inflation. Due to the
symmetry only the cases with $\phi_{0}>0$ are shown.

In Fig. 6, we present typical phase space trajectories in the presence of the
vector fields. It can be seen that the dynamics of the inflaton shows an
oscillatory nature during the early stage in which the effect of the spatial
curvature is large. This pre-inflationary dynamics is caused by the interplay
among the inflaton, the vector fields, and ${\cal K}$. In this phase, the
inflaton equation of motion (59) is given approximately by

$\displaystyle\ddot{\phi}+3H\dot{\phi}\simeq
3ff^{\prime}\left(\dot{\psi}^{2}-\frac{4{\cal K}}{a^{2}}\psi^{2}\right),$ (64)

and the sign of the right-hand side flips several times because $\psi$
oscillates around zero, yielding the oscillatory dynamics as shown in Fig. 6.

Figure 6: Trajectories of the inflaton in the phase space. Black points
represent the initial conditions.

##  IV Conclusions and discussion

In this paper, we have studied the dynamics of a homogeneous, isotropic, and
positively curved universe in the presence of the SU(2) gauge field or a
triplet of mutually orthogonal vector fields, and obtained the following
results.

  * •

We have obtained the consistent ansatz for a triplet of the generalized Proca
fields in a closed universe, which corrects the previous analysis of a certain
vector-filed model of inflation in a curved universe. Our ansatz can also be
used to the SU(2) gauge field and reproduces the known result up to field
redefinition.

  * •

In addition to the usual curvature terms in the cosmological equations, new
curvature-dependent terms appear through the vector fields. In the cases of
axion-SU(2) inflation [19] and inflation with the vector fields having
U(1)$\times$U(1)$\times$U(1) symmetry [23], we have found that the vector
fields support the closed universe against collapse, though the effect is not
so significant.

  * •

In both models we have found nontrivial pre-inflationary dynamics caused by
spatial curvature.

Let us discuss a possible extension of the present study. It is easy to see
that the ansatz we have introduced in Sec. III can be generalized
straightforwardly to include spatial anisotropies:

$\displaystyle{\rm d}s^{2}=-N^{2}(t){\rm d}t^{2}+h_{ab}\omega^{a}\omega^{b},$
(65) $\displaystyle h_{ab}=\frac{a^{2}}{4}\times{\rm
diag}(e^{-4\sigma_{+}},e^{2\sigma_{+}+2\sqrt{3}\sigma_{-}},e^{2\sigma_{+}-2\sqrt{3}\sigma_{-}}),$
(66) $\displaystyle A_{i}^{a}{\rm
d}x^{i}=\sqrt{h_{aa}}\psi_{a}\omega^{a}\quad(\text{no summation over }a),$
(67)
$\displaystyle\psi_{a}=\psi(t)\times(e^{-2\beta_{+}},e^{\beta_{+}+\sqrt{3}\beta_{-}},e^{\beta_{+}-\sqrt{3}\beta_{-}}),$
(68)

where $\sigma_{\pm}=\sigma_{\pm}(t)$ and $\beta_{\pm}=\beta_{\pm}(t)$
characterize anisotropies. It would therefore be interesting to explore the
Bianchi type-IX dynamics in the presence of mutually orthogonal vector fields,
which is left for further study.

###### Acknowledgements.

TK was partially supported by JSPS KAKENHI Grant No. JP20H04745 and No.
JP20K03936.

##  Appendix A Ansatz (46) applied to the SU(2) gauge field

In this Appendix let us confirm that the ansatz (46) can also be used for the
SU(2) gauge field and the same result as in Sec. II is reproduced up to field
redefinition (for ${\cal K}>0$). Using Eqs. (42) and (46) we find

$\displaystyle F_{\mu\nu}^{a}F_{a}^{\mu\nu}$
$\displaystyle=-6\left[\frac{1}{N^{2}}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)^{2}-\psi^{2}\left(g_{A}\psi+\frac{2\sqrt{{\cal
K}}}{a}\right)^{2}\right],$ (69)
$\displaystyle\widetilde{F}^{a}_{\mu\nu}F_{a}^{\mu\nu}$
$\displaystyle=-\frac{12}{N}\left(\dot{\psi}+\frac{\dot{a}}{a}\psi\right)\psi\left(g_{A}\psi+\frac{2\sqrt{{\cal
K}}}{a}\right).$ (70)

These are apparently different from Eqs. (15) and (16). It is easy, however,
to see that in terms of the new variable

$\displaystyle\psi_{\rm new}:=-g_{A}\psi-\frac{\sqrt{{\cal K}}}{a},$ (71)

Eqs. (69) and (70) coincide with Eqs. (15) and (16), respectively.

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