aainstitutetext: Institute for Theoretical Physics, Kanazawa University,
Kanazawa 920-1192, Japanbbinstitutetext: Max-Planck-Institute for
Astrophysics, Karl-Schwarzschild-Str. 1, 85741 Garching,
Germanyccinstitutetext: Kavli Institute for the Physics and Mathematics of the
Universe (Kavli IPMU, WPI), UTIAS, The University of Tokyo, Chiba, 277-8583,
Japan

# Axion-Gauge Field Dynamics with Backreaction

Koji Ishiwata b,c Eiichiro Komatsu b Ippei Obata
<EMAIL_ADDRESS><EMAIL_ADDRESS>obata@mpa-
garching.mpg.de

###### Abstract

Phenomenological success of inflation models with axion and SU(2) gauge fields
relies crucially on control of backreaction from particle production. Most of
the previous study only demanded the backreaction terms in equations of motion
for axion and gauge fields be small on the basis of order-of-magnitude
estimation. In this paper, we solve the equations of motion with backreaction
for a wide range of parameters of the spectator axion-SU(2) model. First, we
find a new slow-roll solution of the axion-SU(2) system in the absence of
backreaction. Next, we obtain accurate conditions for stable slow-roll
solutions in the presence of backreaction. Finally, we show that the amplitude
of primordial gravitational waves sourced by the gauge fields can exceed that
of quantum vacuum fluctuations in spacetime by a large factor, without
backreaction spoiling slow-roll dynamics. Imposing additional constraints on
the power spectra of scalar and tensor modes measured at CMB scales, we find
that the sourced contribution can be more than ten times the vacuum one.
Imposing further a constraint of scalar modes non-linearly sourced by tensor
modes, the two contributions can still be comparable.

##  1 Introduction

Detection of a stochastic background of primordial gravitational waves (tensor
modes) Grishchuk:1974ny ; Starobinsky:1979ty  provides a strong evidence for
cosmic inflation Guth:1980zm ; Sato:1980yn ; Linde:1981mu ; Albrecht:1982wi .
Statistical properties of the density fluctuations (scalar modes) measured by
cosmic microwave background (CMB) experiments agree well with the basic
predictions Mukhanov:1981xt ; Hawking:1982cz ; Starobinsky:1982ee ;
Guth:1982ec ; Bardeen:1983qw  of the simplest models of inflation based on a
single scalar field (“inflaton” field) rolling down on its potential slowly
Komatsu:2014ioa ; Planck:2018jri .

Polarization data of CMB experiments place the strongest constraint on the
amplitude of primordial tensor modes Tristram:2020wbi ; SPIDER:2021ncy ;
BICEP:2021xfz . The current upper bound on the ratio of squared amplitudes of
tensor and scalar modes (tensor-to-scalar ratio) is $r<0.036$ (95% C.L.)
BICEP:2021xfz . Future experiments aim at discovering $r$ at the level of
$r=10^{-3}$ or below SimonsObservatory:2018koc ; CMB-S4:2016ple ;
Hazumi:2019lys ; NASAPICO:2019thw . However, even when $r$ is detected,
interpretation of the result regarding the origin of tensor modes may not be
unique.

What generated primordial tensor modes? The leading idea is the quantum vacuum
fluctuation in spacetime Grishchuk:1974ny ; Starobinsky:1979ty , which relates
$r$ to the Hubble expansion rate during inflation, $H$. The current bound on
$r$ gives $H<1.9\times 10^{-5}M_{pl}\simeq 4.7\times 10^{13}$ GeV ($M_{pl}$ is
the reduced Planck mass). Future detection of $r$ has profound implications
for the fundamental physics behind inflation at such a high energy Lyth:1996im
; Lyth:1998xn . This relation, however, does not hold when tensor modes are
sourced by matter fields during inflation. In this paper, we study tensor
modes sourced by non-Abelian (SU(2)) gauge fields coupled to an axion field,
which evade the above relationship Adshead:2013qp ; Dimastrogiovanni:2012ew .

Unlike scalar- and tensor-mode perturbations described above, massless free
gauge fields are conformally coupled to gravitation and cannot be excited
during inflation. Conformal invariance can be broken if a non-Abelian gauge
field has a coupling $(F^{a}_{\mu\nu}\tilde{F}^{a\,\mu\nu})^{2}$, where
$F^{a}_{\mu\nu}$ and $\tilde{F}^{a\,\mu\nu}$ are the field strength tensor of
the gauge field with the color index $a$ and its dual field, respectively.
Then an isotropic and homogeneous gauge configuration is established and leads
to a slow-roll inflationary dynamics called “Gauge-flation” Maleknejad:2011sq
; Maleknejad:2011jw . One can also break conformal invariance by coupling a
pseudo-scalar axion field, $\chi$, to a non-Abelian gauge field via the Chern-
Simons interaction, $\chi F^{a}_{\mu\nu}\tilde{F}^{a\mu\nu}$. This set up, in
which $\chi$ acts as the inflaton field with a cosine potential Freese:1990rb
is called “Chromo-natural inflation” Adshead:2012kp . We can obtain the
$(F_{\mu\nu}^{a}\tilde{F}^{a\,\mu\nu})^{2}$ term of Gauge-flation by
integrating out the inflaton field of Chromo-natural inflation near the
minimum of the potential Adshead:2012qe ; Sheikh-Jabbari:2012tom . Both models
share similar cosmological features Maleknejad:2012fw  and reach the isotropic
gauge field configuration as an attractor, even if the initial configuration
is highly anisotropic Maleknejad:2013npa ; Adshead:2018emn ; Wolfson:2020fqz ;
Wolfson:2021fya .

These models have unique cosmological consequences compared to single real-
scalar inflaton models. During the slow-roll regime, a copious amount of
spin-2 particles are produced, which source gravitational waves. Such
gravitational waves are strongly scale-dependent and chiral Adshead:2013qp ;
Dimastrogiovanni:2012ew ; Maleknejad:2012fw , and highly non-Gaussian
Agrawal:2017awz ; Agrawal:2018mrg ; Dimastrogiovanni:2018xnn ; Fujita:2018vmv
; Fujita:2021flu . Since the production of gravitational waves is very
efficient, the original Gauge-flation and Chromo-natural inflation models are
excluded by the lack of detection of $r$ Adshead:2013nka ; Namba:2013kia .
Several models have been proposed to avoid this constraint; modification to
the inflaton potential Maleknejad:2016qjz ; Caldwell:2017chz  or the kinetic
term Watanabe:2020ctz ; introduction of another scalar field as the inflaton
field Dimastrogiovanni:2016fuu ; Iarygina:2021bxq ; mass of the gauge field
via spontaneous symmetry breaking Adshead:2016omu ; Adshead:2017hnc ; and
delaying the period of Chromo-natural inflation to be out of the observable
scale of CMB observation Obata:2014loa ; Obata:2016tmo ; Domcke:2018rvv . The
scale-dependent and chiral gravitational wave spectrum is testable not only
with CMB experiments but also with direct detection experiments (such as laser
interferometers) across a wide range of frequencies Thorne:2017jft ;
Campeti:2020xwn .

Such particle production causes backreaction on dynamics of the axion-gauge
field system during inflation; the energy and momentum of the produced
particles affect the equations of motion for axion and gauge fields
Dimastrogiovanni:2016fuu ; Fujita:2017jwq . Ref. Maleknejad:2018nxz  derives
analytical formulae of the backreaction terms. Dynamics of the axion-gauge
system can be altered significantly for a large value of the background gauge
field. Most of the previous study Dimastrogiovanni:2016fuu ;
Maleknejad:2018nxz ; Papageorgiou:2019ecb  discuss impacts of the backreaction
terms by comparing them to the other terms in the equations of motion on the
basis of order-of-magnitude estimation. Ref. Fujita:2017jwq  solves the
equations of motion numerically including backreaction, but the analysis is
limited to one particular choice of the model parameters.

In this paper, we study dynamics of the spectator axion-SU(2) model of Ref.
Dimastrogiovanni:2016fuu  with backreaction in the comprehensive parameter
space. We solve the equations of motion for the background axion and gauge
fields to find slow-roll solutions. We find two solutions; one is a known
solution that has already been studied in the literature, while the other is
new. We then find that the dynamics becomes unstable if the backreaction terms
dominate in the equations of motion. We provide accurate conditions for
avoiding this instability.

This paper is organized as follows. In Sec. 2, the Lagrangian of the model is
given, leading to the equations of motion for the background axion and gauge
fields. In Sec. 3, we present two slow-roll solutions, conditions for the
parameters to give stable solutions, and an example of the time evolution of
the axion-gauge system. In Sec. 4, we calculate the tensor power spectra
sourced by the gauge fields and compare them to those of the vacuum
fluctuation. We conclude in Sec. 5.

For the rest of the paper we take $M_{pl}=1$ and use a convention of the
metric tensor, $g_{\mu\nu}={\rm diag}(-1,1,1,1)$, in the flat limit.

##  2 Spectator axion-SU(2) model

###  2.1 Field equations without backreaction

The action of the model Dimastrogiovanni:2016fuu  is given by $S=\int
d^{4}x\sqrt{-g}({\cal L}_{\chi\mathchar 45\relax{\rm gauge}}+{\cal L}_{\rm
inf})$, where $g$ is the determinant of the metric tensor and

$\displaystyle{\cal L}_{\chi\mathchar 45\relax{\rm gauge}}$
$\displaystyle=\frac{1}{2}R-\frac{1}{2}(\partial_{\mu}\chi)^{2}-V_{\chi}(\chi)-\frac{1}{4}F^{a}_{\mu\nu}F^{a\,\mu\nu}-\frac{\lambda}{4f}\chi\tilde{F}^{a\,\mu\nu}F^{a}_{\mu\nu}\,,$
(1) $\displaystyle{\cal L}_{\rm\inf}$
$\displaystyle=-\frac{1}{2}(\partial_{\mu}\phi)^{2}-V_{\phi}(\phi)\,.$ (2)

Here, $\phi$ is the inflaton field with a potential $V_{\phi}(\phi)$, $\chi$
is the pseudo scalar (axion) field with a potential $V_{\chi}(\chi)$,
$F_{\mu\nu}^{a}$ ($a=1,2,3$) is the field strength tensor of the SU(2) gauge
field, and we define
$\tilde{F}^{a\,\mu\nu}=\epsilon^{\mu\nu\rho\sigma}F^{a}_{\rho\sigma}/(2\sqrt{-g})$
with $\epsilon^{0123}=1$. The axion decay constant and the coupling constant
are given by $f$ and $\lambda$, respectively. When computing the Ricci scalar,
$R$, we use the Friedmann-Lemaître-Robertson-Walker metric for $g_{\mu\nu}$,
$ds^{2}=g_{\mu\nu}dx^{\mu}dx^{\nu}=-dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}$,
with $a(t)$ being the scale factor.

The SU(2) gauge field $A^{a}_{\mu}$ acquires an isotropic and homogeneous
background solution $\bar{A}^{a}_{\mu}$ given by $\bar{A}_{0}^{a}(t)=0$ and
$\bar{A}_{i}^{a}(t)=a(t)\psi(t)\delta^{a}_{i}$ for $i=1,2,3$ Maleknejad:2011sq
; Maleknejad:2011jw . We have introduced a field $\psi(t)$ that describes the
background gauge field configuration. Here we assume that the axion and
inflaton fields are also homogeneous. Then, in terms of $\psi$, the Lagrangian
densities are written as

$\displaystyle{\cal L}_{\chi\mathchar 45\relax{\rm gauge}}$
$\displaystyle=\frac{1}{2}R+\frac{1}{2}\dot{\chi}^{2}-V_{\chi}(\chi)+\frac{3}{2}\left[\frac{1}{a^{2}}\left(\partialderivative{(a\psi)}{t}\right)^{2}-g_{A}^{2}\psi^{4}\right]-3g_{A}\frac{\lambda}{f}\chi\frac{\psi^{2}}{a}\partialderivative{(a\psi)}{t}\,,$
(3) $\displaystyle{\cal L}_{\rm\inf}$
$\displaystyle=\frac{1}{2}\dot{\phi}^{2}-V_{\phi}(\phi)\,,$ (4)

where $g_{A}$ is the gauge coupling constant of the SU(2) field and the dot
denotes the time derivative.

The equations of motion for $\chi$ and $\psi$ derived from the action are
given by

$\displaystyle\ddot{\psi}+3H\dot{\psi}+(\dot{H}+2H^{2})\psi+2g_{A}^{2}\psi^{3}=\frac{g_{A}\lambda}{f}\dot{\chi}\psi^{2}\,,$
(5)
$\displaystyle\ddot{\chi}+3H\dot{\chi}+V_{\chi,\chi}(\chi)=-\frac{3g_{A}\lambda}{f}\psi^{2}(\dot{\psi}+H\psi)\,,$
(6)

where $V_{\chi,\chi}(\chi)=dV_{\chi}(\chi)/d\chi$ and $H$ is the Hubble
expansion rate given by

$\displaystyle
3H^{2}=\frac{1}{2}\dot{\phi}^{2}+V_{\phi}(\phi)+\frac{1}{2}\dot{\chi}^{2}+V_{\chi}(\chi)+\frac{3}{2}\left[(\dot{\psi}+H\psi_{A})^{2}+g_{A}^{2}\psi_{A}^{4}\right]\,.$
(7)

The equation of motion for the inflaton field,
$\ddot{\phi}+3H\dot{\phi}+V_{\phi,\phi}=0$, with
$V_{\phi,\phi}(\phi)=dV_{\phi}(\phi)/d\phi$ is decoupled from the other field
equations.

Instead of using $\psi$, we use the following dimensionless parameter,

$\displaystyle\xi_{A}\equiv\frac{g_{A}\psi}{H}\,,$ (8)

when studying dynamics of the background gauge field.

###  2.2 Field equations with backreaction

There are spin-2 fluctuations around the homogeneous and isotropic background
solution for the gauge field Maleknejad:2011sq ; Maleknejad:2011jw . The
energy and momentum densities of the fluctuations backreact on the background
equations of motion for $\psi$ and $\chi$ Dimastrogiovanni:2016fuu ;
Fujita:2017jwq . Namely, Eqs. (5) and (6) are modified to111The sign of ${\cal
J}_{A}$ term is different from Eq. (6.9) of Ref. Maleknejad:2018nxz . This is
a typo and their conclusions are not affected. We thank A. Maleknejad for the
confirmation.

$\displaystyle\ddot{\psi}+3H\dot{\psi}+(\dot{H}+2H^{2})\psi+2g_{A}^{2}\psi^{3}=\frac{g_{A}\lambda}{f}\dot{\chi}\psi^{2}-{\cal
J}_{A}\,,$ (9)
$\displaystyle\ddot{\chi}+3H\dot{\chi}+V_{\chi,\chi}(\chi)=-\frac{3g_{A}\lambda}{f}\psi^{2}(\dot{\psi}+H\psi)+{\cal
P}_{\chi}\,,$ (10)

where ${\cal J}_{A}$ and ${\cal P}_{\chi}$ are evaluated as

$\displaystyle{\cal J}_{A}$ $\displaystyle=\frac{g_{A}H^{3}}{6\pi^{2}}{\cal
K}_{\rm reg}[\dot{\bar{\alpha}}/H;\,\xi,\xi_{A}]\,,$ (11) $\displaystyle{\cal
P}_{\chi}$ $\displaystyle=\frac{3\lambda H^{4}}{4\pi^{2}f}{\cal K}_{\rm
reg}[\xi_{A};\,\xi,\xi_{A}]\,,$ (12)

with ${\cal K}_{\rm reg}[X;\,\xi,\xi_{A}]$ given in Appendix D of Ref.
Maleknejad:2018nxz .

For the spectator axion-SU(2) model, $\dot{\bar{\alpha}}$ and $\xi$ are given
by

$\displaystyle\frac{\dot{\bar{\alpha}}}{H}=\xi\equiv\frac{\lambda\dot{\chi}}{2fH}\,.$
(13)

If we ignore $\ddot{\psi}$, $\dot{\psi}$, $\dot{H}$ and the backreaction term
in Eq. (9), we obtain the slow-roll result
$\dot{\bar{\alpha}}/H\simeq\xi_{A}+\xi_{A}^{-1}$ . However, we find that this
approximation (especially neglecting $\dot{\psi}$) is not always valid in
describing the dynamics of the axion-gauge system. In the later analysis, we
solve the differential equations without using this slow-roll result.

So far, the discussion is general and applicable to any $V_{\phi}(\phi)$ and
$V_{\chi}(\chi)$. In this paper, we do not specify the inflaton sector, hence
$V_{\phi}$, but simply assume that it provides (quasi) de Sitter background
with a given value of $H$ Dimastrogiovanni:2016fuu . Specifically, the energy
density $\rho$ of the universe is dominated by that of the slowly-rolling
inflaton field, i.e., $\rho\simeq\rho_{\phi}\simeq V_{\phi}(\phi)\simeq{\rm
constant}$, which implies

$\displaystyle\dot{H}\simeq 0\,,~{}~{}~{}~{}~{}V_{\chi}(\chi)\ll H^{2}\,.$
(14)

Then, using Eq. (8) and $\tilde{\chi}\equiv\chi/f$, Eqs. (9) and (10) are
rewritten as

$\displaystyle\xi_{A}^{\prime\prime}+3\xi^{\prime}_{A}+2\xi_{A}(1+\xi^{2}_{A})=\lambda\xi_{A}^{2}\tilde{\chi}^{\prime}-\tilde{{\cal
J}}_{A}\,,$ (15)
$\displaystyle\tilde{\chi}^{\prime\prime}+3\tilde{\chi}^{\prime}+\frac{V_{\chi,\chi}(\chi)}{H^{2}f}=-\frac{3}{\lambda\kappa}\xi_{A}^{2}(\xi_{A}+\xi_{A}^{\prime})+\tilde{{\cal
P}}_{\chi}\,,$ (16)

where the primes denote derivatives with respect to $x\equiv Ht$ and

$\displaystyle\tilde{{\cal J}}_{A}$
$\displaystyle\equiv\frac{g_{A}^{2}}{6\pi^{2}}{\cal K}_{\rm
reg}[\dot{\bar{\alpha}}/H;\,\xi,\xi_{A}]\,,$ (17) $\displaystyle\tilde{{\cal
P}}_{\chi}$ $\displaystyle\equiv\frac{3g_{A}^{2}}{4\pi^{2}\kappa\lambda}{\cal
K}_{\rm reg}[\xi_{A};\,\xi,\xi_{A}]\,.$ (18)

Here we have introduced a dimensionless parameter $\kappa$,

$\displaystyle\kappa\equiv\left(\frac{g_{A}f}{\lambda H}\right)^{2}\,.$ (19)

We still keep our discussion general for $V_{\chi}(\chi)$, but will eventually
use the usual cosine potential Freese:1990rb ,
$V_{\chi}(\chi)=\mu^{4}[1+\cos(\chi/f)]$, when presenting numerical solutions.

##  3 Backreaction

In this section, we solve the equations of motion given in Eqs. (15) and (16).
The background axion and gauge fields may not be rolling slowly, depending on
the model parameters. We study a generic picture of the axion-gauge dynamics
and discuss the impact of the backreaction effects.

###  3.1 Stationary points

Although the slow-roll conditions are not always satisfied for axion and gauge
fields, they may be satisfied around possible stationary point(s), if any.
Therefore, it is legitimate to assume slow-roll motions for both fields for
the moment to find stationary points. Neglecting $\xi^{\prime\prime}_{A}$ and
$\chi^{\prime\prime}$, Eqs. (15) and (16) are

$\displaystyle
3\xi^{\prime}_{A}+2\xi_{A}(1+\xi_{A}^{2})\simeq\lambda\xi_{A}^{2}\tilde{\chi}^{\prime}-\tilde{{\cal
J}}_{A}\,,$ (20) $\displaystyle
3\tilde{\chi}^{\prime}+\frac{V_{\chi,\chi}(\chi)}{H^{2}f}\simeq-\frac{3}{\lambda\kappa}\xi_{A}^{2}(\xi_{A}+\xi_{A}^{\prime})+\tilde{{\cal
P}}_{\chi}\,.$ (21)

They are easily diagonalized as

$\displaystyle\xi^{\prime}_{A}$ $\displaystyle=F_{A}(\xi_{A},\chi)\,,$ (22)
$\displaystyle\tilde{\chi}^{\prime}$ $\displaystyle=F_{\chi}(\xi_{A},\chi)\,,$
(23)

where

$\displaystyle
F_{A}(\xi_{A},\chi)=\frac{-\kappa\lambda\xi_{A}^{2}V_{\chi,\chi}(\chi)/(H^{2}f)-3\xi_{A}^{5}-6\kappa\xi_{A}(1+\xi_{A}^{2})+\kappa\lambda\xi_{A}^{2}\tilde{\cal
P}_{\chi}-3\kappa\tilde{J}_{A}}{3(3\kappa+\xi_{A}^{4})}\,,$ (24)
$\displaystyle F_{\chi}(\xi_{A},\chi)=\frac{-\lambda\kappa
V_{\chi,\chi}(\chi)/(H^{2}f)-\xi_{A}^{3}+2\xi_{A}^{5}+\kappa\lambda\tilde{\cal
P}_{\chi}+\xi_{A}^{2}\tilde{J}_{A}}{\lambda(3\kappa+\xi_{A}^{4})}\,.$ (25)

Then the stationary points for $\xi_{A}$ are given by $F_{A}(\xi_{A},\chi)=0$.

Let us first consider the case in which the backreaction terms are negligible.
We find stationary points for $\xi_{A}$ by solving

$\displaystyle\kappa\beta\xi_{A}^{2}-3\xi_{A}^{5}-6\kappa\xi_{A}(1+\xi_{A}^{2})=0\,,$
(26)

where

$\displaystyle\beta(\chi)\equiv-\frac{\lambda
V_{\chi,\chi}(\chi)}{H^{2}f}(>0)\,.$ (27)

The solutions can be separated into two cases:

  * (a)

$\xi_{A0}^{4}\geq 2\kappa(1+\xi_{A0}^{2})$

  * (b)

$\xi_{A0}^{4}\leq 2\kappa(1+\xi_{A0}^{2})$

where $\xi_{A0}$ is the solution of Eq. (26). For $\xi_{A0}\sim\order{1}$, we
expect that $\kappa\ll 1$ and $\kappa\gg 1$ roughly correspond to the case (a)
and (b), respectively, which we will check later. If we further take a limit
$\xi_{A0}^{4}\gg 2\kappa(1+\xi_{A0}^{2})$ in the case (a), the solution is
obtained analytically as

$\displaystyle\xi_{A0}(\chi)\simeq\left[\frac{\kappa\beta(\chi)}{3}\right]^{1/3}\,.$
(28)

This is the known solution and is equal to $g_{A}\psi_{0}/H$ with
$\psi_{0}=[\mu^{4}\sin(\chi/f)/(3g\lambda H)]^{1/3}$ given in Ref.
Adshead:2012kp  for $V_{\chi}(\chi)=\mu^{4}[1+\cos(\chi/f)]$. In the other
limit, $\xi_{A0}^{4}\ll 2\kappa(1+\xi_{A0}^{2})$, in the case (b), the root of
Eq. (26) is given by222There is another solution; however, it is unstable.

$\displaystyle\xi_{A0}(\chi)\simeq\frac{1}{12}\left[\beta(\chi)+\sqrt{\beta(\chi)^{2}-144}\right]\,.$
(29)

This is a new solution. In summary, there are two types of stationary points
for $\xi_{A}$, which are given in Eqs. (28) and (29), respectively, depending
on the parameters. We note that there are solutions that satisfy Eq. (26) but
are not given by Eqs. (28) and (29) which are valid only in the limits
$\xi_{A0}^{4}\gg 2\kappa(1+\xi_{A0}^{2})$ and $\xi_{A0}^{4}\ll
2\kappa(1+\xi_{A0}^{2})$, respectively. We take into account all solutions by
solving Eq. (26) numerically and classify them by $\xi_{A0}^{4}\gtrless
2\kappa(1+\xi_{A0}^{2})$. Then the time evolution of $\xi_{A}$ is expected to
be stable and given by

$\displaystyle\xi_{A}(t)\simeq\xi_{A0}(\chi(t))\,.$ (30)

However, the backreaction terms may change the trajectory of $\xi_{A}(t)$ or
break the slow-roll dynamics.

We now include the backreaction terms in the calculation. To understand
dynamics of the two fields, it is useful to define an effective potential for
$\psi$,

$\displaystyle V^{\rm
eff}_{A}(\xi_{A},\chi)=-\int^{\xi_{A}}dy~{}F_{A}(y,\chi)\,.$ (31)

In Fig. 1, we show two examples of the (arbitrarily normalized) effective
potentials of case (a) (left panel) and case (b) (right panel) as a function
of $\xi_{A}$. The values of the other parameters, $\lambda$, $g$, $H$, $f$ and
$\mu$, are given in the figure caption. As expected, $\kappa$ is smaller
(larger) than unity in the left (right) panel; we find $\kappa=6.6\times
10^{-1}$ and $5.4\times 10^{1}$ for the left and right panels, respectively.
We find that the backreaction terms deform the effective potential, depending
on $\xi\,(=\dot{\bar{\alpha}}/H=\lambda\tilde{\chi}^{\prime}/2)$. We show
various values of $\xi$ in the range of $-1.5\leq\log_{10}\xi\leq 0.5$ in
color.

Which value of $\xi$ is relevant in the figure? We can estimate $\xi$ from
Eqs. (13) and (23) and find that it is of order unity for the parameters used
in the figure. This is expected from $\xi\simeq\xi_{A}+\xi_{A}^{-1}$ when the
slow-roll approximation is valid. Therefore, if the initial value of $\xi_{A}$
is around the stationary point and the initial velocity of $\chi$ is within
the slow-roll approximation, the effect of the backreaction is expected to be
subdominant or negligible. We obtained similar results for smaller and larger
values of $\lambda$.

Figure 1: Arbitrarily normalized effective potential as a function of
$\xi_{A}$. Color of each line shows the value of $\xi\,(=\dot{\alpha}/H)$. For
comparison, the potential without the backreaction terms is shown in the
dotted line. We use $\lambda=10^{2}$ and $\chi/(f\pi)=0.3$. The other
parameters are $g=1.1\times 10^{-2}$, $f=6.8\times 10^{-2}$, $H=9.6\times
10^{-6}$ and $\mu=8.3\times 10^{-4}$ (left panel; case (a)), and $g=1.3\times
10^{-2}$, $f=4.4\times 10^{-1}$, $H=7.9\times 10^{-6}$ and $\mu=1.3\times
10^{-3}$ (right panel; case (b)).

Although we will solve the differential equations without the slow-roll
approximation, the effective potential for $\xi_{A}$ is useful for
understanding the impact of the backreaction. First, the backreaction terms
modify the effective potential significantly for $\xi_{A}\gtrsim\order{10}$,
and the modification is highly sensitive to $\xi$. Therefore, even if we find
a stationary solution for $\xi_{A}\gtrsim\order{10}$, the stationary solution
might be merely temporal and lead to an unstable behavior for the two fields.
The conditions to avoid large contributions of the backreaction terms are
roughly estimated as

$\displaystyle\frac{\\{\kappa\lambda\xi_{A}^{2}\tilde{{\cal
P}}_{\chi},\,3\kappa\tilde{J}_{A}\\}}{3\xi_{A}^{5}}\sim\frac{g_{A}^{2}{\cal
K}_{\rm
reg}[\xi_{A};\xi,\xi_{A}]}{4\pi^{2}\xi_{A}^{3}}\,\\{1,\,\kappa/\xi_{A}^{2}\\}\ll
1\,,$ (32)

for case (a) and

$\displaystyle\frac{\\{\kappa\lambda\xi_{A}^{2}\tilde{{\cal
P}}_{\chi},\,3\kappa\tilde{J}_{A}\\}}{6\kappa\xi_{A}^{3}}\sim\frac{g_{A}^{2}{\cal
K}_{\rm
reg}[\xi;\xi,\xi_{A}]}{12\pi^{2}\xi_{A}^{3}}\\{\xi_{A}^{2}/\kappa,\,1\\}\ll
1\,,$ (33)

for case (b) by assuming $\xi_{A}\gtrsim 1$. Since $\kappa$ is expected to be
small (large) for case (a) ((b)), the above two conditions are satisfied when

$\displaystyle\frac{g_{A}^{2}{\cal K}_{\rm reg}}{2\pi^{2}\xi_{A}^{3}}\ll 1\,,$
(34)

where we have suppressed the argument of ${\cal K}_{\rm reg}$. This leads to
the following conditions:

  * •

Stationary solution with $\xi_{A}\gtrsim\order{1}$ needs a small $g_{A}$

  * •

Stationary solution with $\xi_{A}\sim\order{1}$ allows for
$g_{A}\sim\order{1}$

To make the criterion (34) more quantitative, we define a parameter ${\cal
I}$,

$\displaystyle{\cal
I}\equiv\frac{g_{A}^{2}}{4\pi^{2}\xi_{A}^{3}}\times\left\\{\begin{array}[]{ll}{\cal
K}_{\rm reg}[\xi_{A};\xi,\xi_{A}]&~{}~{}~{}~{}~{}{\rm
case~{}(a)}\\\\[5.69054pt] \frac{1}{3}{\cal K}_{\rm
reg}[\xi;\xi,\xi_{A}]&~{}~{}~{}~{}~{}{\rm case~{}(b)}\end{array}\right.\,,$
(37)

and we will check whether ${\cal I}\ll 1$ is satisfied or not to get the
stable solutions. Note that only the case (a) was considered in the
literature. Refs. Dimastrogiovanni:2016fuu ; Fujita:2017jwq ;
Maleknejad:2018nxz ; Papageorgiou:2019ecb  discussed the conditions for the
backreaction terms to be subdominant for the case (a).

###  3.2 Parameters of stable solutions

We perform a parameter search to find stable slow-roll solutions for $\xi_{A}$
and $\chi$. Here, by “stable slow-roll” we mean that both $\xi_{A}$ and $\chi$
relax to the global minima ($\xi_{A}\to 0$ and $\chi\to f\pi$) at $x\to\infty$
satisfying Eq. (26).

We check the stability by solving the differential equations (15) and (16).
Here is the concrete procedure of the analysis:

  * 1.

Find stationary point $\xi_{A0}(\chi)$ from the effective potential
$V_{A}^{\rm eff}(\xi_{A},\chi)$ for $\chi_{0.3}\equiv 0.3\pi f$ and
$\chi_{0.5}\equiv 0.5\pi f$. Here we take $\xi=1$.

  * 2.

For given initial values of $\xi_{A}=\xi_{A,{\rm ini}}$ and $\chi=\chi_{\rm
ini}$ at $x=0$, we compute $\xi_{A}^{\prime}$ and $\chi^{\prime}$ at $x=0$
from Eqs. (22) and (23) without backreaction. Inclusion of the backreaction
terms at this stage is not possible because Eqs. (24) and (25) are functions
of $\chi^{\prime}$. We take $(\chi_{\rm ini},\xi_{A,\rm
ini})=(\chi_{0.3},1.1\xi_{A0}(\chi_{0.3}))$ and
$(\chi_{0.5},1.1\xi_{A0}(\chi_{0.5}))$.

  * 3.

Using them as the initial conditions, we solve Eqs. (15) and (16) and find the
solutions that satisfy the slow-roll conditions.

We have checked that dynamics of $\chi(t)$ and $\xi_{A}(t)$ is not affected
significantly by taking different initial conditions, unless we take very
large values for $\xi_{A,{\rm ini}}$ for which the backreaction terms become
extremely enhanced and inevitably give rise to unstable solutions. For
reasonable initial conditions, $\xi_{A}(t)$ soon reaches the local minimum
$\xi_{A0}(\chi(t))$ and finally relaxes to the global minimum.

The ranges of the parameters used in this analysis are

$\displaystyle g_{A}:$ $\displaystyle~{}10^{-10}\,{\rm\mathchar
45\relax\mathchar 45\relax}\,1$ (38) $\displaystyle f:$
$\displaystyle~{}10^{-10}\,{\rm\mathchar 45\relax\mathchar 45\relax}\,1$ (39)
$\displaystyle H/f:$ $\displaystyle~{}10^{-10}\,{\rm\mathchar
45\relax\mathchar 45\relax}\,10^{-1}$ (40) $\displaystyle\mu^{4}/H^{2}:$
$\displaystyle~{}10^{-10}\,{\rm\mathchar 45\relax\mathchar 45\relax}\,10^{-1}$
(41)

We explore three values of $\lambda=50$, 100 and 500.

Fig. 2 shows the parameters of stable slow-roll solutions on
$(\xi_{A0}(\chi_{0.5}),g_{A})$ plane for $\lambda=100$. In Appendix A the
results for $\lambda=50$ and 500 are shown in Figs. 10 and 11, respectively.
Here we have chosen $\xi_{A0}(\chi_{0.5})$ to give the maximum value for
$\xi_{A0}$. As expected from Eq. (34), upper bounds on $g_{A}$ as a function
of $\xi_{A0}$ are obtained. They are approximately given by an analytic
function

$\displaystyle g_{A}^{\rm max}=e^{a\xi_{A0}^{2}+b\xi_{A0}+c}\,,$ (42)

with $a$, $b$ and $c$ given in Table 1. In the calculation we use data points
in the range of $\xi_{A}>\sqrt{2}$, as the scalar modes in the model would
have a negative frequency-squared that may cause instability for
$\xi_{A}<\sqrt{2}$ Dimastrogiovanni:2012ew ; Adshead:2013nka . For both cases,
we find lower bounds on $g_{A}$ as well as upper bounds. This is because
$g_{A}$ can not be arbitrary small due to the lower limit on $H/f$ in the
current parameter search to obtain $\xi_{A0}\gtrsim\order{1}$. In addition,
for case (b), the lower bound is more stringent since a too small $g_{A}$ can
not give a large $\kappa$.

Figure 2: Scatter plots of the parameters of stable slow-roll solutions on $(\xi_{A0}(\chi_{0.5}),g_{A})$ plane for case (a) (left) and (b) (right). We use $\lambda=100$. We show upper bounds on $g_{A}$ (orange, dashed), given in Eq. (42) and Table 1. The shaded region shows $\xi_{A}<\sqrt{2}$ which we discard in the current study. For comparison, we show Eq. (5.5) in Ref. Papageorgiou:2019ecb  (black, dashed).  | $a$ | $b$ | $c$
---|---|---|---
(a) $\lambda=50$ | $-0.0133$ | $-1.71$ | 0.831
(a) $\lambda=100$ | $0.000351$ | $-1.89$ | 1.29
(a) $\lambda=500$ | $-0.0101$ | $-1.81$ | 1.13
(b) $\lambda=50$ | $-0.0594$ | $-0.981$ | $-0.232$
(b) $\lambda=100$ | $-0.0378$ | $-1.32$ | 0.846
(b) $\lambda=500$ | $-0.0652$ | $-1.13$ | 0.718
Table 1: Parameters of the fitting function for the upper bound on $g_{A}$,
given in Eq. (42), for $\lambda=50$, 100, and 500.

Figure 3: Scatter plots of the parameters of stable slow-roll solutions on
$(\xi_{A0}(\chi_{0.5}),H)$ and $(\xi_{A0}(\chi_{0.5}),f)$ planes. We use
$\lambda=100$.

We find ${\cal I}\lesssim 0.1$ and $\kappa/\xi_{A0}^{2}<1$ for case (a) and
${\cal I}\lesssim 0.2$ and $\kappa/\xi_{A0}^{2}\gtrsim 1$ for case (b). While
the discussion given at the end of Sec. 3.1 is qualitatively correct, we find
that the condition ${\cal I}\ll 1$ is too stringent; ${\cal I}<0.1$ is
sufficient for obtaining stable slow-roll solutions. In summary, we have
quantitatively confirmed that dynamics of the axion-gauge field system becomes
unstable when the backreaction terms become larger than the other terms in the
equations of motion, as argued in Refs. Dimastrogiovanni:2016fuu ;
Fujita:2017jwq ; Maleknejad:2018nxz ; Papageorgiou:2019ecb  for case (a), and
sharpened the required condition, ${\cal I}<0.1$. The results for case (b) are
new.

Fig. 3 shows $H$ and $f$ of stable slow-roll solutions (see Figs. 10 and 11 in
Appendix A for $\lambda=50$ and 500, respectively). We find upper bounds on
$H$, which become more restrictive as $\xi_{A0}$ increases. On the other hand,
$f$ can be as large as $\order{1}$. For case (b), more stringent lower bounds
on $f$ are obtained. This is because too small $f$ can not give a large
$\kappa$ required for case (b).

###  3.3 Dynamics of the axion and gauge fields

We now discuss dynamics of the fields. Figs. 4 and 5 show the time evolution
of the axion-gauge system for case (a) and (b), respectively, for
$\lambda=100$ (see Figs. 12 and 13 in Appendix A for $\lambda=50$ and 500,
respectively). The parameters are the same as those in Fig. 1 and we take an
initial condition $\chi_{\rm ini}=\chi_{0.3}$ and $\xi_{A,\rm
ini}=1.1\xi_{A0}(\chi_{0.3})$. For case (a), we find that the time evolution
of $\chi$ and $\xi_{A}$ coincides precisely with that without backreaction
(see “BR” and “w/o BR” in the figure). This can be understood from the values
of ${\cal I}$ and $\kappa/\xi_{A0}^{2}$ in the present choice of parameters:
Since $({\cal I},\,\kappa/\xi_{A0}^{2})\simeq(0.014,0.10)$ and $(0.014,11)$
for case (a) and (b), respectively, the backreaction terms are subdominant.
Here we have taken $\chi=\chi_{0.3}$. We also find that the solution agrees
with that in the slow-roll approximation.

Figure 4: Time evolution of the axion ($\chi/(f\pi)$, left panel) and gauge
($\xi_{A}=g_{A}\psi/H$, right panel) fields for case (a). The initial
conditions are $\chi_{\rm ini}=\chi_{0.3}=0.3\pi f$ and $\xi_{A,\rm
ini}=1.1\xi_{A0}(\chi_{0.3})$. The other parameters are the same as in the
left panel of Fig. 1. In both panels, we show numerical solutions of Eqs. (15)
and (16) with backreaction (“BR”, blue solid); those without backreaction
(“w/o BR”, green dashed); and Eqs. (20) and (21) without backreaction (“w/o BR
(SR apprx.)”, red dot-dashed). All curves lie on top of each other. In the
right panel, we also show the trajectory given by Eq. (28) (“stationary”,
orange dotted).

Figure 5: Same as Fig. 4 but for case (b).

Let us turn our attention to the new solution, i.e., case (b). As $\xi_{A}$
relaxes to the stationary point, which is approximately given in Eq. (29),
$\chi$ slowly rolls down on the potential. We find that the qualitative
behaviors of the fields are similar to those for case (a), while they give
quantitatively different observational consequences, e.g., the spectrum of the
gravitational waves, which we discuss in the next section.

We find that the slow-roll approximation tends to be unreliable for smaller
values of $\lambda$. Even in the case where the slow-roll approximation is
valid, both fields relax too quickly to the global minimum after $\order{1}$
Hubble time. This is true for both case (a) and (b). Recalling that the
effective potential and the allowed parameter space are less dependent on the
value of $\lambda$, one may wonder why dynamics of $\chi$ and $\xi_{A}$
depends on $\lambda$. This can be understood from a factor of $1/\lambda$ in
$F_{\chi}(\xi_{A},\chi)$ (Eq. (25)). We have also numerically checked that
$F_{\chi}(\xi_{A},\chi)$ is smaller for larger $\lambda$, for which $\chi$
tends to slowly roll to the minimum and the slow-roll behavior of $\xi_{A}$ is
obtained.

Such large values of $\lambda$ are difficult to realize in a more fundamental
theory Agrawal:2018mkd . This issue is specific to the cosine potential and
the canonical kinetic term for $\chi$, and reasonable values of
$\lambda\simeq\order{1}$ are possible for axion monodromy potentials
Maleknejad:2016qjz  or non-canonical kinetic terms Watanabe:2020ctz .

##  4 Gravitational waves

Figure 6: Power spectra of the gravitational waves sourced by the gauge field,
${\cal P}_{T}^{\rm(s)}$. The parameters are the same as those in Fig. 4 (left)
and Fig. 5 (right). $k_{*}$ is some reference scale.

The power spectrum of primordial gravitational waves is given by the sum of
two contributions: the quantum vacuum fluctuation in spacetime, ${\cal
P}_{T}^{\rm(v)}$ Grishchuk:1974ny ; Starobinsky:1979ty , and the sourced
contribution, ${\cal P}_{T}^{\rm(s)}$ Adshead:2013qp ; Dimastrogiovanni:2012ew
; Maleknejad:2012fw :

$\displaystyle{\cal P}_{T}={\cal P}_{T}^{\rm(v)}+{\cal P}_{T}^{\rm(s)}\,,$
(43)

where Lyth:1998xn

$\displaystyle{\cal P}_{T}^{\rm(v)}=\frac{2H^{2}}{\pi^{2}}\,.$ (44)

In Refs. Maleknejad:2018nxz ; Fujita:2018ndp  ${\cal P}_{T}^{\rm(s)}$ is
computed by both analytical and numerical methods. For instance, the maximum
amplitude for the case (a) is estimated as

$\displaystyle{\cal P}_{T\,{\rm
est}}^{\rm(s)\,max}\approx\frac{H^{4}\xi_{A}^{4}e^{p\xi_{A}}}{\pi^{2}g_{A}^{2}}\,,$
(45)

where $p\approx 3.6$ – 4 and $\xi_{A}$ is to be the maximum value in the time
evolution.

In Fig. 6, we show ${\cal P}_{T}^{\rm(s)}$ computed by numerically solving the
equations of the tensor perturbation given in Appendix of Ref. Fujita:2018ndp
. We find that the the peak value obtained from the numerical calculation can
be different from the estimate given in Eq. (45) as $\xi_{A}$ becomes large.
Although it does not change the order of magnitude, the estimate is not
accurate enough for quantitative study. In Appendix A, we show the power
spectra for $\lambda=50$ and 500 in Fig. 14.

In our analysis, we solve the equations for the tensor perturbation and
calculate the maximum tensor spectrum sourced by the gauge field. After
computing the tensor spectrum, we select the data points that give ${\cal
P}_{T}^{\rm(s)\,max}\geq 10^{-12}$, which is a reference value for the
possibility to detect in future experiments SimonsObservatory:2018koc ;
CMB-S4:2016ple ; Hazumi:2019lys ; NASAPICO:2019thw . In Fig. 7, we show the
resultant scatter plot on $(\xi_{A0}(\chi_{0.5}),g_{A})$ plane. We find that
${\cal P}_{T}^{\rm(s)\,max}\geq 10^{-12}$ is possible in wide parameter space
for both cases (a) and (b).333For case (a) the data points are denser in the
region where $g_{A}\gtrsim 10^{-3}$, as we performed an additional parameter
search to obtain more points in this region.

Figure 7: Values of ${\cal P}_{T}^{\rm(s)\,max}$ on
$(\xi_{A0}(\chi_{0.5}),g_{A})$ plane. The shaded region shows
$\xi_{A}<\sqrt{2}$ which we discard in the current study. The dark and light
blue shaded regions show $R_{s}^{\rm est}<1$ for the vacuum tensor-to-scalar
ratios of $r_{\rm v}={\cal P}_{T}^{\rm(v)}/{\cal P}_{\zeta}^{\rm obs}=0.1$ and
$0.01$, respectively. The dotted green line shows $g_{A}>g_{A}^{\rm min}$
required from the observed scalar power spectrum on CMB scales, ${\cal
P}_{\zeta}^{\rm obs}\simeq 2.1\times 10^{-9}$. The dot-dashed and dashed
purple lines show lower bounds on $g_{A}$ from ${\cal R}_{\delta\phi}<0.1$ for
$r_{\rm v}=0.1$ and 0.01, respectively. We take $\epsilon_{B}>\epsilon_{\phi}$
and $N_{k}=10$ when computing ${\cal R}_{\delta\phi}$.

Fig. 8 shows the scatter plots with different vertical axes, $H$, $f$ and
$\mu$. We find larger sourced gravitational waves for larger values of $H$ and
$\mu$. Regarding $f$, on the other hand, we find $f\sim 10^{-2}$ – $1$ and
$f\sim 10^{-1}$ – $1$ for the case (a) and (b), respectively, and the
amplitude of the gravitational waves is independent on $f$ (see Figs. 15 – 18
in Appendix A for $\lambda=50$ and 500).

Figure 8: Same as Fig. 7 but on $(\xi_{A0}(\chi_{0.5}),H)$ ,
$(\xi_{A0}(\chi_{0.5}),f)$, and $(\xi_{A0}(\chi_{0.5}),\mu)$ planes.

We now discuss the viable parameter space for large sourced gravitational
waves on the scales constrained by CMB observations. We require the amplitude
of the scalar perturbation ${\cal P}_{\zeta}$ coincide with the observed value
${\cal P}_{\zeta}^{\rm obs}\simeq 2.1\times 10^{-9}$ Planck:2018vyg . In the
current model, ${\cal P}_{\zeta}$ is given by

$\displaystyle{\cal P}_{\zeta}$
$\displaystyle\simeq\frac{H^{2}\epsilon_{\phi}}{8\pi^{2}\epsilon_{H}^{2}}\simeq\frac{H^{2}\epsilon_{\phi}}{8\pi^{2}(\epsilon_{\phi}+\epsilon_{B})^{2}}\,,$
(46)

where $\epsilon_{H}=\epsilon_{\phi}+\epsilon_{\chi}+\epsilon_{B}+\epsilon_{E}$
and

$\displaystyle\epsilon_{H}=-\frac{\dot{H}}{H^{2}}\,,~{}~{}\epsilon_{\phi}=\frac{\dot{\phi}^{2}}{2H^{2}}\,,~{}~{}\epsilon_{\chi}=\frac{\dot{\chi}^{2}}{2H^{2}}\,,~{}~{}\epsilon_{B}=\frac{H^{2}\xi_{A}^{4}}{g_{A}^{2}}\,,~{}~{}\epsilon_{E}=\frac{H^{2}(\xi_{A}+\xi_{A}^{\prime})^{2}}{g_{A}^{2}}\,.$
(47)

Here we have checked that $\epsilon_{\chi}$ and $\epsilon_{E}$ are subdominant
in $\epsilon_{H}$. Therefore,

$\displaystyle\frac{g_{A}^{2}}{8\pi^{2}\xi_{A}^{4}}\frac{\epsilon_{B}/\epsilon_{\phi}}{(1+\epsilon_{B}/\epsilon_{\phi})^{2}}={\cal
P}_{\zeta}^{\rm obs}\,,$ (48)

is satisfied, and a lower bound on $g_{A}$,

$\displaystyle g_{A}\geq g_{A}^{\rm min}\simeq\sqrt{32\pi^{2}{\cal
P}_{\zeta}^{\rm obs}}\xi_{A}^{2}\,,$ (49)

should be satisfied Papageorgiou:2019ecb  for $\epsilon_{B}/\epsilon_{\phi}$
to have a solution. We show this bound by the green dotted line in Fig. 7. We
find that this constraint is satisfied in the upper left corner of the
parameter space shown in the figure for both cases (a) and (b).

There is another constraint coming from scale modes non-linearly sourced by
tensor modes Papageorgiou:2018rfx ; Papageorgiou:2019ecb . Namely, the quantum
loop correction to the inflaton fluctuation ${\cal R}_{\delta\phi}$ given by

$\displaystyle{\cal R}_{\delta\phi}\simeq\frac{5\times
10^{-12}}{(1+\epsilon_{B}/\epsilon_{\phi})^{2}}e^{7\xi_{A}}\xi_{A}^{11}N_{k}^{2}r_{\rm
v}^{2}\,,$ (50)

should be small. Here, $r_{\rm v}={\cal P}_{T}^{\rm(v)}/{\cal P}_{\zeta}^{\rm
obs}$ is the tensor-to-scalar ratio of the vacuum contribution and $N_{k}$ is
the number of $e$-folds of inflation during which $\xi_{A}$ rolls slowly. This
expression is valid for $2.5\leq\xi_{A}\leq 3.5.$ Demanding ${\cal
R}_{\delta\phi}<0.1$, we obtain lower bounds on $g_{A}$ as a function of
$\xi_{A}$. Note that $\epsilon_{B}/\epsilon_{\phi}$ is determined from Eq.
(48) depending on $\epsilon_{B}/\epsilon_{\phi}>1$ or
$\epsilon_{B}/\epsilon_{\phi}<1$. In Fig. 7, we show the constraints from
${\cal R}_{\delta\phi}<0.1$ for $r_{\rm v}=0.1$ (dot-dashed purple line) and
$0.01$ (dashed purple line), assuming $N_{k}=10$ and
$\epsilon_{B}/\epsilon_{\phi}>1$. This restricts the parameter space further
in the upper left corner.

Can the sourced contribution exceed the vacuum one? In Fig. 7, we show the
parameter space in which the ratio of the sourced and vacuum tensor modes,
$R_{s}^{\rm est}$, defined by

$\displaystyle R_{\rm s}^{\rm est}\equiv\frac{{\cal P}_{T\,{\rm
est}}^{\rm(s)\,max}}{{\cal P}_{T}^{\rm(v)}}\,,$ (51)

is less than unity. Here we use an estimation for ${\cal P}^{(s)\,{\rm
max}}_{T\,{\rm est}}$ given in Eq. (45), instead of the numerical solution. We
find a parameter space in which $R_{s}^{\rm est}>1$ and all the constraints
discussed above are satisfied. This is the parameter space most relevant to
future CMB experiments SimonsObservatory:2018koc ; CMB-S4:2016ple ;
Hazumi:2019lys ; NASAPICO:2019thw , which we focus now.

Figure 9: Same as Fig. 7 but on $(\xi_{A},R_{s})$ plane (top). The middle and
bottom panels are the same as the top ones but for ${\cal P}_{\zeta}={\cal
P}_{\zeta}^{\rm obs}$ and $r_{\rm v}<0.036$ (middle) and ${\cal
P}_{\zeta}={\cal P}_{\zeta}^{\rm obs}$, ${\cal R}_{\delta\phi}<0.1$ and
$r_{\rm v}<0.036$ (bottom).

We compute the correct ratio of the sourced and vacuum tensor modes,

$\displaystyle R_{\rm s}\equiv\frac{{\cal P}_{T}^{\rm(s)\,max}}{{\cal
P}_{T}^{\rm(v)}}\,,$ (52)

by solving the tensor mode equations numerically, rather than using an
estimate given in Eq. (45). The results are shown in Fig. 9 for $\lambda=100$
(see Figs. 19 and 20 in Appendix A for $\lambda=50$ and $500$, respectively).
We find that the parameter space giving $R_{\rm s}>1$ decreases as we impose
more constraints coming from observations. The scalar power spectrum amplitude
constraint, ${\cal P}_{\zeta}^{\rm obs}\simeq 2.1\times 10^{-9}$
Planck:2018vyg , and the upper bound on the vacuum tensor-to-scalar ratio,
$r_{\mathrm{v}}<0.036$ BICEP:2021xfz , allow for $R_{\rm s}\sim\order{10}$.
This is the most robust constraint applicable to CMB observations at all
multipoles. The tensor power spectrum from the gauge field with a cosine
potential has a bump as shown in Fig. 6, whereas the vacuum contribution is
nearly scale invariant. Therefore, the vacuum contribution must satisfy the
current upper bound on $r$ from the CMB polarization measurement at
$\ell\simeq 80$. On the other hand, the sourced contribution can avoid
$\ell\simeq 80$ and make a significant contribution at lower (e.g.,
“reionization bump” at $\ell\lesssim 10$) or higher multipoles, without
violating the current upper bound on $r$.

The non-linearly sourced scalar modes reduce the allowed value of $R_{\rm s}$
further to $\order{1}$. This constraint applies to CMB measurements at high
multipoles, where the power spectrum and non-Gaussiainity of scalar modes are
constrained precisely. The constraint is much weaker at $\ell\lesssim 10$ due
to large cosmic variance of CMB temperature anisotropy data: for ${\cal
R}_{\delta\phi}<1$, we find that $R_{\rm s}\simeq 5$ is still allowed. We thus
conclude that $R_{\rm s}\gtrsim 1$ is possible on the CMB scales without
violating slow-roll dynamics and the current observational constraints on
scalar and tensor modes.

These stringent constraints do not apply to other probes of primordial
gravitational waves such as pulsar timing arrays and laser/atomic
interferometers, as they probe much higher frequencies than accessible by CMB
observations. Therefore, $R_{\mathrm{s}}\gg 1$ is possible for these probes
Thorne:2017jft ; Campeti:2020xwn .

##  5 Conclusions

We have studied the axion-gauge field dynamics with backreaction in the
spectator axion-SU(2) model of Ref. Dimastrogiovanni:2016fuu . We solved the
equations of motion for the axion and gauge fields with backreaction. Unlike
the previous study based on the order-of-magnitude estimation or a numerical
calculation for a particular choice of the parameters, we provided the
accurate condition, ${\cal I}<0.1$ (Eq. (37)), which guarantees stable slow-
roll dynamics in the comprehensive parameter space.

We found a new slow-roll solution, “case (b)” (Eq. (29)), in addition to the
known one, “case (a)” (Eq. (28)). The new solution shares similar features
with the known one, but quantitative results are different. We then computed
the power spectra of gravitational waves sourced by the gauge field. We found
that the sourced gravitational waves can be much larger than the vacuum
contribution, $R_{\rm s}\gg 1$, in a wide parameter space without violating
slow-roll dynamics.

If we further assume that the axion-gauge dynamics occurs at around the last
50 – 60 $e$-folds before the end of inflation, which is relevant to CMB
experiments, the parameter space that gives $R_{\rm s}>1$ is constrained by
the observed power spectra of scalar and tensor modes and the scalar modes
non-linearly sourced by tensor modes. We found that $R_{\rm s}\sim\order{10}$
is possible for the observed scalar and tensor amplitudes, and $R_{\rm
s}\sim\order{1}$ is still possible for the additional constraint on non-
linearly sourced scalar modes at high multipoles. At low multipoles $R_{\rm
s}\simeq 5$ would be allowed.

Such sourced gravitational waves are scale-dependent, chiral and non-Gaussian,
and can be probed in future CMB experiments SimonsObservatory:2018koc ;
CMB-S4:2016ple ; Hazumi:2019lys ; NASAPICO:2019thw . Measurements at low
multipoles, which are accessible only to space-borne experiments
Hazumi:2019lys ; NASAPICO:2019thw , are especially important because the
sourced contribution can exceed the vacuum contribution. A large enhancement
of the sourced contribution is also possible for non-CMB probes of
gravitational waves such as pulsar timing arrays and laser/atomic
interferometers Thorne:2017jft ; Campeti:2020xwn . Discovery of the sourced
gravitational waves opens up new windows into the particle physics behind
inflation, with profound implications for fundamental physics.

## Acknowledgments

We thank A. Maleknejad for discussion. KI and EK thank S. Ando and the
institutes of Physics and Astronomy at the University of Amsterdam for
hospitality, where this work was initiated. EK thanks the Stichting Van der
Waals Fonds for the Johannes Diderik van der Waals rotating chair which
enabled the visit. The work of KI is supprted by JSPS KAKENHI Grant No.
JP18H05542, JP20H01894, and JSPS Core-to-Core Program Grant No.
JPJSCCA20200002. The work of EK is supported in part by JSPS KAKENHI Grant No.
JP20H05850 and JP20H05859, and the Deutsche Forschungsgemeinschaft (DFG,
German Research Foundation) under Germany’s Excellence Strategy - EXC-2094 -
390783311. The work of IO is supported by JSPS Overseas Research Fellowship.
The Kavli IPMU is supported by World Premier International Research Center
Initiative (WPI), MEXT, Japan.

##  Appendix A Additional figures

In this appendix, we give the results for $\lambda=50$ and 500. All the
figures are shown in the same manner as those in the main text. The additional
parameters are given in the figure caption when needed.

Figure 10: Same as Figs. 2 and 3 but for $\lambda=50$.

Figure 11: Same as Figs. 2 and 3 but for $\lambda=500$.

Figure 12: Same as Fig. 4 but for $\lambda=50$, $g=9.3\times 10^{-3}$,
$f=5.8\times 10^{-2}$, $H=8.5\times 10^{-6}$ and $\mu=7.2\times 10^{-4}$
(top), and $\lambda=500$, $g=6.1\times 10^{-3}$, $f=7.4\times 10^{-1}$,
$H=1.1\times 10^{-5}$ and $\mu=1.8\times 10^{-3}$ (bottom).

Figure 13: Same as Fig. 5 but for $\lambda=50$, $g=7.5\times 10^{-3}$,
$f=4.3\times 10^{-1}$, $H=1.0\times 10^{-5}$ and $\mu=1.7\times 10^{-3}$
(top), and $\lambda=500$, $g=1.5\times 10^{-2}$, $f=8.6\times 10^{-1}$,
$H=9.1\times 10^{-6}$ and $\mu=1.4\times 10^{-3}$ (bottom).

Figure 14: Same as Fig. 6 but $\lambda=50$ (top) and 500 (bottom) and the
other parameters are the same as Figs. 12 and 13 for case (a) and (b),
respectively.

Figure 15: Same as Fig. 7 but for $\lambda=50$.

Figure 16: Same as Fig. 7 but for $\lambda=500$.

Figure 17: Same as Fig. 8 but for $\lambda=50$.

Figure 18: Same as Fig. 8 but for $\lambda=500$.

Figure 19: Same as Fig. 9 but for $\lambda=50$.

Figure 20: Same as Fig. 9 but for $\lambda=500$.

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