# Higgs Inflation with a Gauss-Bonnet term

Seoktae Koh  Seong Chan Park,11footnotetext: Corresponding author. Gansukh
Tumurtushaa

###### Abstract

Higgs inflation with a Gauss-Bonnet term is studied in the Einstein frame. Our
model features two coupling functions, $\Omega^{2}(\phi)$ and $\omega(\phi)$,
coupled to the Ricci scalar and Gauss-Bonnet combinations. We found a special
relation $\Omega^{2}\propto\omega$ sets the system a lot more simplified;
therefore we take it for granted in our analytical studies. As a result of a
Weyl transformation to the Einstein frame, we notice the emergence of new
interactions: a non-minimal kinetic coupling between the scalar field and
gravity and a derivative self-interaction of the scalar field. In the Einstein
frame, we investigate the cosmological implications of these interactions by
deriving the background equation of motion and observable quantities. Our
numerical result on $n_{S}$ vs. $r$ suggests our model is consistent with the
observational data for a wide range of the model parameter, $-1.4\times
10^{4}\lesssim\alpha\equiv\frac{\omega}{\Omega^{2}}\lesssim 8\times 10^{3}$,
where both the positive and negative values of $\alpha$ are allowed. As the
Gauss-Bonnet contributions decay away with time after inflation, the
propagation speed of gravitational waves is turned out to be consistent with
the recent constraints on the propagation speed of gravitational waves (GWs)
without inducing ghost instability.

##  1 Introduction

Cosmic inflation, an idea of accelerated exponential expansion of the early
universe, is a successful paradigm that not only solved the flatness and
horizon problems but also made definite predictions for primordial
cosmological perturbations that observations can directly test; see Ref. [1]
for review. However, there is no conclusive solution to the problem of how to
embed inflation into a particle physics framework. The most common approach
for embedding inflation into the particle physics framework is to couple the
gravity sector to a scalar field, such as the Higgs field. Driven by the Higgs
field $\phi$, which is non-minimally coupled to gravity, Higgs inflation is a
minimal model of inflation without introducing additional scalar degrees of
freedom to those appearing in the Standard Model (SM) of particle physics [2,
3, 4, 5, 6, 7, 8]. This model agrees with data from Cosmic Microwave
Background (CMB) experiments on the bounds of the scalar spectral index
$n_{S}$ and the tensor-to-scalar ratio $r$ [9, 10, 11, 12, 13]. What makes
Higgs inflation consistent with the observational data is the non-minimal
coupling function between the Higgs field and the gravitational sector, which
flattens the potential in the Einstein frame in the large-field regime,
allowing the slow-roll conditions for inflation to be realized [14, 15, 16].
222See [15, 16] for the non-minimal coupling of assistant field(s).  As a
result, at first order in slow-roll approximation, the Higgs inflation model
predicts a $n_{S}$ value consistent with data and a $r$ value to be
comfortably below the experimental limits [17, 18]; see [19] for the recent
review.

While the non-minimal coupling between gravity and the Higgs field is well-
motivated by consideration of the renormalization of a scalar field in curved
space, it is feasible to expect additional interactions to be present. From
the effective field theory viewpoint $R^{2}$ term [20, 21, 22, 24, 23, 25, 26,
27, 28, 29, 30, 31], especially the Gauss-Bonnet combination
$R_{GB}^{2}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$,
are expected to arise [32]. Higher curvature terms, $R^{2+p}$ terms (of mass
dimension $4+2p$) may also arise, but they are supposed to be suppressed [33,
34]. The Gauss-Bonnet term, in isolation, is purely topological and therefore
does not impact the dynamics of inflation. However, it can introduce
intriguing phenomenological effects when coupled with the inflaton field.
Therefore, in the present work, we are motivated to study inflation in the
context of a scalar field non-minimally coupled to the Ricci scalar and the
Gauss-Bonnet combination in the action. Such motivations for adding the Gauss-
Bonnet term are also complemented by the string theory perspective, where
particular couplings between the Gauss-Bonnet term and scalar fields have been
found [35, 36]. We note that many authors have studied phenomenological
aspects of the Gauss-Bonnet combination, including cosmic inflation [52, 51,
43, 50, 49, 41, 48, 47, 46, 45, 44, 42, 40, 39, 38, 35, 36, 37, 53],
primordial black holes [54, 55], gravitational-wave leptogenesis [56], dark
energy [57, 58, 59, 60, 61, 62, 63, 64], blackholes [65, 66], and wormholes
[67, 68], in the Einstein frame version of a theory, where a generic function
of a scalar field coupled to the Gauss-Bonnet combination is often considered
in addition to the Einstein-Hilbert term.

The paper is organized as follows. Section 2 begins with our setup formulated
in the Jordan frame, where we have a scalar (or Higgs) field coupled to the
Ricci scalar and the Gauss-Bonnet combination. At the end of the section, we
obtain the Einstein frame action using the so-called conformal transformation.
From the Einstein frame action, we derive the background equations of motion
and the observable quantities in Section 3 following Ref. [69]. In the same
section, we provide our numerical results and discuss the consequent findings
of our work. Finally, we conclude our work in Section 4.

##  2 Setup and Conformal transformation

Let us begin with an action given in the Jordan frame as

$\displaystyle S^{J}=\int
d^{4}x\sqrt{-g^{J}}\left[\frac{M_{p}^{2}}{2}\Omega^{2}(\phi)R^{J}-\frac{1}{2}g^{J}_{ab}\nabla^{a}\phi\nabla^{b}\phi-V(\phi)+\omega(\phi)R_{GB}^{2^{J}}\right]\,.$
(2.1)

Here, the superscript $J$ denotes quantities in the Jordan frame, where the
scalar field $\phi$ and the Ricci scalar $R$ are coupled through
$\Omega(\phi)$, which is also known as the non-minimal coupling function,
$M_{p}$ is the reduced Planck mass, and $V(\phi)$ is the scalar field
potential. The $\omega(\phi)$ is the coupling function between the $\phi$ and
the Gauss-Bonnet combination,
$R_{GB}^{2}=R^{2}-4R_{\mu\nu}R^{\mu\nu}+R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$.
Many properties of the physically interesting quantities become more apparent
and easier to present in the Einstein frame. Thus, using the so-called Weyl
transformation, a local conformal transformation, one moves from the Jordan
frame to the Einstein frame, where gravity is minimally coupled to the scalar
field. The spacetime metric and the square root of its determinants change
under the conformal transformation as

$\displaystyle
g_{ab}^{J}=\Omega^{-2}g_{ab}\,,\quad\sqrt{-g^{J}}=\Omega^{-4}\sqrt{-g},$ (2.2)

where the metric $g_{ab}$ without the superscript $J$ represents the metric in
the Einstein frame.

The first three terms in Eq. (2.1) are well known in the context of Higgs
inflation and, in the Einstein frame, they can be written as [2, 4, 3, 6, 7]

$\displaystyle S=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-\frac{1}{2}g_{ab}\nabla^{a}s\nabla^{b}s-V(s)\right]\,,$
(2.3)

where

$\displaystyle\frac{s}{M_{p}}\equiv\sqrt{\frac{3}{2}}\ln\Omega^{2}(\phi(s))\,,\quad
V(s)=\frac{V(\phi(s))}{\Omega^{4}(\phi(s))}\,.$ (2.4)

It is, therefore, interesting to see how the last term in Eq. (2.1) changes
under the conformal transformation and to investigate what consequent dynamics
would be apparent in the Einstein frame that otherwise does not come into
sight in the Jordan frame. Thus, let us now focus on the last part of the
action,

$\displaystyle S^{J}_{GB}=\int
d^{4}x\sqrt{-g^{J}}\omega(\phi)R_{GB}^{2^{J}}\,,$ (2.5)

and transform it into the Einstein frame. The Gauss-Bonnet combination changes
under the conformal transformation as [70]

$\displaystyle R_{GB}^{2^{J}}$
$\displaystyle=\Omega^{4}\left[R_{GB}^{2}-8\Omega^{-1}G_{ab}\nabla^{a}\nabla^{b}\Omega-4R\Omega^{-2}\nabla_{a}\Omega\nabla^{a}\Omega+8\Omega^{-2}\left(\nabla_{a}\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega\right.\right.$
$\displaystyle\left.\quad\left.-\nabla_{b}\nabla_{a}\Omega\nabla^{b}\nabla^{a}\Omega\right)-24\Omega^{-3}\nabla_{a}\Omega\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega+24\Omega^{-4}\left(\nabla_{a}\Omega\nabla^{a}\Omega\right)^{2}\right]\,,$
(2.6)

where $G_{ab}\equiv R_{ab}-g_{ab}R/2$ is the Einstein tensor. Substituting Eq.
(2) into Eq. (2.5) and using Eq. (2.2), we obtain the action in the Einstein
frame as

$\displaystyle S_{GB}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,\omega(\phi)\left[R_{GB}^{2}-8\Omega^{-1}G_{ab}\nabla^{a}\nabla^{b}\Omega-4R\Omega^{-2}\nabla_{a}\Omega\nabla^{a}\Omega+8\Omega^{-2}\left(\nabla_{a}\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega\right.\right.$
$\displaystyle\left.\quad\left.-\nabla_{b}\nabla_{a}\Omega\nabla^{b}\nabla^{a}\Omega\right)-24\Omega^{-3}\nabla_{a}\Omega\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega+24\Omega^{-4}\left(\nabla_{a}\Omega\nabla^{a}\Omega\right)^{2}\right]\,.$
(2.7)

The coupling functions $\omega(\phi)$ can generally be either a constant or a
generic function of a scalar field. In the Appendix A, we show that, if
$\omega=\text{const.}$, no accountable effect comes from the Gauss-Bonnet term
in both frames. Thus, from now on, we regard the Gauss-Bonnet coupling as a
function of the scalar field.

With the use of the integration by parts, the second and the third terms in
Eq. (2) can be simplified as

$\displaystyle-8\int
d^{4}x\sqrt{-g}\,\omega\left[\Omega^{-1}G_{ab}\nabla^{a}\nabla^{b}\Omega+\frac{1}{2}g_{ab}R\Omega^{-2}\nabla^{a}\Omega\nabla^{b}\Omega\right]$
$\displaystyle\qquad=-8\int
d^{4}x\sqrt{-g}\left[\omega\Omega^{-2}R_{ab}\nabla^{a}\Omega\nabla^{b}\Omega-\Omega^{-1}G_{ab}\nabla^{a}\omega\nabla^{b}\Omega\right]\,.$
(2.8)

The fourth term in Eq. (2) becomes

$\displaystyle 8\int
d^{4}x\sqrt{-g}\omega\Omega^{-2}\left(\nabla_{a}\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega-\nabla_{b}\nabla_{a}\Omega\nabla^{b}\nabla^{a}\Omega\right)=8\int
d^{4}x\sqrt{-g}\left[\omega\Omega^{-2}R_{ab}\nabla^{a}\Omega\nabla^{b}\Omega\right.$
$\displaystyle\qquad\left.-\omega\Omega^{-2}\left(\omega^{-1}\nabla_{a}\omega-2\Omega^{-1}\nabla_{a}\Omega\right)\left(\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega-\nabla_{b}\Omega\nabla^{a}\nabla^{b}\Omega\right)\right]\,,$
(2.9)

where Eq. (A) is used. Consequently, Eq. (2) can be rewritten as

$\displaystyle S_{GB}$ $\displaystyle=\int d^{4}x\sqrt{-g}\left[\omega
R_{GB}^{2}+8\Omega^{-1}G_{ab}\nabla^{a}\omega\nabla^{b}\Omega-8\omega\Omega^{-2}\left(\omega^{-1}\nabla_{a}\omega-2\Omega^{-1}\nabla_{a}\Omega\right)\right.$
$\displaystyle\quad\left.\times\left(\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega-\nabla_{b}\Omega\nabla^{a}\nabla^{b}\Omega\right)-24\omega\Omega^{-3}\nabla_{a}\Omega\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega+24\omega\Omega^{-4}\left(\nabla_{a}\Omega\nabla^{a}\Omega\right)^{2}\right]\,,$
(2.10)

where the first term in Eq. (2) is canceled with that of Eq. (2). It is
interesting to note that the third term in Eq. (2) vanishes when the two non-
minimal couplings are proportional to each other, maintaining the following
relation:

$\displaystyle\omega=\alpha\Omega^{2}\,,$ (2.11)

where $\alpha\in\mathbb{R}$. Although the coupling functions
$\Omega^{2}(\phi)$ and $\omega(\phi)$ have the flexibility to be arbitrary
functions of a scalar field, we would assume Eq. (2.11) as a part of our
model. For further elaboration, including the form of action and the case of
an arbitrary power relationship $\omega\propto\Omega^{p}$, please refer to
Appendix B. Now the action is greatly simplified as

$\displaystyle S_{GB}$ $\displaystyle=\int
d^{4}x\sqrt{-g}\,\alpha\Omega^{2}\left[R_{GB}^{2}+4G_{ab}\nabla^{a}\ln\Omega^{2}\nabla^{b}\ln\Omega^{2}-3\nabla^{b}\nabla_{b}\ln\Omega^{2}\nabla_{a}\ln\Omega^{2}\nabla^{a}\ln\Omega^{2}\right]\,.$
(2.12)

In terms of the scalar field $s$ defined in Eq. (2.4), the action Eq. (2.12)
reads

$\displaystyle S_{GB}$ $\displaystyle=\int d^{4}x\sqrt{-g}\,\alpha
e^{\sqrt{\frac{2}{3}}\frac{s}{M_{p}}}\left[R_{GB}^{2}+\frac{8}{3M_{p}^{2}}G_{ab}\nabla^{a}s\nabla^{b}s-\frac{1}{M_{p}^{3}}\sqrt{\frac{8}{3}}\nabla^{b}\nabla_{b}s\nabla_{a}s\nabla^{a}s\right]\,.$
(2.13)

Combining Eq. (2.13) with Eq. (2.3), we can write the full action in the
Einstein frame as

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\left[\frac{M_{P}^{2}}{2}R-\frac{1}{2}g_{ab}\nabla^{a}s\nabla^{b}s-V(s)\right.$
$\displaystyle\qquad\left.-\frac{1}{2}\xi(s)\left(c_{1}R_{GB}^{2}+\frac{c_{2}}{M_{p}^{2}}G_{ab}\nabla^{a}s\nabla^{b}s+\frac{c_{3}}{M_{p}^{3}}\nabla_{a}s\nabla^{a}s\nabla^{b}\nabla_{b}s\right)\right]\,,$
(2.14)

where

$\displaystyle\xi(s)\equiv-2\alpha
e^{\sqrt{\frac{2}{3}}\frac{s}{M_{p}}}\,,\quad c_{1}=1\,,\quad
c_{2}=\frac{8}{3}\,,\quad c_{3}=-\sqrt{\frac{8}{3}}\,.$

Note that the scalar-field potential has not been specified yet. Thus, Eq. (2)
applies to any potential and the coupling functions that adhere to the
relationship given in Eq. (2.11). The Einstein frame potential $V(s)$ can be
determined through Eq. (2.4). For instance, consider the non-minimal coupling
function and the Jordan frame potential:

$\displaystyle\Omega^{2}=1+\frac{\sigma}{M_{p}^{2}}\phi^{2}\,\quad\text{and}\quad
V(\phi)=\frac{\lambda}{4}\phi^{4}\,,$

then, from Eq. (2.4), the Einstein frame potential reads:

$\displaystyle V(s)=\frac{\lambda
M_{p}^{4}}{4\sigma^{2}}\left(1-e^{-\sqrt{\frac{2}{3}}\frac{s}{M_{p}}}\right)^{2}\,,$
(2.15)

where $\sigma$ and $\lambda$ are the coupling constant and the potential
parameters, respectively [2, 4, 3, 6, 7]. In addition to the expected Gauss-
Bonnet term, Eq. (2) presents new interactions, the kinetic coupling between
the scalar field and gravity and a derivative self-interaction of the scalar
field, that were not apparent in the Jordan frame. Such interactions discussed
in Ref. [71, 72, 73] as a particular subclass of Horndeski’s theory [74], the
most general scalar-tensor theory of gravity, or equivalently the generalized
Galileons [75]. Moreover, multiplied by $\xi(s)$, the last term in Eq. (2) is
discussed in Ref. [69] as a string correction in the Einstein frame because
the general relativity (GR) limit can be reached as $\alpha\rightarrow 0$. In
the GR limit, the inflationary dynamics of our model would be dictated mainly
by the potential shape, $V(s)$. Thus, in this work, the $\alpha$ does not
necessarily need to be small and may have either a negative or positive sign.
The determination of the sign should be based on the observational data. From
now on, we will specify the last term of Eq. (2) as the Gauss-Bonnet
combinations.

##  3 Higgs inflation with a Gauss-Bonnet term in the Einstein frame

In this section, we investigate Higgs inflation with the Gauss-Bonnet
combination in the Einstein frame with potential presented in Eq. (2.15). From
Eq. (2), we derive gravitational and field equations of motion as [69]

$\displaystyle
G_{ab}=\nabla_{a}s\nabla_{b}s-\frac{1}{2}g_{ab}\left(\nabla_{c}s\nabla^{c}s-2V\right)-\frac{1}{2}T_{ab}^{GB}\,,$
(3.1a) $\displaystyle\nabla_{a}\nabla^{a}s-V_{,s}=\frac{1}{2}T^{GB}\,,$ (3.1b)

where

$\displaystyle T_{ab}^{GB}$
$\displaystyle=c_{1}\left[4g_{ab}\left(2\nabla_{c}\nabla_{d}\xi
R^{cd}-\nabla_{c}\nabla^{c}\xi R\right)\right.$
$\displaystyle\qquad\left.-4\left(2\nabla^{c}\nabla^{d}\xi
R_{acbd}-2\nabla_{c}\nabla^{c}R_{ab}+4\nabla_{c}\nabla_{(b}\xi
R_{a)}^{c}-\nabla_{a}\nabla_{b}\xi R\right)\right]$
$\displaystyle\quad+\frac{c_{2}}{M_{p}^{2}}\left[\xi(R_{ab}\nabla^{c}s\nabla_{c}s+R\nabla_{a}s\nabla_{b}s-4R^{c}_{(a}\nabla_{b)}s\nabla_{c}s)-\nabla_{c}\nabla^{c}\left(\xi\nabla_{a}s\nabla_{b}s\right)\right.$
$\displaystyle\qquad-\nabla_{a}\nabla_{b}\left(\xi\nabla^{c}s\nabla_{c}s\right)+2\nabla_{c}\nabla_{(b}(\xi\nabla^{c}s\nabla_{a)}s)$
$\displaystyle\qquad\left.+g_{ab}\left(\xi
G^{cd}\nabla_{c}s\nabla_{d}s-\nabla_{d}\nabla_{c}(\xi\nabla^{c}s\nabla^{d}s)+\nabla_{d}\nabla^{d}(\xi\nabla_{c}s\nabla^{c}s)\right)\right]$
$\displaystyle\quad+\frac{c_{3}}{M_{p}^{3}}\left[2\nabla_{(a}(\xi\nabla^{c}s\nabla_{c}s)\nabla_{b)}s-2\xi\nabla^{c}\nabla_{c}s\nabla_{a}s\nabla_{b}s-g_{ab}\nabla_{d}(\xi\nabla^{c}s\nabla_{c}s)\nabla^{d}s\right]\,,$
$\displaystyle T^{GB}$
$\displaystyle=c_{1}\xi_{,s}R^{2}_{GB}-\frac{c_{2}}{M_{p}^{2}}G^{ab}\left(\xi_{,s}\nabla_{a}s\nabla_{b}s+2\xi\nabla_{a}\nabla_{b}s\right)$
$\displaystyle\quad+\frac{c_{3}}{M_{p}^{3}}\left[\xi_{,s}\nabla_{b}\nabla^{b}s\nabla^{a}s\nabla_{a}s+\nabla_{b}\nabla^{b}\left(\xi\nabla_{a}s\nabla^{a}s\right)-2\nabla_{a}\left(\xi\nabla_{b}\nabla^{b}s\nabla^{a}s\right)\right]\,.$

with “$V_{,s}=\partial V/\partial s$” and “$\xi_{,s}=\partial\xi/\partial s$.”
In the spatially flat FRW universe with metric

$\displaystyle ds^{2}=-dt^{2}+a(t)^{2}\delta_{ij}dx^{i}dx^{j}\,,$

where $a(t)$ is the scale factor, the background equations of motion yield
from Eq. (3.1) [69]

$\displaystyle 3M_{p}^{2}H^{2}$
$\displaystyle=\frac{1}{2}\dot{s}^{2}+V+12c_{1}\dot{\xi}H^{3}-\frac{9}{2}\frac{c_{2}}{M_{p}^{2}}\xi\dot{s}^{2}H^{2}+\frac{1}{2}\frac{c_{3}}{M_{p}^{3}}(\dot{\xi}-6\xi
H)\dot{s}^{3}\,,$ (3.2a) $\displaystyle M_{p}^{2}(2\dot{H}+3H^{2})$
$\displaystyle=-\frac{1}{2}\dot{s}^{2}+V+4c_{1}\left[\ddot{\xi}H^{2}+2\dot{\xi}H(\dot{H}+H^{2})\right]$
$\displaystyle\quad-\frac{1}{2}\frac{c_{2}}{M_{p}^{2}}\dot{s}\left[\xi\dot{s}(2\dot{H}+3H^{2})+4\xi\ddot{s}H+2\dot{\xi}\dot{s}H\right]-\frac{1}{2}\frac{c_{3}}{M_{p}^{3}}\dot{s}^{2}(2\xi\ddot{s}+\dot{\xi}\dot{s})\,,$
(3.2b) $\displaystyle\ddot{s}+3H\dot{s}+V_{,s}$
$\displaystyle=-12c_{1}\xi_{,s}H^{2}(\dot{H}+H^{2})+\frac{3}{2}\frac{c_{2}}{M_{p}^{2}}\left[H^{2}(\dot{\xi}\dot{s}+2\xi\ddot{s})+2H\xi\dot{s}(2\dot{H}+3H^{2})\right]$
$\displaystyle\quad-\frac{1}{2}\frac{c_{3}}{M_{p}^{3}}\dot{s}\left[\ddot{\xi}\dot{s}+3\dot{\xi}\ddot{s}-6\xi(\dot{H}\dot{s}+2H\ddot{s}+3H^{2}\dot{s})\right]\,,$
(3.2c)

where $H\equiv\dot{a}/a$ is the Hubble parameter and the over-dot denotes the
derivative with respect to time $t$. The imprints of the Gauss-Bonnet
contributions in the Einstein frame can be easily identified by examining the
equation of motion for the presence of the $\xi(s)$ function. Thus, the terms
containing $\xi(s)$ are clear indicators of the Gauss-Bonnet contributions in
the Einstein frame and should not be overlooked.

In the context of slow-roll inflation, it is often assumed that the
acceleration of the scalar field is negligible with respect to the
gravitational friction, $\ddot{s}\ll 3H\dot{s}$, and the potential energy
dominates over the kinetic energy, $V\gg\dot{s}^{2}/2$; together they are
known as the slow-roll approximations. Thus, in light of the slow-roll
approximations, the above equations can be simplified even further as

$\displaystyle 3M_{p}^{2}H^{2}\simeq V\,,$ (3.3) $\displaystyle
3H\dot{s}\simeq-\frac{\mathcal{B}\mp\sqrt{\mathcal{B}^{2}-4\mathcal{A}\mathcal{C}}}{2\mathcal{A}}\,,$
(3.4)

where

$\displaystyle\mathcal{A}\equiv\frac{c_{3}}{M_{p}^{3}}\xi\,,\quad\mathcal{B}\equiv
1-\frac{3c_{2}}{M_{p}^{2}}\xi H^{2}\simeq 1-\frac{c_{2}}{M_{p}^{4}}\xi
V\,,\quad\mathcal{C}\equiv V_{,s}+12c_{1}\xi_{,s}H^{4}\simeq
V_{,s}+\frac{4c_{1}}{3M_{p}^{4}}\xi_{,s}V^{2}\,.$

In obtaining Eqs. (3.3) and (3.4), we assumed $\dot{\xi}/(2\xi H)\ll 1$ with
$\xi(s)\neq 0$. Without any loss of generality, one can rewrite Eq. (3.4) as

$\displaystyle 3H\dot{s}\simeq-V_{,s}\left[1+\delta(s)\right]\,,$ (3.5)

where

$\displaystyle\delta(s)\equiv\frac{\mathcal{B}\mp\sqrt{\mathcal{B}^{2}-4\mathcal{A}\mathcal{C}}}{2\mathcal{A}V_{,{s}}}-1\,.$

From Eq. (3.5), one can also get

$\displaystyle\frac{\dot{s}}{H}\simeq-M_{p}^{2}\frac{V_{,s}}{V}(1+\delta)\,,$
(3.6)

which is an important quantity to estimate the duration of inflation. The
duration of inflation is measured by the so-called number $N$ of the
$e$-folds, which is defined as

$\displaystyle
N\equiv\int_{t_{i}}^{t_{e}}Hdt=\int_{s_{i}}^{s_{e}}\frac{H}{\dot{s}}ds\,.$
(3.7)

where the variables $t_{i}$ and $t_{e}$ represent the initial and end times of
inflation, while $s_{i}$ and $s_{e}$ refer to the beginning and ending scalar
field values of inflation, respectively. It is evident from Eqs. (3.6) and
(3.7) that the contributions of the Gauss-Bonnet in the Einstein frame impact
the number of $e$-folds.

Furthermore, to reflect the aforementioned slow-roll approximations, it is
useful to introduce the following so-called slow-roll parameters

$\displaystyle\epsilon_{1}\equiv\frac{\dot{H}}{H^{2}}\simeq-\epsilon_{V}(1+\delta)\,,\quad\epsilon_{2}\equiv\frac{\ddot{s}}{H\dot{s}}\simeq\left[\epsilon_{V}-\eta_{V}-\sqrt{2\epsilon_{V}}M_{p}\ln(1+\delta)_{,s}\right](1+\delta)\,,$
(3.8)

where

$\displaystyle\epsilon_{V}\equiv\frac{M_{p}^{2}}{2}\left(\frac{V_{,s}}{V}\right)^{2}\,,\quad\eta_{V}\equiv
M_{p}^{2}\frac{V_{,ss}}{V}\,.$

Following Ref. [69], we also introduce the following additional slow-roll
parameters to take the effects of the Gauss-Bonnet combination into account

$\displaystyle\epsilon_{3}\equiv\frac{\dot{E}}{2EH}=\frac{E_{,s}}{2E}\frac{\dot{s}}{H}\,,\quad\epsilon_{4}\equiv\frac{Q_{a}}{4M_{p}^{2}HQ_{t}}\,,\quad\epsilon_{5}\equiv\frac{\dot{Q}_{t}}{2Q_{t}H}=\frac{Q_{t,s}}{2Q_{t}}\frac{\dot{s}}{H}\,,$
(3.9)

where

$\displaystyle
E\equiv\frac{1}{\dot{s}^{2}}\left(\dot{s}^{2}+\frac{3Q_{a}^{2}}{2M_{p}^{2}Q_{t}}+Q_{c}\right)\simeq
1-\frac{c_{2}}{M_{p}^{4}}\xi
V\left(1-\frac{2c_{3}}{c_{2}}\sqrt{2\epsilon_{V}}\right)\,,$

with

$\displaystyle
Q_{a}\equiv-4c_{1}\dot{\xi}H^{2}+\frac{2c_{2}}{M_{p}^{2}}\xi\dot{s}^{2}H+\frac{c_{3}}{M_{p}^{3}}\xi\dot{s}^{3}\,,\qquad
Q_{b}\equiv-8c_{1}\dot{\xi}H+\frac{c_{2}}{M_{p}^{2}}\xi\dot{s}^{2}\,,$
$\displaystyle
Q_{c}\equiv-\frac{3c_{2}}{M_{p}^{2}}\xi\dot{s}^{2}H^{2}+\frac{2c_{3}}{M_{p}^{3}}\dot{s}^{3}(\dot{\xi}-3\xi
H)\,,\qquad Q_{t}\equiv 1+\frac{Q_{b}}{2M_{p}^{2}}\,.$

For inflation to occur successfully, these slow-roll parameters must be
smaller than unity, $|\epsilon_{1,2,3,4,5}|\ll 1$, during inflation. In terms
of the slow-roll parameters, Eq. (3.6) reads

$\displaystyle
N=\frac{1}{M_{p}}\int_{s_{e}}^{s_{i}}\frac{ds}{\sqrt{2\epsilon_{V}}(1+\delta)}\,,$

where the field value $s_{e}$ at the end of inflation can be estimated from
the condition $\epsilon_{1}(s_{e})\equiv 1$. One can notice that the $s_{e}$
is also affected by the Gauss-Bonnet contributions. Following the linear
perturbation analyses carried out in Ref. [69], we obtain the spectral indices
for scalar and tensor fluctuation modes [69]

$\displaystyle n_{S}-1=2(2\epsilon_{1}-\epsilon_{2}-\epsilon_{3})\,,\quad
n_{T}=2(\epsilon_{1}-\epsilon_{5})\,,$ (3.10)

and the tensor-to-scalar ratio

$\displaystyle r$
$\displaystyle=16\left|\frac{1}{Q_{t}}\left(\frac{c_{A}}{c_{T}}\right)^{3}\left(\epsilon_{1}-\frac{1}{4M_{p}^{2}H^{2}}\left(2Q_{c}+Q_{d}-HQ_{e}+H^{2}Q_{f}\right)\right)\right|$
$\displaystyle\simeq
16\left|\epsilon_{1}-\frac{1}{4M_{p}^{2}H^{2}}\left(2Q_{c}+Q_{d}-HQ_{e}+H^{2}Q_{f}\right)\right|\,.$
(3.11)

Here, the squared propagation speeds of the scalar and tensor perturbation
modes are given [69, 76] by

$\displaystyle c_{A}^{2}\equiv
1+\frac{Q_{d}+\frac{Q_{a}}{2M_{p}^{2}Q_{t}}Q_{e}+\left(\frac{Q_{a}}{2M_{p}^{2}Q_{t}}\right)^{2}Q_{f}}{\dot{s}^{2}+\frac{3Q_{a}^{2}}{2M_{p}^{2}Q_{t}}+Q_{c}}\,,\quad
c_{T}^{2}\equiv 1-\frac{Q_{f}}{2M_{p}^{2}Q_{t}}\,,$ (3.12)

where

$\displaystyle
Q_{d}\equiv-\frac{2c_{2}}{M_{p}^{2}}\xi\dot{s}^{2}\dot{H}-\frac{2c_{3}}{M_{p}^{3}}\dot{s}^{2}(\dot{\xi}\dot{s}+\xi\ddot{s}-\xi\dot{s}H)\,,\quad$
$\displaystyle
Q_{e}\equiv-16c_{1}\dot{\xi}\dot{H}+\frac{2c_{2}}{M_{p}^{2}}\dot{s}(\dot{\xi}\dot{s}+2\xi\ddot{s}-2\xi\dot{s}H)-\frac{4c_{3}}{M_{p}^{3}}\xi\dot{s}^{3}\,,$
$\displaystyle Q_{f}\equiv
8c_{1}(\ddot{\xi}-\dot{\xi}H)+\frac{2c_{2}}{M_{p}^{2}}\xi\dot{s}^{2}\,.$

In the GR limit, where $\alpha\rightarrow 0$, the $Q_{a,b,c,d,e,f}$ quantities
become zero, while $Q_{t}$ becomes unity. Consequently,
$\\{c_{A},c_{T}\\}\rightarrow 1$ and the canonical case is restored. When
$\alpha\neq 0$, on the other hand, the propagation speeds deviate from the
unity. However, if the $c_{A}$ is either a negative ($c_{A}<0$) or
superluminal ($c_{A}>1$), one must worry about the ghost instability [69, 76].
We perform full numerical analyses for the values of the $c_{A}$ and $c_{T}$
of our model later in this section. Now that we have the key observable
quantities, we will conduct numerical analyses in the following using Eqs.
(3.10) and (3) and put constraints on the model parameters. In general, we
have three free parameters, including $\alpha$, $\lambda$, and $\sigma$.
However, if we adopt the Planck normalization [77] for
$\lambda/\sigma^{2}\sim\mathcal{O}(10^{-9})$ in our numerical study, our model
becomes a one-parameter model effectively. This adaptation, in the Gauss-
Bonnet contributions, i.e., $\alpha=0$, allows us to recover well-known
results of Higgs inflation in the Einstein frame.

Figure 1: Numerical plot of $n_{S}$ vs. $r$ (top) and their number of $e$–fold
dependence (bottom) from Eqs. (3.10)–(3). The two ends of each solid line
denote $N_{\ast}=50$(squares) and $N_{\ast}=60$ (disks). The solid black line
($\alpha=0$) indicates the absence of the Gauss-Bonnet contribution. The model
parameter $\alpha$ varies along the black-dotted lines between $-1.4\times
10^{4}\text{(red)}\leq\alpha\leq 8\times 10^{3}\text{(blue)}$.

Figure 1 presents the theoretical predictions of our model in the $n_{S}$ vs.
$r$ plane, along with the observational data. The background dark- and light-
blue contours represent the $1\sigma$ and $2\sigma$ confidence level (C.L.) of
the Planck TT, TE, EE+lowE+lensing+BK15+BAO data, respectively. At the same
time, the blue, black, and red lines show theoretical predictions of our model
for $\alpha=8\times 10^{3}$, $\alpha=0$, and $\alpha=-1.4\times 10^{4}$,
respectively. The orange squares and disks denote the $N_{\ast}=50$ and
$N_{\ast}=60$ $e$-folds, respectively. 333To plot Figure 1, we first solve the
background equations of motion Eq. (3.2) for $\\{s,\dot{s},H\\}$, with the
initial conditions
$\\{s_{0},\dot{s}_{0},H_{0}\\}=\\{5.6M_{p},0,\sqrt{V(s_{0})/(3M_{p}^{2})}\\}$,
and use the result in Eqs. (3.10) and (3). We use the number of $e$-folds,
which is related to time $t$ via $dN=Hdt$, as a time parameter, and the
duration of inflation is counted from the end of inflation. The solid black
line in the figure indicates the absence of the Gauss-Bonnet contributions
($\alpha=0$), and in this case, we recover theoretical predictions of Higgs
inflation in the GR case.

In the presence of the Gauss-Bonnet contributions ($\alpha\neq 0$), the
theoretical predictions of our model shift along the dotted-black lines.
Moreover, the $n_{S}$ value decreases (increases), while the $r$ value
slightly increases (decreases) for the positive (negative) $\alpha$. These
shifts can be seen more clearly in the embedded inset, where the $\alpha$
parameter varies between $-1.4\times 10^{4}\leq\alpha\leq 8\times 10^{3}$
along the dotted lines. The wiggles in the plot are simply a byproduct of the
numerical solution to the background equations of motion and do not indicate
any specific significance. The limiting values of $\alpha$ being relatively
large, i.e., $|\alpha|\gg\mathcal{O}(1)$, mean our model requires relatively
large values of $|\alpha|$ to deviate from the GR predictions because the
Gauss-Bonnet contributions without $\xi(s)$ are, in general, quite small in
comparison to the effects of a scalar field in GR. Thus, the relatively large
value of $|\alpha|$ plays the role of compensation here.

Figure 2: Numerical plot of $\epsilon_{i}(N)$ from Eqs. (3.8) and (3.9), where
$i=1,2,3,4,5$, for $\alpha=-1.4\times 10^{4}$(red) and $\alpha=8\times
10^{3}$(blue). The red and blue disks mark the end time of inflation for each
case.

Since our analyses in this section are purely numerical, we need to ensure the
smallness of the slow-roll parameters during inflation. Figure 2 shows the
slow-roll parameters ($\epsilon_{i}$ with $i$ ranging from $1$ to $5$) are
indeed small during inflation: $\epsilon_{i}\lesssim 1$ for the values of
$\alpha$ that are favored by observational data. The red and blue disks in
each sub-figure mark the end time $N_{\text{end}}$ of inflation, which is
determined from $\epsilon(N_{\text{end}})=1$. As expected, the absence of
Gauss-Bonnet contributions results in the $\epsilon_{3,4,5}$ value becoming
zero, as demonstrated in the figure. From Eq. (3.10), it is evident that the
tensor spectral index $n_{T}$ can be negative if the
$\epsilon_{5}>\epsilon_{1}$. However, as is explicit in Figure 2, the
$\epsilon_{1}$ significantly outweighs the $\epsilon_{5}$ throughout
inflation. As a result, we notice that the power spectrum of the tensor
fluctuations is red-tilted with $n_{T}>0$.

The direct detections of GWs from a neutron star merger GW170817 [78], as well
as its associated electromagnetic counterpart GRB170817A [79], allows us to
constrain the GWs propagation speed with remarkable precision: $-3\times
10^{-15}\leq c_{T}/c_{\gamma}-1\leq 7\times 10^{-16}$ where $c_{\gamma}$ is
the speed of light. We normalize $c_{\gamma}=1$. This bound indicates that the
difference in the propagation speed between light and gravitational waves is
less than about one part in $10^{15}$. However, the bound corresponds to the
late-time universe, where the scalar field value in our model must have
reached zero, i.e., $s=0$.

Figure 3: Numerical plot from Eq. (3.12) where $c_{\gamma}(=1)$ is the speed
of light. The red and blue lines denote $\alpha=-1.4\times 10^{4}$ and
$\alpha=8\times 10^{3}$, respectively. The red and blue disks mark the end
time of inflation for each case. The horizontal black solid lines at “$1$”
indicate the GR limit where $c_{T}=1=c_{A}$.

When $s\neq 0$, which is the case for the early universe, one can expect
significant deviations out of this bound induced by the Gauss-Bonnet
contributions. Such deviations are subject to future probes. In Figure 3, we
plot $c_{T}/c_{\gamma}$ and $c_{A}$ as functions of $N$. The red and blue
lines denote $\alpha=-1.4\times 10^{4}$ and $\alpha=8\times 10^{3}$,
respectively, and the ends of inflation for each case are marked with the red
and blue disks. The figure, especially the insets, shows that the Gauss-Bonnet
contributions gradually decay away and become negligible a few $e$–folds after
the end of inflation. As a result, we conclude that Gauss-Bonnet contributions
play a significant role in letting the GWs propagate at a speed different from
the speed of light during inflation and become negligible over time such that
the GW propagation speed converges to that of the speed of light after
inflation.

##  4 Conclusion

We have investigated Higgs inflation with a Gauss-Bonnet term in the Einstein
frame for the model given in Eq. (2.1). Our model in the Jordan frame has two
coupling functions, $\Omega^{2}(\phi)$ and $\omega(\phi)$, coupled
respectively to the Ricci scalar and the Gauss-Bonnet combinations. We assumed
these two coupling functions to hold a relation presented in Eq. (2.11) to
simplify the delivery of results in the Einstein frame. Our key analytic
result of the current work is derived in Eq. (2), where additional
interactions, including a non-minimal kinetic coupling between the scalar
field and gravity, as well as a derivative self-interaction of the scalar
field, emerged in the Einstein frame as a result of a conformal transformation
from the Jordan to the Einstein frame.

From Eq. (2), the background equations of motion are derived in Eq. (3.2), and
the observable quantities are obtained in Eq. (3.10) and (3), where we have
followed Ref. [69] closely. Although there are three free parameters in the
model, including $\lambda$, the potential parameter, $\sigma$, the parameter
representing the non-minimal coupling between the scalar field and the Ricci
scalar, and $\alpha$, the coupling parameter of the Gauss-Bonnet
contributions, we showed that our model becomes effectively the one-parameter
model if we adopt the Planck normalization for
$\lambda/\sigma^{2}\sim\mathcal{O}(10^{-9})$. The key numerical result of our
current work is presented in Fig. 1, where the theoretical predictions
$\\{n_{S},r\\}$ of our model are plotted together with the observational data.

Without the Gauss-Bonnet contributions, where $\alpha=0$, our result recovers
the predictions of Higgs inflation in the GR. Once the Gauss-Bonnet
contributions are turned on with $\alpha\neq 0$, the $n_{S}$ and $r$
predictions deviate from the GR case. The $n_{S}$ value decreases (increases),
while the $r$ value increases (decreases) for the positive (negative) $\alpha$
values, as is seen in the bottom panels of Figure 1. The observational data
favor the broad range model parameter, $-1.4\times
10^{4}\lesssim\alpha\lesssim 8\times 10^{4}$. In Figure 3, our analysis
reveals a remarkable phenomenon: the propagation speed of gravitational waves
(GWs) deviates from the speed of light during the inflationary period,
influenced by Gauss-Bonnet contributions on the order of a few parts in
hundreds of thousand. These Gauss-Bonnet effects gradually dissipate after the
inflation, leading the GWs to progressively align with the speed of light. We
have also shown the validity of the slow-roll approximation in Figure 2 by
showing the slow-roll parameters are small, much smaller than unity, during
inflation.

In our future research, we plan to relax our assumption made in Eq. (2.11) and
explore post-inflationary cosmology and its implications for (p)reheating. It
also remains to be determined whether these newly emerged interactions in the
Einstein frame can adequately account for the observed late-time accelerating
expansion of the universe.

## Acknowledgments

The authors acknowledge that this work was supported by the Basic Science
Research Program through the National Research Foundation of Korea (NRF),
funded by the Ministry of Education (grant numbers) (NRF-2022R1I1A1A01053784)
(GT), (NRF-2021R1A2C1005748) (SK) and by (NRF-2021R1A4A2001897),
(NRF-2019R1A2C1089334) (SCP).

##  Appendix A Constant coupling to the Gauss-Bonnet term

Let us consider the constant Gauss-Bonnet coupling, i.e.,
$\omega(\phi)=\text{const.}$ in Eq. (2.1), and simplify the Einstein frame
action in Eq. (2). For the fourth term in Eq. (2), we use

$\displaystyle\nabla_{a}\left[\Omega^{-2}\left(\nabla_{b}\nabla^{b}\Omega\nabla^{a}\Omega-\frac{1}{2}\nabla^{a}(\nabla\Omega)^{2}\right)\right]=\Omega^{-2}\left[(\nabla_{a}\nabla^{a}\Omega)^{2}-(\nabla_{a}\nabla_{b}\Omega)^{2}\right]-R^{ab}\Omega^{-2}\nabla_{a}\Omega\nabla_{b}\Omega$
$\displaystyle\qquad\qquad-2\Omega^{-3}(\nabla\Omega)^{2}\nabla_{b}\nabla^{b}\Omega+2\Omega^{-3}\nabla_{a}\Omega\nabla_{b}\Omega\nabla^{a}\nabla^{b}\Omega\,,$
(A.1)

and integration by parts to obtain

$\displaystyle 8\omega\int
d^{4}x\sqrt{-g}\,\Omega^{-2}\left(\nabla_{a}\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega-\nabla_{b}\nabla_{a}\Omega\nabla^{b}\nabla^{a}\Omega\right)=\omega\int
d^{4}x\sqrt{-g}$
$\displaystyle\quad\times\left[2R^{ab}\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2}+4(\Omega^{-1}\nabla_{b}\nabla^{b}\Omega)(\nabla\ln\Omega^{2})^{2}-4(\Omega^{-1}\nabla^{a}\nabla^{b}\Omega)(\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2})\right]\,,$
(A.2)

where the following relations are used

$\displaystyle\nabla_{a}\ln\Omega=\Omega^{-1}\nabla_{a}\Omega\,,$
$\displaystyle\Omega^{-1}\nabla^{a}\nabla_{a}\Omega=\nabla^{a}\nabla_{a}\ln\Omega+\nabla^{a}\ln\Omega\nabla_{a}\ln\Omega=\frac{1}{2}\nabla^{a}\nabla_{a}\ln\Omega^{2}+\frac{1}{4}\nabla^{a}\ln\Omega^{2}\nabla_{a}\ln\Omega^{2}\,.$
(A.3)

The third term in Eq. (2) can also be rewritten as

$\displaystyle-4\omega\int
d^{4}x\sqrt{-g}R\Omega^{-2}\nabla_{a}\Omega\nabla^{a}\Omega=-\omega\int
d^{4}x\sqrt{-g}g^{ab}R\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2}\,.$ (A.4)

Then, the first term on the right-hand side of equality in Eq. (A) is combined
with Eq. (A.4) to give

$\displaystyle 2\omega\int
d^{4}x\sqrt{-g}G^{ab}\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2},$ (A.5)

which is then canceled with the second term in Eq. (2). 444The integration by
parts of the second term in Eq. (2)  $\displaystyle-8\omega\int
d^{4}x\sqrt{-g}\Omega^{-1}G_{ab}\nabla^{a}\nabla^{b}\Omega=-2\omega\int
d^{4}x\sqrt{-g}G_{ab}\nabla^{a}\ln\Omega^{2}\nabla^{b}\ln\Omega^{2}\,.$ Thus,
the remaining terms in Eq. (2) reads

$\displaystyle S=\int
d^{4}x\sqrt{-g}\,\omega\left[R_{GB}^{2}-2(\Omega^{-1}\nabla_{b}\nabla^{b}\Omega)(\nabla\ln\Omega^{2})^{2}-4(\Omega^{-1}\nabla^{a}\nabla^{b}\Omega)(\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2})\right.$
$\displaystyle\qquad\quad+\left.\frac{3}{2}\left(\nabla_{a}\ln\Omega^{2}\nabla^{a}\ln\Omega^{2}\right)^{2}\right]\,.$
(A.6)

Let us rewrite Eq. (A) once again using Eq. (A.3)

$\displaystyle S=\int
d^{4}x\sqrt{-g}\,\omega\left[R_{GB}^{2}-(\nabla_{b}\nabla^{b}\ln\Omega^{2})\left(\nabla_{a}\ln\Omega^{2}\nabla^{a}\ln\Omega^{2}\right)\right.$
$\displaystyle\qquad\left.-2(\nabla^{b}\nabla^{a}\ln\Omega^{2})(\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2})-(\nabla_{a}\ln\Omega^{2}\nabla_{b}\ln\Omega^{2})^{2}+\left(\nabla_{a}\ln\Omega^{2}\nabla^{a}\ln\Omega^{2}\right)^{2}\right]\,.$
(A.7)

The second and third terms are canceled after integration by parts. Thus, we
obtain

$\displaystyle S=\int
d^{4}x\sqrt{-g}\,\omega\left[R_{GB}^{2}-\frac{4}{9}(\nabla_{a}s\nabla_{b}s)^{2}+\frac{4}{9}\left(\nabla_{a}s\nabla^{a}s\right)^{2}\right]=\int
d^{4}x\sqrt{-g}\,\omega R_{GB}^{2}\,,$ (A.8)

where $s\equiv\sqrt{3/2}\ln\Omega^{2}$. It is well-known in the literature
that the Gauss-Bonnet term is topological in $4$–dimensions if the Gauss-
Bonnet coupling is a constant. Thus, for the $\omega=\text{const.}$ case, we
conclude that no dynamical contributions emerge from the Gauss-Bonnet term in
the Jordan and Einstein frames.

##  Appendix B Power-law coupling to Gauss-Bonnet term

Let us now assume the coupling functions in Eq. (2.1) hold a more general
relation as $\omega=\alpha\Omega^{p}$. When $p=2$, the action in the Jordan
frame reads

$\displaystyle S^{J}=\int
d^{4}x\sqrt{-g^{J}}\left[\Omega^{2}(\phi)\left(\frac{M_{p}^{2}}{2}R^{J}+\alpha
R_{GB}^{2}\right)-\frac{1}{2}g^{J}_{ab}\nabla^{a}\phi\nabla^{b}\phi-V(\phi)\right].\,$
(B.1)

The Einstein frame action is presented in Eq. (2). For
$\omega=\alpha\Omega^{p}$ with arbitrary power of $p$, the third term of Eq.
(2) can be written as

$\displaystyle
8\omega\Omega^{-2}\left(\omega^{-1}\nabla_{a}\omega-2\Omega^{-1}\nabla_{a}\Omega\right)\left(\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega-\nabla_{b}\Omega\nabla^{a}\nabla^{b}\Omega\right)$
$\displaystyle\qquad=8\alpha(p-2)\Omega^{p-3}\nabla_{a}\Omega\left(\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega-\nabla_{b}\Omega\nabla^{a}\nabla^{b}\Omega\right)\,.$
(B.2)

Consequently, Eq. (2) becomes

$\displaystyle S$ $\displaystyle=\int
d^{4}x\sqrt{-g}\alpha\Omega^{p}\left[R_{GB}^{2}+8p\Omega^{-2}G_{ab}\nabla^{a}\Omega\nabla^{b}\Omega-8(p+1)\Omega^{-3}\nabla_{a}\Omega\nabla^{a}\Omega\nabla_{b}\nabla^{b}\Omega\right.$
$\displaystyle\left.\quad\qquad+8(p-2)\Omega^{-3}\nabla_{a}\Omega\nabla_{b}\Omega\nabla^{a}\nabla^{b}\Omega+24\Omega^{-4}\left(\nabla_{a}\Omega\nabla^{a}\Omega\right)^{2}\right]\,.$
(B.3)

Applying Eqs. (A.3) to the last term in the first line, we obtain

$\displaystyle S$ $\displaystyle=\int d^{4}x\sqrt{-g}\alpha
e^{\frac{p}{2}\sqrt{\frac{2}{3}}\frac{s}{M_{p}}}\left[R_{GB}^{2}+\frac{4p}{3M_{p}^{2}}G_{ab}\nabla^{a}s\nabla^{b}s-\frac{p+1}{3M_{p}^{3}}\sqrt{\frac{8}{3}}\nabla_{b}\nabla^{b}s\nabla_{a}s\nabla^{a}s\right.$
$\displaystyle\quad\left.+\frac{p-2}{3M_{p}^{3}}\sqrt{\frac{8}{3}}\nabla_{a}s\nabla_{b}s\nabla^{a}\nabla^{b}s\right]\,,$
(B.4)

where $(s/M_{p})\equiv\sqrt{3/2}\ln\Omega^{2}$. The last term vanishes for
$p=2$ and we get Eq. (2.13). As a result of conformal transformation from the
Jordan frame to the Einstein frame, we notice the emergence of new
interactions such as the kinetic coupling between the scalar field and gravity
and the derivative self-interactions of the scalar field. These interactions
certainly would contribute both to the background and the perturbation
dynamics.

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# Coexistence of polar and antipolar phases in ferroelectric halide perovskite
CsGeBr3

Ravi Kashikar<EMAIL_ADDRESS>Department of Physics, University of South
Florida, Tampa, Florida 33620, USA  S. Lisenkov  I. Ponomareva
<EMAIL_ADDRESS>Department of Physics, University of South Florida, Tampa,
Florida 33620, USA

###### Abstract

Recently ferroelectricity has been demonstrated in the family of halide
perovskites: CsGeX3 (X=Cl, Br, I). We develop a first-principles-based
computational approach to probe finite-temperature properties of CsGeBr3. Our
computations reveal the existence of a dynamic antipolar phase above the Curie
temperature. The phase can be stabilized below the Curie temperature through
quenching. Furthermore, competition between the polar and antipolar phases
results in the formation of rare polar-antipolar domains. We demonstrate that
such polar-antipolar domains can result in the formation of unusual
polarization fields with head-to-head and tail-to-tail dipole arrangements,
which not only offers an explanation for the recent experimental findings but
could also lead to the discovery of novel topological polar structures.

##  I Introduction

Inorganic halide perovskites of the form ABX3, where A=Cs or Rb; B=Ge, Sn, Pb
and X=Cl, Br, I have gained significant attention because of their
optoelectronic propertiesWang  _et al._ (2021); Fakharuddin  _et al._ (2022);
Xiao  _et al._ (2015). These perovskites exhibit direct bandgap, low thermal
conductivity, high absorption coefficient, high carrier mobility, and long
carrier diffusion lengthLi  _et al._ (2021). These properties find application
in solar cells, light-emitting diodes, semiconductor lasers, photocatalysts,
and memristors. These properties are further tunable with composition
alterations and external fields, such as pressure, uniaxial strain, and
biaxial strainKim  _et al._ (2020); Moloney  _et al._ (2020); Yang  _et al._
(2022). The general structural arrangement consists of BX6 octahedral cages
surrounded by A site entity. Many halide perovskites undergo phase transitions
under pressure and/or temperature to lower symmetry phasesAlaei  _et al._
(2021); Yang  _et al._ (2020). Among these, Sn and Pb-based crystal phases are
centrosymmetric in nature and have multiple phase transitions with
temperatureYang  _et al._ (2020). For example, CsSnX3 and CsPbX3 exhibit cubic
to tetragonal and tetragonal to orthorhombic phase transitions due to
octahedral rotations because their Goldschmidt tolerance factor is less than
1. For CsGeBr3 and CsGeCl3, the tolerance factor is greater than 1, which
favors the B site displacements and leads to stabilization of the polar phase.
Ge-based perovskites have been proposed to exhibit order-disorder single-phase
transition from cubic $Pm\bar{3}m$ phase to polar $R3m$ phase at 428 K, 511 K,
and 550 K for X=Cl, Br, and I, respectivelyThiele  _et al._ (1987). Recent
experiments have demonstrated ferroelectricity in CsGeX3 (X = Cl, Br, I) and
reported spontaneous polarization of 12-15 $\mu$C/cm2 for X=Br, 20 $\mu$C/cm2
for X=IZhang  _et al._ (2022). So far, there are no experimental polarization
reports for X=Cl. With this, CsGeX3 are the only inorganic halide perovskites
where the ferroelectricity has been demonstrated experimentally. Furthermore,
the bandgap of these materials lies between 1.6 to 3.3 eV and is suitable for
ferroelectric semiconducting devicesZhang  _et al._ (2022).

At the moment, the understanding of the ferroelectric phase transitions in
these materials is lacking. How do the order parameters emerge and evolve with
temperature? What is the character of the phase transition and its order? How
does the material’s response to the electric field depend on the temperature?
Does the phase transition in this family have some distinct features? Are
there any hidden phases or unusual domains/structures? Do the phase
transitions have unique dynamical signatures? Atomistic first-principles-based
simulations, such as effective HamiltonianZhong  _et al._ (1994, 1995), is an
excellent tool to probe the atomistic nature of the phase transitions in
complex ferroics and answer some of the aforementioned fundamental questions.
Such methodology has previously been used to study phase transitions in
ferroelectricsZhong  _et al._ (1994); Waghmare and Rabe (1997), ferroelectric
alloysWalizer  _et al._ (2006); Bellaiche  _et al._ (2000); Akbarzadeh  _et
al._ (2012); Mentzer  _et al._ (2019), multiferroicsWang  _et al._ (2012),
antiferroelectricsMani  _et al._ (2015a) and predict novel phases and
phenomena such as ferroelectric vorticesFu and Bellaiche (2003); Naumov  _et
al._ (2004), nanostripesLai  _et al._ (2007), bubblesZhang  _et al._ (2017);
Govinden  _et al._ (2023), ferroelectric phases in antiferroelectricsMani  _et
al._ (2015b), and electromagnonsChang  _et al._ (2016) to name a few.

In this work, we aim to: (i) develop an effective Hamiltonian approach to
study finite temperature properties of CsGeBr3; (ii) utilize this approach to
probe ferroelectricity and associated phase transition in this material in
order to answer some of the fundamental questions raised above; (iii) to
predict the existence of both polar and antipolar phases in CsGeBr3; (iv) to
reveal the possibility of these phase coexistence leading to the formation of
polar-antipolar domains, which result in unusual polarization fields.

We begin by investigating phonon instabilities in the cubic phase of CsGeBr3
using density functional theory (DFT) computations as implemented in VASP
package Kresse and Furthmüller (1996, 1996). Technically, we use projector-
augmented basis set (PAW) Blöchl (1994) within GGA-PBE Perdew  _et al._ (1996)
approximation. Plane wave energy cutoff of 550 eV and k-point grid density 0.2
$\AA^{-1}$ were used for all our ground state calculations. However, for
computing phonon dispersions, elastic constants and macroscopic dielectric
tensor higher energy cutoff (800 eV) and denser k-point mesh ($16\times
16\times 16$) were used to ensure convergence of the above-mentioned
properties. Figure 1(a) reports the phonon dispersion for cubic $Pm\bar{3}m$
phase of CsGeBr3 computed using density functional perturbation theory (DFPT)
and post-processed using PHONOPY package Togo and Tanaka (2015). The phonon
dispersion predicts the existence of an unstable branch with the strongest
instabilities in $\Gamma$, $X$, and $M$ points. This is in agreement with
previous computational findingsYang  _et al._ (2020).

Note, that we have also used the LDA exchange-correlation functional Ceperley
and Alder (1980). However, we found that it predicts the cubic phase to be
more stable than $R3m$ (which contradicts to experiment), and computed phonon
dispersion within LDA approximation does not posses unstable modes. Therefore,
we use the PBE approach for our methodological developments. Next, we compute
the energy profile along the eigenvector direction of the unstable phonons at
$\Gamma$, $X$, and $M$ points. The associated energy landscape is given in
Fig. 1(b) and suggests the existence of metastable phases in this material,
that is, the phases of local minima at the energy surface. The structures that
correspond to the minima for $\Gamma$, $X$, and $M$ point instabilities in
Fig. 1 are subjected to full structural relaxation, which resulted in the
following phases - $R3m$ (R), $Pmma$ (O) and $P4/nmm$ (T) with the energies
with respect to the cubic one of -86.1, -29.7, and -25.9 meV/f.u.,
respectively. These structures are associated with antipolar arrangements Ge
and Br ions and are provided in Ref. our .

Figure 1: (a) Phonon dispersion of CsGeBr3. (b) Energy along the directions of
the CsGeBr3 unstable modes at $\Gamma$, M and X.

Next, we use DFT calculations to parameterize the effective Hamiltonian model
of Ref.Zhong  _et al._ (1995). The degrees of freedom for the effective
Hamiltonian are local modes (which are proportional to the local dipole moment
of the unit cell), homogeneous, and inhomogeneous strains. We use the
eigenvector of unstable mode at $\Gamma$ point to parameterize our
Hamiltonian. The vector is dominated by Ge displacement, so we center local
modes on Ge (see Table 1). The Hamiltonian includes the following
interactions: local mode self energy (harmonic and anharmonic contributions),
a long-range dipole-dipole interaction, a short-range interaction between
local modes, elastic energy, and the interaction between the local modes and
strains. Effective Hamiltonian can be utilized in the framework of Monte
Carlo, Molecular Dynamics (MD), and other techniques to simulate finite-
temperature properties and has a long success history Zhong  _et al._ (1995);
Waghmare and Rabe (1997); Mani  _et al._ (2015a); Naumov  _et al._ (2004);
Prosandeev  _et al._ (2008); Mani  _et al._ (2015b); Wang  _et al._ (2012);
Ponomareva  _et al._ (2012).

The parameters of the effective Hamiltonian that we derived for CsGeBr3 are
given in Table 1 and can be used with the publicly available software FERAM
Nishimatsu  _et al._ (2008); fer . We note that the set of parameters
reproduces very well the lowest energy phonon branch shown in Fig. 1(a). In
particular, it captures competing instabilities in $\Gamma$, $X$, and $M$
points and their relative strength. Table 2 compares predictions of the
effective Hamiltonian with DFT and some available experimental data and
demonstrates its accuracy, including in capturing the dynamics of the soft
mode.

Table 1: First-principles-based parameters for CsGeBr3 in atomic units using the notations of Ref.Zhong  _et al._ (1995) derived from PBE/r2SCAN functionals.  On-site | $\kappa_{2}$ | 3.29$\times$10-3/2.50$\times$10-3 | $\alpha$ | 5.61$\times$10-2/6.62$\times$10-2 | $\gamma$ | -9.01$\times$10-2/-10.6$\times$10-2
---|---|---|---|---|---|---
| j1 | -2.56$\times$10-3/-3.09$\times$10-3 | j2 | 7.34$\times$10-4/10.34$\times$10-4 |  |
Intersite | j3 | 8.96$\times$10-4/10.79$\times$10-4 | j4 | -4.62$\times$10-4/-5.62$\times$10-4 | j5 | 4.67$\times$10-4/8.19$\times$10-4
| j6 | 2.12$\times$10-4/2.58$\times$10-4 | j7 | 0.00/0.00 |  |
Elastic | B11 | 1.91/2.15 | B12 | 0.42/0.45 | B44 | 0.39/0.46
Coupling | B1xx | -0.48/-0.51 | B1yy | -0.16/-0.19 | B5xz | -0.01/-0.01
Other | Z∗ | 7.69/8.25 | $\epsilon_{\infty}$ | 7.11/7.00 | m | 53.52/48.31
|  a ($\AA$) | 5.61/5.54 |  |  |  |
Eigenvector | (-0.01,-0.75,0.66,0.05,0.05)/(-0.01,-0.75,0.66,0.03,0.03)
Table 2: Some ground state properties of CsGeBr3 as obtained from DFT, effective Hamiltonian (Heff), and experiment (Exp). The experimental data are reported for room temperature and taken from Zhang  _et al._ (2022). The energies are reported with respect to the cubic phase, while ⋆ indicates the estimate for the energy difference between the two phases from the Curie point.  |  $\Delta$ET |  $\Delta$EO |  $\Delta$ER | a | PS | $\omega_{E}$ | $\omega_{A_{1}}$
---|---|---|---|---|---|---|---
| (K) | (K) | (K) | (Å) | ($\mu$C/cm2) | (cm-1) | (cm-1)
DFT | -301 | -345 | -999 | 5.75 | 19.9 | 130.4 | 160.8
Heff | -242 | -285 | -862 | 5.72 | 23.4 | 141.5 | 160.8
Exp. | - | - | -511∗ | 5.63 | 12-15 (RT) | 138 | 163
Figure 2:  Polarization components (a) and lattice parameters (b) as a
function of temperature. In (a) $P_{x,y,z}$ are the Cartesian components of
spontaneous polarization computed from simulated annealing. Black data give
one Cartesian component of remnant polarization obtained from hysteresis
loops. Green shading outlines the region where the polar phase is stabilized
by the electric field. Hysteresis loops computed at 1 GHz frequency in 200 K
to 600 K temperature range (c); Coercive field as a function of temperature
(d). The inset compares the hysteresis loops at T=280 K with the RT
experimental loop reported in Ref.Zhang  _et al._ (2022).

We now use the effective Hamiltonian in the framework of classical MD
simulations to model finite-temperature properties of bulk CsGeBr3. We use a
supercell of 30$\times$30$\times$30 unit cells of CsGeBr3 repeated
periodically along three Cartesian directions. The integration step is 0.5 fs.
The Evans-Hoover Rapaport and Rapaport (2004) thermostat is used to simulate
constant temperature. A simulated annealing technique is chosen to obtain
equilibrium phases at different temperatures. Technically, we start
simulations at 600 K and proceed in steps of 10 K down to 10 K. For each
temperature, we performed 500,000 MD steps (0.25 ns) and used half of them for
equilibration and half for computing thermal averages. The polarization is
computed as the average dipole moment of the supercell per its volume. Its
temperature evolution is given in Fig. 2(a) and predicts phase transition from
nonpolar $Pm\bar{3}m$ cubic phase to ferroelectric $R3m$ phase at 270 K. The
only experimental report to date reports the transition between $Pm\bar{3}m$
and $R3m$ phase at 511 K on the basis of X-ray diffraction and Raman
Spectroscopy Thiele  _et al._ (1987); Zhang  _et al._ (2022). It appears that
our effective Hamiltonian underestimates transition temperature, which is
rather common Zhong  _et al._ (1995); Nishimatsu  _et al._ (2008). At least
part of it could be due to the underestimation of the energy difference
between the ground state and cubic phase as compared with the DFT data (see
Table 2). In fact, the difference of 150 K between the effective Hamiltonian
and DFT values could account for the underestimation. Another contribution
could be due to the dependence of the energy prediction on the exchange-
correlation functional. Indeed, parametrization with r2SCAN functional Furness
_et al._ (2020) resulted in an increase of TC by 70 K. We notice that the
steep onset of the order parameter (polarization) at the Curie point is
suggestive of the first-order character of the phase transition.

Figure 3: Probability density functions for the components of the local modes
$u_{i}$ (a); and for the average components $<u_{i}>$ (b) computed at
different temperatures. Probability density functions for the components of
the local modes $u_{i}$ computed for the quenched structures at 100 K (c). The
temperature from which quenching has started is given on the labels. Dipole
pattern of CsGeBr3 at 100 K exhibiting polar (black boxes) and antipolar
(yellow box) domains.

To elucidate the response of polarization to the electric field, we have
computed electric hysteresis loops by applying an ac electric field with
frequencies of 1.0, 2.5, 5.0, and 10.0 GHz for all temperatures investigated
and show some representative ones in Fig 2(c). For an ac field frequency of 1
GHz, we find ferroelectric loops up to 295 K, paraelectric response in the
range of 300 K - 400 K, and linear dielectric response for higher
temperatures. The loops are used to compute remnant polarization as a function
of temperature and added to Fig. 2(a). From the comparison between spontaneous
and remnant polarization in Fig. 2(a), we notice that ferroelectricity can be
stabilized by the electric field in the temperature range of 270-300 K. The
coercive field as a function of temperature is reported in Fig. 2(d). We note
that the coercive field in computations typically overestimates the
experimental one due to the lack of defects in simulated supercells, their
finite size, and high-frequency of the applied fields. Nevertheless, the
comparison between the experimental and our computational coercive field is
good (see Fig. 2(d)), which further validates our computational approach. The
inset to Fig. 2(d) shows the comparison between computational and experimental
hysteresis loops.

The temperature evolution of the lattice parameters is given in Fig. 2(b) and
predicts somewhat unusual dependence. Firstly, it does not follow the first-
order-like trend exhibited by polarization. The second feature is the
dependence on temperature in the $Pm\bar{3}m$ phase. In our model, the
temperature evolution of the lattice parameters originates from the
temperature evolution of the local modes owing to the on-site coupling between
the local modes and local strains. Therefore, the computational data suggest
that the local modes are nonzero even above TC and even cooperate with each
other in some way, which, however, does not result in macroscopic
polarization. To elucidate the origin of this unusual behavior, we compute
probability density functions, $\rho(u_{i})$, for the Cartesian components of
the local modes. Technically, we ran additional 50,000 MD steps on the
equilibrated supercells in the temperature range of 250 K to 600 K and used
them to obtain probability density functions. Some representative functions
are shown in Fig. 3(a) and reveal the presence of two maxima in the functions.
Next, we compute the probability density functions for the average local
modes, $\rho(<u_{i}>)$, where the average is taken over 50,000 MD steps. The
representative functions are shown in Fig. 3(b) and reveal that the two peaks
have now merged into a single one, which suggests their dynamic nature. We
note that this finding also means that experimentally the phase would indeed
appear as $Pm\bar{3}m$. To probe further into the origin of the peaks in
$\rho(u_{i})$, we quench the equilibrium supercells that correspond to
temperatures from the 350-600 K range down to 100 K. The probability density
functions at 100 K are given in Fig. 3(a) and exhibit two well-defined peaks,
whose presence proves that quenching stabilizes the dynamical phase that
exists above $T_{C}$. Inspection of the quenched dipole patterns reveals
antipolar domains, which suggests that the dynamic high-temperature phase is
the antipolar one. Recall that antipolar Pmma(O) and P4/nmm(T) phases are also
predicted in DFT computations. Furthermore, the stabilization of antipolar
domains as opposed to single-domain phase may suggest that the high-
temperature dynamical phase is made up of antipolar nanoregions, the concept
similar to the famous polar nanoregions in relaxors Shvartsman  _et al._
(2009); Shvartsman and Lupascu (2012). Interestingly, such dynamical antipolar
domains, or antipolar nanoregions, could explain why the transition in
CsGeBr3is believed to be of order-disorder typeThiele  _et al._ (1987).

So far, we learned from simulations that annealing results in the
stabilization of the polar ground state, while ultra-fast quenching yields the
antipolar phase with domains. Could the two phases coexist? To answer this
question, we carried out quenching with a lower cooling rate; that is, we
quenched the supercell from 600 K down to 100 K in steps of 50 K, using 50,000
MD steps for each temperature. Indeed, in this case, we find the coexistence
of polar and antipolar domains as shown in Fig. 3(d), which to the best of our
knowledge, is extremely rare. The possibility of the coexistence of polar and
antipolar domains has been considered in hafnia filmsCheng  _et al._ (2022).
Moreover, such coexistence leads to the formation of head-to-head and tail-to-
tail domains, as shown in Fig. 3(d), which are energetically unfavored in
traditional ferroelectrics. Interestingly, such domains are in good agreement
with experimentally reported ones from Ref.Zhang  _et al._ (2022). Moreover,
the stabilization of such unusual domains could also mean that new topological
polar phases could be discovered in the materials with polar/antipolar phase
competition. Experimentally, the stabilization of polar-antipolar domains may
be induced by the thermal history. For example, the samples that have not been
annealed or not annealed to high enough temperatures may exhibit polar-
antipolar domains similar to the ones predicted in simulations. We note, that
all the main findings have been reproduced with parameters obtained with
r2SCAN functional.

In summary, we developed a first-principles-based computational methodology to
investigate the finite-temperature properties of ferroelectric halide
perovskite CsGeBr3. The application of the methodology to bulk CsGeBr3
predicts a single-phase transition from paraelectric cubic $Pm\bar{3}m$ phase
to ferroelectric rhombohedral phase $R3m$ in agreement with experimental
observation. However, it also reveals a dynamical antipolar phase that exists
above the Curie temperature. Such an antipolar phase can be stabilized through
rapid quenching from high temperatures and can compete with the polar phase.
The latter competition was found to result in the formation of polar-antipolar
domains, which allow for exotic domain configurations that are energetically
prohibitive in traditional ferroelectrics. In fact, the only known mechanism
to stabilize such domains is through free-charge compensation. The presence of
competing antipolar phases offers a different route to such domains and may
lead to new forms of topological polar structures. We believe that our work
will promote discoveries of novel properties and functionality associated with
polar-antipolar phase competitions in halide perovskites.

This work was supported by the U.S. Department of Energy, Office of Basic
Energy Sciences, Division of Materials Sciences and Engineering under Grant
No. DE-SC0005245. Computational support was provided by the National Energy
Research Scientific Computing Center (NERSC), a U.S. Department of Energy,
Office of Science User Facility located at Lawrence Berkeley National
Laboratory, operated under Contract No. DE-AC02-05CH11231 using NERSC award
BES-ERCAP-0025236.

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