# Power Spectra of Slow-Roll inflation in the consistent $D\to 4$ Einstein-
Gauss-Bonnet gravity

Tian-Chen Lia,b<EMAIL_ADDRESS>Tao Zhua,b<EMAIL_ADDRESS>(Corresponding author) Anzhong Wangc<EMAIL_ADDRESS>aInstitute for
Theoretical Physics & Cosmology, Zhejiang University of Technology, Hangzhou,
310023, China
b United Center for Gravitational Wave Physics (UCGWP), Zhejiang University of
Technology, Hangzhou, 310023, China c GCAP-CASPER, Physics Department, Baylor
University, Waco, TX 76798-7316, USA

###### Abstract

The slow-roll inflation which took place at extremely high energy regimes is
in general believed to be sensitive to the high-order curvature corrections to
the classical general relativity (GR). In this paper, we study the effects of
the high-order curvature term, the Gauss-Bonnet (GB) term, on the primordial
scalar and tensor spectra of the slow-roll inflation in the consistent $D\to
4$ Einstein Gauss-Bonnet (4EGB) gravity. The GB term is incorporated into
gravitational dynamics via the re-scaling of the GB coupling constant
$\alpha\to\alpha/(D-4)$ in the limit $D\to 4$. For our purpose, we calculate
explicitly the primordial scalar and tensor power spectra with GB corrections
accurate to the next-to-leading order in the slow-roll approximation in the
slow-roll inflation by using the third-order uniform asymptotic approximation
method. The corresponding spectral indices and their runnings of the spectral
indices for both the scalar and tensor perturbations as well as the ratio
between the scalar and tensor spectra are also calculated up to the next-to-
leading order in the slow-roll expansions. These results represent the most
accurate results obtained so far in the literature. In addition, by studying
the theoretical predictions of the scalar spectral index and the tensor-to-
scalar ratio with Planck 2018 constraint in a model with power-law potential,
we show that the second-order corrections are important in future
measurements.

##  I Introduction

The inflationary theory provides a successful solution to the problems of the
standard big bang cosmology for instance the flatness problem and the horizon
problem. It also explains successfully the almost scale-invariant and nearly
Gaussian spectra of primordial density perturbations Guth:1980zm ;
Starobinsky:1980te ; Sato:1980yn ; Baumann:2009ds , which eventually evolute
to generate the large-scale structure (LSS) observed today in the universe and
the cosmic microwave background (CMB) temperature anisotropies, which have
been detected with high precision by WMAP WMAP:2010qai ; Larson:2010gs ,
PLANCK Planck:2015sxf ; Planck:2015zfm , and other CMB experiments.

Inflation took place at an extremely high energy regime in the early universe.
In this regime, the standard theory of the slow-roll inflation in GR also
suffers from several conceptional problems, such as the trans-Planckian
problem Martin:2000xs ; Brandenberger:2012aj  and the initial singularity
problem Borde:1993xh ; Borde:2001nh . This is because the classical GR is
usually expected to be broken down at such an extremely high energy regime.
Thus, the inflationary theory in GR with some corrections can be regarded as
the effective theory of the complete UV quantum gravity. This has led to a
large amount of research works for considering possible high-order curvature
corrections to slow-roll inflation that arise from radiative corrections of
quantum gravity, for example, Horava-Lifshitz gravity Wang:2017brl  and
string/M-theory Baumann:2014nda .

The most important high-order curvature terms are the GB term and its Lovelock
generalization, which have been extensively studied in various alternative
theories beyond GR. However, because the GB term is a topological invariant in
four dimensions, this term can have contributions to the gravitational
dynamics only when it is coupled to a matter field. Latterly, Glavan and Lin
formulated a new theory with GB term in four dimensions (4EGB) by rescaling
the GB coupling constant $\alpha\rightarrow\alpha/(D-4)$ in the limit
$D\rightarrow 4$ Glavan:2019inb . With this scaling, it is argued that in the
limit $D\rightarrow 4$, the GB term can make nontrivial contributions to the
gravitational dynamics. The 4EGB gravity has been extended to higher-order
Lovelock gravity in Hennigar:2020lsl ; Fernandes:2020nbq . However, there are
also certain criticisms of the new 4EGB theory Ai:2020peo ; Gurses:2020ofy ;
Lu:2020iav ; Kobayashi:2020wqy ; Hennigar:2020lsl ; Fernandes:2020nbq ;
Shu:2020cjw ; Bonifacio:2020vbk ; Mahapatra:2020rds . Someone argued that this
theory lacks an intrinsically four-dimensional description in terms of a
covariantly-conserved rank-2 tensor in four dimensions Gurses:2020ofy . The
vacua of the model are ill-defined too Shu:2020cjw . Several variants of the
4EGB theory are proposed for the purpose of curing these problems, It is shown
that by compressing a higher dimensional theory down to a $D-4$ dimensional
maximally symmetric space and redefining a few parameters, the 4EGB theory can
be reformulated to a specific class of the Horndeski theory Lu:2020iav . Its
reformulation and Lovelock generalization as a scalar-tensor theory have been
deduced Kobayashi:2020wqy ; Lu:2020iav . In these realizations, the original
4EGB theory is reformulated to a scalar-tensor theory, with a coupling between
the scalar field and the GB term. And thus such theories, in general,
propagate three degrees of freedom. Note that in such a scalar-tensor
extension, it is also shown that the scalar field could be strongly coupled in
a cosmological background Kobayashi:2020wqy  and in a flat background
Bonifacio:2020vbk . Another way to cure the pathologies of the original 4EGB
theory is to break time diffeomorphism invariance but preserve the spatial one
Aoki:2020lig . It is shown that this spatial covariant 4EGB gravity (i.e. the
consistent $D\to 4$ Einstein-Gauss-Bonnet gravity) can only have two degrees
of freedom due to a Lagrangian multiplier term in the gravitational action
Aoki:2020lig .

In this paper, we will especially pay attention to the spatial covariant 4EGB
gravity in the early universe, and their corrections to the standard slow-roll
inflationary perturbations. It is interesting to note that the GB corrections
to slow-roll inflationary spectra in the scalar-Gauss-Bonnet gravity have
already been investigated and expanded at length in several works
Jiang:2013gza ; Guo:2010jr ; Koh:2014bka ; Satoh:2010ep ; Satoh:2008ck ;
vandeBruck:2015gjd ; Satoh:2007gn ; Chakraborty:2018scm . These papers have
studied the primordial perturbation spectra with the GB corrections, and
compared it with observed values. It is interesting to note that in the slow-
roll inflation, the coupling between the scalar field and the GB term
generates a time-dependent sound speed related to the equation of motion for
scalar and tensor perturbations. In the spatial covariant 4EGB theory, it not
only predicts time-dependent sound speeds associated with the tensor
perturbations in the slow-roll inflation but also modifies the linear
dispersion relation of the tensor modes. The primordial power spectra and non-
Gaussianities with both the effects of the time-dependent sound speeds and
modified dispersion have been explored in Aoki:2020ila . However, for
calculating the perturbation spectra, the previous works have supposed this
time-dependent sound speed as well as the slow-roll quantities as constants.
In fact, this approach will be invalid when it is used to calculate the
primordial spectra above the first-order in the slow-roll approximation.

With the emergence of new high-accuracy cosmological data, such as the Planck
data, one expects to get tighter constraints on the theory. Thus, it is
important to compare the theoretical predictions of specific inflation models
with the observational data. With the rapid development of technology, the CMB
observations and the incoming large-scale structures surveys will become more
and more accurate. For example, the data from the stage-4 CMB experiments and
the LiteBird satellite provide will not only increase the sensitivity of
primordial gravitational waves ($\epsilon_{1}$) but also explore much higher
multipoles CMB-S4:2016ple ; SimonsObservatory:2018koc ; Mallaby-Kay:2021tuk ;
LiteBIRD:2022cnt ; Auclair:2022yxs . The Euclid satellite and other ground
surveys are going to measure the small scales in the matter power spectrum
LSSTScience:2009jmu ; Lacasa:2019flz ; Euclid:2021qvm ; Auclair:2022yxs . All
these measurements will be sensitive to the high-order corrections to the
slow-roll perturbation spectra beyond the leading-order in the slow-roll
approximations Adshead:2010mc ; Martin:2014rqa ; Sprenger:2018tdb . In this
situation, in order to analyze the data with future observations, the
theoretical predictions must be more precise and go beyond the leading slow-
roll orders Auclair:2022yxs ; Martin1 ; Martin2 .

With these considerations, it is highly demanded to consider the slow-roll
approximation beyond the leading-order in the calculations of the inflationary
perturbation spectra. For this purpose, one has to consider both the effects
of the time variation of the sound speeds and the modified dispersion relation
that arise in the spatial covariant 4EGB theory. Both the time-dependent
propagating sound speeds and the modified dispersion relation can make
important corrections in the primordial scalar and tensor perturbation
spectra. However, considerations of the time variation of the sound speed and
the modified dispersion relation make it very difficult to calculate the
corresponding power spectra. In this paper, in order to calculate the
primordial perturbation spectra and their spectral indices to desired
precision, we employ the uniform asymptotic approximation developed in a
series of papers Zhu:2013fha ; Zhu:2013upa ; Zhu:2014wfa ; Zhu:2015ata ;
Zhu:2016srz ; Zhu:2014wda ; Zhu:2014aea ; Zhu:2015xsa ; Zhu:2015owa . This
approximation provides a better treatment to equations with turning points and
poles, and has been widely applied in calculating primordial spectra for
various inflation models Zhu:2013fha ; Zhu:2013upa ; Zhu:2014wfa ; Zhu:2015ata
; Zhu:2016srz ; Zhu:2014wda ; Zhu:2014aea ; Zhu:2015xsa ; Zhu:2015owa  and
applications in studying the reheating process Zhu:2018smk  and quantum
mechanics Li:2019cre . The main goal of this paper is to use the uniform
asymptotic approximation to calculate the inflationary observables of the
slow-roll inflation in the spatial covariant 4EGB theory with high accuracy.

The content of the paper is arranged as follows. In Sec. II, we present a
brief introduction of spatial covariant 4EGB gravity, and in Sec. III, we
consider the cosmological perturbations in a flat FRW background, including
the linear scalar perturbations and tensor perturbations. Then we calculate
explicitly the power spectra, spectral indices, and runnings of the spectral
indices of both scalar and tensor perturbations and the ratio between the
scalar and tensor spectra in the slow-roll inflation with the GB correction in
Sec. IV. With the obtained expressions of the scalar spectral index and the
tensor-to-scalar ratio, we study their predictions with Planck 2018 constraint
in a model for specific power-law potential in Sec. V. Our main conclusions
and outlook are summarized in Sec. VI. We also give the most general formulas
for calculating the primordial power spectra in the high-order uniform
asymptotic approximations in appendixes A and B.

##  II spatial covariant 4EGB gravity

In this section, we will briefly introduce the spatial covariant 4EGB gravity.
The details of this theory can be found in Glavan:2019inb ; Aoki:2020lig . In
this theory, the dynamical variables are the shift vector $N^{i}$, lapse
function $N$ and spatial metric $\gamma_{ij}$. We can write the metric of a
spacetime in the Arnowitt-Deser-Misner (ADM) form,

$\displaystyle
ds^{2}=-N^{2}dt^{2}+\gamma_{ij}(dx^{i}+N^{i}dt)(dx^{j}+N^{j}dt).$ (2.1)

The action of this theory is as follows,

$\displaystyle S_{\rm EGB}^{\rm 4D}$ $\displaystyle=$ $\displaystyle\int
dtd^{3}x\sqrt{\gamma}N{\cal{L}}^{\rm 4D}_{\rm EGB}$ (2.2)

here $\gamma$ is the determinate of the spatial metric $\gamma_{ij}$ with
$\gamma={\rm det}(\gamma_{ij})$ , and

$\displaystyle{\cal{L}}^{\rm 4D}_{\rm EGB}$ $\displaystyle=$
$\displaystyle\frac{M_{\rm Pl}^{2}}{2}\Bigg{[}2R-{\cal
M}+\frac{\tilde{\alpha}}{2}\Big{(}8R^{2}-4R{\cal M}-{\cal M}^{2}$
$\displaystyle-\frac{8}{3}\left(8R_{ij}R^{ij}-4R_{ij}{\cal M}^{ij}-{\cal
M}_{ij}{\cal M}^{ij}\right)\Big{)}\Bigg{]},$

with

$\displaystyle{\cal M}_{ij}$ $\displaystyle\equiv$ $\displaystyle R_{ij}+{\cal
K}^{k}_{k}{\cal K}_{ij}-{\cal K}_{ik}{\cal K}^{k}_{j},$ (2.4)
$\displaystyle{\cal M}$ $\displaystyle\equiv$ $\displaystyle{\cal M}^{i}_{i},$
(2.5) $\displaystyle{\cal K}_{ij}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2N}\left(\partial_{t}\gamma_{ij}-2D_{(i}N_{j)}-\gamma_{ij}D^{2}\lambda_{\rm
GF}\right).$ (2.6)

In the above, $M_{\rm Pl}^{2}=1/(8\pi G)$ is the reduced Planck mass, where
$G$ being the Newton gravitational coupling constant. $R$, $R_{ij}$ here are
respectively the Ricci scalar and the Ricci tensor refer to the spatial-metric
$\gamma_{ij}$. $D_{i}$ is the covariant derivative and can be compatible with
the spatial metric. $\lambda_{\rm GF}$ is similar to the Lagrange multipliers,
and it imposes a primary constraint on the theory such that it can propagate
only two degrees of freedom Glavan:2019inb ; Aoki:2020lig . The coupling
constant $\tilde{\alpha}$ is the rescaled GB coupling constant
$\tilde{\alpha}\to\alpha/(D-4)$ with the limit $D\to 4$.

One important feature of the spatial covariant 4EGB gravity is that it don’t
admit the full diffeomorphism invariance of the four-dimensional spacetime,
but break the temporal diffeomorphism. Then this theory only has the time
reparametrization symmetry and the three-dimensional spatial diffeomorphism,

$\displaystyle t$ $\displaystyle\to$ $\displaystyle\tilde{t}=t-f(t),$ (2.7)
$\displaystyle x^{i}$ $\displaystyle\to$
$\displaystyle\tilde{x}^{i}=x^{i}-\zeta^{i}(t,x^{j}).$ (2.8)

where $f(t)$ and $\zeta^{i}(t,x^{j})$ are the the infinitesimal generators of
the time reparametrization and the spatial diffeomorphism, respectively. With
this gauge transformation, the dynamical variables $N$, $N_{i}$, and
$\gamma_{ij}$ can transform as

$\displaystyle\delta N$ $\displaystyle=$
$\displaystyle\zeta^{k}D_{k}N+\dot{N}f+N\dot{f},$ $\displaystyle\delta N_{i}$
$\displaystyle=$ $\displaystyle
N_{k}D_{i}\zeta^{k}+\zeta^{k}D_{k}N_{i}+\gamma_{ik}\dot{\zeta}^{k}+\dot{N}_{i}f+N_{i}\dot{f},$
$\displaystyle\delta\gamma_{ij}$ $\displaystyle=$ $\displaystyle
D_{i}\zeta_{j}+D_{j}\zeta_{i}+f\dot{\gamma}_{ij}.$ (2.9)

Here a dot is the derivative of respect to the time $t$.

##  III Cosmological perturbations in a flat FRW background

In this section, we present a brief introduction of the scalar and tensor
perturbations of the slow-roll inflation in the spatial covariant 4EGB
gravity.

###  III.1 Slow-roll inflation

For later convenience to introduce the cosmological perturbations, we first
consider a flat Friedmann-Robertson-Walker (FRW) background,

$\displaystyle ds^{2}$ $\displaystyle=$ $\displaystyle-
dt^{2}+a^{2}(t)\delta_{ij}dx^{i}dx^{j}$ (3.1) $\displaystyle=$ $\displaystyle
a^{2}(\tau)(-d\tau^{2}+\delta_{ij}dx^{i}dx^{j}),$

where $a(t)$ is the scalar factor of the universe , $t$ is the cosmic time,
and $\tau$ is the conformal time. For studying the inflationary cosmology and
cosmological perturbations, we can consider the theory in (2.2) with a scalar
field $\phi$ with a potential $V(\phi)$, that is

$\displaystyle S=S_{\rm EGB}^{\rm 4D}+S_{\phi}$ (3.2)

where

$\displaystyle S_{\phi}=\int
dtd^{3}x\sqrt{\gamma}N\left[\frac{1}{2}g^{\mu\nu}\partial_{\mu}\phi\partial_{\nu}\phi+V(\phi)\right].$
(3.3)

Here the scalar field $\phi$ is canonical and coupled to gravity minimally,
$\phi$ is the inflation field with potential $V(\phi)$, and $g^{\mu\nu}$ is
the metric of the 4-dimensional spacetime. Thus the modified Friedmann and the
Klein-Gordon equations in the spatial covariant 4EGB gravity can be written as
Glavan:2019inb ; Aoki:2020lig

$\displaystyle 3M_{\rm
Pl}^{2}(H^{2}+\tilde{\alpha}H^{4})=\frac{1}{2}\dot{\phi}^{2}+V(\phi),$ (3.4)
$\displaystyle-2M_{\rm Pl}^{2}(1+2\tilde{\alpha}H^{2})\dot{H}=\dot{\phi}^{2},$
(3.5)

and

$\displaystyle\ddot{\phi}+3H\dot{\phi}+V_{,\phi}=0,$ (3.6)

where $H\equiv\dot{a}/a$ is the Hubble parameter with a dot is the derivative
with respect to the cosmic time $t$, and $V_{,\phi}=dV(\phi)/d\phi$. With
these equations of the background evolutions, we will respectively consider
the cosmological scalar and tensor perturbations in the following subsections.

To consider the slow-roll inflation, we also need to impose the following
slow-roll conditions

$\displaystyle\frac{1}{2}\dot{\phi}^{2}\ll
V,\;\;|\ddot{\phi}|\ll|3H\dot{\phi}|.$ (3.7)

Then we can easily introduce the Hubble flow slow-roll parameters
$\epsilon_{n}$. The definitions of the Hubble flow slow-roll parameters are

$\displaystyle\epsilon_{n+1}\equiv\frac{d\ln\epsilon_{n}}{d\ln
a},\;\;\epsilon_{1}\equiv-\frac{\dot{H}}{H^{2}}.$ (3.8)

The conformal time $\tau$ is defined as

$\displaystyle\tau(t)=\int^{t}_{t_{\text{end}}}\frac{dt^{\prime}}{a(t^{\prime})},$
(3.9)

where $t_{\text{end}}$ is the time of the slow-roll inflation ends. We also
introduce a new parameter to represent the GB effects and simplify the
calculation, which is defined as

$\displaystyle f_{\rm GB}=2\tilde{\alpha}H^{2}.$ (3.10)

During the slow-roll inflation, this parameter is also slow-varying , and it
characterizes the correction of the GB term in the spatial covariant 4EGB
gravity. In the slow-roll approximation, we treat this parameter as a new
slow-roll parameter in the slow-roll expansion. In principle, the value of the
paramater $f_{\rm GB}$ is not necessary to be at the order to of
$\mathcal{O}(\epsilon)$. However, in orde to simplfy the calcualtions of the
power spectra later, we assume that $f_{\rm\rm GB}\sim\mathcal{O}(\epsilon)$.
Thus the calculations presented in this paper are valid only with this
condition.

###  III.2 Scalar perturbations

Let us consider the linear scalar perturbations around the flat FRLW spacetime
in the spatial covariant 4EGB gravity. It is more convenient to use the
conformal time $\tau$ to achieve this purpose, and the background variables
$(N,N^{i},\gamma_{ij})$ is as follows

$\displaystyle\hat{N}=a(\tau),\;\;\hat{N}^{i}=0,\;\;\hat{\gamma}_{ij}=a^{2}(\tau)\delta_{ij},$
(3.11)

where the quantities with a hat denote the background fields only depend on
$\tau$. Under this background, we can introduce the scalar perturbations
$(A,B,\psi,E,\delta\phi,\delta\lambda)$ as,

$\displaystyle N=a(\tau)(1+A),\;\;N^{i}=\delta^{ij}\partial_{j}B,$
$\displaystyle\gamma_{ij}=a^{2}(\tau)[(1+2\psi)\delta_{ij}+2\partial_{i}\partial_{j}E],$
$\displaystyle\phi=\hat{\phi}+\delta\phi,\;\;\;\lambda_{\rm
GF}=\hat{\lambda}_{\rm GF}+a^{2}\delta\lambda.$ (3.12)

Here $\hat{\phi}$ and $\hat{\lambda}_{\rm GF}$ are the background fields of
$\phi$ and $\lambda_{\rm GF}$ respectively, and only depends on $\tau$. With
the gauge transformation (2.9), the scalar perturbations can transform in the
form

$\displaystyle A\to\tilde{A}$ $\displaystyle=$ $\displaystyle
A+f^{\prime}+\mathcal{H}f,$ (3.13) $\displaystyle B\to\tilde{B}$
$\displaystyle=$ $\displaystyle B+\zeta^{\prime},$ (3.14)
$\displaystyle\psi\to\tilde{\psi}$ $\displaystyle=$
$\displaystyle\psi+\mathcal{H}f,$ (3.15) $\displaystyle E\to\tilde{E}$
$\displaystyle=$ $\displaystyle E+\zeta.$ (3.16)

Here we split $\zeta^{i}=\zeta^{,i}+\zeta_{\perp}^{i}$ with
$\zeta_{\perp}^{i}$ being the transverse part of the vector $\zeta^{i}$.
Because the sptaial covariant 4EGB gravity breaks the time diffeomorphism, one
can chose a gauge like neither $A=0$ nor $\psi=0$. From gauge transformation
has been discussed above, there are only two gauges can be chosen:

$\displaystyle(i)\;\;B=0,\;\;{\rm or}\;\;(ii)\;\;E=0.$ (3.17)

Thus, of the four scalar type metric perturbations $(A,B,\psi,E)$, only one of
$(E,B)$ can be eliminated by choosing $\zeta(t,x^{i})$ in the gauge
transformation. This is different from the theory of full diffeomorphism in
4-dimensional spacetime, where either $A$ or $\psi$ can also be eliminated by
choosing $f$ freely.

Then we consider the comoving curvature perturbation $\cal{R}$, which is
defined as

${\cal{R}}=-\psi+\frac{\cal{H}}{\hat{\phi^{\prime}}}\delta\phi.$ (3.18)

Under the above gauge transformation, it is not difficult to verify that
$\mathcal{R}$ is still gauge-invariant even in the current setup. With this
definition and a series of tedious calculations, we can obtain the second
order action of the terms of $\mathcal{R}$ Glavan:2019inb ; Aoki:2020lig ,

$\displaystyle S^{(2)}=M_{\rm Pl}^{2}\int d\tau
d^{2}xa^{2}\epsilon_{1}(1+f_{\rm GB})\Big{[}{\cal R}^{\prime
2}+\partial^{2}{\cal R}^{2}\Big{]}.$

With variation of this double-dip action with respect to ${\cal R}$, the
equation of motion for the comoving curvature perturbation ${\cal R}$ is
obtained as follows,

$\displaystyle\mathcal{R}^{\prime\prime}+\left(2\mathcal{H}+\frac{\epsilon_{1}^{\prime}}{\epsilon_{1}}+\frac{f_{\rm
GB}^{\prime}}{1+f_{\rm
GB}}\right)\mathcal{R}^{\prime}-\partial^{2}\mathcal{R}=0.$ (3.20)

We can define the mode function $u_{\mathcal{R}}$ in terms of the Fourier
modes of curvature perturbations as
$u_{\mathcal{R}}=z_{\mathcal{R}}\mathcal{R}_{k}$. Then the equation of motion
(3.20) can be transformed into the form

$\displaystyle
u^{\prime\prime}_{\mathcal{R}}(\tau)+\left(k^{2}-\frac{z^{\prime\prime}_{\mathcal{R}}}{z_{\mathcal{R}}}\right)u_{\mathcal{R}}(\tau)=0.$
(3.21)

Here a prime is the derivative with respect to the conformal time $\tau$.
$z_{\mathcal{R}}$ and $\frac{z^{\prime\prime}_{\mathcal{R}}}{z_{\mathcal{R}}}$
can be written as follows

$\displaystyle z_{\mathcal{R}}$ $\displaystyle=$ $\displaystyle
a\sqrt{2\epsilon_{1}(1+f_{\rm GB})}M_{\rm Pl},$ (3.22)

and

$\displaystyle\frac{z^{\prime\prime}_{\mathcal{R}}}{z_{\mathcal{R}}}$
$\displaystyle=$ $\displaystyle 2a^{2}H^{2}\Bigg{[}1-\frac{1+4f_{\rm
GB}}{2(1+f_{\rm GB})}\epsilon_{1}+\frac{3}{4}\epsilon_{2}$ (3.23)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{(3+2f_{\rm GB})f_{\rm
GB}}{2(1+f_{\rm GB})^{2}}\epsilon_{1}^{2}-\frac{1+5f_{\rm GB}}{4(1+f_{\rm
GB})}\epsilon_{1}\epsilon_{2}$
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}+\frac{\epsilon_{2}^{2}}{8}+\frac{1}{4}\epsilon_{2}\epsilon_{3}\Bigg{]}.$

###  III.3 Tensor perturbations

The cosmological tensor perturbations $h_{ij}$ is defined as,

$\displaystyle\delta\gamma_{ij}=a^{2}h_{ij},\;\delta N^{i}=0,\;\;\delta N=0,$
(3.24)

where $h_{ij}$ denotes the traceless and transverse metric perturbations,
namely,

$\displaystyle\partial^{i}h_{ij}=0=h_{i}^{i}.$ (3.25)

With the gauge transformation (2.9), the tensor perturbations are found to be
gauge invariant. The above definition requires us to derive equations of
motion for tensor perturbations. To this end, the metric perturbation can be
first substituted into Eq. (3.2) and then expanded to the quadratic order of
$h_{ij}$. After a series of complicated calculations, one obtains
Glavan:2019inb ; Aoki:2020lig ,

$\displaystyle S^{(2)}_{h}$ $\displaystyle=$ $\displaystyle\frac{M_{\rm
Pl}^{2}}{8}\int d\tau
d^{3}xa^{2}(\tau)\Big{[}(1+2\tilde{\alpha}H^{2})h^{\prime}_{ij}h^{\prime ij}$
(3.26)
$\displaystyle\;\;\;\;-\left(1+2\tilde{\alpha}H^{2}+4\tilde{\alpha}\dot{H}\right)\partial_{k}h_{ij}\partial^{k}h^{ij}$
$\displaystyle~{}~{}~{}~{}-\frac{4\tilde{\alpha}}{a^{2}(\tau)}\partial^{2}h_{ij}\partial^{2}h^{ij}\Big{]}.$

Then one can take the variation of the quadratic action of$h_{ij}$ to obtain
the equations of motion of the tensor perturbations, which gives,

$\displaystyle
h_{ij}^{\prime\prime}+2\mathcal{H}\left(1+\frac{2\alpha\dot{H}}{1+2\alpha
H^{2}}\right)h_{ij}^{\prime}$
$\displaystyle-\left(1+\frac{4\alpha\dot{H}}{1+2\alpha
H^{2}}\right)\partial^{2}h_{ij}+\frac{4\alpha/a^{2}}{1+2\alpha
H^{2}}\partial^{4}h_{ij}=0,$

where $\mathcal{H}\equiv a^{\prime}/a$ with a _prime_ representing the
derivative with respect to the conformal time $\tau$.

In order to study the evolution of $h_{ij}$, we expand it to the spatial
Fourier harmonics,

$\displaystyle h_{ij}(\tau,x^{i})=\sum_{A={\rm
R,L}}\int\frac{d^{3}k}{(2\pi)^{3}}h_{A}(\tau,k^{i})e^{ik_{i}x^{i}}e_{ij}^{A}(k^{i}),$

where $e_{ij}^{A}$ represent the circular polarization tensors. $e_{ij}^{A}$
satisfy the following relation

$\displaystyle\epsilon^{ijk}n_{i}e_{kl}^{A}=i\rho_{A}e^{jA}_{~{}l},$ (3.29)

where $\rho_{\rm L}=-1$ and $\rho_{\rm R}=1$. It is worth noting that the
propagation equations of these two modes are decoupled, and can be transformed
into

$\displaystyle h_{A}^{\prime\prime}+\left(2\mathcal{H}+\frac{f_{\rm
GB}^{\prime}}{1+f_{\rm GB}}\right)h_{A}^{\prime}+\omega_{k}^{2}h_{A}=0.$
(3.30)

The quantity $\frac{f_{\rm GB}^{\prime}}{1+f_{\rm GB}}$ in parentheses gives a
description of the correction of the friction term, while $\omega_{k}^{2}$
describes the correction of the dispersion relation for the tensor
perturbations, which reads

$\displaystyle\omega_{k}^{2}$ $\displaystyle\equiv$
$\displaystyle\left(1-\frac{8\alpha\dot{H}}{M_{\rm Pl}^{2}+4\alpha
H^{2}}+\frac{4\tilde{\alpha}/a^{2}}{1+2\tilde{\alpha}H^{2}}k^{2}\right)k^{2}$
(3.31) $\displaystyle=$ $\displaystyle\left(1+\frac{2f_{\rm
GB}\epsilon_{1}}{1+f_{\rm GB}}+\frac{2f_{\rm GB}}{1+f_{\rm
GB}}\frac{k^{2}}{a^{2}H^{2}}\right)k^{2}$ $\displaystyle=$ $\displaystyle
c_{h}^{2}k^{2}+\frac{2f_{\rm GB}}{1+f_{\rm GB}}\frac{k^{2}}{a^{2}H^{2}}k^{2},$

with

$\displaystyle c_{h}^{2}\equiv 1+\frac{2f_{\rm GB}\epsilon_{1}}{1+f_{\rm
GB}}.$ (3.32)

We then define a new variable ${u_{h}=\frac{1}{2}z_{h}h_{A}}$ and change Eq.
(3.30) to

$\displaystyle
u_{h}^{\prime\prime}+\left[\omega_{k}^{2}-\frac{z_{h}^{\prime\prime}}{z_{h}}\right]u_{h}=0,$
(3.33)

where

$\displaystyle{z_{h}\equiv a\sqrt{1+f_{\rm GB}}M_{\rm{pl}}},$ (3.34)

and

$\displaystyle\frac{z_{h}^{\prime\prime}}{z_{h}}$ $\displaystyle=$
$\displaystyle 2a^{2}H^{2}\Bigg{[}1-\frac{1+4f_{\rm GB}}{2(1+f_{\rm
GB})}\epsilon_{1}+\frac{(3+2f_{\rm GB})f_{\rm GB}}{2(1+f_{\rm
GB})^{2}}\epsilon_{1}^{2}$ (3.35)
$\displaystyle~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}-\frac{f_{\rm
GB}}{2(1+f_{\rm GB})}\epsilon_{1}\epsilon_{2}\Bigg{]}.$

##  IV Scalar and Tensor Perturbation Spectra in the uniform asymptotic
approximation

In this section, we calculated the scalar and tensor spectra of the slow-roll
inflation in the spatial covariant 4EGB gravity. The power spectrum
calculation method used in this paper is the uniform asymptotic approximation,
and the third-order approximation of it are developed in a series papers
Zhu:2013fha ; Zhu:2013upa ; Zhu:2016srz ; Zhu:2014wfa ; Zhu:2014wda ;
Zhu:2014aea ; Zhu:2015xsa ; Zhu:2015owa ; Zhu:2015ata ; Zhu:2018smk . The
general formulas for calculating the primordial power spectra from the slow-
roll inflation are described in appendix A and B. The primordial power spectra
for both the scalar and tensor perturbations are calculated up to the second-
order in the slow-roll approximation in the following subsections.

###  IV.1 Scalar Spectrum

We first consider the scalar perturbations. In order to employ the uniform
asymptotic approximation, we need to map the equation of motion of $u_{\cal
R}$ in (3.21) into the standard form in (A.1). We have

$\displaystyle\lambda^{2}\hat{g}(y)+q(y)=\frac{\nu_{\cal R}^{2}}{y^{2}}-1,$
(4.1)

with

$\displaystyle\lambda^{2}\hat{g}(y)=\tau^{2}\frac{z^{\prime\prime}_{\cal
R}}{z_{\cal R}}\frac{1}{y^{2}},$ (4.2) $\displaystyle q(y)=-\frac{1}{4y^{2}},$
(4.3)

where

$\displaystyle\nu_{\cal R}^{2}$ $\displaystyle=$
$\displaystyle\frac{1}{4}+\tau^{2}\frac{z^{\prime\prime}_{\cal R}}{z_{\cal
R}}$ (4.5) $\displaystyle=$
$\displaystyle\frac{1}{4}+2\tau^{2}a^{2}H^{2}\Bigg{[}1-\frac{1+4f_{\rm
GB}}{2(1+f_{\rm GB})}\epsilon_{1}+\frac{3}{4}\epsilon_{2}$
$\displaystyle+\frac{(3+2f_{\rm GB})f_{\rm GB}}{2(1+f_{\rm
GB})^{2}}\epsilon_{1}^{2}-\frac{1+5f_{\rm GB}}{4(1+f_{\rm
GB})}\epsilon_{1}\epsilon_{2}$
$\displaystyle+\frac{\epsilon_{2}^{2}}{8}+\frac{1}{4}\epsilon_{2}\epsilon_{3}\Bigg{]}$

It is easy to see that the function $\lambda^{2}\hat{g}(y)$ has a single
turning point $y_{0}(\bar{\tau}_{0})=-k\bar{\tau}_{0}$, which related to
$\nu_{\cal R}$ as

$\displaystyle y_{0}(\bar{\tau}_{0})=-k\bar{\tau}_{0}=\nu_{\cal
R}(\bar{\tau}_{0}).$ (4.6)

Hereafter we use a bar over the quantities denoting the quantities evaulated
at the turning point.

In order to match the accuracy of forthcoming observations, we need to
calculate the scalar spectrum and the corresponding spectral indices up to the
next-to-leading order (second-order) in the expansions of the slow-roll
approximation. For this purpose, we have to consider the time variations of
the slow-roll quantities such as $\nu_{\cal R}$. Considering $\nu_{\cal R}$ is
slowly varying, it is convenient to expand it up to the second-order in the
slow-roll expansions as that presented in Eq. (B.2) in appendix B. For
$\nu_{\mathcal{R}}$, the expansion coefficients $\bar{\nu}_{\mathcal{R}0}$,
$\bar{\nu}_{\mathcal{R}1}$, and $\bar{\nu}_{\mathcal{R}2}$ in (B.2) can be
calculated from Eqs.(4.5) and (3.22), we find

$\displaystyle\bar{\nu}_{\mathcal{R}0}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{2}+\bar{\epsilon}_{1}+\frac{\bar{\epsilon}_{2}}{2}-\bar{f}_{\text{GB}}\bar{\epsilon}_{1}+\bar{\epsilon}_{1}^{2}+\frac{11\bar{\epsilon}_{1}\bar{\epsilon}_{2}}{6}+\frac{\bar{\epsilon}_{2}\bar{\epsilon}_{3}}{6},$
$\displaystyle\bar{\nu}_{\mathcal{R}1}$ $\displaystyle\simeq$
$\displaystyle-\bar{\epsilon}_{1}\bar{\epsilon}_{2}-\frac{\bar{\epsilon}_{2}\bar{\epsilon}_{3}}{2},$
(4.8)

and

$\displaystyle\bar{\nu}_{\mathcal{R}2}\equiv\frac{d^{2}\nu_{\mathcal{R}}}{d\ln^{2}(-\tau)}$
$\displaystyle=$ $\displaystyle\mathcal{O}(\bar{\epsilon}_{i}^{3}).$ (4.9)

Here a letter with an over bar means that the quantity is at the turning point
$\bar{y}_{0}$.

Then, using the above expansions, the power spectrum for the curvature
perturbation $\mathcal{R}$ can be calculated via Eq.(Appendix A: General
formulas of primordial spectra in the uniform asymptotic approximation). After
tedious calculations we obtain,

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{181\bar{H}^{2}}{72e^{3}M_{\rm{pl}}^{2}\pi^{2}\bar{\epsilon}_{1}}\Big{[}1-\bar{f}_{\text{GB}}+\bar{f}_{\text{GB}}^{2}+\Big{(}2\ln{2}-\frac{496}{181}+\frac{630\bar{f}_{\text{GB}}}{181}-4\bar{f}_{\text{GB}}\ln{2}\Big{)}\bar{\epsilon}_{1}+\Big{(}\frac{67}{181}-\ln
2\Big{)}\Big{(}\bar{f}_{\text{GB}}-1\Big{)}\bar{\epsilon}_{2}$ (4.10)
$\displaystyle+$
$\displaystyle\Big{(}\frac{293}{181}-\frac{630\ln{2}}{181}+2\ln{2}\Big{)}\bar{\epsilon}_{1}^{2}-\Big{(}\frac{11}{362}+\frac{67\ln
2}{181}-\frac{\ln^{2}2}{2}\Big{)}\bar{\epsilon}_{2}^{2}$ $\displaystyle+$
$\displaystyle\Big{(}\frac{\pi^{2}}{12}-\frac{4231}{1629}+\frac{47\ln{2}}{181}+\ln^{2}2\Big{)}\bar{\epsilon}_{1}\bar{\epsilon}_{2}+\Big{(}\frac{\pi^{2}}{24}-\frac{86}{1629}+\frac{67\ln
2}{181}-\frac{\ln^{2}2}{2}\Big{)}\bar{\epsilon}_{2}\bar{\epsilon}_{3}\Big{]},$

where $e$ is the natiral constant. From the scalar spectrum (4.10), we find
that the parameter $\bar{f}_{\rm GB}$ enters into the scalar spectrum at the
leading-order in the slow-roll expansion.

Then with the scalar power spectrum given above, the scalar spectral index is,

$\displaystyle n_{\mathcal{R}}-1$ $\displaystyle\simeq$
$\displaystyle-2\Big{(}1-\bar{f}_{\text{GB}}\Big{)}\bar{\epsilon}_{1}-\bar{\epsilon}_{2}-2\bar{\epsilon}_{1}^{2}$
$\displaystyle-\Big{(}\frac{677}{181}-2\ln{2}\Big{)}\bar{\epsilon}_{1}\bar{\epsilon}_{2}-\Big{(}\frac{67}{181}-\ln{2}\Big{)}\bar{\epsilon}_{2}\bar{\epsilon}_{3},$

and the running of the scalar spectral index is expressed as

$\displaystyle\alpha_{\mathcal{R}}$ $\displaystyle\simeq$
$\displaystyle-2\bar{\epsilon}_{1}\bar{\epsilon}_{2}-\bar{\epsilon}_{2}\bar{\epsilon}_{3}-4\bar{f}_{\text{GB}}\bar{\epsilon}_{1}^{2}+2\bar{f}_{\text{GB}}\bar{\epsilon}_{1}\bar{\epsilon}_{2}-6\bar{\epsilon}_{1}^{2}\bar{\epsilon}_{2}$
(4.12) $\displaystyle-\Big{(}\frac{677}{181}-2\ln
2\Big{)}\bar{\epsilon}_{1}\bar{\epsilon}_{2}^{2}-\Big{(}\frac{858}{181}-2\ln
2\Big{)}\bar{\epsilon}_{1}\bar{\epsilon}_{2}\bar{\epsilon}_{3}$
$\displaystyle-\Big{(}\frac{67}{181}-\ln
2\Big{)}\Big{(}\bar{\epsilon}_{2}\bar{\epsilon}_{3}^{2}+\bar{\epsilon}_{2}\bar{\epsilon}_{3}\bar{\epsilon}_{4}\Big{)}.$

For scalar spectral index, the new effects denoted by the slow-roll parameter
$\bar{f}_{\rm GB}$ appears at the next-to-leading order, while for the running
of the index, they only contribute to the third-order of the slow-roll
approximation.

###  IV.2 Tensor Spectrum

Now we consider the tensor spectrum. First we need to derive the expressions
of $\bar{\nu}_{h0},\bar{\nu}_{h1},\bar{b}_{h0},\bar{b}_{h1}$, and
$\bar{c}_{h0}$, which are the expansion coefficients presented in Eqs. (B.2,
B.3, B.4). Repeating similar calculations for scalar perturbations, we obtain

$\displaystyle\bar{\nu}_{h0}$ $\displaystyle\simeq$
$\displaystyle\frac{3}{2}+\bar{\epsilon}_{1}-\bar{f}_{\text{GB}}\bar{\epsilon}_{1}+\frac{4}{3}\bar{\epsilon}_{1}\bar{\epsilon}_{2}+\bar{\epsilon}_{1}^{2},~{}~{}~{}~{}~{}~{}~{}$
(4.13) $\displaystyle\bar{\nu}_{h1}$ $\displaystyle\equiv$
$\displaystyle\frac{d\nu_{h}}{d\ln(-\tau)}\simeq-\bar{\epsilon}_{1}\bar{\epsilon}_{2},$
(4.14)

and

$\displaystyle\bar{b}_{h0}$ $\displaystyle\simeq$ $\displaystyle
2\bar{f}_{\text{GB}}-2\bar{f}_{\text{GB}}^{2}-4\bar{f}_{\text{GB}}\bar{\epsilon}_{1},~{}~{}~{}~{}~{}~{}~{}~{}$
(4.15) $\displaystyle\bar{b}_{h1}$ $\displaystyle\equiv$
$\displaystyle\frac{db_{h}}{d\ln(-\tau)}\simeq 4\bar{f}_{\text{GB}}.$ (4.16)

and

$\displaystyle\bar{c}_{h0}$ $\displaystyle\simeq$ $\displaystyle
1+\bar{f}_{\text{GB}}\bar{\epsilon}_{1}.$ (4.17)

Then, the power spectrum for the tensor perturbation $h_{k}$ reads

$\displaystyle\Delta_{h}^{2}(k)$ $\displaystyle\simeq$
$\displaystyle\frac{181\bar{H}^{2}}{36{M_{\rm{pl}}^{2}}e^{3}\pi^{2}}\Big{[}1+\frac{179}{181}\bar{f}_{\text{GB}}-\frac{6695}{181}\bar{f}_{\text{GB}}^{2}$
(4.18)
$\displaystyle-\Big{(}\frac{496}{181}-2\ln{2}\Big{)}\bar{\epsilon}_{1}+\Big{(}\frac{321}{362}-\frac{913\ln{2}}{181}\Big{)}\bar{f}_{\text{GB}}\bar{\epsilon}_{1}$
$\displaystyle+\Big{(}\frac{293}{181}-\frac{630\ln{2}}{181}+2\ln^{2}2\Big{)}\bar{\epsilon}_{1}^{2}$
$\displaystyle+\Big{(}\frac{\pi^{2}}{12}-\frac{4636}{1629}+\frac{496\ln{2}}{181}-\ln^{2}2\Big{)}\bar{\epsilon}_{1}\bar{\epsilon}_{2}\Big{]},$

Also the tensor spectral index and its running are given by

$\displaystyle n_{h}$ $\displaystyle\simeq$
$\displaystyle-2\bar{\epsilon}_{1}-\frac{358}{181}\bar{f}_{\text{GB}}\bar{\epsilon}_{1}-2\bar{\epsilon}_{1}^{2}$
(4.19)
$\displaystyle-\Big{(}\frac{496}{181}-2\ln{2}\Big{)}\bar{\epsilon}_{1}\bar{\epsilon}_{2},$

and

$\displaystyle\alpha_{h}$ $\displaystyle\simeq$
$\displaystyle-2\bar{\epsilon}_{1}\bar{\epsilon}_{2}-\frac{716}{181}\bar{f}_{\text{GB}}\bar{\epsilon}_{1}^{2}-\frac{358}{181}\bar{f}_{\text{GB}}\bar{\epsilon}_{1}\bar{\epsilon}_{2}-6\bar{\epsilon}_{1}^{2}\bar{\epsilon}_{2}$
(4.20) $\displaystyle-\Big{(}\frac{496}{181}-2\ln
2\Big{)}\Big{(}\bar{\epsilon}_{1}\bar{\epsilon}_{2}^{2}+\bar{\epsilon}_{1}\bar{\epsilon}_{2}\bar{\epsilon}_{3}\Big{)}.$

###  IV.3 Expressions at Horizon Crossing

In the last two subsections, all the results are expressed in terms of
quantities that are evaluated at the turning point. However, usually those
expressions were expressed in terms of the slow-roll parameters which are
evaluated at the time $\tau_{\star}$ when scalar or tensor perturbation modes
cross the horizon, i.e.,
$a(\tau_{\star})H(\tau_{\star})=c_{\mathcal{R}}(\tau_{\star})k$ for scalar
perturbations and $a(\tau_{\star})H(\tau_{\star})=c_{h}(\tau_{\star})k$ for
tensor perturbations. Consider modes with the same wavenumber $k$, it is easy
to see that the scalar and tensor modes left the horizon at different times if
$c_{\mathcal{R}}(\tau)\neq c_{h}(\tau)$. When
$c_{\mathcal{R}}(\tau_{\star})>c_{h}(\tau_{\star})$, the scalar mode leaves
the horizon later than the tensor mode, and for
$c_{\mathcal{R}}(\tau_{\star})<c_{h}(\tau_{\star})$, the scalar mode leaves
the horizon before the tensor one.

As we have two different horizon crossing times, it is reasonable to rewrite
all our results in terms of quantities evaluated at the later time, i.e., we
should evaluate all expressions at scalar horizon crossing time
$a(\tau_{\star})H(\tau_{\star})=c_{\mathcal{R}}(\tau_{\star})k$ for
$c_{\mathcal{R}}(\tau_{\star})>c_{h}(\tau_{\star})$ and at tensor-mode horizon
crossing $a(\tau_{\star})H(\tau_{\star})=c_{h}(\tau_{\star})k$ for
$c_{\mathcal{R}}(\tau_{\star})<c_{h}(\tau_{\star})$.

####  IV.3.1 $c_{\mathcal{R}}(\tau_{\star})>c_{h}(\tau_{\star})$

We shall re-write all the expressions in terms of quantities evaluated at the
time when the scalar-mode horizon crossing
$c_{\mathcal{R}}(\tau_{\star})>c_{h}(\tau_{\star})$. Skipping all the tedious
calculations, we find that scalar spectrum can be written in the form

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{181H_{\star}^{2}}{72e^{3}M_{\rm{pl}}^{2}\pi^{2}\epsilon_{\star}1}\Big{[}1-f_{\star{\rm
GB}}+f_{\star{\rm
GB}}^{2}+\Big{(}2\ln{3}-\frac{496}{181}+\frac{630}{181}f_{\star{\rm
GB}}-4\ln{3}f_{\star{\rm GB}}\Big{)}\epsilon_{\star 1}$ (4.21)
$\displaystyle+\Big{(}\ln{3}-\frac{67}{181}+\frac{67}{181}f_{\star{\rm
GB}}-\ln{3}f_{\star{\rm GB}}\Big{)}\epsilon_{\star
2}+\Big{(}\frac{517}{543}-\frac{630\ln{3}}{181}+2\ln^{2}3\Big{)}\epsilon_{\star
1}^{2}$
$\displaystyle+\Big{(}\frac{\pi^{2}}{12}-\frac{3688}{1629}+\frac{47\ln{3}}{181}+\ln^{2}3\Big{)}\epsilon_{\star
1}\epsilon_{\star
2}+\Big{(}\frac{329}{1086}-\frac{67\ln{3}}{181}+\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star
2}^{2}$
$\displaystyle+\Big{(}\frac{\pi^{2}}{24}-\frac{86}{1629}+\frac{67\ln{3}}{181}-\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star
2}\epsilon_{\star 3}\Big{]}.$

where the subscript “$\star$” denotes evaluation at the horizon crossing. For
the scalar spectral index, one obtains

$\displaystyle n_{\mathcal{R}}-1$ $\displaystyle\simeq$
$\displaystyle-2\epsilon_{\star 1}-\epsilon_{\star 2}-2\epsilon_{\star
1}^{2}+2f_{\star{\rm GB}}\epsilon_{\star 1}$ (4.22)
$\displaystyle-\Big{(}\frac{677}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}$ $\displaystyle-\Big{(}\frac{67}{181}-\ln
3\Big{)}\epsilon_{\star 2}\epsilon_{\star 3}.$

The running of the scalar spectral index reads

$\displaystyle\alpha_{\mathcal{R}}$ $\displaystyle\simeq$
$\displaystyle-2\epsilon_{\star 1}\epsilon_{\star 2}-\epsilon_{\star
2}\epsilon_{\star 3}-4f_{\star{\rm GB}}\epsilon_{\star 1}^{2}+2f_{\star{\rm
GB}}\epsilon_{\star 1}\epsilon_{\star 2}$ (4.23)
$\displaystyle-6\epsilon_{\star 1}^{2}\epsilon_{\star
2}-\Big{(}\frac{677}{181}-2\ln 3\Big{)}\epsilon_{\star 1}\epsilon_{\star
2}^{2}$ $\displaystyle-\Big{(}\frac{858}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}\epsilon_{\star 3}$
$\displaystyle-\Big{(}\frac{67}{181}-\ln 3\Big{)}\epsilon_{\star
2}\epsilon_{\star 3}^{2}.$

Similar to the scalar perturbations, now let us turn to consider the tensor
perturbations, which yield

$\displaystyle\Delta_{h}^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{181H_{\star}^{2}}{36e^{3}{M_{\rm{pl}}^{2}}\pi^{2}}\Big{[}1-\Big{(}\frac{496}{181}-2\ln
3\Big{)}\epsilon_{\star 1}+\frac{179}{181}f_{\star{\rm
GB}}-\frac{6695}{181}f_{\star{\rm GB}}^{2}-\Big{(}\frac{654}{181}+9\ln
2-\frac{716}{181}\ln 3\Big{)}f_{\star{\rm GB}}\epsilon_{\star 1}$ (4.24)
$\displaystyle+\Big{(}\frac{517}{543}-\frac{630}{181}\ln
3+2\ln^{2}3\Big{)}\epsilon_{\star
1}^{2}+\Big{(}\frac{\pi^{2}}{12}-\frac{4636}{1629}+\frac{496}{181}\ln
3-\ln^{2}3\Big{)}\epsilon_{\star 1}\epsilon_{\star 2}\Big{]}.$

For the tensor spectral index, we find

$\displaystyle n_{h}$ $\displaystyle\simeq$ $\displaystyle-2\epsilon_{\star
1}-\frac{358}{181}f_{\star{\rm GB}}\epsilon_{\star 1}-2\epsilon_{\star 1}^{2}$
(4.25) $\displaystyle-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}.$

Then, the running of the tensor spectral index reads

$\displaystyle\alpha_{h}$ $\displaystyle\simeq$
$\displaystyle-2\epsilon_{\star 1}\epsilon_{\star 2}-6\epsilon_{\star
1}^{2}\epsilon_{\star 2}+\frac{716}{181}f_{\star{\rm GB}}\epsilon_{\star
1}^{2}-\frac{358}{181}f_{\star{\rm GB}}\epsilon_{\star 1}\epsilon_{\star 2}$
$\displaystyle-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}^{2}-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}\epsilon_{\star 3}.$

Finally with both scalar and tensor spectra given above, we can evaluate the
tensor-to-scalar ratio at the horizon crossing time$(\tau_{\star})$, and find
that

$\displaystyle{r}$ $\displaystyle\simeq$ $\displaystyle{16\epsilon_{\star
1}}\Bigg{[}1+\frac{360}{181}f_{\star{\rm GB}}-36f_{\star{\rm
GB}}^{2}+\Big{(}\frac{67}{181}-\ln 3\Big{)}\epsilon_{\star
2}-\Big{(}\frac{53844}{32761}+9\ln 2-\frac{720}{181}\ln 3\Big{)}f_{\star{\rm
GB}}\epsilon_{\star 1}$ (4.27)
$\displaystyle+\Big{(}\frac{24120}{32761}-\frac{360}{181}\ln
3\Big{)}f_{\star{\rm GB}}\epsilon_{\star
2}-\Big{(}\frac{32615}{196566}+\frac{67}{181}\ln
3-\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star
2}^{2}+\Big{(}\frac{42500}{98283}-\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}$
$\displaystyle+\Big{(}\frac{86}{1629}-\frac{\pi^{2}}{24}-\frac{67}{181}\ln
3+\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star 2}\epsilon_{\star 3}\Bigg{]}.$

####  IV.3.2 $c_{\mathcal{R}}(\tau_{\star})<c_{h}(\tau_{\star})$

For $c_{\mathcal{R}}(\tau_{\star})<c_{h}(\tau_{\star})$, as the scalar mode
leaves the horizon before the tensor mode does,we shall rewrite all the
expressions in terms of quantities evaluated at the time when the tensor mode
leaves the Hubble horizon
$a(\tau_{\star})H(\tau_{\star})=c_{h}(\tau_{\star})k$. Skipping all the
tedious calculations, we find that the scalar spectrum can be written in the
form

$\displaystyle\Delta_{\mathcal{R}}^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{181H_{\star}^{2}}{72e^{3}M_{\rm{pl}}^{2}\pi^{2}\epsilon_{\star
1}}\Big{[}1-f_{\star{\rm GB}}+f_{\star{\rm
GB}}^{2}-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}-\Big{(}\frac{67}{181}-\ln 3\Big{)}\epsilon_{\star
2}+\Big{(}\frac{630}{181}-4\ln 3\Big{)}f_{\star{\rm GB}}\epsilon_{\star 1}$
(4.28) $\displaystyle+\Big{(}\frac{67}{181}-\ln 3\Big{)}f_{\star{\rm
GB}}\epsilon_{\star 2}+\Big{(}\frac{517}{543}-\frac{630}{181}\ln
3+2\ln^{2}3\Big{)}\epsilon_{\star
1}^{2}+\Big{(}\frac{\pi^{2}}{12}-\frac{4774}{1629}+\frac{47}{181}\ln
3+\ln^{2}3\Big{)}\epsilon_{\star 1}\epsilon_{\star 2}$
$\displaystyle-\Big{(}\frac{11}{362}+\frac{67}{181}\ln
3-\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star
2}^{2}+\Big{(}\frac{\pi^{2}}{24}-\frac{86}{1629}+\frac{67}{181}\ln
3-\frac{\l^{2}3}{2}\Big{)}\epsilon_{\star 2}\epsilon_{\star 3}\Big{]}.$

For the scalar spectral index, one obtains

$\displaystyle n_{\mathcal{R}}-1$ $\displaystyle\simeq$
$\displaystyle-2\epsilon_{\star 1}-\epsilon_{\star 2}-2\epsilon_{\star
1}^{2}+2f_{\star{\rm GB}}\epsilon_{\star 1}$ (4.29)
$\displaystyle-\Big{(}\frac{677}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}$ $\displaystyle-\Big{(}\frac{67}{181}-\ln
3\Big{)}\epsilon_{\star 2}\epsilon_{\star 3}.$

The running of the scalar spectral index reads

$\displaystyle\alpha_{\mathcal{R}}$ $\displaystyle\simeq$
$\displaystyle-2\epsilon_{\star 1}\epsilon_{\star 2}-\epsilon_{\star
2}\epsilon_{\star 3}-4f_{\star{\rm GB}}\epsilon_{\star 1}^{2}+2f_{\star{\rm
GB}}\epsilon_{\star 1}\epsilon_{\star 2}$ (4.30)
$\displaystyle-6\epsilon_{\star 1}^{2}\epsilon_{\star
2}-\Big{(}\frac{677}{181}-2\ln 3\Big{)}\epsilon_{\star 1}\epsilon_{\star
2}^{2}$ $\displaystyle-\Big{(}\frac{858}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}\epsilon_{\star 3}$
$\displaystyle-\Big{(}\frac{67}{181}-\ln 3\Big{)}\epsilon_{\star
2}\epsilon_{\star 3}^{2}.$

Similar to the scalar perturbations, now let us turn to consider the tensor
perturbations, which yield

$\displaystyle\Delta_{h}^{2}(k)$ $\displaystyle=$
$\displaystyle\frac{181H_{\star}^{2}}{36e^{3}{M_{\rm{pl}}^{2}}\pi^{2}}\Big{[}1-\Big{(}\frac{496}{181}-2\ln
3\Big{)}\epsilon_{\star 1}+\frac{179}{181}f_{\star{\rm
GB}}-\frac{6695}{181}f_{\star{\rm GB}}^{2}-\Big{(}\frac{654}{181}+9\ln
2-\frac{716}{181}\ln 3\Big{)}f_{\star{\rm GB}}\epsilon_{\star 1}$ (4.31)
$\displaystyle+\Big{(}\frac{517}{543}-\frac{630}{181}\ln
3+2\ln^{2}3\Big{)}\epsilon_{\star
1}^{2}+\Big{(}\frac{\pi^{2}}{12}-\frac{4636}{1629}+\frac{496}{181}\ln
3-\ln^{2}3\Big{)}\epsilon_{\star 1}\epsilon_{\star 2}\Big{]}.$

For the tensor spectral index, we find

$\displaystyle n_{h}$ $\displaystyle\simeq$ $\displaystyle-2\epsilon_{\star
1}-\frac{358}{181}f_{\star{\rm GB}}\epsilon_{\star 1}-2\epsilon_{\star 1}^{2}$
(4.32) $\displaystyle-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}.$

Then, the running of the tensor spectral index reads

$\displaystyle\alpha_{h}$ $\displaystyle\simeq$
$\displaystyle-2\epsilon_{\star 1}\epsilon_{\star 2}-6\epsilon_{\star
1}^{2}\epsilon_{\star 2}+\frac{716}{181}f_{\star{\rm GB}}\epsilon_{\star
1}^{2}-\frac{358}{181}f_{\star{\rm GB}}\epsilon_{\star 1}\epsilon_{\star 2}$
$\displaystyle-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}^{2}-\Big{(}\frac{496}{181}-2\ln 3\Big{)}\epsilon_{\star
1}\epsilon_{\star 2}\epsilon_{\star 3}.$

Finally with both scalar and tensor spectra given above, we can evaluate the
tensor-to-scalar ratio at the horizon crossing time $(\tau_{\star})$, and find
that

$\displaystyle{r}$ $\displaystyle\simeq$ $\displaystyle{16\epsilon_{\star
1}}\Bigg{[}1+\frac{360}{181}f_{\star{\rm GB}}-36f_{\star{\rm
GB}}^{2}+\Big{(}\frac{67}{181}-\ln 3\Big{)}\epsilon_{\star
2}-\Big{(}\frac{53844}{32761}+9\ln 2-\frac{720}{181}\ln 3\Big{)}f_{\star{\rm
GB}}\epsilon_{\star 1}$ (4.34)
$\displaystyle+\Big{(}\frac{24120}{32761}-\frac{360}{181}\ln
3\Big{)}f_{\star{\rm GB}}\epsilon_{\star 2}+\Big{(}\frac{108022}{98283}-\ln
3\Big{)}\epsilon_{\star 1}\epsilon_{\star
2}+\Big{(}\frac{10969}{65522}-\frac{67}{181}\ln
3+\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star 2}^{2}$
$\displaystyle+\Big{(}\frac{86}{1629}-\frac{\pi^{2}}{24}-\frac{67}{181}\ln
3+\frac{\ln^{2}3}{2}\Big{)}\epsilon_{\star 2}\epsilon_{\star 3}\Bigg{]}.$

##  V A Special Model with Power-law potential

With the accuracy of future observations becoming more and more precise, slow-
rolling inflation theory requires to be accurate to the higher order. And this
is the most important reason why we need to calculate the above results in the
second order. In order to show the difference between the second-order
corrections and the first-order results, we can use a specific model with a
power-law potential. In this section, we study the predicted results of the
relationship between the scalar spectral index $n_{\cal R}$ and the tensor-to-
scalar ratio $r$ in the first order and the second order in this specific
model.

Firstly, the correspondence between the Hubble flow parameters and the
potential slow-roll parameters $(\epsilon_{V},\eta_{V},\xi_{V}^{2})$ can be
written as

$\displaystyle\epsilon_{V}$ $\displaystyle\equiv$
$\displaystyle\frac{V_{,\phi}^{2}M_{\rm{pl}}^{2}}{2V^{2}}$ $\displaystyle=$
$\displaystyle\epsilon_{1}\frac{\Big{[}1+f_{\rm{GB}}-\frac{\epsilon_{1}(1+f_{\rm{GB}})}{3}+\frac{\epsilon_{2}(1+f_{\rm{GB}})}{6}\Big{]}^{2}}{(1+f_{\rm{GB}})\Big{[}1+\frac{f_{\rm{GB}}}{2}-\frac{\epsilon_{1}(1+f_{\rm{GB}})}{3}\Big{]}^{2}},$
$\displaystyle\eta_{V}$ $\displaystyle\equiv$
$\displaystyle\frac{V_{,\phi\phi}M_{\rm{pl}}^{2}}{V}$ $\displaystyle=$
$\displaystyle\frac{1}{(1+f_{\rm{GB}})^{2}(1+\frac{f_{\rm{GB}}}{2}-\frac{\epsilon_{1}(1+f_{\rm{GB}})}{3})}$
$\displaystyle\;\;\times\Bigg{[}2\epsilon_{1}(1+\frac{5f_{\rm{GB}}}{2}+\frac{3f_{\rm{GB}}^{2}}{2})-\frac{\epsilon_{2}}{2}(1+2f_{\rm{GB}}+f_{\rm{GB}}^{2})$
$\displaystyle\;\;\;\;-\frac{2\epsilon_{1}^{2}}{3}(1+\frac{9f_{\rm{GB}}}{2}+3f_{\rm{GB}}^{2})-\frac{\epsilon_{2}^{2}}{12}(1+2f_{\rm{GB}}+f_{\rm{GB}}^{2})$
$\displaystyle\;\;\;\;+\frac{5\epsilon_{1}\epsilon_{2}}{6}(1+\frac{14f_{\rm{GB}}}{5}+\frac{9f_{\rm{GB}}^{2}}{5})$
$\displaystyle\;\;\;\;-\frac{\epsilon_{2}\epsilon_{3}}{6}(1+2f_{\rm{GB}}+f_{\rm{GB}}^{2})\Bigg{]},$

and

$\displaystyle\xi_{V}$ $\displaystyle\equiv$
$\displaystyle\frac{V_{,\phi}V_{,\phi\phi\phi}M_{\rm{pl}}^{4}}{V^{2}}$
$\displaystyle=$
$\displaystyle\frac{1+f_{\rm{GB}}-\frac{\epsilon_{1}(1+2f_{\rm{GB}})}{3}+\frac{\epsilon_{2}(1+f_{\rm{GB}})}{6}}{(1+f_{\rm{GB}})^{4}(1+\frac{f_{\rm{GB}}}{2}-\frac{\epsilon_{1}(1+f_{\rm{GB}})}{3})^{2}}$
$\displaystyle\times\Bigg{[}4\epsilon_{1}^{2}(1+4f_{\rm{GB}}+\frac{9f_{\rm{GB}}^{2}}{2}+\frac{3f_{\rm{GB}}^{3}}{2})$
$\displaystyle-3\epsilon_{1}\epsilon_{2}\Big{(}1+\frac{10f_{\rm{GB}}}{3}+\frac{11f_{\rm{GB}}^{2}}{3}+\frac{4f_{\rm{GB}}^{3}}{3}\Big{)}$
$\displaystyle+\frac{\epsilon_{2}\epsilon_{3}}{2}(1+3f_{\rm{GB}}+3f_{\rm{GB}}^{2}+f_{\rm{GB}}^{3})$
$\displaystyle-\epsilon_{1}\epsilon_{2}^{2}\Big{(}1+\frac{11f_{\rm{GB}}}{3}+\frac{13f_{\rm{GB}}^{2}}{3}+\frac{5f_{\rm{GB}}^{3}}{3}\Big{)}$
$\displaystyle+3\epsilon_{1}^{2}\epsilon_{2}\Big{(}1+5f_{\rm{GB}}+\frac{19f_{\rm{GB}}^{2}}{3}+\frac{7f_{\rm{GB}}^{3}}{3}\Big{)}$
$\displaystyle-\frac{4\epsilon_{1}^{3}}{3}(1+8f_{\rm{GB}}+9f_{\rm{GB}}^{2}+3f_{\rm{GB}}^{3})$
$\displaystyle-\frac{7\epsilon_{1}\epsilon_{2}\epsilon_{3}}{6}\Big{(}1+\frac{25f_{\rm{GB}}}{7}+\frac{29f_{\rm{GB}}^{2}}{7}+\frac{11f_{\rm{GB}}^{3}}{7}\Big{)}$
$\displaystyle+\frac{1}{6}(\epsilon_{2}^{2}\epsilon_{3}+\epsilon_{2}\epsilon_{3}^{2}+\epsilon_{2}\epsilon_{3}\epsilon_{4})(1+3f_{\rm{GB}}+3f_{\rm{GB}}^{2}+f_{\rm{GB}}^{3})\Bigg{]}.$

Then it is easy to express the Hubble flow parameters in terms of the
potential $V(\phi)$ and the 4EGB coupling $f_{\rm{GB}}$ with slow-roll
approximation as

$\displaystyle\epsilon_{1}$ $\displaystyle\simeq$
$\displaystyle\frac{M_{\rm{pl}}^{2}V_{,\phi}^{2}}{6V^{4}}\Big{(}3V^{2}-2M_{\rm{pl}}^{2}V_{,\phi}^{2}+2M_{\rm{pl}}^{2}VV_{,\phi\phi}\Big{)},$
$\displaystyle\epsilon_{2}$ $\displaystyle\simeq$
$\displaystyle-\frac{M_{\rm{pl}}^{2}}{3V^{4}}\Big{[}6M_{\rm{pl}}^{2}V_{,\phi}^{4}+3(2+f_{\rm{GB}})V^{3}V_{,\phi\phi}$
$\displaystyle-3(2+f_{\rm{GB}})V^{2}V_{,\phi}^{2}+2M_{\rm{pl}}^{2}V^{2}V_{,\phi\phi}^{2}$
$\displaystyle+2M_{\rm{pl}}^{2}V^{2}V_{,\phi}V_{,\phi\phi\phi}-10M_{\rm{pl}}^{2}VV_{,\phi}^{2}V_{,\phi\phi}\Big{]},$
$\displaystyle\epsilon_{3}$ $\displaystyle\simeq$
$\displaystyle\Big{[}-6M_{\rm{pl}}^{2}V_{,\phi}(2V_{,\phi}^{3}-3VV_{,\phi}V_{,\phi\phi}+V^{2}V_{,\phi\phi\phi})\Big{]}$
(5.4)
$\displaystyle\Big{[}6M_{\rm{pl}}^{2}V_{,\phi}^{4}+3(2+f_{\rm{GB}})V^{3}V_{,\phi\phi}-3(2+f_{\rm{GB}})V^{2}V_{,\phi}^{2}$
$\displaystyle+2M_{\rm{pl}}^{2}V^{2}V_{,\phi\phi}^{2}+2M_{\rm{pl}}^{2}V^{2}V_{,\phi}V_{,\phi\phi\phi}$
$\displaystyle-10M_{\rm{pl}}^{2}VV_{,\phi}^{2}V_{,\phi\phi}\Big{]}^{-1}.$

With the slow-roll conditions, the number of e-folds can be changed as

$\displaystyle
N=\int_{t}^{t_{\rm{end}}}Hdt\simeq\int_{\phi_{\rm{end}}}^{\phi}\frac{V(2-f_{\rm{GB}})}{2M_{\rm{pl}}^{2}V_{,\phi}}d\phi,$
(5.5)

where $\phi_{\rm{end}}$ can get from by $\epsilon_{1}(\phi_{\rm{end}})=1$,
which means the value at the end of inflation.

To proceed, it is necessary to consider a specific model with a power-law
potential,

$\displaystyle V(\phi)$ $\displaystyle=$ $\displaystyle V_{0}\phi^{n},$
$\displaystyle f_{\rm{GB}}$ $\displaystyle\simeq$ $\displaystyle\frac{2\alpha
V(\phi)}{3M_{\rm{pl}}^{2}}=\frac{2\alpha V_{0}\phi^{n}}{3M_{\rm{pl}}^{2}},$
(5.6)

where the value of $V_{0}$ can be determined by Planck 2018 data with a
specific value of $n$. Then the Hubble flow can be expressed as

$\displaystyle\epsilon_{1}$ $\displaystyle\simeq$
$\displaystyle\frac{M_{\rm{pl}}^{2}n^{2}(-2M_{\rm{pl}}^{2}n+3\phi^{2})}{6\phi^{4}},$
$\displaystyle\epsilon_{2}$ $\displaystyle\simeq$
$\displaystyle\frac{M_{\rm{pl}}^{2}n[-2M_{\rm{pl}}^{2}n+(2+f_{\rm{GB}})\phi^{2}]}{\phi^{4}},$
$\displaystyle\epsilon_{3}$ $\displaystyle\simeq$
$\displaystyle\frac{4M_{\rm{pl}}^{2}n}{-2M_{\rm{pl}}^{2}n+(2+f_{\rm{GB}})\phi^{2}}.$
(5.7)

The scalar field at the end of the inflation $\phi_{\rm{end}}$ can be get from

$\displaystyle\phi_{\rm{end}}^{2}=\frac{M_{\rm{pl}}^{2}n^{2}}{2}.$ (5.8)

Then we can easily get the expression of the e-fold number with Eq.(5.5),

$\displaystyle
N=\frac{\phi^{2}}{2M_{\rm{pl}}^{2}n}-\frac{\phi^{2}f_{\rm{GB}}}{2M_{\rm{pl}}^{2}n(n+2)}-\frac{n}{4}+\frac{n}{4(n+2)}f_{\rm{GB}}^{\rm{end}}.$

where $f_{\rm{GB}}^{\rm{end}}$ the value of $f_{\rm{GB}}$ at the end of the
inflation, and when $N$ and $n$ are fixed values, $f_{\rm{GB}}^{\rm{end}}$ can
be treated as constant.

In the case of $c_{\cal R}(\eta_{\star})>c_{h}(\eta_{\star})$, substituting
the above results into Eqs. (4.22) and (4.27), it is easy to find that
$n_{\cal R}$ depends on the parameter $f_{\rm{GB}}$, which is different from
the first-order results. For the tensor-to-scalar ratio $r$, it is directly
related to both $n$ and $f_{\rm{GB}}$. In Fig. 1, we plot the theoretical
predictions of $n_{\cal R}-r$ relation with $n=4/3$ for different values of
$f_{\rm{GB}}$ in comparison with the observational data. The contours are
taken from the Planck 2018 TT+TE+EE+lowP data Planck:2018vyg . For different
values of $f_{\rm{GB}}$, we respectively plot them with the first and the
second corrections.

Figure 1: Marginalized constraint (68% and 95% confidence level) from
$(n_{\cal R},r)$ using Planck 2018 data, compared to the theoretical
predictions of the model with $n=4/3$. For both $f_{\rm{GB}}=0.02$ and
$f_{\rm{GB}}=-0.01$, the solid lines give the results with the first-order
corrections while the dash gives the predictions with the second-order
corrections, respectively.

Considering about the smaller errors on $n_{R}$ and $r$ in the future, the
second-order corrections to both $n_{R}$ and $r$ in the slow-roll
approximation are important. In the forthcoming experiments, especially the
Stage IV experiments CMB-S4:2016ple ; SimonsObservatory:2018koc ; Mallaby-
Kay:2021tuk ; LiteBIRD:2022cnt , the errors of the measurements on both
$n_{R}$ and $r$ will be smaller than $10^{-3}$, namely $\sigma(n_{\cal
R},r)<10^{-3}$. That means it is necessary to calculate the second-order
corrections of $n_{\cal R}$ and $r$ to the magnitude of $\mathcal{O}(10^{-3})$
for getting more accurate constraints on the parameters of the models. On this
order of magnitude, the first-order and second-order results will be easily
distinguished by future experiments. It is worth noting that the contributions
of second-order corrections are affected by the parameters of the models.

Figure 2: Difference between the $n_{\cal R}-r$ relations with the first- and
the second-order corrections. The gray shaded $1-\sigma$ contour corresponds
to a futuristic measurement with $\sigma_{n_{s-1}}=\sigma_{r}=10^{-3}$.

In Fig. 2, the first-order and the second-order results of $n_{\cal R}$ and
$r$ with Planck 2018 constraints in the models for different values of $N$ and
$f_{\rm{GB}}$ are compared. The upper panel in Fig. 2 is for
$f_{\rm{GB}}=0.02$, while the bottom panel is for $f_{\rm{GB}}=-0.01$, in
which the shaded contour means a possible futuristic measurement with
$\sigma(n_{\cal R},r)<10^{-3}$. It is easy to find that for both
$f_{\rm{GB}}=0.02$ and $f_{\rm{GB}}=-0.01$, the difference between the first-
order and second-order predictions is larger than the futuristic experimental
sensitivity on $n_{\cal R}$ and $r$, and this is even more obvious for
$f_{\rm{GB}}=-0.01$ . This would make it convincing that the second-order
corrections in the slow-roll approximation of $n_{\cal R}$ and $r$ in 4EGB
inflation are necessary for fitting future experimental data.

##  VI Conclusions and Outlook

The uniform asymptotic approximation method is an error-controlled and
systematically improvable method for constructing exact analytical solutions
of linear perturbations. The effectiveness of this method has been verified in
many applications, for example, in calculating primordial spectra for various
inflation models Zhu:2013fha ; Zhu:2013upa ; Zhu:2014wfa ; Zhu:2015ata ;
Zhu:2016srz ; Zhu:2014wda ; Zhu:2014aea ; Zhu:2015xsa ; Zhu:2015owa  and
applications in studying the reheating process Zhu:2018smk  and quantum
mechanics Li:2019cre .

In this paper, we applied the third-order uniform asymptotic approximation to
derive the inflationary observables for scalar and tensor perturbations in the
slow-roll inflation in the spatial covariant 4EGB gravity. With both scalar
and tensor perturbations in terms of the flow of the Hubble flow slow-roll
paramaters and the GB coupling constant, we obtained explicitly the analytical
expressions of the primordial power spectra, spectral indices, running of
spectral indices for both scalar and tensor perturbations, and the ratio
between the tensor and scalar spectra up to the second-order in the expansions
of the slow-roll approximation. Compared with the results obtained in the
previous literature, the results presented in this paper represent the most
accurate results obtained so far in the literature.

As we have mentioned in Sec. I, the high-order corrections to the inflationary
observables in the slow-roll expansions are very important in the future
analysis of different inflation models with precise experimental data. For
example, it is pointed out in Martin:2014rqa  that the future CMB experiments
can measure both $n_{\mathcal{R}}$ and $r$ to be accurate with errors $\gtrsim
10^{-3}$. Thus, the contributions from the GB coupling constant to the scalar
spectral index and the tensor-to-scalar ratio in general can not be ignored if
they are at the magnitude of $\mathcal{O}(10^{3})$. We expect the more precise
forthcoming experimental data in the future can make significant constraint on
the GB coupling constant in the spatial covariant 4EGB gravity, which could
help us to understand the physics of the early universe.

## Acknowledgements

This work is supported in part by the National Key Research and Development
Program of China Grant No.2020YFC2201503, and the Zhejiang Provincial Natural
Science Foundation of China under Grant No. LR21A050001 and LY20A050002, the
National Natural Science Foundation of China under Grant No. 12275238, No.
11975203, No. 11675143, and the Fundamental Research Funds for the Provincial
Universities of Zhejiang in China under Grant No. RF-A2019015.

## Appendix A: General formulas of primordial spectra in the uniform
asymptotic approximation

In this section, we present a brief introduction of the general formulas of
primordial perturbations with a slow-varying sound speed and parameter
$b(\tau)$ arsing from the nonlinear dispersion relation.

In the uniform asymptotic approximation, we first write Eqs.(3.21) and (3.33)
in the standard form Zhu:2013upa

$\displaystyle\frac{d^{2}\mu(y)}{dy^{2}}=\\{\lambda^{2}\hat{g}(y)+q(y)\\}\mu(y),$
(A.1)

where the new variables $y=-k\tau$, $\mu(y)=\mu_{\mathcal{R}}(y)$ and
$\mu_{h}(y)$ correspond to scalar and tensor perturbations respectively, and

$\displaystyle\lambda^{2}\hat{g}(y)+q(y)=\frac{\nu^{2}(\tau)-1/4}{y^{2}}-c^{2}(\tau)-b(\tau)y^{2}.$
(A.2)

where $c(\tau)=c_{s,t}(\tau)$ is the effective sound speed for scalar and
tensor perturbation modes respectively. For scalar perturbation, we have
Aoki:2020iwm

$\displaystyle\nu_{\mathcal{R}}^{2}(\tau)=\tau^{2}\frac{z^{\prime\prime}(\tau)}{z(\tau)}+\frac{1}{4}\;\;{\rm
with}\;\;c(\tau)=1\;\;{\rm and}\;\;b(\tau)=0,$

and for tensor modes,

$\displaystyle\nu_{h}^{2}(\tau)=\tau^{2}\frac{a^{\prime\prime}(\tau)}{a(\tau)}+\frac{1}{4}.$
(A.4)

It’s worth noting that $\lambda$ is an assumed large parameter which is used
to label the different approximate orders of the uniform asymptotic
approximation, as we will see later in the square bracket of the general
formulas of power spectrum in Eq. (Appendix A: General formulas of primordial
spectra in the uniform asymptotic approximation) in the square bracket.
$\lambda=1$ can be set to $\lambda=1$ for simplicity in the ending
calculation. Now we need to use the uniform asymptotic approximation to
construct the approximate solutions of the above equation. In order to ensure
the convergence of the errors of the approximate solutions, the only option is
Zhu:2013upa

$\displaystyle q(y)=-\frac{1}{4y^{2}}.$ (A.5)

Then, Eq. (A.2) can be written as

$\displaystyle\lambda^{2}\hat{g}(y)=\frac{\nu^{2}(\tau)}{y^{2}}-c^{2}(\tau)-b(\tau)y^{2},$
(A.6)

Here for scalar perturbation, $b(\tau)>0$ must meet the condition $b(\tau)>0$
for a healthy UV limit .

Since $b(\tau)$ is a small quantity, it is easy to find that the function
$\lambda^{2}\hat{g}(y)$ has a unique turning point, which is expressed as
follows

$\displaystyle y_{0}^{2}(\bar{\tau}_{0})$ $\displaystyle=$
$\displaystyle\frac{-c_{0}^{2}(\bar{\tau}_{0})+\sqrt{\bar{c}_{0}^{4}(\tau_{0})+4b(\bar{\tau}_{0})\bar{\nu}_{0}^{2}(\bar{\tau}_{0})}}{2b(\bar{\tau}_{0})}.$
(A.7)

Then with reference to Zhu:2014wfa , the general formula of the power spectrum
can be written as follows

$\displaystyle\Delta^{2}(k)$ $\displaystyle\equiv$
$\displaystyle\frac{k^{3}}{4\pi^{2}}\left|\frac{u(y)}{z(\tau)}\right|^{2}_{y\to
0^{+}}$ $\displaystyle=$
$\displaystyle\frac{k^{2}}{8\pi^{2}}\frac{-k\tau}{z^{2}(\tau)\nu(\tau)}\exp{\left(2\lambda\int_{y}^{\bar{y}_{0}}\sqrt{\hat{g}(y^{\prime})}dy^{\prime}\right)}$
$\displaystyle\times\left[1+\frac{\mathscr{H}(+\infty)}{\lambda}+\frac{\mathscr{H}^{2}(+\infty)}{2\lambda^{2}}+\mathcal{O}\left(\frac{1}{\lambda^{3}}\right)\right].$

The order of the parameter $\lambda$ shows that the formula is in third order
approximation. The integral of $\sqrt{g}$ in this formulas is given by Eqs.
(B.9) where $I_{0}$ and $I_{1}$ are given by Eqs. (B.10) and (B.11), and the
error control function is given by (B.13).

Then we need to calculate the spectral indices. For this purpose, we primarily
specify the $k$-dependence of $\bar{\nu}_{0}(\tau_{0})$,
$\bar{\nu}_{1}(\tau_{0})$ through $\tau_{0}=\tau_{0}(k)$. From the relation
$-k\tau_{0}=\bar{y}_{0}$, we obtain

$\displaystyle\frac{d\ln(-\tau_{0})}{d\ln k}$ $\displaystyle=$
$\displaystyle-1+\frac{d\ln\bar{y}_{0}}{d\ln(-\tau_{0})}\frac{d\ln(-\tau_{0})}{d\ln
k},$ (A.9)

which approximation is

$\displaystyle\frac{d\ln(-\tau_{0})}{d\ln k}$ $\displaystyle\simeq$
$\displaystyle-1-\frac{d\ln\bar{y}_{0}}{d\ln(-\tau_{0})}.$ (A.10)

Then with this relation, the spectral index can be written as

$\displaystyle n_{\mathcal{R}}-1,\;n_{h}$ $\displaystyle\equiv$
$\displaystyle\frac{d\ln\Delta^{2}_{s,t}}{d\ln k}$ (A.11)
$\displaystyle\simeq$ $\displaystyle
3-2\bar{\nu}_{0}-\frac{2\bar{b}_{1}\bar{\nu}_{0}}{3\bar{c}_{0}^{4}}+\frac{2\bar{c}_{1}\bar{\nu}_{0}}{\bar{c}_{0}}+\frac{2\bar{b}_{1}\bar{\nu}_{0}^{3}}{3\bar{c}_{0}^{4}}$
$\displaystyle-2\bar{\nu}_{1}\ln 2+\frac{\bar{\nu}_{1}}{6\bar{\nu}_{0}^{2}}.$

The running of the spectral index $\alpha\equiv dn/d\ln k$ can be similarly
written as

$\displaystyle\alpha_{\mathcal{R},\;h}(k)$ $\displaystyle\simeq$
$\displaystyle
2\bar{\nu}_{1}+\frac{2\bar{b}_{2}\bar{\nu}_{0}}{3\bar{c}_{0}^{4}}-\frac{2\bar{c}_{2}\bar{\nu}_{0}}{\bar{c}_{0}}-\frac{2\bar{b}_{2}\bar{\nu}_{0}^{3}}{3\bar{c}_{0}^{4}}$
(A.12) $\displaystyle+2\bar{\nu}_{2}\ln
2-\frac{\bar{\nu}_{2}}{6\bar{\nu}_{0}^{2}}.$

In the above, all the formulas (Eqs.(Appendix A: General formulas of
primordial spectra in the uniform asymptotic approximation), (A.11), and
(A.12)) can be used to calculate the primordial perturbation spectra with
different inflation models. It is easy to use these formulas, because they are
determined only by the quantities $H(\tau)$, ($c_{0},c_{1},c_{2}$),
($\nu_{0},\nu_{1},\nu_{2}$) which are evaluated at the turning point. And
these quantities can be easily obtain from Eqs. (3.22, Appendix A: General
formulas of primordial spectra in the uniform asymptotic approximation) for
scalar perturbations and Eqs. (3.34, A.4) for tensor perturbations. In section
IV, we apply these formulas to calculate the slow-roll power spectra for both
scalar and tensor perturbations in the spatial covariant 4EGB gravity.

## Appendix B: Integrals of $\sqrt{g}$ and the error control function

From Eq.(A.6) with $\lambda=1$, we have

$\displaystyle g(y)=\frac{\nu^{2}(\tau)}{y^{2}}-c^{2}(\tau)-b(\tau)y^{2}.$
(B.1)

where

$\displaystyle\nu(\tau)\simeq\bar{\nu}_{0}+\bar{\nu}_{1}\ln\frac{y}{\bar{y}_{0}}+\bar{\nu}_{2}\ln^{2}\frac{y}{\bar{y}_{0}},$
(B.2) $\displaystyle
c(\tau)\simeq\bar{c}_{0}+\bar{c}_{1}\ln\frac{y}{\bar{y}_{0}}+\bar{c}_{2}\ln^{2}\frac{y}{\bar{y}_{0}},$
(B.3) $\displaystyle
b(\tau)\simeq\bar{b}_{0}+\bar{b}_{1}\ln\frac{y}{\bar{y}_{0}}+\bar{b}_{2}\ln^{2}\frac{y}{\bar{y}_{0}}.$
(B.4)

with

$\displaystyle\bar{\nu}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{d\nu(\tau)}{d\ln(-\tau)}\Big{|}_{\tau_{0}},\bar{\nu}_{2}\equiv\frac{d^{2}\nu(\tau)}{d\ln^{2}(-\tau)}\Big{|}_{\tau_{0}},$
(B.5) $\displaystyle\bar{c}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{dc(\tau)}{d\ln(-\tau)}\Big{|}_{\tau_{0}},\bar{c}_{2}\equiv\frac{d^{2}c(\tau)}{d\ln^{2}(-\tau)}\Big{|}_{\tau_{0}},$
(B.6) $\displaystyle\bar{b}_{1}$ $\displaystyle\equiv$
$\displaystyle\frac{db(\tau)}{d\ln(-\tau)}\Big{|}_{\tau_{0}},\bar{b}_{2}\equiv\frac{d^{2}b(\tau)}{d\ln^{2}(-\tau)}\Big{|}_{\tau_{0}}.$
(B.7)

Substitute Eqs.(B.2), (B.3) and (B.4) into Eq.(B.1), we have

$\displaystyle\sqrt{g(y)}=$
$\displaystyle\frac{\sqrt{\bar{\nu}_{0}^{2}-\bar{c}_{0}^{2}y^{2}-\bar{b}_{0}y^{4}}}{y}$
(B.8) $\displaystyle+$
$\displaystyle\frac{2\bar{\nu}_{0}\bar{\nu}_{1}-2\bar{c}_{0}\bar{c}_{1}y^{2}-b_{1}y^{4}}{2y\sqrt{\bar{\nu}_{0}^{2}-\bar{c}_{0}^{2}y^{2}-\bar{b}_{0}y^{4}}}\ln\frac{y}{\bar{y}_{0}}.$

Then we divided the integral $\int\sqrt{g(y)}dy$ into two parts,

$\displaystyle\int_{y}^{\bar{y}_{0}}\sqrt{g(y)}dy\simeq I_{0}+I_{1},$ (B.9)

where

$\displaystyle\lim_{y\rightarrow 0}I_{0}$ $\displaystyle\simeq$
$\displaystyle-\frac{\bar{\nu}_{0}}{2}-\frac{\bar{c}_{0}^{2}}{4\sqrt{\bar{b}_{0}}}\arctan\Big{(}\frac{2\sqrt{\bar{b}_{0}}\bar{\nu}_{0}}{\bar{c}_{0}^{2}}\Big{)}$
(B.10)
$\displaystyle-\frac{1}{4}\bar{\nu}_{0}\ln(\bar{c}_{0}^{4}+4\bar{b}_{0}\bar{\nu}_{0}^{2})-\bar{\nu}_{0}\ln\frac{y}{2\bar{\nu}_{0}},$
$\displaystyle\lim_{y\rightarrow 0}I_{1}$ $\displaystyle\simeq$
$\displaystyle\bar{b}_{1}\bar{\nu}_{0}^{3}\frac{5-6\ln{2}}{18\bar{c}_{0}^{4}}+\bar{\nu}_{0}\bar{c}_{1}\frac{1-\ln{2}}{\bar{c}_{0}}$
(B.11)
$\displaystyle+\bar{\nu}_{1}\Big{(}-\frac{\pi^{2}}{24}+\frac{\ln^{2}2}{2}-\frac{1}{2}\ln^{2}\frac{y}{\bar{y}_{0}}\Big{)}.$

The integral form of error control function is

$\displaystyle\mathscr{H}(\xi)$ $\displaystyle\simeq$
$\displaystyle\frac{5}{36}\Big{\\{}\int_{\bar{y}_{0}}^{y}\sqrt{g{y}dy}\Big{\\}}^{-1}\Big{|}_{\bar{y}_{0}}^{y}$
(B.12)
$\displaystyle-\int_{\bar{y}_{0}}^{y}\Big{\\{}\frac{q}{g}-\frac{5{g^{\prime}}^{2}}{16g^{3}}+\frac{g^{\prime\prime}}{4g^{2}}\Big{\\}}\sqrt{g}dy.$

When $y\rightarrow 0$, the error control function can be written as

$\displaystyle\mathscr{H}(+\infty)$ $\displaystyle\simeq$
$\displaystyle\frac{\bar{c}_{0}^{4}+8\bar{b}_{0}\bar{\nu}_{0}^{2}}{6\bar{c}_{0}^{4}\bar{\nu}_{0}+24\bar{b}_{0}\bar{\nu}_{0}^{3}}$
(B.13)
$\displaystyle+\frac{\bar{c}_{1}}{6\bar{c}_{0}\bar{\nu}_{0}}-\frac{23+12\ln{2}}{72\bar{\nu}_{0}^{2}}\bar{\nu}_{1}$
$\displaystyle+\frac{(1+4\ln{2})\bar{\nu}_{0}}{6\bar{c}_{0}}\bar{b}_{1}.$

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