# 4D Gauss-Bonnet gravity: cosmological constraints, $H_{0}$ tension and large
scale structure

Deng Wang<EMAIL_ADDRESS>National Astronomical Observatories, Chinese
Academy of Sciences, Beijing, 100012, China  David Mota  Institute of
Theoretical Astrophysics, University of Oslo, P.O. Box 1029 Blindern, N-0315
Oslo, Norway

###### Abstract

We perform correct and reasonable cosmological constraints on the newly
proposed 4D Gauss-Bonnet gravity. Using the joint constraint from cosmic
microwave background, baryon acoustic oscillations, Type Ia supernovae, cosmic
chronometers and redshift space distortions, we obtain, so far, the strongest
constraint $\tilde{\alpha}=(1.2\pm 5.2)\times 10^{-17}$, namely
$\alpha=(2.69\pm 11.67)\times 10^{48}$ eV-2, among various observational
limitations from different information channels, which is tighter than
previous bound from the speed of gravitational wave by at least one order of
magnitude. We find that our bound is well supported by the observations of
temperature and lensing potential power spectra of cosmic microwave background
from the Planck-2018 final release. Very interestingly, the large $H_{0}$
tension between the local measurement from the Hubble Space Telescope and
global derivation from the Planck-2018 final data under the assumption of
$\Lambda$CDM can be greatly resolved from $4.4\sigma$ to $1.94\sigma$ level in
the 4D Gauss-Bonnet gravity. In theory, we find that this model can partly
relieve the coincidence problem and the rescaling Gauss-Bonnet term, which
needs the help of the cosmological constant to explain current cosmic
acceleration, is unable to serve as dark energy alone.

##  I Introduction

Up to now, general relativity (GR) is believed to be the most successful
gravity theory to describe the physical and cosmological phenomena over a
large range of energy from large scales to small scales Einstein:1916vd .
However, it is not a perfect theory based on observational limitations and
theoretical considerations. Specifically, two dark clouds of modern physics,
dark matter Young:2016ala  and dark energy Weinberg:1988cp ; Riess:1998cb ;
Perlmutter:1998np , can not be well explained in the framework of GR plus
$\Lambda$-cold dark matter ($\Lambda$CDM), where $\Lambda$ is the cosmological
constant. Meanwhile, the quantum version of Einstein’s gravity, which meets at
least the re-normalization problem, can not be reasonably constructed
tHooft:1974toh ; Deser:1974cy ; Deser:1974cz ; Goroff:1985sz ; Goroff:1985th ;
vandeVen:1991gw . This implies that the underlying gravity theory governing
the gravitational dynamics of the universe may not be GR and could be an
alternative gravitational scenario, which can help understand the dark sector
better at least.

The Lovelock’s theorem Lanczos:1938sf ; Lovelock:1971yv ; Lovelock:1972vz
indicates that GR is the sole gravity theory under the following four
assumptions: (i) metricity; (ii) diffeomorphism invariance; (iii) equations of
motion must be second order; (iv) spacetime must be $3+1$ dimension. In
general, one can construct a new gravitational theory by relaxing these
assumptions. As an elegant variant or modification of GR, Gauss-Bonnet gravity
Zumino:1985dp  satisfies the former three assumptions but just lives in high
dimensions, and its corresponding action in $D$ dimensions is written as

$S=\int d^{D}x\sqrt{-g}\left(\frac{R-2\Lambda}{16\pi
G}+\alpha\mathcal{G}\right)+S_{m},$ (1)

where $R$ is the Ricci Scalar, $G$ is the Newton gravitational constant,
$S_{m}$ is the action of matter fields, $\alpha$ is a constant and
$\mathcal{G}$ is the so-called Gauss-Bonnet invariant read as

$\mathcal{G}=R^{\mu\nu\alpha\beta}R_{\mu\nu\alpha\beta}-4R^{\mu\nu}R_{\mu\nu}+R^{2}.$
(2)

As we know, Gauss-Bonnet invariant is the unique form constructed from
quadratic products of Riemann tensor that does not introduce any terms with
more than two derivatives into the gravitational field equations
Lovelock:1971yv . In 4-dimensional spacetime, since it is a total derivative,
it can not make contributions to gravitational dynamics. Nonetheless, this
does not means that Gauss-Bonnet invariant is completely useless in 4
dimensions. Because of its topological properties, it can be used to classify
topologies when doing the path-integral quantization of gravity. Recently, in
light of the fact that the contribution of Gauss-Bonnet term to equations of
motion is proportional to $(D-4)$ Mardones:1990qc ; Torii:2008ru , a new
4-dimensional Gauss-Bonnet theory (hereafter GB) is proposed in
Ref.Glavan:2019inb  by rescaling the coupling constant
$\alpha\rightarrow\alpha/(D-4)$ in order to produce nontrivial contributions
to gravitational dynamics. Some interesting results have been obtained based
on this new GB model Haghani:2020ynl ; Narain:2020qhh ; Gurses:2020ofy ;
Aoki:2020iwm ; Clifton:2020xhc ; Feng:2020duo ; Garcia-Aspeitia:2020uwq . In
particular, several groups have placed observational constraints on the free
parameter $\alpha$ of this GB model. Clifton et al. Clifton:2020xhc  find that
the inflation of the early universe rules out almost all the negative values
of $\alpha$ except extremely small negative ones, and that the system of
binary black holes gives the constraint $0\lesssim\alpha\lesssim
10^{21}\mathrm{eV}^{-2}$. However, Feng et al. Feng:2020duo  point out that
the GB model with a bare vanishing $\Lambda$ has been ruled out by current
cosmological and gravitational wave observations. They give $-3.49\times
10^{50}\mathrm{eV}^{-2}\leqslant\alpha\leqslant 1.49\times
10^{51}\mathrm{eV}^{-2}$ according to the tight constraint on the speed of
gravitational waves from the first detection of an electromagnetic counterpart
(GRB 170817A) to the gravitational wave signal (GW170817). Specially, Garcá-
Aspeitia et al. Garcia-Aspeitia:2020uwq  give the estimated value
$\alpha\approx 1.604^{+0.017}_{-0.018}\times 10^{62}\mathrm{eV}^{-2}$ by using
the joint constraint of five cosmological probes at the background level, but
their result is very inconsistent with the constraint from the speed of
gravitational waves. Therefore, we are aiming at giving a correct and
reasonable constraint on the typical parameter $\alpha$ of the GB model in
light of current cosmological observations. Our constraint will be compatible
with observational limitations from Refs.Clifton:2020xhc ; Feng:2020duo . By
using a data combination of cosmic microwave background (CMB), baryon acoustic
oscillations (BAO), Type Ia supernovae (SNe Ia), cosmic chronometers and
redshift space distortions (RSD), we obtain, so far, the strongest constraint
$\alpha=(2.69\pm 11.67)\times 10^{48}$ eV-2 among various observational
limitations from different information channels.

This study is outlined in the following manner. In Section II, we display the
cosmological equations of the GB model. In section III, we discuss the
coincidence problem and investigate the behaviors of the effective equation of
state of dark energy in the GB model. In Section IV, we describe the
observational datasets and implement the cosmological constraints. In Section
IV, we investigate the behaviors of large scale structure of the GB model. The
discussions and conclusions are presented in the final section.

##  II Cosmological equations

In 4-dimensional spacetime, this new GB model makes the Gauss-Bonnet term
produce a nontrivial contribution to the background evolution of the universe.
By inserting Eq.(2) into Eq.(1) and varying the action Eq.(1), the
D-dimensional modified Einstein equation is shown as

$R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}+\Lambda
g_{\mu\nu}+\frac{\alpha}{D-4}\left(4R_{\mu\alpha\beta\sigma}R^{\;\,\alpha\beta\sigma}_{\nu}-8R_{\mu\alpha\nu\beta}R^{\alpha\beta}-8R_{\mu\alpha}R^{\;\,\alpha}_{\nu}+4RR_{\mu\nu}-g_{\mu\nu}\mathcal{G}\right)=8\pi
GT_{\mu\nu},$ (3)

where $g_{\mu\nu}$ and $T_{\mu\nu}$ are the spacetime metric and energy-
momentum tensor of matter fields, respectively. In a spatially flat
4-dimensional ($D\rightarrow 4$) Friedmann-Robertson-Walker (FRW) universe
(see also Glavan:2019inb ; Haghani:2020ynl ; Narain:2020qhh ) Friedmann
equation of the GB model is expressed as

$H^{2}+2\alpha H^{4}=\frac{8\pi
G}{3}(\rho_{b}+\rho_{cdm}+\rho_{r})+\frac{\Lambda}{3},$ (4)

where $H$ denotes the Hubble parameter and $\rho_{b}$, $\rho_{cdm}$ and
$\rho_{r}$ are the energy densities of baryons, CDM and radiation components,
respectively. Assuming a perfect fluid with the equation of state (EoS)
$p_{i}=\omega_{i}\rho_{i}$, where $i=b$, $cdm$ and $r$, the energy
conservation equation of the GB model can be written as

$\dot{\rho}_{i}+3H(p_{i}+\rho_{i})=0,$ (5)

where the dot denotes the derivative with respect to the cosmic time $t$.
Notice that for the independent component $\Lambda$, the EoS reads as
$p_{\Lambda}=-\rho_{\Lambda}$. In order to constrain the GB model at
background level, it is convenient to write its dimensionless Hubble parameter
$E(z)$ as follows

$E^{2}(z)+2\alpha
H_{0}^{2}E^{4}(z)=(\Omega_{b0}+\Omega_{cdm0})(1+z)^{3}+\Omega_{r0}(1+z)^{4}+\Omega_{\Lambda
0},$ (6)

where $E(z)\equiv H(z)/H_{0}$, $H_{0}$ is the Hubble constant, and
$\Omega_{b0}$, $\Omega_{cdm0}$, $\Omega_{r0}$ and $\Omega_{\Lambda 0}$ are the
present-day energy density ratio of baryons, CDM, radiation and cosmological
constant, respectively. Note that the present-day density ratio of matter
$\Omega_{m0}=\Omega_{b0}+\Omega_{cdm0}$. To carry out constraints more
conveniently, we define a dimensionless parameter $\tilde{\alpha}\equiv
2\alpha H_{0}^{2}$ here. Solving the above equation with respect to $E(z)$ at
a given redshift $z$, one can easily obtain

$E(z)=\left[\frac{\sqrt{X(z)}-1}{2\tilde{\alpha}}\right]^{\frac{1}{2}},$ (7)

where

$X(z)=1+4\tilde{\alpha}\left[\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}+\Omega_{\Lambda
0}\right].$ (8)

Using the condition $E(0)=1$, one can obtain

$1+\tilde{\alpha}=\Omega_{m0}+\Omega_{r0}+\Omega_{\Lambda 0}.$ (9)

It is worth noting that this equation indicates that $\tilde{\alpha}$ is a
very small constant close to zero. In the following context, this relation can
help determine the range of the model parameter $\tilde{\alpha}$.

Figure 1: The density ratio between dark matter and dark energy is shown as a
function of scale factor $a$. The solid (red), dash-dotted (orange) and dotted
(magenta) lines denote $\tilde{\alpha}=0$ ($\Lambda$CDM), $-10^{-5}$ and
$-10^{-8}$, respectively. The horizontal dashed (blue) line represents dark
matter and dark energy share the same energy density.

Figure 2: The effective EoS of dark energy $\omega_{de}(z)$ is shown as a
function of redshift $z$. The solid (red) lines denote the $\Lambda$CDM model.
Left: The dash-dotted (orange), dashed (blue) and dotted (magenta) lines are
$\tilde{\alpha}=-10^{-3}$, $10^{-8}$ and $10^{-16}$ in the GB model,
respectively. Right: The dash-dotted (orange), dash-dotted (cyan), dashed
(blue) and dotted (magenta) lines are $\tilde{\alpha}=-10^{-3}$, $10^{-5}$,
$10^{-8}$ and $10^{-16}$ in the GB model, respectively.  Figure 3: The
marginalized constraint on the PGB model is shown by using the “C” dataset.
Figure 4: The marginalized constraint on the NGB model is shown by using the
“C” dataset. Figure 5: The marginalized constraints on the GB model are shown
by using the “C”, “R” and “CBSHR” datasets, respectively. The dashed magenta
line denotes $\tilde{\alpha}=0$, i.e., the $\Lambda$CDM model. Figure 6: The
$\Omega_{m0}$-$H_{0}$ parameter spaces constrained by the Planck-2018 CMB data
are shown in the PGB (pink), NGB (purple) and GB (blue) models, respectively.
The horizontal cyan bands represents the direct measurement $H_{0}=74.03\pm
1.42$ km s-1 Mpc-1 from the HST project Riess19 .

##  III The coincidence problem and effective equation of state

It is interesting to study whether the coincidence problem can be resolved in
the GB model. First of all, according to Eq.(4), we derive the effective
energy density of dark energy as follows

$\rho_{de}=\frac{1}{8\pi G}(\Lambda-6\tilde{\alpha}H^{4}),$ (10)

where the second term is from the contribution of Gauss-Bonnet invariant.
Then, substituting Eq.(10) into Eq.(5), we obtain the effective pressure of
dark energy

$p_{de}=\frac{1}{8\pi
G}(8\tilde{\alpha}H^{2}\dot{H}+6\tilde{\alpha}H^{4}-\Lambda).$ (11)

Furthermore, the density ratio between dark matter and dark energy is
expressed as

$\frac{\rho_{cdm}}{\rho_{de}}=\frac{3H_{0}^{2}\Omega_{cdm0}(1+z)^{3}}{3H_{0}^{2}\Omega_{\Lambda
0}-6\tilde{\alpha}H^{4}}.$ (12)

It is easy to find that only when $\tilde{\alpha}<0$, can the densities of
dark matter and dark matter reach the same order of magnitude earlier than
those do in $\Lambda$CDM. The analysis result is shown in Fig.1. We find that
when $\tilde{\alpha}=-10^{-5}$ dark matter shares the same density as dark
energy in GB earlier than in $\Lambda$CDM. But if we choose a smaller
$\tilde{\alpha}$ such as $-10^{-8}$, the density ratio of GB will behaves the
same as that of $\Lambda$CDM. As a consequence, we conclude that, in theory,
the GB model is able to partly resolve the coincidence problem.

Furthermore, it is also interesting to study the effects of Gauss-Bonnet term
on the effective EoS of dark energy. Combining Eq.(10) with (11), we have

$\omega_{de}(z)=-1+\frac{8\tilde{\alpha}(1+z)H^{2}H^{\prime}}{6\tilde{\alpha}H^{4}-\Lambda},$
(13)

where the prime denotes the derivative with respect to $z$. From Eq.(6), it is
easy to see that $\tilde{\alpha}$ can be either positive or negative.
Consequently, in GB, we should discuss the effective EoS of dark energy for
the cases of $\tilde{\alpha}>0$ and $\tilde{\alpha}<0$, respectively. The
corresponding numerical results are presented in Fig.2. We find that
$\tilde{\alpha}$ can not be too large but very small when $\tilde{\alpha}>0$.
Specifically, when $\tilde{\alpha}=10^{-3}$, to a large extent, the EoS
deviates from $\Lambda$CDM. If $\tilde{\alpha}=10^{-8}$, it is always the same
as $\Lambda$CDM at low redshifts and exhibits a clear deviation starting from
$z=2$. If choosing a much smaller value $\tilde{\alpha}=10^{-16}$, the EoS
tends to be same as $\Lambda$CDM. This provides a clue that $\tilde{\alpha}$
should be very small for us to constrain the GB model with observations.
Similarly, for the case of $\tilde{\alpha}<0$, we also conclude that the
absolute value of $\tilde{\alpha}$ should be very small in order to deviate
small from $\Lambda$CDM. Very interestingly, we find that when
$\tilde{\alpha}=-10^{-8}$, the EoS first becomes a phantom state starting from
$z=0$ and then returns to $\Lambda$CDM at about $z=0.3$. If $\tilde{\alpha}$
is larger in the negative direction, e.g., $\tilde{\alpha}=-10^{-16}$, the EoS
reduces to $\Lambda$CDM. This can also be viewed as a clue to implement
cosmological constraints.

##  IV Cosmological constraints

In general, to construct a new modified gravity model, one should use the
modified terms in the Lagrangian to replace the role of cosmological constant
$\Lambda$. However, for the purpose of constructing a viable 4D GB gravity,
the authors in Ref.Glavan:2019inb  include the small modified term
$\alpha\mathcal{G}/(D-4)$ together with $\Lambda$ in the Lagrangian. Hence, we
are very interested in whether the rescaling Gauss-Bonnet term can explain the
cosmic acceleration alone. Base on this concern, Eq.(8) with a vanishing
$\Lambda$ is expressed as

$X(z)=1+4\tilde{\alpha}\left[\Omega_{m0}(1+z)^{3}+\Omega_{r0}(1+z)^{4}\right],$
(14)

and the relation Eq.(9) reads as

$1+\tilde{\alpha}=\Omega_{m0}+\Omega_{r0}.$ (15)

On the one hand, according to Eq.(7), we have the condition $X(z)\geqslant 0$.
If we calculate $X(z)$ at the decoupling redshift $z_{\star}=1090.3$
Aghanim:2018eyx  and fix $\Omega_{m0}=0.3$ and $\Omega_{r0}=8.47\times
10^{-5}$, then we obtain

$\tilde{\alpha}\geqslant-4.91\times 10^{-10}.$ (16)

One the other hand, one can easily find that $\alpha\approx-0.7$ derived from
Eq.(15), which is strongly inconsistent with the inequality (16). Therefore,
from the background level only, we conclude that the rescaling Gauss-Bonnet
term can not serves as dark energy alone. This explanation is the basic reason
of the conclusion in Ref.Feng:2020duo  that the GB model with vanishing bare
$\Lambda$ is roughly ruled out by the current observational limits of EoS of
dark energy.

In this study, we shall consider the case of nonvanishing $\Lambda$. Observing
Eqs.(7-9), one can easily find that $\tilde{\alpha}$ should be a very small
positive or negative value. Based on this concern, we consider three models to
implement the Bayesian analysis, i.e., $\tilde{\alpha}>0$ (PGB),
$\tilde{\alpha}<0$ (NGB) and free $\tilde{\alpha}$ (GB). Hereafter, we denote
these three models as PGB, NGB and GB, respectively. Since we do not know the
possible bounds of $\tilde{\alpha}$, the main purpose of constructing the PGB
and NGB models is to determine the possible limitations of $\tilde{\alpha}$,
in order to constrain $\tilde{\alpha}$ better in the GB model.

Table 1:  The results of marginalized constraints on the cosmological parameters of PGB, NGB and GB models are shown, respectively. The symbols “$\diamondsuit$” denote the parameters that cannot be well constrained by observational datasets.  Parameter | $\Omega_{m0}$ | $\Omega_{b0}$ | $\alpha$ | $H_{0}$ | $\sigma_{8}$
---|---|---|---|---|---
PGB | C | $0.315\pm 0.026$ | $0.0492\pm 0.0041$ | $(0.20^{+1.00}_{-0.19})\times 10^{-16}$ | $67.6^{+2.6}_{-3.2}$ | $\diamond$
R | $0.275^{+0.066}_{-0.098}$ | $\diamond$ | $0.00030^{+0.00140}_{-0.00028}$ | $70\pm 20$ | $0.786^{+0.058}_{-0.070}$
CBSHR | $0.3008\pm 0.0087$ | $0.0473\pm 0.0012$ | $(0.08^{+0.68}_{-0.07})\times 10^{-16}$ | $68.88\pm 0.90$ | $0.749\pm 0.028$
NGB | C | $0.334^{+0.034}_{-0.047}$ | $0.0522^{+0.0052}_{-0.0073}$ | $(-2.5^{+2.4}_{-2.0})\times 10^{-21}$ | $65.8^{+4.2}_{-3.7}$ | $\diamond$
R | $0.27^{+0.11}_{-0.14}$ | $\diamond$ | $-0.0070^{+0.0069}_{-0.0702}$ | $69\pm 20$ | $0.777\pm 0.064$
CBSHR | $0.2997\pm 0.0085$ | $0.0471\pm 0.0012$ | $(-3.6^{+3.58}_{-5.40})\times 10^{-21}$ | $68.99\pm 0.89$ | $0.751\pm 0.028$
GB | C | $0.340^{+0.041}_{-0.050}$ | $0.0533^{+0.0064}_{-0.0079}$ | $(4.2^{+3.3}_{-4.9})\times 10^{-17}$ | $65.2\pm 4.3$ | $\diamond$
R | $0.278^{+0.088}_{-0.092}$ | $\diamond$ | $0.0004\pm 0.0016$ | $68\pm 21$ | $0.790\pm 0.066$
CBSHR | $0.3004\pm 0.0087$ | $0.0473\pm 0.0012$ | $(1.2\pm 5.2)\times 10^{-17}$ | $68.80\pm 0.90$ | $0.750\pm 0.028$

We will adopt the background and perturbation data to perform cosmological
constraints on these models. Specifically, the background part consists of
CMB, BAO and SNe Ia. Since the contribution of Gauss-Bonnet term is
proportional $H^{4}$ in Eq.(4), one can place a strong constraint on
$\tilde{\alpha}$ with high redshift background data. Very interestingly, the
CMB data includes the distance information of comoving sound horizon at the
decoupling redshift $z_{\star}$, and is able to give a tight restriction. Here
we use the Planck-2018 distance prior from TTTEEE$+$lowl$+$lowE$+$lensing
data, i.e., compressed CMB data obtained in Ref.Zhai:2018vmm  to implement
constraints. This dataset is denoted as “C”. BAO as a standard cosmological
ruler can constrain the expansion history of the universe after decoupling by
measuring the position of the oscillations in the matter power spectrum at
different redshifts, while breaking the degeneracies between parameters
better. It is a clean signature and unaffected by other systematic
uncertainties. Here we use the BOSS DR12 sample at three effective redshifts
$z_{eff}=$ 0.38, 0.51 and 0.61 Alam:2016hwk , the 6dFGS one at $z_{eff}=$
0.106 Beutler:2011hx  and the SDSS-MGS one at $z_{eff}=$ 0.15 Ross:2014qpa .
This dataset is identified as “B”. SNe Ia, the so-called standard candle, is a
powerful tool to probe the expansion history of the universe. Up to now, the
largest SNe Ia dataset is the “Pantheon” sample consisting of 1048
spectroscopically confirmed SNe Ia and covering the redshift range
$z\in[0.01,2.3]$. It is worth noting that under the assumption of a flat
universe, this sample can be compressed into six $E^{-1}(z)$ points and is
also able to constrain the background expansion well Riess:2017lxs . For
simplicity, we shall employ these compressed SNe Ia points in the numerical
analysis and denote this dataset as “S”. As a complementary background probe,
we also include 31 cosmic chronometers Moresco:2016mzx , which measure the
Hubble parameter at different redshifts, in our constraint. We refer to this
dateset as “H”.

For the first time, we would like to constrain these 4D Gauss-Bonnet models
using the perturbation data, i.e., RSD, which includes the information of
large scale structure. Here we adopt the so-called “Gold-2018” growth-rate
dataset Basilakos:2016nyg  and denote this dataset as “R”. To determine the
value of $\tilde{\alpha}$ in a detailed way, we constrain the above three
models using the dataset “C”, “R” and a data combination of “CBSHR”,
respectively.

In order to perform the Bayesian analysis and obtain the posterior
distributions of model parameters, we employ the Affine Invariant Markov chain
Monte Carlo Ensemble sampler EMCEE ForemanMackey:2012ig  and analyze the
chains with the package GetDist Lewis:2019xzd .

The results of marginalized constraints on the PGB, NGB and GB models are
presented in Figs.3-5 and Tab.1. By constraining the PGB and NGB models with
Planck-2018 CMB data alone, we obtain a very tight bound $-4.5\times
10^{-21}\leqslant\tilde{\alpha}\leqslant 1.2\times 10^{-16}$ at the $1\sigma$
confidence level, which exhibits a non-symmetric limitation by about five
orders of magnitude in both the positive and negative directions. Enlarging
this bound by six orders of magnitude as a prior for the GB model, we obtain
the estimated value $\tilde{\alpha}=(4.2^{+3.3}_{-4.9})\times 10^{-17}$ for
the case of free $\tilde{\alpha}$, while the non-symmetric property of
$\tilde{\alpha}$’s error disappears. Very interestingly, we find that the
$4.4\sigma$ $H_{0}$ tension between the directly local measurement from the
Hubble Space Telescope (HST) Riess19  and indirectly global derivation from
the Planck-2018 final release under $\Lambda$CDM Aghanim:2018eyx  can be
greatly relieved to $2.17\sigma$, $1.86\sigma$ and $1.94\sigma$ level in the
PGB, NGB and GB models, respectively (see also Fig.6). We refer the readers to
Ref.DiValentino:2021izs , which is a complete review of $H_{0}$ tension.
Subsequently, using the combined dataset CBSHR, we obtain a relatively
conservative value $\tilde{\alpha}=(1.2\pm 5.2)\times 10^{-17}$, which is
consistent with $\Lambda$CDM within $1\sigma$ level. Note that this is, so
far, the strongest constraint on the typical parameter $\tilde{\alpha}$. In
natural units $h=c=k=1$, this constraint can be translated into
$\alpha=(2.69\pm 11.67)\times 10^{48}$ eV-2, which is tighter than the result
obtained in Ref.Feng:2020duo  by at least one order of magnitude. From Fig.5
and Tab.1, in the GB model, one can find that the constraint on
$\tilde{\alpha}$ from RSD alone is much looser than those from CMB and CBSHR,
and it gives a poor constraint on $H_{0}$. Furthermore, the constraints on the
amplitude of matter clustering $\sigma_{8}$ are all compatible with
Planck-2018 CMB final release at the $1\sigma$ confidence level
Aghanim:2018eyx  but with about a ten times larger uncertainty. Meanwhile, we
observe that the introduction of CBSH into R dataset leads to a smaller matter
clustering effect in the large scale structure.

##  V Large scale structure

In the framework of conformal Newtonian gauge, when considering the scalar
perturbations only, the perturbed FRW metric is expressed as Ma1995

$ds^{2}=a^{2}(\tau)\left[-(1+2\Psi)d\tau^{2}+(1-2\Phi)\gamma_{ij}dx^{i}dx^{j}\right],$
(17)

where $\Psi$ and $\Phi$ denote perturbed metric potentials and $\tau$ is the
conformal time. The components of perturbed energy-momentum tensor are written
as

$\delta T_{0}^{0}=-\tilde{\delta}\rho,$ (18) $\delta
T_{0}^{i}=-(1+c_{s}^{2})\rho v^{i},$ (19) $\delta T_{1}^{1}=\delta
T_{2}^{2}=\delta T_{2}^{2}=c_{s}^{2}\tilde{\delta}\rho,$ (20)

where $\rho$ is the density of the fluid, $\tilde{\delta}=\delta\rho/\rho$ is
the dimensionless density perturbation, $v$ is the velocity perturbation, and
$c_{s}$ is the adiabatic sound speed of the fluid.

For the non-relativistic matter component, its equation of state $\omega_{m}$
and squared adiabatic sound speed $c^{2}_{s(m)}$ both equal zero, namely
$\omega_{m}=c^{2}_{s(m)}=\delta P_{m}/\delta\rho_{m}=0$. Since the energy
conservation equation in the GB model is identical to that in GR, the
perturbed conservation equation also remains the same as that in GR.
Subsequently, for the matter component, the temporal and spatial components of
perturbed Einstein field equation can be shown as

$\delta_{m}^{\ast}=3\Phi^{\ast}-\theta_{m},$ (21)
$\theta_{m}^{\ast}=k^{2}\Psi-\mathcal{H}\theta_{m},$ (22)

where $\delta_{m}$ and $\theta_{m}$ are, respectively, the density and
velocity perturbations of matter, $\mathcal{H}$ is the conformal Hubble
parameter and the symbol “$\ast$” denotes the derivative with respect to the
conformal time. The spatial off-diagonal component of perturbed field equation
is given by

$(2\tilde{\alpha}\mathcal{H}^{2}+a^{2}H_{0}^{2})\Psi+(2\tilde{\alpha}\mathcal{H}^{2}-a^{2}H_{0}^{2}-4\tilde{\alpha}\mathcal{H}^{\ast})\Phi=0.$
(23)

One can easily find that this equation reduces to $\Psi=\Phi$ when
$\tilde{\alpha}=0$. Substituting Eq.(23) into Eqs.(21) and (22), in the sub-
horizon limit $k\gg\mathcal{H}$ ($k$ is the comoving wave number), we obtain a
second order differential equation for the density perturbation as

$\delta_{m}^{\prime\prime}+(\frac{H^{\prime}}{H}-\frac{1}{1+z})\delta_{m}^{\prime}+\frac{3\Omega_{m0}(1+z)\left\\{2\tilde{\alpha}E(z)\left[2(1+z)\frac{H^{\prime}}{H_{0}}-E(z)\right]-1\right\\}}{2E^{2}(z)\left[2\tilde{\alpha}E^{2}(z)+1\right]}\delta_{m}=0,$
(24)

where the prime denotes the derivative with respect to the redshift $z$.

To investigate the behaviors of large scale structure in the GB model, we
define the following two perturbation quantities

$f\sigma_{8}(z)=\sigma_{8}(z)\frac{\delta_{m}^{\prime}}{\delta_{m}},$ (25)
$S_{8}(z)=\sigma_{8}(z)\left(\frac{\Omega_{m0}}{0.3}\right)^{\frac{1}{2}},$
(26)

where $\sigma_{8}(z)\equiv\sigma_{8}\delta_{m}(z)/\delta_{m}(0)$. By choosing
appropriate initial conditions at the beginning of dark matter dominated
epoch, we solve numerically Eq.(24) and the corresponding results are shown in
Fig.7. We find that a large value of $\tilde{\alpha}$ such as $10^{-2}$ in the
GB model leads to a lower $f\sigma_{8}$ value at all redshifts than
$\Lambda$CDM, can not explain the growth-rate dataset well, and gives a
stronger effect of matter clustering at the same redshift than $\Lambda$CDM.
The GB model with the constrained parameter $\tilde{\alpha}=0.0004$ by RSD
data alone or a smaller value $10^{-8}$ almost exhibits the same evolutionary
behaviors as $\Lambda$CDM. This implies that the constraining power of RSD
data, which just can give restrictions on $\tilde{\alpha}\sim\mathcal{O}(-3)$
(see also Tab.1), is weak.

Figure 7: Two perturbation quantities $f\sigma_{8}(z)$ and $S_{8}(z)$ are
shown as a function of redshift $z$, respectively. The solid (red) lines
denote the $\Lambda$CDM model. The dash-dotted (orange), dashed (blue) and
dotted lines are $\tilde{\alpha}=10^{-2}$, $0.0004$ and $10^{-8}$ in the GB
model, respectively. The points with errors represent the RSD data.  Figure 8:
The matter power spectra are shown for $\Lambda$CDM (solid),
$\tilde{\alpha}=10^{-15}$ (dotted) and $10^{-17}$ (dashed), respectively.

Figure 9:  The CMB temperature power spectra and lensing potential power
spectra ranging from multipole $\ell=2$ to 2200 are shown for $\Lambda$CDM
(solid), $\tilde{\alpha}=10^{-15}$ (dotted) and $10^{-17}$ (dashed),
respectively. The points with errors represent the Planck-2018 temperature
power spectrum data ranging from $\ell=2$ to 30.

To study further the matter power spectra, CMB temperature power spectra and
CMB lensing potential power spectra, we modify the background and perturbation
equations of the GB model in the public Boltzmann code CAMB Lewis2013  and the
results are displayed in Figs.8-9. Note that we adopt the amplitude of
primordial matter spectrum $A_{s}=2.1\times 10^{-9}$, scalar spectral index
$n_{s}=0.96$, pivot scale $k_{pivot}=0.05$ Mpc-1, $\Omega_{b0}h^{2}=0.0226$,
$\Omega_{cdm0}h^{2}=0.112$ and $H_{0}=70$ km s-1 Mpc-1 in CAMB. Here $h\equiv
H_{0}/(100$ km s-1 Mpc${}^{-1})$.

From Fig.8, we find a clear deviation from $\Lambda$CDM in the case of
$\tilde{\alpha}=10^{-15}$, which has a larger power than $\Lambda$CDM at all
scales. In the left panel of Fig.9, we find that when
$\tilde{\alpha}=10^{-15}$, although the CMB temperature power spectrum gives
lower values than $\Lambda$CDM, it seems to be still compatible with the data
points at large angular scales. However, the whole CMB temperature power
spectrum has a obvious shift towards a larger multipole, which presents a
clear deviation from $\Lambda$CDM. This indicates that one can rule out this
value at small angular scales (see also Fig.1 in Ref.Aghanim:2018eyx ).
Furthermore, we also analyze the lensing potential power spectrum for the case
of $\tilde{\alpha}=10^{-15}$, in the right panel of Fig.9, we find that this
value has been ruled out by Planck-2018 lensing potential data (see also Fig.3
in Ref.Aghanim:2018eyx ). The above results are all consistent with our best
constraint $(0.2\pm 7.0)\times 10^{-17}$ and clearly rule out the GB model
with $\tilde{\alpha}\geqslant 10^{-15}$. It is worth noting that when
$\tilde{\alpha}=10^{-17}$, the GB model always tends to be very close to
$\Lambda$CDM in the analysis of matter, CMB temperature and lensing potential
power spectra. In light of this, we need more high precision data to break the
parameter degeneracy better, give tighter constraints, and then distinguish
cosmological models in a more efficient way.

##  VI Discussions and conclusions

A new Gauss-Bonnet gravity makes the Gauss-Bonnet invariant contribute
nontrivially to gravitational dynamics in 4D spacetime by a regularization-
like method. We are motivated by giving a correct and reasonable constraint on
the free parameter $\alpha$ of the GB model in light of current cosmological
observations. Using the joint constraint from cosmic microwave background,
baryon acoustic oscillations, Type Ia supernovae, cosmic chronometers and
redshift space distortions, we obtain the tightest constraint
$\tilde{\alpha}=(1.2\pm 5.2)\times 10^{-17}$, i.e., $\alpha=(2.69\pm
11.67)\times 10^{48}$ eV-2 among various observational bounds from different
information channels, which is tighter than previous limitation from the speed
of gravitational wave by at least one order of magnitude.

By studying the temperature and lensing potential power spectra of cosmic
microwave background, we find that our constrained value of $\alpha$ is also
supported by the Planck-2018 final data. However, due to large errors, the
growth-rate data is unable to help improve the constraint.

The improvement of our constraint mainly originates from the high precision
distance data of cosmic microwave background at a high redshift, namely the
decoupling redshift $z_{\star}=1090.3$ (see also Fig.5 and Tab.1). Since the
Gauss-Bonnet term $2\alpha H^{4}$ in Eq.(4) must be small and the Hubble
parameter $H$ becomes very large at $z_{\star}$, we obtain a very strong
constraint. One can also find that combining other datasets with cosmic
microwave background data just ameliorates the constraint a little.

It is interesting that the serious $H_{0}$ tension between the local
measurement from the Hubble Space Telescope (HST) Riess19  and global
derivation from the Planck-2018 final data under the assumption of
$\Lambda$CDM Aghanim:2018eyx  can be greatly alleviated from $4.4\sigma$ to
$1.94\sigma$ level in the GB model.

In theory, the 4D GB gravity can partly resolve the coincidence problem. But
if using the constrained small value of $\alpha$, the coincidence problem can
not be resolved obviously. We also verify that the rescaling Gauss-Bonnet term
is unable to serve as dark energy alone. It needs the help of the cosmological
constant to explain the cosmic acceleration. Interestingly, one can also give
a relatively large constraint by only using the behaviors of background
evolution such as the effective equation of state of dark energy.

In future, we expect that high precision data from the enhanced measurements
of background expansion and large scale structure can help explore the nature
of dark energy better.

##  VII Acknowledgements

DW warmly thanks Yuan Sun and Changjun Gao for useful communications and
discussions on gravitational theories. This work is supported by the Ministry
of Science and Technology of China under Grant No.2017YFB0203300, National
Nature Science Foundation of China under Grants No.11988101 and No.11851301.
DFM thanks the Research Council of Norway for their support. Computations were
performed on resources provided by UNINETT Sigma2 – the National
Infrastructure for High Performance Computing and Data Storage in Norway.

## References

  * (1) A. Einstein, “The Foundation of the General Theory of Relativity,” Annalen Phys. 49, no.7, 769-822 (1916).
  * (2) B. L. Young, Front. Phys. (Beijing) 12, no.2, 121201 (2017).
  * (3) S. Weinberg, Rev. Mod. Phys. 61, 1-23 (1989).
  * (4) A. G. Riess et al. [Supernova Search Team], Astron. J. 116, 1009-1038 (1998).
  * (5) S. Perlmutter et al. [Supernova Cosmology Project], Astrophys. J. 517, 565-586 (1999).
  * (6) G. ’t Hooft and M. J. G. Veltman, Ann. Inst. H. Poincare Phys. Theor. A 20, 69-94 (1974).
  * (7) S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10, 401 (1974).
  * (8) S. Deser and P. van Nieuwenhuizen, Phys. Rev. D 10, 411 (1974).
  * (9) M. H. Goroff and A. Sagnotti, Phys. Lett. B 160, 81-86 (1985).
  * (10) M. H. Goroff and A. Sagnotti, Nucl. Phys. B 266, 709-736 (1986).
  * (11) A. E. M. van de Ven, Nucl. Phys. B 378, 309-366 (1992).
  * (12) C. Lanczos, Annals Math. 39, 842-850 (1938).
  * (13) D. Lovelock, J. Math. Phys. 12, 498-501 (1971).
  * (14) D. Lovelock, J. Math. Phys. 13, 874-876 (1972).
  * (15) B. Zumino, Phys. Rept. 137, 109 (1986).
  * (16) A. Mardones and J. Zanelli, Class. Quant. Grav. 8, 1545-1558 (1991).
  * (17) T. Torii and H. A. Shinkai, Phys. Rev. D 78, 084037 (2008).
  * (18) D. Glavan and C. Lin, Phys. Rev. Lett. 124, no.8, 081301 (2020).
  * (19) Z. Haghani, [arXiv:2005.01636 [gr-qc]].
  * (20) G. Narain and H. Q. Zhang, [arXiv:2005.05183 [gr-qc]].
  * (21) M. Gürses, T. Ç. Şişman and B. Tekin, Eur. Phys. J. C 80, no.7, 647 (2020).
  * (22) K. Aoki, M. A. Gorji and S. Mukohyama, [arXiv:2005.08428 [gr-qc]].
  * (23) T. Clifton, P. Carrilho, P. G. S. Fernandes and D. J. Mulryne, [arXiv:2006.15017 [gr-qc]].
  * (24) J. X. Feng, B. M. Gu and F. W. Shu, [arXiv:2006.16751 [gr-qc]].
  * (25) M. A. García-Aspeitia and A. Hernández-Almada, [arXiv:2007.06730 [astro-ph.CO]].
  * (26) N. Aghanim et al. [Planck], [arXiv:1807.06209 [astro-ph.CO]].
  * (27) Z. Zhai and Y. Wang, JCAP 1907, 005 (2019).
  * (28) S. Alam et al. [BOSS], Mon. Not. Roy. Astron. Soc. 470, no.3, 2617-2652 (2017).
  * (29) F. Beutler et al., Mon. Not. Roy. Astron. Soc. 416, 3017-3032 (2011).
  * (30) A. J. Ross et al., Mon. Not. Roy. Astron. Soc. 449, no.1, 835-847 (2015).
  * (31) A. G. Riess et al., Astrophys. J. 853, no.2, 126 (2018).
  * (32) M. Moresco et al., JCAP 05, 014 (2016).
  * (33) S. Basilakos and S. Nesseris, Phys. Rev. D 94, 123525 (2016).
  * (34) D. Foreman-Mackey, D. W. Hogg, D. Lang and J. Goodman, Publ. Astron. Soc. Pac. 125, 306 (2013).
  * (35) A. Lewis, arXiv:1910.13970 [astro-ph.IM].
  * (36) A. G. Riess, S. Casertano, W. Yuan, L. M. Macri and D. Scolnic, Astrophys. J. 876, 85 (2019).
  * (37) E. Di Valentino et al., [arXiv:2103.01183 [astro-ph.CO]].
  * (38) C. P. Ma and E. Bertschinger, Astrophys. J. 455, 7 (1995).
  * (39) A. Lewis, Phys. Rev. D 87, 103529 (2013).