# CPL effective dark energy from the backreaction effect

Yan-Hong Yao<EMAIL_ADDRESS>Xin-He Meng<EMAIL_ADDRESS>Department
of Physics, Nankai University, Tianjin 300071, China

###### Abstract

In this paper, we interpret the dark energy as an effect caused by small scale
inhomogeneities of the universe with the use of the spatial averaged approach
of Buchertbuchert2000average ; buchert2001average . The model considered here
adopts the Chevallier-Polarski-Linder(CPL) parameterizations of the equation
of state of the effective perfect fluid from the backreaction effect. Thanks
to the effective geometry introduced by Larena et. al.larena2009testing  in
their previous work, we confront such backreaction model with latest type Ia
supernova and Hubble parameter observations, coming out with results that
reveal the difference between the Friedmann-Lemaître-Robertson-Walker model
and backreaction model.

Keywords:backreaction; CPL parameterization

##  I Introduction

According to recent observations of type Ia supernovae, the universe is in a
state of accelerated expansion riess1998 ; perlmutter1999measurements , and
the simplest scenario to account for these observations is the so called
Lambda cold dark matter $(\Lambda CDM)$ model, which is also named standard
cosmology which inclues the simplest dark energy model. However, because of
the huge discrepancy between the theoretical expected value of the
cosmological constant and the observed one, other alternative scenarios have
been proposed, including scalar field models such as
quintessenceCaldwell1998Cosmological , phantomCaldwell1999A ,
dilatonicPiazza2004Dilatonic , tachyonPadmanabhan2002Accelerated and
quintomBo2006Oscillating  etc. and modified gravity models such as braneworlds
maartens2010brane , scalar-tensor gravity esposito2001scalar , higher-order
gravitational theories capozziello2005reconciling ; das2006curvature . Whereas
the so called fitting problem that how well is our universe described by a
standard Friedmann-Lemaître-Robertson-Walker(FLRW) model is not solved yet,
recently a third alternative has been considered to explain the dark energy
phenomenon as backreaction effect(a Large scale effect caused by small scale
inhomogeneities of the universe rasanen2004dark ; kolb2006cosmic .).

In order to consider backreaction effect, it is necessary to answer a
longstanding question that how to average a general inhomogeneous spacetime.
To date the macroscopic gravity (MG) approach zalaletdinov1992averaging ;
zalaletdinov1997averaging ; zalaletdinov1993towards ; mars1997space  is the
only approach that gives a prescription for the correlation functions which
emerge in an averaging of the Einstein’s field equations, however, so far it
required a number of assumptions about the correlation functions, which make
the theory less convictive. Therefore, in this paper we adopt another averaged
approach put forward by Buchert buchert2000average ; buchert2001average , in
despite of its foliation dependent nature, such approach is quite simple and
hence becomes the most well studied averaged approach. Since the final
averaged equations in such approach do not form a closed set, one needs to
make some assumptions about the backreaction term appeared in the averaged
equations in order to track the averaged evolution of the universe with
certain initial condition. In buchert2006correspondence , Buchert proposes a
backreaction model by taking the assumption that the backreaction term ${\cal
Q}_{\cal D}$ and the averaged spatial Ricci scalar $\left\langle{\cal
R}\right\rangle_{\cal D}$ obey the scaling laws of the volume scale factor
$a_{\cal D}$. Although simple, such model is able to describe a accelerated
universe without introducing strange negative pressure matter. Of course,
there is no conclusive evidences that scaling laws assumption is absolutely
correct, thus other supposition about the behaviors of the backreaction term
can be make. In this paper, we propose a new backreaction model by assuming
that the equation of state of the effective perfect fluid follows the CPL form
and using a relation appeared in QING2013CONSTRAINTS  to reduce a parameter
for simplicity reason. after then, we apply the template metric presented in
buchert2006correspondence  to confront the new model with observations and
perform a likelihood analysis. We use the natural unit c=1 through the paper.

The paper is organized as follows. In Section II, the spatial averaged
approach of Buchert is demonstrated with presentation of the averaged
equations for the volume scale factor $a_{\cal D}$. In Section III, we
introduce the template metric, which is a necessary tool to test the
theoretical preditions with observations, and computation of observables. In
Section IV, we apply a likelihood analysis of the backreaction models by
confronting it with latest type Ia supernova and Hubble parameter
observations. After analysis of the results in Section IV, we summarize
ourresults in the last section.

##  II The backreaction models

In buchert2000average , Buchert considers a universe filled with irrotational
dust with energy density $\varrho$. By foliating space-time with the use of
Arnowitt-Deser-Misner(ADM) procedure and defining an averaging operator that
acts on any spatial scalar $\Psi$ function as

$\left\langle\Psi\right\rangle_{\cal D}:=\frac{1}{V_{{\cal D}}}\int_{{\cal
D}}\Psi Jd^{3}X,$ (1)

where $V_{{\cal D}}:=\int_{{\cal D}}Jd^{3}X$ is the domain’s volume, Buchert
obtains two averaged equations here we need, the averaged Raychaudhuri
equation

$3\frac{{\ddot{a}}_{\cal D}}{a_{\cal D}}+4\pi
G\left\langle\varrho\right\rangle_{\cal D}={{\cal Q}}_{\cal D},$ (2)

and the averaged Hamiltonian constraint

$3\left(\frac{{\dot{a}}_{\cal D}}{a_{\cal D}}\right)^{2}-8\pi
G\left\langle\varrho\right\rangle_{\cal D}=-\frac{\left\langle{\cal
R}\right\rangle_{\cal D}+{{\cal Q}}_{\cal D}}{2}.$ (3)

In these two equations, $a_{\cal D}(t)=\left(\frac{V_{\cal D}(t)}{V_{{{\cal
D}_{\rm\bf 0}}}}\right)^{1/3}$ is the volume scale factor, where $V_{{{\cal
D}_{\rm\bf 0}}}=|{{{\cal D}_{\rm\bf 0}}}|$ denotes the present value of the
volume, and ${\cal Q}_{\cal D}$, $\left\langle{\cal R}\right\rangle_{\cal D}$
represent the backreaction term and the averaged spatial Ricci scalar
respectively, which are related by the following integrability condition

$\frac{1}{a_{\cal D}^{6}}\partial_{t}\left(\,{{\cal Q}}_{\cal D}\,a_{\cal
D}^{6}\,\right)\;+\;\frac{1}{a_{\cal
D}^{2}}\;\partial_{t}\left(\,\left\langle{\cal R}\right\rangle_{\cal D}a_{\cal
D}^{2}\,\right)\,=0\;.$ (4)

Now we can define the effective prefect fluid by

$\displaystyle\varrho_{eff}^{{\cal D}}:$ $\displaystyle=$
$\displaystyle-\frac{1}{16\pi G}({{\cal Q}}_{\cal D}+\left\langle{\cal
R}\right\rangle_{\cal D})$ (5) $\displaystyle p_{eff}^{{\cal D}}:$
$\displaystyle=$ $\displaystyle-\frac{1}{16\pi G}({{\cal Q}}_{\cal
D}-\frac{\left\langle{\cal R}\right\rangle_{\cal D}}{3})$ (6)

the averaged Raychaudhuri equation and the averaged Hamiltonian constraint
then can formally be recast into standard Friedmann equations for a total
perfect fluid energy momentum tensor

$3\frac{{\ddot{a}}_{\cal D}}{a_{\cal D}}+4\pi
G(\left\langle\varrho\right\rangle_{\cal D}+\varrho_{eff}^{{\cal
D}}+3p_{eff}^{{\cal D}})=0$ (7) $3\left(\frac{{\dot{a}}_{\cal D}}{a_{\cal
D}}\right)^{2}=8\pi G(\left\langle\varrho\right\rangle_{\cal
D}+\varrho_{eff}^{{\cal D}})$ (8)

given the effective energy density and pressure, the effective equation of
state reads

$w_{eff}^{{\cal D}}:=\frac{p_{eff}^{{\cal D}}}{\varrho_{eff}^{{\cal
D}}}=\frac{{{\cal Q}}_{\cal D}-\frac{\left\langle{\cal R}\right\rangle_{\cal
D}}{3}}{{{\cal Q}}_{\cal D}+\left\langle{\cal R}\right\rangle_{\cal D}}$ (9)

One can then obtain a specific backreaction model with an extra ansatz about
the form of $w_{eff}^{{\cal D}}$, for example, in this paper, we assume that
effective equation of state follows the Chevallier-Polarski-Linder(CPL)form,
i.e.

$w_{eff}^{{\cal D}}=w_{0}^{{\cal D}}+w_{a}^{{\cal D}}(1-a_{\cal D})$ (10)

so the Eq. (8) can be rewrite as

$H_{{\cal D}}^{2}=H_{{{\cal D}_{\rm\bf 0}}}^{2}[\Omega_{m}^{{{\cal D}_{\rm\bf
0}}}a_{{\cal D}}^{-3}+(1-\Omega_{m}^{{{\cal D}_{\rm\bf 0}}})a_{{\cal
D}}^{-3(1+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}e^{-3w_{a}^{{\cal D}}(1-a_{{\cal
D}})}]$ (11)

here $H_{\cal D}:={\dot{a}}_{\cal D}/a_{\cal D}$ denotes the volume Hubble
parameter and $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}:=\frac{8\pi G}{3H_{{{\cal
D}_{\rm\bf 0}}}^{2}}\left\langle\varrho\right\rangle_{{\cal D}_{\rm\bf 0}}$.
Combine the above equations, we have the following formulas for the
backreaction term and the averaged spatial Ricci scalar

${\cal Q}_{\cal D}=-\frac{9}{2}(1-\Omega_{m}^{{{\cal D}_{\rm\bf 0}}})H_{{{\cal
D}_{\rm\bf 0}}}^{2}e^{-3w_{a}^{{\cal D}}(1-a_{{\cal
D}})}[(\frac{1}{3}+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})a_{{\cal
D}}^{-3(1+w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}-w_{a}^{{\cal D}}a_{{\cal
D}}^{-2-3(w_{0}^{{\cal D}}+w_{a}^{{\cal D}})}]$ (12) $\left\langle{\cal
R}\right\rangle_{\cal D}=-\frac{9}{2}(1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})H_{{{\cal D}_{\rm\bf 0}}}^{2}e^{-3w_{a}^{{\cal D}}(1-a_{{\cal
D}})}[(1-w_{0}^{{\cal D}}-w_{a}^{{\cal D}})a_{{\cal D}}^{-3(1+w_{0}^{{\cal
D}}+w_{a}^{{\cal D}})}+w_{a}^{{\cal D}}a_{{\cal D}}^{-2-3(w_{0}^{{\cal
D}}+w_{a}^{{\cal D}})}]$ (13)

In QING2013CONSTRAINTS , Qing Gao and Yungui Gongand show that quinteesence
dark energy model can be approximated as the CPL parametrization at low
redshift with a degeneracy relationship, here we use this relationship to
reduce a parameter in our backreaction model,i.e. we assume

$w_{a}^{{\cal D}}=6(1+w_{0}^{{\cal D}})\frac{(1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})^{-\frac{1}{2}}-(1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})^{\frac{1}{2}}-((1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})^{-1}-1)tanh^{-1}((1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})^{\frac{1}{2}})}{(1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})^{-\frac{1}{2}}-((1-\Omega_{m}^{{{\cal D}_{\rm\bf
0}}})^{-1}-1)tanh^{-1}((1-\Omega_{m}^{{{\cal D}_{\rm\bf 0}}})^{\frac{1}{2}})}$
(14)

so there are only two parameters($\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$ and
$w_{0}^{{\cal D}}$ )remain in the model we propose.

##  III Effective geometry

###  III.1 The template metric

In larena2009testing , a template metric was proposed by Larena et. al. as
follows,

${}^{4}{\bf g}^{\cal D}=-dt^{2}+L_{H_{D_{0}}}^{2}\,a_{D}^{2}\gamma^{\cal
D}_{ij}\,dX^{i}\otimes dX^{j}\;\;,$ (15)

where $L_{{H_{\rm\bf 0}}}=1/H_{{{\cal D}_{\rm\bf 0}}}$ is the present size of
the horizon introduced so that the coordinate distance is dimensionless, and
the domain-dependent effective three-metric reads:

$\gamma^{\cal D}_{ij}\,dX^{i}\otimes dX^{j}=\frac{dr^{2}}{1-\kappa_{{\cal
D}}(t)r^{2}}+r^{2}d\Omega^{2}$ (16)

with $d\Omega^{2}=d\theta^{2}+\sin^{2}(\theta)d\phi^{2}$, this effective
three-template metric is identical to the spatial part of a FLRW metric at any
given time, but its scalar curvature $\kappa_{{\cal D}}$ can vary from time to
time. As was pointed out by Larena et. al., $\kappa_{{\cal D}}$ cannot be
arbitrary, more precisely, they argue that it should be related to the true
averaged scalar curvature $\left\langle{\cal R}\right\rangle_{\cal D}$ in the
way that

$\left\langle{\cal R}\right\rangle_{\cal D}=\frac{\kappa_{{\cal
D}}(t)|\left\langle{\cal R}\right\rangle_{{\cal D}_{\rm\bf 0}}|a_{{{\cal
D}_{\rm\bf 0}}}^{2}}{a_{{\cal D}}^{2}(t)}$ (17)

###  III.2 Computation of observables

The computation of effective distances along the light cone defined by the
template metric is very different from that of distances in FLRW models.
Firstly, let us introduce an effective redshift $z_{{\cal D}}$ defined by

$1+z_{{\cal D}}:=\frac{(g_{ab}k^{a}u^{b})_{S}}{(g_{ab}k^{a}u^{b})_{O}}\mbox{
,}$ (18)

where the letters O and S denote the evaluation of the quantities at the
observer and at the source respectively, $g_{ab}$ in this expression
represents the template metric, while $u^{a}$ is the four-velocity of the dust
which satisfies $u^{a}u_{a}=-1$, $k^{a}$ the wave vector of a light ray
travelling from the source S towards the observer O with the restrictions
$k^{a}k_{a}=0$. Then, by normalizing this wave vector such that
$(k^{a}u_{a})_{O}=-1$ and introducing the scaled vector $\hat{k}^{a}=a_{{\cal
D}}^{2}k^{a}$, we have the following equation:

$1+z_{{\cal D}}=(a_{{\cal D}}^{-1}\hat{k}^{0})_{S}\mbox{ ,}$ (19)

with $\hat{k}^{0}$ obeying the null geodesics equation
$k^{a}\nabla_{a}k^{b}=0$ which leads to

$\frac{1}{\hat{k}^{0}}\frac{d\hat{k}^{0}}{da_{{\cal D}}}=-\frac{r^{2}(a_{{\cal
D}})}{2(1-\kappa_{{\cal D}}(a_{{\cal D}})r^{2}(a_{{\cal
D}}))}\frac{d\kappa_{{\cal D}}(a_{{\cal D}})}{da_{{\cal D}}}\mbox{ .}$ (20)

As usual, the coordinate distance can be derived from the equation of radial
null geodesics:

$\frac{dr}{da_{{\cal D}}}=-\frac{H_{{{\cal D}_{\rm\bf 0}}}}{a_{{\cal
D}}^{2}H_{{\cal D}}(a_{{\cal D}})}\sqrt{1-\kappa_{{\cal D}}(a_{{\cal
D}})r^{2}}$ (21)

Solving these two equations with the initial condition
$\hat{k}^{0}(1)=1,r(1)=0$ and then plugging $\hat{k}^{0}(a_{\cal D})$ into Eq.
(19), one finds the relation between the redshift and the scale factor. With
these results, we can determine the volume Hubble parameter $H_{{\cal
D}}(z_{{\cal D}})$ and the luminosity distance $d_{L}(z_{{\cal D}})$ of the
sources defined by the following formula

$\displaystyle d_{L}(z_{{\cal D}})$ $\displaystyle=$
$\displaystyle\frac{1}{H_{{{\cal D}_{\rm\bf 0}}}}(1+z_{{\cal D}})^{2}a_{{\cal
D}}(z_{{\cal D}})r(z_{{\cal D}}).$ (22)

Having computed these two observables , it is then possible to compare the
backreaction model predictions with type Ia supernova and Hubble parameter
observations.

##  IV Constraints from supernovae data and OHD

In this section, we perform a likelihood analysis on the parameters of the
backreaction model mentioned above with the combination of datasets from type
Ia supernova and Hubble parameter observations. The constraints placed by the
recently released Union2.1Suzuki2012The  compilation on $(\Omega_{m}^{{{\cal
D}_{\rm\bf 0}}},w_{0}^{{\cal D}})$ can be obtained by minimizing

$\chi_{SNIa}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal
D}})=R-\frac{S^{2}}{T}$ (23)

here $R$, $S$ and $T$ are defined as

$\displaystyle R$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{580}\frac{\left(5\log_{10}\left[H_{{{\cal D}_{\rm\bf
0}}}d_{L}({z_{{\cal D}}}_{i})\right]-\mu_{obs}({z_{{\cal
D}}}_{i})\right)^{2}}{\sigma_{\mu}^{2}({z_{{\cal D}}}_{i})},$ (24)
$\displaystyle S$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{580}\frac{\left(5\log_{10}\left[H_{{{\cal D}_{\rm\bf
0}}}d_{L}({z_{{\cal D}}}_{i})\right]-\mu_{obs}({z_{{\cal
D}}}_{i})\right)}{\sigma_{\mu}^{2}({z_{{\cal D}}}_{i})},$ (25) $\displaystyle
T$ $\displaystyle=$
$\displaystyle\sum_{i=0}^{580}\frac{1}{\sigma_{\mu}^{2}({z_{{\cal D}}}_{i})}.$
(26)

where $\mu_{obs}$ represents the observed distance modulus and $\sigma_{\mu}$
denotes its statistical uncertainty.

For the observed Hubble parameter dataset in Table 1, the best-fit values of
the parameters $(H_{{{\cal D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf
0}}},w_{0}^{{\cal D}})$ can be determined by a likelihood analysis based on
the calculation of

$\chi^{2}_{H}(H_{{{\cal D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf
0}}},w_{0}^{{\cal D}})=\sum_{i=0}^{30}\frac{(H_{{\cal D}}({z_{{\cal
D}}}_{i};H_{{{\cal D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf
0}}},w_{0}^{{\cal D}})-H_{obs}({z_{{\cal
D}}}_{i}))^{2}}{\sigma_{H}^{2}({z_{{\cal D}}}_{i})}.$ (27)

As Ma et. al.ma2011power  stated, the marginalized probability density
function determined by integrating $\rm e^{-\frac{\chi^{2}_{H}(H_{{{\cal
D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}})}{2}}$
over $H_{{{\cal D}_{\rm\bf 0}}}$ from $x$ to $y$ with a Gaussian prior reads

${\rm e^{-\frac{\chi^{2}_{H}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal
D}})}{2}}}=\frac{1}{\sqrt{A}}[erf(\frac{B}{\sqrt{A}})+1]e^{\frac{B^{2}}{A}}$
(28)

where

$A=\frac{1}{2\sigma_{H}^{2}}\sum_{i=0}^{30}\frac{H_{{\cal D}}^{2}({z_{{\cal
D}}}_{i};H_{{{\cal D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf
0}}},w_{0}^{{\cal D}})}{2H_{{{\cal D}_{\rm\bf 0}}}^{2}\sigma_{H}^{2}({z_{{\cal
D}}}_{i})},B=\frac{\mu_{H}}{2\sigma_{H}^{2}}\sum_{i=0}^{30}\frac{H_{{\cal
D}}({z_{{\cal D}}}_{i};H_{{{\cal D}_{\rm\bf 0}}},\Omega_{m}^{{{\cal D}_{\rm\bf
0}}},w_{0}^{{\cal D}})H_{obs}({z_{{\cal D}}}_{i})}{2H_{{{\cal D}_{\rm\bf
0}}}\sigma_{H}^{2}({z_{{\cal D}}}_{i})},$ (29)

Finally, the total $\chi^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal
D}})$ for the combined observational dataset are given by
$\chi^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal
D}})=\chi_{SNIa}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal
D}})+\chi_{H}^{2}(\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}})$.

The fitting results attained from analyzing $\chi^{2}(\Omega_{m}^{{{\cal
D}_{\rm\bf 0}}},w_{0}^{{\cal D}})$ by using functions Findminimum and
ContourPlot in mathematica are presented in Fig.1, Table 2. First of all,
comparing with 0.29 111This best-fit value is obtained from a similar
likelihood analysis which uses the same data sets and assumes the same prior
for Hubble parameter. in the standard cosmology model($\Lambda CDM$ model),
the best-fit value of $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$ in our model is
smaller, this is because the template metric we use to confront the model with
observations is not the same form as the FLRW metric, i.e. we assume the
universe is not homogeneous-isotropic at all scale but at large-scale, at
smaller scales, the universe feature strong inhomogeneities and anisotropies,
and such inhomogeneities and anisotropies influence the propagation of light
hence the relation between volume scale factor and effective redshift, leading
to a lower values of $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$ for the models to be
compatible with data. Also, different from the standard cosmology model, the
model we propose predicts the existence of effective dynamical dark energy,
whose effective equation of state is plotted in Fig.2, which suggests that the
density of the effective dynamical dark energy decreases in the distant past
and becomes to increase in time when $a_{{\cal D}}$ approach about 0.15, this
increasing behavior maybe the consequence of the formation of large-scale
structure. Although both the smaller $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$ and
late-time increasing density make the volume Hubble parameter in the our model
smaller than that in the standard cosmology model at the same volume scale
factor, which is shown in the left picture of Fig.3, at the same effective
redshift, However, the the situation is the opposite, as is shown in the right
picture of Fig.3, this is because in these two different models, the relations
between volume scale factor and effective redshift are also different, it is
easy to realize that by having a look on the Fig.4.

$z$ | $H(z)$ | Ref.
---|---|---
$0.0708$ | $69.0\pm 19.68$ | Zhang et al. (2014)-Zhang2014
$0.09$ | $69.0\pm 12.0$ | Jimenez et al. (2003)-Jimenez2003
$0.12$ | $68.6\pm 26.2$ | Zhang et al. (2014)-Zhang2014
$0.17$ | $83.0\pm 8.0$ | Simon et al. (2005)-Simon2005
$0.179$ | $75.0\pm 4.0$ | Moresco et al. (2012)-Moresco2012
$0.199$ | $75.0\pm 5.0$ | Moresco et al. (2012)-Moresco2012
$0.20$ | $72.9\pm 29.6$ | Zhang et al. (2014)-Zhang2014
$0.27$ | $77.0\pm 14.0$ | Simon et al. (2005)-Simon2005
$0.28$ | $88.8\pm 36.6$ | Zhang et al. (2014)-Zhang2014
$0.352$ | $83.0\pm 14.0$ | Moresco et al. (2012)-Moresco2012
$0.3802$ | $83.0\pm 13.5$ | Moresco et al. (2016)-Moresco2016
$0.4$ | $95\pm 17.0$ | Simon et al. (2005)-Simon2005
$0.4004$ | $77.0\pm 10.2$ | Moresco et al. (2016)-Moresco2016
$0.4247$ | $87.1\pm 11.2$ | Moresco et al. (2016)-Moresco2016
$0.4497$ | $92.8\pm 12.9$ | Moresco et al. (2016)-Moresco2016
$0.4783$ | $80.9\pm 9.0$ | Moresco et al. (2016)-Moresco2016
$0.48$ | $97.0\pm 62.0$ | Stern et al. (2010)-Stern2010
$0.593$ | $104.0\pm 13.0$ | Moresco et al. (2012)-Moresco2012
$0.68$ | $92.0\pm 8.0$ | Moresco et al. (2012)-Moresco2012
$0.875$ | $125.0\pm 17.0$ | Moresco et al. (2012)-Moresco2012
$0.88$ | $90.0\pm 40.0$ | Stern et al. (2010)-Stern2010
$0.9$ | $117.0\pm 23.0$ | Simon et al. (2005)-Simon2005
$1.037$ | $154.0\pm 20.0$ | Moresco et al. (2012)-Moresco2012
$1.3$ | $168.0\pm 17.0$ | Simon et al. (2005)-Simon2005
$1.363$ | $160.0\pm 33.6$ | Moresco (2015)-Moresco2015
$1.43$ | $177.0\pm 18.0$ | Simon et al. (2005)-Simon2005
$1.53$ | $140.0\pm 14.0$ | Simon et al. (2005)-Simon2005
$1.75$ | $202.0\pm 40.0$ | Simon et al. (2005)-Simon2005
$1.965$ | $186.5\pm 50.4$ | Moresco (2015)-Moresco2015
Table 1:  The current available OHD dataset.

Figure 1: The best-fit point and $1\sigma$, $2\sigma$, $3\sigma$ confidence regions of the parameters $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}}$ for this backreaction model, along with their own probability density function. The prior for $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}\varpropto$ H(0.4-$\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$)H($\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$-0.1), The prior for $w_{0}^{{\cal D}}\varpropto$ H(-0.9-$w_{0}^{{\cal D}}$)H($w_{0}^{{\cal D}}$-(-1.3)), H(x)is the step function.  $\Omega_{m}^{{{\cal D}_{\rm\bf 0}}}$ |  | $0.23_{-0.04}^{+0.04}$
---|---|---
$w_{0}^{{\cal D}}$ |  | $-1.07_{-0.04}^{+0.03}$
$\chi^{2}_{min}$ |  | $-1345.84$
Table 2: Value of the $\chi^{2}_{min}$ and the fitting results of the
parameters ($\Omega_{m}^{{{\cal D}_{\rm\bf 0}}},w_{0}^{{\cal D}}$).

Figure 2: The equation of state of the effective perfect fluid on the Y-axis
and volume scale factor on the X-axis.

Figure 3: Volume Hubble parameter with respect to volume scale factor on the
left picture and with respect to effective redshift on the right picture. The
blue line corresponding to our model and the orange line corresponding to the
standard cosmology model.

Figure 4: Relation between volume scale factor and effective redshift in the
standard cosmology model and our model. The blue line corresponding to our
model and the orange line corresponding to the standard cosmology model.

##  V conclusion

In this paper, we follow the third way to handle the dark energy problem by
proposing a backreaction model in which the effective dynamical dark energy’s
equation of state having the CPL form. By using a formula appeared in
QING2013CONSTRAINTS , we reduce a model parameter. Equipping with the template
metric, we confront the model with latest type Ia supernova and Hubble
parameter observations, and we find out that the small-scale inhomogeneities
and anisotropies influence the evolution of the universe and the propagation
of light, leading to the reduction of the value of $\Omega_{m}^{{{\cal
D}_{\rm\bf 0}}}$, what’s more, observational results show that the effective
dark energy decreases in the distant past and becomes to increase in time when
$a_{{\cal D}}$ approach about 0.15. Both of these features indicate that the
volume Hubble parameter in the our model smaller than that in the standard
cosmology model at the same volume scale factor, although our model predicts a
larger observational expanding rate.

## Acknowledgments

The paper is partially supported by the Natural Science Foundation of China.

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