Scholarly article on topic 'Observational constraints on unified dark matter including Hubble parameter data'

Observational constraints on unified dark matter including Hubble parameter data Academic research paper on "Physical sciences"

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{"Unified dark matter" / "Affine equation" / "Hubble parameter"}

Abstract of research paper on Physical sciences, author of scientific article — Kai Liao, Shuo Cao, Jun Wang, Xiaolong Gong, Zong-Hong Zhu

Abstract We constrain a unified dark matter (UDM) model from the latest observational data. This model assumes that the dark sector is degenerate. Dark energy and dark matter are the same component. It can be described by an affine equation of state P X = p 0 + α ρ X . Our data set contains the newly revised H ( z ) data, type Ia supernovae (SNe Ia) from Union2 set, baryonic acoustic oscillation (BAO) observation from the spectroscopic Sloan Digital Sky Survey (SDSS) data release 7 (DR7) galaxy sample, as well as the cosmic microwave background (CMB) observation from the 7-year Wilkinson Microwave Anisotropy Probe (WMAP7) results. By using the Markov Chain Monte Carlo (MCMC) method, we obtain the results in a flat universe: Ω Λ = 0.719 − 0.0305 + 0.0264 ( 1 σ ) − 0.0458 + 0.0380 ( 2 σ ) , α = 1.72 − 4.79 + 3.92 ( 1 σ ) − 7.30 + 5.47 ( 2 σ ) ( × 10 − 3 ) , Ω b h 2 = 0.0226 − 0.0011 + 0.0011 ( 1 σ ) − 0.0015 + 0.0016 ( 2 σ ) . Moreover, when considering a non-flat universe, Ω Λ = 0.722 − 0.0447 + 0.0362 ( 1 σ ) − 0.0634 + 0.0479 ( 2 σ ) , α = 0.242 − 0.775 + 0.787 ( 1 σ ) − 1.03 + 1.10 ( 2 σ ) ( × 10 − 2 ) , Ω b h 2 = 0.0227 − 0.0014 + 0.0015 ( 1 σ ) − 0.0018 + 0.0021 ( 2 σ ) , Ω k = − 0.194 − 1.85 + 2.02 ( 1 σ ) − 2.57 + 2.75 ( 2 σ ) ( × 10 − 2 ) . These give a more stringent results than before. We also give the results from other combinations of these data for comparison. The observational Hubble parameter data can give a more stringent constraint than SNe Ia. From the constraint results, we can see the parameters α and Ω k are very close to zero, which means a flat universe is strongly supported and the speed of sound of the dark sector seems to be zero.

Academic research paper on topic "Observational constraints on unified dark matter including Hubble parameter data"

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Physics Letters B

www.elsevier.com/locate/physletb

Observational constraints on unified dark matter including Hubble parameter data

Kai Liao, Shuo Cao, Jun Wang, Xiaolong Gong, Zong-Hong Zhu *

Department of Astronomy, Beijing Normal University, Beijing 100875, China

ARTICLE INFO

Article history:

Received 24 December 2011

Received in revised form 22 February 2012

Accepted 24 February 2012

Available online 2 March 2012

Editor: S. Dodelson

Keywords:

Unified dark matter Affine equation Hubble parameter

ABSTRACT

We constrain a unified dark matter (UDM) model from the latest observational data. This model assumes that the dark sector is degenerate. Dark energy and dark matter are the same component. It can be described by an affine equation of state PX = p0 + apX. Our data set contains the newly revised H(z) data, type Ia supernovae (SNe Ia) from Union2 set, baryonic acoustic oscillation (BAO) observation from the spectroscopic Sloan Digital Sky Survey (SDSS) data release 7 (DR7) galaxy sample, as well as the cosmic microwave background (CMB) observation from the 7-year Wilkinson Microwave Anisotropy Probe (WMAP7) results. By using the Markov Chain Monte Carlo (MCMC) method, we obtain the results in a flat universe: iA = 0.7 1 9+0 03o4(1a)+0 0450(2a), a = 1.72+4-79(1a)+5'40(2a)(x10-3), ibh2 = 0 .0226—0. 0011 (1^)+0 '0015(2^). Moreover, when considering a non-flat universe, iA = 0 . 7 22—0.0447(1+0.0644(2^), a = 0 .242—0.777(1^)—1.03(2ct)(x10—2), i2bh2 = 0.0227+0o014(1CT)—00018(2^), i = —0.194+182(1^)—2'5?(2^)(x 10-2). These give a more stringent results than before. We also give the results from other combinations of these data for comparison. The observational Hubble parameter data can give a more stringent constraint than SNe Ia. From the constraint results, we can see the parameters a and i are very close to zero, which means a flat universe is strongly supported and the speed of sound of the dark sector seems to be zero.

© 2012 Elsevier B.V. All rights reserved.

1. Introduction

Cosmic acceleration is one of the most striking discoveries in modern cosmology [1]. Many works have been done to explain this unexpected phenomenon. There are different mechanisms for the universe acceleration. The most popular idea is assuming a new component with negative pressure known as dark energy. Other mechanisms contain various modifications of gravitation theory [3] and violation of cosmological principle [2]. There are all kinds of dark energy models including quintessence [4], holographic dark energy [5], quintom [6], phantom [7], generalized Chaplygin gas model [8] and so on. Among these, the simplest and most successful model is ACDM model [9] while it has to face some theoretical problems. For example, the "coincidence" problem and the "fine-tuning" problem [10]. From this standard model, the flat universe consists of ~4% baryons, ~23% dark matter, ~73% dark energy [11]. However, we still know little about dark matter and dark energy [12]. All we can feel about them are their gravitational effects. Observationally, they are degenerate. Based on these, a unified dark matter model was proposed which assumes that dark matter and dark energy are from a same dark component [13].

Since the equation of state of the unified dark component is unknown, we assume that it has a constant speed of sound c2s = a which is equivalent to the affine form [14]. In order to break the degeneracies, different kinds of observational data are used to constrain cosmological model, among which SNe Ia, CMB, and BAO are usually used to constrain cosmological parameters [15]. Recently, other observations are included. For example, the gamma-ray burst [16], cluster gas mass fraction [17], lensing [18] and so on. But these processes need to integrate the Hubble parameter to obtain the distance scale. The more information about H(z) cannot be embodied. Therefore, it is meaningful to study the H(z) data directly. The expression of the Hubble parameter can be written in this form

H(z) = -

1 dz 1 + zdt '

Corresponding author.

E-mail address: zhuzh@bnu.edu.cn (Z.-H. Zhu).

which depends on the differential age as a function of redshift. Based on Jimenez et al. [20], Simon et al. [21] used the age of evolving galaxies and got nine H(z) data. Many works based on these nine data were done [22]. Recently, Stern et al. [25] revised these data at 11 redshifts from the differential ages of red-envelope galaxies. Moreover, Gaztanaga et al. [26] took the BAO scale as a standard ruler in the radial direction, obtained three more additional data.

0370-2693/$ - see front matter © 2012 Elsevier B.V. AH rights reserved. doi:10.1016/j.physletb.2012.02.079

In this Letter, we use the Markov Chain Monte Carlo (MCMC) method to give constraints on the unified dark matter model from the latest data including the observational Hubble parameter data. In Section 2, we give a brief introduction of the UDM model. In Section 3, we introduce the observational data set we use. The full parameter space using different combinations of data is exhibited in Section 4. Finally, we give a conclusion in Section 5.

2. Unified dark matter with constant speed of sound

In this section, we give a brief introduction to unified dark matter model. For details of this model, please refer to [13].

Considering the universe is described by the FriedmannRobertson-Walker metric and assuming a single UDM besides baryons and radiation, the Friedmann equation is expressed as

H (a) = H o E (a),

where H0 represents the Hubble constant today, and

E (a)2 = Qr (a) + Qb (a) + Qx (a) + Qk (a),

where Qi (a) = pi (a)/pc, pc is the critical density today. The subscripts r, b, k and X represent for radiation, baryons, curvature and UDM component respectively. Each component satisfies the energy-momentum conservation equation

Pi = -3H (Pi + Pi ).

For the radiation and baryons, the equations of state are 1/3 and 0, respectively. The conservation equation gives Qb(a) a a-3 and Qr(a) a a-4.

For the UDM, it is considered as an approximation to a barotropic fluid. Since the equation of state Px = PX (PX) is unknown, we assume a constant speed of sound

dPx/dpx - a,

which leads to the 2-parameter affine form Px - Po + apx.

Combined with the conservation equation, we can regard this component as an effective cosmological constant plus a generalized dark matter which has a constant equation of state a. The density evolves as

âx (a) = £2a + (âm )a-3(1+a),

where âm = âxo — âA stands for the generalized dark matter, and âA stands for the effective cosmological constant component, Pa = 0.

The Friedmann equation now can be expressed as E (a)2 = âra—4 + âba—3 + âA + (1 — âb — âr — âA)a—3(1+a)

+ âka

where Qr, Qb and represent dimensionless radiation, baryon matter and curvature today. QA and a are the parameters we introduced. We notice that when a = 0, the UDM model is equivalent to ACDM. In UDM model, we can equivalently think that the generalized dark matter may has a non-zero equation of state. Naturally, because the property of the effective cosmological constant component is the same as the one in ACDM, when the property of generalized dark matter is equivalent to the usual dark matter in ACDM, UDM model totally becomes ACDM.

The speed of sound characterizes the perturbations to the dark matter density and pressure which are related to the microphysics. Since we know litter about dark sector in our universe, the equation of state is introduced to describe the property of dark sector.

However, it cannot give us the information about the microphysics and other properties. The speed of sound can give information about the internal degrees of freedom. For example, quintessence has cs = 1. If we can confirm the sound speed that is below the speed of light, we can get further degrees beyond a canonical, minimally coupled scalar field. If the sound speed is small, the perturbations can be detected on a proper scale. These perturbations will change the gravitational potential, thus affecting the geodesic of photons. As a result, the cosmic microwave background will be changed through the Integrated Sachs-Wolfe (ISW) effect. In order to show the differences between UDM model and ACDM which corresponds to a = 0, we give the CMB temperature angular power spectrum in Fig. 7. Our calculations are based on CAMB [19].

3. Current observational data

Now, we introduce the data set using to constrain the UDM model.

3.1. Observational Hubble parameter data

The observational Hubble parameter as a function of redshift has been used to constrain cosmological parameters [23]. For the observational data, we choose 11 data from [25]. These data were obtained through the measurement of differential ages of red-envelope galaxies known as "differential age method". The aging of stars can be regarded as an indicator of the aging of the universe. The spectra of stars can be converted to the information of their ages, as the evolutions of stars are well known. Since we cannot observe stars one by one at cosmological scales, we take the spectra of galaxies which contain relatively uniform star population. The surveys including the Gemini Deep Deep Survey [24], VIMOS-VLT Deep Survey and SDSS [25] are related to obtaining data. We consider the red-envelope galaxies which are massive with fairly homogeneous star population. Moreover, the data can be obtained from the BAO scale as a standard ruler in the radial direction known as "Peak Method" [26]. Though this method, we have three additional data [26]: H(z = 0.24) = 79.69 ± 2.32, H(z = 0.34) = 83.8 ± 2.96, and H(z = 0.43) = 86.45 ± 3.27, which are independent. Plus the measurement of Hubble constant [27], we totally have 15 data now. The x2 value of the H (z) data can be expressed as

x22 = E

[ H (Zi ) — Hobs (zi )]2

where ai is the 1-a uncertainty of the observational H (z) data.

3.2. Type la supernovae

Since the SNe Ia fist revealed the acceleration of the universe, it has been a well-established method for probing dark energy.

Recently, the Supernova Cosmology Project (SCP) released the Union2 data set including 557 samples [28]. It covers a redshift region 0-1.4. These data have been widely used to investigate dark energy models [29]. The distance modules are given by

Mtheory(z) = 5log10[di (z)] + 1X0, (10)

where dL(z) is the luminosity distance. It is defined as

dL (z) = (1 + z)r(z), c

r(z) =

: sinn

Table 1

The best-fit values of parameters Qbh2, a, QA for UDM model in a flat universe with the 1-ct and 2-a uncertainties, as well as x^im' f°r the data sets H (z) + CMB + BAO, SNe + BAO + CMB, and H (z) + SNe + BAO + CMB, respectively.

Flat universe

H(z) + CMB + BAO

SNe + BAO + CMB

H(z) + SNe + CMB + BAO

a âbh2

0 7 1 7+0.0285(1a)+0.0404 (2a) 0.'1'—0.0338 (1° ) —0.0506(2° )

I.61—3.59 (1a)—5.95(2a)(x10—3)

n nnc+0.0012,^ \+0.0017

0.02 26—0 0010(1a)—0.0017 C2a)

II.592

0 719+°.°282(1a)+0.0412(2a) 0.' 19—0.0353 (1° )—0.0532 (2° )

1.81+8;10(1a)+11.65(2a)(x10—3 ) 0 0227+a0022(1a)+0.0035(2a)

0.022^0.0016(1° ^0.0022 (2° )

532.142

0 7 1 9+0.0264(1a)+0.0380(2a) °.' 19 —0.0305 ( 10 )—0.0458 (2° )

1.72+3.72(10 )+5.3o (2a)(x10—3)

n nnc+0.0011 \+0.0016

0.0226—0.0011 (1o)—0.0015 (20)

542.354

0.0210.0220.0230.024 x 10~3

0.021 0.0220.0230.024

Fig. 1. The 2-D regions and 1-D marginalized distribution with the 1-ct and 2-ct contours of parameters Qbh2, a, QA in a flat universe, for the data sets H(z) + CMB + BAO.

The marginalized nuisance parameter [30] for x2 is expressed

X 2 = A---+ ln

A = (^data — /heory)2 /a2,

B = J2 (Mdata — Mtheory) /a2, C = J21 /aI

ai is the 1-a uncertainty of the observational data.

Note that the expression

XSNe = A —

which coincides to Eq. (12) up to a constant [31], is used in this Letter. The constraint results are the same.

3.3. Cosmic microwave background

For CMB, the shift parameters R and la are important indicators which are related to the positions of the CMB acoustic peaks. These quantities are affected by the cosmic expansion history from the decoupling epoch to today. The shift parameter R is given by [32]

0.02 0.022 0.024 0.026

0.76 0.74 < 0.72

0.02 0.022 0.024 0.026

0.7 0.68 0.66

0.02 0.022 0.024 0.026 801

0.02 0.022 0.024 0.026

Fig. 2. The 2-D regions and 1-D marginalized distribution with the 1-a and 2-a contours of parameters ibh2, a, iA in a flat universe, for the data sets SNe + CMB + BAO.

R = 1/2 sinn

where im0 = im (hereafter), since we have defined the generalized dark matter. The redshift z* corresponding to the decoupling epoch of photons is given by [33]

z* = 1048 [1 + 0.00124(ibh2)—0738(1 + g1 (im0h2)g2)], (15) with

g1 = 0.0783 (ibh2)—0238 (1 + 39.5(ibh2)—0763)—1, g2 = 0.560(1 + 21.1 (ibh2)181)—1.

Moreover, the acoustic scale is related to the distance ratio. It is given by

ß-1/2sinn[ß,1/2 f0Z* d)]/H0

rs (z*)

where the comoving sound horizon at photo-decoupling epoch is expressed as

rs (z*) = H o

Cs(z)/E(z) dz.

From WMAP 7 measurement, the best-fit values of the data set are given by [35]

I5 CMB = R \z*.

302.09 ± 0.76 1.725 ± 0.018 1091.3 ± 0.91

where APcmb = Pcmb — I>cmb, and the corresponding inverse co-variance matrix is

CCMB =1 29.698

29.698

6825.270 -113.180 -1.333 -113.180 3.414

With these, we can calculate the x2 by [35]

xCMB = apCmbccimbapcmb.

3.4. Baryon acoustic oscillation

The baryon acoustic oscillations we use measure the distance-redshift relation at two redshifts. One is in the clustering of the combined 2dFGRS and SDSS main galaxy samples at z = 0.2, the other is in the clustering of the SDSS luminous red galaxies at z = 0.35. Calculated from these samples, the observed scale of the

0.021 0.022 0.023 0.024

x 10"'

Fig. 3. The 2-D regions and 1-D marginalized distribution with the 1-ct and 2-a contours of parameters Qbh2, a, QA in a flat universe, for the data sets H(z) + SNe + CMB + BAO.

BAO is analyzed to constrain the form of the distance scale. The distance scale of BAO is defined as [34]

Dv (z) = c

âkH (z)

The peak positions of the BAO depend on the ratio of Dv (z) to the sound horizon size at the drag epoch. The redshift Zd corresponding to the drag epoch when baryons were released from photos is given by

1291 (^m0h2)0251

r[(1 + b1 (âbh2)b2 )],

[1 + 0.659(Qm0h2)0 828]

where b1 = 0.313(Qm0h2)-0419[1 + 0.607(Qm0h2)0 674]-1 and b2 = 0.238(Qm0h2)0 223 [36]. In this Letter, we choose the measurement of the distance radio (dz) at z = 0.2 and z = 0.35 [37]. This quantity is given by

rs(zd)

Dv (z)'

where rs (zd) is the comoving sound horizon. From the SDSS data release 7 (DR7) galaxy sample, the best-fit values of the data set (d0.2, d0.35)[37]

1 matrix —

0.1905 ± 0.0061 0.1097 ± 0.0036

where the corresponding inverse covariance matrix is _i i 30124 -17227'

matrix '

-17227 86977

The x2 value of this BAO observation from SDSS DR7 can be calculated as

xBAO = APmatrixCmatrixAPmatrix. where APmatrix = Pmatrix _ Pmatrix.

4. Constraint results

The statistics we use to constrain parameters is the usual maximum likelihood method of x2 fitting with the Markov Chain Monte Carlo (MCMC) method. Our code is based on CosmoMCMC [38]. The corresponding convergence of the chains R-1 is set to be less than 0.003 which can guarantee the accuracy well. We combine H (z), SNe Ia, CMB and BAO data to give a global

Table 2

The best-fit values of parameters ibh2, a, iA, ik for UDM model in a non-flat universe with the 1-a and 2-a uncertainties, as well as x^im' f°r the data sets H(z) + CMB + BAO, SNe + BAO + CMB, and H (z) + SNe + BAO + CMB, respectively.

Non-flat universe

H (z) + CMB + BAO

SNe + BAO + CMB

H (z) + SNe + CMB + BAO

Qbh2 Qk

0 72l+°.°414(lff)+0.0549(2ff)

721 -0.0535 (1° )—0.0779 ( )

2.21-7;90(1ff)-10.9?(2ff)(x10-3 )

0;022/-0.0013 ( 10 )-0;00119(2u )

-0.208+1;88aa)-3;6() (2o)(x10-2)

0 734+°;°609 ( 10 ) +°;°799 (2o) 0./34-0;0815(1° )-0;124 (20)

0.662-02802(1o)+33;63 (2o)(x10-3) 0 0220+0;0039(1o)+0;0056 (2o)

0.02 20-0 0030(10 )-0;0035 (20 )

-1.41+6;33 (1o)+6'46(2o)(x10-2)

0 722+0;0362(1o)+0;0479(2o) 0./22-0;0447(1° )-0;634 (20)

2.42-7;85(1o)-10;29(2o)(x10-3) 0 0227+0;0015(1o)+0;0021 (2o)

0.02 2/ -0 0014 ( 10 )-0;0018 (20 )

-0.194+1.85 (1o)-2;55(2o)(x10-2)

11.578

531.870

542.309

0.022 0.024

1-0.01

0.022 0.024 -0.02 0 0.02

0.022 0.024

-0.02 0

0.022 0.024 Oh2

74 72 70 68

0.65 0.7 0.75

Fig. 4. The 2-D regions and 1-D marginalized distribution with the 1-a and 2-a contours of parameters ibh2, a, iA, ik in a non-flat universe, for the data sets H(z) + CMB + BAO.

fitting on determining the parameters of UDM model. The total x2 is given by

2 2 2 2 2 \J 2 - V I V I V I V

Atot = AH + ASNe + ACMB + ABAO;

First, we consider a flat universe (see Table 1). In order to embody the H(z) data, we choose three combinations of data sets and show the 1-D probability distribution of each parameter (ibh2, iA, a) and 2-D plots for parameters between each other. Fitting

results from the joint data of H(z) + CMB + BAO are given in Fig. 1, the best-value for a is

a = 1.61 -4.98(1 o)-5.92 (2o) (x10-3).

For comparison, we give the results from SNe + CMB + BAO in Fig. 2. The best-fit value is

= 1.81-8;15(1a)-11.99(2a)( x!0-3).

Fig. 5. The 2-D regions and 1-D marginalized distribution with the 1-ct and 2-a contours of parameters Qbh2, a, Qa, Qk in a non-flat universe, for the data sets SNe + CMB + BAO.

We can see the observational Hubble parameter data gives a more stringent constraint than SNe Ia in this model. In Fig. 3, the total combinations of them result in

a = 1.72-3;99(1ct)-5.30(2ct)(x10-3).

From the result, the parameter a which stands for the speed of sound of the unified dark sector seems to be vanishing or slightly larger than 0. So ACDM model is strongly supported. We further consider a non-flat universe with curvature (see Table 2). For H(z) + CMB + BAO in Fig. 4, the results are

a = 2.21-7;68(1a)+";96(2a)(x10-3),

Qk = -0.208-2.27 (1ct)+3.6o(2ct)(x10-2).

Using SNe + CMB + BAO in Fig. 5,

a = 0.662-02802 (1 a)+33 63 (2a) (x 10-3), Qk = -1.41 +6.H (1 a )+|4|(2ct) ( x 10-2).

These results are consist with the situation in a flat universe. The H (z) data also gives a much more stringent constraint on Qk. The total joint data in Fig. 6 give

a = 2.42_7:87(1ct)_|j:29(2CT)(x10-3),

-7.75 v

Qk = -0 . 194-1 :s5(1oO_2 . 57(

^°.77(2ct)( x10-2) ,

which suggest both the speed of sound and the curvature are very little. Our results are much more stringent than previous work [14]. Our results seem to be consistent with a = 0 using the data sets we select. Does that mean there is no need to introduce a new parameter? We think although the UDM model is contrived, the sound speed of the unified dark matter is related to the micro-physics of the dark component while equation of state does not. The sound speed leads to the inhomogeneities of dark matter, thus affecting CMB and matter power spectra. With the development of technique, more information in CMB, matter power spectra and so on may distinguish UDM model from ACDM. Moreover, we assume a constant sound speed, the whole effective value is very small, however, if when we consider the evolving speed of sound, the results may be changed.

0.022 0.024

-0.02 0 0.02

Fig. 6. The 2-D regions and 1-D marginalized distribution with the 1-o and 2-o contours of parameters Qbh2, a, Qa , Q in a non-flat universe, for the data sets H (z) + SNe + CMB + BAO.

5. Conclusions

In this Letter, we constrain the unified dark matter with constant speed of sound from the latest data: the newly revised 15 Hubble data, type Ia supernovae (SNe Ia) from Union2 set, bary-onic acoustic oscillation (BAO) observation from the spectroscopic Sloan Digital Sky Survey (SDSS) data release 7 (DR7) galaxy sample, as well as the cosmic microwave background (CMB) observation from the 7-year Wilkinson Microwave Anisotropy Probe (WMAP7) results. We consider two cases. In a flat universe, the constraint results are

o n "71 q+0.0264/-] ^+0.0380/-,^ qa = °.719—0.0305(1a)—°.°458(2a),

a = 1.72+3.92(1a)—7.40(2a)(x10—3),

ibh2 = 0.0226+0;00H(1a)—0.0015(2a).

In a non-flat universe, the constraint results are

QA = O.722+0.0i577(1a)+0;05з79(2a),

a = 0.242+0.787 (1a)+1.J3(2aH x10—2),

2 +0.0015 +0.0021 Qbh = 0.0227—00014(1ct)—00018(2CT),

Qk = -0.194+2.82(1ст)+2;75(2ст)(x10—2).

These constraint results are much stringent than previous work [14], which suggest both the speed of sound of the UDM and the curvature are very little and a flat ACDM model is still a good choice. In order to know the effects of the observational Hubble data, we also consider the combinations H(z) + CMB + BAO and SNe + BAO + CMB. From the results, the H(z) data gives a better constraint especially for a, Qk and Qbh2 than SNe Ia in this model. We think the H (z) data will play an important role in cosmology in the future [39].

Acknowledgements

This work was supported by the National Natural Science Foundation of China under the Distinguished Young Scholar Grant 10825313, the Ministry of Science and Technology National Basic

6000 5000 4000

g- 3000 +

~ 2000 1000 0

10° 101 102 103 104

Fig. 7. The theoretical CMB temperature angular power spectrum of UDM model. a = 0.01 (red), 0 (black) and -0.01 (green). Qbh2 = 0.022, QA = 0.71, H0 = 70. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this Letter.)

Science Program (Project 973) under Grant No. 2012CB821804, the Fundamental Research Funds for the Central Universities and Scientific Research Foundation of Beijing Normal University. We are grateful to Lixin Xu for introducing the MCMC method.

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