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Physics Letters B

www.elsevier.com/locate/physletb

Cardassian universe constrained by latest observations

Chao-Jun Feng *, Xin-Zhou Li

Shanghai United Center for Astrophysics (SUCA), Shanghai Normal University, 100 Guilin Road, Shanghai 200234, PR China

ARTICLE INFO

Article history:

Received 23 April 2010

Received in revised form 28 June 2010

Accepted 14 July 2010

Available online 23 July 2010

Editor: A. Ringwald

Keywords:

Cosmology

Viscosity

Cardassian universe

ABSTRACT

Several Cardassian universe models including the original, modified polytropic and exponential Cardassian models are constrained by the latest Constitution Type Ia supernova data, the position of the first acoustic peak of CMB from the five years WMAP data and the size of baryonic acoustic oscillation peak from the SDSS data. Both the spatial flat and curved universes are studied, and we also take into account the possible bulk viscosity of the matter fluid in the flat universe case.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

Nowadays, there are many dark energy models and modified gravity theories proposed to explain the current accelerating expansion of the universe, which has been confirmed by the observations like Type Ia supernovae (SNe Ia), CMB and SDSS, etc. The dark energy models assume the existence of an energy component with negative pressure in the universe, and it dominates and accelerates the universe at late times. The cosmological constant seems the best candidate of dark energy, but it suffers the fine tuning problem and coincidence problem, and it may even have the age problem [1]. To alleviate these problems, many dynamic dark energy models were proposed. However, people still do not know what is dark energy.

Since the Einstein general gravity theory has not been checked in a very large scale, then one does not know whether this gravity theory is suitable or not for studying the observational data like SNe Ia, and maybe the accelerating expansion of universe is due to the gravity theory that differs from the general gravity. Thus, many modified gravity theories like f (R), DGP, etc., are proposed to explain the accelerating phenomenology. The Cardassian model is a kind of model in which the Friedmann equation is modified by the introduction of an additional nonlinear term of energy density, and we will give briefly review on this model in the next section.

Dissipative processes in the universe including bulk viscosity, shear viscosity and heat transport have been conscientiously stud-

* Corresponding author.

E-mail addresses: fengcj@shnu.edu.cn (C.-J. Feng), kychz@shnu.edu.cn (X.-Z. Li).

0370-2693/$ - see front matter © 2010 Elsevier B.V. AH rights reserved. doi:10.1016/j.physletb.2010.07.028

ied [2], and these dissipative effects are supposed to play a very significant role in the astrophysics [3] and nuclear physics [4]. Therefore, it is also important to study these effects in the Cardas-sian cosmology, and in this Letter, we will consider the Cardassian models with and without matter viscosity and constrain the relevant parameters in these models by using the latest data, and for works on viscous dark energy models, see Ref. [5]. The general theory of dissipation in relativistic imperfect fluid was put on a firm foundation by Eckart [6], and, in a somewhat different formulation, by Landau and Lifshitz [7]. This is only the first order deviation from equilibrium and it may have a causality problem, the full causal theory was developed by Israel and Stewart [8,9], and also has been studied in the evolution of the early universe [10]. However, the character of the evolution equation is very complicated in the full causal theory. Fortunately, once the phenomena are quasi-stationary, namely slowly varying on space and time scale characterized by the mean free path and the mean collision time of the fluid particles, the conventional theory is still valid. In the case of isotropic and homogeneous universe, the dissipative process can be modeled as a bulk viscosity Z within a thermodynamical approach, while the shear viscosity n can be neglected, which is consistent with the usual practice [11].

The bulk viscosity introduces dissipation by only redefining the effective pressure, peff, according to peff = p — 3% H where % is the bulk viscosity coefficient and H is the Hubble parameter. The condition % > 0 guaranties a positive entropy production, consequently, no violation of the second law of the thermodynamics [12]. The case % = tH, implying the bulk viscosity is proportional to the fluid's velocity vector, is physically natural, and has been considered earlier in an astrophysical context, see the review article of Gr0n [13].

Table 1

Summary of Cardassian models with different functions of g(pm) for the late-time evolution of the universe. Here, pcard is a characteristic energy density.

g(Pm ) Model E2 = H2/H2

pm[1+)n-1 ] FOC OC VOC ßm0(1 + Z)3 + (1 - ßm0)(1 + Z)3n ßm0(1 + Z)3 + (1 - Qm0 - ßk0)(1 + Z)3n + ßko(1 + Z)2 ßm0(1 + Z)3<1-T)[]T/(T-n)[1 + FVOc(Z)(&-0 - 1)]

Pm E1 + (fc ^"]1 FMPC MPC VMPC ßm0(1 + Z)3 [1 + (ttm0 - 1)(1 + z)3q(n-1)]1/q ßm0(1 + Z)3 [1 + ((1 - Vk0)q- 1)(1 + Z)3q(n-1)]1/q + ßk0(1 + Z)2 ßrnOÜ + Z)3(1-T)[ T2+^1c(Z) ] ^ [1 + Fvmpc(Z)(aJ 1)]1

Pm exp )-n] FEC EC VEC ßmo(1 + Z)3 exp [-(1 + Z)-3n ln ßmo] ßmo(1 + Z)3 exp [-(1 + Z)-3n (ln Qm0 - ln(1 - Qko))] + ^ko (1 + Z)2 ßmo(1 + Z)3[ J+C]T/<n<1-T» exp (-F-1 lnQm0)

In this Letter, we will focus on several Cardassian models including the original, modified polytropic and exponential Cardassian models and constrain their parameters by the latest Constitution Type Ia supernova data (SNe Ia), the position of the first acoustic peak of the cosmic microwave background (CMB) from the five years WMAP data and the size of baryonic acoustic oscillation (BAO) peak from the SDSS data. We have consider the case of spatial flat and curved universe and the case of flat universe with the bulk viscosity. After a lengthy numerical calculation, we obtain the best fit values of the parameters in each Cardassian model.

This Letter is organized as follows: In Section 2, we present a brief review of Cardassian models, and derive the Hubble parameter in terms of the redshift and some parameters for several models. In Section 3, we analysis each model with statistical method and constrain their parameters with the observational data. In the last section, we give some conclusions and discussions.

2. The Cardassian model

In the following, we will assume that the universe is homogeneous and isotropic, which is described by the FRW metric:

ds = -dt2 + a(t)

1- kr2

+ r2 (d9

2 + sin2 Ö d4>2

where k is the spatial curvature. Cardassian models were introduced in [14] as a possible alternative to explain the acceleration of the universe by a model that has no energy components in addition to ordinary matter. Necessarily, the Friedmann equation must be modified: k

+ T = a2

g3)=3+^ (p),

where p is the total energy density of matter and radiation and we will neglect the contribution of radiation for the late-time evolution of the universe, namely p & pm. In Eq. (2.2), the function g(p) reduces to p in the early universe, then Eq. (2.2) reduces to the ordinary Friedmann equation during early epochs such as primordial nucleosynthesis. However, it differs from the FRW universe at the redshift z < O(1), during which it will gives rise to accelerated expansion. The appearance of the extra term F(p) on the right-hand side of Eq. (2.2) was originally motivated by brane cosmology [15]. Instead, one can view the Cardassian models as a class of phenomenological models that attempt to explain the accelerated expansion of the universe without explicitly resorting to adding new cosmic fluids, such as a cosmological constant, to the Friedmann equation [16]. Different forms of the function g(p) corresponds to different Cardassian models, and we will focus on the original Cardassian model (OC) [14], the modified polytropic Cardassian model (MPC) [17], the exponential model (EC) [18], their

flat versions (FOC, FMPC, FEC), in which the spatial curvature is neglected, and their viscous versions (VOC, VMPC, VEC) [19], in which the bulk viscosity of the matter is taken into account while the spatial curvature is neglected. We summaries these models in Table 1. For recent works on constraining the Cardassian universe, see Ref. [20].

Energy conservation of pressureless matter is given by

Pm + 3H(Pm - 3$mH) = 0, (2.3)

where $m is the bulk viscosity for the matter pm. Following [21], the function g could be written as g = pm + pk, where pk is so-called Cardassian term, which may indicate that our observable universe as (3 + 1)-dimensional brane in extra dimensions. Thus, the total energy density can be written as

g + 3H(g + pt — 3$H) = 0, (2.4)

where $ is the bulk viscosity for the total energy density g(pm). Here, pT is defined as the effective pressure of total fluid without bulk viscosity, and the first law of thermodynamics in an adiabatic expanding universe gives

PT = Pm~--g-

Therefore, one can get $ = jp^$m from Eqs. (2.3) and (2.4). In the following, we will choose $ = tH, in which the cosmological dynamics can be analytically solvable [19] and t is a constant. Then, the conservation law (2.4) becomes

'+dy >y+3

^ + (1 - T) f d y

where f = g/pm, y = ln(pm/pcard), the prime denotes the derivative with respect to x = ln(a) = — ln(1 + z), and z is the redshift. For the VOC model, f = 1 + e(n—V)y, and the solution is

Pm = Pm0 (1 + z)3(1-T)

1 - T + (n - T)(pmo/Pcard)n 1 _ 1 - T + (n - T)(pm/Pcard)

where pm0 is the present value of the matter's energy density, and the Hubble parameter E2 = H2/H2, is given by

E2 = ßm0(1 + z)3(1-T)

T1 + 1

_T1 + FVoc (z)_ x [1 + Fvoc(z)(am1 -1)]

where Qm0 = pm0/(3H2), H0 is the present value of the Hubble parameter and

1 \1 - i2m0j \n - t)

T— n

T— n

Here the function Fvoc(z) = (pm/Pmo)n 1 satisfies fi + 1

fvoc = (1 + zfO-rX"-1)

Ti + Fv.

T(n-1) T -n

(2.10)

from which one can get the solution for Fvoc and substitute it into Eq. (2.8), then one obtains the Hubble parameter in terms of z and parameters Qm0, n, t. When t = 0, the solution is rather simple, and the Hubble parameter (2.8) becomes

E2 = ßmo(1 + z)3 + (1 - ßmo)(1 + Z)3n.

(2.11)

For the VMPC model, f = (1 + eq(n-1)y)1/q, and the solution is

Pm = Pm0 (1 + z)3(1-T)

-1 - T + (n - T)(pmo/Pcard)q(n-1y

q(T-n)

L 1 - T + (n - T)(pm/Pcard)q(n-1) ] and the Hubble parameter is given by

(2.12)

E2 = ßmo(1 + Z)3(1-T)

T2 + 1

T2 + Fvmpc (z) 1

q(T-n)

X [1 + Fvmpc(z)(Vj - 1)] q

(2.13)

1 - q q

T(n-1) T-n

n — T /

Here the function Fvmpc(z) = (pm/pm0)q(n—1) satisfies Fvmpc = (1 + z^^J^

L T2 + Fvmpc _

and when t = 0, the Hubble parameter (2.13) becomes

E2 = Qm0(1 + z)3[1 + (Q— — 1)(1 + z)3q(n—1)]1.

For the VEC model, f = exp (e—ny), and the solution is

Pm = Pm0 (1 + z)3

n - (Pmo/Pcard)n(1 - T) n - (Pm /Pcard)n (1 - T)

n(1-T)

(2.15)

(2.16)

(2.17)

and the Hubble parameter is given by

E2 = Qmo(1 + Z)3

. T3 + Fvec (z) _

n(1-T)

x exp (- FveC ln ßmo)

ln Qmo.

Fvec = (1 + Z)

T3 + 1

,T3 + Fvec.

and when t = 0, the Hubble parameter (2.18) becomes E2 = i2mo(1 + z)3 exp (-(1 + z)-3n ln S2mo).

(2.18)

(2.19)

(2.2o)

(2.21)

We summarize all the solutions of Hubble parameters for each model in Table 1.

3. Statistical analysis with the observational data

In general, the expansion history of the universe H (z) or E (z) can be given by a specific cosmological model or by assuming an arbitrary ansatz, which may be not physically motivated but just designed to give a good fit to the data for the luminosity distance dL or the 'Hubble-constant free' luminosity distance DL defined by

where the light speed c is recovered to show that DL is dimension-less. In the following, we will take the first strategy that assuming the Hubble parameter H (z; a1,...,an ) with some parameters (a1,...,an) predicted by the class of Cardassian models could be used to describe the universe, and then we obtain the predicted

value of Df by

nth (1 + z) ..

DL' = .. _ Sinn

VIQxol

vOwi j

E (z'; a1,...,an )

where Q^o = —k/(aoH2) and Sinn(x) = sin(x), x, sinh(x) for respectively a spatially closed (Qk < 0), flat (Qk = 0) and open (Qk > 0) universe.

On the other hand, the apparent magnitude of the supernova is related to the corresponding luminosity distance by

(2Л4) p(z) = m(z) - M = 5log1o

dL (z) Mpc

where /x(z) is the distance modulus and M is the absolute magnitude which is assumed to be constant for standard candles like Type Ia supernovae. One can also rewrite the distance modulus in terms of Dl as

p(z) = 5log1o Dl (z) + po,

po = 5log1o

AHo-1\

v Mpc J

+ 25 = -5log1o h + 42.38

is the zero point offset, which is an additional model independent parameter. Thus, we obtain the predicted value of jxth by using the value of Df and the observational value of jxobs we used is the latest data called the constitution data [22], which contains 397 data points including the 307 Union data set [23] and 90 CFA data set.

There are also some constraints from CMB and BAO observations. We will take the parameter R from the CMB data [24] and the parameter A from the SDSS data [25] as well as the supernova data to constrain parameters of the Cardassian models. The parameter R is defined as

zis f -

where zis = 1090 is the redshift of the last scattering surface and the observational value is given by Robs = 1.170 ± 0.019. While the parameter A is defined as

E(z1 ) Okol

where z1 = 0.35 and the observational value is given by Aobs = 0.469(0.95/0.98)-0 35 ± 0.017.

Here the function Fvec(z) = (pm/pmo)n satisfies

Table 2

Result: The minimum value of x2 and the best fit parameters with 1a confidence level in each model.

Model x 2 Amin x2mJDOF Best fit parameters (1a )

FOC 474.083 1.194 &m0 - 0 2 70+0-023 0 -0-021 ' „ — 0 053T0 070 0 - 053 -0 075

OC 473.084 1.195 &m0 0 283+0-037 — 0-283_0033, „ — 0 -023+0-013, ßk0 --0 010+0 -018 0 -010-0-019

VOC 473.101 1.195 &m0 0 2 81+0-029 „— 0 0l0+0 -110 T-0 -010-0-150, L — -0 -004-0-150

FMPC 473.746 1.196 &m0 - 0 273+0-027 „— -0 600+0-980 a- 0 -600-0 -450, a - — 0 -48,0-J080

MPC 473.072 1.198 &m0 0.285T0.030, — 0-285_0035, „— 0 - 200+0 -200, a — -0 015+0-030 0 - 015-0 -015

VMPC 473.205 1.198 &m0 - 0 2 79+0-026 — 0 - 279—0 -02Q, „— -0 -050-0-900, a- 0 900+2 000, T - 0 -900 - 0700, I — -0 003+0-008 0 - oo3—0-007

FEC 474.128 1.194 &m0 — 0 2 7 7+0-024 — u- z/ /_0 020, „— n cit:+0 -059 0 -625-0-051

EC 474.127 1.197 &m0 - 0 2 7 7+0-033 — 0 - 2//-0 027, „— 0 -626+0-076, ßk0 - -0 0004+0 0213 — 0 -0004-0 0196

VEC 474.127 1.197 &m0 - 0 2 76+0-034 — 0 - 276-0 026, „— 0 -623+0-117, T — 0 0002 +0-009 0 - 0002-0 0082

In order to determine the best value of parameters (with 1a error at least) in the Cardassian models, we will use the maximum likelihood method and need to minimize the following quantity

Contours for FOC Model

2 ~ 2 2 2 ■.j 2 — \j 2 I \j 2 I \j 2

x — xSN + ACMB + ABAO, where

2 _ iR - 1 -710\

Xcmb — ( nmo )

Xbao —

A - 0-469(0-96/0-98)-0-35 0-0172

X Su (ai,---,an )

[ßobs(Zi ) - 5logi0 D?(Zi ; ai,---,an ) - M0]2

(3.10)

where a is the 1a error of the observation value )z0bs(zi). Since the nuisance parameter is model-independent, then we analytically marginalize it by using a flat prior P (j^0) = 1:

J e-x2/2 P fa) dto)

b2 ( c

— a---T ln —

397 r. ,obS(,.\ r>th(v-- n ^n2

[ßobs(Zi ) - 5logi0 Df(Zi ; ai,---,an)]

[fiobs(Zi ) - 5logi0 Df(Zi ; ai,---,a„)]

(3.11)

(3.12)

(3.13)

(3.14)

then, from now on, we will work with xSN and to minimize x2 in Eq. (3.8). The best fit parameter values and the corresponding x^nn and xmin/DOF will be summarized for each model in Table 2. Here DOF is the degree of freedom defined as

DOF = N - v, (3.15)

where N is the number of data points, and v is the number of free parameters.

c 0.00

Fig. 1. FOC: Constraints on Qm0 and n from 1a to 3a confidence level obtained by using 397 SNe Ia + CMB + BAO for the FOC model. The best fit point corresponds to Qm0 = 0.270, n = 0.053.

We now apply the maximum likelihood method for each model in Table 1, and we summarize the results in Table 2 including the minimum values of x2 and the best fit parameters with 1a confidence level for each model. Since both the FOC and FEC model contain two parameters, we also plot the contours from 1a to 3a confidence levels for them, see Figs. 1 and 2. For each model, the predicted dimensionless luminosity is plotted in Fig. 3, from which one can see that these Cardassian models predict almost the same luminosity distance with their best fit parameters.

4. Discussion

We have used the 397 SNe Ia, CMB and SDSS data to constrain several Cardassian models. We have summarized these model in Table 1, in which different forms of the function g(pm) have been chosen and the corresponding Hubble parameters are also given. In particular, we discuss the viscous Cardassian models in Section 2, in which we rewrite the Hubble parameter in a continent way to do the statistical analysis.

The fitting results are presented in Table 2, in which we have shown the minimum value of x2 and the minimum value x^iin per degree of freedom. The best fit parameters with 1a confidence

Contours for FEC Model

Best Fit +

Fig. 2. FEC: Constraints on Qm0 and n from 1a to 3a confidence level obtained by using 397 SNe Ia + CMB + BAO for the FEC model. The best fit point corresponds to Qm0 = 0.277, n = 0.625.

Luminosity Distance

FOC -OC VOC FM PC MPC VMPC FEC ■

EC VEC

Data: 397 SNe la y i

Fig. 3. The observed 397 SNe Ia distance modulus along with the theoretically predicted curves in the Cardassian models with best fit parameters, and we have taken a priori that current dimensionless Hubble parameter h = 0.70.

level for each model are also presented in Table 2, from which one can see that, the latest observational data cannot distinguish these models at this classical level. In other words, they predict almost the same evolution history of the universe and we need to take the perturbation of universe into account that will be studied in our further work.

In fact, the minimal of x2 in Eq. (3.8) is very sensitive to the observational error of the distance modulus. Once the error is smaller in the future data than that at present, not every model will fit the data well, then one can distinguish these models and even rule out some of them. Thus, more precise data are very needed.

Since in Cardassian universe one can explain the accelerating expansion without introducing any dark energy component, it is very interesting and worth further studying. We also hope that future observation data could give more stringent constraints on the parameters in the Cardassian model.

Acknowledgements

We would like to thank the referee for his comments to improve this work. We thank Dao-Jun Liu and Ping Xi for useful discussions on the analysis of the data. This work is supported by National Education Foundation of China grant No. 2009312711004 and Shanghai Natural Science Foundation, China grant No. 10ZR1422000.

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