Reduced modified Chaplygin gas cosmology

In this paper, we study cosmologies containing the reduced modified Chaplygin gas (RMCG) fluid which is reduced from the modified Chaplygin gas p = Aρ − Bρ−α for the value of α = −1/2. In this special case, dark cosmological models can be realized for different values of model parameter A. We investigate the viabilities of these dark cosmological models by discussing the evolutions of cosmological quantities and using the currently available cosmic observations. It is shown that the special RMCG model (A = 0 or A = 1) which unifies the dark matter and dark energy should be abandoned. For A = 1/3, RMCG which unifies the dark energy and dark radiation is the favorite model according to the objective Akaike information criteria. In the case of A < 0, RMCG can achieve the features of the dynamical quintessence and phantom models, where the evolution of the universe is not sensitive to the variation of model parameters.


Introduction
Observations indicate some challenges to the standard Big Bang model of cosmology. Several invisible components what we have to search in universe are hinted. For example, the observations on rotation curve of galaxy [1] directly relate to the amount of pressureless matter, proposing dark matter (DM) in our Universe; the observations on supernovae of type Ia [2,3] point out an accelerating universe at late time, which is usually interpreted as the existence of a new ingredient called dark energy (DE); the Wilkinson microwave anisotropy probe (WMAP) provides precise measurement of the cosmic microwave background radiation. Combining the 9-year WMAP results with the Hubble constant measured from the Hubble Space Telescope (HST) and the baryons acoustic oscillations (BAO) from the SDSS puts a constraint on the effective number of relativistic degrees of freedom N eff = 3.84 ± 0.4 which implies the presence of an extra dark radiation (DR) component at 95% confidence level [4,5]. 1 It is interesting to search origins of these dark sectors. In the past years, efforts were made to study these dark sectors comprising the DM, DE 1 Recently, ref. [6] studied the effect of H0 prior on the value of N eff . In the ΛCDM model, the evidence of DR is weakened to ∼ 1.2 standard deviations (N eff = 3.52 ± 0.39 at 68% confidence level) [6] by taking the median statistics (MS) prior H0 = 68 ± 2.8 km s −1 Mpc −1 to replace the HST prior H0 = 73.8 ± 2.4 km s −1 Mpc −1 . This result tends to show that the evidence for DR is not pressing any more.
In addition to these dark sectors (DM, DE, DR), baryon and radiation as visible constituents naturally exist in our Universe. Current cosmic observations suggest that our Universe contains about 70% the negative-pressure DE, 30% the pressureless matter (or called dust) including the DM and baryon, and a small fraction of radiation components which are composed of the photon, neutrino as well as additional relativistic species [28]. Someone proposed an economical model which can unify the DM and DE in a single fluid, say the generalized Chaplygin gas [29][30][31][32] and the modified Chaplygin gas (MCG) [33,34] for instances. In this paper we will perform new search of dark sectors from the reduced MCG (RMCG) fluid. We study the RMCG fluid using the analyses of theoretical constraints and the comparisons with the observational data, and obtain several interesting properties such as the DE and DR can be uniformly described by this single fluid, the evolutions of the cosmological quantities in the dynamical RMCG model are not sensitive to the variation of model-parameter values, and so on.
This paper is organized as follows. In the next section, we introduce the dark models in the RMCG cosmology. In section III, we examine the evolutions of growth factor and Hubble parameter in the RMCG model, and compare them with the current observational data. The parameter evaluation and model comparison for the RMCG model are performed in section IV. section V is the conclusions.

Dark models in RMCG cosmology
The MCG model was widely studied for explaining the cosmic inflation [35][36][37][38] or providing an unified model of the DM and DE [39][40][41][42]. We consider the equation of state (EoS) p = Aρ − Bρ 1/2 , (2.1) dubbed as the RMCG, which is reduced from the modified Chaplygin gas p = Aρ − Bρ −α for the constant model parameter α = −1/2. This model (2.1) can produce a emergent universe without the time singularity [43][44][45][46]. But in this paper, we will take this RMCG fluid as the dark components in our Universe. Using the energy conservation equation dρ/dt = −3H(ρ + p), we obtain the energy density of the RMCG fluid, where C is an integration constant, A s = Bρ −1/2 0RM CG /(1 + A). ρ 1 , ρ 2 and ρ 3 are current values of three energy densities in the RMCG fluid. According to eq. (2.2), some unified JHEP02(2015)071 where Ω 0i is the current dimensionless energy density beyond the dark sectors, Ω 01 , Ω 02 and Ω 03 correspond to three current dimensionless energy densities given by the RMCG fluid. a is the scale factor that is related to cosmic redshift by a = 1/(1 + z). In the following, we show expressions of some basic cosmological parameters in the RMCG model: (1) The adiabatic sound speed for the RMCG fluid, c 2 s = δp/δρ = A − 1 2 (1+A)As As+(1−As)a − 3 2 (1+A) . A small non-negative sound speed for matter component is necessary for forming the large scale structure of our Universe.
(2) Equation of state for the RMCG fluid, w = p/ρ = A − (1+A)As To obtain a late time accelerating expansion universe, it should be respected that the current value of EoS w 0 < − 1 3 . Table 1 lists the theoretical constraints on model parameter A s in the RMCG cosmology by locating the w 0 at the quintessence region or phantom region, where the different values or intervals for model parameter A are adopted.
(3) Deceleration parameter q(a) = −ä/(aH 2 ). An expanding universe having a transition from deceleration to acceleration is consistent with the current cosmic observations.

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Density parameter Explicit form Parameter value EOS Ω 0r Ω 0r    figure 1, one receives the result that the negative pressure is provided by the RMCG fluid at late time of our Universe. For the evolutions of c 2 s (z), the unexpected negative sound speed is appeared in this RMCG fluid. Since this unified fluid includes dust component, the negative sound speed will induce the classical instability to the system at structure form, where the perturbations on small scales will increase quickly with time and the late time history of the structure formations will be significantly modified [47]. Then it seems that the RMCG (A=0) model is not a good one.
One from eq. (2.5) gains 55, 0.79) and Ω 0s ∈ (0.01, 0.05), which are listed in table 3. For this case, Ω 0m = 0.3 gives Ω Λ = 0.67. Figure 2 illustrates the evolutions of the adiabatic sound speed, EoS, deceleration parameter and dimensionless energy density in the RMCG (A = 1). As we can see, the value of EoS is transited from the positive to the negative. Correspondingly, a transition from decelerating-expansion universe to accelerating-expansion universe can be realized. Meanwhile, this RMCG unified fluid at hand would not bring the negative value of the adiabatic sound speed. But it has other problems we have to face, such as (1) deceleration parameter is q > 1 2 at high redshift, which is not satisfied with q ≤ 1 2 in the matter-dominate universe. Matter-dominate universe is necessary for structure formation; (2) Radiationdominate universe will not appear in this RMCG universe, because of stiff matter. From these points, it seems that this model is not consistent with the current observational universe.

A unified model of dark energy and dark radiation in RMCG cosmology
Combined analysis of several cosmological data hints the existence of an extra relativisticenergy component (called dark radiation) in the early universe, in addition to the wellknown three neutrino species predicted by the standard model of particle physics. The total amount of this extra DR component is often related to the parameter N eff denoting the effective number of relativistic degrees of freedom, which has relation to the energy density of relativistic particles via ρ ν = 7 8 (4/11) 4/3 ρ γ N eff . Here ρ ν and ρ γ represent the fractional energy density for neutrino and CMB photon, respectively. The entropy transfer between neutrinos and thermal bath modifies this number to N eff = 3.046 [48,49]. However, larger values of N eff are reported by the cosmic observations. Depending on the datasets, constraint results on N eff are qualitatively changed. For instance, it is pointed out that the observational deuterium abundance D/H favors the presence of extra radiation [50,51]: N eff = 3.90 ± 0.44. The combining analysis of CMB data from the 7-year WMAP and the Atacama Cosmology Telescope (ACT) gives an excess N eff = 5.3±1.3 [52], and the addition of BAO and H 0 data decreases the value N eff = 4.56 ± 0.75 [52,53]. CMB data from the 9-year WMAP combining with the South Pole Telescope (SPT) and the 3-year Supernova Legacy Survey (SNLS3) provides a non-standard value, N eff = 3.96±0.69 [54,55]. Ref. [28] shows that N eff = 3.62 +0. 50 −0.48 for using the Planck+WP+highL+H 0 and N eff = 3.52 +0.48 −0.45 for JHEP02(2015)071 using the Planck+WP+highL+BAO+H 0 , whose analysis suggests the presence of a dark radiation at 95% confidence level. For more limits on N eff , one can see refs. [56][57][58].
The above urgency to search source of DR is relieved by the study in ref. [6]. Given that N eff is degenerate with the value of H 0 , ref. [6] focuses on how the H 0 prior changes the value of N eff , and obtains the result that a lower prior for H 0 moves the limits to lower N eff . It is pointed out in ref. [6] that there is no longer that much evidence supporting the existence of DR, since this evidence is partially driven by the larger value H 0 = 73.8±2.4 km s −1 Mpc −1 from the HST while several measurements suggest the lower value of H 0 , such as H 0 = 68 ± 2.8 km s −1 Mpc −1 from the median statistics (MS) analysis of the 537 non-CMB measurements [59], H 0 = 67.3 ± 1.2 km s −1 Mpc −1 from the Planck+WP+highL [28] and H 0 = 68.1 ± 1.1 km s −1 Mpc −1 from the 6dF+SDSS+BOSS+WiggleZ BAO data sets [28]. For model-dependent results, ref. [6] shows that in the ΛCDM it indicates the presence of DR with the HST H 0 prior, while there is no significant statistical evidence for existence of DR with the MS H 0 prior [6]; in XCDM parametrization of time-evolving DE it brings the result: the evidence for DR is significant for both the HST H 0 prior and the MS H 0 prior [6].
In this section, we explore the RMCG model that apparent extra DR directly links to the physics of the cosmological-constant (CC) DE. Fixing A = 1/3, RMCG fluid unifies the DE and DR, where the Friedmann equation becomes (2.6) Here 0k a −2 dilutes as a −2 jus like the curvature density in the nonflat geometry, called effective curvature density. In the non-flat universe, then the current curvature density is modified as Ω 0k + Ω eff 0k . Besides the RMCG fluid, we supplement the matter and radiation components in eq. (2.6).
Eq. (2.6) shows that the dimensionless density parameters (DE, DR and effective curvature density) relate to the RMCG model parameter A s . The values of these density parameters should be consistent with observations. Given that relativistic particle includes the photon, neutrino and dark radiation, the total dimensionless density parameter of relativistic particle is written as Ω tot where the photon density parameter Ω 0γ = 2.469 × 10 −5 h −2 [60]. Writing N eff = N SM eff + ∆N eff and N SM eff = 3.04, one reads Ω 0dr = 7 8 ( 4 11 ) 4/3 Ω 0γ ∆N eff . On the other hand, in the RMCG Taking Ω 0m = 0.3 and ∆N eff = [0.5, 1, 2], we can calculate the values of A s and the dimensionless density parameters, which are listed in table 4. It is found from this table that the values of Ω Λ and Ω eff 0k are compatible to the cosmic observations [4,28], where Ω Λ is around 0.7 and Ω 0k ∼ 0. And corresponding to A s < 1 (or A s > 1), we have Ω eff 0k > 0 (or Ω eff 0k < 0).   The negative c 2 s for DE is in fact necessary if one requires the negative pressure to produce the accelerating universe. This is not inconsistent with the structure formation. For the behavior of w, at late time we can get w < 0 which can be responsibility to the accelerating universe, and at early time we obtain w ∼ 1/3. According to the analysis above, the behaviors of cosmological quantities in the RMCG (A = 1/3) model are accordant with the current observational universe. Then the RMCG (A = 1/3) model can be considered as a candidate for the DE and DR. At last, we note that we do not discuss the case of A s > 1 for the RMCG (A = 1/3), since the c 2 s and w will be divergent at some points (when A s = −(1 − A s )a − 3 2 (1+A) ).

RMCG fluid as dark energy
The unification of the DE and DM (or DR) have been discussed in above parts. In the following, we investigate other possible properties of the RMCG fluid by taking values of A (except A = 0, 1 and 1/3). For A > 0 (A = 0, 1, 1/3), eq. (2.2) states that the RMCG fluid contains the CC and other positive-pressure or negative-pressure components (depending on the concrete value of A). We know nothing about these indeterminate components, such as their function in universe or their responsibility to observations. So, here we do not discuss the case of A > 0. For A < 0, the RMCG fluid plays a role as the dynamical phantom or dynamical quintessence DE, where the Friedmann equation is written as where So, the RMCG fluid comprises the CC and phantom DE, which plays a role as the phantom-type DE; for 0 > A > −1, the RMCG fluid includes the CC and quintessence DE; for A = −1 or A s = 1, the RMCG fluid reduces to the CC. Since we in theory have 0 ≤ A s ≤ 1 due to the constraint on the current dimensionless density parameter 0 < Ω 0j < 1, we can get the limit −1 < A < −1/3 for the quintessence-type DE; we can obtain the theoretical limit A < −1 with 0 < A s < 1 for the phantom-type DE. For A s > 1, the phantom-type DE (−1 < A < 0) and the quintessence-type DE (A < −1) are non-physical, which should be ruled out.  figure 4, the value of more near to A = −1, the less influence on w from A s . Also, from four lower figures in figure 4, we obtain the result that the value of more near to A s = 1, the less influence on w from A.
Trajectories of q(z) in the RMCG (A < 0) model are drew in figure 5, which describe a universe transiting from decelerating expansion to accelerating expansion. One can also see an interesting property for q(z) from figure 5. The behavior of q(z) is almost the same for using the different value of A s (or A), when the value of another model parameter A (or A s ) is near to −1 (or 1). For example, q(z) is not sensitive to the change of value for A s (or A), when we take A = −0.9 and A = −1.1 (or, A s = 0.9 and A s = 0.95). By the way, figure 6 illustrates the evolutions of c We apply the f and H to test the RMCG models by comparing them with the observational data. Growth factor is defined as f ≡ d ln δ/d ln a, which complies with the following equation       deriving by the perturbation equationδ + 2Hδ − 4πGρ m δ = 0. Here δ ≡ δρ m /ρ m is the matter density contrast and "dot" denotes the derivative with respect to cosmic time t. Usually , it is hard to find the analytical solutions to eq. (3.1). The approximation f Ω γ m has been used in many papers, which provides an excellent fit to the numerical form of f (z) for various cosmological models [76][77][78][79][80][81]. Growth index γ can be given by considering the zeroth order and the first order terms in the expansion for γ [82] . We illustrate the Ω γ m versus z in figure figure 7), where an increasing function versus z is consistent with the current observations. But, Ω γ m (z) in the RMCG (A = 1) much deviates from that in the ΛCDM model at the higher redshift. For clarity, we plot the trajectories of H(z)/(1 + z) for the discussional models, and compare them with the 29 observational H(z) data listed in table 6. The difference of pictures between the RMCG (A = 1) model and ΛCDM model is apparent at high redshift. And at the high redshift, the evolutions of Ω γ m (z) and H(z)/(1 + z) in the RMCG (A = 1) obviously deviate from the observational data. From above, it is shown that the RMCG (A = 1) fluid as the unification of dark matter and dark energy is not well accordant with the f data and the Hubble data. But, the RMCG (A = 1/3) and RMCG (A < 0) model are well consistent with these two cosmic datasets.

Parameter evaluation and model comparison
In this section, we investigate the parameter space of the RMCG model. It can be known from the analysis above that the RMCG unified model of the DE and DM are not favored, which have some questions on structure formation. For the RMCG (A=0) unified model, a negative sound speed will introduce the instability at structure formation. For the RMCG (A=1) unified model, perturbation quantity f is not compatible with cosmic data, and a super-deceleration (q > 1 2 ) expanded universe is not satisfied with the matter-dominate universe. So, these two cases will not studied in the following. We discuss the cosmic constraint on the RMCG models with A = 1/3 (RMCG1) and A < 0 (RMCG2). The data we use

Case model
Free parameters includes: baryon acoustic oscillation (BAO) data from the WiggleZ [83], 2dfGRs [84] and SDSS [85] survey, X-ray cluster gas mass fraction [86], Union2 dataset of type supernovae Ia (SNIa) [87] and 29 Hubble data listed in Obviously, A s is near to 1 and has the small confidence level. This calculation result for A s is approximatively equal to the cosmic constraint on ∆N eff ∈ (0, 1), which is consistent with other combining constraints on N eff [28]. By the analysis of error-propagation, we calculate the DE density Ω Λ = 0.713 +0.012+0.024 −0.012−0.024 . For RMCG2, we find that Ω 0m = 0.297 +0.015+0.031 −0.016−0.028 and H 0 = 68.25 +1.46+2.92 −1.45−3.02 , while the model parameters A and A s are not convergent. The results are illustrated in figure 8. From eq. (2.7), we notice tha RMCG2 DE model reduces to the popular CC model by fixing A = −1 (or A s = 1), whatever value of A s (or A) is taken. The non-convergent results on A and A s may be interpreted that the RMCG2 model can not be distinguished from the CC model by the cosmic data used in this paper.
Next we use the objective information criteria (IC) to estimate the quality of above RMCG models. Akaike information criteria (AIC) is defined as [88,89] AIC = −2 ln L max + 2K, (4.1) where L max is the highest likelihood in the model with −2 ln L max = χ 2 min , K is the number of free parameters to interpret the complexity of model. Usually, candidate model which minimizes the AIC is usually considered the best. Comparing with the best one, one can calculate the difference for other model ∆AIC = ∆χ 2 min + 2∆K. The rules for judging the strength of models are as follows. For 0 ≤ ∆AIC i ≤ 2, model i almost gains the same data support as the best model; for 2 ≤ ∆AIC i ≤ 4, model i gets the less support; and with ∆AIC i > 10 model i is practically irrelevant [88]. Since several observations imply the existence of DR, we take the DR density Ω 0dr as an additional free parameter in the ΛCDM, RMCG2 and MCG models. But, Ω 0dr is naturally included in the RMCG1 model by the relation between Ω 0dr and model parameter A s and Ω 0m . According to the calculation results in   [91].

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One can notice that the other criticism mechanism--Bayesian information criteria (BIC) that is defined as BIC = −2 ln L max + K ln n [90] is not studied in this paper. Here n is the number of datapoints in the fitting. As we can see from the BIC definition, the BIC value not only depends on the number of free parameter K and the value of χ 2 , but also depends on the number of datapoints n. So, for the same models the different evaluation results would be given by the BIC analysis (induced by the different values of ln n) when one uses the different datapoints. For instance, the value of ln n is obviously different for case of including or not including SNIa data in combining constraint, since the SNIa data have the large number. Given that the datapoint are always increasing, it seems that the calculation result from BIC is not "fair" for more-parameter model when the more datapoints are given. Quantitatively, the AIC and BIC method can give the same result for ln n = 2 (n 7.4). For datapoints used in our analysis, it has ln n = 6.452. Seeing that the BIC is not "absolutely objective", i.e. its value much depends on the number of datapoints one use, here we do not apply the BIC criticism method to evaluate the RMCG models.

Conclusions
The RMCG models are from the subclass of the famous MCG model that has been studied in great detail over the years. But, most of them were studied as a unification of DM and DE in the past. In this paper, we study the RMCG cosmology from a different point of view. We discuss the different cases in which the RMCG is regarded as the DE or the unified model. New interesting physical results are obtained in the RMCG dark models. The results show that (1) the RMCG unified model of the dark energy and dark matter (with model parameter A = 0 or A = 1) tends to be ruled out by analysing the behaviors of cosmological quantities. For example, the RMCG (A=0) unified model appears a negative sound speed which leads to the instability of the structure formation, growth factor f in the RMCG (A=1) unified model is not consistent with cosmic observational data. In addition, a super-deceleration expanded universe (q > 1/2) is not satisfied at the matterdominate epoch and a radiation-dominate universe will not appear in the RMCG (A=1) model, due to the stiff matter; (2) the RMCG (A = 1/3) unified model of the DE and DR is a candidate to interpret the accelerating universe. It produces the good behaviors of cosmological quantities and the good fits to the current observational data: growth factor and Hubble parameter. In addition, it provides an origin of the DE and DR. The energy densities of these two dark components are self-consistent; (3)  at 68% and 95% confidence levels, which are consistent with other constraint result on N eff ∈ (0, 1). Meanwhile, the RMCG (A = 1/3) model almost has the same support as the most popular ΛCDM model via the AIC calculation. In case of fitting the cosmic data to the RMCG (A < 0) model, model parameters A and A s are not convergent. The theoretical predictions on the RMCG (A < 0) model parameters are 0 < A s < 1 with −1 < A < −1/3 for the quintessence DE, and 0 < A s < 1 with A < −1 for the phantom DE. But by the analysis of AIC, the RMCG (A < 0) model has the less support from the observational data.

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proper angular diameter distance which relates to other two distance quantities D L and D V by Here p s is the theoretical model parameters, sinn( |Ω k |x) denotes sin( |Ω k |x), |Ω k |x and sinh( |Ω k |x) for Ω k < 0, Ω k = 0 and Ω k > 0, respectively.

A.1 BAO
BAO data can be extracted from the WiggleZ Dark Energy Survey (WDWS) [83], the Two Degree Field Galaxy Redshift Survey (2dFGRS) [84] and the Sloan Digitial Sky Survey (SDSS) [85]. One can construct V −1 is the inverse covariance matrix [85,92]. X is a column vector which is given by theoretical values minus observational values, and X t denotes its transpose. r s (z) = c

A.2 X-ray gas mass fraction
In observation of the X-ray gas mass fraction, one can define a parameter [86] is the angular correction factor. η = 0.214 ± 0.022 is the slope of the f gas (r/r 2500 ) data [86]. Parameter γ denotes permissible departures from the assumption of hydrostatic equilibrium, due to non-thermal pressure support. Bias factor b(z) = b 0 (1 + α b z) accounts for uncertainties in the cluster depletion factor. s(z) = s 0 (1+α s z) accounts for uncertainties of the baryonic mass fraction in stars, and a Gaussian prior for s 0 is employed with s 0 = (0.16 ± 0.05)h 0.5 70 [86]. Factor K is utilized to describe the combining effects of the residual uncertainties, and a Gaussian prior K = 1.0±0.1 is used [86]. Adopting the datapoints published in ref. [86] and following the method introduced in refs. [86], we can constrain theoretical model by calculating where σ fgas (z i ) is the statistical uncertainties. As pointed out in [86], the acquiescent systematic uncertainties have been considered via the parameters η, b(z), s(z) and K.

A.4 H(z) data
Using the H(z) data listed in table 6, we can determine the model parameters by minimizing [104][105][106][107][108][109][110][111]  where H th is the theoretical value and H obs is the observational value for the Hubble parameter.
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