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 $p=A\rho-B\rho^{-\alpha}$ for the value of $\alpha=-1/2$. 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.

Besides dark sectors deduced from cosmic observations, visible constituents of baryon and radiation naturally exist in universe. Observations [27] suggest that in universe about 70% energy density are negative-pressure DE, about 30% are pressureless matter (or called dust) including DM and baryon, and there are a small quantity of radiation components including photon, neutrino as well as additional relativistic species. Someone propose that unified models composed by DE and DM can be expressed in one cosmic fluid, such as generalized Chaplygin gas [28][29][30], modified Chaplygin gas (MCG) [31,32] and so on. In this paper we perform a new search for dark sector from reduced MCG model. Here the unified model of dark sectors and the interested properties of dark energy can be given, such as the DE and DR can be unified in one fluid, and the evolutions of dynamical dark-energy model are not sensitive to the variation of model-parameter values, etc.

II. Dark models in RMCG cosmology
MCG model has been widely studied. In this model, MCG fluid can be used to explain cosmic inflation [33][34][35][36] or seen as a unified model of DE and DM [37][38][39][40]. We consider an equation of state (EOS) that is reduced from the modified Chaplygin gas p = Aρ − Bρ −α , with constant model parameter α = − 1 2 . We call this model as reduced modified Chaplygin gas (RMCG). Using Eq. (1) to study a no time-singularity emergent universe has been done in Refs. [41][42][43][44]. Here RMCG fluid is studied by regarding it as the dark components.
Using the energy conservation equation dρ dt = −3H(ρ + p), we have the energy density of RMCG fluid, where C is an integration constant, A s = B 3 ), we read a unified model of DE and other unknown components; and for A as a free parameter with A < 0, RMCG fluid plays a role of dynamical DE (for A < −1, its behavior is a phantom; for 0 > A > −1, it is a quintessence.). Considering that there are the RMCG fluid and other known components (baryon, radiation) in universe, one has the Friedmann equation where Ω 0i is the current dimensionless energy density of anyone known ingredient in universe, Ω 01 , Ω 02 and Ω 03 correspond to three current dimensionless energy densities given by 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 RMCG model.
(2) Equation of state for RMCG fluid, w = p ρ = A− (1+A)As . To obtain an accelerating expanded universe at present, the current value of EOS for RMCG fluid w 0 < − 1 3 should be appeared. Assuming w 0 in quintessence region or in phantom region, the theoretical constraint on model parameter A s in RMCG cosmology is listed in table I, following taking the different values or intervals for model parameter A.
(3) Deceleration parameter q(a) = −ä aH 2 . An expanded universe from deceleration to acceleration is consistent with the current cosmic observations.
Ω 0b and Ω 0r represent the fractional energy densities for baryon and radiation (including all relativistic particles, such as CMB photon Ω 0γ , neutrino Ω 0ν , etc.). From Eq. (4), obviously it has current dimensionless dark-energy Adopting current values Ω 0r ∼ 0, Ω 0b = 0.05 and Ω 0m ∈ (0.2, 0.4), after calculation one has A s ∈ (0.39, 0.6), 15, 0.34) and Ω 0u ∈ (0.45, 0.46). It is obvious that, for existence of Ω 0u the value of DE density is smaller than observations. Taking a = 1 in Eq. (4), one has √ According to this relation, the values of Ω Λ and Ω 0m are illustrated in Fig. (1 Ω0r Analyzing the evolution of deceleration parameter in RMCG (A=0) model, a universe from decelerated expansion to accelerated expansion can be emerged, with the larger current value q 0 ∈ (−0.355, −0.056) than in the standard ΛCDM cosmology q 0 ∈ (−0.7, −0.4), as plotted in Fig. 1. In Fig. 1 Fig. 1, a negative sound speed is appeared in this RMCG fluid. Since this unified fluid include 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 [45]. Then it seems that, this RMCG (A=0) model is not a good one.

Density parameter
Explicit form Parameter value EOS

Ω0r
Ω0r  problems we have to face, such as (1) deceleration parameter is q > 1 2 at high redshift, which is not satisfied with the matter dominated universe q ≤ 1 2 . Matter dominated universe is necessary for requirement of structure formation. (2) Radiation dominated universe will not appear in this RMCG universe, for the existence of stiff matter. From these points, it seems that this model is not consistent with our understanding on current observational universe.

B.
A unified model of dark energy and dark radiation in RMCG cosmology Combined analysis of recent cosmological data hint the existence of an extra relativistic energy component (called dark radiation) in the early universe, in addition to the well-known 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 ef f . N ef f denotes the effective number of relativistic degrees of freedom, which is related to the energy density of relativistic particles by ρ ν = ρ γ 7 8 ( 4 11 ) 4/3 N ef f , where ρ ν and ρ γ represent the fractional energy densities for neutrino and CMB photon. The inclusion of entropy transfer between neutrinos and the thermal bath modifies this number to about N ef f = 3.046 [46,47]. However, larger values of N ef f are reported by the recent cosmic observations. Depending on the datasets used, constraint results on N ef f are qualitatively changed. For instance, it is pointed out that the observed deuterium abundance D/H favors the presence of extra radiation [52,53]: N ef f = 3.90 ± 0.44. The analysis of Refs. [48,49] that combined CMB data from 7-year WMAP with Atacama Cosmology Telescope (ACT) reports an excess N ef f = 5.3 ± 1.3, and the addition of BAO and H 0 data improves this constraint, N ef f = 4.56 ± 0.75 [48,49].
CMB data from the 9-year WMAP combining with the South Pole Telescope (SPT) and the 3-year Supernova Legacy Survey (SNLS3) gives a non-standard value of N ef f = 3.96 ± 0.69 [50,51]. Recently, Planck Collaboration finds N ef f = 3.52 +0. 48 −0.45 [27] for using Planck+BAO+H 0 . And a combination of the Planck-satellite data combining with previous CMB data and Hubble constant measurements, Ref. [54] shows N ef f is 3.62 +0. 50 −0.48 , whose analysis is clearly suggesting the presence for dark radiation at 95% confidence level, here previous CMB data include low multipole polarization measurements from the 9-year WMAP data and high multipole CMB data from both the SPT and the ACT [54]. For more constraint values of N ef f , one can see Refs. [55][56][57].
In this part, we explore a model of the apparent extra DR being linked directly to the physics of cosmological- where 0k a −2 dilutes as a −2 jus like the curvature density in a non-flat geometry, called effective curvature density. Thus, in a non-flat universe the current curvature density is modified as Ω 0k + Ω ef f 0k . Besides the RMCG fluid, in Eq. (6) we supplement matter and radiation components, according to the current view on cosmic ingredients.
We note that this RMCG fluid is composed of CC dark energy, dark radiation and effective curvature component.
As shown in Eq. (6), dimensionless density parameters are related to the RMCG model parameter A s . For values of these density parameters, they should be self-consistent with observed allowable values. Considering relativistic particle including photon, neutrino and dark radiation component, the total dimensionless density parameter is written [58].
Taking Ω 0m = 0.3 and ∆N ef f = (0.5, 1, 2), the values of model parameter A s and dimensionless density parameters can be calculated. The results are listed in table IV. From this table, it is found that these values of density parameters including dark energy and curvature density are compatible to the current observations [4,27], where Ω 0de is about 0.7 and Ω 0k is around zero. And for model parameter A s < 1 (or A s > 1), correspondingly, it has the effective curvature density Ω ef f 0k > 0 (or Ω ef f 0k < 0).   unified fluid do not include matter, the negative value of c 2 s will not destroy the structure formation. Just as for cosmological constant dark energy, it has c 2 s = −1. With requiring the negative pressure to produce an accelerating universe, a negative value of c 2 s for dark energy is necessary, which is not inconsistent with the structure formation. For EOS w of RMCG fluid in Fig.3, at late time we have w < 0 which can be responsibility to an accelerating universe, and at early time w approaches to a value of radiation component w = 1 3 . According to above analysis the behaviors of these cosmological quantities in RMCG (A = 1 3 ) model are accordant with the current understanding on our universe, which is a candidate for dark energy and dark radiation. At last, we note the values of A s > 0 is not discussed for RMCG model at hand, since adiabatic sound speed and EOS will be divergent at some points (when = Ω 0m a −3 + Ω 0r a −4 + Ω 01 + Ω 02 a −3(1+w2) + Ω 03 a −3(1+w3) , where Ω 0RMCG = 1 − Ω 0m − Ω 0r , Ω 01 = Ω 0RMCG A 2 s , Ω 02 = Ω 0RMCG (1 − A s ) 2 and Ω 03 = 2Ω 0RMCG A s (1 − A s ). Obviously, for taking A < −1 one has w 2 = A < −1 and w 3 = A−1 2 < −1, so RMCG fluid is composed of CC and phantom dark energy, i.e. it plays a role of phantom-type dark energy; for taking 0 > A > −1, RMCG is composed of CC and quintessence dark energy. For taking A s = 1 or A = −1, a CC model is obtained from RMCG fluid. Since the theoretical constraint on current dimensionless density parameter is 0 < Ω 0j < 1, we have 0 ≤ A s ≤ 1 for RMCG fluid. Combining with the theoretical constraint results on model parameter listed in table I, we get −1 < A < − 1 3 for 1+3A 3(1+A) < A s < 1 (requiring 0 < A s < 1), in which EOS of RMCG locates at the quintessence region. For A < −1 and 0 < A s < 1, EOS is the phantom type. For A s > 1 in table I, the phantom-type DE (−1 < A < 0) and the quintessence-type DE (1 < A s < 1+3A 3(1+A) , equivalently 1 < A s < ∞) are non-physical, which should be ruled out. (3) As we can see from four upper figures in Fig.4, the value of more near to A = −1 is taken (such as two middle figures relative to two-side figures), the less influence on EOS from parameter A s is happened. Also, from four lower figures in Fig. 4, we have the results that the value of more near to A s = 1 is taken, the less influence on EOS from parameter A is happened (it has the smaller varied region of w for using different values of A).
Trajectories of deceleration parameter in RMCG dark energy model are drew in Fig. 5, with describing an universe from decelerated expansion to accelerated expansion. Here an interested property for q can be found. When we take the value being more near to A = −1 (or A s = 1), such as for A = −0.9 or A = −1.1 (two upper-middle figures in Ref. [59,60] derived by the perturbation equationδ + 2Hδ − 4πGρ m δ = 0. δ ≡ δρm ρm is the matter density contrast, and "dot" denotes the derivative with respect to cosmic time t. Analytical solutions to Eq. (8) are usually hard to find. An approximation to f ≃ Ω γ m has been used in many papers, which provides an excellent fit to the numerically obtained form of f (z) for various cosmological models [72][73][74][75][76][77]. Growth index γ can be given by considering the zeroth order and the first order terms in the expansion for γ [78] . Taking  In this part, we investigate the parameter space of RMCG model. One knows that the RMCG unified model of DE and DM are not favored by above analysis. For example, they have some questions on structure formation.
For RMCG (A=0) unified model, a negative sound speed will introduce the instability at structure formation. For RMCG (A=1) unified model, perturbation quantity f is not compatible to cosmic data, and a super-deceleration (q > 1 2 ) expanded universe is not satisfied with the matter dominated universe. So, these two cases are not studied in this part. We discuss the cosmic constraint on RMCG models with A = 1 3 (RMCG1) and A < 0 (RMCG2). The used data includes: baryon acoustic oscillation (BAO) data from WiggleZ [79], 2dfGRs [80] and SDSS [81] survey, X-ray cluster gas mass fraction [82], Union2 dataset of type supernovae Ia (SNIa) [83]  Next we use the objective information criteria (IC) to estimate the quality of above RMCG models. Akaike information criteria (AIC) is defined as [84,85] AIC = −2 ln L max + 2K,  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 that interprets model complexity. For candidate models, the one that minimizes the AIC is usually considered the best.
Comparing with the best one, the difference for other model is expressed as ∆AIC = ∆χ 2 min + 2∆K. The rules for judging the strength of models are as follows. When 0 ≤ ∆AIC i ≤ 2 model i has almost the same support from the data as the best model, for 2 ≤ ∆AIC i ≤ 4, model i is supported considerably less, and with ∆AIC i > 10 model i is practically irrelevant [84].
Since the current observations indicate the existence of dark radiation, we take dark radiation density Ω 0dr as an additional free parameter in ΛCDM, RMCG2 and MCG models. But in RMCG1 model, the DR density is naturally included by relating to model parameter A s and Ω 0m . According to the calculation results in  and H 0 = 69.97 +1.64+3.20 −1.64−3.13 for MCG model. In addition, one can notice that the other criticism mechanism--Bayesian information criteria (BIC) that is defined as BIC = −2 ln L max + K ln n, is not studied in this paper. n is the number of data points in the fit. As we can see from the BIC definition, the BIC value not only depends on the numbers of free parameters K and χ 2 , but also depends on the numbers of data points n. So for the same models, the different evaluation results could be given by the BIC method (induced by the different value of ln n) when one use the different data points. For instance, especially for combined constraint including or not including SNIa data, the value of ln n is obviously different, since the SNIa data have the large number. Considering that the data points are always increasing, it seems that the calculation result from BIC is not "fair" for more-parameter model when the more data points are appeared. Quantitatively, for ln n = 2 (n ≃ 7.4), AIC and BIC can give the same result. For the used data in our analysis, it has ln n = 6.444.
Seeing that the BIC is not "absolutely objective", i.e. its value much depends on the number of used data points, here we do not discuss the BIC criticism method to above RMCG models.

V. Conclusions
RMCG models are from a 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 studies on RMCG cosmology are performed from different angles, where the RMCG fluid as dark energy or as unified model of dark components are discussed. New interesting physical results are given in RMCG dark models. Studies show, (1) the RMCG unified model of dark energy and dark matter (with model parameter A = 0 or A = 1) tends to be ruled out, according to the behaviors of some cosmological quantities. For example, for RMCG (A=0) unified model, a negative sound speed appears, which will introduce the instability at structure formation. For RMCG (A=1) unified model, growth factor f is not consistent with cosmic data, and a super-deceleration (q > 1 2 ) expanded universe is not satisfied with the matter dominated epoch. Also, a radiation dominated universe will not appear in RMCG (A=1) model, for existence of the stiff matter. (2) the RMCG (A = 1 3 ) unified model of dark energy and dark radiation, is a candidate for interpreting the accelerating universe, which is well accordant with the current understanding on our universe, for examples, the good behaviors of cosmological quantities and good fits to the current observational data: growth factor and Hubble parameter. In addition, it provide an origin of dark radiation and dark energy. And energy densities of these two dark components are self-consistent; (3) the RMCG (A < 0) fluid as dark energy also has some attractive features, such as the CC, the quintessence and the phontom dark energy can be realized in this RMCG fluid, at some situations the evolutions of cosmological quantities are not much sensitive to the variation of model-parameters values.
At last, for RMCG (A = 1 3 ) and RMCG (A < 0) model, we investigate their parameter space by using the recent cosmic data. Fitting the observational data to the RMCG (A = 1 3 ) model, it is found that the constraint result on RMCG (A = 1 3 ) model parameter is A s = 0.9995 +0.0018+0.0030 −0.0018−0.0030 at 68% and 95% confidence levels, which is consistent with other cosmic constraint result on the effective number of relativistic degrees of freedom with △N ef f ∈ (0, 1).
By the AIC calculation, it is shown that RMCG (A = 1 3 ) model almost has the same support as the most popular ΛCDM model. For RMCG (A < 0) model, the model parameters A and A s are not convergent by comparing this model with the combined observational data. The theoretical constraint on RMCG (A < 0) model parameters are 0 < A s < 1 with −1 < A < − 1 3 for EOS at quintessence region, and 0 < A s < 1 with A < −1 for EOS at phantom region. But by the analysis of AIC, the RMCG (A < 0) model has the less support from the observational data.
Acknowledgments The research work is supported by the National Natural Science Foundation of China (11205078,11275035,11175077).

VI. Appendix
Here we describe the used cosmic data including the BAO, the f gas , the SNIa, and the H(z) data. Firstly, we introduce three distance parameter in the following. D A (z) is the proper angular diameter distance It relates with other two distance quantities D L and D V by p s denotes the theoretical model parameters, sinn( |Ω k |x) respectively denotes sin( |Ω k |x), |Ω k |x and sinh( |Ω k |x) for Ω k < 0, Ω k = 0 and Ω k > 0.

A. BAO
BAO data are extracted from the WiggleZ Dark Energy Survey (WDWS) [79], the Two Degree Field Galaxy Redshift Survey (2dFGRS) [80] and the Sloan Digitial Sky Survey (SDSS) [81]. The χ 2 BAO (p s ) is given by with V −1 is the inverse covariance matrix [81,87]. X is a column vector formed from theoretical values minus observational values, and X t denotes its transpose. r s (z) is the comoving sound horizon size r s (z) = c

B. X-ray gas mass fraction
In observation of the X-ray gas mass fraction, for the reference model ΛCDM, parameter f gas is presented as [82] f ΛCDM gas (z) = KAγb(z) 1 + s(z) A is the angular correction factor, A = H(z)DA(z) [H(z)DA(z)] ΛCDM η . Index η is the slope of the f gas (r/r 2500 ) data with η = 0.214 ± 0.022 [82]. 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 [82]. factor K is used to describe the combined effects of the residual uncertainties, such as the instrumental calibration and certain X-ray modelling issues, and a Gaussian prior for the 'calibration' factor is considered by K = 1.0 ± 0.1 [82]. Adopting the datapoints published in Ref. [82] and following the method in Refs. [82], χ 2 for the X-ray gas mass fraction analysis is expressed as where σ fgas (z i ) is the statistical uncertainties. As pointed out in [82], the acquiescent systematic uncertainties have been considered according to the parameters η, b(z), s(z) and K.