Cosmological constraints on alternative model to Chaplygin fluid revisited

In this work we explore an alternative phenomenological model to Chaplygin gas proposed by H. Hova et. al., consisting on a modification of a perfect fluid, to explain the dynamics of dark matter and dark energy at cosmological scales immerse in a flat or curved universe. Adopting properties similar to a Chaplygin gas, the proposed model is a mixture of dark matter and dark energy components parameterized by only one free parameter denoted as $\mu$. We focus on contrasting this model with the most recent cosmological observations of Type Ia Supernovae and Hubble parameter measurements. Our joint analysis yields a value $\mu = 0.843^{+0.014}_{-0.015}\,$ ($0.822^{+0.022}_{-0.024}$) for a flat (curved) universe. Furthermore, with these constraints we also estimate the deceleration parameter today $q_0=-0.67 \pm 0.02\,(-0.51\pm 0.07)$, the acceleration-deceleration transition redshift $z_t=0.57\pm 0.04\, (0.50 \pm 0.06)$, and the universe age $t_A = 13.108^{+0.270}_{-0.260}\,\times (12.314^{+0.590}_{-0.430})\,$Gyrs. We also report a best value of $\Omega_k = 0.183^{+0.073}_{-0.079}$ consistent at $3\sigma$ with the one reported by Planck Collaboration. Our analysis confirm the results by Hova et al, this Chaplygin gas-like is a plausible alternative to explain the nature of the dark sector of the universe.

hand, studies based on the matter power spectrum without baryons effects [30,31] rule out the GCG model. However, when the LSS analysis includes a baryon component, the GCG reproduces the 2dF mass power spectrum [32]. In addition, some authors consider models including an extra DM component to the GCG to obtain a suitable mass power spectrum [8,33,34]. Motivated by these results, we aim to revisit GCG alternative models with the latest cosmological observations.
In this work we investigate an extension of a recent proposal [1] to a non-flat geometry; where a generalized perfect fluid model that follows the Chaplygin gas-like scheme is studied as an alternative to GCG. The model supposes a mixture of unclustered DE and DM with stable sub-horizon fluctuations and conservation of the scale invariance instead of an unified dark sector context. Considering the good agreement between the CC and the observational data at present times, this model modifies the Equation of State (EoS) of the CC by adding an extra term which is a function of the energy density of the fluid at present. Therefore, the free parameters of the theory are constrained by the observational Hubble data (OHD) from differential age (DA) technique [35] and the joint-light-analysis (JLA) sample of SNIa [36].
The papers is organized as follows. In Sec. 2 we state the theoretical framework of the model presented in Ref. [1]. Section 3 provides a description of the dataset and methods used to constrain the parameters of the Chaplygin gas-like model. In Sec. 4 we discuss the results obtained and finally, in Sec. 5 the remarks and conclusions are presented.

Theoretical Background
The traditional form to obtain the Chaplygin gas is through the scalar field Lagrangian written in the form: where V (φ) is the scalar potential usually written in the form V (φ) = φ 2 ln φ 2 + V 0 and the action is associated with the tachyonic scalar field, φ, which couples with the U (1) gauge field living on the world volume theory of the non-BPS brane (see [37] for details). In the same sense, Hova and Yang [1] establish the connection of the Chaplygin gas with the tachyon scalar field through the assumption of a constant potential in the form V (φ) ∼ A 1/2 = V 0 , where A is related to the Chaplygin EoS; similarly happens for a generalized Chaplygin gas EoS (see also [1]). Following the recipe of [1], the generalized Chaplygin gas-like EoS is expressed as p df = −ρ df + ρ df sinc(µπρ df 0 /ρ df ), being sinc(x) ≡ sin(x)/x and ρ df the dark fluid density, which plays the role of the mixture of DE and DM densities. In this case µ is a dimensionless parameter and ρ df 0 is the present energy density of this fluid. It behaves as a CC in the late times of the universe evolution and as DM at the matter domination epoch. The evolution of the EoS of the dark fluid is given by where ξ(z) ≡ arctan[(z +1) −3 tan λ] and λ ≡ µπ/2. In order to explore the universe dynamics in this context, we consider a general Friedmann-Lemaître-Robertson-Walker (FLRW) metric including baryonic and radiation components, hence we write the Friedmann and acceleration equations as where H ≡ȧ/a is the Hubble parameter, k is the curvature parameter which depends on the universe geometry, and the index i runs over baryonic and radiation components.
here Ω df 0 ≡ 8πGρ df 0 /3H 2 0 is the density parameter associated with the Chaplygin gas-like fluid, Ω i0 and ω i are the density parameters and the EoS for baryonic matter and radiation 2 , Ω k ≡ −k/H 2 0 is the curvature density parameter and H 0 = h×100 km s −1 Mpc −1 . In addition, we have the constraint Ω df 0 +Ω b0 +Ω r0 = 1−Ω k . The deceleration parameter, q(z), is written in the form [1] where q(z) is computed by the definition q ≡ −äa/ȧ 2 , which written in terms of redshift and E(z) results q(z) ≡ −1 + (z + 1)E −1 (z)(dE(z)/dz). As a complement, we compute the jerk parameter which is dimensionless and defined as j = ... a /aH 3 : where E(z) and q(z) come from Eqs. (2.5) and (2.6) respectively, and , . (2.11) Note that we have followed the positive sign definition of the jerk parameter as [40]. Commonly, this quantity provide information on the possible evolution of any DE component. Thus, if its value is j = 1, the DE behaves as CC, otherwise it is a dynamical dark energy fluid. In addition, from Eqs. (2.5) and (2.6) it is possible to calculate an effective EoS containing the contributions of the Chaplygin gas-like and the standard fields like baryons, radiation and the curvature term Finally, using the following expression we estimate the age of the universe for the Chaplygin gas-like model.

Data and methodology
In this section we introduce the observational data and methodology used to constrain the free parameters of the Chaplygin-like model.

Measurements of H(z) from cosmic chronometers
Some of the current estimation of the Hubble measurements are obtained from cosmic chronometers. In the literature, a cosmic chronometer is a passive-evolving galaxy, i.e. without ongoing star formation. The difference in age (related to H) is obtained by considering two of these galaxies with similar metallicities and separated by a small redshift interval [35]. The data provided by the DA method are cosmological-model-independent and can be used to probe alternative cosmological models. Here, we use the latest OHD obtained from DA, which contains 31 data points covering 0 < z < 1.97, compiled by [41] and references therein. The chi-square for the OHD is written as where H(z i ) is the theoretical Hubble parameter related to Eq. (2.5), H DA (z i ) is the observational one at redshift z i , and σ H i its uncertainty. Notice that in the chi-square formula we also consider the measurement of H 0 = 73.24 ± 1.74 Kms −1 Mpc −1 [42] as a Gaussian prior.

Type Ia Supernovae
We use the JLA compilation by Ref. [36] consisting in 740 SNIa in the range 0.01 < z < 1.2. The observational distance modulus is computed as where m B is the observed peak magnitude in rest-frame B band, X 1 is the time stretching of the light-curve, and C is the supernovae color at maximum brightness. The M B parameter is defined as Thus, we have two free parameter, M 1 b , and δ M . The quantities a, and b are nuisance parameters in the distance estimate. On the other hand, the theoretical distance modulus is given by µ th = 5 log 10 (d L / 10 pc), being d L = (1 + z)D M , the luminosity distance predicted by the Chaplygin-like model and D M is for Ω k > 0 for Ω k = 0 for Ω k < 0 The chi-square for SNIa data can be calculated as where C η is the covariance matrix of the measurements provided by [36].

Joint analysis
To provide stronger constraints, we also perform a joint statistical analysis by combining the OHD and SNIa datasets. The chi-square function results as In the following section, we present our results of the parameter estimation for the Chaplyginlike gas models.

Results
We test two models: one is a flat universe and the other one has a curvature term Ω k = 0.
To estimate the free model parameters we perform a Bayesian analysis employing an Affineinvariant Markov chain Monte Carlo (MCMC) method provided in the emcee Python module [43] for three data sets: OHD, SNIa and its joint analysis (i.e. OHD+SNIa). We consider a burn-in phase which is stopped when the converge is achieved, which is done by requesting that the Gelman-Rubin test is less than 1.07 for all parameters [44]. Then, we set 6000 MCMC steps with 500 walkers. We consider Gaussian priors for h and Ω b0 h 2 centered at h = 0.723 ± 0.017 and Ω b0 h 2 = 0.02202 ± 0.00046, and flat priors over µ and Ω k in the range 0.60 < µ < 1.0 and −1.0 < Ω k < 1.0 respectively. The lower limit for µ is established to be consistent with bounds on the age of the universe of t A > 11 − 12 Gyrs [1]. Table 1 provides the best fit values and their corresponding uncertainties at 68% CL for both geometries of the universe. The different data sets estimate consistent values on the µ parameter and the chi-square values (χ 2 min ) indicate a good-fit of the data. The joint constraint, µ = 0.843 +0.014 −0.015 , is within 2.4σ to the value chosen as initial condition by [1] to obtain late cosmic acceleration. On the other hand, our constraints on the curvature term under this Chaplygin-like cosmology are consistent, within 3σ, with the estimated Ω k = −0.052 +0.049 −0.055 from the Planck measurements of the CMB temperature spectra [45]. Figure 1 (Figure 5) shows the 1D marginalized posterior distributions and the 2D 68%, 95%, 99.7% confidence levels (CL) for the Ω b0 , h, µ, and (Ω k ) parameters for a flat (curved) universe. In the flat universe, the correlations of µ with Ω b0 and h are ρ(µ, Ω b0 ) = −0.37 and ρ(µ, h) = 0.44. The corresponding correlations in the non-flat universe are ρ(µ, Ω b0 ) = −0.17, ρ(µ, h) = −0.16, and ρ(µ, Ω k ) = −0.70. Following the notation in [46], the effects of µ over Ω b0 and h are negligible when the universe is curved, but with noticeable influence over the curvature component.
Taking into account the best fit values of the model parameters obtained from the joint analysis, we compare the H(z) and the q(z) reconstruction between the spatially flat and curved universes and found that there is an agreement (within 2σ) in the region 0 < z < 2.0 (see Fig. 2). For the flat (curved) universe, the deceleration parameter at the present epoch is q 0 = −0.67 ± 0.02 (−0.51 ± 0.07), which is consistent with the concordance model q ΛCDM 0 = −0.54 ± 0.07, calculated from the ΛCDM mean values obtained by Ref. [15]. We obtain a similar redshift, z t = 0.57 ± 0.04 (0.50 ± 0.06), for the deceleration-acceleration transition in both geometries, which is consistent within 2.5σ with z t = 0.64 +0. 11 −0.06 obtained by [47] from cosmic chronometers and baryonic acoustic oscillations data for an open universe. Based on the EoS reconstruction of the dark fluid (Eq. 2.2), its behavior for both flat and non-flat cases at recent times is consistent with quintessence region and also confirms the Universe acceleration (see Fig. 3). In addition, for both models, the effective EoS (Eq. 2.12) at z 2 is achieved for ω ef f → 0, indicating that the dynamics of the universe is dominated by a non-relativistic fluid, which is consistent with our hypothesis of the Chaplygin gaslike. Moreover, the ω df behavior at z 0.5 also confirms that the dark fluid behaves like a quintessence field, which dominates the dynamics of the universe. On the other hand, the jerk parameter, presented in Fig. 4, shows a clear deviation, more than 3σ CL, with respect to a perfect fluid (jerk equal to one) in a flat universe; this reinforces the idea of a dynamical DE. However, for a non-flat universe the jerk parameter may mimic the perfect fluid within 3σ CL.
We estimate the universe age by using the expression (2.13) and the joint analysis, obtaining t A = 13.108 +0.270 −0.260 Gyrs for a flat geometry and t A = 12.314 +0.500 −0.430 Gyrs for a curved one. The results are, as expected, in agreement with the values reported by [45], t Planck A = 13.799 ± 0.021 Gyrs, assuming a ΛCDM model.
To statistically compare both, flat and non-flat models, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC) are given in Table 2. We also provide the difference with respect to the minimum value for each data set. From the joint analysis, the minimum AIC and BIC values are those for the non-flat model. Thus, if the universe is filled with a Chaplygin-like fluid instead of DM and DE, a non-flat geometry is preferred for this combination of data. However, the model in both geometries are in good agreement with the observational data used.

Conclusions
This paper is focused on the viability of a Chaplygin gas-like fluid in a curved space-time to resemble the current Universe dynamics. Inspired in the scheme of a Chaplygin gas, i.e. a unique fluid formed with the mixing of the DM and DE components, the model proposed by [1] is a modified perfect fluid that behaves as dust in the early epochs of the universe and as DE (CC) at recent times. An advantage of this kind of models is their ability to reproduce the Universe dynamics, without the need of a DE component of unknown nature, by adding an extra term on the perfect fluid EoS. We used the latest observational Hubble data from cosmic chronometers and the type Ia SN JLA compilation to constraint the cosmological parameters under this cosmology. We showed that in a flat universe the acceleration is variable, presenting a phase change at z t = 0.57 ± 0.04. The jerk parameter    with those expected for ΛCDM. The effective EoS has a dust behavior at redshifts higher than ∼ 1.5, acting as dark matter and behaving like a fluid that fulfills the relation ω < −1/3 at redshift below ∼ 0.57. It is worth to notice that ω ef f (0) ∼ −0.8, entering the quintessence regime. We also report an estimate of universe age of about 13.108 Gyrs for a flat geometry. In the context of a curved geometry of the universe, the jerk parameter of Chaplygin gas-like fluid is consistent at 3σ with j = 1 for CC in the region of 0 < z < 2. We observe a consistent (within 1σ) behavior of the dark fluid EoS (and also of the universe) between both geometries, i.e., the dark fluid also enters to the quintessence regime about z ∼ 0.57 and we estimate an universe age of 12.314 Gyrs. Our best value Ω k = 0.183 +0.073 −0.079 is compatible within 3σ to the one reported by the Planck Collaboration. Finally, our results underscore the importance of the Chaplygin-like gas model as a plausible alternative to shed light onto the DE and DM nature.