Is the cosmological dark sector better modeled by a generalized Chaplygin gas or by a scalar field?

Both scalar fields and (generalized) Chaplygin gases have been widely used separately to characterize the dark sector of the Universe. Here we investigate the cosmological background dynamics for a mixture of both these components and quantify the fractional abundances that are admitted by observational data from supernovae of type Ia and from the evolution of the Hubble rate. Moreover, we study how the growth rate of (baryonic) matter perturbations is affected by the dark-sector perturbations.


I. INTRODUCTION
The standard cosmological model, the ΛCDM model (Λ denotes the cosmological constant, CDM stands for cold dark matter), assumes the presently observed cosmic substratum mainly to consist of a cosmological-constant type dark-energy (DE) together with pressureless CDM. These dark components make up about 95% of the cosmic energy budget. The simple cosmological-constant model has been "dynamized" in several ways. Even before the advent of the observations of supernovae of type Ia (SNIa) by [3][4][5] which supported the idea of a universe in accelerated expansion, a scalar field (SF) has been suggested as an agent that might drive the cosmological dynamics [6]. Further studies along this line were performed in [7][8][9][10][11][12][13][14][15][16].
A fluid dynamical description which is able to account both for an early matter-dominated phase and for effects similar to those generated by a cosmological constant has been established in terms of (generalized) Chaplygin gases. The original Chaplygin gas [17] is characterized by an equation of state (EoS) p = − A ρ . It was applied to cosmology in [18] followed by [19,20]. A phenomenological generalization to an EoS p = − A ρ α with a constant α > −1 was introduced in [21], where also its relation to a scalar-field Lagrangian of a generalized Born-Infeld type was clarified. For α = 1 this generalization reduces to the original Chaplygin gas, for α = 0 it is related to the ΛCDM model. An appealing feature of the (generalized) Chaplygin gas (GCG) is its capability of a unified description of the dark sector. Its energy density is changing smoothly from that of nonrelativistic matter at high redshift to an almost constant far-future value. Thus it interpolates between an early phase of decelerated expansion, necessary for successful structure formation, and a late period in which it acts similarly as a cosmological constant, generating an accelerated expansion. Cosmological models relying on the dynamics of generalized Chaplygin gases have been widely studied in the literature [22][23][24][25][26][27][28][29][30][31][32][33].
While both SF based models and models aiming at a unified description of the dark sector of the type of GCGs have separately attracted ample attention, our aim in this paper is to investigate a model in which a GCG and a SF are simultaneously present (GCSF model) in addition to a pressureless matter component which is supposed to describe the baryon fraction of the Universe. By suitable parameter choices the GCSF model has two ΛCDM limits which allows us to investigate deviations from the latter in various directions. We use SNIa and H(z) data to test whether the observations admit a dark sector of the GCSF type. The SF dynamics will be described with the help of the CPL parametrization [34].
While this may seen as a loss of generality, it has the advantage of providing us with an explicit analytic expression for the Hubble rate. The existence of an analytic solution of the background dynamics is essential for the perturbation analysis. This solution determines the coefficients of the system of coupled first-order perturbation equations.
The best-fit values of the background analysis are then used for a study of the growth rate of the (baryonic) matter perturbations. With the help of a simplifying parametrization of the dark-sector perturbations we investigate the impact of the latter on the matter-perturbation growth.
In Sec. II we recall the basic properties of the GCG and SF components of the dark sector and find the Hubble rate of the GCSF model. The background data analysis and its interpretation is the subject of Sec. IV. Section V is devoted to the sub-horizon dynamics of matter perturbations, while Sec. VI summarizes our results.

A. Cosmic medium as a whole
We assume a perfect-fluid structure of the cosmic medium as a whole, described by the energy-momentum tensor where ρ = T ik u i u k is the total energy density, p = 1 3 T ik h ik is the total pressure and u i is the four-velocity of the cosmic substratum as a whole, normalized to u i u i = −1.

B. Decomposition into 3 components
The total energy-momentum tensor in (1) is split into a GCG (subindex c), a SF component (subindex s) and a matter component (subindex m), We assume perfect-fluid structures of each of the components as well (A = c, s, m) and separate energy-momentum conservation where ρ A = T ik A u Ai u Ak . In general, the 4 velocities of the components are different from each other and from the total four velocity u i as well.

C. Equations of state
The equation of state for the GCG is A simple scalar field (quintessence) is characterized by an EoS parameter ω q , This parameter is restricted to −1 ≤ ω q ≤ 1. Under more general circumstances, e.g., for non-minimally coupled scalar fields or scalar fields with a non-standard kinetic term a phantom-type EoS is possible as well. For the SF we use the effective fluid description which is supposed to cover both the quintessence and the phantom cases. As far as the matter component with is concerned, our main interest here is baryonic matter, but in some special cases below also CDM will be included.

A. Conservation equations
For a a homogeneous, isotropic and spatially flat universe with a Robertson-Walker metric, the total energy-momentum conservation in (1) reduces tȯ where H =˙a a is the Hubble rate and a is the scale factor of the Robertson-Walker metric.
In this background all the four-velocities are assumed to coincide, The energy conservation equations for the components arė B. Energy densities

Chaplygin gas
With the EoS (4) we obtain the energy density ρ c , where B is a non-negative constant, or where ρ c0 is the energy density for a = 1. An EoS parameter ω c is introduced via and the adiabatic sound speed byṗ ForĀ = 0 the GCG reduces to a pure matter component.

Scalar field in CPL parametrization
For the general EoS (4) one has which can either be quintessence with (5) or phantom matter with ω s < −1. The adiabatic For the purpose of this paper we adopt the frequently used CPL [34] parametrization ω s = with the help of which we have the explicit formula In this manner the scalar-field dynamics is reduced to a two-parameter fluid description.
Such approximation is expected to make sense close to the present time, i.e., for small redshift. The total EoS parameter of the cosmic substratum is

C. The Hubble rate
Friedmann's equation reads Introducing the fractional quantities the Hubble rate is given by With the explicit expression (21) the background dynamics is analytically known. The ΛCDM model is recovered both for α = 0 together with ρ s0 = 0 (no SF, the GCG accounts both for DE and CDM) and for A = 0 together with ω 0 = −1 and ω 1 = 0 (vanishing kinetic term of the SF, the GCG accounts for CDM only).

IV. OBSERVATIONS AND STATISTICAL ANALYSIS
Now we confront the CGSF model with data from SNIa and H(z) data. The five free parameters of the model are α, Ω s0 ,Ā, ω 1 and h, where h is defined by H 0 = 100h kms −1 Mpc −1 .
The parameter ω 0 will be fixed to either ω 0 = −1.0 or, alternatively, to ω 0 = −1.05 or To get a better understanding of the combined model, we shall also evaluate the limiting cases of a vanishing SF contribution (the GCG describes the entire dark sector) as well as the case of a SF with a certain amount of pressureless matter (no GCG).
We use the binned set of supernovae data from the JLA compilation [35]. This test relies on the observed distance modulus µ obs (z) of each binned SN Ia data at some redshift z, where the luminosity distance d L in a spatially flat Robertson-Walker metric, is given by the formula The binned JLA data set contains 31 data points. The corresponding χ 2 function is constructed according to where C is the covariance matrix [35].
As a second observational source we consider the evaluation of differential age data of old galaxies that have evolved passively [36][37][38][39][40]. Here we use the 36 measurements of H(z) listed in [41] which consist of 30 differential age measurements and 6 data from an analysis of baryon acoustic oscillations (BAO). The relevant relation here is The spectroscopic redshifts of galaxies are known with very high accuracy. A differential measurement of time dt at a given redshift interval allows one to obtain values for H(z).
The chi-square function for the analysis of the H(z) data is where N H is the number of data points and σ i is the observational error associated to each observation H obs while H th is the theoretical value predicted by the GCSF model.
Combining the information from both tests, we construct the total chi-square function as The one-dimensional probability distribution functions (PDF) are obtained from the likelihood function     A robust picture is obtained in terms of the one-particle PDFs in FIGS. 1-5. FIG. 1 shows the one-dimensional PDFs for α, Ω s0 ,Ā and ω 1 where h was fixed to its corresponding bestfit value and we assumed ω 0 = −1. The same analysis for ω 0 = −0.95 has been performed   for ω 1 . The circumstance that Ω s0 > ∼ 0.6 seems to indicate that the background data prefer a SF dominated dark sector over a GCG dominated configuration.
In the following section we use the best-fit values listed in the second line of TABLE II

V. MATTER PERTURBATIONS
In the absence of anisotropic stresses and in the longitudinal gauge scalar metric perturbations are described by the line element Denoting first-order variables by a hat symbol, the perturbed time components of the fourvelocities areû Under this condition the total first-order energy perturbation is the sum of the first-order perturbations of the components of the cosmic medium, ρ =ρ m +ρ c +ρ s .
Introducing the fractional quantities one has where with H 2 0 H 2 from (21). The equation for the fractional matter perturbations δ m in the quasi-static, sub-horizon approximation isδ where k is the comoving wave number. Einstein's field equations relate φ to the (comoving) energy-density perturbations via the Poisson equation Combining (37) and (38) does not result in a closed equation for the matter perturbations, since δ m is generally coupled to the perturbations of the other components. Formally, we may write Only for known ratios δc δm and δs δm there would be a a closed second-order equation for δ m . This means, to obtain δ m one has to solve the entire coupled dynamics of δ m , δ c and δ s .
To get a rough idea of how the fluctuations of the GCG and the SF components affect the matter fluctuations we introduce the simple parametrizations equivalent to with where ω C is defined in (13) and ω S = ω 0 +ω 1 (1 − a). All background coefficients in equation (42)   Apparently, any non-vanishing s leads to unacceptable strong deviations from the ΛCDM model (solid curve).

VI. SUMMARY
In this paper we allowed for a competition between a generalized Chaplygin gas (GCG) and a scalar field (SF) to find out which of these is preferred by the data as the dynamically dominating component of the cosmic substratum. Both GCG and SF based models have found ample use separately to describe the dark sector with the intention to "dynamize" the cosmological constant. To assess the background dynamics we used SNIa data from the JLA sample and H(z) data both from the differential age of old galaxies that have evolved passively and from BAO. Our combined CGSF model is characterized by 5 free parameters, where we used the CPL parametrization ω s = ω 0 + ω 1 (1 − a) for the scalar field. We fixed ω 0 and the baryon fraction. The general analysis with all 5 parameters left free indicates degeneracies in the parameter space. A two-step analysis in which the Hubble constant is fixed to its best-fit values by a suitable marginalization procedure results in a viable twocomponent model of the dark sector. The limiting cases of pure Chaplygin-gas and pure scalar-field dark sectors are consistently recovered. A robust picture is obtained in terms of the one-particle distribution functions for each of the parameters. The observation seem to prefer a SF fraction of more than 60%, leaving less than 40% for the GCG. Moreover we found a GCG parameter α close to zero and a present EoS parameter for the GCG between −0.6 and −0.7. The CPL parameter ω 1 is positive and of the order of one.
An approximate perturbation analysis reveals that for fluctuations of the GCG energy density that are of the same order as fluctuations of the matter density the growth rate of the latter remains close to the result for the ΛCDM model. it is the GCG component which governs the matter growth. A more advanced analysis with a detailed gauge-invariant perturbation theory will be the subject of a forthcoming paper.