Exploring the low redshift universe: two parametric models for effective pressure

Astrophysical observations have put unprecedentedly tight constraints on cosmological theories. The $\Lambda$CDM model, mathematically simple and fits observational data-sets well, is preferred for explaining the behavior of universe. But many basic features of the dark sectors are still unknown, which leaves rooms for various nonstandard cosmological hypotheses. As the pressure of cosmological constant dark energy is unvarying, ignoring contributions from radiation and curvature terms at low redshift, the effective pressure keeps constant. In this paper, we propose two parametric models for non-constant effective pressure in order to study the tiny deviation from $\Lambda$CDM at low redshift. We recover our phenomenological models in the scenarios of quintessence and phantom fields, and explore the behavior of scalar field and potential. We constrain our model parameters with SNe Ia and BAO observations, and detect subtle hints of $\omega_{de}<-1$ from the data fitting results of both models, which indicates possibly a phantom dark energy scenario at present.


I. INTRODUCTION
Since the discovery of the current acceleration of our universe expansion in 1998, maybe the greatest mystery in cosmology today is the deceptive nature of the dark energy. Recent observational results [1] have put tight constraints on the properties of dark energy, but there is still no theoretical or observational indication to pin down the nature of the dark energy. On one hand, although the simple cosmological constant Λ can accommodate the accelerating expansion, it faces two serious problems. The first one is the fine tuning problem: the measured energy of the vacuum so much smaller than the estimated value ρ obs vac ≪ ρ theo vac , which is the famous 120-orders-of-magnitude discrepancy that makes the vacuum explanation suspecious. The second one, that is, why the dominance of the cosmological constant over the matter component at the present epoch. These two basic problems prompt us to propose some alternatives, which include an evolving scalar field called quintessence, noncanonical scalar field (such as K-essence [2][3][4], phantom [5][6][7][8]), modified gravity [9][10][11][12], coupled dark energy [13,14] or decaying dark energy [15] and so on. On the other hand, as we know the equation of state (EoS) parameter of the cosmological constant is precisely ω de = −1.
Recent observational data show that the EoS parameter of modeled dark energy is ω de = −1.13 0. 23 −0.25 , which mildly favour a lower value ω de < −1. Anyhow the small deviations from the cosmological constant Λ allow one to consider models with ω de = −1. So one can make efforts to construct new models to explain the deviations which may be detectable at the precision of current and future observations. Parameterization is an useful tool towards a more complete characterization of dark energy modelling and has been routinely employed to analyze dataset. Most parameterizations for dark energy models involve the EoS parameter ω de for the dark energy behavior. Several well-known parameterizations for the EoS of dark energy have been proposed so far. We can write such parameterizations in the form ω de (z) = n=0 ω n x n (z) generally, where the expansions can be given by the following forms, (i)Redshift: x n (z) = z n , (ii)Scale factor: x n (z) = (1 − a a0 ) n = ( z 1+z ) n , (iii)Logarithmic: x n (z) = [ln(1 + z)] n . Parameterization (i) was proposed by Huterer and Turner [16] and Weller and Albrecht [17] with n ≤ 1. Parameterization (ii) with n ≤ 1 was introduced by Chevalier, Polarski and Linder [18,19], which is the most famous Chevallier-Polarski-Linder(CPL) parameterization ω de = ω 0 + ω 1 (1 − a) = ω 0 + ω 1 z 1+z , which behaves as ω de → ω 0 + ω 1 for z → ∞ and ω de → ω 0 for z → 0, later a more general form with ω de = ω 0 + ω 1 z (1+z) p was proposed by Jassal, Bagla and Padmanabhan [21]. Parameterization (iii) with n ≤ 1 was introduced by Efstathiou [20]. In recent years, some new parameterizations have been proposed, such as using Padé parameterizations for the EoS of dark energy [22], namely ω de = ω0+ωa(1−a) 1+ω b (1−a) , ω de = ω0+ω1 ln a 1+ω2 ln a . It is worth mentioning that several years ago Sen proposed a parameterization for the pressure of dark energy model [23,24]:P Λ = −P 0 + P 1 (1 − a) + .... to study the small deviations from the cosmological constant. Different from parameterizations concentrating on the EoS of dark energy mentioned previously, in this paper we aim to make parameterizations for the total pressure functions of dark energy, that is two parametric models for the total pressure written by hand are proposed in order to explore the universe at the late-time evolution stage. This paper is organized as follows: in section II, we propose two new parametric models for the total pressure: P (z) = P a + P b z and P (z) = P c + P d 1+z . In Section III, we mimic our models in the scenario of two type of scalar fields-quintessence and phantom. In Section IV, we constrain our model parameters with Supernova type Ia dataset and BAO dataset. In Section V, We end the paper by discussions and conclusions.

II. TWO PARAMETRIC MODELS
A fluid with a constant negative pressure behaves like the cosmological constant Λ. This can be easily obtained by substituting the constant pressure P 0 into the energy conservation equatioṅ ρ + 3H(P + ρ) = 0 (1) then integrate it out and obtain the expression of ρ. In this section, two parametric models for the total pressure at low redshift are proposed.
A. Model 1 In this subsection, we propose a model which reads: where P a and P b are model parameters, z is the redshift.
Consider the relation between the scale factor a and the redshift z: Here we set a 0 = 1 corresponding to the value today. Substitute Eqs. (2)and (3) to Eq. (1), we can obtain the expression of the total energy density as a function of scale factor: where C 1 is an integration constant. If we set the total energy density ρ 0 today, then the integration constant is , not difficult to find term C 1 a −3 corresponds to dust matter, and term −(P a − P b ) is a constant which seems like the cosmological constant in the ΛCDM model. As to term -3 2 P b a −1 , which is nonexistent in the ΛCDM model, its physical nature will be explored in next section.
In order to make data fitting more clearly and conveniently, we need to introduce new parameters. First, we define new dimensionless density and pressure as follow: The expressions of the total density Eq. (4) and total pressure Eq. (2) can be rewritten as: We propose another model which reads: where P c and P d are model parameters. Substitute Eqs. (3) and (11) to Eq. (1), one can obtain the expression of the total energy density for model 2: where C 2 is the integration constant. Set the present energy density ρ 0 , then C 2 = ρ 0 + P c + 3 4 P d . Still one can find term C 2 a −3 corresponds to dust matter, and term −P c is a constant which seems like the cosmological constant in the ΛCDM model, compare model 2 with model 1, we find the difference between the two models appears in term − 3 2 P b a −1 and term − 3 4 P d a. We will explore their physical nature in next section. Like model 1, we need to introduce new model parameters in model 2. With Eqs. (5) and (6), we can obtain the expressions of total density and total pressure for model 2: where

III. IN THE PHYSICAL SCENARIO OF SCALAR FIELDS
Deviation from the ΛCDM model in our models can be mimicked through different physical scenarios. Scalar fields is a mainstream approach to explain the acceleration of the universe expansion, in the scenario of scalar fields the cosmological constant Λ is exactly zero and the dark energy evolves with time. In this section, we will take "quintessence" and "phantom" as two examples to mimic our models.
Quintessence: "Quintessence" denotes a canonical scalar field φ with a potential V 1 (φ) that interacts with all the other components only through standard gravity, its EoS parameter ω de > −1. The quintessence is described by the action: where κ 2 = 8πG, R is the Ricci scalar and S M is the matter action. The variation of the action Eq. (17) with respect to φ givesφ where V 1 (φ) is the potential of the quintessence field, the prime denotes the derivative with respect to φ. In a FLRW background, the energy density ρ de and the pressure P de of the quintessence field are which give the EoS Phantom: Minimally coupled phantom model is also a possible realization, its EoS parameter ω de < −1. The action of the phantom field minimally coupled to gravity and matter sources is Calculate the variation of the action of Eq. (22), then we obtain the equation of motion corresponding to such a scalar field:φ where V 2 (φ) is the potential of the phantom field, the prime denotes the derivative with respect to φ. The energy density and pressure of the phantom are given by(assuming flat FRW metric) The EoS of the phantom field is One has ω de < −1 for 1 2φ 2 < V 2 (φ).
A. Model 1 Consider a system with two fluids: matter and scalar fields, then one can write down the EoS of the scalar fields for model 1: Note that in above equation, there will be a singularity coming out when z = − α β − 1, in this paper we aim to consider the universe at low redshift, so we need not to worry about that situation, besides in section IV data fitting will support our argument.
Model mimicry demands that the expression of the total pressure and total density in one model can be realized in another model. In the scenario of quintessence, assume the cosmic components consist of matter and quintessence, compare Eq. (19) and Eq. (20) with Eq. (4) and Eq. (2), and then we have Simplify the above two equations, refer to Eqs. (5) (6) (7) (8) (9) (10) replace model parameters (P a , P b ) with redefined parameters (α , β), then: By the above two equations, one can construct the kinetic energy 1 2φ 2 and potential V 1 (φ) of the quintessence field with parameters (α , β) of model 1.
In the case of phantom scenario, assuming the cosmological components consist of matter and phantom, compare Eq. (24) and Eq. (25) with Eq. (4) and Eq. (2), then we have Simplify the above two equations, replace model parameters (P a , P b ) with redefined parameters (α , β), we have By the above two equations, one can construct the kinetic energy 1 2φ 2 and potential V 2 (φ) of the phantom field with model parameter (α , β), then realize model 1 in a minimally coupled phantom scenario. Compare Eq. (30) and Eq. (31) with Eq. (34) and Eq. (35), one can find that the difference between quintessence scenario and phantom scenario appears in the expression of kinetic energy 1 2φ 2 , β > 0 corresponds to a quintessence dark energy scenario and a negative value of β < 0 corresponds to a phantom dark energy scenario, no matter in a quintessence scenario or a phantom scenario the expression of potential has the same form.

B. Model 2
Write down the EoS of the scalar fields for model 2: It is obvious that only when parameters (γ, δ) are opposite sign, there will be a singularity coming out when z = −1 − δ γ , in section IV data fitting results will show that such a singularity would not come out at low shift. In the scenario of quintessence, compare Eq. (19) and Eq. (20) with Eq. (12) and Eq. (11), we can obtain Simplify the above two equations, refer to Eqs. (5) (6) (13) (14) (15) (16) replace model parameters (P c , P d ) with redefined parameters (γ , δ), then: By the above two equations, the kinetic energy 1 2φ 2 and potential V 1 (φ) of the quintessence field are constructed with parameters (γ , δ) of model 2.
In order to mimic model 2 in a phantom scenario, compare Eq. (24) and Eq. (25) with Eq. (12) and Eq. (11), we can obtain Simplify the above two equations, replace model parameters (P c , P d ) with redefined parameters (γ , δ), we have By the above two equations, one can construct the kinetic energy 1 2φ 2 and potential V 2 (φ) of the phantom field with parameters (γ , δ) of model 2. Compare Eq. (39) and Eq. (40) with Eq. (43) and Eq. (44), one can find that corresponding to a quintessence dark energy scenario one has δ < 0 and corresponding to a phantom dark energy scenario one has δ > 0, the expression of potential have the same form.

IV. ASTROPHYSICAL DATA CONSTRAINTS
In this section, we will use the observational SNe Ia dataset and BAO dataset to obtain constraints for model parameters.
The corresponding χ 2 SN function is calculated from where µ obs (z i ) and σ i are the observed value and the corresponding 1σ error of distance modulus for each supernova.
The minimization with respect to µ 0 can be made trivially by expanding χ 2 SN as where Thus µ 0 is minimized as µ 0 = B C by calculating the following transformed χ 2 : Baryon Acoustic Oscillations The baryon acoustic oscillation (BAO) dataset listed in Table (I), we use the parameter A measuring the BAO peak in the distribution of SDSS luminous red galaxies, following A is defined as where z b = 0.35. The χ 2 for BAO data is The total χ 2 is given by The best fitting model parameters are determined by minimizing the total χ 2 . The best fitting results and the reduced χ 2 for model 1 and model 2 are listed in Table (II), besides the confidence levels of parameter (α , β) and (γ , δ) are shown in Fig. (1) and Fig. (2), respectively.

V. CONCLUSION
Since cosmological observations have confirmed the late time acceleration of the Universe expansion more than ten years ago, different kinds of models have been proposed to explain the source of this acceleration. Parameterization is a widely used method to help us understand a more complete characterization of dark energy and is routinely employed to analyze data, to optimize survey design and to compare results. In this paper, we have studied two parametric models for the total pressure P (z) = P a + P b z and P (z) = P c + P d 1+z at low redshift. Deviation from the ΛCDM model can be mimicked through different physical scenarios. Simply there are two ways to consider: one way is to assume there exists the cosmological constant Λ plus a small but nonzero component, after all we do not have any reasons to make the cosmological constant Λ vanish, in this case different kinds of models have been studied, such as viscous cosmology [26][27][28][29], cosmic strings [24,30] and so on. The other way is to assume the cosmological constant Λ is exactly zero and the dark energy evolves with time, scalar fields is a mainstream approach to explain the acceleration of the universe. In this paper, we consider the second, for which scarlar fields possibility. We mimicked model 1 and model 2 in the scenario of two type of scalar fields-quintessence and phantom, the kinetic energy term 1 2φ 2 and potential V (φ) have been constructed with model parameters (α , β) and (γ , δ) respectively. We have constrained model parameters (α , β) and (γ , δ) with Supernova type Ia dataset and BAO dataset. The best fitting results together with 1σ and 2σ confidence levels for parameters of the two models show that model 1 and model 2 both mildly support that the EoS parameter of dark energy ω de < −1 which corresponds to a phantom dark energy scenario at present, still we can not eliminate a quintessence dark energy scenario or a Λ dark energy scenario.
Though our parametric models can fit astrophysical observations very well, the model 1 and model 2 are just two parametric models at low redshift. Next we will study some more suitable parametric models to explore the evolution of the universe at the late age.