Exploring the deviation of cosmological constant by a generalized pressure dark energy model

We bring forward a generalized pressure dark energy (GPDE) model to explore the evolution of the universe. This model has covered three common pressure parameterization types and can be reconstructed as quintessence and phantom scalar fields, respectively. We adopt the cosmic chronometer (CC) datasets to constrain the parameters. The results show that the inferred late-universe parameters of the GPDE model are (within $1\sigma$): The present value of Hubble constant $H_{0}=(72.30^{+1.26}_{-1.37})$km s$^{-1}$ Mpc$^{-1}$; Matter density parameter $\Omega_{\text{m0}}=0.302^{+0.046}_{-0.047}$, and the universe bias towards quintessence. While when we combine CC data and the $H_0$ data from Planck, the constraint implies that our model matches the $\Lambda$CDM model nicely. Then we perform dynamic analysis on the GPDE model and find that there is an attractor or a saddle point in the system corresponding to the different values of parameters. Finally, we discuss the ultimate fate of the universe under the phantom scenario in the GPDE model. It is demonstrated that three cases of pseudo rip, little rip, and big rip are all possible.

of the GPDE model. The discussion of fixed points under the GPDE model is analyzed in Sec. IV. In Sec. V, we exhibit the end of the universe under the phantom case. The last section Sec. VI is the conclusion.

II. THEORETICAL MODEL
Pressure parameterization describes our universe in the following ways: First, hypothesize a relationship between the pressure P and the redshift z. Then the expression of the density ρ can be derived from the conservation equatioṅ ρ + 3(ȧ/a)(ρ + P ) = 0. Finally, by utilizing the Friedmann equations H 2 = 3/(8πG) i ρ i and the EoS ω = P/ρ, we can get the form of the Hubble parameter H and ω, respectively. Here we take the speed of light as c = 1. At this point, a closed system of cosmic evolution has been established which is described by the Friedmann equations, the conservation equation, and the EoS form. It is worth noting that there are still some deviations between the ΛCDM and actual (e.g., H 0 tension), but the physical mechanism behind it is not clear. Taking advantage of this kind of handwritten model, we can probe the possible deviations further between the dynamic case and the cosmological constant case without a specific premise. In this work, we propose a generalized pressure dark energy model of the total energy components in a spatially flat Fridenmann-Robertson-Walker (FRW) universe Where P 1 , P 2 and β are free parameters. Notice that P 1 is the current value of the total pressure in the universe, and P 2 represents the deviation of P (z) − z. The model degenerates into the ΛCDM model as P 2 = 0. To mention, this parametric form of P (z), i.e. Eq. (1), is inspired by a generalized equation of state for dark energy [34]. When specific limits are given to β, this model returns to the three models mentioned in Sec. I, i.e.
By using Eq. (1), the relationship of scale factor a (a = 1/(1+z)) and the conservation equation (ρ+3(ȧ/a)(ρ+P ) = 0), we can get the density as Where C is the integral constant. We assume ρ 0 is the current total density, i.e. ρ(a = 1) = ρ 0 . Finally, the total density and total pressure can be respectively sorted into the following form Parameters P 1 and P 2 have been replaced here by new parameters P a and Ω m0 , where P a ≡ 3P 2 /((3 + β)βρ 0 ), Ω m0 ≡ (βρ 0 + P 2 + βP 1 − 3P 2 /(3 + β))/(βρ 0 ). In the density expression (4), the item ρ 0 Ω m0 a −3 is corresponding to the matter density ρ m . So Ω m | a=1 = ρ m /ρ = Ω m0 signifies that the physical meaning of the parameter Ω m0 is the present-day matter density parameter. The term ρ 0 (P a a β + 1 − P a − Ω m0 ) accords with the dark energy density ρ de , and the constant part 1 − P a − Ω m0 looks similar to the ΛCDM case. The term P a a β makes ρ de change with time: The larger the |P a | is, the more deviation from the ΛCDM model will be; The larger the β is, the faster the dark energy density will change. Accordingly, this cosmological model only includes matter and dark energy components, and the pressure of dark energy P de is the total pressure P . From (ρ de /ρ m )| a→0,β>−3 → 0 we can also know that for the case of β > −3, the DE accounts for a small percentage in the early universe. Note that when β = −3, the density of the part of the dark energy is expressed as the matter density, and the total density ρ(a) of our GPDE model is equivalent to the ΛCDM model. Suppose the dark energy is a scalar field φ that changes with time. The corresponding pressure and density are equivalent to ρ de = (n/2)φ 2 + V (φ) and P de = (n/2)φ 2 − V (φ), separately, where n = 1 or −1 corresponds to the quintessence and phantom scalar field, respectively. The calculation shows thatφ 2 = −(nβ/3)ρ 0 P a a β , so βP a < 0 fits quintessence, and βP a > 0 matches for phantom.
Additionally, for the GPDE model, the EoS of the dark energy ω de , the dimensionless Hubble parameter E, the deceleration parameter q and the jerk parameter j respectively take the form as q ≡ −ä III. RESULTS OF THE DATA ANALYSIS By measuring the age difference between two galaxies under different redshifts, we can get the Hubble constant H(z), called cosmic chronometer data. In this section, We constrain our parameter by 33 unrelated cosmic chronometer data listed in table I, spanning the redshift range 0 < z < 2. The optimal values of the parameters can be obtained by taking the minimum value of χ 2 , which is expressed as with the corresponding four-dimensional parameter space {H 0 , Ω m0 , P a , β}.  As the first attempt, we adopt the Monte Carlo Markov chain (MCMC) method and use the python package emcee [53] to produce a MCMC sample with CC data. The results are displayed as a contour map by another python package pygtc [54]. We list the priors and initial seeds on the parameter space in Table II. Figure 1 shows the 1-dimensional and 2-dimensional marginalized probability distributions of the GPED model. In the meantimethe best-fit values and 1σ confidence level for the H 0 , Ω m0 , P a and β are listed in Table III. From the constraint results, P a β < 0 in 1σ, which indicates our universe is under quintessence situation and has some deviation from the ΛCDM model in a point of view of data. The differences of H 0 between our results and SHoES [13] and Planck base-ΛCDM [10] are 0.9 σ and 3.5 σ, respectively. The evolution of DE EoS parameter ω de , DE density parameter Ω de , deceleration parameter q and jerk parameter j with 1 σ error propagation from data fitting (Table III) are shown in figure 2.
From figure 2 we find that these results are acceptable, except for the DE density parameter Ω de , which is too high at the beginning of the universe and contradicts the facts we now know. For this reason, we make a further try: while     figure 3. Table IV lists the best-fit values and 1σ confidence level for the P a and Ω m0 under CC data and the H 0 data joint constraints. We discover that for the six kinds of circumstances, the best values of Ω m0 are all around 0.33, and βP a > 0 indicates that the universe is slightly biased toward phantom but still includes quintessence within 1σ confidence level. The value of P a is small, meaning in this case the deviation of this model from the ΛCDM model is not significant. The minimum χ 2 of these six cases are very close, implying that this model is not very sensitive to the selected values of β, that is, the degeneracy is high. Figure 4 shows the difference in export parameters ω de , Ω de , q and j between the GPDE model and the ΛCDM model with the best-fit values for the considered values of β. It can be concluded from figure 4 that the distinction between these two models is almost indistinguishable. The exceptions are for the cases of β = −1.5 and −1, whose ω de rapidly increase and decrease at the beginning of the universe, and then stabilize at a position slightly less than −1. Unlike the first attempt, these fitting results show that our model is consistent with the ΛCDM model. The reason is estimated to be that in the second try, the data of H 0 is from Planck, which depends on the ΛCDM model.  Here we also quote χ 2 min in every case of β. Line 3 and line 5 are 1σ errors of Pa and Ωm0, respectively.

IV. DYNAMIC ANALYSIS
In this section, we will construct a self-consistent dynamical system to analyze the cosmological evolution of the GPDE model. Select Ω de and E as independent variables. By Friedmann equation, we can get the self-consistent dynamic system as where P b ≡ 1 − P a − Ω m0 , " " represents the derivative of ln(a). The following are common methods for finding fixed points and its stability of a system. Let Ω de = E = 0, then for P b ≥ 0, there is a fixed point {Ω de = 1, E = P 1/2 b } in the dark energy dominant period. Do the perturbation expansion of the system near the fixed point and then we can get the Jacobian matrix Its eigenvalues are λ 1 = −3 and λ 2 = β. The stability of the system depends on the sign of the eigenvalues λ 1 and λ 2 . For the GPDE model, β is a real number and not equal to zero, so there are two situations: When β < 0, point {1, P 1/2 b } is an attractor, and when β > 0, point {1, P 1/2 b } is a saddle point. While for P b < 0, we can determine the properties of the fixed point by observing its figure. On the other side, for the case of ΛCDM, the parameter P a = 0, so βΩ de − P b β E 2 = β Paa β E 2 = 0, the dynamic system goes to Figure 5 illustrates the evolution of the dynamics system for β = −1, 1 and P b = −1, 0, 1. There also exhibits the evolutional trajectories of the ΛCDM model as a comparison in the last row. From figure 5, it can be found that there is a saddle point {1, 0} for P b < 0. When β > 0 and P b < 0, it shows that E → ∞ and Ω de → 1, which is the case of phantom. While for β < 0 and P b < 0, one can see Ω de → 0 which corresponds to quintessence. When P b ≥ 0, whether the universe is biased towards quintessence or phantom depends on β and initial conditions which cannot be judged directly. Based on the above analysis, we summarize the stability of the GPDE model in table V.  Table IV. The shaded region and blue lines represent the 1σ level regions and corresponding boundaries in the GPDE model.

V. UNIVERSE FATE UNDER THE PHANTOM FIELD
In recent years, various data have shown that the dark energy state equation is close to 1: the WMAP9 have found ω = −1.073 +0.090 −0.089 based on WMAP+CMB+BAO+H 0 [4]. Planck-2018 constrained the EoS as ω = −1.03 ± 0.03 with SNe [10]. Moreover DES suggests ω = −0.978 ± 0.059 in the joint analysis of DES-SN3YR+CMB [55]. Using existing data, it is still impossible to distinguish between phantom (ω < −1) and non-phantom(ω ≥ −1), that is, the phantom case cannot be excluded. Phantom has gradually increased energy density, and eventually, the universe accelerates so fast that the particles lost contact with each other and rip apart. Based on the various evolutionary behaviors of H(t), the final fate of the universe can be divided into the following three categories [56]: 1 Big rip: H(t) → ∞ as t → constant, so the rip will happen at a certain time.
2 Little rip: H(t) → ∞ as t → ∞. This situation has no singularities in the future.
3 Pseudo rip: H(t) → constant. This is the case with the de-Sitter universe and little rip.
In this section, we are interested in the rip case of our GPED model. For the GPED model, Hubble constant is written as In section II, we have pointed out that βP a > 0 for phantom case of the the GPED model. Next we discuss each situation separately.
1 β < 0, P a < 0: With the growth of cosmic time, a → ∞ and i.e. H(t) tends to be a constant, and the universe will approach the de-Sitter universe infinitely. So this case is pseudo rip.
2 β > 0, P a > 0: When a → ∞, By solving the above differential equation, we can get the relation between scale factor a and time t, which can be written as Where t 0 is the present value of time. Substitute Eq. (19) for Eq. (18), one obtains So when t → 2 βP 1/2 a H0 + t 0 , H(t) → ∞ which means the universe will have a big rip after t − t 0 = 2 βP 1/2 a H0 . Thus, the lifetime of the universe is determined by two parameters, β and P a , regardless of the matter density Ω m0 . Let us make a rough estimate. Take β = 1, P a = 0.02, H 0 = 70km s −1 Mpc −1 , then the universe will be torn apart after 198Gyr. If 13.8Gyr is the current age of the universe, then the universe has spent only 6.5% of its life.
3 β → 0: In this case, the GPDE model degenerates into the model in [43]. According to the discussion in [43], the ultimate fate of the universe is the little rip.
To sum up, there are three possible fates under the phantom of the GPDE model: 1 Big rip for β > 0, P a > 0; 2 Little rip for β → 0; 3 Pseudo rip for β < 0, P a < 0.

VI. CONCLUSIONS
In principle, it is interesting to insert models or theories into a more general framework to test their validity. Not only does this reveal a new set of solutions, but it may also enable more accurate consistency checks for the original model. This paper has made this attempt by expanding the ΛCDM model to a generalized pressure dark energy (GPDE) model. The GPED model has three independent parameters: The present value of matter density parameter Ω m0 , the parameter P a which represents the deviation from the ΛCDM model, and the parameter β. Picking different values of parameter β can produce three common pressure parametric models. By using the cosmic chronometer (CC) datasets to constrain parameters, it shows that Hubble constant is H 0 = (72.30 +1.26 −1.37 )km s −1 Mpc −1 . And the differences of H 0 between our results and SHoES [13] and Planck base-ΛCDM [10] are 0.9 σ and 3.5 σ, respectively. In addition, for the GPDE model, the matter density parameter is Ω m0 = 0.302 +0.046 −0.047 , and the universe bias towards quintessence in 1σ error. While when we combine CC datasets and the H 0 data from Planck, the constraint implies that our model matches the ΛCDM model well. Then we explore the fixed point of this model and find that there is an attractor or a saddle point corresponding to the different values of parameters. Next, we analyze the rip of the universe under phantom case and draw the conclusion that there are three possible endings of the universe: Pseudo rip for β < 0, P a < 0, big rip for β > 0, P a > 0 and little rip for β → 0. Finally, we estimate that for the big rip case, the universe has a life span of 198Gyr.
Dark energy has been proposed for twenty years, but its nature remains unknown. With this model, we can probe the possible deviation further between the dynamic case and the cosmological constant condition through existing data.