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

TL;DR

This paper introduces a generalized pressure dark energy (GPDE) model to study universe evolution, showing it can mimic quintessence or phantom fields, with observational constraints indicating a universe leaning towards quintessence and potential future rip scenarios.

Contribution

The paper develops a unified GPDE model covering common pressure parameterizations, reconstructs it as scalar fields, and constrains it using cosmic data, revealing its compatibility with $ ext{Lambda}$CDM and possible future fates.

Findings

GPDE model fits late-universe data within 1$\sigma$

Model aligns with $ ext{Lambda}$CDM when combined with Planck data

Future universe scenarios include pseudo rip, little rip, and big rip

Abstract

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.

Tables

Table 1. Table 1: Cosmic chronometers data.

$z$	$H\left(z\right)$	${\sigma }_{H\left(z\right)}$	Ref.	$z$	$H\left(z\right)$	${\sigma }_{H\left(z\right)}$	Ref.	$z$	$H\left(z\right)$	${\sigma }_{H\left(z\right)}$	Ref.
0.07	69	19.68	Zhang et al. (2014)	0.36	81.2	5.9	Moresco et al. (2012)	0.7812	105	12	Moresco et al. (2012)
0.09	69	12	Jimenez et al. (2003)	0.3802	83	13.5	Moresco et al. (2016)	0.8754	125	17	Moresco et al. (2012)
0.1	69	12	Stern	0.4	95	17	Simon et al. (2005)	0.88	90	40	Stern
0.12	68.6	26.2	Zhang et al. (2014)	0.4004	77	10.2	Moresco et al. (2016)	0.9	117	23	Simon et al. (2005)
0.17	83	8	Simon et al. (2005)	0.4247	87.1	11.2	Moresco et al. (2016)	1.037	154	20	Moresco et al. (2012)
0.1791	75	4	Moresco et al. (2012)	0.4497	92.8	12.9	Moresco et al. (2016)	1.3	168	17	Simon et al. (2005)
0.1993	75	5	Moresco et al. (2012)	0.47	89	50	Wang et al. (2017b)	1.363	160	33.6	Moresco (2015)
0.2	72.9	29.6	Zhang et al. (2014)	0.4783	80.9	9	Moresco et al. (2016)	1.43	177	18	Simon et al. (2005)
0.27	77	14	Simon et al. (2005)	0.48	97	62	Stern	1.53	140	14	Simon et al. (2005)
0.28	88.8	36.6	Zhang et al. (2014)	0.5929	104	13	Moresco et al. (2012)	1.75	202	40	Simon et al. (2005)
0.3519	83.0	14	Moresco et al. (2012)	0.6769	92	8	Moresco et al. (2012)	1.965	186.5	50.4	Moresco (2015)

Table 2. Table 2: The priors and initial seeds of parameters used in the pasterior analysis.

Parameter	Prior	Initial seed
$H_0$	$[0,100]$	69
${\Omega }_{\text{m0}}$	$[0,1]$	0.33
$P_a$	$[-1,1]$	$0$
$\beta$	$[-5,5]$	$-0.02$

Table 3. Table 3: Best-fit parameters H 0 subscript 𝐻 0 H_{0} , Ω m0 subscript Ω m0 \Omega_{\text{m0}} , P a subscript 𝑃 𝑎 P_{a} and β 𝛽 \beta from CC datasets. Here we also list χ m ​ i ​ n 2 subscript superscript 𝜒 2 𝑚 𝑖 𝑛 \chi^{2}_{min} .

Parameter	Best-fit value with 1$\sigma$ error
$H_0$	${72.30}_{-1.37}^{+1.26}$
${\Omega }_{\text{m0}}$	${0.302}_{-0.047}^{+0.046}$
$P_a$	${0.249}_{-0.180}^{+0.160}$
$\beta$	$-{3.87}_{-1.66}^{+1.77}$
${\chi }_{min}^2$	$14.4053$

Table 4. Table 4: Best-fit parameters P a subscript 𝑃 𝑎 P_{a} and Ω m0 subscript Ω m0 \Omega_{\text{m0}} from CC datasets (in combination with the local value of the Hubble parameter H 0 subscript 𝐻 0 H_{0} ) for different choices of β 𝛽 \beta . Here we also quote χ m ​ i ​ n 2 superscript subscript 𝜒 𝑚 𝑖 𝑛 2 \chi_{min}^{2} in every case of β 𝛽 \beta . Line 3 and line 5 are 1 ​ σ 1 𝜎 1\sigma errors of P a subscript 𝑃 𝑎 P_{a} and Ω m0 subscript Ω m0 \Omega_{\text{m0}} , respectively.

$\beta$	$-1.5$	$-1$	$-0.15$	$0.15$	$1$	$1.5$
$P_a$	$-8.47$e$-3$	$-1.71$e$-2$	$-1.35$e$-1$	$1.37$e$-1$	$-1.93$e$-2$	$1.15$e$-2$
$1\sigma$	$0.6\sim 0.7$	$0.8\sim 0.9$	$5.2\sim 5.4$	$5.2\sim 5.4$	$0.8\sim 0.9$	$0.5\sim 0.6$
${\Omega }_{\text{m0}}$	$0.333$	$0.333$	$0.333$	$0.333$	$0.332$	$0.332$
$1\sigma$	$0.107$	$0.133$	$0.103$	$0.097$	$0.085$	$0.080$
${\chi }_{min}^2$	$14.6368$	$14.6363$	$14.6358$	$14.6357$	$14.6360$	$14.6363$

Table 5. Table 5: The stability of the GPDE model.

	$\{{\Omega }_{\text{de}},E\}$	Stability
$P_b\ge 0$,	$\{1,P_b^{1/2}\}$	attractor
$P_b\ge 0$,$\beta >0$	$\{1,P_b^{1/2}\}$	saddle point
	$\{1,0\}$	saddle point

Equations

$$P(z)=P_1-P_2\left[\frac{(1+z{)}^{-\beta }-1}{\beta }\right],\beta \ne 0,$$

$$P(z)=\left\{\begin{array}[]{ll}P_{a}+P_{b}z,&$\text{for}$\ \beta=-1\\ \\ P_{a}+P_{b}\ln(1+z),&$\text{for}$\ \beta\rightarrow 0\\ \\ P_{a}+P_{b}{\left(\frac{z}{1+z}\right)},&$\text{for}$\ \beta=+1\end{array}\right..$$

$$\rho (a)=\frac{3a^{\beta }{P}_2}{(3+\beta )\beta }-\frac{P_2}{\beta }-P_1+C{a}^{-3},$$

$$\rho (a)={\rho }_0\left(P_a{a}^{\beta }+{\Omega }_{\text{m0}}{a}^{-3}+1-P_a-{\Omega }_{\text{m0}}\right),$$

$$P(a)={\rho }_0\left[-(1-P_a-{\Omega }_{\text{m0}})-\frac{3+\beta }{3}{P}_a{a}^{\beta }\right].$$

$${\omega }_{\text{de}}=-1-\frac{\frac{1}{3}\beta P_a{a}^{\beta }}{1-P_a-{\Omega }_{\text{m0}}+P_a{a}^{\beta }},$$

$$E={\left({\Omega }_{\text{m0}}{a}^{-3}+P_a{a}^{\beta }+1-P_a-{\Omega }_{\text{m0}}\right)}^{1/2},$$

$$q\equiv -\frac{\ddot{a}}{a{H}^2}=-1+\frac{3}{2}{\Omega }_m-\frac{\beta P_a}{2E^2}{a}^{\beta },$$

$$j\equiv \frac{\stackrel{...}{a}}{a{H}^3}=1+2(q+1)(2q-1)-\frac{a^{\beta }{P}_a\beta ((\beta +3)}{2E^2}.$$

$${\chi }^2=\sum _{i=1}^{33}{\left[\frac{H_{\text{obs}}(z_i)-H_{0}E(z_i)}{{\sigma }_i^2}\right]}^2,$$

$${\Omega }_{\text{de}}^{\prime }=(1-{\Omega }_{\text{de}})\left(\beta {\Omega }_{\text{de}}-\frac{P_b\beta }{E^2}+3{\Omega }_{\text{de}}\right),$$

$$E^{\prime }=\frac{E}{2}\left[\beta {\Omega }_{\text{de}}-\frac{P_b\beta }{E^2}-3(1-{\Omega }_{\text{de}})\right].$$

$$M\equiv \left(\begin{array}{cc}\frac{\partial {\Omega }_{\text{de}}^{\prime }}{{\Omega }_{\text{de}}}& \frac{\partial {\Omega }_{\text{de}}^{\prime }}{\partial E}\\ \frac{\partial E^{\prime }}{\partial {\Omega }_{\text{de}}}& \frac{\partial E^{\prime }}{\partial E}\end{array}\right)=\left(\begin{array}{cc}-3& 0\\ \frac{\beta +3}{2}{P}_b^{1/2}& \beta \end{array}\right)$$

$${\Omega }_{\text{de}}^{\prime }=3{\Omega }_{\text{de}}(1-{\Omega }_{\text{de}}),$$

$$E^{\prime }=-\frac{3}{2}E(1-{\Omega }_{\text{de}}).$$

$$H=\frac{\dot{a}}{a}=H_0{\left({\Omega }_{\text{m0}}{a}^{-3}+P_a{a}^{\beta }+1-P_a-{\Omega }_{\text{m0}}\right)}^{1/2}.$$

$$H\to H_0(1-P_a-{\Omega }_{\text{m0}}{)}^{1/2},$$

$$H\to H_0(P_a{a}^{\beta }{)}^{1/2}.$$

$$a={\left(1-\frac{\beta P_a^{1/2}{H}_0}{2}(t-t_0)\right)}^{-2/\beta },$$

$$H=\frac{H_0{P}_a^{1/2}}{1-\frac{\beta P_a^{1/2}{H}_0}{2}(t-t_0)}.$$

Full text

I Introduction

Over the past two decades, a large number of cosmological observations has confirmed that the expansion of the late universe is speeding up Riess et al. (1998); Perlmutter et al. (1999); Eisenstein et al. (2005); Bennett et al. (2013); Ade et al. (2014), which has become one of the greatest challenges of cosmology. In order to explain the accelerated expansion, there are two main approaches: modifying gravity (MG) and adding dark energy (DE). The former means modifies the geometric parts of General relativity (GR), such as scalar-tensor theory Dvali et al. (2000), $f(R)$ gravity Carroll et al. (2004) and brane cosmology Brans and Dicke (1961); Sahni and Shtanov (2003). The other method is to add the dark energy which breaks the strong energy condition and produces a mysterious repulsive force to make the universe accelerate the expansion. The simplest DE model is the $\Lambda$CDM model, where the cosmological constant $\Lambda$ is related to DE, and its equation of state (EoS) is $\omega_{\text{de}}=-1$. The $\Lambda$CDM model provides a fairly good explanation for current cosmic observations. Recently, Planck-2018 reaffirmed the validity of the 6-parameter $\Lambda$CDM model in describing the evolution of the universe Aghanim et al. (2018). Nonetheless, there are two long-term problems with the $\Lambda$CDM model. One is the fine-tuning problem: The observation of dark energy density is 120 orders of magnitude smaller than the theoretical value in quantum field theory Carroll (2001); Weinberg (1989); The second is the coincidence problem: At the beginning of the universe, the proportion of DE is especially tiny while now the ratio of the dark energy density and matter density are exactly on the same magnitude. Besides, in recent years, the tension of Hubble constant between the Planck datasets and SHoES has reached to 4.4$\sigma$ Riess et al. (2019), which are under the $\Lambda$CDM model and the cosmic distance ladder, respectively. So as to alleviate these problems, many dynamic dark energy models with the time-variation EoS are proposed, including scalar field models (such as quintessence Caldwell et al. (1998); Kamenshchik et al. (2001); Amendola (2000); Chiba et al. (2000); Zlatev et al. (1999), phantom Caldwell (2002); Caldwell et al. (2003); Nojiri and Odintsov (2003), k-essence Armendariz-Picon et al. (2001); Deffayet et al. (2011); Scherrer (2004), quintom Cai et al. (2010); Guo et al. (2005) and tachyon Bagla et al. (2003)), holographic model Li (2004), agegraphic model Wei and Cai (2008), chaplygin gas model Bento et al. (2002); Gorini et al. (2003) and so on.

The model presented in this paper is also a dynamic dark energy model, which parameterizes the total pressure of the universe. Parameterization of the observable is an effective method to explore the characteristics of DE, such as the parameterization of EoS Chevallier and Polarski (2001); Linder (2003); Barboza Jr et al. (2009), luminosity distance Cattoën and Visser (2007); Gruber and Luongo (2014), dark energy density Alam et al. (2004), pressure Sen (2008); Kumar et al. (2013); Zhang et al. (2015); Yang et al. (2016); Wang et al. (2017a); Wang and Meng (2018) and deceleration factors Akarsu and Dereli (2012). Taking the pressure parameterization as an example. In general, we can write the pressure parameter equation as $P=\sum_{n=0}P_{n}x_{n}(z)$, where $x_{n}(z)$ expands at the late universe as the following forms (i) Redshift: $x_{n}=z^{n}$, (ii) Scale factor: $x_{n}(z)=(1-a)^{n}=(z/(1+z))^{n}$, (iii) Logarithmic form: $x_{n}(z)=(\ln(1+z))^{n}$. The form corresponding to $n=1$ in (i) and (ii) was proposed by Zhang, Yang, Zou, et al. Zhang et al. (2015). Case (iii) for $n=1$ was given by Wang and Meng Wang and Meng (2018). In order to unify these mainstream parameterization methods, we suggest a three-parameter pressure parameterization model to explore the evolution of the universe.

The content of this paper is organized as follows: Sec. II presents a generalized pressure dark energy (GPDE) model of the total pressure and discusses its feature. In Sec. III, we use CC datasets to impose constraints on the parameters 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 $\rho$ can be derived from the conservation equation $\dot{\rho}+3(\dot{a}/a)(\rho+P)=0$. Finally, by utilizing the Friedmann equations $H^{2}=3/(8\pi G)\sum_{i}\rho_{i}$ and the EoS $\omega=P/\rho$, we can get the form of the Hubble parameter $H$ and $\omega$, 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 $\Lambda$CDM 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

[TABLE]

Where $P_{1}$, $P_{2}$ and $\beta$ 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 $\Lambda$CDM 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 Barboza Jr et al. (2009). When specific limits are given to $\beta$, this model returns to the three models mentioned in Sec. I, i.e.

[TABLE]

By using Eq. (1), the relationship of scale factor $a$ ($a=1/(1+z)$) and the conservation equation ($\dot{\rho}+3(\dot{a}/a)(\rho+P)=0$), we can get the density as

[TABLE]

Where $C$ is the integral constant. We assume $\rho_{0}$ is the current total density, i.e. $\rho(a=1)=\rho_{0}$. Finally, the total density and total pressure can be respectively sorted into the following form

[TABLE]

[TABLE]

Parameters $P_{1}$ and $P_{2}$ have been replaced here by new parameters $P_{a}$ and $\Omega_{\text{m0}}$, where $P_{a}\equiv 3P_{2}/((3+\beta)\beta\rho_{0})$, $\Omega_{\text{m0}}\equiv(\beta\rho_{0}+P_{2}+\beta P_{1}-3P_{2}/(3+\beta))/(\beta\rho_{0})$. In the density expression (4), the item $\rho_{0}\Omega_{\text{m0}}a^{-3}$ is corresponding to the matter density $\rho_{m}$. So $\Omega_{m}|_{a=1}=\rho_{m}/\rho=\Omega_{\text{m0}}$ signifies that the physical meaning of the parameter $\Omega_{\text{m0}}$ is the present-day matter density parameter. The term $\rho_{0}(P_{a}a^{\beta}+1-P_{a}-\Omega_{\text{m0}})$ accords with the dark energy density $\rho_{\text{de}}$, and the constant part $1-P_{a}-\Omega_{\text{m0}}$ looks similar to the $\Lambda$CDM case. The term $P_{a}a^{\beta}$ makes $\rho_{\text{de}}$ change with time: The larger the $|P_{a}|$ is, the more deviation from the $\Lambda$CDM model will be; The larger the $\beta$ 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 $P_{\text{de}}$ is the total pressure $P$. From $(\rho_{\text{de}}/\rho_{m})|_{a\rightarrow 0,\beta>-3}\rightarrow 0$ we can also know that for the case of $\beta>-3$, the DE accounts for a small percentage in the early universe. Note that when $\beta=-3$, the density of the part of the dark energy is expressed as the matter density, and the total density $\rho(a)$ of our GPDE model is equivalent to the $\Lambda$CDM model.

Suppose the dark energy is a scalar field $\phi$ that changes with time. The corresponding pressure and density are equivalent to $\rho_{\text{de}}=(n/2)\dot{\phi}^{2}+V(\phi)$ and $P_{\text{de}}=(n/2)\dot{\phi}^{2}-V(\phi)$, separately, where $n=1$ or $-1$ corresponds to the quintessence and phantom scalar field, respectively. The calculation shows that $\dot{\phi}^{2}=-(n\beta/3)\rho_{0}P_{a}a^{\beta}$, so $\beta P_{a}<0$ fits quintessence, and $\beta P_{a}>0$ matches for phantom.

Additionally, for the GPDE model, the EoS of the dark energy $\omega_{\text{de}}$, the dimensionless Hubble parameter $E$, the deceleration parameter $q$ and the jerk parameter $j$ respectively take the form as

[TABLE]

[TABLE]

[TABLE]

[TABLE]

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 1, spanning the redshift range $0<z<2$. The optimal values of the parameters can be obtained by taking the minimum value of $\chi^{2}$, which is expressed as

[TABLE]

with the corresponding four-dimensional parameter space $\{H_{0},\Omega_{\text{m0}},P_{a},\beta\}.$

As the first attempt, we adopt the Monte Carlo Markov chain (MCMC) method and use the python package emcee Foreman-Mackey et al. (2013) to produce a MCMC sample with CC data. The results are displayed as a contour map by another python package pygtc Bocquet and Carter (2016). We list the priors and initial seeds on the parameter space in Table 2.

Figure 1 shows the 1-dimensional and 2-dimensional marginalized probability distributions of the GPED model.

In the meantime，the best-fit values and $1\sigma$ confidence level for the $H_{0}$, $\Omega_{\text{m0}}$, $P_{a}$ and $\beta$ are listed in Table 3.

From the constraint results, $P_{a}\beta<0$ in 1$\sigma$, which indicates our universe is under quintessence situation and has some deviation from the $\Lambda$CDM model in a point of view of data. The differences of $H_{0}$ between our results and SHoES Riess et al. (2019) and Planck base-$\Lambda$CDM Aghanim et al. (2018) are 0.9 $\sigma$ and 3.5 $\sigma$, respectively. The evolution of DE EoS parameter $\omega_{\text{de}}$, DE density parameter $\Omega_{\text{de}}$, deceleration parameter $q$ and jerk parameter $j$ with 1 $\sigma$ error propagation from data fitting (Table 3) are shown in figure 2.

From figure 2 we find that these results are acceptable, except for the DE density parameter $\Omega_{\text{de}}$, 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 using CC data, we also combine the data of Planck-2018 Aghanim et al. (2018) with the Hubble constant $H_{0}=67.4$km s*-1* Mpc*-1* and pick $\beta=-1.5,-1,-0.15,0.15,1$ and $1.5$, respectively to constrain the parameter pair $\{P_{a}$, $\Omega_{\text{m0}}\}$. The results are shown in figure 3. Table 4 lists the best-fit values and $1\sigma$ confidence level for the $P_{a}$ and $\Omega_{\text{m0}}$ under CC data and the $H_{0}$ data joint constraints. We discover that for the six kinds of circumstances, the best values of $\Omega_{\text{m0}}$ are all around $0.33$, and $\beta P_{a}>0$ indicates that the universe is slightly biased toward phantom but still includes quintessence within $1\sigma$ confidence level. The value of $P_{a}$ is small, meaning in this case the deviation of this model from the $\Lambda$CDM model is not significant. The minimum $\chi^{2}$ of these six cases are very close, implying that this model is not very sensitive to the selected values of $\beta$, that is, the degeneracy is high. Figure 4 shows the difference in export parameters $\omega_{\text{de}},\Omega_{\text{de}},q$ and $j$ between the GPDE model and the $\Lambda$CDM model with the best-fit values for the considered values of $\beta$. It can be concluded from figure 4 that the distinction between these two models is almost indistinguishable. The exceptions are for the cases of $\beta=-1.5$ and $-1$, whose $\omega_{\text{de}}$ 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 $\Lambda$CDM model. The reason is estimated to be that in the second try, the data of $H_{0}$ is from Planck, which depends on the $\Lambda$CDM model.

IV DYNAMIC ANALYSIS

In this section, we will construct a self-consistent dynamical system to analyze the cosmological evolution of the GPDE model. Select $\Omega_{\text{de}}$ and $E$ as independent variables. By Friedmann equation, we can get the self-consistent dynamic system as

[TABLE]

[TABLE]

where $P_{b}\equiv 1-P_{a}-\Omega_{\text{m0}}$, ”′” represents the derivative of $\ln(a)$. The following are common methods for finding fixed points and its stability of a system. Let $\Omega_{\text{de}}^{\prime}=E^{\prime}=0$, then for $P_{b}\geq 0$, there is a fixed point $\{\Omega_{\text{de}}=1,E=P_{b}^{1/2}\}$ 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

[TABLE]

Its eigenvalues are $\lambda_{1}=-3$ and $\lambda_{2}=\beta$. The stability of the system depends on the sign of the eigenvalues $\lambda_{1}$ and $\lambda_{2}$. For the GPDE model, $\beta$ is a real number and not equal to zero, so there are two situations: When $\beta<0$, point $\{1,P_{b}^{1/2}\}$ is an attractor, and when $\beta>0$, point $\{1,P_{b}^{1/2}\}$ 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 $\Lambda$CDM, the parameter $P_{a}=0$, so $\beta\Omega_{\text{de}}-\frac{P_{b}\beta}{E^{2}}=\beta\frac{P_{a}a^{\beta}}{E^{2}}=0$, the dynamic system goes to

[TABLE]

[TABLE]

Figure 5 illustrates the evolution of the dynamics system for $\beta=-1$, $1$ and $P_{b}=-1$, [math], $1$. There also exhibits the evolutional trajectories of the $\Lambda$CDM 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 $\beta>0$ and $P_{b}<0$, it shows that $E\rightarrow\infty$ and $\Omega_{\text{de}}\rightarrow 1$, which is the case of phantom. While for $\beta<0$ and $P_{b}<0$, one can see $\Omega_{\text{de}}\rightarrow 0$ which corresponds to quintessence. When $P_{b}\geq 0$, whether the universe is biased towards quintessence or phantom depends on $\beta$ and initial conditions which cannot be judged directly. Based on the above analysis, we summarize the stability of the GPDE model in table 5.

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 $\omega=-1.073^{+0.090}_{-0.089}$ based on WMAP+CMB+BAO+$H_{0}$ Bennett et al. (2013). Planck-2018 constrained the EoS as $\omega=-1.03\pm 0.03$ with SNe Aghanim et al. (2018). Moreover DES suggests $\omega=-0.978\pm 0.059$ in the joint analysis of DES-SN3YR+CMB Abbott et al. (2019). Using existing data, it is still impossible to distinguish between phantom ($\omega<-1$) and non-phantom($\omega\geq-1$), 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 Frampton et al. (2012):

• 1

Big rip: $H(t)\rightarrow\infty$ as $t\rightarrow$ constant, so the rip will happen at a certain time.

• 2

Little rip: $H(t)\rightarrow\infty$ as $t\rightarrow\infty$. This situation has no singularities in the future.

• 3

Pseudo rip: $H(t)\rightarrow\text{constant}$. 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

[TABLE]

In section II, we have pointed out that $\beta P_{a}>0$ for phantom case of the the GPED model. Next we discuss each situation separately.

• 1

$\beta<0,P_{a}<0$:

With the growth of cosmic time, $a\rightarrow\infty$ and

[TABLE]

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

$\beta>0,P_{a}>0$:

When $a\rightarrow\infty$,

[TABLE]

By solving the above differential equation, we can get the relation between scale factor $a$ and time $t$, which can be written as

[TABLE]

Where $t_{0}$ is the present value of time. Substitute Eq. (19) for Eq. (18), one obtains

[TABLE]

So when $t\rightarrow\frac{2}{\beta P_{a}^{1/2}H_{0}}+t_{0}$, $H(t)\rightarrow\infty$ which means the universe will have a big rip after $t-t_{0}=\frac{2}{\beta P_{a}^{1/2}H_{0}}$. Thus, the lifetime of the universe is determined by two parameters, $\beta$ and $P_{a}$, regardless of the matter density $\Omega_{\text{m0}}$. Let us make a rough estimate. Take $\beta=1$, $P_{a}=0.02$, $H_{0}=70$km s*-1* Mpc*-1*, then the universe will be torn apart after 198Gyr. If $13.8$Gyr is the current age of the universe, then the universe has spent only 6.5% of its life.

• 3

$\beta\rightarrow 0$:

In this case, the GPDE model degenerates into the model in Wang and Meng (2018). According to the discussion in Wang and Meng (2018), 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 $\beta>0,P_{a}>0$;

• 2

Little rip for $\beta\rightarrow 0$;

• 3

Pseudo rip for $\beta<0,P_{a}<0$.

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 $\Lambda$CDM model to a generalized pressure dark energy (GPDE) model. The GPED model has three independent parameters: The present value of matter density parameter $\Omega_{\text{m0}}$, the parameter $P_{a}$ which represents the deviation from the $\Lambda$CDM model, and the parameter $\beta$. Picking different values of parameter $\beta$ 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 Riess et al. (2019) and Planck base-$\Lambda$CDM Aghanim et al. (2018) are 0.9 $\sigma$ and 3.5 $\sigma$, respectively. In addition, for the GPDE model, the matter density parameter is $\Omega_{\text{m0}}=0.302^{+0.046}_{-0.047}$, and the universe bias towards quintessence in $1\sigma$ error. While when we combine CC datasets and the $H_{0}$ data from Planck, the constraint implies that our model matches the $\Lambda$CDM 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 $\beta<0$, $P_{a}<0$, big rip for $\beta>0$, $P_{a}>0$ and little rip for $\beta\rightarrow 0$. 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.

ACKNOWLEDGEMENTS

Jun-Chao Wang thanks Wei Zhang for the helpful discussions and code guidance about MCMC.