Exploring the deviation of cosmological constant by a generalized pressure parameterization
Abstract
We bring forward a generalized pressure (GP) parameterization for dark energy to explore the evolution of the universe. This parametric 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 GP parameterization are (within \(1\sigma \)): the present value of Hubble constant \(H_{0}=(72.30^{+1.26}_{-1.37}) \ \hbox {km s}^{-1}\hbox { Mpc}^{-1}\); the matter density parameter \(\Omega _{\text {m0}}=0.302^{+0.046}_{-0.047}\), and the bias of the universe towards quintessence. Then we perform a dynamic analysis on the GP parameterization and find that there is an attractor or a saddle point in the system corresponding to the different values of the parameters. Finally, we discuss the ultimate fate of the universe under the phantom scenario in the GP parameterization. It is demonstrated that the three cases of pseudo rip, little rip, and big rip are all possible.
1 Introduction
Over the past two decades, a large number of cosmological observations have confirmed that the expansion of the late universe is speeding up [1, 2, 3, 4, 5], 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 modifying the geometric parts of general relativity (GR), such as scalar-tensor theory [6], f(R) gravity [7, 8] and brane cosmology [9, 10]. 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 six-parameter \(\Lambda \)CDM model in describing the evolution of the universe [11]. 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 [12, 13]; the second is the coincidence problem: At the beginning of the universe, the proportion of DE is especially tiny while now the dark energy density and matter density are exactly of the same magnitude. Besides, in recent years, the tension of Hubble constant between the Planck datasets and SHoES has reached 4.4\(\sigma \) [14], which are brought about by the \(\Lambda \)CDM model and the cosmic distance ladder method, 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 [15, 16, 17, 18, 19], phantom [20, 21, 22], k-essence [23, 24, 25], quintom [26, 27, 28] and tachyon [29]), holographic model [30], agegraphic model [31], Chaplygin gas model [32, 33].
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 [34, 35, 36], the luminosity distance [37, 38], dark energy density [39], pressure [40, 41, 42, 43, 44, 45] and deceleration factors [46]. Take 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 for the late universe in 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 et al. [42]. Case (iii) for \(n=1\) was given by Wang and Meng [45]. In order to unify these mainstream parameterization methods, we suggest a three-parameter pressure parameterization model to explore the evolution of the universe.
This paper is organized as follows: Sect. 2 presents a generalized pressure (GP) parameterization of the total pressure and discusses its feature. In Sect. 3, we use CC datasets to impose constraints on the parameters of the GP parameterization. The discussion of fixed points using the GP parameterization is analyzed in Sect. 4. In Sect. 5, we exhibit the end of the universe under the phantom case. Section 6 is for the conclusion.
2 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 \(c=1\). 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.
Cosmic chronometers data used in this paper
z | H(z) | \(\sigma _{H(z)}\) | References | z | H(z) | \(\sigma _{H(z)}\) | References | z | H(z) | \(\sigma _{H(z)}\) | References |
---|---|---|---|---|---|---|---|---|---|---|---|
0.07 | 69 | 19.68 | [47] | 0.36 | 81.2 | 5.9 | [48] | 0.7812 | 105 | 12 | [48] |
0.09 | 69 | 12 | [49] | 0.3802 | 83 | 13.5 | [50] | 0.8754 | 125 | 17 | [48] |
0.1 | 69 | 12 | [51] | 0.4 | 95 | 17 | [52] | 0.88 | 90 | 40 | [51] |
0.12 | 68.6 | 26.2 | [47] | 0.4004 | 77 | 10.2 | [50] | 0.9 | 117 | 23 | [52] |
0.17 | 83 | 8 | [52] | 0.4247 | 87.1 | 11.2 | [50] | 1.037 | 154 | 20 | [48] |
0.1791 | 75 | 4 | [48] | 0.4497 | 92.8 | 12.9 | [50] | 1.3 | 168 | 17 | [52] |
0.1993 | 75 | 5 | [48] | 0.47 | 89 | 50 | [53] | 1.363 | 160 | 33.6 | [54] |
0.2 | 72.9 | 29.6 | [47] | 0.4783 | 80.9 | 9 | [50] | 1.43 | 177 | 18 | [52] |
0.27 | 77 | 14 | [52] | 0.48 | 97 | 62 | [51] | 1.53 | 140 | 14 | [52] |
0.28 | 88.8 | 36.6 | [47] | 0.5929 | 104 | 13 | [48] | 1.75 | 202 | 40 | [52] |
0.3519 | 83.0 | 14 | [48] | 0.6769 | 92 | 8 | [48] | 1.965 | 186.5 | 50.4 | [54] |
The parameters \(P_1\) and \(P_2\) have been replaced here by the 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 \(\rho _{0}\Omega _{\text {m0}}a^{-3}\) corresponds to the matter density \(\rho _{m}\). So \(\Omega _m|_{a=1}=\rho _{m}/\rho =\Omega _{\text {m0}}\) signifies the physical meaning of the parameter \(\Omega _{\text {m0}}\); it is the present-day matter density parameter. The term \(\rho _{0}(P_aa^{\beta }+1-P_a-\Omega _{\text {m0}})\) 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 \(P_aa^{\beta }\) makes \(\rho _{\text {de}}\) change with time: The larger \(|P_a|\) is, the higher the deviation from the \(\Lambda \)CDM model will be; the larger \(\beta \) is, the faster the dark energy density will change. Accordingly, this cosmological model only includes matter and dark energy components, and the pressure of the dark energy \(P_{\text {de}}\) is the total pressure P. 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 GP parameterization has the same form as 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 \({\dot{\phi }}^2=-(n\beta /3)\rho _0P_aa^{\beta }\), so \(\beta P_a<0\) fits quintessence, and \(\beta P_a> 0\) matches the phantom model.
3 Results of the data analysis
The priors and initial seeds of parameters used in the posterior analysis
Parameter | Prior | Initial seed |
---|---|---|
\(H_0\) | [0, 100] | 69 |
\(\Omega _{\text {m0}}\) | [0, 1] | 0.33 |
\(P_a\) | \([- \ 1,1]\) | 0 |
\(\beta \) | \([- \ 5,5]\) | \(- \ 0.02\) |
We adopt the Monte Carlo Markov chain (MCMC) method and use the python package emcee [55] to produce a MCMC sample with CC data. The results are displayed as a contour map by another python package, pygtc [56]. We list the priors and initial seeds on the parameter space in Table 2. Figure 1 shows the one-dimensional and two-dimensional marginalized probability distributions of the GP parameterization. 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 that our universe is in a quintessence situation and has some deviation from the \(\Lambda \)CDM model from the point of view of the data. The differences of \(H_0\) between our results and SHoES [14] and the Planck base-\(\Lambda \)CDM [11] are 0.9 \(\sigma \) and 3.5 \(\sigma \), respectively. This suggests that the GP parameterization is more supportive of the results of SHoES. The \(\Lambda \)CDM model is not the best form in this general framework and shows some difference from the best form.
4 Dynamic analysis
Best-fit parameters \(H_0\), \(\Omega _{\text {m0}}\), \(P_a\) and \(\beta \) from CC datasets. Here we also list \(\chi ^2_{min}\)
Parameter | Best-fit value with 1\(\sigma \) error |
---|---|
\(H_0\) | \(72.30^{+1.26}_{-1.37}\) |
\(\Omega _{\text {m0}}\) | \(0.302^{+0.046}_{-0.047}\) |
\(P_a\) | \(0.249^{+0.160}_{-0.180}\) |
\(\beta \) | \(-3.87^{+1.77}_{-1.66}\) |
\(\chi ^2_{min}\) | 14.4053 |
5 Fate of the universe under the phantom field
- 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.
- 1.
\(\beta<0, P_a< 0\):
- 2.
\(\beta>0, P_a> 0\):
- 3.
\(\beta \rightarrow 0\):
- 1.
Big rip for \(\beta>0, P_a > 0\).
- 2.
Little rip for \(\beta \rightarrow 0\).
- 3.
Pseudo-rip for \(\beta<0, P_a<0\).
The stability of the GP parameterization
\(\{\Omega _{\text {de}},E\}\) | Stability | |
---|---|---|
\(P_b\ge 0\), \(\beta <0\) | \(\{1,P_b^{1/2}\}\) | Attractor |
\(P_b\ge 0\), \(\beta >0\) | \(\{1,P_b^{1/2}\}\) | Saddle point |
\(P_b< 0\) | \(\{1,0\}\) | Saddle point |
6 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 (GP) parameterization. The GP parameterization has three independent parameters: The present value of the 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 the parameter \(\beta \) can produce three common pressure parametric models. By using the cosmic chronometer (CC) datasets to constrain the parameters, it shows that the Hubble constant is \(H_{0}=(72.30^{+1.26}_{-1.37})\) \(\hbox {km s}^{-1}\) \(\hbox {Mpc}^{-1}\). The differences of \(H_0\) between our results and SHoES [14] and Planck base-\(\Lambda \)CDM [11] are 0.9 \(\sigma \) and 3.5 \(\sigma \), respectively. In addition, for the GP parameterization, the matter density parameter is \(\Omega _{\text {m0}}=0.302^{+0.046}_{-0.047}\), and the universe shows bias towards quintessence. 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 the parameters. Next, we analyze the rip of the universe in the 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 198 Gyr.
Dark energy has been proposed for 20 years, but its nature remains unknown. With this parametric model, we can probe the possible deviation further between the dynamic case and the cosmological constant condition through existing data.
Notes
Acknowledgements
Jun-Chao Wang thanks Wei Zhang for helpful discussions and code guidance about MCMC.
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