The data on cosmic microwave background anisotropies [1], especially, Plank collaboration result provides high precision on the parameters in the universe. More specifically, the significant parameter H0which represents the age and scale of the universe has been determined from the observational numbers. On the other hand, supernova type Ia, the local distance measurement [2], also provides important result on H0which, controversially, gives 3σ tension away from CMB anisotropies. Since the H0tension might be the most considerable argument in present cosmological observation, there are several hypothetical models, different from cosmological constant, trying to present superior explanation on the scenario of the universe [3, 4, 5, 6].
The equation of state on cosmological-constant model persuades that w should remain on -1. However, recent researches [7] have found out that the value of the equation of state on dark energy has the tendency to be lower than it used to be. In addition, more and more evidences show that cosmological constant might not be the only solution of the universe. In fact, not only the scenario of quintessence has been involved to picturized the evolution of the universe but also the utilization of phantom seems indispensable.
In some cases, dynamical dark energy does have advantages on describing the mechanism of cosmic acceleration [8].
Dynamically evolving dark energy field is not an innovative idea. It has already been applied on certain models, for instance, quintessence. There are numerous studies determining different categories of scalar potential V (φ) based on varies motivations, for example, pseudo Nambu-Goldstone boson, an inverse power law, an exponential, the tracking characteristic, the oscillating feature, and so on [9, 10, 11, 12, 13].
Those researches aim to cover the tensions on cosmological observations and provide reasonable solutions.
As the equation of state has decreased below the value of −1, phantom has started to become popular.
More and more papers attempt to reconstruct scalar field as a w < −1 and more rapidly evolving dark energy such as vacuum phase transition, Dvali-Gabadadze-Porrati (DGP) branes, Dirac-Born-Infeld (DBI), galileon, kinetic braiding, and other scalar tensor varieties [14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25].
To be acquaint with the dark energy, we should start from the basic theories.
2.1.1 Equation of state
Equation of state is a beneficial way to categorize the dark energy. To study the equation of state we should start from Einstein field equation
Gµν = Rµν−1
2Rgµν = 8πG
c4 Tµν , (2.1)
which infers that energy and space-time are relate to each other’s. Also introduce the Rodertson-Walker metric:
for 3 types of spcetime curvature κ. By utilizing the concept of general relativity starting from Einstein’s field equation, Friedmann derived his eponymous equation:
˙a
which straightforwardly describes the temporal variation in the cosmic energy density ε(t). However, it is still impossible to illustrate how the scale factor a(t) evolves with time only by Friedmann equation. As the matter of fact, the fluid equation must be involved:
˙ ρ + 3˙a
a(ρ + P ) = 0 , (2.5)
where the P stands for pressure, and ρ = ε/c2denotes the mass density. In order to solve the equation, the equation of state should be assumed as a relation between the energy density ε and the pressure P which the equation can be written in a simple linear form:
P = wε , (2.6)
where w is an important dimensionless factor which represent different values in different components.
More specific, the equation of state of radiation wr = 1/3, matter wm = 0 and also another wφ for dark energy. Various categories of dark energy will be described in the following subsection which have different values on the equation of state.
2.1.2 Cosmological Constant
The cosmological constant Λ has first been introduced by Einstein. Despite Einstein claimed his idea of cosmological constant as "The Biggest Bluder" after Hubble’s 1929 discovery that the galaxies are moving away from each other’s, the perspective of cosmological constant still exist since the supernova observation gives the fact that our universe is accelerating expanding. By adding an extra term in the Einstein field equation
Rµν−1
2Rgµν− Λgµν = 8πG
c4 Tµν. (2.7)
The Friedmann equation will become
We can see that the last term of the eq. 2.8 is like adding an additional component in our universe.
Therefore, the Λ in the energy density term will be ρΛ ≡ Λ
8πG = −PΛ. (2.9)
If Λ remains constant in the evolution, the relation between energy density εΛand pressure PΛin the fluid equation (eq. 2.5) will be axiomatic.
PΛ= −εΛ= wΛεΛ, (2.10)
in which the eqution of state of the cosmological constant is specified as wΛ = −1. Due to the strong observational evidence for the accelerating universe, the standard ΛCDM model has been fully developed, in which the universe is flat with an energy density made up of about 27% of matter and 73% of dark energy [26]. However, the ΛCDM model has its own unresolved issues such as cosmological constant problem, coincident problem and the Hubble constant problem. We will discuss the Hubble constant problem in the following section. As the matter of fact, there are several modified models aim to interpret the missing energy, quintessence and phantom for instance.
2.1.3 Quintessence
Quintessence is a canonical scalar field that used to describe the accelerating universe. The first example of quintessence is brought by Ratra and Peebles (1988) [27]. As we avert in the previous subsection that the ΛCDM model has the problem on vacuum energy, two classified solutions have been introduced. The first solution is quintessence and the second is modification of gravity [28, 29, 30, 31]. In both researches the equation of state of dark energy evolve dynamically with time in which it is the distinguished aspect from ΛCDM model. In order to study more about the dynamic scalar field, we should consider a quintessential tracker field φ with time-varying equation of state wφto give rise to the cosmic magnetic field through the mechanism of spinodal instability.
where Vλk(η) (λ = 1, 2) are the mode functions, η is the conformal time, c is the coupling constant between φ field and the EM field. The information needed from the background evolution as input to solve the above equations is (dφ/dη). Assuming the quintessence field φ is in a matter-dominated flat Friedmann universe with energy density ρφ, the kinetic and potential energy can be prescribed as
φ˙2 = (1 + wφ)ρφ, (2.13)
2V (φ) = (1 − wφ)ρφ. (2.14)
The dark energy eqution of state is given as wφ≡ Pφ
ρφ = φ˙2/2 − V (φ)
φ˙2/2 + V (φ) , (2.15)
where we can see that the equation of state is various with time. The dark energy equation of state has often been written as the form of wφ= w0+ w(a) which w0indicates the basic value of cosmological constant w0 = −1 and the w(a) is the time varying part. Base on different motivation, there will be different kinds of w(a). In addition, there used to be a boundary on the equation of state w > −1 comes from the condition of positive kinetic energy. However, scientists have broken down this boundary since the other scenario, phantom, has been brought out.
2.1.4 Phantom
The motivation to generate the idea of phantom comes from the observational data. Planck collaboration 2015 gave the constrain on w = −1.006 ± 0.045 [1], and the value in Planck collaboration 2013 is w = −1.13+0.13−0.10 [32]. The combinational data from the Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP9), the cosmic microwave background (CMB), baryonic acoustic oscillations (BAO), supernova measurements, and H0 measurements, is w = −1.084 ± 0.063 [33]. Considering those constrains, the phantom-like (w < −1) dark energy seems possible. Because the phantom-like dark energy has the concept of the increasing energy density, the universe under this model will end up in a big rip [34] which means that the scalar factor a will attain infinity in the future. As we mention in the previous subsection that the equation of state has the constrain of w > −1, the sign in front of the kinetic energy should be flipped to allow for compatibility. The equation of state will become:
wφ= −| ˙φ|2/2 − V (|φ|)
−| ˙φ|2/2 + V (|φ|) . (2.16)
And w < −1 will be mathematically acceptable. However, the negative kinetic energy is still required in this hypothesis which is a controversial issue in this model. On the other hand, some studies show that the dynamically evolving dark energy can solve the tension in Hubble constant [35] which the problem on Hubble constant will be talk in the following section.