H_0 tension in dark energy cosmology

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(1)H0 Tension in Dark Energy Cosmology. Yu-Ping Teng. A thesis submitted in partial fulfillment for the degree of Master of Physics in the Physics Department National Taiwan Normal University. Supervisor: Prof. Wolung Lee. July 2018.

(2) Contents 1. abstract. 3. 2. Introduction. 4. 2.1. Recent dark energy research . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 4. 2.1.1. Equation of state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.1.2. Cosmological Constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 5. 2.1.3. Quintessence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 6. 2.1.4. Phantom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. Hubble constant problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 7. 2.2 3. Method. 9. 3.1. Basic equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 9. 3.2. Characteristic parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 3.2.1. Time-average hwi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 10. 3.2.2. Acoustic scale lA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 11. 3.2.3. η? approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. 3.2.4. η0 approximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 12. Numerical code . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 13. 3.3 4. Result and Conclusion. 15. 4.1. Phantom model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 4.2. H0 -hwi diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 15. 4.3. Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 18. Bibliography. 19. 2.

(3) Chapter 1. abstract Recently, the equation of state on dark energy has dropped lower than the edge of -1. Based on this outcome, the nature of dark energy may be far away from cosmological constant. As the adjustment on the equation of state keeps decreasing, the introduction of phantom seems necessary. By employing the observational constraints on supernovas and the acoustic scale of the cosmic microwave background(CMB), where the accuracy has become extraordinary, we apply a phenomenological method to mimic the evolution of the universe. The demonstration on the constraining evolution of phantom will show the model has high consistency with current observation. Key word: Dark energy, Hubble constant problem, dymamically evolving dark energy, quintessence, phantom.. 3.

(4) Chapter 2. Introduction 2.1. Recent dark energy research. 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 H0 which 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 H0 which, controversially, gives 3σ tension away from CMB anisotropies. Since the H0 tension 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.. 4.

(5) 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 1 8πG Gµν = Rµν − Rgµν = 4 Tµν , 2 c. (2.1). which infers that energy and space-time are relate to each other’s. Also introduce the Rodertson-Walker metric: h. i. ds2 = −c2 dt2 + a(t)2 dr2 + Sκ (r)2 dΩ2 ,. (2.2). where. Sκ (r) =.    . R0 sin(r/R0 ) r.   . (κ = +1) (κ = 0). (2.3). R0 sinh(r/R0 ) (κ = −1),. 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: 2. a˙ a. =. 8πG κc2 1 ε(t) − , 3c2 R02 a(t)2. (2.4). 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: a˙ ρ˙ + 3 (ρ + P ) = 0 , a. (2.5). where the P stands for pressure, and ρ = ε/c2 denotes 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 1 8πG Rµν − Rgµν − Λgµν = 4 Tµν . 2 c. (2.7). 5.

(6) The Friedmann equation will become 2. a˙ a. =. 8πG κc2 Λ ε(t) − + . 2 2 2 3c 3 R0 a. (2.8). 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 = −PΛ . 8πG. (2.9). If Λ remains constant in the evolution, the relation between energy density εΛ and pressure PΛ in the fluid equation (eq. 2.5) will be axiomatic. PΛ = −εΛ = wΛ εΛ ,. (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 wφ to give rise to the cosmic magnetic field through the mechanism of spinodal instability. ". ". d2 dφ + k 2 − 4ck 2 dη dη. #. d2 dφ + k 2 + 4ck 2 dη dη. #. . . V1k (η) = 0 ,. (2.11). V2k (η) = 0 ,. (2.12). 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). 6.

(7) The dark energy eqution of state is given as wφ ≡. Pφ φ˙ 2 /2 − V (φ) = , ρφ 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 w0 indicates 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 =. ˙ 2 /2 − V (|φ|) −|φ| . ˙ 2 /2 + V (|φ|) −|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.. 2.2. Hubble constant problem. Hubble parameter is one of the most representative factor in the cosmology which indicates the expansion rate of the universe. Furthermore, the Hubble parameter in today H0 is called the Hubble constant. The Hubble constant can be calculated by the distance measurement using the relationship of v = H0 d. Where the v is the recession velocity and the d is the proper distance to the object. This relation has been introduced by Edwin Hubble in 1929 [36]. Generally, Hubble constant can be measured by two various methods: model independent and dependent. The first category will be using the relationship of distance and velocity to investigate H0 , such as the local distance measure on type Ia supernova [2]. This kind of measurement doesn’t change under different cosmological models which is our preferred choice. The second category will be model-dependent method such as the Planck collaboration result [1]. These methods create their fiducial model and give the constrain on H0 , ΛCDM model for instance.. 7.

(8) The value provided by Planck collaboration 2015 is 67.74 ± 0.46 km s−1 Mpc [1]. On the other hand, the value offered by the local distance measurement is 73.24 ± 1.74 km s−1 Mpc [2]. It is obvious to see the tension between these data values which is usually been mention as a 3.4σ [2, 37] tension between local and global measurement. To be more specific, the two categories of measurement are one depends on the ΛCDm model and the other is not. Consequently, the tension on H0 illustrate the physics beyond standard ΛCDM model. Combining the fact that the equation of state can be lower than negative one and the tension on Hubble constant, bring up the model of phantom seems indispensable. There are several phantom models base on different motivations are aiming to explain the dark energy. Although every theoretical hypothesis seems logical, without the exact evidence, those assumptions are only plausible. To be honest, the maturity on dark energy theories is still insufficient to give a comprehensive interpretation. Despite the difficulties on the identification of dark energy, we have not given up on the investigation of this project. The following chapter will show the method that we use to study the dark energy and the evolution of the universe.. 8.

(9) Chapter 3. Method 3.1. Basic equations. By evading the reconstruction on actual potential field and focusing on the evolution expression, a phenomenological method could be offered to assist with the discussion on dark energy under certain observational numbers. In particular, there is a convenient way which has already been employed known as the generic quintessence [38]. First of all, we define φ field as a dynamically evolving dark energy – without regards to quintessence, cosmological constant, phantom or others. Through its equation of state p = wρ (used to have the boundary of −1 < w < 1), the evolution of universe can be distinctly demonstrated without presuming hypothesis. Consider a flat universe where the total density parameter Ω0 is Ω0 = Ωm,0 + Ωr,0 + Ωφ,0 = 1 with negligible Ωr,0 , Ωm,0 ∼ 0.3, and Ωφ,0 ∼ 0.7. [1] The evolution of the cosmic background can be ruled by the eqution of motion for the φ field: ε˙i + 3H(1 + wi )εi = 0 .. (3.1). And also the eqution of motion for the background expansion: X wi 3 H˙ + H 2 + εi = 0 , 2 2Mp2 i. (3.2). where i is equal to r, m and φ, which is assumed as a spatially inhomogeneous field. We also need the cosmic scale factor a(t): a˙ = aH.. (3.3). By using the reduce Planck mass Mp = (8πG)−1/2 , conformal time dη = H0 a−1 dt, dimensionless Hubble constant H = H/H0 , and rescaling the energy density ρi = εi /(Mp H0 )2 , the evolution can be expressed under four equations. dρi dη dH dη da dη dτ dη. = −3aH(1 + wi )ρi ,. (3.4). X wi 3 = − aH2 − aρi , 2 2 i. (3.5). = a2 H,. (3.6). = a.. (3.7) 9.

(10) Figure 3.1: An example of dynamically evolving dark energy as presented in [38], where the error on θ? is 3.18% and time-average hwi is -0.898. As the equations of evolution have been determined, a numerical equation of state w(η) is given to test the different models. Applying the initial conditions in present time, the problem will be simplified to a set of first-order ordinary differential equations. The result of the ordinary quintessence model by solving the ODE is in Fig. 3.1. The point to utilize this method is that it can easily illustrate the whole picture of the evolution with specific constrains under certain models. It is also benificial to take a look in the ΛCDM model which will be demonstrated in Fig. 3.2.. 3.2 3.2.1. Characteristic parameter Time-average hwi. To be clear about the distinction of different dynamically evolving dark energy, the parameter, time-average hwi, should be defined as the following equation. R η0. hwφ i =. η?. Ωφ (η)wφ (η)dη. R η0 η?. Ωφ (η)dη. .. (3.8). 10.

(11) Figure 3.2: The demonstration of ΛCDM model. The error on θ? is 2.59% and time-average hwi is -1. Where η0 is the conformal time at today and η? is the time at last scattering surface. Since each model has its own characteristic on hwi, the approximation using constant number hwi is effective in each model. [39, 40, 41, 42]. 3.2.2. Acoustic scale lA. Additionally, the decisive parameter, acoustic scale lA , illustrates the consistency in different models. It is defined by lA = π/θ? ≡ πDA /r? , where DA = η0 − η? is the commoving distance to last scattering surface, and r? =. (3.9) R η? 0. cs dη is the. sound horizon at decoupling epoch with sound speed cs . The accuracy on the acoustic scale has become significantly high since the acoustic peak location has already been studied in detail. According to the baseline model, the study has shown [43] that the power spectrum peaks at lA = pπ/θ? = 302p. Where θ? is the angular size of sound horizon at z? = 1089. θ? will be R η?. θ? =. cs dη = 1.0402 × 10−2 . η0 − η? 0. (3.10) 11.

(12) This equation only base on the theory before last scattering surface and the observational result on CMB anisotropies. It is relatively insensitive to distinct dark energy models. Consequently, the error on the angular size, ∆θ? = (θ?,cal − θ?,obs )/θ?,obs , will guide us to the superior model of dynamically evolving dark energy. Here the cal stands for the calculation value of our numerical method and obs is the actual observational value. In order to employ the high accuracy acoustic scale in our method, the calculation of equation 3.10 needs to be done within different dark energy models. The sound speed cs in the equation is equal to cs = 1/(3 + R) where R is approximate [11]to N (Ωb h2 )(1 + z)−1 , (N = 30230). Furthermore, the commoving distance DA requires the estimation of η? and η0 . we will use the approximation on η? and η0 in which the conformal times are evaluated by substituting the Ωφ with the energy density of φ around last scattering surface Ωφ,? and the equation of state w with time-average hwi. The analytical formulas for η? and η0 can be obtained as following: Considering the Friedmann equation with the reduce Planck mass Mp2 ≡ (8πG)−1 H2 =. 1 (ρm + ρr + ρφ ), 3Mp2. (3.11). where ρφ represents the energy density of the dark energy field in flat universe. Using the reduce Planck mass the Friedmann equation will become . 3Mp2 H 2 (1 − Ωφ (t)) = 3Mp2 H02 Ωm,0 a−3 + Ωr,0 a−4. . (3.12). 3.2.3 η? approximation Substituting Ωφ (t) with the constant average Ωφ,? for the period around last scattering surface . H H0. 2. =. Ωm,0 a−3 + Ωr,0 a−4 . 1 − Ωφ,?. (3.13). Transfer the time coordinate to the conformal time η . da dη. 2. = H02 (1 − Ωφ,? )−1 (Ωm,0 a + Ωr,0 ),. (3.14). where the equation will become Z η? 0. q. dη = H0−1 1 − Ωφ,?. Z a? 0. p. da . Ωm,0 a + Ωr,0. (3.15). As the result, the approximation of η? will become p. 2 1 − Ωφ,? p η? ' H0 Ωm,0. s. Ωr,0 a? + − Ωm,0. s. Ωr,0 Ωm,0. !. .. (3.16). 3.2.4 η0 approximation To dirive the analytical form of η0 , first introduce the time-average hwi. For a constant hwi, the scaling law of the dark energy component is approximated by Ωφ (t) ' Ωφ,0 a−3(1+hwi) . The Friedmann eqution in conformal time coordinate becomes . da dη. 2. = H02 (Ωm,0 a + Ωφ,0 a(1−3hwi) + Ωr,0 ),. (3.17) 12.

(13) where the conformal time can be calculated by Z η0 0. dη = H0−1. Z 1 0. da q. .. (3.18). Ωm,0 a + Ωφ,0 a(1−3hwi) + Ωr,0. As the result, the approximation of η0 becomes 1 p η0 ' H0 Ωm,0. 3.3. Z 1( 0. Ωφ,0 (1−3hwi) Ωr,0 a+ a + Ωm,0 Ωm,0. )−1/2. da.. (3.19). Numerical code. In this project, we use python as the coding langue on the jupyter notebook platform. We modified the code used in Wolung Lee and King-Wang’s paper and transferred to the phantom model. Also note that the numerical methods that are applied on our method are runge-kutta method of order 5 due to Dormand and Prince, and Real-valued Variable-coefficient Ordinary Differential Equation solver which provides implicit Adams method and a method based on backward differentiation formulas (for stiff problems). The numerical code will be in Fig. 3.3 Fig. 3.4 Fig. 3.5. Figure 3.3: The runge-kutta method solving the four first order differential equations. 13.

(14) Figure 3.4: The Real-valued Variable-coefficient Ordinary Differential Equation solver solving the acoustic scale. Figure 3.5: calculating the deceleration parameter q and the time-average hwi. 14.

(15) Chapter 4. Result and Conclusion 4.1. Phantom model. Collecting the result from Planck collaboration [1] and the local distance measurement [2], the constrains show [3] a preference for w < −1 and deeply phantom. Also, the parameterization of the equation of state offers a superior fit to a wide range of scalar field. [44, 35] So as to fit the observational data, the research [45] yeld a reasonably motivated model, vacuum metamorphosis (from now on call VM). [14, 15, 16] Despite the model has their own stably fiducial foundation, here we are going to skip the reconstruction of the scalar field and focus on the properties that the equation of state w has. In the original VM model, the dark energy rapidly evolves in fairly recent redshift which is between z = 1.3 to z = 0.7. We replicate the equation of state numerically in this respect. In Fig. 4.1 we demonstrate the whole picture of the evolution under VM model. The initial conditions are established by present observational data which include local H0 constant [13], energy density on Planck collaboration [1]. More specifically, the energy densities are Ωb h2 = 0.02230 ± 0.00014, Ωm h2 = 0.14170 ± 0.00097, ΩΛ = 0.6911 ± 0.0062 which change slightly in the VM model as from ΛCDM model. In the other hand, the Hubble constant change more significantly in different models. Under the circumstances, we pick the value of H0 in local distance measurement to examine the consistency on acoustic scale. The result shows high accuracy on CMB acoustic scale (at the 0.2% level) where the feature of this model hwi = −1.29. Also note that H0 t represents the rescaling present time and q is the deceleration parameter.. 4.2 H0 -hwi diagram The outcome for VM model already provides great conformity on CMB acoustic scale, however, we are more interested about why the involvement of phantom is necessary. Therefore, we plot the relation between H0 values, due to the tension on Hubble constant, and different phantom models characterized by time average hwi. The result is in Fig. 4.2 which the depth of the color indicates the accuracy on acoustic scale. The average energy density of dark energy around last scattering surface Ωφ,? has been ignored due to the rapidly evolving equation of state. The approximation in Eq. 3.16 of η? can be reduced to an equation dominated by H0 (also effects the value of energy densities distribution on each component). Here we note that in some quintessence models (the part in Fig. 4.2 where the time-average hwi > −1) Ωφ,? is not negligible. In Lee‚s paper [38] we can see that Ωφ,? is another parameter which will impact the consistency on acoustic scale (also see in Eq. 3.16). Here we still keep Ωφ,? negligible in the fitting 15.

(16) Figure 4.1: Evolution of universe under vacuum metamorphosis model using numerical method. The error on θ? is 0.22% and time-average hwi is −1.29 Table 4.1: The result on three typical categories of dark energy under numerical methods. ∆θ? hwi Ωφ,?. GQ1 +Planck3 1.555% -0.902 0.294. GQ +R164 3.182% -0.898 0.347. ΛCDM 1.118% -1.0 1.27 × 10−9. VM2 +Planck 0.723% -1.261 2.08 × 10−10. VM +R16 0.220% -1.290 2.65 × 10−10. test because the possibilities of quintessence have already been eliminated by calculating the evolution without ignoring Ωφ,? . Table 4.1 will provide the contrast between different types of dark energy. In addition, we can see that in the middle of Fig. 4.2 there is a darkest part where the error of acoustic scale is minimal. As the matter of fact, we can infer two meaningful results from this particular area. Firstly, if we 1. the quintessence model, from Ref. [38]. the phantom model, from Ref. [45]. 3 Planck collaboration data, from Ref. [1]. 4 distance measurement data, from Ref. [2]. 2. 16.

(17) Figure 4.2: Hubble constant and time-average hwi diagram with angular size error. The two vertical lines stand for the two representative value of the Hubble constant, CMB anisotropies and local distance measurement on supernova.. are going to accept the supernova’s data, the VM model seems sensible to explain the dark energy. While other phantom models could also be a reasonable interpretation if their time-average hwi shut on the appropriate values. Secondly, even if we utilize the Hubble constant on CMB anisotropies, the equation of state is still less thean negative one. The prediction on w will be situated between −1.15 to −1.35. As the result, it is essential to comprise the phantom as a hypothesis of the dynamically evolving dark energy. However, these works are all done by phenomenological methods. The actual physics is still a missing. 17.

(18) piece of the dark energy puzzle. Since we could compose the result without reconstructing the scalar field, the only supportive motivation is the decreasing equation of state. Considering the contradiction on negatively kinetic energy, phantom models need a more comprehensive perspective.. 4.3. Conclusion. In this paper, we resume a numerical method which has already been applied on quintessence models in order to be acquainted with a relatively new type of dark energy, phantom. It is a quite benificial way to avoid reconstructing the actual dark energy scalar field and jump directly into the expression of large scale evolution. The characteristics of each various models can be easily illustrated in this numerical method. We choose to employ specific parameter, acoustic scale lA , to indicate the consistency in each model because of the high accuracy in recent CMB data and the invariance due to the background theories before last scattering surface. In the first targeted phantom model, vacuum metamorphosis, the error on the acoustic scale has reduced to 0.22% which highly supports the indispensable involvement of phantom. However, in the H0 -hwi diagram Fig. 4.2 we discovered that phantom-like hwi is model independently necessary under present observational data, Hubble constant H0 especially. As the result, phantom might be a misleading issue which its authenticity is hard to distinguish. It also brings up the coincident problem why the missing energy density can evolve so rapidly to two-third in present days. New phantom model should give a prediction on future observation but not only shows the consistency on recent data. Phenomenologically speaking, we already used our numerical method to tell that our univers has a preference on w < −1 and more rapidly evolving dark energy. In conclusion, we still rely on the next generational observations, such as gravitational wave, to resolve the tension on Hubble constant and hopefully offer new perspective on dark energy. We will need more fundamental theories to interpret the essence of the missing energy.. 18.

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