Dark energy models from a parametrization of H : A comprehensive analysis and observational constraints
DDark energy models from a parametrization of H : A comprehensive analysis andobservational constraints S. K. J. Pacif ∗ Department of Mathematics, School of Advanced Sciences,Vellore Institute of Technology, Vellore 632014, Tamil Nadu, India (Dated: June 18, 2020)The presented paper is a comprehensive analysis of two dark energy (DE) cosmological modelswherein exact solutions of the Einstein field equations (EFEs) are obtained in a model-independentway (or by cosmological parametrization). A simple parametrization of Hubble parameter ( H ) isconsidered for the purpose in the flat Friedmann-Lemaitre-Robertson-Walker (FLRW) background.The parametrization of H covers some known models for some specific values of the model parametersinvolved. Two models are of special interest which show the behavior of cosmological phase transitionfrom deceleration in the past to acceleration at late-times. The model parameters are constrainedwith 57 points of Hubble datasets together with the 580 points of Union 2 . φ and their dynamics are discussed on the geometrical basebuilt. The geometrical and physical interpretations of the two models in consideration are discussedin details and the evolution of various cosmological parameters are shown graphically. The age ofthe Universe in both models are also calculated. Various cosmological parametrization schemes usedin the past few decades to find exact solutions of the EFEs are also summarized at the end whichcan serve as a unified reference for the readers. I. INTRODUCTION
Late-time cosmic acceleration is an essential constituent of precision cosmology at present. The idea of cosmicacceleration was first evidenced by the observations of high redshift supernova of type Ia [1, 2]. The idea of cosmicacceleration was later accepted quickly by the scientific community largely because of the independent observationswith different methodology adopted by the supernova search teams lead by Perlmutter and Riess and also the CMBand the large scale structure data were providing substantial evidence for a cosmological constant, indirectly [3–5].Later on some robust analysis and precise observations strengthen the idea of cosmic acceleration and a flat Universeconsistent with Ω Λ = 1 − Ω m = 0 .
75 [6–9]. What causing the accelerating expansion is still a mystery and we aremostly in dark in this context. However, the theoretical predictions and advanced surveys in observational point ofview indicating the presence of a weird form of energy in the Universe with high negative pressure with increasingdensity. The mysterious energy is named as dark energy [10] as it’s nature, characteristics is speculative only withoutany laboratory tests. Also, the candidature of dark energy is a debatable topic at present cosmological studies.Moreover, the age crisis in the standard model need cosmic acceleration [11]. Although, the modification of gravitytheory at infra red scale attracted attention to explain the late-time acceleration without invoking any extra sourceterm [12, 13], but the theory of dark energy became quite popular [14, 15].Very recently, gravitational wave detection and the picture of black hole shadow strenghten the Einstein’s generaltheory of relativity and any modifications in the Einstein’s theory (specifically to the geometry part) is not worthappreciated. However, Einstein himself was not convinced with the matter distribution in the Universe i.e. the righthand side of his field equations (representing matter sector) is considered to made up of low grade wood while thegeometry part is of solid marble (representing the space-time). Any extra source term such as Einstein’s cosmologicalconstant (representing energy density of vacuum) could be added into the energy momentum tensor and serve as acandidate of dark energy. The most favoured candidate of dark energy is the well known cosmological constant Λ.Also, ΛCDM models have the best fit with many observational datasets. However, with this significant Λ, due to itsnon dynamical and the long standing fine tuning problem, researchers thought beyond it for a better candidate ofdark energy. So, scalar field models were discussed after the cosmological constant for which Λ could also be generatedfrom particle creation effect [16]. The dynamically evolving scalar field models have been utilized for the purpose ∗ Electronic address: [email protected] a r X i v : . [ phy s i c s . g e n - ph ] J un are quintessence [17–19], K-essence [20–22], phantom [23] and tachyonic field [24–26]. The exotic fluid is also servethe purpose to explain the cosmic acceleration phenomenology that considered an equation of state producing largenegative pressure e.g. Chapligyn gas equation of state [27], Polytropic gas equation of state [28] etc.Soon after the formulation of EFEs, theoreticians worked on finding exact solutions. The first exact solution ofthe EFEs is the Schwarzschild exterior solution [29] wherein the prefect fluid equation of state was considered as asuplementary condition. Despite of the high non linearity of the EFEs, various exact solutions are obtained for staticand spherically symmetric metrics. Einstein’s static solution [30], de-Sitter solution [31], Tolman’s solutions [32],Adler’s solutions [33], Buchdahl’s solution [34], Vaidya and Tikekar solution [35], Durgapal’s solutions [36], Knutsen’ssolutions [37] and many more well-known solutions of EFEs are obtained which are summarized in the literature [38]and also discussed in [39]. Milne’s model [40], steady state model [41] are some different models proposed. All thosephenomenological cosmological models explain the Universe theoretically very well. However, observations play amajor role in modern cosmology which validate or discard a model. Now, numerical computations are also playingbig role in modern cosmology and estimating cosmological parameters and also parameters of a model. In this study,an important discussion is given on a technique of finding exact solution of EFEs known as model independent waystudy. Moreover, two models are discussed and analyzed comprehensively with current trendz in theoretical comology.The paper is organized as follows. The first section is an introduction to present cosmological scenario. The secondsection describes the Einstein’s field equations in general relativity in presence of dark energy. The third section is amotivation to the idea of model independent way or the cosmological parametrization study to obtain exact solutionsto EFEs. A simple parametrization of Hubble parameter is considered in the light of cosmographical study in thefourth section. In the fifth section, observational constraints have been found for the model parameters involved inthe functional form of H for the two models obtained. The sixth section is devoted to the geometrical dynamics andanalysis of some important cosmological parameters describing the geometrical behavior of the Universe for both themodels. In the seventh section, two candidates of dark energy is explored, cosmological constant and a general scalarfield and the physical parameters such as energy density, density parameter, potential of scalar field and equation ofstate parameter are discussed for both the models under considerations. In the eighth section, the age of the Universefor the obtained models are calculated. The final section summarizes the physical insights of the results obtained andconcluded. A brief summary of the various parametrization schemes used in the past few decades are given in theappendix. II. EFES IN PRESENCE OF DARK ENERGY
The nature of dark energy and its candidature is a mystery and it is a matter of speculation to express it as asource term into the Einstein field equations. However, DE is speculated to be homogeneous permeating all over thespace for which the energy momentum tensor can be represented in the form of a perfect fluid as T DEij = ( ρ DE + p DE ) U i U j + p DE g ij , (1)with its equation of state in the form p DE = ω DE ρ DE , where ω DE is the equation of state (EoS) parameter and isa function of time in general satisfying the inequality ω DE <
0. There is hot debate going on for a suitable valueof ω DE and the analysis of some observational data shows that its value lies in the range − . < ω DE < − . ω DE [44, 45]. The different values of ω DE in certain ranges gives rise to different candidates and can broadly be be classified as follows. For (cid:73) ω DE = −
1, the case is for the cosmological constant; (cid:73) ω DE = constant (cid:54) = −
1, the case is for cosmic strings, domain walls, etc.; (cid:73) ω DE (cid:54) = constant , the cases for scalar fields (quintessence, k-essence etc.), braneworlds, Dirac-Born-Infeld(DBI)action, Chaplygin gas etc.; (cid:73) ω DE < −
1, the case is for phantom models.For a broader list of dark energy models see [14, 15] (and refs. therein). There are interesting cases in each of themwith some problems though. For example, Cosmological constant Λ is the most consistent model for dark energyexplaining observations but is plagued with fine tuning problem. Similarly, the phantom models are interesting wherethe weak energy condition ( ρ > , ρ + p >
0) is violated with the feature of finite time singularity [23].In general relativity, dark energy can be introduced by supplementing the energy momentum tensor T DEij into theEinstein field equations G ij = − πGT ij together with the matter source T Mij as a perfect fluid, T totij = T mij + T deij = ( ρ tot + p tot ) U i U j + p tot g ij , (2)with ρ tot = (cid:80) ρ + ρ de and p tot = (cid:80) p + p de denoting the total energy densities and total pressure due to all typesof matter (baryonic matter, dark matter and radiation) and dark energy respectively. U i is the usual four velocityvector and g ij is the metric tensor. Now, the modified Einstein Field Equations for a flat FLRW metric ds = − dt + a ( t ) (cid:2) dr + r ( dθ + sin θdφ ) (cid:3) , (3)where a ( t ) is the scale factor of the Universe, can be written as M − pl ρ tot = 3 (cid:18) ˙ aa (cid:19) = 3 H , (4) M − pl p tot = − aa − (cid:18) ˙ aa (cid:19) = (2 q − H . (5)The conservation of energy-momentum (or from Eqs. (4) and (5) yields˙ ρ tot + 3( p tot + ρ tot ) ˙ aa = 0 . (6)The continuity equation (6) play significant role in the evolution as it deals with the matter and its interaction. Incurrent cosmology, two kinds of dark energy models generally discussed; interacting models of dark energy (consideringthe interaction between cold dark matter and dark energy) [46–48] and non-interacting models of dark energy whereall the matters allowed to evolve separately [49–52]. Up to date, there are no known interaction other than gravitybetween the matter and dark energy. The present study refers to non interacting models only. The system of equationsare non linear ordinary differential equations and is difficult to find exact solutions. There are tremendous efforts tofind both the exact and numerical solutions to EFEs in the past. In the next section, the solution techniques of theabove system of equations will be discussed elaborately. III. COSMOLOGICAL PARAMETRIZATION
The above system of equations (4), (5) and (6) possesses only two independent equations with five unknowns a , ρ , p , ρ de , p de (or ω de ). Due to the homogeneous distribution of matter in the Universe at large scale, it is customaryto consider the barotropic equation of state p = ωρ , ω ∈ [0 , ω thatincludes baryonic matter ( ω = 0), dark matter ( ω = 0), radiation ( ω = 1 / ω = 1), etc. This additionalequation provides the third constraint equation. Another constraint equation can be the consideration of equation ofstate of dark energy ( ω de = constant or a function of time t or function of scale factor a or function of redshift z ) -best known as parametrization of dark energy equation of state. These four equations can explain the cosmologicaldynamics of the Universe where all the geometrical parameters (Hubble parameter H , deceleration parameter q , jerkparameter j , etc.) or physical parameters (densities ρ , ρ de , pressures p , p de , EoS parameter ω de , density parameterΩ i , etc.) are expressed as functions of either scale factor a or the redshift z (= a a − a being the present value ofscale factor generally normalized to a = 1). But, there is still one more equation short to close the system for thecomplete determination of the system; the time evolution of scale factor a is yet to be determined. In literature, thereare various schemes of parametrization of the scale factor and it’s higher order derivatives ( H , q , j etc.) providingthe complete solution of the EFEs i.e. the explicit forms of cosmological parameters as a function of cosmic time t .In fact, a critical analysis of the solution techniques of EFEs in general relativity theory or in modified theorieshas two aspects; one is the parametrization of geometrical parameters a , H , q , j giving the time dependent functionsof all the cosmological parameters; another is the parametrization of the physical parameters ρ , p , ρ de , p de (or ω de )giving the scale factor dependence or redshift dependence of all the cosmological parameters. See the appendix for abroad list of different schemes of parametrization of geometrical parameters and physical parameters and also somephenomenological ansatzs used in the past few decades to find the exact solutions of Einstein field equations. If we,observe carefully, we can say that the first kind of parametrization schemes (of geometrical parameters) are consideredto find exact solutions that discusses the expansion dynamics of the Universe and provides the time evolution of thephysical parameters ρ , p , ρ de , p de (or ω de ). This method is generally known as model independent way study ofcosmological models or the cosmological parametrization [53–55]. The method do not affect the background theoryanyway and provide solutions to the EFEs explicitly and also has an advantage of reconstructing the cosmic historyof the Universe explaining some phenomena of the Universe. Also, this method provides the simplest way to resolvesome of the problems of standard model e.g. the initial singularity problem, cosmological constant problem, etc.and also explain the late-time acceleration conundrum, theoretically. While the second kind of parametrization(of physical parameters) are generally considered to discuss all the physical aspects (thermodynamics, structureformation, nucleosynthesis etc.) of the Universe. However, both the schemes of parametrization are adhoc choices orsome phenomenological ansatzs (e.g. Λ-varying cosmologies). All parametrization schemes (see appendix-1) containsome arbitrary constants, referred to as model parameters which are constrained through any observational datasets.The purpose here is to obtain an exact solution of the Einstein field equations in standard general relativity theorywith a simple parametrization of the Hubble parameter H and discuss the reconstructed cosmic evolution. IV. PARAMETRIZATION OF H & THE MODELS The cosmographic analysis provide clues to study the evolution of the observable Universe in a model independentway in terms of the kinematic variables [56]. Moreover, analysis of cosmographic parameters helps in studying thedark energy without any assumption of any particular cosmological model except only the cosmological principle. Inthe standard approximation the scale factor can be expanded in Taylors series around the present time t (which is thecurrent age of the Universe also) and is the simple strategy adopted in cosmographical analysis. Here and afterwardsa suffix 0 denotes the value of the parameter at present time t . The Taylor’s series expansion can be written as: a ( n ) = 1 + H ( t − t ) − q H ( t − t ) + 13! j H ( t − t ) + 14! s H ( t − t ) + 15! l H ( t − t ) + ..... (7)where H ( t ) = a dadt is the Hubble parameter measuring velocity, q ( t ) = − a d adt (cid:2) a dadt (cid:3) − is the deceleration parametermeasuring acceleration, j ( t ) = a d adt (cid:2) a dadt (cid:3) − jerk parameter measuring jerk, s ( t ) = a d adt (cid:2) a dadt (cid:3) − is the snap param-eter and l ( t ) = a d adt (cid:2) a dadt (cid:3) − is the lerk parameter. All of these parameters play significant roles in the cosmographicanalysis of the Universe (specifically the H , q and j ) and distinguish various dark energy models.Motivated by the above discussion, in this paper, a simple parametrization of the Hubble parameter ( H ) is consideredas an explicit function of cosmic time ‘ t ’ in the form [55] H ( t ) = k t m ( t n + k ) p (8)where k , k (cid:54) = 0 , m, n, p are real constants (or model parameters). k and k both have the dimensions of time.Some specific values of the parameters m, n, p suggest some distinguished models which are elaborated by Pacif et al[55]. It is easy to see that, the single parametrization (8) generalizes several known models e.g. ΛCDM model, powerlaw model, hybrid expansion model, bouncing model, linearly varying deceleration parameter model and some more.Out of the twelve models deduced for some integral or non integral values of m , n , p in the functional form of HP in(8), two models (with m = − p = 1, n = 1 and with m = − p = 1, n = 2) show the possibility of describing thephenomena of cosmological phase transition for negative k & k and is described as in the following Table-1.Table-1: The modelsModels H ( t ) a ( t ) q ( t )M1 k t ( k − t ) β (cid:16) tk − t (cid:17) k k − k k − k t M2 k t ( k − t ) β (cid:16) t k − t (cid:17) k k − k k − k t where, β is an integrating constant which also play an important role in the evolution. Pacif et al. obtained solutionsfor these two models in a scalar field background and also found the observational constraints on model parameterswith 28 points of H ( z ) datasets. The present paper is an extension of the same study for these two models M1 andM2 wherein much deeper analysis have been done.One can see, for both the models M1 and M2, the Hubble parameter and scale factor both diverge in finite timeand show a big rip singularity in near future at t = t s = k for model M1 and at t = t s = √ k for model M2. Thephase transition occurs at time t tr = k − k for model M1 and at time t tr = (cid:113) k − k for model M2 and suggest that k must be greater than k . With some suitable choice of model parameters k , k and β , rough sketches for thetime-evolution of scale factor (SF) and the Hubble parameter (HP) are made and are shown graphically in the figuresFIG. 1 and FIG. 2 respectively showing that a ( t ) diverges in finite time and the H ( t ) becomes asymptotic showingbig rip in near future. ( k , k , β )(
3, 0.5, 1.0 )(
3, 1.0, 1.0 )(
3, 2.1, 1.5 )(
3, 2.9, 1.5 )(
4, 1.1, 1.0 )(
4, 1.9, 1.0 ) t a ( t ) ( k , k , β )(
3, 0.5, 1.0 )(
3, 1.0, 1.0 )(
3, 2.1, 1.5 )(
3, 2.9, 1.5 )(
4, 1.1, 1.0 )(
4, 1.9, 1.0 ) t a ( t ) ( a ) ( b )FIG. 1: Figures (a) and (b) respectively show rough sketches of the evolotion of the scale factor w.r.t. cosmic time ‘ t ’ for bothmodels M1 and M2 with some arbitrary values of the model parameters k , k , β . ( k , k )(
3, 0.5 )(
3, 1.0 )(
3, 2.1 )(
3, 2.9 )(
4, 1.1 )(
4, 1.9 ) t H ( t ) ( k , k )(
3, 0.5 )(
3, 1.0 )(
3, 2.1 )(
3, 2.9 )(
4, 1.1 )(
4, 1.9 ) t H ( t ) ( a ) ( b )FIG. 2: Figures (a) and (b) respectively show rough sketches of the evolotion of the Hubble parameter w.r.t. cosmic time ‘ t ’for both models M1 and M2 with some arbitrary values of the model parameters k , k , β . In order to check the consistency of the theoretical models obtained here with the observations, some availabledatasets are used in the next section. The model parameters are constrained through these datasets.
V. OBSERVATIONAL CONSTRAINTS
Three datasets are considered here for our analysis namely Hubble datasets ( Hz ), Type Ia supernovae datasets( SN ) and Baryon Acoustic Oscillations datasets ( BAO ). The detailed datasets and the method used are explainedbelow.In the study of late-time Universe and the observational studies, it is convenient to express all the cosmologicalparameters as functions of redshift z . As the cosmological parameters here are functions of cosmic time t , the time-redshift relationship must be established. The t - z relations are obtained as: t ( z ) = k (cid:104) { β (1 + z ) } k k (cid:105) − , (9)for model M1 and t ( z ) = (cid:112) k (cid:104) { β (1 + z ) } k k (cid:105) − (10)for model M2. The above expressions (9) and (10) contain three parameters β , k and k but actually two modelparameters are sufficient to describe these models by taking k k = α which is also beneficial for further analysis andnumerical computations for which the expressions for the Hubble parameter for both the models M1 and M2 arewritten in terms of redshift z as follows: H ( z ) = H (1 + β α ) − (1 + z ) − α [1 + { β (1 + z ) } α ] , (11)for model M1 and H ( z ) = H (cid:0) β α (cid:1) − (1 + z ) − α (cid:104) { β (1 + z ) } α (cid:105) (12)for model M2. The different datasets are described below. A. H(z) datasets
It is well known that the Hubble parameter ( H = ˙ aa ) directly probes the expansion history of the Universewhere ˙ a is the rate of change of the scale factor a of the Universe. Hubble parameter is also related to the differentialredshift as, H ( z ) = − z dzdt , where dz is obtained from the spectroscopic surveys and so a measurement of dt providesthe Hubble parameter which will be independent of the model. In fact, two methods are generally used to measurethe Hubble parameter values H ( z ) at certain redshift and are extraction of H ( z ) from line-of-sight BAO data anddifferential age method [57]-[75] estimating H ( z ). Here, in this paper, an updated list of 57 data points are usedas listed in Table-2 out of which 31 data points measured with DA method and 26 data points are obtained withBAO and other methods in the redshift range 0 . (cid:54) z (cid:54) .
42 [76]. Moreover, the value of H is taken as priorfor our analysis as H = 67 . Km/s/M pc [77]. The chi square function to determine the mean values of the modelparameters α & β (which is equivalent to the maximum likelihood analysis) is, χ H ( α, β ) = (cid:88) i =1 [ H th ( z i , α, β ) − H obs ( z i )] σ H ( z i ) , (13)where, H th is the theoretical, H obs is the observed value and σ H ( z i ) is the standard error in the observed value ofthe Hubble parameter H . The 57 points of Hubble parameter values H ( z ) with errors σ H from differential age (31points) method and BAO and other (26 points) methods are tabulated in Table-2 with references.Table-2: 57 points of H ( z ) datasets31 points from DA method 26 points from BAO & other method z H ( z ) σ H Ref. z H ( z ) σ H Ref. z H ( z ) σ H Ref. z H ( z ) σ H Ref.0 .
070 69 19 . . .
24 79 .
69 2 .
99 [64] 0 .
52 94 .
35 2 .
64 [66]0 .
90 69 12 [58] 0 .
480 97 62 [57] 0 .
30 81 . .
22 [65] 0 .
56 93 .
34 2 . .
120 68 . . .
593 104 13 [59] 0 .
31 78 .
18 4 .
74 [66] 0 .
57 87 . . .
170 83 8 [58] 0 . .
34 83 . .
66 [64] 0 .
57 96 . . . . .
35 82 . . .
59 98 .
48 3 .
18 [66]0 . . .
36 79 .
94 3 .
38 [66] 0 .
60 87 . . .
200 72 . . .
880 90 40 [57] 0 .
38 81 . . .
61 97 . . .
270 77 14 [58] 0 .
900 117 23 [58] 0 .
40 82 .
04 2 .
03 [66] 0 .
64 98 .
82 2 .
98 [66]0 .
280 88 . . .
037 154 20 [59] 0 .
43 86 .
45 3 .
97 [64] 0 .
73 97 . . . .
300 168 17 [58] 0 .
44 82 . . .
30 224 8 . . . .
363 160 33 . .
44 84 .
81 1 .
83 [66] 2 .
33 224 8 [73]0 .
400 95 17 [58] 1 .
430 177 18 [58] 0 .
48 87 .
79 2 .
03 [66] 2 .
34 222 8 . . . .
530 140 14 [58] 0 .
51 90 . . .
36 226 9 . . . . .
750 202 40 [58]0 . . . .
965 186 . . .
470 89 34 [62]
B. SN Ia datasets
The first indication for the accelerating expansion of the Universe is due to observations of supernovae oftype Ia . Since then, several new SN Ia datasets have been published. In this analysis, the Union 2 . µ i = µ obsi for a particular redshift z i in the interval 0 < z i ≤ .
41. The model parameters of the models areto be fitted with, comparing the observed µ obsi value to the theoretical µ thi value of the distance moduli which are thelogarithms µ thi = µ ( D L ) = m − M = 5 log ( D L ) + µ , where m and M are the apparent and absolute magnitudesand µ = 5 log (cid:0) H − /M pc (cid:1) + 25 is the nuisance parameter that has been marginalized. The luminosity distance isdefined by, D l ( z ) = c (1 + z ) H S k (cid:18) H (cid:90) z H ( z ∗ ) dz ∗ (cid:19) , where S k ( x ) = sinh( x √ Ω k ) / Ω k , Ω k > x , Ω k = 0sin x (cid:112) | Ω k | ) / | Ω k | , Ω k < . Here, Ω k = 0 (flat space-time). For our cosmological models M1 and M2 with theoretical value H ( z ) which aredepending on the model parameters α & β , the distance D L ( z ) is calculated and the corresponding chi squarefunction measuring differences between the SN Ia observational data and values predicted by the models is given by, χ SN ( µ , α, β ) = (cid:88) i =1 [ µ th ( µ , z i , α, β ) − µ obs ( z i )] σ µ ( z i ) , (14) σ µ ( z i ) is the standard error in the observed value. Following [79] after marginalizing µ , the chi square function iswritten as, χ SN ( α, β ) = A ( α, β ) − [ B ( α, β )] /C ( α, β )where A ( α, β ) = (cid:80) i =1 [ µ th ( µ =0 ,z i ,α,β ) − µ obs ( z i )] σ µ ( zi ) ,B ( α, β ) = (cid:80) i =1 [ µ th ( µ =0 ,z i ,α,β ) − µ obs ( z i )] σ µ ( zi ) , C ( α, β ) = (cid:80) i =1 1 σ µ ( zi ) . C. BAO datasets
Baryonic acoustic oscillations is an analysis dealing with the early Universe. It is known that the early Universefilled with baryons, photons and dark matter. Moreover, baryons and photons together act as single fluid (coupledtightly through the Thompson scattering) and can not collapse under gravity rather oscillate due to the large pressureof photons. These oscillations are termed a Baryonic acoustic oscillations (BAO). The characteristic scale of BAO isgoverned by the sound horizon r s at the photon decoupling epoch z ∗ is given as, r s ( z ∗ ) = c √ (cid:90) z ∗ daa H ( a ) (cid:112) b / γ ) a , where Ω b stands for the baryon density and Ω γ stands for the photon density at present time.The BAO sound horizon scale is also used to derive the angular diameter distance D A and the Hubble expansionrate H as a function of z . If (cid:52) θ be the measured angular separation of the BAO feature in the 2 point correlationfunction of the galaxy distribution on the sky and the (cid:52) z be the measured redshift separation of the BAO feature inthe 2 point correlation function along the line of sight then, (cid:52) θ = r s d A ( z ) where d A ( z ) = (cid:82) z dz (cid:48) H ( z (cid:48) ) and (cid:52) z = H ( z ) r s .In this work, BAO datasets of d A ( z ∗ ) /D V ( z BAO ) from the references [80–85] is considered where the photon decou-pling redshift is z ∗ ≈ d A ( z ) is the co-moving angular diameter distance and D V ( z ) = (cid:0) d A ( z ) z/H ( z ) (cid:1) / isthe dilation scale. The data used for this analysis is given in the Table-3Table-3: Values of d A ( z ∗ ) /D V ( z BAO ) for distinct values of z BAO z BAO .
106 0 . .
35 0 .
44 0 . . d A ( z ∗ ) D V ( z BAO ) . ± .
46 17 . ± .
60 10 . ± .
37 8 . ± .
67 6 . ± .
33 5 . ± . χ BAO = X T C − X , (15)where X = d A ( z (cid:63) ) D V (0 . − . d A ( z (cid:63) ) D V (0 . − . d A ( z (cid:63) ) D V (0 . − . d A ( z (cid:63) ) D V (0 . − . d A ( z (cid:63) ) D V (0 . − . d A ( z (cid:63) ) D V (0 . − . , and C − is the inverse covariance matrix defined in [85]. C − = . − . − . − . − . − . − . . − . − . − . − . − . − . . − . − . − . − . − . − . . − . . − . − . − . − . . − . − . − . − . . − . . With the above samples of Hubble ( Hz ), supernovae of type Ia ( SN ) and baryon acoustic oscillations ( BAO )datasets, the chi square functions (13), (14) and (15) are minimized to get the average values of the model parameters α & β . The maximum likelihood contours for the model parameters α & β are shown in the following figures FIG. 3,FIG. 4, FIG. 5, and FIG. 6 for independent Hz datasets and combined Hz + SN , SN + BAO and Hz + SN + BAO datasets respectively with 1- σ , 2- σ and 3- σ error contours in the α - β plane. α β α β ( a ) ( b )FIG. 3: Figures (a) and (b) are contour plots for Hubble datasets ( Hz ) for models M1 and M2 respectvely. α β α β ( a ) ( b )FIG. 4: Figures (a) and (b) are contour plots for combined Hz + SN datasets for models M1 and M2 respectvely. α β α β ( a ) ( b )FIG. 5: Figures (a) and (b) are contour plots for combibed SN + BAO datasets for models M1 and M2 respectvely. α β α β ( a ) ( b )FIG. 6: Figures (a) and (b) are contour plots for combined Hz + SN + BAO datasets for models M1 and M2 respectvely.
The average mean values (constrained values) of the model parameters and the minimum chi square values aretabulated in Table-4 for independent Hz datasets and combined Hz + SN , SN + BAO and Hz + SN + BAO datasets. Table-4: Constrained values of model parameters and chi square valuesDatasets Models α β χ χ /dofH ( z ) M1M2 1 . . . . . . . . H ( z ) + SN M1M2 1 . . . . . . . . SN + BAO
M1M2 1 . . . . . . . . H ( z ) + SN + BAO
M1M2 1 . . . . . . . . Hz and 580 points of Union 2 . VI. GEOMETRICAL DYNAMICS OF THE MODELSA. Deceleration parameter & Phase transition
The expressions for the deceleration parameter can be written in terms of redshift z as: q ( z ) = − α − α [1 + { β (1 + z ) } α ] − (16)for model M1 and q ( z ) = − α − α (cid:104) { β (1 + z ) } α (cid:105) − (17)for model M2. The behavior of q is shown in the following FIG. 9 and the important values assumed by the decelerationparameter q in the course of evolution are tabulated in the following Table-5 for different sets of α & β as obtained.The initial value of the deceleration parameter ( q i as z −→ ∞ ), the present value of the deceleration parameter ( q as z −→
0) and the far future value of the deceleration parameter ( q f as z −→ −
1) are calculated together with the1
HzHz + SNSN + BAOHz + SN + BAO z H ( z ) HzHz + SNSN + BAOHz + SN + BAO z H ( z ) ( a ) ( b )FIG. 7: Figures (a) and (b) are the error bar plots for 57 data points from Hubble datasets together with the models M1 andM2 shown in solid red lines respectively. The dashed lines in both the figures are ΛCDM model shown for comparision. HzHz + SNSN + BAOHz + SN + BAO z μ ( z ) HzHz + SNSN + BAOHz + SN + BAO z μ ( z ) ( a ) ( b )FIG. 8: Figures (a) and (b) are the error bar plots for Union 2 . phase transition redshift ( z tr for which q = 0, the redshift at which the Universe transited from decelerating expansionto accelerating one) using the constrained (numerical) values of the model parameters α & β (see Table-4) for boththe models M1 and model M2 in the following Table-5 and Table-6 respectively.Table-5: Values of q at different epochs & phase transition redshift for model M1redshift formula Hz Hz + SN SN + BAO Hz + SN + BAOz −→ ∞ ( q i ) q i = − α . . . . z −→ q ) q = − α − α β α − . − . − . − . z −→ − q f ) q f = − − α − . − . − . − . z tr ( q = 0) z tr = − β (cid:16) α +1 α − (cid:17) α . . . . - - - - HzHz + SNSN + BAOHz + SN + BAO - - - - - - z q ( z ) - - - - HzHz + SNSN + BAOHz + SN + BAO - - - - - z q ( z ) ( a ) ( b )FIG. 9: Figures (a) and (b) show the evolution of deceleration parameter from past ( z = 4) to far future with a phase transitionfor models M1 and M2 respectively. Table-6: Values of q at different epochs & phase transition redshift for model M2redshift formula Hz Hz + SN SN + BAO Hz + SN + BAOz −→ ∞ ( q i ) q i = − α . . . . z −→ q ) q = − α − α β α − . − . − . − . z −→ − q f ) q f = − − α − . − . − . − . z tr ( q = 0) z tr = − β (cid:16) α +1 α − (cid:17) α . . . . z tr ≈ .
72 in model M1 for all numerical constrained values of model parameters α & β while in model M2, thephase transition occurs at z tr ≈ . Hz and Hz + SN constrained values of α & β and z tr ≈ . SN + BAO and Hz + SN + BAO constrained values of α & β . The present values of the deceleration parameter q found in bothmodels can be seen in the above tables consistent with predicted values. In the future, the Universe enter into superacceleration phase ( q < −
1) for both the models M1 and M2 and finally attains maximum values q f < − .
58 in modelM1 and q f < − .
61 in model M2 for all the values of α & β . B. Statefinder diagnostics
Statefinder diagnostics [86–89] is a technique generally used to distinguish various dark energy models and comparetheir behavior using the higher order derivatives of the scale factor. The parameters are s & r and calculated usingthe relations: r = ... aaH , s = r − q − ) . (18)The statefinder diagnostics pairs are constructed as { s, r } and { q, r } wherein different trajectories in the s - r and q - r planes are plotted to see the temporal evolutions of different dark energy models. The fixed points in thiscontexts are generally considered as { s, r } = { , } for ΛCDM model and { s, r } = { , } for SCDM (standard colddark matter) model in FLRW background and the departures of any dark energy model from these fixed points areanalyzed. The other diagnostic pair is { q, r } and the fixed points considered are { q, r } = {− , } for ΛCDM modeland { q, r } = { . , } for SCDM model. The statefinder parameters for the considered model M1 are calculated as, r ( z ) = 1 + α (2 α −
3) + 6 α { β (1 + z ) } α (cid:20) − α + α { β (1 + z ) } α (cid:21) (19)3 Λ CDM SCDMQUINTESSENCECHALPYGINGAS
HzHz + SNSN + BAOHz + SN + BAO - - s r Λ CDM SCDMQUINTESSENCECHALPYGINGAS
HzHz + SNSN + BAOHz + SN + BAO - s r ( a ) ( b )FIG. 10: Figures (a) and (b) are the s - r plots for model M1 and M2 respectively showing the different trajectories of themodels. Λ CDM SCDMdS de - S i tt e r li ne P ha s e t r an s i t i on li ne M a tt e r - do m i na t ede r a li ne QUINTESSENCECHALPYGINGAS
HzHz + SNSN + BAOHz + SN + BAO - - q r Λ CDM SCDMdS de - S i tt e r li ne P ha s e t r an s i t i on li ne M a tt e r - do m i na t ede r a li ne QUINTESSENCECHALPYGINGAS
HzHz + SNSN + BAOHz + SN + BAO - - q r ( a ) ( b )FIG. 11: Figures (a) and (b) are q - r plots for models M1 and M2 respectively showing different trajectories of the models. s ( z ) = 2 α − α { β (1 + z ) } α + α (3 + 2 α )3 [ − − α + (2 α − { β (1 + z ) } α ] (20)and for model M2 r ( z ) = 1 − α + 2 α + 12 α (cid:104) { β (1 + z ) } α (cid:105) + 3 α (3 − α )1 + { β (1 + z ) } α (21) s ( z ) = 23 α − α { β (1 + z ) } α + α (3 + 4 α ) (cid:104) − − α + (2 α − { β (1 + z ) } α (cid:105) (22)In the above figure FIG. 10, one can see the diverge evolutions of the model M1 and model M2 in the s - r and q - r planes. Both the models showing distinctive features as compared to the other standard models. One can observe4that at early times, model M1 presumes values in the range r > s < { , } during evolution. But, the model M2 is different and evolutes from quintessenceregion in the past and goes to Chaplygin gas region intermediating the ΛCDM fixed point { , } during it’s evolutionfor all cases. The figure FIG. 11 depicts the temporal evolution of the models M1 and M2 in the { q, r } plane providingadditional information about the models M1 and M2 wherein the dashed lines describe the evolution of the ΛCDMmodel below which quintessence region and the upper one is Chaplygin gas region are shown. The evolution of modelM1 and M2 are clearly observed. Both the models M1 and M2 deviates from de Sitter point ( − , C. Om diagnostic Om diagnostic is another tool introduced in [90–93], using the Hubble parameter and serving the purpose ofproviding a null test of the ΛCDM model. Like, statefinder diagnostic, Om diagnostic is also an effective methodto discriminate various DE models from ΛCDM model according to the slope variation of Om ( z ). Positive slope ofdiagnostic implies a Quintessence nature ( ω > − ω < − ω = − Om ( z ) for a flat Universe is defined as: Om ( z ) = (cid:16) H ( z ) H (cid:17) − z ) − Om ( z ) = Ω m + (1 − Ω m ) (1+ z ) ω ) − z ) − . For a constant EoS parameter ω imply Om ( z ) = Ω m and different values of Om ( z ) suggest whether the model is a ΛCDM model or quintessence or phantommodels. For the models of consideration here, the expressions for Om ( z ) for models M1 and M2 are obtained as, Om ( z ) = [1+ { β (1+ z ) } α ] (1+ β α ) (1+ z ) α − z ) − Om ( z ) = [ { β (1+ z ) } α ] (1+ β α ) (1+ z ) α − z ) − Om ( z ) vs. z are shown in the following figure FIG. 12 for models M1 and M2. For boththe models M1 and M2 and for all values of α & β , the Om ( z ) values is less than Ω m in the redshift range z > z < Om ( z ) values decreasessharply and becomes negative implying the both the models enter into phantom region. D. Jerk, Snap and Lerk parameters
Likewise, the Hubble and the deceleration parameters, the other cosmographic parameters, jerk, snap and lerkparameters also play significant roles in analyzing a cosmological model. The cosmic jerk j ( z = 0) (cid:39) α & β , j ∈ (1 . , .
4) for model M1 and j ∈ (0 . , .
2) formodel M2. The increasing values of jerk, snap and lerk parameters in the future ( z <
0) showing the deviation fromthe ΛCDM model which can also be interpreted from the statefinder diagrams FIG. 9 and FIG. 10. Similary, theevolution of snap and lerk parameters are shown in FIG. 14 and FIG. 15 respectively for both the models M1 andM2. From the figures, it can be seen that for all values of model parameters α & β , s ∈ (1 . , .
2) for model M1 and s ∈ ( − . , − .
7) for model M2 and l ∈ (6 ,
9) for model M1 and l ∈ (5 ,
13) for model M2. These values of j , s , l are in good agreement with the expected values. One can also interpret that the model M2 has better fit to theobservational datasets as compared to model M1 which can also be seen from FIG. 14 and FIG. 15.5 HzHz + SNSN + BAOHz + SN + BAO - - - - z O m ( z ) HzHz + SNSN + BAOHz + SN + BAO - - - - z O m ( z ) ( a ) ( b )FIG. 12: Figures (a) and (b) show the plots for Om ( z ) vs. z for models M1 and M2 respectively. HzHz + SNSN + BAOHz + SN + BAO - - - z j ( z ) HzHz + SNSN + BAOHz + SN + BAO - - - - z j ( z ) ( a ) ( b )FIG. 13: Figures (a) and (b) show the plots for jerk parameter j ( z ) vs. z for models M1 and M2 respectively. HzHz + SNSN + BAOHz + SN + BAO - - z s ( z ) HzHz + SNSN + BAOHz + SN + BAO - - - - - - - z s ( z ) ( a ) ( b )FIG. 14: Figures (a) and (b) show the plots for snap parameter s ( z ) vs. z for models M1 and M2 respectively. HzHz + SNSN + BAOHz + SN + BAO - - - z l ( z ) HzHz + SNSN + BAOHz + SN + BAO - - - - z l ( z ) ( a ) ( b )FIG. 15: Figures (a) and (b) show the plots for lerk parameter l ( z ) vs. z for models M1 and M2 respectively. VII. PHYSICAL DYNAMICS OF THE MODELS
The geometrical part of the Einstein field equations is discussed elaborately and now the physical interpretationscan be discussed for the obtained models once the matter content of the Universe is specified. In the introduction, itis mentioned that the candidate of dark energy is still unknown and it is a matter of speculation only to choose anycandidate described in literature. However, the most discussed candidate and having best fit with some observations isthe Einstein’s cosmological constant. So, in the following, the cosmological constant will be considered as a candidateof dark energy for further analysis.So, let us consider the two fluid Universe, cold dark matter and dark energy only, since the radiation contributionat present is negligible. The matter pressure is p = p m = 0 for cold dark matter and for dark energy the equation ofstate is p DE = ω DE ρ DE . In the following, the physical behavior of the matter and dark energy densities and pressuresare found out and their evolutions are shown graphically. A. Cosmological constant
When the candidate of dark energy is the cosmological constant implying ρ DE = ρ Λ = M − pl Λ and for whichthe equation of state parameter ω DE reduces to −
1. Solving equations (4) and (5), it is easy to obtain the explicitexpressions for the matter energy density and the energy density of cosmological constant as, ρ m M pl H = 2 α [1 + { β (1 + z ) } α ] − α [1 + { β (1 + z ) } α ] (1 + β α ) (1 + z ) α , (26) ρ Λ M pl H = Λ H = (3 − α ) [1 + { β (1 + z ) } α ] + 4 α [1 + { β (1 + z ) } α ] (1 + β α ) (1 + z ) α . (27)for model M1 and ρ m M pl H = 2 α (cid:104) { β (1 + z ) } α (cid:105) − α (cid:104) { β (1 + z ) } α (cid:105) (1 + β α ) (1 + z ) α , (28) ρ Λ M pl H = Λ H = (3 − α ) (cid:104) { β (1 + z ) } α (cid:105) + 6 α (cid:104) { β (1 + z ) } α (cid:105) (1 + β α ) (1 + z ) α . (29)for model M2. The evolution of these physical parameters are shown in the FIG. 16 and FIG. 17.7 HzHz + SNSN + BAOHz + SN + BAO - - z ρ m / M p l H HzHz + SNSN + BAOHz + SN + BAO - - z ρ m / M p l H ( a ) ( b )FIG. 16: Figures (a) and (b) show the evolution of the matter energy densities ( ρ m ) for models M1 and M2 respectively. HzHz + SNSN + BAOHz + SN + BAO - z ρ Λ / M p l H HzHz + SNSN + BAOHz + SN + BAO z ρ Λ / M p l H ( a ) ( b )FIG. 17: Figures (a) and (b) show the evolution of the energy densities of the cosmological constant ( ρ Λ ) for models M1 andM2 respectively. The density parameters for matter (cid:16) Ω m = ρ m Mpl H (cid:17) and density parameter for cosmological constant (cid:0) Ω Λ = Λ3 H (cid:1) can also be computed for both the models M1 and M2 as,Ω m = 2 α − α { β (1 + z ) } α ] , Ω Λ = 1 − α α { β (1 + z ) } α ] (30)for model M1 and Ω m = 2 α − α (cid:104) { β (1 + z ) } α (cid:105) , Ω Λ = 1 − α α (cid:104) { β (1 + z ) } α (cid:105) (31)for model M2. One can see from the above expressions that the sum total of the density parameters with thesecomponents is equal to 1. The evolution of the density parameters are shown in FIG. 18. B. Scalar field
Since the equation of state for cosmological constant is non dynamical and observations reveal it’s dynamicalcharacteristics, other candidates such as a general scalar field came into picture for a suitable candidate of dark energy.For an ordinary scalar field φ for the action can be represented as,8 Ω Λ Ω m HzHz + SNSN + BAOHz + SN + BAO - - z Ω HzHz + SNSN + BAOHz + SN + BAO Ω m Ω Λ - - - z Ω ( a ) ( b )FIG. 18: Figures (a) and (b) show the evolution of the density parameters for matter (Ω m ) and cosmological constant (Ω Λ )for models M1 and M2 respectively. S = (cid:90) d x √− g (cid:40) M p R − ∂ µ φ∂ µ φ − V ( φ ) + L Matter (cid:41) . (32)The term V ( φ ) is the potential function for the scalar field φ . In the considered FLRW background the energy density ρ φ will take the form ρ φ = ˙ φ + V ( φ ) and pressure p φ will take the form p φ = ˙ φ − V ( φ ). For a two componentUniverse, scalar field and cold dark matter with minimal interaction between them (i.e. they conserve separatelygiving ρ = ca − = c (1 + z ) , c is a constant of integration), then the solutions obtained from Eqs. (4) and (5) are, V ( φ ) M pl H = (3 − α ) [1 + { β (1 + z ) } α ] + 2 α [1 + { β (1 + z ) } α ] (1 + β α ) (1 + z ) α − c M pl H (1 + z ) (33) ρ φ M pl H = 3 [1 + { β (1 + z ) } α ] (1 + β α ) (1 + z ) α − cM pl H (1 + z ) (34)and the expression for the scalar field φ ( z ) can be calculated by integrating, φ − φ √ M pl = − (cid:90) (cid:34) α [1 + { β (1 + z ) } α ] − α [1 + { β (1 + z ) } α ] (1 + β α ) (1 + z ) α − c M pl H (1 + z ) (cid:35) (1 + β α ) (1 + z ) α − [1 + { β (1 + z ) } α ] dz (35)for model M1. Here, φ is an integrating constant. Similarly, the potential and energy densities for model M2 areobtained as, V ( φ ) M pl H = (3 − α ) (cid:104) { β (1 + z ) } α (cid:105) + 3 α (cid:104) { β (1 + z ) } α (cid:105) (1 + β α ) (1 + z ) α − c M pl H (1 + z ) (36) ρ φ M pl H = 3 (cid:104) { β (1 + z ) } α (cid:105) (1 + β α ) (1 + z ) α − cM pl H (1 + z ) (37)and the expression for the scalar field φ ( z ) can be calculated by integrating,9 HzHz + SNSN + BAOHz + SN + BAO - - - - - z ρ ϕ / M p l H HzHz + SNSN + BAOHz + SN + BAO - - - - z ρ ϕ / M p l H ( a ) ( b )FIG. 19: Figures (a) and (b) show the evolution of the scalar field energy density ( ρ φ ) for models M1 and M2 respectively. φ − φ √ M pl = − (cid:90) α (cid:104) { β (1 + z ) } α (cid:105) − α (cid:104) { β (1 + z ) } α (cid:105) (1 + β α ) (1 + z ) α − c M pl H (1 + z ) (1 + β α ) (1 + z ) α − (cid:104) { β (1 + z ) } α (cid:105) dz .(38)The density parameters for matter (cid:16) Ω m = ρ m M plH (cid:17) and density parameter for the scalar field (cid:16) Ω φ = ρ φ M plH = ˙ φ + V ( φ )3 M plH (cid:17) can be computed for both the models M1 and M2 as,Ω φ = 1 − Ω m , Ω m = c (1 + z ) (1 + β α ) (1 + z ) α M pl H [1 + { β (1 + z ) } α ] (39)for model M1 and Ω φ = 1 − Ω m , Ω m = c (1 + z ) (cid:0) β α (cid:1) (1 + z ) α M pl H (cid:104) { β (1 + z ) } α (cid:105) (40)for model M2. From equations (39) and (40), one obtains Ω m = c M pl H = ⇒ c = Ω m M pl H . The equations ofstate parameter ( ω φ = p φ ρ φ ) are given by, ω effφ = 13 (2 α −
3) [1 + { β (1 + z ) } α ] − α [1 + { β (1 + z ) } α ] [1 + { β (1 + z ) } α ] − Ω m (1 + β α ) (1 + z ) α +3 (41) ω effφ = 13 (2 α − (cid:104) { β (1 + z ) } α (cid:105) − α (cid:104) { β (1 + z ) } α (cid:105) (cid:104) { β (1 + z ) } α (cid:105) − Ω m (1 + β α ) (1 + z ) α +3 (42)The evolution of the Scalar field energy density, scalar field potential and the density parameters are shown in FIG.19, FIG. 20 and FIG. 21 respectively for models M1 and M2.The evolution of the equation of state parameter ( ω φ ( z )) vs. redshift z is plotted by neglecting the matter contri-bution and shown in the FIG. 22 for models M1 and M2.0 HzHz + SNSN + BAOHz + SN + BAO - - - z V ( ϕ ) / M p l H HzHz + SNSN + BAOHz + SN + BAO - - - - z V ( ϕ ) / M p l H ( a ) ( b )FIG. 20: Figures (a) and (b) show the evolution of the scalar field potential V ( φ ) ∼ z for models M1 and M2 respectively. HzHz + SNSN + BAOHz + SN + BAO Ω m Ω ϕ z Ω ( z ) HzHz + SNSN + BAOHz + SN + BAO Ω m Ω ϕ - - z Ω ( z ) ( a ) ( b )FIG. 21: Figures (a) and (b) show the evolution of the density parameters Ω φ & Ω m w.r.t. redshift z for models M1 and M2respectively. HzHz + SNSN + BAOHz + SN + BAO - - - - - - - - - - - - - - - - z ω ϕ ( z ) HzHz + SNSN + BAOHz + SN + BAO - - - - - - - - - - - - z ω ϕ ( z ) ( a ) ( b )FIG. 22: Figures (a) and (b) show the evolution of equation of state parameter vs. redshift ( ω φ ( z ) ∼ z ) for models M1 andM2 respectively. VIII. AGE OF THE UNIVERSE
The calculation of the age of the Universe is associated to the values of the cosmological parameters, specificallythe Hubble parameter. In general, using the Friedmann equation one can obtain the relation as t = H F (Ω x ), x =radiation, matter, dark energy, neutrino etc. The functional F contributes a fraction and largely the term 1 /H inthe age calculation e.g. for H = 69 km/s/M pc , one obtains 1 /H ≈ . Gyr (Giga years) and the factor F = 0 . m , Ω Λ ) = (0 . , . t and F = 0 .
666 for Einstein-de-Sitter model with (Ω m , Ω Λ ) = (1 ,
0) giving much smaller value of t . So, the introduction of cosmological constant issignificant as matter-only Universe was not enough to explain the globular clusters in the Milky Way which appearedto be older than the age of the Universe calculated then. According to the Planck2015 results age of the universe isestimated to be 13 . ± . Gyr with H = 67 . ± .
46 within 68% confidence limits for ΛCDM model constrainedby combined CMB power spectra, Planck polarization data, CMB lensing reconstruction and external data of BAO,JLA (Joint light curve analysis) and Hubble datasets.Here, the present work is a model independent study wherein the geometrical parameter H is parametrized forwhich the calculation of the age is unaffected by the matter content and solely depend on the functional form ofthe Hubble parameter H ( t ). We have already established the t - z relationships for models M1 and M2 which can berewritten as, t ( z ) = (1 + β α ) αβ α [1 + { β (1 + z ) } α ] 1 H and t ( z ) = (cid:0) β α (cid:1) αβ α (cid:104) { β (1 + z ) } α (cid:105) H respectively. By considering the present value of the Hubble parameter, H = 67 . Km/Sec/M pc , the terms multipliedto 1 /H are calculated and for both the models are greater than 0 .
96 for all constrained values of α & β and givepretty good estimate for the present age of the Universe and is larger than the standard model. The age calculationis tabulated in the following Table-7 for all the constrained numerical values of α & β for both the models M1 andM2. Table-7Models M1 M2Datasets ( α, β ) Factor Age (in Gyr ) ( α, β ) Factor Age (in
Gyr ) Hz (1 . , . . . . , . . . Hz + SN (1 . , . . . . , . . . SN + BAO (1 . , . . . . , . . . Hz + SN + BAO (1 . , . . . . , . . . IX. RESULTS AND CONCLUSION
To summarize the results, the philosophy behind writing this present paper is to discuss the phenomenology ofcosmological parametrization to obtain exact solutions of Einstein field equations. As an exemplification, a simpleparametrization of Hubble parameter is considered with some model parameters which reduce to some known models(see [55]) for some specific values of the model parameters involved. Two models discussed here in details and both themodels M1 and M2 exhibit a phase transition from deceleration to acceleration. Also, both the models diverges in finitetime and show big rip singularity. For consistency of the models obtained here, some observational datasets namely, H ( z ) datasets with updated 57 data points, Supernovae datasets from union 2.1 compilation datasets containing 580data points and BAO datasets with 6 data points are considered and compared with the standard ΛCDM model. Boththe models M1 and M2 contain two model parameters α & β which are constrained through these datasets and somenumerical values are obtained in pairs with independent Hz , combined Hz + SN , SN + BAO and Hz + SN + BAO datasets which are then used for further analysis for geometrical and physical interpretations of the models. Thepresent values of the deceleration parameters obtained for these constrained values of model parameters α & β arecalculated which are tabulated in Table-5 and Table-6 together with the phase transition redshifts and are in certainstandard estimated range. In the future the Universe in both the models enters into super acceleration phases anddiverges in finite times. The other geometrical parameters such as jerk, snap and lerk parameters are also discussedand their evolutions are shown graphically. The statefinder diagnostics and om diagnostics are also presented tocompare the obtained models with the standard ΛCDM model and the models behavior are shown in plots compared2with the standard ΛCDM model and SCDM model. After the brief cosmographic analysis, the physical interpretationof the models are discussed by considering the cosmological constant and scalar field as candidates of dark energy.The matter energy density ( ρ m ), energy density of ( ρ Λ ), the density parameters Ω Λ and Ω m are also calculated andtheir dynamical behavior w.r.t. redshift z are shown graphically for both the models M1 and M2 using the numericalconstrained values of the model parameters α & β . Similarly, the evolution of energy density ( ρ φ ), the potential( V ( φ )) of the scalar field ( φ ), density parameters Ω Λ and Ω m and also the equation of state parameter ω φ of thescalar field are shown graphically for both the models M1 and M2. The geometrical and physical analysis for both themodels M1 and M2 interpret that both models M1 and M2 have the quintessence behavior in the past and phantomlike behavior in the future. Finally, the age of the Universe for both the models M1 and M2 are calculated for theconstrained numerical values of model parameters α & β . It is found that the age found for both the models aregreater than the standard model and consistent with the age constraints of ΛCDM model.The conclusion is that the model M2 which is a quadratic varying deceleration parameter model has better fit to theobservational datasets (see FIG. 7 and FIG. 8) and shows better approximation to the present cosmological scenario ongeometrical as well as physical grounds as compared to the model M1 which is a linearly varying deceleration parametermodel. The presented study is an example of doing a comprehensive analysis of any cosmological model that describea simple methodology of finding exact solution of the Einstein field equations, comparing to the observations andestimating model parameters from the observational datasets. A brief list of various schemes of parametrization ofdifferent geometrical and physical parameters used in the past few decades to obtain the exact solutions of EFEs arealso summarized here which will help the readers for their studies in cosmological modelling. X. APPENDIX
A brief list of various parametrization schemes of parametrization of geometrical and physical parameters used inthe past few decades to find exact solutions of Einstein Field Equations is given below.
A. PARAMETRIZATIONS OF GEOMETRICAL PARAMETERS
Scale factor a ( t )Given below a list of different expansion laws of the scale factor those have been extensively studied in differentcontexts. a ( t ) = constant [30] (Static model) a ( t ) = ct [40, 94] (Milne model or Linear expansion) a ( t ) ∼ exp( H t ) [95] (ΛCDM model or Exponential expansion) a ( t ) ∼ exp (cid:104) − αt ln (cid:16) tt (cid:17) + βt (cid:105) [96] (Inflationary model) a ( t ) ∼ exp [ − αt − βt n ] [96] (Inflationary model) a ( t ) ∼ [exp( αt ) − β exp( − αt )] n [96] (Inflationary model) a ( t ) ∼ exp (cid:0) tM (cid:1) (cid:104) (cid:16) ς ( t ) N (cid:17)(cid:105) [97] (quasi steady state cosmology, Cyclic Universe) a ( t ) ∼ t α [98] (Power law Cosmology) a ( t ) ∼ t n exp( αt ) [99] (Hybrid expansion) a ( t ) ∼ exp [ n (log t ) m ] [100] (Logamediate expansion) a ( t ) ∼ cosh αt [95] (Hyperbolic expansion) a ( t ) ∼ (sinh αt ) n [101] (Hyperbolic expansion) a ( t ) ∼ (cid:16) tt s − t (cid:17) n [102] (Singular model) a ( t ) ∼ t n exp [ α ( t s − t )] [102] (Singular model) a ( t ) ∼ exp (cid:16) α t t ∗ (cid:17) [103] (Bouncing Model) a ( t ) ∼ exp (cid:16) βα +1 ( t − t s ) α +1 (cid:17) [103] (Bouncing Model) a ( t ) ∼ (cid:0) ρ cr t + 1 (cid:1) [103] (Bouncing Model) a ( t ) ∼ (cid:16) t s − tt ∗ (cid:17) [103] (Bouncing Model) a ( t ) ∼ sin (cid:16) α tt ∗ (cid:17) [103] (Bouncing Model) Hubble parameter H ( t ) or H ( a ) H ( a ) = Da − m [104]3 H ( a ) = e − γa αa [105] H ( a ) = α (1 + a − n ) [106] H ( t ) = mαt + β [107] H ( t ) = αt αt ) / [108] H ( t ) = m + nt [109] H ( t ) = αt R t ( t R − t ) [110] H ( t ) = α ( t + T ) − β ( t + T ) + γ [111] H ( t ) = αe λt [112] H ( t ) = α + β ( t s − t ) n [112] H ( t ) = α − βe − nt [113] H ( t ) = f ( t ) + f ( t )( t s − t ) n [114] H ( t ) = βt m ( t n + α ) p [55] H ( t ) = nα tanh( m − nt ) + β [115] H ( t ) = α tanh (cid:16) tt (cid:17) [15] H ( z ) = [ α + (1 − α ) (1 + z ) n ] n [116] Deceleration parameter q ( t ) or q ( a ) , q ( z ) q ( t ) = m − q ( t ) = − αt + m − q ( t ) = α cos( βt ) − q ( t ) = − αt t [120] q ( t ) = − α (1 − t )1+ t [120] q ( t ) = − αt + β − q ( t ) = (8 n − − nt + 3 t [122] q ( a ) = − − αa α a α [123] q ( z ) = q + q z [124] q ( z ) = q + q z (1 + z ) − [125] q ( z ) = q + q z (1 + z )(1 + z ) − [126] q ( z ) = + q (1 + z ) − [127] q ( z ) = q + q [1 + ln(1 + z )] − [128] q ( z ) = + ( q z + q )(1 + z ) − [129] q ( z ) = − (cid:16) (1+ z ) q q +(1+ z ) q (cid:17) [131] q ( z ) = − (cid:104) q + 1 − q + 1) (cid:16) q e q z ) − e − q z ) q e q z ) + e − q z ) (cid:17)(cid:105) [132] q ( z ) = − + (cid:18) q e q z √ z − e − q z √ z q e q z √ z + e − q z √ z (cid:19) [132] q ( z ) = q f + q i − q f − qiqf ( zt z ) τ [133] q ( z ) = q − q (cid:16) (1+ z ) − α − α (cid:17) [134] q ( z ) = q + q (cid:104) ln( α + z )1+ z − β (cid:105) [135] q ( z ) = q − ( q − q )(1 + z ) exp (cid:2) z c − ( z + z c ) (cid:3) [136] Jerk parameter j ( z ) j ( z ) = − j f ( z ) E ( z ) , where f ( z ) = z , z z , z (1+ z ) , log(1 + z ) and E ( z ) = H ( z ) H [137] j ( z ) = − j f ( z ) h ( z ) , where f ( z ) = 1, 1 + z , (1 + z ) , (1 + z ) − and h ( z ) = H ( z ) H [138] B. PARAMETRIZATIONS OF PHYSICAL PARAMETERS
Pressure p ( ρ ), p ( z )The matter content in the Universe is not properly known but it can be categorized with its equations of states p = p ( ρ ). Following is a list of some cosmic fluid considerations with their EoS. Also, some dark energy pressureparametrization are listed. p ( ρ ) = wρ (Perfect fluid EoS)4 p ( ρ ) = wρ − f ( H ) [139] (Viscous fluid EoS) p ( ρ ) = wρ + kρ n [140] (Polytropic gas EoS) p ( ρ ) = wρ − ρ − ρ [141] (Vanderwaal gas EoS) p ( ρ ) = − ( w + 1) ρ ρ P + wρ + ( w + 1) ρ Λ [142] (EoS in quadratic form) p ( ρ ) = − Bρ [143] (Chaplygin gas EoS) p ( ρ ) = − Bρ α [144] (Generalized Chaplygin gas EoS) p ( ρ ) = Aρ − Bρ α [145] (Modified Chaplygin gas EoS) p ( ρ ) = Aρ − B ( a ) ρ α [146] (Variable modified Chaplygin gas EoS) p ( ρ ) = A ( a ) ρ − B ( a ) ρ α [147] (New variable modified Chaplygin gas EoS) p ( ρ ) = − ρ − ρ α [148] (DE EoS) p ( z ) = α + βz [149] (DE EoS) p ( z ) = α + β z z [149] (DE EoS) p ( z ) = α + β (cid:16) z + z z (cid:17) [150] (DE EoS) p ( z ) = α + β ln(1 + z ) [151] (DE EoS) Equation of state parameter w ( z ) w ( z ) = w + w z [152] (Linear parametrization) w ( z ) = w + w z (1+ z ) [153] (JBP parametrization) w ( z ) = w + w z (1+ z ) n [154] (Generalized JBP parametrization) w ( z ) = w + w z z [155] (CPL parametrization) w ( z ) = w + w (cid:16) z z (cid:17) n [154] (Generalized CPL parametrization) w ( z ) = w + w z √ z [156] (Square-root parametrization) w ( z ) = w + w sin( z ) [157] (Sine parametrization) w ( z ) = w + w ln(1 + z ) [158] (Logarithmic parametrization) w ( z ) = w + w ln (cid:16) z z (cid:17) [159] (Logarithmic parametrization) w ( z ) = w + w z (1+ z )1+ z [160] (BA parametrization) w ( z ) = w + w (cid:16) ln(2+ z )1+ z − ln 2 (cid:17) (MZ parametrization) w ( z ) = w + w (cid:16) sin(1+ z )1+ z − sin 1 (cid:17) [161] (MZ parametrization) w ( z ) = w + w z z (FSLL parametrization) w ( z ) = w + w z z [162] (FSLL parametrization) w ( z ) = − z α +2 β (1+ z ) γ +2 α (1+ z )+ β (1+ z ) [163] (ASSS parametrization) w ( z ) = ( z zs ) α w + w ( z zs ) α [164] (Hannestad Mortsell parametrization) w ( z ) = − α (1 + z ) + β (1 + z ) [165] (Polynomial parametrization) w ( z ) = − α [1 + f ( z )] + β [1 + f ( z )] [166] (Generalized Polynomial parametrization) w ( z ) = w + z (cid:0) dwdz (cid:1) [167] w ( z ) = − z ) d (cid:48)(cid:48) c − d (cid:48) c (cid:104) d (cid:48) c − Ω M (1+ z ) ( d (cid:48) c ) (cid:105) where d (cid:48) c = z (cid:82) H dzH ( z ) [168] w x ( a ) = w exp( a −
1) [169] w x ( a ) = w a (1 − log a ) [169] w x ( a ) = w a exp(1 − a ) [169] w x ( a ) = w a (1 + sin(1 − a )) [169] w x ( a ) = w a (1 + arcsin(1 − a )) [169] w de ( z ) = w + w q [170] w de ( z ) = w + w q (1 + z ) α [170] w de ( z ) = w [1+ b ln(1+ z )] [171] w x ( z ) = w + b { − cos [ln(1 + z )] } [172] w x ( z ) = w + b sin [ln(1 + z )] [172] w x ( z ) = w + b (cid:104) sin(1+ z )1+ z − sin 1 (cid:105) [172] w x ( z ) = w + b (cid:16) z z (cid:17) cos(1 + z ) [172]5 w ( z ) = w + w a (cid:104) ln(2+ z )1+ z − ln 2 (cid:105) [173] w ( z ) = w + w a (cid:104) ln( α +1+ z ) α + z − ln( α +1) α (cid:105) [174] Energy density ρρ = ρ c [175], [176] ρ ∼ θ [177] ρ = Aa √ a + b [178]( ρ + 3 p ) a = A [179] ρ + p = ρ c [180] ρ de ( z ) = ρ de (0) (cid:104) α (cid:16) z z (cid:17) n (cid:105) [181] ρ de ( z ) = ρ φ (cid:16) dρ φ dφ (cid:17) = − αa ( β + a ) [182] ρ de ( z ) = αH ( z ) [183] ρ de ( z ) = αH ( z ) + βH ( z ) [183] ρ de ( z ) = κ (cid:2) α + βH ( z ) (cid:3) [183] ρ de ( z ) = κ (cid:104) α + β ˙ H ( z ) (cid:105) [183] ρ de ( z ) = κ (cid:104) αH ( z ) + β ˙ H ( z ) (cid:105) [183] ρ de ( z ) = κ (cid:104) α + βH ( z ) + γ ˙ H ( z ) (cid:105) [183] ρ de ( z ) = ρ φ (1 + z ) α e βz [184] Cosmological constant ( Λ ) In order to resolve the long standing cosmological constant problem, authors have considered some variation lawsfor the cosmological constant in the past forty years, commonly known as “Λ-varying cosmologies” or “Decayingvacuum cosmologies”. Later the idea was adopted to explain the accelerated expansion of the Universe consideringvarying Λ. Following is list of such decay laws of Λ.Λ ∼ a − n [185]Λ ∼ H n [186]Λ ∼ ρ [185]Λ ∼ t n [186]Λ ∼ q n [186]Λ ∼ e − βa [187]Λ = Λ( T ) [188] T is TemperatureΛ ∼ C + e − βt [189]Λ = 3 βH + αa − [190]Λ = β ¨ aa [191]Λ = 3 βH + α ¨ aa [192] d Λ dt ∼ β Λ − Λ [193] Scalar field Potentials V ( φ ) V ( φ ) = V φ n [14] (Power law) V ( φ ) = V exp (cid:104) − αφM pl (cid:105) [14] (exponential) V ( φ ) = V cosh[ φ/φ ] [14] V ( φ ) = V [cosh ( αφ/M pl )] − β (hyperbolic) [14] V ( φ ) = αφ n (Inverse power law) [194] V ( φ ) = V β exp( − ακφ ) (Woods-Saxon potential) [195] V ( φ ) = αc (cid:104) tanh φ √ α (cid:105) ( α -attractor) [196] V ( φ ) = V (1 + φ α ) [197] V ( φ ) = V exp( αφ ) [197] V ( φ ) = ( φ − [198]Note: All the parametrization listed above contain some arbitrary constants such as α , β , γ , m , n , p , q , q , q , w , w , A , B are model parameters which are generally constrained through observational datasets or through any6analytical methods and also some arbitrary functions f ( t ), f ( t ). t s denote the bouncing time or future singularitytime and t ∗ some arbitrary time. [1] S. Perlmutter et al., Astrophys. J.
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