Reconstruction of cosmic history from a simple parametrization of H
aa r X i v : . [ g r- q c ] S e p Reconstruction of cosmic history from a simpleparametrization of H S. K. J. Pacif , R. Myrzakulov , S. Myrzakul Centre for Theoretical Physics,Jamia Millia Islamia,New Delhi 110025, India Eurasian International Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan Department of Theoretical and Nuclear Physics,Al-Farabi Kazakh National University, Al-Farabi av. 71, 050040,Almaty, [email protected] , [email protected] , [email protected] October 1, 2018
Abstract
In this paper, we propose a simple parametrization of the Hubble parameter H inorder to explain the late time cosmic acceleration. We show that our proposal coversmany models obtained in different schemes of parametrization under one umbrella. Wedemonstrate that a simple modification in the functional form of Hubble parameter cangive rise to interesting cosmological phenomena such as big rip singularity, bounce andothers. We have also constrained the model parameters using the latest 28 points of H ( z )data for three cases which admit transition from deceleration to acceleration. Keywords: Hubble parameter, parametrization, acceleration, dark energy
One of the aims of cosmology is to determine a mathematical model of the large scalestructure of the Universe which can explain the results of astronomical observations and whosedynamics can be determined by the physical laws describing the behavior of matter on largerscale. According to Alan Sandage [1], in cosmology at the background level, one search for twonumbers: H and q (suffix ’0’ stands for the present value of the quantity), where H (Hubbleparameter (HP)) and q (deceleration parameter(DP)) are two dynamical quantities which tell1bout the expansion rate of the Universe. But the present day cosmology use around four totwenty parameters to explain the Universe. Still, H and q play the central role in the Einstein’sfield equations (EFEs) explaining the observations. They can be defined naturally in the linearand non linear derivatives of scale factor a ( t ) in the Taylor series expansion of a ( t ) in the vicinityof the present time t as a ( t ) = a ( t ) + ˙ a ( t ) [ t − t ] + 12 ¨ a ( t ) [ t − t ] + · · · . (1)An overhead dot ‘ · ’ represents derivative w.r.t. cosmic time ‘ t ’. From equation (1), we obtain a ( t ) a ( t ) = 1 + H [ t − t ] − q H [ t − t ] + · · · , (2)where H ( t ) = ˙ aa , q ( t ) = − a ¨ a ˙ a . (3)Till today the most successful theory explaining the Universe is the big bang theory whichis based on general relativity. After Hubble’s work, cosmologists made attempts to measure thedeceleration of the expansion with the belief that the expansion of the Universe must slow downcaused by gravity. However, the observations of distant supernovae of type Ia by SupernovaCosmology Project [2] and the High-Z Supernova Search team [3] gave totally unexpected resultto the fact that the expansion of the Universe is accelerating. Since then further searchespresented convincing evidence for accelerating expansion with greater accuracy [4], [5], [6], [7],[8], [9]. The fact is also supported by some other observations such as CMB [10], [11], BAO [12],[13], SDSS [14], [15] etc. For both cosmology and physics, the cosmic acceleration is probablyan important discovery. It raised a lot of questions on the fundamental principles funded withcosmology. Based on the accelerating expansion of the Universe, the past few years produced aplethora of cosmological models either by modifying the energy momentum tensor in the righthand side of Einstein’s field equation (EFE) or by modifying the gravity theory (modifyingthe LHS of EFE). Alternative theories are also there such as the inclusion of inhomogeneity,back reaction, averaging etc. Recently, a series of papers by Vishwakarma [16], [17], [18], [19]explained this fact in a simple and viable way which raised some questions on the geometrizationof gravity theory.Although there are several ways to describe this cosmic acceleration, but it is generallyattributed to the presence of dark energy (DE) throughout the Universe. Obviously it givesrise to the question of what this mysterious DE really is, what is its nature and why it startsdominating the Universe so recently. The literature contains numerous models of DE but thesimplest and popular candidate of DE is the Einstein’s cosmological constant (Λ) [20], [21],[22]. However, it suffers from the well known cosmological constant problem [23] which can bealleviated by considering a dynamically decaying Λ. On the other hand primordial inflation hastaken a special status in explaining the origin of the anisotropies in the CMB radiation and theformation of large scale structures. This motivates theorists to invoke scalar field to explainthe early and the late time acceleration together. So far, a wide variety of scalar field models2f DE have been proposed in the past few years including quintessence [24], [25], [26], [27],K-essence [28], [29], spintessence [30], tachyon [31], [32], quintom [33], [34], [35], [36], chameleon[37], [38], [39] and many more. Though these scalar field models give the equation of state(EoS) parameter (cid:16) w = pρ (cid:17) , − < w <
0, observational data also allow models of DE with EoSparameter crossing − z ∼ .
5. The existence of deceler-ated expansion phase in the Universe is also supported by the gravitational instability theory ofstructure formation and of big bang nucleosynthesis. This implies that the Universe must haveundergone from decelerated to accelerated phase of evolution. This motivates the theorists inmodeling the Universe with deceleration to acceleration phase transition. The kinematic ap-proach is discussed in [55] to explain the cosmic acceleration which do not assume the validityof general relativity or any particular gravitational theory (see [56]). This method do not effectthe physical or geometrical properties of DE and is known as the model-independent way tostudy the DE i.e. by a parametrized EoS of dark energy. For a review on parametrization ofequation of state parameter w , one can see [57]. Another model-independent way to study theDE is by parametrizing the DP. For a brief review on DP, one can see [58], [59]. Also, thereare several parametrization of HP considered by Nojiri and Odintsov (and collaborators) [60] tostudy the future cosmological singularities. Here, in this paper, we have studied the evolution ofthe Universe by parametrizing the functional form of H and see how it reduces to some knownobtained models and explains the late time cosmic acceleration. We would like to stress on the cosmographic parameter H describing the expansion of theUniverse and its role in generating some interesting mathematical models of the Universe inEinstein’s theory of gravitation. In FRW cosmology, there are three variables namely a ( t ) , ρ ( t ) ,p ( t ) with two independent equations which can be solved by supplementing the equation ofstate(s), w = pρ of the energy component(s). In this case the parameter w is a constant. For adynamical Λ, one needs one more constrain equation to close the system. This extra constrainequation (or the supplementary equation) has been chosen in various ways in order to explain thestandard cosmological problems such as to solve the cosmological constant problem, singularityproblem, horizon problem, flatness problem, density fluctuation problem, dark matter problem,exotic relics problem, thermal state problem etc. during the past forty-fifty years. With theaddition of the DE component into the field equations the EoS parameter w becomes dynamical( w ( t )). Thus, there are many traditional ways to choose this supplementary equation relatingany two variables involved in the field equations e.g. p ∼ ρ , Λ ∼ a − , Λ ∼ ρ etc. Also, one canparametrize any variable to get this extra constrain equation to close the system. The variousparametrization used in literature relating to a ( t ) , ρ ( t ) , p ( t ) , Λ( t ) , q ( t ) or w ( t ) are summarizedhere in detail (see table-6 to table-11 in appendix-1).3lso there are various parametrization of Hubble parameter H ( t ) in literature used to explainsome problems of standard cosmology and are listed in the below table-1. Table-1Parametrization of HP ( H ) Ref. H ( a ) = Da − m ( D and m are constants) [61] H ( a ) = e − γa αa ( γ and α are constants) [62] H ( a ) = α (1 + a − n ) ( α and n are constants) [63] H ( t ) = mαt + β ( α, β, m are constants) [64], [65] H ( t ) = αt R t ( t R − t ) ( α is a constant, t R is big Rip time) [66] H ( t ) = α ( t + T ) − β ( t + T ) + γ, [67], [39] γ = − α T + βT ( α, β, T are constants) H ( t ) = H e λt ( H , λ are constants) [68] H ( t ) = c + b ( t s − t ) α ( c , b , α are constants) [69] H ( t ) = H − H e − βt ( H > , H > , β are constants) [70] H ( t ) = f ( t ) + f ( t )( t s − t ) α [60]( f ( t ) & f ( t ) are arbitrary functions, α constant) One can find some more parametrization of HP in [60].From equation (3), we find a ( t ) = Ce R H ( t ) dt , where C is a constant of integration. (4) q ( t ) = − ddt (cid:18) H ( t ) (cid:19) , (5)EFEs can also be expressed as X ρ i ( t ) = 3 M P l (cid:20) ( H ( t )) + ka (cid:21) , (6) X [1 + 3 w i ( t )] ρ i ( t ) = 6 M P l ( H ( t )) (cid:20) − ddt (cid:18) H ( t ) (cid:19)(cid:21) . (7)In the above equations all the physical variables are in terms of H ( t ). Now, it is easy to seethat a simple integrable form of H ( t ) will determine all the physical variables smoothly. Weprefer to parametrize the HP because the variation of Hubble’s law assumed is not inconsistentwith observations and has the advantage of providing simple functional form of the time evolu-tion of the scale factor and so as dynamics. Motivated by the above discussions, we propose asimple and convenient form of HP as an explicit function of cosmic time ‘ t ’ in the form H ( t ) = βt m ( t n + α ) p (8)where α, β = 0 , m, n, p are real constants (better call them model parameters). α and β bothhave the dimensions of time. The specific values of m, n, p will suggest the different forms ofHP and produce interesting cosmologies. Our parametrization generalizes several known models4hich were obtained by the parametrization of any cosmological parameters a ( t ) , H ( t ) , q ( t ) , Λ( t ) , ρ ( t ) or w ( t ) in different contexts. In the next section, we formulate the Einstein’s fieldequations for a general scalar field cosmology and solve the system with the help of our mainansatz. We know scalar fields are extremely important in modern physics being invariant undercoordinate transformations. There have been a great activity in modelling the Universe witha motivation to explain both the early and late time acceleration of the Universe with scalarfields. We know, the nature of DE remain matters of speculation, but it is generally believed tobe homogeneous, not very dense and is not known to interact through any of the fundamentalforces other than gravity. So, it can be represented as large scale scalar field φ . For an ordinaryscalar field φ minimally coupled to gravity with Lagrangian density L = − g µν ∂ µ φ∂ ν φ − V ( φ ),the action is given by S = Z d x √− g (cid:20) − g µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) , (9)where V ( φ ) is the potential of the field. The stress-energy tensor of the field φ take the formof a perfect fluid as [21] T φµν = ( ρ φ + p φ ) U µ U ν + p φ g µν , (10)where the density and pressure of scalar field are expressed as ρ φ = ˙ φ + V ( φ ) and p φ = ˙ φ − V ( φ ), with the understanding that φ is spatially homogeneous. The evolution of the scalarfield is governed by the wave equation ¨ φ +3 H ˙ φ + V ′ ( φ ) = 0 , where a prime denotes differentiationwith respect to φ . The state equation of scalar field w φ can be represented as w φ = p φ ρ φ = − ˙ φ V ˙ φ V .This give rise to several candidates for DE, which depends upon the dynamics of the field φ andits potential energy V ( φ ). For a slow roll scalar field ˙ φ ≪ V ( φ ), it reduces to the case of mostfavoured cosmological constant Λ for which w φ = −
1. For − < w φ < w φ crossing −
1, phantom field is observed. To introduce DE into EFEs, we replace theenergy momentum tensor T µν by T T otalµν = T µν + T φµν = ( ρ T otal + p T otal ) U µ U ν + p T otal g µν (11)with the understanding that ρ T otal = ρ eff = ρ + ρ φ and p T otal = p eff = p + p φ .With the fluid described here by (11), the EFEs reduce to ρ eff = ρ + ρ φ = 3 M P l (cid:18) H + ka (cid:19) , (12) p eff = p + p φ = − M P l (cid:18) aa + H + ka (cid:19) , (13)with the state equations 5 = wρ (0 w
1) and p φ = w φ ρ φ . (14)Meanwhile we consider the minimal interaction between matter and dark energy which yield˙ ρ + 3 H (1 + w ) ρ = 0 , ˙ ρ φ + 3 H (1 + w φ ) ρ φ = 0 , (15)leading to ρ ∼ a − w ) and ρ φ ∼ a − w φ ) (for constant w φ (such as cosmological constant)).But, w φ must be a function of time in general.With our main ansatz (8) the general expressions for the time variation of all the CP areobtained as follows a ( t ) = Ce β R tmdt ( tn + α ) p , C is an arbitrary constant of integration, (16) q ( t ) = − β { ( np − m ) t n − mα } ( t n + α ) p − t m +1 , (17) ρ eff ( t ) = 3 M P l (cid:20) β t m ( t n + α ) p + kC e − β R tmdt ( tn + α ) p (cid:21) , (18) p eff ( t ) = M P l " − β { ( np − m ) t n − mα } ( t n + α ) p − t m +1 ! β t m ( t n + α ) p − kC e − β R tmdt ( tn + α ) p , (19) ρ ( t ) = (cid:2) DC − w ) (cid:3) e − w ) β R tmdt ( tn + α ) p , D is an arbitrary constant of integration. (20)We observe that some particular values of m, n, p will give explicit solutions of EFEs. Ingeneral there are one, two or three model parameters in all the parametrization considered (seetable-1 and table-6 to table-11). But, we have five parameters α, β, m, n, p in the functionalform of HP. Without the loss of generality, we reduce the number of model parameters by givingsome specific values to m, n, p which will be helpful to analyze the physical and geometricalbehavior of our obtained models. For some suitable choice of integral values of m, n, p (and onenon-integral value of p ), we obtain some specific models leaving α and β generic. The variousmodels thus obtained with two model parameters α and β are given in the following table-8.The two model parameters α and β can be constrained from any observational data (e.g. SneIa data, H(z) data or BAO data). However, one can also constrain all five model parameterssimultaneously but in this work we confined to two model parameters α, β by specifying m, n,p to see how our parametrization of HP can reproduces some particular models.6able-2Models Specific Values of HP SF DP m, n, p H ( t ) a ( t ) q ( t )I m = 0 , p = 0 , ∀ n β Ce βt − m = − , p = 0 , ∀ n βt Ct β − β III m = 0 , p = 1 , n = 1 βt + α C ( t + α ) β − β IV m = 1 , p = 0 , ∀ n βt Ce β t − − β t V m = 0 , p = 1 , n = 2 βt + α Ce β √ α tan − t √ α − β t VI m = 0 , p = , n = 1 β √ t + α Ce β √ t + α − β √ t + α VII m = 0 , p = , n = 2 β √ t + α C (cid:0) t + √ t + α (cid:1) β − β t √ t + α VIII m = 1 , p = 1 , n = 1 βtt + α Ce βt ( t + α ) − αβ − − αβ t IX m = 1 , p = 1 , n = 2 βtt + α C ( t + α ) β − β − αβ t X m = 1 , p = , n = 2 βt √ t + α Ce β √ t + α − − αβ t √ t + α XI m = − , p = 1 , n = 1 βt ( t + α ) C (cid:0) tt + α (cid:1) βα − αβ + β t XII m = − , p = 1 , n = 2 βt ( t + α ) C (cid:16) t t + α (cid:17) β α − αβ + β t In table-2, we see Λ
CDM model (model-I, where β plays the role of Λ), power law cosmology(PLC) [71] (model-II), Berman’s model of constant deceleration parameter (BM) [72] (model-IIIwith β = m ), Abdel Rahman’s model (AR) (model-IX) [73] (with β = 1) and its generalizedmodel [74] are obtained here. Model-XI imitate the linearly varying deceleration parametermodel (LVDPt) of Akarsu [75] (where β = − k and αβ = m ). Thus, we can see all these modelscome under our scheme of parametrization of HP for some specific choice of model parameters.We note that, many solutions obtained here are non-singular bouncing solutions, where thebounce occur at some finite value of the scale factor a ( t ).For α = 0, the form of H ( t ) becomes H ( t ) = βt γ , ( γ = ( m − np ) is a new constant) givingthe same result as p = 0 (Models-I,II,IV). The case for negative α is that for which our mainansatz (8) will take the form H ( t ) = βt m ( t n − α ) p , α >
0. In this case the behavior of scale factorwill differ greatly in some models and so as dynamics e.g. with this form of HP, all the models( p = 0) have collapsing nature at t = α n . If we take H ( t ) = βt m ( α − t n ) p , α > t then this form of HPwill lead to models with future singularity at t = α n . One can study these future singularitiesfor different models so obtained. Modifying the form of HP to H ( t ) = βt m + η ( α − t n ) p , α > t and η isanother parameter then for p = 0, one can obtain the hybrid scale factor cosmology [76], [77]and for p = − H ( t ) also gives rise to different evolutions of scale factor that is being studied in this work.We shall make note that, α & β are two model parameters and the dynamics of obtainedmodels (in table-2) or the behavior of cosmological parameters a ( t ) , H ( t ) , q ( t ) , ρ ( t ) , w ( t ) allheavily depend on these two. In the next section we shall discuss the behavior of differentcosmological parameters in view of the positive value of the model parameters α & β and7iscuss their analytical bounds. A lot of studies have been done on model-I (ΛCDM), model-II(PLC) and model-III (BM). So, we do not dwell in these known models and try to explore theother models only i.e. models-IV–XII. The expressions for the scale factor, Hubble parameter and deceleration parameter formodel-IV–XII are given in table-2. For the positive values of α, β , their behavior near thesingularities (at t = 0 and t → ∞ ) are obtained as Table-3CP ↓ models IV V VI VII VIII IX X XI XII a ( t ) t = 0 C C Ce β √ α Cα β/ Cα − αβ Cα β/ Ce β √ α t → ∞ ∞ Ce πβ √ α ∞ ∞ ∞ ∞ ∞ C CH ( t ) t = 0 0 βα β √ α β √ α ∞ ∞ t → ∞ ∞ β β q ( t ) t = 0 −∞ − − β √ α − −∞ −∞ −∞ − αβ − αβ t → ∞ − ∞ − − β − − β − ∞ + ∞ From the above table we can see that the models-IV–X are free from initial singularityand starts with a finite initial radius while models-XI,XII have big bang origin. As t → ∞ models-IV,VI,VII,VIII,IX,X diverge while the scale factor takes finite values in models-V,XI,XII.Models-IV,VI,VIII,X collapse in near future showing these models have finite time singularitybut the singularity can be delayed by larger (smaller in case of model-IV) values β . Similarly,looking at the values of HP and DP near singularities, we can conclude that in case of models-IV,VIII,IX,X, the Universe starts with zero velocity and infinite acceleration. In case of Model-V,VI,VII the Universe starts with finite velocity and finite acceleration while in models-XI,XIIthe Universe starts with infinite velocity and finite acceleration. The rate of initial velocitiesand initial accelerations for these models depend upon the choice of model parameters α & β .We can observe that in the models-V,XI,XII, the Universe ceases as t → ∞ where the velocitybecomes zero and DP becomes + ∞ (no acceleration).As the observations reveal that the total energy budget of the Universe is dominated by DE( ∼ q = 0 (or ¨ a = 0).So, at present theorists take interest in modelling the Universe with phase transition from earlydeceleration to present acceleration. These kinds of models are considered as viable models asthere is an obvious provision for the structure formation in the Universe during the deceleratedphase and also they can explain the result of observation of Type Ia supernovae at present. Onthe other hand, before the discovery of late time acceleration, theorists were taking interest inmodelling the Universe with early inflation and late time deceleration of the Universe as earlyinflation is necessary to explain the origin of the large scale structure of the cosmos. So, the8 eceleration-acceleration phase transition is important in current picture while the acceleration-deceleration phase transition is important in the very early Universe. In conclusion, we can saya model which has initial acceleration, middle deceleration and late-time acceleration scenariocan be treated as a better model that can explain all the phenomena explained by observations.Out of the twelve models listed in table-2, the DP comes out to be constant in models-I,II,IIIwhere as the DP is time-dependent in models-IV–XII. For α, β >
0, models-IV,VIII,X exhibiteternal acceleration and models-V,VII,IX,XI,XII show transition from initial acceleration todeceleration; or may accelerate for ever for certain choice of α and β . Only model-VI shows aphase transition from deceleration to acceleration. The various cases in view of phase transitionare analyzed in the following table. Table-4Models Transition type Phase transition time t tr constrain on α , β IV Ever accelerating — α, β > V Acceleration → Deceleration β α, β > VI Deceleration → Acceleration β − α α, β > , β √ α < Ever accelerating — α, β > , β √ α > VII Acceleration → Deceleration √ αβ √ − β α, β > , β √ α > Ever accelerating — α, β > , β √ α < VIII Ever accelerating — α, β > IX Acceleration → Deceleration q α − β α, β > , β + α > Ever accelerating — α, β > , β + α < X Ever accelerating — α, β > XI Acceleration → Deceleration β − α α, β > , β > α Ever Decelerating — α, β > , β < α XII Acceleration → Deceleration q β − α α, β > , β > α Ever Decelerating — α, β < , β < α Observations suggest the present value of DP is somewhere in the neighborhood of − . α and β appropriately such that the presentvalue of DP q will be in the neighborhood − .
55 or with a very small positive value of q (fordecelerating models). It may be noted that the observations favour accelerating models butthe decelerating models are also in agreement with these observations [79]. The deceleratingmodels also show nice fit to some data even with zero cosmological constant if one considers theextinction of light by the metallic dust ejected from the supernovae explosions [79]. With someindependent analysis, we have chosen the values of α and β in model-VI such that q ≈ − . t tr ≈
3. Thetime evolution of q ( t ) for models-IV–XII are 9igure 1: Plots for the Deceleration parameter for models-IV–XII. We can see model-IV (with β = 2), model-VIII (with α = 1 , β = 1), model-X (with α = 1 , β = 1) show eternal accelerationwhile model-V (with α = 4 , β = 2), model-VII (with α = 2 , β = 0 . α =1 , β = 0 . α = 1 , β = 2), model-XII (with α = 1 , β = 2) show transitionfrom acceleration to deceleration. Only model-VI (with α = 0 . , β = 0 .
28) shows transitionfrom deceleration to acceleration. We can also see that the model-VI (with α = 0 . , β = 2),model-VII (with α = 2 , β = 1 . α = 1 , β = 1 .
1) show eternal accelerationand model-XI (with α = 2 , β = 1), model-XII (with α = 2 , β = 1) show eternal deceleration asdiscussed in table-4. For the plots we have chosen the units suitably e.g. In case of model-IV,the time axis is scaled such that 0 . ρ ( t ) , ρ eff ( t ) , w eff ( t ) for models-IV–XII are obtainedas Model-IV Model-V ρ ( t ) = (cid:2) DC − w ) (cid:3) e − w ) β t ρ ( t ) = (cid:2) DC − w ) (cid:3) e − w ) β √ α tan − t √ α ρ eff ( t ) = 3 M P l h β t + kC e − βt i ρ eff ( t ) = 3 M P l h β ( t + α ) + kC e − β √ α tan − t √ α i w eff ( t ) = ( − − β t ) β t − kC e − βt [ β t + kC e − βt ] w eff ( t ) = ( − β t ) β ( t α ) − kC e − β √ α tan − t √ αβ ( t α ) + kC e − β √ α tan − t √ α Model-VI Model-VII ρ ( t ) = (cid:2) DC − w ) (cid:3) e − w ) β √ t + α ρ ( t ) = (cid:2) DC − w ) (cid:3) (cid:0) t + √ t + α (cid:1) − w ) β ρ eff ( t ) = 3 M P l h β t + α + kC e − β √ t + α i ρ eff ( t ) = 3 M P l (cid:20) β t + α + kC ( t + √ t + α ) β (cid:21) w eff ( t ) = (cid:16) − β √ t + α (cid:17) β t + α − kC e − β √ t + αβ t + α + kC e − β √ t + α w eff ( t ) = (cid:18) − β t √ t α (cid:19) β t α − kC ( t + √ t α ) ββ t α + kC ( t + √ t α ) β Model-VIII Model-IX ρ ( t ) = (cid:2) DC − w ) (cid:3) e − w ) βt ( t + α ) w ) αβ ρ ( t ) = (cid:2) DC − w ) (cid:3) ( t + α ) − w ) β ρ eff ( t ) = 3 M P l h β t ( t + α ) + kC e βt ( t + α ) − αβ i ρ eff ( t ) = 3 M P l h β t ( t + α ) + kC ( t + α ) β i w eff ( t ) = ( − − αβ t ) β t t + α )2 − kC e βt ( t + α ) − αββ t t + α )2 + kC e βt ( t + α ) − αβ w eff ( t ) = ( − β − αβ t ) β t ( t α ) − kC ( t α ) ββ t ( t α ) + kC ( t α ) β Model-X Model-XI ρ ( t ) = (cid:2) DC − w ) (cid:3) e − w ) β √ t + α ρ ( t ) = (cid:2) DC − w ) (cid:3) (cid:0) tt + α (cid:1) − w ) βα ρ eff ( t ) = 3 M P l h β t ( t + α ) + kC e β √ t α i ρ eff ( t ) = 3 M P l " β t ( t + α ) + kC ( tt + α ) βα w eff ( t ) = (cid:18) − − αβ t √ t α (cid:19) β t t α − kC e β √ t αβ t t α + kC e β √ t α w eff ( t ) = ( − αβ + β t ) β t t + α )2 − kC ( tt + α ) βαβ t t + α )2 + kC ( tt + α ) βα Model-XII ρ ( t ) = (cid:2) DC − w ) (cid:3) (cid:16) t t + α (cid:17) − w ) β α ρ eff ( t ) = 3 M P l " β t ( t + α ) + kC (cid:16) t t α (cid:17) βα w eff ( t ) = ( − αβ + β t ) β t ( t α ) − kC (cid:18) t t α (cid:19) βαβ t ( t α ) + kC (cid:18) t t α (cid:19) βα .1 Negative β consideration We discuss the possibility of taking negative value of β together with negative α in certainmodels giving rise to some new cosmologies. In this work, we consider negative β togetherwith negative α in models-XI,XII only. In other models one can work out for negative α, β inmodels-III,V,VIII,IX where α > t . This kind of analysis have been done by Nojiri and Odintsov[69], [70], [60] to study the future finite time singularity where they have taken α = t s → futuresingularity time.So, for negative α, β we obtain the cosmological parameters for models-XI,XII as Model-XI ∗ Model-XII ∗ H ( t ) = βt ( α − t ) H ( t ) = βt ( α − t ) a ( t ) = C (cid:0) tα − t (cid:1) βα a ( t ) = C (cid:16) t α − t (cid:17) β α q ( t ) = − αβ − β t q ( t ) = − αβ − β t ρ ( t ) = (cid:2) DC − w ) (cid:3) (cid:0) tα − t (cid:1) − w ) βα ρ ( t ) = (cid:2) DC − w ) (cid:3) (cid:16) t α − t (cid:17) − w ) β α ρ eff ( t ) = 3 M P l " β t ( α − t ) + kC ( tα − t ) βα ρ eff ( t ) = 3 M P l " β t ( α − t ) + kC (cid:16) t α − t (cid:17) βα w eff ( t ) = ( − αβ − β t ) β t α − t )2 − kC ( tα − t ) βαβ t α − t )2 + kC ( tα − t ) βα w eff ( t ) = ( − αβ − β t ) β t ( α − t ) − kC (cid:18) t α − t (cid:19) βαβ t ( α − t ) + kC (cid:18) t α − t (cid:19) βα For both these models, Hubble parameter and scale factor both diverge in finite time andshow big rip singularity in near future. → ACC transi-tion
To constrain the model parameters α and β and to compare our results with observation,we also re-write the DP and HP that are given as functions of cosmic time t , in terms of redshift z (= a a −
1, where a is the value of scale factor at present time t = t ) using the relation between t and z for the models with deceleration → acceleration transition i.e. for models-VI,XI ∗ ,XII ∗ . Model-VI t ( z ) = − α + h √ t + α − β ln(1 + z ) i q ( z ) = − β h √ t + α − β ln(1 + z ) i − H ( z ) = H h − ln(1+ z )2 β √ t + α i − Model-XI ∗ t ( z ) = α h (cid:16) αt − (cid:17) (1 + z ) αβ i − q ( z ) = − αβ − αβ h (cid:16) αt − (cid:17) (1 + z ) αβ i − H ( z ) = H t α n (1 + z ) − α β + (cid:16) αt − (cid:17) (1 + z ) α β o Model-XII ∗ ( z ) = √ α h (cid:16) αt − (cid:17) (1 + z ) αβ i − q ( z ) = − αβ − αβ h (cid:16) αt − (cid:17) (1 + z ) αβ i − H ( z ) = H t α n (1 + z ) − α β + (cid:16) αt − (cid:17) (1 + z ) α β o We find the observational constraints on both of the model parameters α and β to the latest28 data points of H ( z ) in the redshift range 0 . z . σ H ) given by Table-5 Hubble parameter vs redshift data z H ( z ) (cid:16) kms − Mpc (cid:17) σ H (cid:16) kms − Mpc (cid:17)
Ref. z H ( z ) (cid:16) kms − Mpc (cid:17) σ H (cid:16) kms − Mpc (cid:17)
Ref. .
100 69 12 [80] .
730 97 . [83] .
170 83 8 [80] .
781 105 12 [81] .
179 75 4 [81] .
875 125 17 [81] .
199 75 5 [81] .
880 90 40 [84] .
270 77 14 [80] .
900 117 23 [80] .
320 79 . . [82] .
037 154 20 [81] .
352 83 14 [81] .
300 168 17 [80] .
400 95 17 [80] .
363 160 33 . [85] .
440 82 . . [83] .
430 177 18 [80] .
480 97 62 [84] .
530 140 14 [80] .
570 100 . . [82] .
750 202 40 [80] .
593 104 13 [81] .
965 186 . . [85] .
600 87 . . [83] .
340 222 7 [86] .
680 92 8 [81] .
360 226 8 [87]
To complete the data set, we take H = 67 . Km/s/M pc . The mean values of modelparameters α and β are determined by minimizing χ OHD ( p s ) = X i =1 [ H th ( p s ; z i ) − H obs ( z i )] σ H ( z i ) where p s denotes the parameters of the model, H th is the theoretical (model based) value forthe Hubble parameter, H obs is the observed one, σ H ( z i ) is the standard error in the observedvalue, and the summation runs over 28 observational data points at redshifts z i .From our analysis, the model-VI show a poor fit for higher redshifts (not shown), but models-XI ∗ & XII ∗ show nice fit to the Hubble data compared with Λ CDM model and are shown infigure-2 . The likelihood contours in the α − β plane with 1 σ and 2 σ error are also obtained forthese models and are shown in figure-3. 13igure 2: This figure corresponds to the latest H ( z ) data with error bars. In both the plots, solidlines corresponds to the best fitted behavior for model-XI ∗ & XII ∗ and dotted lines correspondsto Λ CDM model. H ( z ) is expressed in unit of Km/s/M pc .Figure 3: This figure shows the plots for 1 σ (dark shaded) and 2 σ (light shaded) likelihoodcontours in the α − β planes, obtained for model-XI ∗ & XII ∗ .14he best fit values of α and β obtained by minimizing the chi-square ( χ ) with 1 σ error areobtained as α β Models–XI ∗ . +0 . − . . +0 . − . Models–XII ∗ . +1 . − . . +0 . − . .With these values of α and β , we plot q ( z ) vs z (see figure-4) and w ( z ) vs z (see figure-5) for models–XI ∗ & XII ∗ for flat ( k = 0) case. For both the models EoS parameter w ( z )crosses the phantom divide line in near future. The transition redshift z tr from deceleration toacceleration and the redshift z ph at which w ( z ) crosses the phantom divide line are also shown.From the figures we can find z tr ≈ .
73 for model-XI ∗ and z tr ≈ .
58 for model-XII ∗ . Similarly,the redshift at which w ( z ) crosses the phantom divide line is z ph ≈ − .
38 for model-XI ∗ and z ph ≈ − .
01 for model-XII ∗ .Figure 4: This figure shows plots for the DP q ( z ) vs redshift z for model-XI ∗ & XII ∗ showingthe transition from deceleration to acceleration.Figure 5: This figure shows plots for the EoS w ( z ) vs redshift z for model-XI ∗ & XII ∗ showingthe phantom divide line crossing. 15 Conclusion
In this paper we proposed a convenient and simple parametrization of H . For certainchoices of model parameters in our scheme, we reproduce several known solutions such asΛCDM cosmology, Power law cosmology, Berman’s model, Aksrsu’s model, Abdel Rahman’smodel and others. Thus, our parametrization covers all these models and also produces somenew cosmologies. The models under consideration either show transition from decelerationto acceleration or vice versa; in some cases we observed eternal acceleration. The variousconstrain equations and parametrization related to a ( t ) , H ( t ) , q ( t ) , Λ( t ) , ρ ( t ) , w ( t ) consideredin literature are also summarized in detail. As the present observations agree with decelerationto acceleration transition, we have analyzed three obtained models which exhibit this importantfeature. Our analysis shows that the model-VI has a poor fit for higher redshifts, but models-XI ∗ & XII ∗ show a nice fit with the Hubble data. The likelihood contours in the α − β planeobtained for models–XI ∗ & XII ∗ are shown in figure-3. The best fit values of α and β are givenby, α = 3 . +0 . − . & β = 2 . +0 . − . for models-XI ∗ and α = 3 . +1 . − . & β = 2 . +0 . − . for models-XII ∗ . For these values of α and β , the evolution of q ( z ) showing the deceleration to accelerationphase transition and the evolution of w ( z ) showing the phantom divide line crossing for thesemodels are shown in figure-4 and figure-5 respectively.It is interesting to note that our parametrization for H can give rise to several interestingfeatures and can further be studied in anisotropic Bianchi space-times in the framework ofgeneral theory of relativity as well as in modified theories of gravity. In particular it would beinteresting to discuss the problem of future singularities in the proposed framework which wedeffer to our future investigations. Acknowledgement
The authors wish to thank M. Sami for his useful comments andsuggestions throughout the work. Author SKJP wish to thank Department of Atomic Energy(DAE), Government of India for financial support through the post-doctoral fellowship of Na-tional Board of Higher Mathematics (NBHM). SKJP is also thankful to J. V. Narlikar, S.Jhingan and R. G. Vishwakarma for discussions on the related theme and thankful to SafiaAhmad and Dharm Veer Singh for their help in some numerical computations.
Appendix-1
Table-6Variations of Scale factor ( a ) Ref. a ∼ t n [71], [88] a ∼ exp ( βt ) [89] a ∼ sinh ( βt ), a ∼ cosh ( βt ) [89] a ∼ (cid:16) tt (cid:17) α e β (cid:16) tt − (cid:17) [76] a ∼ e αt t n [77] a ( t ) = e α ( t − t s ) β ) [90] a ( t ) = a (cid:16) tt s − t (cid:17) γ [91] where n , α , β , γ are constants. t is the present time. t s future singularity time. able-7Variations ofEnergy density ( ρ ) Ref. ρ = ρ c [92], [93], [94] ρ ∼ θ [95], [96] ρ = Aa √ a + b [97], [98] ( ρ + 3 p ) a = A [99], [100] ρ + p = ρ c [101], [102] A , b are constants.Table-8Pressure ( p ) considerations EoS Ref. p = wρ perfect fluid p = wρ − f ( H ) viscous fluid [117], [118] p = − ρ − ρ α DE [119] p = wρ + kρ n polytropic [120], [121] p = − ( w + 1) ρ ρ P + wρ + ( w + 1) ρ Λ quadratic [122] p = − Bρ Chaplygin Gas [46], [47] p = − Bρ α generalized CG [123], [124] p = Aρ − Bρ α modified CG [125], [126] p = Aρ − B ( a ) ρ α variable MCG [127] p = A ( a ) ρ − B ( a ) ρ α new variable MCG [128] ρ P -Plank density ρ Λ -vacuum density w k A > , B > areconstants. A ( a ) & B ( a ) arefunctions of scalefactor. Note:
In literature there exist numerous solutions to Einstein field equations with the ansatz σ ∝ θ , where σ is the energy density associated with anisotropy and θ is the volume expansionscalar in homogeneous anisotropic Bianchi models.Table-9Variations of cosmological constant ( Λ) Ref. Λ ∼ a − n [103], [104], [105] Λ ∼ H n [78], [105], [107] Λ ∼ ρ [105], [106], [108] Λ ∼ t n [78], [105] Λ ∼ q n [78] Λ ∼ e − βa [109] Λ = Λ( T ) , T is Temperature [110] Λ ∼ C + e − βt [111] Λ = 3 βH + αa − [112], [113] Λ = β ¨ aa [114], [106] Λ = 3 βH + α ¨ aa [115] d Λ dt ∼ β Λ − Λ [116] where n, α, β, C appearing inthe expressions are constants.For a complete set of decaylaws of Λ one can see [78]. able-10Parametrization of DP ( q ) Ref. q = m − [72], [129] q ( t ) = − αt + m − [75] q ( t ) = − αt + β − [74] q ( a ) = − − αa α a α [130] q ( z ) = q + q z [131], [132], [133], [134] q ( z ) = q + q z (1 + z ) − [134], [135], [136] q ( z ) = q + q z (1 + z )(1 + z ) − [137] q ( z ) = + q (1 + z ) − [134] q ( z ) = q + q [1+ ln (1 + z )] − [136] q ( z ) = +( q z + q )(1 + z ) − [138], [140], [139] q ( z ) = − (cid:16) (1+ z ) q q +(1+ z ) q (cid:17) [141] q ( z ) = − h q + 1 − q + 1) (cid:16) q e q z ) − e − q z ) q e q z ) + e − q z ) (cid:17)i [142] q ( z ) = − + (cid:18) q e q z √ z − e − q z √ z q e q z √ z + e − q z √ z (cid:19) [142] q ( z ) = q f + q i − q f − qiqf ( zt z ) τ [143] m , α , β , q , q , q appearing in the above expressions are constants.Table-11Parametrization of EoS ( w ) Ref. w ( z ) = w + w z [144], [145] Linear w ( z ) = w + w z (1+ z ) [146] JBP w ( z ) = w + w z z [147], [148] CPL w ( z ) = w + w z √ z [149] sqrt w ( z ) = w + w z (1+ z )1+ z [150] BA w ( z ) = w + w z z [151] FSLL Model 1 w ( z ) = w + w z z [151] FSLL model 2 w ( z ) = w + w sin ( z ) [152] Sine w ( z ) = w + w ln (1 + z ) [153] Logarithmic w ( z ) = w + w (cid:16) ln(2+ z )1+ z − ln 2 (cid:17) [154] MZ Model 1 w ( z ) = w + w (cid:16) sin(1+ z )1+ z − sin 1 (cid:17) [154] MZ Model 2 w ( z ) = w + w (cid:0) z z (cid:1) n [155] nCPL w ( z ) = w + w z (1+ z ) n [155] nJBP w ( z ) = w + w ln (cid:0) z z (cid:1) [156] Modified Logarithmic w , w appearing in the above expressions are constants. 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