Cosmology with hybrid expansion law: scalar field reconstruction of cosmic history and observational constraints
Ozgur Akarsu, Suresh Kumar, R. Myrzakulov, M. Sami, Lixin Xu
aa r X i v : . [ g r- q c ] J a n Cosmology with hybrid expansion law: scalar field reconstruction ofcosmic history and observational constraints ¨Ozg¨ur Akarsu a , Suresh Kumar b , R. Myrzakulov c , M. Sami c , Lixin Xu d a Department of Physics, Ko¸c University, 34450 Sarıyer, ˙Istanbul, Turkey. b Department of Mathematics, BITS Pilani, Pilani Campus, Rajasthan-333031, India. b Centre of Theoretical Physics, Jamia Millia Islamia, New Delhi-110025, India. c Institute of Theoretical Physics, Dalian University of Technology, Dalian, 116024, P. R. China.
E-Mail: [email protected], [email protected], [email protected], [email protected],[email protected]
Abstract.
In this paper, we consider a simple form of expansion history of Universe referred to as the hybridexpansion law − a product of power-law and exponential type of functions. The ansatz by construction mimicsthe power-law and de Sitter cosmologies as special cases but also provides an elegant description of the transitionfrom deceleration to cosmic acceleration. We point out the Brans-Dicke realization of the cosmic history underconsideration. We construct potentials for quintessence, phantom and tachyon fields, which can give rise to thehybrid expansion law in general relativity. We investigate observational constraints on the model with hybridexpansion law applied to late time acceleration as well as to early Universe a la nucleosynthesis . Keywords. dark energy theory, supernova type Ia - standard candles, big bang nucleosynthesis
Since the first observation on late time cosmic acceleration in 1998 [1, 2], attempts have been made to understandthe cause of this remarkable phenomenon within the framework of Einstein general relativity and beyondit. Broadly, the model building undertaken in the literature to capture the essential features of cosmologicaldynamics can be classified in two categories: Models based on dark energy [3, 4, 5, 6, 7, 8, 9, 10] and scenariosrelated to modified theories of gravity [11, 12].The candidates of dark energy include cosmological constant and a variety of scalar field models; the latterwere invoked to alleviate the problems associated with cosmological constant. Unfortunately, the scalar fieldmodels are plagued with similar problems. The models based upon modified theories of gravity are faced withchallenges posed by the local physics. Large scale modification of gravity essentially involves extra degree(s)of freedom which might influence local physics where Einstein theory of gravity is in excellent agreement withobservations. One then needs to invoke mass screening mechanisms to hide these degrees of freedom. To befair, these scenarios do not perform better then the ones based upon dark energy. As for the latter, one canreconstruct the cosmic history referring to FRW background or by making use of the growth of perturbations onsmall scales. Given a priori a cosmic history specifying either the equation of state (EoS) or the scale factor a , onecan always construct a scalar field potential which would mimic the desired result [13, 14, 15, 16, 17, 18, 19, 20].Similar reconstruction can be carried out in scalar tensor theories.On phenomenological grounds, a number of parametrization schemes have been investigated with the require-ment of their theoretical consistency and observational viability. In particular, parametrization of EoS/Hubbleparameter/pressure have been extensively used in the literature [21, 22, 23, 24]. The dynamics of realisticUniverse is described by an EoS parameter which behaves differently at different epochs. For instance, in gen-eral relativistic description of the dynamics of the spatially flat RW spacetime, the fluids with constant EoSparameter w > − a ∝ t w ) ) of the Universe and to an exponentialexpansion a ∝ e k t , where k > w = −
1. The solution of the Einstein’s field equation in thepresence of single fluid with a constant EoS parameter gives the relation between the EoS parameter of a fluidand the deceleration parameter (DP) of the Universe as, q = − ¨ aa ˙ a = w . Obviously, a fluid with a constantEoS cannot give rise to realistic cosmic history. he realistic Universe should be dominated by stiff fluid (which was suggested as the most probable EoS fordescribing the very early Universe [25, 26]), radiation, pressureless matter and cosmological constant (or a fluidexhibiting a similar behavior with the cosmological constant in the present Universe) respectively as it evolves.In other words, the EoS parameter of the effective fluid ( ρ eff = P ρ i ) that drives the expansion of Universe willnot be yielding a constant EoS, once we consider a mixture of fluids with different EoS parameters at differentepochs.A variety of scalar field models including quintessence, K-essence, tachyons and phantoms investigated inthe literature can give rise to a time dependent EoS parameter a la dynamical dark energy. Hence, the trajectoryof the evolution of the DP would be dependent on the characteristics of the dark energy model. For instance, inmost of the scalar field dark energy models, the effective EoS parameter of the dark energy evolves from w = 1to w = − q = 2 to q = − In what follows, we shall consider the following ansatz referred to, hereafter, as hybrid expansion law (HEL): a ( t ) = a (cid:18) tt (cid:19) α e β (cid:16) tt − (cid:17) , (1)where α and β are non-negative constants. Further, a and t respectively denote the scale factor and age ofthe Universe today. The cosmological parameters; Hubble parameter, DP and jerk parameter are respectivelygiven by: H = ˙ aa = αt + βt , (2) q = − ¨ aaH = αt ( βt + αt ) − , (3) j = ... aaH = 1 + (2 t − βt − αt ) αt ( βt + αt ) . (4)In particular cases, one obviously obtains power-law and exponential expansion from (1) choosing α = 0and β = 0 respectively. It is evident that one may choose the constants such that the power-law term dominatesover the exponential term in the early Universe, namely, at the time scales of the primordial nucleosynthesis( t ∼ second). Accordingly, for t ∼
0, the cosmological parameters approximate to the following: a ∼ a (cid:18) tt (cid:19) α , H ∼ αt , q ∼ − α and j ∼ − α + 2 α . (5)Similarly, the exponential term dominates at late times, such that in the limit t → ∞ we have a → a e β (cid:16) tt − (cid:17) , H → βt , q → − j → . (6)It may be observed that the parameter α determines the initial kinematics of the Universe while the verylate time kinematics of the Universe is determined by the parameter β . When α and β both are non-zero,Universe evolves with variable DP given by (3) and the transition from deceleration to acceleration takes placeat tt = √ α − αβ , which puts α in the range 0 < α < n the next section, we shall focus on HEL realization in the framework of Brans-Dicke theory while in thefollowing section we shall study the corresponding effective fluid as well as single scalar field reconstruction ingeneral theory of relativity. While doing these investigations, inspired by the inflationary scenarios, we shallconsider only the spatially homogeneous and isotropic RW spacetime as the background geometry for describingthe Universe. As it is demonstrated in Ref.[11], it is always possible to carry out reconstruction program in the framework ofscalar-tensor theory giving rise to a desired cosmic history. However, in what follows, we directly show that theparticular case α = of HEL is a particular solution of the Brans-Dicke field equations in the presence of dustfluid.The action for the Jordan-Brans-Dicke (Brans-Dicke in Jordan frame) theory can be given as S = Z d x √− g (cid:20) − ϕ ω R + 12 ∇ α ϕ ∇ α ϕ (cid:21) + S M , (7)where S M is the matter action, ϕ is the Jordan field and ω is the Brans-Dicke coupling parameter/constant.The field equations obtained from this action for spatially flat RW spacetime are as follows:3 ˙ a a + 6 ˙ aa ˙ ϕϕ − ω ˙ ϕ ϕ = 4 ωϕ ρ, (8)˙ a a + 2 ¨ aa + 2 ¨ ϕϕ + (2 + 2 ω ) ˙ ϕ ϕ + 4 ˙ aa ˙ ϕϕ = − ωϕ p, (9) − ω ¨ aa − ω ˙ a a + ¨ ϕϕ + 3 ˙ aa ˙ ϕϕ = 0 , (10)where ρ and p are the energy density and pressure of matter, respectively. This system consists of threedifferential equations (8)-(10) that should be satisfied by four unknown functions of a , ϕ , ρ and p and thereforeis not fully determined. At this point, considering the fact that cold dark matter (CDM) and ordinary matterhave zero pressure, we further assume pρ = 0 , (11)as an additional constraint to fully determine the system.For the particular case ω = − , the solution of the system is (with the adjustment a ( t ) = 0 at t = 0) a = c c t e c t , ϕ = c t − e − c t and ρ = c c t − e − c t , (12)where c , c and c are integration constants. We note that the scale factor we obtained here in (12) correspondsto the particular case α = of our HEL ansatz. It behaves as a ∼ t at t ∼ | ω | ∼ ω = −
1. Interestingly, the value ω = − that we used above for the particular solution, corresponds to the four-dimensional spacetime with0-brane ( d = 1) in the d -branes string model [27, 28].However, we should point out that Brans-Dicke theory with parameter | ω | ∼ The behavior of the scale factor under consideration in this paper was studied in the context of inflation inthe early Universe by Parsons and Barrow in Ref. [29] (where a simple mathematical method for generating ew inflationary solutions of Einstein’s field equations from old ones was provided). They pointed out that theEinstein’s field equations in the presence of self-interacting scalar field are invariant under constant rescaling ofthe scalar field, and then they generated the HEL behavior from power-law expansion. They also showed thatsuch an expansion of the Universe can be represented as a Friedmann Universe in the presence of imperfectfluid that can be described by an EoS parameter of a perfect fluid with an added constant bulk viscous stress.In this paper, on the other hand, we study HEL expansion in the context of the history of the Universe afterthe inflation took place, and mainly investigate whether this law could be used for describing the evolution ofthe Universe starting from the radiation- or matter-dominated Universe to the currently accelerating Universe.Accordingly, we next discuss the effective fluid by interpreting it as a mixture of different sources that wouldlead to the HEL expansion in general relativity in Section 4.1. We also study the single scalar field realizationof the HEL expansion in Section 4.2. In general relativity, one can always introduce an effective source that gives rise to a given expansion law.Accordingly, we obtain the energy density and EoS parameter of the effective source, which is assumed todescribe the mixture of different types of sources such as matter, radiation and dark energy in general relativity.Hence, using the ansatz (1) in the Friedmann equations in general relativity, we obtain the energy density andthe EoS parameter of the effective fluid as follows: ρ eff = 3 (cid:18) αt + βt (cid:19) and w eff = 2 α t (cid:18) αt + βt (cid:19) − − . (13)The EoS parameter of the effective fluid starts with w eff = − α at t = 0, and evolves to w eff → − t → ∞ . One may observe that α is the parameter that allows us to determine the effective fluid at early epochs.Accordingly, choosing α = we can set a beginning with a stiff fluid domination, i.e., w eff = 1 at t ∼ α = we can set a beginning with a radiation domination, i.e., w eff = at t ∼ α = , the EoS parameter w eff evolves from 1 to − tt = √ β − β . Again choosing α = , the EoS parameter w eff evolves from to −
1, and the accelerated expansion starts at tt = √ β − β .In this study, the value of α is not fixed to a certain value, but is left as a free parameter to be constrainedusing the latest data from H ( z ) and SN Ia observations. Hence, the observational constraints on α shalldetermine the starting era of Universe within the framework of HEL model, while the ΛCDM model describesthe Universe starting from the matter-dominated era. If it is found that α ∼ , then the effective fluid may beinterpreted as a mixture of radiation, matter and dark energy, and be written as ρ eff = ρ r + ρ m + ρ DE , where ρ r ∝ a − and ρ m ∝ a − stand for the radiation, matter (baryonic matter+CDM) constituents respectively, and ρ DE stands for the unknown dark energy source that gives rise to HEL expansion together with radiation andmatter. In this case, HEL model may be considered as a candidate for describing the Universe starting fromthe radiation dominated era, and hence one can further investigate the HEL model by discussing the primordialnucleosynthesis times, and checking whether the matter-dominated era would be recovered properly or not. Onthe other hand, if it is found that α ∼ , then the effective fluid may be interpreted as a mixture of matter anddark energy, and be written as ρ eff = ρ m + ρ DE , like in the ΛCDM model for which dark energy is given bycosmological constant Λ. Hence, in this case, HEL model may be considered as a candidate for describing theUniverse starting from the matter-dominated era as in the ΛCDM model.We note here that the dark energy fluid can be obtained by subtracting the known constituents (such asmatter and radiation) from the effective fluid. We shall stick to this approach in our discussions that followthe observational analysis. On the other hand, it might be interesting and useful to see the single scalar fieldcorrespondence of the HEL expansion before doing the observational analysis. We can always construct a scalar field Lagrangian which can mimic a given expansion law. Accordingly, inthis section, assuming that the effective energy density and EoS parameter given in (13) correspond to a singlescalar field, we obtain the potentials for the quintessence, tacyhon and phantom fields, which are the most ommonly considered scalar field candidates for dark energy. One can use these potentials for describing darkenergy sources. Most of the dark energy studies are carried out within the quintessence paradigm of a slowly rolling canonicalscalar field with a potential. Therefore, we first consider the quintessence realization of the HEL. The energydensity and pressure of the quintessence minimally coupled to gravity can be given by ρ = 12 ˙ φ + V ( φ ) and p = 12 ˙ φ − V ( φ ) , (14)where φ is the canonical scalar field with a potential V ( φ ). Using these with the HEL ansatz (1), we obtain φ ( t ) = √ α ln( t ) + φ and V ( t ) = 3 (cid:18) αt + βt (cid:19) − αt , (15)where φ is the integration constant. The potential as a function of the scalar field φ is then given by thefollowing expression: V ( φ ) = 3 β e − √ α ( φ − φ ) + α (3 α − e − √ α ( φ − φ ) + 6 αβe − √ α ( φ + φ − φ ) , (16)where φ = φ + √ α ln( t ).One may observe that this potential can be seen as the summation of three different potentials, i.e., aconstant potential and two exponential potentials. Choosing α = 0 the potential reduces to a constant, henceto a cosmological constant, that would give rise to exponential expansion. Choosing β = 0, the potential reducesto a single exponential potential that would give rise to power-law expansion (see for instance [30]) and maydescribe a matter field with a constant EoS parameter. For α = 0 = β , on the other hand, the potential containsa constant on the left, an exponential potential in the middle and an additional potential term depending onboth α and β , which may be interpreted as an interaction term between the first two potentials.One may observe that the condition for non-negativity of the potential is α ≥ . Under this condition, wehave ˙ φ → ∞ and V → ∞ as t →
0, and ˙ φ → V → β e − √ α ( φ − φ ) as t → ∞ . We see thatthe scalar field approaches positive cosmological constant at late times of the Universe.
Quintessence paradigm relies on the potential energy of scalar fields to drive the late time acceleration of theUniverse. On the other hand, it is also possible to relate the late time acceleration of the Universe with thekinetic term of the scalar field by relaxing its canonical kinetic term. This idea is known as k-essence [31]. Inthis section we are interested in a special case of k-essence that is known as Tachyon. Tachyon fields can betaken as a particular case of k-essence models with Dirac-Born-Infeld (DBI) action and can also be motivatedby string theory [32, 33]. It has been of interest to the dark energy studies due to its EoS parameter w = ˙ φ − ρ = V ( φ ) q − ˙ φ and p = − V ( φ ) q − ˙ φ , (17)where φ is the Tachyon field with potential V ( φ ). Using these with the HEL ansatz (1), we find φ ( t ) = s αt β ln( βt + αt ) + φ and V ( t ) = 3 (cid:18) αt + βt (cid:19) s − αt βt + αt ) , (18)where φ is an integration constant. The tachyon potential that drives the HEL Universe is given by V ( φ ) = 3 β t e r β αt ( φ − φ ) s − αt e r β αt ( φ − φ ) αt − e r β αt ( φ − φ ) ! − . (19)We see that the tachyon potential is real subject to the condition α ≥ . Also the EoS parameter w = ˙ φ − −
1, as it should be for the tachyon field, during the evolution of Universe. .2.3 Phantom field correspondence Quintessence and tacyhon fields considered in the previous two subsections can yield EoS paremeters w ≥−
1. However, the observations at present allow slight phantom values for the EoS parameter, i.e., w < − S -braneconstructions in string theory [41, 42, 43, 44, 45]. On the other hand, the phantom energy, in general, can besimply described by a scalar field with a potential V ( φ ) like the quintessence dark energy but with a negativekinetic term [46]. Accordingly, the energy density and pressure of the phantom field are respectively given by ρ = −
12 ˙ φ + V ( φ ) and p = −
12 ˙ φ − V ( φ ) , (20)where φ is the phantom field with potential V ( φ ).In case of the phantom scenario, the HEL ansatz (1) must be slightly modified in order to acquire selfconsistency. In particular, we rescale time as t → t s − t , where t s is a sufficiently positive reference time. Thus,the HEL ansatz (1) becomes a ( t ) = a (cid:18) t s − tt s − t (cid:19) α e β (cid:16) ts − tts − t − (cid:17) , (21)and the Hubble parameter, its time derivative and DP read as H = − αt s − t − βt s − t , ˙ H = − α ( t s − t ) and q = α ( t s − t ) [ β ( t s − t ) + α ( t s − t )] − . (22)We observe that α < q < H > t → t s and thus exposing the Universe to Big Rip. Further, we find φ ( t ) = √− α ln( t s − t ) + φ and V ( t ) = 3 (cid:18) αt s − t + βt s − t (cid:19) − α ( t s − t ) , (23)where φ is an integration constant. The phantom potential in terms of the phantom field reads as V ( φ ) = 3 β e − √ − α ( φ − φ ) + α (3 α − e − √ − α ( φ − φ ) + 6 αβe √ − α ( φ + φ − φ ) , (24)where φ = φ + √− α ln( t s − t ). In the phantom HEL cosmology, we find that at late times w = − α ,which lies below the phantom divide line ( w = −
1) for α < w = −
1) in a single field model. We then needto tune phantom field such that it remains sub-dominant during matter phase and shows up only at late times.In case of quintessence and tachyon, the field can formally mimic dark matter like behavior which is impossiblein case of a phantom field. H ( z ) +SN Iadata For observational purposes, we use the following relation between scale factor and redshift: a ( t ) = a z . (25)Invoking HEL (1) into the above equation and solving for t , we have t = αt β f ( z ) , (26)where f ( z ) = LambertW (cid:18) βα e β − ln(1+ z ) α (cid:19) . H ( z )(km s − Mpc − ) measurements with 1 σ errors. z H ( z ) σ H ( z ) Reference0.090 69 12 [47]0.170 83 8 [47]0.270 77 14 [47]0.400 95 17 [47]0.900 117 23 [47]1.300 168 17 [47]1.430 177 18 [47]1.530 140 14 [47]1.750 202 40 [47]0.24 79.69 3.32 [48]0.43 86.45 3.27 [48]0.480 97 62 [49]0.880 90 40 [49]0.179 75 4 [50]0.199 75 5 [50]0.352 83 14 [50]0.593 104 13 [50]0.680 92 8 [50]0.781 105 12 [50]0.875 125 17 [50]1.037 154 20 [50]0.07 69.0 19.6 [51]0.12 68.6 26.2 [51]0.20 72.9 29.6 [51]0.28 88.8 36.6 [51]
The Hubble parameter in terms of redshift for the HEL cosmology reads as H ( z ) = H βα + β (cid:20) f ( z ) + 1 (cid:21) , (27)where H = α + βt . We see that the parameter space of HEL cosmology consists of three parameters namely α , β and H to be constrained by the observations.The authors of Ref.[47] obtained nine H ( z ) data points from the relative dating of 32 passively evolvinggalaxies. Using the BAO peak position as a standard ruler in the radial direction, H ( z ) was estimated for z = 0 .
24 and z = 0 .
45 in Ref.[48]. Two determinations for H ( z ) were given in Ref.[49] using red-envelopegalaxies while a reliable sample of eight H ( z ) points was derived in Ref.[50] using the differential spectroscopicevolution of early-type galaxies as a function of redshift. The authors of Ref.[51] presented four H ( z ) pointsadopting the differential age method and utilizing selected 17832 luminous red galaxies from Sloan Digital SkySurvey (SDSS) Data Release Seven. We compile all the 25 H ( z ) data points spanning in the redshift range0 . < z < .
750 in Table 1. It may, however, be noted that though the H ( z ) data points derived from differentmethods/sources are frequently used in the literature for constraining cosmological parameters but these areprone to systematics. Following the methodology given in Ref.[52], we utilize these 25 observational H ( z ) datapoints in addition to the SN Ia Union2.1 sample [53] that contains 580 SN Ia data points spanning in the redshiftrange 0 . < z < .
414 for constraining the parameters of HEL cosmology. We use the Markov Chain MonteCarlo (MCMC) method, whose code is based on the publicly available package cosmoMC [54], for the dataanalyses. We have also constrained the standard ΛCDM model parameters with the same observational datasets of H ( z ) and SN Ia for the sake of comparison with the HEL models (See Appendix A for the dynamics ofthe ΛCDM cosmology).The 1D marginalized distribution on individual parameters and 2D contours with 68.3 %, 95.4 % and 99.73% confidence limits are shown in Fig.1 for the HEL model. The mean values of the HEL model parameters α , β nd H constrained with H ( z )+SN Ia data are given in Table 2. The 1 σ , 2 σ and 3 σ errors, χ min and χ min /dofare also given in Table 2. H β α H β Figure 1:
The 1D marginalized distribution on individual parameters of HEL model and 2D contours with 68.3 %,95.4 % and 99.7 % confidence levels are obtained by using H ( z )+SN Ia data points. The shaded regions show the meanlikelihood of the samples. Table 2: Mean values with errors of the HEL model parameters constrained with H ( z )+SN Ia data. Parameters Mean values with errors α . +0 . . . − . − . − . β . +0 . . . − . − . − . H (km s − Mpc − ) 69 . +2 . . . − . − . − . χ min . χ min /dof 0 . σ errors that we obtained using theHEL and ΛCDM models, viz., age of the present Universe t ; Hubble constant H , current values of the DP q and jerk parameter j , time passed since the accelerating expansion started t − t tr , redshift of the onset ofthe accelerating expansion z tr , energy density ρ and the EoS parameter w of the effective fluid at the presentepoch of evolution of the Universe. We also give χ min and χ min /dof in Table 3 to compare the success of themodels on fitting the data. We notice that both the models fit observational data with a great success but theΛCDM model fits the data slightly better than the HEL model does.Next, for comparing the ΛCDM and HEL models we make use of statistical tools such as Akaike InformationCriterion (AIC), Kullback Information Criterion (KIC) and Bayes Information Criterion (BIC) , which arecommonly used in modern cosmology for model selection among competing models. For instance, in a recentpaper [55], Melia and Maier used these information criteria to compare ΛCDM model and R h = ct Universe.The three information criteria are defined as follows (see Ref.[55] and references therein for details):AIC = χ + 2 k, KIC = χ + 3 k, BIC = χ + k ln n , σ errors of some important cosmological parameters related to HEL andΛCDM models. χ min /dof, AIC, KIC and BIC values are also displayed. Parameters HEL ΛCDM t (Gyr) 13 . ± .
096 13 . ± . H (km s − Mpc − ) 69 . +2 . − . . +1 . − . q − . ± . − . ± . j . ± .
156 1 t − t tr (Gyr) 6 . ± .
558 6 . ± . z tr . +0 . − . . ± . ρ (10 − kg m − ) 9 . ± .
528 9 . ± . w − . ± . − . ± . χ min .
161 556 . χ min /dof 0 . . .
16 560 . .
16 562 . .
37 569 . k is number of model parameters and n is number of data used in fitting. A model with lower value ofAIC, KIC or BIC is considered to be closest to the real model. So these information criteria provide relativeevidence of better model among the models under consideration. Further, the difference of the AIC values oftwo models is denoted by ∆AIC. A rule of thumb used in the literature is that if ∆AIC .
2, the evidence isweak; if ∆AIC ≈ ≈
4, it is mildly strong and in case ∆AIC &
5, it is quite strong. Similar rule of thumbis followed for testing the strength of evidence while dealing with KIC. In case of BIC, the evidence is judgedpositive for the values of ∆BIC between 2 and 6. A value of greater than 6 indicates strong evidence. In thecase in hand, we have two models namely ΛCDM and HEL. The corresponding values of AIC, KIC and BICare given in Table 3. We immediately find that ∆AIC = 2 .
66, ∆KIC = 3 .
66 and ∆BIC = 7 .
06. These figuresfrom H ( z )+SN Ia data only suggest that ΛCDM model is favored over the HEL model. One may see that AICdoes not offer a strong evidence against the HEL model. However, the HEL model pays penalty in KIC andBIC cases because it carries one additional parameter in comparison to the ΛCDM model.We note all the cosmological parameters related with the present day Universe as well as with the onset ofthe acceleration given in Table 3 for the HEL and ΛCDM models are consistent within the 1 σ confidence level.Only exception is that the present values of the jerk parameter do not coincide in the two models within the1 σ confidence level. However, we should recall that jerk parameter is determined to be a constant j ΛCDM = 1in ΛCDM, and hence doesn’t involve error, while it is function of time with two free parameters α and β in theHEL model. We additionally note that jerk parameter involves the third time derivative of the scale factor, andconsequently it is constrained observationally rather weakly [56, 57, 58, 59, 60, 61, 62]. Hence, we are not ableto decide which model describes the expansion of the Universe well considering the jerk parameter. It is, on theother hand, a very useful parameter to see how the HEL model deviates from the ΛCDM model. In accordancewith this at the end of this section, we shall also compare the evolution trajectories of the HEL and ΛCDMmodels in the plane of DP and jerk parameter.In Table 4, we give the values of some important cosmological parameters with 1 σ errors for the HELand ΛCDM models at three different epochs: early epoch ( z → ∞ ), present epoch ( z = 0) and future epoch( z → − σ errors of various parameters pertaining to theHEL and ΛCDM models. Model → HEL ΛCDMParameter z → ∞ z = 0 z → − z → ∞ z = 0 z → − H (km s − Mpc − ) ∞ . +2 . − . . ± . ∞ . +1 . − . . ± . q . ± . − . ± . − . − . ± . − j . ± .
060 0 . ± .
156 1 1 1 1 ρ (10 − kg m − ) ∞ . ± .
528 2 . ± . ∞ . ± .
481 6 . ± . w . ± . − . ± . − − . ± . − z → ∞ limit of the ΛCDM model isalready determined as the dust dominated Universe in general relativity, and in fact this model can be used fordescribing the actual Universe for redshift values less than z ∼ q z →∞ ∼ w z →∞ ∼ ) in the HEL model is a very good approximation tothe radiation dominated Universe in general relativity (where, q = 1 and w = ). This motivates us to furtherinvestigate the early Universe behavior in the HEL model, that we shall do in the next section.In the aforesaid, we observed and discussed the particular values of the cosmological parameters in HELand ΛCDM models for the present Universe and for two extremes z → ∞ and z → −
1. Next, we would like toconclude this section by comparing the continuous evolution of these models. A very useful way of comparing anddistinguishing different cosmological models, that have similar kinematics, is to plot the evolution trajectoriesof the { q, j } and { j, s } pairs. Here, q and j have the usual meaning and s is a parameter defined as s = j − q − ) . (28)In the above definition s , there is in the place of in the original definition s = j − q − ) by Sahni et al. [56].This is to avoid the divergence of the parameter s when the HEL model passes through q = 1 or q = as donein Ref. [52]. The parameter s was originally introduced to characterize the properties of dark energy, and hencethe evolution of the Universe was considered starting from dust dominated era in general relativity, which gives q = and j = 1. However, in accordance with the HEL model, here we are also interested in the possibilityof describing the Universe starting from primordial nucleosynthesis times, where the expansion of the Universecan be best described by q ∼
1. Accordingly, using (3) and (4) in (28) we get s = 2 αt [3 βt + (3 α − t ]3( βt + αt ) [5( βt + αt ) − αt ] (29)for the HEL model.We plot evolution trajectories of the HEL and ΛCDM models in the j − q plane in Fig. 2(a) and in the j − s plane in Fig. 2(b) in the range − ≤ q < { q, j } and { j, s } pairs while the black dots show the matter dominated phases of the models.We observe that all the models have different evolution trajectories but the values of q , j and s do notdeviate a lot in different models for q . . { j, s } = { , } . In the HEL model, on the other hand, the evolution trajectory doesnot start at { j, s } = { , } . GPGalileonChaplygin Gas L CDM d S L i n e T r a n s iti on L i n e M a tt e r - do m i n a t e d E r a L i n e R a d i a ti on - do m i n a t e d E r a L i n e HEL - - j q (a) HEL DGPGalileonChaplygin Gas L CDM - - - - j s (b) Figure 2: (a)
Variation of q versus j . Vertical Purple line stands for the de Sitter (dS) state q = − (b) Variationof s versus j . Horizontal and vertical dashed lines intersect at the ΛCDM point (0 , s, j ) or ( q, j ) pair while the black dots show the matter dominated phases ofthe models. The dark red dot on the radiation-dominated line corresponds to the HEL model. Thus, it starts from theradiation-dominated phase and evolves to de Sitter phase while all other models under consideration evolve from theSCDM phase to the de Sitter phase. In the following, using the values of the model parameters obtained from the 25 + 580 data points from thelatest H ( z ) and SN Ia compilations spanning in the redshift range 0 . < z < .
750 in the previous section, wediscuss whether the HEL model makes successful predictions for high redshift values ( z ∼ − ) consideringBBN in Section 6.1, for low redshift values ( z ∼
0) considering BAO in Section 6.2 and then for intermediateredshift values considering, particularly, CMB in Section 6.3.
It is showed in Section 5 that HEL law predicts the value of the DP at z → ∞ , i.e., in the early Universe, as q z →∞ = 1 . ± .
590 (1 σ ), which can be maintained by the presence of an effective fluid that yields an EoSparameter w z →∞ = 0 . ± .
393 in general relativity. It is interesting that this predicted early Universe in theHEL model using the cosmological data related with the present day Universe is in good agreement with ourconventional expectations on the early Universe, viz., it should have been dominated by radiation ( w ∼ ) andexpanding with a DP q ∼
1. Hence, we first discuss the early Universe prediction in the HEL model which canbe done through the BBN processes that occur at redshift range z ∼ − (when the temperature rangesfrom T ∼ T ∼ . t ∼ t ∼ He mass ratio Y p ≡ n n n + n p ≈ n n n n + n p (here n He , n n and n p are the number densities of the neutrons,protons and He respectively) is a very useful tool for studying the expansion rate of the Universe at the timeof BBN, since it is very sensitive to temperature and hence to the expansion rate of the Universe at the timeneutron-proton ratio freezes-out. In the standard BBN (SBBN) for which it is assumed that the standardmodel of particle physics is valid (i.e., there are three families of neutrinos N ν ≈
3) and that the effective EoSof the physical content of the Universe during that time interval can be described by p = ρ/
3, which givesthe expansion rate of the Universe as H SBBN = . t through the Friedmann equations. We can utilize a goodapproximation for a primordial He mass fraction in the range 0 . . Y p . .
27 given by Steigman [64, 65] topredict Y p values for non-standard expansion rates during BBN. Accordingly, if the assumption of the SBBNmodel expansion rate is relaxed, both BBN and CMB will be affected and the approximation to Y p in this case s given as follows: Y p = 0 . ± . . η −
6) + 100 ( S − . (30)where S = H/H
SBBN is the ratio of the expansion rate to the standard expansion rate and η = 10 n B /n γ isthe ratio of baryons to photons in a comoving volume. We can safely ignore the term η − η ∼ S −
1. One may check that in the HEL model H z ∼ ∼ = H z →∞ → αt , (31)is a very good approximation. Hence, (30) can safely be written as Y p = 0 . ± . .
16 (2 α − , (32)for the HEL model. Note that S = 2 α and SBBN is recovered provided that S = 2 α = 1. Using this equationwith the value α = 0 . +0 . − . (1 σ ) from Table 2, that is obtained using the H ( z )+SNe Ia data, we find thatthe predicted He abundance in the HEL model is Y p = 0 . ± . σ ) . (33)We note that this value covers both the SBBN value prediction Y SBBNp = 0 . ± . Y p = 0 . ± . τ n ∼
887 s)when the deuterium bottleneck would be broken, i.e., the CMB temperature drops down to T ∼
80 keV.Otherwise the BBN model would not work properly and then our prediction given in (33) would not be valid.This can be done using the standard relation between the CBR temperature T and the scale factor a of theUniverse in the HEL model: aa = T ηT = (cid:18) tt (cid:19) α e β (cid:16) tt − (cid:17) , (34)where η stands for any non-adiabatic expansion due to entropy production. In standard cosmology, the instan-taneous e ± annihilation is assumed at T = m e . The heating due to this annihilation is accounted by η where η = 1 for T < m e while η = (11 / / for T > m e . It is enough for us to check whether the time scales areconsistent and hence we simply consider η = 1. Now using age of the present Universe in HEL model fromTable 3 and the present temperature of the Universe T ∼ = 2 . × − eV ( T = 2 .
728 K) [67], we find that thetemperature T ≈
80 keV was reached when the age of the Universe was t T =80 keV = 3 . ± . . (35)We note that this value is less than τ n ∼
887 seconds and also very close to ∼ T = 80 keV in the conventional SBBN scenario.It is interesting that using the cosmological data related with the expansion rate of the present day Universe(spanning in the redshift range 0 . < z < . z ∼ − ) with a great success. This shows that although the HEL model fits the H ( z )+SNe Iadata with a slightly less success compared to ΛCDM model, it has an advantage of describing the history of theUniverse starting from the BBN epoch to the present day Universe whereas ΛCDM as well as many other darkenergy models can describe the Universe only starting from the dust dominated epoch of the Universe. BAO observations provide a completely independent way from the supernova observations for investigating theexpansion properties of the universe at low redshift values and give us the opportunity to compare and test thepredictions of cosmological models at different redshift values. The imprint of the primordial baryon-photonacoustic oscillations in the matter power spectrum provides a standard ruler via the dimensionless quantity[68, 69] A ( z ) = p Ω m [ H ( z ) /H ] − / (cid:20) z Z z H H ( z ) d z (cid:21) / . (36) he BAO data set from the current surveys 6dFGS [70], SDSS [71], and WiggleZ [72] spanning in the redshiftrange 0 . < z < .
73, is shown in Table 5, where we also give the predicted A HEL ( z ) and A ΛCDM ( z ) for theHEL and ΛCDM models respectively using the values for the parameters that we constrained using H ( z )+SNIa data (see Table 2). We observe that the predicted A HEL ( z ) and A ΛCDM ( z ) values are consistent with eachother as well as with the values from the BAO surveys. Table 5: The BAO data points from different surveys and their comparison with the A ( z ) valuespredicted in HEL and ΛCDM models in our study. z A ( z ) from survey A HEL ( z ) A ΛCDM ( z )0.106 0 . ± .
028 (6dFGS) 0 . ± .
001 0 . ± . . ± .
016 (SDSS) 0 . ± .
002 0 . ± . . ± .
016 (SDSS) 0 . ± .
003 0 . ± . . ± .
034 (WiggleZ) 0 . ± .
003 0 . ± . . ± .
020 (WiggleZ) 0 . ± .
003 0 . ± . . ± .
021 (WiggleZ) 0 . ± .
004 0 . ± . It is well known that the Universe should have transited from radiation- to matter-dominated era at z ∼ z ∼ w ≃
0. This process should be achieved properly in a realistic description of thehistory of the Universe. A plot of the evolution of the effective EoS parameter in terms of redshift may be usefulfor a discussion in this respect. We plot the effective EoS parameters of HEL model (green curves) and ΛCDMmodel (red curves) versus redshift for 0 < z < in Figure 3. The solid green and red curves correspond tothe mean values of the EoS parameters while the shaded regions between the dotted curves are 1 σ error regions.We note first that there is, at high redshift values, a broad error region in HEL model but almost negligible - - w ( z ) zw = w = − Figure 3:
The effective EoS parameters of HEL model (Green curves) and ΛCDM model (Red curves) are shownversus redshift with logarithmic scale on z-axis. The solid Green and Red curves correspond to the mean values of theEoS parameters while the shaded regions between the dotted curves are 1 σ error regions. The Pink and Blue coloredhorizontal lines stand for w = and w = − respectively. error region in the ΛCDM model. In the later model, the error region shrinks as z increases since the matterdomination is the only possible past in this model. In the HEL model, on the other hand, the error regionbroadens as z increases because error of the parameter α (determines essentially the early Universe) is largerthan that of the parameter β (determines essentially the late Universe) (see Table 2 and eqns. (5) and (6)). e note that the mean value of the effective EoS parameter vanishes at z ∼ . ∼ for the red-shift values higher than z ∼
12. This is obviously not in favor of the HEL model. Onthe other hand, within 1 σ error region, it excludes neither a start with matter dominated era nor a start withradiation dominated era at z ∼ α from H ( z )+SN Ia data. We would evade this error completely by setting α = so that w ∼ = at z ∼ − in accordance with the SBBN. However, in this case, although the HEL model could describe thetimes of the BBN ( z ∼ − ) as well as the present times z ∼ R = p Ω m Z z dec H H ( z ) d z, (37)where Ω m is the usual matter density parameter at the present time Universe, and the integral term is thecomoving distance of the redshift z dec at decoupling in a spatially flat Universe. This parameter describes thescaled distance to recombination, and is a useful tool for constraining and comparing models, which do notdeviate lot from ΛCDM model [73, 74, 69]. We adopt z dec = 1090 for consistency with the latest observations(e.g., Planck experiment [75]) in our calculations. Using Ω m = 0 . +0 . − . (the value obtained from the H ( z )+SNIa data in our study and consistent with Planck experiment), we find R ΛCDM = 1 . ± .
07 for the ΛCDMmodel. In HEL model, we do not have explicit contribution from matter but we make use of the flat valueΩ m = 0 .
29 since the ΛCDM and HEL models behave very close to each other at z ∼ α and β from Table 2, we find R HEL = 1 . ± .
71. Theshift parameter we found for the ΛCDM model is consistent with the measured value R Planck = 1 . ± . α .In the following section, we summarize the work done in this paper and then conclude it by discussing thepossible directions for improving the HEL model in the light of the investigations done in the current section. We have examined the hybrid form of scale factor, namely, a product of power law and an exponential function,which provides a simple mechanism of transition from decelerating to accelerating phase. We showed thatsuch an expansion history for the Universe can be obtained in the presence of dust for the particular case ofBrans-Dicke theory of gravity. We also carried out the effective fluid and the single scalar field reconstructionusing quintessence, tachyon and phantom fields, which can capture HEL in the framework of general relativity.We constrained the parameters of HEL model using the 25 + 580 data points from the latest H ( z ) and SN Iacompilations spanning in the redshift range 0 . < z < . z ∼ − , where BBN proccesses areexpected to occur. We find that the HEL model predicts the He abundance and time scale of the energy scalesof the BBN processes with a great success. It is indeed interesting that the model is consistent with nucleo-synthesis which tells us that the simple expansion law under consideration can successfully describe thermalhistory as well as the late time transition to accelerating phase. The HEL model successfully passes the BAOtest.We conclude that the HEL and ΛCDM models are indistinguishable at low red-shift values. Also, the HELmodel is good at describing the early radiation dominated era and the current accelerating phase of the Universe t the same time. However, from the CMB test we find that it does not accommodate the matter-dominatedera properly unless we consider the parameter α with its large errors. Thus, with the current form of HEL,one should choose either to start the model with radiation domination that faces inconsistencies related withmatter domination or start the model with matter domination at the expense of probing back to the radiationdomination. The second case is not interesting since we know that ΛCDM model is doing pretty well. If onepursues the first option, one then needs to improve the model by modifying HEL ansatz such that a correctionto it would cure the issues related with the intermediate matter-dominated era. In this respect two differentroutes may be followed: (a) One can use two power laws multiplying with an exponential term. This willbring additional free parameters, which is not fine as we have seen in AIC, KIC and BIC analysis in Section 5.However, one can choose, for instance, one of the powers such that the Universe will be dominated by radiationin the early times (minutes time scale). (b) One can use the potentials we obtained for a single scalar fielddriving the HEL expansion for describing the dark energy component in a cosmological model, where the wellknown components of the universe, such as matter and radiation, are given explicitly. In this case, the presenceof matter and/or radiation in addition to the dark energy source described by the potential will give rise to adeviation from HEL ansatz. The state of our understanding of current acceleration of the Universe argues forkeeping an open mind. Obtaining new forms of potential for describing dark energy source through the scalefactor with some interesting properties like the HEL ansatz may provide us an opportunity to generate newclasses of solutions that may fit the cosmological observations successfully. Acknowledgments
We thank to J.D. Barrow, P. Diego and S.Y. Vernov for fruitful comments on the paper. ¨O.A. acknowledgesthe postdoctoral research scholarship he is receiving from The Scientific and Technological Research Council ofTurkey (T ¨UB˙ITAK-B˙IDEB 2218). ¨O.A. appreciates also the support from Ko¸c University. S.K. is supported bythe Department of Science and Technology, India under project No. SR/FTP/PS-102/2011. M.S. is supportedby the Department of Science and Technology, India under project No. SR/S2/HEP-002/2008. L.X.’s workis supported in part by NSFC under the Grant No. 11275035 and “the Fundamental Research Funds for theCentral Universities” under the Grant No. DUT13LK01. The authors are thankful to the anonymous refereefor critical and fruitful comments on the paper.
A The scalar field dynamics of Λ CDM model
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