A Parametric Reconstruction of the Deceleration Parameter
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Eur. Phys. J. C (2017) 77:495 DOI 10.1140/epjc/s10052-017-5066-4 Regular Article - Theoretical Physics A parametric reconstruction of the deceleration parameter Abdulla Al Mamon1,2,a, Sudipta Das2,b 1 Manipal Centre for Natural Sciences, Manipal University, Manipal 576104, India 2 Department of Physics, Visva-Bharati, Santiniketan 731235, India Received: 25 March 2017 / Accepted: 12 July 2017 / Published online: 25 July 2017 © The Author(s) 2017. This article is an open access publication Abstract The present work is based on a parametric recon- known as the “fine tuning” problem [3], “coincidence” prob- struction of the deceleration parameter q(z) in a model for lem [4] and the “age” problem [5]. To overcome these issues, the spatially flat FRW universe filled with dark energy and it is quite natural to think for some alternative possibilities non-relativistic matter. In cosmology, the parametric recon- to explain the origin and nature of DE. Models with scalar struction technique deals with an attempt to build up a model fields (both the canonical and non-canonical scalar field) play by choosing some specific evolution scenario for a cosmo- a major role in current description of the evolution of the logical parameter and then estimate the values of the param- universe. Motivated by the scalar field theories, over the last eters with the help of different observational datasets. In this decade, numerous DE models were explored, which include paper, we have proposed a logarithmic parametrization of quintessence, K-essence, phantom, tachyon, chaplygin gas q(z) to probe the evolution history of the universe. Using the and so on (for review see [6] and the references therein). type Ia supernova, baryon acoustic oscillation and the cosmic However, we do not yet have a concrete and satisfactory DE microwave background datasets, the constraints on the arbi- model. trary model parameters q0 and q1 are obtained (within 1σ As mentioned before, the cosmological observations also and 2σ confidence limits) by χ 2-minimization technique. indicate that the observed cosmic acceleration is a recent phe- We have then reconstructed the deceleration parameter, the nomenon. So, in the absence of DE or when its effect is sub- total EoS parameter ωtot, the jerk parameter and have com- dominant, the same model should have decelerated phase in pared the reconstructed results of q(z) with other well-known the early epoch of matter era to allow the formation of struc- parametrizations of q(z). We have also shown that two model ture (as gravity holds matter together) in the universe. For this selection criteria (namely, the Akaike information criterion reason, a cosmological model requires both a decelerated and and Bayesian information criterion) provide a clear indica- an accelerated phase of expansion to describe the whole evo- tion that our reconstructed model is well consistent with other lution history of the universe. In this context, the deceleration popular models. parameter plays an important role, which is defined as aa¨ q =− (1) 1 Introduction a˙2 where a(t) is the scale factor of the universe. For an acceler- Recent observations strongly suggest that the universe is ating universe, a¨ > 0, i.e., q < 0 and vice versa. The most undergoing an accelerated expansion in the present epoch popular way of achieving such a scenario is to consider a [1,2]. The matter content responsible for such a certain stage parametrization for the deceleration parameter as a function of evolution of the universe is popularly referred to as “dark of the scale factor (a) or the redshift (z) or the cosmic time (t) energy” (DE), which accounts for almost 70% of the current (see Refs. [7–19]). It should be noted that, for most of these energy budget of the universe. In this regard, various DE mod- parametrizations, the q-parametrization diverges in the far els have been proposed to match with recent observed data future and others are valid at low redshift (i.e., z << 1) [10– and the CDM model is the most simplest one in this series. 15]. Also such parametric methods may be misleading as to But this model suffers from some other problems which are the true nature of dark energy due to the assumed paramet- a e-mails: [email protected]; ric form. The non-parametric method is more advantageous [email protected] than the parametric method in the literature, since it avoids b e-mail: [email protected] parametrizing cosmological quantities and finds the evolu- 123 495 Page 2 of 9 Eur. Phys. J. C (2017) 77 :495 tion of our universe directly from observational data [20– on these model parameters using various datasets and present 24]. However, such approaches also have drawbacks [25]. our results. Finally, in Sect. 4, there is a brief conclusion as Till now, there is no well-motivated theoretical model of the regards the results obtained in this work. universe which can describe the entire evolution history of the universe. So, it is reasonable to adopt a parametric approach to measure the transition from a decelerating to an accelerat- 2 Basic equations and solutions ing phase of the universe. In addition to this, the parametric approach also helps to improve efficiency of the future cos- Let us consider a spatially flat, homogeneous and isotropic mological surveys. Motivated by these facts, in the present FRW universe work, we have chosen a special form of deceleration param- ( ) eter in such a way that q z will provide the desired property ds2 = dt2 − a2(t)[dr 2 + r 2d2] (4) for sign flip from a decelerating to an accelerating phase. The properties of this parametrization are mentioned in Sect. 2. composed of two perfect fluids, namely ordinary matter with The constraints on the model parameters of our toy model negligible pressure and canonical scalar field (as a candidate have also been obtained using the SNIa, BAO and CMB of dark energy). In this case, the Einstein field equations datasets. We have also reconstructed the redshift evolutions become (choosing 8πG = c = 1) of q(z) and the total EoS parameter and have compared their evolution behaviors with other well-known models, such as 2 3H = ρ + ρφ, (5) ∝ ∝ z m q z [8], q 1+z [11–15] and CDM, to study the dif- ferent properties of this model. We have shown that for this ˙ 2 2H + 3H =−pφ, (6) choice of q(z), the present model describes the evolution of the universe from an early decelerated phase (where mat- = a˙ ρ where H a is the Hubble parameter and m is the matter ter dominates over DE) to an accelerated phase (where DE energy density. Here, ρφ and pφ are the contributions of the dominates over matter) at the current epoch for each dataset. scalar field to the energy density and pressure, respectively, Another important kinematical quantity, related to the and are given by expansion of the universe, is the jerk parameter (the dimen- ( ) 1 ˙2 sionless third derivative of the scale factor a t with respect ρφ = φ + V (φ), (7) to cosmic time t), which is defined as [26–29] 2 1 ˙2 pφ = φ − V (φ), (8) d3a 2 t3 j = d , (2) where an overhead dot denotes a derivative with respect to aH3 the cosmic time t and V (φ) is the potential associated with and in terms of q the scalar field φ. Also, the conservation equations for the scalar field and dq matter field read j = q(2q + 1) + (1 + z) , (3) dz ρ˙φ + 3H(ρφ + pφ) = 0, (9) which will be useful when the parametric form of q(z) is given. A powerful feature of the jerk parameter is that for ρ˙m + 3Hρm = 0. (10) the standard CDM model j = 1 (constant) always, which provides a simple test for departure from the CDM sce- Now one can easily solve the above equation to find the nario. It deserves mention here that Sahni et al. [30] and energy density for the normal matter as Alam et al. [31] drew attention to the importance of j for 3 discriminating various models of dark energy, because any ρm = ρm0(1 + z) (11) deviation from 1 in the value of j would favor a dynami- cal dark energy model, instead of the CDM model. In this where ρm0 is an integration constant (which represents the paper, we have also investigated the evolution of j for the matter energy density at the present epoch) and z is the red- = 1 − present parametrized model. shift parameter, defined as z a 1. The present paper is organized as follows. The basic equa- The corresponding equation of state (EoS) parameter is tions for the phenomenological DE model are presented in now given by the next section. We then try to obtain some accelerating ˙ solutions for this toy model using a specific parametrization pφ (2H + 3H 2) ωφ = =− , ( ) 2 (12) of q z . In Sect. 3, we obtain the observational constraints ρφ (3H − ρm) 123 Eur. Phys. J. C (2017) 77 :495 Page 3 of 9 495 q − q κ, for z →+∞(early epoch), which further leads to q(z) = 0 1 q0 + q1(ln N − κ), for z = 0 (present epoch). 2q − 1 ωφ = (18) 2 (13) − ( + z)3 H0 3 3 m0 1 H Clearly, one can realize the history of the cosmic evolu- tion with this new parametrization. Similarly, at low redshift ρ = m0 where m0 2 is the matter density parameter at the (z << 1), the form of q(z) turns out to be 3H0 present epoch. ln N z From Eqs.