An FLRW interacting dark energy model of the Universe
aa r X i v : . [ phy s i c s . g e n - ph ] F e b An FLRW interacting dark energy model of the Universe
Anirudh Pradhan , G. K. Goswami , A. Beesham , Archana Dixit Department of Mathematics, Institute of Applied Sciences and Humanities, G L A University,Mathura-281 406, Uttar Pradesh, IndiaE-mail: [email protected] Department of Mathematics, Kalyan P G College, Bhilai-490006, IndiaEmail: [email protected] Department of Mathematical Sciences, University of Zululand, Kwa-Dlangezwa 3886, South AfricaE-mail: [email protected] Department of Mathematics, Institute of Applied Sciences and Humanities, G L A University,Mathura-281 406, Uttar Pradesh, IndiaE-mail: [email protected] Astronomy 78 (2020) 101368
Abstract
In this paper, we have presented an FLRW universe containing two-fluids (baryonic and dark energy),with a deceleration parameter (DP) having a transition from past decelerating to the present acceleratinguniverse. In this model, dark energy (DE) interacts with dust to produce a new law for the density. Asper our model, our universe is at present in a phantom phase after passing through a quintessence phasein the past. The physical importance of the two-fluid scenario is described in various aspects. The modelis shown to satisfy current observational constraints such as recent Planck results. Various cosmologicalparameters relating to the history of the universe have been investigated.
PACS No. : 98.80.Jk; 95.36.+x; 98.80.-k
Keywords : FLRW universe; Observational parameters; Phantom; Quintessence.
A cosmological model must satisfy the basic cosmological principle (CP) which says that at any time the uni-verse is spatially homogeneous and isotropic. There is no privileged position in the universe. The Friedmann-Lemaitre-Robertson-Walker (FLRW) model satisfies the CP. This was manifest in an expanding and deceler-ating universe filled with a perfect fluid. However the latest findings on observational grounds during the lastthree decades by various cosmological missions [1] − [17] confirm that universe has an accelerating expansionat present. It is believed that there is a bizarre form of dark energy (DE) with negative pressure prevailingall over the universe which is responsible for the said acceleration. In Λ CDM cosmology [18, 19], the Λ-termis used as a candidate of DE with equation of state p Λ = − ρ Λ = − Λ c πG . However, the model suffers from, interalia, fine tuning and cosmic coincidence problems [20]. Any acceptable cosmological model must explain theaccelerating universe.As of now, many models and theories such as quintessence, phantom, k -essence, holographic DE models, f ( R ) and f ( R, T ) theories have been proposed to explain the acceleration in the universe. One may refer tothe review article [18] for a brief introduction to these models and theories.Of late, many authors [21] − [27] presented DE models in which the DE is considered in a conventionalmanner as a fluid with an EoS parameter ω de = p de ρ de . It is assumed that our universe is filled with two typesof perfect fluids of which one is a baryonic fluid (BF) which has positive pressure and creates deceleration inthe universe. The other is a DE fluid which has negative pressure and creates acceleration in the universe.Both fluids have different EoS parameters. The EoS for baryonic matter has been solved by cosmologists1y providing the phases of the universe like stiff matter, radiation dominated and present dust dominateduniverse, but the determination of the EoS for DE is an important problem in observational cosmology atpresent. The present value of ω de is observationally estimated nearly equal to −
1. In the quintessence model, − ≤ ω de < ω de ≤ −
1. Latest surveys [28] − [32] rule out the possibility of ω de ≪ −
1, but ω de may be little less than −
1. But we are facing fine tuning and coincidence problems [33].So we need a dynamical DE with an effective EoS, ω ( de ) = p ( de ) /ρ ( de ) < − /
3. The two types of surveysSDSS and WMAP [9] and [34] provide limits on ω ( de ) as − . < ω de < − .
62 and − . < ω de < − . − [39] proposed that DE may interact withBF, so they have developed both types of interacting and non-interacting models of the universe. Recently ithas been discovered that allowing and interaction between DE and dark matter(DM) offers an attractive al-ternative to the standard model of the cosmology [40, 41]. In these works the motivation to study interactingDE model arises from high energy physics. In recent work Risalti and Lusso [42] and Riess et al [43] statedthat a rigid Λ is ruled out by 4 σ and allowing for running vacuum favored phantom type DE ( ω < −
1) andΛ CDM is claimed to be ruled out by 4 . σ motivating the study of interacting DE models. Interacting DEmodels [44] − [51] lead to the idea that DE and DM do not evolve separately but interact with each othernon gravitationally (see recent review [52] and references there in.).Motivated from above discussion, in this paper, we have presented an FLRW universe containing two-fluids(baryonic and dark energy), with a deceleration parameter (DP) having a transition from past deceleratingto the present accelerating universe As per our model, universe is at present in a phantom phase after passingthrough a quintessence phase in the past. The model is shown to satisfy current observational constraintssuch as Planck’s latest observational results [17]. Various cosmological parameters relating to the history ofthe universe have been investigated.Our paper is structured as follows: In Sec. 2, we set the initial field equations. In Sec. 3, we havedescribed the results and physical properties of interacting DE model. Finally, Sec. 4 is devoted to ourconclusions. The FLRW space-time (in units c = 1) is given by ds = dt − a ( t ) (cid:20) dr (1 + kr ) + r ( dθ + sin θdφ ) (cid:21) , (1)where a ( t ) stands for the scale factor and k is the curvature parameter.The stress-energy tensor T ij = T ij ( m ) + T ij ( de ) , where T ij ( m ) = ( ρ m + p m ) u i u j − p m g ij and T ij ( de ) =( ρ de + p de ) u i u j − p de g ij . We assume that DE interacts with and transforms energy to baryonic matter. Wefollow arXiv:1905.10801 and 1906.00450 to get Einstein field equations (EFEs) for the FLRW metric (1) areas follows. H (1 − Ω de ) = H (cid:20) (Ω m ) (cid:16) a a (cid:17) − σ ) + (Ω k ) (cid:16) a a (cid:17) (cid:21) , (2)and 2 q = 1 + 3 ω de Ω de + 3 H H ω k (Ω k ) (cid:16) a a (cid:17) , (3)where symbols have their usual meanings. In the above, we have found two field equations (2) and (3) in five unknown variables a, H, q, Ω de and ω de .Therefore, for a complete solution, we need three more relations involving these variables. Many researchers[53] − [55] have considered constant DP which is not valid from present observations. The DP q may be takenas time dependent as supported by many observations like SN Ia [5, 6, 28] and CMB anisotropies [7, 8]. From2hese observations, we observe that z < . z > . z t = 0 . ± .
07 by (1 σ ) c.1. [8] from z t = 0 . ± . σ ) c.1. [28] as of late found by the High-Z Supernova Search (HZSNS) group. The Supernova LegacySurvey (SNLS) [29], and additionally the one as of late incorporated by Knop et al [33], yields z t ∼ . σ )in better concurrence with the flat ΛCDM model ( z t = (2Ω Λ / Ω m ) − ∼ . − [60]. From thesediscussions, q may not be taken as a constant, but it should be time-dependent. Recently, many researchers[35] − [39] & [60] − [63] have used the time-dependent DP for solving various cosmological problems. So weconsider q as a linear function of the Hubble function parameter which was earlier used by [64] − [66] indifferent context of cosmological models. q = βH + α (4)Here α , and β are arbitrary constants and unit of β is Gyr as H is expressed in Gyr − and q is dimensionless quantity.From above equation, we have a ¨ a ˙ a + β ˙ aa + α = 0, which on solving, yields a = exp (cid:20) − (1 + α ) β t − α ) + lβ (cid:21) , provided α = − . Here l is a constant of integration.From this, we calculate ˙ a = − (cid:18) αβ (cid:19) exp (cid:20) − (cid:18) αβ (cid:19) t − α ) + lβ (cid:21) , ¨ a = (cid:18) αβ (cid:19) exp (cid:20) − (cid:18) αβ (cid:19) t − α ) + lβ (cid:21) . Putting above values in Eq. (4), we obtain the DP value as q = −
1. Similarly we also observed that q = − α = 0.For α = −
1, we have to find another solution. In this case Eq. (12) reduces to q = − a ¨ a ˙ a = − βH, which yields the following differential equation: a ¨ a ˙ a + β ˙ aa − . The solution of above equation is found to be a = exp (cid:20) β p βt + k (cid:21) , (5)where k is an integrating constant.Since we are interested to study the cosmic decelerated-accelerated transit universe, so we only considerthe later case for which α = − β and k on the basisof the latest observational findings due to Planck [17]. The values of the cosmological parameters at presentare as follows. (Ω m ) = 0 .
30 (Ω k ) = ± . , ( ω de ) = − , (Ω de ) = 0 . ± . , H = 0 . Gyr − q ≃− . , t = 13 . Gyr.
Eq. ( ?? ) provides following differential equation(1 + z ) H z = βH = H (1 + q ) = β βt + k (6)where we have used a a = 1 + z, ˙ z = − (1 + z ) H and H z = dHdz . From Eq. (6) and the Planck results, we getthe value of constants β and k as k = 27 . Gyr , β = 6 . Gyr (7)3ntegrating Eq. (6), we get H − = A − βlog (1 + z ), where A is constant of integration. As H = 0 . A = 100 / . So, we get following solution H = 7100 − log (1 + z ) Gyr − , q = 45100 − log ( z + 1) − . (8) (i) Hubble function H : The determination of the two physical quantities H and q plays an important role to describe the evolutionof the universe. H provides us the rate of expansion of the universe which in turn helps in estimating theage of the universe, whereas the DP q describes the decelerating or accelerating phases during the evolutionof the universe. From the last two decades, many attempts have been made to estimate the value of theHubble function [27], [67] − [69]. For detailed discussions, readers are referred to Kumar [27]. We present thefollowing figures 1, 2 & 3 to illustrate the solution Eq. (8) . Various researchers [15, 16], [70] − [76] have z HG y r - timetGyr re d - s h i ft z Acceleration Deceleration z t = 2.395 - z q Figures 1,2 and 3: Plot of Hubble function ( H ) versus red shift ( z )(left),Variation of ( z )versus ( t )(middle) and Variation of q with z (right) estimated values of the Hubble function at different red-shifts using a differential age approach and galaxyclustering method [see [76] for list of 38 Hubble function parameters ]. We obtain χ from the followingformula χ = i =38 X i =1 [( Hth ( i ) − Hob ( i )) /σ ( i ) ] , where Hth (i)’s are theoretical values of Hubble function parameter as per Eq. (8) and σ ( i )’s are errors inthe observed values of H ( z ). It comes to χ = 33 .
22 i.e. 87 .
43 over 38 data’s, which shows best fit in theoryand observation. From figure 1, we observe that H increases with the increase of red shift. In this figure,cross signs are 31 observed values of the Hubble function H ob with corrections, whereas the linear curve isthe theoretical graph of the Hubble function H as per our model. Figure 2 plots the variation of red shift z with time t , which shows that in the early universe the red shift was more than at present. (ii) Transition from deceleration to acceleration: Now we can obtain the DP ‘ q ’ in term of red shift ‘ z ’ by using Eq.(8). We present figure 3 to illustrate thesolution. This describes the phase variation of the universe from deceleration to acceleration. We see thatat present our universe is undergoing an accelerating phase. It has begun at the transit red shift z t = 2 . T t = 1 .
034 Giga year. It was decelerating before time T t (iii) DE Parameter Ω de and EoS ω de Now, from Eqs. (2), (3) and energy conservation equations, the density parameter Ω de and EoS parameter ω de for DE are given by the following equations and are solved numerically. H Ω de = H − (Ω m ) H (1 + z ) − σ ) (9) ω de = H (2 αH + 2 β − H − H (Ω m ) (1 + z ) − σ ) ] . (10)where we have taken (Ω k ) = 0 for the present spatially flat universe. We would take σ = 0 .
243 for numericalsolutions to match with latest observations. We solve Eqs. (9) and (10) with the help of Eq. (8) and present4 z W d e Quintessence EraDeceleration EraAcceleration Era - - - - - z Ω d e Figures 4 and 5: Plot of Ω de versus red shift ( z )(left) and plot of ω de versus z (right).Phantom phase (0 ≤ z ≤ . . ≤ z ≤ .
74 and decelerationphase z ≥ . the following figures 3 and 4 to illustrate the solution.Our model envisages that at present we are living in a phantom phase ω ( de ) ≤ −
1. In the past at z = 2 . ω ( de ) = − . ≤ z ≤ . z = 3 .
665 where ω de comes up to − . . ≤ z ≤ . . DE favors deceleration at z ≥ . z tr = 2 . ≤ z ≤ . z ≥ . ω de have no physical rolls. We may say that the validity of Figs.4 and 5 is only during the said tenure. During this DE always increases with time. As per our model, thepresent ratio of DE is 0 .
7. It decreases over the past, attains a minimum value Ω de = 0 .
005 at z = 2 . (iv) Distance modulus µ and Apparent Magnitude m b : The distance modulus µ and apparent magnitude m b [18] are derived as µ = m b − M = 5 log (cid:18) D L M pc (cid:19) + 25 = 25 + 5 log (cid:20) c (1 + z ) H Z z dzh ( z ) (cid:21) (11) m b = 16 .
08 + 5 log (cid:20) z. Z z dzh ( z ) (cid:21) . (12)We solve Eqs. (11) − (12) with the help of Eq. (8). Our theoretical results have been compared with SNe Iarelated union 2 . χ from the following formula χ = LengthSN aData X i =1 µ th ( i ) − µ obs ( i )) σSN a ( i ) where µ th (i)’s are theoretical values of distance modulus as per Eq. (12) and σSN a ( i )’s are errors in theobserved values of µ . It comes to χ = 562.227 i.e. 96.7% over 581 data’s, which shows best fit in theory andobservation. 5 .0 0.2 0.4 0.6 0.8 1.0 1.2 1.4343638404244 z Μ Figure 6: Plot of distance modulus ( µ = M − m b ) versus red shift ( z ). Crosses are SNe Iarelated union 2 . In the present paper, we have presented an FLRW universe filled with two fluids (baryonic and dark energy),by assuming a scale factor as a linear function of the Hubble function . This results in a time-dependentDP having a transition from past decelerating to the present accelerating universe. The main findings of ourmodel are itemized point-wise as follows. • The expansion of the universe is governed by a expansion law a ( t ) = ( βH −
1) = exp √ βt + kβ , where β = 6 . Gyr and k = 27 . Gyr . This describes the transition from deceleration to acceleration. • Our model is based on the recent observational findings due to the Planck results [17]. The modelagrees with present cosmological parameters.(Ω m ) = 0.30 (Ω k ) = ± . ω de ) = −
1, (Ω de ) = 0 . ± . H = 0 .
07 Gy − , q = 0 .
055 andpresent age t = 13 .
72 Gy. • At present our universe is undergoing an accelerating phase. It has begun at the transit red shift z t = 2 . T t = 1 .
034 Gigayear. It was decelerating before time T t • Our model has a variable EOS ω de for the DE density. Our model envisages that at present we areliving in the phantom phase ω ( de ) ≤ −
1. In the past at z = 2 . ω ( de ) = − . ≤ z ≤ . z = 3 .
665 where ω de comes up to − . . ≤ z ≤ . . DE favors deceleration at z ≥ . de = 0 .
005 at z = 2 . • The DE interacts with dust matter in our model, giving rise to a new density law for dust as ρ m =( ρ m ) (cid:0) a a (cid:1) − σ ) , where σ is a constant which has been assigned the value 0 .
243 to match with obser-vations.In a nutshell, we believe that our study will pave the way to more research in future, in particular, in thearea of the early universe, inflation and galaxy formation, etc. The proposed hybrid expansion law may helpin investigations of hidden matter like dark matter, dark energy and black holes.
Acknowledgement
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