A flat FRW model with dynamical Λ as function of matter and geometry
aa r X i v : . [ phy s i c s . g e n - ph ] O c t A flat FRW model with dynamical Λ as function of matter and geometry Anirudh Pradhan , De Avik , Tee How Loo , Dinesh Chandra Maurya Department of Mathematics, Institute of Applied Sciences & Humanities, GLA University,Mathura -281 406, Uttar Pradesh, India Department of Mathematical and Actuarial Sciences, Universiti Tunku Abdul Rahman, Jalan SungaiLong, 43000 heras, Malaysia Institute of Mathematical Sciences, University of Malaya, 50603 Kuala Lumpur, Malaysia Department of Mathematics, Faculty of Engineering & Technology, IASE (Deemed to be University),Sardarshahar-331 403, Rajsthan, India Email:[email protected] Email:[email protected] Email:[email protected] Email:[email protected]
Abstract
We revisit the evolution of the scale factor in a flat FRW spacetime with a new generalized decay rulefor the dynamic Λ-term under modified theories of gravity. It analyses certain cosmological parameters andexamines their behaviours in this generalized setting which includes several decay laws in the literature.We have also obtained observational constraints on various model parameters and estimated the presentvalues of cosmological parameters { Ω m , Ω Λ , q , t , ω } and have discussed with various observationalresults. Finite time past and future singularities in this model are also discussed. Keywords
FLRW universe, Modified gravity, Dynamical cosmological constant, Observational parame-tersPACS No.: 98.80.Jk; 98.80.-k; 04.50.-h
The idea of the cosmological constant Λ was first introduced by Einstein in 1917 to obtain a static universebecause of Einstein’s field equations in general relativity were showing a non-static universe i.e. the universehas either contracting or expanding nature and at that time observational studies supported an static universedue to presence limited observational data sets. Therefore, he revised his original field equations proposed in1915 to remove this feature of size-changing. But the discovery of Hubble in 1929 suggested that our universeis in expanding nature which was already obtained by the Friedmann-Lamatire-Rabortson-Walker in 1922in their solution of Einstein’s field equations using the same metric and this finding was once removed thenewly added cosmological term Λ.After the discovery of accelerating universe in 1998, the cosmological term came back in its place butat this time it did not prevent the expanding nature of universe, but in order to explain why it did notcontinues expanding nature with a constant rate. Although it should be possible for the known amount ofmatter present in the universe to exert adequate gravitational force to slow down the expansion, initiatedwith Big Bang. However, modern studies of standard candles showed that instead of decreasing, the rate atwhich it is expanding actually increases. The new generation of cosmologists proposes the existence of an1nknown, exotic, homogeneous energy density with high negative pressure referred to as dark energy (DE) toexplain these observational findings. And the idea of cosmological term Λ is the simplest contributing factorof this dark energy.We still face the critical question of whether Λ is always a fundamental constant or a mild dynamical vari-able constant after almost two decades. It turns out that the selection of Λ as dark energy faces some seriousissues in terms of (a) the “old” cosmological constant problem (or fine-tuning problem) and (b) the cosmiccoincidence problem, if we consider it as a constant rather than a specific dynamical variable. Moreover,the overall fit to the cosmological observables SNIa+BAO+H(z)+LSS+BBN+CMB has also been shown tosupport the class of unique dynamic models of the so-called fundamental constant Λ as well. Dynamic modelof Λ may ultimately be necessary not only as a new epitome to improve the EFE, but also phenomenologi-cally to unwind a number of tensions between the concordance model and the observation that is indicatingtowards DE. There are innumerable Λ( t ) laws available in the literature although all of which might not beviable under theoretical and observational ground and they might not follow covariant action, but they arestill interesting to study from phenomenological ground. For example, Carvalho et al. [1] and Waga [2] firstconsidered the case Λ ∝ ( ˙ aa ) as a generalization of Chen and Wu’s result advocating the possibility that the(effective) cosmological constant Λ varies in time as a − . The case Λ ∝ ρ was first studied by Viswakarma [3]and the case Λ ∝ ¨ aa by Arbab [4]. Ray et al. [5] and Viswakarma [6] showed that if we consider these threecases separately, the free parameters satisfy certain relations. Recently, Pan [7] investigated dynamical Λ as afunction of several form of polynomials of H and ˙ H , discussing the finite time future singularities for each case.Although several studies being done on these particular phenomenological models in the past, with morepowerful observational analysis of the present days, most of these are detected to be suffering from certaintheoretical and observational limitations. The Λ ∝ H and Λ ∝ ˙ H cases are largely overruled now since thecorresponding linear growth of cosmic perturbations is strongly disfavored by the observational data [8], [9].If Λ ∝ ρ alone, one can trivially absorb it to the constant and hence there will be no dark energy scenario inthe cosmological evolution. This is indeed very clear for a perfect fluid with equation of state p = ωρ . Theintroduction of Λ ∝ ρ effectively modifies the proportionality constant ω and the model becomes equivalentto one with merely a new fluid only. We tried to overcome this issue by introducing a new decay law whichcombines both matter and geometry into the dynamical cosmological term. Moreover, we can show thata motivation towards the present decay law of Λ can be drawn from the f(R)-gravity or equivalently fromBrans-Dicke scalar-tensor theory derived from a covariant action.In cosmology, the fundamental variables are either constructed from a spacetime metric directly (geomet-rical), or depend upon properties of physical fields. Naturally the physical variables are model-dependent,whereas the geometrical variables are more universal. The most popular geometric variables are the Hubbleconstant H and the deceleration parameter q whose values depend on the first and second time derivativesof the scale factor, respectively. We use these two parameters extensively to resolve the mystery of the infla-tionary universe. However, a more sensitive discrimination of the expansion rate and hence dark energy canbe performed by considering a third time derivative in the general form for the scale factor of the Universe a ( t ) = a ( t ) + ˙ a | ( t − t ) + ¨ a | t − t ) + ... a | t − t ) + . . . This led Sahni et al. [10] and Alam et al. [11] to introduce the statefinder parameters, a dimensionless pair { r, s } from the scale factor and its third order time derivatives in concordance cosmology. In [12] this resultis generalized to show that the hierarchy, A n = a ( n ) aH n , can take into account the deceleration parameter, thestatefinder parameters, the snap parameter, the lerk parameter etc and further it can be expressed in termsof the deceleration parameter or the matter density parameter.In observational cosmology, the beginning of the 21st century saw yet another significant milestone.BOOMERanG in the year 2000 collaboration (Balloon Measurements of Millimetric Extragalactic Radiationand Geophysics), a research on cosmic microwave history (CMB) a cosmos of flat spatial geometry ([13], [14])was recorded using balloon-borne instruments. The MAXIMA (Millimeter Anisotropy Experiment ImagingArray) collaboration [15, 16], DASI (Degree Angular Scale Interferometer) [17], CBI (Cosmic Background2mager) [18] and WMAP (Wilkinson Microwave Anisotropy Probe) [19] have also reported a similar re-sult. These observations were an significant landmark in modern cosmology; assuming astronomy’s valueof Ω M ∼ . , the data directly pointed to a positive cosmological constant with a contribution of energydensity of the order of ΩΛ ∼ . . The observations from the supernova probes and the Hubble Space Tele-scope were ideally matched. The precise values of these parameters Ω M and Ω Λ as obtained by Sievers etal. [18] and Spergel et al. [20] are [0 . ± . , . +0 . − . ] and [0 . +0 . − . , . +0 . − . ], respectively. Morerecently, the Lyman- α forest measurement of the baryon acoustic oscillations (BAO) by the Baryon Oscilla-tion Spectroscopic Survey preferred an even smaller value of the matter density Ω M than that obtained bycosmic microwave background data [21]. We show that our model perfectly adapts to these observational data.The present paper is organized in the following format: a brief review of literature is given in section-1,section-2 contains field equations with cosmological term Λ and some specific cosmological solutions likescale factor a ( t ), Hubble function H ( t ), energy density parameters Ω m and Ω Λ . An observational constraintson various model parameters using available observational data sets like H ( z ), union 2.1 compilation ofSNe Ia data sets and Joint Light Curve Analysis (JLA) etc. by applying R -test formula are obtained anddiscussed with various observational results in section-3. Some finite time singularities of the current modelare discussed in the section-4 and in section-5 we have discussed some more cosmological parameters likestatefinder with deceleration parameters. Finally conclusions are given in section-6. Λ( t ) We consider an action [22] S = 116 πG Z f ( R ) √− gd x + Z L m √− gd x, where f ( R ) is an arbitrary function of the Ricci scalar R , L m is the matter Lagrangian density, and wedefine the stress-energy tensor of matter as T µν = − √− g δ ( √− gL m ) δg µν .Assuming that the Lagrangian density of matter Lm depends only on the components of the metric tensor gµν and not on its components, we obtain derivatives, T µν = g µν L m − ∂L m ∂g µν . By varying the action S of the gravitational field with respect to the metric tensor components g µν andusing the least action principle we obtain the field equation f R ( R ) R µν − f ( R ) g µν + ( g µν ✷ − ∇ µ ∇ ν ) f R ( R ) = 8 πG T µν , (1)where ✷ = ∇ = ∇ µ ∇ µ represents the d’Alembertian operator and f R = ∂f ( R ) ∂R . EFE can be reawakenedby putting f ( R ) = R . The contracted equation is given by Rf R ( R ) − f ( R ) + 3 ✷ f R ( R ) = 8 πGT, (2)which transforms (1) to R µν − R g µν + 16 f R ( R ) [2 kT + f ( R ) + Rf R ( R )] g µν = 8 πG T eff µν , (3) T eff µν = f R ( R ) [ T µν + π G ∇ µ ∇ ν f R ( R )].The Friedmann-Robertson-Walker (FRW) metric is given by the line element ds = − dt + a ( t ) (cid:20) dr − kr + r ( dθ + sin θdφ ) (cid:21) , (4)3here the curvature parameter k = − , , a ( t ) is the scale factor shaping the universe. Since current observations from CMB detectors such asBOOMERanG, MAXIMA, DASI, CBI and WMAP confirm a spatially flat universe, k = 0. Hence theRicci scalar becomes R = 6(( ˙ aa ) + ¨ aa ). Comparing the equation (3) with the Einstein’s field equations withcosmological constant R µν − R g µν + Λ g µν = 8 πGT µν , (5)we conclude that the dynamic cosmological term Λ( t ) is possibly a function of ¨ aa , ( ˙ aa ) and ρ and weconsider the simplest model as a linear combinationΛ = l ¨ aa + λ (cid:18) ˙ aa (cid:19) + 4 πGηρ, (6)where l , λ and δ are arbitrary constants.As usual, the Friedmann and Raychaudhuri equations for flat FRW line element are given by3 ¨ aa + 4 πG (1 + 3 w ) ρ = Λ , (7)¨ aa + 2 (cid:18) ˙ aa (cid:19) − πG (1 − w ) ρ = Λ , (8)where w = pρ .From Eqs. (7) and (8) we obtain (cid:18) ˙ aa (cid:19) − πGρ . (9)Using Eq. (6), in terms of Hubble parameter H = ˙ aa , Eq. (9) transforms into3 H − πGρ = l ˙ H + ( l + λ ) H + 4 πGηρ. (10)Therefore, 4 πGρ = − l η ˙ H + 3 − l − λ η H (11)and Λ = 2 l η ˙ H + 2 l + 2 λ + 3 η η H . (12)From Eq. (8) we obtain ˙ HH = − (1 + w )(3 − l − λ )(2 + η ) − l (1 + w ) , (13)which gives us H = H [(2 + η ) − l (1 + w )] H (1 + w )(3 − l − λ )( t − t ) + (2 + η ) − l (1 + w ) . (14)Integrating again, we obtain a ( t ) a = (cid:20) H (1 + w )(3 − l − λ )(2 + η ) − l (1 + w ) ( t − t ) (cid:21) (2+ η ) − l (1+ w )(1+ w )(3 − l − λ ) . (15)Therefore, the vacuum energy density parameter (Ω Λ = Λ3 H ) and the cosmic matter density parameter(Ω m = πGρ H ) are calculated as 4Ω Λ = 2 l + 2 λ + 3 η − l (1 + w )(2 + η ) − l (1 + w ) (16)and 3Ω m = 2 3 − l − λ (2 + η ) − l (1 + w ) , (17)respectively, which confirms Ω m + Ω Λ = 1 . The Hubble parameter H is one of the key parameter in observational cosmology as well as in theoreticalcosmology and it measures expansion rate of the universe. In our paper, we have obtained the Hubblefunction using Eqs. (14) & (15) as H ( t ) = H (cid:18) a a ( t ) (cid:19) (1+ ω )(3 − l − λ )2+ η − l (1+ ω ) (18)Several observational data sets are available in terms of redshift z and hence, to obtain constraints onvarious model parameters we have to convert the above relationship in terms of the redshift. Therefore, usingthe relation a a ( t ) = 1 + z between scale factor a ( t ) and redshift z , we obtain: H ( z ) = H (1 + z ) (1+ ω )(3 − l − λ )2+ η − l (1+ ω ) (19)We can also obtain the expression for luminosity distance D L and apparent magnitude m ( z ) in the purposeof observational constraints (Union 2.1 compilation of SNe Ia data sets and Joint light curve analysis datasets) respectively as D L = cH η − l (1 + ω )2 + η + ( λ − ω ) (1 + z ) η +( λ − l − ω )2+ η − l (1+ ω ) (20)and m ( z ) = 16 .
08 + 5 × log (cid:18) .
026 2 + η − l (1 + ω )2 + η + ( λ − ω ) (1 + z ) η +( λ − l − ω )2+ η − l (1+ ω ) (cid:19) (21)Now, we consider the 29 Hubble data sets H ( z ) [23]-[29], 580 data sets of apparent magnitude m ( z ) fromunion 2.1 compilation of SNe Ia data sets [30] and 51 data sets of m ( z ) from Joint Light Curve Analysis (JLA)data sets [31] and using the R -test formula, we obtain the best fit values of various model parameters l , λ , η , ω and H for the best fit curve of Hubble function H ( z ) and apparent magnitude m ( z ). The case R = 1means the curve of the Hubble function is exact match with observational curve and using this technique offitting we have obtained the constraints on model parameters for maximum R values. The best fit valuesof l , λ , η , ω and H for the various data sets are mentioned in below Table-1 & 2 with their maximum R value and root mean square error (RMSE).Using the above estimated values of various parameters (as mentioned in Table-1 & 2) we have calculatedthe values of matter energy density parameter Ω m , dark energy density parameter Ω Λ , deceleration parameter q , age of the present universe and statefinder parameters r , s for two types of universes one is dusty universeand second is the case of 0 < ω < which are mentioned in Table-3 & 4 given below.Table-3 & 4 shows the present values of various cosmological parameters { Ω m , Ω Λ , q , t , r , s } and wesee that for the deceleration in expansion of the universe dark energy parameter must be Ω Λ > . m < . q lies between − . ≤ q ≤ . ω = 0) H ( z ) SNe Ia JLA l .
435 0 . . λ . . . η . .
119 0 . H . − − R . . . RM SE . . . l , λ , η and H H ( z ) and the apparent magnitude m ( z ) with different observationaldata sets ( H ( z ), JLA, SNe Ia) with confidence level of constraints of 95%.Parameters H ( z ) SNe Ia JLA ω . . . l .
764 0 . . λ . . . η . . . H . − − R . . . RM SE . . . < ω < : The best fit values for the model parameters l , λ , η and H H ( z ) and the apparent magnitude m ( z ) for different observational datasets ( H ( z ), JLA, SNe Ia) for confidence level of boundaries of 95%.Figures 1 a & 1 b represent the best fit curve of the Hubble function H ( z ) for the best fit values of the modelparameters as well as the best fit values of the model parameters where the curvature parameter k = − , , a & 2 b depict the behaviour of cosmological constant Λ( z ) over redshift z and we seethat it is an increasing function of z which is consistent with observational findings. Figures 3 a & 3 b showthe age of the universe defined as ( t − t ) = R z dz (1+ z ) H ( z ) for both cases of ω and we have calculated the ageof the present universe t as 14 . ≤ t ≤ . t = 13 . . ≤ Ω ≤
1. Thesefindings are consistent with the observational constraints obtained by MAXIMA-1 flight and COBE-DMRexperiments [32], Ω = 1 . ± . . ± . . +0 . − .
30. Ω = 1 . ± . l , λ and η (the values of l, λ, η in the present epoch) in the dust case( w = 0) from the above equations. The first flight of the MAXIMA combined with COBE-DMR resulted in0 . < Ω m < .
50 and 0 . < Ω Λ < .
75. Observations of SNeIa combined with the total energy densityconstraints from CMB and combined gravitational lens and steller dynamical analysis lead to Ω m ∼ . Λ ∼ .
7. Sievers et al. and Sperger et al. obtained the values of [Ω m , Ω Λ ] = [0 . ± . , . +0 . − . ] and[0 . +0 . − . , . +0 . − . ] respectively [18, 20]. Considering the varied observational dataset, many researchersprefer a range of 0 . ≤ Ω m ≤ .
365 [5].For dust case, from (17) we obtain 3Ω m = 2 − l − λ η − l . If we consider the case λ = 0 = η , we obtain3 . ≤ l ≤ . l smaller than thatof Arbab’s [34] which was (3 , .
5) and Ray’s [5] which was [3 . , . . ≤ λ ≤ .
115 and 3 . ≤ η ≤ . H ( z ) versus redshift z for the best fit values of model parameters l , λ , η and H in both cases of ω .a. b.Figure 2: The plot of Cosmological constant Λ( z ) versus redshift z for the best fit values of model parameters l , λ , η and H in both cases of ω .a. b.Figure 3: The plot of Cosmic time ( t − t ) H versus redshift z for the best fit values of model parameters l , λ , η and H in both cases of ω . 7arameters (for ω = 0) H ( z ) SNe Ia JLAΩ m . . . Λ . . . q . − . − . t . . . r . − . − . s . . . H ( z ) SNe Ia JLA ω . . . m . . . Λ . . . q − . − . − . t . . . r − . − . − . s . . . < ω < : The present values of various parameters obtained for different data sets. We investigate equation (15) for possible finite time past or future cosmic singularities present in the currentmodel. We consider distinct cases depending on the sign of (2 + η ) − l (1 + w ).Case 1: ηl > w .If (1 + w )(3 − l − λ ) >
0, then there is a finite past singularity at t p = t − η − l (1+ w ) H ((1+ w )(3 − l − λ )) < t whereas if(1 + w )(3 − l − λ ) <
0, then the scale factor diverges at some finite future time t f = t − η − l (1+ w ) H ((1+ w )(3 − l − λ )) > t .Case 2: ηl < w .If (1 + w )(3 − l − λ ) <
0, then there is a finite past singularity at t p = t − η − l (1+ w ) H ((1+ w )(3 − l − λ )) < t whereas if(1 + w )(3 − l − λ ) >
0, then the scale factor diverges at some finite future time t f = t − η − l (1+ w ) H ((1+ w )(3 − l − λ )) > t .From this discussion it is clear that we can control the free parameters l, λ, η to avoid any future singularityin the system. The deceleration parameter q = − a ¨ a ˙ a = − ˙ HH − r, s ) is defined originally by r = ... aaH , s = r − q − / . We may link the ( r, s ) parameters to the Hubble parameters H and the deceleration parameter q for ourpresent model as r = 2 ˙ HH ! + 3 ˙ HH + 1 = q (2 q + 1) (22) s = − H H = 23 ( q + 1) . (23)8e also obtain a relation between r and s as r = 92 s ( s −
1) + 1 . (24)In general, the statefinder hierarchy A n = a ( n ) aH n for our present model can be calculated as A k +1 = q (2 q + 1)(3 q + 2) . . . (2 kq + 2 k − ,A k = − q (2 q + 1)(3 q + 2) . . . ((2 k − q + 2 k − . (25)In this regard, it is to be mentioned that these handy relations between the various cosmological pa-rameters as obtained in this section are valid for any time independent w and with any of the decay lawsconsidered in [1, 2, 3, 4] or any of their linear combinations.For a ΛCDM model the statefinder pair ( r, s ) have the value (1 , ˙ HH = 0 which means the scale factor a ( t ) increases exponentially with time. It also implies that q = −
1, which is true if α + β = 3, but this condition also provide Λ is truly a constant. Dust Case:
Below we obtain some important results for the particular case w = 0.First and foremost, we have q = 3 − l − λ η − l − . (26)Hence for an accelerating universe ( q <
0) we obtain a relation λ + η > m = 2 q is now replaced by Ω m = q + ,same as the result of Arbab [4]. However, both models give q = 1 / ρ c = H πG . We alsoobtain s = Ω m .Furthermore, using equations (13) and (16) we obtain that the deceleration parameter q and the vacuumdensity parameter Ω Λ are connected by the relation3Ω Λ = 1 − q + 2 w − l − λ (2 + η ) − l (1 + w ) , (27)and so for dust case, an accelerating universe ( q <
0) requires Ω Λ > /
3, which perfectly fits the modernobservational data as the present accepted value of Ω Λ is 0.7, much larger than 1/3. Hence our model fits anaccelerating universe. We have considered a spatially homogeneous and isotropic spacetime in the presence of a dynamic cosmolog-ical term Λ( t ) satisfying Λ = l ( ˙ aa ) + λ ¨ aa + 4 πGηρ motivated by modification of the Einstein-Hilbert action.This generalizes several results of the previous models in flat FRW spacetime and also enables us to find theranges for l, λ, η from the observational data apart from the interrelation we obtain between these parametersfor accelerating universe ( q < w . Furthermore, we show that ourmodel fits in perfectly with the modern observational datasets.We have found that the scale factor a ( t ) vanishes at t = t − { (2+ η ) − l (1+ w ) } H (1+ w )(3 − l − λ ) while for this value of timethe Hubble function H and the cosmological term Λ get infinitely large values. We have found that withincreasing time or decreasing redshift the scale factor increases but the H and Λ decreases to a finite smallvalues in late time universe. These behaviours of cosmological parameters reveal that our universe model9tarts with a finite time big-bang singularity stage and goes on expanding till late time. We have also noticedthat the total energy density parameter Ω ≤ Λ > to provide an acceleration in expansion of the universe which is compatiblewith the result Ω Λ > . We have also calculated the age of the present universe in the range 14 < t < Acknowledgments
De Avik and Tee How Loo are supported by the grant FRGS/1/2019/STG06/UM/02/6. A. Pradhan wouldlike to express his appreciation to the Inter-University Centre for Astronomy & Astrophysics (IUCAA), Indiafor supporting under the visiting associateship.
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