A Complete Cosmological Scenario in Teleparallel Gravity
AA Complete Cosmological Scenario in Teleparallel Gravity
Sanjay Mandal ∗ and P.K. Sahoo
1, † Department of Mathematics, Birla Institute of Technology and Science-Pilani,Hyderabad Campus,Hyderabad-500078, India
Teleparallel gravity is a modified theory of gravity in which the Ricci scalar R of the Lagrangianreplaced by the general function of torsion scalar T in action. With that, cosmology in teleparallelgravity becomes profoundly simplified because it is second-order theory. The article present a com-plete cosmological scenario in f ( T ) gravity with f ( T ) = T + β ( − T ) α , where α , and β are modelparameters. We present the profiles of energy density, pressure, and equation of state (EoS) parame-ter. Next to this, we employ statefinder diagnostics to check deviation from the Λ CDM model as wellas the nature of dark energy. Finally, we discuss the energy conditions to check the consistency of ourmodel and observe that SEC violates in the present model supporting the acceleration of the Universeas per present observation.
Keywords: f ( T ) gravity, EoS parameter, statefinder diagnostics, energy conditions. PACS numbers: 04.50.Kd
I. INTRODUCTION
Several observations confirm that the universe is ac-celerating in every second [1–15]. The main gradientwhich is responsible for the acceleration of the universeis unknown so-called ‘dark energy’. Nowadays, one ofthe difficult problems of modern cosmology and particlephysics is to identify the properties of dark energy. Thefirst time, it was described correctly by adding a cosmo-logical constant ( Λ ) to Einstein’s equation, but its mag-nitude is unwanted and unmotivated by the fundamen-tal physics. Also, it has agreed that approximately 75%of the total energy of the universe covered with dark en-ergy. Furthermore, according to observations, it is rep-resented by an EoS parameter ω (cid:39) −
1. Looking at allthese things, cosmologists have been proposed a lot ofproposals to overcome this problem [16–27].The modification of general relativity was a great idea todescribe dark energy. As a result, several modified theo-ries have proposed in the literature. Because it presentsthe nature of dark energy as a geometrical property ofthe universe, also, all the modified gravity theories areconnected to the modification of Einstein-Hilbert action[28]. The modified theories of gravity such as f ( R ) gravity, f ( R , T ) gravity, Gauss-Bonnet gravity, etc. arewidely used in modern cosmology (some interesting re-sults reported in [29–45]).A well established modified theory of gravity, whichhas attracted the interests of the cosmologists, is so-called f ( T ) teleparallel gravity [46–49]. Teleparallel ∗ Electronic address: [email protected] † Electronic address: [email protected] gravity motivated by the generalization of f ( R ) grav-ity in which the arbitrary function f of Ricci scalar R replaced by the arbitrary function of torsion scalar T inaction. We used the conventional torsionless Levi-Civitaconnection in general relativity, whereas the curvature-less Weitzenb ¨ o ck connection used in teleparallel gravityto describe the effects of gravitation in terms torsion in-stead of curvature [28, 50–52].The linear forms of f ( T ) are the teleparallel equivalentof general relativity (TEGR) [53]. However, the physi-cal interpretation of f ( T ) gravity is different from f ( R ) gravity. Also, the Ricci scalar R of f ( R ) gravity con-tains the second-order derivatives of the metric tensor,whereas the torsion scalar T of f ( T ) gravity containsonly the first-order derivatives of the vierbeins. There-fore, it is easy to find the exact solutions of the cos-mological models in f ( T ) gravity than the other mod-ified theories of gravity. Although it is a simple mod-ified theory, there are few exact solutions proposed inthe literature. Some power-law solutions in Friedmann-Lemaˆıtre-Robertson-Walker spacetime and anisotropicspacetime have found in [54–56]. Solution for staticspherically symmetric spacetime and Bianchi I space-time reported in [28, 57, 58].In f ( T ) gravity, the study of cosmological scenarios iseasier in comparison to other modified theories of grav-ity. So, it has incorporated to study the cosmologi-cal scenarios such as big bounce [59–62], inflationarymodel [63], late time cosmic acceleration [46, 64, 65].Recently, spherical and cylindrical solution [66], confor-mally symmetric traversable wormholes [67], noethercharge and black hole entropy [68] in f ( T ) gravity havebeen discussed.The manuscript represented as follows: In section II, a r X i v : . [ phy s i c s . g e n - ph ] M a y we discuss the overview of f ( T ) gravity and presentthe gravitational field equations in a spatially flat FLRWspacetime. In section III, we discuss some kinematicvariables such as Hubble parameter, deceleration pa-rameter, which is followed by the construction of thecosmological model in f ( T ) gravity in section IV. Also,we present the expressions and behaviours for energydensity, pressure, and EoS parameter. In section V, wediscuss some geometrical diagnostics to distinguish ourmodel from other dark energy models. The energy con-ditions for our model are discussed in section VI. Andfinally, in section VII, we present our results and conclu-sions. II. TELEPARALLEL GRAVITY
In this section, we briefly discuss the f ( T ) gravity.The vierbein fields, e µ ( x i ) act as a dynamical variablefor the teleparallel gravity. As usual, x i use to runover the space-time coordinates, and µ denotes the tan-gent space-time coordinates. At each point of the mani-fold, the vierbein fields form an orthonormal basis forthe tangent space, which is presented by the line el-ement of four-dimensional Minkowski space-time i.e., e µ e ν = η µν = diag (+ − − − ) . In vector compo-nent, the vierbein fields can be expressed as e i µ ∂ i , andthe metric tensor can be written as g µν = η ij e i µ ( x ) e j ν ( x ) . (1)Moreover, the vierbein basis follow the general relation e i µ e µ j = δ ij and e i µ e ν i = δ νµ . In f ( T ) gravity, the curvature-less Weitzenb ¨ o ck connection [69] defined asˆ Γ γµν ≡ e γ i ∂ ν e i µ ≡ − e i µ ∂ ν e γ i . (2) Using Weitzenb ¨ o ck connection one can write the non-zero torsion tensor as T γµν ≡ ˆ Γ γµν − ˆ Γ γνµ ≡ e γ i ( ∂ µ e i ν − ∂ ν e i µ ) . (3)The contracted form of the above torsion tensor can bewritten as [50, 52, 70] T ≡ S µνγ T γµν ≡ T γµν T γµν + T γµν T νµγ − T γγµ T νµν , (4)where S µνγ = ( K µνγ + δ µγ T ανα − δ νγ T αµα ) (5)be the superpotential tensor and the difference betweenthe Levi-Civita and Weitzenb ¨ o ck connections is the con-tortion tensor which is defined as K µνγ = − ( T µνγ − T νµγ − T µνγ ) . (6)The extension of Einstein-Hilbert Lagrangian of f ( T ) theory of gravity [49] (which is similar to f ( R ) gravityextension from the Ricci scalar R to R + f ( R ) in the ac-tion), namely the teleparallel gravity term T is replacedby T + f ( T ) , where f ( T ) is an arbitrary function of T as S = π G (cid:90) [ T + f ( T )] ed x , (7)where e = det ( e i µ ) = √− g and G is the gravitationalconstant. Assume k = π G = M − p , where M p is thePlanck mass. By the variation of the total action S + L m ,here L m is the matter Lagrangian gives us the field equa-tion for f ( T ) gravity as e − ∂ µ ( ee γ i S µνγ )( + f T ) − ( + f T ) e λ i T γµλ S νµγ + e γ i S µνγ ∂ µ ( T ) f TT + e ν i [ T + f ( T )] = k e γ i T ( M ) νγ , (8)where f T = d f ( T ) / dT , f TT = d f ( T ) / dT , T ( M ) νγ rep-resents the energy-momentum tensor to the matter La-grangian L m .Now we consider a flat FLRW universe with the metricas ds = dt − a ( t ) dx µ dx ν , (9)where a ( t ) is the scale factor, which gives us e i µ = diag ( a , a , a ) . (10) Moreover, assuming the energy-momentum tensor forthe perfect fluid which takes the form T ( M ) µν = ( ρ + p ) u µ u ν − pg µν , (11)where ρ , p and u µ be the energy density, pressure andthe four velocity of the matter fluid, respectively.Using equation (9) into the field equation (8), we get themodified field equations as follows6 H + f + T f T = π G ρ , (12)˙ H ( + f T + T f TT ) = − π G ( ρ + p ) , (13)where H ≡ ˙ a / a be the Hubble parameter and ‘dot’ rep-resents the derivative with respect to t . Additionally, wehave used the relation T = − H , (14)which holds for a FLRW Universe according to equation(4).Using equation (12), (13) and (14) we can write the equa-tion of state parameter (EoS) as follows ω = p ρ = − − H ( + f T − H f TT )( H + f + H f T ) (15)where 8 π G =
1. Also, the matter fluid satisfies the con-tinuity equation ˙ ρ + H ( + ω ) ρ =
0, (16)which can be used to study the dynamics of matter fluid.
III. KINEMATIC VARIABLES
The cosmological parameters such as scale factor a ( t ) ,Hubble parameter H ( t ) , deceleration parameter q ( t ) have a very significant role in describing the evolutionof the Universe. And, these are the key parameters ofmost of the cosmological models in modified gravitytheories. For analysis, we have presumed the scale fac-tor presented by Moraes and Santos [71] as follows a ( t ) = e ct [ sech ( n − mt )] d . (17)Using (17), one can get the Hubble parameter H ( t ) anddeceleration parameter q ( t ) as follows H ( t ) = ˙ aa = c + dm tanh ( n − mt ) , (18) q = − − ˙ HH = − + dm [ c cosh ( n − mt ) + dm sinh ( n − mt )] .(19)Currently, our universe is undergoing an accelerated ex-pansion phase for that the second derivative of the scalefactor i.e., ¨ a must be positive or ˙ a is an increasing func-tion over the cosmic time evolution. Additionally, theHubble parameter H ( t ) is a decreasing function overthe growth of time. From Fig. 1, one can easily see that H ( t ) keeps its value approximately the same in the earlystage of the universe. After that, it gradually decreases and maintains a constant behaviour during the late timeof the universe. According to standard cosmology, theHubble parameter is proportional to the energy densityin the late time of the cosmic evolution. Luckily, we getthe same behaviour of the Hubble parameter for the latetime of cosmic evolution.The evolution of the deceleration parameter as a func-tion of cosmic time presented in Fig. 2. From Fig. 2,one can observe that the evolution of the decelerationparameter starts at q = −
1, which represents the de-Sitter expansion phase, and then it goes to the deceler-ation phase through accelerating power-law expansionphase. After that, it again returns to the de-Sitter expan-sion phase in the late time. t H FIG. 1: Plot of Hubble parameter ( H ) as a function of cosmictime ( t ) for c = d = m = n = - - - - - t q FIG. 2: Plot of deceleration parameter ( q ) as a function of cos-mic time ( t ) for c = d = m = n = IV. COSMOLOGICAL MODEL IN f ( T ) GRAVITY
In this section, we presume the general function of f ( T ) to analyse the cosmological model in teleparallelgravity as follows f ( T ) = T + β ( − T ) α . (20) Using (20), (17) in (12), (13) and (15) we can find the en-ergy density ρ , pressure p and EoS parameter ω as fol-lows ρ = [ c + dm tanh ( n − mt )] − β α ( α − ) [ c + dm tanh ( n − mt )] α , (21) p = α − ( α − ) β [ c + dm tanh ( n − mt )] α − (cid:104) c + dm tanh ( n − mt ) { c + m ( α + d ) tanh ( n − mt ) } − α dm (cid:105) − (cid:104) c + dm tanh ( n − mt ) { c + ( d + ) m tanh ( n − mt ) } − dm (cid:105) , (22)and ω = − − dm sech ( n − mt ) (cid:110) α α ( α − ) β [ c + dm tanh ( n − mt )] α − [ c + dm tanh ( n − mt )] (cid:111) [ c + dm tanh ( n − mt )] − α α + ( α − ) β [ c + dm tanh ( n − mt )] α + . (23)The profile of energy density ρ , pressure p , and equationof state parameter ω for the cosmic time shown in Fig.3, 4, and 5, respectively. From Fig. 3, one can easily ob-serve that the energy density is high in the early time ofthe universe and then gradually decreases to null. Thepressure p lies in the negative range to suffice the accel-eration of the universe in Fig. 4. We got an interestingbehaviour of the EoS parameter ω , as shown in Fig. 5. Inthe early phase of the evolution of the universe ω takesits values − ω ∼ , and finally converging to − t ρ FIG. 3: Plot of energy density ( ρ ) as a function of cosmic time ( t ) for c = d = m = n = α =
2, & β = − - - - - - t p FIG. 4: Plot of pressure ( p ) as a function of cosmic time ( t ) for c = d = m = n = α =
2, & β = − - - - - - t ω FIG. 5: Plot of EoS parameter ( ρ ) as a function of cosmic time ( t ) for c = d = m = n = α =
2, & β = − V. STATEFINDER DIAGNOSTICS
The unknown nature of dark energy (DE) arises manyproblems in modern cosmology. To understand the na- ture of dark energy, many dark energy models such as Λ CDM, HDE, SCDM, CG, quintessence have proposedin the literature. Also, these DE models have differentbehaviours in comparison to each other. Therefore, the { r , s } parametrization technique proposed in [72, 73] isalso used to distinguish all these DE models. The r and s take the value as follows r = ˙¨ aaH , (24) s = r − q − q (cid:54) = . The pair { r , s } represents different darkenergy models. These are discussed in following• The pair { r = s = } → Λ CDM .• The pair { r = s = } → SCDM.• The pair (cid:8) r = s = (cid:9) → HDE.• The pair { r > s < } → CG.• The pair { r < s > } → Quintessence.The main idea is to study the convergence and diver-gence nature of the trajectory of the r − s parametriccurve corresponding to the Λ CDM model. The devia-tion from {
1, 0 } represents the deviation from Λ CDMmodel. Furthermore, the values of r and s could be con-cluded from the observation [74, 75]. Therefore, it isworthy of describing the DE models in the near future.Using (17) in (24), (25) we can rewrite the r , s as follows r = c + dm tanh ( n − mt ) (cid:8) c + ( d + ) m tanh ( n − mt )[ c + ( d + ) m tanh ( n − mt )] − ( d + ) m (cid:9) − cdm [ c + dm tanh ( n − mt )] , (26) s = dm sech ( n − mt )[ c + ( d + ) m tanh ( n − mt )] [ c + dm tanh ( n − mt )] { c + dm tanh ( n − mt )[ c + ( d + ) m tanh ( n − mt )] − dm } . (27)In Fig. 6, the parametrization of r and s shown in ( r , s ) plane and the arrow mark represents the direction of the trajectory. From Fig. 6, one can observe that the trajec-tory diverges from Λ CDM model, initially and later, itconverges to Λ CDM model. Also, the evolution of thetrajectory completely lies in the quintessence. Addition-ally, we have shown the parametrization of r and q inFig. 7. From Fig. 7, we observed that our model startswith the de-Sitter universe, and initially, it goes to Chap-lygin gas (which is represented by r > Λ CDM SCDM
QUINTESSENCEHDE - - s r FIG. 6: r − s parametric plot for c = d = m = n = SCDM Λ CDMdS - - q r FIG. 7: r − q parametric plot for c = d = m = n = VI. ENERGY CONDITIONS
There are many more advantages to study the energyconditions such as to understand singularity theorem ofspace-like and time-like curves, to describe geodesics,black holes, and wormholes based on the well knownRaychaudhuri equation [76, 77]. Energy conditions inteleparallel gravity have been studied in [78–81]. Also,in the presence of singularity, energy conditions provideus the edge of the parameters. The energy conditionsare the linear relations between energy density and pres-sure. They are presented follows• SEC : ρ + p ≥ ρ ≥ ρ + p ≥ ρ + p ≥ ρ ≥ | p | ≤ ρ . NECDECSEC
13 14 15 16 17 18 - - - t - - t FIG. 8: ECs as a function of t for c = d = m = n = α =
2, & β = − In the above figure, the portrait of all Energy conditions(ECs) shown. From Fig. 8, we observed that the WEC,NEC satisfies in our present model, whereas SEC vio-lates. In detail, if one can see close to the behaviour ofSEC, then he will observe for a time interval, it satisfiesby our model. As we discussed the profile of ω previ-ously, it starts with acceleration, then smoothly goes tothe deceleration phase and finally returns to the acceler-ated phase. The exciting thing is getting the same resultsfor SEC. VII. RESULTS AND CONCLUSIONS
The article represents a complete cosmological sce-nario of the FLRW universe in teleparallel gravity. Tofind the exact solution of the field equations, we consid-ered the general function f ( T ) = T + β ( − T ) α which isproposed in [82]. In the below paragraphs, we shall dis-cuss the cosmological feasibility of the fundamental of f ( T ) gravity.The EoS parameter ω is a key parameter to describe thedifferent matter-dominated eras in the evolution of theuniverse. The portrait of the EoS parameter with time t have shown in Fig. 5. By analyzing Fig. 5, we observedthat in the early phase of the evolution of the universe,the EoS parameter takes its value as ω ∼ −
1. This resultis in harmony with some limitations of EoS, which haveput to the inflationary EoS, recently [83–85]. After infla-tion, ω smoothly rises to ∼ , and this is the maximumvalue of ω during its evolution. Moreover, ω ∼ repre-sents the radiation-dominated phase of the evolution ofthe universe [86, 87]. Also, we noted that the EoS param-eter ω converges to − ω = − ω = − ω present the acceleration of the uni-verse), then smoothly it takes its value to positive i.e.,deceleration phase. Finally, it returns to the acceleratedphase which is the second phase of the acceleration ofthe universe in the present time.In section VI, we show the temporal evolution of the en-ergy conditions (ECs). It is to keep in mind that to servethe late-time acceleration of the universe, the SEC has to violate [89]. As we discussed in the previous sectionfor the late time acceleration ω takes its value as −
1, sothe SEC ρ ( + ω ) < ω .In Fig. 3, 4, we show the behaviour of energy densityand pressure for our model. The energy density shouldbe positive to fulfil WEC, and pressure should be neg-ative for cosmic acceleration. The evolution of pressureand energy density of our model satisfies WEC. And,this type of property only achieved by the modificationof general relativity or exotic matter.In section V, we investigate the difference between thedark energy models with our model by statefinder di-agnostic. From Fig. 6, we observed that the trajectoryof { r , s } parametric curve deviates from Λ CDM modelinitially. However, at the late time, it coincides with the Λ CDM model, which is consistent with standard cos-mology. Addition to this, we show { r , q } parametricplot in Fig. 7. It is also consistent with the standardcosmology. Acknowledgments
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