On The Temporal Evolution of Particle Production in f(T) Gravity
JJanuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333
Modern Physics Letters A © World Scientific Publishing Company
On The Temporal Evolution of Particle Production in f ( T ) Gravity Sanjay Mandal
Department of Mathematics,Birla Institute of Technology and Science-Pilani,Hyderabad Campus, Hyderabad-500078, [email protected]
P.K. Sahoo
Department of Mathematics,Birla Institute of Technology and Science-Pilani,Hyderabad Campus, Hyderabad-500078, [email protected]
Received (22 July 2020)Accepted (05 October 2020)The thermodynamical study of the universe allow particle production in modified f ( T )( T is the torsion scalar) theory of gravity within a flat FLRW framework for line element.The torsion scalar T plays the same role as the Ricci scalar R in the modified theoriesof gravity. We derived the f ( T ) gravity models by taking f ( T ) as the sum of T andan arbitrary function of T with three different arbitrary function. We observe that theparticle production describes the accelerated expansion of the universe without a cos-mological constant or any unknown “quintessence” component. Also, we discussed thesupplementary pressure, particle number density and particle production rate for threecases. Keywords : Modified f ( T ) gravity; Particle creation; ThermodynamicsPACS Nos.: 04.50.kd
1. Introduction
At the beginning of the 19th century, the General Theory of Relativity broughtthe revolution to the modern cosmology proposed by Albert Einstein. The Rie-mannian space-time formulates this theory based on the Levi-Civita connection, atorsion-free, and metric compatibility connection. It also helps us to understandthe geodesic structure of the Universe. Later on, it faced problems like fine-tuning,cosmic co-incidence, initial singularity, cosmological constant, and flatness, sincemodern cosmology is growing by a prominent number of accurate observations. Be-sides, the cosmological observation, such as type Ia supernovae,
2, 3 cosmic microwavebackground (CMB) radiation,
4, 5 large scale structure,
6, 7 baryon acoustic oscilla-tions, and weak lensing confirms that currently, our Universe is going through an a r X i v : . [ g r- q c ] J a n anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo accelerated expansion phase that happens because of the highly negative pressureproduced by the unknown form of matter and energy, called dark energy and darkmatter. To overcome the above issues, researchers started to modify the Einstein’stheory of relativity, and they ended up with several modified theories of gravity suchas f ( R ) gravity, f ( R, T ) gravity, f ( T ) gravity, f ( Q ) gravity, f ( Q, T ) grav-ity, etc. As a result, cosmologists found many interesting results such as Yousaf etal., studied the self-gravitating structures, gavastars, Sahoo et al. studied thewormhole geometry, bouncing cosmology, and accelerated expansion of the universeusing modified theories of gravity. The main advantage of the modified theories ofgravity is that it successfully describes the late-time cosmic acceleration and theearly time inflation. In the early stage of the Universe, there is a possibility of parti-cle creation. In this study, we are focusing on particle production in the teleparallelgravity.The f ( T ) theories of gravity are the generalization of the Teleparallel Equivalent ofGeneral Relativity (TEGR), where T is the torsion scalar. TEGR was first pre-sented by Einstein. In the context of f ( T ) theories, the Reimann-Cartan space-timerequires the torsional curvature to vanish. Furthermore, this type of space-time isconstructed by the Weitzenb¨ock connection.
19, 20
TEGR is equivalent to GR; thereason is that both cases’ action is the same except the surface term in TEGR.But, the physical interpretation is different from each other. The construction ofgravitational Lagrangian in TEGR formulation was done in.
21, 22
TEGR is Lorentzinvariant theories, whereas the modified f ( T ) theories are not Lorentz invariant.Also, the motion equations in f ( T ) gravity are not necessarily Lorentz invariant, be-cause the certainty is that this theory’s property explains the recent interests. Moreover, the modification of TEGR was motivated by the f ( R ) gravity theory.In the teleparallel gravity theory, we use the Weitzenb¨ock connection instead ofthe Levi-Civita connection, which uses f ( R ) gravity to vanish the non-zero torsioncurvature. Here, we would like to mention that the f ( T ) gravity does not needthe Equivalence principle because the Weitzenb¨ock connection describes its grav-itational interaction. It is a simple modified theory compared to other modifiedtheories because the torsion scalar T contains only the first-order derivatives of thevierbeins. In contrast, Ricci scalar R contains the second-order derivatives of themetric tensors. Recently, Mandal et al., studied the acceleration expansion of theUniverse using the parametrization technique with presuming exponential and log-arithmic form of f ( T ) in f ( T ) gravity. They also studied a complete cosmologicalscenario of the Universe in f ( T ) gravity, where they discussed the difference betweenGeneral Relativity and Teleparallel gravity in f ( T ) gravity. M. Sharif and S. Ranistudied the dynamical instability ranges in Newtonian as well as post-Newtonianregimes considering power-law f ( T ) model with anisotropic fluid in f ( T ) gravity. Cai et al., studied the matter bounce cosmology using perturbation technique andthey found a scale-invariant power spectrum, which is consistent with cosmologicalobservations in f ( T ) gravity. In, the wormhole solutions with non-commutativegeometry have been studied assuming power-law f ( T ) model and a particular shapeanuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 On The Temporal Evolution of Particle Production in f ( T ) Gravity function in teleparallel gravity. Inflationary universe studied using power-law f ( T )function and logamediate scale factor in, and constant-roll inflation studied in. In the early stage of the universe, the possibility of particle creation has been dis-cussed for curved space-time by Schrodinger, Dewitt, Imamura. Later, the firstever particle creation was treated by an external gravitational field by Parkar.
37, 38
In flat space-time, the unique vacuum state is identifying by the guidance of Lorentzinvariance. Moreover, we do not have Lorentz symmetry in curved space-time. Ingeneral, there are more than one vacuum state exists in a curved space-time. There-fore, the particle creation idea becomes open to discuss, but it’s physical interpre-tation becomes more difficult.
39, 40
The interaction between the dynamical externalgradients causes the particle creation from the vacuum. The particle creation pro-duces negative pressure, so it is considered to explain the accelerated expansion ofthe universe and got some unexpected outcomes. Also, it might play the role ofunknown gradients of the universe. In
41, 42 studied the particle creation with SNe Iadata. Singh, and Singh and Beesham
44, 45 studied the particle creation with somekinematical tests in FLRW cosmology. The continuous creation of particle predictsthe assumptions of standard Big Bang cosmology.The thermodynamical study of black hole gives the fundamental relation betweenthermodynamics and gravitation.
In GR, the relation between the entropy andthe horizon area with the Einstein equation derives from the Clausius relation inthermodynamics.
51, 52
This idea is also used for other theories, mainly, the general-ized thermodynamics laws and modified theories of gravity which are derived fromthe GR.
53, 54
Among the modified theories of gravity, f ( R ) gravity got more atten-tion on this framework. Thereby, one can obtained the gravitational field equationthrough the non-equilibrium feature of thermodynamics by using the Clausius ap-proach. There are some work have been done in the thermodynamics of particlecreation in f ( T ) gravity theory.
50, 55–58
In this work, we study the theoretical significance of particle creation in f ( T ) gravitytheory considering a flat FRW model. Assuming f ( T ) as the sum of torsion scalar T and an arbitrary function of torsion scalar T , we studied the thermodynamics ofparticle creation with f ( T ) = 0 is a simple teleparallel gravity, f ( T ) = A ( − T ) q aspower law gravity and f ( T ) = A (1 − e − qT ) as exponential gravity. After that wediscussed the behaviour of supplementary pressure p c , particle number density n ,and the particle creation rate ψ for three models. Also we compared the effect of thecosmological pressure p m with the supplementary pressure p c for different values ofequation of state parameter ω on particle creation.This work is organised as follows. In Sec. 2, we discussed the thermodynamics ofparticle creation, which is followed by the overview of f ( T ) gravity and it’s fieldequations in Sec. 3. In Sec. 4, we discussed three f ( T ) gravity models. Finally, theresults are summarized in Sec. 5anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo
2. Thermodynamics of particle creation
If we assume the total number of particles in the universe to be conserved, the lawsof thermodynamics can be expressed as dQ = d ( ρ m V ) + p m dV (1)and T dS = p m dV + d ( ρ m V ) (2)where p m , ρ m , V , T and S denote respectively the cosmological pressure, density,volume, temperature and entropy. Also, dQ represent the heat exchange in the timeinterval dt . From (1) and (2), we further obtain, dQ = T dS (3)Eq. (3) reflects the fact that the entropy is a conserved quantity, since for loseadiabatic system dQ = 0. We now consider a scenario in which the total number ofparticles in the universe is not constant. Under this condition, Eq (1) gets modifiedto dQ = d ( ρ m V ) + p m dV + ( h/n ) d ( nV ) (4)where N = nV , n being the number density of the particles and h = ( p m + ρ m ) theenthalpy per unit volume of the system. For an adiabatic system where dQ = 0, (4)reads d ( ρ m V ) + p m dV = ( h/n ) d ( nV ) (5)In, the authors stated that in cosmology this change in the total number of par-ticles in the universe can be understood as a transformation of gravitational fieldenergy to the matter.For an open thermodynamic system, Eq. (5) can be expressed as d ( ρ m V ) = − ( p m + p c ) dV (6)where p c = − ( h/n )( dN/dV ) (7)represents supplementary pressure associated with the creation of particles. Notethat negative p c indicate production of particles whereas positive p c implies particleannihilation and finally for p c = 0 the total number of particles is constant. UsingEq (2) and (5), it can also be shown that S = S (cid:18) NN (cid:19) (8)where S and N represent current values of these quantities.Additionally, we assume the particles follow a barotropic equation of state andtherefore can be written as p m = ( ω ) ρ m (9)anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 On The Temporal Evolution of Particle Production in f ( T ) Gravity where − ≤ ω ≤ ρ m as n = n (cid:18) ρ m ρ (cid:19) ω (10)where ρ ≥ n ≥ ψ ( t ) = 3 βnH (11)where 0 ≤ β ≤ ψ ( t ) represent the rate of particlecreation and has a dimension of t − . ψ can either be positive or negative dependingon the creation or annihilation of particles. ψ = 0 indicate particle number beingconserved in the universe. For cosmological matter following barotropic equation ofstate (Eq. 9), the supplementary pressure p c can be expressed as p c = − β ( ω + 1) ρ m (12)
3. Overview of f ( T ) Gravity Let us consider the extension of Einstein-Hilbert Lagrangian of f ( T ) theory ofgravity (which is similar to f ( R ) gravity extension from the Ricci scalar R to R + f ( R ) in the action), namely the teleparallel gravity term T to T + f ( T ), where f ( T )is an arbitrary function of T as S = 116 πG (cid:90) [ T + f ( T )] ed x, (13)where e = det ( e iµ ) = √− g and G is the gravitational constant. Assume k = 8 πG = M − p , where M p is the Planck mass.The gravitational field is defined by the torsionone as T γµν ≡ e γi ( ∂ µ e iν − ∂ ν e iµ ) . (14)The contracted form of the above torsion tensor is T ≡ T γµν T γµν + 12 T γµν T νµγ − T γγµ T νµν . (15)By the variation of the total action S + L m , here L m is the matter Lagrangian givesus the field equation for f ( T ) gravity as e − ∂ µ ( ee γi S µνγ )(1 + f T ) − (1 + f T ) e λi T γµλ S νµγ + e γi S µνγ ∂ µ ( T ) f T T + 14 e νi [ T + f ( T )] = k e γi T ( M ) νγ , (16)where f T = df ( T ) /dT , f T T = d f ( T ) /dT , the ”superpotential “ tensor S µνγ writtenin terms of cotorsion K µνγ = − ( T µνγ − T νµα − T µνα ) as S µνγ = ( K µνγ + δ µγ T ανα − δ νγ T αµα )anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo and T ( M ) νγ represents the energy-momentum tensor to the matter Lagrangian L m .Now we consider a flat FLRW universe with the metric as ds = dt − a ( t ) dx µ dx ν , (17)where a ( t ) is the scale factor, which gives us e iµ = diag (1 , a, a, a ) . (18)Using equation (18) into the field equation (16), we get the modified field equationas follows H = 8 πG ρ m − f T f T , (19)˙ H = − (cid:20) πG ( ρ m + p m + p c )1 + f T + 2 T f
T T (cid:21) , (20)where H ≡ ˙ a/a be the Hubble parameter and ”dot“ represents the derivative withrespect to t . Here, ρ m and p m be the energy density and pressure of the mattercontent, p c be the supplementary pressure. Also, we have used T = − H , (21)which holds for a FLRW Universe according to equation (15). f ( T ) gravity models In this section we shall investigate the temporal evolution of particle productionin radiation ( ω = 1 /
3) and dust universe ( ω = 0) for various f ( T ) gravity modelswith model parameters constrained from cosmological observations related to grav-itational baryogenesis.For the purpose of analysis, we shall assume a power law evolution of scale factorof the form a ( t ) = a t [ ω ) ] (22)where a > Simple Teleparallel gravity
In simple teleparallel equivalent of general relativity, where f ( T ) = 0 , for auniverse composed of perfect fluid, the field equations (19) and (20) becomes H = 8 πG ρ m (23)˙ H = − πG ( ρ m + p m + p c ) (24)Substituting (22) in (23), we obtain the expression of density ρ m as ρ m = 43 (cid:18) t (1 + ω ) (cid:19) (25)anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 On The Temporal Evolution of Particle Production in f ( T ) Gravity The expression of supplementary pressure p c , particle number density n and particlecreation rate ψ are obtained respectively as p c = 43 (cid:18) β (1 + ω ) t (1 + ω ) (cid:19) (26) n = (cid:20) (cid:18) t (1 + ω ) (cid:19)(cid:21) ω ) (27) ψ = 3 β (cid:20) t (1 + ω ) (cid:21) × (cid:20) (cid:18) t (1 + ω ) (cid:19)(cid:21) ω ) (28) ω = / ω =
00 2 4 6 8 10 - - - - t p c Fig. 1. The behaviour of supplementary pressure p c with respect to cosmic time t for ω = , ω = 0and β = 1. anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo ω = / ω =
00 2 4 6 8 100.00.20.40.60.8 t n Fig. 2. The behaviour of particle number density n with respect to cosmic time t for ω = , ω = 0and β = 1. ω = / ω =
00 2 4 6 8 100.00.20.40.60.8 t ψ Fig. 3. The behaviour of particle creation rate ψ with respect to cosmic time t for ω = , ω = 0and β = 1. anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 On The Temporal Evolution of Particle Production in f ( T ) Gravity Power Law Gravity
The power law model of Bengochea and Ferraro reads f ( T ) = A ( − T ) q (29)where A is a constant and q >
1. In, the authors reported viable baryon-to-entropy ratio for A = − − or − − and q (cid:38) .
8. However, other values ofthe model parameters could also yield viable estimates of baryon-to-entropy ratio.Nonetheless, we restrict ourselves to the values A = − − and q = 5 for the presentanalysis. Substituting (22) and (29) in (19) and (20), the expression of density ρ m reads ρ m = A ( q − (1 − q ) t − q (cid:20) ω ) (cid:21) q (30)The expression of supplementary pressure p c , particle number density n and particlecreation rate ψ for the power law gravity are obtained respectively as p c = − β (1 + ω ) A ( q − (1 − q ) t − q (cid:20) ω ) (cid:21) q (31) n = (cid:34) A ( q − (1 − q ) t − q (cid:20) ω ) (cid:21) q (cid:35) / (1+ ω ) (32) ψ = 3 β (cid:20) t (1 + ω ) (cid:21) × (cid:34) A ( q − (1 − q ) t − q (cid:20) ω ) (cid:21) q (cid:35) / (1+ ω ) (33) ω = / ω =
00 2 4 6 8 10 - - - - - t p c Fig. 4. The behaviour of supplementary pressure p c with respect to cosmic time t for ω = , ω = 0and β = 1 , q = 5 , A = − − anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo ω = / ω =
00 2 4 6 8 100.000000.000050.000100.00015 t n Fig. 5. The behaviour of particle number density p c with respect to cosmic time t for ω = , ω = 0and β = 1 , q = 5 , A = − − . ω = / ω =
00 2 4 6 8 100.000000.000050.000100.00015 t ψ Fig. 6. The behaviour of particle creation rate ψ with respect to cosmic time t for ω = , ω = 0and β = 1 , q = 5 , A = − − . Exponential Gravity
The exponential f ( T ) model given in reads f ( T ) = A (1 − e − qT ) (34)anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 On The Temporal Evolution of Particle Production in f ( T ) Gravity where A and q are model parameters. In the authors reported a wide range ofvalues of A and q for which a viable baryon-to-entropy ratio could be realized.However, we shall work with A = 1 and q = 10 − as these values were used in to fit the baryon-to-entropy ratio with observations. Substituting (22) and (34) in(19) and (20), the expression of density ρ m reads ρ m = A (cid:104) ω ) (cid:105) qe [ ω ) ] qt t (35)The expression of supplementary pressure p c , particle number density n and particlecreation rate ψ for the exponential gravity are obtained respectively as p c = − β (1 + ω ) A (cid:104) ω ) (cid:105) qe [ ω ) ] qt t (36) n = A (cid:104) ω ) (cid:105) qe [ ω ) ] qt t / (1+ ω ) (37) ψ = 3 β (cid:20) t (1 + ω ) (cid:21) × A (cid:104) ω ) (cid:105) qe [ ω ) ] qt t / (1+ ω ) (38)anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo ω = / ω =
00 2 4 6 8 10 - × - - × - - × - - × - - × - t p c Fig. 7. The behaviour of supplementary pressure p c with respect to cosmic time t for ω = , ω = 0and β = 1 , q = 10 − , A = 1. ω = / ω =
00 2 4 6 8 1001. × - × - × - × - t n Fig. 8. The behaviour of particle number density n with respect to cosmic time t for ω = , ω = 0and β = 1 , q = 10 − , A = 1. anuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 On The Temporal Evolution of Particle Production in f ( T ) Gravity ω = / ω =
00 2 4 6 8 1005. × - × - × - × - × - t ψ Fig. 9. The behaviour of particle creation rate ψ with respect to cosmic time t for ω = , ω = 0and β = 1 , q = 10 − , A = 1.
5. Discussion of Outcomes and Conclusions
In this article, we have studied the thermodynamics of an open system with par-ticle creation of a flat FLRW universe in f ( T ) theory of gravity. We have con-structed three cosmological models by assuming suitable functions for f ( T ) as f ( T ) = 0 , f ( T ) = A ( − T ) q , f ( T ) = A (1 − e − qT ) and the particle creation rate ψ . To analyze our models, we have considered the power law evolution of the scalefactor and studied the behaviour of physical quantities (i.e. the supplementary pres-sure p c , particle number density n , and the particle creation rate ψ ) through theirgraphical representations with respect to cosmic time t and some fixed values of β in various phase of the evolution of the universe. And, details of our cosmologicalmodels discussed in the following.In our simple Teleparallel gravity model, we have considered the minimal cou-pling between matter and geometry. In Fig. 3, profiles of ψ have been shown. FromFig. 3, one can easily observe that the rate of particle creation is high in the earlytime and tends to zero when t tends to infinity. But, the number of particle in theuniverse increases with cosmic time t shown in Fig. 2. The supplementary pressure p c has higher negative which shows that the particle production is high during theearly stage and tends to zero when t tends infinity, in Fig. 1. From this model wehave concluded that the evolution of the universe depends on the contribution ofthe particle production.In power law gravity and exponential gravity models, we have considered thenon-minimal coupling between matters. The profiles of ψ, n and p c have been shownfor the corresponding models. In Fig. 6,9, the particle creation rate ψ is high inthe early stage and it tends to zero as cosmic time t tends to infinity. Also, theanuary 6, 2021 1:33 WSPC/INSTRUCTION FILE MPLA-D-20-00333 Sanjay Mandal, P.K. Sahoo particle number density n in Fig. 5,8 goes to zero as cosmic time t goes to infinitywhich concluded that the expansion rate overcomes the rate particle creation asthe supplementary pressure in Fig. 4,7 is negative throughout the evolution of theuniverse in different phases. The density parameters in three models shows that theuniverse is open in the presence of particle creation in f ( T ) theory of gravity.In summary, we have studied the cosmological models with particle produc-tion in f ( T ) theory of gravity to explore the current accelerated phenomenon ofthe universe. We have found that the particle creation produces negative pressurewhich may derive the accelerated expansion of the universe and play the role ofunknown matter called “dark energy” in f ( T ) theory of gravity. We may expectthat the particle creation process be a constraint for the unexpected observationaloutcomes. The new fact about this article is that the particle creation is studied bythe thermodynamics approach in f ( T ) theory of gravity. Acknowledgements
S.M. acknowledges Department of Science & Technology(DST), Govt. of India, New Delhi, for awarding Junior Research Fellowship (File No.DST/INSPIRE Fellowship/2018/IF180676). PKS acknowledges DST, New Delhi,India for providing facilities through DST-FIST lab, Department of Mathematics,BITS-Pilani, Hyderabad Campus where a part of this work was done. The authorsthank S. Bhattacharjee for stimulating discussions. We are very much grateful tothe honorable referee and the editor for the illuminating suggestions that havesignificantly improved our work in terms of research quality and presentation.
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