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Bouncing Cosmology in Gravity with Logarithmic Trace Term

Bouncing Cosmology in Gravity with Logarithmic Trace Term Hindawi Advances in Astronomy Volume 2021, Article ID 8852581, 12 pages https://doi.org/10.1155/2021/8852581 Research Article Bouncing Cosmology in f(G, T) Gravity with Logarithmic Trace Term M. Farasat Shamir National University of Computer and Emerging Sciences, Islamabad, Lahore Campus, Lahore, Pakistan Correspondence should be addressed to M. Farasat Shamir; farasat.shamir@nu.edu.pk Received 20 September 2020; Revised 11 January 2021; Accepted 2 February 2021; Published 3 March 2021 Academic Editor: Rafael Correa Copyright © 2021 M. Farasat Shamir. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. &is study is devoted to explore bouncing cosmology in the context of f(G, T) theory of gravity. For this purpose, a Gauss–Bonnet cosmological model with logarithmic trace term is considered. In particular, the possibility of obtaining bouncing solutions by considering two equations of state parameters is investigated. A graphical analysis is provided for analyzing the obtained bouncing solutions. &e energy conditions are discussed in detail. It is interesting to notice that null and strong energy conditions are violated near the neighborhood of bouncing points justifying the accelerating universe in the light of the recent observational data. &e behavior of the scale factor, red shift function, deceleration parameter, and Hubble parameter is also debated. An important feature of the current study is the discussion of conservation equation in f(G, T) gravity. &e possibility of some suitable constraint equations which recover the standard conservation equation is discussed, and all the free parameters are assumed accordingly. All the results in this study suggest that the proposed f(G, T) gravity model provides good bouncing solutions with the chosen EoS parameters. perturbations having spectrum consistent with the obser- 1. Introduction vational data can be produced [7]. &e so-called BKL in- Bouncing universes are possible alternatives to standard big stability can be witnessed after the contracting phase making bang cosmology. Unfortunately, some of the older studies of the universe anisotropic [8]. &e possible ways of avoiding bouncing universes raise the issues such as cusp and angular- BKL instability and issues of the bounce in the ekpyrotic momentum barrier, thus leaving this turnaround of nature scenario have been studied [9–12]. It has been explicitly quite ambiguous [1]. Interest in bouncing universes was confirmed that spatially flat nonsingular bouncing cos- declined after the onset of first cosmological singularity mologies corresponds to effective theories of gravity [13]. theorem [2, 3]. In particular, some new types of isotropic Libanov et al. [14] studied flat bouncing cosmological cosmological models were proposed without singularity [4]. models with the early time Genesis epoch in the context of However, a standard big bang cosmological model has also generalized Galileon theories. &ey found that the bouncing some shortcomings such as horizon problem, flatness issues, models either possessed these instabilities or had some transplanckian problem, and entropy problem. &erefore, in singularities. &e presence of a nonsingular bounce in a flat recent era, the investigation about bouncing cosmological universe may imply violation of the null energy condition solutions is an interesting and attractive topic for re- (NEC), which can be obtained via a ghost condensation searchers. &ese solutions involve the cosmological models phase. Kobayashi [15] argued that the NEC could be violated that replace the big bang cosmological singularity with a “big in the generalized Galileon theories supporting the possi- bounce” scenario, a smooth transition from contraction to bility of nonsingular cosmology. However, it was argued that the expansion phase [5, 6]. In this situation, the contraction on many occasions, cosmological solutions were plagued phase of the universe is dominated by matter, and also, a with instabilities. Furthermore, within the framework of the nonsingular bounce is occurred. Moreover, the density effective field theory and beyond Horndeski theory, it has 2 Advances in Astronomy brief structure of the study is as follows: in Section 2, some been shown that stable bouncing cosmologies can be con- structed [16–20]. &e investigation of some cosmological basic important preliminaries to develop a required back- ground for the analysis are provided. Section 3 is devoted to issues in extended gravity seems interesting because the big bang singularity could be replaced by a big bounce using provide cosmological solutions and a detailed graphical modified gravity models [21–25]. analysis. Last section provides a brief summary of the work Barragan et al. [26] studied oscillating cosmology in and conclusive remarks. f(R) Palatini formalism, and it was shown that the big bang singularities could be replaced by a big bounce without 2. Some Important Preliminaries violating energy conditions. In fact, the bounce is possible even for pressureless dust in f(R) gravity background. &e &e modified GB gravity can be described using the action [32] metric version of f(R) gravity has also been used to find the 1 √�� � √�� � 4 4 behavior of bouncing cosmology, and it was concluded that A � 􏽚 d x − g[R + f(G, T)] + 􏽚 d x − gL , (1) 2κ this behavior could solve the singularity problem in standard big bang cosmology [27]. Singh and his collaborators [28] where G and T represent the GB term and trace of the stress- investigated a cosmological model in a flat homogeneous energy tensor, respectively. L stands for the standard and isotropic background in the context of f(R, T) gravity. matter Lagrangian, R being the Ricci scalar, κ is a coupling &ey proposed the Hubble parameter in a functional form parameter, and g is the determinant of metric tensor. &e such that it fulfilled the successful bouncing criteria to in- modified field equations are obtained by metric variation of vestigate the solution of the gravitational field equations action above [32]. without any initial singularity. Oikonomou [29] showed how 2 μ] a cosmological bounce could be evolved by a vacuum f(G) R − g R + 􏽨2Rg ∇ − 2R∇ ∇ − 4g R ∇ ∇ ηζ ηζ ηζ η ζ ηζ μ ] gravity model, and the stability of the obtained solutions was addressed by analyzing a dynamical system of equations of 2 μ μ μ ] − 4R ∇ + 4R ∇ ∇ + 4R ∇ ∇ + 4R ∇ ∇ 􏽩f (G, T) ηζ η ζ μ η μ ημζ] G motion. Analytic bounce in nonlocal Einstein- –Gauss–Bonnet cosmology has been discussed, and the bouncing solutions are found to be stable during the bounce − g f(G, T) + 􏽨T + Θ 􏽩f (G, T) + 􏽨2RR − 4R R ηζ ηζ ηζ T ηζ η μζ phase [30]. Escofet and Elizalde [31] investigated some μ] μ]δ 2 Gauss–Bonnet (GB) extended gravity models exhibiting the − 4R R + 2R R 􏽩f (G, T) � − κ T , ημζ] ζμ]δ G ηζ bouncing behavior. &ey argued that how the addition of a (2) GB term to a viable gravity model could influence some properties and even the physical nature of the obtained where all the symbols involved have their usual meanings. cosmological solutions. In particular, some new dark energy Moreover, the subscript G and T appearing in the functions models can be proposed in which the equation of the state are to denote the partial derivatives and (EoS) parameter leads either to a big rip singularity or to a μ] Θ � g (δT /δg ). &e trace of equation (2) turns out to ζη μ] ζη bouncing solution evolving into a de Sitter spacetime. A new be theory by coupling GB term and trace of energy momentum R + κ T − (T + Θ)f (G, T) + 2f(G, T) + 2Gf (G, T) tensor has been proposed and named as f(G, T) gravity T G [32], and it was proved that due to the presence of extraforce, 2 ζη − 2R∇ f (G, T) + 4R ∇ ∇ f (G, T) � 0. G ζ η G the massive test particles could follow nongeodesic lines of geometry. &ough f(R, T) theories of gravity are the sim- (3) plest modifications with matter coupling, the construction of It is worth noticing that if f(G, T) � 0 is replaced in some viable f(R, T) gravity models is not an easy task. &e equation (3), GR is recovered as main reason behind this is that Ricci modified gravity may produce a strong coupling between dark energy and a R + κ T � 0. (4) nonrelativistic matter in the Einstein frame [33]. While, some modified GB models may be consistent with solar &e important aspect of equation (3) is that it relates R, system barriers under certain constraints [34]. Nojiri et al. G, and T differentially. However, as evident by the corre- [35] explored some f(G) gravity nonminimally coupled sponding GR version in equation (4), R and G are ma- models with matter Lagrangian and concluded that theories nipulated algebraically. &is clearly suggests that the with such coupling may unify the inflationary era with modified field equations will have more solutions than usual current cosmic expansion. &us, modified GB gravity with GR. &e covariant divergence of equation (2) is given by matter couplings can be more fascinating to study the f (G, T) ζ ζ universe in comparison with f(R, T) gravity. ∇ T � 􏽨􏼐T + Θ 􏼑∇ ln f (G, T)􏼁 ζη ζη ζη T κ − f (G, T) Motivated from the above discussions, this study fo- cussed to examine bouncing cosmology in the context of ζη ζ ζ f(G, T) theory of gravity. For this purpose, a GB cosmo- + ∇ Θ − ∇ T􏼕. ζη logical model with logarithmic trace term is considered. &e (5) analysis is based upon two important EoS parameters. A Advances in Astronomy 3 It can be seen that the conservation equation is not importantly, the SEC is violated on cosmological scales in covariantly divergent here as in the case of GR. It is due to the light of the recent observational data regarding the accelerating universe [37]. Also, the minimal condition for a the involvement of higher order derivatives of the matter components in the modified field equations due to the cosmological bounce rather than a big bang singularity matter coupling. &us, a little drawback is that this theory requires the violation of SEC [12]. Moreover, violation of might be suffered by divergences at cosmological scales. &is DEC is typically associated with either a large negative is an issue with other theories as well that includes higher cosmological constant or superluminal acoustic modes [49]. order terms of energy momentum tensor, such as the f(R, T) theory of gravity. However, to deal with the issue, 2.1. f(G, T) Gravity Model with Logarithmic Trace Term. some constraints are put to equation (5) to recover the Now, for FLRW spacetime (6) with perfect fluid, the field standard conservation equation. In present work, cosmology equation (2) takes the form in this modified theory by considering the flat Fried- 2 3 mann–Lemaitre–Robertson–Walker (FLRW) spacetime is a _ a _ (8) 6 − 24 f + Gf − f − 2(ρ + p)f � 2κ ρ, investigated: G G T 2 3 a a 2 2 2 2 2 2 ds � dt − a (t)􏽨dx + dy + dz 􏽩, (6) € _ _ € a _ a a aa _ € − 2 2 + + 16 f + 8 f − Gf + f � 2κ p, 􏼠 􏼡 G G G 2 2 and assumed that the universe is filled with perfect fluid: a a a T � (ρ + p)u u − pg , (9) (7) μ] μ ] μ] where f ≡ f(G, T) and f ≡ f (G, T) are considered for where a, ρ, and p represent the cosmic scale factors of G G the sake of simplicity. Due to complicated and highly universe, energy density, and pressure of the fluid, respec- nonlinear nature of field equations, sometimes it becomes tively. &e study of energy conditions involving these im- very difficult to choose a particular f(G, T) model which portant parameters (energy density and pressure) has many could provide some viable results both analytical and nu- significant applications in cosmology. For example, one can merical. &e simplest choice is to consider a linear com- easily investigate the validity of the second law of black hole bination [51]: thermodynamics and Hawking–Penrose singularity theo- rems by using energy conditions [36]. In relativistic cos- f(G, T) � f (G) + f (T). (10) 1 2 mology, many interesting constraints have been described by the use of energy conditions [37–46]. Mainly five different In present study, f (G) � G + λG , with λ being a real types of energy bounds are found in the literature: constant is proposed. &is choice is important as the similar power law f(G) gravity model has been studied with some Trace energy condition (TEC), now abandoned interesting results [50]. Most importantly, following the Null energy condition (NEC) work of Elizalde et al. [21], consider f (T) � 2βLog(T), Weak energy condition (WEC) where β is an arbitrary model parameter. &us, the proposed f(G, T) model takes the form Strong energy condition (SEC) Dominant energy condition (DEC) (11) f(G, T) � G + λG + 2βLog(T). &e TEC suggests that the trace of the energy-mo- To the best of our knowledge, this is the first such at- mentum tensor should always be negative (or positive tempt to consider logarithmic trace term in the study of depending on metric conventions). &is condition was f(G, T) cosmology. &us, in this case, TEC must be satisfied popular among the researchers during the decade of 1960. to obtain realistic results from the logarithmic term. However, once it was shown that stiff EoS, such as those which are appropriate for neutron stars, violates the TEC [47, 48]. &us, the study of this energy condition was not 2.2. Bouncing Cosmology and Equation of State. In recent further encouraged, and it is now completely abandoned, in years, there has been an increasing interest of the researchers fact no longer cited in the literature. However, the remaining in cosmological models that replace the big bang cosmo- four energy constraints are known as a necessary feature for logical singularity with a “big bounce,” a smooth transition the cosmological discussions. For an acceptable cosmolog- from contraction to the expansion phase. In order to resolve ical model, these constraints should be validated. For this the fundamental problems in cosmology, the study of purpose, the most important requirement is the positivity of cosmological dynamics in modified gravity seems interesting energy density. However, the negative pressure may indicate because the big bang singularity could be avoided by a big the presence of exotic matter. In fact, the violation of energy bounce using modified gravity models [21–25]. &e behavior conditions may lead to some fascinating cosmological fea- of the bouncing universe can be judged by the evolution of tures. &e violation of these conditions may yield some the scale factor and Hubble parameter. One of the indica- instabilities and ghost pathologies in the presence of a ca- tions is that the size of the scale factor gets contracted to nonical scalar field. &e SEC is currently the most heated some finite volume not necessarily zero and then shows an subject of discussions. It has been argued that the SEC increasing trend. Another possibility to indicate a bounce is should be violated in the inflationary era [48]. Most when the Hubble parameter becomes zero and then blows 4 Advances in Astronomy up. Mathematically, there must exist some finite points of analytic form of h(t) which could provide the nonzero value time at which the size of the universe attains a minimum at these points can be considered. &us, a specific form of value. Another indication is the violation of NEC for some h(t) is given by period of time near the neighborhood of the bounce point in ζ t h(t) � e (15) the context of FLRW spacetime. Moreover, the EoS pa- rameter goes in the negative range, especially a bouncing where ζ is any arbitrary real parameter. &us, the complete cosmology with ω ≈ − 1 justifies the current cosmic ex- parameterized form of the Hubble parameter turns out to be pansion [52–54]. &e most interesting aspect of studying ζ t bouncing cosmology is that the cosmic singularity problem H(t) � αe Sin(ξt). (16) can be avoided, geodesically complete evolution can be &is form of the Hubble parameter is important as it enabled, chaotic mixmaster behavior can be eliminated, the allows to obtain the behavior of the cosmic scale factor in the horizon problem can be resolved, the smoothness and later stages of the evolution of the universe. &is form of the flatness issues can be tackled, and the small entropy at the Hubble parameter provides the scale factor: onset of the expanding phase can be naturally explained [55]. EoS parameter is an important constituent to study ζ t αe (ζSin(ξt) − ξSin(ξt)) cosmological dynamics, in particular in the context of ⎝ ⎠ ⎛ ⎞ a(t) � κ exp , (17) 2 2 modified gravity. In this analysis, two interesting proposals ζ + ξ were discussed [21]. First, the possibility of obtaining a where κ is an arbitrary integration constant. Now, investi- bouncing solution in f(G, T) gravity described by the following EoS parameter is considered: gate bouncing cosmology in the below subsections for the abovementioned two different EoS cases. kLog(t + ε) (12) ω (t) � − − 1, 3.1. f(G, T) Cosmology: ω (t) � − (kLog (t + ε)/t) − 1. where k is any arbitrary constant, and ε is the very small real Here, the universe is assumed to be dominated by the matter parameter. It is interesting to notice from equation (12) that with the EoS given by (12). &e field equations (8) and (9) are ω varies from negative infinity as t ⟶ 0 to ω � − 1 (cosmic simplified using equation (12), for details see Appendix. &e expansion phase) when t � 1 − ε. Second, the bouncing evolution of energy density and pressure of the universe solution of f(G, T) gravity models by considering the using the scale factor (17) are shown in Figure 1. &e left plot following EoS is investigated: indicates that the energy density is negative within the ω (t) � − s, neighborhood of the bouncing point t � 0. &ough, it is not (13) Log t physical and one can get a better result (positive in the immediate neighborhood) by manipulating the parameters where r is a negative parameter, while s is a positive pa- involved. Since this negative trend is for a very small du- rameter. In this case, witness that ω varies from negative (r/s− 1) ration and positive energy density is witnessed very soon, infinity as t ⟶ 1 to the cosmic expansion era at t � e leave as it is. &e right plot indicates that the pressure is and moves on, eventually coming back to again the same negative, which might be an indication of accelerated ex- phase as t approaches positive infinity and s � 1. Elizalde pansion of the universe. &e left plot of Figure 2 depicts that et al. [21] obtained viable cosmological solutions in the NEC is violated near the neighborhood of the bouncing framework of f(R, T) theory of gravity using these two point. interesting choices of the EoS parameter. In this study, their &is justifies the indication of bouncing universe with work in the context of f(G, T) gravity is extended. violation of NEC for some period of time near the neigh- borhood of the bounce point in the context of FLRW 3. Cosmological Solutions and spacetime. Similarly, WEC and SEC are also violated as Numerical Analysis evident from left plots of Figures 1 and 2. &is also ensures the fact that the SEC is violated on cosmological scales in the In this section, the main focus is to discuss the evolution of light of the recent observational data regarding the accel- energy density and pressure profile by using some suitable erating universe [37]. Figure 3 describes that DEC and TEC choice of the scale factor. For this purpose, the possible are satisfied with the chosen values of parameters. &is choice of the Hubble parameter which could provide viable provides the justification that the chosen cosmological bouncing cosmology is first discussed. A well-known model is well behaved. In particular, the choice of param- functional form of the Hubble parameter as described in the eters strictly depends upon the evolution of energy density following equation is considered [21]. and TEC. It is worthwhile to mention here that TEC sometimes gets violated; however, in our case, it must be H(t) � αh(t)Sin(ξt), (14) satisfied due to the involvement of the logarithmic function where h(t) is an arbitrary smooth function, and α and ξ are in the f(G, T) gravity model. &e well behaved behavior of the real constants. Mathematically, it is an interesting form the scale factor and red shift function is shown in Figure 4. of the Hubble parameter as the trigonometric sine function According to a successful bouncing model, the Hubble vanishes at some periodic values of t. Furthermore, such an parameter passes through zero from H< 0 when the Advances in Astronomy 5 0.20 0.10 0.15 0.05 0.10 0.00 0.05 ρ p –0.05 0.00 –0.10 –0.05 –0.15 –0.10 –0.20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 1: Energy density and pressure profiles for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. 0.00004 0.2 0.00003 0.1 0.00002 0.0 0.00001 –0.1 0.00000 –0.2 –0.3 –0.00001 –0.4 –0.00002 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) (b) Figure 2: Evolution of NEC and SEC for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.0 0.0 –0.2 –0.1 –0.2 –0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 3: Evolution of DEC and TEC for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. universe contracts to H> 0 when the universe expands and € aa q � − . (18) H � 0 when the bouncing point occurs. &is feature of a 2 bouncing model in our case is evident from the left plot of &e negative trend of the deceleration parameter is Figure 5. &e deceleration parameter q in cosmology is shown in the right plot of Figure 5. &e behavior of the EoS defined as ρ + p ρ – p ρ – 3p ρ + 3p 6 Advances in Astronomy 0.47585 0.677684 0.47580 0.677683 0.677683 0.47575 0.677683 0.47570 0.677683 0.47565 0.677683 0 2468 10 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 4: Evolution of the scale factor and red shift function for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. 0.00004 –20000 0.00002 –40000 0.00000 –0.00002 –60000 –0.00004 –80000 –4 –2 0 2 4 02468 10 t t (a) (b) Figure 5: Evolution of Hubble and deceleration parameters for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. –0.99965 Conservation equation does not hold –0.99970 –0.99975 4 Conservation –0.99980 equation valid –0.99985 –0.99990 –0.99995 0 2468 10 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 6: Evolution of EoS parameter ω (t) � − (kLog(t + ε)/t) − 1 and behavior of conservation equation with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. parameter is depicted in the left plot of Figure 6. It is shown evolution of the universe, the value of EoS parameter is that the EoS parameter remains negative after the bouncing negative and closer to zero (as evident in our case). point, in particular ω ≈ − 1, which justifies the current One of the important aspects of this study is the dis- cosmic expansion [52–54]. In particular, bouncing universe cussion of conservation equation. Modified theories of in this case supports the lambda cold dark matter (Λ-CDM) gravity which involve matter curvature coupling do not act model; since when the cold dark matter dominates the as per usual conservation of GR. In case of f(G, T) gravity, H (t) a (t) z (t) q (t) Conservation equation Advances in Astronomy 7 this issue is evident from equation (5). Perhaps, there is a are similar to already obtained bouncing solutions in the little drawback that the theory might be plagued by diver- context of modified GB gravity without matter coupling [23]. gences at astrophysical scales. However, one can put the following constraint on equation (5) to deal with the issue and standard conservation equation can be recovered. 4. Outlook ζη ζ ζ ζ In present study, bouncing cosmology in the context of 􏼐T + Θ 􏼑∇ ln f (G, T)􏼁 + ∇ Θ − ∇ T � 0. ζη ζη T ζη f(G, T) theory of gravity is examined. For this purpose, a (19) GB cosmological model with logarithmic trace term, i.e., f(G, T) � G + λG + 2βLog(T), is considered. In this study, &e exact solution of this equation is very difficult to the possibility of obtaining bouncing solutions by consid- obtain due to highly nonlinear terms. However, in the ering two EoS parameters is investigated. A detailed current study, the parameters are in such a way that this graphical analysis is provided for discussing the obtained equation is partially satisfied. It can be seen in the right plot bouncing solutions. To best of our knowledge, this is the first of Figure 6 that conservation equation is satisfied in the such attempt in the frame-work of f(G, T) gravity. &e neighborhood of the bouncing point but deviates as the time main results of present study are itemized as follows. passes. All the above discussions and graphical analyses suggest that the proposed f(G, T) gravity model provides &e analysis is based upon two EoS parameters, i.e., good bouncing solutions with the chosen EoS parameters. ω (t) � − (kLog(t + ε)/t) − 1 and ω (t) � (r/Log t) − s. 1 2 &e evolution of energy density and pressure profiles of the universe for both these cases are shown in Figures 1 3.2. f(G, T) Cosmology: ω (t) � (r/Log t) − s. Now, the and 7. &e energy density is positive within the universe is considered as dominated by the matter with the neighborhood of bouncing points while the pressure EoS given in (13). Here, also the field equations (8) and (9) profiles are negative, which might be an indication of are simplified using equation (13) as in the previous case, for accelerated expansion of the universe. details see Appendix. &e evolution of energy density and As evident from Figures 2 and 8, NEC is violated near pressure of the universe is shown in Figure 7. Left plot the neighborhood of bouncing points. &is justifies the indicates that the energy density is positive within the indication of bouncing universe with violation of NEC neighborhood of the bouncing point t � 0. However, the for some period of time near the neighborhood of right plot indicates that the pressure is negative, which might bounce point in the context of FLRW spacetime. be an indication of accelerated expansion of the universe. Similarly, WEC and SEC are also violated for both these Left plot of Figure 8 depicts that NEC is violated near the cases. &is also ensures the fact that the SEC is violated neighborhood of the bouncing point. In this case, the in- on cosmological scales in the light of the recent ob- dication of bouncing universe with violation of NEC for servational data regarding the accelerating universe some period of time near the neighborhood of the bounce [37]. Figures 3 and 9 describe that DEC and TEC are point in the context of FLRW spacetime is also justified. satisfied with the chosen values of parameters. &is Here, WEC and SEC are also violated as evident from the left provides the justification that our chosen cosmological plot of Figures 7 and 8. Figure 9 describes that DEC and TEC model is well behaved. In particular, the choice of are satisfied with the chosen values of parameters. &is parameters strictly depends upon the evolution of provides the justification that the chosen cosmological energy density and TEC. It is worthwhile to mention model is well behaved. In particular, the choice of param- here that TEC sometimes gets violated; however, in our eters strictly depends upon the evolution of energy density case, it must be satisfied due to the involvement of and TEC. It is worthwhile to mention here that TEC logarithmic function in the f(G, T) gravity model. sometimes gets violated; however, in our case, it must be &e well behaved behavior of the scale factor and red satisfied due to the involvement of logarithmic function in shift function is shown in Figures 4 and 10. From the f(G, T) gravity model. &e well behaved behavior of the Figures 5 and 12, it is evident that for t< 0, H< 0, while scale factor and red shift function is shown in Figure 10. for t> 0, H> 0, so that around t � 0, i.e., in the early &e negative trend of deceleration parameter is shown in universe, the bouncing behavior of the universe can be the right plot of Figure 11, while the behavior of the EoS justified. Two bouncing points are shown in Figure 12 parameter is shown in the left plot of Figure 11. Here, the (a magnified view is also inserted in the figure for better EoS parameter also remains negative after the bouncing understanding), one around t ≈ − 7 and the other point. From Figure 12, it is evident that for t< 0, H< 0, while around t ≈ 6. &ese results are similar to already ob- for t> 0, H> 0, so that around t � 0, i.e., in the early uni- tained bouncing solutions in the context of modified verse, the bouncing behavior of the universe can be justified. GB gravity without matter coupling [23]. Two bouncing points are shown in Figure 12 (a magnified view is also inserted in the figure for better understanding), &e deceleration parameter q in cosmology is the one around t ≈ − 7 and the other around t ≈ 6. &ese results measure of the cosmic acceleration of the universe 8 Advances in Astronomy 0.095 –0.34 –0.36 0.090 –0.38 0.085 –0.40 ρ p 0.080 –0.42 0.075 –0.44 –0.46 0.070 –0.48 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 7: Energy density and pressure profiles for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. –0.28 –1.0 –0.30 –1.1 –0.32 –0.34 –1.2 –0.36 –1.3 –0.38 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 8: Evolution of NEC and SEC for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. 1.5 0.55 1.4 0.50 1.3 0.45 1.2 1.1 0.40 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 9: Evolution of DEC and TEC for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. ρ + p ρ – p ρ + 3p ρ – 3p Advances in Astronomy 9 0.7 1.0 0.6 0.9 0.5 0.8 0.4 0.3 0.7 0.2 0.6 0.1 0.5 0.0 (a) (b) Figure 10: Evolution of the scale factor and red shift function for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. –4.998 –4.999 –5 –5.000 –10 –15 –5.001 –20 –5.002 012 345 012345 (a) (b) Figure 11: Evolution of EoS and deceleration parameters with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. 0.00 –0.02 –1 –0.04 –0.06 –2 –7 –6 –5 –4 –3 –2 –1 0 –6 –4 –2 0 24 6 Figure 12: Evolution of the Hubble parameter for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. a (t) H (t) z (t) q (t) 10 Advances in Astronomy expansion. &e positive deceleration parameter cor- be plagued by divergences at astrophysical scales. responds to a decelerating model, while the negative However, by putting some suitable constraints on value provides cosmic expansion. &e negative trend of equation (5), the standard conservation equation has the deceleration parameter can be seen in the right plots been tried to be recovered. &e exact solution of the of Figures 5 and 11. &e behavior of EoS parameters is constraint equation is very difficult to obtain due to depicted in the left plots of Figures 6 and 11. It is shown highly nonlinear terms. However, in the current study, that EoS parameter remains negative after the bouncing the parameters have been set in such a way that this point, in particular ω ≈ − 1, which justifies the current equation is partially satisfied. It can be seen in the right cosmic expansion [52–54]. Moreover, the bouncing plot of Figure 6 that conservation equation is satisfied universe in the case of ω (t) � − (kLog(t + ε)/t) − 1 in the neighborhood of the bouncing point but deviates supports the Λ-CDM model; since when the cold dark as the time passes. matter dominates the evolution of the universe, the All the above itemized discussions suggest that the value of EoS parameter is negative and closer to zero (as proposed f(G, T) gravity model provides good bouncing evident in our case). solutions with the chosen EoS parameters. One of the important features of the current study is the discussion of conservation equation. Modified theories Appendix of gravity which involve matter curvature coupling do not act as per usual conservation of GR. In case of Simplified field equations for the case f(G, T) gravity, this issue is evident from equation (5). ω (t) � − (kLog(t + ε)/t) − 1: Perhaps, there is a little drawback that the theory might 2t 6 6 5 2 5 2 ρ � × 􏼐3ka a €Log(t + ε) + 4ta a € − 3ka a _ Log(t + ε) − 4ta a _ ka Log(t + ε)(3kLog(t + ε) + 4t) ˙ ˙ t t 5 5 2 2 3 4 2 + 864kλaaa _ Log(t + ε) + 1152λtaaa _ − 1152kλa a _ a € Log(t + ε) + 3456kλaa _ a € Log(t + ε) 6 2 2 3 4 2 6 2 3 − 2592kλa _ a €Log(t + ε) − 1536λta a _ a € + 4608λtaa _ a € − 3456λta _ a € − 576kλaa a _ a €Log(t + ε) 2 3 7 − 768λtaa a _ a € + β(− k)a Log(t + ε)􏼓, (A.1) 2(kLog(t + ε) + t) 6 6 5 2 € € _ p � − × 􏼐3ka aLog(t + ε) + 4ta a − 3ka a Log(t + ε) ka Log(t + ε)(3kLog(t + ε) + 4t) ˙ ˙ t t 5 2 5 5 2 2 3 4 2 − 4ta a _ + 864kλaaa _ Log(t + ε) + 1152λtaaa _ − 1152kλa a _ a € Log(t + ε) + 3456kλaa _ a € Log(t + ε) 6 2 2 3 4 2 6 2 3 − 2592kλa _ a €Log(t + ε) − 1536λta a _ a € + 4608λtaa _ a € − 3456λta _ a € − 576kλaa a _ a €Log(t + ε) 2 3 7 − 768λtaa a _ a € + β(− k)a Log(t + ε)􏼓. Advances in Astronomy 11 Simplified field equations for the case ω (t) � (r/Log t) − s: 2Log(t) ρ � − 7 2 2 2 2 2 a[t] 3r + 2rLog[t] − 6rsLog[t] − Log[t] − 2sLog[t] + 3s Log[t] 􏼐 􏼐 􏼑􏼑 6 6 6 5 2 5 2 5 2 × 3ra a € − 3sa Log(t)a € − a Log(t)a € − 3ra a _ + 3sa Log(t)a _ + a Log(t)a _ ˙ ˙ ˙ t t t 5 5 5 2 2 3 4 2 + 864λraaa _ − 864λsaaLog(t)a _ − 288λaaLog(t)a _ − 1152λra a _ a € + 3456λraa _ a € 6 2 2 3 4 2 6 − 2592λra _ a € + 1152λsa Log(t)a _ a € − 3456λsaLog(t)a _ a € + 2592λsLog(t)a _ a € 2 2 3 4 2 6 2 3 + 384λa Log(t)a _ a € − 1152λaLog(t)a _ a € + 864λLog(t)a _ a € − 576λraa a _ a € ˙ ˙ t t 2 3 2 3 7 7 7 _ € _ € + 576λsaa Log(t)a a + 192λaa Log(t)a a + β(− r)a + βsa Log(t) − βa Log(t)􏼓, (A.2) 2(r − sLog(t)) p � 7 2 2 2 2 2 a 􏼐3r − 6rsLog(t) + 2rLog(t) + 3s Log (t) − 2sLog (t) − Log (t)􏼑 6 6 6 5 2 5 2 5 2 5 × 􏼒− 3ra a € + 3sa Log(t)a € + a Log(t)a € + 3ra a _ − 3sa Log(t)a _ − a Log(t)a _ − 864λraaa _ ˙ 5 ˙ 5 t t 2 2 3 + 864λsaaLog(t)a _ + 288λaaLog(t)a _ + 1152λra a _ a € 4 2 6 2 2 3 4 2 6 − 3456λraa _ a € + 2592λra _ a € − 1152λsa Log(t)a _ a € + 3456λsaLog(t)a _ a € − 2592λsLog(t)a _ a € ˙ ˙ t t 2 2 3 4 2 6 2 3 2 3 − 384λa Log(t)a _ a € + 1152λaLog(t)a _ a € − 864λLog(t)a _ a € +576λraa a _ a € − 576λsaa Log(t)a _ a € 2 3 7 7 7 − 192λaa Log(t)a _ a € + βra − βsa Log(t) + βa Log(t)􏼓. 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Bouncing Cosmology in Gravity with Logarithmic Trace Term

Advances in Astronomy , Volume 2021 – Mar 3, 2021

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Hindawi Advances in Astronomy Volume 2021, Article ID 8852581, 12 pages https://doi.org/10.1155/2021/8852581 Research Article Bouncing Cosmology in f(G, T) Gravity with Logarithmic Trace Term M. Farasat Shamir National University of Computer and Emerging Sciences, Islamabad, Lahore Campus, Lahore, Pakistan Correspondence should be addressed to M. Farasat Shamir; farasat.shamir@nu.edu.pk Received 20 September 2020; Revised 11 January 2021; Accepted 2 February 2021; Published 3 March 2021 Academic Editor: Rafael Correa Copyright © 2021 M. Farasat Shamir. &is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. &is study is devoted to explore bouncing cosmology in the context of f(G, T) theory of gravity. For this purpose, a Gauss–Bonnet cosmological model with logarithmic trace term is considered. In particular, the possibility of obtaining bouncing solutions by considering two equations of state parameters is investigated. A graphical analysis is provided for analyzing the obtained bouncing solutions. &e energy conditions are discussed in detail. It is interesting to notice that null and strong energy conditions are violated near the neighborhood of bouncing points justifying the accelerating universe in the light of the recent observational data. &e behavior of the scale factor, red shift function, deceleration parameter, and Hubble parameter is also debated. An important feature of the current study is the discussion of conservation equation in f(G, T) gravity. &e possibility of some suitable constraint equations which recover the standard conservation equation is discussed, and all the free parameters are assumed accordingly. All the results in this study suggest that the proposed f(G, T) gravity model provides good bouncing solutions with the chosen EoS parameters. perturbations having spectrum consistent with the obser- 1. Introduction vational data can be produced [7]. &e so-called BKL in- Bouncing universes are possible alternatives to standard big stability can be witnessed after the contracting phase making bang cosmology. Unfortunately, some of the older studies of the universe anisotropic [8]. &e possible ways of avoiding bouncing universes raise the issues such as cusp and angular- BKL instability and issues of the bounce in the ekpyrotic momentum barrier, thus leaving this turnaround of nature scenario have been studied [9–12]. It has been explicitly quite ambiguous [1]. Interest in bouncing universes was confirmed that spatially flat nonsingular bouncing cos- declined after the onset of first cosmological singularity mologies corresponds to effective theories of gravity [13]. theorem [2, 3]. In particular, some new types of isotropic Libanov et al. [14] studied flat bouncing cosmological cosmological models were proposed without singularity [4]. models with the early time Genesis epoch in the context of However, a standard big bang cosmological model has also generalized Galileon theories. &ey found that the bouncing some shortcomings such as horizon problem, flatness issues, models either possessed these instabilities or had some transplanckian problem, and entropy problem. &erefore, in singularities. &e presence of a nonsingular bounce in a flat recent era, the investigation about bouncing cosmological universe may imply violation of the null energy condition solutions is an interesting and attractive topic for re- (NEC), which can be obtained via a ghost condensation searchers. &ese solutions involve the cosmological models phase. Kobayashi [15] argued that the NEC could be violated that replace the big bang cosmological singularity with a “big in the generalized Galileon theories supporting the possi- bounce” scenario, a smooth transition from contraction to bility of nonsingular cosmology. However, it was argued that the expansion phase [5, 6]. In this situation, the contraction on many occasions, cosmological solutions were plagued phase of the universe is dominated by matter, and also, a with instabilities. Furthermore, within the framework of the nonsingular bounce is occurred. Moreover, the density effective field theory and beyond Horndeski theory, it has 2 Advances in Astronomy brief structure of the study is as follows: in Section 2, some been shown that stable bouncing cosmologies can be con- structed [16–20]. &e investigation of some cosmological basic important preliminaries to develop a required back- ground for the analysis are provided. Section 3 is devoted to issues in extended gravity seems interesting because the big bang singularity could be replaced by a big bounce using provide cosmological solutions and a detailed graphical modified gravity models [21–25]. analysis. Last section provides a brief summary of the work Barragan et al. [26] studied oscillating cosmology in and conclusive remarks. f(R) Palatini formalism, and it was shown that the big bang singularities could be replaced by a big bounce without 2. Some Important Preliminaries violating energy conditions. In fact, the bounce is possible even for pressureless dust in f(R) gravity background. &e &e modified GB gravity can be described using the action [32] metric version of f(R) gravity has also been used to find the 1 √�� � √�� � 4 4 behavior of bouncing cosmology, and it was concluded that A � 􏽚 d x − g[R + f(G, T)] + 􏽚 d x − gL , (1) 2κ this behavior could solve the singularity problem in standard big bang cosmology [27]. Singh and his collaborators [28] where G and T represent the GB term and trace of the stress- investigated a cosmological model in a flat homogeneous energy tensor, respectively. L stands for the standard and isotropic background in the context of f(R, T) gravity. matter Lagrangian, R being the Ricci scalar, κ is a coupling &ey proposed the Hubble parameter in a functional form parameter, and g is the determinant of metric tensor. &e such that it fulfilled the successful bouncing criteria to in- modified field equations are obtained by metric variation of vestigate the solution of the gravitational field equations action above [32]. without any initial singularity. Oikonomou [29] showed how 2 μ] a cosmological bounce could be evolved by a vacuum f(G) R − g R + 􏽨2Rg ∇ − 2R∇ ∇ − 4g R ∇ ∇ ηζ ηζ ηζ η ζ ηζ μ ] gravity model, and the stability of the obtained solutions was addressed by analyzing a dynamical system of equations of 2 μ μ μ ] − 4R ∇ + 4R ∇ ∇ + 4R ∇ ∇ + 4R ∇ ∇ 􏽩f (G, T) ηζ η ζ μ η μ ημζ] G motion. Analytic bounce in nonlocal Einstein- –Gauss–Bonnet cosmology has been discussed, and the bouncing solutions are found to be stable during the bounce − g f(G, T) + 􏽨T + Θ 􏽩f (G, T) + 􏽨2RR − 4R R ηζ ηζ ηζ T ηζ η μζ phase [30]. Escofet and Elizalde [31] investigated some μ] μ]δ 2 Gauss–Bonnet (GB) extended gravity models exhibiting the − 4R R + 2R R 􏽩f (G, T) � − κ T , ημζ] ζμ]δ G ηζ bouncing behavior. &ey argued that how the addition of a (2) GB term to a viable gravity model could influence some properties and even the physical nature of the obtained where all the symbols involved have their usual meanings. cosmological solutions. In particular, some new dark energy Moreover, the subscript G and T appearing in the functions models can be proposed in which the equation of the state are to denote the partial derivatives and (EoS) parameter leads either to a big rip singularity or to a μ] Θ � g (δT /δg ). &e trace of equation (2) turns out to ζη μ] ζη bouncing solution evolving into a de Sitter spacetime. A new be theory by coupling GB term and trace of energy momentum R + κ T − (T + Θ)f (G, T) + 2f(G, T) + 2Gf (G, T) tensor has been proposed and named as f(G, T) gravity T G [32], and it was proved that due to the presence of extraforce, 2 ζη − 2R∇ f (G, T) + 4R ∇ ∇ f (G, T) � 0. G ζ η G the massive test particles could follow nongeodesic lines of geometry. &ough f(R, T) theories of gravity are the sim- (3) plest modifications with matter coupling, the construction of It is worth noticing that if f(G, T) � 0 is replaced in some viable f(R, T) gravity models is not an easy task. &e equation (3), GR is recovered as main reason behind this is that Ricci modified gravity may produce a strong coupling between dark energy and a R + κ T � 0. (4) nonrelativistic matter in the Einstein frame [33]. While, some modified GB models may be consistent with solar &e important aspect of equation (3) is that it relates R, system barriers under certain constraints [34]. Nojiri et al. G, and T differentially. However, as evident by the corre- [35] explored some f(G) gravity nonminimally coupled sponding GR version in equation (4), R and G are ma- models with matter Lagrangian and concluded that theories nipulated algebraically. &is clearly suggests that the with such coupling may unify the inflationary era with modified field equations will have more solutions than usual current cosmic expansion. &us, modified GB gravity with GR. &e covariant divergence of equation (2) is given by matter couplings can be more fascinating to study the f (G, T) ζ ζ universe in comparison with f(R, T) gravity. ∇ T � 􏽨􏼐T + Θ 􏼑∇ ln f (G, T)􏼁 ζη ζη ζη T κ − f (G, T) Motivated from the above discussions, this study fo- cussed to examine bouncing cosmology in the context of ζη ζ ζ f(G, T) theory of gravity. For this purpose, a GB cosmo- + ∇ Θ − ∇ T􏼕. ζη logical model with logarithmic trace term is considered. &e (5) analysis is based upon two important EoS parameters. A Advances in Astronomy 3 It can be seen that the conservation equation is not importantly, the SEC is violated on cosmological scales in covariantly divergent here as in the case of GR. It is due to the light of the recent observational data regarding the accelerating universe [37]. Also, the minimal condition for a the involvement of higher order derivatives of the matter components in the modified field equations due to the cosmological bounce rather than a big bang singularity matter coupling. &us, a little drawback is that this theory requires the violation of SEC [12]. Moreover, violation of might be suffered by divergences at cosmological scales. &is DEC is typically associated with either a large negative is an issue with other theories as well that includes higher cosmological constant or superluminal acoustic modes [49]. order terms of energy momentum tensor, such as the f(R, T) theory of gravity. However, to deal with the issue, 2.1. f(G, T) Gravity Model with Logarithmic Trace Term. some constraints are put to equation (5) to recover the Now, for FLRW spacetime (6) with perfect fluid, the field standard conservation equation. In present work, cosmology equation (2) takes the form in this modified theory by considering the flat Fried- 2 3 mann–Lemaitre–Robertson–Walker (FLRW) spacetime is a _ a _ (8) 6 − 24 f + Gf − f − 2(ρ + p)f � 2κ ρ, investigated: G G T 2 3 a a 2 2 2 2 2 2 ds � dt − a (t)􏽨dx + dy + dz 􏽩, (6) € _ _ € a _ a a aa _ € − 2 2 + + 16 f + 8 f − Gf + f � 2κ p, 􏼠 􏼡 G G G 2 2 and assumed that the universe is filled with perfect fluid: a a a T � (ρ + p)u u − pg , (9) (7) μ] μ ] μ] where f ≡ f(G, T) and f ≡ f (G, T) are considered for where a, ρ, and p represent the cosmic scale factors of G G the sake of simplicity. Due to complicated and highly universe, energy density, and pressure of the fluid, respec- nonlinear nature of field equations, sometimes it becomes tively. &e study of energy conditions involving these im- very difficult to choose a particular f(G, T) model which portant parameters (energy density and pressure) has many could provide some viable results both analytical and nu- significant applications in cosmology. For example, one can merical. &e simplest choice is to consider a linear com- easily investigate the validity of the second law of black hole bination [51]: thermodynamics and Hawking–Penrose singularity theo- rems by using energy conditions [36]. In relativistic cos- f(G, T) � f (G) + f (T). (10) 1 2 mology, many interesting constraints have been described by the use of energy conditions [37–46]. Mainly five different In present study, f (G) � G + λG , with λ being a real types of energy bounds are found in the literature: constant is proposed. &is choice is important as the similar power law f(G) gravity model has been studied with some Trace energy condition (TEC), now abandoned interesting results [50]. Most importantly, following the Null energy condition (NEC) work of Elizalde et al. [21], consider f (T) � 2βLog(T), Weak energy condition (WEC) where β is an arbitrary model parameter. &us, the proposed f(G, T) model takes the form Strong energy condition (SEC) Dominant energy condition (DEC) (11) f(G, T) � G + λG + 2βLog(T). &e TEC suggests that the trace of the energy-mo- To the best of our knowledge, this is the first such at- mentum tensor should always be negative (or positive tempt to consider logarithmic trace term in the study of depending on metric conventions). &is condition was f(G, T) cosmology. &us, in this case, TEC must be satisfied popular among the researchers during the decade of 1960. to obtain realistic results from the logarithmic term. However, once it was shown that stiff EoS, such as those which are appropriate for neutron stars, violates the TEC [47, 48]. &us, the study of this energy condition was not 2.2. Bouncing Cosmology and Equation of State. In recent further encouraged, and it is now completely abandoned, in years, there has been an increasing interest of the researchers fact no longer cited in the literature. However, the remaining in cosmological models that replace the big bang cosmo- four energy constraints are known as a necessary feature for logical singularity with a “big bounce,” a smooth transition the cosmological discussions. For an acceptable cosmolog- from contraction to the expansion phase. In order to resolve ical model, these constraints should be validated. For this the fundamental problems in cosmology, the study of purpose, the most important requirement is the positivity of cosmological dynamics in modified gravity seems interesting energy density. However, the negative pressure may indicate because the big bang singularity could be avoided by a big the presence of exotic matter. In fact, the violation of energy bounce using modified gravity models [21–25]. &e behavior conditions may lead to some fascinating cosmological fea- of the bouncing universe can be judged by the evolution of tures. &e violation of these conditions may yield some the scale factor and Hubble parameter. One of the indica- instabilities and ghost pathologies in the presence of a ca- tions is that the size of the scale factor gets contracted to nonical scalar field. &e SEC is currently the most heated some finite volume not necessarily zero and then shows an subject of discussions. It has been argued that the SEC increasing trend. Another possibility to indicate a bounce is should be violated in the inflationary era [48]. Most when the Hubble parameter becomes zero and then blows 4 Advances in Astronomy up. Mathematically, there must exist some finite points of analytic form of h(t) which could provide the nonzero value time at which the size of the universe attains a minimum at these points can be considered. &us, a specific form of value. Another indication is the violation of NEC for some h(t) is given by period of time near the neighborhood of the bounce point in ζ t h(t) � e (15) the context of FLRW spacetime. Moreover, the EoS pa- rameter goes in the negative range, especially a bouncing where ζ is any arbitrary real parameter. &us, the complete cosmology with ω ≈ − 1 justifies the current cosmic ex- parameterized form of the Hubble parameter turns out to be pansion [52–54]. &e most interesting aspect of studying ζ t bouncing cosmology is that the cosmic singularity problem H(t) � αe Sin(ξt). (16) can be avoided, geodesically complete evolution can be &is form of the Hubble parameter is important as it enabled, chaotic mixmaster behavior can be eliminated, the allows to obtain the behavior of the cosmic scale factor in the horizon problem can be resolved, the smoothness and later stages of the evolution of the universe. &is form of the flatness issues can be tackled, and the small entropy at the Hubble parameter provides the scale factor: onset of the expanding phase can be naturally explained [55]. EoS parameter is an important constituent to study ζ t αe (ζSin(ξt) − ξSin(ξt)) cosmological dynamics, in particular in the context of ⎝ ⎠ ⎛ ⎞ a(t) � κ exp , (17) 2 2 modified gravity. In this analysis, two interesting proposals ζ + ξ were discussed [21]. First, the possibility of obtaining a where κ is an arbitrary integration constant. Now, investi- bouncing solution in f(G, T) gravity described by the following EoS parameter is considered: gate bouncing cosmology in the below subsections for the abovementioned two different EoS cases. kLog(t + ε) (12) ω (t) � − − 1, 3.1. f(G, T) Cosmology: ω (t) � − (kLog (t + ε)/t) − 1. where k is any arbitrary constant, and ε is the very small real Here, the universe is assumed to be dominated by the matter parameter. It is interesting to notice from equation (12) that with the EoS given by (12). &e field equations (8) and (9) are ω varies from negative infinity as t ⟶ 0 to ω � − 1 (cosmic simplified using equation (12), for details see Appendix. &e expansion phase) when t � 1 − ε. Second, the bouncing evolution of energy density and pressure of the universe solution of f(G, T) gravity models by considering the using the scale factor (17) are shown in Figure 1. &e left plot following EoS is investigated: indicates that the energy density is negative within the ω (t) � − s, neighborhood of the bouncing point t � 0. &ough, it is not (13) Log t physical and one can get a better result (positive in the immediate neighborhood) by manipulating the parameters where r is a negative parameter, while s is a positive pa- involved. Since this negative trend is for a very small du- rameter. In this case, witness that ω varies from negative (r/s− 1) ration and positive energy density is witnessed very soon, infinity as t ⟶ 1 to the cosmic expansion era at t � e leave as it is. &e right plot indicates that the pressure is and moves on, eventually coming back to again the same negative, which might be an indication of accelerated ex- phase as t approaches positive infinity and s � 1. Elizalde pansion of the universe. &e left plot of Figure 2 depicts that et al. [21] obtained viable cosmological solutions in the NEC is violated near the neighborhood of the bouncing framework of f(R, T) theory of gravity using these two point. interesting choices of the EoS parameter. In this study, their &is justifies the indication of bouncing universe with work in the context of f(G, T) gravity is extended. violation of NEC for some period of time near the neigh- borhood of the bounce point in the context of FLRW 3. Cosmological Solutions and spacetime. Similarly, WEC and SEC are also violated as Numerical Analysis evident from left plots of Figures 1 and 2. &is also ensures the fact that the SEC is violated on cosmological scales in the In this section, the main focus is to discuss the evolution of light of the recent observational data regarding the accel- energy density and pressure profile by using some suitable erating universe [37]. Figure 3 describes that DEC and TEC choice of the scale factor. For this purpose, the possible are satisfied with the chosen values of parameters. &is choice of the Hubble parameter which could provide viable provides the justification that the chosen cosmological bouncing cosmology is first discussed. A well-known model is well behaved. In particular, the choice of param- functional form of the Hubble parameter as described in the eters strictly depends upon the evolution of energy density following equation is considered [21]. and TEC. It is worthwhile to mention here that TEC sometimes gets violated; however, in our case, it must be H(t) � αh(t)Sin(ξt), (14) satisfied due to the involvement of the logarithmic function where h(t) is an arbitrary smooth function, and α and ξ are in the f(G, T) gravity model. &e well behaved behavior of the real constants. Mathematically, it is an interesting form the scale factor and red shift function is shown in Figure 4. of the Hubble parameter as the trigonometric sine function According to a successful bouncing model, the Hubble vanishes at some periodic values of t. Furthermore, such an parameter passes through zero from H< 0 when the Advances in Astronomy 5 0.20 0.10 0.15 0.05 0.10 0.00 0.05 ρ p –0.05 0.00 –0.10 –0.05 –0.15 –0.10 –0.20 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 1: Energy density and pressure profiles for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. 0.00004 0.2 0.00003 0.1 0.00002 0.0 0.00001 –0.1 0.00000 –0.2 –0.3 –0.00001 –0.4 –0.00002 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 (a) (b) Figure 2: Evolution of NEC and SEC for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. 0.4 0.8 0.3 0.6 0.2 0.4 0.1 0.2 0.0 0.0 –0.2 –0.1 –0.2 –0.4 0.0 0.2 0.4 0.6 0.8 1.0 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 3: Evolution of DEC and TEC for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. universe contracts to H> 0 when the universe expands and € aa q � − . (18) H � 0 when the bouncing point occurs. &is feature of a 2 bouncing model in our case is evident from the left plot of &e negative trend of the deceleration parameter is Figure 5. &e deceleration parameter q in cosmology is shown in the right plot of Figure 5. &e behavior of the EoS defined as ρ + p ρ – p ρ – 3p ρ + 3p 6 Advances in Astronomy 0.47585 0.677684 0.47580 0.677683 0.677683 0.47575 0.677683 0.47570 0.677683 0.47565 0.677683 0 2468 10 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 4: Evolution of the scale factor and red shift function for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. 0.00004 –20000 0.00002 –40000 0.00000 –0.00002 –60000 –0.00004 –80000 –4 –2 0 2 4 02468 10 t t (a) (b) Figure 5: Evolution of Hubble and deceleration parameters for ω (t) � − (kLog(t + ε)/t) − 1 with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. –0.99965 Conservation equation does not hold –0.99970 –0.99975 4 Conservation –0.99980 equation valid –0.99985 –0.99990 –0.99995 0 2468 10 0.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 6: Evolution of EoS parameter ω (t) � − (kLog(t + ε)/t) − 1 and behavior of conservation equation with α � 0.001; κ � 0.5; ε � 0.001; ζ � 0.01; ξ � 0.01; k � − 0.001; β � 0.005; λ � 0.005. parameter is depicted in the left plot of Figure 6. It is shown evolution of the universe, the value of EoS parameter is that the EoS parameter remains negative after the bouncing negative and closer to zero (as evident in our case). point, in particular ω ≈ − 1, which justifies the current One of the important aspects of this study is the dis- cosmic expansion [52–54]. In particular, bouncing universe cussion of conservation equation. Modified theories of in this case supports the lambda cold dark matter (Λ-CDM) gravity which involve matter curvature coupling do not act model; since when the cold dark matter dominates the as per usual conservation of GR. In case of f(G, T) gravity, H (t) a (t) z (t) q (t) Conservation equation Advances in Astronomy 7 this issue is evident from equation (5). Perhaps, there is a are similar to already obtained bouncing solutions in the little drawback that the theory might be plagued by diver- context of modified GB gravity without matter coupling [23]. gences at astrophysical scales. However, one can put the following constraint on equation (5) to deal with the issue and standard conservation equation can be recovered. 4. Outlook ζη ζ ζ ζ In present study, bouncing cosmology in the context of 􏼐T + Θ 􏼑∇ ln f (G, T)􏼁 + ∇ Θ − ∇ T � 0. ζη ζη T ζη f(G, T) theory of gravity is examined. For this purpose, a (19) GB cosmological model with logarithmic trace term, i.e., f(G, T) � G + λG + 2βLog(T), is considered. In this study, &e exact solution of this equation is very difficult to the possibility of obtaining bouncing solutions by consid- obtain due to highly nonlinear terms. However, in the ering two EoS parameters is investigated. A detailed current study, the parameters are in such a way that this graphical analysis is provided for discussing the obtained equation is partially satisfied. It can be seen in the right plot bouncing solutions. To best of our knowledge, this is the first of Figure 6 that conservation equation is satisfied in the such attempt in the frame-work of f(G, T) gravity. &e neighborhood of the bouncing point but deviates as the time main results of present study are itemized as follows. passes. All the above discussions and graphical analyses suggest that the proposed f(G, T) gravity model provides &e analysis is based upon two EoS parameters, i.e., good bouncing solutions with the chosen EoS parameters. ω (t) � − (kLog(t + ε)/t) − 1 and ω (t) � (r/Log t) − s. 1 2 &e evolution of energy density and pressure profiles of the universe for both these cases are shown in Figures 1 3.2. f(G, T) Cosmology: ω (t) � (r/Log t) − s. Now, the and 7. &e energy density is positive within the universe is considered as dominated by the matter with the neighborhood of bouncing points while the pressure EoS given in (13). Here, also the field equations (8) and (9) profiles are negative, which might be an indication of are simplified using equation (13) as in the previous case, for accelerated expansion of the universe. details see Appendix. &e evolution of energy density and As evident from Figures 2 and 8, NEC is violated near pressure of the universe is shown in Figure 7. Left plot the neighborhood of bouncing points. &is justifies the indicates that the energy density is positive within the indication of bouncing universe with violation of NEC neighborhood of the bouncing point t � 0. However, the for some period of time near the neighborhood of right plot indicates that the pressure is negative, which might bounce point in the context of FLRW spacetime. be an indication of accelerated expansion of the universe. Similarly, WEC and SEC are also violated for both these Left plot of Figure 8 depicts that NEC is violated near the cases. &is also ensures the fact that the SEC is violated neighborhood of the bouncing point. In this case, the in- on cosmological scales in the light of the recent ob- dication of bouncing universe with violation of NEC for servational data regarding the accelerating universe some period of time near the neighborhood of the bounce [37]. Figures 3 and 9 describe that DEC and TEC are point in the context of FLRW spacetime is also justified. satisfied with the chosen values of parameters. &is Here, WEC and SEC are also violated as evident from the left provides the justification that our chosen cosmological plot of Figures 7 and 8. Figure 9 describes that DEC and TEC model is well behaved. In particular, the choice of are satisfied with the chosen values of parameters. &is parameters strictly depends upon the evolution of provides the justification that the chosen cosmological energy density and TEC. It is worthwhile to mention model is well behaved. In particular, the choice of param- here that TEC sometimes gets violated; however, in our eters strictly depends upon the evolution of energy density case, it must be satisfied due to the involvement of and TEC. It is worthwhile to mention here that TEC logarithmic function in the f(G, T) gravity model. sometimes gets violated; however, in our case, it must be &e well behaved behavior of the scale factor and red satisfied due to the involvement of logarithmic function in shift function is shown in Figures 4 and 10. From the f(G, T) gravity model. &e well behaved behavior of the Figures 5 and 12, it is evident that for t< 0, H< 0, while scale factor and red shift function is shown in Figure 10. for t> 0, H> 0, so that around t � 0, i.e., in the early &e negative trend of deceleration parameter is shown in universe, the bouncing behavior of the universe can be the right plot of Figure 11, while the behavior of the EoS justified. Two bouncing points are shown in Figure 12 parameter is shown in the left plot of Figure 11. Here, the (a magnified view is also inserted in the figure for better EoS parameter also remains negative after the bouncing understanding), one around t ≈ − 7 and the other point. From Figure 12, it is evident that for t< 0, H< 0, while around t ≈ 6. &ese results are similar to already ob- for t> 0, H> 0, so that around t � 0, i.e., in the early uni- tained bouncing solutions in the context of modified verse, the bouncing behavior of the universe can be justified. GB gravity without matter coupling [23]. Two bouncing points are shown in Figure 12 (a magnified view is also inserted in the figure for better understanding), &e deceleration parameter q in cosmology is the one around t ≈ − 7 and the other around t ≈ 6. &ese results measure of the cosmic acceleration of the universe 8 Advances in Astronomy 0.095 –0.34 –0.36 0.090 –0.38 0.085 –0.40 ρ p 0.080 –0.42 0.075 –0.44 –0.46 0.070 –0.48 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 7: Energy density and pressure profiles for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. –0.28 –1.0 –0.30 –1.1 –0.32 –0.34 –1.2 –0.36 –1.3 –0.38 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 8: Evolution of NEC and SEC for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. 1.5 0.55 1.4 0.50 1.3 0.45 1.2 1.1 0.40 0.2 0.4 0.6 0.8 1.0 0.2 0.4 0.6 0.8 1.0 t t (a) (b) Figure 9: Evolution of DEC and TEC for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. ρ + p ρ – p ρ + 3p ρ – 3p Advances in Astronomy 9 0.7 1.0 0.6 0.9 0.5 0.8 0.4 0.3 0.7 0.2 0.6 0.1 0.5 0.0 (a) (b) Figure 10: Evolution of the scale factor and red shift function for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. –4.998 –4.999 –5 –5.000 –10 –15 –5.001 –20 –5.002 012 345 012345 (a) (b) Figure 11: Evolution of EoS and deceleration parameters with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. 0.00 –0.02 –1 –0.04 –0.06 –2 –7 –6 –5 –4 –3 –2 –1 0 –6 –4 –2 0 24 6 Figure 12: Evolution of the Hubble parameter for ω (t) � (r/Log t) − s with α � 0.2222; κ � 0.0256; ζ � 0.5; ξ � 0.5; β � − 0.04858; λ � 0.5; r � − 0.0005; s � 5. a (t) H (t) z (t) q (t) 10 Advances in Astronomy expansion. &e positive deceleration parameter cor- be plagued by divergences at astrophysical scales. responds to a decelerating model, while the negative However, by putting some suitable constraints on value provides cosmic expansion. &e negative trend of equation (5), the standard conservation equation has the deceleration parameter can be seen in the right plots been tried to be recovered. &e exact solution of the of Figures 5 and 11. &e behavior of EoS parameters is constraint equation is very difficult to obtain due to depicted in the left plots of Figures 6 and 11. It is shown highly nonlinear terms. However, in the current study, that EoS parameter remains negative after the bouncing the parameters have been set in such a way that this point, in particular ω ≈ − 1, which justifies the current equation is partially satisfied. It can be seen in the right cosmic expansion [52–54]. Moreover, the bouncing plot of Figure 6 that conservation equation is satisfied universe in the case of ω (t) � − (kLog(t + ε)/t) − 1 in the neighborhood of the bouncing point but deviates supports the Λ-CDM model; since when the cold dark as the time passes. matter dominates the evolution of the universe, the All the above itemized discussions suggest that the value of EoS parameter is negative and closer to zero (as proposed f(G, T) gravity model provides good bouncing evident in our case). solutions with the chosen EoS parameters. One of the important features of the current study is the discussion of conservation equation. Modified theories Appendix of gravity which involve matter curvature coupling do not act as per usual conservation of GR. In case of Simplified field equations for the case f(G, T) gravity, this issue is evident from equation (5). ω (t) � − (kLog(t + ε)/t) − 1: Perhaps, there is a little drawback that the theory might 2t 6 6 5 2 5 2 ρ � × 􏼐3ka a €Log(t + ε) + 4ta a € − 3ka a _ Log(t + ε) − 4ta a _ ka Log(t + ε)(3kLog(t + ε) + 4t) ˙ ˙ t t 5 5 2 2 3 4 2 + 864kλaaa _ Log(t + ε) + 1152λtaaa _ − 1152kλa a _ a € Log(t + ε) + 3456kλaa _ a € Log(t + ε) 6 2 2 3 4 2 6 2 3 − 2592kλa _ a €Log(t + ε) − 1536λta a _ a € + 4608λtaa _ a € − 3456λta _ a € − 576kλaa a _ a €Log(t + ε) 2 3 7 − 768λtaa a _ a € + β(− k)a Log(t + ε)􏼓, (A.1) 2(kLog(t + ε) + t) 6 6 5 2 € € _ p � − × 􏼐3ka aLog(t + ε) + 4ta a − 3ka a Log(t + ε) ka Log(t + ε)(3kLog(t + ε) + 4t) ˙ ˙ t t 5 2 5 5 2 2 3 4 2 − 4ta a _ + 864kλaaa _ Log(t + ε) + 1152λtaaa _ − 1152kλa a _ a € Log(t + ε) + 3456kλaa _ a € Log(t + ε) 6 2 2 3 4 2 6 2 3 − 2592kλa _ a €Log(t + ε) − 1536λta a _ a € + 4608λtaa _ a € − 3456λta _ a € − 576kλaa a _ a €Log(t + ε) 2 3 7 − 768λtaa a _ a € + β(− k)a Log(t + ε)􏼓. Advances in Astronomy 11 Simplified field equations for the case ω (t) � (r/Log t) − s: 2Log(t) ρ � − 7 2 2 2 2 2 a[t] 3r + 2rLog[t] − 6rsLog[t] − Log[t] − 2sLog[t] + 3s Log[t] 􏼐 􏼐 􏼑􏼑 6 6 6 5 2 5 2 5 2 × 3ra a € − 3sa Log(t)a € − a Log(t)a € − 3ra a _ + 3sa Log(t)a _ + a Log(t)a _ ˙ ˙ ˙ t t t 5 5 5 2 2 3 4 2 + 864λraaa _ − 864λsaaLog(t)a _ − 288λaaLog(t)a _ − 1152λra a _ a € + 3456λraa _ a € 6 2 2 3 4 2 6 − 2592λra _ a € + 1152λsa Log(t)a _ a € − 3456λsaLog(t)a _ a € + 2592λsLog(t)a _ a € 2 2 3 4 2 6 2 3 + 384λa Log(t)a _ a € − 1152λaLog(t)a _ a € + 864λLog(t)a _ a € − 576λraa a _ a € ˙ ˙ t t 2 3 2 3 7 7 7 _ € _ € + 576λsaa Log(t)a a + 192λaa Log(t)a a + β(− r)a + βsa Log(t) − βa Log(t)􏼓, (A.2) 2(r − sLog(t)) p � 7 2 2 2 2 2 a 􏼐3r − 6rsLog(t) + 2rLog(t) + 3s Log (t) − 2sLog (t) − Log (t)􏼑 6 6 6 5 2 5 2 5 2 5 × 􏼒− 3ra a € + 3sa Log(t)a € + a Log(t)a € + 3ra a _ − 3sa Log(t)a _ − a Log(t)a _ − 864λraaa _ ˙ 5 ˙ 5 t t 2 2 3 + 864λsaaLog(t)a _ + 288λaaLog(t)a _ + 1152λra a _ a € 4 2 6 2 2 3 4 2 6 − 3456λraa _ a € + 2592λra _ a € − 1152λsa Log(t)a _ a € + 3456λsaLog(t)a _ a € − 2592λsLog(t)a _ a € ˙ ˙ t t 2 2 3 4 2 6 2 3 2 3 − 384λa Log(t)a _ a € + 1152λaLog(t)a _ a € − 864λLog(t)a _ a € +576λraa a _ a € − 576λsaa Log(t)a _ a € 2 3 7 7 7 − 192λaa Log(t)a _ a € + βra − βsa Log(t) + βa Log(t)􏼓. 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