Bounce cosmology in f(\mathcal{R}) gravity
aa r X i v : . [ g r- q c ] F e b Bounce cosmology in f ( R ) gravity M. Ilyas ∗ and W. U. Rahman † Institute of Physics, Gomal University,Dera Ismail Khan, 29220, Khyber Pakhtunkhwa, Pakistan Department of Physics, Abdul Wali khan University Mardan,Mardan, 23200, Pakistan
Abstract
In this paper, we analyze the modified f ( R ) gravity models in Friedmann–Lemaˆıtre–Robertson–Walker (FLRW) background. The actions of bouncing cosmology are stud-ied under consideration of different viable models in f ( R ) gravity theory that canresolve the difficulty of singularity in standard Big-Bang cosmology. Under differentviable models in f ( R ) gravity theory, the cosmological constraints are plotted in pro-visions of cosmic-time, then investigated the bounce circumstance. In addition, thered-shift parameter is used to reconstruct the modified gravity, and compile the cos-mological parameters that infer accelerated universe expansion. Finally, the situationstability is evaluated with a sound speed feature, which illustrates late-time stability. General theory of relativity has explained many specification of universe by the various theo-ries and observational evidence. The observational components of Λ-Cold dark space modelis consistent with all cosmological observations, but it suffers from such differences such ascosmic coincidence or tuning [1–3]. Since an unusual drop in the observed energy streamsfrom cosmic radiation base radiation, massive systems, red-shift as well as supernovae TypeIa assessments the rapid expansion of the earth has become apparent [4–6]. The reasonsbehind this curious and riddling phenomena referred to such discoveries as dark energy (DE) ∗ ilyas [email protected] † [email protected] H ( t ), transitfrom H ( t ) < H ( t ) > H ( t ) = 0 [62–65].Recently, several articles have been studied due to various theories and observationalevidence about the splitting of universe while this growing is undergoing to moving an ac-celerating stage. This study was arise due to the discovery in the type Ia supernova [66]connected with long scale arrangement [67], and background with cosmic microwave [68] .We have studied the accelerated growing is appropriate to a mysterious energy (known as theDE) that is approximately 70% of the whole universe. As the negative pressure in universewith perfect fluid and the equation of state (EoS) parameter, ω , which is not greater than − f ( R ) gravity, in which Ricci scalar, R , is replaced by an arbitrary function, f ( R ),in the Hilbert-Einstein gravitational action [86]. f ( R ) is actually a family of theories, each one defined by a different arbitrary function.The accelerated expansion and structure formation of the Universe were studied withoutadding unknown forms of DE or DM. Some models in f ( R ) gravity may be inspired by cor-rections arising from the reconciliation of GR with quantum mechanics (e.g. string theories2tc.). The modified f ( R ) gravity was suggested in 1970 and a wide range of phenomenacan be produced from this theory by adopting different functions; however, many functionalforms can now be ruled out on observational grounds, or because of pathological theoreticalproblems.The modified f ( R ) gravity is a best alternative way of instead of the standard model ofgravity which is known as a basis of DE (for reviews on modified f ( R ) theory of gravity,see [87–92]).In this study, in the big bang theory, we will avoid the starting singularity by bouncingmodel with the advantage of different models in modified f ( R ) theory. Furthermore, we willprove that there is one of the speeded phase-shift from a starting contracting-phase to thegrowing-phase. The problem is well explain by the temporal derivative of the scale-factorwhich is ˙ a ( t ), during contraction decreases ( ˙ a ( t ) <
0) and in the increasing phase ( ˙ a ( t ) > a ( t ) = 0). The already men-tioned story will be confirmed by the corresponding figures. from the above discussion we saythat the dark energy that is reconstructed by red shift parameter is a source of f ( R ) gravity.Therefore parametrization for f ( R ) will be found, with the help of motivation acceleratedgrowing of the world can be showed. Finally, the consistency of the model is analyzed suchthat the subjectivity of adiabatic disorder is assumed to be a thermodynamic framework.Sound speed is used to research the stability of the system as a specific purpose.We attempt to explain the following the f ( R ) Gravity to recreate by red-shift factor asan origin of DE. We would to discuss a parameterizations for f ( R ) gravity than we candefine in this inspiration as the expanding of the accelerated universe expansion. Finally,the consistency of the study will be tested such that the University is seen under adiabaticperturbation as either a thermodynamic device. So, the stabilization of the model is studiedusing a valuable feature called the sound speed.In this paper, we study the modified f ( R ) gravity and solve the field equations for FLRWmetric in the framework of perfect fluid in Sec. 2. While in Sec. 4, we analyze the bouncingbehavior by the Hubble parameter and scale-factor, by assuming different models in f ( R )gravity. Then in Sec. 5, the existing red-shift parameter model is reconstructed. Then,effective pressure and effective energy density is redefined in term of red-shift parameterand also the cosmological parameters will also be specified. In Sec. 6, We investigate andexamine the stability of the models. The brief description and conclusion of our results aresummarized in last section. 3 The Modified f ( R ) gravity theory Let us begin by analyzing the implications of the modified f ( R ) gravity A f ( R ) = Z √− g (cid:2) f ( R ) + 2 κ L m (cid:3) d x, (1)where f ( R ) is an arbitrary function of Ricci scalar while κ ≡ πG is coupling constant, and L m is Lagrangian matter density. By variational principles, the field equation becomes f R R λρ − f g λρ − [ ∇ λ ∇ ρ − g λρ (cid:3) ] f R = κ T ( m ) λρ , (2)where ∇ λ , (cid:3) ≡ ∇ λ ∇ λ , T ( m ) λρ are the covariant derivative, D’Alembert’s operator and theenergy-momentum tensor respectively, while f R = dfd R . The quantity f R contains the secondorder derivatives of the metric variables and the trace of Eq.(2) gives3 (cid:3) f R + Rf R − f ( R ) − κ T = 0 , (3)where T ≡ T ( m ) ρρ . The above equation is a second order differential equation in f R , unlikelythe trace of field equation in general theory of relativity is reduces to R + κ T = 0. Thispoints f R as a source of generating scalar degrees of freedom in f ( R ) theory. As T = 0this conditions doesn’t essentially indicates the constant value (or vanishing) of R , is thedynamics and Eq.(3) is a fruitful mathematical methods to study various interesting andhidden cosmic arena, e.g. stability, Newtonian limit and so on. The constraint, constantlyRicci scalar as well as T αβ = 0, reduces Eq.(3) as R f R − f ( R ) = 0 , (4)which is known as the Ricci algebraic equation after selecting any viable formulations ofmodified f ( R ) gravity model. By the roots of the above mention equation, i.e., R = Λ(assume), so Eq.(3) gives R αβ = g αβ Λ4 , (5)We let FRLW background metric as ds = a dr + a [ dθ + r sin θdφ ] − dt , (6)here a is the scale-factor that is the function of t . Then the Ricci scalar R for above metric(Eq. 6) is R = 6 ˙ H + 12 H , (7)4ere H = ˙ aa is known as the Hubble parameter while dot represent derivative w.r.t cosmic-time.By solving the field Eq. (2) for metric Eq. (6), we get3 H f R = κ ρ m + 12 [ R f R − f ] − H ˙ R f RR , (8a) − [2 ˙ H + 3 H ] f R = κ p m −
12 [ R f R − f ] + ˙ R f RRR + 2 H ˙ R f RR + ¨ R f RR , (8b)here ρ m , p m are energy density and pressure respectively. By using conservation equation, ∇ α T ( m ) αβ = 0, along with EoS Parameter, ω m = p m ρ m , one find˙ ρ m + 3 ρ m ( ω m + 1) H = 0 , (9)the above equation yields the solution as ρ m = ρ m a − ω m +1) . (10)Furthermore, by using the standard Friedmann equations to compare the present ap-proach ρ eff = 3 κ H and p eff = − κ (3 H + 2 ˙ H ) , we can rewrite Eqs. (8) as ρ eff = ρ m + ρ f ( R ) = κ − (cid:20) κ ρ m + 3 H (1 − f R ) −
12 ( f − R f R ) − H ˙ R f RR (cid:21) , (11a) p eff = p m + p f ( R ) = κ − (cid:20) κ p m − (3 H + 2 ˙ H )(1 − f R )+ 12 ( f − R f R ) + ˙ R f RRR + 2 H ˙ R f RR + ¨ R f RR (cid:21) , (11b)here ρ eff , p eff are effective energy density and pressure, while ρ f ( R ) = κ − h H (1 − f R ) − ( f − R f R ) − H ˙ R f RR i , (12a) p f ( R ) = κ − h − (2 ˙ H + 3 H )(1 − f R ) + ( f − R f R ) + ˙ R f RRR + 2 H ˙ R f RR + ¨ R f RR i . (12b)5n the case of effective terms, the conservation equation can be rewritten with the helpof using Eqs. (11) as ˙ ρ eff + 3 ρ eff H (1 + ω eff ) = 0 , (13)Here ω eff = p eff ρ eff = − H H ! , (14)is the parameter of effective EoS . By using the method in Refs [93, 94], we reconstruct the modified gravitational models withthe help of introducing proper functions P ( t ) and Q ( t ) of a scalar field t , which is interpretedas the cosmic time, the action in Eq. (1) with the absence of matter is written as I = Z dx √− g [ P ( t ) R + Q ( t ) + L m ] (15)then by solving this equation with eq. (8) we get Q ( t ) = − H P ′ ( t ) − P ( t ) H ( t ) (16)while the second equation becomes P ′′ ( t ) − H ( t ) P ′ ( t ) + 2 P ( t ) H ′ ( t ) = 0 (17)We solve the above differential equation for the power-law scale factor and then we usethe solution to write down the general form of f ( R ) as follow f ( R ) = P ( R ) R + Q ( R ) (18)The scale factor for power law model is given by a ( t ) = βt (2 n ) (19)Where β is constant and n is an integer. The behavior of this type of bouncing is shown infig.(1)By using this scale factor and solving for P ( t ) and Q ( t ), we get. P ( t ) = t n − √ n ( n +5)+1+ (cid:16) c t √ n ( n +5)+1 + c (cid:17) , (20)6 - - - H t L Figure 1: Hubble parameter for Power Law7 ( t ) = 6 nt (cid:16) n − √ n ( n +5)+1 − (cid:17) h c (cid:16) − n + p n ( n + 5) + 1 − (cid:17) (21) − c (cid:16) n + p n ( n + 5) + 1 + 1 (cid:17) t √ n ( n +5)+1 i (22)The cosmic time in term of ricci scalar is written as t ± = " ± √ p n ( − n ) √ R (23)By putting these equations in Eq. (18), we get f ( R ) = 14 n − (cid:16) n − √ n ( n +5)+1 − (cid:17) (cid:16) n − √ n ( n +5)+1+1 (cid:17) (24) R [ p n (4 n − √ R ] n − √ n ( n +5)+1+ [ c (2 n + p n ( n + 5) + 1 −
3) (25) − c √ n ( n +5)+1 √ n ( n +5)+1 {− n + p n ( n + 5) + 1 + 3 o (26)[ p n (4 n − √ R ] √ n ( n +5)+1 (27)Here, c , c are constants. We can see that this is valid for every n except 0 ≤ n ≤ − n = 1 we get f ( R ) = 19 c R / − c R (28) f ( R ) gravity Here, we will investigate the rebound conditions in modified f ( R ) gravity. As an evolvingworld emerges, the world transitions fluctuations into an expanding stage from an earliercontracting period. This stage transition leads to a non-singular outcome in the Big Bangcosmological norm [62–65].Consequently, the Hubble parameter moves for a good bounce from H ( t ) < H ( t ) > H ( t ) = 0. We may also claim the dilemma of bouncing World as regardsthe scale-factor, i.e. that we have a decline in scale-factor even during contracting stage as˙ a ( t ) <
0, and we have a development in the expanding process ˙ a ( t ) > a ( t ) = 0 and this point around ¨ a ( t ) > H ( t ) < H ( t ) >
0, since its derivativemust be greater than zero in the bounce stage as regards time evolution for its bouncingworld ˙ H bounce = − κ ω eff ) ρ eff > , (29)We deduce the condition ρ eff > ω eff < − H bounce > H bounce = 0, such that the bouncing stage (29) in thebounce stage by Eqs. (11) be assumed as˙ H bounce = (2 f R ) − h − κ ( ρ m + p m ) + ˙ R f RRR + ¨ R f RR i > . (30)In the following, we will investigate bouncing behavior with the advantage of particularchoices in f ( R ) models • Model 1We consider the model with quadratic corrections firstly proposed by Starobinsky [95] f ( R ) = R + α R (31)where α is constants. • Model 2we take the Ricci scalar exponential corrections to GR [96] f ( R ) = R + α R (cid:16) e ( − R γ ) − (cid:17) (32)where α and γ are constants. • Model 3One of the cubic correction to GR [97] f ( R ) = R + α R (1 + γ R ) (33)where γ and α are arbitrary constants and if γ >> R , Therefore the quadratic modelis suitable model. This one. the case in doubt α R (1 + γ R ) << R can be employedto maximise the influence of cubic terms on quadratic terms.9igure 2: The scale-factor, a ( t ), in terms of cosmic-time under different viable models.Figure 3: The Hubble parameter, H ( t ), in terms of cosmic-time under different viable models. • Model 4 f ( R ) = R + α R (cid:18) γLog [ R µ ] (cid:19) (34)This model contain the logarithm term which describe the universe evolution in theabsence of DE [98].Moreover, by inserting models (31),(32),(33) and (34) into Eqs. (11) separately, andsolving numerically. By plotting the numerical solutions, the cosmological parameters caneasily be drawn in terms of cosmic-time as shown in Figs. 2-3.The bouncing behavior under consideration of four different viable models are:In case of model 1, we can detect the bouncing action in the first part of Fig. 2. TheHubble parameter moves from the point of bounce ( H ( t ∼ − . H ( t ) < ρ eff , and pressure, p eff , in terms of cosmic-time under differentviable models.Figure 5: The energy density, ρ eff , and pressure, p eff , in terms of cosmic-time under differentviable models.Figure 6: The EoS parameter, ω eff , in terms of cosmic-time under different viable models.11 ( t ) >
0, while the lowest scale-factor is or ˙ a ( t ∼ − . H ( t ∼ − . H ( t ) < H ( t ) >
0, and in other hand side, we see that the minimal element for the scale is or˙ a ( t ∼ − . H ( t ∼ − . H ( t ) < H ( t ) >
0, and In another hand, we can see how the minimal element for thescale or ˙ a ( t ∼ − . H ( t ∼ − . H ( t ) < H ( t ) >
0, and in otherhand, we can see how the minimal element for the scale or ˙ a ( t ∼ − . ρ eff > p eff <
0, These illustrate an acceleratinguniverse. We may also note EoS variations with reference to galactic time in fig. 7, andone indicates that the problem relates to late-time observational data crossing from overphantom-divide-line [99, 100].We will rebuild the above model using the definition of red-shift during the next part.
In this part, we will analyze the red-shift parameter model. In view of the scale-factor, onewill be added as z + 1 = a a ( t ) , here a in the value in the current time. Further, we assumethe dimensionless parameters r ( z ) = H ( z ) H where H = 71 ± km s − M pc − , is the Hubblevalue in the present time (universe today). From the above, one can relate cosmic-time tored-shift, the differential form appears ddt = dadt dzda ddz = − H ( z + 1) ddz , (35)Furthermore, the Ricci scalar (Eq. (7)) and the effective energy density and pressure (Eq.(11)) can be rewritten in terms of z as R = 12 H r − z + 1) H r ′ , (36)12 eff = κ − h κ ρ m + 3 H r ( f R ( z ) + 1) − H ( z + 1) r ′ f R ( z ) + 3 H ( z + 1) rf ′R ( z ) − f ( z ) i , (37) p eff = κ − h κ p m + 12 H ( z + 1) r ′ ( f R ( z ) + 2) − H r ( f R ( z ) + 1) + 12 f ( z )+ H ( z + 1) rf ′′R ( z ) + 12 H ( z + 1) r ′ f ′R ( z ) − H ( z + 1) rf ′R ( z ) i , (38)here derivative is denoted by the prime with regard to z . The energy density of matter isidentified as ρ m = ρ m (cid:18) a z (cid:19) − ω m +1) . (39)Currently, in order to recreate the model as a source of DE, we need to add the r ( z )feature to be equipped with supernova descriptive statistics [101, 102]. In form of red-shift,one of the suitable choice [103, 104], written as following r ( z ) = C + C ( z + 1) + C ( z + 1) + Ω m ( z + 1) , (40)where C = 1 − C − C − Ω m . It should be remembered that the above parametrizationcorresponds to ΛCDM model for C = C = 0 along with C = 1 − Ω m . The criteria for theright fit are as good as Ω m = 0 . C = − . ± .
53 and C = 1 . ± .
03 [99]. It is to benoted that the freely parameters of the model play a crucial role in this work.By substituting, Eqs. (40) into Eqs. (37) and (38), we get the cosmological parametersin terms of red-shift with the results shown in Fig. 7 and 8. The variance of the successfulEoS tells us that its value is around ∼ − z = 0), as shown in Fig. 9, and ω eff < − We will address the stability of the modified f ( R ) gravity. As the ideal fluid has beenfilled by the Cosmos, we can accept it as a thermodynamic device. We will use the sum ofsound-velocity for the ideal fluid device to this end. As we observe, a useful function such as C s = dp eff dρ eff induces sound-velocity, in which p eff and ρ eff are the Universe’s efficient powerdensity and effective strain. In a thermodynamic system, because the sound-velocity C s is13igure 7: The energy density ρ eff and pressure p eff in terms of red-shift under differentviable models.Figure 8: The energy density ρ eff and pressure p eff in terms of red-shift under differentviable models. 14igure 9: The parameter of EoS is ω eff in terms of red-shift under different viable models.positive, the stability state thus occurs when the parameter C s becomes greater than zero.We observe that a thermodynamic system can be represented by quantities of effective powerdensity, entropy and effective pressure, with adiabatic and non-adiabatic disruptions.We now assume the related scheme to be p eff = p eff ( S, ρ eff ), And disturbing ourselveswith regard to the successful burden that we have δp eff = (cid:18) ∂p eff ∂S (cid:19) ρ eff δS + (cid:18) ∂p eff ∂ρ eff (cid:19) S δρ eff = (cid:18) ∂p eff ∂S (cid:19) ρ eff δS + C s δρ eff , (41)In which the first phrase in the cosmological dilemma is compared to a non-adiabatic method,the second term is compared to the adiabatic process. Because adiabatic disruption isconsidered in cosmology, so for the cosmological process, the variety of entropy appears zero,i.e. δS = 0. Therefore, we are continuing our study that only involves adiabatic processes.Now, to get the C s feature, to separate the Eqs. With regard to red-shift, (37) and (38),we get C s in terms of red-shift. In that case, by numerical estimation, as seen in Fig. 10,we map the speed of sound function in the process of red-shift. Hence, the Fig. 10 tells usthat there is a busy time stability, since the C s value is positive for the z = 0 event.In case of Model 1, the stability condition satisfied in late-time e.g. at z = 0, C s = 2 .
49 andalso C s > z > .
5, while in case of model 2 and 3, the condition is fulfilled at z > . C s >