From a Bounce to the Dark Energy Era with F(R) Gravity
aa r X i v : . [ g r- q c ] S e p From a Bounce to the Dark Energy Era with F ( R ) Gravity
S. D. Odintsov, , V.K. Oikonomou, , , Tanmoy Paul , ICREA, Passeig Luis Companys,23, 08010 Barcelona, Spain Institute of Space Sciences (IEEC-CSIC) C. Can Magrans s/n,08193 Barcelona, Spain Department of Physics,Aristotle University of Thessaloniki,Thessaloniki 54124, Greece Laboratory for Theoretical Cosmology,Tomsk State University of Control Systems and Radioelectronics,634050 Tomsk, Russia (TUSUR) Tomsk State Pedagogical University,634061 Tomsk, Russia Department of Physics,Chandernagore College, Hooghly - 712 136, India. (7)
Department of Theoretical Physics,Indian Association for the Cultivation of Science,2A &
2B Raja S.C. Mullick Road,Kolkata - 700 032, India
In this work we consider a cosmological scenario in which the Universe contracts initially havinga bouncing-like behavior, and accordingly after it bounces off, it decelerates following a matterdominated like evolution and at very large positive times it undergoes through an acceleratingstage. Our aim is to study such evolution in the context of F ( R ) gravity theory, and confrontquantitatively the model with the recent observations. Using several reconstruction techniques, weanalytically obtain the form of F ( R ) gravity in two extreme stages of the universe, particularlynear the bounce and at the late time era respectively. With such analytic results and in addition byemploying appropriate boundary conditions, we numerically solve the F ( R ) gravitational equation todetermine the form of the F ( R ) for a wide range of values of the cosmic time. The numerically solved F ( R ) gravity realizes an unification of certain cosmological epochs of the universe, in particular,from a non-singular bounce to a matter dominated epoch and from the matter dominated to alate time dark energy epoch. Correspondingly, the Hubble parameter and the effective equationof state parameter of the Universe are found and several qualitative features of the model arediscussed. The Hubble radius goes to zero asymptotically in both sides of the bounce, which leadsto the generation of the primordial curvature perturbation modes near the bouncing point, becauseat that time, the Hubble radius diverges and the relevant perturbation modes are in sub-Hubblescales. Correspondingly, we calculate the scalar and tensor perturbations power spectra near thebouncing point, and accordingly we determine the observable quantities like the spectral index ofthe scalar curvature perturbations, the tensor-to-scalar ratio, and as a result, we directly confrontthe present model with the latest Planck observations. Furthermore the F ( R ) gravity dark energyepoch is confronted with the Sne-Ia+BAO+H(z)+CMB data. I. INTRODUCTION
A major challenge in modern theoretical cosmology is to reveal whether the Universe emerged from an initialsingularity or the Universe started its expansion from a non-singular bounce-like phase. In other words, whether theUniverse evolve according to the standard Big-Bang cosmology which must end in a singularity (known as Big-Bangsingularity) when extrapolated backwards, or, the Universe passes through a bouncing stage leading to a non-singularbehavior of the spacetime curvature. The inflationary description is an appealing early Universe scenario, due to thefact that it can produce an almost scale invariant power spectrum, which is consistent with the latest observations.The inflationary scenario [1–8] requires an early period of accelerated expansion in order to solve the horizon andflatness problems. However, most inflationary scenarios have inherent connection with an initial singularity, at leastmost of the canonical scalar models of inflation. Moreover, the observations to date do not undoubtedly confirmthat inflation indeed took place. One of the attractive alternatives to inflation is the bouncing cosmology scenario[9–53]. Apart from generating an observationally compatible power spectrum in some bouncing cosmology scenarios,bouncing cosmology has the additional advantage that it leads to a singular free evolution of the Universe. However,the Big-Bang singularity may be a manifestation for the failure of classical gravity theory which may be unable todescribe the evolution of the Universe at such small scales. Quite possibly, the quantum generalization of the gravitytheory may resolve the singularity issue; just like the classical electrodynamics fails to remove the Coulomb potentialsingularity at the origin, which, however is resolved by the quantum electrodynamics. However, in the absence ofa fully accepted quantum gravity theory, bouncing cosmology is the most promising one to deal with the initialsingularity issue.Among the various bouncing models proposed so far, the matter bounce scenario [14, 22, 23, 54–70] has attracted alot of attention since it produces a scale invariant power spectrum and moreover in a matter bounce scenario (MBS),the Universe passes through a matter dominated epoch at the late-time. The MBS is characterized by a symmetricscale factor where the Hubble radius diverges to infinity at late times both in the contraction and expanding erasof the bounce. This indicates that the perturbation modes are generated far away from the bouncing point, deeplyinside the Hubble radius. Despite the aforementioned successes that the MBS is able to produce a scale invariantpower spectrum, the matter bounce model has some serious drawbacks: (1) in scalar-tensor MBS model, the scalarpower spectrum becomes exactly scale invariant leading to the spectral index for curvature perturbations as unity,which is, in fact, not compatible with the Planck observations. Such inconsistency was also confirmed in [64, 67]from a different point of view, in particular in the context of F ( R ) gravity theory. However, it is well known thata F ( R ) model can be equivalently mapped to a scalar-tensor model by a suitable conformal transformation of thespacetime metric and thus the inconsistencies of the scalar power spectrum (with regard to the Planck observationaldata) in matter bounce scenario for both the scalar-tensor and F(R) theories are well justified. (2) The running ofthe scalar spectral index is constrained to lie within − . ± . r h = aH , in the matter bounce scenario monotonically increases with the cosmic time in the lateera and asymptotically diverges to infinity, due to the reason that the scale factor far away from the bounce behavesas ∼ t / . Such increasing behavior of the Hubble radius confirms a decelerating evolution of the Universe at latetimes. On other hand, the supernovae observations [71–73] confirm that the present Universe is dominated by somenegative pressure matter component, known as dark energy, which generates the accelerated expansion. Thus thedark energy dominated epoch, which is responsible for the late time acceleration, is not well described by the matterbounce scenario.It is shown in the literature, that in the so-called quasi-matter bounce scenario, where the scale factor evolves as t w ) (with w = 0, note for w = 0, it becomes similar to the exact MBS), it is possible to recover the consistencyof the spectral index and of the running index even in a single scalar field model [14]. However, the tensor-to-scalarratio is found to be problematic in the case of a quasi-matter bounce model; actually the tensor and the scalarperturbation amplitudes in the quasi-matter bounce scenario are found to be comparable to each other leading tothe tensor-to-scalar ratio as order of unity. On other hand, modified gravity theories [74–81] may provide successfuldescriptions for both the primordial and the late-time era of our Universe. In most modified gravity descriptionsof bouncing cosmology, the Hubble radius monotonically increases at late times and thus the problem to describethe dark energy epoch of the Universe still persists, because an increasing behavior of the Hubble radius reveals adecelerating phase of the universe. Motivated by this problem, in this paper, we intend to generalize the bouncingscenario which is also compatible with the late-time acceleration as well, in the context of F ( R ) gravity theory. Itis clear that in order to get a dark energy dominated phase, the Hubble radius must decrease with respect to thecosmic time at late times. This in turn indicates that the primordial curvature perturbation modes generate near thebounce, in contrast to the usual matter bounce scenario where the perturbation modes generate far away from thebounce deeply in the contracting era. Keeping these issues in mind, we try to provide a cosmological scenario whichunifies certain cosmological epochs of the universe, in particular, from a non-singular bounce to a matter dominatedera and from the matter dominated to a late time dark energy epoch, in the context of F ( R ) gravity. In this regard,we would like to mention that the merging of bounce with the dark energy epoch was studied earlier [82], howeverin a different context, where the universe evolution was not considered to be symmetric with respect to the bouncepoint, unlike to our present work where we will consider a symmetric behavior of the scale factor as a function of thecosmic time and consequently the Hubble radius asymptotically goes to zero in both sides of the bounce.The outline of this paper is as follows : after briefly describing the essential features of F ( R ) gravity in Sec.[II], wewill demonstrate the behavior of F ( R ) gravity near the bounce and during the late-era respectively in Sec.[III] andconsequently in Sec.[IV], we will perform the scalar and tensor perturbations. Some numerical treatment of severalqualitative aspects of the theory at hand are presented in Sec.[V]. Finally the conclusions follow in the end of thepaper. II. ESSENTIAL FEATURES OF F ( R ) GRAVITY
Let us briefly recall some basic features of F ( R ) gravity, which are necessary for our presentation, for reviews onthis topic see [74, 75, 77]. The gravitational action of F ( R ) gravity in vacuum is equal to, S = 12 κ Z d x √− gF ( R ) (1)where κ stands for κ = 8 πG = M and also M Pl is the reduced Planck mass. By using the metric formalism, wevary the action with respect to the metric tensor g µν , and the gravitational equations read, F ′ ( R ) R µν − F ( R ) g µν − ∇ µ ∇ ν F ′ ( R ) + g µν ✷ F ′ ( R ) = 0 (2)where R µν is the Ricci tensor constructed from g µν . Since the present article is devoted to cosmological context, thebackground metric of the Universe will be assumed to be a flat Friedmann-Robertson-Walker (FRW) metric, ds = − dt + a ( t ) (cid:2) dx + dy + dz (cid:3) (3)with a ( t ) being the scale factor of the Universe. For this metric, the temporal and spatial components of Eq.(2)become, 3 H = − f ( R )2 + 3 (cid:0) H + ˙ H (cid:1) f ′ ( R ) − (cid:0) H ˙ H + H ¨ H (cid:1) f ′′ ( R )0 = F ( R )2 − (cid:0) H + ˙ H (cid:1) F ′ ( R ) + 6 (cid:0) H ˙ H + 4 ˙ H + 6 H ¨ H + ... H (cid:1) F ′′ ( R ) + 36 (cid:0) H ˙ H + ¨ H (cid:1) F ′′′ ( R ) (4)respectively, where H = ˙ a/a is the Hubble parameter of the Universe and f ( R ) is the deviation of F ( R ) gravity fromthe Einstein gravity, that is F ( R ) = R + f ( R ). Comparing the above equations with the usual Friedmann equations,it is easy to understand that F ( R ) gravity provides a contribution in the energy-momentum tensor, with its effectiveenergy density ( ρ eff ) and pressure ( p eff ) being given by, ρ eff = 1 κ (cid:20) − f ( R )2 + 3 (cid:0) H + ˙ H (cid:1) f ′ ( R ) − (cid:0) H ˙ H + H ¨ H (cid:1) f ′′ ( R ) (cid:21) (5) p eff = 1 κ (cid:20) f ( R )2 − (cid:0) H + ˙ H (cid:1) f ′ ( R ) + 6 (cid:0) H ˙ H + 4 ˙ H + 6 H ¨ H + ... H (cid:1) f ′′ ( R ) + 36 (cid:0) H ˙ H + ¨ H (cid:1) f ′′′ ( R ) (cid:21) (6)respectively. Thus, the effective energy-momentum tensor (EMT) depends on the form of F ( R ), as expected. There-fore, different forms of F ( R ) will lead to different evolution of the Hubble parameter. In the present context, we willuse such effective EMT of F ( R ) gravity to realize the cosmological evolution of the Universe. III. RECONSTRUCTION OF F ( R ) GRAVITY: REALIZATION OF BOUNCE AND DARK ENERGYERA
In this section, we reconstruct the F ( R ) gravity from Eq.(4) by considering a certain form of the scale factor of theUniverse. In particular, we reconstruct the F ( R ) gravity in the two distinct eras, namely near the bouncing pointand at the late-time epoch, which are the primary subjects in the following two subsections respectively. Clearly suchforms of F ( R ) will describe the gravity theory in the two extreme stages of the Universe and thus will not be able toreveal the Universe evolution as a whole. However later in Sec.[V], we will numerically solve Eq.(4) and will determinethe form of F ( R ) for a wide range of cosmic times, which will provide an unification of certain cosmological epochs ofthe universe, particularly from a bounce to matter dominated era, followed by a late time accelerating phase. Duringthe numerical solution of Eq.(4), the form of F ( R ) in the two distinct eras, which will be analytically evaluated inthe following two subsections, will act as boundary conditions. A. Reconstruction near the bounce
In this section, we reconstruct the form of the F ( R ) gravity near the bounce of the Universe. The Universe’sevolution in a general bouncing cosmology, consists of two eras, an era of contraction and an era of expansion. Someof the well known functional forms of the scale factor which correspond to a non-singular bounce, have the form, a ( t ) = e αt , a ( t ) = cosh t , a ( t ) = ( a t + 1) n and so on. In general, the non-singular bounce of a scale factor ischaracterized by a ( t b ) = 0 , ˙ a ( t b ) = 0 , ¨ a ( t b ) > t b is the cosmic time when the bounce occurs. The above conditions lead to a finite Kretschmann scalar( K = R µναβ R µναβ ) and thus the Universe becomes free of the initial singularity (known as Big-Bang singularity).Keeping these conditions in mind, we consider the scale factor near the bounce as, a b ( t ) = 1 + αt (7)(the suffix ’b’ stands for near bounce scale factor) with α being a free parameter having mass dimension [+2] andthe bounce happens at t = 0. The above form of the scale factor can be thought as a Taylor series expansion of a ( t )around t = 0 and keeping up-to quadratic order in cosmic time (t). We neglect the higher orders of t in the Taylorexpansion of a ( t ) as, in this section, we are interested to reconstruct the F ( R ) gravity near the bouncing point. Thelinear order of t in the Taylor expansion vanishes due to the condition ˙ a = 0, necessary for a bounce. Moreover thecondition ¨ a (0) > α must be positive and thus we take α > H = ˙ aa ) and Ricci scalar ( R ( t )) as, H ( t ) = 2 αt αt ≃ αt ,R ( t ) = 12 H + 6 ˙ H = 12 α (1 + 3 αt )(1 + αt ) ≃ α + 12 α t (8)respectively, with the H ( t ) and R ( t ) being considered up to O ( t ), similar to the case of the scale factor. HoweverEq.(8) clearly indicates that the Hubble parameter varies linearly with t and goes to zero at the bouncing point,while the Ricci scalar, on the other hand, becomes R (0) = 12 α . Later, during the calculations of scalar and tensorperturbations, we will show that the parameter α should be at the order ∼ − /κ to make the model compatiblewith the Planck constraints and thus the Ricci scalar becomes ∼ GeV (with κ = M = 10 GeV ) at thebouncing point. In order to reconstruct the F ( R ) gravity, we will use Eq.(4), for which we determine the followingquantities (appearing in Eq.(4)) with the scale factor given in Eq.(7), H + ˙ H = 2 α αt ≃ α − α t , H ˙ H + H ¨ H = 8 α t (1 − αt )(1 + αt ) ≃ α t With the above expressions, Eq.(4) becomes,24 α ( R − α ) f ′′ b ( R ) + ( R − α ) f ′ b ( R ) + f b ( R ) + 2( R − α ) = 0 (9)Solving the above equation for f ( R ), we get, f b ( R ) = (cid:18) αD √ e − (cid:19) R − D √ απ e − R α (cid:0) R − α (cid:1) / Erf i (cid:20) √ R − α √ α (cid:21) (10)where Erf i [ z ] is the imaginary error function defined as Erf i [ z ] = − iErf [ iz ] with Erf [ z ] being the error functionand ’ i ’ is the imaginary unit. Moreover D is an integration constant having mass dimension [-2]. The above solutionof f b ( R ) immediately leads to the form of F b ( R ) = R + f b ( R ) as, F b ( R ) = 12 αD √ e R − D √ απ e − R α (cid:0) R − α (cid:1) / Erf i (cid:20) √ R − α √ α (cid:21) (11)The presence of e − R α and Erf i (cid:20) √ R − α √ α (cid:21) in the right hand side of Eq.(11) confirm that the F b ( R ) contains allthe positive integer powers of R where the coefficient of the linear Ricci scalar is other than unity. This indicates adeviation from Einstein gravity. However recall, the form of F ( R ) in Eq.(11) represents the gravity theory near thebounce and in a later Sec.[V], during the numerical reconstruction of F ( R ) gravity for a wide range of cosmic times,we will show that the late time F ( R ) indeed matches with the Einstein gravity. B. Reconstruction in the late time epoch
In this section, we apply a reconstruction scheme to describe the behavior of the Universe at late times. Recall, inthe previous section during the reconstruction near the bounce, we considered a scale factor (suitable for bounce) andthen, by solving Eq.(4), we determine the corresponding form of F ( R ). However the late-time reconstruction methodwhich we are going to apply in this section, is slightly different in comparison to that of the earlier one, in particular,we will start with a form of F ( R ) (rather than a scale factor) suitable for a dark energy model and then reconstructthe corresponding Hubble parameter from the gravitational equation of motion.In order to reproduce the current acceleration of the Universe, several versions of viable modified gravity includingthe so-called one-step models have been proposed in Ref. [83–87]. The simplest one is given by the so-called exponentialgravity which includes an exponential function of the Ricci scalar in the action. We consider such exponential F ( R )model in the present section, which is known to provide a good dark energy model as described in [88, 89]. Inparticular, we consider, F l ( R ) = R − (cid:18) − e − βR (cid:19) (12)with β and Λ are two model parameters having mass dimensions [0] and [+2] respectively and the suffix ’l’ is for “late”time epoch. The exponential correction over the usual Einstein-Hilbert action becomes important at cosmologicalscales and at late-times, providing an alternative to the dark energy problem. Due to to the Supernovae Ia (Sne-Ia) [90, 91], Baryon Acoustic Oscillations (BAO) [92–95], Cosmic Microwave Background (CMB) [96–99] and H ( z )[100–103] data, the parameters β and Λ are well constrained, particularly the F ( R ) model (12) is best fitted withSne-Ia+BAO+H(z)+CMB data for the parametric regimes given by : β = 3 . + ∞− . and Λ = 1 . × − GeV [88].Such parametric ranges also make the model compatible with Sne-Ia+BAO+H(z) data. In effect, we stick with β = 4and Λ = 1 . × − GeV throughout the paper. With these viable ranges of the model parameters along with the F ( R ) of Eq.(12), now we are going to solve the gravitational equation of motion to reconstruct the evolution of theHubble parameter during the dark energy epoch. Plugging the above form of F ( R ) into Eq.(4), we get, − H l + Λ (cid:0) − e − βR (cid:1) − βe − βR (cid:0) H l + ˙ H l (cid:1) − β Λ e − βR (cid:0) H l ˙ H l + H l ¨ H l (cid:1) = 0 (13)where the “dot” represents ddt . However getting an analytic solution of Eq.(13) is troublesome, so we proceed tosolve it numerically and for this purpose, we introduce a dimensionless time scale ( t r ) and a dimensionless Hubbleparameter ( H r ) defined as follows: t r = tt s , H r = 1 a l da l dt r with t s = 10 sec i.e at the order of the present age of the Universe. The actual Hubble parameter ( H l ( t )) is relatedwith the rescaled one as H l ( t ) = t s H r ( t r ) = 10 − H r (sec) − , moreover the derivative of the Hubble parameter in thetwo time coordinates are connected as ˙ H l ( t ) = t s dH r dt r (note, ddt is represented by a “dot” while the ddt r is mentionedby as it is). In view of these relations, Eq.(13) can be expressed in terms of the rescaled quantities as follows, − H r + Λ t s (cid:0) − e − βR (cid:1) − βe − βR (cid:0) H r + dH r dt r (cid:1) − β Λ t s e − βR (cid:0) H r dH r dt r + H r d H r dt (cid:1) = 0 (14)where R Λ = t s (cid:0) H r + 6 dH r dt r (cid:1) . With β = 4 and Λ t s = 1 . × − (the conversion 1 sec. = 1 . × GeV − maybe useful), Eq.(14) is solved numerically and is given in the left part of Fig.[1], where the Hubble parameter is plottedfor 60 ≤ t r ≤
90 (or equivalently 6 × ≤ t ≤ × sec . ) during the late era. Moreover we use H r ( t r = 60) = 0 . dH r dt r (cid:12)(cid:12)(cid:12)(cid:12) t r =60 = − . t r and by using the relation H l = 10 − H r sec − , we give the actual Hubble parameter H l ( t ) (in the unit of (sec) − ) along the y-axis. The solution of the the Hubble parameter immediately leads to theevolution of the Hubble radius ( r ( l ) h = H l exp (cid:2) R H l ( t ) dt (cid:3) ) as shown in the right part of Fig.[1] where once again, thex-axis is the rescaled time coordinate and the y-axis represents the actual Hubble radius i.e r ( l ) h ( t ) = (cid:0) a l ( t ) H l ( t ) (cid:1) − .Fig.[1] clearly reveals that the Hubble radius decreases with time and leading to an accelerating stage of the Universe
60 65 70 75 80 85 904. × - × - × - × - t / t s H ( t )
60 65 70 75 80 85 90248024902500251025202530 t / t s r h = ( a H ) - FIG. 1:
Left plot : H ( t ) (along y-axis) vs. tt s (along x-axis) for 6 × ≤ t ≤ × sec . The Hubble parameter along they-axis is in the unit of sec − . Right plot : r ( l ) h ( t ) (along y-axis) vs. tt s for 6 × ≤ t ≤ × sec, where the Hubble radiusis in the unit of sec. The decreasing behavior of r ( l ) h ( t ) clearly indicates an accelerating stage of the Universe during late time. at late time. Therefore as a whole, the F ( R ) (12) successfully provides a viable dark energy model in respect toSne-Ia+BAO+H(z)+CMB data where the evolution of the Hubble parameter and the Hubble radius are given inFig.[1]. Moreover, here it may be mentioned that the exponential F ( R ) model has a Schwarzschild de-Sitter solutionin the regime R ≫ Λ ∼ − GeV . For example, in the Solar System regime R ∗ ≃ − GeV , the exponential F ( R ) model can be approximated as F ( R ∗ ) = R ∗ −
2Λ (note that R ∗ is greater than the Λ) and thus in this case,the Schwarzschild de-Sitter solution can be obtained, which is indeed a stable solution as depicted by some of ourauthors in [86].Therefore Eqs.(11) and (12) represent the F ( R ) gravity in the two situations, one depicting the early bounce of theUniverse where the scale factor behaves as a ( t ) = 1 + αt (= a b ( t )) and the other providing a late time acceleratingphase when the scale factor ( a l ( t )) or more explicitly the Hubble parameter H l ( t ) goes according to the Fig.[1] (thecorresponding scale factor can be easily obtained by using a l ( t ) = e R H l ( t ) dt ). Till now, we have described the bounceand the dark energy (DE) era in a disjoint manner i.e F b ( R ) describes the Universe near the bounce but not thewhole cosmological epochs, while the F l ( R ) reveals the behavior only at present time. Thus the immediate questionthat springs to mind is, what is the form of F ( R ) for the entire range of cosmic time, which will further demonstratethe unification of bounce and dark energy era followed by a deceleration stage in the intermediate region? As adeceleration era, we may consider a matter dominated epoch in between the bounce and the present era. With thisconsideration, in Sec.[V], we will reconstruct the full F ( R ) gravity, however numerically, by taking the forms of F b ( R )and F l ( R ) as boundary conditions. In the method of numerical reconstruction, we need an appropriate scale factor(or equivalently Hubble radius) describing the Universe evolution for a wide range of cosmic time. For this purpose,we may consider an ansatz of the scale factor as follows, a ( t ) = e αt tanh (cid:20) β (cid:18) − tt p (cid:19)(cid:18) − tt I (cid:19)(cid:21) + (cid:18) tT (cid:19) / tanh (cid:20) β (cid:18) tt b − (cid:19)(cid:18) − tt p (cid:19)(cid:21) + a l ( t ) tanh (cid:20) β (cid:18) tt b − (cid:19)(cid:18) tt I − (cid:19)(cid:21) (15)where t b is the bouncing time or even the horizon crossing time, t p is the present time instance and t I is anintermediate time. The above scale factor behaves as a b ( t ) = 1 + αt near t ≃ t b ≪ t I ≪ t p i.e near the bounce,however goes as a I ( t ) = (cid:0) t/T (cid:1) / near t b ≪ t = t I ≪ t p and as a l ( t ) = e R H l ( t ) dt near the present time i.e t ≃ t p .The parameters β i can be fixed by considering the corresponding hyperbolic tangent function tending to unity at therespective time.Though the scale factor (15) provides the correct evolution of the universe near the bounce and at the late stage andalso at a certain instance of the intermediate stage ( t ≃ t I ), but it does not lead to correct evolution of the universefor the whole intermediate epoch. Thus to address the universe evolution smoothly for a wide range of cosmic time,we may consider a ( t ) as, a ( t ) = 1 + αt = a b ( t ) , near the bounce, f or t ≃ + a ( t ) = (cid:18) tT (cid:19) / = a I ( t ) , in the intermediate regime, f or + . t . × seca ( t ) = e R H l ( t ) dt = a l ( t ) , during late time, f or t & × sec (16)(with T being an arbitrary parameter with dimension sec ) and the transition from the bounce to the matter dominatedand from matter dominated to the dark energy era will be obtained by the method of numerical interpolation. However,the process of interpolation requires appropriate choices for the values of the free parameters present in our model. Itmay be observed that there are three parameters hanging around in the theory, α , Λ and β . The parameter α appearsfrom the near-bounce scale factor and thus can be named as “early stage parameter”, while Λ and β appear duringthe reconstruction in the late-time and thus the name - “late stage parameters”. The late stage parameters have beenalready estimated in view of the viability of the exponential F ( R ) as a successful dark energy model. However, theearly stage parameter α is still remaining to be determined and in the next section, we will estimate α from the latestPlanck 2018 observational results. IV. COSMOLOGICAL PERTURBATION: ESTIMATION OF EARLY STAGE PARAMETERS
Being the early stage parameter, α can be determined from various primordial observable quantities like the scalarspectral index ( n s ), the tensor-to-scalar ratio ( r ) by directly confronting their theoretical expectations with the Planck2018 observations. In this section, we consider the spacetime fluctuations over the FRW metric and consequentlycalculate various observable quantities like the scalar spectral index and tensor-to-scalar ratio. In a bouncing Universe,the primordial perturbation modes (relevant to the present day observation) generate either near the bounce or at adistant past far away from the bouncing point, depending upon the late-time behavior of the Hubble radius. Therebybefore moving to the explicit perturbation calculation, it is important to analyze when the perturbation modes generatein the present context of bouncing Universe. In all the bouncing models, the Hubble parameter becomes zero and thusthe comoving Hubble radius, defined by r h = aH (where H is the Hubble parameter), diverges at the bouncing point.However, the asymptotic behavior of the comoving Hubble radius makes a difference in various bouncing models. Inthis regard, (1) for some bouncing scale factors, the Hubble radius decreases monotonically at both sides of the bounceand finally shrinks to zero size asymptotically, which corresponds to an accelerating late-time Universe. Therefore,in such cases, the Hubble horizon goes to zero at large values of the cosmic time, and only for cosmic times near thebouncing point the Hubble horizon has an infinite size. So the primordial perturbation modes relevant for presenttime era generate for cosmic times near the bouncing point, because only at that time all the primordial modes arecontained in the horizon. As the horizon shrinks, the modes exit the horizon and become relevant for present timeobservations. On the other hand, (2) some bouncing scale factors lead to a divergent Hubble radius asymptotically,which corresponds to a decelerating Universe at late times. In such cases, the perturbation modes generate at verylarge negative cosmic times, corresponding to the low curvature regime of the contracting era, unlike to the previoussituations, where the perturbation modes generate near the bouncing era. More explicitly, in the latter case, thecomoving wave number k begins its propagation through spacetime at large negative cosmic times, in the contractingphase on sub-Hubble scales, and exits the Hubble radius during this phase, and re-enters the Hubble radius duringthe low-curvature regime in expanding phase and thus being relevant for present time observations. Therefore, thephysical picture in the two cases is very different with regard to when the perturbation modes generate. However asmentioned earlier, in the present work, we will deal with an unified scenario of bounce with late time accelerationconnected by an intermediate deceleration epoch. Thus, the late-time Universe is characterized by an acceleratingscale factor and hence the Hubble radius should shrink to zero asymptotically at both “sides” of the bounce (as thescale factor is considered to be symmetric around the bounce). This indicates that the perturbation modes generatenear the bounce, rather than far away from the bouncing point, because near the bouncing regime the Hubble radiushas an infinite size and all the perturbation modes are contained inside the horizon. Hence we solve the perturbationequations near the bounce i.e near t = 0, which is the primary subject in the remaining part of this section. A. Scalar perturbation
In principle, perturbations should always be expressed in terms of gauge invariant quantities. In the present work,we shall work in the comoving gauge defined by the vanishing of the momentum density δT i = 0 (where T µν is theeffective energy-momentum tensor and the symbol ’ δ ’ denotes the corresponding perturbation). In this gauge, thescalar metric perturbation is expressed as, δg ij = a ( t ) (cid:2) − ℜ (cid:3) δ ij (17)where ℜ ( t, ~x ) denotes the scalar perturbation and known as the comoving curvature perturbation which is indeeda gauge invariant quantity. The additional metric perturbations δg and δg i can be obtained in terms of ℜ fromperturbed gravitational equations and as a result, the second order action of ℜ ( t, ~x ) is given by [104–106], δS ℜ = Z dtd ~xa ( t ) z ( t ) (cid:20) ˙ ℜ − a ( ∂ i ℜ ) (cid:21) , (18)with z ( t ) has the following expression, z ( t ) = a ( t ) κ (cid:18) H ( t ) + F ′ ( R ) dF ′ ( R ) dt (cid:19) s F ′ ( R ) (cid:18) dF ′ ( R ) dt (cid:19) (19)Eq.(18) clearly indicates that the sound speed of the scalar perturbation ( c s ) is unity, which in turn guarantees theabsence of superluminal modes or equivalently one may argue that the model is free from gradient instabilities. Theunit sound speed for scalar perturbation is, in fact, a generic nature of F ( R ) theory and also of a scalar-tensor theory.This equivalence, in respect of sound speed, between F ( R ) and scalar tensor theory is however expected as both thetheories can be mapped to one another (in the action level) by a conformal transformation of the metric. Comingback to the action (18), the scalar perturbation has positive kinetic terms if z ( t ) > F ′ ( R ) > F ( R ) obtained in the earlier sections indeedsatisfies F ′ ( R ) > δS ℜ leads to the equation for theperturbed variable ℜ ( ~x, t ) as, 1 a ( t ) z ( t ) ddt (cid:20) a ( t ) z ( t ) ˙ ℜ (cid:21) − a ∂ i ∂ i ℜ = 0 (20)In terms of the Fourier transformed scalar perturbation variable ℜ k ( t ) = R d~xe − i~k.~x ℜ ( ~x, t ), the above equation canbe written as, 1 a ( t ) z ( t ) ddt (cid:20) a ( t ) z ( t ) ˙ ℜ k (cid:21) + k a ℜ k ( t ) = 0 (21)where k is the wave number for the k -th perturbation mode. As mentioned earlier, we solve the above equation forcosmic times near the bouncing point as the perturbation modes themselves are generated close to the bounce andthereby we can use the form of F ( R ) reconstructed in Eq.(11). With this expression of near-bounce- F ( R ), we willdetermine z ( t ) (an essential ingredient of perturbation equation (21)) and for this purpose, we first need to evaluate F ′ ( R ) and F ′′ ( R ) which are given by, F ′ ( R ) = 12 αD √ e + D (cid:20) − (cid:0) R − α (cid:1) √ e + √ πe − R α (cid:0) R − α (cid:1) √ R − α Erf i (cid:2) √ R − α √ α (cid:3) √ α (cid:21) (22)and F ′′ ( R ) = D α / (cid:20) √ α (cid:0) R − α (cid:1) √ e − √ πe − R α (cid:0) R − αR + 1440 α (cid:1) Erf i (cid:2) √ R − α √ α (cid:3) √ R − α (cid:21) (23)respectively. Recall, the Ricci scalar scalar at the bounce becomes R (0) = 12 α and thus the function Erf i (cid:20) √ R − α √ α (cid:21) can be expressed as Taylor series expansion in the powers of √ R − α near the bounce. However the terms with O (( R − α ) / ) in the Taylor expansion can be neglected as such terms will eventually lead to a higher power of t in comparison to t and all the near-bounce quantities have been or will be evaluated up to O ( t ), as we also did forthe near-bounce scale factor in Sec.[III A]. Using the Taylor expansion of Erf i (cid:20) √ R − α √ α (cid:21) along with the expressionof R ( t ) (see Eq.(8)), we evaluate the F ′ ( R ) and F ′′ ( R ) up to O ( t ) as follows, F ′ ( R ( t )) = 12 αD √ e − αD √ e αt (24)and F ′′ ( R ( t )) = − α (cid:20) αD √ e − αD √ e αt (cid:21) (25)respectively. Eq.(24) indicates that F ′ ( R ) is positive near the bounce and hence ensures the stability of the scalarperturbation. On other hand, F ′′ ( R ) becomes negative near the bounce, which is clear from Eq.(25). Actually thecondition F ′′ ( R ) < F ( R ) gravity theory and thussuch condition should hold in order for a bounce to be generated, and the demonstration goes as follows: recall,as mentioned in Sec.[II] that the F ( R ) gravity contributes an effective energy-momentum tensor where the effectiveenergy density ( ρ eff ) and the pressure ( p eff ) are given in Eqs.(5) and (6) respectively. Using these expressions of ρ eff and p eff along with the Hubble factor H ( t ) = 2 αt , it is easy to show that at the bounce ρ eff + p eff becomes= 2 ˙ H (cid:0) F ′ ( R ( t )) − (cid:1) + 24 ˙ H F ′′ ( R ( t )). The F ′ ( R ) and F ′′ ( R ), as we obtained near t = 0 in the present context,leads to ρ eff + p eff < t = 0.Plugging the expressions of F ′ ( R ), F ′′ ( R ) into Eq.(19), we get, a ( t ) z ( t ) = 72 αDκ √ e (cid:20) αt (cid:21) = U + V αt (26)where U = αDκ √ e and V = αDκ √ e . With the above expression of a ( t ) z ( t ), the scalar perturbed equation (21) takesthe following form, U ¨ ℜ k + 2 αV ˙ ℜ k t + k U ℜ k ( t ) = 0 (27)at leading order in t . Solving Eq.(27) for ℜ k ( t ), we get, ℜ k ( t ) = b ( k ) e − V U αt H (cid:20) − k α U V , r αV U t (cid:21) (28)with H [ n, x ] is the nth order Hermite polynomial. b ( k ) is the integration constant which can be determined by theinitial state of the canonical Mukhanov-Sasaki variable v k ( t ) defined by v k ( t ) = z ( t ) ℜ k ( t ), which is the adiabaticvacuum state. Thus near the bounce (i.e near t ≃ v k ( t ) satisfieslim t → v k = 1 √ k e − ikτ (29)where τ is the conformal time defined by dτ = dta ( t ) . However near t ≃
0, the conformal and cosmic time become sameas, τ = Z dta ( t ) = Z dt (1 + αt ) ≃ t . Furthermore for cosmic times that satisfy t ≃
0, the Hubble horizon is infinitely large and thus the primordial modesare well inside the Hubble horizon i.e the perturbation modes satisfy k ≫ aH . As a result, the equation for the field v k ( t ) becomes d v k dτ + k v k = 0 . (30)This is the equation of a simple harmonic oscillator with time-independent frequency. For this case, the solutionfor the minimum energy state is given by v k = √ k e − ikτ . Hence we impose the initial condition as shown in0Eq.(29). This justifies the choice of the adiabatic vacuum state in the present context. On other hand, the function H (cid:20) − k U αV , q αV U t (cid:21) has a limiting value for t → H (cid:20) − k U αV , r αV U t (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) t → = 2 k U αV √ π (cid:0) − k U αV (cid:1) with Γ( z ) symbolizes the “Gamma function“. Therefore, by considering the adiabatic vacuum as the initial conditionof ℜ k ( t ), the integration constant turns out to be, b ( k ) = 1 z ( t → (cid:20) (cid:0) − k U αV (cid:1) √ πk k U αV (cid:21) = 1 √ U (cid:20) (cid:0) − k U αV (cid:1) √ πk k U αV (cid:21) (31)where in the second equality, we used z ( t →
0) = √ U from Eq.(26). Thus as a whole, the solution for the scalarperturbation variable becomes, ℜ k ( t ) = (cid:18) (cid:0) − k U αV (cid:1) √ πk k U αV √ U (cid:19) e − V U αt H (cid:20) − k U αV , r αV U t (cid:21) = (cid:18) κ Γ (cid:0) − k α (cid:1) √ πk k α √ αD (cid:19) e (cid:2) − αt (cid:3) H (cid:20) − k α , r α t (cid:21) (32)where we have used U = αDκ √ e and V = αDκ √ e in the second equality. Consequently, the above solution of ℜ k ( t )immediately leads to the scalar power spectrum for k -th mode as follows, P ℜ ( k, t ) = k π (cid:12)(cid:12)(cid:12)(cid:12) ℜ k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = k αDπ (cid:18) κ Γ (cid:0) − k α (cid:1)(cid:19) k α e (cid:2) − αt (cid:3) (cid:26) H (cid:20) − k α , r α t (cid:21)(cid:27) (33)Using the horizon crossing relation k = aH , the scalar power spectrum can be expressed in terms of the quantities athorizon crossing as, P ℜ ( k, t ) (cid:12)(cid:12)(cid:12)(cid:12) h.c = αt h Dπ (cid:18) κ Γ (cid:0) − αt h (cid:1)(cid:19) αt h e (cid:2) − αt h (cid:3) (cid:26) H (cid:20) − αt h , r α t h (cid:21)(cid:27) (34)where t h is the horizon crossing time. With Eq. (33), we can determine the spectral index of the primordial curvatureperturbations. However before proceeding to calculate the spectral index, we will perform first the tensor perturbation,which is necessary for evaluating the tensor-to-scalar ratio. B. Tensor perturbation
Let us now focus on the tensor perturbations, and the tensor perturbation on the FRW metric background is definedas follows, ds = − dt + a ( t ) ( δ ij + h ij ) dx i dx j , (35)where h ij ( t, ~x ) is the tensor perturbation. The variable h ij ( t, ~x ) is itself a gauge invariant quantity, and the tensorperturbed action up to quadratic order is given by, δS h = Z dtd ~xa ( t ) z T ( t ) (cid:20) ˙ h ij ˙ h ij − a ( ∂ l h ij ) (cid:21) , (36)1where z T ( t ) has the following form [104], z T ( t ) = a ( t ) κ p F ′ ( R ) , (37)Therefore the speed of the tensor perturbation is c T = 1 i.e the gravitational waves propagate with the speed of lightwhich is unity in the natural units.Coming back to Eq.(37), the stability of the tensor perturbation will be guaranteed by the condition F ′ ( R ) >
0- the same condition by which the scalar perturbation is ensured to be stable. Recall, in the previous section, weshowed the positivity of F ′ ( R ) in Eq. (24) and thus the tensor perturbation is indeed stable in the present context.The action (36) leads to the equation of the tensor perturbed variable as,1 a ( t ) z T ( t ) ddt (cid:20) a ( t ) z T ( t ) ˙ h ij (cid:21) − a ∂ l ∂ l h ij = 0 (38)The Fourier transformed tensor perturbation variable is defined as h ij ( t, ~x ) = R d~k P γ ǫ ( γ ) ij h ( γ ) ( ~k, t ) e i~k.~x , where γ = ′ + ′ and γ = ′ × ′ represent two polarization modes. Moreover ǫ ( γ ) ij are the polarization tensors satisfying ǫ ( γ ) ii = k i ǫ ( γ ) ij = 0. In terms of the Fourier transformed tensor variable h k ( t ), Eq.(38) can be expressed as,1 a ( t ) z T ( t ) ddt (cid:20) a ( t ) z T ( t ) ˙ h k (cid:21) + k a h k ( t ) = 0 (39)The two polarization modes obey the same Eq.(39) and thus we omit the polarization index. Moreover, both thepolarization modes even follow the same initial condition and that is why they have the same solution. So in thefollowing, what we will do in the context of tensor perturbation is applicable for both the polarization modes. Finally,in the expression of the tensor power spectrum, we will introduce a multiplicative factor 2 due to the contributionfrom both the polarization modes. Using the expression of F ′ ( R ) from Eq.(24), we determine a ( t ) z T ( t ) as, a ( t ) z T ( t ) = 12 αDκ √ e (cid:20) αt (cid:21) = U + V αt (40)with U = αDκ √ e and V = αDκ √ e . Plugging back this expression of a ( t ) z T ( t ) into Eq.(39) and after some algebra, weget the following equation, U ¨ h k + 2 αV ˙ h k t + k U h k ( t ) = 0 (41)at leading order in t , which is a viable consideration because the perturbation modes generate near the bouncingphase in the present scenario. Solving Eq.(41) for h k ( t ) , we get, h k ( t ) = b ( k ) e − V U αt H (cid:20) − k α U V , r V U α t (cid:21) (42)where b ( k ) is an integration constant and can be determined from an initial condition. As an initial condition, weconsider that the tensor perturbation field starts from the adiabatic vacuum, more precisely the initial configurationis given by, lim t → (cid:2) z T ( t ) h k ( t ) (cid:3) = √ k . This immediately leads to the expression of b ( k ) as, b ( k ) = 1 z T ( t → (cid:20) (cid:0) − k U αV (cid:1) √ πk k U αV (cid:21) = 1 √ U (cid:20) (cid:0) − k U αV (cid:1) √ πk k U αV (cid:21) . (43)In the second equality of the above equation, we use z T ( t →
0) = 1 / √ U from Eq.(40). Plugging this expression of b ( k ) into Eq.(42) yields the final solution of h k ( t ) as follows, h k ( t ) = (cid:18) (cid:0) − k U αV (cid:1) √ πk k U αV √ U (cid:19) e − V U αt H (cid:20) − k U αV , r V U α t (cid:21) = (cid:18) κ Γ (cid:0) − k α (cid:1) √ πk k α √ αD (cid:19) e (cid:2) − αt (cid:3) H (cid:20) − k α , √ α t (cid:21) (44)2where we have used U = αDκ √ e and V = αDκ √ e . Eq.(44) represents the solution of the tensor perturbation for boththe polarization modes. The solution of h k ( t ) immediately leads to the tensor power spectrum as, P h ( k, t ) = k π X γ (cid:12)(cid:12)(cid:12)(cid:12) h ( γ ) k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) = k αDπ (cid:18) κ Γ (cid:0) − k α (cid:1)(cid:19) k α e (cid:2) − αt (cid:3) (cid:26) H (cid:20) − k α , √ α t (cid:21)(cid:27) (45)It may be noticed that γ = ′ + ′ and γ = ′ × ′ modes contribute equally to the power spectrum, as expected becausetheir solutions behave similarly. At the horizon crossing, the tensor power spectrum turns out to be, P h ( k, t ) (cid:12)(cid:12)(cid:12)(cid:12) h.c = 2 αt h Dπ (cid:18) κ Γ (cid:0) − αt h (cid:1)(cid:19) αt h e (cid:2) − αt h (cid:3) (cid:26) H (cid:20) − αt h , √ α t h (cid:21)(cid:27) (46)Now we can explicitly confront the model at hand with the latest Planck observational data [107], so we shallcalculate the spectral index of the primordial curvature perturbations n s and the tensor-to-scalar ratio r , which aredefined as follows, n s − ∂ ln P ℜ ∂ ln k (cid:12)(cid:12)(cid:12)(cid:12) H.C , r = P h ( k, t ) P ℜ ( k, t ) (cid:12)(cid:12)(cid:12)(cid:12) H.C (47)As evident from these expressions, n s and r are evaluated at the time of the first horizon exit near the bouncingpoint, for positive times (symbolized by ’H.C’ in the above equations), when k = aH i.e. when the mode k crossesthe Hubble horizon. Using Eq.(33), we determine ∂ ln P ℜ ∂ ln k as follows, ∂ ln P ℜ ∂ ln k = 2 − k α ψ (0) (cid:20) − k α (cid:21) − k α (ln 2) + 3 k α (cid:26) H (1 , (cid:20) − k α , q α t (cid:21) H (cid:20) − k α , q α t (cid:21) (cid:27) (48)where ψ (0) [ z ] is the zeroth order Polygamma function or equivalently the digamma function and H (1 , [ z , z ] is thederivative of H [ z , z ] with respect to its first argument. Moreover the following expressions were used to calculateEq.(48), ∂∂k (cid:26) Γ (cid:18) − k α (cid:19)(cid:27) = − k α Γ (cid:18) − k α (cid:19) ψ (0) (cid:18) , − k α (cid:19) ∂∂k (cid:26) H (cid:20) − k α , r α t (cid:21)(cid:27) = 3 k α H (1 , (cid:20) − k α , r α t (cid:21) (49)Thereby Eq.(48) immediately leads to the spectral index as, n s = (cid:20) − k α ψ (0) (cid:18) , − k α (cid:19) − k α (ln 2) + 3 k α (cid:26) H (1 , (cid:18) − k α , q α t (cid:19) H (cid:18) − k α , q α t (cid:19) (cid:27)(cid:21) H.C (50)As mentioned earlier, the perturbation modes are generated and also cross the horizon near the bounce. Thus we cansafely use the near-bounce scale factor in the horizon crossing condition to determine k = aH = 2 αt h (where t h is thehorizon crossing time). Using this relation, Eq.(50) turns out to be, n s = 3 − αt h ψ (0) (cid:18) , − αt h (cid:19) − αt h (ln 2) + 12 αt h (cid:26) H (1 , (cid:18) − αt h , q α t h (cid:19) H (cid:18) − αt h , q α t h (cid:19) (cid:27) (51)3Furthermore, the tensor-to-scalar ratio is given by, r = (cid:12)(cid:12)(cid:12)(cid:12) h k ( t ) ℜ k ( t ) (cid:12)(cid:12)(cid:12)(cid:12) k = a ( t h ) H ( t h ) (52)where the solutions of h k ( t ) and ℜ k ( t ) are shown in Eqs.(44) and (32) respectively. Eqs.(51) and (52) clearly indicatethat n s and r depend on the dimensionless parameter αt h which is further connected to the Ricci scalar at horizoncrossing by αt h = (cid:0) R h α − (cid:1) . Thereby, we can argue that the observable quantities n s and r depend on R h /α . Withthis information, we now directly confront the theoretical expectations of spectral index and tensor-to-scalar ratiowith the Planck 2018 constraints [107]. In Fig. [2] we present the estimated spectral index and tensor-to-scalar ratioof the present scenario for three choices of R h α , on top of the 1 σ and 2 σ contours of the Planck 2018 results [107]. Aswe observe, the agreement with observations is efficient, and in particular well inside the 1 σ region. At this stage itis worth mentioning that in an F ( R ) gravity theory, the matter bounce scenario, in which case the perturbations aregenerated far away from the bouncing point deeply in the contracting regime, is not consistent with the Planck resultsas shown in [64]. However, here we demonstrate that a F ( R ) gravity model indeed leads to a viable bouncing modelwhere the primordial perturbations are generated near the bounce. Thereby, we can argue that the viability of a F ( R )model (with respect to the Planck constraints) is interrelated with the generation era of the perturbations modes.Furthermore the scalar perturbation amplitude ( A s ) is constrained by ln (cid:2) A s (cid:3) = 3 . ± .
014 due to the Planck n s r FIG. 2: 1 σ (yellow) and 2 σ (light blue) contours for Planck 2018 results [107], on n s − r plane. Additionally, we present thepredictions of the present bounce scenario with R h α = 14 (blue point), R h α = 16 (black point) and R h α = 20 (red point). results [107]. Eq.(33) along with the consideration of the integration constant D = α (recall D has mass dimension [-2])depict that the scalar perturbation amplitude depends on the dimensionless parameters R h α and ακ . We may choose R h α = 16 (which is within the range that makes the n s and r compatible with the Planck results) and consequently thescalar perturbation amplitude in the present context becomes A s = π ακ . Therefore the theoretical expectationof A s will match with the Planck observations provided ακ lies within ακ = [1 . × − , . × − ]. Thus asa whole, the observable quantities n s , r and A s are simultaneously compatible with the Planck constraints for theparameter ranges : 14 ≤ R h α ≤
20 and ακ = [1 . × − , . × − ] respectively. However the viable range of ακ depends on the consideration D = 1 /α , i.e. a different D will eventually lead to a different range of viability ofthe parameter ακ . As an example, for D = α , the scalar perturbation amplitude becomes A s = π ακ while n s and r have the same form as given in Eqs.(51) and (52) respectively and therefore the parameters should lie within14 ≤ R h α ≤
20 and ακ = [3 . × − , . × − ] in order to make the theoretical values of the observablequantities compatible with the latest Planck 2018 results. Such parametric ranges make the horizon crossing Ricciscalar of the order R h ∼ − /κ = 10 GeV .Thereby, the viable ranges of the early and late stage parameters are given by 14 . R h α . ακ = [3 . × − , . × − ], β = 3 . + ∞− . and Λ = 1 . × − GeV respectively. As a result, wecan interpolate the scale factor or the Hubble radius in the two transition eras (i.e from the bounce to matterdominated and from the matter dominated to dark energy epoch), leading to a smooth evolution of the Hubbleradius for a large range of cosmic time.Before moving to the next section we would like to mention that in the present context, the non-singular bounceuniverse is time symmetric and the initial adiabatic vacuum condition is set at the bouncing point, which seems4that the model is more similar to a scenario of the creation of the universe from nothing. The spontaneous birthof the universe from “nothing” has been discussed earlier in the context of quantum cosmology where the universeis described by a wave function satisfying the well known Wheeler-DeWitt equation [108]. With the developmentof quantum cosmology theory, it has been suggested that the universe can be created spontaneously from nothing,where “nothing” means there is neither matter nor space or time, and the problem of singularity can be avoidednaturally [109, 110]. In particular, the author of [109] showed that a quantum universe of zero size, or, indeed, thecosmological “nothing” may quantum mechanically tunnel to a universe of a finite size. However the argument of[109] was established for a closed type universe, unlike to the present bounce scenario which has been presented for aflat FRW universe. V. NUMERICAL SOLUTIONS: AN UNIFIED DESCRIPTION FROMBOUNCE-TO-DECELERATION-TO-LATE TIME ACCELERATION
Given the structure of the scale factor in the early and late stages of the Universe (as a b ( t ) = 1 + αt in Eq.(7)and a l ( t ) = e R H l ( t ) dt respectively, where the H l ( t ) is given in Fig.[1]), we would like to provide a complete picture byconsidering the intermediate region to be a matter dominated epoch, and moreover the transitions from the bounce tomatter dominated (MD) and from the matter dominated to the DE era will be obtained by the method of numericalinterpolation. Due to complicated nature of the equations governing the evolution of the scale factor in a generalcontext, we will determine the interpolating function using numerical techniques and will illustrate the same. Letus briefly point out the methods one may use in order to generate such interpolating solutions. In the transitionregions (i.e. from the bounce-MD and from the MD-DE), one approximates the behavior of the physical quantity ofinterest (e.g., the scale factor a ( t ) or the Hubble radius r h = aH ) by a polynomial function of time, with degree of thepolynomial kept arbitrary. Then, in the early bounce epoch one uses the given behavior of the desired physical quantity(for example for the scale factor, the behavior near the bounce is a b ( t ) = 1 + αt ) to generate numerical estimates ofthe respective quantity at various time instants till the description is reliable. Similar numerical estimations are beingmade at the intermediate matter dominated and at the late stage as well. With these sets of data and the polynomialfunction one can use any standard interpolation software package to end up getting the desired plots. The structureof the plot, of course, depends on the degree of the polynomial and desired accuracy level. All the plots in this paperare for an accuracy level of O (10 − ). Since our main aim is to merge certain cosmological epochs of the Universe, agood physical quantity to start with is the Hubble radius ( r h ( t )) rather than the scale factor, because the acceleratingor decelerating stage of the Universe is easily realized by the decreasing or increasing behavior of the Hubble radius(with respect to cosmic time) respectively. Following Eq.(16), we may construct the Hubble radius as, r h ( t ) = (cid:2) ˙ a b ( t ) (cid:3) − = 12 αt , near the bounce, f or t ≃ + r h ( t ) = (cid:2) ˙ a I ( t ) (cid:3) − = 3 t (cid:18) Tt (cid:19) / , in the intermediate regime, f or + . t . × sec.r h ( t ) = (cid:2) ˙ a l ( t ) (cid:3) − ( see F ig. [1]) , during late time, f or t & × sec. (53)and then with the procedure described above, we numerically interpolate the Hubble radius in the two transitionsi.e. from the bounce to matter dominated and from the matter dominated to DE era respectively. Moreover theHubble radius in the contracting regime can also be obtained by assuming a symmetric behavior of (cid:12)(cid:12) r h ( t ) (cid:12)(cid:12) around thebounce. However, the details of the interpolation of the curve connecting the early bounce to the late acceleratingstage is an artifact of the procedure followed and admits possible variations depending on the process of interpolationby numerical techniques. However such indeterminacy in determining the interpolating function would not affect ourmain conclusions in the present context. Thus, having explained the details of the interpolating procedure, we nowturn to the corresponding implications and present the variations of all the relevant parameters with time.As mentioned earlier, we start with the Hubble radius ( r h = a ), in particular, taking the reduced Planck mass tobe M Pl = 10 GeV and using the aforementioned construction of the Hubble radius, we obtain r h ( t ) as a functionof time in the expanding phase of the Universe. Further the Hubble radius in the contracting regime is obtained byconsidering a symmetric character of (cid:12)(cid:12) r h ( t ) (cid:12)(cid:12) in the both sides of the bounce. As a result, we get the Hubble radiusfor − × ≤ t ≤ × sec., which is presented in the left part of Fig.[3], while the right part is a zoomed-inversion of the left one near t = 0. The x-axis of the Fig.[3] corresponds to a “rescaled” time coordinate obtained as tt s (with t s = 10 sec.) which is dimensionless, while the y axis represents the rescaled Hubble radius ¯ r h = 10 − r h which is in the unit of “sec”. Fig.[3] clearly demonstrates that the Hubble radius depicts a bounce at t = 0 (asexpected, because it is constructed from r ( b ) h ( t ) = (cid:0) a b H b (cid:1) − near t = 0), then it increases monotonically with time up5to tt s ≃
50 (or t = 5 × sec.) leading to a decelerating phase of the Universe and finally after t & × sec., theHubble radius starts to decrease, which in turn indicates the dark energy epoch of the Universe. This may providea merging of a non-singular bounce to matter dominated epoch, followed by a late time dark energy era. It may benoticed that the Hubble radius goes to zero asymptotically at both “sides” of the bounce, which in turn confirmsthe fact that the relevant primordial perturbation modes generate near the bouncing point as we have consideredin Sec.[IV] during the calculation of scalar and tensor perturbations. The generation era of perturbation modes inthe present context makes the model different than the usual matter bounce scenario where the perturbation modesgenerate in the distant past far away from the bounce. Having this Hubble radius, we are now going to solve the scale -
50 0 5005101520 t / t s | r h | - × - - × - × - × - / t s | r h | FIG. 3:
Left plot : The modulus of the rescaled Hubble radius (cid:12)(cid:12) ¯ r h (cid:12)(cid:12) (along y-axis, in the unit of second) is being plottedagainst tt s (along x-axis) for a wide range of cosmic time, in particular for − × ≤ t ≤ × sec. Such construction ofthe Hubble radius depends on Eq.(53) and the numerical interpolation in the transition eras i.e from the bounce-MD and theMD-dark energy epoch. During numerical interpolation, we take ακ = 3 . × − , β = 4 and T = 1 sec. The curve explicitlyshows a smooth unification of the Universe evolution from the bounce-to-deceleration-to-late time acceleration. Right plot : Azoomed-in version of the left plot near t = 0. factor and Hubble parameter, however numerically and then by using the numerical solution of H ( t ) we reconstructthe form of F ( R ) from the gravitational Eq.(4). The scale factor can be obtained from ˙ a ( t ) = r h ( t ) which is a firstorder differential equation and thus the solution of the same requires one boundary condition. We choose a (0) = 1,because we want the scale factor to behave like a b ( t ) near t = 0 and recall a b (0) = 1. With such initial conditionalong with the r h ( t ) given in Fig.[3], we solve a ( t ) numerically, which is presented in Fig.[4] where once again, the xaxis is rescaled by t r = t/t s .It is evident that the scale factor is smooth everywhere and the graph in the inset shows that the scale factor acquires -
50 0 5002.0 × × × × × × t / t s a ( t ) - - × × × × FIG. 4: a ( t ) (along y-axis) vs. tt s (along x-axis). The scale factor gets a minimum at t = 0 and thus indicates a non-singularbounce at that point of time. To get a better view of what is happening near the bounce, we give a zoomed-in version by theinset-graph depicting the behavior of the scale factor near t = 0. a minimum at t = 0. Thus we can argue that the Universe transits from a contracting phase to an expanding one6through a non-singular bounce at t = 0 and thus the Universe evolution becomes free of the initial singularity. Theabove numerical solutions of the scale factor can be immediately differentiated providing the Hubble parameter as afunction of time, shown in Fig.[5]. The Hubble parameter becomes zero and shows an increasing behavior with time -
50 0 50 - - - / t s H ( t )
70 75 80 85 902.5 × - × - × - × - × - × - t / t s H ( t ) FIG. 5:
Left plot : H ( t ) (along y-axis, in the unit of sec − ) vs. tt s (along x-axis) for − × ≤ t ≤ × sec., whichis obtained from the solution of the scale factor shown in Fig.[4]. As evident, H ( t ) becomes zero at t = 0 and ˙ H (0) > Right plot : A zoomed-in version of H ( t ) vs. tt s for7 × ≤ t ≤ × sec. (i.e ˙ H >
0) at t = 0, however this is expected as the scale factor itself acquires a minimum at t = 0 i.e. a bounce.Moreover, the Hubble parameter acquires a maximum very near to bounce and then decreases rapidly, and continuedto decrease until the late-time era. A zoomed-in version of the late time behavior of H ( t ) is shown in the right plot ofFig. [5], which clearly demonstrates that at the late epoch i.e. for t & × sec., the Hubble parameter becomesof the order ∼ − sec − . The occurrence of the maximum of H ( t ) near the bounce in the present context is inagreement with [82] where the authors explained the merging of bounce with late-time acceleration from a differentviewpoint, namely from the holonomy generalizations of def ormed matter bounce scenario. In this regard, it may bementioned that a deformed matter bounce scenario is represented by an asymmetric scale factor where the Hubbleradius diverges at t → −∞ and asymptotically goes to zero at t → + ∞ . Thus the primordial perturbation modes ina deformed matter bounce model generate deeply in the contracting regime far away from the bounce, unlike to ourpresent case where the Hubble radius goes to zero asymptotically at both “sides” of the bounce (see Fig.[3]) leadingto the generation of the perturbation modes near the bouncing regime.With the Hubble parameter at hand, our next task is to solve the gravitational equation (4) to reconstruct theform of F ( R ) which can realize such evolution of the Hubble parameter. For this purpose, we need two boundaryconditions as Eq.(4) is a second order differential equation with respect to the Ricci scalar. As mentioned earlier, weuse the forms of F b ( R ) and F l ( R ) obtained in Sec.[III] for such boundary conditions and thus the boundary conditionsare given by: F ( R = 0) = F l (0) and F ( R b = 12 α ) = F b (12 α ) respectively (recall, R b is the Ricci scalar at thebounce). As a result, we solve Eq. (4) numerically and the form of F ( R ) we get is depicted in the left part of Fig.[6].Actually the F ( R ) is demonstrated by the red curve, while the yellow one represents the Einstein gravity. Beforegoing into the discussion regarding the behavior of the F ( R ) gravity, let us briefly comment about the range of the“rescaled Ricci scalar” that we take along the x-axis in Fig.[6]. Recall, the Ricci scalar near the bounce gets thevalue R ( t → ≃ α where the parameter α lies within [3 . × − /κ , . × − /κ ] to make the primordialobservable quantities compatible with the Planck constraints. Thereby, the Ricci scalar near the bounce becomes ofthe order ∼ × GeV , while in the present epoch, the scalar curvature is generally considered to take 10 − GeV .Thus in order to correctly describe the epochs from bounce to late time acceleration, we have to cover the range of R ( t ) from ∼ × GeV to 10 − GeV . However in Fig.[6], we consider a “rescaled Ricci scalar” given by ¯ R = Rα along the x-axis, which is dimensionless as α has the mass dimension [+2]. Due to such rescaling, the x-axis of the plotruns from the value x i = 10 − (or we may take x i = 0) to x f = 12. Moreover, the y-axis of Fig.[6] is also rescaledby ¯ F ( ¯ R ) = F ( R ) /α so that the Einstein gravity can be described by a straight line having slope of unity even inthese rescaled coordinates. The left part of the Fig. [6] clearly demonstrates that the solution of F ( R ) matches withthe Einstein gravity as the Ricci scalar approaches to the present value, while the F ( R ) seems to deviate from theusual Einstein gravity, when the scalar curvature takes larger and larger values. It is evident that near the bounce, F ′ ( R ) is positive, which in turn indicates the stability for the scalar and tensor perturbations. This finding from thenumerical solution, in fact, resembles with the analytic results obtained in Sec.[IV]. The comparison of F ( R ) gravity7 R F ( R ) × - × - × - × - × - × - × - × - R F ( R ) FIG. 6:
Left plot : ¯ F ( ¯ R ) (along y axis) vs. ¯ R (along x axis). The red curve depicts the numerical solution of the F ( R ) andthe yellow curve represents the Einstein gravity. The figure reveals that the F ( R ) matches with the Einstein gravity in the lowcurvature regime, while the F ( R ) seems to deviate from the usual General Relativity when the scalar curvature takes largerand larger values Right plot : A zoomed-in version of the left plot near R → F ( R ) during the present epoch. with the Einstein one in the low-curvature regime is explicitly demonstrated in the right part of Fig.[5], which revealsthat the form of F ( R ) reconstructed in the present context is indeed comparable with Einstein gravity during thepresent epoch. Thus, we can safely argue that the numerical solution of F ( R ) obtained above indeed matches withthe analytic results determined near the bounce and at the dark energy era. Using the form of F ( R ) and the Hubble - - - - - -
20 t / t s w e ff FIG. 7: w eff (along y axis) vs. tt s (along x axis) for 0 . t ≤ × sec. The red curve represents the w eff for the presentmodel while the yellow one is for the constant value − . Thereby the successive crossing between the red and yellow curvesdepicts the transition of the Universe from acceleration to deceleration or vice-versa. parameter from Fig.[6] and [5] respectively, we determine the remaining bit i.e the effective equation of state (EoS)parameter defined by, w eff = (cid:20) f ( R )2 − (cid:0) H + ˙ H (cid:1) f ′ ( R ) + 6 (cid:0) H ˙ H + 4 ˙ H + 6 H ¨ H + ... H (cid:1) f ′′ ( R ) + 36 (cid:0) H ˙ H + ¨ H (cid:1) f ′′′ ( R ) (cid:21)(cid:20) − f ( R )2 + 3 (cid:0) H + ˙ H (cid:1) f ′ ( R ) − (cid:0) H ˙ H + H ¨ H (cid:1) f ′′ ( R ) (cid:21) , as shown in Fig. [7], where the x-axis is the rescaled time coordinate i.e tt s with t s = 10 sec. Actually the red curveof the plot represents the w eff for the present model while the yellow one is for the constant value − (we will keepthe yellow graph to investigate the accelerating or decelerating era of the Universe). Fig.[7] clearly demonstrates thatat the bounce i.e. at t = 0, the EoS parameter diverges from the negative side, however this is expected because at thebounce the Hubble parameter itself becomes zero and in turn makes the w eff = − − H H → −∞ . Then immediatelyafter the bounce, w eff crosses the value − leading to a transition from a bounce to a decelerating phase of the8Universe, and during the deceleration epoch, the EoS parameter remains almost constant and near to zero indicatinga quasi-matter dominated Universe which continues till t ≃ × sec, when the EoS parameter again crossesthe value − and thus the Universe transits from a stage of deceleration to a stage of acceleration continued untilthe late-time era. Therefore, the present model may provide an unified scenario of certain cosmological epochs frombounce to late-time acceleration followed by a quasi-matter dominated epoch in the intermediate regime. Moreover,the EoS parameter during the dark energy era approaches to the value − . w eff is, however, in agreement with [86] where the authors considered the exponential F ( R ) as adark energy model, similar to the present case. VI. LOGARITHMIC CORRECTED EXPONENTIAL F ( R ) GRAVITY AS DARK ENERGY MODEL: ADIFFERENT F ( R ) FOR UNIFICATION OF BOUNCE AND LATE TIME ACCELERATION
In this section, we consider a different F ( R ) dark energy model in comparison to the one we considered in theprevious sections, in particular, we incorporate an additional logarithmic correction to the exponential F ( R ) gravity.Thus the form of F ( R ) is given by [88], F ( R ) = R − (cid:18) − e − βR (cid:19)(cid:20) − γR
2Λ log (cid:0) R (cid:1)(cid:21) (54)where the correction over Einstein gravity is given by F ( R ) − R = − (cid:18) − e − βR (cid:19)(cid:20) − γR
2Λ log (cid:0) R (cid:1)(cid:21) . It may be observed that such correction goes to zero at R → F ( R ) model or equivalently to avoid the fine tuning problemassociated with the cosmological constant, the correction over the Einstein-Hilbert term of the F ( R ) gravity requiresto vanish in the low curvature limit i.e lim R → (cid:0) F ( R ) − R (cid:1) = 0 . (55)As just mentioned, this condition is satisfied for the considered F ( R ) model in Eq.(54) and thus this F ( R ) modelcan produce the vacuum solution as the Minkowski spacetime. Thereby we may argue that the model (54) is freefrom the problem of including an implicit cosmological constant, unlike to the scenario with the action given by S = R d x √− g (cid:2) R − (cid:3) . The condition (55) for avoiding a cosmological constant is also satisfied for the exponential F ( R ) gravity as considered earlier in Eq.(12) i.e without the logarithmic corrections. However as described in [88], thepresence of the logarithmic correction, modelled by the free parameter γ by Eq.(54), leads to a better fit (in respectto a viable dark energy model) than in absence of the logarithmic correction and also better than the ΛCDM model.Hence the inclusion of the logarithmic correction provides a test for this type of F ( R ) theory and motivates to checkhow a deviation is allowed in comparison to the ΛCDM model. Note that the logarithmic correction introduces a newparameter γ i.e for γ = 0, the F ( R ) model (54) becomes similar to the exponential model considered earlier in Eq.(12).This new F ( R ) model with the logarithmic term still satisfies the viability conditions (under some conditions of thefree parameter) and provides an extra term in the action that evolutes smoothly along the cosmological evolution (farfrom the pole obviously), as addressed in the Ref. [88]. In particular, the logarithmic corrected exponential F ( R )gravity comes as a viable dark energy model with respect to the Sne-Ia+BAO+H(z)+CMB data for the parametricchoices : γ = 0 . +0 . − . , Λ = 1 . × − GeV and β = 4 . + ∞− . respectively. Such parametric regimes makes the F ( R ) model free from antigravity effects due to the condition γ log (cid:0) R (cid:1) < γ leads tobetter fit than in the absence of logarithmic correction (i.e. the pure exponential F ( R ) model), obviously at the priceof introducing a new parameter γ . Moreover the model (54) avoids the presence of large corrections on the Newton’slaw as well as the appearance of large instabilities at local systems, leading to a suitable model that recovers the wellknown results of General Relativity at the appropriate scales. In view of this new dark energy model, we propose thefollowing F ( R ) gravity, F ( R ) = R − (cid:18) − e − βR (cid:19)(cid:20) − γR
2Λ log (cid:0) R (cid:1)(cid:21) + exp (cid:26) − cosh (cid:18) RR b + R b R − (cid:19)(cid:27)(cid:20) (cid:18) αD √ e − − γ log (cid:0) R (cid:1)(cid:19) R − D √ απ e − R α (cid:0) R − α (cid:1) / Erf i (cid:2) √ R − α √ α (cid:3)(cid:21) (56)9which can merge the bounce and late-time acceleration as we will show in the following, where the late time behavioris described by the F ( R ) of Eq.(54) and the bouncing scenario is still depicted by the F b ( R ) determined in Eq.(11).In the above expression, the R b denotes the scalar curvature at the bounce i.e R b = 12 α ∼ GeV (see Sec. [IV]for the estimation of α ). We introduce an additional cosine hyperbolic factor (let us call it “fixing factor“) with thesecond term of the right hand side of Eq.(56) in order to avoid the effects of the second term during the late-time era.With the above F ( R ), the demonstration of universe’s evolution at different curvature regimes goes as follows: (1)for R ≫ Λ when the Ricci scalar is given by R ( t ) = 12 α + 12 α t (i.e near the bounce), the cosine hyperbolic termbehaves as,cosh (cid:18) RR b + R b R − (cid:19) = cosh (cid:18) α t (1 + αt ) (cid:19) = 12 (cid:2) e α t αt + e − α t αt (cid:3) = 1 + α t αt ) + ......... = 1 + O ( t )and therefore the fixing factor can be expressed by e −O ( t ) = 1 −O ( t ). However, recall that the near bounce quantitiesare determined up to O ( t ) and thus the factor e −O ( t ) can be approximated to unity near R ≃ R b . As a result alongwith the fact R b ≫ Λ, the F ( R ) of Eq.(56) turns out to be, F ( R ) = 12 αD √ e R − D √ απ e − R α (cid:0) R − α (cid:1) / Erf i (cid:20) √ R − α √ α (cid:21) = F b ( R ) (57)in the regime R ≃ R b , which realizes a bouncing Universe as discussed in Sec. [III A]. However in order to ensure abounce, it is also necessary to check F ′ ( R ) and F ′′ ( R ) as they are connected to the issues of energy conditions. Beingthe evolution of the fixing factor as e −O ( t ) = 1 − O ( t ) and since dRdt is proportional to t around the bounce, onecan immediately write F ′ ( R ) = F ′ b ( R ) and F ′′ ( R ) = F ′′ b ( R ) up to O ( t ) near R ≃ R b . Thereby, the F ( R ) we proposein Eq.(56) as well as its first and second derivatives match with that of F b ( R ) up to O ( t ), which clearly indicates abouncing Universe for R ≃ R b . Here it may be mentioned that a different fixing factor like e − (cid:0) RRb + RbR − (cid:1) behaves as1 + O ( t ) near the regime R ≃ R b , unlike to our considered fixing factor where the leading order is proportional to t due to the additional cosine hyperbolic term. However in effect of fourth order term (i.e t ) being the leading orderone, the fixing factor e − (cid:0) RRb + RbR − (cid:1) makes the F ′ ( R ) different in comparison to F ′ b ( R ) even up to O ( t ), which in turnmay violate the bounce at R ≃ R b . In view of these arguments, we stick to the “fixing factor” as proposed in Eq.(56)rather than e − (cid:0) RRb + RbR − (cid:1) . On other hand, (2) for R ≪ R b i.e in the low curvature regime (or equivalently R ∼ Λ), R b R becomes very much larger in comparison to RR b and as a consequence the factor exp (cid:26) − cosh (cid:18) RR b + R b R − (cid:19)(cid:27) can be approximated to exp (cid:2) − e R b /R (cid:3) which further tends to zero in the regime RR b ≪
1. Thereby, in the lowcurvature regime, the form of F ( R ) can be written as F ( R ) = R − (cid:18) − e − βR (cid:19)(cid:20) − γR log (cid:0) R (cid:1)(cid:21) leading toa viable dark energy dominated era in respect to Sne-Ia+BAO+H(z)+CMB observations for a suitable parametricspaces, as mentioned earlier. Moreover the F ( R ) model (56) satisfies lim R → (cid:0) F ( R ) − R (cid:1) = 0 i.e the correction overthe Einstein-Hilbert term vanishes in the limit R →
0. This indicates that the F ( R ) of Eq.(56) is able to provide theMinkowski solution in vacuum case and thus is free from the problem of including an implicit cosmological constant.Thus, the F ( R ) proposed in Eq. (56) appropriately provides a non-singular bounce and a late acceleration near R ≃ R b and during R ∼ Λ respectively, and hence is able to merge a bounce with dark energy (DE) epoch. Thedeceleration stage in between the bounce and the dark energy epochs can be obtained by the numerical interpolation,similarly as we did for the earlier case in Sec.[V].At this stage it deserves mentioning that we have taken the bottom-up approach to reconstruct the F ( R )gravity in the present context, in which we attempt to figure out the functional form of F ( R ) by demanding thatit should explain the observational results. However, various terms in the F ( R ) of Eq.(56) may be thought tohave a fundamental origin, like - near the bouncing regime, the F ( R ) (denoted by F b ( R )) is given by Eq.(57).Thereby if we expand the Taylor series of the function Erf i (cid:20) √ R − α √ α (cid:21) (present in Eq.(57)) around the bouncingpoint i.e around R = 12 α and keeping the leading order term, then the near-bounce expression of F ( R ) willcontain linear and quadratic power in curvature R . Such quadratic correction in the Ricci scalar plays a rolefor renormalizability of General Relativity (GR) in quantum gravity. On other hand, during the late time, the F ( R ) behaves as of Eq.(54). The logarithmic correction present in the late time form of F ( R ) may be thoughtas one-loop effects of quantum gravity. Despite these arguments, it is true that due to the complicated nature0of the full F ( R ) presented in Eq.(56), it is hard to realize some fundamental origin of this full form of F ( R ). Ofcourse, the complete gravitational action should be defined by a fundamental theory, which, however, remains tobe the open problem of modern high-energy physics. In the absence of fundamental quantum gravity, the modi-fied gravity approach in this work is a phenomenological one that is constructed by complying with observational data.Before concluding, we would like to mention that the present scenario merges certain cosmological epochs of theuniverse, in particular, from a non-singular bounce to a matter dominated epoch and from the matter dominated to alate time accelerating epoch; i.e the model is similar to a matter bounce model which is also compatible to a late darkenergy phase of the cosmic evolution as well. However this is not the full evolution history of the universe mainly dueto the absence of the radiation era. Thereby in order to unify the entire evolutionary epochs of the universe in thecontext of bouncing cosmology, one should show the unification of an early bounce with the radiation domination,followed by the photon decoupling, followed by matter domination, all the way to the late-time dominance of darkenergy stage. The unification for the full evolution of the universe is a longstanding problem in cosmology and wehope that our present paper may enlighten some part(s) of this bigger problem. VII. CONCLUSION
In the present work, we proposed a cosmological model in the context of F ( R ) gravity, which merges a non-singularbounce to a matter dominated epoch and from the matter dominated to a late time accelerating epoch; i.e the modelis similar to a generalized matter bounce model which is also compatible to a late dark energy dominant phase of thecosmic evolution. Using the reconstruction schemes, we reconstructed the form of F ( R ) near the bounce and at thelate-time era respectively. However, the reconstruction techniques applied in these two eras are slightly different, inparticular: near the bouncing regime, we first considered a scale factor (suitable for bounce) and then determined theform of F ( R ) which can realize such bouncing scale factor by using the gravitational equation of motion, while on otherhand in the case of late times, we have used a reverse reconstruction procedure in comparison to that of the earlier onei.e. we started with a form of F ( R ) (rather than a scale factor) viable for dark energy model, namely the exponential F ( R ) gravity, and then reconstructed the corresponding Hubble parameter from the gravitational equations of motion.During such reconstructions, we got early and late stage model parameters which have been estimated from variousobservational constraints. The early stage parameters have been obtained from the primordial perturbations and weconfronted the theoretical results of the observable quantities like the scalar spectral index, the tensor-to-scalar ratiowith the latest Planck 2018 constraints. On other hand, the late stage parameters are estimated by investigatingthe viability of the exponential F ( R ) gravity as a dark energy model with respect to the Sne-Ia+BAO+H(z)+CMBdata. Due to a late accelerating phase, the Hubble radius decreases monotonically at late times and asymptoticallygoes to zero, which in turn leads to the generation of the primordial perturbation near the bounce (because at thattime the relevant perturbation modes are within the horizon), unlike to the usual matter bounce scenario where theperturbation modes generate deeply in the contracting regime far away from the bouncing point time. Thus we haveperformed the perturbations near the bounce and as a result the primordial observable quantities like the spectralindex for curvature perturbation, and the tensor-to-scalar ratio, are found to be simultaneously compatible with thelatest Planck 2018 observations. Moreover, the scalar and tensor perturbations are stable as the condition F ′ ( R ) > F ( R ) gravitational equation to determine the form of F ( R ) for a widerange of cosmic time, which clearly depicts that the F ( R ) matches with the Einstein gravity in the low curvatureregime, while it deviates from the usual Einstein gravity as the scalar curvature acquires larger and larger values.Correspondingly, the Hubble parameter and the effective EoS parameter of the Universe have been determined. As aresult the EoS parameter is found to successively cross the value − twice indicating the transition of the Universefrom bounce to a quasi-matter dominated phase and from the quasi-matter dominated to a late-time acceleratingstage respectively. Moreover, the EoS parameter during the dark energy era approaches to the value − . F ( R ) gravity form which is able to merge a non-singular bounce with adark energy epoch, however the DE epoch is described by a logarithmic corrected exponential F ( R ) gravity, unliketo the earlier case where the dark energy model is depicted by the usual exponential F ( R ) gravity i.e. without thelogarithmic corrections. Similar to the exponential F ( R ) model, the logarithmic generalized exponential F ( R ) gravityis also known to provide a viable dark energy model with respect to Sne-Ia+BAO+H(z)+CMB observations, in fact,1the presence of logarithmic corrections leads to a better fitted model than the usual exponential F ( R ) model. Acknowledgments
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