Bounce and cyclic cosmology in new gravitational scalar-tensor theories
aa r X i v : . [ g r- q c ] J u l Bounce and cyclic cosmology in new gravitational scalar-tensor theories
Emmanuel N. Saridakis,
1, 2, 3, ∗ Shreya Banerjee, † and R. Myrzakulov ‡ Chongqing University of Posts & Telecommunications, Chongqing, 400065, China Department of Physics, National Technical University of Athens, Zografou Campus GR 157 73, Athens, Greece Eurasian International Center for Theoretical Physics,Eurasian National University, Astana 010008, Kazakhstan Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
We study the bounce and cyclicity realization in the framework of new gravitational scalar-tensor theories. In these theories the Lagrangian contains the Ricci scalar and its first and secondderivatives, in a specific combination that makes them free of ghosts, and transformed into theEinstein frame they are proved to be a subclass of bi-scalar extensions of general relativity. Wepresent analytical expressions for the bounce requirements, and we examine the necessary qualitativebehavior of the involved functions that can give rise to a given scale factor. Having in mind thesequalitative forms, we reverse the procedure and we construct suitable simple Lagrangian functionsthat can give rise to a bounce or cyclic scale factor.
PACS numbers: 98.80.-k, 95.36.+x, 04.50.Kd
I. INTRODUCTION
Although inflation is considered to be an importantpart of the universe history [1], the “problem of the ini-tial singularity” is still present in the standard model ofthe universe. In particular, since such a singularity isunavoidable if inflation is driven by a scalar field in theframework of general relativity [2], a lot of effort has beendevoted in resolving it through quantum gravity consid-erations or effective field theory applications.Non-singular bouncing cosmologies may offer a poten-tial solution to the cosmological singularity problem [3].Modified gravities are an ideal framework for their re-alization, since they allow for the necessary violationof the null energy condition [4, 5]. In particular, onecan obtain bouncing solutions in Pre-Big-Bang [6] andEkpyrotic [7, 8] models, f ( R ) gravity [9, 10], f ( T ) grav-ity [11], gravity actions with higher order corrections[12, 13], braneworld scenarios [14, 15], non-relativisticgravity [16, 17], Lagrange modified gravity [18], massivegravity [19], loop quantum cosmology [20–22] etc. Alter-natively, bouncing cosmology can be realized introduc-ing matter fields that violate the null energy condition[23–25], or constructing non-conventional mixing terms[26, 27]. Furthermore, one may extend bouncing cosmol-ogy to the paradigm of cyclic cosmology [28], in whichthe universe experiences a sequence of expansions andcontractions [29, 30] (see [34] for a review). This offersalternative insights for the origin of the observable uni-verse [31–33], and can explain the scale invariant powerspectrum [34, 35] and possible non-Gaussianities [36].One recently constructed class of modified gravity isthe so-called “new gravitational scalar-tensor theories” ∗ Electronic address: Emmanuel [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] [37, 38]. In these theories one uses a Lagrangian withthe Ricci scalar and its first and second derivatives, how-ever in a specific combination that makes the theory freeof ghosts. Transforming into the Einstein frame, one canshow that these constructions propagate 2 + 2 degreesof freedom, and thus they fall outside Horndeski [39],Galileon [40, 41] and beyond Horndeski theories [42].Nevertheless, although these theories can be seen as asubclass of bi-scalar extensions of general relativity, theycan still be expressed in pure geometrical terms, and thatis why the authors called them “new gravitational scalar-tensor theories”. Due to the presence of extra degrees offreedom, these theories can lead to very interesting cos-mological behavior [38].In the present work, we are interested in studying therealization of bounce and cyclicity in the framework ofnew gravitational scalar-tensor theories. The plan of thework is as follows: In Section II we review the new grav-itational scalar-tensor theories, and we apply them in acosmological framework. In Section III we construct spe-cific subclasses of Lagrangian functions that give rise tobouncing and cyclic scale factors. Finally, we summarizeour results in section IV.
II. NEW GRAVITATIONAL SCALAR-TENSORTHEORIES
In this section we briefly review the gravitational theo-ries that include higher derivative curvature terms, whichwere named new gravitational scalar-tensor theories in[37]. Transformed into the Einstein frame these theoriespresent two extra scalar degrees of freedom comparingto general relativity, and thus they fall in the class ofbi-scalar modifications.The action of the new gravitational scalar-tensor the-ories is S = Z d √− g f (cid:0) R, ( ∇ R ) , (cid:3) R (cid:1) , (1)with ( ∇ R ) = g µν ∇ µ R ∇ ν R , and where for simplicity,here and in the following, we set the Planck mass M pl toone. Despite the presence of higher derivatives, these ac-tions are ghost free and can be transformed into bi-scalartheories in the Einstein frame through double Lagrangemultipliers. Although one can consider models, namely f -forms, nonlinear in (cid:3) R = β , in the present work wefocus on theories with [38] f ( R, ( ∇ R ) , (cid:3) R ) = K (( R, ( ∇ R ) ) + G ( R, ( ∇ R ) ) (cid:3) R. (2)In this case, action (1) can be transformed to S = Z d x p − ˆ g (cid:20)
12 ˆ R −
12 ˆ g µν ∇ µ χ ∇ ν χ − √ e − √ χ ˆ g µν G∇ µ χ ∇ ν φ + 14 e − √ χ K + 12 e − √ χ G ˆ (cid:3) φ − e − √ χ φ (cid:21) , (3)where K = K ( φ, B ) , G = G ( φ, B ) , (4)with B = 2 e √ χ g µν ∇ µ φ ∇ ν φ. (5)In the above expressions we have introduced the χ -fieldthrough the conformal transformation g µν = e − √ χ ˆ g µν (the hat denotes the conformally related frame), whilethe φ -field enters through ϕ ≡ f β . We mention herethat the above action contains two scalar fields, i.e. χ and φ , however in the specific combination that makes itequivalent to the original higher-derivative gravitationalaction. Thus, these theories consist pure gravitationalformulations of standard multi-scalar-tensor theories.We will work in the Einstein-frame version of the abovetheories, namely with action (3), and for simplicity weneglect the hats. Variation of (3) with respect to themetric leads to the field equations E µν = 12 G µν + 14 g µν g αβ ∇ α χ ∇ β χ − ∇ µ χ ∇ ν χ + 14 g µν r e − √ χ g αβ G∇ α χ ∇ β φ − r e − √ χ G∇ ( µ χ ∇ ν ) φ − r g αβ ∇ α χ ∇ β φ G B ∇ µ φ ∇ ν φ − g µν e − √ χ G (cid:3) φ + G B ( (cid:3) φ ) ∇ µ φ ∇ ν φ + 12 e − √ χ G∇ µ ∇ ν φ − ∇ κ (cid:16) e − √ χ G δ λ ( µ δ κν ) ∇ λ φ (cid:17) + 14 ∇ κ (cid:16) e − √ χ G g µν ∇ κ φ (cid:17) − g µν e − √ χ K + 12 e − √ χ K B ∇ µ φ ∇ ν φ + 18 g µν e − √ χ φ = 0 , (6) where the parentheses in spacetime indices denote sym-metrization, and the subscripts in G and K mark partialderivatives (e.g. G B = ∂ G ( φ,B ) ∂B etc). Furthermore, vari-ation of (3) with respect to χ and φ gives rise to theirequations of motion, namely E χ = (cid:3) χ + 13 e − √ χ g µν G∇ µ χ ∇ ν φ − g µν ∇ µ χ ∇ ν φ G B g αβ ∇ α φ ∇ β φ + 12 r ∇ µ (cid:16) e − √ χ g µν G∇ ν φ (cid:17) − r e − √ χ G (cid:3) φ + r G B ∇ µ φ ∇ ν φ g µν (cid:3) φ − r e − √ χ K + 12 e − √ χ K B r g µν ∇ µ φ ∇ ν φ + 14 r e − √ χ φ = 0 , (7)and E φ = − r e − √ χ g µν G φ ∇ µ χ ∇ ν φ +2 r ∇ β (cid:0) g µν G B g αβ ∇ α φ ∇ µ χ ∇ ν φ (cid:1) + 12 r ∇ ν (cid:16) e − √ χ g µν G∇ µ χ (cid:17) + 12 e − √ χ G φ (cid:3) φ − G B ( (cid:3) φ ) − ∇ ν G B (cid:3) φ ∇ ν φ − r ∇ µ (cid:16) e − √ χ ∇ µ χ G (cid:17) + 12 ∇ µ (cid:16) e − √ χ G φ ∇ µ φ (cid:17) − r e − √ χ ∇ µ χ G B ∇ µ B + 12 e − √ χ ∇ µ G B ∇ µ B + r e − √ χ G B ∇ µ (cid:16) e √ χ ∇ µ χ ∇ ν φ ∇ ν φ (cid:17) +2 e − √ χ G B ∇ µ (cid:16) e √ χ ∇ ν φ (cid:17) ∇ µ ∇ ν φ +2 G B R µν ∇ µ φ ∇ ν φ + 14 e − √ χ K φ −∇ ν (cid:16) e − √ χ K B g µν ∇ µ φ (cid:17) − e − √ χ = 0 . (8)As mentioned above, we do verify that all field equationsdo not contain problematic higher-derivative terms, andthus theory (1) is indeed healthy as it is constructed tobe. Finally, note that in the scenario at hand generalrelativity is reproduced when K = φ/ G = 0, andin this case the triviality of the conformal transformationgives χ = − q ln 2.We proceed by applying the above theories into a cos-mological framework. We add the matter sector straight-away in the Einstein frame and we consider the totalaction S tot = S + S m [38]. Therefore, the metric fieldequations (6) become E µν = 12 T µν , (9)with T µν = − √− g δS m δg µν the energy-momentum tensor ofthe matter sector considered to correspond to a perfectfluid. Moreover, we consider a flat Friedmann-Robertson-Walker (FRW) geometry with metric ds = − dt + a ( t ) δ ij dx i dx j , (10)with a ( t ) is the scale factor, and hence the two scalars aretime-dependent only. With these considerations, equa-tions (6) lead to the two Friedmann equations:3 H − ρ m −
12 ˙ χ + 14 e − √ χ K + 23 ˙ φ h ˙ φ (cid:16) √ χ − H (cid:17) − φ i G B − e − √ χ (cid:20) ˙ B ˙ φ G B + φ φ ( G φ − K B ) (cid:21) = 0 , (11)3 H + 2 ˙ H + p m + 12 ˙ χ + 14 e − √ χ K + 12 e − √ χ (cid:18) − φ B ˙ φ G B + ˙ φ G φ (cid:19) = 0 , (12)with B ( t ) = 2 e √ χ g µν ∇ µ φ ∇ ν φ = − e √ χ ˙ φ , H = ˙ a/a the Hubble parameter, and where dots denote differenti-ation with respect to t . Additionally, we have introducedthe energy density ρ m and pressure p m of the matterfluid. Similarly, the two scalar field equations (7) and (8)lead to the scalar evolution equations: E χ = ¨ χ + 3 H ˙ χ −
13 ˙ φ h ˙ φ (cid:16) √ H − χ (cid:17) + √ φ i G B + 12 √ e − √ χ h B ˙ φ G B − φ + 2 ˙ φ ( K B + G φ ) i + 1 √ e − √ χ K = 0 , (13)and E φ = 13 e − √ χ h ˙ φ (cid:16) − H + √ χ (cid:17) − φ i K B + 16 ˙ B n e − √ χ ˙ B + 4 ˙ φ h ˙ φ (cid:16) H − √ χ (cid:17) + 3 ¨ φ io G BB + 13 e − √ χ h ˙ φ (cid:16) H − √ χ (cid:17) + 3 ¨ φ i G φ + (cid:26) e − √ χ ˙ B ˙ φ + 23 ˙ φ h ˙ φ (cid:16) H − √ χ (cid:17) + 3 ¨ φ i(cid:27) G Bφ − e − √ χ ˙ φ K Bφ + 12 e − √ χ ˙ φ G φφ − e − √ χ ˙ B ˙ φ K BB + (cid:20)
43 ˙ φ (cid:16) H − √ χ (cid:17) ¨ φ − √ e − √ χ ˙ B ˙ χ + ˙ φ (cid:18) H + 6 ˙ H − √ H ˙ χ −
23 ˙ χ − √ χ (cid:19)(cid:21) G B − e − √ χ K φ + 14 e − √ χ = 0 , (14) with G Bφ = G φB ≡ ∂ G ∂B∂φ , etc. We mention that amongstthe above four equations, namely (11),(12), (13),(14),only three are independent, due to the fact that the totalaction is diffeomorphism invariant, i.e. we have a conser-vation equation [38]: ∇ µ E µν + 12 E χ ∇ ν χ + 12 E φ ∇ ν φ = 12 ∇ µ T µν = 0 . (15)Thus, the matter energy density and pressure satisfy thestandard evolution equation˙ ρ m + 3 H ( ρ m + p m ) = 0 . (16)The investigation of the above cosmological scenarioat late times was performed in [38], where it was indeedshown that one can obtain interesting phenomenology. Inthe present work we are interested in studying the early-time phases, and in particular to examine whether thebounce realization is possible. This is performed in thefollowing sections. III. BOUNCING AND CYCLIC SOLUTIONS
In this section we proceed to the investigation ofbounce and cyclicity realization in cosmology driven bynew gravitational scalar-tensor theories. As it is known,in order to obtain a bounce we need a contracting phase(
H <
0) succeeded by an expanding one (
H > H = 0 and ˙ H > H = 0 and˙ H <
0. Although these conditions cannot be fulfilledin the framework of general relativity [13], observing theform of the two Friedmann equations (11),(12), as wellas of the scalar-field equations (13),(14) we deduce thatfor suitable choices of the free functions K and G one canobtain the necessary violation of the null energy condi-tion and thus satisfy the aforementioned bouncing andcyclic conditions. A. Reconstruction of a bounce
Let us now proceed to the investigation of the bouncerealization. Suppose that we impose a given form ofa bouncing scale factor, in which case H ( t ) and ˙ H ( t )are also known. Substitution of this bouncing scalefactor into the three independent equations (11),(13)and (14), and recalling that in FRW geometry B ( t ) = − e √ χ ( t ) ˙ φ ( t ) , we obtain a system of three differentialequations for the four functions φ ( t ) and χ ( t ) and for K and G (and their derivatives) considered as functions of t .Thus, we have the freedom to further consider the formof one of K and G . In the following paragraphs we ex-amine two such cases separately, having in mind that inorder to be able to obtain a bounce we need to go beyondthe simple K and G forms investigated in [38], which wereadequate to describe the late-time universe.
1. Model I: K = φ/ and G = G ( B ) Since general relativity is re-obtained for K = φ/ G = 0, one class of viable models of new gravitationalscalar-tensor theories is the one with K = φ , (17)and with G = G ( B ) , (18)i.e. G is independent of φ . Concerning the explicit bounc-ing scale factor, without loss of generality we consider thematter bounce form [11] a ( t ) = a b (1 + qt ) / , (19)with a b denoting the value of scale factor at the bouncepoint t = 0, and q a positive parameter which determineshow fast the bounce is realized. From this scale factorwe immediately find H ( t ) = 2 qt qt ) (20)˙ H ( t ) = 2 q (cid:20) − qt (1 + qt ) (cid:21) . (21)Inserting these considerations into equations (11),(13)and (14), and replacing G B ( t ) = G ′ ( t ) /B ′ ( t ), and G BB ( t ) = ( G ′′ ( t ) B ′ ( t ) − B ′′ ( t ) G ′ ) / ( B ′ ( t )) , we obtain asystem of differential equations for φ ( t ), χ ( t ) and G ( t ).Unfortunately, this system cannot be solved analytically,however it can easily be elaborated numerically, leadingto the extraction of the φ ( t ), χ ( t ) and G ( t ). Since B ( t )can then be found, we can acquire the function G ( B ) ina parametric form. Hence, it is this reconstructed G ( B )that generates the input scale factor (19).In Fig. 1 we present the G ( B ) that is reconstructedfrom the given bouncing scale-factor form (19), accord-ing to the above procedure, where we have neglected thematter sector in order to investigate the pure effect of thenovel terms of the present theory. As we observe fromthe above procedure, and in particular from Fig. 1, inorder to obtain a bouncing scale factor in the case where K = φ/
2, we need a G ( B ) function that resembles anexponential function of B .The explicit example of the above reconstruction pro-cedure offered us qualitative information for the G ( B )form that leads to a bouncing scale factor. Thus, we cannow reverse the reconstruction procedure and impose theform of G ( B ) a priori, and then extract the induced scalefactor, which is the physical procedure. Having the qual-itative requirements for G ( B ) in mind, we choose its formto be G ( B ) = G e G B , (22) FIG. 1:
The reconstructed G ( B ) that generates the bouncingscale factor (19), in the case where K = φ/ . The bouncingparameters have been chosen as a b = 0 . , q = 10 − . Allquantities are measured in M pl units. where G and G are parameters. Substituting it intoequations (11), (12) and (13), considering once again K = φ/
2, we acquire three second order differential equa-tions for a ( t ), φ ( t ) and χ ( t ). Elaborating the system nu-merically we extract the scale factor, and we depict itin Fig. 2. Hence, we do verify that new gravitational - - a ( t ) FIG. 2:
The evolution of the scale factor a ( t ) that is gen-erated by the the exponential form G ( B ) = G e G B , in thecase where K = φ/ , with G = − and G = 0 . in M pl units. scalar-tensor theories with K = φ/ G ( B ) = G e G B lead to the realization of a cosmological bounce.The above procedure can easily be repeated in thepresence of the matter sector which gives rise to mat-ter bounce. We find that the same forms of the functions K and G in the presence of pressureless matter, i.e. with p m = 0 and ρ m ∝ a , give rise to a matter bounce,namely to a scale-factor similar to that of Fig. 2.
2. Model II: K = φ/ f ( B ) and G = ξB In this paragraph we investigate the bounce realizationin a different subclass of new gravitational scalar-tensortheories. In particular, we choose K = φ f ( B ) G = ξB, (23)where f ( B ) is an unknown function of B and ξ is a pa-rameter. Similarly to the previous paragraph, firstly weconsider the bounce scale factor (19), in order to numer-ically reconstruct f ( B ) and acquire a qualitative pictureof its form. Indeed, substituting these into equations(11),(13) and (14), and replacing K B = f ′ ( t ) /B ′ ( t ) and K BB ( t ) = ( f ′′ ( t ) B ′ ( t ) − B ′′ ( t ) f ′ ) / ( B ′ ( t )) , we obtain asystem of differential equations for φ ( t ), χ ( t ) and f ( t ).Since this system cannot be solved analytically, we elab-orate it numerically and we extract φ ( t ), χ ( t ) and f ( t ).Since B ( t ) can then be found as B ( t ) = − e √ χ ( t ) ˙ φ ( t ) ,we can finally acquire the function f ( B ) in a paramet-ric form. Hence, this reconstructed f ( B ) generates theinput scale factor (19).In Fig. 3 we present the f ( B ) that is reconstructed ac-cording to the above procedure, in the absence of mattersector. As we observe, in order to obtain a bouncing scale - - - - -
10 001020304050 B f FIG. 3:
The reconstructed f ( B ) that generates the bouncingscale factor (19), in the case where K = φ/ f ( B ) and G = ξB . The bouncing parameters have been chosen as a b = 0 . , q = 10 − , while ξ = 0 . . All quantities are measured in M pl units. factor in the case where K = φ/ f ( B ) and G = ξB ,we need a f ( B ) form that resembles a parabolic functionof B .Having in mind the qualitative information for theform of f ( B ) obtained through the above reconstructionprocedure, we can now reverse the procedure and imposethe form of f ( B ) a priori, and then extract the inducedscale factor. We choose K ( φ, B ) = φ/ B − B ) , (24)where B is a constant. Substituting it into equations(11), (12) and (13), alongside with G = ξB , we result to three second order differential equations for a ( t ), φ ( t ) and χ ( t ). Elaborating the system numerically we extract thescale factor, which is presented in Fig. 4. Thus, we doverify that new gravitational scalar-tensor theories with K ( φ, B ) = φ/ B − B ) and G = ξB lead to therealization of a cosmological bounce. - - a ( t ) FIG. 4:
The evolution of the scale factor a ( t ) that is gen-erated by K ( φ, B ) = φ/ B − B ) and G = ξB , with B = 25 . and ξ = 0 . in M pl units. Finally, we mention that the above procedure can berepeated in the presence of the matter sector, and wefind that the same forms of the functions K and G in thepresence of dust matter give rise to a matter bounce.
3. General conditions for a bounce
We close this subsection by examining analytically theconditions for the bounce realization in the theories andmodels at hand, namely H = 0 and ˙ H > H + bH + c = 0 , (25)where b = − φ G B (26) c = −
16 ˙ χ + 112 e − √ χ K + 29 ˙ φ ( ˙ φ √ χ − φ ) G B − e − √ χ (cid:20) ˙ B ˙ φ G B + φ φ ( G φ − K B ) (cid:21) . (27)The general solution of the above quadratic equation is H = − b ± √ b − c . (28)Hence, in order for the first bounce condition, namely H b = 0 (the subscript “b” denotes the value at thebounce point), to be satisfied, we deduce that at thebounce point we need c = 0. In the case of Model Iabove, i.e. for K = φ/ G = G ( B ), this conditionbecomes − χ + 3 φ e − √ χ + 4 ˙ φ ( ˙ φ √ χ − φ ) G B − e − √ χ (cid:18) ˙ B ˙ φ G B + φ (cid:19) = 0 (29)at the bounce point, while for Model II above, i.e. for K = φ/ f ( B ) and G = ξB , this condition becomes − χ + 3 e − √ χ (cid:20) φ f ( B ) (cid:21) + 8 ˙ φ ( ˙ φ √ χ − φ ) ξ − e − √ χ (cid:18) ˙ B ˙ φξ + φ − φ f B (cid:19) = 0 (30)at the bounce point. The above conditions simplify fur-ther once we consider the forms of the functions G ( B )and f ( B ) obtained above.Concerning the second bounce condition, namely ˙ H > χ + e − √ χ K + 2 e − √ χ (cid:18) ˙ B ˙ φ G B − φ φ G φ (cid:19) > . (31)Thus, in the cases of Model I and Model II respectivelywe obtain4 ˙ χ + e − √ χ φ + 4 e − √ χ (cid:18) ˙ B ˙ φ G B − φ (cid:19) > χ + e − √ χ (cid:20) φ f ( B ) (cid:21) + 2 e − √ χ (cid:18) ˙ B ˙ φξ − φ (cid:19) > . (33)We mention here that in the analysis of the previousparagraphs we took the above general requirements intoaccount in order to determine the initial conditions forthe differential equations at the bounce point. Indeed,as we mentioned above, in order to be able to satisfythe above requirements and obtain a bounce, one mustgo beyond the simple K and G forms of [38] that wereadequate to describe the late-time universe. B. Reconstruction of cyclic evolution
In this subsection we extend the above analysis in or-der to construct a sequence of bounces and turnarounds,namely in order to obtain a cyclic cosmological evolu-tion. As a first step we will consider a specific cyclicscale factor and we will reconstruct the corresponding K and G forms that generate it. Then, having obtained in-formation for their qualitative behavior, we will considerspecific K and G forms and show that they lead to cyclicevolution. In the following paragraphs we examine twosuch cases separately.
1. Model I: K = φ/ and G = G ( B ) As a first model we consider K = φ/ G = G ( B ).We start considering an oscillating scale factor of theform a ( t ) = A sin( ωt ) + a c , (34)with a c − A > A + a c the scale factor value at the turnaround.Inserting these into (11),(13) and (14), and replacing G B ( t ) = G ′ ( t ) /B ′ ( t ), and G BB ( t ) = ( G ′′ ( t ) B ′ ( t ) − B ′′ ( t ) G ′ ) / ( B ′ ( t )) , we obtain a system of differentialequations for φ ( t ), χ ( t ) and G ( t ). We mention thatsince cyclic cosmology describes the whole universe evo-lution, and not only the phase around the bounce, wemust necessarily include the matter sector. Without lossof generality we focus on the dust matter case, where p m = 0 and thus the continuity equation (16) leads to ρ m = ρ mb ( a c − A ) /a .Since the above system of differential equations cannotbe solved analytically, we elaborate it numerically and weobtain the solutions for φ ( t ), χ ( t ) and G ( t ), and thus for B ( t ) too, and therefore we acquire the function G ( B ) in aparametric form. In Fig. 5 we depict this reconstructed G ( B ), which is the one that gives rise to the cyclic scalefactor (19). - × - × - × - × - × - × - - B G FIG. 5:
The reconstructed G ( B ) that generates the cyclic scalefactor (34), in the case where K = φ/ . The parameters havebeen chosen as A = 0 . , a c = 10 , ω = 1 . All quantities aremeasured in M pl units. As we observe from the above procedure, and in partic-ular from Fig. 5, in order to acquire a cyclic scale factorin the case where K = φ/
2, we need a G ( B ) functionthat resembles an exponential function of B . Thus, wecan now reverse the reconstruction procedure and imposean exponential form of G ( B ), namely G ( B ) = G e G B , (35)where G and G are parameters. Substituting it into(11), (12) and (13), with K = φ/
2, we obtain threesecond-order differential equations for a ( t ), φ ( t ) and χ ( t ).Elaborating the system numerically we extract the scale - - a ( t ) FIG. 6:
The evolution of the scale factor a ( t ) that is gen-erated by the exponential form G ( B ) = G e G B , in the casewhere K = φ/ , with G = 2 . and G = 0 . in M pl units. factor, and we present it in Fig. 6. Thus, we can see thatnew gravitational scalar-tensor theories with K = φ/ G ( B ) = G e G B can indeed produce a cyclic uni-verse. We mention here that the exponential G ( B ) mayalso lead to a single bounce realization, as we saw in theprevious subsection, however what distinguishes the twopossibilities are the parameter values.
2. Model II: K = φ/ f ( B ) and G = ξB We now study cyclicity in a different model, namelyin the case where K = φ/ f ( B ) and G = ξB , with ξ a parameter. Similarly to the previous paragraph,we first consider the cyclic scale factor (34), in orderto numerically reconstruct f ( B ) and acquire a qualita-tive picture of its form. Substituting these into equa-tions (11),(13) and (14), replacing K B = f ′ ( t ) /B ′ ( t ) and K BB ( t ) = ( f ′′ ( t ) B ′ ( t ) − B ′′ ( t ) f ′ ) / ( B ′ ( t )) , and consider-ing a dust matter sector, we obtain a system of differen-tial equations for φ ( t ), χ ( t ) and f ( t ). Solving it numer-ically we extract φ ( t ), χ ( t ) and f ( t ), and therefore B ( t )too, and thus we obtain the function G ( B ) in a paramet-ric form. In Fig. 7 we show this reconstructed G ( B ),which is the one that gives rise to the cyclic scale factor(19).As we observe from Fig. 7, in order to acquire a cyclicscale factor in the case where K = φ/ f ( B ) and G = ξB , we need a f ( B ) function that resembles an parabolicfunction. Having this in mind we can now consider as aninput the parabolic form K ( φ, B ) = φ/ B − B ) , (36)with B a constant. Inserting into (11), (12) and (13),with G = ξB , we obtain a system of differential equa-tions for a ( t ), φ ( t ) and χ ( t ). Solving the equations nu-merically we extract the scale factor, and we depict itin Fig. 8. Hence, we deduce that the theories at handwith K ( φ, B ) = φ/ B − B ) and G = ξB can indeedproduce a cyclic universe. Note that, as we saw in the - - - - - - - - B f FIG. 7:
The reconstructed f ( B ) that generates the cyclic scalefactor (34), in the case where K = φ/ f ( B ) and G = ξB .The parameters have been chosen as A = 0 . , a c = 10 , ω = 1 and ξ = 0 . . All quantities are measured in M pl units. - - a ( t ) FIG. 8:
The evolution of the scale factor a ( t ) that is gen-erated by K ( φ, B ) = φ/ B − B ) and G = ξB , with B = 19 . and ξ = 0 . in M pl units. previous subsection, this parabolic form for K ( φ, B ) mayalso lead to a single bounce solution, however what dis-tinguishes the two possibilities are the parameter values. IV. CONCLUSIONS
In this work we investigated the bounce and cyclicityrealization in the framework of new gravitational scalar-tensor theories. In particular, in these theories one con-siders a Lagrangian with the Ricci scalar and its first andsecond derivatives, however in a specific combination thatmakes the theory free of ghosts. Transforming into theEinstein frame, one can show that these constructionspropagate 2 + 2 degrees of freedom, and thus they are asubclass of bi-scalar extensions of general relativity. Nev-ertheless, the fact that these theories can be expressed inpure geometrical terms is a significant advantage.We studied bouncing and cyclic solutions in variouscases, reconstructing the forms of the functions K ( φ, B )and G ( φ, B ) that can give rise to a given scale factor.Thus, having in mind the necessary qualitative form ofthese functions, we were able to reverse the procedurein the more physical base, namely we considered suit-able simple functions K ( φ, B ) and G ( φ, B ) exhibiting thisqualitative form, and we showed that they can give riseto a bounce or cyclic scale factor.We close this work by referring to the perturbationsof the obtained background solutions. In every bounc-ing scenario the analysis of perturbations is necessary,since they are related to observables such as the tensor-to-scalar ratio. While in inflationary cosmology the gen-eration of the primordial power spectrum requires thatthe cosmological fluctuations emerge initially inside theHubble horizon, then they exit it, and later on they re-enter, in a bounce scenario the situation is radically dif-ferent. In particular, in bouncing cosmology the quan-tum fluctuations around the initial vacuum state are gen-erated in the contraction phase before the bounce, theyexit the Hubble radius as contraction continues, since theHubble horizon decreases faster than the wavelengths ofthe primordial fluctuations, then the bounce happens,and finally they re-enter inside the horizon at later timesin the expanding phase. Definitely, at the bounce pointthe background evolution could affect the perturbationsscale dependence, however one expects this effect to beimportant only in the UV regime, where the gravitationalmodification effects play role, while the IR regime, whichis responsible for the primordial perturbations related tothe large-scale structure, remains almost unaffected [43–45].Although the generation of perturbations in bouncingmodels with one extra scalar degree of freedom is wellunderstood and studied [34, 46, 47], in the case wherethe underlying theory has more than one extra scalar de-grees of freedom, where both of them contribute to thebounce, the perturbation generation has not been stud- ied in detail. In particular, the examined scenarios inthis subclass assume that one of the two extra scalar de-grees of freedom is the dominant one at some point [48–53]. However, this approach cannot be straightforwardlyfollowed in scenarios where both fields have more or lessequal contribution to the bounce realization, and one cansee that the present scenario lies in this category. Hence,the analysis of perturbation generation in the bouncingscenario at hand has to be performed in a thorough andsystematic way, through the full and detailed perturba-tion generation analysis of general two-field bounces. Forthis investigation one could use concepts and techniquesof the perturbation generation in two-field inflation [54–58] (which is different from single-field inflation with asecond sub-dominant field such as in cases of hybrid in-flation [59–61]). Nevertheless, this detailed analysis ofperturbation generation in two-field bouncing models isa separate work that lies beyond the scope of the presentproject, and it is left for a future investigation.In summary, we showed that the new gravitationalscalar-tensor theories, namely a subclass of bi-scalar ex-tension of general relativity that can be constructed bypure geometrical terms, can naturally give rise to bounc-ing and cyclic behavior. This capability acts as an addi-tional advantage for these theories. Acknowledgments
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