Classical defocussing of world lines -- Cosmological Implications
RRevised VersionClassical defocussing of world lines - Cosmological Implications
R.Parthasarathy The Chennai Mathematical InstituteH1, SIPCOT IT Park, SiruseriChennai 603103, India.
Abstract
We have extended our result on defocussing of world lines [1] by modifyinggravity in the early epoch by a 5-d theory with scalar ψ ( r ). The accelerationterm in the Raychaudhuri equation has been shown to be positive for flatFRW metric. The scalar ψ ( r ) satisfies a non-linear differential equation whichis solved. Though singular, the acceleration term turns out to be finite. Withthis, the equations for the Hubble parameter H ( t ) and the scale factor a ( t ) areobtained. These are analyzed using ’fixed point analysis’. Without the scalarfield, the age of the universe is finite showing a beginning of the universe andwith out bounce. With the contribution of the scalar field included, the ageof the universe is shown to be infinite, thereby resolving the singularity. Thescale factor a ( t ) exhibits classical bounce, the bounce being proportionalto the effect of the scalar field. The effect of the 5-d gravity in the earlyuniverse is to cause defocussing of the world lines, give infinite age of theuniverse thereby resolving the big bang singularity and classical bounce forthe FRW scale factor. [email protected] a r X i v : . [ phy s i c s . g e n - ph ] F e b . Introduction: In our earlier communication [1], we have shown classical defocussing ofworld lines in 5-dimensional Kaluza theory by modifying gravity in the earlyuniverse epoch by 5-d gravity with Kaluza scalar. The result obtained wasgeneral in the sense that no specific metric, other than spherical symmetry orexplicit form for the Kaluza scalar ψ ( r ) were used. When there is defocussingof world lines, we pointed out the possible avoidance of big bang singularity.This implies that the universe exists for ever and there should be bounce inthe FRW scale factor. It is the purpose of this paper to examine these twoissues for flat FRW universe. Bounce in the cosmology of early universe hasbeen proposed to replace inflation as the mechanism for addressing issues inthe standard big bang cosmology. Considered as an alternative to standardcosmological model without the initial singularity, bouncing cosmology is anattempt of addressing the early universe.Bounce cosmologies have been proposed based upon stringy effects [2, 3,4, 5], path integral methods [6,7], loop gravity approaches [8, 9], group fieldtheory [10, 11, 12], from f ( T ) gravity [13], from f ( R ) gravity [14] and Gauss-Bonnet gravity [15]. Emergent cosmological models with bouncing scenarioof the early universe has been considered in [16]. Bouncing cosmologies asalternatives to cosmological inflation for providing a description of the earlyuniverse has been studied in [17]. Big bang singularities are avoided at theclassical level in Friedmann universe by introducing constrained scalar fieldsin [18]. A class of non-singular bouncing cosmologies that evade singularitytheorems through vorticity in compact extra dimensions, the vorticity com-bating the focusing of geodesics has been proposed in [19]. The list aboveis not exhaustive but indicates the recent surge of activity in avoiding thesingularities in the early universe. In regions of spacetime where gravity isstrong, modifications of the coupling between gravity and electromagneticfield with non-minimal coupling involving curvature has been considered in[20]. From these investigations the consensus is that certain modificationsof Einstein theory are expected during the early universe epoch where gravityis strong . We have considered in [1] one such modification, namely, duringthe early universe the spacetime is 5-dimensional and the modified gravityis taken to be 5-dimensional Kaluza gravity with the metric scalar ψ ( r ). Wepoint out here that we do not introduce a potential for the scalars and theyare massless. 2e summarize our results. We first show that the acceleration term inthe Raychaudhuri equation is positive for flat FRW metric. The differentialequation for the scalar ψ ( r ) which is the classical equation of motion for ψ ( r ), is solved. From the Raychaudhuri equation, the differential equationsfor Hubble parameter H ( t ) and for the FRW scale factor a ( t ) are obtained.These are analyzed and shown that the age of the universe is infinite, theuniverse existing for ever with out initial singularity. The scale factor a ( t )exhibits classical bounce.In Section.2, we briefly review our earlier [1] results. In Section.3, wegive the results using Friedmann - Walker - Robertson flat metric and inSection.4, we present the differential equations for the Hubble parameter H ( t ) and the FRW scale factor a ( t ). In Section.5, the differential equationfor H ( t ) is analyzed by iterative method and ’fixed point analysis’. Theage of the universe now has been shown to be infinite, consistent with thedefocussing of world lines and avoiding the big bang singularity. In Section.6,the differential equation for FRW scale factor a ( t ) is analyzed and shownto exhibit classical bounce consistent with the avoidance of the big bangsingularity. The results are summarized in Section.7.
2. Brief review of classical defocusing of world lines:
A 5-dimensional gravity theory with Kaluza scalar has been considered[1] at the early universe where the gravity is expected to be strong, thespacetime being five dimensional, that is( ds ) = g µν dx µ dx ν − g ( dx ) . A remarkable consequence is that the world line equation has an accelerationterm from the ’55’ component of the 5-d metric g ( r ) = ψ ( r ), namely d x µ ds + (cid:52) µνλ dx ν ds dx λ ds = 12 a ψ g µλ ( ∂ λ ψ ) , (1)where µ, ν, λ are four dimensional indices and a is a constant along theworldline, a consequence of the independence of the metric components on thefifth coordinate x . (cid:52) µνλ are the 5-d connection coefficients restricting to 4-dindices. In the Raychaudhuri equation [21] in 5-d spacetime describing theevolution of a collection of particles following their worldline (1) characterized3heir volume Θ = u µ ; µ the particles having the 4-velocity u µ ˙Θ = − Θ − σ + 2 ω − R µν u µ u ν + ( ˙ u µ ) ; µ , (2)where 2 σ = σ µν σ µν and 2 ω = ω µν ω µν . R µν is the 4-d Ricci tensor and thesubscript ; stands for covariant derivative using (cid:52) . σ µν is the symmetric sheartensor and ω µν is the antisymmetric vorticity tensor. The last term involves˙ u µ = u µ ; ν u ν , the possible acceleration (orthogonal to u µ ) of the collectionof particles. The 5-d Raychaudhuri equation restricting to 4-d, namely (2),follows from Ehlers identity R αβ u α u β = − ˙Θ −
14 Θ − σ + 2 ω + ( ˙ u α ) ; α , with α, β = 0 , , , , g αβ [22, 23]. Restricting to4-d, as R = 0 (by the equation of motion for ψ ( r ) shown in the Appendix),and as none of the quantities depend on x , (2) follows from Ehlers identity.In view of (1), the last term in (2) exists now and it was shown in [1] that( ˙ u µ ) ; µ = − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) , (3)where D µ stands for the covariant derivative D µ ( ∂ ρ ψ ) = ∂ µ ∂ ρ ψ − (cid:52) σµρ ( ∂ σ ψ ).Thus (2) becomes, by replacing R µν u µ u ν → πG ( ρc + 3 p ),˙Θ = − Θ − σ + 2 ω − πG c ( ρc + 3 p ) − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) , (4)for the density ρ and pressure p of the collection of particles. The last termin (4) was shown to be [1], − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) = 3 a e − ν ( ψ (cid:48) ) ψ > , (5)for a spherically symmetric metric( ds ) = e µ c ( dt ) − e ν ( dr ) − r { ( dθ ) + sin θ ( dφ ) } − ψ ( r )( dx ) , (6)with µ, ν as functions of r = √ x + y + z . Since ψ ( r ) > e − ν is positive, the above result45) exhibits defocussing of world lines classically. In obtaining the result (5),we made use of ˆ R AB = 0 (in particular ˆ R = 0), the 5-d vacuum Einsteinequations. The ˆ R = 0 equation is shown in the Appendix to be the classicalequation of motion for ψ ( r ).
3. FRW metric and the ’acceleration term’:
In obtaining the classical defocussing of world lines in [1], we used spher-ically symmetric metric (6) and ˆ R = 0, the 55-component of 5-d Einsteinvacuum equations. In this section, we use a specific metric, namely flat FRWmetric as( ds ) = c ( dt ) − a ( t ) { ( dr ) + r ( dθ ) + r sin θ ( dφ ) } − ψ ( r )( dx ) , (7)where a ( t ) is the three dimensional spatial scale factor and the non-vanishingconnection coefficients are: (cid:52) trr = a ( t ) c da ( t ) dt ; (cid:52) tθθ = r c a ( t ) da ( t ) dt ; (cid:52) tφφ = r c sin θ a ( t ) da ( t ) dt ; (cid:52) rtr = a ( t ) da ( t ) dt ; (cid:52) rθθ = − r ; (cid:52) rφφ = − r sin θ ; (cid:52) r = − ψ (cid:48) a ( t ) ; (cid:52) θtθ = a ( t ) da ( t ) dt ; (cid:52) θrθ = r ; (cid:52) θφφ = − sin θ cos θ ; (cid:52) φtφ = a ( t ) da ( t ) dt ; (cid:52) φrφ = r ; (cid:52) φθφ = cotθ ; (cid:52) r = ψ (cid:48) ψ , (8)where ψ (cid:48) = dψdr . In (8) a ( t ) is FRW scale factor. The aim of using (7) is tocalculate the ’acceleration term’ in (4) (the last term) and to show that it is positive for the FRW metric (7). The 5-d curvature tensor˜ R AB = ∂ C (cid:52) CAB − ∂ B (cid:52) CAC + (cid:52) CDC (cid:52)
DAB − (cid:52)
CDB (cid:52)
DCA , (9)satisfies the 5-d vacuum Einstein equation˜ R AB = 0 . (10)In particular ˜ R = 0, an ingredient in our result in [1] is˜ R = 1 a ( t ) (cid:32) − ψ (cid:48)(cid:48) − ψ (cid:48) r + ( ψ (cid:48) ) ψ (cid:33) = 0 , (11)where ψ (cid:48)(cid:48) = d ψdr . 5ow, we consider the ’acceleration term’ (3) in 5-d flat FRW metric (7).It is − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) = − a ψ g rr ( ψ (cid:48) ) + a ψ g rr ψ (cid:48)(cid:48) − a ψ g µρ (cid:52) rµρ ψ (cid:48) , (12)and from (8), we have g µρ (cid:52) rµρ = 2 a ( t ) r . (13)Therefore using (13) in (12), we find − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) = a a ( t ) ψ (cid:32) − ψ (cid:48)(cid:48) ψ (cid:48) ) ψ − ψ (cid:48) r (cid:33) . (14)From ˜ R = 0 equation (11), ψ (cid:48)(cid:48) = − ψ (cid:48) r + ( ψ (cid:48) ) ψ and so − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) = 3 a a ( t ) ( ψ (cid:48) ) ψ > . (15)Thus, the result that the ’acceleration term’ in the Raychaudhuri equation ispositive shown in [1], holdsgood for 5-d flat FRW metric (7) as well, as ψ ( r )is positive so as to preserve the sign convention for the metric in (7).It is observed that the result of ˜ R = 0 (11) for ψ is consistent withthe equation of motion for the scalar field ψ ( r ) for the action (15) of [1](see Appendix) and agrees with that obtained by Overduin and Wesson [24],using FRW metric (7).Using (11), the scalar field ψ ( r ) in FRW metric satisfies a non-lineardifferential equation ψ (cid:48)(cid:48) ψ (cid:48) r − ( ψ (cid:48) ) ψ = 0 . (16)Although (16) is non-linear, an exact solution is possible. It is ψ ( r ) = 1 r . (17)6f course ψ ( r ) = constant is a solution to (16). With ψ = constant , the5-d world will be same as 4-d world save for trivial changes and we do notconsider this case.With (17), the acceleration term in (15) for this solution is positive and finite as − a g µρ D µ (cid:32) ∂ ρ ψ (cid:33) = 3 a a ( t ) ( ψ (cid:48) ) ψ = 3 a a ( t ) . (18)In the next section we explore the equation governing a ( t ) using Raychaud-huri equation.
4. Raychaudhuri equation and the equation for a ( t ) : We now consider the Raychaudhuri equation (2) for homogeneous andisotropic space-time. The vorticity can be assumed to be vanishing. Sheardescribes kinematic anisotropy. Requiring spatial homogeneity and isotropyimplies σ µν = 0 [25]. Further, CMB anisotropies have been studied exten-sively in [26] in homogeneous cosmology and the study concludes σ = 0 isclearly allowed at 95 percent confidential level. So we consider (2) with noshear and vortcity. It is˙Θ = − Θ − R µν u µ u ν − a g µρ D µ ( ∂ ρ ψ ) , (19)where the last term has been shown to be a a ( t ) in view of (18). In (19),˙Θ = d Θ ds . We are considering the motion of particles with speeds much lessthan the speed of light, that is, the non-relativistic motion of the particles.In this case we can replace dds by ddt . This can be seen by considering (7) as( ds ) = c ( dt ) − a ( t ) { ( dr ) + r ( dθ ) + r sin θ ( dφ ) } − ψ ( r )( dx ) , = c ( dt ) (cid:18) − a ( t ) { c (cid:16) drdt (cid:17) + r c (cid:16) dθdt (cid:17) + r sin θc (cid:16) dφdt (cid:17) } − ψ ( r ) c (cid:16) dx dt (cid:17) (cid:19) , (cid:39) c ( dt ) , for non-relativistic speeds of the particles c terms can be neglected. Thiscan be seen using the geodesic equation with FRW metric (with k = 0) aswell. As the spatial part is homogeneous and isotropic, the geodesic passesthrough some origin (say r = 0). Then by writing the geodesic equation as7 u µ = ( ∂ µ g νσ ) u ν u σ , with { x µ } = { t, r, θ, φ } , it is seen that ˙ u = 0 so that u is constant along the geodesic. But u = − a ( t ) r sin θu so that u = 0at the origin. As ˙ u = 0 along the path, u = 0 along the path. Similarly u = 0 along the path. Then it is seen that u = u = 0; u = u = 0 and u is a constant. Using u = ˙ t and u = − a ( t ) ˙ r along with the normalization u µ u µ = c for massive particles (we use w (cid:54) = 0), it can be shown that (cid:16) dtds (cid:17) =1 + a ( t ) c ˙ r [27]. In the early times, a ( t ) is small and so a ( t ) c can be neglected.Then we can replace dds by ddt . Further, s can be taken as cosmic time. Anycoordinate system of the type t = f ( s ) and r (cid:48) = g ( x, y, z ) would not changethe description of the universe; the sets { s, r } and { t, r (cid:48) } will be equivalent[28]. Spatial homogeneity and isotropy then identify r with r (cid:48) . Also, the scalefactor a ( t ) operates on the whole spatial part. By allowing each galaxy tocarry its own clock measuring its own proper time s , these clocks may ideallybe synchronized at some initial time. Because the universe is homogeneousand isotropic there is no reason for clocks in different places to differ in themeasurement of their proper time. If we tie the coordinate system ( t, r, θ, φ )to the galaxies so that their world lines are given by ( r, θ, φ ) = constant ,then we have a comoving coordinate system and the time t is nothing morethan the proper time s [29]. So, dds can be replaced by ddt generally. Then d Θ dt = − Θ − R µν u µ u ν + 3 a a ( t ) . (20)The second order Friedmann equation can be obtained from (20) by consid-ering a case of hyper surfaces orthogonal to the world lines and by replacingΘ by a ( t ) da ( t ) dt and R µν u µ u ν by πG c ( ρc + 3 p ) [30, 31] where ρ and p stand forthe matter density and pressure of the collection of particles. So, (20) gives1 a ( t ) d a ( t ) dt = − πG c ( ρc + 3 p ) + 3 a a ( t ) , (21)a differential equation for a ( t ).We introduce Hubble parameter H as H ( t ) = 1 a ( t ) da ( t ) dt , (22)so that ˙ H = dH ( t ) dt = 1 a ( t ) d a ( t ) dt − H ( t ) , (23)8sing (22). Then the equation for a ( t ) (21) gives˙ H = − H − πG c ( ρc + 3 p ) + 3 a a ( t ) . (24)From (22), it follows that a ( t ) = e (cid:82) H ( t ) dt and then (24) can be expressed as˙ H = − H ( t ) − πG c ( ρc + 3 p ) + 3 a e − (cid:82) H ( t ) dt , (25)a first order non-linear differential equation for H ( t ).Thus the Raychaudhuri equation (19) upon setting Θ = a ( t ) da ( t ) dt gives(21), an equation for a ( t ) the scale factor in FRW metric and (25), an equa-tion for the Hubble parameter. Both the differential equations are non-linear.Apart from ρ and p , the density and the pressure of the collection of particlesand a the geodesic constant, there are no free parameters thus far.The distribution of matter in the visible universe on scales of about 300Mpc or higher is found to be homogeneous and isotropic to a high degree ofaccuracy. One can assume following [32] that this matter to be perfect fluidcollection of particles, described by the equation of state p = wρc , (26)where w is a constant characterizing the fluid of particles; − . < w ≤ . πG = c = 1 system of units. Then (25) becomes˙ H = − H ( t ) − ρ
24 (1 + 3 w ) + 3 a e − (cid:82) H ( t ) dt . (27)We have considered flat FRW metric and here the Friedmann equation gives ρ = 3 H . Then (27) becomes˙ H = − H w ) + 3 a e − (cid:82) H ( t ) dt , ≡ F ( H ) . (28)Similarly, the equation for a ( t ), (21) becomes using ρ = 3 H = 3 a ( t ) (cid:16) da ( t ) dt (cid:17) , a ( t ) d a ( t ) dt = −
18 (1 + 3 w ) (cid:32) da ( t ) dt (cid:33) + 3 a . (29)9quations (28) and (29) are to be analyzed.
5. Analysis of the equation (28) for H ( t ) : Suppose the contribution of the scalar ψ ( r ) is neglected by setting thegeodesic constant a to zero, then (28) becomes˙ H = − H ( t )8 (9 + 3 w ) . (30)This is used to find the ’age of the universe’ T as T = (cid:90) T (cid:90) dt = (cid:90) H P H dH ˙ H = − (cid:90) H P H dHH (9 + 3 w ) , (31)where H signifies the current epoch. Then T = 8(9 + 3 w ) (cid:18) H P − H (cid:19) , (32)showing finite T . This implies that the universe had a beginning before T .With the contribution from ψ ( r ) included ( a (cid:54) = 0), we have T = (cid:90) H P H dH ˙ H = (cid:90) H P H dHF ( H ) , (33)where F ( H ) given in (28) as F ( H ) = − H ( t )8 (9 + 3 w ) + 3 a e − (cid:82) H ( t ) dt . (34)This is evaluated iteratively. As a first approximation, neglecting the secondterm in (34), (28) gives ˙ H = − αH where α = (9 + 3 w ). Then − (cid:90) Hdt = − (cid:90) H (cid:32) dtdH (cid:33) dH = − (cid:90) H ˙ H dH, = 2 α (cid:90) H ( t ) H P dHH = 2 α (cid:96)og (cid:32) H ( t ) H P (cid:33) . (35)10t is tempting to use (35) in (34) to write F ( H ) = − αH ( t ) + 3 a (cid:32) H ( t ) H P (cid:33) α , which will be the result of first iteration. Instead, we write (35) as − (cid:90) H ( t ) dt = 2 α (cid:96)og (cid:32) H ( t ) − H P H P (cid:33) , (cid:39) α (cid:32) H ( t ) − H P H P − ( H ( t ) − H P ) H P + · · · (cid:33) , so as to effectively take in to account higher iterations. Then (34) becomes F ( H ) (cid:39) − αH ( t ) + 3 a e α (cid:18) H ( t ) − HPHP − ( H ( t ) − HP )22 H P + ··· (cid:19) . (36)The advantage is that second and higher iterations are expected to producea polynomial in ( H ( t ) − H P ). The contribution from the scalar ψ changesthe structure of F ( H ).The farthest zero of F ( H ) in (36) corresponds to H = H P with F ( H P ) =0 = − αH P + a and so H P = a α . It is to be noted that when the contributionfrom the scalar ψ is neglected (by setting a = 0), F ( H ) = − αH and thishas no non-trivial zero. With the contribution from the scalar ψ included, F ( H ) has non-trivial fixed point. To see this, we follow the fixed pointanalysis of Awad [33] and note that F ( H ) is continuous and differentiable.By introducing dimensionless variable x = HH P , we see F ( H ) H P = y = − αx + α e α { x − − ( x − } , (37)keeping the first two terms in the exponent for the sake of illustration. InFig. 1, the stable fixed point is exhibited for α ∼ . w ∼ − . a = 1 for representative purposes.11igure 1: a = 1 is chosen.The stable fixed point occurs when 0 < H < H P , F ( H ) has a ’future fixedpoint’ H < H P . Since F ( H ) is differentiable, the slope of the tangent at anyfixed point is finite. Near the stable fixed point, F ( H ) = F (cid:48) ( H )( H − H )following [33]. Then the age of the universe T = (cid:90) H H dH ˙ H = (cid:90) H H dHF (cid:48) ( H )( H − H ) = ∞ , (38)showing the universe had no beginning, consistent with the defocussing ofworld lines shown in [1]. Other allowed values of α and choice for a arefound to be qualitatively similar without affecting the conclusion in (38).However, for other values of α , the stable fixed point moves towards H P with H coinciding with H P . This is exhibited in Fig.2.12igure 2: a = 1 is chosen.In these cases also, the conclusion that T → ∞ is obtained. Such aconclusion has been reached in [32] by considering quantum effects to Ray-chaudhuri equation. Here, the same conclusion is obtained using classical5-d gravity for the description of the early universe.
6. Analysis of for a ( t ) : The scale factor a ( t ) in the flat FRW metric (7) satisfies (29), that is a ( t ) d a ( t ) dt + A (cid:32) da ( t ) dt (cid:33) − a , (39)where A = (1 + 3 w ). We wish to solve the above equation for a ( t ). Solution.1:
By letting a ( t ) = ψ ( t ) γ , (39) becomes γ ( γ − ψ ( t ) γ − (cid:32) dψdt (cid:33) + γψ ( t ) γ − d ψdt + Aγ ψ ( t ) γ − (cid:32) dψdt (cid:33) = 3 a . Now, suppose we choose γ = A , then the above equation simplifies to ψ ( t ) − A A d ψdt = 3 a A ) . (40)13y letting ψ ( t ) = A ( t + (cid:15) ) ρ and taking ρ = 1 + A , we find a ( t ) = (cid:115) a A ( t + (cid:15) ) . (41)It is to be noted with this exact solution of (39), the second derivative d a ( t ) dt vanishes. This solution exhibits classical bounce as a (0) = (cid:113) a A (cid:15) . Further,this solution gives ˙ H = − H ( t ) = F ( H ) and the universe had a beginning. Solution.2
By letting W ( t ) = a ( t ) A +1 , the (39) becomes d W ( t ) dt − a A + 1) W ( t ) A − A +1 = 0 . (42)Although the non-linearity in (39) is softened, it is still non-linear. As inthe case of (28) for H ( t ), we use iterative method. By neglecting the secondterm in (42), the solution of d W ( t ) dt = 0 gives W ( t ) = βt + γ where β and γ are constants. using this for the second term in (42), we obtain d W ( t ) dt = 3 a A + 1) ( βt + γ ) A − A +1 . (43)This equation is integrated to give W ( t ) = 3 a ( A + 1) A (3 A + 1) β ( βt + γ ) A +1 A +1 + C t + C , (44)where C , C are constants. Then, the FRW scale factor is a ( t ) = (cid:32) a ( A + 1) A (3 A + 1) β ( β t + γ ) A +1 A +1 + C t + C (cid:33) A +1 , (45)iterative solution for a ( t ). When the scalar contribution is neglected (setting a = 0), we see that a ( t ) = ( C t + C ) A +1 . With a = 0, we should havestandard cosmology for which a (0) = 0 and so we choose C = 0. With thischoice, we have a ( t ) = (cid:32) a ( A + 1) A (3 A + 1) β ( βt + γ ) A +1 A +1 + C t (cid:33) A +1 , (46)14he behavior of the scale parameter a ( t ). Now from this, a (0) = (cid:32) a ( A + 1) A (3 A + 1) β (cid:33) A +1 γ A +1( A +1)2 , (47)showing classical bounce . It is to be noted that the bounce is proportional to a , the effect of the scalar in 5-d gravity in the early universe. Further from(46), we have expanding universe. Solution.3. (Series solution),
In this method, the second derivative term in (39) is maintained. A seriessolution consists in taking a ( t ) = b + b t + b t + b t + b t + · · · (48)where b , b , b · · · are constants. Substituting in (39) and equating like pow-ers of t , we obtain relations among these coefficients. From them, we considerthree classes of solutions.(1) If b = 0, then, a solution a ( t ) = (cid:113) a A t which correspond to theSolution.1 with (cid:15) = 0.(2) If we choose, b (cid:54) = 0; b (cid:54) = 0; b = 0, then a solution of a ( t ) = b + (cid:113) a A t is obtained which is similar to Solution.1. Both these give ˙ H = − H sameas in Solution.1.(3) The third case corresponds to b (cid:54) = 0; b = 0 and then the seriessolution corresponds to a ( t ) = b + b t + b t + · · · , all even powers of t . Thecoefficients in this case are: b = a b , b = a b − a (1+2 A )128 b and so on. Forillustrative purpose, we take b = 1; a = 1 and A = − correspondingto w = 0 .
5. Then, H = t +0 . t . t +0 . t and ˙ H = . t . t +0 . t − H , keepingupto t in a ( t ). From these, the graph connecting ˙ H with H is drawn andthis qualitatively gives Fig.2. In this third case, there is classical bounce as a (0) (cid:54) = 0.Thus, the series method ensures departure from standard cosmology withclassical bounce in agreement with the earlier analysis.
7. Summary:
We have extended our result on defocussing of world lines [1] by modifyinggravity in the early epoch by a 5-d theory with scalar ψ ( r ). The acceleration15erm in the Raychaudhuri equation has been shown to be positive for flatFRW metric. The scalar ψ ( r ) satisfies a non-linear differential equation whichis solved. Though singular, the acceleration term turns out to be finite. Withthis, the equations for the Hubble parameter H ( t ) and the scale factor a ( t ) areobtained. These are analyzed using ’fixed point analysis’. Without the scalarfield, the age of the universe is finite showing a beginning of the universe andwith out bounce. With the contribution of the scalar field included, the ageof the universe is shown to be infinite, thereby resolving the singularity. Thescale factor a ( t ) exhibits classical bounce. Acknowledgements:
We are thankful to Sonakshi Sachdev for help in drawing the figures. Use-ful discussions with Govind Krishnaswamy, B.V. Rao and K.S. Viswanathanare acknowledged with thanks.
References:
1. R.Parthasarathy, K.S.Viswanathan and Andrew DeBenedictis, Ann.Phys. (2018) 1.2. R.Brandenberger and C.Vefa, Nucl.Phys.
B316 (1989) 391.3. M.Gasperini and G.Veneziano, Astro Particle Physics. (1993) 317.4. J.Khoury, B.A.Ovrut, P.J.Steinhardt and N.Turok, Phys.Rev. D64 (2001) 123522.5. F.Finelli and R.Brandenberger, Phys.Rev.
D65 (2002) 103522.6. J.B.Hartle and S.W.Hawking, Phys.Rev.
D28 (1983) 2960.7. S.Gielen and N.Turok, Phys.Rev.Lett. (2016) 021301.8. M.Bojowald, Phys.Rev.Lett. (2001) 5227.9. A.Ashtekar, T.Pawlowski and P.Singh, Phys.Rev. D74 (2006) 084003.160. S.Gielen and L.Sindoni,
Quantum cosmology from group field theorycondensates , arXiv 1602.08104.11. M.de Caesare and M.Sakellariadov, Phys.Lett.
B764 (2017) 49.12. D.Oriti, L.Sindoni and E.Wilson-Ewing, Class.Quant.Gravity. (2017)04LT01.13. M.Hohmann, L.Jarv and V.Valiklianova, Phys.Rev. D96 (2017) 043508,14. S.D.Odintsov and V.K.Oikonomou, Int.J.Mod.Phys.
D26 (2017) 1750085.15. V.K.Oikonomov, Phys.Rev.
D92 (2015) 124027.16. K.Martineau and A.Barrau,
Primordial power spectra from an emergentuniverse; basic results and clarfications , gr-qc/1812.05522.17. R.Brandenberger and P.Peter,
Bouncing cosmologies; Progress and Prob-lems , hep-th/ 1603.05834.18. A.M.Chamseddine and V.Mukhanov,
Resolving cosmological singular-ities , gr-qc/1612.05860.19. P.W.Graham, D.E.Kaplan and S.Rajendran, Phys.Rev.
D97 (2018)044003.20. L.Annulli, V.Cardoso and L.Gualtieri,
Electromagnetism and hiddenvector fields in modified gravity theories , gr-qc/1901.02461.21. A.K.Raychaudhuri, Phys,Rev. (1955) 1123.22. E.G. Mychelkin and M.A. Makukov, Unified geometrical basis for thegeneralized Ehlers identities and Raychaudhuri equations. gr-qc/1707.00862.23. S. Ghosh, A. Dasgupta and S. Kar, Phys.Rev.
D83 (2011) 084001.24. J.M.Overduin and P.S.Wesson, Phys.Rep. (1993) 303.25. J. Borgman and L.H. Ford, Phys.Rev.
D70 (2004) 064032.26. E.F. Bunn, P. Ferriera and J. Silk, Phys.Rev.Lett. (1996) 2883.27. M.P. Hobson, G. Efstathiou and A.N. Lasenby, General Relativity: AnIntroduction for physicists , Cambridge University Press, 2006. Page.367.178. F. De Felice and C.J.S. Clarke,
Relativity on curved manifilds.
Cam-bridge University Press, 1990.29. J. Foster and J.D. Nightingle,
A short course in General Relativity.
Springer, 2006.30. A.K.Raychaudhuri,
Theoretical Cosmology , Oxford, U.K., 1979.31. S.Das, Phys.Rev.
D89 (2014) 084068.32. A.F.Ali and S.Das, Phys.Lett.
B741 (2016) 276.33. A. Awad, Phys.Rev.
D87 (2013) 103001. S.H. Strogatz,
NonlinearDynamics and Chaos: Applications to Physics, Biology, Chemistry andEngineering , CRC Press, Taylor and Francis Group, A Chapman andHall Book., 2018.
AppendixClassical equation of motion for ψ ( r ) : We start from the action (15) of [1]. After a partial integration of thesecond term (classical equation of motion will not be affected by having atotal derivative), we have S = 116 πG (cid:90) d x √− g (cid:18) √ g R − g − g µν ( ∂ µ g )( ∂ ν g ) (cid:19) , ( A L = √− g √ g R − √− gg − g µν ( ∂ µ g )( ∂ ν g ) . ( A ∂ L ∂ ( ∂ λ g ) = −√− gg − g λµ ( ∂ µ g ) . Using ∂ λ ( √− gg λµ ) = −√− g Γ µαβ g αβ , it is seen that ∂ λ (cid:32) ∂ L ∂ ( ∂ λ g ) (cid:33) = 32 √− gg − g λµ ( ∂ λ g )( ∂ µ g ) − √− gg − g αβ D α ( ∂ β g ) , D α is the covariant derivative. Next, ∂ L ∂g = 12 √− gg − R + 34 √− gg − g µν ( ∂ µ g )( ∂ ν g ) . ( A g is34 g − g µν ( ∂ µ g )( ∂ ν g ) − g − g µν D µ ( ∂ ν g ) − g − R = 0 . ( A R = g g µν D µ ( ∂ ν g ). Usingthis in above, the classical equation of motion for g becomes g µν D µ ( ∂ ν g ) −
12 1 g g µν ( ∂ µ g )( ∂ ν g ) = 0 . ( A This agrees with Overduin and Wesson [22].
Now, using FRW metric, theequation of motion becomes with g = ψ ( r ),12 a ( t ) 1 ψ ( ψ (cid:48) ) − a ( t ) ψ (cid:48)(cid:48) − g αβ Γ rαβ ψ (cid:48) = 0 . ( A g αβ Γ rαβ = ra ( t ) and therefore, the above classical equation ofmotion becomes2 a ( t ) (cid:32) ( ψ (cid:48) ) ψ − ψ (cid:48)(cid:48) − ψ (cid:48) r (cid:33) = 0 . ( A R = 0. So we need not impose R = 0 as this becomes automatically zero by virtue of classical equation ofmotion for ψ ( rr