Variety of (d + 1) dimensional Cosmological Evolutions with and without bounce in a class of LQC -- inspired Models
aa r X i v : . [ g r- q c ] J u l IMSc/2017/06/04
Variety of ( d + 1) dimensionalCosmological Evolutions with and without bouncein a class of LQC – inspired Models S. Kalyana Rama
Institute of Mathematical Sciences, HBNI, C. I. T. Campus,Tharamani, CHENNAI 600 113, India.email: [email protected] bouncing evolution of an universe in Loop Quantum Cos-molgy can be described very well by a set of effective equations,involving a function sin x . Recently, we have generalised theseeffective equations to ( d + 1) dimensions and to any function f ( x ) . Depending on f ( x ) in these models inspired by LoopQuantum Cosmolgy, a variety of cosmological evolutions are pos-sible, singular as well as non singular. In this paper, we studythem in detail. Among other things, we find that the scale factor a ( t ) ∝ t q (2 q −
1) (1+ w ) d for f ( x ) = x q , and find explicit Kasner–typesolutions if w = 2 q − f ( x ) = √ x , the evolution is non singularand the scale factor a ( t ) grows exponentially at a rate set, not bya constant density, but by a quantum parameter related to thearea quantum. 1 . Introduction Consider a homogeneous expanding universe whose constituents have thedensity ρ and the pressure p obeying the condition ρ + p > ∞ in the infinitepast. The curvature invariants remain finite, the density remains boundedfrom above and, therefore, the evolution of the universe in LQC has a bounceand is non singular.It turns out that the quantum dynamics of such a non singular evolutionin LQC can be described very well by a set of effective equations [3, 4, 8, 9,10, 11]. These equations reduce to Einstein’s equations in the classical limit.These effective equations, in our notation, involve a certain function f ( x ) ,see equations (22) and (23) where it will be first introduced. The variable x will turn out to be related to the time derivative of the scale factor, andthe function f will dictate the precise relation between them. In LQC, thisfunction f ( x ) = sin x and in the classical limit which leads to Einstein’sequations, f ( x ) = x . In fact, Einstein’s equations follow upto a scaling oftime whenever f ( x ) vanishes linearly. Thus, in LQC where f ( x ) = sin x ,the evolution is same as in Einstein’s theory in the limits x → x → π . These limits constitute the two ends of the bouncing evolution inLQC where the size of the universe evolves to infinity, and the evolution inthese two asymptotic limits is same as in Einstein’s theory.In a recent paper [12], we have generalised the effective LQC equations.Our generalisations are empirical and not derived from any underlying theory.But they are simple, straightforward, and natural. We generalised from(3+1) to ( d +1) dimensions where d ≥ and generalised the trigonometric There exists a ( d + 1) dimensional LQG formulation, given in [13, 14, 15]. Our pre- µ function which appear in the effective LQC equations.These generalised equations describe the cosmological evolution of a ( d + 1)dimensional homogeneous, anisotropic universe and may be considered as aclass of LQC – inspired models.In this paper, we will consider only the generalisation of the trigonometricfunction, keeping the ¯ µ function similar to that in the so-called ¯ µ − scheme.In LQC, the function f ( x ) = sin x is determined by the underlying theory.In the present LQC – inspired models, the generalisation is empirical andno underlying theory is invoked which may determine f ( x ) . This feature isa shortcoming of the present models. But taking it as a strength, one mayconsider a variety of functions f ( x ) from a completely general point of view,study the corresponding evolution, and gain insights into the various typesof singular and non singular evolutions possible.Here, we follow this approach and study a variety of ( d + 1) dimen-sional cosmological evolutions in the LQC – inspired models, correspondingto a variety of possible behaviours of the function f ( x ) . Assuming that p = w ρ where w is a constant and (1 + w ) > f ( x ) , we also find the potential V ( σ ) for a minimally coupled scalarfield σ which may give rise to the equation of state p = w ρ . See [18] foranalogous potential V ( σ ) in LQC.In general, given a function f ( x ) , it is not possible to obtain explicitsolutions to the relevant equations of motion. However, these equations havea shift and a scaling symmetry which may be used to understand severalimportant features of the evolutions. In this paper, we first consider the caseswith non trivial functions f ( x ) where explicit solutions may be obtained.For the isotropic case, explicit solutions may be obtained for the functions f ( x ) = sin x , x q , and e x . For the anisotropic case, explicit solutions maybe obtained for the function f ( x ) = x q if w = 2 q − f ( x ) → x as x → liminary analysis [16], see [17] also, suggests that one can derive the LQC analogs of theeffective equations in ( d + 1) dimensions, with f ( x ) = sin x . t → ∞ , the scale factor a → ∞ , and the universe is expanding.As x increases, t decreases, a decreases, and the universe decreases in size.Assuming that f ( x ) remains positive and all its derivatives remain boundedfor 0 < x < x r , we study possible asymptotic behaviours in cases where f ( x ) → ∞ in the limit x → x r , and x r itself may be finite or infinite.We study the cases where the function f ( x ) ∝ ( x r − x ) q in the limit x → x r and also the cases where, in the limit x → ∞ , the function f ( x ) ∝ x q ,or f ( x ) → ( const ) , or f ( x ) ∝ e − b x . Such asymptotic behaviours arequite natural and, hence, they may apply to a wide class of functions f ( x ) .Together with the shift and the scaling symmetries, the explcit solutionsobtained earlier may now be used to describe the asymptotic evolutions inall of these cases.The main results of our study are the following. The explicit solutionpresented in this paper for f ( x ) = sin x generalises the (3 + 1) dimensionalLQC solution given in [18] to ( d + 1) dimensions. The explicit solutionspresented in this paper for f ( x ) = x q generalise the evolution of the scalefactor a in Einstein’s theory to a ( t ) ∝ t q (2 q −
1) (1+ w ) d . When w = 2 q − x → f ( x ) → x , thescale factor a → ∞ , the time t → ∞ , and a and t decrease as x increases. • For the case where f ( x ) ∝ ( x r − x ) q as x → x r and 2 q ≥ x → x r , onehas t → −∞ and a → ∞ . The function f ( x ) = sin x falls under thiscase and corresponds to x r = π and q = 1 . • For the case where f ( x ) ∝ x q as x → ∞ and 2 q ≤ x → ∞ , one has t → −∞ ; and a → < q ≤ a → ( const ) if q = 0 , and a → ∞ if q < q = 0 case in this limit is similar to that expected inthe Hagedorn phase of string/M theory [19] – [24].We find that the case where f ( x ) = x q and 0 < q < q = 1 case where f ( x ) = √ x is particularly fascinating.The evolution now is non singular and it straddles the border between thesingular and non singular evolution : in the limit x → q > q < x → ∞ , theevolution is singular for 2 q > q < q = 1 , the scale factor a grows exponentially and the expo-nential rate is set, not by a constant density, but by a quantum parameterwhich is related to the area quantum as in LQC. The density ρ ∝ a − (1+ w ) d and is not constant. This exponential growth of the scale factor is, therefore,unlike that which occurs in Einstein’s theory due to a positive cosmologicalconstant for which w = − , we present the equationsof motion for a ( d + 1) dimensional homogeneous anisotropic universe inEinstein’s theory. In Section , we present the effective equations of motionin LQC. In Section , we present the generalised effective equations of motionfor our LQC – inspired models. In Section , we describe the general featuresof these models. In Section , we present explicit isotropic solutions and usethem to study other asymptotic evolutions. In Tables I and II, we have alsotabulated the asymptotic behaviours of the scale factor in the isotropic casesso that the results may be seen at a glance. In Section , we present explicitanisotropic solutions and study other anisotropic asymptotic evolutions. InSection , we present a brief summary and conclude by mentioning severalissues for further studies. ( d + 1) dimensional Einstein’s equations Let the spacetime be ( d + 1) dimensional where d ≥ x i , i = 1 , , · · · , d , denote the spatial coordinates. Also, let the d − dimensionalspace be toroidal and let L i denote the coordinate length of the i th direction.Consider a homogeneous and anisotropic universe whose line element ds is5iven by ds = − dt + X i a i ( dx i ) (1)where the scale factors a i depend on t only. Here and in the following, wewill explicitly write the indices to be summed over since the convention ofsumming over repeated indices is not always applicable. Einstein’s equationsare given, in the standard notation with κ = 8 πG d +1 , by R AB − g AB R = κ T AB , X A ∇ A T AB = 0 (2)where A, B = (0 , i ) and T AB is the energy momentum tensor. We assumethat T AB is diagonal and that its diagonal elements are given by T = ρ , T ii = p i (3)where ρ is the density and p i is the pressure in the i th direction. Defining thequantities λ i , Λ , a, G ij , and G ij by e λ i = a i , e Λ = Y i a i = a d −→ Λ = X i λ i ,G ij = 1 − δ ij , X i G ij G jk = δ ik −→ G ij = 1 d − − δ ij , and after a straightforward algebra, Einstein’s equations (2) give X ij G ij λ it λ jt = 2 κ ρ (4) λ itt + Λ t λ it = κ X j G ij ( ρ − p j ) (5) ρ t + X i ( ρ + p i ) λ it = 0 (6)where the t − subscripts denote derivatives with respect to t . It follows fromequations (5) that (cid:16) λ it − λ jt (cid:17) t + Λ t (cid:16) λ it − λ jt (cid:17) = κ ( p i − p j ) . (7)6f the pressures are isotropic then p i = p for all i , and we get λ it − λ jt = ( const ) e − Λ (8)and ρ t + Λ t ( ρ + p ) = 0 . (9)If the scale factors are also isotropic then a i = a and λ it = a t a ≡ H for all i ,and equations (4) and (5) give H = 2 κ ρd ( d −
1) (10) H t = − κ ( ρ + p ) d − . (11)Note that if p i = w i ρ where w i are constants then equations (4) – (6) canbe solved exactly [12]. Also note that for a minimally coupled scalar field σ ,which has a potential V ( σ ) and depends on t only, it is a standard resultthat the density ρ and the pressure p are given by ρ = ( σ t ) V , p i = p = ( σ t ) − V . (12)The equation of motion for the field σ is given by σ tt + Λ t σ t + dVdσ = 0 (13)which also follows from equations (9) and (12). Furthermore, in the isotropiccase, one can construct a potential V for the scalar field σ such that p = wρ where w is a constant: Writing σ t = (1 + w ) ρ and 2 V = (1 − w ) ρ and aftersome manipulations involving equations (10) and (11), see comments belowequation (41), it can be shown that the required potential is given by V ( σ ) ∝ (1 − w ) e − c w σ , c w = vuut (1 + w ) κ d d − . (14)7 . (3 + 1) dimensional Effective LQC equations In this section, we mention briefly the main steps involved in obtainingthe effective equations of motion in the (3 + 1) dimensional Loop QuantumCosmology (LQC). A detailed derivation and a complete description of vari-ous terms and concepts mentioned below are given in the review [4]. Below,we present the LQC expressions in a form which can be readily generalised.Let the three dimensional space be toroidal and let the line element ds be given by equation (1) where d = 3 now, and i = 1 , , L i and a i L i be the coordinate and the physical lengths of the i th direction. In theLoop Quantum Gravity (LQG) formalism, the canonical pairs of phase spacevariables consist of an SU (2) connection A ia = Γ ia + γK ia and a triad E ai of density weight one. Here Γ ia is the spin connection defined by the triad e ai , K ia is related to the extrinsic curvature, and γ > ≈ . ds is given in equation (1) with d = 3, one has A ia ∝ ˆ c i and E ai ∝ ˆ p i where ˆ c i will turn out to be related to the time derivative of a i , andˆ p i is given by ˆ p i = Va i L i , V = Y j a j L j (15)with V being the physical volume. The full expressions for A ia and E ai containvarious fiducial triads, cotriads, and other elements, and are given in [4, 9].The non vanishing Poisson brackets among ˆ c i and ˆ p j are given by { ˆ c i , ˆ p j } = γ κ δ ij (16)where κ = 8 πG . The effective equations of motion are given by the‘Hamiltonian constraint’ C H = 0 and by the Poisson brackets of ˆ p i and ˆ c i with C H which give the time evolutions of ˆ c i and ˆ p i : namely, by C H = 0 , (ˆ p i ) t = { ˆ p i , C H } , (ˆ c i ) t = { ˆ c i , C H } . (17)Given that Einstein’s action for gravity is known, it is to be expectedthat there exists a classical C H , the Poisson brackets with which lead tothe classical dynamics given by Einstein’s equations. Non trivially, and asreviewed in detail in [4], there also exists an effective C H , the Poisson brackets8ith which lead to the equations of motion which describe very well thequantum dynamics of LQC. The effective C H reduces to the classical one ina suitable limit.The expression for the C H is of the form C H = H grav (ˆ p i , ˆ c i ) + H mat (ˆ p i ; { φ mat } , { π mat } ) (18)where H grav denotes the effective gravitational Hamiltonian and H mat denotesa generalised matter Hamiltonian. In the matter sector, the density ρ andthe pressure p i in the i th direction are defined by ρ = H mat V , p i = − a i L i V ∂H mat ∂ ( a i L i ) . (19)The pressure p i is thus, as to be physically expected, proportional to thechange in energy per fractional change in the physical length in the i th direc-tion. As indicated in equation (18), H mat is assumed to be independent ofˆ c i . Since ˆ c i will turn out to be related to ( a i ) t , this assumption is equiva-lent to assuming that matter fields couple to the metric fields but not to thecurvatures. This assumption can also be shown to lead to the conservationequation (6), namely to ρ t = (cid:18) H mat V (cid:19) t = − X i ( ρ + p i ) λ it , (20)irrespective of what H grav is [12].In the gravitational sector, the effective H grav , from which the LQC dy-namics follow, is given in the so–called ¯ µ − scheme by H grav = − Vγ λ qm κ (cid:16) sin (¯ µ ˆ c ) sin (¯ µ ˆ c ) + cyclic terms (cid:17) (21)where V = √ ˆ p ˆ p ˆ p is the physical volume, λ qm = q γκ is the quantumof area, and ¯ µ i = λ qm ˆ p i V in the ¯ µ − scheme. Classical H grav follows in the limit¯ µ i ˆ c i → sin (¯ µ i ˆ c i ) → ¯ µ i ˆ c i . 9 . ( d + 1) dimensional LQC – inspired models In a recent paper [12], we generalised the effective LQC equations. Ourgeneralisations are empirical and not derived from any underlying theory.But they are simple, straightforward, and natural. And, they may be usedto model a variety of non singular cosmological evolutions. In [12], we gen-eralised from (3 + 1) to ( d + 1) dimensions where d ≥ µ functions appearing in the effective H grav inequation (21). In this paper, we will consider only the generalisation of thetrigonometric function, keeping the ¯ µ function as in the ¯ µ − scheme.We now present the generalised effective equations of our LQC – inspiredmodels, expressing them so that they resemble equations (4) and (5) as closelyas possible. For the purpose of this generalisation, we have already presentedthe LQC expressions in a form which can be readily taken over. Upon gen-eralisation, we have the following. • The index i = 1 , , · · · , d now in the LQC expressions. • The canonical pairs of phase space variables are given by ˆ c i which willbe related to ( a i ) t , and ˆ p i which is given by equation (15). The nonvanishing Poisson brackets among ˆ c i and ˆ p j are given by equation (16)where now κ = 8 πG d +1 and γ may characterise the quantum of the( d −
1) dimensional area given by λ d − qm ∼ γκ [13, 14, 15, 25]. • The effective equations of motion are given by equation (17) where C H is of the form given in equation (18). In the matter sector, the density ρ and the pressures p i are given by equations (19), and they satisfy thestandard conservation equation (20). • In the gravitational sector, the effective H grav in equation (21) is nowgeneralised to H grav = − V G γ λ qm κ , G = 12 X ij G ij f i f j = X ij ( i Equations of motion can now be obtained using the generalised H grav given in equation (22). They will describe the evolution of a ( d + 1) dimen-sional homogeneous anisotropic universe in our LQC – inspired models. Therequired algebra is straightforward but involved, see [12] for details. In thispaper, we present only the final equations which suffice for our purposes here.The resulting equations of motion, expressed so that they resemble equations(4) and (5) as closely as possible, are given by X ij G ij f i f j = 2 γ λ qm κ ρ (24)( m i ) t + X j ( m i − m j ) X j ( d − γλ qm = − γλ qm κ X j G ij ( ρ + p j ) (25)( γλ qm ) λ it = X j G ij X j (26)where we have defined X i = g i X j G ij f j , g i = df ( m i ) dm i . (27)Equations (24) – (26) give the conservation equation (6). Also, equation (26)gives ( d − 1) ( γλ qm ) Λ t = P j X j , and equation (25) then gives( m i − m j ) t + Λ t ( m i − m j ) = γλ qm κ ( p i − p j ) . (28)If the pressures are isotropic then p i = p for all i , and we get m i − m j = µ ij e − Λ (29)11here µ ij are constants. In the completely isotropic case, we have( p i , m i , f i , a i ) = ( p, m, f, a ) (30)and, hence, λ it = a t a = H , X i = ( d − gf , g i = g = dfdm . Equations (24) – (26) then give f = 2 γ λ qm κ ρd ( d − 1) = ρρ qm (31) m t = − γλ qm κ d − ρ + p ) (32) H = g fγλ qm (33)where ρ qm = d ( d − γ λ qm κ . Equations (31) – (33) give the conservation equation(9). We will assume that ρ + p > m t < m will increase monotonically as t decreases. Also, equations (31)and (33) give H = 2 κ ( ρ g ) d ( d − . (34)If g can be expressed in terms of f then, using f = ρρ qm , one can express H in terms of ρ alone. For example, f ( x ) = sin x = ⇒ H ∝ ρ − ρρ qm ! (35)and f ( x ) = f ( x ; n ) where n is a positive integer and f ( x ; n ) = 1 − (cid:18) − xx ∗ (cid:19) n = ⇒ H ∝ ρ − s ρρ qm ! n − n . (36)Note that f ( x ; n ) is a class of functions parametrised by n , that n indicatesthe flatness of f near its maximum at x ∗ , and that f ( x ; n ) → nx ∗ (2 x ∗ − x )in the limit x → x ∗ . 12 . General Features in the LQC – inspired models Obtaining the cosmological evolution of the universe in the LQC – in-spired models further requires specifying the equations of state which givethe pressures p i in terms of ρ . Once the equations of state are given, orassumed, equations (24) – (26) can be solved for a given set of initial val-ues ( m i , λ i ) at t = t . Here and in the following, the 0 − subscripts willdenote the initial values at some initial time t . Given m i , the values( f i , g i , X i ) follow. Equation (24) then gives ρ ; equations of state give p i ; and, equation (25) gives ( m i ) t from which the value of m i at t ± δt follows. Repeating this procedure will give m i and ( f i , g i , X i ) for all t .The initial value λ i and equation (26) for λ it then determine λ i for all t .Thus, equations (24) – (26) can always be solved numerically.However, in general, it is not possible to obtain analytical solutions ex-plicitly. Nevertheless, it is possible to understand several features of theevolution, as we now describe in a series of remarks. Remark (1) : Let f ( m i ) = m i . Then g i = 1 , and equation (26) andthe definition of m i give γλ qm λ it = m i , ˆ c i = γL i ( a i ) t . This shows that ˆ c i is related to ( a i ) t . After a little algebra, equations (24)and (25) give equations (4) and (5), the Einstein’s equations for a ( d + 1)dimensional homogeneous anisotropic universe. Remark (2) : Equations (24) – (26) remain invariant under the shift m i → ˜ m i = m i + m s where m s is constant and same for all i . Hence, f ( m i )and f ( ˜ m i ) will lead to the same evolution. Remark (3) : Under the scaling m i → ˜ m i = α m i and t → ˜ t = α t where α is constant and same for all i , and ρ and p i remain unchanged, wehave λ it → λ i ˜ t = λ it α , ρ t → ρ ˜ t = ρ t α , and, from equations (27), g i → ˜ g i = df ( ˜ m i ) d ˜ m i = g i α , X i → ˜ X i = X i α . 13t then follows that equations (6) and (24) – (26) remain invariant underthese scalings. The invariances under the shift and the scaling then implythat { f ( m i ) , t } and { f ( αm i + m s ) , αt } will lead to the same evolution.Note that these invariance properties are accidental, are not present evenfor the more general models presented in [12], and arise from the structureof the equations (24) – (26) considered here. Nevertheless, these propertiesare useful practically. The scaling with α = − t → − t and, hence, this scaling may be thought ofas reversing the direction of time and the corresponding evolution may bethought of as that seen when one goes back in time. Remark (4) : The density ρ and the expansion rates λ it remain finiteif the functions f i and their first derivatives g i are finite. If all the higherderivatives d n f ( m i ) d ( m i ) n are finite then all the higher time derivatives d n λ i dt n will alsobe finite. The evolution will then be non singular. Remark (5) : In our models, we require that f ( x ) → x as x → < x < x r , let f remain positive and bounded, let all thederivatives of f also remain bounded, and let f ( x ) ∝ ( x r − x ) as x → x r .It then follows from the above Remarks that the universe will evolve as inEinstein’s theory when m i → i , and as its time reversed version when m i → x r for all i , and the evolution will remain non singular in between. Notethat the properties mentioned above are satisfied by f ( x ) = sin x for which x r = π , and also by the class of functions f ( x ; n ) given in equation (36)for which x r = 2 x ∗ . Many such examples may be constructed easily. Remark (6) : Consider a function f ( x ) which → x as x → f be positive and bounded, and let all its derivatives also be bounded, in theinterval 0 < x < x r . Then the universe will evolve as in Einstein’s theorywhen m i → i and the evolution will remain non singular until m i approach x r . The nature of the evolution in the limit m i → x r will dependon the behaviour of f ( x ) as x → x r .In LQC, the function f ( x ) = sin x is determined by the underlyingtheory. In the present LQC – inspired models, the generalisation is empiricaland no underlying theory is invoked which may determine f ( x ) . This featureis a shortcoming of the present models. But it may be taken as a strength14lso. One may then consider possible asymptotic behaviours of f ( x ) froma completely general point of view, obtain a variety of asymptotics of theevolution, and thereby gain insights into the various types of singular andnon singular evolutions possible. From such a perspective, it is quite naturalto consider the case where f ( x ) ∝ ( x r − x ) q in the limit x → x r . Note thatit is also possible that x r is infinite. One may then also consider the caseswhere, in the limit x → ∞ , the function f ( x ) ∝ x q , or f ( x ) → ( const ) , or f ( x ) ∝ e − b x . Remark (7) : Consider the isotropic case. Let the equation of state begiven by p = w ρ where w is a constant and 1 + w > − subscripts,it follows from equations (9) and (31) that ρρ = f f = (cid:18) aa (cid:19) − (1+ w ) d , ρ = ρ qm f , (37)which gives a in terms of m . Note that if f has a maximum f mx then thedensity ρ has a maximum ρ mx and the scale factor a has a minimum a mn which are given by ρ mx = ρ qm f mx , a mn = a f f mx ! w ) d . (38)Equations (31) and (32) give F = − Z dmf = c qm ˜ t , ˜ t = t − t + F c qm (39)where c qm = (1+ w ) d γλ qm . Equation (39) defines F ( m ) and gives t in terms of m .Inverting t ( m ) then gives m ( t ) which, in turn, gives a ( t ) . Remark (8) : Consider the isotropic case with p = w ρ as above. Now,let these p and ρ be due to a minimally coupled scalar field σ with a potential V ( σ ) , see equation (12). Writing σ t = (1 + w ) ρ , equations (31) and (32)give σ t = (cid:18)q (1 + w ) ρ qm (cid:19) f , dmdσ = m t σ t = − γλ qm κ d − σ t . 15t then follows that S = − Z dmf = c w ˜ σ , ˜ σ = σ − σ + S c w (40)where c w = r (1+ w ) κ d d − . Equation (40) defines S ( m ) and gives σ in terms of m , hence in terms of t if m ( t ) is known. Inverting σ ( m ) gives m ( σ ) , andthe scalar field potential V ( σ ) then follows from2 V = (1 − w ) ρ qm f . (41)Note that V given in equation (14) follows by taking f ( m ) = m here, or byapplying the above manipulations to equations (10) and (11). Remark (9) : Consider the isotropic case and consider the evolutionnear a maximum of the function f ( m ) . Let m b be a maximum of f and,near its maximum, let f ( m ) ≃ f mx (cid:16) − f ( m b − m ) n (cid:17) (42)where f mx and f are positive constants and, as in equation (36), n is apositive integer which indicates the flatness of f near its maximum. Let t b be the time when f reaches its maximum. Then, as t → t b , it follows fromequations (39), (37), and (40) that m b − m ≃ f mx c qm ( t − t b ) (43) a ≃ a mn (cid:16) a ( t − t b ) n (cid:17) (44) c w ( σ − σ b ) ≃ f mx c qm ( t − t b ) (45)where σ b is a constant, a mn = a (cid:16) f f mx (cid:17) w ) d , and a = f (1+ w ) d ( f mx c qm ) n .16 . Isotropic evolutions Consider the isotropic case, with the equation of state given by p = w ρ where w is a constant and 1 + w > p and ρ are due to a minimally coupled scalar field σ with a potential V ( σ ) . Wewill first study the examples of functions f ( m ) for which equations (31) –(33) can be solved explicitly. Using these examples, we will then study theasymptotic evolutions for the functions f ( m ) and the limits given in Remark (6) . As clear from Remarks (7) and (8) , obtaining explicit solutions dependson whether the integrations in equations (39) and (40), and the consequentfunctional inversions, can be performed explicitly. Example I : f ( m ) = sin m It turns out that all the required integrations and functional inversionscan be performed for the function f ( m ) = sin m . As can be checked easily, F and S are given by F = cot m , e S = cot m . Also, f and S may be expressed in terms of F : f = 1 √ F , Cosh S = 1 sin m = √ F . Equations (37), (39), and (40) now give the solutions (cid:18) aa (cid:19) (1+ w ) d = 1 + c qm ˜ t c qm ˜ t , Cosh ( c w ˜ σ ) = q c qm ˜ t . (46)The potential V ( σ ) for the scalar field given in equation (41) now becomes V ( σ ) = (1 − w ) ρ qm Cosh ( c w ˜ σ ) . (47)We note, in passing, that the above expressions are the ( d + 1) dimensionalgeneralisation of the (3 + 1) dimensional LQC solution given in [18].17 xample II : f ( m ) = m q , m ≥ The function f ( m ) = m q with m ≥ f has no finite maximum and, generically, the evolution will besingular. However, for this example, all the integrations in equations (39) and(40), and the consequent functional inversions, can be performed which willlead to explicit solutions. Moreover, together with the shifting and the scalingof m described in Remarks (2), (3), and (5) , the solutions for this examplecan be used to understand the asymptotics of the nonsingular evolutions ina variety of cases which may be of interest. With this purpose in mind, weconsider this example and present the explicit isotropic solutions. It alsoturns out that the solutions for the 0 < q < q = case are quite intriguing.Before proceeding with the isotropic case, we first note that when thefunction f ( x ) = x q , we have f i = ( m i ) q , m i g i = qf i , and hence X j m j X j = q X jk G jk f j f k . Equation (25) then gets simplified and becomes( m i ) t + Λ t m i = γλ qm κ X j G ij { (2 q − ρ − p j } . (48)If p i = p = w ρ then the above equation becomes( m i ) t + Λ t m i = γλ qm κ d − q − − w ) ρ . (49)Now consider the completely isotropic case given by equation (30), andthe corresponding equations of motion (9) and (31) – (33). Note that thefunction g = q m q − now. Hence, equations (31) and (33) give H = q γ λ qm ρρ qm ! q − q . (50)Note also that for any quantity Z ( m ) which varies as m α , it follows fromequations (31) and (32), and from f = m q , that Z t = Z m m t ∼ m q − α −→ d n Zdt n ∼ m n (2 q − α . (51)18f Z = ( ln a ) then α = 0 formally and d n ( ln a ) dt n ∼ m n (2 q − . In the following,we will take the evolution to be singular if any of the time derivatives of( ln a ) diverges, and to be non singular otherwise.The general explicit solution for f = m q follows upon performing the inte-grations in equations (39) and (40), and the consequent functional inversions.We will now present these solutions. q = For q = or 1 , equations (39) and (40) give m − q = (2 q − c qm ( t − t ) + m − q ≡ T (52) m − q = ( q − c w ( σ − σ ) + m − q ≡ A σ + B (53)where the constants A and B can be read off easily. It then follows fromequation (37) that the scale factor a is given by aa = (cid:18) mm (cid:19) − q (1+ w ) d = (cid:18) TT (cid:19) q (2 q − 1) (1+ w ) d (54)where T = m − q . And, it follows from equation (41) that the scalar fieldpotential V is given by2 V ( σ ) = (1 − w ) ρ qm m q = (1 − w ) ρ qm ( A σ + B ) q − q . (55)For q = 1 , one obtains the standard Einstein’s equations as described inRemark (1) . Thus, for example, one obtains from equations (37) – (40) that m = 1 c qm ˜ t = e − c w ˜ σ , a ∼ m − w ) d = (cid:16) c qm ˜ t (cid:17) w ) d . The scalar field potential is given by V ∼ m ∼ e − c w ˜ σ , see equation (14).The evolution of the universe follows from equations (54), (52), and (51). > : In this case, as m → a → ∞ , the time t → ∞ , and the time derivatives of ( ln a ) will not diverge. The evolution is19on singular in this limit, and proceeds smoothly as m increases further. As m → ∞ , the scale factor a → t → t s from above where t s is finite and its value can be read off easily but is not important here. Also,the time derivatives of ( ln a ) diverge and, hence, the evolution is singular inthis limit. Thus, for the 2 q > < < : In this case, as m → a → ∞ andthe time t → t s from below where t s is finite and its value can be read offeasily but not important here. Also, the time derivatives of ( ln a ) divergeand, hence, the evolution is singular in this limit. The evolution proceedssmoothly as m increases further. As m → ∞ , the scale factor a → t → −∞ , and the time derivatives of ( ln a ) will not diverge.The evolution is non singular in this limit. Thus, for the 0 < q < q < : In this case, as m → a → t → t s from below where t s is finite and its value can be read off easily butnot important here. Also, the time derivatives of ( ln a ) diverge and, hence,the evolution is singular in this limit. The evolution proceeds smoothly as m increases further. As m → ∞ , the scale factor a → ∞ , the time t → −∞ ,and the time derivatives of ( ln a ) will not diverge. The evolution is nonsingular in this limit. Thus, for the q < q = Consider now the q = case. Equations (39) and (37) give m = m e − c qm ( t − t ) (56)20 a = (cid:18) mm (cid:19) − q (1+ w ) d = e t − t γλqm (57)where we have used c qm = (1+ w ) d γλ qm in the last equality. Equations (53) and(55) remain valid, and note that equation (55) gives V ∝ ( Aσ + B ) .In this case, as m → a → ∞ and the time t → ∞ .As m → ∞ , the scale factor a → t → −∞ . Also, clearly,the time derivatives of ( ln a ) will not diverge and, hence, the evolution isnon singular. Thus, for the q = case, the universe starts with a zero sizein the infinite past, and expands to infinite size in the infinite future with nosingularities.Note that the evolution in the q = case straddles the border betweenthe singular and non singular evolution : in the limit m → q > q < m → ∞ ,the evolution is singular for 2 q > q < q = 1 , the evolution is non singular in both the asymptoticlimits.Also note that, for q = , we have an exponentially growing scale factorwith the exponential rate set by the parameter λ qm alone, which is relatedto the quantum of the ( d − 1) dimensional area as in LQC. The density isgiven by equation (37), ρ ∼ f ∼ a − (1+ w ) d ∼ e − c qm t , and does not remain constant. Also, there is no restriction on w , the equationof state parameter. So, this exponential growth of the scale factor is unlikethat which occurs in Einstein’s theory due to a positive cosmological constant.See equation (50), now with 2 q = 1 , to see how this comes about. Theseresults are intriguing and fascinating but their significance, if any, is not clearto us at present. Example III : f ( m ) = e m We consider the function f ( m ) = e m for the same reasons as given forExample II : solutions can be obtained explicitly. They may then be used21o understand the asymptotics of the nonsingular evolutions in other caseswhich may be of interest. For this example, equations (31) and (33) give H = 1 γ λ qm ρρ qm ! . (58)Equations (39) and (40) give e − m = 2 c qm ( t − t ) + e − m ≡ T (59) e − m = c w ( σ − σ ) + e − m ≡ A σ + B (60)where the constants A and B can be read off easily. It then follows fromequation (37) that the scale factor a is given by aa = e − m − m w ) d = (cid:18) TT (cid:19) w ) d (61)where T = e − m . And, it follows from equation (41) that the scalar fieldpotential V is given by2 V ( σ ) = (1 − w ) ρ qm e m = (1 − w ) ρ qm ( A σ + B ) . (62)Also, for any quantity Z ( m ) which varies as m α , it follows from equations(31) and (32), and from f = e m , that d n Zdt n ∼ n X k =1 A k m α − k ! e n m (63)where A k are some constants. The evolution of the universe follows fromequations (61), (59), and (63).As m → −∞ , the scale factor a → ∞ , the time t → ∞ , and the timederivatives of ( ln a ) will not diverge. The evolution is non singular in thislimit, and proceeds smoothly as m increases further. As m → ∞ , the scalefactor a → t → t s from above where t s is finite and its valuecan be read off easily but not important here. Also, the time derivatives of( ln a ) diverge and, hence, the evolution is singular in this limit. Thus, the22niverse starts with a zero size and a singularity at a finite time in the past,and expands to infinite size in the infinite future with no further singularities.In Table I below, we tabulate the asymptotic behaviour of the scalefactor a ( t ) in Examples I – III, described above in detail. In the Table, wegive the forms of the function f ( m ) and the asymptotic values of a ( t ) in thelimit t → ±∞ or t s where t s is finite. Also, we use the letter S or N S todenote whether the evolution in this limit is singular or non singular. f ( m ) t → − ∞ t → t s t → ∞ I sin m ∞ , N S → a min → ∞ , N S q > , S ∞ , N S q = 1 0 , N S −→ ∞ , N S II m q < q < , N S ∞ , Sq < ∞ , N S , S III e m , S ∞ , N S Table I : Asymptotic behaviours of the scale factor a ( t ) . Tabulated here are the forms of the function f ( m ) in Examples I – III, andthe asymptotic values of the scale factor a ( t ) in the limit t → ±∞ or t s where t s is finite. The accompanying letter S or N S denotes whether the volution in this limit is singular or non singular. The entire evolution isnon singular only in Example I, where it has a bounce, and in Example IIwith q = 1 , where it has no bounce. The values of m in these limits followstraightforwardly and, hence, are not tabulated here. Other Examples Using Examples II and III, we now study the asymptotic evolutions for thefunctions f ( m ) and the limits given in Remark (6) . For these functions, weassume that f ( m ) → m in the limit m → m approachesthe limit of interest, f remains positive and bounded, and all its derivativesalso remain bounded. It then follows that, in the limit m → a → ∞ , the time t → ∞ , andthe time derivatives of ( ln a ) will not diverge. As m increases from 0 , thetime t decreases from ∞ , the scale factor a decreases from ∞ but remainsnon zero since f remains bounded from above, and the evolution proceedssmoothly with no singularity until m approaches the limit of interest.We characterise the evolution in these limits of interest as follows. If thetime derivatives of ( ln a ) do not diverge then the evolution will be referred toas non singular; if the scale factor a → ∞ then the evolution will be said tohave a bounce; when there is a bounce, if the evolution as a → ∞ is same asthat in Einstein’s theory then the evolution will be referred to as symmetric;and, when there is a bounce, if the evolution as a → ∞ is different from thatin Einstein’s theory then evolution will be referred to as asymmetric.Note that the words symmetric and asymmetric refer not to the actualshape of a ( t ) , but refer only to whether or not a ( t ) evolves as in Einstein’stheory at both the ends of a bounce where a → ∞ . Thus, for example, if f ( m ) = sin m , or if f ( m ) = f ( m ; n ) given in equation (36), then theevolution is non singular, has a bounce, and is symmetric. It turns out that,because of the symmetric shapes of sin m and f ( m ; n ) , the shape of a ( t )is also symmetric but, generically, this need not be the case.24 xample IV : f ( m ) ∝ ( m s − m ) q as m → m s We now consider the Example where the function f ( m ) ∝ ( m s − m ) q in the limit of interest m → m s . The evolution in the limit m → m s canbe read off from the asymptotic behaviour in Example II as m → t there since f ∝ ( m s − m ) q now. Itthen follows that if 2 q ≥ m → m s , the scale factor a → ∞ , thetime t → −∞ , and the time derivatives of ( ln a ) will not diverge. Hence,the evolution is non singular and is different from that in Einstein’s theoryunless q = 1 . Thus, as m increases from 0 to m s , the universe evolves asin Einstein’s theory in the limit t → ∞ , the scale factor a remains non zerothroughout, increases to ∞ and, unless q = 1 , evolves asymmetrically in thelimit t → −∞ . The evolution has a bounce, is asymmetric unless q = 1 ,and remains non singular throughout.If 0 < q < m → m s , the scale factor a → ∞ , the time t → t s which is finite, the time derivatives of ( ln a ) diverge and, hence, the evolutionis singular. Thus, as m increases from 0 to m s , the universe evolves as inEinstein’s theory in the limit t → ∞ , the scale factor a remains non zerothroughout and increases to ∞ at a finite time t s in the past. The evolutionhas a bounce, and is singular as t → t s .If q < m → m s , the scale factor a → t → t s whichis finite, the time derivatives of ( ln a ) diverge and, hence, the evolution issingular. Thus, as m increases from 0 to m s , the universe evolves as inEinstein’s theory in the limit t → ∞ , the scale factor a decreases from ∞ to 0 at a finite time t s in the past, and the evolution is singular as t → t s . Example V : f ( m ) ∝ m q as m → ∞ We now consider the Example where the function f ( m ) ∝ m q in thelimit of interest m → ∞ . The evolution in the limit m → ∞ can be readoff from the asymptotic behaviour in Example II as m → ∞ there. It thenfollows that if 2 q > m → ∞ , the scale factor a → t → t s which is finite, the time derivatives of ( ln a ) diverge and, hence, theevolution is singular. Thus, as m increases from 0 to ∞ , the universe evolves25s in Einstein’s theory in the limit t → ∞ , the scale factor a decreases from ∞ to 0 at a finite time t s in the past, and the evolution is singular as t → t s .If 0 < q ≤ m → ∞ , the scale factor a → t → −∞ , the time derivatives of ( ln a ) will not diverge and, hence, theevolution is non singular. Thus, as m increases from 0 to ∞ , the universeevolves as in Einstein’s theory in the limit t → ∞ , the scale factor a decreasesfrom ∞ to 0 as t → −∞ , and the evolution remains non singular throughout.If q < m → ∞ , the scale factor a → ∞ , the time t → −∞ ,the time derivatives of ( ln a ) will not diverge and, hence, the evolution isnon singular. Thus, as m increases from 0 to ∞ , the universe evolves asin Einstein’s theory in the limit t → ∞ , the scale factor a decreases from ∞ to some non zero value and then increases again to ∞ as t → −∞ , theevolution has a bounce, and remains non singular throughout. Example VI : f ( m ) ∝ → ∞ We now consider the Example where the function f ( m ) ∝ m → ∞ . This Example can be thought of as a special case ofExample V with q = 0 . It follows straightforwardly from equations (37)and (39) that the scale factor a → ( const ) and the time t → −∞ . Clearly,the time derivatives of ( ln a ) will not diverge and, hence, the evolution isnon singular. Thus, as m increases from 0 to ∞ , the universe evolves as inEinstein’s theory in the limit t → ∞ , the scale factor a decreases from ∞ tosome non zero constant value as t → −∞ , and the evolution remains nonsingular throughout.The density ρ also approaches a non zero constant value as t → −∞ .This phase of the evolution is then similar to what is expected in string/Mtheory where, as one goes back in time, the ten/eleven dimensional earlyuniverse is believed to enter and remain in a Hagedorn phase in which itstemperature is of the order of l − s and its density is of the order of l − ( d +1) s where l s is the string length scale [19] – [24].26 xample VII : f ( m ) ∝ e − bm as m → ∞ ; b > We now consider the Example where the function f ( m ) ∝ e − bm with b > m → ∞ . The evolution in the limit m → ∞ herecan be read off from the asymptotic behaviour in Example III as m → −∞ there. One also needs to change the sign of t there since f ∝ e − bm now. Itthen follows that, as m → ∞ , the scale factor a → ∞ , the time t → −∞ ,the time derivatives of ( ln a ) will not diverge and, hence, the evolution isnon singular. Thus, as m increases from 0 to ∞ , the universe evolves asin Einstein’s theory in the limit t → ∞ , the scale factor a decreases from ∞ to some non zero value and then increases again to ∞ as t → −∞ , theevolution has a bounce, and remains non singular throughout. Summary of Examples IV – VII In Table II below, we tabulate the asymptotic behaviour of the scale factor a ( t ) in Examples IV – VII, described above in detail. In the Table, we givethe forms of the function f ( m ) in the limits of interest and the asymptoticvalues of a ( t ) in the limit t → ±∞ or t s where t s is finite. Also, we use theletter S or N S to denote whether the evolution in this limit is singular ornon singular.We now highlight the results of Examples IV – VII by specifically pointingout the cases of non singular evolutions. In all these Examples, by assump-tion, we have that as m increases from 0 , the universe evolves as in Einstein’stheory in the limit t → ∞ , the scale factor a decreases from ∞ and remainsnon zero until m → m s , or m → ∞ as the case may be. • In the Example where f ( m ) ∝ ( m s − m ) q as m → m s , the evolutionis non singular and has a bounce if 2 q ≥ q = 1 . Also, t → −∞ and a → ∞ in the limit m → m s . • In the Example where f ( m ) ∝ m q as m → ∞ , the evolution is nonsingular and asymmetric if 2 q ≤ t → −∞ ; and a → < q ≤ a → ( const ) if q = 0 , and a → ∞ if q < m → ∞ . The evolution for the q = 0 case in this limit is similar tothat expected in the Hagedorn phase of string/M theory.27 In the Example where f ( m ) ∝ e − bm with b > m → ∞ , theevolution is non singular, has a bounce, and is asymmetric. Also, t →−∞ and a → ∞ in the limit m → ∞ . f ( m ) t → − ∞ , t → t s t → ∞ all N S IV ( m s − m ) q , 2 q ≥ ∞ , N S → a min → ∞ m → m s , 0 < q < ∞ , S ∞ m s : f inite q < , S ∞ V m q , 2 q > , S ∞ m → ∞ < q ≤ , N S −→ ∞ q < ∞ , N S → a min → ∞ VI q = 0 const , N S −→ ∞ VII e − bm b > ∞ , N S → a min → ∞ m → ∞ Table II : Asymptotic behaviours of the scale factor a ( t ) . abulated here are the forms of the function f ( m ) in the limits of interestin Examples IV – VII, and the asymptotic values of the scale factor a ( t ) inthe limit t → ±∞ or t s where t s is finite. In the limit m → in theseExamples, f ( m ) → m , t → ∞ , a ( t ) → ∞ as in Einstein’s theory, andthe evolution is non singular. The accompanying letter S or N S denoteswhether the evolution in this limit is singular or non singular. The entireevolution is non singular in several cases in these Examples. 7. Anisotropic evolutions Consider the anisotropic evolutions. The scale factors a i = e λ i are nowdifferent for different i . Let the equations of state be given by p i = p = w ρ where w is a constant and 1+ w > f ( x ) . Therefore, we consider Example II where f ( x ) = x q and study mainly the asymptotic evolutions in the limit m i → i , and in the limit m i → ∞ for all i . Other Examples and otherlimits may be studied similarly.When f ( x ) = x q and p i = p = w ρ , we have f i = f ( m i ) = ( m i ) q and ρ = ρ e − (1+ w ) (Λ − Λ ) , X j = q ( m j ) q − X k G jk ( m k ) q (64)which follow from equations (9) and (27). Equations (24), (25) or (49), and(26) become X ij G ij ( m i ) q ( m j ) q = 2 γ λ qm κ ρ e − (1+ w ) (Λ − Λ ) (65)( m i ) t + Λ t m i = γλ qm κ d − q − − w ) ρ e − (1+ w ) (Λ − Λ ) (66)( γλ qm ) λ it = q X jk G ij G jk ( m j ) q − ( m k ) q . (67)One also has m i − m j = µ ij e − Λ where µ ij are constants, see equation (29).29 = − Explicit solutions can be obtained for the w = 2 q − m i = m i e − (Λ − Λ ) (68)where m i are constants. Consider equation (65). When 2 q = 1 + w , the e Λ − dependent factors cancel in this equation and one obtains X ij G ij ( m i ) q ( m j ) q = 2 γ λ qm κ ρ . (69)Consider equation (67). Using equation (68), one obtains λ it = λ it e − (2 q − 1) (Λ − Λ ) , Λ t = Λ t e − (2 q − 1) (Λ − Λ ) (70)where λ it and Λ t are given by( γλ qm ) λ it = q X jk G ij G jk ( m j ) q − ( m k ) q , Λ t = X i λ it . (71)Solving the equation for Λ t in (70), one obtains e (2 q − 1) (Λ − Λ ) = (2 q − 1) Λ t ˜ t , ˜ t = t − t + 1(2 q − 1) Λ t . (72)Then it follows that ˜ t = q − t , λ it = α i ˜ t , e λ i − λ i = ˜ t ˜ t ! α i , α i = λ it (2 q − 1) Λ t , (73)and that e Λ − Λ = ˜ t ˜ t ! α , α = X i α i = 12 q − . (74)These are the explicit anisotropic solutions for the w = 2 q − m i and λ i , which then determine theremaining initial values ρ , Λ , λ it , and Λ t . The following features ofthese solutions can now be seen easily. • Setting m i = m for all i in the above expressions gives λ it = Λ t d and α i = q − d , and leads to the solutions given in Example II with2 q = 1 + w now; compare equations (54) and (73).30 When 2 q = 1 , hence w = 0 , equation (70) leads to e λ i − λ i = e λ it ( t − t ) , e Λ − Λ = e Λ t ( t − t ) where λ it and Λ t are given by equation (71). Also, if m i = m for all i then λ it = qγλ qm . The solutions given in equation (57) then follow. • Vacuum solutions follow upon setting ρ = 0 in equation (69). • The above anisotropic solutions for the w = 2 q − q = 1 , we have w = 1 and equations (71), (74), (69), and (73) give γλ qm λ it = m i , X i α i = 1 , − X i ( α i ) = 2 κ ρ Λ t . For the vacuum case, ρ = 0 and one obtains P i α i = P i ( α i ) = 1 . w = − When w = 2 q − m i → i , and in the limit m i → ∞ for all i . In the asymptoticlimits, depending on the value of q and depending on whether m i → m i → ∞ for all i , the time t → ± ∞ or, after incorporating a finite shift, t → q = 1 and where t → ∞ or t → m i , e λ i , and λ it begiven by the ansatz m i = c i t b , e λ i ∝ t α i ←→ λ it = α i t (75)where b , c i , and α i are constants which must be determined consistentlyby equations (65) – (67). The present ansatz gives f i = ( c i ) q t b q , X j = x j t b (2 q − , Λ t = αt (76)31here x j = q ( c j ) q − X k G jk ( c k ) q , α = X i α i . (77)We then have, since ρ ∝ e − (1+ w )Λ , e Λ = c Λ t α , ρ = c ρ t (1+ w ) α (78)where c Λ and c ρ are constants. Equations (29) and (65) – (67) now give c i − c j t b = µ ij c Λ t α (79) P ij G ij ( c i ) q ( c j ) q t bq = 2 γ λ qm κ c ρ t (1+ w ) α (80)( α − b ) c i t b = γλ qm κ (2 q − − w ) c ρ ( d − t (1+ w ) α (81)( γλ qm ) α i t = P j G ij x j t b (2 q − . (82)Equation (82) determines b in terms of q and relates α i and x i , equivalently α i and c i : b = 12 q − , ( γλ qm ) α i = X j G ij x j . (83)We now analyse equations (79) – (81). Note that 1 + b = 2 bq . (cid:16) (cid:17) ≫ (cid:16) ( + w ) α (cid:17) −→ Asymptotic Anisotropy Consider first the case where (cid:16) t bq (cid:17) ≫ (cid:16) t (1+ w ) α (cid:17) in the asymptoticlimits. Then, since 1 + b = 2 bq , the right hand sides of both the equations(80) and (81) can be neglected and, hence, the resulting solution is equivalentto the vacuum solutions obtained earlier. Thus, one now obtains α − b = X ij G ij ( c i ) q ( c j ) q = 0 , (84)see equations (69) with ρ = 0 and (74). Since α − b = 0 , equation (81) doesnot impose any further restriction on c i . Also, with α = b , equation (79)32imply determines the constants µ ij and does not restrict ( c i − c j ) . Thismeans that c i , and hence m i and λ it , are generically different for different i and, therefore, the evolution is anisotropic. The volume factor e Λ is nowgiven by e Λ ∝ t α ∝ t q − (85)and, depending on the value of q , it may vanish or diverge asymptotically. • Note that the condition (cid:16) t bq (cid:17) ≫ (cid:16) t (1+ w ) α (cid:17) and the consequentanisotropic evolution are possible in the asymptotic limit t → bq > (1 + w ) α , and possible in the asymptotic limit t → ∞ only if2 bq < (1 + w ) α . Also, note that α = b = q − and we have assumedthat 1 + w > • Thus, anisotropic evolution is possible in the limit t → bq > (1 + w ) b . If 2 q > b > w < q − < q < b < w > q − • Similarly, anisotropic evolution is possible in the limit t → ∞ only if2 bq < (1 + w ) b . If 2 q > b > w > q − < q < b < w < q − • If q < b < , bq > bq > (1 + w ) α always. Thismeans that, when q < t → • Note that equations (84) are the analogs of Kasner’s solutions in Ein-stein’s theory. Indeed, for q = 1 , equations (83) give b = 1 and( γλ qm ) α i = c i ; and, equation (84) then gives α − , α − X i ( α i ) = 0 . Also, since 2 q − t → w < t → ∞ if w > (cid:17) = (cid:16) ( + w ) α (cid:17) −→ Asymptotic Isotropy Consider next the case where (cid:16) t bq (cid:17) = (cid:16) t (1+ w ) α (cid:17) in the asymptoticlimits. Since 1 + b = 2 bq , the right hand sides of both the equations (80)and (81) are now comparable. One then obtains 2 bq = (1 + w ) α and α = 2 q (2 q − 1) (1 + w ) , α − b = 2 q − − w (2 q − 1) (1 + w ) . (86)Note that this value of α follows in the isotropic case also, see equation (54).Equation (80) gives X ij G ij ( c i ) q ( c j ) q = 2 γ λ qm κ c ρ . (87)Consider equation (81). If 2 q = 1 + w then α − b = 0 , equation (81) isidentically satisfied and, therefore, imposes no restriction on c i . Also, since α = b now, equation (79) simply determines the constants µ ij and does notrestrict ( c i − c j ) . This means that m i are generically different for different i and, hence, the evolution is anisotropic. This case where 2 q = 1 + w hasbeen studied in the earlier part of this section and explicit solutions havealso been presented.Let 2 q − − w = 0 . Then α − b = 0 , these factors now cancel each otherin equation (81), and one obtains c i = γλ qm κ d − q − 1) (1 + w ) c ρ . Thus c i , and hence m i and λ it , are same for all i . Then α i = αd , see equation(54) also. This means that the evolution is isotropic. Now, this conclusioncan be consistent with equation (79) only if (cid:16) t b (cid:17) ≫ (cid:16) t α (cid:17) in the asymptoticlimits. Then, the right hand side of equation (79) can be negelcted whichthen implies that c i − c j = 0 for all i and j and, hence, that the evolution isisotropic. The volume factor e Λ is now given by e Λ ∝ t α ∝ t q (2 q − 1) (1+ w ) (88)and, depending on the value of q , it may vanish or diverge asymptotically.34 Note that the condition (cid:16) t b (cid:17) ≫ (cid:16) t α (cid:17) and the consequent isotropicevolution are possible in the asymptotic limit t → b > α ,and possible in the asymptotic limit t → ∞ only if b < α . Also, notethat b = q − and α = bq w and we have assumed that 1 + w > • Thus, isotropic evolution is possible in the limit t → b > bq w .If 2 q > b > w > q − < q < b < w < q − • Similarly, isotropic evolution is possible in the limit t → ∞ only if b < bq w . If 2 q > b > w < q − < q < b < w > q − • If q < b < , α > b < α always. This meansthat, when q < t → ∞ . 8. Conclusion We now present a brief summary and conclude by mentioning severalissues for further studies. In this paper, we considered the LQC – inspiredmodels which generalise the effective equations of LQC to ( d + 1) dimensionsand the function sin x to an arbitrary function f ( x ) , see equations (22) and(23). Then, assuming that p = w ρ and (1 + w ) > d + 1) dimensional cosmological evolutions in these models correspondingto a variety of possible behaviours of the function f ( x ) . We found explicitsolutions for the isotropic cases when f ( x ) = sin x , x q , and e x . For thesefunctions, we also found the potential V ( σ ) for a minimally coupled scalarfield σ which may give rise to the equation of state p = w ρ . We foundanisotropic Kasner–type solutions when f ( x ) = x q and w = 2 q − f ( x ) : examples where f ( x ) → x in the limit x → f ( x ) ∝ ( x r − x ) q in the limit x → x r or, in the limit x → ∞ , the35unction f ( x ) ∝ x q , or f ( x ) → ( const ) , or f ( x ) ∝ e − b x . Such asymptoticbehaviours are quite natural and, therefore, they may apply to a wide classof functions f ( x ) .We find that, depending on f ( x ) in the LQC – inspired models, a varietyof cosmological evolutions are possible, singular as well as non singular. Evenin the cases where there is a bounce and no singularities, the asymptoticevolutions are generically different from that in Einstein’s theory. We alsofound an intriguing and fascinating result that, for f ( x ) = √ x , the evolutionis non singular and a ( t ) grows exponentially at a rate set by λ qm , the quantumparameter related to the area quantum. But its significance, if any, is notclear to us at present.We now conclude by mentioning several issues for further studies. It isworthwhile to understand whether, how, and in what fundamental theories,the effective equations of the LQC – inspired models and, in particular, thefunction f ( x ) may arise. One may also study whether similar effective equa-tions can be constructed, even if only empirically, and applied to black holesingularities. It will indeed be interesting if black hole singularities may alsobe resolved in a variety of ways depending on some empirical function(s)in such models. The present LQC – inspired models may also be appliedin the context of M theory cosmology where, due to U duality symmetriesand appropriate intersecting brane configurations, (10 + 1) dimensional earlyuniverse evolves to a (3 + 1) dimensional universe, with the remaining sevendirections remaining constant in size [24, 26, 27, 28].Note that one way to adapt our LQC–inspired model to study blackhole singularities would be to construct the covarint version of the effectiveequations (24) – (26) which would generalise Einstien’s equations (2). Forvarious proposals for covariantising the effective equations in LQC, see [29] –[34]. Such a covariant formulation is likely to have other applications too. 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