Primeval symmetry breaking. R. Aldrovandi, 1 R.R. Cuzinatto 1,2 and L.G. Medeiros 1. Abstract

Transcription

1 Primeval symmetry breaking R. Aldrovandi, 1 R.R. Cuzinatto 1, and.g. Medeiros 1 Abstract arxiv:gr-qc/ v1 1 Sep 005 Addition of matter and/or radiation to a de Sitter Universe breaks the symmetry generated by four of its Killing fields. Detailed examination of the Killing equations in Robertson Walker coordinates shows that this symmetry-breaking is rather peculiar: the Killing fields timelike components simply vanish in the Friedmann Robertson Walker (FRW) case. It is of common usage to relate the generators responsible for homogeneity to the space section itself which is homogeneous: the latter is identified with the very space of translation parameters. In that case, the present-day Universe (the space section of cosmological spacetime) is not a simple deformation or modification of the pure de Sitter space. The picture of a de Sitter Universe with cosmological constant Λ which, by insertion of matter, would reduce to the Λ+FRW model fails, because the point sets are not the same. That picture of natural transition would be suggested by the Generalized Robertson Transformation here constructed step-by-step and which converts the de Sitter static interval into the standard FRW line element. Keywords: Killing fields, Standard model, Cosmological constant Running head: Primeval symmetry breaking 1 Instituto de Física Teórica, Universidade Estadual Paulista, Rua Pamplona 145, São Paulo SP, Brazil. To whom all correspondence should be addressed. rodrigo@ift.unesp.br. 1

2 1 Introduction Symmetry considerations are enough to fix the general form ] dr ds = c dt a (t) 1 κr + r dθ + r sin θdφ of the Friedmann Robertson Walker (FRW) interval 1, ]. Homogeneity and isotropy of the space section ensure a minimum of 6 symmetry generators, but this number can be higher for particular expressions of the scale parameter a(t). The expression above holds for any spacetime whose space section is homogeneous and isotropic and includes, consequently, de Sitter spacetimes. There are coordinate systems which are specifically more convenient for the latter, such as the static line element ) ds = (1 r c d t ( d r ) r dθ r sin θdφ, () 1 r but expression (1) provides a simpler, unified view of all the homogeneous isotropic Universes. Our objective here will be to analyse the isometries in the general case, with emphasis on the relationship between the de Sitter 10 parameter group of motions and the FRW 6 parameter symmetry. Two points of view are possible. We either start from a standard cosmological Friedmann model with smaller symmetry and add a cosmological constant, or start from an inflationary maximal-symmetry pure de Sitter model and add matter+radiation to observe a partial breaking of that symmetry. The latter standpoint has formal and, nowadays, observational advantages and will be taken up here. A single parameter U = a Ḣ, involving the time-derivative of the Hubble function, will be found whose time behavior encapsulates the interplay of the Killing fields for the two kinds of symmetry. The basic formulae and notation are introduced in Section. The explicit transformations taking the static de Sitter coordinates into the FRW variables which we shall call Generalized Robertson Transformation are given in Section 3. Section 4 introduces the Killing equations for metric (1). It turns out that the timelike components of the Killing vectors require a special consideration, and to them is devoted Section 5. The vanishing or not of these components is characterized by the constancy or not of the parameter U. The Killing fields change accordingly, in number and aspect. Their number is ten when U is time-independent, and correspond to the pure de Sitter case. When U is not a constant, their number drops to six, corresponding to the Friedmann Robertson Walker model with a cosmological constant (Λ- FRW Universe). The Killing fields are given in Section 6 for all cases. The final Section 7 includes a brief discussion of the Killing ie algebras. (1)

3 A standard overview Metric (1) leads, once inserted into Einstein s equations with a cosmological term and energy content modelized by a perfect fluid, R µν 1 R g µν Λg µν = 8πG c 4 T µν, T µν = (p + ρc ) u µ u ν p g µν, to the two Friedmann equations for the scale parameter a(t): ( ) ] 4πG ȧ = ρ + Λc a κc ; (3) 3 3 Λc ä = 3 4πG ( ρ + 3p )] a(t). (4) 3 c In terms of the Hubble function H(t) = ȧ(t) a(t) (5) and the dark energy density these equations become ǫ Λ = Λc4 8πG, (6) H Ḣ = ä a H = κc a (t) 3 = 8πG 3c ǫ m + ǫ Λ ] κc a (t) ; (7) 8πG 3c ǫ m + p m ]. (8) The matter energy density ǫ m = ρ m c = ρ b c + ρ γ c includes both nonrelativistic matter ( baryons ) and radiation. The equations actually acquire their most convenient form in terms of dimensionless variables, obtained by dividing by the present-day value H0. It is interesting to introduce the parameters Ω b (t) = ρ b(t) ρ crit ; Ω m = Ω b + Ω γ = ρ b ρ crit + ρ γ ρ crit (9) Ω Λ = Λc 3H 0 = ρ Λ ρ crit ; Ω κ (t) = κc a H 0 3. (10)

4 With the total matter mass-equivalent density ρ m, the first Friedmann equation becomes H H 0 = ρ m ρ crit κc a H 0 + Λc 3H 0 Ω m (t) + Ω κ (t) + Ω Λ, (11) which leads to the frequently used present-day values relation Ω m0 + Ω κ0 + Ω Λ = 1. (1) Notation can be further simplified by measuring time in units of H 1 0 and by introducing the parameters γc = Ω γ0 a 4 0 (13) ) Mc = Ω b0 a 3 0 = (1 Ω Λ γc + κc a 3 a 0, (14) 0 in terms of which the equations exhibit clearly the role of each contribution: a 4 0 ȧ + κc = Ω a Λ + γc a + Mc ; 4 a 3 (15) a Ḣ = κc γc a 3 Mc a. (16) Analytic solutions for the whole cosmic time span in a Standard Model have been presented elsewhere 4] for all the combinations of Λ = 0, Λ 0, κ = 0 and κ 0, in the presence and absence of radiation and nonrelativistic (baryonic and/or dark) matter. For the problem we are concerned with here, only the equations above are of importance. It will be of some help to say a little more on the simplest, maximally symmetric cases, de Sitter spacetimes. These are solutions of the sourceless Einstein s equations with a cosmological constant related to a pseudo-radius and the scalar curvature by Λ = 3 = R 4. (17) In that case and in our units, Ω Λ = c. As they are spatially homogeneous and isotropic, such spacetimes satisfy (3-4) with vanishing ρ and p: ȧ = c a κc ; (18) ä = c a(t). (19) 4

5 The solution with a(0) = A allowing for inflation 3 is ( ) ct a(t) = A cosh + A κ sinh = e c t A + A κ The Hubble function will be ( ) ct = + e c t A A κ. (0) H = c κc κ + e ct (A + A κ ). (1) The general case including radiation plus matter (with Λ 0, κ 0) is, we repeat, described by Eq.(15), ȧ + κc γc a Mc a c a = 0. () We shall, in what follows, frequently use the variable T = ct. The nonvanishing Christoffel symbols for line element (1) are: a H Γ T rr = 1 κr ; ΓT θθ = a Hr ; Γ T φφ = a Hr sin θ Γ r Tr = H ; Γ r rr = κ r 1 κr ; Γ r θθ = r (1 κr ) ; Γ r φφ = r (1 κr ) sin θ (3) Γ θ Tθ = Γ θ θt = H ; Γ θ rθ = Γ r θr = 1 r ; Γθ φφ = sin θ cosθ Γ φ Tφ = Γ φ φt = H ; Γ φ rφ = Γ φ φr = 1 r ; Γφ θφ = Γ φ φθ = cotθ 3 The generalized Robertson transformation The simultaneous use of de Sitter static line element () and the standard form (1) of the FRW interval has led to very interesting results 5, 6]. We shall here give explicit expressions for the coordinate transformations relating them. We recall that, in common practice, the radial FRW coordinate r is dimensionless, the dimension being carried by the scale factor a(t); the radial coordinate r in Eq.(), on the other hand, has dimension of length, just like the de Sitter horizon pseudo-radius. 3 We are not considering the Anti-de Sitter case, for which Ω Λ = c. 5

6 The simple Robertson transformations 7] or r = ( r/a) 1 r t = t + 1 c ln e c t/, (4) (1 ) r, (5) r = r ( Ae ct/), (6) t = t 1 1 c ln r ( ) ] Ae ct/, (7) convert the de Sitter static line element into the stationary form ds = c dt ( Ae ct/) dr + r dθ + r sin θdφ ], (8) which is exactly the FRW interval (1) with the particular value κ = 0 for the curvature parameter if we identify a(t) = Ae ct/. This is the paradigmatic inflationary solution. Our purpose is to extend the above transformations so as to include cases κ = ±1. Inspired in Eqs.(6) and (7), we look for a transformation r = rf(t), t = g(t) + h(t, r), (9) where f(t), g(t) and h(t, r) are functions to be determined. Inserting these expressions in the static interval () and imposing equality with the FRW interval leads to a system of partial differential equations, 1 (rf) 1 (rf) ] 1 1 (rf) ]f ] ( ) ġ + ḣ 1 ]r 1 f = 1, (30) (rf) ) h (ġ + ḣ 1 ]rff 1 = 0, (31) (rf) 1 (rf) ] h = f (t) 1 κr, (3) f (t) = a (t), (33) where the dots and primes indicate derivations with respect to t and r, respectively. Eliminating the factor (ġ + ḣ) from the first two equations, it is 6

7 possible to determine h in terms of f and f. Substituting the result into Eq.(3) leads to f = c f κc, (34) which is, in view of Eq.(33), just the Friedmann equation (18) for the de Sitter case with κ = 0, ±1. This is a curious fact, since the Friedmann equations come from the gravitational field dynamics dictated by Einstein s equations and the FRW interval, while relation (34) rises solely as a coordinate transformation effect. Its solution is, of course, just Eq.(0): f(t) = A cosh ( ) ct The next step is to isolate h in Eq.(3), + A κ sinh ( ) ct. (35) h (t, r) = c rf ( f ) κ 1 (rf) ] 1 κr, (36) and integrate directly, thereby fixing the radial dependence of h(t, r). It is more secure to solve for each value of κ separately and afterwards gather the results in a single formula, which is h(t, r) = c ln ( f ) κ + f 1 (rf) 1 κr + z(t). (37) For the time being, z(t) is arbitrary and t is determined up to a function of t, say, ḡ(t) g(t) + z(t). The function ḡ(t) will be fixed as soon as we evaluate ḣ(t, r). The derivative of Eq.(37) with respect to t gives, after some algebra, 1 κr ḣ(t, r) = ż(t) +. (38) 1 (rf) This expression gives, once inserted with Eq.(36) into Eq.(31), i.e., dḡ dt = 0, (39) ḡ = g + z = constant t 0. (40) 7

8 Using Eqs.(35), (37) and (40), Eqs.(9) can be rewritten as ( ) ct r(t, r) = ra(t), a(t) = A cosh + A κ sinh a(t) ] t(t, r) = t 0 + κ + a(t) 1 κr ln c 1 ra(t)] ( ) ct ; (41). (4) The constant t 0 can be fixed by requiring that these expressions coincide with the particular Robertson transformations (7) and (6) when κ = 0. The first equation above reduces to Eq.(6), and the second reduces to Eq.(7) if we choose t 0 = ( ) A c ln. (43) From this choice follows the generalized Robertson transformations, ( ) ct r(t, r) = ra(t), a(t) = A cosh + ( ) ct A κ sinh ; (44) t(t, r) = ln 1 a (t) κ + a(t) 1 κr. (45) c A 1 ra(t)] At first sight, the existence of such a general transformation would seem to say that the original de Sitter vacuum-like state does not select the type of Friedmann model: all the three values of κ are permitted. Equation (44), however, requires A κ 0 for a real a(t). For κ = +1 this would impose A. This initial value larger than the horizon pseudo-radius is unacceptable, and we are forced to choose: we either keep only κ = 0, 1 or information is lost beyond the horizon for κ = The Killing equations A Killing vector field ξ (P) with P = 1,,..., n, n being the number of parameters of the symmetry group, will be here indicated by its covariant components ξ α (P) = ξ α (P) (x 0, x 1, x, x 3 ) = ξ α (P) (T = ct, r, θ, φ), ξ (P) = {ξα P } = (ξ(p) 0, ξ (P) 1, ξ (P), ξ (P) 3 ), in terms of which the Killing equations 3] take on the simple forms α ξ (P) β + β ξ (P) α Γ γ αβ ξ (P) γ = 0. (46) 8

9 Notice that, as metric (1) is diagonal, the timelike or spacelike characters (though not the expressions) are the same for covariant and contravariant components. Each Killing field will then be a vector field 8, 9] ξ (P) = ξ (P)α α. (47) There will be one for each independent integration constant. Equations (46) become, once use is made of the Christoffeln (3), I 0 ξ 0 = T ξ 0 = 0 ξ 0 = ξ 0 (r, θ, φ) II 1 ξ ξ 1 H ξ 1 = 1 ξ 0 + ξ 1 0 ln ( ξ 1 ) a = 0 III ξ ξ H ξ = 0 IV 3 ξ ξ 3 H ξ 3 = 0 V ξ ξ r ξ = ξ 1 + ξ 1 ln ( ξ r ) = 0 VI 3 ξ ξ 3 r ξ 3 = 0 VII 3 ξ + ξ 3 cotθ ξ 3 = 3 ξ + ξ 3 ln ( ξ 3 sin θ) = 0 ] VIII 1 κr 1 1 κr ξ 1 a Hξ 0 = 0 IX ξ + r (1 κr ) ξ 1 a Hr ξ 0 = 0 X 3 ξ 3 + r (1 κr ) sin θ ξ 1 + cosθ sin θ ξ a Hr sin θ ξ 0 = 0. The algebraic index (P) was omitted for economy. 5 The timelike components The fastidious process to find the analytic solutions follows the standard procedure of taking crossed derivatives of the equations and eliminating unwanted terms. We shall here only comment on a particular case, which leads to an important result. A second-order equation for the r-dependence of ξ 0 can be obtained from equations I, II and VIII: it is enough to take 0 in VIII and then use again the three equations, retaining only the terms involving ξ 0, to arrive at where (1 κr ) 1 1 ξ 0 κ r 1 ξ 0 + U(T) ξ 0 = 0, (48) U(T) = a Ḣ = a ä ȧ. (49) In terms of the deceleration function q(t) = äa ȧ = 1 H (t) ä, U(T) = a a H (t) (1 + q(t)). Parameter U will play an important role in what follows. Indeed, taking 0 of (48) leads to U ξ 0 = 0. (50) 9

10 We have thus either ξ 0 = 0 or U = 0, or both. It is possible, consequently, to have a nonvanishing solution for ξ 0 provided U is a constant. Now, U = a (T) 0 Ḣ + H ], so that the whole question reduces to whether or not parameter C = Ḣ + H = ä (51) a is a constant. Equation (19) shows that C = ä is a constant in the de Sitter a case. Equation (16), on the other hand, tells us why U is not a constant in the general case: it reads U = a Ḣ = κc γc a 3 Mc. (5) a U will not be constant in the presence of matter (M 0) and/or radiation (γ 0). In these cases Eq.(50) implies that component ξ 0 vanishes for all Killing fields, and as will be seen in next section four of these vanish completely. The 10-generator symmetry becomes lesser, it is partially broken by matter and/or radiation. If we introduce the total (matter + cosmological constant) energy ǫ = ǫ m + ǫ Λ and pressure p = p m + p Λ, the Friedmann equations are written H 8πG = 3c ǫ κc (53) a (t) C = Ḣ + H = 8πG ǫ + 3p. (54) 3c The solution of equation C = ä with constant C is a at] = A cosh CT + AH(0) sinh CT (55) C with U = A C H (0)]. This just (0) with C = c and U = kc. We see thus that ξ 0 0 is precisely the case of de Sitter solutions. Summing up: { = 0 when U is not a constant (FRW spacetime) ξ 0 0 when U is a constant (de Sitter spacetime). 6 The Killing fields We are now in condition to exhibit the solutions for the Killing equations. The covariant components are: ξ 0 (r, θ, φ) = C 0 1 κ r +κ r (C 1 cosφ + C sin φ)sin θ+c 3 cosθ ] (56) 10

11 ξ 1 (T, r, θ, φ) = a (T) 1 A (K 1 cosφ + K sin φ)sin θ + K 3 cosθ] 1 κr + a C 0 r (T)H(T) (C 1 cosφ + C sin φ) + C 3 cosθ sin θ ]] (57) 1 κr ξ (T, r, θ, φ) = a (T) r 1 κr A (K 1 cosφ + K sin φ) cosθ K 3 sin θ ] ] + r ( 1 sin φ cosφ) a (T)H(T) r (C 1 cosφ + C sin φ)cosθ C 3 sin θ] (58) ξ 3 (T, r, θ, φ) = a (T) r 1 κr A (K cos φ K 1 sin φ) sin θ ] + r sin θ cosθ ( 1 cosφ + sin φ) 3 r sin θ a (T)H(T) r sin θ(c cosφ C 1 sin φ) (59) Case U = 0 contributes the whole of Eq.(56), and the last lines of Eqs.(57, 58, 59). Case U 0 corresponds to C 0 = C 1 = C = C 3 = 0, vindicating the statement of last Section: four of the Killing vector fields vanish completely. Each Killing vector field ξ (P) will be obtained by choosing a convenient, simple value (such as ±1) for one of the constants and putting all the other = 0. This means, in particular, that we have complete freedom in the choice of overall signs. Of course, linear combinations of such fields will also be solutions. For example, the Killing vector ξ (P) related to 1, the eighth constant of integration (for with P = 8), is ξ (8) = {ξ ( 1) α } = (0, 0, a sin φ, a sin θ cosθ cosφ), A r A r where we have freely chosen We could also have written the Killing differential equations corresponding to I X in de Sitter coordinates using directly the static line element (). 11

12 The solutions for Killing components in de Sitter coordinates would then be ξ 0 ( T, r, θ, φ) = r {( 1 r K 1 cosh T ( + K 3 cosh T + K 4 sinh T ( + K 5 cosh T + K 6 sinh T + K sinh T ) cosφ ) ] sin φ sin θ+ } ) + K 0 (1 r ) cosθ (60) ( ) 1 ξ 1 T, r, θ, φ = 1 r ( + {( K 1 sinh T + K cosh T ) cosφ K 3 sinh T + K 4 cosh T ( + ) ] sin φ sin θ+ K 5 sinh T + K 6 cosh T ) } cosθ (61) {( ( ) ξ T, r, θ, φ = r 1 r K 1 sinh T + K cosh T ) cosφ ( + K 3 sinh T + K 4 cosh T ) ] sin φ cosθ+ ( K 5 sinh T + K 6 cosh T ) } sin θ + r ( 1 sin φ cosφ ) (6) ( ( ) ξ 3 T, r, θ, φ = r 1 r sin θ K 3 sinh T + K 4 cosh T ) cosφ ( K 1 sinh T + K cosh T ) ] sin φ + + r sin θ cosθ ( 1 cosφ + sin φ ) 3 r sin θ (63) with T = c t. Obviously, the number of integration constants is ten in both coordinate systems: the maximum possible number in a four dimensional space. Equations (56 59) can be obtained from the components (60 63) by the Robertson generalized transformation, Eqs.(44,45). et us go back to the FRW Killing fields (47). Two steps are necessary to arrive at their most convenient form: 1

13 First, we need the contravariant components, got by applying to the components (56 59) the inverse metric (g µν ) = (1 κr ) a a r a r sin θ. (64) Fields are above written in basis { T, r, θ, φ }. Simpler, more recognizable expressions obtain if we change to cartesian variables in the space section, x = r sin θ cosφ, y = r sin θ sin φ, z = r cos θ and the corresponding x, y, z. It will be also convenient to introduce the space section dilation generator D = r r = x x + y y + z z. It is then immediate to replace φ = x y y x ; r = 1 r D; θ = z(cos φ x + sin φ y ) r z z = (zd r z) r z. We shall also profit to absorb the factors in A in the constants K 1,,3 and 1,,3. The final form is then found to be ξ = 10 P=1 10 { } ξ (P) = ξ (P) T T + ξ (P)r r + ξ (P) θ θ + ξ (P) φ φ P=1 = 1 y z z y ] + z x x z ] + 3 x y y x ] 1 κr K 1 x + K y + K 3 z ] + C 1 κ x( T HD) + H x ] + C κ y( T HD) + H y ] + C 3 κ z( T HD) + H z ] + C 0 1 κ r T HD]. There are then two categories of fields: the first six (corresponding to K i 0, i 0) remain in all cases. The last four (related to C α 0) vanish when U 0. Notice that the Hubble function H only appears in the latter. It will consequently be given by (1) whenever it turns up. Convenient choices of constants and notation will put the ten fields into the forms T 1 = 1 κr x ; T = 1 κr y ; T 3 = 1 κr z ; (65) J 1 = y z z y ; J = z x x z ; J 3 = x y y x ; (66) 13

14 S 0 = 1 κr ( T HD) ; S 1 = κ x ( T H D) + H x ; S = κ y ( T H D) + H y ; S 3 = κ z ( T H D) + H z. (67) Fields (65) are responsible for the space section homogeneity, and (66) for its isotropy. Together, they generate the isometry group of the space section. Fields (67) are those which are only present in case U = 0. The above notation for the constants is biased: 1,,3 are associated with rotations, K 1,,3 with space translations, and C 0,1,,3 with spacetime translations. Comparing with the constants in the ξ µ s, we find that the constants 1,,3 are indeed rotation-related. Nevertheless, the constants K 1,...,7 play together the role performed by C 0,...,3 and K 1,,3 in FRW co-moving system, indicating that space translations and temporal symmetry are merged in de Sitter coordinates. 7 Final comments The algebra of the above Killing fields has the following commutation table: J i, J j ] = ǫ ijk J k (68) J i, T j ] = ǫ ijk T k (69) T i, T j ] = κ ǫ ijk J k (70) S i, S j ] = κ c ǫ ijk J k (71) S 0, S k ] = c T k (7) S 0, T k ] = S k (73) J i, S j ] = ǫ ijk S k (74) J i, S 0 ] = 0 (75) T i, S j ] = κδ ij S 0 (76) Commutators (68-70) involve the generators of homogeneity and isotropy of the space section, which are present in any FRW spacetime and remain when U 0. Commutators (71-73) involve those generators which only are present when U = 0, the de Sitter case. Equation (68) just represents the ie algebra of the rotation group SO(3) the J k s are responsible for space section isotropy. Generators {T k } answer for its homogeneity and are, according to Eq.(69), vectors of the rotation 14

15 group. Case κ = 0 refers to usual translations on Euclidean space in that case (68-70) is the commutation table of the ie algebra of the Euclidean group SO(3) T 3. When κ = + 1 the T k s engender the modified translations responsible for the homogeneity of a 3-dimensional sphere S 3. Commutators (68-70) are then related to the ie algebra of group SO(4) = SO(3) SO(3). Finally, in case κ = 1 the T k s generate translations responsible for the homogeneity of a hyperbolic 3-space H,1 and the whole set coincides with the ie algebra of the orentz group SO(3, 1). It is of common usage to relate the generators responsible for homogeneity with the space itself which is homogeneous. This corresponds, for example, to identifying the Euclidean space E 3 with the space of parameters of its translations. With that identification, the present-day Universe (the space section of cosmological spacetime) is not a simple deformation or modification of the pure de Sitter space. The image of a de Sitter Universe which, by addition of matter+radiation would reduce to the Λ-FRW Universe fails, because the point sets are not the same. The de Sitter space section is generated as in Eq.(71), while our present-day Universe is generated by (70). Acknowledgement The authors thank FAPESP-Brazil and CNPq-Brazil for financial support. References 1] J. V. Narlikar, Introduction to Cosmology (Cambridge University Press, Cambridge, 1993). ] S. Weinberg, Gravitation and Cosmology (J. Wiley, New York, 197). 3]. P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, 1949). 4] R. Aldrovandi, R.R. Cuzinatto and.g. Medeiros, Analytic solutions for the Λ-FRW Model ArXiv:gr-qc/ ] É. B. Gliner, Soviet Physics DOKADY (1970). 6] I. G. Dymnikova, Soviet Physics JETP (1986). 7] R C Tolman, Relativity, Thermodynamics and Cosmology, Dover Publications, Inc., New York,

16 8] B. Doubrovine, S. Novikov and A. Fomenko, Géométrie Contemporaine, 3 vols. (MIR, Moscow, 1979). 9] R. Aldrovandi and J. G. Pereira, An Introduction to Geometrical Physics (World Scientific, Singapore, 1995). 10] J. Katz, J. Bicak and D. ynden-bell, Phys.Rev. D , ]. D. andau and E. M. ifschitz, The Classical Theory of Fields (Pergamon Press, Oxford, 1975). 1] P. Coles and G. F. R. Ellis, Is the universe open or closed? (Cambridge University Press, Cambridge, 1997). 13] Ya. B. Zeldovich & I. D. Novikov, Relativistic Astrophysics II: The Structure and Evolution of the Universe (University of Chicago Press, 1981). 14] S. W. Hawking and G. F. R. Ellis, The arge Scale Structure of Space- Time (Cambridge University Press, Cambridge, 1973). 15] See, for example,. D. andau and E. M. ifshitz, The Classical Theory of Fields (Butterworth-Heinemann, Oxford, 1999), 4th Revised Edition. 16] A. inde, Phys. ett. 108B, 389 (198). 17] A. Albrecht and P. Steinhardt, Phys. Rev. ett 48, 10 (198). 16