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Spatially Homogeneous String Cosmologies 7 John D. Barrow and Kerstin E. Kunze 9 Astronomy Centre 9 University of Sussex 1 Brighton BN1 9QH n a U.K. J 9 February 1, 2008 2 2 v Abstract 5 4 We determine the most general form of the antisymmetric H-field tensor derived from 0 a purely time-dependent potential that is admitted by all possible spatially homogeneous 8 cosmological models in 3+1-dimensional low-energy bosonic string theory. The maximum 0 number of components of the H field that are left arbitrary is found for each homogeneous 6 9 cosmologydefinedbythe Bianchigroupclassification. Therelativegeneralityofthesestring / cosmologies is found by counting the number of independent pieces of Cauchy data needed h to specify the general solution of Einstein’s equations. The hierarchy of generality differs t - significantlyfromthatcharacteristicofvacuumandperfect-fluidcosmologies. The degreeof p e generality of homogeneous string cosmologies is compared to that of the generic inhomoge- h nous solutions of the string field equations. : v i PACS numbers: 9880C,1125, 0420J,0450 X r I. INTRODUCTION a The low-energy effective action of the bosonic sector of string theory provides cosmological models that might be applicable just below the Planck (or string) energy scale in the very early universe [1]. A number of studies have been made of these cosmologies in order to ascertain the behaviour of simple isotropic and anisotropic universes, investigate the implications of duality, andsearchforinflationarysolutions[2-6]. Manyofthetraditionalquestionsofgeneralrelativistic cosmology can be asked of the cosmological models defined by string theory: do they possess space-time singularities?, whatis the generic behaviour of the solutions at late and early times?, whatexactsolutionscanbefoundinclosedform?,andwhatrelationdoparticularexactsolutions have to the general cosmological solution? Since this theory is to be applied at times very close to the Planck epoch it would be unwise to make special assumptions about the form of the 1 cosmological solutions. Anisotropies and inhomogeneities could play an important role in the evolution. Indeed, any dimensional reduction process could be viewed as an extreme form of anisotropic evolution in D dimensions in which three spatial dimensions expand whilst the rest remain static. Because of these irreducible uncertainties about the very early Universe, one would like to understand the general behaviour of wide classes of solution in order to ascertain the relative generality of any particular solution that may be found. A number of studies have focused on obtaining particular solutions for 3+1 dimensional space-times in cases wherespatial homogeneity (and sometimes also isotropy) is assumed for the metric of space-time, where the H field is set to zero [4], or where the H field is included by assuming that it takes a particular form which satisfies its constraints and its equation of motion [5]. For example, Copeland et al, [2], discussed Friedmann and Bianchi type I universes, allowing H to be time-dependent or ∗ space-dependent, respectively. In a second paper, [3], they discussed Bianchi I solutions with a homogeneous antisymmetric tensor field. In [6] (see also [5]) Batakis presented an overview of all possible configurations of a (spatially) homogeneous H-field in diagonal Bianchi models with a metric ds2 = dt2+a (t)2(ω1)2+a (t)2(ω2)2+a (t)2(ω3)2 1 2 3 − where dt,ωα isthestandardbasis. However, inthispapertheBianchimodelsarenotassumed { } to be diagonal. The form of the H-field derived from a time-dependent potential will be determined in all four-dimensionalspace-times withhomogeneousthree-spaces. Thesethree-spaces werefirstclas- sifiedbyBianchi[7]andhavebeenextensivelystudiedinthecosmological contextfollowingtheir introduction into cosmology by Taub [8]. They provide us with the general class of cosmological models whose solutions are determined by ordinary differential equations in time. By gener- alising a procedure used to study electromagnetic fields in spatially homogeneous cosmological models by Hughston and Jacobs [10], we can determine the maximum number of components permitted for the H field in each of the Bianchi cosmologies. This enables us to determine the number of degrees of freedom which definethe string cosmology of each case. The results are in- teresting. The Bianchi types containing the most general geometries place the most restrictions upon the presence of the H field. The string world sheet action for a closed bosonic string in a background field including all the massless states of the string as part of the background is given by, [1], 1 S = d2σ √hhαβ∂ Xµ∂ Xνg (Xρ)+ǫαβ∂ Xµ∂ XνB (Xρ)+α′√hφ(Xρ)R(2) (1) −4πα′ Z { α β µν α β µν } where hαβ is the 2-dimensional worldsheet metric, R(2) the worldsheet Ricci scalar, ǫαβ the worldsheet antisymmetric tensor, B (Xρ) the antisymmetric tensor field, g (Xρ) the back- µν µν ground space-time metric (graviton), φ(Xρ) the dilaton, α′ is the inverse string tension, and the functionsXρ(σ)mapthestringworldsheetintothephysicalD-dimensionalspace-timemanifold. 2 For the consistency of string theory it is essential that local scale invariance holds. Imposing this condition results in equations of motion for the fields g , B and φ which can be derived µν µν to lowest order in α′ from the low-energy effective action 1 S = dDx√ ge−φ(R+gab∂ φ∂ φ HabcH Λ). (2) Z − a b − 12 abc− In this paper we assume a vanishing cosmological constant, Λ. Inacosmological context itisgenerally assumedthatbysomemeans allbutfourof the10or 26 dimensionsof space-time are compactified, leaving an expanding3+1-dimensional space-time (D = 4). Sinceweareinterested incosmological solutions of thefieldequations derivedfromthe variation of this action, we adopt the Einstein frame by making the conformal transformation g e−φg . (3) ab ab → In this frame the 4-dimensional string field equations and the equations of motion are given by (indices run 0 a,b,c 3 and 1 α,β 3), ≤ ≤ ≤ ≤ 1 R g R = κ2((φ)T +(H)T ), (4) ab ab ab ab − 2 (e−2φHabc) = 0, (5) a ∇ 1 2φ+ e−2φH Habc = 0, (6) abc 6 where κ2 = 8πG is the 4-dimensional Einstein gravitational coupling and 1 1 (φ)T (φ φ g φ φ,c), (7) ab ,a ,b ab ,c ≡ 2 − 2 1 1 (H)T e−2φ(3H H cd g H Hmlk). (8) ab ≡ 12 acd b − 2 ab mlk The3-geometries oftheninespatially homogeneouscosmological solutionsoftheseequations are defined by the Bianchi classification of homogeneous spaces (with the exception of the Kantowski-Sachs universe, [9], which has a four-dimensional group of motions but no three- dimensional subgroup). In these Bianchi models (e.g. [13]) the spacelike hypersurfaces are invariant under the group G of isometries whose generators are 3 Killing vectors ξ . These 3 α hypersurfaces can be described by an invariant vector basis X satisfying α { } X = [ξ ,X ] = 0 Lξβ α β α where is the Lie derivative in the direction of ξ . The timelike direction X is chosen to be Lξβ β 0 orthogonal to the invariant spacelike hypersurfaces obeying X = [ξ ,X ]= 0. Lξβ 0 β 0 3 Dual to X is the basis of one-forms ωµ satisfying α { } { } 1 ωµ = Cµ ωκ ωλ. 2 κλ ∧ Spatial homogeneity is expressed by the following conditions on φ, g and H φ= 0, Lξα g = 0, Lξα H = 0 ( H) = 0. Lξα ⇒ Lξα ∗ The definition and properties of the Lie derivative imply that φ = ξ φ = 0. Expanding Lξα α H in the invariant basis (that is, H = V0X +VαX ), and using its properties, implies then 0 α ∗ ∗ ξ V0 = 0 and ξ Vβ = 0. The Killing vectors in the Bianchi models are spacelike and time α α independentand this then implies that φ and H are functions of time only in the standard basis dt,ωα . Furthermore, the antisymmetric tensor potential B where H = dB will be assumed to { } be a function of time only. We would like to know the general algebraic form of the H field with a time-dependent potential B in these models, determine which Bianchi universes are the most general, and discover whether the assumption of spatial homogeneity reduces the number of independent pieces of Cauchy data below the number needed to specify a generic inhomogeneous solution of the field equations (4)-(8). This analysis of the allowed components of the H-field is most economically performed using differential forms. II. THE ANTISYMMETRIC TENSOR FIELD AS A 2-FORM There are three equations determining the antisymmetric tensor field: the definition of its field strength (for a closed bosonic string) H = dB, (9) which implies the second equation dH = 0, (10) and there is the equation of motion, (5), d( H) 2(dφ) ( H) = 0. (11) ∗ − ∧ ∗ 4 Spatially homogeneous models are described by choosing an orthonormal tetrad, ds2 = η σaσb, (12) ab where η = diag( 1,1,1,1), and specifying the 1-forms σa [10,13] as ab − σ0 = N(Ω)dΩ, σα = e−Ωbαωβ. (13) β Here, the ωα obey the algebra 1 dωα = Cα ωβ ωγ, (14) 2 βγ ∧ where Cα are the structure constants of the possible isometry groups which define the homo- βγ geneous 3-spaces, and the bα are symmetric matrices which depend only on the time coordinate β Ω. Since B is a 2-form, it can be decomposed as B = B σ0 σα+B σα σβ = Q dΩ ωκ+S ωκ ωµ, (15) 0α αβ 0κ κµ ∧ ∧ ∧ ∧ where Q (Ω) NB e−Ωbα, (16) 0κ 0α κ ≡ S (Ω) e−2ΩB bαbβ. (17) κµ αβ κ µ ≡ Hence, H = dB is given by 1 1 H = (S Cκ Q )dΩ ωα ωβ + S C[κ ωµ] ωα ωβ. (18) αβ|Ω − 2 αβ 0κ ∧ ∧ 2 κµ αβ ∧ ∧ This expression can be analysed further if we introduce the Ellis-MacCallum [11] decomposition of the structure constants into the matrix m and the vector a , αβ β Cγ = ǫ mµγ +δγa δγa , (19) αβ αβµ β α− α β so (18) becomes 1 H = (S Cκ Q )dΩ ωα ωβ +2a S ωµ ωα ωκ. (20) αβ|Ω− 2 αβ 0κ ∧ ∧ α κµ ∧ ∧ γ The structure constants satisfy a Jacobi identity which leaves C with a maximum of 6 αβ independent components. Since the lagrangian is invariant under the gauge transformation 5 B B +∂ Λ , we can always choose Λ such that Q = ∂ Λ = ∂ Λ , and set Q to ab ab [a b] 0κ [0 κ] 0 κ 0κ → − − be zero. The nine Bianchi-type universes fall into two classes, A and B, distinguished by whether the constant a is zero or non-zero respectively [11]. From (20) we see that H has no purely spatial components in Class A models. H is also given by H = H σa σb σc abc ∧ ∧ = H σ0 σα σβ +H σα σβ σγ 0αβ αβγ ∧ ∧ ∧ ∧ = X dΩ ωκ ωλ+Y ωκ ωλ ωµ, (21) 0κλ κλµ ∧ ∧ ∧ ∧ where X = X (Ω) Ne−2ΩbαbβH , (22) 0κλ 0κλ ≡ κ λ 0αβ Y = Y (Ω) e−3ΩbαbβbγH . (23) κλµ κλµ ≡ κ λ µ αβγ Therefore dH = 0 implies 1 Y dΩ ωκ ωλ ωµ+ X (Cκ ωλ Cλ ωκ) ωα ωβ dΩ = 0. (24) κλµ|Ω ∧ ∧ ∧ 2 0κλ αβ − αβ ∧ ∧ ∧ Using the expression (19) for the structure constants, and noting that the 3-dimensional Levi-Civita symbol is defined by ǫ = detg ω1 ω2 ω3, eq. (24) becomes αβ ∧ ∧ p (Y 2X a )dΩ ωκ ωλ ωµ = 0. (25) κλµ|Ω 0µκ λ − ∧ ∧ ∧ The dual, λ, of an n-dimensional p-form λ is defined by the Levi-Civita symbol as [12] ∗ 1 λ = λa1...apǫ . ∗ b1...bn−p p! a1...apb1...bn−p Hence, H is a 1-form given by ∗ 1 H = Hbcdǫ σa = UdΩ+V ωα, (26) bcda α ∗ 6 where 1 U U(Ω) Hαβγǫ N, (27) αβγ0 ≡ ≡ 6 6 1 V V (Ω) Habcǫ bκe−Ω, (28) α ≡ α ≡ 6 abcκ α and so 1 d H = V dΩ ωα+ V Cα ωβ ωγ. (29) ∗ a|Ω ∧ 2 α βγ ∧ Since φ= φ(Ω), we have dφ = φ dΩ and equation (11) reads |Ω 1 (V 2φ V )dΩ ωα+ V Cα ωβ ωγ = 0; (30) α|Ω − |Ω α ∧ 2 α βγ ∧ hence, V 2φ V = 0, (31) α|Ω |Ω α − 1 V Cα = 0. (32) 2 α [βγ] Notice that the constraint (32) is preserved in time. Contracting (31) with Cα gives [βγ] (V Cα ) = 0 so that if (32) is satisfied at one time it holds at all times. Eqn. (32) implies α [βγ] |Ω ǫβγδV Cα = 0, (33) α βγ which can be rewritten as V (mδα +a ǫβαδ) = 0, (34) α β and so, by (20), we have H = X dΩ ωκ ωλ+Y ωµ ωα ωκ, (35) 0κλ κλµ ∧ ∧ ∧ ∧ with 1 X = S Cν Q and Y = 2a S . (36) 0κλ [κλ]|Ω− 2 [κλ] 0ν κλµ [α κµ] Eqn. (25) implies Cα a Q = 0, (37) [µκ λ] 0α and (31) can be integrated to give V = e2φK , (38) α α 7 where K is a constant spatial 3-vector of integration. α Since ( H) = H, we have ∗ ∗ X0αβ = ǫ0αβγV γ − = ǫ0αβγe2φK , (39) γ where the minus sign has been absorbed into the constant spatial 3-vector K . γ TABLE 1 Table 1 displays the restrictions on the spatial components of H imposed by the constraint ∗ equation (34) for the different Bianchi types [11,13], together with the components of the homo- geneousantisymmetrictensorfieldstrengthH inthestandardbasis dΩ,ωα whicharegiven by { } eqn. (20). Note that in Class A, eqn. (23) implies Y = 0, and the contravariant components 123 of Y are obtained by raising the indices using g given by αβγ ab g = N2(Ω) (40) 00 − g = e−2Ω bγbγ. (41) αβ α β Xγ In Class B, eqn.(23) implies that Y = 2a S = 2a S . The matrix α which specifies the 123 [2 31] 3 12 Ellis-MacCallum symbol m = m is defined by, [11], the matrix αβ 0 1 0 α=  1 0 0  (42)  0 0 0    III. COUNTING DEGREES OF FREEDOM Consider first the question of how many independently arbitrary spatial functions are required to specify generic initial data for the system of string field equations (4)-(8). In a syn- chronous frame we require 6 g , 6 g˙ , 3 components of the H field, together with values of φ αβ αβ and φ˙. This amounts to 17 functions, but we can remove 4 by using the coordinate covariance of the theory, another 4 by using the R constraint equations, and another 1 by using the φ 0a equation, (6). This leaves 8 independent functions of three spatial variables to specify a general solution of the field equations (4)-(8). If special symmetries are assumed for the solutions of the field equations then some of the metric components and their time derivatives may be absent but some of the algebraic R constraints may be identically satisfied. As a result, the number 0a 8 of functions characterising the most general solution compatible with some symmetry may be specified by fewer functions (or by lower-dimensional functions) than the general solution. Spatially homogeneous cosmological models will be determined by some number of independently arbitrary constants rather than spatial functions. If spatially homogeneous string cosmologies are representative of the most general inhomogeneous string cosmologies then it is necessary (although not necessarily sufficient) that they becharacterised by 8 independentarbi- trary constants. When the H field vanishes in eqns. (4)-(8), so they reduce Einstein’s equations for a free scalar field, the number of arbitrary functions is required to characterise the general inhomogeneous solution equals the number of constants required for the general homogeneous solution. ThisequivalencealsoholdsforEinstein’sequations withaperfectfluid(orinvacuum), where 8 (or 4) functions specify a general inhomogeneous solution and 8 (or 4) constants specify Bianchi types VI , VII , VIII,and IX, [15,16]. We shall now investigate the degree of generality h h of the different Bianchi type solutions of the string field equations when the H field is present. In order to determine how many free parameters are allowed in the different Bianchi models, consider the field equations, (4), for spatially homogeneous universes in the standard basis dΩ,ωα . The components of the Ricci tensor are given by [14] { } R = θ˙ θ θαβ, (43) 00 αβ − − R = 3a θγ a θ+ǫ mτβθγ, (44) 0α γ α− α γατ β R = θ˙ +θθ 2θ θγ +Γγ Γλ Γγ Γλ +Cκ Γγ , (45) αβ αβ αβ − αγ β λγ αβ − λβ αγ γβ ακ where θ = 1g , θ θα and the Ellis-MacCallum parametrization, (19), has been used to αβ 2 αβ|Ω ≡ α express the spatial curvature terms in (44) and (45). Thestringfieldequationsgive10equationsforthe6componentsofthesymmetricmetricg , αβ so there are at most 4 constraint equations. The initial data for g consist of 12 independent αβ constants: 6 g and 6 g˙ . These are reduced by (9 p+1) due to the fact that there are αβ αβ − 9 p+1 parameters of triad freedom to put the group structure constants into their canonical − Ellis-MacCallum form [14]. The parameter p is the number of independent group structure constants and 0 p 6. Their values are given below, and in Table 1, for each Bianchi group ≤ ≤ type. The number of independent constants is reduced by a further 4 r due to the constraint − equations, where r counts the number of field equations satisfied identically. Hence, a total of 12 (9 p + 1) (4 r) = p + r 2 independent constants specify the general solution − − − − − to equations (4)-(8) for spatially homogeneous universes. To calculate r we must check if any of the field equations are identically satisfied due to a particular choice of the group structure parameters a and m . From eqn. (7) it is clear that the dilaton’s contribution to the R β αβ 0α equations vanishes identically. The contribution by the H field is determined by H H cd, but 0cd α we know from (35)-(36) that H H cd = X0βγY , hence R = 0 for all Class A models. 0cd α αβγ 0α − TheequationsofmotionforH andtheconstraintstheyimposehavebeendiscussedinsection II; Table 1 gives the number of free parameters, 3 u, to specify initial data for H for each − 9 group type. The initial conditions for the dilaton φ require 2 further independent constants: φ andφ˙,whileeqn.(6) determines thedynamicsofφ. Thereforethegeneral spatially homogeneous solution(s) to equations (4)-(8) contain 3+p+r u (46) N ≡ − independent arbitrary constants. Using the constraint equations (34) and (44) we can evaluate p,r,u, and (Bianchi type) explicitly as follows (the values of these parameters are summarised N in Table 1). A. Class A models Bianchi I: R = 0, hence r = 3, p = 0, u= 0 and (I) = 6 0α N Bianchi II: R = 0,R = θ3,R = θ2, hence r = 1, p = 3, u= 1, and (II) = 6 01 02 − 1 03 1 N Bianchi VI : R = θ3, R = θ3,R = θ1 θ2, hence r = 0, p = 5, u = 2, and −1 01 − 1 02 2 03 1 − 2 (VI )= 6 −1 N Bianchi VII : R = θ3, R = θ3, R = θ1 θ2 , hence r = 0, p = 5, u = 2, and 0 01 − 2 02 1 03 2 − 1 (VII ) = 6 0 N Bianchi VIII: R = θ3 θ2,R = θ3+θ1,R = θ2 θ1, hence r = 0, p = 6, u = 3 and 01 2 − 3 02 1 3 03 − 1 − 2 (VIII)= 6 N Bianchi IX: R = θ3 θ2, R = θ1 θ3 ,R = θ2 θ1, hence r = 0, p = 6, u = 3, and 01 2 − 3 02 3 − 1 03 1 − 2 (IX) =6. N Hence, all Class A models are equally general according to the parameter-counting criterion. B. Class B models Bianchi III: R = 2θ3,R = θ3,R = θ1 θ3, hence r = 0, p = 5, u = 0, and 01 − 1 02 − 2 03 1 − 3 (III) = 8 N Bianchi IV: R = 3θ3,R = 3θ3 θ3,R = θ3+θ+θ2, hence r = 0, p = 5, u = 1, 01 − 1 02 − 2 − 1 03 − 3 1 and (IV) = 7 N Bianchi V: R = 3θ3,R = 3θ3,R = θ 3θ3, hence r = 0, p = 3, u = 1, and 01 − 1 02 − 2 03 − − 3 (V)= 5 N Bianchi VI : R = (h + 2)θ3,R = (2h + 1)θ3,R = θ1 + hθ2 (h + 1)θ3. For h6=−1 01 − 1 02 − 2 03 1 2 − 3 special choices of h, either R or R can be made to vanish identically. The two choices are 01 02 either h = 2 or h= 1, [15], so that r(h= 2) = r(h = 1/2) = 1. Therefore, we have three − −2 − − cases, (i) h = 0 :r = 0, p = 5,u = 0, and (VI )= 8, 0 N (ii) h = 0, h = 2 and h = 1 :r = 0, p = 5, u= 1, and (VI ) = 7, 6 6 − 6 −2 N h (iii) h= −2 or h = −12 :r = 1, p = 5, u = 1, and N(VIh6=−2,−21)= 8. 10