PPre-big bang scenario and the WZW model
Marcel JaconMay 21, 2019
Abstract
Extensive studies of pre-big bang scenarios for Bianchi-I type universehave been made, at various approximation levels. Knowing the solutionof the equations for the post-big bang universe, the symmetries of theequations (“time reversal and scale dual transformations”) allow the studyof pre-big bang solutions. However, they exhibit singularitites in both thecurvature and the dilaton kinetic energy.Calculating the β equations for the Non-Linear Sigma model, at thefirst loop approximation and imposing conformal invariance at this level,lead to equations of motion that simply state that the curvature must benil , which in turn allows the utilization of groups to solve the β equations.This is what is done in the Weiss-Zumino-Witten (WZW) model.In this article, we will show that using the WZW model on SU ,someof the difficulties encountered in the determination of the pre and postbig-bang solutions are eliminated, leading to realistic solutions for theevolution of the universe and giving an explanation to the actually ob-served acceleration of the expansion. Extensive studies of pre-big bang scenarios for Bianchi-I type universe havebeen made at various approximation levels [1,2] and references herein. Thegravitational (massless, bosonic) sector of the string action contains not onlythe metric, but also ,at least one more fondamental field, the dilation φ .Thecorresponding tree-level action lead to cosmological equations which have beenestablished in the case where the (NS-NS) two form B µν = , but including thecontribution of perfect fluid sources.These equations are invarariant under “timereversal transformation, but also under “scale dual transformations”. Knowingthe solution of the equations for the post-big bang universe, these symmetryallow the study of pre-big bang solutions. The exact integration of the stringcosmology equations in the fully anisotropic case can be performed but lead tosolutions which exhibit singularitites in both the curvature and the dilaton ki-netic energy. The solutions associated with the pre and post-big bang branches,being disconnected by a singularity, are not appropriate to describe the wholetransition between the two regimes. Particular examples of regular solutions1 a r X i v : . [ phy s i c s . g e n - ph ] J un ay be obtained, but one may also expect that the regularisation of the bigbang singularity need also to introduce the effects of higher order loop and α (cid:48) corrections.The Polyakov action may be modified to incorporate the effect of masslessexcitation, and leads to theNon-Linear Sigma model [3,4]. Calculating the β equations for the Non-Linear Sigma (NLS) model at the first loop approximationand imposing conformal invariance at this level, leads to equations of motionthat simply state that the curvature must be nil, which in turn allows theutilization of group manifolds to solve the β equations. This is what is done inthe Weiss-Zumino-Witten (WZW) model [5]In this article, we will show how, using the WZW model on SU ,some ofthe difficulties encountered in the determination of the pre and post big-bangsolutions are eliminated, leading to realiistic solutions for the evolution of theuniverse. SU . An element of SU is parametrized by means of the well-known Euler angles ( α, β, γ ) : g = exp( ατ )exp( βτ ) exp( γτ ) where the τ i matrices are given in terms of Pauli matrices τ k = σ k i , and theEuler angles have values in the ranges 0 ≤ α <2 π , 0 ≤ β < π , 0 ≤ γ <2 π Using the expressions for the Maurer-Cartan forms [6] , one deduces the lineelement:ds =d α +d β +d γ +2d α d γ cos β which determines the g matrix g = β β (2.0.1)whose invere is g − = β − cos β β − cos β (2.0.2)From the elements of these matrix, one easily obtain the Christoffel symbolsand finally the Ricci tensors elements and the curvature scalar R = R µν g µν = (2.0.3)A space of constant curvature is a conformally flat space, so that its Weyltensor vanishes identically. 2 .1 The evolution of the universe: When studying the evolution on the universe, we work in the
S-frame represen-tation of the string effective action[2]. Assuming that B µν is vanishing and thatthe matter sources can be represented by a fluid, in the synchronous gauge ofthe comoving frame, we have: ˜ g µν = diag (1 , - ˜ a δ ij ) , ˜ a =˜ a ( t ) , ˜ φ = ˜ φ ( t )˜ T νµ =diag( ˜ ρ − ˜ pδ ij ) , ˜ ρ = ˜ ρ ( t ) , ˜ p = ˜ p ( t ) , ˜ σ = ˜ σ ( t ) These definitions and the use of the cosmic time as the time coordinate leadsto the well-known Roberson-Walker metric ds = dt − ˜ a ( t ) (cid:2) ( dr / − Kr ) + r ( dθ + sin ϑdφ ) (cid:3) In this formula, K has dimension L − and ˜ a (t) is dimensionless.In order to preserves the SU character of the 3-dimensional spatial subgroupand keep its interesting symmetry properties, such as a constant curvature ( i.e.independent of the spatial coordinates), for a fixed t, we will work in the EulerAngles frame which is described now.We look for a coordinate system ( ξ, η, ζ ) which diagonalizes the matrix diag(-a ,-a ,-a ) × g (2.1.1)where a depends on t .One has the following transformation: √ − √ √ √ : − a − a cos β − a − a cos β − a √ √ − √ √ = − a (1 − cos β ) 0 00 − a
00 0 − a (1 + cos β ) From which it follows: − a ( dα + dβ + dγ + 2 cos βdαdγ ) = − a ((1 − cos β ) dξ + dη + (1 + cos β ) dζ ) with (cid:2) dξ = ( dα − dγ ) / √ , dη = dβ, dζ = ( dα + dγ ) / √ (cid:3) The 4-dimensional line element is (cid:2) ds = dt − a ((1 − cos β ) dξ + dη + (1 + cos β ) dζ ) (cid:3) from which one sees that the scale factor a has the dimension L In the S-frame, fixing r = r , an arbitrary value, gives ds = dt − ˜ a ( t ) (cid:2) r ( dθ + sin θdφ ) (cid:3) .On the other hand, defining dθ = dη and sin θdφ = 2(sin ( β/ dξ +cos ( β/ dζ ) gives the expected result a = ˜ a ( t ) × r .32.1.2)In this coordinate system, we will see that the curvature scalar is given bythe following constant, generalizing (2.0.3): R=-(3/ a ) (2.1.3)As a consequence, the β -equations for a fixed t at the first loop in stringperturbation theory, imposing that the theory is conformal invariant at thisorder, are satisfied . (2.1.4)We study the graviton-dilaton system , setting B µν = 0 , but including perfectfluid sources. Therefore φ = φ (t) ,T νµ = diag ( ρ, − p i δ ji ) , ρ = ρ ( t ) , p i = p i ( t ) , σ = σ ( t ) , whereT µν represente the tenseur current density of the matter sources and σ thescalar charge density.We first calculate the components of the Christoffell connection Γ , and de-duce the components of the Ricci tensor:From Γ kij = g kl ( ∂ i g jl + ∂ j g il - ∂ l g ij )we deduce Γ i i = ˙ a i a i = H i ; Γ ii = ˙ a i a i ; Γ = Γ = ∂ a a = a a ; Γ = Γ = ∂ a a = − a a : Γ = − a a ∂ a = − a a a ; Γ = − a a ∂ a = a a a One obtains the following expressions, which apart from a constant, aresimilar to those calculated by M. Gasperini in the
S-frame (see Eqns. 4.10 of[2]) R =- (cid:80) i ( ˙ H i + H i ),R ii =- ˙ H i - H i ( (cid:80) k H k )-1/2a (2.1.5)The associated scalar curvature is: R =- (cid:80) i (2 ˙ H i + H i )-( (cid:80) i H i ) -3/2 a showing the above result (Eqn 2.1.3) for the three dimensional spatial man-ifold. (2.1.6)4e retain also the following equations for the dilaton field:( ∇ φ ) = ˙ φ , ∇ φ = ¨ φ + ˙ φ (cid:80) i H i , ∇ ∇ φ = ¨ φ , ∇ i ∇ j φ = ˙ φH i δ ij We have also the following relations between the quantities in the E and S frames: H = H = H = ˙ aa = ˙˜ a ˜ a = ˜ H So the equations for the theory are the same, in the S frame or in the Eframe, except the remaining constant in the R ii .it follows that we can use the results already obtained by M. Gasperini inthe S-frame. (2.1.7)Setting λ sd − =1 ,we have the Euler-Lagrange equation for the dilaton equa-tion (analogous of Eqn 2.24 of [2]. ¨ φ - ˙ φ +2 ˙ φ (cid:80) i H i - (cid:80) i (2 ˙ H i + H i )-( (cid:80) i H i ) -3/2a + V - ∂V∂φ = e φ σ (2.1.8)The ( , component of (Eqn.2.24) in [2] gives ˙ φ -2 ˙ φ (cid:80) i H i +( (cid:80) i H i ) - (cid:80) i ( H i ) +(3/2 a )- V = e φ (cid:37) (2.1.9)While the diagonal part of the space component gives: ˙ H i - H i ( ˙ φ - (cid:80) k H k )+ ∂V∂φ - (1 / a )= e φ ( p i − σ ) (2.1.10)These equations are simplified, introducing the “shifted dilaton variable “ ¯ φ ¯ φ = φ - ln ( a ), ˙¯ φ = ˙ φ - (cid:80) i H i (2.1.11)and the shifted variables for the fluid: ¯ (cid:37) = ρ ( a ), ¯ p = p ( a ), ¯ σ = σ ( a )then, we obtain the analogous of Eqns 4.39,4.40,4.41 in [2]: ˙¯ φ - (cid:80) i ( H i )- a - V = ¯ ρe ¯ φ ˙ H i - H i ˙¯ φ + ∂V∂ ¯ φ - a = e ¯ φ ( ¯ p i - ¯ σ )5 ¨¯ φ - ˙¯ φ - (cid:80) i ( H i )- a + V - ∂V∂ ¯ φ = ¯12 ¯ σe ¯ φ (2.1.12)From these equations, we deduce the following conservation equation: ˙¯ ρ + (cid:80) i H i p i = σ ( ˙¯ φ + (cid:80) i H i ) + e ¯ − φ (cid:104) ∂V∂ ¯ φ (cid:80) i H i + a ¯ φ − a (cid:80) i H i − ddt a (cid:105) However V is not a scalar under general coordinates transformations, andit is impossible to define a potential which can be directly inserted as a scalarinto the covariant action. However, it has been shown that the action and thecorresponding equations of motion can be written in a generalized form whichis invariant under general coordinates transformations using for the potential anon-local variable..The result of the calculations is that the second Eqn. (2.1.12) is replacedby the simpler one (2.1.13) [2,7]: ˙ H i - H i ˙¯ φ - a = e ¯ φ ¯ p i (2.1.13)leading to the modified conservation equation: ˙¯ ρ + (cid:80) i H i p i = σ ˙¯ φ + e ¯ − φ (cid:104) ˙¯ φ a − a (cid:80) i H i − ddt a (cid:105) as well as the Eqn for the Dilaton field ¯ φ ( t ) : (Eqn 4A.27 of ref.2) ¨ φ + (cid:80) i H i ¯ φ - ˙ φ + λ d − s ( V − ∂V∂φ ) + λ d − s e φ ( ρ − (cid:80) i p i )+ ( d − ) λ d − s e φ σ = (2.1.14)Using these equations, we can now study models for the evolution of theUniverse. Instead of introducing a time parameter x as in [2] and get a full analyticalresult, we calculate a numerical solution of the differential equations of motion[ ? ]. However the relations between both methods are easily found. It has beenshown by M. Gasperini and Veneziano [2,7] that the cosmological equationsare rigorously solved, in the isotropic case and for the vacuum ( T µν = 0 = σ )by the solutions A ( t ) = A (cid:34) tt + (cid:18) t t (cid:19) / (cid:35) / √ d (2.2.1)6nd ¯ φ = −
12 ln (cid:20)(cid:112) V t (cid:18) t t (cid:19)(cid:21) Our equations ( ?? ) are slightly different with the presence of the factors a , ,leading to the following conditions, which must relate the scale factor to thesources properties: a = − e ¯ φ ¯ σ = − e ¯ φ ¯ p = − e ¯ φ ¯ ρ = ⇒ ¯ ρ = 3¯ p, ¯ σ = 2¯ ρ (2.2.2)This shows that the solutions (2.2.2) are even valid in the presence of matter,provided the conditions [2] γ = 1 / , γ = 2 are satisfied, i.e. in the case of apure radiation field :this is an improvement over previous results.The curves representing H ( t ) and ¯ φ are similar to those of [2] , (Fig. 4.7)with a bell-like shape for the curvature and the dilaton kinetic energy. Theyshow a pre-big bang inflationary evolution, followed by a decelerated expansion.These solutions exhibit no possible acceleration of the universe expansion,which is actually observed. This demonstrates that such an acceleration is dueto the presence of another kind of field, such as dark energy.The most recent experimental results on the observation of an accelerationof the actual expansion of the universe is found in [ ? ]. From the results pub-lished in this report, we can perform a fitting of the continuous curve of ˙˜ A ( t ) represented on Fig.21 of [ ? ]. For that, we must first numerically solve the firstorder differential equation which determine ˜ A ( t ) :˙˜ A ( t ) ˙= ˜ A ( t ) × H (Ω Λ + Ω M (1 + z ) + (1 − Ω Λ − Ω M )(1 + z )) / (2.2.3)Performing this operation is not easy, because this is a stiff equation whichmust be solved by efficient algorithms. This is done using the Mathematicapackage [ ? ]. Concerning the variables, we will use either t or the redshift pa-rameter z , which taking arbitrarily ˜˜ A (0) = 1 , are related by z ( t ) = 1 y ( t ) , y ( t ) = ˜ A ( t ) (2.2.4)A numerical fitting of z ( t ) Taking arbitrarily ˜ A (0) = 1 , is: / ˜ A ( t ) = 1 . − . t + 3740 . t − . t − . × t − . × t + 8 . × t +1 . × t − . × t − . × t − . (cid:54) t (cid:53) . , t = 0 being the present epoch. In factthe unit for t is very large: relatively to the present epoch, (cid:52) z = 0 . corresponds to (cid:52) t = 0 . s. If we consider that the expansion of the universebegan to accelerate at (cid:52) T = 5 M y (cosmic time)[ ? ], it follows (cid:52) T / (cid:52) t = 2 . × . So the time interval between t = − . and t = 0 . is (cid:52) T = 14 . M y and covers the whole life of the universe. ˜ A ( t ) is an increasing function of t. The curve ˙˜ A ( t ) , when expressed in termsof the redshift parameter z ( t ) , reproduces the continuous curve of Fig. 20 [ ? ].The equation which determines ¯ φ ( t ) is deduced from ( ?? ): ¨¯ φ + (cid:16) − γ (cid:17) γ H ˙¯ φ − ∂V∂ ¯ φ = dH + (cid:16) − γ (cid:17) γ ˙ H + 12 a (cid:20) − γ (cid:16) − γ (cid:17)(cid:21) (2.2.6)Now, the solutions that we have adopted for A ( t ) , ¯ φ ( t ) , V ( ¯ φ ) are not rig-orous solutions of ( ?? ). To get a rigorous solution, we must add a correction δV to the potential to compensate for the other components of the cosmic field,such that − a − δV = e ¯ φ (¯ ρ − ¯ ρ r ) − a = 12 e ¯ φ (¯ p − ¯ p r ) = 12 e ¯ φ γ (¯ ρ − ¯ ρ r ) − a + δV − ∂∂ ¯ φ ( δV ) = 12 e ¯ φ (¯ σ − ¯ σ r ) (2.2.7)From which results: − a = 2 e ¯ φ γ (¯ ρ − ¯ ρ r ) δV = e ¯ φ (3 γ − ρ − ¯ ρ r ) ∂∂ ¯ φ ( δV ) = e ¯ φ (cid:16) γ − − γ (cid:17) (¯ ρ − ¯ ρ r ) (2.2.8)It follows that Eqn ( ?? ) is now ¨¯ φ + (cid:16) − γ (cid:17) γ H ˙¯ φ = dH + (cid:16) − γ (cid:17) γ ˙ H − e ¯ φ γ ρ − ¯ ρ r ) = dH + (cid:16) − γ (cid:17) γ ˙ H + 14 a γ γ with 8 igure 1: The density ρ as a function of zγ = 2 , γ = 13 (2.2.9)With these values for the two constant, Eqn ( ?? ) takes the remarkable simpleform ¨¯ φ = dH + 14 a ( γ /γ ) (2.2.10)We have the following values for the density [ ? ]: ρ = Ω M (1 + z ) + Ω Λ + (1 − Ω Λ − Ω M )(1 + z ) A plot of the density as a function of z is given in Fig. 2.We make a numerical integration of Eqn. ( ?? ), we need to evaluate H ( t ) and ˙ H ( t ) which are given by the following expressions: H ( t ) = ˙ a ( t ) a ( t ) , ˙ H ( t ) = (cid:18) ¨ a ( t ) a ( t ) (cid:19) − (cid:18) ˙ a ( t ) a ( t ) (cid:19) / ˜ A ( t ) being given by ( ?? ).We have also: H ( z ) = 69 . . × z + 22 . × z − . × z We obtain the curve given in Fig. 2.2, showing the evolution of ¯ φ ( t ) with theinitial condition ¯ φ ( t = − . s ) = 0 , fixing the beginning of an acceleratingexpansion at T = − M y. ¯ φ ( t ) , correlating to the evolution of the expansion,is decreasing during the decelerating expansion, and then increasing during theaccelerating expansion phase. Matching of the solutions:
To get a full description of the universe evolu-tion, we must first propagate Gasperini solution as soon as the radiation fielddominates,- as we have seen that the bell-shaped GV-solution is still valid. Forlater time, we must use the Λ CDM solution , matching both at the junctionpoint.This is done in the following way.The GV-solution (Gasperini-Veneziano) is9 igure 2: The dilaton ¯ φ ( t ) igure 3: The global solution for the dilaton field ¯ φ ( t ) . H ( t ) = (cid:0) d × ( t + t ) (cid:1) − . , ˙¯ φ ( t ) = − t/ ( t + t ) Matching H (Λ CDM ) and H(GV) at t = − . fixes t = 0 . . Itresults that ˙¯ φ ( t = − . . . Taking ¯ φ ( − . we get the curvegiven in Fig. 3, showing the increasing of ¯ φ ( t ) . In this article, I have developped models for the evolution of the universe, us-ing the Weiss-Zumino-Witten method on the SU group. Evolution equationsfor the fields variables have been established in the simplest case where thegraviton-dilaton system is described by the dilaton field φ ( t ) . Solution arenumerically computed, taking care of the stiffness of the equations . These solu-tions describe well the actual acceleration of the expansion which, according tothe most present measurements began around 5 billion years ago. So this sim-ple models show the ability of string theory, to describe the actually observedevolution of the universe, giving an interpretation in terms of the dilaton field.11 eferences [1] M.Gasperini and G. Veneziano, String Theory and Pre-Big bangCosmology. Contribution to the book: Beyond the Big Bang, editedby Ruediger Vaas (Frontier Collection Series, Springer Verlag, Hei-delberg, 2007)[2] M. Gasperini, Elements of String Cosmology (Cambridge Univer-sity Press, Cambridge 2007)[3] E.S. Fradkin , A.A. Tseytlin: Quantum String Theory effectiveaction , Nucl. Phys. B , 1 (1985)[4] C.G. Callen, D. Freedan, E.I.Martinec, M.J. Perry: String in back-ground fields, Nucl. Phys. B , 593 (1985)[5] E. Witten: Non-Abelian bosonization in two dimensions, Comm.Math. Phys.93