Accelerating Cosmologies and Black Holes in the Dilatonic Einstein-Gauss-Bonnet (EGB) Theory

Transcription

1 Accelerating Cosmologies and Black Holes in the Dilatonic Einstein-Gauss-Bonnet (EGB) Theory Zong-Kuan Guo Fakultät für Physik, Universität Bielefeld Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November 008 / 35

2 Based on collaboration with Kazuharu Bamba, Nobuyoshi Ohta, Takashi Torii arxiv: , Prog. Theor. Phys. 8 (007) 879 arxiv: , Prog. Theor. Phys. 0 (008) 58 arxiv: , submitted to Prog. Theor. Phys. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November 008 / 35

3 Outline Introduction Accelerating cosmologies EGB theory without a dilaton EGB theory with a dilaton Conclusions and outlook 3 Black holes Asymptotically flat Black Holes Asymptotically AdS Topological Black Holes 4 Conclusions Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

4 Introduction Inflation: Some cosmological puzzles, such as the horizon problem and the flatness problem, can be explained in the inflation scenario. The most important property of inflation is that it can generate irregularities in the Universe, which may lead to the formation of structure. However, so far the nature of the inflaton has been an open question. Higher order corrections: It is known that there are correction terms of higher orders in the curvature to the lowest effective supergravity action coming from superstrings. The simplest correction is the Gauss-Bonnet (GB) term coupled to a dilaton. (We ignore other gauge fields and forms for simplicity.) Thus it is an attempt to obtain inflationary solutions in the EGB theory with a dynamical dilaton. Black holes in EGB theories have been studied much but without a dilaton! In order to understand properties of black holes in string theories, we should include a dilaton! Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

5 Accelerating cosmologies in the Dilatonic EGB theory Abstract: We study cosmological solutions in the low-energy effective heterotic string theory, which is the Einstein gravity with GB term and the dilaton. We show that the field equations are cast into an autonomous system for flat internal and external spaces, and derive all the fixed points in the system. We also examine the time evolution of the solutions and whether the solutions can give accelerating expansion of our four-dimensional space in the Einstein frame. Our action: S = κ D d D x g e φ [ R + 4( µ φ) + α R GB ], κ D : a D-dimensional gravitational constant, φ: a dilaton field, α = α /8: in terms of the Regge slope parameter, R GB = R µνρσ Rµνρσ 4 R µν Rµν + R : the GB correction. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

6 In the Einstein frame S = κ d D x g D [ R ] ( µφ) + α e γφ RGB, g µν = e 4 φ/(d ) g µν, φ = 8/(D ) φ, γ = /(D ). Metric: ds D = eu 0(t) dt + e u (t) ds p + e u (t) ds q, D = + p + q. The external p- and internal q-dimensional spaces (ds p and ds q) are chosen to be maximally symmetric with the signature of the curvature given by σ p and σ q, respectively. Field equations: F F + F = 0, Constraint Solution space ( ) ( ) F (p) f (p) + f (p) + X g (p) + g (p) + Y h (p) + h (p) Z i (p) = 0, ( ) ( ) F (q) f (q) + f (q) + Y g (q) + g (q) + X h (q) + h (q) Z i (q) = 0, F φ Z + α γe u 0 γφ R GB = 0, Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

7 where Z = φ + ( u 0 + p u + q u ) φ, F = p A p + q A q + pq u u φ, f (p) g (p) = (p ) A p + q A q + (p )q u u + φ, = (p ), g (q) = (q ), h (p) = q, h (q) f (q) = p A p + (q ) A q + p(q ) u u + φ, = p, f (p) f (q) F = α e u 0 γφ{ p 3 A p + p q A pa q + q 3 A q + 4(p qa p + pq A q + p q u u ) u u 4γ φ [ (p u + p q u )A p + (pq u + q u )A q + (p q u + pq u ) u u ] }, = α e u 0 γφ{ (p ) 4 A p + (p ) q A pa q + q 3 A q + 4 [ (p ) 3 qa p + (p )q A q + (p ) q u u ] u u +4γ φ [ ((p ) A p + q A q + (p )q u u )( u + γ φ) + ((p ) u A p + q u A q + (p )q u u ( u + u )) ]}, = α e u 0 γφ{ p 3 A p + p (q ) A pa q + (q ) 4 A q + 4 [ p (q )A p + p(q ) 3 A q + p (q ) u u ] u u +4γ φ [ (p A p + (q ) A q + p(q ) u u )( u + γ φ) + (p u A p + (q ) u A q + p(q ) u u ( u + u )) ]}, g (p) = 4(p )α e u 0 γφ[ (p ) 3 A p + q A q + (p )q u u γ((p ) u + q u ) φ ], g (q) = 4(q )α e u 0 γφ[ ] p A p + (q ) 3 A q + p(q ) u u γ(p u + (q ) u ) φ, h (p) = 4qα e u 0 γφ[ ] (p ) A p + (q ) A q + (p )(q ) u u γ((p ) u + (q ) u ) φ, h (q) = 4pα e u 0 γφ[ (p ) A p + (q ) A q + (p )(q ) u u γ((p ) u + (q ) u ) φ ], i (p) = α e u 0 γφ ] 4γ [(p ) A p + q A q + (p )q u u, i (q) = α e u 0 γφ ] 4γ [p A p + (q ) A q + p(q ) u u, Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

8 and (p m) n (p m)(p m )(p m ) (p n), A p u + σpe(u 0 u ), A q u + σqe(u 0 u ), X ü u 0 u + u, Y ü u 0 u + u. We only consider flat internal and external spaces, i.e., the cases σ p = σ q = 0; Setting p = 3 and q = 6, so γ = /; α = and u 0 = 0 can be chosen by time reparametrization; We introduce a new time variable t = e φ/4 T, i.e. dt dt = e φ/4, to remove the exponential factors of the dilaton in the field equations. Field equations an autonomous system (x = du dt, y = du dt, z = dφ dt ) Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

9 Condition for accelerating expansion Question: Is there any region where the accelerating expansion is realized in the four-dimensional Einstein frame? So ds D = e q p u ds E + eu ds q. ds E = e q p u ( dt + e u ds p) = dτ + a (τ)ds p, dτ dt = e q p u, a(τ) = e u + q p u. For p = 3, q = 6, the condition for expansion is da dτ = dt da = ( u + 3 u )e u > 0, dτ dt and the condition for acceleration is d a dτ = dt d ( ) ( u + 3 u )e u = [ü + 3ü + ( u + 3 u ) u ]e u 3u > 0. dτ dt Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

10 EGB theory without a dilaton Figure: Solution space and flow. The dots indicate fixed points. The fixed points, (0, 0), ( , 0.385) and (0.4830, 0.344): unstable. The fixed points ( , 0.344) and (0.8860, 0.385): stable. Only the last one gives accelerating expansion. The behavior of the scale factor: a τ.3 for negative τ, super-inflation. H. Ishihara, Phys. Lett. B 79 (986) 7. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

11 EGB theory with a dilaton x y Figure: Solution space and flow in the case with a dynamical dilaton. The solid (red) lines correspond to d a/dτ > 0 and the dashed (green) lines correspond to d a/dτ < 0. 0 z (x du dt, y du dt, z dφ ) = M(0, 0, 0), dt P ( , ± , ± ), P (±0.98, , ± ), P 3 (±0.6307, ±0.6307, ), Only P gives accelerating expansion. a(τ) τ.75, super-inflation Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November 008 / 35

12 Conclusions and outlook Higher order terms are important at early epoch, and they give accelerating expansion. This super-inflation eventually hits the singularity, which might be resolved by stringy effects. The solutions give enough e-folding. There is a certain range of initial conditions that leads to accelerating expansion. In this sense, these solutions may be capable of explaining naturalness of the accelerating expansion. The internal space is shrinking. There remains the problem how to realize graceful exit. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November 008 / 35

13 Asymptotically flat Black Holes Abstract: We study spherically symmetric, asymptotically flat black hole solutions in the low-energy effective heterotic string theory, which is the Einstein gravity with Gauss-Bonnet term and the dilaton, in various dimensions. We derive the field equations for suitable ansatz for general D dimensions and construct black hole solutions of various masses numerically in D = 4, 5, 6 and 0 dimensional spacetime with (D )-dimensional hypersurface with positive constant curvature. A detailed comparison with the non-dilatonic solutions is made. We also examine the thermodynamic properties of the solutions. It is found that the dilaton has significant effects on the black hole solutions, and we discuss physical consequences. Our action: S = κ D d D x [ g R ] ( µφ) + α e γφ RGB, Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

14 Line element: ( dsd = k Gm ) ( r D 3 e δ dt + k Gm ) r D 3 dr + r h ijdx i dx j, where h ij dx i dx j represents the line element of a (D )-dimensional hypersurface with constant curvature of signature k and volume Σ k for k = ±, 0. The mass function m = m(r) and the lapse function δ = δ(r) depend only on the radial coordinate r. Basic equations: m D r D 4 h 4 B r φ (D ) γφ (k B) 4 e r + (D ) 3 γe γφ B(k B)(φ γφ ) +(D ) 3 γe γφ (k B)[(D 3)k (D )B] φ = 0, r δ (D ) rh + r φ (D ) 3 γe γφ (k B)(φ γφ ) = 0, (e δ r D Bφ ) = γ(d ) 3 e γφ δ r D 4[ (k B) (D 4) 5 r + (B δ B)B 4(k B)BU( r) 4 D 4 ] (B δ B)(k B), r Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

15 where we have defined r r, m Gm α α (D 3)/, B k m r D 3, h + (D 3)e γφ[ (D 4) k B r + γφ 3B k ], r h + (D 3)e γφ[ (D 4) k B r + γφ B r ], [ k B U( r) (D 3) 4 h r B D 3 ( B r B δ ) φ [ +(D 3)e γφ (k B) k B ( B (D 4) 6 r 4 4(D 4) 5 B r 3 B δ γφ ) 4(D 4)γ k B ( r γφ + D ) φ Φ + 8 γφ {( B )( r r δ B γφ δ + r ) D 4 B } ( B + 4(D 4) r B δ ) B r 4 γ r ] ] Φ(B δ B), ( B Φ φ + B δ + D ) φ. r t ir, ds p dt, p =, σ p = 0, q = D, σ q = k, u 0 ln(k Gm/r D 3 )/, u ln(k Gm/r D 3 )/ δ, u ln r. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

16 Symmetry: φ φ φ, r e γφ r, δ δ, m e D 3 γφ m, shift the asymptotic value of the dilaton to zero; δ δ δ, t e δ t, shift the asymptotic value of δ to zero. Boundary conditions (k=): Asymptotic flatness at spatial infinity ( r ): m( r) M <, δ( r) 0, φ( r) 0. The existence of a regular horizon r H : m H = r D 3 H, δ H <, φ H <. 3 The event horizon is the outermost one and the regularity of spacetime for r > r H : m( r) < r D 3, δ( r) <, φ( r) <. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

17 Given the boundary conditions at the horizon, φ H is determined by [ Cγ (D 3) + (D 4)(3D )C + (D 4)C { (D 4) 5 + (D )(3D )γ } + (D ) 5 C 3 γ ] r H φ H [ + (D ) (D 4)C { + C (D 4) 5 C } γ { + (D 4)C} ] {(D 3) + (D 4) 5 C} r H φ H [ +(D ) C (D ) 4(D 4)C (D 4) (D + )C ] γ = 0, where we have defined C = (D 3)e γφ H r H. Only the smaller solution gives regular black holes. We choose γ = / in any dimensions. Looking at the last field equation for the dilaton, we see that it appears singular at the horizon B = 0 if we solve for φ. In order to deal with this, we expand the equations and field variables in the power series of r r H to guarantee the regularity at the horizon. From the first-order terms of r r H, we can express the second derivative φ H by φ H and φ H, and use their analytic solution for the first step of integration. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

18 Non-dilatonic black hole solutions: D = 4: the GB term is total divergence and does not give any contribution. D 5: the field equations can be integrated to yield B = m, δ = 0, r D 3 r D m = 4(D 3) 4 [ ± + 8(D 3) 4 M r D ], M: an integration constant corresponding to the asymptotic value m( ) for the plus sign, in which the Schwarzschild solutions are reduced in the α 0 limit. The M- r H relation for the black hole without the dilaton field M = [ ] D 5 r H r H + (D 3) 4. D. G. Boulware and S. Deser, Phys. Rev. Lett. 55 (985) 656 Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

19 Black hole solutions in 4 dimensions m r r r H r M The regular black hole solutions exist only for r H.476. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

20 Black hole solutions in 5 dimensions m r r r H r M The mass of the dilatonic black holes approaches a non-zero constant M = as r H 0. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

21 Black hole solutions in 6 dimensions r m r r r H M The regular black hole solutions exist for all r H > 0. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November 008 / 35

22 Black hole solutions in 0 dimensions m r r r H r M The regular black hole solutions exist for all r H > 0. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November 008 / 35

23 Thermodynamics Hawking temperature: Entropy: ( T H = e δ H 4πr H S = π [ = A H 4 Σ D 3 m H r D 4 H L R µνρσ ɛ µν ɛ ρσ, ). ] α e γφ H + (D ) 3, where Σ is the event horizon (D )-surface, ɛ µν denotes the volume element binormal to Σ, A H is the area of the event horizon. r H Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

24 Mass-temperature diagram /( ) / /( ) / /( ) / /( ) / M M M M D = 4: The GB term has the tendency to raise the temperature compared to the non-dilatonic solution. The black hole will evaporate until the solution reaches the minimum mass solution. D = 5: The sign of the heat capacity changes at M =.97607, same as the Reissner-Nordström black hole, the second order phase transition. Since the temperature becomes extremely low, the solution cannot reach the singularity with zero horizon radius. The temperature blows up at the singular solution. D 6: The behavior of the temperature is qualitatively the same as that in the non-dilatonic case. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

25 Mass-entropy diagram S/ S/( ) 3/ M M S/( ) S/( ) M M Entropy of the dilatonic black hole is always larger than that of the non-dilatonic black hole with the same mass. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

26 Remaining problems The global structures: Our numerical analysis was limited to outer spacetime of the event horizon. The global structures of the solutions such as the existence of the inner horizon and (central or branch) singularity have not been clarified. This may be done by integrating field equations inward numerically. The ambiguity of the frames: We have studied the solution in the Einstein frame. There is, however, a possibility that the properties of solutions changes drastically by transforming to the string frame. In particular, the conformal transformation may become singular. Stability: The stability of our solutions is another important subject to study. Charged solution: It would be also interesting to extend our analysis to dilatonic black holes (large and small) with charges. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

27 Asymptotically AdS Topological Black Holes Our action: S = κ D d D x g [R ] ( µφ) + α e γφ R GB Λe λφ Line element: Field equations: ds D = Be δ dt + B dr + r h ij dx i dx j [ D 3 ] D (k B) r r D 4 h B r φ γφ (k B) (D ) 4 e r + 4(D ) 3 γe γφ B(k B)(φ γφ ) +(D ) 3 γe γφ (k B)[(D 3)k (D )B] φ r λφ Λe = 0, r δ (D ) rh + r φ (D ) 3 γe γφ (k B)(φ γφ ) = 0, (e δ r D Bφ ) = γ(d ) 3 e γφ δ r D 4[ (k B) (D 4) 5 r + (B δ B)B 4(k B)BU( r) 4 D 4 (B δ ] B)(k B) + e δ r D λ Λe λφ, r Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

28 where r r, Λ = α Λ = (D ), (D m) n (D m)(d m )(D m ) (D n) α l h + (D 3)e γφ[ (D 4) k B r + γφ 3B k r h + (D 3)e γφ[ (D 4) k B r + γφ B r ], [ U( r) ( h) k B (D 3) 4 r B D 3 ( B r B k B ( B 4(D 4) 5 r 3 B +8 γφ [ ( B )( r δ B γφ δ + r Φ φ ( B + B δ + D ) φ. r ], δ ) { φ + (D 3)e γφ (k B) (D 4) 6 r 4 B δ γφ ) 4(D 4)γ k B r ( ) D 4 Symmetry and scaling: (k = 0) r ] B ( B + 4(D 4) γφ + D r B δ ) B r γ γ, λ λ, φ φ restrict γ > 0 B a B, r a r different r H φ ) Φ 4γγ Φ(B δ B) r 3 φ φ φ, Λ e (λ γ)φ Λ, B e γφ B different Λ 4 δ δ δ, t e δ t δ = 0 } Λe ], λφ B Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

29 Boundary conditions: Asymptotic AdS at spatial infinity ( r ): B( r) b r M r µ, δ( r) δ 0 + δ r σ, φ( r) φ 0 + φ r ν. The existence of a regular horizon r H : B( r H ) = 0, δ H <, φ H <. 3 The event horizon is the outermost one and the regularity of spacetime for r > r H : B( r) > 0, δ( r) <, φ( r) <. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

30 The effective potential: φ dṽeff dφ = 0, Ṽ eff = e γφ R GB + Λe λφ = (D) 3 b e γφ + Λe λφ V eff Φ Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

31 the leading term: the next leading term: If ν = σ = µ +, ν = ν ± = D If ν = σ > µ +, Asymptotic expansion: b = λ Λ [ D(D 3) ( Λ) γ ( + (D) 3 γ (D ) λ [ e φ D(D 3) 0 = ( Λ) γ ( + (D ) λ (D 4)λ Dγ (D 4)λ Dγ ) ] γ λ. ) ] γ+λ γ λ, ± 4(D) λγ(λ γ) [ (D 4)λ + Dγ ] (D ) [ (D 4) λ D γ 8(D ) λ γ ]. µ = D 3. φ φ 0 + φ r ν + φ + r ν+ +, B b r M r ν M + r ν+ M 0 r D 3 +, δ δ 0 + δ r ν + δ + r ν+ +. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

32 the Breitenlohner and Freedman (BF) bound: m BF < m < 0 m BF (D ) = b, m = (D) λγ(λ γ) [ (D 4)λ + Dγ ] b 4 (D 4) λ D γ 8(D ) λ γ Γ.5 Γ.5 Γ.5 Γ Λ Λ Λ Λ Figure: The allowed regions for D = 4, D = 5, D = 6 and D = 0. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

33 Since the ν mode is non-normalizable, we eliminate it by tuning the value of φ H. Once a solution for one r H and a fixed cosmological term is obtained, we can get solutions for different r H but with the same Λ by M 0 r D H, Given a solution for a cosmological constant, we can generate solutions for different cosmological constants but the same r H by M 0 Λ γ/(γ λ). It is convenient to define the mass function m g ( r) by g tt = Be δ b r M 0 r D 3 + b r m g( r) r D 3. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

34 (γ = /, λ = /3, φ = 0, δ 0 = 0, r H =, l = ) Φ mg r r r Φ m g r r r Φ 3.84 mg r r r Φ mg r r r 4 5 Figure: Configurations of dilaton, mass and lapse functions in D = 4, 5, 6, 0. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35

35 Conclusions The GB correction coupled with a dilaton can yield an accelerating expansion in the early universe. There exist spherically symmetric, asymptotically flat back hole solutions in the Dilatonic EGB theory. Compared with the non-dilatonic black hole solutions, the dilaton has significant effects on the solutions. 3 Asymptotically AdS topological black hole solutions with k = 0 (plane symmetric) are constructed in Dilatonic EGB theory. We hope that these results are useful for examining properties of field theories via AdS/CFT correspondence. Zong-Kuan Guo (Universität Bielefeld) Dilatonic Einstein-Gauss-Bonnet Theory 6 November / 35