Cosmological solutions of Double field theory

Transcription

1 Cosmological solutions of Double field theory Haitang Yang Center for Theoretical Physics Sichuan University USTC, Oct / 28

2 Outlines 1 Quick review of double field theory 2 Duality Symmetries in String Cosmology 3 Relations between string cosmology and DFT 4 Definitions, notations and computational rules 5 Cosmological solutions 6 Comments and discussions 7 References Joint work with Houwen Wu, arxiv:[ ] 2 / 28

3 Motivation T-duality: R α R, np nw Conjugation pairs: p i x i w i x i Closed string fields are dependent on the doubled coordinates. φ ( x i φ ( x i, x j The action of CSFT ought to be written in doubled form formally: S = dxd xl (φ i (x, x Currently, only consider the massless sector g ij (x, x, b ij (x, x, d (x, x 3 / 28

4 Motivation Double the fluctuation field h ij (x g ij (x η ij to h ij (x, x (Verified from CSFT: ( 2κ 2 S = [ 1 [dxd x] 4 hij 2 h ij 1 4 h 2 h + 1 ( i 2 1 h ij h i j h ij ( ] 2 1 i h ij + 2 h i j h ij 1 4 hij 2 h ij 1 4 h 2 h Double diffeomorphisms of h ij linearised gauge transformations δh ij = i ɛ j + j ɛ i + i ɛ j + j ɛ i Invariance requires an additional Kalb-Ramond field and a dilaton. linearised gauge transformations ( δb ij = i ɛ j j ɛ i ( i ɛ j j ɛ i δd = 1 2 ɛ + 1 ɛ 2 4 / 28

5 Motivation The fields are constrained: weak constraint(level matching : strong constraint(gauge invariance : x k φ = 0 x k x k (φ i φ j = 0 x k Define O (D, D vectors ( ( X M x i i =, x M = i i and generalized metric ( g ij g H MN = ik b kj b ik g kj g ij b ik g kl b lj 5 / 28

6 Spacetime action O (D, D matrices h satisfying ( a b h = c d ( O (D, D, h 1 ηh = η, with η = 1 0 The O (D, D invariant spacetime action S = ( 1 dxd xe 2d 8 HMN M H KL N H KL 1 2 HMN N H KL L H MK M d N H MN + 4H MN M d N d The O (D, D scalar dilaton d is defined by diffeomorphism scalar dilaton Φ, e 2d = ge 2Φ To obtain the EOM, vary δ d S, δ H S 6 / 28

7 Equations of motion However, δ H S = dxd xe 2d δh MN K MN K MN = 0 is not the field equation, since H = H + δh must satisfy the condition H ηh = η 1. The field equation is R MN = 1 2 K MN 1 2 SP M K P QS Q N = 0 where S M N HM N = ηmp H P N = H MP η P N 7 / 28

8 Equations of motion The dilaton EOM: 1 8 HMN M H KL N H KL 1 2 HMN M H KL K H NL M N H MN 4H MN M d N d + 4 M H MN N d + 4H MN M N d = 0 The graviton EOM: where R MN = 1 2 K MN 1 2 SP M K P QS Q N K MN 1 8 M H KL N H KL 1 4 ( L 2 L d (H LK K H MN + 2 M N d 1 2 (N H KL L H MK ( L 2 L d H KL (N H MK ( L 2 L d H K (M KH L N 8 / 28

9 String Cosmology The tree-level action (no Kalb-Ramond S = d D x [ ge 2φ R + 4 ( µφ 2] EOM R µν + 2 µ νφ = 0 2 φ 2 ( µφ 2 = 0 FRW metric form ds 2 = dt 2 + a (t 2 δ ij dx i dx j Component form of EOM φ 2 φ 2 + (D 1 H φ = 0 2 φ (D 1 (Ḣ + H 2 = 0 Ḣ + (D 1 H 2 2H φ = 0 9 / 28

10 String Cosmology Define shifted dilaton, ψ = 2φ (D 1 ln a, the EOM are simplified 2 ψ ψ 2 (D 1 H 2 = 0 (D 1 H 2 + ψ = 0 Ḣ ψh = 0 These EOM are invariant under two transformations: Scale factor duality ã = a 1, H = H, ψ = 2 φ (D 1 ln a 1 = ψ Time reversal, t t ψ ψ, ψ ψ, H H, Ḣ Ḣ 10 / 28

11 String Cosmology The solutions are ( t 1/ D 1 ( t 1 : a (t =, ψ = ln ( t 1/ D 1 ( t 2 : ã (t =, ψ = ln ( 3 : a ( t = t 1/ D 1 (, ψ = ln t ( 4 : ã ( t = t 1/ D 1 (, ψ = ln t where 1 ȧ (t > 0, expansion decelerated decreasing curvature post-big bang 2 ã (t < 0, contraction decelerated decreasing curvature post-big bang 3 ȧ ( t < 0, contraction accelerated increasing curvature pre-big bang 4 ã ( t > 0, expansion accelerated increasing curvature pre-big bang 11 / 28

12 Symmetry relations between string cosmology and DFT Scale factor duality as a special realization of O(D 1, D 1 The low energy effective action S = d D x ge 2φ [ R + 4 ( µφ H ijkh ijk under synchronous gauge g tt = 1 and g ti = b tµ = 0, can be recast to a manifestly O(D 1, D 1 invariant form with S O = [ 1 dte (ṀṀ 2d 8 Tr 1 ] + 4d 2, ( G 1 G M = 1 B BG 1 G BG 1 B and G and B are spatial parts of g ij (t and b ij (t., ], 12 / 28

13 Symmetry relations between string cosmology and DFT S O is invariant under the O (D 1, D 1 transformations with d d, M M = Ω T MΩ. Ω T ηω = η. When choose G ij = δ ij a 2 (t, B = 0, ( G 1 0 M = 0 G The O (D 1, D 1 transformation, is precisely the scale factor duality:. ( G 0 M = 0 G 1, a(t a(t / 28

14 Symmetry relations between string cosmology and DFT Questions: DFT was originally inspired by T-duality, which requires compactifications. However, in string cosmology, no compactification is necessary. In DFT, O (D, D rotates x and x. Whereas in string cosmology, one has only x. Where does the O (D 1, D 1 come from? 14 / 28

15 Symmetry relations between string cosmology and DFT Answers and conjectures: In DFT, O (D, D can formally be a continuous symmetry. After compactifying d dimensions, O (D, D O(D d, D d O(d, d, Z = O(D d, D d T d Scale factor duality is not T-duality! O (D, D is more fundamental. Under certain backgrounds (one coordinate dependence, EOM of barred and unbarred systems are indentical, while related by O (D, D. Even if unbarred system removed, barred one itself still has dual solutions. (Conjecture Is there a hidden symmetry in the low energy effective action? 15 / 28

16 Definitions, notations and computational rules We define the dual vector ( dx M d xi dx i rewritten the vector, dual vector and generalized metric ( 1 M =, dx M = 2 ( dx 1 dx 2, H MN = ( H11 H 12 H 21 H 22 The components of generalized metric can be divided into four parts, each of them defines the metric of space and its dual space. Metric H 11 ( x i, H 21 ( x i, x j = g ij, H 12 ( x i, x j ( = b ik g kj, H 22 x i, x j x j = g ik b kj = g ij b ik g kl b lj 16 / 28

17 Definitions, notations and computational rules There are two contractions: 1. H MN H MN = H 11 H 11 + H 12 H 12 + H 21 H 21 + H 22 H H 1(i1(k H 1(k1(j = g ik g kj The generalized line element becomes Line element ds 2 = H MN dx M dx N = H 11 dx 1 dx 1 + H 12 dx 1 dx 2 + H 21 dx 2 dx 1 + H 22 dx 2 dx 2 = g ij d x i d x j g ik b kj d x i dx j + b ik g kj dx i d x j + (g ij b ik g kl b lj dx i dx j 17 / 28

18 Cosmological solutions Our FRW-like metric ansatz is: ds 2 = g ij d x i d x j + g ij dx i dx j = d t 2 + ã ( t, t 2 δ ij d x i d x j dt 2 + a ( t, t 2 δij dx i dx j The relationship between the three dilatons for the metric ansatz is Relationship between the three dilatons d = φ D 1 2 ln a = φ + D 1 2 ln a Thus 2d is the shifted dilaton in string cosmology. φ is the scal-factor dual dilaton in string cosmology. 18 / 28

19 Cosmological solutions The dilaton EOM is ( (D 1 H 2 4 (D 1 H φ + 4 φ2 + 2 (D 1 H + (D 1 H2 4 φ + ((D 1 H 2 4 (D 1 H φ + 4 φ (D 1 Ḣ + (D 1 H2 4 φ = 0 Non-vanishing graviton EOM R 2(t2(t = (D 1 (Ḣ + H φ ( + (D 1 H + H2 2 φ R 2(i2(i = 1 ( ã 2 H (D 1 H2 + 2 φ H a 2 Ḣ (D 1 H2 + 2 φh with Hubble parameters defined as H = ȧ a and H = ã ã. Two constraints: ã(t, t = a 1 (t, t, φ = φ + (D 1 ln a 19 / 28

20 Cosmological solutions The equations become Equations of motion ( ( dt 2 ( d 4d 2 (D 1 H 2 + 4d d2 t d t d t 2 = 0 ( ( dt 2 ( 1 (D 1 H d + 2d d2 t d t 2 = 0 ( ( dt d t d t (Ḣ 2 dh + H d2 t d t 2 = 0 The special solutions can be achieved by where C is a constant. dt d t = C 20 / 28

21 Strong constraint solutions (C = 0 Strong constraint solutions (C = 0 C = 0 implies that t and t are independent. Then there are four solutions: ( t ±1/ D 1 a ± (t = ã ± (t 1 =, d(t = 1 2 ln t, t > 0, ( t ±1/ D 1 a ± ( t = ã ± ( t 1 =, d( t = 1 t ln, t < 0, 2 Strong constraint is respected ( = 0 a + (t/a (t represents a decelerated expanding/contracting post-big bang with decreasing curvature. a + ( t/a ( t represents an accelerated contracting/expanding pre-big bang with increasing curvature. Note the argument of ã(t is t other than t. 21 / 28

22 Strong constraint solutions (C = 0 There are two interpretations: Expanding/contracting universes The line element can take the form ( ds 2 = dt 2 + a ± (t 2 δ ij dx i dx j + a (t 2 δ ij d x i d x j d t 2, t > 0. Features: 1 No fields depend on t = d t 2 leads no physical consequence and can be neglected. 2 (D 1-dimensional decelerated expanding space accompanied with (D 1-dimensional decelerated contracting space. 3 The curvatures always decrease = post-big bang scenario. 4 Pre-big bang description is achieved by t t. Applications(? 1 Explain our current accelerated expanding universe. 2 Investigate the physics of extra dimensions. 22 / 28

23 Strong constraint solutions (C = 0 Pre-big bang and post-big bang solution Cover the whole spacetime in one single line element ds 2 = dt 2 + a + (t 2 δ ij dx i dx j θ(t + a ( t 2 δ ij d x i d x j θ( t. Properties: 1 d t 2 leads no physical consequence and was abandoned. 2 Naturally unifies the pre- and post-big bangs altogether in one single line element. 3 Possible resolution of cosmic amnesia when Kalb-Ramond and matter are included (dxd x crossing terms appear. 4 Investigate the phase transition at t = / 28

24 Constraint violation solutions (C 0 Constraint violation solutions (C 0 For nonvanishing C, we can scale t to set t = ±t. Four pairs of T-dual solutions are obtained { } { } { } { } { } a(t, ã( t = a + (t, a (t, a (t, a + (t, a (t, a + ( t, a + (t, a ( t, and d(t = 1 ( t 2 ln, Our current universe selects the pair ( 1/ a (t = a + (t = t ã ( t ( 1/ = a ( t = t Since t = t, the complete line element is D 1, d(t = 1 D 1, d( t = 1 ( 2 ln t, t > 0 ( 2 ln t, t < 0. ds 2 = dt 2 + a + (t 2 δ ij dx i dx j θ(t + a ( t 2 δ ij d x i d x j θ( t. This solution differs from the strong constraint one only by the abandoned physically irrelevant extra time d t 2, and therefore has similar physical interpretation. and two others by replacing t t. 24 / 28

25 Cosmological solutions Our current universe selects ( 1/ D 1 a (t = t t, d = 1 0 ã ( t ( 1/ D 1 = t t, d = ln ( t ( 2 ln t Since t > 0 and t = t < 0, ã( t represents accelerated, curvature increasing expansion pre-big bang. a(t represents decelerated, curvature decreasing expansion post-big bang. The full line element naturally unify the pre- and post-big bang scenarios: ds 2 = dt 2 +a(t 2 dx i dx i (as t > 0+a( t 2 d x i d x i (as t < 0+(terms around t = 0 25 / 28

26 Comments and discussions In string cosmology, the four solutions can be selected or grouped arbitrarily. In our results, the physical one is unique and does predict a pre-big bang scenario. cosmic amnesia: pre-big bang leaves no footprint. However, once including Kalb-Ramond, from our current progress, two stages have interplay and evidence of the existence of pre-big bang may arise. Introducing matter sources may remove the singularity around t = 0. How to relax the level match induced constraint? dimension reduction... If not, how to interpret the solution more physically? Other non-trivial solutions? 26 / 28

27 References B. Zwiebach, Glimpses of Double Field Theory Geometry, Strings 2012, Munich. C. Hull and B. Zwiebach, Double Field Theory, JHEP 0909, 099 (2009 [arxiv: [hep-th]]. C. Hull and B. Zwiebach, The Gauge algebra of double field theory and Courant brackets, JHEP 0909, 090 (2009 [arxiv: [hep-th]]. O. Hohm, C. Hull and B. Zwiebach, Background independent action for double field theory, JHEP 1007, 016 (2010 [arxiv: [hep-th]]. O. Hohm, C. Hull and B. Zwiebach, Generalized metric formulation of double field theory, JHEP 1008, 008 (2010 [arxiv: [hep-th]]. B. Zwiebach, Double Field Theory, T-Duality, and Courant Brackets, arxiv: [hep-th]. O. Hohm, S. K. Kwak and B. Zwiebach, Unification of Type II Strings and T-duality, Phys. Rev. Lett. 107, (2011 [arxiv: [hep-th]]. O. Hohm, S. K. Kwak and B. Zwiebach, Double Field Theory of Type II Strings, JHEP 1109, 013 (2011 [arxiv: [hep-th]]. 27 / 28

28 Thank you! Author: Address: Haitang Yang Center for Theoretical Physics Sichuan University Sichuan, , China 28 / 28