Brane annihilation in curved space-time Dan Isra el, iap From D. - PowerPoint PPT Presentation

Brane annihilation in curved space-time Dan Isra¨ el, iap From D. I. & E. Rabinovici, hep-th/0609087

D. Isra¨ el, Brane annihilation in curved space-time 1 Outline of the Talk 1. Quick review of brane reheating 2. Brane annihilation in flat space-time 3. Brane decay in AdS 4. Closed and open string emission 5. Brane decay in non-critical strings 6. Lessons for brane inflation setups

D. Isra¨ el, Brane annihilation in curved space-time 2 A popular scenario of brane inflation ✔ Natural setting of string cosmology: flux compactification of type ii string theory, with stabilized moduli R 3,1 x M 6 x d(t, ) D D ➥ generically warped throats develop 5x M 5 AdS ✔ AdS 5 geometry, capped both in the UV (compact 6-manifold) and in the IR (tip of the throat) [Giddings, Kachru, Polchinski ’03] ✔ D-brane/ anti D-brane pair in the throat: Coulombian attraction redshifted by AdS 5 metric ➥ slow-roll inflation (inflaton d ( t, x ) ) [Kachru, Kallosh, Linde, Maldacena, McAllister, Trivedi ’03]

① Brane Reheating 3 ① Brane Reheating D annihilation ➥ open string tachyon for d 2 < 8 π 2 ℓ 2 ✔ End of inflation: D- ¯ s d(t) V(T) T(t) T D D 2 2π l s ⋆ String theory realization of hybrid inflation ✔ Tachyon condensation: involves all the massive string modes ( m > 1 / ℓ s ) ➥ string corrections important ⋆ One can use exact tree-level string computations [Sen ’02] ➥ one gets a non-relativistic ”tachyon dust” of massive closed strings

① Brane Reheating 4 ✔ Reheating of the standard model [Barnaby, Burgess, Cline ’04] ⋆ Fast decay in Kaluza-Klein modes (` a la Randal-Sundrum) |ψ(φ)| φ ✔ Tunneling to the Standard Model Throat: reheating of the sm Hot gas of KK modes −> matter−dominated 1/M i tunneling Standard Model D−branes 1/M EW −>radiation−dominated ⋆ In all these computations, ℓ local ≫ ℓ s due to the gravitationnal redshift of the s d s 2 = d φ 2 + e 2 φ d x µ d x µ ℓ s ( φ 0 ) = e 2 φ 0 ℓ s AdS metric ➥

② Brane Annihilation: Flat Space-Time 5 ② Brane Annihilation: Flat Space-Time equivalent to coincident D- ¯ ✔ Decay of an unstable D-brane: D pair with no relative velocity (using ( − ) FL orbifold) ➥ solvable worldsheet string model [Sen ’02] � δS = λ d τ exp { X 0 ( τ ) / ℓ s } Wick rotation of boundary Liouville σ τ � V E � λ = ( πλ ) − iE π ✔ Couplings to closed strings (grav. sector) sinh πE ➥ time-dependent source for all closed string modes ⋆ Closed strings production (coherent state) R d E 2 E ρ ( E ) |� V E � λ | 2 [Lambert, Liu, Maldacena ’03] Number of emitted strings (tree-level) : N =

② Brane Annihilation: Flat Space-Time 6 ✔ Density of closed strings oscillators ρ ( N ) ➥ exponentially growing (cf. Hagedorn transition at high temperature) √ √ N with E = 2 ⋆ In flat space-time, ρ ( N ) ∼ N α e +4 π N/ℓ s d E E 2 α − 1 e 2 πE sinh − 2 ( πE ) � N ∼ ✔ Amplitude ➥ divergent for D0-branes ( α = 0 ) (D3-branes: instable to inhomogeneous decay) ⋆ Divergence signals breakdown of string perturbation theory ➥ Large gravitational back-reaction from the brane decay! ⋆ mass of a D0-brane m d0 ∝ 1 / ℓ s g s ➥ energy conservation not ”built-in” the (tree-level) computation ✔ One needs a uv cutoff at E ∼ m d0 ⋆ fraction of total energy in strings of mass m ∼ cst. (up to m d0 ) ➥ most energy in strings m ∼ m d0 , non-relativistic ( p ∝ 1 / ℓ s √ g s ): tachyon dust

② Brane Annihilation: Flat Space-Time 7 Sen’s Conjecture 1. The closed string description of the brane decay breaks down after t ∼ ℓ s √ g s ➥ all energy is converted into tachyon dust of massive closed strings 2. However the open string description of the process remains valid ➥ may be spoiled by open string pair production (more later) 3. The open string description is holographically dual to the closed strings description, hence is complete 4. One can use the tachyon low-energy effective action √ 2) − 1 p d d x cosh( T/ R S t = − det ( η µν + ∂ µ T ∂ ν T + · · · ) ➥ late-time ”dust” 5. Conjecture has been checked in 2D string theory

② Brane Annihilation: Flat Space-Time 8 What Should be Modified? ✔ Cosmological context: D/ ¯ D in a curved space-time (e.g. capped AdS 5 ) ➥ is the physics of the decay similar? (in string theory, uv-ir relation) ✔ In particular cancellation between asympt. density of closed string states & closed string emission amplitude may not be true anymore ⋆ In cft with minimal dimension ∆ m , ρ ( E ) ∼ exp {√ 1 − ∆ m 2 πE } → uv finite? ✔ Is the process still well-described by the curved background generalization of the open string tachyon effective action? d p +1 x √− g cosh( T 2 ) − 1 p R R − det { ( g + B + 2 πℓ 2 s F ) µν + ∂ µ T ∂ ν T } + W ( T )d T ∧ C [ p ] S t = √ ⋆ In particular, if all the brane energy is not radiated into massive closed strings, the whole picture may be challenged

③ Decay in Curved Space (I): Anti-de Sitter 9 ③ Decay in Curved Space (I): Anti-de Sitter ✔ Brane inflation setup: Approx. AdS 5 geometry ➥ However, despite recent progress AdS 5 string theory not solvable ✔ Solvable ”toy model”: three-dimensional AdS ➥ conformal field theory on the string worldsheet: Wess-Zumino Witten model for the group manifold SL(2, R ) d s 2 = ℓ 2 d ρ 2 + sinh 2 ρ d φ 2 − cosh 2 ρ d τ 2 ˜ , with a B-field B = ℓ 2 ˆ s k cosh 2 ρ d τ ∧ d φ s k t t Two types of string modes: short strings trapped in AdS (exponentially decreasing wave-functions) long strings , macroscopic solutions winding w -times around φ φ φ ✔ Unstable D0-brane of type iib superstrings in AdS 3 ×M 7 : localized at the origin ➥ decay of the brane solvable (equivalent to D- ¯ ρ = 0 (infrared) D annihilation)

③ Decay in Curved Space (I): Anti-de Sitter 10 Closed Strings Emission by the brane decay ✔ Open string sector on the D0-brane: tachyon + tower of string modes built on the identity representation of SL (2 , R ) ➥ decay described by the same boundary � � deformation as in flat space δS = λ ∂ Σ d x I × exp { k/ 2 τ ( x ) } ⋆ One gets the couplings of closed string modes to the brane, e.g. for long strings with radial momentum p ρ and winding w : � sinh 2 πp ρ sinh 2 πpρ » p 2 ρ +1 – � ∝ � � 1 � � V p ρ ,w,E � λ k 2 k | with E = kw 2 + 2 4 + N + · · · cosh 2 πρ +cos π ( E − kw ) | sinh πE w k √ ➥ also coupling to discrete states (i.e. localized strings) ⋆ Total number of emitted closed strings given by the imaginary part of the annulus one-loop amplitude, using optical theorem + open/closed channel duality �� d s 2 s Tr open e − πs H � N = Im

③ Decay in Curved Space (I): Anti-de Sitter 11 ✔ As in flat space, an important input is the asymptotic density of string states � w ➥ ρ ( E ) ∼ E α exp { 2 π 2 k ) wE } ( while |� V E �| 2 ∼ exp {− q ⋆ E ∼ 2 N (1 − 1 2 k πE } ) ⋆ Like a 2D field theory (cf. AdS 3 /CFT 2 ) ⋆ for given winding w , long strings emission is (exponentially) uv -finite! • Displacement of p ρ due to non-perturbative corrections in ℓ 2 s (worldsheet instantons) ➥ not seen in sugra limit • For large w , ¯ E ∼ kw ✔ Summation over spectral flow: N long ∼ � ∞ w =1 1 / w ➥ divergence at high energies ⋆ Needs non-perturbative uv cutoff: • w � 1 / g 2 s ( ns-ns charge conservation ) • w � 1 / g s ( energy conservation ) ⋆ On the contrary, emission of short strings (localized strings) stays finite

③ Decay in Curved Space (I): Anti-de Sitter 12 ✔ Conclusion: most of the energy converted into highly excited long strings of winding w ∼ 1 / g s , √ expanding at speed d ρ / d t ∼ 1 / ℓ s k t D−particle ⋆ Closed string emission fails to be convergent because of non-perturbative effects in α ′ = ℓ 2 s ⋆ Production of short strings negligible in the perturbative regime g s ≪ 1 (since it does not depend on the coupling constant) ✔ AdS 3 /CFT 2 correspondence string theory on AdS 3 dual to a symmetric product 2D cft ➥ dual description of tachyon decay? ⋆ Difficult since 2D cft is singular (unstable to fragmentation ↔ long strings emission)

③ Decay in Curved Space (I): Anti-de Sitter 13 Remarks on Open String Pair Production ✔ Open string point of view: time-dependent Hamiltonian ➥ pair production t + λe t + p 2 + N − 1 ∂ 2 � � Mini-superspace limit : ψ ( t ) = 0 [Gutperle, Strominger ’03] ⋆ String theory naturally ”chooses” (from Liouville theory) the | out � vacuum: √ t →−∞ e − iEt + R ( E ) e iEt (R(E): reflection coefficient) ψ ∝ H (2) λe t/ 2 ) − 2 iE (2 ∼ α E ↔ open string two-point function � e iEt ( τ ) e − iEt ( τ ′ ) � ➥ Bogolioubov coefficient γ = β E ⋆ Tension with Sen’s conjecture in flat space? d Eρ ( E ) e − 2 πE � Rate of pair production W = − Re ln � out | in � ∼ ➥ power-law convergent only (divergent for D p> 22 in bosonic strings)