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Gauge theories with time-dependent couplings and cosmological singularities K. Narayan Chennai Mathematical Institute (CMI), Chennai [ hep-th/0602107, hep-th/0610053, Sumit Das, Jeremy Michelson, KN, Sandip Trivedi; arXiv:0711.2994, Adel Awad, Das, KN, Trivedi; arXiv:0807.1517, Awad, Das, Suresh Nampuri, KN, Trivedi.] • basic setup: AdS/CFT with cosmological singularities • gauge theories with time-dep coupling sources • Spacelike cosmological singularities, BKL etc Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.1/25 Related references: Craps, Hertog, Turok, arXiv:0712.4180 [hep-th] , Chu, Ho, hep-th/0602054, arXiv:0710.2640 [hep-th], Hertog, Horowitz, hep-th/0503071 : cosmological generalizations of AdS/CFT framework. B. Craps, S. Sethi, E. Verlinde, hep-th/0506180, and followup work by various people: Matrix theory duals of cosmological singularities. ... Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.2/25 Cosmology, time dependence, . . . Tempting to think very early Universe has deep repercussions on various aspects of physics. • Big Bang singularities, time, in string theory models? Understand spacelike, null singularities — events in time. General Relativity breaks down at singularities: curvatures, tidal forces divergent. Want “stringy” description, eventually towards smooth quantum (stringy) completion of classical spacetime geometry. Previous examples: “stringy phases” in e.g. 2-dim worldsheet (linear sigma model) descriptions (including time-dep versions, e.g. tachyon dynamics in (meta/)unstable vacua), dual gauge/Matrix theories, . . . In what follows, we’ll use the AdS/CFT framework. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.3/25 AdS/CFT and deformations Nice stringy playground: AdS/CFT. Bulk string theory on AdS5 × S 5 with dilaton (scalar) Φ = const, and metric ds2 = z12 (ηµν dxµ dxν + dz 2 ) + ds2S 5 , (Poincare coords) with 5-form field strength, dual to boundary d = 4 N =4 (large N ) SU (N ) Super Yang-Mills theory. Want: time-dependent deformations of AdS/CFT. Bulk subject to time-dependent sources classically evolves in time (thro Einstein eqns), eventually giving rise to a cosmological singularity, and breaks down. Avoid any bulk investigation near singularity. Boundary: Gauge theory dual is a sensible Hamiltonian quantum system in principle, subject to time-dependent sources. Response ? Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.4/25 AdS cosmologies Start with AdS5 × S 5 and turn on non-normalizable deformations for the metric and dilaton (also nontrivial 5-form): ds2 = 1 µ dxν (g̃ dx µν 2 z + dz 2 ) + ds2S 5 , Φ = Φ(xµ ) . This is a solution in string theory if R̃µν = 1 2 ∂µ Φ∂ν Φ , √1 −g̃ √ ∂µ ( −g̃ g̃ µν ∂ν Φ) = 0 , i.e. if it is a solution to a 4-dim Einstein-dilaton system. Time dep: Φ = Φ(t) or Φ = Φ(x+ ) . More later on cosmological solutions. General family of solutions: (Z(xm ) harmonic function) ds2 = Z −1/2 g̃µν dxµ dxν + Z 1/2 gmn dxm dxn , Φ = Φ(xµ ), gmn (xm ) is Ricci flat, and g̃µν = g̃µν (xµ ). [µ = 0123, m = 4 . . . 9.] Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.5/25 AdS cosmologies cont’d In many cases, possible to find new coordinates such that boundary metric ds24 = limz→0 z 2 ds25 is flat, at least as an expansion about the boundary (z = 0) if not exactly. These are Penrose-Brown-Henneaux (PBH) transformations: subset of bulk diffeomorphisms leaving metric invariant (in Fefferman-Graham form), acting as Weyl transformation on boundary. 1 f (x+ ) η dxµ dxν + (e µν z2 w2 f ′ −f /2 − − ze , y =x − 4 , E.g. null cosmologies ds2 = dz 2 ) , Φ(x+ ). The coord. transf. w = gives ds2 = 1 + dy − [−2dx 2 w + dx2i + 1 2 ′ )2 (dx+ )2 ] w (Φ 4 + dw2 w2 , using R++ = 21 (f ′ )2 − f ′′ = 12 (Φ′ )2 , the constraint on these solutions. Now boundary at w = 0 manifestly flat 4D Minkowski spacetime. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.6/25 Gauge theories with time-dep couplings Thus dual gauge theory lives on flat space. So sharp sub-question: Gauge theory with time-dependent coupling gY2 M = eΦ . Response? We would like to study sources that are trivial in the far past (bulk is AdS5 × S 5 ) and smoothly turn on: this means the gauge theory begins in vacuum state and is subject to Hamiltonian time evolution through this external time-dependent source. Basic expectation: time-dep source excites vacuum to higher energy state. Want to consider sources that approach eΦ → 0 at some finite point in time: e.g. gY2 M = eΦ → (−t)p , p > 0 [t < 0]. We’d specially like to understand gauge theory response near t = 0. This point in time corresponds to a singularity in the bulk: Rtt = 12 Φ̇2 ∼ t12 . Curvatures, tidal forces diverge near t = 0. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.7/25 Gauge theories, time-dep couplings Gauge theory kinetic terms e−Φ F 2 not canonical. In usual perturbation theory, we absorb the coupling gY2 M into the definition of the gauge field Aµ so that gY M appears only in the interaction terms. Let’s do something similar here. R First, consider a simpler toy scalar theory L = −e−Φ 21 (∂ X̃)2 + X̃ 4 . Redefining X̃ = eΦ/2 X gives L = −(∂X)2 − m2 (Φ)X 2 − eΦ X 4 , dropping a boundary term. The mass term is m2 (Φ) = 14 ∂µ Φ∂ µ Φ − 21 ∂µ ∂ µ Φ . Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.8/25 Null time-dependence Null cosmologies: Φ = Φ(x+ ) . No nonzero contraction so the mass term vanishes i.e. m2 (Φ) = 0. Similar story for gauge theory using lightcone gauge for convenience. Suppressing many details, but briefly, cubic/quartic interaction terms: multiplied by powers of gY M = eΦ/2 , unimportant near eΦ → 0. Thus we obtain weakly coupled Yang-Mills theory at the location in null time (x+ = 0) corresponding to the bulk singularity [e.g. eΦ = gs (−x+ )p ]. This suggests that lightcone Hamiltonian time evolution of the gauge theory is sensible. In other words, null cosmological singularities seem to be an artifact of using bad (classical bulk sugra) variables. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.9/25 Null time-dependence Is energy pumped in by null-time-dependent source ? No. Since x− -translations are symmetries, there is in fact no particle production. This suggests that continuing past singularity at x+ = 0 is OK, and late-time state is vacuum. Thus late-time bulk is AdS5 × S 5 (dual to vacuum state of N =4 gauge theory, with Φ → const for large x+ ). Bulk: since eΦ → 0 near singularity, no large gs effects. Preliminary calculations suggest stringy (α′ ) effects (beyond GR) are becoming important. In some still simpler toy models with no RR-flux, dilaton, the singularity is purely gravitational so possibly more tractable. I am studying these to understand worldsheet effects near null singularities. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.10/25 Time-dependent couplings The story is very different with time-dependent couplings. 1 2 + X̃ 4 . The redefinition (∂ X̃) 2 −(∂X)2 − m2 (Φ)X 2 − eΦ X 4 , Toy scalar theory L = −e−Φ X̃ = eΦ/2 X gives L= dropping a boundary term, the mass term being m2 (Φ) = − 41 (Φ̇)2 + 21 Φ̈ . . For gY2 M = eΦ → (−t)p , p > 0 [t < 0] , we have m2 = − p(p+2) 4 t2 Tachyonic mass term, divergent as t → 0. * X variables canonical: analysing them shows that the mass term 1 , so that extra information as X → ∞ is required: forces X ∼ tp/2 X description not good. * X̃ variables finite near t = 0: interaction terms e−Φ X̃ 4 |t∼0 large (unlike the null case). Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.11/25 Time-dep quantum mechanics In more detail: first ignore interactions, quantize quadratic theory. For a single momentum-k mode, this is time-dep quantum mechanics: R Sk = dt ( 21 Ẋ 2 − ω 2 (t)X 2 ) , ω 2 (t) = k 2 + m2 (t) −→t→−∞ ω02 . √ Generic classical solutions: X = −t [AJν (−t) + BNν (−t)] , p+1 2 . Diverge as t → 0 : i.e. generic trajectory driven to large X. p πω0 √ 1 ¨ + ω2 f = 0 , −tH as the solution of f Take f (t) = (−ω t) 0 ν 2 ν= with f → e−iω0 t , t → −∞ . Expand X = √ 1 [af (t) 2ω0 + a† f ∗ (t)] . Using the Schrodinger equation: the ground state wave-function is ψ(t, x) = √ A∗ f (t) ˙∗ e 2 i( ff ∗ ) x2 . Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.12/25 Time-dep quantum mechanics X: wave-fn ψ(t, x) = √ A∗ Wave-fn phase ∼ f (t) 1 t ˙∗ e 2 i( ff ∗ ) x2 , , “wildly” oscillating near t = 0. t → 0− : f ∼ (−t)−p . Probability density: |ψ(t, x)|2 = |A|2 |f | e ω0 x2 − |f |2 . Gaussian, width |f |2 → ∞ as t → 0. Wave packet infinitely spread out as t → 0. X variables spread out infinitely: need extra information at X ∼ ∞. X description not good. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.13/25 Time-dep quantum mechanics Original X̃ = eΦ/2 X variables better defined: finite near t = 0. √ Φ/2 X̃ = e −t[AJν (−t) + BNν (−t)] ∼t→0 tp/2 t1/2 t−ν/2 . Wave-fn, probability: A e f ∗ (t)eΦ/2 ψ(t, x̃) = √ f˙∗ x̃2 i( f ∗ + Φ̇ ) 2 2eΦ t → 0− : f ∼ (−t)−p , X̃: wave-fn phase ∼ , |ψ(t, x̃)|2 = |A|2 |f |eΦ/2 e ω0 x̃2 − 2 Φ |f | e . |f |2 eΦ ∼ const . 1 (−t)p−1 , prob. width const . p > 1: wave-fn ill-defined near t ∼ 0. “Wildly” oscillating phase. p < 1: X̃ wave fn phase regular near t ∼ 0 , |ψ(t, x̃)|2 finite. Quadratic approximation shows interactions are important near t = 0. Perturbation theory insufficient. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.14/25 Time-dep field theory wave-fn More general Schrodinger picture analysis in full field theory is possible near t = 0. R 3 −Φ 1 L = d x e ( 2 (∂t X̃)2 − 21 (∂i X̃)2 − X̃ 4 ) . Lagrangian R 3 1 δ2 −Φ Φ Field theory Hamiltonian: H = e V [X̃] + e d x(− 2 δX̃ 2 ) , R 3 1 where V [X̃] = d x ( 2 (∂i X̃)2 + X̃ 4 ) [replacing Π(x) → 1i δδX̃ ] . Schrodinger eqn: i∂t ψ[X̃(x), t] = Hψ[X̃(x), t] . Near t = 0, the potential term e−Φ V dominates in the Hamiltonian ⇒ i∂t ψ = e−Φ(t) V [X̃(x)]ψ . This gives the wave-fn (generic state) ψ[X̃(x), t] = e Phase as before ∼ (t → 0). −i( dt e−Φ(t) )V [X̃(x)] R (−t)1−p 1−p V ψ0 [X̃(x)] . [X̃(x)] . If p > 1, “wildly” oscillating Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.15/25 Energy divergence Analyzing KE terms shows they are indeed subleading near t = 0. This means energy pumped in by time-dep source diverges as R 1 −Φ hHi ≃ e hV i = (−t)p DX̃ V [X̃] |ψ0 [X̃(x)]|2 , since no time-dep in hV i. Oscillating phase cancels in |ψ0 |2 . This holds for generic states. For special states with hV i = 0 , energy may be finite (subleading KE terms do not diverge unless p > 2). Even for these special states, hH 2 i will diverge (if hV i = 0, generically hV 2 i does not vanish). Thus fluctuations non-negligible about states with hV i = 0 . Note: this is not perturbation theory. Interactions important. Diverging energy since coupling strictly vanishes near t = 0. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.16/25 The gauge theory Scalars, fermions: no dilaton coupling in KE terms. Fermion Yukawa and scalar quartic terms come with powers of gY M = eΦ/2 , vanish near t = 0. R −Φ Gauge fields: KE terms have dilaton coupling e TrF 2 . Since eΦ = (−t)p near t = 0, the gauge field terms determine the behaviour of the system near t ∼ 0. Focus on this. Consider non-interacting theory first. Convenient (Coulomb) gauge A0 = 0 , ∂j Aj = 0 (longitudinal part of gauge field time-indep from Gauss law: ∂0 (∂j Aj ) = 0 ). Residual action for two physical transverse components Ai becomes R −Φ e (∂Ai )2 , (i.e. two copies of the scalar theory earlier). Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.17/25 The gauge theory Cubic/quartic interactions: no time derivatives. Contribute only to potential energy terms (from magnetic field), not to R 3 1 i KE terms (from electric field). PE V [A (x)] = 4 d x TrFij2 . R Lg = 41 d3 x e−Φ Tr (∂t Ai )2 − Fij2 . Schrodinger quantization: E i → i ψ[A (x), t] = e 1 δ i δAi R −i( dt e−Φ )V [Ai (x)] . Then wave-fn near t = 0: ψ0 [Ai (x)] . Wave-fn phase as before: “wildly” oscillating as t → 0 (for p > 1). R −Φ Energy diverges hHi ≃ e DAi V [Ai (x)] |ψ0 [Ai ]|2 (if hV i = 6 0) . Thus if eΦ → 0 strictly, gauge theory response singular. For cutoff eΦ , large energy production due to time-dep source. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.18/25 AdS cosmologies with spacelike singularities Recall: 1 µ dxν + dz 2 ) + ds2 , Φ = Φ(xµ ) . (g̃ dx µν 2 S5 z √ 1 1 R̃µν = 2 ∂µ Φ∂ν Φ , √−g̃ ∂µ ( −g̃ g̃ µν ∂ν Φ) = 0 ds2 = Solution if: . Solutions with spacelike Big-Bang (Crunch) singularities: h i P * ds2 = z12 dz 2 − dt2 + 3i=1 t2pi (dxi )2 , √ P 2 P p ) 2(1− Φ i i , [Kasner cosmologies] e = |t| i pi = 1 . i h 2 dr 2 2 2 2 * ds2 = z12 dz 2 + | sinh(2t)|(−dt2 + 1+r 2 + r (dθ + sin θdφ )) , √ eΦ = gs | tanh t| 3 . [k = −1 (hyperbolic) FRW boundary] Dilaton bounded, approaching constant at early/late times: asymptotic spacetime is AdS5 × S 5 (using a coord transformation). The k = 0 (flat) FRW is the same as symmetric Kasner (pi = 31 ). (There is also a k = +1 (spherical) FRW solution.) Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.19/25 AdS BKL-cosmologies In fact, larger family of cosmological solutions where spatial metric is one of the homogenous spaces in the Bianchi classification: i h ds2 = z12 dz 2 − dt2 + ηab (t)(eaα dxα )(ebβ dxβ ) , eΦ = eΦ(t) . eaα dxα are a triad of 1-forms defining symmetry directions. Spatially a vanish, and R0 = 1 (∂ Φ)2 . homogenous dilaton means spatial R(a) 0 2 0 i h Bianchi-IX: ds2 = z12 dz 2 − dt2 + ηi2 (t)eiα eiβ dxα dxβ , eΦ = |t|α . Approximate Kasner-like solution ηi (t) ≃ tpi with P 2 P α2 i pi = 1 − 2 . i pi = 1 , If all pi > 0, cosmology “stable”. Else, spatial curvatures force BKL bounces between distinct Kasner regimes. With each bounce, α increases — dilaton-driven attractor-like behaviour. Attractor basin: generic Kasner-like solution with all pi > 0. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.20/25 More on AdS BKL-cosmologies Bianchi IX: symmetry algebra of Xa = eαa ∂α is SU (2). a Spatial Ricci, decomposing along triad R(a) = Rαa eαa : 1 = R(1) ∂t (η2 η3 ∂t η1 ) η1 η2 η3 − 1 2 [(η 2 2 2(η1 η2 η3 ) − η32 )2 − η14 ] = 0 , .... Say p1 < 0: then η14 ∼ t−4|p1 | non-negligible at some time. This forces metric to transit from one Kasner regime to another. As long as some pi < 0, these bounces continue as: (n+1) pi (n) = −p− (n) 1+2p− , (n+1) pj (n) = (n) p+ +2p− (n) 1+2p− , α(n+1) = αn (n) 1+2p− , for the bounce from the (n)-th to the (n + 1)-th Kasner regime. −2p− If p− < 0 , then αn+1 > αn . Also αn+1 − αn = αn ( 1+2p ), − i.e., α increases slowly for small α: attractor-like behaviour. Finite number of bounces. If all pi > 0, no bounce: cosmology “stable”. For no dilaton (α = 0), BKL bounces purely oscillatory. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.21/25 More on AdS BKL-cosmologies Parametrization: p1 = x, p2,3 = √ 2 1−x 2 ± √ 1−α2 +2x−3x2 2 . 4 2 . Solution existence forces α ≤ Lower bound: p1 ≥ 1− 4−3α 3 3 . Under bounces, α increases, window of allowed pi shrinks. Lower bound hits p1 ≥ 0 ⇒ α2 ≥ 1. Bounces stop, cosmology “stabilizes”. Attractor-like behaviour: e.g.: {p01 = x0 = 0.3, α0 = 0.001}, flows (initially slowly) to {pi > 0} after 15 oscillations (α15 = 1.0896). 9 33 5 7 19 3 5 9 E.g.: (− 15 , 35 , 35 ) → (− 21 , 21 , 21 ) → (− 11 , 11 , 11 ) → (− 15 , 35 , 53 ) → ( 31 , 13 , 31 ) . [multiple flows with same endpoint] Chaotic behaviour: 7% change to smallest exponent − 15 gives 13 9 65 2 13 39 3 2 23 1 3 17 (− 70 , 35 , 70 ) → (− 11 , 44 , 44 ) → (− 28 , 7 , 28 ) → ( 11 , 22 , 22 ) , drastically different endpoint. Note also that dilatonic (α 6= 0) [attractor-like] and non-dilatonic (α = 0) [oscillatory] flows drastically different. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.22/25 Universal behaviour near singularities Consider symmetric Kasner-like AdS BKL-cosmologies. Near singularity, spatial curvatures unimportant. Leading singular behaviour is essentially dilaton-driven, symmetric Kasner spacetime. Holographic N2 stress tensor has similar leading behaviour (Tµν ∼ t4 ). Consider families of such AdS cosmologies which are of the form of the symmetric Kasner-like solution i.e. pi = 13 : (ds23 spatial metric) √ 2 1 2 2 2 Φ ds = z 2 dz + |2t|(−dt + ds3 ) , e = |t| 3 . Ignoring subleading curvature effects, spatial metric approximately flat i.e. ds23 ∼ f lat. Then boundary metric is conformally flat, to leading order. [we’ve used a different time coordinate here.] Can use PBH transformations to recast boundary metric to be flat spacetime. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.23/25 The gauge theory Now gY2 M = eΦ √ = (−t) 3 √ (t < 0). That is, p = 3 > 1 . From earlier: wave-fn phase “wildly” oscillating, ill-defined. Energy production divergent if coupling vanishes strictly near t = 0. * In gauge theory, deform gauge coupling so that gY2 M = eΦ is small but nonzero near t = 0. Now finite but large phase oscillation, finite but large energy production. Eventual gauge theory endpoint ? Depends on details of energy production at coupling O(1). On long timescales, expect gauge theory thermalizes: then reasonable to imagine that late-time bulk is AdS-Schwarzschild black hole. Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.24/25 Conclusions, open questions * If gY2 M (t) → 0 strictly, then gauge theory response singular: energy diverges. Deform gY2 M to be small but nonzero near t = 0. Now finite but large phase oscillation and energy production. finite now, so bulk also nonsingular. Φ̇ ∼ gġYY M M Sugra may still not be valid of course. * Gauge theory on S 3 : We’re investigating this and other issues currently (in part with Archisman Ghosh, Jae Oh). * Explore AdS BKL-cosmologies/duals further ... Gauge theories with time-dependent couplings and cosmological singularities, K. Narayan, CMI – p.25/25