Stability in and of de Sitter space

Transcription

1 Stability in and of de Sitter space arxiv : (hep-th) Benjamin Shlaer Tufts Institute of Cosmology

2 Outline Review of instantons and exponential decay The asymmetric double well Field theory in flat space & finite volume Coleman - De Luccia Canonical quantization methods and the failure of the probe approximation Outlook

3 exponential decay quantum mechanical potential exhibi Ψ 0 exp( iht) Ψ 0 2 exp( Γ t ) We will calculate this decay rate big difference: Ψ 0 Ψ 0 = 1 E fv E fv = 0 V 0 There is no energy eigenstate which resembles the false vacuum. E fv x E fv = false vacuum 0 Ψ 0 1 = ground state of perturbative Hamiltonian

4 Early time behavior: Probability current flows outward.

5 Early time behavior: Probability current flows outward. animation Late time behavior: Survival probability is zero. (not pictured)

6 x0!0 instanton method S. Coleman The uses of instantons FIG. 1: A quantum mechanical potential exhibitin " V0!Ψ0 exp( iht) Ψ0 # 2 exp( Γ t ). tial decay.! " x the We can calculate the decay rate using iht is[x] x 1 [dx]e xf e x = i path integral formalism: #! " FIG. 1: A quantum mechanical potential exhibiting exponen HT SE [x] Thisxis0 e the definition of the path integral x = [dx] e. 0 2!! x0 x1 # t it = T (consistent tial decay.!ψ exp( Γ t ). pole prescription) with Feynman s 0 exp( iht) Ψ0 #! T /2 Evaluating Euclidean action for the sim $ " 1 the # 2 instanton is (x) equivalent integrating p =trivial x + V dτ We can calculatesethe decay using the to Euclidean 2rate path the Tbarrier and back. Thus the bounce actio /2 V (x) V (x) integral formalism:by # # x1 %! " S [x] S[dx] V0 ), B =e 2 E dx. 2m(V (x) (1) x0 e HT x0 = x0 and the rate obtained by of examining On decay the right: Weiswill use method steepest T On the left: Large picks Evaluating the Euclidean action for the simplest non-grou instanton correction to the perturbative descent to calculate the late time behavior of $ out the lowest energy states the low energy pstates. energy: to integrating trivial instanton is equivalent dx across

7 Assume no degeneracies in x0 e HT x 0 = H is unbounded below If we treat like an eigenstate of H, its energy has an imaginary part. Ψ 0 Actually, has offdiagonal terms which preserve unitarity. H = H pert. + large T n H pert. e (E fv+δ fv )T x 0 Ψ 0 2 (no resonances) x 0 e (H pert.+ )T n n x 0 δ fv = i 2 Γ Degeneracies mean H pert. basis has arbitrariness which may be totally destroyed by effects of. non-perturbative correction to false-vacuum energy quantum mechanical potential exhibi Ψ 0 exp( iht) Ψ 0 2 exp( Γ t )

8 S E [x] x cl Method of steepest descent S E [x cl ] δx δ2 S E δx 2 expand about the local minima of [dx]e S E[x] x cl δx xcl S E [x] e S E[x cl ] δ 2 S E δx 2 xcl = 2 τ + V (x cl (τ)) Fluctuations = measure 1 [ ] det δ2 S E δx 2 S E many minima many Gaussians e S E dx π e λx2 = 1 λ

9 S E = T/2 T/2 What are the? x cl ( 1 2ẋ2 + V (x) ) dτ x 0 [dx]e S E[x] x cl any number of bounces. 0 instantons e S E[x cl ] 1 [ ] det δ2 S E δx 2 S E [x 0 ] = V 0 T x 0 first result: E = V (x 0 ) + 2 V (x 0 ) + O( 2 ) perturbative S E [x 1 ] = V 0 T + S B x 1 The Bounce non-perturbative S E [x n ] = V 0 T + ns B n n zero modes!. negative modes! T/2 T/2 x 2 zero mode negative mode S B = 2 2 (V (x) V0 )dx x 3

10 Each zero mode contributes a large, but finite factor. T/2 T/2 S E e S E e 0τ 2 dτ = T benign negative modes Each dangerous negative mode contributes a tiny imaginary factor. Dangerous negative modes allow for exponential decay e + λ x2 dx = i 2 Sagredo forgot this factor of 1/2. T π λ Zero modes make a small effect observable at macroscopic time (extensive). When a physical quantity appears divergent, it is defined via analytic continuation. We must choose amongst the various branches by appealing to the physics. In the case at hand, the unstable state is defined as the ground state of an analytically deformed potential. In the figure above, every maxima marks a crossroad. Only the one leading to an actual divergence must be avoided.

11 S E (z) benign dangerous negative mode e + λ x2 dx = i 2 π λ red = real & positive Factor of i comes from measure dz. all extrema are saddle points in z. e S E(z) Analytic definitions correspond to integration along real sections (red contours). z benign

12 x n n widely separated bounces det [ δ 2 S E δx 2 S E [x n ] = S E [x 0 ] + ns B S B = 2 2 (V (x) V0 )dx = ]xn T n ( ) n [ i det δ 2 S E n! 2 δx 2 = ]xn T n ( ) n [ i det δ 2 ] S E n [ δ 2 ] 2 S E n! 2 δx 2 det x B δx 2 1 [ ] det δ 2 S E δx 2 x 0 = e 1 2 ωt +O( 2 )T ω = V (x 0 ) x 0 [dx]e S E[x] x cl = n=0 e S E[x cl ] 1 [ det δ2 S E e (V 0 + ω 2 +O( 2 ))T = e V 0+ ω 2 i 2 det δx 2 ] ( T n n!» δ 2 S E δx 2 [ ( i n 2) det δ 2 S E δx x B e S B! T ] n 2 x B e ns B )

13 V x At large T e (E fv+δ fv )T V 0+ ω 2 i 2 det δ» 2 S E δx 2 = e δ fv = i 2 Γ Γ = det [ δ 2 S E δx 2 ] 1 2 x B e S B 1 2 x B e S B! T 0 quantum mechanical potential exhibi Ψ 0 exp( iht) Ψ 0 2 exp( Γ t ) V 0 x 0 x 1 x

14 V (x) ground state E 0 E n small lifting compared to...zero? The asymmetric double well not a perturbation of the symmetric double well!

15 A Stable False Vacuum! E 1 1. Fix 2 boundary conditions here 3. We are left with no free parameters. All wave functions will have a growing and a decaying mode with O(1) coefficients. One of them will thus always dominate, and all solutions are primarily supported on the left, or the right. 2. Choose energy for convergence here

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24 The asymmetric double well has a stable false vacuum 0 The (LHS) perturbative vacuum is an approximate energy eigenstate when S B 1. Ψ 0 E n

25 x e iht Ψ 2 0

26 x e iht Ψ 2 0

27 The false vacuum cannot decay because there are no excited true vacuum states with overlapping energy. This is found experimentally in cavity QED. V x E Ψ0 Γ e S B E tv 1/l The true vacuum must be very large for instability to be generic.

28 E Ψ 0 (t) 2 time independent Γ e S B 0.1 stability exponential decay E E resonance Γ E 0.01 approximate exponential decay E density of states set by volume peak width Γ E set by e S B

29 The instanton formalism appears to predict exponential decay. The resolution is that the single negative mode is rendered benign by compactifying the true vacuum. φ negative mode S B convergent path integral 0 S E = T/2 T/2 ( 1 2ẋ2 + V (x) ) dτ Gaussian approximation V x 0 The double well has Euclidean action bounded below! did we even look over here? x

30 For the double well, the bounce has a benign negative mode. The lower action solutions to either side of the bounce are both finite. The contribution to the partition function from the bounce is negligible. S E benign x 0 xb xgm x global min e S E V x x B x T/2 0 T/2

31 The Schwinger model ( 1 S = 4 F D µφd µ Φ + m2 ΦΦ ) 2 vacuum electric θe field =. d 2 x + e 2π θ 2π F E E + e + m a meson m E E = F There is no photon: low energy mesons are stable (a scalar) Confinement: All quarks undergo piecewise uniform acceleration

32 Pair production (vacuum instability) Exponential decay of the false vacuum is calculated precisely as it was in quantum mechanics: E out 1,1 2 ɛ = 1 2 E in E out E in = ±e The Schwinger model S E = The Bounce = S 1 domain wall in 2 zero modes (translations in ) Negative mode corresponds to dilatations of the bubble S E = 2πrm πr 2 ɛ 2 Σ m 2 Σ ɛ ( E 2 out E 2 in) r B = m ɛ

33 Nucleation rate per unit length E e HXT E = S E = m E in E out = ±e [dσ]e S E[Σ] Σ ds + E 2 2 H = 1 E 2 2 i e E ( 4π exp πm 2 /ee ) Σ d 2 x decay rate per unit volume Γ = 1 [ ] 2 det δ 2 S E δ Σ e S B = e E ( 2π exp πm 2 /ee ) Sagredo s missing factor

34 What happens if we compactify the Schwinger model on a circle? As we dilatate the instanton on the cylinder, it will overlap itself. In the probe approximation, this is irrelevant: This can be confirmed by the method of Bogolyubov coefficients. But if we include the back-reaction on the electric field, the action is bounded below. S E = 2πrm πr 2 ɛ Γ > 0 The negative mode is actually benign Γ = 0 This can be confirmed by explicit computation of the discrete spectrum when The negative mode is still dangerous Γl 1/l

35 3+1 d QFT in flat space Integrate out the UV: only domain walls remain a.k.a. Brown - Teitelboim Exponential decay of the false vacuum is calculated precisely as it was in the Schwinger model: 1,3 4 1 S E = 2 µφ µ φ + V (φ)d 4 x 4 S E = µ d 3 x ɛ d 4 x S 3 B 4 ds 2 dτ 2 + dx 2 + dy 2 + dz 2 The Bounce = S 3 4 domain wall in 4 4 zero modes (translations in ) Negative mode corresponds to dilatations of the bubble It is dangerous. Γ e S B S B = 27π2 µ 4 2ɛ 3

36 QFT in de Sitter space ds 2 = (1 r2 l 2 )dτ 2 + (1 r2 l 2 ) 1 ( dr 2 + r 2 dω 2) false vacuum B 4 S 4 see, however, recent work by Polyakov S 3 bubble wall true vacuum B 4

37 Coleman - De Luccia action as a function of bubble radius S E Θ negative mode F 4 S CDL Π Π 2 Π Θ Gaussian approximation probe approximation 0 F 4 = const. The Coleman - De Luccia negative mode is benign in de Sitter space.

38 Caveats M P l De Sitter space is not a box: there is certainly a continuous spectrum, so our intuition about finite volumes is not useful. Just because there is a stable de Sitter false vacuum doesn t mean it is the relevant one (for, say, eternal inflation). I have argued in favor of the existence of a de Sitter invariant false vacuum. (There is no Poincare invariant false vacuum in flat space.) Formalism predicts no exponential decay in ds for any parameter range. We can be sure this breaks down at low curvature just like finite volume QFT, but explicit verification is difficult. What about the Gibbons Hawking temperature? Is that an external heat bath, making QFT in de Sitter space (static patch) non-unitary? Throughout my analysis I assume GH entropy is entanglement entropy, not an external heat bath. A toy model...

39 An example where back-reaction can be ignored: semi-bounded 1+1d Rindler space with a uniform electric field. no firm boundary between: Virtual pair production thermal activation quantum tunneling by an argument similar to Brown & Weinberg Schwinger pairs are produced at rest w.r.t. the initial conditions (instantaneously generated electric field). Even a low acceleration Rindler observer will not 2 experience vacuum decay. 3 1 E = e E = 0

40 Vacuum instability vs. Black hole instability an analogy Flat False vacuum is always unstable (neglecting gravitational back-reaction) Coleman Black holes eat up space at any nonzero temperature. Gross - Perry - Yaffe unstable Anti- de Sitter False vacuum unstable if energy difference is large enough Coleman - De Luccia Black hole instability only if temperature is large enough depends Hawking - Page De Sitter False vacuum is stable. Black holes are insignificant fluctuations Ginsparg - Perry stable

41 Conclusion At semi-classical level, de Sitter space is stable CDL instanton does not mediate decay in ds Method of Bogolyubov coefficients does not predict decay unless backreaction is ignored. de Sitter space is known to be stable against black hole nucleation This is not a question of quantum gravity, which we ignore The global picture of the Landscape is complex, but it is far from clear that the Coleman - De Luccia instanton is applicable. * I assume QFT in de Sitter space is unitary, i.e., there is no external heat bath. *