Bispectrum from open inflation

Transcription

1 Bispectrum from open inflation φ φ Kazuyuki Sugimura (YITP, Kyoto University) Y TP YUKAWA INSTITUTE FOR THEORETICAL PHYSICS K. S., E. Komatsu, accepted by JCAP, arxiv:

2 Bispectrum from a inflation model with quantum tunneling before slow-roll era φ φ Kazuyuki Sugimura (YITP, Kyoto University) Y TP YUKAWA INSTITUTE FOR THEORETICAL PHYSICS K. S., E. Komatsu, accepted by JCAP, arxiv:

3 Contents Introduction In-in formalism on tunneling background Bispectrum from open inflation Conclusion

4 Contents Introduction open inflation, string landscape, non-gaussianity, non-bunch Davies vacuum In-in formalism on tunneling background Bispectrum from open inflation Conclusion

5 Beginning of slow-roll inflation p slow-roll inflation in good consistency with observation gives the initial condition for big-bang What happens before slow-roll inflation? p quantum tunneling tunneling may trigger slow-roll inflation φ long enough false vacuum inflation makes the earlier history irrelevant complete description of our universe φ Is there any observational signature from this event?

6 Open inflation p String landscape (Susskind, 2003) attempt to explain physical origin of inflation DOFs in the shape of extra-dimension (many scalar fields)& local potential minima p Living in a nucleated bubble potential for scalar fields quantum tunneling is 1st order phase transition it proceeds via bubble nucleation We are in the future of the phase transition our universe is inside the bubble Bubble inflaton constant surface inside the bubble is 3-hyperboloid (I will explain later) open inflation

7 Non-Gaussianity (NG) p Gaussian variable: φ G G(k 1 ) G (k 2 ) =(2 ) 3 (k 1 + k 2 )P (k 1 ) power-spectrum(2pt function) the simplest inflation model predicts almost Gaussian fluctuations full statistical property of φ G

8 Non-Gaussianity (NG) p Gaussian variable: φ G G(k 1 ) G (k 2 ) =(2 ) 3 (k 1 + k 2 )P (k 1 ) power-spectrum(2pt function) p Non-Gaussian variable: Φ the simplest inflation model predicts almost Gaussian fluctuations (k 1 ) (k 2 ) (k 3 ) =(2 ) 3 (k 1 + k 2 + k 3 )B(k 1,k 2,k 3 ) full statistical property of φ G power-spectrum(2pt func.), bi-spectrum(3pt func.), tri-spectrum(4pt func.), full statistical property of Φ

9 Non-Gaussianity (NG) p Gaussian variable: φ G G(k 1 ) G (k 2 ) =(2 ) 3 (k 1 + k 2 )P (k 1 ) power-spectrum(2pt function) p Non-Gaussian variable: Φ (k 1 ) (k 2 ) (k 3 ) =(2 ) 3 (k 1 + k 2 + k 3 )B(k 1,k 2,k 3 ) power-spectrum(2pt func.), bi-spectrum(3pt func.), tri-spectrum(4pt func.), full statistical property of Φ p Form of B(k 1,k 2,k 3 ) depends on its physical origin Ex) local-type NG for grav. potential Φ(x) (x) = [ G(x)] G(x)+f (local) NL the simplest inflation model predicts almost Gaussian fluctuations 2 G(x) B(k 1,k 2,k 3 )=f (local) NL (P (k 1 )P (k 2 )+P (k 2 )P (k 3 )+P (k 3 )P (k 1 )) const. Note: In general coefficient f NL has k-dependence full statistical property of φ G 2 G(x) +

10 Non-Gaussianity (NG) p Gaussian variable: φ G G(k 1 ) G (k 2 ) =(2 ) 3 (k 1 + k 2 )P (k 1 ) power-spectrum(2pt function) p Non-Gaussian variable: Φ (k 1 ) (k 2 ) (k 3 ) =(2 ) 3 (k 1 + k 2 + k 3 )B(k 1,k 2,k 3 ) power-spectrum(2pt func.), bi-spectrum(3pt func.), tri-spectrum(4pt func.), full statistical property of Φ Ex) local-type NG for grav. potential Φ(x) (x) = [ G(x)] G(x)+f (local) NL the simplest inflation model predicts almost Gaussian fluctuations p Form of B(k 1,k 2,k 3 ) depends on its physical origin G(x) G(x) ± 5.8 (68% C.L., Planck2013) B(k 1,k 2,k 3 )=f (local) NL (P (k 1 )P (k 2 )+P (k 2 )P (k 3 )+P (k 3 )P (k 1 )) const. Note: In general coefficient f NL has k-dependence 2 full statistical property of φ G 2

11 Bispectrum from non-bunch Davies (NBD) vacuum (Agullo and Parker 2011, Ganc 2011, Ghialva 2012) p BD vacuum state ˆ(x) = p NBD vacuum state ˆ(x) = dk (2 ) 3 ) u k( )e ikx â k + v k ( )e ikx â k dk (2 ) 3 ) â k 0 BD =0 ˆã k 0 NBD =0 ũ k ( )e ikx ˆã k +ṽ k ( )e ikx ˆã k ũ k ( )= k u k ( )+ k v k ( ) u k ( )=H 1+ik 2k 3 v k ( )=u k ( ) ṽ k ( )=ũ k ( ) e ik Bogolubov transformation k 2 k 2 =1

12 Bispectrum from non-bunch Davies (NBD) vacuum (Agullo and Parker 2011, Ganc 2011, Ghialva 2012) p BD vacuum state ˆ(x) = p NBD vacuum state ˆ(x) = dk (2 ) 3 ) u k( )e ikx â k + v k ( )e ikx â k dk (2 ) 3 ) â k 0 BD =0 ˆã k 0 NBD =0 ũ k ( )e ikx ˆã k +ṽ k ( )e ikx ˆã k p Bispectrum from NBD vacuum state (in squeezed limit: k 3 k 1 k 2 ( k) ) f NBD NL k/k 3 ũ k ( )= k u k ( )+ k v k ( ) u k ( )=H 1+ik 2k 3 v k ( )=u k ( ) ṽ k ( )=ũ k ( ) e ik Bogolubov transformation k 2 k 2 =1 k 2 k 3 k 1 observationally interesting effect (Ganc & Komatsu 2012, Agullo & Shandera 2012) In former works, they determine NBD vacuum (α k,β k ) by hand. Quantum tunneling naturally modify the state away from BD vacuum

13 Contents Introduction In-in formalism on tunneling background CDL instanton, open inflation, in-in formalism on CDL instanton Bispectrum from open inflation Conclusion

14 Method of instanton p 1-dim quantum mechanics example wave function inside the barrier S E ( ) (q) exp instanton Euclidean Action (dq/d ) 2 S E ( )= d + V (q) V F 2 = q( ) q F dq V (q) V F Euclidean EOM d 2 q (q) ( )=dv d 2 dq boundary conditions q(± )=q F q(0) = q N

15 Method of instanton p 1-dim quantum mechanics example wave function outside the barrier classical solution (q) exp S E ( ) analytical continuation τ=it (q) exp is(t) Lorentzian action is(t) S E ( = it) = i q( =it) q F dq V F V (q) Lorentzian EOM d 2 q dv (q) ( = it) = dt2 dq initial conditions dq q(0) = q dt (0) = 0 N

16 Coleman- De Luccia instanton p Tunneling of a scalar field with gravity φ (Coleman and De Luccia(CDL), 1980) 4dim Euclidean sphere embedded in 5dim p Instanton φ φ O(4)-symmetric config. contributes most Euclidean metric φ τ end 0 ds 2 = d 2 + a 2 ( )d 2 3 Euclidean EOM 1 d 2 a a d 2 = d 2 d a 8 G 3 da d d d d d 2 dv ( ) d + V ( ) =0 τ end τ end

17 Coleman- De Luccia instanton p Tunneling of a scalar field with gravity φ (Coleman and De Luccia(CDL), 1980) 4dim de Sitter spacetime embedded in 5dim φ φ p Expanding bubble solution φ analytical continuation

18 4-dim de Sitter spacetime (ds 4 ) p Embedding in 5-dim Minkowski spacetime (Mink 5 ) (2-dim projected ds 4 in 3-dim projected Mink 5 ) ds 2 = dx x i=1 4 i=1 x i 2 = 1 H 2 dx i 2 x i x 0 x j

19 Why open inflation is open? p open inflation (Gott 1982, Gott and Statler 1984) tunneling described by CDL instanton triggers slow-roll φ φ inflaton constant surface (dashed line) is 3-dim hyperboloid (open FLRW, no bubble wall!!) If slow-roll inflation is long enough (e-folds 60) time null no contradiction to observations ( Ω K 0.01) space

20 4-dim de Sitter spacetime (ds 4 ) p Embedding in 5-dim Minkowski spacetime (Mink 5 ) (2-dim projected ds 4 in 3-dim projected Mink 5 ) ds 2 = dx x i=1 4 i=1 x i 2 = 1 H 2 dx i 2 x i x 0 x j p Penrose diagram (2-dim projected ds 4 ) causal structure is conserved (space-like, null, time-like) x 0 null x i

21 Slicing of ds 4 (Penrose diagram) x 0 null p Flat slicing ds 2 = p Closed slicing dt 2 f + e2ht f H 2 dr 2 f + r 2 f d 2 d 2 = d 2 + sin 2 d 2 x i ds 2 = dt 2 c + cosh2 Ht 2 c H 2 dr 2 c + sin 2 r c d 2 ds 2 = p Open slicing dt 2 o + sinh2 Ht 2 o H 2 dr 2 o + cosh 2 r o d 2

22 In-in formalism (standard case) p Separating Lagrangian to free and interaction parts L = L 0 + L int p Equal-time N-pt function (in-in formalism) (x 1 ) (x 2 ) (x N ) = (x 10 = x 2 0 = = x N 0 = t 0 ) C: λ-integration (Maldacena 2002) Σ λ : dx 3 -integration C 1 Im λ 0 P (x 1 ) (x 2 ) (x N )e i R C d d 3 x L int (x) 0 Re λ 0 Pe i R C d d 3 x L int (x) 0 p Region of 4-dim volume integral: C x Σ λ (1 i ) P: path ordering operator flat 3-dim volume (1 + i ) C 2 C = C 1 + C 2 t 0 E 3

23 In-in formalism on tunneling background p equal-time N-pt function (same form as the standard case) (x 1 ) (x 2 ) (x N ) = 0 P (x 1 ) (x 2 ) (x N )e i R C d d 3 x L int (x) 0 0 Pe i R C d d 3 x L int (x) 0 p Region of 4-dim volume integral: C x Σ λ

24 In-in formalism on tunneling background p equal-time N-pt function (same form as the standard case) (x 1 ) (x 2 ) (x N ) = 0 P (x 1 ) (x 2 ) (x N )e i R C d d 3 x L int (x) 0 0 Pe i R C d d 3 x L int (x) 0 p Region of 4-dim volume integral: C x Σ λ 4-dim volume where CDL instanton is defined (incl. Lorentzian and Euclidean regions) C = C 1 + C 2

25 integration in the In-in formalism p example interaction Lagrangian L int (x) = g int (t) 3 (x) p Tree contribution (x 1 ) (x 2 ) (x 3 ) = 0 P (x 1 ) (x 2 ) (x 3 )e i R C d d 3 x L int (x) 0 = 2Re i C 1 d 4 x g int (t)g + (x, x 1 )G + (x, x 2 )G + (x, x 3 ) + (perms.) p Wightman functions and mode functions G + (x, x ) 0 (x) (x ) 0 = k C = C 1 + C 2 (integration along C1 and C2 give values complex conjugate to each other) u k (x)v k (x ) inflation model dependent If we know interaction Lagrangian and mode functions, we can calculate 3-pt function in open inflation

26 Contents Introduction In-in formalism on tunneling background Bispectrum from open inflation Conclusion

27 Open inflation model p background geometry ds 2 = dt 2 + a 2 (t) ij dx i dx j p Assumptions for simplicity V ( F ) V ( W ) V ( N ) V ( I ) V I (dv/d ) 2 (φ < φ N ) 2V 2 1 φ metric of unit 3-hyperboloid scale factor = physical curvature radius current horizon scale: k H0 = a 0 H 0 > k curv ( = 1 ) sub-curvature apprx.: (t) H 2 I t (0 <t 1/H I ) H I (1/H I t< ) k obs >> k curv φ W φ φ quasi-de Sitter universe a(t) sinh(h I t)/h I t (0 <t 1/H I ) e H I t /(2H I ) (1/H I <t< ) conformal time: φ I = t φ H 0 1 a 0 H 0 K 0.1 dt a(t ) t =0 = t 1/H I 1 t = =0

28 3rd order action Closed Universe case (K=+1) Clunan and Seery, JCAP1001, 032 (2010) p Original action and scalar field perturbation S[,g µ ]= d 4 R[g µ ] 1 x g 2 2 (t, x) = (t)+ (t, x) p reduced 3rd order action V ( ) g ij = a 2 e 2 solving Hamiltonian- and momentum-constraints after taking ζ=0 gauge (uniform curvature gauge) S 3 = dtd 3 x γ a 5 [ φ ( 2 3) 1 ϕ ] L ϕ 2 int { + dtd 3 x γ a 5 [ φ ( 2 3) 1 ϕ ] ϕ 2 + 3a3 φ [ ( 2 3) 1 ϕ ] ( ϕ ) } 4H a 2 iϕ i ϕ { ( + dtd 3 x a2 φ [( 2 3) 1 ϕ )] 4H a 2 iϕ i ϕ a2 [ ( 2 3) 1 ϕ ] ( ) } δl φ ϕ φϕ 2H δϕ ij suppressed by (k curv /k obs ) irrelevant in squeezed limit

29 u p ( )= mode functions ũ p ( )=H I cosh ṽ p ( )=ũ p ( ) 1 1 e 2 p ũp( )+ e p 2i p 1 e 2 p ṽp( ) + ip sinh 2p(1 + p 2 ) e ip Y plm (x) =f pl (r)y lm ( ) f pl (r) = (ip + l + 1) (ip + 1) p field operator expansion with open harmonics Y plm (x) p evolution of the state ˆ(x) = plm u p ( )Y plm (x)â plm + v p ( )Y plm (x)â plm during false vacuum inflation: BD vacuum at nucleation time: modified away from BD vacuum p the state at nucleation time (Yamamoto, Sasaki and Tanaka1996) p sinh r P l 1/2 ip 1/2 (cosh r) φ (p curv = 1) φ NBD vacuum effect note: high momentum cut-off is automatically introduced φ (complex phase δ p can be obtained but not important) φ

30 <φ(x 1 )φ(x 2 )φ(x 3 )>, B(p 1,p 2,p 3 ) and B(k 1,k 2,k 3 ) p real space 3pt function (x 1 ) (x 2 ) (x 3 ) = 2Re i 0 d d 3 x a 6 ( 2 1 3) 1 d dt G+ (x, x 1 ) integration outside the open universe gives negligible contribution O(e (k/k curv ) ) d dt G+ (x, x 2 ) d dt G+ (x, x 3 ) + (perms.) p bispectrum for multi-poles plm( )= d 3 x Y plm (x) (, x) p 1 l 1 m 1 p 2 l 2 m 2 p 3 l 3 m 3 = B(p 1,p 2,p 3 ) d 3 x Y p1 l 1 m 1 (x)y p2 l 2 m 2 (x)y p3 l 3 m 3 (x) B(p 1,p 2,p 3 ) = 2Re iv p1 (0)v p2 (0)v p3 (0) 0 d a 6 p u p 1 ( ) u p2 ( ) u p3 ( ) + (perms.) p bispectrum for Fourier modes For sub-curvature modes (p i >> p curv ), B(p 1,p 2,p 3 ) in open universe corresponds to B(k 1,k 2,k 3 ) for Fourier modes (in flat universe) with p i k i

31 NBD vacuum effect on bispectrum (in squeezed limit: k 3 k 1 k 2 ( k) ) k 2 k 3 p effect of NBD vacuum state B(k 1,k 2,k 3 ) = 2Re iv k1 (0)v k2 (0)v k3 (0) 0 d a 6 k u k 1 ( ) u k2 ( ) u k3 ( ) k 1 + (perms.) u k ( )= p resulted bispectrum B (NBD) (k, k, k 3 ) 1 1 e 2 (k/k curv) ũk( )+ e (k/kcurv) 2i k 1 e 2 (k/k curv) ṽk( ) e (k/k curv ) k k 3 P (k)p (k 3 ) there exists known enhancement factor k/k 3 (Agullo and Parker 2011, Ganc 2011, Ghialva 2012) exponential suppression by e -πk/kcurv (k/k curv >> 1, sub-curvature modes)

32 Bispectrum from open inflation (in squeezed limit: k 3 k 1 k 2 ( k) ) p leading bispectrum of φ B(k 1,k 2,k 3 ) = 2Re iv k1 (0)v k2 (0)v k3 (0) = 2 P (k)p (k 3 ) 0 d a 6 k u k 1 ( ) u k2 ( ) u k3 ( ) p from fluctuation of φ@horizon exit to ζ on φ=0 gauge = Ḣ 1 2 p bispectrum of ζ H = 2 + O( 3 ) same as flat case (Maldacena 2003) B (k 1,k 2,k 3 )= 1 n s + O k curv P (k)p (k 3 ) k i k 2 k 3 k 1 + (perms.) new term in open inflation is suppressed by the factor (k curv /k i ). (k curv /k i << 1, sub-curvature modes)

33 Contents Introduction In-in formalism on tunneling background Bispectrum from open inflation Conclusion

34 Conclusions p We have calculated the bispectrum from open inflation, where quantum tunneling from false vacuum triggers slow-roll inflation p Open inflation gives complete description of our part of universe, and is partially motivated by string landscape p The leading order bispectrum is same as the flat case and the correction appears in higher order in the sub-curvature approx. p High-momentum cut-off makes the known enhancement due to non-bunch Davies vacuum exponentially suppressed p This is the first calculation of bispectrum from open inflation and we are working on more general cases (with D. Yamauchi, T. Tanaka, and M. Sasaki)

35 Appendix

36 Derivation of In-in formalism on tunneling background (in simplified systems) p 2-dim quantum mechanics (QM) p QFT and 2-dim QM correspondence (t) ȳ(t) i = (x i ) i j k = (t) n K.S., Phys. Rev. D88, (2013) y=ȳ(t) ȳ(t) instanton: ( (t) =0) fluctuation: quantum fluctuation at given y (or t): n y=ȳ(t) separation of instanton and fluctuation parts in QFT (t, x) = (t)+ (t, x)

37 WKB analysis for 2-dim tunneling wave function K.S., Phys. Rev. D88, (2013) p time-independent 2-dim Schrodinger equation Ĥ (y, )=E (y, ) H = p2 y 2 + p2 2 + V (y, ) p instanton as a clock y t : p expansion around 1-dim WKB wave function (y, ) y =ȳ(t) e S y(y)/ (t, ) p emergent time -dependent Schrodinger eq. (lowest WKB order) i t (t, )=Ĥ(t) (t, ) Ĥ(y) = ˆp2 2 + V (y, )

38 In-in formalism on tunneling background in 2-dim QM p Formal solution in Schrodinger picture K.S., Phys. Rev. D88, (2013) (t) = P exp i +i 0 t H(t )dt (t) (t) = P exp i t 0 i H(t )dt (t) C evolution along time path: C p equal time N-pt function in interaction picture n = 1 y=ȳ(t) N 0 P n i I (t) exp normalization factor C H I (t )dt 0

39 Integration volume (Penrose diagram) t = t 0 p standard in-in formalism p in-in formalism on CDL instanton background Σ=Σ 0

40 bispectrum in the folded limit p Folded limit k 1! k 2 + k 3 p Usual flat case result k 2 k 3 k 1 (If initial state is given at past infinity) B (NBD) (k 1,k 2,k 3 ) / k 1 k 1 + k 2 + k 3 p open inflation case result B (NBD) 1 e ( p 1+p 2 +p 3 /p curv ) (p 1,p 2,p 3 ) / p1 exact folded limit is still finite p 1 + p 2 + p 3 (the correspondence to Fourier modes doesn t hold in this regime) 40