Towards a nonsingular inflationary Universe

Transcription

1 Toward a noningular inflationary Univere Taotao Qiu Intitute of Atrophyi, Central China Normal Univerity Baed on Phy.Rev. D88 (013) 04355, JCAP 1110: 036, 011, JHEP 0710 (007) 071, 1 and reent paper 141.XXXX

2 Standard Big Bang Comology (Alpher/Bethe/Gamow 1948) Ralph Alpher (α) Han Bethe (β) George Gamow (γ) The keth plot of Big Bang

3 Standard Big Bang Comology The big bang theory annot olve: Flatne problem Horizon problem Unwanted reli problem Singularity problem 3

4 Tremendou Complementarity: Inflation Guth/Albreht/Linde/Staro binky/sato/fang an olve flatne problem, Horizon problem and Unwanted reli problem. However, a inflation happen after Big-Bang exploion point, it will do nothing to do with the Big-Bang Singularity! 4

5 5 The Singularity Problem Singularity Theorem: The univere will meet a ingularity when (1) it i deribed by General Relativity; 4 R S = d x g[ + Lm ] 16πG () it atifie Null Energy Condition; T nn = [( ρ + Puu ) + g Pnn ] µ ν µ ν µν µ ν µν Where at finite time point S. Hawking R. Penroe µ µ = ( ρ + P)( uµ n ) + Pnµ n un µ = ( ρ + P) 0 for any null vetor n µ µ = 1 : nn µ 0 a u (t) 0, ρ (t) µ = S.W. Hawking, G.F.R. Elli, Cambridge Univerity Pre, Cambridge, 1973; Borde and Vilenkin, Phy.Rev.Lett.7, 3305 (1994). (Khoury et al., 01) u

6 What if the univere ome from a A non-ingular boune Boune? doen t reah 0 at it minimum 6

7 Sound good, but To uefully add a boune before inflation, we need to take are of two more problem: Ghot intability; omi aniotropy in the ontrating phae. 7

8 Ghot Intability Condition for non-ingular boune to happen For a non-ingular boune: Contration: Expanion: H < 0 H > 0 Bouning Point: Nearby: H = ρ = H > 0 0 From Friedmann Equation: whih make H = 4 πg( ρ+ p) w< 1 T nn µ ν 0 µν < Null Energy Condition violated! 8 Y. Cai, T. Qiu, Y. Piao, M. Li and X. Zhang, JHEP 0710:071, 007, ite 100+

9 Ghot Intability NEC violation will generally aue ghot mode! Example: Phantom Energy Lagrangian: Lphantom Ghot mode! Hamiltonian (denity): 1 µ = µφ φ V( φ) 3? 1 phantom φ ω k k H =Π L= dkaa ( + ) unbounded energy! 9 S. Carroll, M. Hoffman, M. Trodden, Phy.Rev. D68 (003) 03509; J. Cline, S. Jeon, G. Moore, Phy.Rev. D70 (004)

10 Solution: Galileon Theorie Horndeki/Galileon theorie (008) L = K( φ, X) L = G ( φ, X) φ 3 3 = φ + φ φ L4 G4 X R G 4, X µ ν (, ) [( ) ( ) ] L G XG G µ ν = 5( φ, ) µν φ 5, X [( φ) 3 φ( µ ν φ) + ( µ νφ) ]/6 Higher derivative in lagrangian but nd order in equation of motion Multi-degree of freedom but only one i dynamial 10 violating NEC free of ghot. e. g. EoM: L= L + L3.... K + [ K + K X ( G + G X) + 6G Hφ+ 6 HG Xφφ ] φ X XX φ Xφ X XX.. 3 HK [ X ( Gφ GXφ X)] φ [KXφ 6 GX( H 3 H) Gφφ ] X = Ghot Free!

11 Comi Aniotropy It i only ontrating phae with w=0 that an generate ale-invariant power petrum, but If the initial metri i not exat iotropi: Friedmann Equation: 11 3 bg i i= 1 Matter 3 β () i t i d = dt + a () t e dx Aniotropy Equation of motion for aniotropy: J. Erikon, D. Weley, P. Steinhardt, N. Turok, Phy.Rev. D69 (004) i= 1 1 3H = ρ + β β + 3 H β = 0 Sale fator w =1 w = 0 Energy denity i Sale fator o we need ontrating phae with w>1! i w ani = 1 ρ ani a = a w >1 w =1 3(1 + w) 6 Energy denity

12 the boune inflation model built from Galileon theory Our New Galileon Boune model: 4 R S = d x g[ + k( φ) X + t( φ) X V( φ) G( φ, X) φ] 16π G where k t0 k φ = t( φ) = + (1 κφ ) 0 ( ) 1 (1 κφ 1 ) G( φ, X) = G-term γ X + (1+ κφ ) 1 we deign the Lagrangian in uh a way that 1. around boune (violating NEC): G-term i important (kill ghot);. away from the boune: anonial field form (implet). T. Qiu et al., in preparation.

13 the boune inflation model built from Galileon theory For region away from boune (ontrating or expanding phae): the ation redue to the form: µ S = d x g[ R µφ φ V( φ)] 16π G ontrating phae: in order to avoid omi aniotropy w > 1 V ( φ ) < 0 expanding phae: in order to get inflation w 1 1 µ V ( φ) >> µ φ φ > 0 13

14 the boune inflation model built from Galileon theory Aording to thi, we hooe the potential funtion to be: φ φ φ V( φ) = V [1 tanh( λ )]e +Λ [1 + tanh( λ )](1 ) φ φb φb v Numerial plot of V ( φ ) and φ() t : 14 T. Qiu et al., in preparation.

15 the boune inflation model built from Galileon theory Numerial plot for H and w (EoS): w>1 (ontration) w=-1 (inflation) oillating w (reheating) 15 T. Qiu et al., in preparation.

16 Obervational Contraint on Bouning Comologie Temperature flutuation(cmb) laial perturbation (primordial) quantum flutuation (vauum) Contraint on primordial perturbation: 16 Spetral Index: n = ± Tenor/alar ratio: r 0.1 (approximately) Plank Collaboration, arxiv:

17 Perturbation of Galileon boune inflation 17 The perturbed ation: where 1 Q δs = dηd xa [ ζ ( ζ)] For far away from the boune, Equation of motion: 3 M X Q = [ k( ) + t( ) X + ( G + G X ) + 4 HG ] ( M H G X ) M 4 p GX X φ φ X XX φ Xφ p X φ p ( M H G X φ) 6G X k t X HG G X Q p X X 1 = [ ( φ) + 6 ( φ) + 6 ( ) ] 4 X + XX φ+ M px M p Q M ε = M H / H p p 1 z u + u u = 0 where u = zζ z z = a Q

18 Perturbation of Galileon boune inflation Initial Condition: Bunh-Davie vauum Sine the univere begin with large and empty phae with infinite horizon, initially all the mode are et inide the horizon. 18 Condition: k Solution: z >> z 1 i u = Ae kη k From anonial quantization relationhip: A The keth plot of the model 1 = k

19 Perturbation of Galileon boune inflation Contrating phae: 19 The mall k (large ale) mode exit the horizon during thi phae, while the large k (mall ale) mode till tay inide the horizon. Condition: a Q z = M w > 1 ε > 3 Solution: p a ε u = η η [ ( kj ) ( k η η ) k B 1 ν B 1 + ( kj ) ( k η η )] ν B The keth plot of the model ~ o( k η ) k η >> 1 ηb 1 ε 1 B ηb ~ ( k η η ) k η << 1 ηb

20 0 Perturbation of Galileon boune inflation Bouning phae: The Hubble parameter tranit from negative to poitive, the ale fator doe not go a power-law any more. Condition: t / a ae α B H αt (parametrization) t X XBe χ from numeri we know χ > α Equation of motion: u + ku + ( χ α) au = 0 Solution: k k B k u = d ( k)o[ l ( η η )] + d ( k)in[ l ( η η )] k 1 B B The keth plot of the model l = k + ( χ α) a B

21 Perturbation of Galileon boune inflation Expanding phae: Inflation The mall k (large ale) mode ha already exit and will tay outide the horizon in thi phae, while the large k (mall ale) mode will exit the horizon during thi phae. 1 Condition: Solution: a Q z = M w 1 ε e 0 p a ε e u = η η [ g( kj ) ( k η η ) k B+ 1 ν B+ + g( kj ) ( k η η )] 1 ν B+ + + The keth plot of the model ~ o( k η η B + ) k η ηb >> 1 1 ε 1 ~ ( k η η ) e k η << 1 B+ ηb

22 Perturbation of Galileon boune inflation d1, 1, The oeffiient 1,, and an be onneted through Hwang- Vihnia/Deruelle-Mukhanov mathing ondition: ζ The power petrum & petral index are defined a: 3 k ζ dln P P ζ n π 1 + ζ dln k Therefore in our model, it turn out to be: P ζ P ζ g = ζ ζ = ζ + εe 1 ε e e 1 ε ε e B+ ν + ν + HB+ (1 + ) ( 1) H k = 16πε Γ (1 + ) + ε 1 ε 1 1 H B e l B lh B l ε + + B k ν + 3 πε ( 3) ε e (1 ν ) H B ( ε 1) [( ε 3)o( η ) in( η )] = Γ + for large-ale perturbation (whih exit the horizon in ontrating phae). ζ n ζ ε e 1+ 1 ε 1 for mall-ale perturbation (whih exit the horizon in expanding phae) ε n ζ 1+ > 1 ε 1 where ηb ηb ηb + e

23 Perturbation of Galileon boune inflation Similar proedure an alo be ued to treat tenor perturbation, exept for the deoupling of gravitational wave from matter, whih aue = 1 z = a during the whole evolution. So the equation for tenor perturbation beome a = 0 v = ah a v v v and the olution in eah phae are: 1 ikη uk = e k u = η η [ ( kj ) ( k η η ) + ( kj ) ( k η η )] k B 1 ν B ν B Initial Condition Contrating phae u = d ( k)o[ l ( η η )] + d ( k)in[ l ( η η )] k 1 B B u = d ( k)e + d ( k)e k l ( η ηb) l ( η ηb) 1 large k mall k Bouning phae 3 u = η η [ g( kj ) ( k η η ) + g( kj ) ( k η η )] k B+ 1 ν B+ ν B+ + + Expanding phae

24 Perturbation of Galileon boune inflation After impoing mathing ondition, we an identify the oeffiient 1,, and. g 1, d1, P ζ 4 The tenor petrum & petral index: In our model: P h k h π 3 εe 1 ε e ε e 1 e H B k ε + ν π ν+ H B+ for large-ale perturbation (whih exit the horizon in ontrating phae), n h dln Ph dln k ( 1) ε e Ph = nh 0 Γ (1 + ) ε e 1 for mall-ale perturbation (whih exit the horizon in expanding phae); ε 1 ε 1 1 H B e l B lh B l ε + + B k ν + 3 πε ( 3) e (1 ν ) H B ( ε 1) [( ε 3)oh( η ) inh( η )] = Γ + where ηb ηb ηb + n ζ ε 1+ > 1 ε 1

25 Perturbation of Galileon boune inflation One an alo alulate the tenor/alar ratio: r P P 3ε h ζ (1 + ) 16ε large k mall k So for large k mode the T/S ratio alo arrie information from boune, then by onidering the oming ontraint on r from PLANCK, we an do ome ontraint on the pre-big-bang evolution. However for mall k mode the T/S ratio ould be large. 5

26 Remark Atually all the above analyi i baed on the onideration that the boune i not far from obervable inflation, i.e. the e-fold number i not too muh, o the information from the boune i detetable. There i alo poibility that if the inflation wa too long, the oberved mode ome from BD even in expanding time (like the purple line), then we have g, intead, o 1 = π / g = i π / everything ould be ompletely the ame a pure inflation and boune beome non-detetable. 6 The keth plot of the model

27 Conluion Inflation i till the main-tream theory, but ome iue remain unolved (the Big Bang Singularity). Boune omology an make the univere non-ingular, but ome iue have to be taken are of: I: ghot intability (olved by Galileon theorie) II: omi aniotropy problem (olved by Ekpyroti ontration) The power petrum for mall ale perturbation i ale invariant (onitent with obervation), The power petrum for mall ale perturbation i blue (reponible for mall l uppreion). 7

28 8 Thank for attention!