Summer School on Cosmology July Inflation - Lecture 1. M. Sasaki Yukawa Institute, Kyoto

Transcription

1 354-1 Summer School on Cosmology 16-7 July 01 Infltion - Lecture 1 M. Ssi Yuw Institute, Kyoto

2 Summer school on cosmology ICT, July 01 Miso Ssi Yuw Institute for Theoreticl hysics Kyoto University

3 1. Infltionry Universe contents horizon & fltness problems slow-roll infltion reheting scenrio. Cosmologicl erturbtions from Infltion curvture (sclr-type) perturbtion grvittionl wve (tensor-type) perturbtion 3. Non-Gussin Curvture erturbtion origin of non-gussinity δn formlism: NG genertion on superhorizon scles other sources of NGs 4. Summry nd outloo

4 3 1. Infltionry Universe horizon problem ds = dt + ( t) dσ ( 3) 4π G ρ ɺɺ = ( ρ + 3) < 0 for > 3 3 n if t, n < 1 grvity=ttrctive η dt ds = ( η)( dη + dσ ( 3) ) dη = : conforml time ( t) now t t 0 0 dt ( t) = finite η = 0 lst scttering surfce x conforml time is bounded from below E prticle horizon

5 4 solution to the horizon problem 4π G ɺɺ = ρ + 3 > 0 η 3 ( ) for sufficient lpse of time in the erly universe now lst scttering surfce t dt 0 = t 0 0 ( t) t η η 0 or lrge enough to cover the present horizon size NB: horizon problem homogeneity & isotropy problem η 8

6 fltness problem (= entropy problem) 8π G K H = ρ ; K < < K if ρ, ρ in the erly universe. conversely if ρ K / t n epoch in the erly universe, the universe must hve either collpsed (if K > 0) or become completely empty ( if K < 0) by now. 5 lterntively, the problem is the existence of huge entropy within the curvture rdius of the universe 3 3 S = T T > T H K K (# of sttes = exp[s])

7 solution to horizon & fltness problems 6 sptilly homogeneous sclr field: 1 1 = ɺ + V ( ), = ɺ V ( ) ρ + 3 = ɺ φ V ( φ) < 0 if ɺ φ < V ( φ) ρ φ φ φ φ ( ) ρ V φ ɺ φ V φ ( ) if ( ) potentil dominted V ~ cosmologicl const./vcuum energy ρ const. decreses rpidly K 8π G H ρ const. infltion 3 vcuum energy converted to rdition fter sufficient lpse of time solves horizon & fltness problems simultneously

8 7 slow-roll infltion single-field slow-roll infltion Linde 8,... V(φ) φ metric i ds = dt + ( t) δ dx dx field eq. ɺɺ φ + 3 Hɺ φ + V ( φ) =0 ɺ V ( φ) φ = 3H ɺ 8π G 1 H = ɺ φ + V ( φ) 3 ij j 3 Hɺ ɺ φ 3ɺ φ = 1 H 1 slow vrition of H ɺ φ + V V ~ e Ht infltion!

9 8 slow-roll conditions 3 Hɺ ɺ φ 3ɺ φ M V ε = = 1 H 1ɺ φ + V V V condition for qusi-de Sitter (infltionry) expnsion ɺɺ φ ɺ ε M V δ ε ε ; δ 1 Hɺ = + φ Hε V sufficient condition on potentil: ε M V M V V V η V η condition for friction-dominted (over-dmped) evolution 1, v ; v 1

10 stndrd scenrio reheting e.g. decy rte: L ~ int gyφψψ Γ ~ gy mφ ; mφ mψ 1 effective eqution of motion: V ( φ) = m φ φ + when mφ H > Γ, dmped oscilltion: ɺɺ φ + 3 ɺ φ + φ = 0 φ / cos( + α) effect of Γ Abbott & Wise 84, Dolgov & Linde 84 3 H mφ mφt ɺɺ φ + 3 Hɺ φ + V ( φ) = Γɺ φ φ d 1 ɺ φ + V = ( 3H + Γ) dt ɺ ρ + 3Hρ = Γρ ɺ φ φ φ φ g Y ψ ψ 9

11 energy conservtion eqns 10 ɺ ρ + 3Hρ = Γρ φ φ φ ɺ ρ + 4Hρ = Γρ r r φ ρ r : produced rdition Γ<H~ t / Γ ρ ρr = ρ φ 1 φ = ρφ, f f 5H f f f f Γ>H ~ t -1 ρ 0 ρ ρ ( t ) f / 5 8 ρr = mx t = φ =, r = r R R 4 t :def by H( t ) = Γ R R

12 reheting temperture & mx temperture 11 log(ρ) ρ f ρ r,mx ρ R ρ φ π T t N T 30 ( ) 4 R : ρ r R = eff R T R 1/ 4 ( ) 1/ Γ M M ~ 1/ 4 N eff N eff 36 Γ = π ρ r log() indep. of ρ f R f mx 1/ 4 1/ 4 ~ H f Γ ~ H f Tmx M T M Γ R dep. on ρ f T mx is importnt for therml history T R is importnt for horizon problem

13 log L comoving scle vs > H subhorizon vs Hubble horizon rdius : comoving wvenumber = H < H superhorizon L = H 1 = H L = > H subhorizon 1 infltion t=t end hot bigbng log (t) Hubble horizon=cusl horizon for locl physics

14 e-folding number: N 13 # of e-folds from φ=φ(t) until the end of infltion redshift ( tend ) ( t) = exp[ N( t t )] end t [ 1 φ ] N = N = Hdt + z end ( φ) ~ ln ( ) t( φ ) log L N=N(φ) L=H -1 ~ t L=H -1 ~ const t=t(φ) t=t end φ determines comoving scle log (t)

15 14 condition on e-folding e number log L ignore vrition of H during infltion. entropy generted within present Hubble volume: H 0-1 S = H e T 3 3N ( φ h ) R 3 f R f 3 φh vlue of φ t which comoving scle of H left horizon 1 0 L H 1 H f -1 H 1 3/ (φ h ) f R log (t)

16 15 3 3/ ρ ρ S = H e T ~ e T 3 3N ( φh ) R 3 f 3N ( φh ) f 3 f R 4 R f M TR 3 M T 1/ e TRρ f H0 h > 3N ( φ ) ~ 10 1/ 4 ρ f TR N( φh ) > 53 + ln + ln GeV GeV N>50-60 solves horizon & fltness problems chnging T R by one order (by 10) chnges N by 1 Q1. Show tht conforml time η h t φ=φ h stisfies η h >η 0, where η 0 is the conforml time tody.

17 preheting Kofmn, Linde & Strobinsy If φ couples to other light sclr (bose) fields e.g. L ~ g, m m int φ χ χ φ ctstrophic χ- prticle cretion cn occur ( ( / ) ) ɺɺ χ + 3Hɺ χ + + gφ χ = 0 3 φ = φ f ( f / ) sin mφt for mφ t 1 ~ > H t oscillting potentil ɺɺ χ + + φ χ = 0 (( / ) g sin m t ) φ ( bcos m t ) ɺɺ χ + χ = 0 φ Mthiew eqn possible prmetric mplifiction of χ

18 instbility bnds 17 b = + m φ b ɺɺ χ ( bcos m t ) + χ = 0 φ b gφ = 4m φ For m, b φ = if b>1 initilly, evolutionry pth psses through unstble region instntneous reheting

19 . Cosmologicl erturbtions from Infltion 18 curvture perturbtion: intuitive derivtion zero-point (vcuum) fluctutions of φ : δφ = ( ) i i δφ x t e π c δφɺɺ + 3 Hδφɺ + ω ( t) δφ = 0 ; ω ( t) = ( t) λ( t) hrmonic oscilltor with friction term nd time-dependent ω δφ physicl wvelength δφ const. frozen when λ > c H -1 (on superhorizon scles) λ(t) ~ (t) grvittionl wve modes lso stisfy the sme eq.

20 19 fluctution mplitude (vcuum fluctutions=gussin) 1 φ = ϕ ϕ e w = H, ~ iwt ; 3/ w H H ( / H) ( = / H) 3 = π / ~ H ϕ ϕ δφ frozen t =/H In the bove, metric perturbtions δg re ignored ~ guge in which δg is minimized = hypersurfce on which δr (3) =0: flt slice R K 6 4 δ = R ( 3) ( 3) =, R δ K = R 3 R: clled curvture perturbtion

21 genertion of comoving curvture perturbtion 0 δφ is frozen on flt (R=0) 3-surfce (t =const. hypersurfce) Infltion ends/dmped osc strts on φ =const. 3-surfce. t T = const., R 0 end of infltion hot bigbng universe φ =const. 3-surfce is clled comoving slice. x i R 0 δφ = 0 R = 0 δφ 0 curvture perturbtion on comoving slices: H guge trnsf. R c = ɺ δφ φ evluted on flt slice

22 conservtion of comoving curvture perturbtion 1 eom ( z ) φ z ɺ R C + R C + R = 0; = ε M ; ' C z H H 3 ε = ɺ 0 = 1 ( + w H ), R + C = z R C ( z ) R 0 C R C R C 1 : z = const. : if R C becomes const., dibtic limit is reched H ( H) RC ( = H) = δφ ( = H) ɺ φ Kodm & MS 84 = d dη = w ε : slow-roll prmeter decying mode growing mode = ρ d dt

23 spectrum Curvture perturbtion spectrum R H ( ) = πφ ɺ = H 1 H = M 1/ π ε = H spectrl index A n 1 ( ) S R = ; n V V V V Liddle & Lyth ( 9) S 1 = M 3 η 6 = V εv spectrum derived by 1 st principle clcultion Muhnov ( 85), MS ( 86) more elegntly derived l Fddeev-Jciw method Grrig, Montes, MS & Tn ( 98) generlized to -infltion: L = ( X, φ); X = g µν µ φ ν φ Grrig & Muhnov ( 99)

24 3 generlized ction for R C z S = dη d x R c R 3 C s C c s c = sound velocity s ; Grrig & Muhnov ( 99) z = 3( 1+ w) M (=1 for cnonicl cse) cnonicl quntiztion: δ S π R = = z c s RC δ R R C C R C, π R = iħ * s icsη r η cs z 3 4π 1 H 3 cs η = 1 π 3cs 1+ w π M c = H = r ( η) + ( ) ; r R ( ) = r = ( ) ( ) 1 c e positive freq fcn ( η ) Q. Derive the bove spectrum by performing cnonicl quntiztion s outlined bove. s

25 δn - formul Strobinsy ( 85) 4 t end φ end H N( φ) = Hdt = dφ t( φ ) φ ɺ φ N H δ N( φ) = δφ = δφ = c φ φ ɺ R = H = H R H N ϕ η ɺ φ = H ( ) = = ; = 1 πφ ϕ H = = δφ π = H geometricl justifiction δ N N = δφ A φ NL generliztion Lyth, Mli & MS ( 04) A A MS & Stewrt ( 96) only nowledge of bcground evolution is necessry

26 5 h = δ h = i TT ij TT ij ij Tensor erturbtion 0 cnoniclly normlized tensor field 4 1 φij S ~ d x g + iii t 1 TT M TT 1 φij hij = hij ; M 3π G 8π G φ ( ; t) = ( ) ϕ ( t) + h. c. σ σ ij ij σ =+, tensor spectrum σ : trnsverse-trceless ϕ ( t ): sme s mssless sclr 4 TT 8 ϕ H hij, σ = φ, 8 ij σ = = M σ M π M Strobinsy ( 79)

27 Tensor-to to-sclr rtio sclr spectrum: sclr spectrum: s ( ) ( π ) ( π ) tensor spectrum: ( ) g tensor spectrl index: b N N H ( φ) φ 3 4π H 1 = N 3 4π 8 = H 3 ( π ) M ( π ) 3 n s n g Hɺ ɺ φ ɺ φ ng = = = H M H M ɺ φ N N b φ 6 H dn dt i = = φ N 1 = M N g 8 s r g s 8 n g vlid for ll slow-roll models with cnonicl inetic term

28 Comprison with observtion Stndrd (single-field, slowroll) infltion predicts scle- invrint Gussin curvture perturbtions. 7 WMA 7yr CMB (WMA) is consistent with the prediction. Liner perturbtion theory seems to be vlid.

29 8 CMB constrints on infltion Komtsu et l. 10 sclr spectrl index: n s = 0.95 ~ 0.98 tensor-to-sclr rtio: r < 0.15

30 However,. 9 Infltion my be non-stndrd multi-field, non-slowroll, DBI, extr-dim s, LANCK, my detect Non-Gussinity (comoving) curvture perturbtion: 3 R = R + + > 5 C guss f guss ; ~ 5? NLR f NL B-mode (tensor) my or my not be detected. -10 energy scle of infltion H < > 10 Mlnc? modified (quntum) grvity? NG signture? Quntifying NL/NG effects is importnt