中央大学セミナー.ppt

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1 String Gas Cosmology

3 References Brandenberger & Vafa, Superstrings in the early universe, Nucl.Phys.B316(1988) 391. Tseytlin & Vafa, Elements of string cosmology, Nucl.Phys.B372 (1992) 443. Brandenberger, String Gas Cosmology, Kanno & Soda, Moduli stabilization in string gas cosmology, Phys.Rev.D72 (2005) Brandenberger, Kanno, Soda, Easson, Khoury, Martineau, Nayeri, Patil, More on the spectrum of perturbations in string gas cosmology, JCAP 0611 (2006) 009.

5 = FLRW k = 0 ds 2 = dt 2 + a 2 (t) dx 2 + dy 2 + dz 2 H 2 = 8πG 3 ρ

6 log ( ) a = 0 a t a t p, p < 1 1 H t 1 H L a H = const. log

7 ρ log ( ) ρ = const. 1 a 4 H = const. ρ = const. 1 a 3 Log

8 V (φ) m 2 1 H 1 m H = a a = const. H = m 8

9 ρ log ( ) ρ = const. 1 a 4 δρ ρ δa a Hδt Hδφ φ Δ 1 a 3 Log

10 log ( ) horizon size Δ = const. Δ inflation log

11 COBE δt T Δ 1992, COBE WMAP

12 12

13 = 1 2πP(k) exp 1 2 Δ 2 (k) P(k) Δ( x) = 1 ( 2π ) 3 2 d 3 k Δ(k) e i k ix Δ(k) Δ(k ') = δ(k + k ')P(k = k ) P(k) const.

14 k GUT scale cm t

15 10-

16 弦理論と宇宙論特異点非インフレーション非幾何学的超弦理論超重力理論 String gas AdS/CFT, Matrix? cosmology Pre-big-bang cosmology Ekpyrotic or cyclic cosmology 幾何学的 D-brane inflation 場の理論インフレーション old inflation new inflation chaotic inflation hybrid inflation 宇宙の精密観測 Primordial GW LSS, CMB, 21cm, etc.

17 T :

19 world line = world sheet

22 string scale in conformal gauge g ab δ ab X µ L (τ + σ ) = x µ + α µ 0 2 (τ + σ ) + i 2 X µ R (τ σ ) = x µ + α µ 0 2 (τ σ ) + i 2 α n µ n e n 0 α n µ n e n 0 in(τ +σ ) in(τ σ ) α µ ν n,α m = nη µν δ n+ m,0 22

23 p µ = 2 α 0 µ = 2 α 0 µ 10, 2,

24 2

27 Kalza-Klein T

28 R p=n/r, w=mr. X L (σ + 2π ) + X R (σ + 2π ) = X L (σ ) + X R (σ ) + 2πw p = 1 2π ( X L + X R ) 2π dσ = α + α 0 0 ( ) 4-d universe α 0 = 1 2 ( p w ) α 0 = 1 2 ( p + w ) N N = nm Target space (T-) duality

29 T- momentum x = p exp( ipx) p x = x + 2π R x = winding p exp( iw x ) w x = x + 2π R 29

30 E 2 1 E 2 na n a n N n=1 a 1 p(n) a 2, ( a ) 2 1 a 3, a 1 a 2, ( a ) 3 1 a 4, a 1 a 3, ( a ) 2 2, a ( 2 a ) 2 1, ( a ) 4 1 p(n) 1 4N 3 exp 2π N N 1

31 E 2 24 na n a n 24N n=1 p(n ) exp 2π 24N exp( 4π N ) 6 E 2 N p(e) exp( β H E) β = 4π H ( ) exp βe i de p(e) i exp( βe)

33 4

34 4 6 4 shifted dilaton

35 T

36 1

40 :

41 ds 2 = dt 2 + e 2λ(t ) δ ij dx i dx j ( ) 2 S = 1 2 d 4 x ge 2φ R + 4 φ ( ) d 4 x F λ,β β = 1 T 3 λ 2 + ϕ 2 = 2e ϕ F + β F β = 2eϕ E λ ϕ λ = e ϕ F β = eϕ P ϕ = 2φ 3λ E + 3 λp = 0 P = 0 E = const. λ = 0 ϕ = 2 t

42 H 1 k H 1 t 42

43 Z(β) = 0 de Ω(E) e βe C β Ω(E) = L +i L i dβ 2πi Z(β) e βe L Heterotic Z(β) = β H β β H β H β β 1 6 β H = β 1 = 1 1 R R 2 Ω(E, R) = β H e β H E 1+ δ Ω(E, R) ( ) 5 [ ] δ Ω = β H E 5! e (β H β 1 )E

44 δρ 2 = ρ 2 ρ 2 = 1 R 6 β F + β F β = T 2 R C 6 V S = logω T = S E = R2 3 log s R 2 3 s T H 1 T T H E 1 T = T H 1+ β β H 1 β H β H β 1 3 s R 2 δ Ω C V = E T = R2 s 3 T 1 T T H 1 δρ 2 = 1 R 4 T s 3 1 T T H 1

45 2 T µ ν = G µα G log Z G αν T i j T i j T i j T i j = 1 βr 3 log R 1 R 3 F log R = 1 p βr 3 R p = T S V = T (β H β 1 ) Eδ Ω V 1 p βr 3 R = 4 3 T 2 1 T R 4 3 s T H T H log R2 s 2 1 T T H 2 k 2 h ij (k) 8πGδT ij (k)

46 2 Φ = 4πGδρ k 1 = R P Φ (k) = k 3 Φ(k) 2 = 16π 2 G 2 k 4 δρ 2 = 16π 2 G 2 T s 3 1 T T H 1 P Φ (k) = p s 4 1 T T H 1 P h (k) = p s 4 1 T T H log 1 2 s k 2 1 T T H 2

51 4 T

52 T 1 1

54 2

meiji_resume_1.PDF

meiji_resume_1.PDF β β β (q 1,q,..., q n ; p 1, p,..., p n ) H(q 1,q,..., q n ; p 1, p,..., p n ) Hψ = εψ ε k = k +1/ ε k = k(k 1) (x, y, z; p x, p y, p z ) (r; p r ), (θ; p θ ), (ϕ; p ϕ ) ε k = 1/ k p i dq i E total = E