String Gas Cosmology
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, 0808.0746 Kanno & Soda, Moduli stabilization in string gas cosmology, Phys.Rev.D72 (2005) 104023. Brandenberger, Kanno, Soda, Easson, Khoury, Martineau, Nayeri, Patil, More on the spectrum of perturbations in string gas cosmology, JCAP 0611 (2006) 009.
= FLRW k = 0 ds 2 = dt 2 + a 2 (t) dx 2 + dy 2 + dz 2 H 2 = 8πG 3 ρ
log ( ) a = 0 a t a t p, p < 1 1 H t 1 H L a H = const. log
ρ log ( ) ρ = const. 1 a 4 H = const. ρ = const. 1 a 3 Log
V (φ) m 2 1 H 1 m H = a a = const. H = m 8
ρ log ( ) ρ = const. 1 a 4 δρ ρ δa a Hδt Hδφ φ Δ 1 a 3 Log
log ( ) horizon size Δ = const. Δ inflation log
COBE δt T Δ 1992, COBE WMAP
12
= 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.
k GUT scale 10 33 cm t
10-
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.
T :
world line = world sheet
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
p µ = 2 α 0 µ = 2 α 0 µ 10, 2,
2
Kalza-Klein T
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 = 1 0 2 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
T- momentum x = p exp( ipx) p x = x + 2π R x = winding p exp( iw x ) w x = x + 2π R 29
E 2 1 E 2 na n a n N n=1 a 1 p(n) 1 2 3 4 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 6 2 3 5 N 1
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)
4
4 6 4 shifted dilaton
T
1
:
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
H 1 k H 1 t 42
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 = 2 + 1 β 1 = 1 1 R 2 + 2 1 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
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
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)
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
4 T
T 1 1
2