La variété de de Sitter et le principe de Mach

SOo(1,4)/SOo(1,3)
ΛCDM
2 109ans
17 ±1 109ans

SOo(1,4)/SOo(1,3)
SOo(1,3)

Hλ=SOo(1,4)/SOo(1,3)
R5
(e0, e1, e2, e3, e4) (e1, e2, e3, e4)
SOo(1,3) (e0, e1, e2, e3)
SOo(1,4)
R3M=HTK
(e1, e2, e3)Hλ
e4
−1
λe41 λe4
atauavSOo(1,1)
(e0, e4)
Hλ
A+
SOo(1,4) Hλ
H−1
λ=HA+H(1
λe4)H0
λ=NAH(1
λe4)H+1
λ=KAH(1
λe4) = Hλ
A+
Hλ
H−1
λds2=dt2−sh2λt
λ2(dα2+sh2αdω2),
H+1
λds2=du2−ch2λu
λ2(dξ2+sin2ξdω2),
H0
λds2=dv2−e2λv(dρ2+ρ2dω2)
dω2=dθ2+sin(θ)2dφ2
H−1
λ
H0
λ
H+1
λ
λΩ(t)
Ω0H0
λ
λ2=H(t)2Ω(t) = −H(t)2q(t),
Ω(t1)<1t1Ω(t)<1
Ω(t1)>1 Ω(t1)=1

H−1
λ
ds2=c2dt2−(dr −r H(t)dt)2
1 + (1 −Ω(t))H2(t)r2−r2dω2,
H(t) = λ / tanh(λ t)λ2= Ω(t)H(t)2λ
HoH−1
λ
Hoqo=−Ωo
ΛCDM HoΩmqo= (3/2)Ωm−
1 = −0.535 Ho= 67.3 Ω
Ωo= 1 −2.1Ωm= 0.349
λ= 67.3√0.349 ≈
39.8km/s/Mpc
qo
Ho

Ho= 63.7±2.3
ΛCDM
dAΛCDM =1
(1 + z)HoZz
0
1
qΩm(1 + x)3+ 1 −Ωm
dx .
dAde Sitter(z) = 1
−Hoqo(1 + z)((1 + z)−q1 + (1 + qo)z(2 + z)) .
z≈0.6
ΛCDM z ≈1.5