Mesure de la constante de Hubble par lentille gravitationnelle

H0
60 80 km.s−1.Mpc−1
W MAP 70.8±1.6km.s−1.Mpc−1

M
ds2=−c2dt2+dr2+r2dΩ2
ds2=−eν(r)dt2+eλ(r)dr2+r2dΩ2
Rµν −1
2Rgµν = 0 ⇔Rµν = 0
ds2=−1−2GM
c2rc2dt2+dr2
(1−2GM
c2r)+r2dΩ2
k=∂t
E=gµν kµuν=1−2GM
c2r˙
t
k0=∂ϕ
Lz=gµν k0µuν=r2sin2θ˙ϕ
θ=π/2˙ θ= 0 Lz=L=r2˙ϕ
||u||2= 0
||u||2=gµν uµuν=−1−2GM
c2rc2˙
t2+˙r2
(1−2GM
c2r)+r2˙ϕ2

˙r2=E2−L2
r21−2GM c2r
r0
δϕ = 2 (ϕ(∞)−ϕ(r0)) = ∆ϕ−π∆ϕ= 2 ´∞
r0
dϕ
dr dr = 2 ´∞
r0
dϕ
dλ
dλ
dr dr = 2 ´∞
r0
L
r2E2−L2 r21−2GM c2r−1
2dr
u=1
r
GM
c2∆ϕ(M) =
∆ϕ(0) + Md∆ϕ
dM mM=0
δϕ =4GM
r0c2
r=p
1+ecosθ δϕNR =
2GM
r0c2=1 2δϕ
S B
O x SB

B
r1/2=1
2nx±px2+ 4n2r2
0'x
2n1±α β
α=pα2
0+β2β=x
nabr0=q4GM
c2nabα0= 2q4GM
c2nab
n=as
as−ab
α'α0
b δϕ =b
f0
f0=c2r2
0
4GM
L LN
Li=n2ri
x
dri
dx LN
dr1/2
dx =1 2n1±x
√x2+4n2r2
0'1
2n1±β α=β α
r1/2
x
1 2
L1=1
42 + α β+β αLNL2=1 4−2 + α β+β αLN
LT=L1+L2LT'α0
2βLNβ1
rSrB
rS<|r1/2|
u=rS
aS
β= 0 u
LT'α0
uLN
0< β < u
LT'α0
u1−1 4
β2
u2LN
β > u LT'α0
2β1 + 1 8
u2
β2LN

t1t+ ∆t2
β1
∆t=c−1´X
0αdX 'naBα0βc−1
L1(t)
L2(t+∆t)=r2
1
r2
2
=α2
1
α2
2
abas
B S
H=zbc
ab=zsc asn=as as−ab=zs zs−zb
∆tc
ab=nα0β ∆t
H=zbzs
zs−zbα0α1−α2 ∆t
L1(t)
L2(t+∆t)=α2
1
α2
2
α1−α2=α√L1/L2−1
√L1/L2+1