Chapter 3: THE FRIEDMANN MODELS
Chapter 3: THE FRIEDMANN MODELS
Chapter 3: THE FRIEDMANN MODELS
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Here S k (u) and C k (u) are analogous to S k (ω) and are sin, sinh, cos, cosh etc.<br />
depending on the value of k. Then<br />
2<br />
Sk<br />
( u)<br />
C<br />
dη<br />
= 2<br />
1−<br />
kS<br />
= 2S<br />
2<br />
k<br />
( u)<br />
du<br />
= k(1<br />
− C<br />
k<br />
⎛ S<br />
η = k⎜u<br />
−<br />
⎝<br />
2<br />
k<br />
( u)<br />
du<br />
( u)<br />
(2u))<br />
du<br />
k<br />
k<br />
(2u)<br />
⎞<br />
⎟<br />
2 ⎠<br />
Now making a final substitution of Θ = 2u, we get parametric solutions<br />
k<br />
η = ( Θ−Sk<br />
( Θ))<br />
2<br />
k<br />
ζ = ( −Ck<br />
2 1 ( Θ))<br />
So, we finally have for R and τ:<br />
(3.46)<br />
kGM<br />
R =<br />
2<br />
Ac<br />
kGM<br />
τ =<br />
3<br />
c<br />
{ 1−<br />
C ( Θ)<br />
}<br />
k<br />
{ Θ − S ( Θ)<br />
}<br />
k<br />
Note that dτ = AR/c dΘ. The parameter Θ is (as noted above) called the "conformal<br />
time" (although we will not make great use of it), and dω is proportional to dΘ for a<br />
light ray.<br />
3.6.2 Determining S k (ω) in the matter-dominated case<br />
We saw in the previous chapter how the effective distance D enters into almost all the<br />
relations between the intrinsic and observed properties of objects. Recall that D = R 0<br />
S k (ω). Thus we need to know S k (ω) or rather S k (z) since the redshift z is the only real<br />
observable related to distance.<br />
We want to get ω(z), which we can get by integrating dω/dz, which in turn can be<br />
obtained by recognizing that:<br />
dω<br />
dz<br />
dω<br />
dt<br />
= × ×<br />
dt dR<br />
We had from equation (3.21):<br />
dR<br />
dz<br />
(3.47)<br />
R&<br />
R<br />
2<br />
2<br />
= H ( 1+Ω<br />
z)<br />
2 0<br />
0<br />
0