314 Highlights in modern observational cosmology
with
m++k= 1 ,tot=m+= 1 −k (11.6)
whereH 0 ≡(R ̇/R)t= 0 =100 km s−^1 Mpc−^1 h =h( 9. 78 × 109 )−^1 years,
is the present value of theHubble constant. The matter density parameter,
m(sometimes denoted as 0 ), can also be written asm=ρ 0 /ρcr,where
ρcr= 3 H 02 /( 8 πG)= 1. 9 × 10 −^29 h^2 gcm−^2 is the critical density, which splits
open and close models in a matter-dominated universe.
Thedeceleration parameteris also often used:
q≡−RR ̈ /R ̇^2 =m/ 2 −. (11.7)With these definitions, the equation (11.4) can be written:
H^2 =H 02
[
m(
R 0
R
) 3
+k(
R 0
R
) 2
+
]
. (11.8)
11.2.2 Observables in cosmology
Suppose we are atr =0 and observe an object at radial coordinater 1 ,when
the expansion factor wasR 1 =R(t 1 )<R 0 , at some lookback timet 1 <t 0.
Quantities liker 1 ,t 1 ,R 1 are not accessible to measurement. However, there are
directly measurable quantities which can be used to test the validity of the FRW
metric and to derive its parameters.
First of all, theredshift. From the spectrum of a distant source we can easily
recognize, say, an emission line whose rest-frame (emitted) wavelength isλe.In
general, we will measure a redshifted emission line at wavelengthλ 0 , so that the
redshift zis defined as
1 +z=λ 0
λe. (11.9)
If the expansion factor of the universe wasRat redshiftz, the following simple
relation holds:
1 +z=R 0
R
. (11.10)
Using this relation, we can now immediately write thelookback time,τ(z),
by integrating equation (11.8) after a change of variable, fromRtoz:
τ(z)=H 0 −^1∫z0( 1 +z′)−^1 [k( 1 +z′)^2 +m( 1 +z′)^3 +]−^1 /^2 dz′.(11.11)τ(z)is plotted in figure 11.1 for three different values of(m,).Theageof
the universe is obtained forz→∞.
We now examine the other measurable quantities.