Master Thesis

4.4 ScalefactorandRedshift
For simplicity the flat space (4.37) will now be discussed. The considerations can also be extended to (4.34) and
(4.36). The space-time metric suggests, that there are comoving coordinates�r, which are blown up to the real physical
coordinates�x by a scale factor a(t):
�x(t)= a(t)�r , (4.38)
where a is set to a(t 0)=1 at the present time t 0. The radial distance and the radial velocity is then:
Both are related by Hubble’s Law:
In conformal time Hubble’s Law gets modified:
The expansion today is given by the Hubble constant:
R=|x(t)|= a(t)|�r|= a(t)r (4.39)
v=˙R= ˙a(t)|�r| . (4.40)
v=HR ⇒ H= ˙a
, (4.41)
a
� := da 1 da
= = Ha, (4.42)
dη a d t
H0 := H(t 0)= 70.1±1.3 km
. (4.43)
s M pc
This value is obtained from [21], including data from WMAP, SN 1a and BAO.
The deceleration at large distances ˙v=−AR is also important for cosmology, where A=−ä/a has to be rescaled by H2 0 ,
to get a dimensionless quantity in (4.79):
q0 :=− A äa
H(t) 2=− . (4.44)
˙a 2
The comoving distance gets stretched. The wavelength (and its frequency) of an emitted photon increases and it gets
red-shifted while it is traveling through the universe until absorption:
The redshift itself is defined by:
z := ν e−ν 0
ν 0
ν e
ν 0
= λ 0−λ e
λ e
= λ0 =
λe ∆r a(t 0)
∆r a(t e) = a(t 0)
a(t e)
= a(t 0)
a(t e) − 1 ⇒ 1+z= a(t 0)
a(t e)
From the frequency shift one also concludes that the proper time of an observed event is increased:
∆t 0
∆t e
= νe =
ν0 a(t 0)
a(t e)
(4.45)
(4.46)
(4.47)
On small distances the velocities given by Hubble’s law are small. The Doppler shift of a photon with wavelengthλ=cT
(T= oscillation period) can then be approximated by∆λ= vT≪λ . From (4.46) one obtains:
z= ∆λ
λ e
≈ ∆λ
λ
= vT
cT
= v
c
(4.48)
If v reaches c and so z reaches 1 this approximation is no longer correct, which corresponds to a length scale called
Hubble radius:
R H(t)= c
H(t)
, RH(t toda y)= c
= 4.28Gpc (4.49)
H0 20 4.4 Scale factor and Redshift