Nearby Supernova Factory: Étalonnage des données de SNIFS et ...

tel-00372504, version 1 - 1 Apr 2009
CHAPTER 1. BIG BANG COSMOLOGY
1.3 The expanding universe
1.3.1 Cosmological redshift
Most methods of measuring the composition of the universe rely on various observations
of electromagnetic radiation. It is therefore important to discuss how the properties of light
detected from a source are connected to a certain cosmological model.
By observing the spectra of starlight of nearby galaxies we can identify known emission lines
and compare their wavelengths λ to the (known) emitted ones. These may appear displaced
towards the blue or the red part of the spectrum, respectively blushifted or redshifted, in what
may be interpreted as a Doppler effect in flat spacetime. That means that these galaxies are
moving towards or away from us, with a velocity v related to the shift in wavelength ∆λ by the
Doppler formula
z ≡ ∆λ
λ
v
c
, (1.9)
where z is the redshift and the second equality is valid for objects in our “neighborhood” (v ≪ c).
If we have now a particle moving in a spacetime geometry described by the time dependent
RW metric (1.2), its energy will change, similarly as it would if it moved in a time-dependent
potential. For a photon whose energy is proportional to frequency, that change in energy is
the cosmological redshift. We consider a photon emitted from a galaxy at comoving coordinate
r = R at time t. It will travel in a geodesic ds 2 = 0, so that in the time between emission and
reception the photon traveled a spatial coordinate distance (in the spatially flat case)
R =
t0
t
dt
a(t)
. (1.10)
If we suppose instead that a series of pulses are emitted with frequency ω = 2π
δt , the time
interval between the pulses at reception δt0 can be calculated from (1.10) since all pulses travel
the same spatial coordinate separation R:
and consequently
which means
t0+δt0
t+δt
t0
dt
= R =
a(t) t
dt
a(t)
, (1.11)
δt0 δt
− = 0 , (1.12)
a(t0) a(t)
ω0
ω
= λ
λ0
= a(t)
a(t0)
. (1.13)
Using (1.9) and a(t0) ≡ 1 we can write the definition of cosmological redshift
1
1 + z
≡ a(t) , (1.14)
and the redshift is thus a direct probe of the scale factor at the time of emission. While this
relation was derived for a spatially flat model, it holds for any homogeneous and isotropic model.
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