and Cosmology

4. Cosmology I: Homogeneous Isotropic World Models
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the angular-diameter distance on the Earth’s surface, we
define D A (D) = L/ϕ = R sin(D/R), in analogy to the
definition (4.45). For values of D that are considerably
smaller than the curvature radius R of the sphere, we
therefore obtain that D A ≈ D, whereas for larger D, D A
deviates considerably from D. In particular, D A is not
a monotonic function of D, rather it has a maximum at
D = πR/2.
There exists a general relation between angulardiameter
distance and luminosity distance,
D L (z) = (1 + z) 2 D A (z) . (4.48)
Fig. 4.11. Angular-diameter distance vs. redshift for different
cosmological models. Solid curves display models with no
vacuum energy; dashed curves show flat models with Ω m +
Ω Λ = 1. In both cases, results are plotted for Ω m = 1, 0.3,
and 0
famous Mattig relation applies,
D A (z) = c 2
H 0 Ωm 2 (1 + (4.47)
[
z)2 (√ )]
× Ω m z + (Ω m − 2) 1 + Ωm z − 1 .
In particular, D A is not necessarily a monotonic function
of z. To better comprehend this, we consider the
geometry on the surface of a sphere. Two great circles
on Earth are supposed to intersect at the North Pole
enclosing an angle ϕ ≪ 1 – they are therefore meridians.
The separation L between these two great circles,
i.e., the length of the connecting line perpendicular to
both great circles, can be determined as a function of
the distance D from the North Pole, which is measured
as the distance along one of the two great circles. If
θ is the geographical latitude (θ = π/2 at the North
Pole, θ =−π/2 at the South Pole), L = Rϕ cos θ is
found, where R is the radius of the Earth. L vanishes
at the North Pole, attains its maximum at the equator
(where θ = 0), and vanishes again at the South Pole;
this is because both meridians intersect there again.
Furthermore, D = R(π/2 − θ), e.g., the distance to the
equator D = Rπ/2 is a quarter of the Earth’s circumference.
Solving the last relation for θ, the distance is then
given by L = Rϕ cos(π/2 − D/R) = Rϕ sin(D/R).For
The reader might now ask which of these distances
is the correct one? Well, this question does not make
sense since there is no unique definition of the distance
in a curved spacetime like our Universe. Instead,
the aforementioned measurement prescriptions must be
used. The choice of a distance definition depends on
the desired application of this distance. For example,
if we want to compute the linear diameter of a source
with observed angular diameter, the angular-diameter
distance must be employed because it is defined just
in this way. On the other hand, to derive the luminosity
of a source from its redshift and observed flux, the
luminosity distance needs to be applied. Due to the definition
of the angular-diameter distance (length/angular
diameter), those are the relevant distances that appear
in the gravitational lens equation (3.48). A statement
that a source is located “at a distance of 3 billion light
years” away from us is meaningless unless it is mentioned
which type of distance is meant. Again, in the
low-redshift Universe (z ≪ 1), the differences between
different distance definitions are very small, and thus
it is meaningful to state, for example, that the Coma
cluster of galaxies lies at a distance of ∼ 90 Mpc.
In Fig. 4.12 a Hubble diagram extending to high redshifts
is shown, where the brightest galaxies in clusters
of galaxies have been used as approximate standard
candles. With an assumed constant intrinsic luminosity
for these galaxies, the apparent magnitude is a measure
of their distance, where the luminosity distance D L (z)
must be applied to compute the flux as a function of
redshift.
Without derivation, we compile several expressions
that are required to compute distances in general
Friedmann–Lemaître models. To do this, we need to