**2.1. Distances of Extragalactic Objects**

Our only means of obtaining information and therefore knowledge of the structure and dynamics of the Universe (on all different scales) are through the electromagnetic radiation that we receive. Therefore it is of primary importance to define a system of measuring luminosities taking also into account that the Universe expands and that light loses energy through a variety of processes.

If we assume that light propagates with no loss of energy, then the
apparent luminosity of a source *l*, is related to its absolute
luminosity *L*, by:

(36) |

where *r* is the distance to the source. We can see the extreme
importance of determining the pair (*l, L*), since such knowledge
would provide the distance of the source, *r*.

Due to historical mostly reasons we use a logarithmic brightness system by which an object with a magnitude of 1 is 100 times brighter than an object with a magnitude 6. We have:

(37) |

where *m* is the apparent magnitude and *M* the absolute
one. Therefore, using (36) we have:

(38) |

where *c*_{1,2} are constants which depend on the filter
used and *c*_{3}
is that value for which *m* = *m* at a distance of 10 parsecs (see
section 2.3) from the Earth, and thus
*c*_{3} = -5. In extragalactic
astronomy, instead of *pc*, we use *Mpc* and therefore (38)
becomes:

(39) |

The above definitions are somewhat *ideal*, since in the real world we
do not observe the total apparent magnitude, but that corresponding to
the particular range of spectral frequencies, that our detector is sensitive
to, and those allowed to pass through Earth's atmosphere. If we detect
*l*_{}
over a range of frequencies
( ±
), then the observed apparent
magnitude is *m* = -2.5 log_{10}
_{}
*l*sub>
d + *c*.
However, neither the atmosphere nor the detectors have a sharp
limit and therefore it is
better to model these effects by a sensitivity mask
*F*_{}, and the
observed apparent magnitude is then:

(40) |

If *F*_{} = 1
then the apparent (or absolute) magnitude of a source is called
the *bolometric* magnitude.

How do the above definitions change by taking into account the fact
that the Universe expands? To answer this we need a metric of
space-time, which in our case is the *Robertson-Walker*
metric. Since light
travels along null-geodesics, a fundamental concept of distance can be
defined by the corresponding light-travel time, which is called
*proper distance*. If a light signal is emitted at a galaxy
_{1} from the
coordinate position (*r*_{1},
_{0},
_{0}) at time
*t* = 0 and received by an observer at
_{0} at
(*r*_{0},
_{0},
_{0}), then these
events are connected only by the light signal and
since all observers must measure the same speed of light, it defines
a very fundamental concept of distance. Obviously, it depends
on the curvature of space, and since d*s* = 0 we have from (1):

(41) |

In the expanding Universe framework, the expressions (36) and (38) change, to:

(42) |

where *d*_{L}
*d*_{pro}(1 + *z*) is the *luminosity
distance*. It is obvious that the
distance measure of an extragalactic object depends on the underlying
Cosmology. A proper derivation of the luminosity distance (cf.
[183],
[112])
provides the following expression:

(43) |

where contains all
the contributions (mass, energy density, curvature).
Another important distance definition is that of the angular-diameter. It
is based on the fact that a length *l*, subtends a smaller angle,
,
the further away it is (*l*
1 /
) and it is given from:
relation to:

(44) |

Note that this notion of distance is used to derive the CMB power spectrum predictions of the different cosmological models (see section 3.1).