The fundamental assumption underlying the construction of cosmological models is that of the cosmological principle: The Universe appears spatially isotropic in all its properties to all observers. The only metric which is consistent with this principle is the Robertson-Walker metric:

(3.1) |

where *r* is the radial comoving coordinate, *a*(*t*) is
the dimensionless
scale factor by which all distances vary as a function of cosmic time,
and *R*_{o}^{-2} is a parameter with dimensions of
inverse length squared
that describes the curvature of the Universe and may be positive, zero,
or negative (see [14]
for a general discussion).

This is the geometry of the Universe, but dynamics is provided by
General Relativity - the Einstein field equations - which yield
ordinary differential equations for *a*(*t*). The time-time
component leads to a second order equation:

(3.2) |

where is the
density, *p* is the pressure and the quantity
in parenthesis is the active gravitational mass density.
Considering conservation of energy for a perfect fluid

(3.3) |

with an equation of state

(3.4) |

we have
*a*^{-1(1+w)}. The equation of state combined with
eq. 3.2 tells us that the Universe is accelerating if *w* < - 1/3.

The space-space components combined with the time-time component yield the usual first-order Friedmann equation

(3.5) |

where
*H* = / *a*
is the running Hubble parameter,
the summation is over the various fluids comprising the Universe and

(3.6) |

with _{k} = -
(*r*_{H} / *R*_{o})^{2}.
We often see eq. 3.5 written in terms of redshift where
*a* = (1 + *z*)^{-1}.
Each component has its own equation of state parameter,
*w*_{i}: *w* = 0 for non-relativistic matter (baryons,
CDM); *w* = 1/3 for radiation or other relativistic fluid;
*w* = -1 for a cosmological constant; and
-1 < *w* < -1/3 for "quintessence", dynamic dark energy
resulting in ultimate acceleration of the universal expansion.
I will not consider *w* < - 1 which has been termed
"phantom" dark energy
[15];
here the effective density increases as the Universe
expands (this could be realized by a ghost field, a scalar with a kinetic
term in the Lagrangian having the wrong sign so it rolls up rather than
down a potential hill).

Given a universe composed of radiation, non-relativistic matter, and quintessence, the Friedmann equation takes its familiar form:

(3.7) |

Here it is evident that radiation drives the expansion at early times
(*a* << 1), non-relativistic matter at later times, a
non-vanishing curvature
(_{k}
0) at later times still,
and, if *w* < -1/3,
the vacuum energy density ultimately dominates. For the purpose of this
lecture, I refer to eq. 3.7 with *w* = -1 (the usual cosmological
constant) as standard FRW cosmology, while
0 > *w* -1 would
represent a minimal modification to FRW
cosmology. Moreover, when *w* = -1, I replace
_{Q}
by _{}.
I will not consider changes to the
Friedmann equation which might result from modified gravitational physics.

Because the subject here is observational cosmology we must discuss the
operational definitions of distance in an FRW Universe. If there exists
a standard meter stick, an object with a known fixed linear size
*d* which does not evolve with cosmic time, then one could obviously
define an angular size distance:

(3.8) |

where would be the
observed angle subtended by this object.
If there exists a standard candle, an object with a
known fixed luminosity *L* which does not vary with cosmic time, then
one could also define a luminosity distance:

(3.9) |

where F is the measured flux of radiation.

For a RW universe both the angular size distance and the luminosity distance are related to the radial comoving coordinate,

(3.10) |

where
= *tH*_{o},
*R*_{o}^{2} = - *r*_{H}^{2} /
_{k}, and

Then it is the case that

(3.11a) |

and

(3.11b) |

It is evident that both the angular size distance and the luminosity
distance depend upon the expansion history (through
*d* /
*a*()) and
the curvature (through
(*x*)).

The same is true of a comoving volume element:

(3.12) |

where here *d*
is an incremental solid angle. Therefore, if there
exists a class of objects with a non-evolving comoving density, then
this leads to another possible cosmological test: simply count those
objects as a function of redshift or flux.

Below, I am going to consider these measures of distance and volume in the form of three classical cosmological tests:

1. Angular size tests which essentially involve the determination of
*D*_{A}(*z*). Here one measure
for objects with a
known and (hopefully) standard linear size (such as compact radio sources).

2. Luminosity distance tests which involve the measurement of
*F*(*z*) for
presumably standard candles (such as supernova type Ia, SNIa).

3. dV/dz test which involve the counts of very faint galaxies as a function of flux and redshift.

But before I come to these classic tests, I want to discuss the evidence supporting the validity of the standard hot Big Bang, as an appropriate description of the early pre-recombination Universe.