19. Big-Bang cosmology
1
19. BIG-BANG COSMOLOGY
Revised September 2009 by K.A. Olive (University of Minnesota) and J.A. Peacock
(University of Edinburgh).
19.1. Introduction to Standard Big-Bang Model
The observed expansion of the Universe [1,2,3] is a natural (almost inevitable) result of
any homogeneous and isotropic cosmological model based on general relativity. However,
by itself, the Hubble expansion does not provide suﬃcient evidence for what we generally
refer to as the Big-Bang model of cosmology. While general relativity is in principle
capable of describing the cosmology of any given distribution of matter, it is extremely
fortunate that our Universe appears to be homogeneous and isotropic on large scales.
Together, homogeneity and isotropy allow us to extend the Copernican Principle to the
Cosmological Principle, stating that all spatial positions in the Universe are essentially
equivalent.
The formulation of the Big-Bang model began in the 1940s with the work of George
Gamow and his collaborators, Alpher and Herman. In order to account for the possibility
that the abundances of the elements had a cosmological origin, they proposed that
the early Universe which was once very hot and dense (enough so as to allow for the
nucleosynthetic processing of hydrogen), and has expanded and cooled to its present
state [4,5].
In 1948, Alpher and Herman predicted that a direct consequence of this
model is the presence of a relic background radiation with a temperature of order a few
K [6,7]. Of course this radiation was observed 16 years later as the microwave background
radiation [8].
Indeed, it was the observation of the 3 K background radiation that singled
out the Big-Bang model as the prime candidate to describe our Universe. Subsequent
work on Big-Bang nucleosynthesis further conFrmed the necessity of our hot and dense
past. (See the review on BBN—Sec. 20 of this
Review
for a detailed discussion of BBN.)
These relativistic cosmological models face severe problems with their initial conditions,
to which the best modern solution is inﬂationary cosmology, discussed in Sec. 19.3.5. If
correct, these ideas would strictly render the term ‘Big Bang’ redundant, since it was
Frst coined by Hoyle to represent a criticism of the lack of understanding of the initial
conditions.
19.1.1.
The Robertson-Walker Universe
:
The observed homogeneity and isotropy enable us to describe the overall geometry
and evolution of the Universe in terms of two cosmological parameters accounting for
the spatial curvature and the overall expansion (or contraction) of the Universe. These
two quantities appear in the most general expression for a space-time metric which has a
(3D) maximally symmetric subspace of a 4D space-time, known as the Robertson-Walker
metric:
ds
2
=
dt
2
−
R
2
(
t
)
·
dr
2
1
−
kr
2
+
r
2
(
dθ
2
+sin
2
θdφ
2
)
¸
.
(19
.
1)
Note that we adopt
c
= 1 throughout. By rescaling the radial coordinate, we can choose
the curvature constant
k
to take only the discrete values +1,
−
1, or 0 corresponding
to closed, open, or spatially ﬂat geometries. In this case, it is often more convenient to