SE6

The Big Bang model does not give a unique description of our present Universe, but rather
leaves quantities such as the present expansion rate, or the present composition of the Uni-
verse, to be fixed by observation.
It is a standard practice to specify cosmological models via a few parameters, which one
then tries to determine observationally to decide which version of the model best describes
our Universe. In this chapter and the next I’ll discuss the most commonly-considered pa-
rameters, including ones we have already seen and new ones.
6.1
The expansion rate H 0
The Hubble constant H 0 tells us the present expansion rate of the Universe, making it a
vital cosmological parameter as measurements of other properties of the Universe often
depend on it. It also ought to be the easiest to measure, since all galaxies are supposed to
obey v = H 0 r. So all we have to do is measure the velocities and distances of as many
galaxies as we can and get an answer. However, each measurement has its problems.
Velocities are given by the redshift of spectral lines, a measurement which is now easy
enough that the velocity of an individual galaxy can be measured to high accuracy. How-
ever, remember that the cosmological principle isn’t perfect, and so, as well as the uniform
expansion we are trying to measure, galaxies also have motions relative to one another, the
so-called peculiar velocity. The peculiar velocities are randomly oriented, and for a given
galaxy we caot split its measured velocity into the Hubble expansion and the peculiar ve-
locity. However, the cosmological principle does tell us that the typical size of the peculiar
velocity should not depend on where in the Universe the galaxy is. It is therefore indepen-
dent of distance, whereas the Hubble velocity is proportional to distance. If we look far
enough away (in practice many tens of megaparsecs) then the Hubble velocity dominates
and the (unknown) peculiar velocity can be ignored.
Given that the expansion velocity can only be accurately distinguished from the peculiar
velocity at large distances, we need to be able to estimate these large distances accurately
in order to carry out the calculation H 0 = v/r. These distances are much harder to obtain,

because galaxies are far too distant to be located by parallax. (Remember that an object
one parsec away has a parallax (i.e. an apparent motion when viewed from different parts
of Earth’s orbit) of one arcsecond, by definition. A galaxy many megaparsecs away will
have an immeasurably small parallax of less than a micro-arcsecond.) The usual method
is known as ‘standard candles’, where some type of object is assumed to have exactly the
same properties in all parts of the Universe. This is the cosmological equivalent of saying
that if one light bulb looks a quarter as bright as another, then from the inverse square law it
must be twice as far away – fine as long as you believe that all light bulbs have precisely the
same brightness. A classic example is the period–luminosity relation in cepheid variable
stars. The period of variability of those stars is readily measured, and there is reasonable
empirical evidence of a relation between the period and the luminosity of the stars which
lets us convert the measured period into a brightness. Other standard candles which have
been used include the brightness of certain types of supernovae, and the brightest galaxies
in galaxy clusters. All these methods have good success at determining relative distances
between two galaxies, which requires that the objects be good standard candles, and relative
distances are all that is needed to confirm the Hubble law.
However, to give an absolute distance, actually measuring the proportionality constant
H 0 , we also need a calibration against an object of known distance, which proves much
harder. In the light bulb analogy, to get relative distances we need only the inverse square
law and the belief that all bulbs have the same brightness; we don’t need to know how bright
the bulbs are. But to be able to say how far away a bulb of a given observed brightness is,
we need to know its absolute brightness.
It is only relatively recently that the calibration problem has begun to come under con-
trol, principally through Hubble Space Telescope observations. The Hubble constant is
usually parametrized as
H 0 = 100 h km s −1 Mpc −1 .
(6.1)
In the 1990s the Hubble Space Telescope Key Project, led by Wendy Freedman, gave
h = 0.72 ± 0.08 ,
(6.2)
where the uncertainty is a one-sigma error (meaning it should be doubled to indicate 95
per cent confidence, at least if the uncertainty is approximately gaussian-distributed). This
was considered the first precision direct measurement of the Hubble constant, and has sub-
sequently been improved to h = 0.75 ± 0.03 by the SH0ES (Supernovae and H 0 for the
Equation of State) project.
On the other hand, indirect measurements, which assess the impact of the expansion
rate on other cosmological observables such as the cosmic microwave background, have
suggested a somewhat lower value, the Planck satellite observations described in Advanced
Topic 6 giving h = 0.673 ± 0.012. There is mild tension between these measurements
which is currently under investigation, but it seems safe to conclude that h is now quite
accurately measured to be 0.70 within at most a few per cent. An object with a recession
velocity of 7000 km s −1 would be expected to be at a distance v/H 0 = 100 Mpc. The long-
standing problem of determining the overall scale of the Universe is therefore essentially
solved!

6.2
51
The density parameter Ω 0
The density parameter is a very useful way of specifying the density of the Universe. Let’s
start with the Friedma equation again. Recalling that H = a/a,
̇
it reads
H 2 =
8πG
k
ρ − 2 .
3
a
(6.3)
For a given value of H, there is a special value of the density which would be required
in order to make the geometry of the Universe flat, k = 0. This is known as the critical
density ρ c , which we see is given by
ρ c (t) =
3H 2
.
8πG
(6.4)
Note that the critical density changes with time, since H does. As we know the present
value of the Hubble constant (at least in terms of h defined in Equation (6.1)), we can com-
pute the present critical density. Since G = 6.67 × 10 −11 m 3 kg −1 sec −2 , and converting
megaparsecs to metres using conversion factors quoted on page xiv, it is
ρ c (t 0 ) = 1.88 h 2 × 10 −26 kg m −3 .
(6.5)
This is a startlingly small number; compare for example the density of water which is
10 3 kg m −3 . If there is any more matter than this apparently tiny amount, it is enough to tip
the balance beyond a flat universe to a closed one with k > 0. So only a very tiny density
of matter is needed in order to provide enough gravitational attraction to halt and reverse
the expansion of the Universe.
However, let us write that another way, since kilograms and metres are rather small and
inconvenient units for dealing with something as big as the Universe. Let’s try measuring
masses in solar masses and distances in megaparsecs. It becomes
ρ c (t 0 ) = 2.78 h −1 × 10 11 M /(h −1 Mpc) 3 .
(6.6)
Suddenly this doesn’t look so small. In fact, 10 11 to 10 12 solar masses is about the mass
of a typical galaxy, and a megaparsec more or less the typical galaxy separation, so the
Universe caot be far away (within an order of magnitude or so) from the critical density.
Its density really must be around 10 −26 kg m −3 .
Be sure to understand that the critical density is not necessarily the true density of the
Universe, since the Universe need not be flat. However, it sets a natural scale for the density
of the Universe. Consequently, rather than quote the density of the Universe directly, it is
often useful to quote its value relative to the critical density. This dimensionless quantity
is known as the density parameter Ω, defined by
Ω(t) ≡
ρ
.
ρ c

Again, in general Ω is a function of time, since both ρ and ρ c depend on time. The present
value of the density parameter is denoted Ω 0 .
With this new notation, we can rewrite the Friedma equation in a very useful form.
Substituting in for ρ in Equation (6.3) using the definitions I have made, Equations (6.4)
and (6.7), leads to
H 2 =
8πG
k
k
ρ c Ω − 2 = H 2 Ω − 2 ,
3
a
a
(6.8)
and rearranging gives
Ω − 1=
k
.
a 2 H 2
(6.9)
We see that the case Ω = 1 is very special, because in that case k must equal zero and since
k is a fixed constant this equation becomes Ω = 1 for all time. That is true independent
of the type of matter we have in the Universe, and this is often called a critical-density
universe. When Ω = 1, this form of the Friedma equation is very useful for analysing
the evolution of the density, as we will see later in the chapter on inflationary cosmology.
Our Universe contains several different types of matter, and this notation can be used
not just for the total density but also for each individual component of the density, so one
talks of Ω mat , Ω rad etc. Some cosmologists even define a ‘density parameter’ associated
with the curvature term, by writing
Ω k ≡ −
k
.
a 2 H 2
(6.10)
This can be positive or negative, and using it the Friedma equation can be written as
Ω + Ω k = 1 .
(6.11)
We’ll return to the observational status of Ω 0 in Chapter 9.
6.3 The deceleration parameter q 0
As we’ve discovered, not only is the Universe expanding, but also the rate at which it
is expanding, given by the Hubble parameter, is changing with time. The deceleration
parameter is a way of quantifying this.
Consider a Taylor expansion of the scale factor about the present time. The general
form of this (with dots as always indicating time derivatives) is
1
a(t) = a(t 0 ) + a(t
̈ (t 0 ) [t − t 0 ] 2 + · · · .
̇ 0 ) [t − t 0 ] + a
2
(6.12)
Let’s divide through by a(t 0 ). Then the coefficient of the [t − t 0 ] term will just be the
present Hubble parameter, and we can write
q 0
a(t)
2
= 1 + H 0 [t − t 0 ] − H 0 2 [t − t 0 ] + · · · ,
a(t 0 )
2
(6.13)

which defines the deceleration parameter q 0 as
q 0 = −
a
̈ (t 0 ) 1
a(t 0 ) ̈
a (t 0 )
= − 2
.
a(t 0 ) H 0 2
a ̇ (t 0 )
(6.14)
The larger the value of q 0 , the more rapid the deceleration.
The simplest situation is if the Universe is matter dominated, p = 0. Remember that by
‘matter’ we mean any pressureless material; it could be a collection of elementary particles,
or equally well a collection of galaxies. Then from the acceleration Equation (3.18) and
the definition of critical density, Equation (6.4), we find
q 0 =
3
4πG
Ω 0
ρ
.
=
3
8πGρ c
2
(6.15)
So in this case, a measurement of q 0 would immediately tell us Ω 0 .
If we know the properties of the matter in the Universe, then q 0 is not independent of the
first two parameters we have discussed, H 0 and Ω 0 . Those two are sufficient to describe all
the possibilities. However, we don’t know everything about the material in the Universe,
so q 0 can provide a new way of looking at the Universe. It can in principle be measured
directly by making observations of objects at very large distances, such as the numbers of
distant galaxies, because the deceleration governs how large the Universe would be at an
earlier time.
In the late 1990s, the first convincing measurements of q 0 were made by two research
groups studying distant supernovae of a class known as type Ia, which are believed to be
good standard candles. To widespread surprise at the time, the result is that the Universe is
accelerating at present, q 0 < 0. 1 None of the cosmological models that we have discussed
so far is capable of satisfying this condition, as can be seen directly from the acceleration
Equation (3.18). This result is becoming firmly established, and is amongst the most dra-
matic observational results in modern cosmology. The following chapter discusses how to
extend our simple cosmological models to account for it.