5.4. COSMOLOGICAL PARAMETERS 79
A crucial assumption is the cosmological principle:
e Universe is assumed to be—on the largest of scales—homogeneous and isotropic.
To say the universe is homogeneous means that it is the same everywhere. To say
that is isotropic is to say that it is the same in every direction.
e cosmological principal is a starting point for doing cosmology. If the consequences
of assuming the cosmological principal to be true turn out to be in serious conflict with obser-
vations, then we will have to drop those assumptions. But so far, and perhaps surprisingly, there
has been no compelling evidence that the cosmological principal is incorrect.
5.4 COSMOLOGICAL PARAMETERS
A particular cosmological model applies the laws of physics, as best we understand them, to the
universe as a whole, following along with the changes that would occur. ese physical laws
alone are not enough, one must still choose the value of various parameters—numerical quan-
tities that have no known particular theoretical value, but that greatly affect the outcome of
the cosmological model. Some of these parameters can be directly measured with astronomical
observations. Others are adjusted such that the model achieves the best agreement with actual
observations of the universe. And so a particular cosmological model is a combination of the
physical laws and a particular set of cosmological assumptions, with parameters adjusted so as
to accommodate the best agreement with many different observations.
In Table 5.2 I list some modern estimates of several of these parameters, and I describe
each of them below. All of the parameters I list here are meant to describe the present state of the
universe. e cosmological model then assumes these values, while applying the laws of physics
to describe the universe in both the past and the future. In the sections that follow, I describe
each of these parameters in turn.
5.4.1 THE HUBBLE PARAMETER (H
0
) AND HUBBLE’S LAW
All galaxies, except for the few that are relatively close to us, show a redshift in their spectrum.
e spectrum of most galaxies show absorption lines (some also show emission lines), and these
lines are inevitably all shifted, by the same percentage, toward longer wavelengths.
is is just what one would expect from the Doppler effect, described in Section 4.5. And
so it would seem to mean that all galaxies are moving away from us. e redshift, z, is defined
in terms of the shift in wavelength, , as follows:
z D
D
observed
´
0
0
; (5.12)
where
0
is the known wavelength as measured in the laboratory—presumably the actual wave-
length emitted by the galaxy. And so z represents the fractional shift in the observed wavelength.
80 5. THE PRESENT
Table 5.2: Cosmological parameters that represent the present state of the universe. e mea-
sured values are taken from Planck Collaboration et al. [2016, Table 4, column 3].
Name Symbol Value
Hubble parameter
H
0
67.9 km s
-1
Mpc
-1
Current Age of Universe
t
0
13.8 × 10
9
years
Baryon Density
Ω
b
0.048
Dark Matter Density
Ω
c
0.26
Dark Energy Density
Ω
λ
0.69
Current CMB Temperature
T
2.73 K
Current average hydrogen abundance
X
0.74
Current average helium abundance
Y
0.23
Current average metallicity
Z
0.02
If we interpret the redshift as arising from the Doppler effect, then we can combine Equa-
tions (5.12) and (4.2) to calculate the speed, v, at which a given galaxy with measured redshift
is receding from us:
v D zc: (5.13)
In the years between 1912 and 1929, Vesto Slipher and Edwin Hubble measured the
redshifts of many galaxies. Combined with estimates of the distances to these galaxies, a famous
relation became apparent, now known as Hubble’s Law:
For galaxies that are outside of our local neighborhood, but still less than several
billion light years away, the redshift of a galaxy is directly proportional to its distance.
And so combining this observation with Equation (5.13), we can express Hubble’s law with the
following simple equation:
v D H
0
d; (5.14)
where d is the distance to the galaxy and H
0
is the Hubble parameter or Hubble constant.
From Equation (5.14) it is clear that the Hubble parameter has dimensions of speed di-
vided by length, and it is usually expressed in km/s of velocity per megaparsec of distance. ese
odd units are chosen to be in line with typical measurements of both v and d for galaxies. In
order to determine the Hubble parameter, one must measure both the redshifts and distances of
many galaxies. ese two values can then be plotted on a graph with v (in km/s) on the vertical
axis and d (in Mpc) on the horizontal axis; H
0
is then the simply the slope of this graph. e
redshift is the easy part; it is the distance that is difficult to measure.
A modern example of this velocity-distance relation or Hubble diagram can be seen in Fig-
ure 5.1. Note that the galaxies in the Virgo cluster show a large spread of velocities for a given
5.4. COSMOLOGICAL PARAMETERS 81
2,000
1,000
0
0 10 20
H
0
= 68 km/s Mpc
Virgo Cluster
Distance (Mpc)
Redshift (km/s)
Figure 5.1: e Hubble diagram for relatively nearby galaxies. e slope gives the Hubble con-
stant. (Graphic by Brews ohare, CC BY-SA 3.0.)
distance. is is because their motions are not only due to Hubble’s law, but also simply because
they are in a large galaxy cluster, and they move according to the gravitational forces they feel
for each other. is brings up an important point. Hubble’s law applies to motions on the large
scale—over small distances galaxies move according to the local gravitational forces they feel for
each other.
Equation (5.14) is commonly used to describe Hubble’s law, and it is the way Hubble
presented it in 1929, before the birth of modern cosmology [see Ryden, 2017, p. 12ff]. But the
modern cosmologist is more careful; it is not the velocity of the galaxy that is directly measured,
but rather its redshift. is redshift is only a velocity if we assume that it is directly caused by
a straightforward Doppler effect. At relatively small distances, this assumption works, but in
general it is incorrect—the redshifts of galaxies are instead caused by the expansion of space
itself.
e value of the Hubble parameter can be measured or inferred by many different meth-
ods. See Figure 5.2 for the recent history of measurements. e most precise measurements
(the smallest error bars) are in reasonable agreement with each other, for a Hubble parameter of
about 70 km/s per Mpc. Values from many different methods vary between about 67–74 km/s
per Mpc.
5.4.2 THE EXPANSION OF THE UNIVERSE AND THE BIG BANG
e simplest explanation for Hubble’s law, in accordance with the cosmological principal, is that
the universe is undergoing a uniform expansion. at is to say, over a given period of time, all
distances in the universe increase by the same percentage.
82 5. THE PRESENT
Figure 5.2: Different measurements of the Hubble constant over recent years. e vertical bars
represent the uncertainty in the measurements. e values are in fair agreement with each other,
even though they use a variety of independent methods. (Graphic by Kintpuash—Own work,
CC0.)
For a good analogy, make some raisin bread. Knead the dough and let it rise. As the
dough expands, it carries the raisins with it, and every raisin would see all of the other raisins
getting farther and farther away. e raisins themselves do not expand; they are simply carried
along by the expanding dough-space. Furthermore, more distant raisins recede faster because
the expansion is proportional. If, for example, the rising dough doubles in size in three hours,
then a raisin initially 1 cm away will now be 2 cm distant—and so it would have receded at a
rate of 2 ´ 1 D 1 cm per three hours, or 0:33 cm h
´1
. By the same logic, however, a raisin that
is 10 cm distant initially would have receded at the proportionally larger speed of 3:3 cm h
´1
.
And so tiny astronomers living on the raisin would measure a raisin bread version of Hubble’s
law with their tiny telescopes.
4
Note that this uniform expansion preserves the cosmological principal. Tiny astronomers
on any rasin would observe the same thing; the raisin bread universe is homogenous. And it only
matters how far apart two raisins are, not their directions, so our expanding-dough universe
would also be isotropic. e model of a uniformly expanding universe is the simplest explanation
for the observation of Hubble’s law, if we assume the cosmological principal is correct.
A flaw in our raisin bread analogy is that the dough has an edge, while the universe does
not. And space and time are not connected together in this raisin bread universe, in the profound
and strange way that they are in the real universe.
Given that the universe is expanding, a simple explanation is that it means just what it
seems to; everything is getting farther apart as time passes. And so in the past everything was
4
Presumably their telescopes would operate at wavelengths that could see through dough.
5.4. COSMOLOGICAL PARAMETERS 83
closer together. Furthermore, there would have been a time when everything was all in the same
place at once. is is the basic idea behind the Hot Big Bang:
e universe was once infinitely hot and dense, and it has been expanding and cooling
ever since. us, there was a t D 0 when the density of the universe was infinite. at
is, the universe has a finite age.
And so our expanding Big Bang universe is the same in every direction, and the same
everywhere. But it is not the same everywhen.
5.4.3 THE CURRENT AGE OF THE UNIVERSE (t
0
)
Although the Hubble parameter is usually expressed in km s
´1
Mpc
´1
, it has units of inverse
time hidden within it; to see this, simply note that it is a length/time per length. We can thus
express the reciprocal of the Hubble parameter, 1/H
0
, in years. is is called the Hubble time,
t
H
, and converting the units in a convenient way, it is given as follows:
t
H
(billions of years) D
976
H
0
(km s
´1
Mpc
´1
)
: (5.15)
For H
0
D 67:7, this gives a Hubble time of t
H
D 14:4 billion years.
e Hubble time represents the age of the universe if it were to expand at a uniform rate
throughout its history. In the absence of additional information, this would be a reasonable first
guess for the actual age of the universe. But we do not expect this to be the actual age, for the
simple reason that we have good reasons to believe the universe did not expand at a uniform
rate. A full cosmological model is required to make a good estimate of the current age of the
universe. We use t
0
to denote this current age, and our best value as of 2019 is t
0
D 13:8 billion
years, just a bit less than the Hubble time.
5.4.4 THE BARYON DENSITY (
b
)
e so-called baryon density refers to the average density of “ordinary” matter in the universe.
is is matter made ultimately of familiar particles such as protons, neutrons and electrons—the
basic constituents of atoms. It is a little bit of a misnomer, because technically electrons are not
baryons; they belong to a different category of particles called leptons. But protons and neutrons
have nearly 2000 times the mass of electrons, so it is perhaps not too much of an oversight to
use a name that, technically, should not include electrons. In practice, “baryon density” simply
means the density of what is usually called “matter.”
As is the case with Hubble’s law, we mean the density on average over a large range of
distances that encompasses not only galaxies or even clusters of galaxies, but superclusters as
well. e current estimate is that the baryon density makes up only about 4.8% of the universe.
And so even though the baryon density represents the kind of matter that matters most to us—it
is related to all that we see and touch—it makes up only 4.8% of the universe. What is the rest?
See Sections 5.4.5 and 5.4.6.
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