Cosm-2

For most of history, astronomers have had to rely on light in the visible part of the spec-
trum in order to study the Universe. One of the great astronomical achievements of the
20th century was the exploitation of the full electromagnetic spectrum for astronomical
measurements. We now have instruments capable of making observations of radio waves,
microwaves, infrared light, visible light, ultraviolet light, X-rays and gamma rays, which
all correspond to light waves of different (in this case increasing) frequency. We are even
entering an epoch where we can go beyond the electromagnetic spectrum and receive in-
formation of other types. A remarkable feature of observations of a nearby supernova in
1987 was that it was also seen through detection of neutrinos, an extraordinarily weakly
interacting type of particle normally associated with radioactive decay. Very high energy
cosmic rays, consisting of highly-relativistic elementary particles, are now routinely de-
tected, though as yet there is no clear understanding of their astronomical origin. And as
I write, experiments appear close to the goal of directly detecting gravitational waves, rip-
ples in space–time itself, and ultimately of using them to observe astronomical events such
as colliding stars.
The advent of large ground-based and satellite-based telescopes operating in all parts
of the electromagnetic spectrum has revolutionized our picture of the Universe. While
there are probably gaps in our knowledge, some of which may be important for all we
know, we do seem to have a consistent picture, based on the cosmological principle, of
how material is distributed in the Universe. My discussion here is brief; for a much more
detailed discussion of the observed Universe, see Rowan-Robinson’s book Cosmology (full
reference in the Bibliography). A set of images, including full-colour versions of the figures
in this chapter, can be found via the book’s Home Page as given in the Preface.
2.1 In visible light
Historically, our picture of the Universe was built up through ever more careful observations
using visible light.

Figure 2.1 If viewed from above the disk, our own Milky Way galaxy would probably resem-
ble the M100 galaxy, imaged here by the Hubble Space telescope. (Figure courtesy NASA.)
Stars: The main source of visible light in the Universe is nuclear fusion within stars. The
Sun is a fairly typical star, with a mass of about 2 × 10 30 kilograms. This is known
as a solar mass, indicated M , and is a convenient unit for measuring masses. The
nearest stars to us are a few light years away, a light year being the distance (about
10 16 metres) that light can travel in a year. For historical reasons, an alternative
unit, known as the parsec and denoted ‘pc’, 1 is more commonly used in cosmology.
A parsec equals 3.261 light years. In cosmology, one seldom considers individual
stars, instead preferring to adopt as the smallest considered unit the conglomerations
of stars known as ...
Galaxies: Our Solar System lies some way off-centre in a giant disk structure known as
the Milky Way galaxy. It contains a staggering hundred thousand million (10 11 ) or so
stars, with masses ranging from about a tenth that of our Sun to tens of times larger.
It consists of a central bulge, plus a disk of radius 12.5 kiloparsecs (kpc, equal to
10 3 pc) and a thickness of only about 0.3 kpc. We are located in the disk about 8
kpc from the centre. The disk rotates slowly (and also differentially, with the outer
edges moving more slowly, just as more distant planets in the Solar System orbit
more slowly). At our radius, the galaxy rotates with a period of 200 million years.
Because we are within it, we can’t get an image of our own galaxy, but it probably
looks not unlike the M100 galaxy shown in Figure 2.1.
1 A parsec is defined as the distance at which the mean distance between the Earth and Sun subtends a second
of arc. The mean Earth–Sun distance (called an Astronomical Unit) is 1.496 × 10 11 m, and dividing that by
tan(1 arcsec) gives 1 pc = 3.086 × 10 16 m.

Our galaxy is surrounded by smaller collections of stars, known as globular clusters.
These are distributed more or less symmetrically about the bulge, at distances of 5–
30 kpc. Typically they contain a million stars, and are thought to be remnants of the
formation of the galaxy. As we shall discuss later, it is believed that the entire disk
and globular cluster system is embedded in a larger spherical structure known as the
Galactic halo.
Galaxies are the most visually striking and beautiful astronomical objects in the Uni-
verse, exhibiting a wide range of properties. However, in cosmology the detailed
structure of a galaxy is usually irrelevant, and galaxies are normally thought of as
point-like objects emitting light, often broken into sub-classes according to colours,
luminosities and morphologies.
The local group: Our galaxy resides within a small concentrated group of galaxies known
as the local group. The nearest galaxy is a small irregular galaxy known as the Large
Magellanic Cloud (LMC), which is 50 kpc away from the Sun. The nearest galaxy
of similar size to our own is the Andromeda Galaxy, at a distance of 770 kpc. The
Milky Way is one of the largest galaxies in the local group. A typical galaxy group
occupies a volume of a few cubic megaparsecs. The megaparsec, denoted Mpc and
equal to a million parsecs, is the cosmologist’s favourite unit for measuring distances,
because it is roughly the separation between neighbouring galaxies. It equals 3.086×
10 22 metres.
Clusters of galaxies, superclusters and voids: Surveying larger regions of the Universe,
say on a scale of hundreds of Mpc, one sees a variety of large-scale structures, as
shown in Figure 2.2. This figure is not a photograph, but rather a carefully con-
structed map of the nearby region of our Universe, on a scale of about 1:10 27 ! In
some places galaxies are clearly grouped into clusters of galaxies.
A famous example of a cluster of galaxies is the Coma cluster, which is about 100
Mpc away from our own Galaxy. The upper panel of Figure 2.3 shows a combined
optical/infrared image of Coma; although the image resembles a star field, almost
every source is a distinct galaxy (the main exception being two bright stars in the
upper right quadrant). Coma contains perhaps 10 000 galaxies, mostly too faint to
show in this image, orbiting in their common gravitational field.
However, most galaxies, sometimes called field galaxies, are not part of a cluster.
Galaxy clusters are the largest gravitationally-collapsed objects in the Universe, and
they themselves are grouped into superclusters, joined by filaments and walls of
galaxies. In between this ‘foamlike’ structure lie large voids, some as large as 50 Mpc
across. Structures in the Universe will be further described in Advanced Topic 5. Fig-
ure 2.4 shows an example of a computer simulation aiming to model the distribution
of material within the Universe.
Large-scale smoothness: Only once we get to scales of hundreds of megaparsecs or more
does the Universe begin to appear smooth, as revealed by extremely large galaxy
surveys such as the 2dF galaxy redshift survey and the Sloan Digital Sky Survey.
Such surveys do not find any huge structures on scales greater than those described
above; the galaxy superclusters and voids are believed to be the biggest structures in
the present Universe.

Figure 2.2 A map of galaxy positions in a narrow slice of the Universe, as measured by the
Sloan Digital Sky Survey. Our galaxy is located at the centre, and the survey radius is around
600 Mpc. The galaxy positions were obtained by measurement of the shift of spectral lines,
as described in Section 2.4. (Figure courtesy M. Blanton and the Sloan Digital Sky Survey,
www.sdss3.org.)
The belief that the Universe does indeed become smooth on the largest scales, the
cosmological principle, is the underpiing of modern cosmology. It is interesting
that while the smoothness of the matter distribution on large scales has been a key
assumption of cosmology for decades now, it is only fairly recently that it has been
possible to provide a convincing observational demonstration.
2.2 In other wavebands
Observations using visible light provide us with a good picture of what’s going on in the
present-day Universe. However, many other wavebands make vital contributions to our
understanding, and in particular our best knowledge of cosmology comes not from visible
light but from microwaves.
Microwaves: For cosmology, this is by far the most important waveband. Penzias & Wil-
son’s accidental discovery in 1965 that the Earth is bathed in microwave radiation,
with a black-body spectrum at a temperature of around 3 Kelvin, was and is one of

Figure 2.3 Images of the Coma cluster of galaxies in visible/infrared light (top) and in
X-rays (bottom), the latter being on a larger angular scale. Colour versions can be found on
the book’s WWW site. (Images courtesy of NASA/Spitzer satellite and ESA/U. Briel/MPE
Garching/XMM-Newton satellite.)

Figure 2.4 A computer simulation showing the predicted distribution of matter in the Uni-
verse on large scales. (Image courtesy V. Springel and the Virgo Consortium.)
the most powerful pieces of information in support of the Big Bang theory, around
which cosmology is based. This is now known as the cosmic microwave background
or simply CMB. Observations by the FIRAS (Far InfraRed Absolute Spectrometer)
experiment on board the COBE (COsmic Background Explorer) satellite have con-
firmed that the radiation is extremely close to the black-body form at a temperature
2.725 ± 0.001 Kelvin. This data is shown in Figure 2.5. Furthermore, the tem-
perature coming from different parts of the sky is astonishingly uniform, and this
presents the best evidence that we can use the cosmological principle as the founda-
tion of cosmology.
Even more importantly, it is possible to identify tiny variations, only one part in a
hundred thousand, between the intensities of the microwaves coming from different
directions. These were first discovered by the COBE satellite in 1992, and Figure 2.6
shows precision measurements published by the Planck Satellite collaboration in
2013. It is believed that these patterns, known as cosmic microwave background
anisotropies, are intimately related to the origin of structure in the Universe, and at
present such measurements give us by far the most accurate understanding of cos-
mology. This topic is explored further in Advanced Topics 5 and 6.
Infrared: Carrying out surveys in the infrared part of the spectrum, as was first done by
the highly-successful IRAS (InfraRed Astronomical Satellite) in the 1980s, is an ex-
cellent way of spotting young galaxies, in which star formation is at an early stage.
Infrared surveys pick up a somewhat different population of galaxies to surveys car-
ried out in optical light, though obviously the brightest galaxies are seen in both.

Figure 2.5 The cosmic microwave background spectrum as measured by the FIRAS experi-
ment on the COBE satellite. The error bars are so small that they have been multiplied by 400
to make them visible on this plot, and the best-fit black-body spectrum at T = 2.725 Kelvin,
shown by the line, is an excellent fit.
Infrared is particularly good for looking through the dust in our own Galaxy to see
distant objects, as it is absorbed and scattered much less strongly than visible radia-
tion. Accordingly, it is best for studying the region close to our Galactic plane, where
obscuration by dust is strongest. Infrared is also vital for studying the most distant
galaxies, whose emission has been shifted predominantly into the infrared part of
the spectrum by the expansion of the Universe after the light was emitted, while
the far infrared can probe emission from interstellar dust grains to learn about dust-
enshrouded environments which are opaque at visible and near-infrared wavelengths.
X-rays: These are a vital probe of clusters of galaxies; in between the galaxies lies gas so
hot that it emits in the X-ray part of the spectrum, corresponding to a temperature
of tens of millions of Kelvin. This gas is thought to be remnant material from the
formation of the galaxies, which failed to collapse to form stars. X-ray emission from
the Coma galaxy cluster is shown in the lower panel of Figure 2.3. The individual
galaxies seen in the visible light image in the upper panel are almost all invisible in X-
rays, with the bright diffuse X-ray emission from the hot gas dominating the image.
Radio waves: A powerful way of gaining high-resolution maps of distant galaxies is by
mapping in the radio part of the spectrum. An important future probe of cosmology
will be observation of the ‘21 cm’ emission line, due to spin-flip of the electron in
a hydrogen atom, enabling maps of the distribution of neutral hydrogen gas in the
distant Universe. This can probe both the early development of structures in the Uni-

Figure 2.6 The cosmic microwave background intensity variations on the sky as measured
by the Planck Satellite. (Image courtesy ESA/Planck collaboration.)
verse, and the transition of the Universe to its present state of high ionization of the
intergalactic medium.
2.3 Homogeneity and isotropy
The evidence that the Universe becomes smooth on large scales supports the use of the
cosmological principle. It is therefore believed that our large-scale Universe possesses two
important properties, homogeneity and isotropy. Homogeneity is the statement that the
Universe looks the same at each point, while isotropy states that the Universe looks the
same in all directions.
These do not automatically imply one another. For example, a universe with a uni-
form magnetic field is homogeneous, as all points are the same, but it fails to be isotropic
because directions along the field lines can be distinguished from those perpendicular to
them. Alternatively, a spherically-symmetric distribution, viewed from its central point, is
isotropic but not necessarily homogeneous. However, if we require that a distribution is
isotropic about every point, then that does enforce homogeneity as well.
As mentioned earlier, the cosmological principle is not exact, and so our Universe does
not respect exact homogeneity and isotropy. Indeed, the study of departures from homo-
geneity is currently the most prominent research topic in cosmology. I’ll introduce this in
Advanced Topic 5, but in the main body of this book I am concerned only with the be-
haviour of the Universe as a whole, and so will be assuming large-scale homogeneity and
isotropy.
2.4 The expansion of the Universe
A key piece of observational evidence in cosmology is that almost everything in the Uni-
verse appears to be moving away from us, and the further away something is, the more

Figure 2.7 A plot of velocity versus estimated distance for a set of 1355 galaxies. A straight-
line relation implies Hubble’s law. The considerable scatter is due to observational uncer-
tainties and random galaxy motions, but the best-fit line accurately gives Hubble’s law. (The
x-axis scale assumes a particular value of H 0 .)
rapid its recession appears to be. These velocities are measured via the redshift, which
is basically the Doppler effect applied to light waves. Galaxies have a set of absorption
and emission lines identifiable in their spectra, whose characteristic frequencies are well
known. However, if a galaxy is moving towards us, the light waves get crowded together,
raising the frequency. Because blue light is at the high-frequency end of the visible spec-
trum, this is known as a blueshift. If the galaxy is receding, the characteristic lines move
towards the red end of the spectrum and the effect is known as a redshift. This technique
was first used to measure a galaxy’s velocity by Vesto Slipher around 1912, and was applied
systematically by one of the most famous cosmologists, Edwin Hubble, in the following
decades.
It turns out that almost all galaxies are receding from us, so the standard terminology
is redshift z, defined by
z =
λ obs − λ em
,
λ em
(2.1)
where λ em and λ obs are the wavelengths of light at the points of emission (the galaxy) and
observation (us). If a nearby object is receding at a speed v, then its redshift is
z =
v
,
c
(2.2)

Figure 2.8 According to Hubble’s law, the further away from us a galaxy is, the faster it is
receding.
where c is the speed of light. 2 Figure 2.7 shows velocity against distance, a plot known as
the Hubble diagram, for a sample of 1355 galaxies.
Hubble realized that his observations, which were of course much less extensive than
those available to us now, showed that the velocity of recession was proportional to the
distance of an object from us:
v = H 0 r .
(2.3)
This is known as Hubble’s law, and the constant of proportionality H 0 is known as Hub-
ble’s constant. Hubble’s law isn’t exact, as the cosmological principle doesn’t hold per-
fectly for nearby galaxies, which typically possess some random motions known as peculiar
velocities. But it does describe the average behaviour of galaxies extremely well. Hubble’s
law gives the picture of our Universe illustrated in Figure 2.8, where the nearby galaxies
have the smallest velocity relative to ours. Over the years many attempts have been made
to find accurate values for the proportionality constant, but, as we will see in Chapter 6, a
consensus is only now being reached.
At first sight, it seems that the cosmological principle must be violated if we observe
everything to be moving away from us, since that apparently places us at the centre of the
Universe. However, nothing could be further from the truth. In fact, every observer sees all
2 This formula ignores special relativity and so is valid only for speeds v
relativity result, of which this is an expansion for small v/c, is
1+ z =
c. If you’re interested, the special
1 + v/c
.
1 − v/c
However, for distant objects in cosmology there are further considerations, concerning the propagation time of
the light and how the relative velocity might change during it, and so this expression should not be used.

objects rushing away from them with velocity proportional to distance. It is perhaps easiest
to convince yourself of this by setting up a square grid with recession velocity proportional
to distance from the central grid-point. Then transform the frame of reference to a nearby
grid-point, and you’ll find that the Hubble law still holds about the new ‘centre’. This only
works because of the linear relationship between velocity and distance; any other law and
it wouldn’t work.
So, although expanding, the Universe looks just the same whichever galaxy we choose
to imagine ourselves within. A common analogy is to imagine baking a cake with raisins
in it, or blowing up a balloon with dots on its surface. As the cake rises (or the balloon is
inflated), the raisin (or dots) move apart. From each one, it seems that all the others are
receding, and the further away they are the faster that recession is.
Because everything is flying away from everything else, we conclude that in the distant
past everything in the Universe was much closer together. Indeed, trace the history back
far enough and everything comes together. The initial explosion is known as the Big Bang,
and a model of the evolution of the Universe from such a begiing is known as the Big
Bang Cosmology. Later on, we will find out why it is commonly called the Hot Big Bang.
2.5
2.5.1
Particles in the Universe
What particles are there?
Everything in the Universe is made up of fundamental particles, and the behaviour of the
Universe as a whole depends on the properties of these particles.
One crucial question is whether a particle is moving relativistically or not. Any particle
has two contributions to its energy, one being the kinetic energy and the other being the
mass–energy, which combine to give
2
E total
= m 2 c 4 + p 2 c 2 ,
(2.4)
where m is the particle rest mass and p the particle momentum. If the mass–energy dom-
inates, the particle will be moving at much less than the speed of light, and we say it is
non-relativistic. In that limit we can carry out an expansion
E total = mc 2 1 +
p 2
m 2 c 2
1/2
≈ mc 2 +
1 p 2
.
2 m
(2.5)
We recognize the first term as Einstein’s famous E = mc 2 , known as the rest mass–energy
as it is the energy of the particle when it is stationary. The second term is the usual kinetic
energy (p = mv in the non-relativistic limit). If the mass–energy does not dominate, the
particle will be moving at a substantial fraction of the speed of light and so is relativistic.
In particular, any particle with zero rest mass is always relativistic and moves at the speed
of light, the simplest example being light itself.
Let’s review the nature of the particles which are believed to exist in our Universe.

Baryons
We ourselves are built from atoms, the bulk of whose mass is attributable to the protons
and neutrons in the atomic nuclei. Protons and neutrons are believed to be made up of more
fundamental particles known as quarks, a proton being made of two up quarks and a down
quark, while a neutron is an up and two downs. A general term for particles made up of three
quarks is baryons. Of all the possible baryons, only the proton and neutron can be stable, 3
and so these are thought to be the only types of baryonic particle significantly represented in
the Universe. Yet another piece of terminology, nucleon, refers to just protons and neutrons,
but I’ll follow the standard practice of using the term baryon. In particle physics units, the
mass–energies of a proton and a neutron are 938.3 MeV and 939.6 MeV respectively, where
‘MeV’ is a Mega-electron volt, a unit of energy equal to a million electron volts (eV) and
rather more convenient than a Joule.
Although electrons are certainly not made from quarks, they are traditionally also in-
cluded under the title baryon by cosmologists (to the aoyance of particle physicists). A
crucial property of the Universe is that it is charge neutral, so there must be one electron
for every proton. Weighing in at a puny 0.511 MeV, well under a thousandth of a proton
mass, the contribution of electrons to the total mass is a tiny fraction, not meriting separate
discussion.
In the present Universe, baryons are typically moving non-relativistically, meaning that
their kinetic energy is much less than their mass–energy.
Radiation
Our visual perception of the Universe comes from electromagnetic radiation, and such ra-
diation, at a large variety of frequencies, pervades the Universe. In the quantum mechanical
view of light, it can be thought of as made up of individual particles – like packets of energy
– known as photons and usually indicated by the symbol γ. Photons propagate, naturally
enough, at the speed of light; since they have zero rest mass their total energy is always
given by their kinetic energy, and is related to their frequency f by
E = hf ,
(2.6)
where h is Planck’s constant.
Photons can interact with the baryons and electrons; for example, a high-energy photon
can knock an electron out of an atom (a process known as ionization), or can scatter off
a free electron (known as Thomson scattering in the non-relativistic case hf
m e c 2 ,
otherwise Compton scattering). The more energetic the photons are, the more devastating
their effects on other particles.
3 The proton lifetime is known to be either infinite, corresponding to the proton being absolutely stable, or much
longer than the age of the Universe so that they are effectively stable. Isolated neutrons are unstable (decaying
into a proton, an electron and an anti-neutrino), but those bound in nuclei may be stable; this will prove crucial
in Chapter 12.

Neutrinos
Neutrinos are extremely weakly interacting particles, produced for example in radioactive
decay. There is now significant experimental evidence that they possess a non-zero rest
mass, but so far there has been no clear evidence that this mass is large enough to have
observable cosmological effects. Hence for now it remains a working assumption in cos-
mology to treat them as effectively massless. I will adopt that assumption for the main body
of this book, and in that case they, like photons, are always relativistic. The combination
of photons and neutrinos makes up the relativistic material in our Universe. Confusingly,
sometimes the term ‘radiation’ is used to refer to all the relativistic material.
There are three types of neutrino, the electron neutrino, muon neutrino and tau neutrino,
and if they are indeed all massless they should all exist in our Universe. Unfortunately, their
interactions are so weak that for now there is no hope of detecting cosmological neutrinos
directly. Originally their presence was inferred on purely theoretical grounds, though we
will see that the existence of the cosmic neutrino background may be inferred indirectly by
some cosmological observations.
Because they are so weakly interacting, the experimental limits on the neutrino masses,
especially of the latter two types, are quite weak, and it is in fact perfectly possible that
they are massive enough to be non-relativistic. The possible effects of neutrino masses are
explored in Advanced Topic 3. Cosmological effect of neutrino masses might well have
been spotted by the time you read this book.
Dark matter
In this book we’ll encounter one further kind of particle that may exist in our Universe,
which is not part of the Standard Model of particle theory. It is known as dark matter, and
its properties are highly uncertain and a matter of constant debate amongst cosmologists.
We’ll return to it in Chapter 9.
2.5.2
Thermal distributions and the black-body spectrum
I end this section with some discussion of the physics of radiation. If this is unfamiliar to
you, the details aren’t all that crucial, though some of the final results will be used later in
the book.
If particles are frequently interacting with one another, then the distribution of their
energies can be described by equilibrium thermodynamics. In a thermal distribution, inter-
actions are frequent, but a balance has been reached so that all interactions proceed equally
frequently in both the forward and backward directions, so that the overall distribution of
particle numbers and energies remains fixed. The number of particles of a given energy
then depends only on the temperature.
The precise distribution depends on whether the particles considered are fermions,
which obey the Pauli exclusion principle, or bosons, which do not. In this book the most
interesting case is that of photons, which are bosons, and their characteristic distribution
at temperature T is the Planck or black-body spectrum. Photons have two possible polar-

Figure 2.9 The Planck function of Equation (2.7). There are far more photons with very low
energy than very high energy.
izations, and each has an occupation number per mode N given by the Planck function
N =
1
,
exp (hf /k B T ) − 1
(2.7)
where h is Planck’s constant and k B is one of the fundamental constants of Nature, the
Boltzma constant, whose value is 1.381 × 10 −23 J K −1 = 8.619 × 10 −5 eV K −1 .
To interpret this equation, remember that hf is the photon energy. The purpose of the
Boltzma constant is to convert temperature into a characteristic energy k B T . Below this
characteristic energy, hf
k B T , it is easy to make photons and the occupation number is
large (as photons are bosons, the Pauli exclusion principle doesn’t apply and there may be
arbitrarily many photons in a given mode). Above the characteristic energy, hf
k B T , it
is energetically unfavourable to make photons and the number is exponentially suppressed,
as shown in Figure 2.9.
More interesting than the number of photons in a mode is the distribution of energy
amongst the modes. We focus on the energy per unit volume, known as the energy den-
sity . Because there are very few photons with hf
k B T there isn’t much energy at
high frequencies. But, despite their large number, there also isn’t much total energy at low
frequencies hf
k B T , both because those photons have less energy each (E = hf ), and
because their wavelength is longer and so each photon occupies a greater volume. With
quite a bit of work, the energy density in a frequency interval df about f can be shown

Figure 2.10 The energy density distribution of a black-body spectrum, given by Equation
(2.8). Most of the energy is contributed by photons of energy hf ∼ k B T .
to be
(f ) df =
f 3 df
8πh
,
c 3 exp (hf /k B T ) − 1
(2.8)
which tells us how the energy is distributed amongst the different frequencies. We see in
2.8 k B T /h, corresponding to
Figure 2.10 that the peak of the distribution is at f peak
an energy E peak = hf peak
2.8 k B T . That is to say, the total energy in the radiation is
dominated by photons with energies of order k B T . Indeed, the mean energy of a photon in
this distribution is E mean 3 k B T .
When we study the early history of the Universe, an important question will be how
this typical energy compares to atomic and nuclear binding energies.
A further quantity of interest will be the total energy density of the black-body radiation,
obtained by integrating Equation (2.8) over all frequencies. Setting y = hf /k B T quickly
leads to
rad
=
4
8πk B
T 4 ×
3
3
h c
∞
0
y 3 dy
.
e y − 1
(2.9)
The integral is not particularly easy to compute, but you might like to try it as a challenge.
The answer is π 4 /15, giving a radiation energy density
rad
= αT 4 ,
(2.10)

where the radiation constant α is defined as
α =
4
π 2 k B
−16
J m −3 K −4 .
3 3 = 7.565 × 10
15 ̄
h c
(2.11)
Here ̄ h = h/2π is the reduced Planck constant.
Problems
2.1. Suppose that the Milky Way galaxy is a typical size, containing say 10 11 stars, and
that galaxies are typically separated by a distance of one megaparsec. Estimate the
density of the Universe in SI units. How does this compare with the density of the
Earth?
1 M
2 × 10 30 kg, 1 parsec
3 × 10 16 m.
2.2. In the real Universe the expansion is not completely uniform. Rather, galaxies exhibit
some random motion relative to the overall Hubble expansion, known as their pecu-
liar velocity and caused by the gravitational pull of their near neighbours. Supposing
that a typical (e.g. root mean square) galaxy peculiar velocity is 600 km s −1 , how
far away would a galaxy have to be before it could be used to determine the Hubble
constant to ten per cent accuracy, supposing
(a) the true value of the Hubble constant is 100 km s −1 Mpc −1 ?
(b) the true value of the Hubble constant is 50 km s −1 Mpc −1 ?
Assume in your calculation that the galaxy distance and redshift could be measured
exactly. Unfortunately, that is not true of real observations.
2.3. What evidence can you think of to support the assertion that the Universe is charge
neutral, and hence contains an equal number of protons and electrons?
2.4. The binding energy of the electron in a hydrogen atom is 13.6 eV. What is the fre-
quency of a photon with this energy? At what temperature does the mean photon
energy equal this energy?
2.8k B T /h
2.5. The peak of the energy density distribution of a black-body at f peak
implies that f peak /T is a constant. Evaluate this constant in SI units (see page xiv
for useful numbers). The Sun radiates approximately as a black-body with
T sun 5800 K. Compute f peak for solar radiation. Where in the electromagnetic
spectrum does the peak emission lie?2.5. PARTICLES IN THE UNIVERSE
19
2.6. The cosmic microwave background has a black-body spectrum at a temperature of
2.725 K. Repeat Problem 2.5 to find the peak frequency of its emission, and also
find the corresponding wavelength and compare to Figure 2.5. Confirm that the peak
emission lies in the microwave part of the electromagnetic spectrum. Finally, com-
pute the total energy density of the microwave background.