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AST 121S
The origin and evolution of the Universe
Mathematical Handout 3: Basic observations about the Universe
Modern scientific cosmology, i.e. the Big Bang models for the Universe, is based on just a few very
simple observations about the Universe. The resulting theory is generally accepted as providing a
good description of the origin and evolution of our Universe because it then explains, or even
predicted, several other basic observational facts about the Universe. In this handout we will look at
these basic observations, concentrating on the first two.
1. Isotropy and homogeneity
The Universe is to a good approximation isotropic which means that it looks the same in all
directions. This is certainly true of the microwave background which we have seen is uniform
across the sky down to a level of roughly one part in 1000 The simple anisotropy that is seen at this
level of 0.001 is most likely due to the fact that our vantage point on the Earth is moving at a speed
of about 350 kms-1 relative to the rest of the Universe and if we account for this then the isotropy is
maintained down to a level of about one part in 100,000.
Clearly, the sky is not isotropic when we look in other wavebands (optical, X-ray etc.) but we can
understand these anisotropies in terms of our location relative to relatively local structures such as
the Sun or the Milky Way galaxy. When we look into the distant Universe, the appearance of
isotropy returns even at these other wavelengths. For instance counting the number of faint (i.e.
distant) galaxies in different parts of the sky shows that there are roughly equal numbers in different
directions.
Unless we are exactly at the center of the Universe, a notion with which we have been
uncomfortable since the time of Copernicus, a Universe that appears isotropic must also be
homogeneous. Homogeneity means that it is the same at all locations. The notion of homogeneity
in the Universe is enshrined in the Cosmological Principle which states that "at any cosmological
epoch, the Universe appears the same in its general properties to all observers regardless of their
location". Clearly the Cosmological Principle can not be absolutely true (if the Universe was
completely uniform you and I could not exist!!). However, it is, as far as we know, a very good
approximation in terms of important things like the microwave background or the distribution of
matter on very large scales. The assumption that the Universe is homogeneous makes the
calculation of the gravitational field and the dynamics of the Universe straightforward.
Note in passing that, although as we look out in distance we look back in time, we do not
necessarily require that the Universe be unchanging with time in order for isotropy to be maintained,
since all observers will look back equal amounts of time when they look to a given distance in each
direction (think about this and you can convince yourself it is true!).
2. The recession of the galaxies
Shortly after he established that the "nebulae" were separate galaxies external to our own Milky
Way Galaxy, Hubble established (1929) that galaxies are receding from us with a speed proportional
to their distances (it was in fact Slipher who measured the velocities - Hubble provided the
distances). How did Hubble and Slipher measure these velocities?
They used the Doppler effect, whereby the waves from a source (either of sound waves or of
electromagnetic radiation) that is moving relative to an observer are shifted in frequency and/or
wavelength. Remember that what we called line emission is emitted at just a few particular
wavelengths/frequencies, so we can easily see if this radiation has been shifted in this way.
For light and other forms of electromagnetic radiation, we define the redshift, z, in terms of the
wavelengths at which the light was emitted, em, and at which it was received, obs:
(3.1)
z
 obs   em
 em
A source moving away from the observer has obs > em and a positive redshift. Hubble actually
observed a redshift-distance relationship:
(3.2)
cz  Hd
where z is the redshift of a particular galaxy, H is Hubble's constant and c is the speed of light. For
small velocities the relationship between speed and redshift is simple:
(3.3)
z
v
c
As an aside, for velocities close to the speed of light, we should use the correct formula from
Special Relativity
v
c 1
z
v
1
c
1
This reduces to equation (3.3) for speeds which are small compared to the speed of light, c.
Interpreting Hubble's redshift-distance relation (3.2) in terms of Doppler effects (3.3), then gives a
velocity distance relation:
(3.4)
v  Hd
A situation in which galaxies are moving away with a recessional velocity that is proportional to
their distance is the signature of an expanding Universe.
The balloon analogy
It is extremely difficult to correctly visualize an expanding Universe. In particular, it is misleading
to think of the Universe as expanding into a surrounding region of empty space. A very useful
mental picture is to consider a two-dimensional Universe, which we can then represent as a twodimensional expandable surface like that of a balloon. Imagine the Universe as the two-dimensional
surface of a balloon. The galaxies, including our Milky Way, are dots painted on the balloon. We
should view ourselves as 2-dimensional astronomers living on one of these dots. I tend to think of
these astronomers as little ants, but strictly speaking they should be two dimensional beings like
shadows!
Notice that if the dot-galaxies are painted on the balloon uniformly, then the "Universe" will appear
isotropic to one of our ant-astronomers. In each direction within the 2-d surface of the balloon,
he/she will see the same general appearance of the Universe (our 2-d ant-astronomers are not
allowed to even think of the third dimension, let alone look in that direction!). Furthermore, that
view of the Universe will again be generally the same for all the ant-astronomers, regardless of
which dot-galaxy they live on. Our 2-d balloon Universe thus satisfies our Cosmological Principle.
As the balloon is blown up, a given dot will observe all the other dots moving away from it with
speeds proportional to their distances, reproducing Hubble's Law (equation 3.4). However, note that
all astronomers (regardless of which dot-galaxy they live on) will see this effect, and no dot can
legitimately say: "I am at the center of the expansion", even though all of them see all of the other
galaxies moving directly away from them..
The balloon analogy leads naturally to the idea that the expansion of the Universe is an expansion of
space itself and not (emphatically not!) an expansion of anything into anything else. No ant can find
a boundary of the balloon-Universe and sit on the edge and see it expanding into anything else.
Rather, by observing that all the dot-galaxies are moving away from each other, they simply infer
that their Universe is expanding. Their 2-d surface is expanding in exactly the same way,
mathematically, that our 3-d space is expanding.
An important idea which we will need in the next handout is that the distance r between any two
points in the Universe (be they real galaxies in our 3-d Universe or the distance, within the surface,
between two dots on the 2-d surface of the balloon) may be written as the combination of a
coordinate distance  that does not change as the Universe expands and a cosmic scale factor R,
which does increase as the Universe expands. The coordinate distance  is usually called the
comoving distance because this coordinate frame moves (expands) with the Universe as a whole.
(3.5)
r  R(t )  
3. Other basic observations about the Universe
It may surprise you to know that the Cosmological Principle and the observed recession of the
galaxies as described by Hubble's Law are all that is required to produce the basic Big Bang model
of the Universe. In fact, Einstein in 1917, just two years after formulating General Relativity,
applied the Cosmological Principle to GR and came within an ace of predicting the expansion of the
Universe some 12 years before it was discovered by Hubble. He actually found that static
Universes were impossible (by static we mean not expanding or contracting). Such a notion was so
outlandish in 1917 that he added the so-called cosmological constant,  so as to permit static
solutions. We now know that the Universe is expanding and that  is extremely small and probably
exactly zero. Einstein later called the introduction of  his "greatest scientific mistake" (a lesson for
us all!).
The reason why we take the Big Bang so seriously is because, based on these very simple
observational facts (isotropy and Hubble's Law), it accounts naturally for several rather interesting
things about the Universe. In this last part of this handout I will describe these and give only an
outline of how the Big Bang theory accounts for them, leaving any further discussion until later
when we will have explored more of the details of the Big Bang theory.
(a) The microwave background
We saw earlier that the Universe is filled with a radiation field that has a precise Planck spectrum
characteristic of a temperature of 2.74 K. This is surprising, because as far as we can tell there is
nothing much in the Universe at this temperature.
The Big Bang naturally produces this radiation field because, in the Big Bang scenario, the Universe
was once in a compressed state at much higher temperature. At early epochs, all of the matter in the
Universe would have been at very high temperatures and could then produce this radiation field. As
the Universe expanded, the matter and radiation would have cooled (see later). By the present
epoch, the radiation would have cooled down to 2.74 K. We might also expect that most of the
matter would have cooled to comparable temperatures. However, most of the matter (at least
visible matter) has been reheated by localized sources of energy, such as stars etc., to the typical
temperatures that we see today.
The uniform microwave radiation field that bathes the Universe is thus a remnant of the initial
fireball of the Big Bang when everything was extremely hot.
(b) The abundance of 4He.
We can, using spectroscopy, estimate the chemical composition of objects in the Universe. For all
objects larger than planets, we find a rather uniform abundance. The following fractional
abundances are observed:
0. 71  H  0. 75
0. 25  4 He  0. 27
(3.6)
0. 00  everything else  0. 02
That most of the Universe should be made of Hydrogen is reasonable since it is the simplest
element, consisting of a single proton and electron. As we'll see in the third part of the course, we
can also understand the very small contamination from "everything else" as due to the fusion
reactions in stars. It turns out that these fusion processes should have only produced about as much
4He as "everything else"(see the last part of the course). What is therefore hard to understand is both
why the abundance of 4He is so much larger than all of the other elements combined and why it
shows so little variation when the abundances of the other elements varies greatly (from nothing to
2%). Both these facts suggest a cosmological or primordial origin for the 4He that produced the
same amount of 4He everywhere
As we'll see later, the Big Bang beautifully explains this curious fact. When the Universe had a
temperature of 1010 K, about a second after the Big Bang, nuclear reactions (via the Weak
interaction) occurred which fixed the neutron/proton ratio in the Universe at such a level that
roughly 25% of the Universe after this time was in the form of 4He. The amount of 4He produced in
this way is sensitive to the details of how the Universe was expanding at this temperature, and,
again as we'll see, the fact that the observed abundance of 4He is exactly as predicted by the standard
Big Bang theory is the strongest evidence that the Big Bang really happened.
(c) Olbers' Paradox and the age of things
This interesting thought experiment has been known for several centuries. It was publicized by
Olbers, though he was not the first to think about it. If the Universe is infinite in extent and if it has
been around "forever" (plus a few other "ifs" which need not concern us here) then it is easy to show
that the night sky should be "infinitely" bright!
Consider a shell of the Universe surrounding us at a distance r with a thickness dr. The volume of
this shell, dV is given by
dV  4r 2 dr
Let's now suppose that the average number density of stars in the Universe is n stars per unit
volume, and that each star has an average luminosity L so that its brightness, f, is
f 
L
4 r 2
The brightness of our shell as seen from Earth, df, is thus
df  n f dV  n L dr
Thus, the total brightness of the sky, F, is given by the integral of df over r from r = 0 to r =


(3.7)
F  nL  dr  nLr 0  

0
This is clearly not the case. In fact, this estimate neglected the finite size of stars - we cannot see to
infinity because the intervening stars block our line of sight. But noticing that the surface brightness
of a source of light is independent of distance (as you know from elementary photography) one can
see that the brightness of the night sky in Olbers' infinite (in space and time) Universe would be the
same as that of the Sun (a typical star) since every possible line of sight would eventually intersect
the surface of a star. This second approach to the question gives a predicted sky brightness that,
while at least not infinite, is still too large by a factor of about 1013!
Big Bang cosmologies explain this paradox because the Universe as we know it came into being a
finite time ago (between 10-20 billion years). There are no stars more distant than the distance that
light can travel in 10-20 billion years. Thus, we would need to truncate our simple-minded integral
above at r = 20 billion light years, or so, avoiding the problems associated with integrating out to
infinity.
(d)
The ages of things in the Universe
As we discussed earlier in the course, we know of nothing in the Universe that is demonstrably
older than the ages of the oldest star-clusters which we estimate to be about 13-16 billion years.
This is consistent with the idea that the Universe came into being in the Big Bang about 10-20
billion years ago (allowing for some uncertainties in the age estimates).