Download Word

AST 121S: The origin and evolution of the Universe
Mathematical Handout 8: The formation of structure in the Universe
We based our dynamical equations that described the evolution of the Universe on the assumption
of homogeneity - that the Universe was uniform in density. This is a reasonable approximation on
the largest scales, but clearly is not satisfactory on smaller scales where we see stars galaxies and
clusters of galaxies that clearly represent significant inhomogeneity in the Universe. In many ways
it is just as well that the Universe "clumped" in this way - the more interesting phenomena that have
occurred in the Universe since the Big Bang (that have led to interesting things such as ourselves)
required substantially higher densities than would have been present in a Universe in which the
mass was completely uniformly distributed. The aim in this handout is to briefly look at the process
by which this clumping occurred.
8.1 Gravitational Instability
Newton himself realized that an initially homogeneous (i.e. uniform) medium is unstable to
fragmentation due to the action of gravity. The reason for this is fairly obvious: any regions with
very slightly higher density will attract gravitationally matter from surrounding regions, thereby
increasing in density, and, correspondingly, any regions of lower than average density will have
matter removed by the gravitational attraction of neighbouring regions. The following sketch shows
the gravitational force on particles due to an inhomogeneous medium after the force that would have
been produced by a perfectly homogeneous medium has been subtracted, i.e. it shows the effect of
the perturbation to the density.
This shows that the gravitational effect of the density perturbations is to make high density regions
increase in density and low density regions to decrease in density. The medium is thus unstable and
will tend to fragment into regions of very high density surrounded by regions of very low density.
Let us describe the density perturbation as the difference between the density and the average
density divided by the average density, and denote this as D.
8.1

   b 

b

The growth of  through gravitational instability was studied by Jeans in 1902 for a static medium
and then and then later for an expanding medium by Lifshitz in 1946. The equation they derived is
quite complicated and need not concern us here. Let me simply state their results. They showed that
density perturbations do not always grow. First, they will oscillate as a sound wave if their size is
less than a critical length scale known as the Jean's Length. We will generally be interested in
density perturbations that have a larger scale. On these larger scales where growth is possible, we
encounter three different solutions appropriate for different situations.
If the medium is not expanding (as in say a gas cloud in the Galaxy), then the growth of the density
perturbation is rapid and  increases exponentially.
8.2    o exp( 4G t )
If the medium is expanding, as in an expanding Universe, it turns out that we get growth of if the
Universe is "critical" with  ~ 1 but no growth at all if  ~ 0 (we have seen that a typical Universe
will spend most of its time with  either 1 or 0, with little time spent in between so we won't worry
about the more complicated behaviour for 0 <  < 1.). Even when we get growth,  increases only
as R, or hence as t2/3.
8.3
  R for   1
  cons tan t for   0
8.2 The formation of large scale structure in the Univedfsrse
As we saw, galaxies are not distributed at random in the Universe but rather are found in clusters
and in large filamentary and sheet like structures. If we look at the distribution of galaxies in the
Universe, we find that  ~ 1 on scales of around 10 Mpc (~ 30 million light-years). What we mean
by this is that if we put down spheres of this radius in the Universe then the r.m.s. number of
galaxies within the spheres is roughly equal to the average number within each sphere.
On larger scales, is smaller (obviously  must tend to zero on the largest scales if our assumption
about the Universe being homogeneous is to have been valid) and a reasonable approximation valid
on scales from 1 to 100 Megaparsecs is that
(8.4)
  length scale 
1
It is generally believed that the lumpy distribution of galaxies seen today is the result of the growth,
through gravitational instability, of initially tiny density perturbations that were present in the early
Universe. These may be the highly amplified result of exceedingly small quantum fluctuations (
10-27) that we would have expected to be present in the field that drove inflation (assuming inflation
occurred). Different ideas for the origin of the fluctuations and for the type of dark matter that
(may) dominates the mass in the Universe lead to different predictions for the strength of the density
fluctuations on different scales. These predictions can in principle be tested and the current "bestbet" (though I wouldn't put money on it) is known as Cold Dark Matter.
We can now begin to test directly the growth of density fluctuations in the Universe using our
observations of the temperature variations of the microwave background. Any density variations
that were present 100,000 years after the Big Bang will produce (through a number of different
mechanisms that need not concern us here) a change in the apparent temperature of the microwave
background as seen on the inside of the "fog-bank". The temperature fluctuations which have been
detected on angular scales of 10o and greater indicate density fluctuations were present at that earlier
time at the 10-5 level on scales which today would be measured to be 3000 million light years.
Given that these would have grown by a factor of 1000 between then and now (due to the change in
R - assuming that  ~ 1) these fluctuations are presumably the precursors of fluctuations in the
present day Universe at the 10-2 level. While such modest density fluctuations have not yet been
detected directly, they are consistent with the larger density fluctuations that we do see on smaller
scales in the galaxy distribution and with the expected change of  with length scale (e.g. eqn 8.4).
8.3 The formation of galaxies
The formation of galaxies is thought to have been the result of similar density fluctuations to those
in the previous section, produced in the early Universe but on much smaller scales. These would be
expected to have had higher initial , and will collapse to produce concentrated objects such as
galaxies at relatively early times. We would expect the formation of typical massive galaxies to
have occurred within the first few billion years after the Big Bang, and indeed we have seen that the
oldest stars in the Galaxy have ages that are not much less than that of the Universe as a whole.
Unfortunately, this fact means that truly young galaxies seen at the time of their birth or formation
will be exceedingly distant, since the light must have traveled for much of the history of the
Universe. Such distant galaxies would be expected to be very faint and their light will be highly
redshifted into the challenging infrared region of the electromagnetic spectrum. Despite intensive
searches, no convincing population of forming galaxies (so-called primeval galaxies) has yet been
discovered. As we study more and more distant galaxies, we find that the population clearly shows
evolutionary changes over cosmic time (there are many more vigourously star-forming galaxies at
earlier epochs) but the objects that we study are already fairly well-formed, and the details of galaxy
formation process are hardly understood at all. It is likely to be complicated, with frequent merging
of young galaxies and extensive feed-back between inflowing gas and the explosions from
supernova explosions produced by massive stars.
8.4 The formation of stars
Stars are observed to be forming out of dense gas clouds in our own Galaxy and in other galaxies. In
contrast to the situation with galaxies, there is an abundance of observational data on star-formation.
There are many regions of star-formation quite close to us in the Milky Way galaxy that can be
studied in some detail. Nevertheless, the problem is complicated by the fact that the timescales for
star-formation are very much longer (typically of order 106 years) than the timescales of modern
astronomy (less than 100 years) Thus, we can only observe each star-formation region at a single
"instant". We must then try to piece together the overall sequence by figuring out where in their
relative histories the different star forming regions belong - a process complicated by the large
number of other variables that will vary from one star-formation region to another.
8.4.1 The collapse of a proto-star
In simple terms, we can view the formation of a star out of the collapse of a local density
enhancement within a gas cloud as a two stage process. They differ as to how easily the
gravitational potential energy liberated as the cloud collapses can escape.
(a) Transparent collapse
As the dense region of the cloud collapses, the gravitational potential energy, which is given for a
region of mass M and radius R approximately as
GM 2
P. E.~ 
R
This potential energy will go into thermal energy of the gas in the collapsing region as it heats up as
it is compressed.
At first the region of the gas cloud in question is still transparent and so the thermal energy thus
produced can be quickly radiated into surrounding space and the temperature of the center of the
cloud increases only a little. The collapsing gas cloud thus steadily shrinks in size and the core
does not heat up significantly and does not provide significant pressure support for the cloud.
(b) Opaque collapse
As the density of the central parts of the gas cloud increases, it becomes increasingly difficult for
radiation to escape easily from the central regions since photons can only travel a short distance
before scattering off of a gas particle and starting off in some new direction - it takes well over 106
years or so for a typical photon to travel from the center of the Sun to the surface due to these
random scatterings). The thermal energy thus becomes trapped in the central core and the central
temperature will start to rise. We would now call the core of the gas-cloud a proto-star
This has the important consequence that thermal pressure from hot gas can start to support the
proto-star and the variation of density, temperature and pressure through the proto-star will now be
governed by the same equations of stellar structure that describe the Sun and other stars. However,
energy will still be being lost from the proto-star's surface and so the proto-star will continue to
contract (as we discussed earlier when talking about the possible sources of power in the Sun).
The transition from transparency to opaqueness as the density increases occurs last for photons of
longer wavelength. Just as the proto-star starts to become opaque (and the core hidden from view)
the dominant cooling mechanism by which thermal energy is lost from the core is through
microwave photons produced by energy transitions in molecules. Studying this microwave
radiation thus gives us the clearest view of proto-stars at this important phase of their evolution an
this is a major motivation for millimeter-wave telescopes such as the JCMT in Hawaii in which
Canada has a 25% share (along with the UK and The Netherlands).
(c) Thermonuclear emission and "failed stars"
As the proto-star continues to slowly contract, the central temperature will rise and may rise high
enough for sufficient fusion reactions to occur so as to replace all the energy being lost from the
surface of the proto-star. This will stabilize the collapse of the proto-star at this size, as long as
there is sufficient fuel for fusion. Such a stabilized object is, of course, a proper star.
However, some objects may never attain high enough central temperatures for fusion to occur. It is,
in principle, easy to calculate that proto-stars whose mass is less than 0.08 times the mass of the Sun
will never have central temperatures higher than the 107 K threshold for fusion. These objects
continue to collapse and radiate away the gravitational energy produced by their collapse. These
failed stars are known as brown dwarfs. They are expected to fade rapidly until they are eventually
supported by non-thermal pressure (see the next handout). They resemble giant planets such as
Jupiter except that they are 10-100 times more massive. A few candidate objects are known, but the
existence of these objects is not proven beyond doubt. However, since we see stars down to this
0.08 Mo limit and since there is no reason to think that objects of slightly lower mass would not be
formed, brown dwarfs are thought to be quite common.
Brown dwarfs are a good example of a possible source of baryonic dark matter.
The above is a simplified view of star-formation that is almost certainly broadly correct. However,
the details of star-formation are poorly understood and are very complicated. We are trying to
understand the evolution of a gas cloud over an enormous range of densities as it collapses. The
complexity of the Earth's weather, where gas densities change only by a few percent, means that we
should not be surprised if we encounter very complicated situations during the birth of a star.