So far we have looked into the theoretical
aspects of structure formations, but in
order to really, reach a deeper
understanding of how it works we have to
turn to numerical simulations. This is a
very common practice in astrophysics and
many other sciences. And the reason is
that many interesting problems simply do
not have analytical solutions. For
example, in simple Newtonian gravity,
which is about as simple as it gets, we
can analytically solve rigorously only a 2
body problem, the Kepler problem. At a
third mass point there are no analytical
solutions except in some very special
circumstances. This was proven
analytically by point array 100 years ago,
that there are no analytical solutions for
more than 2-body problem when talking
gravity. Now consider a galaxy with a
couple hundred billion mass points of
stars. Considered in hundreds of billions
of galaxies in the universe. Clearly, we
cannot solve a problem analytically, and
we can resort to numerical simulations.
Thus, numerical simulations are essential.
They're not a sign of weakness. It's not
just hard to do this, it's simply
impossible to do it in any other way than
numerically, and to simulate using the
right kind of input physics inside a
computer what mother nature does out there
in the universe. Now, as far as the
gravity part is concerned, we're in luck
because the physics is relatively simple.
It's well understood. We even have good
understanding of the initial conditions
from primordial density perturbation
spectrum. Simulations of purely
gravitating systems, just to name gravity,
nothing else are relatively
straightforward. However, if you start
adding dissipative component such as gas,
or radiation, things become vastly more
complicated and infact, that's where
cutting edge of numerical simulations in
astrophysics is today. And, then finally
we can compare output of the numberical
simulations to the actually observations
of the large scale structure. The first
astrophysical simulation using N body
interacting gravitationa lly was done as
early as 1941 and not using a computer or
rather an analog computer by Eric Homberg.
What he did was he used light bulbs and
photocells and since their intensity to
light Decreases the same proportion as
gravity, as inverse square law. You can
then measure relative acceleration from
each point to each other point. Compute
the net accelerations. Move the points.
And then repeat the whole thing. A very
tedious way to do it. Things really
started moving on in the 1970's, with the.
Onset of modern digital computers. Here is
one of the earliest simulations by Jim
Peebles, the famous theoretical
cosmologist. Soon enough, people were
starting to add more realistic initial
conditions and here is another early
simulation by Simon White. Notice that
these only have few hundred particles,
which is obviously vastly less than, than
is needed for really realistic simulation.
The milestone paper was by Alar and Juri
Toomre, who did simulations of colliding
disc galaxies using a lot of
approximations, but with a great physical
intuitions, and they already reproduced
interesting Morphological features like
tidal distortions, tidal tails and bridges
and things like that. With a small number
of particles say planets in the solar
system gravity works like a clock work and
it's fairly simply to see what it happens.
But, once you start having many, many
bodies new phenomena come into the play.
There is a phenomenon o dynamical friction
which is when a massive body passes
through a sea of utter massive particles,
giving away some of its kinetic energy by
accelerating them all. In effect, the
gravitational field of all target
particles acting with viscosity. So, this
is what Toomre and Toomre saw. And in fact
this a, explained many puzzling
observations. We see strange looking
galaxies or pairs of galaxies and now we
understand that they're all product of
galaxy collisions and galaxy mergers. More
modern simulations carry out essentially
the same thing and show that Toomre and
Toomre pretty much g ot everything right.
These are from Josh Barnes and his
collaboraters and show simple collisions
of galaxy discs using more modern
computations. So what are simulations used
for? First, we can use them to understand
the evolution of large scale galaxy
clustering. Then we can compare the
simulations with actual observations like
surveys.
We can also use them to make predictions
of what should be seen. And help interpret
things like surveys.
The basic process that goes on is
hierarchical merging of smaller pieces
into the larger ones. And that is, again,
something that can be studied in detail,
and compared to observations. Since the
dark matter dominates gravitationally, by
studying what happens even with just
visible components, we can understand a
little more about how dark matter is
distributed. And thus, get some clues
about what it might consist of. And once
dissipative effects are understood, we can
then study behavior of cosmic gas,
radiation feedback by star formation, and
so on. Different kinds of physics have to
be put in, obviuosly, first the gravity
thing, that's something we understand
fairly well. But once we start talking
about gas and dissipation we have to put
hydrodynamics, energy dissipation
processess, radiation losses, feedback and
so on. So far, at least, magnetic fields
do not play an important role in
cosmology. But we just don't know really
what happens in very early universe and
what the origin of these magnetic fields
is. Possibly the most challenging at this
time is question of the radiation
transfer. The photons are much smaller
than galaxies and it's an scale.
So if we really need to understand how the
radiation propagates through universe,
interacts with gas. Then we need a very
precise More sophisticated simulations
that we've been, able to do until
recently. But let us focus on
gravitational simulations. The N-body
method. Once we have sufficiently large
number of particles, we can consider them
as a fluid. Not the gas that you know but
it's a very much compressible fluid
consisting of interactive particles,
mostly dark matter but also things like
stars and gas and sun. To the extent that
we look at gravitational interaction alone
this is a dissipation-less fluid. Energy
is exchanged between particles and systems
of particles, but there is no net energy
lost to radiation. And so there, there is
numerical methods that have been deployed.
Direct integration of gravitational
accelerations from each particle to each
other one, is Very practical and time
consuming. So a number of approximations
that have been developed that can actually
convey master important physics and be
done much faster. In some situations like
evolutional globular star clusters where
we're talking about 10's or 100's or
1000's of mass points, direct simulations
are actually viable. However, people do
make special computer hardware. They
design chips that are optimized to perform
gravitational N-Body Simulation. A group
in Tokyo is, in particular, being
producing machines like that. They call
them grape off of gravity pipe. Beyond
that large glass of models, uses so called
particle mesh simulations, where by
collective effect of all other particles
except one, is smoothed into a density
field, a mesh, and then You can compute
interaction of that particle with a
density field. Much faster than you would
if you say, consider all particles
individually. So that reduces compute time
from n squared, where n is number of
particles, to n log n, which is much
easier. There is another class of methods
called tree methods or hierarchical
methods, which just use clever grouping of
particles to compute things again at
similar faster pace. Which techniques is
used is depending on the actual problem
that's being attacked. In often times many
of them are used at once for different
aspects of the. Problem.
Now let's take a look at the simulation by
Andre Kravtsov of what a chunk of universe
might look like as it's evolving. What's
shown here is evolution of a cubical
region of universe ab out hundred
megaparsec in size and it's being rotate
to show you what structure happens in 3D.
Let's try it again. You can see that,
first, dark matter condenses in smaller
clumps. Which then merge into ever larger
ones Forming the fila-, filamentary
structure that we see, the cosmic web.
Now, if we look at slightly smaller scale,
we can see more detail. And here, we show
formation of a group of galaxies. Again,
using same thing. 'Kay.
At the beginning, things are very smooth.
Those are the initial density
perturbations, very low amplitude. But
then they grow, condensing into little
dark halos, which keep merging into every
larger ones. Let's take a look. Very
smooth at first, but soon enough, things
condense. Under their own gravity and then
keep merging hierarchically into larger
structures. Let's look at this again, and
this is where you see hierarchical
structure formation in action. Smaller
things condensing under their own gravity,
merging together to make larger ones. Here
is a similar thing, but in this case
simulation of a formation of a cluster of
galaxies, done by John Dubinsky, again we
start with very smooth sample conditions,
and you see objects condensing and merging
violently together with each other making
larger and larger structures in the
middle. Let's take a look again. Note,
this is not stars. This is dark matter. If
you, if you could see dark matter, this is
what you would see.