So far, we've been discussing a little bit
of modern cosmology. It's a fascinating subject.
We can't do it the full justice in this course.
In fact, it would be nice to have one entire
course devoted to that particular subject.
There are many things that we don't have time
to go into. We saw that, according to the
Big Band Theory, the universe started in an
explosion where all the matte and space in
time were created some 13.7 to 14 billion
years ago, and it's been expanding ever since.
We looked for the evidence for such an event
and it's the cosmic microwave background radiation.
When the universe became transparent to light
some 380,000 years after it's creation, the
radiation was in equilibrium with matter and
the temperature of the universe at that time
was about 3,000 degrees centigrade.
It was cool enough so that the protons could
capture the electrons and remove the electrons
from this dense soup so that the photons could
propagate freely among neutral hydrogen atoms.
The universe has been expanding since then.
It has expanded roughly by a factor of a thousand.
The wavelength corresponding to the temperature
of 3,000 degrees has now increased by the
same factor of a 1,000 and is now in the microwave
range and the corresponding temperature is
about three degrees Kelvin.
Sometimes, people refer to that microwave
background radiation as three-degree background
radiation. It was first observed in the '60s
by two physicists from Bell Labs, Penzias
and Wilson. Since then, there were satellite
probes launched to examine it's details.
And certainly, [inaudible 0:02:34] variations
in temperature are detected, but they are
still minute. Overall, it's three-degree background.
That is pretty strong evidence that the Big
Bang indeed took place as proposed by the
theory.
There is another piece of evidence that I
didn't discuss, because there is no real point.
That is if you apply the laws of nuclear physics
to the conditions in that early universe,
then, you indeed find out that the relative
abundance of different chemical elements would
be roughly three-quarters hydrogen, one-quarter
helium, and small amounts of all other chemical
elements.
That's precisely what we see today. There
are two very strong pieces of evidence, microwave
background radiation and the distribution
of chemical elements in the universe, that
fit the scenario of the Big Bang. Then, what
is the fate of the universe? What is going
to happen? Will it stop expanding? Would the
matter in the universe through it's gravitational
attraction hold it's expansion or will it
continue to expand forever?
The answer depends on what is the density
of matter and energy in the universe. That
density is characterized by a parameter that
cosmologists or astrophysicists like to label
with the Greek letter Omega.
It gives the ratio of the actual density of
matter in the universe over so-called critical
density and it turns out that that critical
density is 9.5 * 10- 30 grams per centimeter
^3, which corresponds to about six hydrogen
atoms per meter ^3.
The density corresponds to sharing about six
hydrogen atoms in a cube of space, one meter
on each side. That's pretty low. In fact,
our best vacuums on earth don't provide such
a low density.
Nevertheless, it turns out that that particular
number is critical and what matters is how
the actual density of matter in the universe
compares to that critical value. There are
different scenarios as to what will happen
in the future, depending on the value of omega.
If omega is bigger than one, if the actual
density of the universe is bigger than the
critical value, then eventually, the universe
will stop expanding and it will start contracting.
This set of diagrams illustrates schematically
what is happening to the universe.
This first flash denotes the creation, the
Big Bang, and then the time goes up, and we
know that universe is expanding, still is.
This particular slice corresponds to the present
time. The size of this disc represents, if
you will, the size of the universe -- its
diameter, right?
If omega is bigger than one, the universe
is expanding now, it will continue expanding,
but then, the gravitational force of matter
in the universe will halt that expansion.
Right? Just as when you throw something in
the air, the Earth's gravity will slow it
down, and eventually, force it to come back
to Earth.
If omega is bigger than one, then, the universe
will stop expanding at some time in the future
and it will start contracting, and it would
end in what is called Big Crunch. It was created
in the Big Bang, and it would end in a Big
Crunch, if omega is bigger than one.
It turns out that that value of omega also
has implications on the global curvature of
space. I'm not talking about the local curvature
of space, such as increased curvature near
a star like our sun. We saw that the sun's
mass is such that it's sufficient to bend
the space around it so that it bends the starlight
from the distant stars that pass the Earth.
I'm not talking about that curvature that
is caused by a local distribution of mass
in the universe, that you have the same positive
curvature, or rather curved space, near a
black hole or a neutron star. I'm talking
about the global, overall curvature of the
universe.
If omega is bigger than one, the universe
has a positive curvature. It's hard for us
to imagine, to see a curvature of our space,
right? Because we developed in a three-dimensional
space. Our nervous system developed in three-dimensional
space, we have hard time visualizing things
in higher dimensions. We can get an idea as
to what is involved by looking at lower-dimensional
space. Then, we can appreciate the idea of
different types of curvatures.
Positive curvature is just like the curvature
of two-dimensional surface of a ball, right?
Here you have a tennis ball, and it has a
positive curvature, and there are several
different things associated with positive
curvature.
If you had two lines that were locally parallel
to each other, like if these two bugs, when
they started crawling, they were moving in
parallel directions, it could happen that
actually these lines would eventually cross,
that these two bugs would, at some point,
be at the same spot. That is the feature of
a space with a positive curvature.
If this was the case, then if I had, here,
a massive object, say, a black hole or a neutron
star or a star, right, it would make a local
indentation around that particular object,
but globally, overall, the universe would
have a positive curvature, just like the two-dimensional
surface of the ball.
If omega was less than one, then, we would
have universe expanding forever, at a constant
rate. This type of expansion is referred to
often as the Coasting Universe. If this was
the case, if omega was less than one, then
the universe will simply continue expanding.
There is not enough mass, there is not enough
matter in the universe to halt that expansion.
It turns out that in that case, if omega is
less than one, overall, the global curvature
of space-time would be negative. If omega
is less than one, we have this so-called Coasting
Universe. It keeps on expanding at a constant
rate, but the global curvature of the space-time
is negative, just like the surface of a saddle.
In the case of space with negative curvature,
unlike the space with positive curvature,
if I had two lines that were initially parallel
to each other, if these two bugs were initially
moving in the same direction, in the space
with negative curvature these two initially
parallel lines could actually diverge from
each other.
These two bugs would be moving at increasing
distance, so that the distance between them
is increasing. OK? That's the feature of a
negative curvature of the space-time.
Finally, if omega was equal to one, which
is this middle panel here, the universe will
continue expanding, but the rate of expansion
would slowly decrease in time, and eventually,
become zero in infinite future. In infinitely
distant future, that expansion would stop.
After the Big Bang, the universe is expanding
and always will be expanding, but the rate
of the expansion, as indicated by the slope
of this line, is slowly decreasing. It will
always expand, but the rate at which it expands
would be slowly decreasing, and it would become
zero at infinite future.
If omega was equal to one, we would have a
so-called "Critical Universe," and the global
curvature of space-time would be zero. The
space-time would be flat.
Here is a familiar case of zero curvature.
Actually, our everyday experience tells us
that we live in such space. Actually, there
are certain postulates of Euclidean geometry
that are intuitively clear, because they are
so close to our experience. In this space
with zero curvature, two lines that were parallel
initially will always stay parallel to each
other.
They would never intersect or diverge from
each other. These two bugs would always be
at the same distance from one another. One
of the Euclidean postulates is that, basically,
two lines that were parallel at a particular
region of space will always remain parallel.
If omega happens to be exactly equal to one,
the universe would keep on expanding, but
the expansion rate would slowly decrease to
become zero, not in any finite interval of
time but at infinite future.
What are the observations? There was a spectacular
discovery in late '90s, by two groups of astronomers.
They used a Type Ia supernova to measure the
distances to the most-distant galaxies in
the universe, because of the reasons we discussed
before. They are very bright, therefore they
can be seen from large distances. They always
have the same luminosity, because the mechanism
is always the same, we can calibrate their
luminosity.
What they found is that about four to five
billion years ago, the rate of expansion started
to increase. This is not quite accurate description.
Basically, this sudden increase in rate started
some four to five billion years in the past.
You get the picture.
What was observed is that, as of four to five
billion years ago, the rate at which the universe
is expanding is increasing. The expansion
is accelerating. For that discovery, which
was fundamental -- it completely changed our
understanding or our picture of our entire
universe -- they were awarded the Nobel Prize
in physics in 2011.
The question is, why? What caused it? Of course,
we physicists and astrophysicists are very
resourceful. We always have, through some
new experimental measurement, something Earth-shattering,
and there is bound to be somebody on the Earth
who already has an idea as to what is happening.
What people did, and that was the simplest
thing to do, is to reintroduce so-called cosmological
constant, which was introduced by Einstein
after he formulated his general theory of
relativity. The reason he introduced it is
because general theory of relativity predicted,
among other things, the possibility that the
universe could be expanding.
In 1916-17, there was no observational evidence
for that. All observations pointed out to
the steady-state universe, where the universe
is neither contracting nor expanding.
Einstein, in order to prevent the universe
-- and basically, at that time, the universe
was largely restricted to our own galaxy -- he
introduced, by fiat, by hand, an additional
term that was not there initially in his equations
of general relativity, and the effect of that
term that he called cosmological constant
was actually to counteract the gravity. Right?
Because, without that term, Einstein's equations
would say that all the mass in the universe
would cause it to collapse. It wasn't collapsing.
The observational evidence at that time was
that it was neither expanding nor contracting.
Right? It was in that steady state.
To reproduce that observation, he introduced
an addition, totally artificially, without
any deep reasons, such as symmetry. Usually
we, in physics, use symmetry arguments as
most fundamental principles in setting up
a theory, because we believe that the universe
is symmetric, beautiful. He introduced this
so-called cosmological constant to prevent
the collapse of the universe under its own
gravity.
The astrophysicists then said, once they discovered
that the expansion is accelerating, that,
"Aha! If we had, in Einstein's equations,
this term, again, that is counteracting gravity,
we can explain, albeit phenomenologically,
the expansion of the universe."
Why did Einstein said later that that was
the greatest blunder of his career was, of
course, the observations by Hubble in 1929
were consistent, as we discussed in great
detail before, with an expanding universe.
The universe wasn't static. It was actually
expanding.
That was clear after Hubble's observations.
That's why Einstein called it the biggest
blunder of his career. That cosmological constant
is now again invoked in order to explain that
there's something that is counteracting gravity
and actually causing the expansion to accelerate.
Basically, this is very qualitative explanation,
but I hope you will get the idea. Suppose
that we have a bunch of galaxies, to here,
right? They are attracting each other with
the force of gravitational attraction, as
I've indicated with these yellow arrows.
Each two galaxies are attracting each other.
This one here is pulling on this one to the
left with the force of gravity, and at the
same time, the one on the left is pulling
on the galaxy on the right with equal force,
but opposite direction. Of course, the same
with these two and these two. They are attracting
each other, right? These forces of gravity
are trying to pull them together.
Gravity caused by mass and energy, as we know,
it decreases with distance. Right? Not only
in Newtonian theory of gravity, but also the
same is true in the general theory of relativity.
On the other hand, this cosmological constant
is the property of space-time itself. It has
no relationship to the matter in the universe.
It is counteracting the gravity. It's trying
to pull the galaxies away from each other.
Moreover, unlike gravity, that decreases with
distance between the two objects, it acts
over all distances. It's totally distance-independent.
Then the idea is, in explaining this sudden
acceleration of the universe, that about four
to five billion years ago -- and as I said,
imagine that this takeoff is put here, in
the past, by about four to five billion years
ago -- that at that time, the distances between
the galaxies became sufficiently large so
that the cosmological constant started to
dominate over gravitational attraction.
If two objects are distant enough, then the
force of gravity between them, the force of
gravitational attraction, would be reduced.
The cosmological constant, that force that
is pulling them apart, does not depend on
the distance.
Four to five billion years ago, the universe
became sufficiently large so that the cosmological
constant started to dominate over gravitational
attraction. OK? At that time in the past,
Lambda -- this is capital Greek letter Lambda,
corresponding to L in other alphabets -- started
to dominate over gravitational attraction,
and the expansion started to accelerate.
This is the way how we now deal with this
observation that the expansion of the universe
started to accelerate a few billion years
in the past. Of course, a natural question
is, what is it? We can understand mass, matter,
right? We represent ordinary matter. We can
also, perhaps, deal with the dark matter.
Right?
We don't know, still, what it is, but based
on how the stars at the edge of a galaxy move,
they move too fast, or how the galaxies in
a galaxy cluster move, they again move too
fast, we know that there is some other stuff
that we can't see that interacts weakly with
ordinary matter. It interacts with ordinary
matter via force of gravity, right? That's
how we know it's there. It's still something
that perhaps we could understand.
What is cosmological constant? There are several
guesses, which I don't want to discuss, because
it would be meaningless. I would have to teach
you a little bit about quantum mechanics to
explain one way of how people approach the
nature of cosmological constant. I think that
honest answer is that nobody knows.
Whatever it is, we call it dark energy. We
have ordinary matter, or baryonic matter,
the stuff we are made of, and stars and galaxies,
and so on and so forth. We also have dark
matter, but we have this new thing called
dark energy. All we know about its properties,
it has to have a positive energy density,
amount of energy per unit volume of space
has to be positive, but it also has to have
negative pressure.
What is negative pressure? I tried to illustrate
that here. I could make my own drawing for
positive pressure. It was easy enough. Imagine
you have a vessel of some sort. You fill it
up with liquid, and you drill a hole on the
side.
What happens, of course, is that the liquid
will start to come out. Why? Because the liquid
exerts positive pressures on the walls of
the vessel. Unless I oppose this pressure
from this side, the positive pressure, by
pressing in the opposite direction, on the
hole, the liquid will come out. I have to
stop it.
Most of the things we are familiar with, like
the pressure of the air in your tires is also
positive. It pushes on the side. If it's not
high enough, if your tire leaks, you have
to pump enough so that your wheels would function
properly. This is positive pressure.
The negative pressure is realized in a stretched
rubber band. When you stretch a rubber band,
right, you now have to pull in the opposite
direction. Here you have to push into the
positive pressure. Now, you have to pull in
the opposite direction. Why? Because now the
pressure in the band acts this way. It tries
to contract it. The case of a fluid, a gas
or a liquid, it expands under its own pressure.
Here the negative pressure would cause the
rubber band to contract.
All we know, that whatever it is, dark energy
has to have positive energy density and it
has to have this negative pressure.
The question is, what is the total omega of
the universe? Remember, the parameter omega
determines, among other things, the curvature
of the universe. It turns out that observations
show that the contribution from ordinary matter
and dark matter to omega is about 0.3. That
observation was there up until late '90s,
when it was discovered that the expansion
of the universe is accelerating.
You can ask the question, how do we know?
How do we know what is the contribution to
omega from ordinary matter and dark matter?
You can take a large assembly of galaxies,
do statistics, count the number of stars,
how many stars on average they contain, the
size of the star, how much mass they contain.
From that, you can get an inference as to
what contribution of ordinary matter to Omega
is.
You can do something similar to the dark matter.
It's more complicated. You can take many galaxy
clusters. From the way the galaxies and the
clusters move, you can estimate how much dark
matter has to be in a galaxy cluster. Then
you do, again, statistics over very many such
clusters, and estimate the contribution of
the dark matter to Omega.
It turned out that, and it's still correct
roughly, that the total contribution of ordinary
matter and dark matter to Omega is about .3.
That was never very satisfactory, and I'll
tell you why without going into the details.
It turns out that the general theory of relativity
predicts that if Omega was less than one,
if it was, say, .3, that it would very quickly
become zero. The same if it was bigger than
one.
There are theoretical reasons, big theoretical
reasons, why Omega should be exactly equal
to one. When it was discovered that the expansion
of the universe is accelerating, then they
realized there should be contribution to Omega
also from dark energy.
Its value is then reduced based on the rate
of the expansion of the universe or the acceleration
in the rate of the universe. It was estimated
that the contribution of dark energy to overall
omega is about .7, so that the total value
of Omega is one.
Now, I mentioned the last time that there
were three satellite missions that examined.
You can tell exactly from homogeneities in
microwave background the different components.
The ordinary matter, dark matter, and dark
energy are responsible for these homogeneities.
By doing the statistics on these things, you
can then calculate. It's a complicated procedure.
You can calculate the percentages of different
components of the universe. The latest mission
was just finished. People are still analyzing
data by this probe, Planck, named after Max
Planck who was one of the founders of quantum
mechanics early in the 20th Century.
It's an effort largely launched by Europeans.
Before that, there were two. There was COBE
and WMAP.
Before Planck, the distribution of different
components of the universe was estimated to
be like this: ordinary matter 4.5 percent,
dark matter just under 23 percent, and dark
energy about 73 percent. After more detail,
more precise measurements done by Planck,
these numbers have been revised.
Now, we know there's a little bit less of
that dark energy compared to what we thought
before, but it's still close to 70 percent.
These numbers that I give here, they are approximate
values of what has been reduced, but we can
round all these numbers to about 70 percent.
There's a little bit more of dark matter than
thought before. Now, it's almost 27 percent.
There's a little bit more of ordinary or baryonic
matter than what thought before. But nevertheless,
it's still roughly dark matter, and ordinary
matter is about 30.3 percent. Dark energy's
about 70.7 percent.
Now, when you think about it, we really understand
very little about the universe. We don't understand
what 95 percent of the universe is because
we don't know what dark matter is. We don't
know what dark energy is. We just know that
they are there, dark matter, to explain the
speed of the stars at the edge of the galaxy
or the speed of galaxies in the galaxy cluster.
We also know that we have to have this dark
energy to account for accelerated expansion
of the universe, but we don't know what either
one of these are. We understand only this
relatively small wedge of the total composition
of the universe.
When people say, "Oh, we are just around the
corner to understanding everything," that
is not the case. There is so much that we
still need to find out.
I find that extremely exciting. It's extremely
exciting. One thing that, perhaps, you are
wondering about, and you can say, "OK, if
the universe is expanding because space and
time are expanding, how come that I'm not
expanding now? How come that I'm not blowing
up if everything else is expanding? Why is
the sun not expanding?"
It turns out that what is holding me together
or what is holding the sun together are ultimately
the forces of electrostatic attraction between
different parts of atoms in my body. It's
that force of electrical attraction that is
actually holding me together to occupy this
same volume.
Unless, of course, I start more eating, then,
I would expand. The space itself is expanding,
but different objects in it, like people,
earth, the stars, they are not because they
are held together by different concrete forces
that resist that expansion.
The space is expanding around us, but objects
like myself, like the earth, like the sun,
they keep, more or less, the same volume.
