So, we now have a description of how in the context of gravity as geometry, we
can describe universe in which space at any time is isotropic and homogeneous.
it's a space of constant curvature and all that changes is a scale.
And we have a natural set of coordinates to describe this.
These are coordinates and which are, what we call co-moving.
You pick where something will be at a given time, say the present, if it is
freely moving and take a collection of freely falling observers that start out
addressed at some point and use their positions today.
If they're freely falling througout the history of the universe, to denote the
name of a point. And that observer, is as far as I'm
concerned, always at the same position. So these are observers are at rest.
And then relative to them you can have peculiar velocities.
It is in these coordinates that the space looks, nice and isotropic and homogenous.
It would be nice if we had such observers, we do! By and large galaxies,
we claim, are such observers. They are at rest, they only, their
average motion is the Hubble motion so galaxies are precisely such an object.
And now, having described the kinematics, we now turn to the dynamics.
In other words, we have to understand the solutions to Einstein's equations and to
we've studied sort of a left hand side, the geometry.
Now we need to study what matter looks like.
You would not expect an isotropic homogeneous universe to contain, for
example, stars, because certainly if you have a star then the place where the star
is, is not the place where the star isn't, so matter in an isotropic
homogeneous universe has to be itself isotropic and homogeneous.
Now one way to do that is to have nothing at all in the universe.
That does not describe our universe, and so what we mean is, imagine, averaging
out over large distances, ignoring little fluctuations, like galaxies and clusters,
on average, we should be able to describe, at large distances, the
distribution of matter in the universe, as completely uniform.
So that means there is a constant density of energy, because mass is energy, so
we're talking relativistically. There's a constant energy density, rho,
or mass, if you want, density that is the same everywhere. Otherwise the universe
wouldn't be homogeneous. And if there is a gas with a constant
density, then we know enough about thermodynamics to assume that there will
be a constant pressure. And these are constants everywhere in
space, but of course they can depend on time.
So all of the crazy right-hand side of Einstein's equation about energy and
momentum is contained in two functions of only time. There's no variation with
position, the equations are going to simplify.
in further, and furthermore, we have density and we have pressure. Density and
pressure are related by the properties of whatever the stuff is that you fill you
space with. for example, if you have an ideal gas,
then we know that pressure and density are related to temperature.
in particular, we will deal with, two extreme cases of equations of state,
relations between pressure and density. We have one case, which in cosmology
lingo is called dust, and so we will be cosmologists and call it dust.
What is, dust has nothing to do with the kind of dust that causes extinction.
the particles, if you will, of this dust are galaxy clusters.
dust is what describes a collection of uniform distribution of slow massive
particles moving freely and interacting only through gravitationally.
Each with each other or more importantly with a whole collection of uniform
distribution because they're moving very slowly we're going to imagine that this
is a 0 temperature object. The pressure is essentially 0.
so this is the canonical example of dust of course is galaxies, slowly moving
objects A collection of dust is isotropic in a
frame that moves with it. In other words, in a frame in which you
measure the velocity zero. This is not something fancy, every pilot
knows that your air speed is constant, whether you're going upwind or downwind.
In other words, the air, the moving air, is isotropic in a frame that's moving
with the air. It's only when you're measuring ground
speed that you realize that you might not be making head, any headway trying to go
upwind. Your air speed of your craft is the same.
The airplane flies the same way upwind as downwind.
It's only relative to the ground that things change, so in a frame, where the
velocity of the fluid is zero, at any given time, at any given place. So at any
event, flew this dust has a preferred rest
frame. That's the rest frame in which the dust
at that event appears to be addressed. And of course since we're going to make
this dust out of galaxies that are comoving in the Robertson Walker
coordinates then what we're going to say is that the preferred Robertson Walker
frame at any point is the prefer, is the preferred, is the rest frame of the dust.
This is be, that way, they will both be isotropic.
Both space and matter will be isotropic in the same frame.
And then the other extreme case is very relativistic particles.
In other words the extreme case is particles moving at the speed of light
and we'll call such particles radiation. This can include photons.
It can include gravitons. It can include essentially neutrinos.
Which even if they're not exactly mass-less are very light and move, under
most circumstances, at very near the speed of light or highly relativistic
speeds, and you can make a calculation for an ideal gas of relativistic
particles. Some of you have followed through in the
context of electron degeneracy learning this and it turns out that for a
relativistic gas, since the speed with which the particles are moving is the
speed of light. density determines the pressure, and even
dimensional analysis will tell you that pressure is the density times the square
of the speed of light, and it turns out, the, the coefficient is 1/3, so the
pressure of a relativistic gas is, related to its density by the square of
the speed light over 3. There's no, in, temperature dependence,
because a relativistic gas, by definition, everything is moving at the
speed of light. And again,
you cannot go to the rest frame of relativistic gas in which the speed of
the particles is zero, because they're always moving at the speed of light in
all different directions. But there will be a frame, a specific
frame, in which the radiation field looks isotropic.
This is the frame in which, if you look to the right and look to the left, you
see exactly the same spectrum. And of course, if you are in that frame
and someone is at the same place you are in the same radiation field but they are
moving, they will observe a blueshift, Dop-, blue Doppler shift in one
direction, a red Doppler shift in the other direction, and then say aha, the
spectrum is not isotropic. Only one observer in any event sees an
isotropic spectrum. That is the instantaneous rest frame.
So we have our description of matter and we're going to assert that the matter is
at rest in the Friedmann coordinates, so that space and matter are isotropic and
homogeneous in one system of coordinates. And with that in mind, we write down our
friend the Einstein's equations, this time written correctly. These, the,
latex error that ruined things, so right here the Einstein equation correctly, and
we insert what we know. On the right hand side we insert that
the, geometry, remember these R's represent curvature, and the geometry is
the geometry of a space of constant curvature with curvature R zero which
scales with when the distance is increased, like 1 over a squared.
And then this coefficient, k, is simply given by 0 or plus or minus 1.
0 means, we're in the flat space case. 1 means we're in the positive curvature
case. -1, negative curvature case,
and in all cases I can give the same Parametrization in the curved cases our 0
represents the curvature at T equals 0. At the presence at T equals T0, when A is
1. At present in the case that K equals 0,
you can put whatever you want for R equals 0, cause it's multiplying 0.
That's the simplification on the left hand, on the right, left hand side.
Whereas on the right hand side, we insert what we know about matter.
And what we know about matter again is that it's the same everywhere and it's
determined by this one function rho, and then the pressure is which appears inside
the energy momentum tensor is a, some given function of rho, depending on what
you put in. Okay, so you take all this data,
you plug it into Einstein's equations, you do the mathematics. This is one of
the cases in this class where I say, you do the calculation.
And the equations that you are, you find are two equations for 1, whatever
variables here. The variables are the time dependance of
the scale constant. There's no space dependance of anything.
So we're trying to determine the time dependance of the scale factor The time
dependence of the energy density, and that's it, we have 2 functions, and we're
trying to find the time dependence of these 2.
And so we indeed find 2 equations, they are physics equations, they are
differential equations, so watch out, I said we would skate near calculus, so
there's this little dot here, this dot means the rate of change of a, with
respect to time and we already said, if a were a position this would be it's
velocity and we already noted that the rate of change of a divided by either
relative change of a is what we call the Hubble constant, only we no longer expect
it to be constant. So in general this will be a function of
time, and this is given by Einsteins equation tells us that this is given by
the right hand side 8 pi G. The c squares end up cancelling,
times rho of t. It's the same everywhere in space, but it varies with time.
And then, this is actually not a part of the right hand side of it, obviously.
This is a curvature piece, it's a another piece of the left hand side, and we've
put it on the right hand side to get an equation for the Hubble Constant.
And so the curvature of space and the density of energy both together determine
the rate at which the scale constant, the scale factor changes.
That's one equation. We need another equation, and so again
this tells us that the curvature of space if you want.
Another way to think about it, is if you take the density and the velocity of
everything. So that gives you the Hubble Constant,
that will determine the curvature of space.
Space curves in response to the density of energy and momentum. And so, the
density of energy rho and the rate at which everything is moving, which is
determined by a dot. put both of those in, and you can
determine the curvature of space. But of course, like all equations, if you
know one thing, you can determine another.
Everything ends up determined. What's the second equation? Second
equation looks even scarier. I have here an a with two dots.
if A were a position, this would be the velocity,
this would be the acceleration. It's the rate of change of the rate of
change of a. And it's related to the Hubble Constant
by this expression, where this is the rate of change of the Hubble Constant,
and this its square. And this is given in term of the matter
distribution by a combination of the matter, the energy density and the
pressure. notice that for a relativistic gas, both
of these are in fact equal because for a relativistic gas, p is equal to rho c
squared over 3, whereas for the case of dust, of slow-moving matter, p was
actually 0 so this is just rho. And now This is really a lot of calculus.
So let's try to understand what it is that we're trying to say.
So imagine that I have some time t* and I expand a times t, and express it for
times near t*, and the first term in the expansion was the Hubble expansion term
that we talked about before. The next term in the order will be a
quadratic term proportional to t minus t* squared, and of course in general there,
you can expand and get higher order terms.
And I will, for dimensional reasons, factor out the factor of H, H squared and
then the coefficient is called q over 2 and notice I put a minus sign here.
Because I understand that gravity will decelerate things and indeed, if I write
the expression in this way. So a of t for t near t star is a of t
star times 1, which is the value when t is equal to t star plus the 1st order
term which, who's coefficient I called H, and then the 2nd order term whose
coefficient I call minus H squared, q over 2.
q is called the deceleration parameter for good reason, because what it
indicates is that if you take since you're subtracting a positive quantity,
it means that if you start as a function of T and you plot a.
It starts at t0, at the value a0. The line determined by the Hubble
expression would be this constant increase, and look what the deceleration
parameter does is it slows it down. It decreases below the straight line,
and in terms of this deceleration parameter, you can rewrite this crazy a
double dot thing as just minus H squared q. It's just twice this coefficient.
And notice that the sign here is well justified, because pressure being
positive, density being positive, this sign being negative, gravity, [SOUND]
decelerates expansion. This is not a surprise, but this
non-surprising result is derivable with great work by following through
Einstein's equations. So Einstein's equations not surprisingly
tell us that both density and pressure which is, the, remember, the pressure
results from the velocity The momentum that the particles are carrying.
This momentum contributes to gravitational attraction because momentum
gravitates just in the same way that energy does.
So remember, notice that the pressure of the relativistic gas actually increases
the deceleration, contributes to deceleration, as opposed to one would
think of pressure leads to, makes thing try to expand.
Well pressure makes things expand, if you put something in a container, but in this
case we're talking about a universe where the pressure is the same everywhere.
There's no pressure gradient between inside and outside for the pressure to
put us, push on. There's no container for the universe, no
edge of the universe for the pressure to push on.
Every bit of the universe experiences the same pressure on both sides.
What is going on is that pressure contributes to the gravitational
attraction, because remember, momentum as well as energy is a source for gravity,
couldn't be otherwise consistent with relativity.
So these are our equations, and they are going to, in the end, determine the
evolution both of the density and of the scale constant.
However in general, we will have differing sources of density.
Remember, we can have a universe that contains both dust, slow moving things,
and, say, photons, relativistic things. And one can imagine a universe in which
these don't interact, and then that means that the energy in dust is conserved.
Because no dust is created, the total mass, remember the energy of the dust is
just is mass, it's addressed, and so the total mass of the dust is conserved.
And separately the total energy in photons is conserved.
And if this is the situation when they do not interact, then I know the time
evolution of rho of t if I know the time evolution of a of t.
and this is because, if you take the given volume,
it has some dust in it, and then as this volume expands, it has the same amount of
dust in it. But if the distances have grown by a factor of 2, the volume of
this region has grown by a factor of 3. So when distance by a factor of 8, and so
the density, mass conservation says density decreases like the third power of
the scale factor, when space doubles in length and all
lengths double, the density decreases by a factor of 8.
How does this affect relativistic particles? Well for relativistic
particles, think again about photons, but of course the theory is universal.
In a, the case of a relativistic particle moving at the speed of light, the same
property of Thames, if I take the volume, containing a certain number of them,
their number is conserved when I, that volume increases, it carries the same
amount, number of particles in twice the, eight times the volume.
However, they're all redshifted, so the energy carried by each of them is also
decreased by a factor of 8. And so the behavior of the density of
nonrelativist-, the density of nonrelativistic matter is that as a
function of time, if there is no interaction and no creation of matter or
loss of matter, like say the loss of stars, if you wish, convert mass dust
energy into radiation energy, but stars are a small perturbation in the
large scale of things. Most of the matter of the universe is
inter cluster of gas, which is not participating in stellar processes, and
even when there are, when dust isn't a star,
only a few percent of the mass of a star over 10 billion years will be converted
to energy. But, so this is a small violation of this conservation of energy.
Ignoring little effects like stars, then the energy density of dust scales with
the scale factor, like the negative 3rd power, whereas the energy density of
radiation scales like the negative 4th power.
Remember, three of those are because of the volume, and then one extra factor
because each photon's energy is decreased by the redshift.
Dense slide. Now, Einstein had understood all of these
equations. it's the usual case that the, what you
get out an equation is what you put into it.
You put in a lot of symmetry, the equation simplifies.
This complicated set of coupled partial differential equations becomes this
relatively simple set of ordinary differential equations for functions of
one variable, time. And Einstein was trying to write the
solution that describes our universe, and what he knew about our universe was that
it's static. So what he wanted was a universe in which
the skill factor didn't change, in which nothing essentially changed.
He had no reason to imagine that distances were changing.
He did not know at the time of Hubble's result, which by the way had previously
been, it turns out, measured by [FOREIGN] and understood in a deep way.
And so he tried to set this to 0, and it's possible to set the rate of change
of A to 0 by making balancing the energy density against curvatures. So you need
to have a curved space, if you have any matter in it, but if you have positive
curvature, and positive energy density, you can balance them to get zero
velocity. Glorious, however, you cannot make the
acceleration zero, so the zero velocity because of course both of these terms are
all negative And so the acceleration is always negative. So the 0 velocity
solution you found out is sort of like the 0 velocity of my little rubber ball
at the top of its flight. It stops for a minute but it's
acceleration is still negative, it won't stay at 0 velocity, it will fall.
So, indeed if you have 0a., then the negative acceleration means that
you have universe that is not expanding but is about to start contracting.
So Einstein was very saddened by his inability to find a description of what
he thought to be the universe, namely a static universe.
But Einstein was nothing if not creative. He looked at his equation, and he
realized that without ruining any of the axioms or the logic that led him, to the
equations of general relativity, he could make a modification that would have
absolutely no impact on the connection, the agreement with Newtonian physics,
but would fix this, would allow a static universe.
And the correction, in terms of Einstein's equation, involved adding one
more term to the left hand side. So, modifying the way that gravity, the,
metric function G is determined by the distribution of energy momentum tensor.
He called this the cosmological term, we today call it the cosmological constant.
the properties of lambda is that for this to be a consistent modification of
Einstein's equation lambda is just a number.
A constant, it has the dimensions if you work through of inverse length squared.
So lamba is measured in 1 over meters squared,
and this is the modified version of Einstein's equation.
When, of course you can alternatively think of taking this same blue thing,
flipping its sign and putting it on the right-hand side of the equation, and
thinking of good old Einstein's equation in the presence of a different kind of
energy momentum distribution. So what kind of crazy energy momentum
distribution is lambda? Well it's a constant, so it does not depend on space,
so it satisfies our isotropic and homogeneous relations. So I should be
able to describe it by a constant energy density.
Indeed, lambda corresponds to a constant energy.
Related to the value of lambda by this, this was what follows from placing lambda
in the position where rho sits here, and then the cosmological constant
corresponds to a pressure which is negative rho times c squared.
So if you have a positive, and the cosmological constant can have either
sign. If it's positive, then it has a positive corresponds to positive energy
density and a negative pressure, whereas a negative cosmological constant
the roles are reversed and then if you now re-do the calculation that led us to
Friedman's equations in the presence of this cosmological term Einstein finds
that the first equation for H squared is modified and the 2nd equation for minus
QH squared. Remember, here we're determining Q, the
acceleration, the deceleration parameter is also modified,
but it is essentially by exactly the same term.
This has to do with the fact that pressure and density are essentially the
same thing. And so, you plug this thing into that.
And what do you discover, what you find is that now I can find a solution where
the left hand side, the rate of change of the scale factor, and its acceleration,
are both 0. How do I do this?
Well, I'm going to pick a universal which p is equal to 0.
So my matter is going to be dust. I'm going to of course need positive
curvature so that the energy density is balanced against curvature. And then,
it's balanced by, curvature, and I'm going to also have a cosmological
constant, which will allow the negative term here, this is going to be zero, to
be balanced by this term. And it turns out that if you set our 0
equal to lambda. Remember both of these have dimensions of
1 over length squared, so that even makes sense,
equal to this concoction, proportional to the energy density, both of these
equation vanish. This is called the static Einstein
universe, and Einstein thought, well, since this is the homogeneous, isotropic,
static universe, this must describe our universe.
He then was informed about Hubble's measurements and realized that our
universe perhaps is not static. And this cosmological term was
unnecessary. Einstein then regretted introducing this
uglification of what was a beautifully elegant equation before. He called it his
greatest blunder. As we shall see, it might not have been a
blunder, Einstein, even when he blundered, had foresight.