Okay, we worked, we learned about how GR describes homogeneous isotropic spaces
like ours, how they can evolve and what the equations are that describe them.
What has all that got to do with us? Can we see how it applies to our universe?
Yes, we can actually link it to astronomy, so here's how we parameterize
our universe. here the 2 freeman equations as we wrote
them the one for velocity and the one for the deceleration parameter and what do we
know? Well, we measure the Hubble constant at present, remember we
parameterize it as a 100 times H traditionally but we will use H as 0.71
There are constant refinements of the precision of our understanding of the
Hubble Constant. It is certainly not 110, it will settle
down somewhere in the vicinity of 70 but we're going to use 71 kilometers per
second per megaparsec for the remainder of this class unless they discover that
it's 30, so. So this is what is known here.
We know this. Can we measure anything else? Well we
certainly can't measure the scale factor. That's something that we know the scale
factor by construction today as 1. What you want is the history of it.
What do you know about. The density, well we can measure the
density of stuff of energy of mass in the universe by our we talked about it the
way we measure the density of the solar system.
We count dust. we can count stars.
we can do better than that. We can actually estimate the mass say of
the Milky Way. Beyond the stars, and the dust, and the
gas, we discover that most of it is dark matter.
We do the same thing for a galaxy cluster by doing the Newtonian orbits of things
and find that most of the mass of a cluster is made of dark matter.
and so we can get some idea about the total density of matter.
And it will turn out that the density of the, the, the, most of the energy density
of matter in the universe is going to be in the form of dust, of slowly moving
heavy particles. There is radiation density of
relativistic particles, but it'll turn out to be much smaller.
Now It turns out that it pays to parameterize the density by, instead of
describing [UNKNOWN] as a function of t, describing something called omega which
is [UNKNOWN] basically scaled at any time by this quantity.
This is called the critical density for a reason that will become clear.
And the reason this is called the critical density is very simply that if
you imagine, ignoring for a moment the cosmological constant term which we are
going to interpret as though it were a kind of energy.
So, we are going to imagine that we put lambda.
As one of the kinds of energy in the universe.
In other words, in the right hand side of the equation.
then if the density is the critical density.
In other words, if A Pi G Rho/3 is H^2. The k is 0.
So if the density is critical, space is flat.
If rho is bigger, s- the curvature is positive, if rho is less than the
critical density the curvature of space is negative.
So critical density, which means density given by this or omega = 1.
Is the condition for space to be flat,okay? So this is why it pays to
scale density to this critical density that is a significant standard density to
compare to and we can evaluate it, since we measure H 0 and you know Newton's
Constant, we can evaluate this in today's universe and the critical density
corresponds. To 10^-28 approximately kilo / m^3 ,
that's about six protons per m^3. This is far smaller than say the density
that we found for the Milky Way. Obviously, most of space is not part of
the Milky Way. And so, if the density of matter and
space is this, then space is of energy and space.
Is this, that space is flat, if it's larger than this it's positively curved,
if it's less it's negatively curved. And as I said, because this, we are
accounting for the cosmological term as part of our energy and momentum, then
this will allow us to write the total. energy momentum or the total scaled
energy momentum dens- energy density of space as a contribution to, to
nonrelativistic objects today plus a contribution due to relativistic
particles, say photons and neutrinos, and a contribution due to the cosmological
term. Which because we've moved it on the right
hand side is a form of energy and because we don't understand what it is, is this
contribution, is what we term dark energy.
It's dark only in the sense that it does not correspond to any form of matter that
we know. We do not see it, it does, like dark
matter does not interact with light, but it's certainly not dark matter, it has
these very bizarre This bizarre equation of state that is not like anything that
we know. Almost at the end of the class we might
discuss things that it could be. And so, Hubble's three months first
equation becomes in terms of this, the equation that says omega just dividing
everything by H-squared I get 1 on the right-hand side.
And here I get omega here I get kR^2, R0c^2 over H s-.
And if I remember that this is included in the total density then this is
Fremont's first equation evaluated today if you val- if you remove the 0's you
just get another H^2. Down here, I've just set a to 1 at the
present and so this is now the current density determines the curvature and then
if I plug in the relation between pressure and density for matter where
pressure is 0 and for radiation. Where, these two terms have the same
magnitude. I find that the second equation simply
reads that the deceleration parameter is 1/2 * , remember I have to divide by h^2
to get q, the deceleration parameter is 1/2 * the, Density, normalized density of
a non-relativistic matter, sometimes I call slip.
We sometimes call dust matter, so you have matter and radiation.
I'll try to remember it's dust and radiation.
Plus, the contribution of radiation does not get a factor of 1/2 because there are
2 equal terms, minus the contribution of the cosmological constant which, as you
recall, shows up with an opposite sign. And so the parameters of the universe are
given if you know this number, that number and that number and age 0, then
you can determine the entire evolution of the universe and the pie chart that you
saw in the . Introductory could, told you, that the
best data that we have, is what we have over here.
The total, fraction of the critical density that is dust is, about a quarter
of this. A smal fraction is what we call bionic
dust. Atoms and the rest, dark matter, which
has you will recall a much larger mass than the bar-ionic mass of the universe.
radiation as I said, is a negligible part of the energy density today.
And About 74%. Again, the pie chart, this, pie chart
didn't say 74%. The numbers are, known to a better
accuracy than what I'm representing, but it's the principle that I'm trying to get
across. You can look up better numbers than I
would have, than I have used, if you want.
About 74% of the energy density Let's just do this crazy dark energy.This is
the meaning of that pie chart. However, what all this tells us is not
that clear. And we already got these numbers is far
from clear. And we're going to spend the rest of the
week figuring that out. But, can we do anything with this.
So, our universe today, ignoring for a moment the cosmological constant.
Before, people discovered that the cosmological constant was nonzero,
When I was a graduate student 20 years ago.
It was assumed that the cosmological constant was 0.
For theoretical reasons that we'll discuss later in the week.
But, so people assumed that the energy density was all non-relativistic matter.
We knew a little bit about dark matter and so imagine a universe in which.
omega r is 0, omega lambda is 0, and we only have a density of dust which with
expansion scales is diluted by the factor of the volume and which has pressure 0.
Plug that into these equations, it turns out that solving these differential
equations Is aa, rather straight forward and here are some solutions so aa, i have
written 3 sort of characteristic solutions.This is a plot over here on the
left of the scale factor as a function of time in billions of years and i have
written the Description of universes with essentially the same hubble constant h 0.
Though to be fair the present time is not exactly the same in all of these graphs.
So there's, these are not exactly extrapolations from our universe but the
blue curve shows a universe. With omega 0 = 0.5, since omega is less
than 1, the curvature of space is negative.
This is one of those hyperbolic space in this universe, is one of those hyperbolic
spaces. The red curve is the critical universe
omega = 1, this is a flat universe. the black curve is the case omega equals
to 2 and if omega equals to 2 notice that the scale factor grows and then shrinks
the universe expands from a big bang and ends in what we used to call a big crunch
these were the kinds of diagrams we drew they assume that the cosmological
constant is 0, the universe is full of dust and
For the case, k=0, in other words, omega=1.
you can actually write, the relevant, which, which will turn out to be the
relevant case as you saw. In the data I gave before.
I didn't actually add them up. But if you add the numbers I provided,
they add up to omega=1. When you add up all together.
And indeed The, our universe we know to a very good approximation is flat, how we
know that we'll figure out as we go along.
But for the relevant case K equal 0, so for the red curve this is in fact the
expression for the scale factor, the scale factor grows with time since the
Big Bang, as t to the 2/3rd and in fact the.
Present time, the time when the scale factor is 1, is related to the Hubble
factor measured today by not T is 0. The age of the universe is not A 0, but
only 2/3 of A 0 inverse and so A 0 inverse.
So, a dust filled universe predicts a slightly younger universe that the 13.8
billion. That we got by just plugging in t zero
equals h zero inverse, but we see that our order of magnitude estimate was is
borne out by these more precise solutions of Einstein's equation.
on the right hand side what we have is a plot of the red shift.
Versus distance in these three universes again normalized so that they have the
same value of H0 so what the same value of H0 of course means is that all of
these graphs agree in the region where we have z = H0 D over c.
For a small this is a D coordinate distance by the way, so this is D0.
And of course at small distance it doesn't matter what distance you use
cause they're all the same and the Hubble expression holds and we see that the
deviation from Hubble. depends on the kind of universe in which
you live. for example the universe with the, the
super critical, the closed universe with positive curvature has departures where
at large distances The red shifts are larger than you would have expected.
What that means is that at, large distances means longs time, long times
ago because the look back time increases with distance.
And it means that the universe, the motion of galaxies was faster.
in the past than one would have expected by extrapolating the Hubble expansion,
and if you imagine that the present is here, then you realize indeed, that in
the past the scale factor was growing faster than it is now, and so that
explains this deviation above what would, one would expect.
But all of these curves are tilting down, so in all of the cases I find A little
upward curvature of the redshift distance relation, how much depends on how much
the scale factor's slowing down. So this is sort of an example in the
context of an easily computable dust universe as we will see, this behavior,
this t to the 2/3rd behavior, in fact does describe our universe.
Universe to a resonable approximation until the recent past, though it's not an
idle academic exercise, but I wanted to get us, us to get a sense for how these
equations determine the actual evolution of the universe.
OK, so now, we have the parameters of our unicerse, right over here, We can use
them to plug in to the equations the kind of calculations are little more
complicated that what it took me to produce these graphs.
But you can get some quality of, of ideas without doing any kind of calculation.
In fact, the data I gave you is recent, I'm talking about what people could do in
the 1930s. So in the 1930s here is what they
realized, first of all that because the universe is expanding, we measure that
it's expanding right now, that means that in the past a of t was smaller, a of t
was smaller, conservation of mass tells us that the universe was denser, and as
you go farther and farther into the past it gets denser and denser.
Moreover because of the way peculiar velocities decrease with expansion it was
also hotter. This is not that shocking what we're
saying is that an expanding gas cools and the gas in question is the gas of
galaxies. We're observing at essentially zero
temperature but if it has any peculiar motion at all.
Then in the past, the motion was faster, there was more random motion, and the gas
was hotter in the past, and has cooled as it expanded.
As you go far enough into the past, temperatures get hot enough, that when
you get, one of the first major events, is that when you get to a red shift of
approximately 1000, in other words When all distances in the universe were about
1,000 times smaller than they are now this turns out to be about 380,000 years
depending on it, in, exactly your model, but in our current model of the universe.
About 350,000 years after the big bang At that point, before that time,
temperatures were so high, that hydrogen atoms were ionized.
And so, if you read the history of the universe forward, you have ionized
hydrogen, and then as temperatures dropped down below the ionization
temperature Hydrogen atoms becomes, atoms become stable against ionization and that
occurs at that time and for the rest of history hydrogen is ionized of course in
stars and in interstellar clouds. But it's not globally everywhere ionized
by the cosmic temperature. this event we will call it ionization.
It's often called by the confusing term recombination.
Recombination is what happens when you have ions.
In, you know, chemistry, in some experiment, and the ions recombine to
form atoms, it becomes logical constant, it's a misguided term, because it's not
recombination, it's, the first combination.
There was not a, previously, combined neutral atom.
These ions, were formed as ions. And then, later, electrons and protons,
managed to produce atoms, as the universe cooled.
before that, before, the time Or in the first 380,000 years after the Big Bang,
baryonic matter and radiation exchange energy rapidly so our conservation
equations might not exactly hold because as long as the universe, the matter in
the universe, the dust, is in the form of hydrogen ions, so we have a dense.
Plasma of charged particles. And remember than, that means the mean
free path of the photon is very short. Remember we talked about a photon taking
100,000 years to get out of the sun. In the process it's absorbed and emitted
many times. So there's a, a, tight interaction
between matter, between dust and radiation.
until Dust forms hydrogen atoms. Once you have neutral hydrogen atoms, and
once the, then the photons, with frequencies too low to ionize hydrogen,
so anything below the ultraviolet energies that are required to excite
hydrogen atoms Are basically then unimpeded.
And the universe which, remember is full of hydrogen, goes from being an opaque
strongly interacting plasma to a transparent gas of neutral hydrogen.
This is bizzare. We often talked about ionized hydrogen
being transparent and. Hydrogen atoms being opaque but that is
because or excited hydrogen ions being opaque, this is true but that is not at
the densities that here we are talking, what we are talking about is very dense
plasma is opaque and strongly interacting photons do not propagate.
Whereas a gas of neutral hydrogen. Of atoms is transparent to low frequency
radiation, below the ultraviolet frequencies required to excite it, and as
we'll see, that was the relevant question.
So the universe becomes transparent at the age of 380,000 years.
We'll see what that means. If you go further back, then you realize
that if there's any radiation, and remember, we said that there's some
radiation. some negligible energy density and
radiation today. As you go farther and farther back in
time d- dust becomes dense, the energy density of dust increases, but the energy
density of radiation increases more. It increases more because in the past
these photons were not only more tightly clumped, they were also not yet red
shifted. And so the energy per photon was larger.
So if you go far in, far, far enough back in time then there's value for the scale
factor given by the appropriate ratio of the energy densities.
Such that when the world was about 10,000 times when distances were about 10,000
times as small as they are now. Corresponding to a redshift about 3,300
and an age for the universe of about 55,000 years.
Note a higher redshift because it's before recombination.
At that point radiation energy density, energy density radiation and energy
density in matter, in mass were about equal and before that time.
The world was dominated, the energy density of the universe, was dominated by
radiation. And the same kind of solution that gave
us T to the two-thirds behavior for the scale factor in a universe containing
only dust. If the universe contains only radiation,
then you find that the scale factor grows with time.
AS the square root of time, and sort of because at the time of this transition if
you sort of imagine that rather than a continuous changeover from radiation to
some equal mixture of radiation and dust to all dust.
The universe went from all radiation suddenly to all dust.
Then I basically match on the two solutions.
By saying that until this time, I had this function and when I got to this
time, I had this value of the scale factor and thereafter, I had the growth
as t^2/3 matched on at that point. Now, I keep talking about radiation.
I said radiation is any relativistic particle.
What's a relativistic particle? Well any particle will become relativistic if it's
moving at the speed of light. The characteristic velocity of a particle
in a thermal gas is determined by the temperature and if the temperature is
such that KB times T is bigger than the rest energy of the particle, then the
average. Energy of a particle being, k b t, this
particle will be relativistic. So at high enough temperatures,
everything becomes relativistic. when you have a gas of relativistic
particles, then, they behave like photons.
Their energy spectrum is determined by the Plank blackbody distribution.
It's a property of thermodynamics. The mean energy of course, of every
particle is kBT. And the enrgy density, well it's the
energy density of a gas of photons. And so, there's some constant G which
we'll describe in a moment. G for a photon.
Is 2. And then, you're not surprised it's
related to the Stefan Boltzmann Law, the sigma T^4 law.
so, and then how do you take a flux and change it, it into an energy density you
need something with the dimensions of velocity, the relevant velocity of course
is the speed of light. Basically, this is the same as the
calculation of the number of winds going through a square meter, you take the
density of wind and multiply by their velocity, you want to know how much
radiation traverses a square meter that's the flux with which the universe is
radiating, you take the energy density of radiation multiply by its speed, you get
the flux and the only Tricky business is this geometric factor
of 2. We're used to our order of magnitudes
missing geometric factors of order 2. Remember, G is 2.
So, for photons the equation reads the energy density of photons is sigma
T^4/2C. Now, of course as the
Universe expands the energies of all relativistic particles are redshifted,
this as I said, preserves the blackbody spectrum but the temperature of a
relativistic gas decreases as the scale factor grows as the inverse power or
increases. Into the past.
If you go far enough into the past, a gets small enough, temperatures get high.
Far enough into the past everything is relativistic.
We'll see where that comes into play soon.