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.