And heavy lifting. 
Now we can put it together with the data 
that I promised you I would explain where 
we got. 
And understand, the history of the 
universe in the past and into the future 
because I said that understanding these 
parameters determines everything. 
Now, let's see how that goes and we'll 
talk along the way 
Early days of the universe, remember, 
early days of the universe would of been 
radiation dominated. 
I've talked about the fact that into the 
past, matter density increases like 1 / 
a^3 as a gets small. 
radiation density decreases like 1 / a^4. 
I didnt' say anything about. 
Dark energy density because remember 
that's a cosmological constant, that 
completely does not change the energy, 
the density of dark energy does not 
change with the scale of, factor of the 
universe and so this tells us, in 
particular, that if it's 74% of the 
energy density in the universe now, in 
the past when it was smaller and 
radiation and dust were denser, it was 
less important. 
So early in the universe the cosmological 
term plays absolutely no role and 
remember that the universe is radiation 
dominated until about 55,000 years of age 
and what does it tell me that it's 
radiation dominated? Radiation dominated 
basically tells me that Rho radiation. 
Behaves with the scale factor like a^-4, 
and so I plug that into the Freedmont 
equations and I find that the scale 
factor as I promised increases like the 
square root of time so the universe 
starts expanding very rapidly and then 
slows down. 
Of course this region over here where the 
universe is very tiny is, The beginning 
of this graph is something we will focus 
on a little bit later, but for the first 
55,000 years of its universe, this is the 
plot of the scale factor as a function of 
time. 
I can sew this on To graph that describes 
the current universe. 
Of course here the scale is billions of 
years, so I've completely ignored the 
little tiny bit here that includes the 
beginning of the other graph. 
And so I've imagined starting at 0 With a 
matter dominated universe, things is, 
matter domination gives me the behavior I 
described. 
The scale factor increases like T ^2/3 
and there's a transition from matter 
domination to domination by dark energy 
which remember, today there is about 3 
times as much dark energy as there is 
dust, remember I talked matter, dust, 
same difference. 
The transition occurs in the past when 
the density of dust was about 3 times 
what it is now. 
The density of dark energy not having 
changed, it would have been equal, and 
prior to that, dust would had been 
denser. 
That happened at a red shift of about. 
0.4 or at an age of the universe of about 
0.6 of what it is now or about 8 billion 
years. 
And we see that over here and this 
heralds a transition from early times 
when the scale factor behaved like t to 
the 2/3, a power smaller than one, so 
that the, 
expansion was decelerating to the 
behavior at large times. 
At large times, as the scale gets larger, 
radiation has already been diluted to 
insignificance. 
dust also gets diluted to insignificance 
and what is left at large values of A. 
If the universe ever gets large And there 
is a cosmological constant. 
It will eventually take over. 
the expansion driven by a cosmological 
constant is an expansion where the right 
hand side of the Freeman equation does 
not depend on a, does not change with 
time at all. 
This gives us the closeset thing to a 
constant expansion. 
it's not a constant, 
expansion in terms of the rate of change 
of A it's a constant expansion in terms 
of the Hubble Constant is in this case 
constant. 
That means, remember the rate of change 
of A has to accelerate. 
In fact, it corresponds to an exponential 
growth of the scale factor and you can 
see that at late times as this term takes 
over. 
The curve turns over to an exponential 
rate, and because we have a non-zero 
cosmological constant or dark energy 
component, we predict that in late times 
we will have exponential expansion of the 
universe. 
We live right about here at 13.8 billion 
years where the scale factor by 
convention is exactly one. 
Notice we are not far from this turnover 
from the transition from dust dominated 
to cosmological dark energy dominated 
universe. 
We're at the region where expansion which 
had been slowing down is already 
accelerating This is the exact form of 
the function I've plotted. 
But more important is that it's 
exponential at one end and to the 2/3 at 
the other end and the remember way down 
here is a little region where it behaves 
like T to the 1/2. 
That, if you want, is the history of the 
universe. 
Note, that understanding the current 
parameter, cosmological parameters, once 
we figure out how to measure them, we can 
[UNKNOWN] the entire past of the universe 
and predict it's future. 
And we will see what all that means as a 
first application of all of these. 
Let's introduce this important issue not 
[UNKNOWN] related to oldest paradox of 
particle horizons. 
So how far can we see in flight space, 
normal space. 
You can see. 
All of the universe, that you just see 
farther and farther, but if you want to 
see things that are farther from you, you 
might need to look farther and farther in 
the past. 
But our universe, as Poe reminds us, has 
a finite age, so the question is, can we 
see the entire universe? What fraction of 
the universe can we see, and That 
fraction of the universe that we can see 
is called our particle horizon. 
And, it turns out that if you take any 
observer, of course all observers are the 
same, take some co-moving observer 
sitting at one point throughout the 
history of the universe, and ask what 
fraction of the universe He can see 
whether universe is infinite and we're 
going to talk here about a flag universe, 
remember i have a strong bias to a flag 
universe and i want to ask which other 
observers has he seen, okay and the aa, 
good way to describe this is, this is 
what we call the size of the observer by 
the universe. 
This is the distance from us to the most 
distant objects we have seen or can in 
principle see. 
These are the objects that we see so long 
ago, so far, that the light left them at 
the time of the big bang. 
So another way to ask the question is at 
any given time, what are the things that 
you see as they were at the time of the 
big? in time. 
And so the objects on the particle 
horizon are the things you see at the 
time of the beginning. 
Now of course in the radiation era for 
example you solve the equations and you 
see that the distance to the particle 
horizon increases with time [UNKNOWN] 
factor increases linearly as. 
Twice c t. 
So if you are looking at the universe a 
second after the big bang you can see two 
light seconds in each direction. 
Note two light seconds despite the fact 
that light is moving at the speed of 
light. 
How is this happening? Well remember this 
is the distance, now Coordinate distance 
at time 1 second. 
To those observers who's light you are 
seeing at time 1 second, from which light 
took 1 second to reach you but of course, 
in that 1 second those observers 
themselves moved so they are now 2 light 
seconds from you because expansion has 
carried them that far. 
In the matter of solving the same 
equations. 
You find that the objects you see, again 
ignoring the little bit of expansion 
during the small radiation era, in a 
matter dust dominated universe the 
distance that you can see all the way 
back to the big bang at time T increases 
with time as 3ct. 
You plug in that complicated function 
that describes. 
The transition region between dust 
dominated and dark energy dominated and 
figure out how far we can see today and 
again this is not how far the light has 
actually traveled, that's an ill defined 
question. 
This is how far today are the objects in 
today's coordinates are the objects which 
we see. 
As they were back at the Big Bang and if 
we could see them they would be 46 
billion light years away. 
Note this is a lot more than 13.8 billion 
light years. 
Light has traveled for 13.8 billion years 
but the objects that it left 13.8 billion 
years ago are now a distance 46 billion 
light years away from us so we can see, 
in principle. 
A big sphere around us whose radius today 
is 46 billion light years and that sphere 
is still growing but that growth stops in 
the sense that when you move to an 
asymptotic expansion, then at large times 
when we're describing an asymptotic 
expansion, the distance of the horizon 
becomes just a constant number. 
Some constant distance times the scale 
factor, so basically you're seeing the 
same objects. 
You no longer see, and once you hit 
exponential expansion more and more of 
the universe the ultimate limit, the 
things that we will see, out to the end 
of time, the most distant things we will 
ever see. 
Are those which are currently 62 light 
years away from us and anything that is 
currently farther than 62 billion light 
years, we will never see because the 
expansion of the universe carries 
themselves far, away farther away from us 
so fast from us that light from them will 
never reach us. 
So if you want this is the size of the 
universe we've observed so far. 
If, you, we had observed the universe, I 
don't know, 6 billion years ago, and you 
imagine that it was dust dominated, we 
would have seen about, half that 
distance. 
Because, in the dust dominated, world, 
the horizon distance grows linearly with 
time. 
And, it is, the growth, is, slowing down 
now. 
When I say half that distance, I mean, we 
would have seen half, that an eighth as 
many galaxies were visible, six billion 
years ago, as are visible now, assuming 
that galaxy density has not changed, 
which is probably false. 
and there's an ultimate limit to how much 
we will see, as I said when I discussed 
Olber's, it's not just that we have not 
yet seen everything, but even if we wait 
til the end of time. 
We will only see a finite extent of the 
universe. 
Only some finite chunk of universe. 
That chunk will grow but it will, we will 
only see a finite list of observers 
asymototically. 
You can ask a different question that 
defines a particle horizon, and here is 
drawn our particle horizon. 
is drawn in this dashed line as a 
function of time. 
And you see that indeed that's this 
dashed line over here as our particle 
horizon. 
at times from the beginning of the 
universe to 25 billion years, so right 
over here the horizontal line is where we 
are now. 
And these are at any given time all of 
the things, all of the parts, all of The 
galaxies if you want co-moving observers 
move, are at rest in this frame. 
They move along vertical lines so the 
dist, horizontal distance measures 
distance in two day's coordinate in 
today's s- universe. 
And so you see that we will see objects. 
bite, 25 billion years into the age of 
the universe we will have seen out a 
little bit past 50 billion light years 
and eventually this asymptotically ends 
at about, I said, 62 billion light years 
and by today we've seen out to about 46 
billion light years despite the fact that 
the universe is only 13.8 billion years 
old and. 
That's one question you can ask. 
The other question you can ask is take 
today's universe, so think of this as a 
space time diagram, here is the universe 
of today, and say, suppose that something 
happens. 
Which of the events that are happening in 
today's universe will I ever see? And 
that answer turns out to be a different 
question. 
That defines something called our event 
horizon, so you can draw, and this is the 
picture of our event horizon. 
It separates, and that it turns out 
narrows with time. 
In other words these are of course. 
these are all of the events inside this 
which at any point in the history of our 
little corner of the universe will be 
visible. 
So if you want, this is the analog of our 
past light cone. 
At any given time. 
This is our light cone now, and if you 
add up all of the light cones into the 
infinite future, you get this event 
horizon. 
So, something that occurs today here, of 
course we can't see it, but if you give 
light time, will eventually, if something 
that occurs, say, here Today, of course 
we can't see it, but eventually someone 
on earth will be able to see it. 
Something that occurs, farther than 
about, what is it? About 15, 16 billion 
light years away from earth. 
Today, will never be seen, we can see the 
things that are 16 billion light years as 
they were in the past But we can not see 
anything that happens 16 billion light 
years away today. 
We will never, ever see. 
And in fact, because of the exponential 
growth, past once the cosmological term 
takes over, you see that our event 
horizon shrinks rather rapidly. 
And into the future eventually, if you 
ask, what are the objects that we will be 
able to see as they are 10 billion years 
from now, which is about here. 
You see, that they form a small fraction 
of the object whose, from today, which we 
will see. 
And, so, over time we get to see less and 
less. 
Of the universe in that sense. 
We get to see more and more of its past 
but if you ask say stars starting forming 
at some period over time we see less and 
less of the current history of the 
universe. 
an object and what you see is that over 
time a given object that is stationary 
say something 20 billion light years from 
us, well we could have see what happened 
to it at the big bang. 
We could have seen what happened to it at 
the age of 5 billion years, but by about 
10 billion years we lost it. 
Anything that happened to this object 
later than this point will never be seen 
at Earth so over time objects are leaving 
our event horizon. 
So what does that mean? Of course, they 
won't it's like an event horizon of a 
black hole it's not that they hit the 
event horizon. 
going to poof disappear what happens is 
that the light from this object 
experience an infinite, experiences an 
infinite cosmological red shift as it 
approaches the event horizon. 
Because essentially the time for light to 
get here becomes infinite, you get an 
infinite cosmological red shift and so we 
will see objects dimming. 
As they approach. 
So as the light from this object starts 
to reach us from bits of its history as 
it approaches the event horizon, we'll 
see it more and more and more and more 
and more red shifted, and it will 
basically dim itself out of existence, 
and of course we'll never see it across 
the event horizon. 
So we have our particle horizon Which is 
all of the bits of the universe that we 
can see. 
Since the big bang let me clear the 
picture up a little bit. 
So we have our particle horizon which 
currently at the present time involves, 
all the parts of the universe that we can 
see. 
As there were, at the Big Bang, this is 
our current particle horizon. 
Right, it touches, here. 
This is our current particle horizon. 
This is what we see is, we see all of 
this part of the universe. 
At some time after the big bang whereas 
in blue over here this is our current 
event horizon, these are all of the 
objects today that we will some day see. 
So in some sense with time we see more 
and more of the ancient universe and less 
and less.