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.