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PROFESSOR: I'd like to
begin building up momentum
00:00:24.050 --> 00:00:26.540
by going over what we've
already done to kind of see
00:00:26.540 --> 00:00:28.380
how it all fits together.
00:00:28.380 --> 00:00:31.840
So I put the last lecture
summarised on transparencies
00:00:31.840 --> 00:00:32.390
here.
00:00:32.390 --> 00:00:33.820
And we'll go through
them quickly,
00:00:33.820 --> 00:00:35.560
and then we'll start
new material, which
00:00:35.560 --> 00:00:38.310
I'll be doing on the blackboard.
00:00:38.310 --> 00:00:42.790
So we've been studying
this mathematical model
00:00:42.790 --> 00:00:47.060
of a universe obeying
Newton's laws of gravity.
00:00:47.060 --> 00:00:50.070
We considered simply a
uniform distribution of mass,
00:00:50.070 --> 00:00:53.740
initially spherical,
and uniformly expanding,
00:00:53.740 --> 00:00:55.940
which means expanding
according to Hubble's law,
00:00:55.940 --> 00:01:00.910
with velocities proportional to
the distance from the origin.
00:01:00.910 --> 00:01:03.850
And Newton's laws then tell
us how that's going to evolve.
00:01:03.850 --> 00:01:06.510
And our job was just to
execute Newton's laws
00:01:06.510 --> 00:01:08.810
to calculate how
it would evolve.
00:01:08.810 --> 00:01:12.810
And we did that by describing
the evolution in terms
00:01:12.810 --> 00:01:16.850
of a function little r, which
is a function of r, i, and t.
00:01:16.850 --> 00:01:21.740
And that's the radius at time t
of the shell that was initially
00:01:21.740 --> 00:01:23.219
a radius r sub i.
00:01:23.219 --> 00:01:25.510
So we're trying to track
every particle in this sphere,
00:01:25.510 --> 00:01:27.970
not just the particles
on the surface.
00:01:27.970 --> 00:01:32.430
And we want to verify that it
will remain uniform in time.
00:01:32.430 --> 00:01:35.110
Matter will not collect
near the edge of the center.
00:01:35.110 --> 00:01:38.160
And we discovered it will
remain uniform if we have a 1
00:01:38.160 --> 00:01:40.810
over r squared force law,
but for any other force law
00:01:40.810 --> 00:01:43.650
it will in fact
not remain uniform.
00:01:43.650 --> 00:01:46.080
So we derived the
equations and found
00:01:46.080 --> 00:01:48.730
that it obeyed this
scaling relationship, which
00:01:48.730 --> 00:01:52.680
is what indicated the
maintenance of uniformity,
00:01:52.680 --> 00:01:55.470
wholesale scale by the
same factor, which we then
00:01:55.470 --> 00:01:57.490
called a of t.
00:01:57.490 --> 00:01:59.250
So the physical
distance of any shell
00:01:59.250 --> 00:02:02.950
from the origin at any time
is just equal to a of t times
00:02:02.950 --> 00:02:05.510
the initial distance
from the origin.
00:02:05.510 --> 00:02:08.720
And furthermore, we were able
to drive equations of motion
00:02:08.720 --> 00:02:11.840
for the scale factor.
00:02:11.840 --> 00:02:15.110
And it obeys the equations,
which in fact were derived
00:02:15.110 --> 00:02:19.640
from general relativity by
Alexander Freedman in 1922,
00:02:19.640 --> 00:02:22.164
and therefore they're called
the Freedman equations.
00:02:22.164 --> 00:02:24.080
There's a second order
equation, a double dot,
00:02:24.080 --> 00:02:27.210
which just tells us how
the expansion slowed down
00:02:27.210 --> 00:02:28.770
by Newtonian gravity.
00:02:28.770 --> 00:02:30.925
a double dot is negative,
so the expansion
00:02:30.925 --> 00:02:33.730
is being slowed by the
gravitational attraction
00:02:33.730 --> 00:02:37.130
of every particle in this
spherical distribution
00:02:37.130 --> 00:02:39.650
towards every other particle.
00:02:39.650 --> 00:02:42.360
And we also were able to
find a first order equation
00:02:42.360 --> 00:02:45.550
by integrating the
second order equation.
00:02:45.550 --> 00:02:47.494
And the first equation
has this form.
00:02:47.494 --> 00:02:49.410
It could be written a
number of different ways
00:02:49.410 --> 00:02:50.743
depending on how you arrange it.
00:02:50.743 --> 00:02:55.050
But this is the way that
I consider most common.
00:02:55.050 --> 00:02:59.720
And many books refer to the
second equation as the Freedman
00:02:59.720 --> 00:03:00.632
equation.
00:03:00.632 --> 00:03:03.090
Both of these equations were
derived by Alexander Freedman.
00:03:03.090 --> 00:03:06.310
I think it's perfect called
both Freedman equations.
00:03:06.310 --> 00:03:08.920
But most books do not do that.
00:03:08.920 --> 00:03:12.450
And in addition to finding
the equations of evolution
00:03:12.450 --> 00:03:16.800
for a of t, we also understood
how rho of t evolves.
00:03:16.800 --> 00:03:21.230
And that really is pretty
trivial to begin with.
00:03:21.230 --> 00:03:24.470
The Newtonian mass of this
sphere stays the same.
00:03:24.470 --> 00:03:28.360
It just spreads out over a
larger volume as a of t grows.
00:03:28.360 --> 00:03:30.220
So if the mass stays
the same and the volume
00:03:30.220 --> 00:03:33.700
grows as a cubed,
then the density
00:03:33.700 --> 00:03:36.820
has to go like one over a cubed.
00:03:36.820 --> 00:03:39.320
And this alone could be
written more precisely
00:03:39.320 --> 00:03:43.450
as an equation by saying the
easiest way to view the logic
00:03:43.450 --> 00:03:45.920
is that this equation
implies that a cubed times
00:03:45.920 --> 00:03:48.880
rho is independent of time.
00:03:48.880 --> 00:03:50.640
And then once you know
that a cubed times
00:03:50.640 --> 00:03:53.280
rho is independent of time,
you can write the equation
00:03:53.280 --> 00:03:55.890
in this form,
which if I multiply
00:03:55.890 --> 00:03:59.344
a of t cubed to the left
hand side of the equation,
00:03:59.344 --> 00:04:03.120
we can just say that a
of t cubed times rho is
00:04:03.120 --> 00:04:07.370
the same at time t as it
is at some other time t1.
00:04:07.370 --> 00:04:10.710
Now in lecture last time,
we wrote this equation
00:04:10.710 --> 00:04:16.450
where t1 was t sub i the initial
time, and a of t sub i was one.
00:04:16.450 --> 00:04:17.826
So we didn't
include that factor.
00:04:17.826 --> 00:04:19.325
So this is slightly
more general way
00:04:19.325 --> 00:04:20.839
of writing it than
we did last time.
00:04:20.839 --> 00:04:24.610
But it still has no more
content than the statement
00:04:24.610 --> 00:04:27.740
that rho of t falls as
1 over a cubed of t.
00:04:33.120 --> 00:04:37.590
We introduced a special set
of units to describe this.
00:04:37.590 --> 00:04:38.780
Question, yes?
00:04:38.780 --> 00:04:41.450
AUDIENCE: I had a question
about the Freedman equations
00:04:41.450 --> 00:04:42.881
and what we're
calling e squared,
00:04:42.881 --> 00:04:44.172
and the interpretation of that.
00:04:46.970 --> 00:04:48.950
PROFESSOR: What we're
calling what squared?
00:04:48.950 --> 00:04:49.783
AUDIENCE: h squared.
00:04:49.783 --> 00:04:52.246
PROFESSOR: h squared, yes.
00:04:52.246 --> 00:04:54.860
AUDIENCE: So in the
spherical universe,
00:04:54.860 --> 00:04:58.960
we showed that if e was
positive that corresponded
00:04:58.960 --> 00:05:02.806
to an open universe that
would expand forever.
00:05:02.806 --> 00:05:09.414
So doing the p set for this week
for the cylindrical universe,
00:05:09.414 --> 00:05:13.190
I think we found that that
universe would collapse.
00:05:13.190 --> 00:05:14.071
PROFESSOR: Yes.
00:05:14.071 --> 00:05:16.526
AUDIENCE: E was also
positive for that one.
00:05:16.526 --> 00:05:19.226
So I was just wondering
about the interpretation
00:05:19.226 --> 00:05:19.963
of [INAUDIBLE]
00:05:23.540 --> 00:05:25.982
PROFESSOR: Right, well I'll
be coming to that issue later.
00:05:25.982 --> 00:05:27.190
Let me come back to that, OK.
00:05:27.190 --> 00:05:29.920
Since I haven't
introduced e yet.
00:05:29.920 --> 00:05:31.926
And the slide is here.
00:05:31.926 --> 00:05:33.050
So an interesting question.
00:05:33.050 --> 00:05:34.120
We'll come back to that.
00:05:38.470 --> 00:05:41.890
So to describe this
system of equations,
00:05:41.890 --> 00:05:44.790
I like to introduce this
notion of a notch, which
00:05:44.790 --> 00:05:50.760
is a special unit used only to
measure co-moving coordinates.
00:05:50.760 --> 00:05:54.790
And in this case, r sub i
is our co-moving coordinate.
00:05:54.790 --> 00:05:56.650
As our shells
move, we label them
00:05:56.650 --> 00:05:58.610
all by where they
were a time t i.
00:05:58.610 --> 00:06:00.150
We don't change those labels.
00:06:00.150 --> 00:06:02.680
So those are the
co-moving coordinates,
00:06:02.680 --> 00:06:05.900
a coordinate system that
expands with the universe.
00:06:05.900 --> 00:06:08.650
So instead of
measuring r i in meters
00:06:08.650 --> 00:06:10.950
or any other physical
length, I like
00:06:10.950 --> 00:06:12.990
to measure them in a
new unit called a notch,
00:06:12.990 --> 00:06:15.250
just to keep things separate.
00:06:15.250 --> 00:06:18.270
And a notch is
defined so that a of t
00:06:18.270 --> 00:06:23.030
is measured in meters per
notch, and at time t i,
00:06:23.030 --> 00:06:24.730
one notch equals 1 meter.
00:06:24.730 --> 00:06:26.430
But at different
times, the relationship
00:06:26.430 --> 00:06:28.055
between notches and
meters is different
00:06:28.055 --> 00:06:32.920
because it's given by this time
dependent scale factor of t.
00:06:32.920 --> 00:06:35.940
If [INAUDIBLE] works to have
things depend on these units,
00:06:35.940 --> 00:06:38.600
one find that this new
quantity that we introduced,
00:06:38.600 --> 00:06:41.220
little k, which all we know
is that it's a constant,
00:06:41.220 --> 00:06:44.290
has the units of 1
over a notch squared.
00:06:44.290 --> 00:06:45.900
And that has some
relevance to us,
00:06:45.900 --> 00:06:48.920
because it means that we can
make little k have any value we
00:06:48.920 --> 00:06:51.880
want by choosing different
definitions for the notch.
00:06:51.880 --> 00:06:53.520
And the notch is up for grabs.
00:06:53.520 --> 00:06:55.640
We are just inventing
a unit to use
00:06:55.640 --> 00:06:58.710
to measure our co-moving
coordinate system.
00:06:58.710 --> 00:07:00.250
So we can always
adjust the meaning
00:07:00.250 --> 00:07:04.380
of a notch so that k has
whatever value we want.
00:07:04.380 --> 00:07:07.250
As long as we can't change its
sign by changing the units.
00:07:07.250 --> 00:07:09.820
And if it's zero we can't
change it by changing its units.
00:07:09.820 --> 00:07:11.920
As long as non-zero we
can make it any value
00:07:11.920 --> 00:07:16.817
we want, and in fact that is
often used in many textbooks.
00:07:16.817 --> 00:07:18.400
And that's I guess
what I want to talk
00:07:18.400 --> 00:07:23.750
about next, the conventions
that are used to define a of t.
00:07:23.750 --> 00:07:28.310
And for us, I'm going to treat
this notch as being arbitrary.
00:07:28.310 --> 00:07:33.090
We've defined the notch
originally so that a of t i
00:07:33.090 --> 00:07:36.250
was one meters per
notch at time t i,
00:07:36.250 --> 00:07:39.390
and that gave the notch more
or less a specific meaning.
00:07:39.390 --> 00:07:43.570
But the specific meaning
depends on what t i is.
00:07:43.570 --> 00:07:45.700
One can take the
same relationship
00:07:45.700 --> 00:07:48.325
and view it as simply
a definition of t i.
00:07:48.325 --> 00:07:50.450
t i is the time
at which the scale
00:07:50.450 --> 00:07:53.520
factor is one meter per notch.
00:07:53.520 --> 00:07:55.935
We take the definition of t
i and we can let the notch
00:07:55.935 --> 00:07:58.210
be anything we want, and
there will be some time t
00:07:58.210 --> 00:08:01.680
i that will still make
that statement true.
00:08:01.680 --> 00:08:05.280
So what I want to do basically
is to think of this equation,
00:08:05.280 --> 00:08:07.700
a of t i equals 1
meter per notch,
00:08:07.700 --> 00:08:09.920
not as a definition
of a notch, which
00:08:09.920 --> 00:08:12.790
I want to leave arbitrary, but
rather as a definition of t sub
00:08:12.790 --> 00:08:13.650
i.
00:08:13.650 --> 00:08:15.715
And after defining
t sub i, I want
00:08:15.715 --> 00:08:17.560
to just forget about t sub i.
00:08:17.560 --> 00:08:21.340
t sub i in fact will not
enter our equations anywhere.
00:08:21.340 --> 00:08:25.250
So we don't need to
remember its definition.
00:08:25.250 --> 00:08:27.980
I decided like the cubit.
00:08:27.980 --> 00:08:30.770
All of us know that the
cubit is some unit distance,
00:08:30.770 --> 00:08:33.900
but we don't care what it is
because we never use cubits.
00:08:33.900 --> 00:08:34.730
Same thing here.
00:08:34.730 --> 00:08:36.447
We'll just never use t sub i.
00:08:36.447 --> 00:08:38.030
And since we're never
going to use it.
00:08:38.030 --> 00:08:41.250
We don't need to remember
how it was initially defined.
00:08:41.250 --> 00:08:43.789
It's only of
historical interest.
00:08:43.789 --> 00:08:46.080
So the bottom line then is
simply that for us the notch
00:08:46.080 --> 00:08:49.710
is just an undefined
unit of distance
00:08:49.710 --> 00:08:52.490
in the co-moving
coordinate system.
00:08:52.490 --> 00:08:54.190
Other people use
different definitions.
00:08:54.190 --> 00:08:56.760
Ryden, for example
uses the definition
00:08:56.760 --> 00:09:01.000
where a of t sub, a of the
present time is equal to 1.
00:09:01.000 --> 00:09:04.220
And we would interpret that
as meaning one meter per notch
00:09:04.220 --> 00:09:05.106
today.
00:09:05.106 --> 00:09:06.730
And that's a perfectly
good definition.
00:09:06.730 --> 00:09:08.380
And we can use it
whenever we want,
00:09:08.380 --> 00:09:11.140
because our notch is
initially undefined.
00:09:11.140 --> 00:09:13.490
So that allows us the
freedom to define it
00:09:13.490 --> 00:09:17.510
in any particular problem
in any way that we want.
00:09:17.510 --> 00:09:19.370
Many other books take
advantage of the fact
00:09:19.370 --> 00:09:23.290
that this quantity k has units
of inverse notch squared,
00:09:23.290 --> 00:09:25.100
even though they don't say that.
00:09:25.100 --> 00:09:29.380
But that means you could rescale
the co-moving coordinate system
00:09:29.380 --> 00:09:32.360
to make k equal to
whatever value you want.
00:09:32.360 --> 00:09:34.750
So in many books k is always
equal to plus or minus 1
00:09:34.750 --> 00:09:36.920
if it's non-zero.
00:09:36.920 --> 00:09:39.570
The co-moving coordinate
systems is just scaled.
00:09:39.570 --> 00:09:43.680
We would do is we scaling of the
notch to make k have magnitude
00:09:43.680 --> 00:09:44.180
1.
00:09:47.390 --> 00:09:51.540
OK, having derived
these equations,
00:09:51.540 --> 00:09:54.020
the next step is
to go about asking
00:09:54.020 --> 00:09:57.700
what do the solutions to
the equations look like.
00:09:57.700 --> 00:09:59.850
And that's where things
start getting interestingly
00:09:59.850 --> 00:10:03.090
when we start getting some
real nontrivial results.
00:10:03.090 --> 00:10:05.590
The Freedman equation,
the first order one,
00:10:05.590 --> 00:10:08.100
could be rewritten this way.
00:10:08.100 --> 00:10:10.820
It's just a rewriting
of rearranging things.
00:10:10.820 --> 00:10:14.470
And I used here the fact
that rho times a cubed
00:10:14.470 --> 00:10:15.830
was a constant.
00:10:15.830 --> 00:10:19.040
If we took our original form
of the Freedman equation,
00:10:19.040 --> 00:10:22.280
this would be rho
sub i times a cubed
00:10:22.280 --> 00:10:24.940
of t sub i, which would be 1.
00:10:24.940 --> 00:10:27.300
But knowing that rho times
a cubed is a constant,
00:10:27.300 --> 00:10:30.180
we can let t argument
here be anything we want,
00:10:30.180 --> 00:10:33.420
which is what I'll do, just
to emphasize that we don't
00:10:33.420 --> 00:10:36.080
care anymore what t sub i was.
00:10:36.080 --> 00:10:37.970
It really has disappeared
from our problem.
00:10:37.970 --> 00:10:41.350
It was just our way
of getting started.
00:10:41.350 --> 00:10:43.000
So this equation holds.
00:10:43.000 --> 00:10:46.920
And we can use it to discuss
how the different classes
00:10:46.920 --> 00:10:48.760
of solutions will behave.
00:10:48.760 --> 00:10:51.550
And what we can see very
quickly from this equation
00:10:51.550 --> 00:10:53.760
is that the behavior
of the solutions
00:10:53.760 --> 00:10:57.350
will depend crucially
on the sign of k.
00:10:57.350 --> 00:10:59.716
And it's useful
here to remember,
00:10:59.716 --> 00:11:01.340
although we don't
need to know anything
00:11:01.340 --> 00:11:03.750
more than this
proportionality, that k
00:11:03.750 --> 00:11:07.500
is proportional to the negative
of something that we call e.
00:11:07.500 --> 00:11:08.920
And that the thing
that we call e
00:11:08.920 --> 00:11:12.567
is related to the overall
energy of this thing.
00:11:12.567 --> 00:11:14.400
So we'll keep that in
the back of our minds.
00:11:14.400 --> 00:11:16.110
But it's really
only for intuition.
00:11:16.110 --> 00:11:17.526
Everything that
we're going to say
00:11:17.526 --> 00:11:19.230
follows directly
from this equation,
00:11:19.230 --> 00:11:23.560
where we don't know
anything about e.
00:11:23.560 --> 00:11:25.380
There are three
types of solutions,
00:11:25.380 --> 00:11:27.670
depending on whether k is
positive, negative, or zero,
00:11:27.670 --> 00:11:30.530
and those are the options
for what k might be.
00:11:30.530 --> 00:11:32.310
It's a real number.
00:11:32.310 --> 00:11:34.420
So first we consider
with the solutions
00:11:34.420 --> 00:11:39.070
where k is less than zero, which
means e is greater than zero.
00:11:39.070 --> 00:11:41.520
And e being greater
than zero means
00:11:41.520 --> 00:11:44.270
the system has more
energy than zero.
00:11:44.270 --> 00:11:46.560
And in this case,
zero energy would
00:11:46.560 --> 00:11:50.550
correspond to having all the
particles infinitely far away.
00:11:50.550 --> 00:11:52.420
So the potential
energies would be zero.
00:11:52.420 --> 00:11:54.740
And all particles the rest,
so the kinetic energies
00:11:54.740 --> 00:11:56.520
would be zero.
00:11:56.520 --> 00:11:59.910
So in particular, zero energy
would correspond to the system
00:11:59.910 --> 00:12:04.520
being completely dispersed,
no longer compact.
00:12:04.520 --> 00:12:08.270
And in this case, our system
has more energy than that.
00:12:08.270 --> 00:12:10.130
And more energy
than that means it
00:12:10.130 --> 00:12:14.109
can blow outward without limit.
00:12:14.109 --> 00:12:15.900
And we see that directly
from the equation.
00:12:15.900 --> 00:12:18.950
If the second term has
a negative value of k,
00:12:18.950 --> 00:12:21.190
then the second term
itself is positive.
00:12:21.190 --> 00:12:23.890
And the first term is
also always positive.
00:12:23.890 --> 00:12:25.742
And that means
that a dot squared,
00:12:25.742 --> 00:12:27.450
no matter what happens
to the first term,
00:12:27.450 --> 00:12:29.408
is always at least bigger
than the second term,
00:12:29.408 --> 00:12:31.320
which is a constant.
00:12:31.320 --> 00:12:33.320
And if a dot is always
bigger than some constant
00:12:33.320 --> 00:12:35.360
means that a grows indefinitely.
00:12:35.360 --> 00:12:37.530
And that's called
an open universe.
00:12:37.530 --> 00:12:39.445
And it goes on
expanding forever.
00:12:42.040 --> 00:12:44.780
Second case we'll
consider is k greater
00:12:44.780 --> 00:12:48.580
than zero, which corresponds
to e less than zero.
00:12:48.580 --> 00:12:51.880
And that means since 0
corresponds to the system
00:12:51.880 --> 00:12:55.320
being completely dispersed,
e less than 0 means
00:12:55.320 --> 00:12:56.960
the system does not
have enough energy
00:12:56.960 --> 00:13:00.130
to ever become
completely dispersed.
00:13:00.130 --> 00:13:02.100
So we'll have some maximum size.
00:13:02.100 --> 00:13:06.170
And the maximum size follows
immediately from this equation.
00:13:06.170 --> 00:13:08.389
a dot squared has
to be positive.
00:13:08.389 --> 00:13:09.430
It can ever get negative.
00:13:09.430 --> 00:13:13.190
It can become zero, but
it can never get negative.
00:13:13.190 --> 00:13:21.960
In this case, the minus k
c squared term is negative.
00:13:21.960 --> 00:13:24.970
And that means that if this
term gets to be too small,
00:13:24.970 --> 00:13:27.550
the sum will be negative,
which is not possible.
00:13:27.550 --> 00:13:30.150
So this term cannot
get to be too small.
00:13:30.150 --> 00:13:31.775
And since a of t is
in the denominator,
00:13:31.775 --> 00:13:34.530
that means a of t cannot
get to be too big.
00:13:34.530 --> 00:13:36.730
And you can easily
derive the inequality
00:13:36.730 --> 00:13:39.140
that a has to obey for
the right hand side
00:13:39.140 --> 00:13:40.306
to always be positive.
00:13:40.306 --> 00:13:41.930
And in that case you
[INAUDIBLE] a max,
00:13:41.930 --> 00:13:44.346
which is what you would get
if you just set the right hand
00:13:44.346 --> 00:13:46.675
side equal to 0, given
by that expression.
00:13:46.675 --> 00:13:49.152
a can never get bigger than
that, because if it did,
00:13:49.152 --> 00:13:50.610
the right hand side
of the equation
00:13:50.610 --> 00:13:54.210
become negative,
which is not possible.
00:13:54.210 --> 00:13:56.600
So this universe will
reach a maximum size,
00:13:56.600 --> 00:13:58.820
which we just calculated,
and then [INAUDIBLE]
00:13:58.820 --> 00:14:00.640
will come back.
00:14:00.640 --> 00:14:03.790
So we already have a very
nontrivial result here.
00:14:03.790 --> 00:14:06.720
Given a description of
a universe of this type,
00:14:06.720 --> 00:14:08.889
we can calculate
how big it will get
00:14:08.889 --> 00:14:10.430
before it turns
around and collapses.
00:14:13.990 --> 00:14:16.410
And this of universe
ultimately undergoes
00:14:16.410 --> 00:14:20.180
a big crunch when it collapses,
where the word big crunch
00:14:20.180 --> 00:14:23.750
was made as an analogy
to the phrase big bang.
00:14:23.750 --> 00:14:26.890
It's called a closed universe.
00:14:26.890 --> 00:14:31.080
And then finally, we've now
considered the k less than zero
00:14:31.080 --> 00:14:32.310
and k greater than zero.
00:14:32.310 --> 00:14:35.102
There are many cases
when k equals zero.
00:14:35.102 --> 00:14:36.560
And that's called
the critical mass
00:14:36.560 --> 00:14:39.570
density or critical universe.
00:14:39.570 --> 00:14:42.480
And we can figure
out what it means
00:14:42.480 --> 00:14:44.370
in terms of the mass density.
00:14:44.370 --> 00:14:47.580
This again is our
Freedman equation.
00:14:47.580 --> 00:14:49.880
If k is zero, this
last term is absent.
00:14:49.880 --> 00:14:53.220
So we just have a relationship
between rho and h, the Hubble
00:14:53.220 --> 00:14:55.250
expansion rate.
00:14:55.250 --> 00:14:56.290
And we can solve that.
00:14:56.290 --> 00:14:59.190
And the value of rho which
satisfies that equation
00:14:59.190 --> 00:15:01.820
is called the critical density.
00:15:01.820 --> 00:15:03.940
As the density is equal
to the critical density,
00:15:03.940 --> 00:15:06.540
it means that k is zero.
00:15:06.540 --> 00:15:09.180
And that's called a flat case.
00:15:09.180 --> 00:15:12.790
We'll figure out in a
minute how it evolves.
00:15:12.790 --> 00:15:15.340
It's not clear if it will be
collapsed or stop or what.
00:15:15.340 --> 00:15:16.340
But we'll find out soon.
00:15:16.340 --> 00:15:18.010
It's really on the
borderline between something
00:15:18.010 --> 00:15:19.860
which we know expands
and something which
00:15:19.860 --> 00:15:22.110
we know collapses.
00:15:22.110 --> 00:15:23.930
And it's called a flat universe.
00:15:23.930 --> 00:15:26.430
The word flat suggests
geometry, and we'll
00:15:26.430 --> 00:15:28.090
be learning about that later.
00:15:28.090 --> 00:15:30.130
General relativity tells
us a little bit more
00:15:30.130 --> 00:15:31.240
than we learn here.
00:15:31.240 --> 00:15:33.010
These equations are
all exactly true
00:15:33.010 --> 00:15:35.090
in the context of
general relativity.
00:15:35.090 --> 00:15:37.290
But general relativity
also tells us
00:15:37.290 --> 00:15:38.800
that these equations
are connected
00:15:38.800 --> 00:15:41.270
to the geometry of space.
00:15:41.270 --> 00:15:46.940
And only for this critical mass
density is the space Euclidean.
00:15:46.940 --> 00:15:52.040
The word flat here is used
in the sense of Euclidean.
00:15:52.040 --> 00:15:55.900
So to summarize what we've said,
if the mass density is bigger
00:15:55.900 --> 00:15:58.350
than this critical value,
we get a closed universe,
00:15:58.350 --> 00:16:01.060
which reaches a maximum
size and then collapses.
00:16:01.060 --> 00:16:03.390
If the mass density is less
than the critical density,
00:16:03.390 --> 00:16:06.680
we get an open universe, which
goes on expanding forever.
00:16:06.680 --> 00:16:08.561
And if the mass
density is exactly
00:16:08.561 --> 00:16:10.810
equal to the critical density,
that's called a flat k.
00:16:10.810 --> 00:16:14.260
So we'll explore a little
bit more in a minute.
00:16:14.260 --> 00:16:16.870
It's interesting to know what
this critical density is.
00:16:16.870 --> 00:16:19.140
It depends on the
expansion rate.
00:16:19.140 --> 00:16:22.530
But the expansion rate has now
been measured quite accurately.
00:16:22.530 --> 00:16:25.760
So if I take the
value of 67.3, which
00:16:25.760 --> 00:16:29.600
is the value that comes from
the Planck satellite combining
00:16:29.600 --> 00:16:33.180
their results with
several other experiments,
00:16:33.180 --> 00:16:35.380
they get a value of 67.3.
00:16:35.380 --> 00:16:37.740
And when we'll put
that into this formula,
00:16:37.740 --> 00:16:39.950
the number one gets
is 8.4 times 10
00:16:39.950 --> 00:16:43.950
to the minus 30 grams
per centimeter cubed,
00:16:43.950 --> 00:16:49.010
which is only about 5 proton
masses per cubic meter.
00:16:49.010 --> 00:16:52.540
It's an unbelievably empty
universe that we live in.
00:16:52.540 --> 00:16:54.020
I say the universe
that we live in
00:16:54.020 --> 00:16:56.290
because in fact the mass
density of our universe
00:16:56.290 --> 00:16:58.610
is very close to
this critical value.
00:16:58.610 --> 00:17:04.000
It's equal to it to within about
half of a percent we now know.
00:17:04.000 --> 00:17:05.501
An important
definition, which we'll
00:17:05.501 --> 00:17:07.125
be continuing to use
through the course
00:17:07.125 --> 00:17:08.689
and which cosmologists
always use,
00:17:08.689 --> 00:17:12.670
is omega, where omega
means capital Greek omega.
00:17:12.670 --> 00:17:15.246
And that's just defined
to be the actual density
00:17:15.246 --> 00:17:17.394
of the universe,
whatever it is, divided
00:17:17.394 --> 00:17:18.435
by this critical density.
00:17:22.329 --> 00:17:25.319
OK, the one remaining thing
that we did last time,
00:17:25.319 --> 00:17:27.220
and we'll summarize
this and go on,
00:17:27.220 --> 00:17:30.080
is we figured out
what the evolution is
00:17:30.080 --> 00:17:32.520
for a flat universe.
00:17:32.520 --> 00:17:34.750
And we can do that just by
solving the differential
00:17:34.750 --> 00:17:38.350
equation, which is a fairly
simple differential equation.
00:17:38.350 --> 00:17:40.520
If we leave out the k
term, the Freedman equation
00:17:40.520 --> 00:17:43.440
becomes a dot over a
squared is equal to 8 pi
00:17:43.440 --> 00:17:47.440
g over 3 times rho And we
know how rho depends on a.
00:17:47.440 --> 00:17:50.397
It's proportional
to 1 over a cubed.
00:17:50.397 --> 00:17:52.230
So the right hand side
here is some constant
00:17:52.230 --> 00:17:54.890
divided by a cubed.
00:17:54.890 --> 00:17:57.880
And by just
rearranging things, we
00:17:57.880 --> 00:18:00.670
can rewrite that
as da over dt is
00:18:00.670 --> 00:18:04.040
equal to some constant
over a to the 1/2.
00:18:04.040 --> 00:18:07.480
In this slide I use this
symbol const a number of times.
00:18:07.480 --> 00:18:09.480
Those constants are not
all equal to each other.
00:18:09.480 --> 00:18:11.145
But they're all
constants, which you
00:18:11.145 --> 00:18:12.892
can keep track of
if you wanted to.
00:18:12.892 --> 00:18:14.600
But there's no need
to keep track of them
00:18:14.600 --> 00:18:18.240
because they have no bearing
on the answer anyway.
00:18:18.240 --> 00:18:23.270
So I just called
these constants const.
00:18:23.270 --> 00:18:25.039
So again, da over dt
equals const over a
00:18:25.039 --> 00:18:27.330
to the one half, which is an
easy differential equation
00:18:27.330 --> 00:18:29.090
to solve.
00:18:29.090 --> 00:18:32.780
We just multiply through by
dt and a to the one half,
00:18:32.780 --> 00:18:37.790
and write it in as a
to the one half times
00:18:37.790 --> 00:18:41.680
da equals constant times
dt, which can easily
00:18:41.680 --> 00:18:43.390
be integrated both
sides of the equation
00:18:43.390 --> 00:18:45.530
as indefinite integrals.
00:18:45.530 --> 00:18:48.840
And then we get 2/3
times a to the 3/2
00:18:48.840 --> 00:18:51.672
is equal to a constant times
t, where this constant happens
00:18:51.672 --> 00:18:54.130
to be the same as that constant,
for whatever that's worth,
00:18:54.130 --> 00:18:57.900
plus a new constant of
integration, c prime.
00:18:57.900 --> 00:19:00.940
Then we argue that
the value of c prime
00:19:00.940 --> 00:19:03.530
depends on how we
synchronize our clocks.
00:19:03.530 --> 00:19:09.060
If we reset our clock by
changing t by a constant
00:19:09.060 --> 00:19:12.180
that would change
the value of c prime.
00:19:12.180 --> 00:19:13.830
And since we haven't
said anything yet
00:19:13.830 --> 00:19:17.080
about how we're
going set our clock,
00:19:17.080 --> 00:19:18.590
we're perfectly
free at this point
00:19:18.590 --> 00:19:23.200
to just say that we're going to
set our clock so that t equals
00:19:23.200 --> 00:19:27.840
0 corresponds to the same
time that a is equal to zero.
00:19:27.840 --> 00:19:29.930
The initial singularity
of the universe
00:19:29.930 --> 00:19:33.700
starts from zero
size with a as zero.
00:19:33.700 --> 00:19:36.170
So if we do that when
a is zero, t is zero.
00:19:36.170 --> 00:19:38.270
That means that c prime is zero.
00:19:38.270 --> 00:19:40.440
So setting c prime
equal to zero is just
00:19:40.440 --> 00:19:43.970
a choice of how
to set our clocks.
00:19:43.970 --> 00:19:44.930
So we do that.
00:19:44.930 --> 00:19:47.390
And then we can take the
2/3 power of this equation.
00:19:47.390 --> 00:19:49.590
And since constants are
just constants that we don't
00:19:49.590 --> 00:19:51.320
care about, we
end up with a of t
00:19:51.320 --> 00:19:54.460
is proportional to t to the 2/3.
00:19:54.460 --> 00:19:57.440
And proportional is
all we need to know,
00:19:57.440 --> 00:19:59.210
because the constant
of proportionality
00:19:59.210 --> 00:20:01.175
would depend on the
definition of the notch.
00:20:01.175 --> 00:20:02.550
And we haven't
defined the notch.
00:20:02.550 --> 00:20:04.410
And we don't need
to define the notch.
00:20:04.410 --> 00:20:08.470
And so anything that depends on
the constant of proportionality
00:20:08.470 --> 00:20:11.610
will never enter
any physical answer.
00:20:11.610 --> 00:20:16.260
It will be relevant to
questions like how many notches
00:20:16.260 --> 00:20:18.410
are a certain
distance on your map.
00:20:18.410 --> 00:20:21.010
But for any physical
answer, we don't care.
00:20:21.010 --> 00:20:22.730
If we want to talk
about our map,
00:20:22.730 --> 00:20:25.271
we could just define a notch to
be whatever we want it to be.
00:20:29.440 --> 00:20:32.320
OK, that's the
end of my summary.
00:20:32.320 --> 00:20:34.100
Any questions about any of that?
00:20:34.100 --> 00:20:34.140
I'm sorry.
00:20:34.140 --> 00:20:35.890
I didn't come back to
answer your question
00:20:35.890 --> 00:20:37.240
about the cylinder.
00:20:37.240 --> 00:20:40.624
The cylinder problem does
end up always giving you
00:20:40.624 --> 00:20:42.790
a closed universe no matter
what the parameters are.
00:20:42.790 --> 00:20:44.800
It always collapses.
00:20:44.800 --> 00:20:50.870
And even though its
energy as you compute it
00:20:50.870 --> 00:20:54.680
would turn out to be positive,
the difference though
00:20:54.680 --> 00:20:58.090
is that for the case
of the cylinder,
00:20:58.090 --> 00:21:01.480
the potential energy
does not go to 0
00:21:01.480 --> 00:21:03.635
as the thing becomes
infinitely big.
00:21:03.635 --> 00:21:07.840
The potential energy has a
logarithmic diversion in it.
00:21:07.840 --> 00:21:09.789
So the zero is just
placed differently
00:21:09.789 --> 00:21:11.330
for the case of the
cylinder problem.
00:21:14.450 --> 00:21:20.950
So it ends up being closed no
matter how fast it's expanding.
00:21:24.070 --> 00:21:24.877
Yes?
00:21:24.877 --> 00:21:25.752
AUDIENCE: [INAUDIBLE]
00:21:33.840 --> 00:21:36.580
PROFESSOR: Well that is true.
00:21:36.580 --> 00:21:38.080
Certainly the
differential equations
00:21:38.080 --> 00:21:39.538
break down when a
is equal to zero.
00:21:39.538 --> 00:21:42.360
The mass density
goes to infinity.
00:21:42.360 --> 00:21:45.900
But we're still certainly
free to set our clocks
00:21:45.900 --> 00:21:49.510
so that the equations
themselves, when extrapolated
00:21:49.510 --> 00:21:51.920
to 0, would have the
property that a equals zero
00:21:51.920 --> 00:21:53.470
when t is equal to zero.
00:21:53.470 --> 00:21:54.910
It's certainly
correct, and I was
00:21:54.910 --> 00:21:56.701
going to be talking
about this in a minute,
00:21:56.701 --> 00:22:00.580
that you should not trust
these equations back
00:22:00.580 --> 00:22:01.990
to t equals zero.
00:22:01.990 --> 00:22:04.880
But that doesn't
stop you for choosing
00:22:04.880 --> 00:22:06.815
whatever you want
as your origin of t.
00:22:06.815 --> 00:22:08.440
And if these are the
equations we have,
00:22:08.440 --> 00:22:10.790
the simplest way to deal
with these equations
00:22:10.790 --> 00:22:14.590
is to use the zero of t when
equations say that a was zero.
00:22:19.067 --> 00:22:19.900
Any other questions?
00:22:32.530 --> 00:22:37.620
OK, in that case, we will
leave the slides for a bit
00:22:37.620 --> 00:22:39.800
and proceed on the blackboard.
00:22:56.845 --> 00:22:58.220
So so far I think
we have learned
00:22:58.220 --> 00:23:00.750
two varied nontrivial things
from this calculation.
00:23:00.750 --> 00:23:02.954
We learned how to calculate
the critical density.
00:23:02.954 --> 00:23:05.620
We learned how to calculate what
density the universe would have
00:23:05.620 --> 00:23:07.390
to have so it would re-collapse.
00:23:07.390 --> 00:23:11.190
And we've also learned that
if the universe is closed,
00:23:11.190 --> 00:23:16.985
we can calculate how large it
will get before it collapses.
00:23:16.985 --> 00:23:19.110
So those are two very
nontrivial results coming out
00:23:19.110 --> 00:23:20.850
of this Newtonian calculation.
00:23:20.850 --> 00:23:22.620
The next question I
want to ask is still
00:23:22.620 --> 00:23:25.000
about the flat universe.
00:23:25.000 --> 00:23:27.870
It's a fairly trivial
extension of what we have.
00:23:27.870 --> 00:23:29.810
Given this formula
for a of t, I would
00:23:29.810 --> 00:23:34.122
like to calculate the
age of a flat universe.
00:23:34.122 --> 00:23:36.330
If you were living in a flat
universe that was matter
00:23:36.330 --> 00:23:39.070
dominated like the
one we're describing,
00:23:39.070 --> 00:23:41.780
how would you determine
how old it was?
00:23:41.780 --> 00:23:45.440
And the answer is that
it's immediately related
00:23:45.440 --> 00:23:47.680
to the Hubble expansion
rate, and the age
00:23:47.680 --> 00:23:50.820
can be expressed in terms of
the Hubble expansion rate.
00:23:50.820 --> 00:23:59.540
To see that-- so we're
calculating the age of a matter
00:23:59.540 --> 00:24:01.125
dominated flat universe.
00:24:12.794 --> 00:24:14.210
And these age
calculations will be
00:24:14.210 --> 00:24:15.793
extending as we go
through the course.
00:24:15.793 --> 00:24:18.250
So in the end, we'll
have the full calculation
00:24:18.250 --> 00:24:21.180
for the real model that
we have of our universe.
00:24:21.180 --> 00:24:22.720
But you have to start somewhere.
00:24:22.720 --> 00:24:24.178
So we're starting
with just a flat,
00:24:24.178 --> 00:24:26.190
matter dominated universe.
00:24:26.190 --> 00:24:32.600
We know that a of t is equal
to some constant, which
00:24:32.600 --> 00:24:40.955
I will call little v
times t to the 2/3 power.
00:24:40.955 --> 00:24:42.580
Previously I just
used proportional to,
00:24:42.580 --> 00:24:44.910
but now it's just more
convenient to give a name
00:24:44.910 --> 00:24:46.490
to the constant of
proportionality.
00:24:46.490 --> 00:24:48.090
We'll never need
to know what it is.
00:24:48.090 --> 00:24:50.781
But v is some constant
of proportionality.
00:24:50.781 --> 00:24:52.780
This by the way already
tells us something else,
00:24:52.780 --> 00:24:54.696
which wasn't obvious
from the beginning, which
00:24:54.696 --> 00:24:59.140
is our flat universe does go
on expanding forever, somewhat
00:24:59.140 --> 00:25:00.630
like an open universe.
00:25:00.630 --> 00:25:06.000
An important difference
is that if you calculate
00:25:06.000 --> 00:25:09.420
da dt for the open
universe, that approaches
00:25:09.420 --> 00:25:12.220
a constant as time
goes to infinity.
00:25:12.220 --> 00:25:14.620
That is, the universe
keeps on expanding
00:25:14.620 --> 00:25:17.610
at some-- minimal rate forever.
00:25:17.610 --> 00:25:22.620
In this case, if you calculate
the adt, it goes to 0 at times.
00:25:22.620 --> 00:25:25.079
So the flat universe
expands forever,
00:25:25.079 --> 00:25:26.870
but at an ever, ever,
ever decreasing rate.
00:25:29.810 --> 00:25:33.370
We know how to
relate a of t to h.
00:25:33.370 --> 00:25:35.810
The Hubble expansion
rate is a dot over a,
00:25:35.810 --> 00:25:38.010
we learned a long time ago.
00:25:38.010 --> 00:25:41.170
And if we know what a
is, we know what this is.
00:25:41.170 --> 00:25:46.055
So this is just 2 over 3t.
00:25:53.930 --> 00:25:55.920
The 2/3 coming from
differentiating the 2/3.
00:25:55.920 --> 00:25:58.150
And that gives you a
t to the minus 1/3,
00:25:58.150 --> 00:25:59.840
but then you're
also dividing by a,
00:25:59.840 --> 00:26:03.100
which turns that t to the minus
1/3 to a t to the minus 1.
00:26:03.100 --> 00:26:06.570
So you get 2 over 3t
is the final answer.
00:26:06.570 --> 00:26:09.430
So this is the relationship
now between h and t,
00:26:09.430 --> 00:26:11.890
and the question we asked
is how to calculate the age.
00:26:11.890 --> 00:26:13.330
The age is t.
00:26:13.330 --> 00:26:16.960
This is all defined as where t
is equal to 0 at the big bang.
00:26:16.960 --> 00:26:21.770
So t really is the time
elapsed since the big bang.
00:26:21.770 --> 00:26:25.250
So we're left immediately
with a simple result,
00:26:25.250 --> 00:26:30.255
that t is equal to
2/3 times h inverse.
00:26:38.620 --> 00:26:41.200
Now this result immediately
makes rigorous contact
00:26:41.200 --> 00:26:44.260
with something that we talked
about in vague terms earlier.
00:26:44.260 --> 00:26:49.600
If you are so unfortunate
as to badly mis-measure h,
00:26:49.600 --> 00:26:52.550
you can get a pretty wild
answer for the h of your model
00:26:52.550 --> 00:26:53.950
universe.
00:26:53.950 --> 00:26:57.620
And Hubble mis-measured
h by about a factor
00:26:57.620 --> 00:27:00.305
of seven comparative to
present modern values.
00:27:00.305 --> 00:27:02.560
He got h to be too high.
00:27:02.560 --> 00:27:05.130
His value of h was too high
by a factor of about seven,
00:27:05.130 --> 00:27:07.410
and that meant that when big
bang theorists calculated
00:27:07.410 --> 00:27:09.280
the age of the
universe were getting
00:27:09.280 --> 00:27:12.250
ages that were too low
by a factor of seven.
00:27:12.250 --> 00:27:13.940
And in particular
that meant they
00:27:13.940 --> 00:27:16.790
were getting ages of the
order of 2 billion years.
00:27:16.790 --> 00:27:20.650
And even back in
the 1920s and '30s,
00:27:20.650 --> 00:27:22.640
there was sufficient
geological evidence
00:27:22.640 --> 00:27:25.000
that the Earth was older
than 2 billion years.
00:27:25.000 --> 00:27:27.010
There was also
significant understanding
00:27:27.010 --> 00:27:28.640
of stellar evolution,
that starts
00:27:28.640 --> 00:27:31.640
took longer to evolve
than 2 billion years.
00:27:31.640 --> 00:27:34.550
So the big bang model was
in trouble from the start,
00:27:34.550 --> 00:27:38.610
largely because of this
very serious mis-measurement
00:27:38.610 --> 00:27:40.990
in the early days of the
Hubble expansion rate.
00:27:51.550 --> 00:27:54.010
If we put in some
numbers, this of course
00:27:54.010 --> 00:27:57.820
is not an accurate model for
our universe, we now know.
00:27:57.820 --> 00:28:01.450
Our universe is now currently
dark energy dominated.
00:28:01.450 --> 00:28:04.140
But nonetheless, just
to see how this works,
00:28:04.140 --> 00:28:07.290
we can put in numbers.
00:28:07.290 --> 00:28:20.640
So h, using this Planck value
that I quoted earlier 67.3
00:28:20.640 --> 00:28:29.720
plus or minus 1.2 kilometers
per second per megaparsec.
00:28:39.570 --> 00:28:41.320
To be able to get
an answer in years,
00:28:41.320 --> 00:28:46.110
one has to be able to convert
this into inverse years.
00:28:46.110 --> 00:28:48.810
h is actually an inverse time.
00:28:48.810 --> 00:28:53.520
And a useful conversion
number is 1 over 10
00:28:53.520 --> 00:28:57.370
to the 10 years is
equal to almost 100,
00:28:57.370 --> 00:29:06.920
but not quite, 97.8 kilometers
per second per megaparsec,
00:29:06.920 --> 00:29:10.640
which allows you to convert
these Hubble expansion rate
00:29:10.640 --> 00:29:14.420
units into inverse years.
00:29:14.420 --> 00:29:21.990
And using that one finds that
the age of the universe using
00:29:21.990 --> 00:29:37.770
the 2/3 h inverse formula
is 9.7 plus or minus
00:29:37.770 --> 00:29:40.625
0.2 billion years.
00:29:51.810 --> 00:29:57.360
Now this number played a role
in the fairly recent history
00:29:57.360 --> 00:29:58.910
of cosmology.
00:29:58.910 --> 00:30:02.170
Before 1998, when
the dark energy
00:30:02.170 --> 00:30:05.430
was discovered, which kind of
settled all these questions,
00:30:05.430 --> 00:30:10.790
but before 1998, we thought the
universe what matter dominated.
00:30:10.790 --> 00:30:11.790
It might have been open.
00:30:11.790 --> 00:30:12.990
It didn't have to be flat.
00:30:12.990 --> 00:30:14.610
That was debated.
00:30:14.610 --> 00:30:17.330
It looked more open and flat.
00:30:17.330 --> 00:30:20.830
But some of us wanted to
hold out for a flat universe
00:30:20.830 --> 00:30:23.310
because we were
fans of inflation
00:30:23.310 --> 00:30:26.290
and admired inflation's
other successes,
00:30:26.290 --> 00:30:28.690
which we'll learn about
later in the course,
00:30:28.690 --> 00:30:31.470
and thought therefore that
the universe should be flat,
00:30:31.470 --> 00:30:34.400
and wanted to try to
reconcile all this.
00:30:34.400 --> 00:30:40.270
And the problem was that like
cosmology in the '20s and '30s
00:30:40.270 --> 00:30:42.250
when the age of the
universe that you calculate
00:30:42.250 --> 00:30:47.920
was too young, the same thing
was happening here before 1998,
00:30:47.920 --> 00:30:50.190
when we thought the universe
was matter dominated.
00:30:50.190 --> 00:30:54.040
This is the age that we got,
modified a little bit by having
00:30:54.040 --> 00:30:57.040
different values of h,
but pretty close to this.
00:30:57.040 --> 00:30:59.550
At the same time,
there were calculations
00:30:59.550 --> 00:31:01.650
about how old the
universe had to be
00:31:01.650 --> 00:31:04.560
to accommodate the oldest stars.
00:31:04.560 --> 00:31:07.760
And in the lecture notes
I quote a particular paper
00:31:07.760 --> 00:31:09.453
by Krauss and Chaboyer.
00:31:20.300 --> 00:31:24.170
Lawrence Krauss is an
MIT PhD by the way.
00:31:24.170 --> 00:31:29.504
And what they decided by
studying globular clusters,
00:31:29.504 --> 00:31:31.170
which are supposed
to contain the oldest
00:31:31.170 --> 00:31:39.430
stars that astronomers know
about, that the oldest stars,
00:31:39.430 --> 00:31:56.060
they said, had an age of
12.6 plus 3.4 minus 2.2
00:31:56.060 --> 00:31:56.670
billion years.
00:32:00.830 --> 00:32:05.340
And this is a 95%
confidence number.
00:32:15.110 --> 00:32:17.680
That is, instead of using
one sigma, which are often
00:32:17.680 --> 00:32:21.200
used to quote errors,
these are two sigma errors,
00:32:21.200 --> 00:32:24.580
which have probabilities
of being wrong by only 5%
00:32:24.580 --> 00:32:28.670
if things work properly
according to the statistics.
00:32:28.670 --> 00:32:32.470
So then we're going to
think about 95% limits.
00:32:32.470 --> 00:32:34.810
So they were willing to
take the lower limit here,
00:32:34.810 --> 00:32:45.750
which was 10.4.
00:32:45.750 --> 00:32:54.350
So they got a minimum
age of 10.4 billion years
00:32:54.350 --> 00:32:56.390
for the oldest stars.
00:32:56.390 --> 00:32:59.070
But they also argued that the
stars really could not possibly
00:32:59.070 --> 00:33:02.460
start to form until
about 0.8 billion years
00:33:02.460 --> 00:33:03.835
into the history
of the universe.
00:33:24.790 --> 00:33:29.190
And doing a little bit of
simple arithmetic there,
00:33:29.190 --> 00:33:31.420
they decided that the
minimum possible age
00:33:31.420 --> 00:33:34.690
for the universe at the
95% confidence level,
00:33:34.690 --> 00:34:07.180
would be 10.4 plus 0.8 or 11.2.
00:34:07.180 --> 00:34:14.679
And 11.2 is older than
9.7, and by a fair number
00:34:14.679 --> 00:34:17.699
of standard deviations,
although [INAUDIBLE]
00:34:17.699 --> 00:34:21.730
were somewhat bigger in
1998 than they are now.
00:34:21.730 --> 00:34:25.210
But in any case, this led to I
think what people at the time
00:34:25.210 --> 00:34:28.570
regarded as a tension between
the age of the universe
00:34:28.570 --> 00:34:32.320
question and the possibility
of having a flat universe.
00:34:32.320 --> 00:34:34.710
A flat universe
seemed to produce
00:34:34.710 --> 00:34:38.050
ages that were too young
to be consistent with what
00:34:38.050 --> 00:34:40.070
we knew about stars.
00:34:40.070 --> 00:34:43.170
Yet there was still
evidence in terms
00:34:43.170 --> 00:34:46.199
of the desired to make
inflationary models
00:34:46.199 --> 00:34:49.580
work to indicate that
the universe was flat.
00:34:49.580 --> 00:34:53.500
And actually it's also
true that by 1998, there
00:34:53.500 --> 00:34:56.942
was evidence from the
Kobe satellite measuring
00:34:56.942 --> 00:34:59.150
fluctuation in the cosmic
background radiation, which
00:34:59.150 --> 00:35:02.180
also suggested
that omega was one,
00:35:02.180 --> 00:35:03.950
that the universe with flat.
00:35:03.950 --> 00:35:07.160
So things didn't fit together
very well before 1998.
00:35:07.160 --> 00:35:10.250
And this was at the
crux of the argument.
00:35:10.250 --> 00:35:12.960
It all got settled with the
discovery of the dark energy,
00:35:12.960 --> 00:35:16.496
which we'll learn how to
account for in a few weeks.
00:35:16.496 --> 00:35:18.620
When we includes dark energy
in these calculations,
00:35:18.620 --> 00:35:22.810
the ages go up,
and everything does
00:35:22.810 --> 00:35:24.557
come into accord
with the idea now
00:35:24.557 --> 00:35:26.265
that the age of the
universe is estimated
00:35:26.265 --> 00:35:28.570
at 13.8 billion years.
00:35:28.570 --> 00:35:31.070
And that's consistent with the
Hubble expansion rate given
00:35:31.070 --> 00:35:34.030
here, as long as one
has dark energy and not
00:35:34.030 --> 00:35:36.110
just relativistic matter.
00:35:39.000 --> 00:35:40.600
OK any question about that?
00:35:46.621 --> 00:35:48.620
OK next thing I wanted
to say a little bit about
00:35:48.620 --> 00:35:51.950
is what exactly we
mean here by age.
00:35:51.950 --> 00:35:56.260
And the question of what we
mean by age does of course
00:35:56.260 --> 00:35:59.850
connect to the question of how
do we think it actually began.
00:35:59.850 --> 00:36:03.730
Because age means time since
the beginning presumably.
00:36:03.730 --> 00:36:05.730
And the answer really
is that we don't
00:36:05.730 --> 00:36:08.170
know how the universe began.
00:36:08.170 --> 00:36:12.740
The big bang is often said to be
the beginning of the universe.
00:36:12.740 --> 00:36:14.942
But I would argue that
we don't know that,
00:36:14.942 --> 00:36:16.400
and I think most
cosmologists would
00:36:16.400 --> 00:36:18.550
agree that we don't know that.
00:36:18.550 --> 00:36:21.220
As we extrapolate
backwards, we're
00:36:21.220 --> 00:36:24.540
using our knowledge of physics
that we measure in laboratories
00:36:24.540 --> 00:36:27.910
and physics that we confirm
with other astrophysical type
00:36:27.910 --> 00:36:31.350
observations, but nonetheless,
as we get closer and closer
00:36:31.350 --> 00:36:35.910
to t equals zero, the mass
density in this approximation
00:36:35.910 --> 00:36:39.430
grows like 1 over the
scale factor cubed, which
00:36:39.430 --> 00:36:42.260
means it goes up
arbitrarily large.
00:36:42.260 --> 00:36:46.720
Later we'll learn that when the
universe gets to be very young,
00:36:46.720 --> 00:36:50.600
we have to include radiation and
not just relativistic matter.
00:36:50.600 --> 00:36:53.152
The dark energy is actually
totally unimportant
00:36:53.152 --> 00:36:54.360
when we go backwards in time.
00:36:54.360 --> 00:36:57.900
It becomes important when
we go forwards in time.
00:36:57.900 --> 00:37:01.110
But when we put in radiation,
it does not solve this problem.
00:37:01.110 --> 00:37:04.800
The universe still
requires a mass density
00:37:04.800 --> 00:37:09.140
that goes to infinity as
we approach t equals zero.
00:37:09.140 --> 00:37:11.470
People had wondered
whether maybe that's
00:37:11.470 --> 00:37:13.890
an idealization associated
with our approximation
00:37:13.890 --> 00:37:16.720
of exact homogeneity
and isotropy
00:37:16.720 --> 00:37:18.940
which after all do break
down at some level.
00:37:18.940 --> 00:37:22.870
Maybe if we put in a slightly
inhomogeneous and slightly
00:37:22.870 --> 00:37:24.980
anisotropic universe
and ran it backwards,
00:37:24.980 --> 00:37:28.300
maybe the mass density
would not climb to infinity.
00:37:28.300 --> 00:37:31.700
Hawking proved that
that was not a way out.
00:37:31.700 --> 00:37:33.785
The universe would
become singular.
00:37:33.785 --> 00:37:36.160
He didn't really prove the
mass density went to infinity,
00:37:36.160 --> 00:37:39.020
but he proved it becomes
singular in other ways
00:37:39.020 --> 00:37:44.100
as t went to zero no matter
what geometry you put in.
00:37:44.100 --> 00:37:49.790
So the bottom line is that
classical general relativity
00:37:49.790 --> 00:37:53.020
does predict a
singularity of some sort
00:37:53.020 --> 00:37:55.740
as we extrapolate
backwards in time.
00:37:55.740 --> 00:38:01.610
But the important
qualification is
00:38:01.610 --> 00:38:04.880
that once the mass density goes
far above any mass densities
00:38:04.880 --> 00:38:06.750
that we've had any
experience with,
00:38:06.750 --> 00:38:09.500
we really don't know how
things are going to behave.
00:38:09.500 --> 00:38:12.230
And we don't really know how
classical general relativity
00:38:12.230 --> 00:38:13.790
holds in that regime.
00:38:13.790 --> 00:38:17.440
And in fact, we have
very strong ways
00:38:17.440 --> 00:38:20.170
is to believe that classical
general relativity will not
00:38:20.170 --> 00:38:21.850
hold in that regime.
00:38:21.850 --> 00:38:23.720
Because classical
general relativity
00:38:23.720 --> 00:38:27.010
is after all a classical
theory, a theory in which one
00:38:27.010 --> 00:38:28.990
talks about fields that
have definite values
00:38:28.990 --> 00:38:33.460
at definite times
without incorporating
00:38:33.460 --> 00:38:37.950
the ideas of the uncertainty
principle of quantum theory.
00:38:37.950 --> 00:38:42.050
So nobody in fact knows how to
build a theory in which matter
00:38:42.050 --> 00:38:47.020
is quantized and gravity
is not quantized.
00:38:47.020 --> 00:38:51.930
So all the smart money bets on
the fact that gravity is really
00:38:51.930 --> 00:38:54.770
a quantum theory, even though
we don't yet quite understand it
00:38:54.770 --> 00:38:55.785
as a quantum theory.
00:38:55.785 --> 00:38:57.910
And that as we go back in
time, the quantum effects
00:38:57.910 --> 00:39:00.200
become more and more important.
00:39:00.200 --> 00:39:03.440
So there's no reason to trust
classical general relativity
00:39:03.440 --> 00:39:06.970
as we approach t equals
zero, and therefore no reason
00:39:06.970 --> 00:39:10.840
to really take this
singularity seriously.
00:39:10.840 --> 00:39:14.550
Furthermore, we'll even see
at the end of the course
00:39:14.550 --> 00:39:20.130
that most inflationary
scenarios imply
00:39:20.130 --> 00:39:22.335
that what we call
the big bang is not
00:39:22.335 --> 00:39:24.770
a unique beginning
of the universe.
00:39:24.770 --> 00:39:27.430
But rather it now
seems pretty likely,
00:39:27.430 --> 00:39:30.250
although we sure don't
know, that our universe
00:39:30.250 --> 00:39:32.570
is part of a multiverse,
where we are just
00:39:32.570 --> 00:39:36.890
one universe in the multiverse,
and that the big bang, what
00:39:36.890 --> 00:39:40.650
we call the big bang, is really
our big bang, the beginning
00:39:40.650 --> 00:39:42.900
of our pocket universe.
00:39:42.900 --> 00:39:45.770
But before that the space
of time already existed.
00:39:45.770 --> 00:39:48.270
The big bang is
just a nucleation
00:39:48.270 --> 00:39:49.410
of a phase transition.
00:39:49.410 --> 00:39:51.960
It's not really a beginning.
00:39:51.960 --> 00:39:54.670
And that there was
other stuff that
00:39:54.670 --> 00:39:58.600
existed before what
we call the big bang.
00:39:58.600 --> 00:40:01.460
I should add though that the
inflationary scenario does not
00:40:01.460 --> 00:40:04.920
provide any answer whatever
to the question of how did it
00:40:04.920 --> 00:40:06.950
all ultimately begin.
00:40:06.950 --> 00:40:08.660
That's still very
much an open question.
00:40:08.660 --> 00:40:10.470
And it's clear that
inflation by itself
00:40:10.470 --> 00:40:14.460
does not even offer an
answer to that question.
00:40:14.460 --> 00:40:16.610
So when we talk about
the age of the universe,
00:40:16.610 --> 00:40:18.100
what are we talking about?
00:40:18.100 --> 00:40:20.750
What we're talking about is
the age, the amount of time
00:40:20.750 --> 00:40:22.500
that has elapsed, since
this event that we
00:40:22.500 --> 00:40:23.499
call the big bang.
00:40:23.499 --> 00:40:26.040
The big bang might not have been
the beginning of everything,
00:40:26.040 --> 00:40:28.590
but certainly the evidence
is overwhelmingly strong
00:40:28.590 --> 00:40:29.799
that it happened.
00:40:29.799 --> 00:40:31.340
And we could talk
about how much time
00:40:31.340 --> 00:40:33.260
has elapsed since it happened.
00:40:33.260 --> 00:40:36.550
And that's the t that we're
trying to calculate here.
00:40:36.550 --> 00:40:39.620
And it will be offset
by a tiny amount
00:40:39.620 --> 00:40:42.760
by changing the history in
the very, very early stages,
00:40:42.760 --> 00:40:45.690
but only by a tiny
fraction of a second.
00:40:45.690 --> 00:40:47.470
So the uncertainties
of quantum gravity
00:40:47.470 --> 00:40:52.181
are not important in calculating
the age of the universe.
00:40:52.181 --> 00:40:54.180
Although they are important
in interpreting what
00:40:54.180 --> 00:40:55.410
you mean by it.
00:40:55.410 --> 00:40:59.580
I think we don't really mean
the origin of space and time,
00:40:59.580 --> 00:41:02.030
but rather simply
the time has elapsed
00:41:02.030 --> 00:41:05.190
since the event
called the big bang.
00:41:05.190 --> 00:41:06.460
OK any questions about that?
00:41:09.320 --> 00:41:14.320
All right, next event
I want to talk about
00:41:14.320 --> 00:41:17.930
is that if the
universe as we know
00:41:17.930 --> 00:41:21.720
it began some 13.8
billion years ago to use
00:41:21.720 --> 00:41:26.550
the actual current number, that
would mean that light could
00:41:26.550 --> 00:41:31.210
only have traveled some finite
distance since the beginning
00:41:31.210 --> 00:41:34.640
of the universe as
we know it, meaning
00:41:34.640 --> 00:41:37.526
the universe since the big
bang, and that would mean there
00:41:37.526 --> 00:41:40.510
would be some maximum distance
that we could see things.
00:41:40.510 --> 00:41:42.844
And beyond that there
might be more things,
00:41:42.844 --> 00:41:45.010
but they'd be things for
which the light has not yet
00:41:45.010 --> 00:41:46.960
had time to reach us.
00:41:46.960 --> 00:41:50.380
So that's an important
concept in cosmology,
00:41:50.380 --> 00:41:52.720
the maximum distance
that you could see.
00:41:52.720 --> 00:41:54.535
It goes by the name
the horizon distance.
00:41:58.964 --> 00:42:00.755
If you're sailing on
the ocean, the horizon
00:42:00.755 --> 00:42:03.780
is the furthest
thing you can see.
00:42:03.780 --> 00:42:06.950
So what we want to do now
is to calculate this horizon
00:42:06.950 --> 00:42:10.810
distance in the model
that we now understand,
00:42:10.810 --> 00:42:14.490
the flat matter
dominated universe.
00:42:32.819 --> 00:42:34.485
And this of course
is also a calculation
00:42:34.485 --> 00:42:36.164
that we will be
generalizing as we
00:42:36.164 --> 00:42:37.580
go through the
course learning how
00:42:37.580 --> 00:42:40.330
to treat more and more
complicated cases and more
00:42:40.330 --> 00:42:43.440
and more realistic cases.
00:42:43.440 --> 00:42:50.000
So this horizon distance, I
should define it more exactly.
00:42:50.000 --> 00:42:58.800
It's the present
distance, and maybe I
00:42:58.800 --> 00:43:01.750
should even stick
the word proper here.
00:43:04.707 --> 00:43:06.790
I've been usually using
the word physical distance
00:43:06.790 --> 00:43:10.000
to refer to the
distance to an object
00:43:10.000 --> 00:43:12.450
as it would be measured
by rulers, which
00:43:12.450 --> 00:43:15.920
are each along the way
moving with the velocity
00:43:15.920 --> 00:43:18.750
of the average matter
at those locations.
00:43:18.750 --> 00:43:20.910
That is also called
the proper distance,
00:43:20.910 --> 00:43:24.680
which is [INAUDIBLE] calls it.
00:43:24.680 --> 00:43:26.860
And this horizon
distance is defined
00:43:26.860 --> 00:43:45.610
as the present proper distance
of the most distant objects
00:43:45.610 --> 00:43:55.160
that can be seen, limited
only by the speed of light.
00:44:10.859 --> 00:44:13.400
So we pretend we have telescopes
that are incredibly powerful
00:44:13.400 --> 00:44:16.764
and could see anything,
any light that
00:44:16.764 --> 00:44:17.870
could have reached us.
00:44:17.870 --> 00:44:20.040
But we know the light has
a finite propagation time,
00:44:20.040 --> 00:44:22.980
so take that into account in
talking about this horizon
00:44:22.980 --> 00:44:23.480
distance.
00:44:26.070 --> 00:44:29.730
OK so what is the horizon
distance going to be?
00:44:29.730 --> 00:44:35.300
Well remember that the
coordinate velocity of light
00:44:35.300 --> 00:44:42.440
is equal to c divided by a of t.
00:44:42.440 --> 00:44:45.030
I should maybe start by
saying before we get down
00:44:45.030 --> 00:44:48.450
to details, that you might think
naively that the answer should
00:44:48.450 --> 00:44:51.130
be the speed of light times
the age of the universe.
00:44:51.130 --> 00:44:53.370
That's how far light can travel.
00:44:53.370 --> 00:44:56.010
And so if the universe was
static and just appeared
00:44:56.010 --> 00:44:59.030
a certain time in the past,
that would be the right answer.
00:44:59.030 --> 00:45:00.770
I would just start
off at the beginning
00:45:00.770 --> 00:45:02.930
and travel at speed c.
00:45:02.930 --> 00:45:05.000
But it's more complicated
because the universe
00:45:05.000 --> 00:45:06.520
has been expanding all along.
00:45:06.520 --> 00:45:09.220
And it started out with
a scale factor of 0.
00:45:09.220 --> 00:45:10.900
And furthermore, what
we're looking for
00:45:10.900 --> 00:45:12.604
is present distance
in these objects,
00:45:12.604 --> 00:45:14.270
and the objects of
course have continued
00:45:14.270 --> 00:45:17.596
to move after the light that
we're now seeing has left them.
00:45:17.596 --> 00:45:18.970
So it's a little
more complicated
00:45:18.970 --> 00:45:20.590
than just c times
the speed of light.
00:45:20.590 --> 00:45:22.650
And we'll see what
it is by tracking it
00:45:22.650 --> 00:45:24.380
through very carefully.
00:45:24.380 --> 00:45:26.410
We'll imagine a light
beam that leaves
00:45:26.410 --> 00:45:29.370
from some distant object.
00:45:29.370 --> 00:45:35.020
And the light beam will get the
furthest if it leaves earliest.
00:45:35.020 --> 00:45:36.860
So we want the earliest
possible light beam
00:45:36.860 --> 00:45:38.950
that could have left
this distant object.
00:45:38.950 --> 00:45:41.241
And that would be a light
beam that left at literally t
00:45:41.241 --> 00:45:42.574
equals zero.
00:45:42.574 --> 00:45:44.240
So the light beam
leaves the object at t
00:45:44.240 --> 00:45:46.570
equals zero and
reaches us today.
00:45:46.570 --> 00:45:48.790
And we want to know how
far away is that object?
00:45:48.790 --> 00:45:50.690
That's the furthest
object that we could see,
00:45:50.690 --> 00:45:53.060
objects for which we can
only see the light that
00:45:53.060 --> 00:45:56.122
was emitted from the
object at t equals zero.
00:45:56.122 --> 00:45:58.330
So we're going to use our
co-moving coordinate system
00:45:58.330 --> 00:45:59.180
to trace things.
00:45:59.180 --> 00:46:01.940
All calculations are done
most straightforwardly
00:46:01.940 --> 00:46:04.190
in the co-moving
coordinate system.
00:46:04.190 --> 00:46:06.780
And we know that light travels
in the co-moving coordinate
00:46:06.780 --> 00:46:11.050
system at the rate of dx dt is
equal to c divided by a of t.
00:46:11.050 --> 00:46:13.580
And this really just says
that as the light passes
00:46:13.580 --> 00:46:16.190
any observer in this
co-moving coordinate system,
00:46:16.190 --> 00:46:19.750
the observer sees speed c, as
special relativity tells us
00:46:19.750 --> 00:46:21.340
he must.
00:46:21.340 --> 00:46:24.680
But we need to convert it
into notches per second
00:46:24.680 --> 00:46:27.180
to be able to trace it through
the co-moving coordinate
00:46:27.180 --> 00:46:28.210
system.
00:46:28.210 --> 00:46:31.150
And the relationship between
notches and meters is a of t.
00:46:31.150 --> 00:46:33.570
So a of t is just a
conversion factor here
00:46:33.570 --> 00:46:38.250
that converts the local speed
of this light pulse from meters
00:46:38.250 --> 00:46:41.840
per second to notches
per second, which
00:46:41.840 --> 00:46:46.010
is what dx dt has
to be measured in.
00:46:46.010 --> 00:46:50.240
So this will be the speed.
00:46:50.240 --> 00:46:52.860
That means that the
maximum distance that light
00:46:52.860 --> 00:46:55.665
will travel, still
measured in notches
00:46:55.665 --> 00:46:57.880
in co-moving
coordinates, will just
00:46:57.880 --> 00:46:59.170
be the integral of the speed.
00:46:59.170 --> 00:47:03.460
The integral of dx
dt is just delta x.
00:47:03.460 --> 00:47:09.440
So it would be the
integral of dx dt dt,
00:47:09.440 --> 00:47:15.390
integrating from 0 up to
t zero, the present time.
00:47:27.790 --> 00:47:30.870
Now this is not the final
answer that we're interested in.
00:47:30.870 --> 00:47:33.530
We want to know the
present physical distance
00:47:33.530 --> 00:47:36.020
or the present proper
distance of this object
00:47:36.020 --> 00:47:39.040
that's the furthest
object that we can see.
00:47:39.040 --> 00:47:41.220
And the way to go from
co-moving distances
00:47:41.220 --> 00:47:44.419
to physical businesses is to
multiply by the scale factor.
00:47:44.419 --> 00:47:45.960
And we are interested
in the distance
00:47:45.960 --> 00:47:48.820
today, so we multiply
by the scale factor
00:47:48.820 --> 00:47:51.650
today, the present value
of the scale factor.
00:47:51.650 --> 00:47:53.630
So the answers to
our problem, which
00:47:53.630 --> 00:47:58.625
I will call l sub p or
sub [INAUDIBLE] of t.
00:48:02.060 --> 00:48:05.320
I want to get the word horizon
into the subscript someplace.
00:48:05.320 --> 00:48:13.830
So I will call it l sub
p comma horizon, which
00:48:13.830 --> 00:48:18.110
means the physical distance
to the horizon at time t
00:48:18.110 --> 00:48:22.110
is just equal to x
max that we have here
00:48:22.110 --> 00:48:25.360
times the present value
of the scale factor.
00:48:25.360 --> 00:48:32.460
So it's a of 2
[INAUDIBLE] times x max.
00:48:32.460 --> 00:48:37.040
Or the final formula, just
substituting in x max,
00:48:37.040 --> 00:48:43.160
will be of a of t naught
times the integral from 0
00:48:43.160 --> 00:48:46.690
to t naught of dx dt.
00:48:46.690 --> 00:48:48.705
I'm going to substitute
c over a of t.
00:49:06.610 --> 00:49:09.460
Let me just remind
you that [INAUDIBLE]
00:49:09.460 --> 00:49:11.007
variable integration
should never
00:49:11.007 --> 00:49:13.090
have the same symbol as
the limits of integration,
00:49:13.090 --> 00:49:14.548
because that just
causes confusion.
00:49:14.548 --> 00:49:16.930
They're never really
the same thing.
00:49:16.930 --> 00:49:18.740
So I called the
limits t sub zero.
00:49:18.740 --> 00:49:22.710
So it's perfectly OK to call
the variable of integration t.
00:49:22.710 --> 00:49:25.880
In the notes, I call
the value of the time
00:49:25.880 --> 00:49:27.630
that we want to
calculate this as t,
00:49:27.630 --> 00:49:30.110
and then I use t prime for
the variable of integration.
00:49:30.110 --> 00:49:31.651
Whatever you do,
you should make sure
00:49:31.651 --> 00:49:33.129
that those are not the same.
00:49:33.129 --> 00:49:34.670
There's one variable
that corresponds
00:49:34.670 --> 00:49:39.140
to the variable that varies
from the initial time
00:49:39.140 --> 00:49:40.120
to the final time.
00:49:40.120 --> 00:49:41.578
And then there's
also another value
00:49:41.578 --> 00:49:43.690
that represents the final time.
00:49:43.690 --> 00:49:47.590
OK, so now all we have
to do is plug in a of t
00:49:47.590 --> 00:49:50.060
is a constant times t to
the 2/3 into this formula
00:49:50.060 --> 00:49:51.800
and we have our answer.
00:49:51.800 --> 00:49:54.430
Notice that it does obey
an important property.
00:49:54.430 --> 00:49:57.650
There's an a in the numerator
and an a in the denominator.
00:49:57.650 --> 00:50:08.490
And that means that when we
put in the formula for a of t,
00:50:08.490 --> 00:50:12.000
the constant of proportionality
b will cancel out.
00:50:12.000 --> 00:50:15.310
And it must, as the
constants proportionality is
00:50:15.310 --> 00:50:20.560
measuring the notches or notches
per seconds to the 2/3 power
00:50:20.560 --> 00:50:22.250
but proportional to the notch.
00:50:22.250 --> 00:50:24.310
And the answer can't
depend on notches
00:50:24.310 --> 00:50:27.790
because notches are not
really a physical unit.
00:50:27.790 --> 00:50:28.930
But it works.
00:50:28.930 --> 00:50:30.280
That's an important check.
00:50:30.280 --> 00:50:33.400
So-- now it's just a
matter of plugging in here,
00:50:33.400 --> 00:50:37.290
and maybe I'll do it explicitly.
00:50:37.290 --> 00:50:38.774
I'll leave out the
b's [INAUDIBLE]
00:50:38.774 --> 00:50:39.690
so you see the cancel.
00:50:39.690 --> 00:50:46.520
We have b times t zero to the
2/3 times the integral from 0
00:50:46.520 --> 00:50:57.430
to t zero, times c over b
times t to the 2/3 times dt.
00:50:57.430 --> 00:51:00.140
The b's cancel as I claimed.
00:51:00.140 --> 00:51:07.740
The integral of t to the minus
2/3 is 3 times t to the 1/3.
00:51:07.740 --> 00:51:11.400
We then subtract the t to
the 1/3 giving it the value t
00:51:11.400 --> 00:51:13.760
zero on the positive
side, and then
00:51:13.760 --> 00:51:15.240
we subtract the
value [INAUDIBLE]
00:51:15.240 --> 00:51:17.530
same expression at zero.
00:51:17.530 --> 00:51:22.090
But t to the 2/3 when
t is zero vanishes.
00:51:22.090 --> 00:51:26.800
So we just get t zero to
the 2/3 from the upper limit
00:51:26.800 --> 00:51:28.610
of integration.
00:51:28.610 --> 00:51:31.010
I'm sorry, to the 1/3 power.
00:51:31.010 --> 00:51:33.310
We integrate minus 2/3.
00:51:33.310 --> 00:51:34.880
We get t to the 1/3.
00:51:34.880 --> 00:51:37.077
The t to the 1/3
multiplies the t
00:51:37.077 --> 00:51:41.290
to the 2/3 giving us
a full one power of t.
00:51:41.290 --> 00:51:49.907
So what we're left with is
just 3 times c times t zero,
00:51:49.907 --> 00:51:50.990
which has the right units.
00:51:50.990 --> 00:51:53.290
It should have units
of physical distance.
00:51:53.290 --> 00:51:56.400
Speed times time
is the distance.
00:51:56.400 --> 00:52:00.070
And it has a surprising
factor of 3 in it.
00:52:00.070 --> 00:52:03.710
The naive answer would have
just been c times t zero,
00:52:03.710 --> 00:52:06.890
saying that the light
at time t travels
00:52:06.890 --> 00:52:10.520
so it travels a
distance c times t zero.
00:52:10.520 --> 00:52:12.890
That would be true, as I said,
in a stationary universe.
00:52:12.890 --> 00:52:14.860
But the universe
is not stationary.
00:52:14.860 --> 00:52:15.860
It's expanding.
00:52:15.860 --> 00:52:17.600
And the fact it's
expanding means
00:52:17.600 --> 00:52:20.640
that you expect this to be
bigger than c times t zero.
00:52:20.640 --> 00:52:23.510
It means that at earlier
times, things were closer.
00:52:23.510 --> 00:52:29.010
So the light can save time by
leaving early and traveling
00:52:29.010 --> 00:52:31.310
a good part of the
distance while distances
00:52:31.310 --> 00:52:33.700
are smaller than they are now.
00:52:33.700 --> 00:52:36.480
And it's a full factor of 3.
00:52:36.480 --> 00:52:41.720
So that is the horizon
distance for a flat,
00:52:41.720 --> 00:52:43.960
[INAUDIBLE] universe.
00:52:43.960 --> 00:52:45.880
We can also, since we
know how to relate t
00:52:45.880 --> 00:52:51.602
sub zero to the
Hubble expansion rate,
00:52:51.602 --> 00:52:53.060
we can express the
horizon distance
00:52:53.060 --> 00:52:55.460
if we want in terms of
the Hubble expansion rate
00:52:55.460 --> 00:52:57.520
by just doing that substitution.
00:52:57.520 --> 00:53:02.610
So this then becomes 2 times
c times the current Hubble
00:53:02.610 --> 00:53:05.650
expansion rate inverse.
00:53:05.650 --> 00:53:07.402
So these are both
valid expressions
00:53:07.402 --> 00:53:09.443
for the horizon distance
in this particular model
00:53:09.443 --> 00:53:10.109
of the universe.
00:53:15.800 --> 00:53:18.254
Any questions about the
meaning of horizon distance?
00:53:18.254 --> 00:53:20.170
There is actually a
subtlety about the meaning
00:53:20.170 --> 00:53:21.820
of horizon, which I
should talk about.
00:53:25.170 --> 00:53:29.270
The initial value of the scale
factor in our model is zero.
00:53:29.270 --> 00:53:33.490
It's t to the 2/3, and t goes to
0, the scale factor goes to 0.
00:53:33.490 --> 00:53:34.990
Things are of course
singular there,
00:53:34.990 --> 00:53:37.640
where you don't really trust
exactly what the equations are
00:53:37.640 --> 00:53:39.560
telling us at t equals zero.
00:53:39.560 --> 00:53:42.020
But that's certainly
how they behave.
00:53:42.020 --> 00:53:45.290
a of t goes to 0 as t goes to 0.
00:53:45.290 --> 00:53:48.270
That means initially everything
was on top of everything else.
00:53:48.270 --> 00:53:52.820
So if everything was on
top of everything else,
00:53:52.820 --> 00:53:54.470
why is there any
horizon distance?
00:53:54.470 --> 00:53:56.940
Couldn't anything have
communicated with anything
00:53:56.940 --> 00:53:59.905
at t equals zero, when the
distance between anything
00:53:59.905 --> 00:54:03.310
and anything was zero?
00:54:03.310 --> 00:54:06.232
The answer to that is
perhaps somewhat ambiguous.
00:54:06.232 --> 00:54:08.440
We of course don't really
understand the singularity.
00:54:08.440 --> 00:54:11.970
We don't claim to
understand the singularity.
00:54:11.970 --> 00:54:14.870
And therefore anything
you want to believe
00:54:14.870 --> 00:54:16.270
about the singularity
at t equals
00:54:16.270 --> 00:54:18.640
zero you are welcome to believe.
00:54:18.640 --> 00:54:22.035
And nobody intelligent is
going to contradict you.
00:54:22.035 --> 00:54:23.910
They might not reinforce
you, but they're not
00:54:23.910 --> 00:54:27.000
going to contradict you
either, because nobody knows.
00:54:27.000 --> 00:54:31.740
So it is conceivable that
everything had a chance
00:54:31.740 --> 00:54:33.510
to communicate with
everything else at t
00:54:33.510 --> 00:54:35.279
equals zero at the singularity.
00:54:35.279 --> 00:54:37.820
And it's conceivable that when
we understand quantum gravity,
00:54:37.820 --> 00:54:38.861
it may even tell us that.
00:54:38.861 --> 00:54:40.730
We don't know.
00:54:40.730 --> 00:54:45.920
What is still the case is
that if you strike out t
00:54:45.920 --> 00:54:50.810
equals zero exactly, then
everything is well defined.
00:54:50.810 --> 00:54:54.100
You can ask what happens if a
photon is sent from one object
00:54:54.100 --> 00:54:56.970
to another leaving that
object at time epsilon,
00:54:56.970 --> 00:54:59.370
when epsilon is slightly
later than zero.
00:54:59.370 --> 00:55:01.290
And you can ask how
long does that photon
00:55:01.290 --> 00:55:05.040
take to go from
object a to object b?
00:55:05.040 --> 00:55:07.450
And that's really exactly
the calculation we did,
00:55:07.450 --> 00:55:09.370
except instead of going
down to t equals zero,
00:55:09.370 --> 00:55:11.399
you go down to t
equals epsilon, where
00:55:11.399 --> 00:55:12.940
epsilon is the
earliest time that you
00:55:12.940 --> 00:55:15.300
trust your classical
calculations.
00:55:15.300 --> 00:55:20.320
Then you'd be asking how far
is the furthest object that we
00:55:20.320 --> 00:55:23.720
could see during this classical
era, the era that starts from t
00:55:23.720 --> 00:55:24.384
equals epsilon?
00:55:24.384 --> 00:55:26.550
The only difference would
be to put an epsilon there
00:55:26.550 --> 00:55:27.670
instead of 0.
00:55:27.670 --> 00:55:30.360
And the answer-- you
can go through it--
00:55:30.360 --> 00:55:34.032
differs only by some
small multiple of epsilon.
00:55:34.032 --> 00:55:36.240
And if epsilon is small, it
doesn't change the answer
00:55:36.240 --> 00:55:37.850
at all.
00:55:37.850 --> 00:55:40.620
Physically what is
going on is that if you
00:55:40.620 --> 00:55:43.910
go to very, very early times
and look at two objects a and b
00:55:43.910 --> 00:55:46.520
and trace them back, the
distance between them
00:55:46.520 --> 00:55:49.240
does get smaller and smaller
as epsilon goes to 0.
00:55:49.240 --> 00:55:53.260
So you might think that
communication would be trivial.
00:55:53.260 --> 00:55:56.870
But at the same time as the
distance is going to zero,
00:55:56.870 --> 00:55:59.460
you can calculate
the velocities.
00:55:59.460 --> 00:56:02.940
h remember is going
like 2 over 3t.
00:56:02.940 --> 00:56:05.110
h is blowing up.
00:56:05.110 --> 00:56:06.824
The velocities between
these two objects
00:56:06.824 --> 00:56:08.990
a and b are going to go to
infinity at the same time
00:56:08.990 --> 00:56:11.940
that the distance between
them goes to zero.
00:56:11.940 --> 00:56:14.160
So even though they
will become very close,
00:56:14.160 --> 00:56:16.054
if one sends a light
beam to the other,
00:56:16.054 --> 00:56:17.720
the other is actually
moving away faster
00:56:17.720 --> 00:56:19.840
than the light would be moving.
00:56:19.840 --> 00:56:21.860
The light would
eventually catch up,
00:56:21.860 --> 00:56:24.630
but the amount of time it would
take for the light to catch up
00:56:24.630 --> 00:56:27.960
is exactly what this
integral is telling us.
00:56:27.960 --> 00:56:30.630
So there is no
widespread communication
00:56:30.630 --> 00:56:35.320
that is possible once
these equations are valid.
00:56:35.320 --> 00:56:37.290
They really do say that
in the early universe,
00:56:37.290 --> 00:56:41.770
things just cannot talk to other
things because the universe is
00:56:41.770 --> 00:56:45.150
expanding so fast.
00:56:45.150 --> 00:56:47.950
And the maximum distance
that you can see
00:56:47.950 --> 00:56:51.020
is more than c times t zero,
but less than infinity.
00:56:55.244 --> 00:56:55.744
Yes?
00:56:55.744 --> 00:56:58.720
AUDIENCE: Why does it
matter how far away
00:56:58.720 --> 00:57:01.396
the furthest galaxies
we can see are now?
00:57:01.396 --> 00:57:02.770
Because we're
seeing them as they
00:57:02.770 --> 00:57:06.034
were a long time ago when
they were closer to us.
00:57:06.034 --> 00:57:08.700
PROFESSOR: Yea, now we certainly
are seeing them a long time ago
00:57:08.700 --> 00:57:09.866
when they were closer to us.
00:57:09.866 --> 00:57:10.860
That's right.
00:57:10.860 --> 00:57:14.000
I would just say that this
is sort of a figure of merit.
00:57:14.000 --> 00:57:18.820
If you want to describe what
you think the universe looks
00:57:18.820 --> 00:57:23.300
like now, you would assume
that those galaxies that you're
00:57:23.300 --> 00:57:26.430
seeing billions of light years
in the past are still there,
00:57:26.430 --> 00:57:28.290
and you'd extrapolate
to the present.
00:57:28.290 --> 00:57:30.637
So it's relevant to
the picture that you
00:57:30.637 --> 00:57:32.720
would draw in your mind
of what the universe looks
00:57:32.720 --> 00:57:36.990
like at this instant, although
that picture would be based
00:57:36.990 --> 00:57:38.935
on things you haven't
actually seen.
00:57:38.935 --> 00:57:39.810
AUDIENCE: [INAUDIBLE]
00:57:42.810 --> 00:57:43.577
PROFESSOR: Yes?
00:57:43.577 --> 00:57:45.160
AUDIENCE: Is the
fact that this number
00:57:45.160 --> 00:57:48.974
is greater than c t zero
proof that the universe is
00:57:48.974 --> 00:57:50.890
the actual space of the
universe is expanding?
00:57:54.390 --> 00:57:57.130
PROFESSOR: I don't think
so because if you just
00:57:57.130 --> 00:58:00.330
have the objects moving
and a space that you regard
00:58:00.330 --> 00:58:03.370
as absolutely fixed
and then you ask
00:58:03.370 --> 00:58:07.090
the present distance of that
object given that you can see
00:58:07.090 --> 00:58:10.510
a light pulse that was
emitted by that object,
00:58:10.510 --> 00:58:13.640
it will still be bigger than
ct, because the object continues
00:58:13.640 --> 00:58:16.710
to move after it
emits the light pulse.
00:58:16.710 --> 00:58:20.330
So I don't think this is
proof of anything like that.
00:58:20.330 --> 00:58:24.340
I think I should add
that certainly we
00:58:24.340 --> 00:58:28.010
think of this as
the space expanding.
00:58:28.010 --> 00:58:29.980
That is certainly
the easiest picture.
00:58:29.980 --> 00:58:34.260
But you can't have any
absolute definition
00:58:34.260 --> 00:58:36.210
of what it means for
the space to expand.
00:58:36.210 --> 00:58:37.585
AUDIENCE: So you're
saying it has
00:58:37.585 --> 00:58:39.354
moved three times the
original distance it
00:58:39.354 --> 00:58:41.490
was when it sent out an impulse?
00:58:41.490 --> 00:58:44.540
PROFESSOR: No it actually was
zero distance when it sent out
00:58:44.540 --> 00:58:49.929
the light pulse, because the
first object we in principle
00:58:49.929 --> 00:58:52.470
could see is an object for which
the light pulse left it at t
00:58:52.470 --> 00:58:55.272
equals zero when it literally
was zero distance from us.
00:58:55.272 --> 00:58:57.730
And the light pulse actually
then gets further away from us
00:58:57.730 --> 00:59:00.770
and comes back and
eventually reaches us.
00:59:00.770 --> 00:59:02.807
We have enough information
here to trace it.
00:59:02.807 --> 00:59:03.890
And that's how it behaves.
00:59:10.517 --> 00:59:11.475
OK any other questions?
00:59:16.120 --> 00:59:18.475
OK, next thing I
want to do, and I
00:59:18.475 --> 00:59:20.350
don't know if we'll
finish this today or not.
00:59:20.350 --> 00:59:22.320
But we'll get ourselves started.
00:59:22.320 --> 00:59:24.900
We'd like to now, having
solved the equation
00:59:24.900 --> 00:59:27.447
for the flat universe
for determining a of t,
00:59:27.447 --> 00:59:29.030
we would now like
to do the same thing
00:59:29.030 --> 00:59:31.177
for the open and
closed universes.
00:59:31.177 --> 00:59:32.760
And we'll do the
closed universe first
00:59:32.760 --> 00:59:34.590
because it's a
little bit simpler.
00:59:34.590 --> 00:59:35.610
I mean they're about
equal, but we're
00:59:35.610 --> 00:59:37.170
going to do the
closed universe first
00:59:37.170 --> 00:59:39.272
because it's first in the
notes that I've written.
00:59:42.050 --> 00:59:44.040
So we know what equation
we're trying to solve.
00:59:44.040 --> 00:59:45.930
This is really now
just an exercise
00:59:45.930 --> 00:59:47.800
in solving
differential equations.
00:59:47.800 --> 00:59:50.610
So one has to be clever
to solve this equation.
00:59:50.610 --> 00:59:52.890
So we will go
through it together
00:59:52.890 --> 00:59:56.360
and see how one can be clever
and find the solution to it.
00:59:56.360 --> 00:59:58.960
The equation is a
dot over a squared
00:59:58.960 --> 01:00:06.290
is equal to 8 pi
over 3 g rho minus k
01:00:06.290 --> 01:00:14.410
c squared over a
squared, with rho of t
01:00:14.410 --> 01:00:30.530
being equal to rho i times a
cubed of t i over a cubed of t.
01:00:30.530 --> 01:00:32.130
And I'm writing i here.
01:00:32.130 --> 01:00:33.460
I could just as well write 1.
01:00:33.460 --> 01:00:34.420
It could be any time.
01:00:37.030 --> 01:00:39.535
Rho times a cubed isn't
dependent of time.
01:00:39.535 --> 01:00:41.160
So you can put any
time you want there.
01:00:41.160 --> 01:00:43.110
And the numerator is
just a fixed number.
01:00:55.040 --> 01:00:59.242
OK, so our goal is to
solve this equation.
01:00:59.242 --> 01:01:00.950
The first thing I want
to do is something
01:01:00.950 --> 01:01:02.408
which is usually
a good thing to do
01:01:02.408 --> 01:01:05.797
when you are given some
physical differential equation.
01:01:05.797 --> 01:01:08.380
Physical differential equations
usually have constants in them
01:01:08.380 --> 01:01:12.380
like g and c squared, which
have different units which
01:01:12.380 --> 01:01:13.690
complicate things.
01:01:13.690 --> 01:01:16.410
And they don't complicate
things in any intrinsic way.
01:01:16.410 --> 01:01:20.590
They just give you extra
factors to carry around.
01:01:20.590 --> 01:01:23.440
So it's usually a little
cleaner to eliminate
01:01:23.440 --> 01:01:28.260
them to begin with by redefining
variables that absorb them.
01:01:28.260 --> 01:01:30.510
One can do that by
defining variables
01:01:30.510 --> 01:01:33.600
that have the simplest
possible units that
01:01:33.600 --> 01:01:36.650
are available for
the problem at hand.
01:01:36.650 --> 01:01:39.340
And in our case, we
have these complications
01:01:39.340 --> 01:01:43.720
that k is measuring in
inverse notch squared,
01:01:43.720 --> 01:01:46.270
and a is meters per notch.
01:01:46.270 --> 01:01:51.560
We could simplify
all that by defining
01:01:51.560 --> 01:01:55.340
some auxiliary quantities.
01:01:55.340 --> 01:01:58.550
So in particular, I'm
going to define an a
01:01:58.550 --> 01:02:08.710
with a tilde above it, which is
sort of like the scale factor,
01:02:08.710 --> 01:02:11.110
but redefined by the
square root of k.
01:02:15.304 --> 01:02:16.970
And this is the case
when k is positive.
01:02:16.970 --> 01:02:19.094
I said we're going to do
the closed universe first.
01:02:23.637 --> 01:02:25.720
I would not want to divide
by the square root of k
01:02:25.720 --> 01:02:26.610
if k were negative.
01:02:26.610 --> 01:02:27.610
That would be confusing.
01:02:30.522 --> 01:02:32.230
Now the nice thing
about this is remember
01:02:32.230 --> 01:02:34.550
k has units of
inverse notch squared.
01:02:34.550 --> 01:02:36.390
a has units of meters per notch.
01:02:36.390 --> 01:02:39.170
That means the
notches just go away.
01:02:39.170 --> 01:02:41.995
So this has units
of length only.
01:02:48.050 --> 01:02:50.969
And that was the
motivation for dividing
01:02:50.969 --> 01:02:53.010
by the square root of k,
to get rid of the notch.
01:02:57.810 --> 01:03:03.046
Similarly, time has units
of meters per second.
01:03:03.046 --> 01:03:04.420
I'm trying to
minimize the number
01:03:04.420 --> 01:03:08.210
of distinct physical quantities
that we have in our problem.
01:03:08.210 --> 01:03:13.960
So I'm going to define
a t twiddle which
01:03:13.960 --> 01:03:16.354
is just equal to c times t.
01:03:16.354 --> 01:03:18.770
This is of course no difference
in just saying the working
01:03:18.770 --> 01:03:21.360
units is c is equal to
1, which a lot of times
01:03:21.360 --> 01:03:22.855
is something people say.
01:03:22.855 --> 01:03:24.230
This is the same
thing except I'm
01:03:24.230 --> 01:03:28.640
a little bit more
explicit about.
01:03:28.640 --> 01:03:31.330
So t twiddle now is
units of length also.
01:03:38.250 --> 01:03:39.890
So the idea is to
translate everything
01:03:39.890 --> 01:03:41.681
so that everything is
the same units, which
01:03:41.681 --> 01:03:44.190
would be meters or
whatever physical unit
01:03:44.190 --> 01:03:45.910
of length you want to use.
01:03:52.074 --> 01:03:53.990
OK, now I'm just going
to rewrite the Freedman
01:03:53.990 --> 01:03:56.570
equation using
these substitutions.
01:03:56.570 --> 01:03:59.280
And to make way for
the substitutions,
01:03:59.280 --> 01:04:01.730
I'm first just going to
divide the Freedman equation
01:04:01.730 --> 01:04:02.570
by kc squared.
01:04:37.680 --> 01:04:38.840
This is k to the 3/2.
01:04:44.630 --> 01:04:46.760
OK now when I divided
by kc squared,
01:04:46.760 --> 01:04:52.139
the kc squared over a squared
there became-- I'm sorry.
01:04:52.139 --> 01:04:53.930
I'm doing more than
dividing by kc squared.
01:04:53.930 --> 01:04:57.300
I'm dividing by kc squared
and multiplying by a squared.
01:04:57.300 --> 01:05:01.070
So I have turned
that last term into 1
01:05:01.070 --> 01:05:03.970
by multiplying by its inverse.
01:05:03.970 --> 01:05:05.900
On the left hand side,
the 1 over a squared
01:05:05.900 --> 01:05:07.630
went away when I
multiplied by a squared.
01:05:07.630 --> 01:05:11.180
So I just have a dot squared
divided by kc squared so far.
01:05:11.180 --> 01:05:13.180
We'll simplify that shortly.
01:05:13.180 --> 01:05:16.190
And the middle term,
I took the liberty
01:05:16.190 --> 01:05:19.270
of multiplying by
the square root of k,
01:05:19.270 --> 01:05:21.350
and then we have a
k to the 3/2 here.
01:05:21.350 --> 01:05:23.230
If we absorb this into
that, it would just
01:05:23.230 --> 01:05:25.230
be the kc squared factor
that we divided by.
01:05:29.765 --> 01:05:36.620
And the a squared
has been also divided
01:05:36.620 --> 01:05:40.370
into two pieces, a cubed and a.
01:05:40.370 --> 01:05:42.167
So together these
two factors make up
01:05:42.167 --> 01:05:44.500
the factor of a squared that
we multiplied that term by.
01:05:44.500 --> 01:05:45.980
So it's the same
thing we had, just
01:05:45.980 --> 01:05:48.280
multiplied by the common factor.
01:05:54.050 --> 01:05:57.440
And now the nice thing
is that this is, in fact,
01:05:57.440 --> 01:06:03.690
our definition of
da tilde over dt.
01:06:03.690 --> 01:06:08.460
The c turns the dt into a
dt tilde, and the 1 over k
01:06:08.460 --> 01:06:11.510
turns the da into da tilde.
01:06:11.510 --> 01:06:14.610
So the left hand
side now is simply
01:06:14.610 --> 01:06:21.260
da tilde over dt tilde squared.
01:06:25.210 --> 01:06:31.075
And the right hand side,
rho times a cubed remember
01:06:31.075 --> 01:06:33.490
is a constant.
01:06:33.490 --> 01:06:36.100
So the only thing that depends
on time on the right hand side
01:06:36.100 --> 01:06:39.290
is the a divided by mu k,
which I've isolated here.
01:06:39.290 --> 01:06:40.750
Earth That's a tilde.
01:06:40.750 --> 01:06:43.000
So I'm going to take all of
this, which is a constant,
01:06:43.000 --> 01:06:46.590
and just give it
a name, constant.
01:06:46.590 --> 01:06:53.810
So this term becomes a constant,
we'll call 2 alpha divided
01:06:53.810 --> 01:06:56.520
by a tilde.
01:06:56.520 --> 01:06:59.580
And then we still have minus 1.
01:06:59.580 --> 01:07:02.330
And this constant,
which I'm calling alpha,
01:07:02.330 --> 01:07:07.010
is just all of this factor
except for a factor of 2.
01:07:07.010 --> 01:07:19.110
So it's 4 pi over 3
g rho a tilde cubed
01:07:19.110 --> 01:07:22.290
divided by c squared.
01:07:22.290 --> 01:07:26.400
a tilde because we had a
cubed divided by k to the 3/2,
01:07:26.400 --> 01:07:29.120
and that's a tilde cubed.
01:07:29.120 --> 01:07:31.610
So I've just rearranged things.
01:07:31.610 --> 01:07:33.980
But now everything has
the units of length.
01:07:33.980 --> 01:07:35.860
a tilde has units of length.
01:07:35.860 --> 01:07:37.260
t tilde has units of length.
01:07:37.260 --> 01:07:39.080
This is dimensionless,
which is good
01:07:39.080 --> 01:07:41.260
because that's
also dimensionless.
01:07:41.260 --> 01:07:43.390
Alpha, if you work
out all this stuff,
01:07:43.390 --> 01:07:45.900
has units of length. a
tilde has units of length.
01:07:45.900 --> 01:07:48.080
This is length divided
by length, dimensionless.
01:07:51.212 --> 01:07:53.170
We haven't really changed
anything significant.
01:07:53.170 --> 01:07:56.060
But at least as far as
keeping track of factors,
01:07:56.060 --> 01:07:57.720
that equation is
the one we'll solve.
01:07:57.720 --> 01:08:01.620
And the factors are now absorbed
into the constant alpha, which
01:08:01.620 --> 01:08:03.514
is the only thing we
have to worry about.
01:08:03.514 --> 01:08:05.680
And we don't need to worry
about that until the end.
01:08:05.680 --> 01:08:06.509
Yes?
01:08:06.509 --> 01:08:07.384
AUDIENCE: [INAUDIBLE]
01:08:12.005 --> 01:08:14.630
PROFESSOR: As long as these two
are evaluated at the same time,
01:08:14.630 --> 01:08:15.370
it doesn't matter.
01:08:15.370 --> 01:08:16.245
AUDIENCE: [INAUDIBLE]
01:08:22.670 --> 01:08:24.077
PROFESSOR: That's right.
01:08:24.077 --> 01:08:25.910
One does need to remember
that these two are
01:08:25.910 --> 01:08:29.430
to be evaluated at the
same time, whatever it is.
01:08:29.430 --> 01:08:33.210
And the product does
not depend on time.
01:08:37.430 --> 01:08:38.910
So I didn't write the arguments.
01:08:38.910 --> 01:08:41.470
I could have, and I could have
just put in an arbitrary time.
01:08:41.470 --> 01:08:43.803
But it would be the same time
for the rho and the tilde.
01:08:49.350 --> 01:08:50.530
I will do that.
01:08:50.530 --> 01:08:54.149
It will be rho of
some t one times
01:08:54.149 --> 01:09:00.069
a tilde cubed, evaluated the
same t one for any t one.
01:09:00.069 --> 01:09:01.590
This product is
independent of time.
01:09:50.830 --> 01:09:52.540
OK, now that is the
kind of equation
01:09:52.540 --> 01:09:55.065
where we could move things
from one side of the equation
01:09:55.065 --> 01:09:59.810
to the other to reduce it
to doing ordinary intervals.
01:09:59.810 --> 01:10:03.810
So I can multiply by
d tilde and divide
01:10:03.810 --> 01:10:06.960
by the expression on
the right hand side
01:10:06.960 --> 01:10:13.590
and get an expression
that says that dt tilde is
01:10:13.590 --> 01:10:20.100
equal to da tilde divided
by the square root of 2
01:10:20.100 --> 01:10:25.740
alpha over a tilde minus 1.
01:10:32.020 --> 01:10:35.570
And since this has an a
tilde in the denominator
01:10:35.570 --> 01:10:38.180
of the denominator, I'm
going to multiply through
01:10:38.180 --> 01:10:42.860
by a tilde to
rationalize things.
01:10:42.860 --> 01:10:49.670
So I'm going to rewrite
this as a tilde da tilde
01:10:49.670 --> 01:11:00.160
over the square root of 2 alpha
a tilde minus a tilde squared.
01:11:00.160 --> 01:11:02.280
OK this looks better
than we've had so far.
01:11:08.100 --> 01:11:11.730
Now in principle, if we imagine
we can do that integral,
01:11:11.730 --> 01:11:13.430
we can just
integrate both sides.
01:11:13.430 --> 01:11:15.320
We get an equation
that says t tilde is
01:11:15.320 --> 01:11:20.639
equal to some expression
involving the final value of a.
01:11:20.639 --> 01:11:22.180
So we're going to
imagine doing that.
01:11:22.180 --> 01:11:25.020
And we will actually be
able to carry it out.
01:11:25.020 --> 01:11:31.170
When we integrate, there are
two ways you can proceed.
01:11:31.170 --> 01:11:33.910
When I solve the
analogous problem
01:11:33.910 --> 01:11:36.160
for the flat case
with the t to the 2/3
01:11:36.160 --> 01:11:37.622
it was a much simpler equation.
01:11:37.622 --> 01:11:39.330
But you might remember
that at one point,
01:11:39.330 --> 01:11:41.140
I had an equation
where I calculated
01:11:41.140 --> 01:11:43.300
the indefinite
integral of both sides,
01:11:43.300 --> 01:11:45.910
and then I got a constant
of integration, which I then
01:11:45.910 --> 01:11:49.050
argued should be set
equal to zero if we wanted
01:11:49.050 --> 01:11:51.760
to find the zero of time
to be the zero of a.
01:11:55.760 --> 01:11:58.150
The situation is really
exactly the same,
01:11:58.150 --> 01:12:00.020
but to show you
an alternative way
01:12:00.020 --> 01:12:02.410
of thinking about
it, this time I'm
01:12:02.410 --> 01:12:05.765
going to apply definite
integrals to both sides.
01:12:05.765 --> 01:12:07.890
And if you apply a definite
integral to both sides,
01:12:07.890 --> 01:12:10.710
the thing to keep in mind is
that you should be integrating
01:12:10.710 --> 01:12:15.150
over the same physical
interval on both sides.
01:12:15.150 --> 01:12:18.190
So on the left hand
side, I'm going
01:12:18.190 --> 01:12:28.580
to integrate from zero
up to some t tilde final.
01:12:28.580 --> 01:12:30.330
t tilde final is just
some arbitrary time,
01:12:30.330 --> 01:12:34.370
but I'm going to give it
the name t tilde final.
01:12:34.370 --> 01:12:36.530
And that integral
of course we can do.
01:12:36.530 --> 01:12:40.160
It's just t tilde final.
01:12:40.160 --> 01:12:42.850
And that should be equal to the
integral of the right hand side
01:12:42.850 --> 01:12:44.634
over the same period of time.
01:12:44.634 --> 01:12:46.300
But the right hand
side is not expressed
01:12:46.300 --> 01:12:47.440
as an integral over time.
01:12:47.440 --> 01:12:50.350
It's expressed as an
integral over a tilde.
01:12:50.350 --> 01:12:52.450
So we have to ask
ourselves, what
01:12:52.450 --> 01:12:55.030
do we call the integral
over a tilde that
01:12:55.030 --> 01:13:01.770
corresponds to the integral over
time from 0 to t tilde sub f.
01:13:01.770 --> 01:13:06.340
And the lower
limits should match.
01:13:06.340 --> 01:13:11.617
And we know what we want a tilde
to be at t tilde equals zero.
01:13:11.617 --> 01:13:13.200
We're going to use
the same dimensions
01:13:13.200 --> 01:13:14.530
we had in the other case.
01:13:14.530 --> 01:13:16.120
We're going to define
the zero of time
01:13:16.120 --> 01:13:19.530
to be the time when the
scale factor is zero.
01:13:19.530 --> 01:13:22.580
So t tilde equals
zero should correspond
01:13:22.580 --> 01:13:24.520
to a tilde equals zero.
01:13:24.520 --> 01:13:27.050
So to get the lower
limits of integration
01:13:27.050 --> 01:13:30.530
to correspond to the same
time, we just put zero her.
01:13:30.530 --> 01:13:32.210
And zero here doesn't
mean time zero.
01:13:32.210 --> 01:13:33.870
It means a tilde equals zero.
01:13:33.870 --> 01:13:35.900
But that's what we want.
01:13:35.900 --> 01:13:40.750
And for the upper limit of
integration, that should just
01:13:40.750 --> 01:13:45.800
be the value of a tilde
at the time t tilde sub f.
01:13:45.800 --> 01:13:52.870
So I will call
that a tilde sub f,
01:13:52.870 --> 01:13:54.820
where I might make
a note on the side
01:13:54.820 --> 01:14:06.200
here that a tilde sub f is
e tilde of t tilde sub f.
01:14:06.200 --> 01:14:09.270
It's just the final
value of a tilde, where
01:14:09.270 --> 01:14:12.125
final means anytime I choose
to stop this integration.
01:14:12.125 --> 01:14:14.460
The interval is valid
over any time period.
01:14:18.960 --> 01:14:20.864
So these are moments
of integration
01:14:20.864 --> 01:14:22.780
where the only thing
that's new on this line--
01:14:22.780 --> 01:14:33.350
now I just copy from the
line above, 2 alpha times
01:14:33.350 --> 01:14:46.210
a tilde minus a tilde squared.
01:15:01.590 --> 01:15:04.940
So t tilde f is equal
to that integral.
01:15:04.940 --> 01:15:08.560
And our goal now would
be to do that integral.
01:15:08.560 --> 01:15:10.510
And if possible-- it
won't quite be possible.
01:15:10.510 --> 01:15:11.885
We can see how
close we can come.
01:15:11.885 --> 01:15:14.960
But if possible we would like
to invert that relationship
01:15:14.960 --> 01:15:16.890
that we get from
doing this integral,
01:15:16.890 --> 01:15:21.217
to determine a tilde as
a function of t tilde.
01:15:21.217 --> 01:15:22.550
That's what we would love to do.
01:15:22.550 --> 01:15:24.590
It turns out that's
not quite possible.
01:15:24.590 --> 01:15:27.740
But we will nonetheless be
able to obtain something called
01:15:27.740 --> 01:15:29.990
a parametric solution
to this problem.
01:15:29.990 --> 01:15:32.110
And we'll stop now.
01:15:32.110 --> 01:15:36.090
But I will tell you next
Tuesday after the quiz
01:15:36.090 --> 01:15:40.820
how we proceed here to get
a solution to this problem.