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PROFESSOR: A quick review of
what we talked about last time.
00:00:24.830 --> 00:00:26.760
So the first thing
we did last time
00:00:26.760 --> 00:00:29.350
was to discuss the
age of universe,
00:00:29.350 --> 00:00:31.280
considering so far
only at this point,
00:00:31.280 --> 00:00:34.950
flat matter-dominated universes
where the scale factor goes
00:00:34.950 --> 00:00:36.540
like t to the 2/3.
00:00:36.540 --> 00:00:40.110
And we were easily able to see
that the age of such a universe
00:00:40.110 --> 00:00:43.320
was 2/3 times h inverse.
00:00:43.320 --> 00:00:45.720
And we did discuss what
happens if you plug-in numbers
00:00:45.720 --> 00:00:50.610
into that formula using the
best current value of h,
00:00:50.610 --> 00:00:54.986
the value obtained by the
Planck team last March.
00:00:54.986 --> 00:01:00.260
The age turns out to be about
9.5 to 9.9 billion years.
00:01:00.260 --> 00:01:03.530
And that can't be the real
age of the universe, we think,
00:01:03.530 --> 00:01:06.410
because there are stars
that are older than that.
00:01:06.410 --> 00:01:08.120
The age of the stars
seems to indicate
00:01:08.120 --> 00:01:10.240
that the university
should be at least,
00:01:10.240 --> 00:01:13.640
according to one paper I cited
last time, 11.2 billion years
00:01:13.640 --> 00:01:16.030
old, and this is younger.
00:01:16.030 --> 00:01:18.880
So the conclusion is
that our universe is not
00:01:18.880 --> 00:01:21.284
a flat, moderate,
matter-dominated universe.
00:01:21.284 --> 00:01:23.700
We do in fact have good evidence
that the universe is very
00:01:23.700 --> 00:01:26.500
nearly flat, so it's the
matter-dominated part that
00:01:26.500 --> 00:01:28.600
has to fail, and it does fail.
00:01:28.600 --> 00:01:33.400
We also have good evidence that
the universe is dominated today
00:01:33.400 --> 00:01:36.510
by dark energy, which we'll
be talking about later.
00:01:36.510 --> 00:01:40.430
But one of the pieces of
evidence for this dark energy
00:01:40.430 --> 00:01:41.900
is this age calculation.
00:01:41.900 --> 00:01:43.500
The age calculation
just does not
00:01:43.500 --> 00:01:46.930
work unless you assume
that the universe has
00:01:46.930 --> 00:01:50.167
a significant component
of this dark energy, which
00:01:50.167 --> 00:01:51.250
we'll be discussing later.
00:01:54.220 --> 00:01:56.310
We then talked about the
big bang singularity,
00:01:56.310 --> 00:01:58.780
which is an important part of
understanding, when you talk
00:01:58.780 --> 00:02:02.470
about the age, what exactly
you mean by the age, age
00:02:02.470 --> 00:02:04.290
since when.
00:02:04.290 --> 00:02:05.860
And the point that
I tried to make
00:02:05.860 --> 00:02:10.470
there is that the big bang
singularity, which gives us
00:02:10.470 --> 00:02:12.990
mathematically the
statement that the scale
00:02:12.990 --> 00:02:16.500
factor at some time which we
call zero was equal to zero,
00:02:16.500 --> 00:02:19.810
and if you put back that
formula into other formulas,
00:02:19.810 --> 00:02:22.350
you discover that the
mass density, for example,
00:02:22.350 --> 00:02:26.810
was infinite at this magical
time that we call zero.
00:02:26.810 --> 00:02:30.350
That singularity is certainly
part of our mathematical model
00:02:30.350 --> 00:02:32.590
and doesn't go away
even when we make
00:02:32.590 --> 00:02:35.010
changes in the
mathematical model.
00:02:35.010 --> 00:02:37.410
But we don't really know
if it's an actual feature
00:02:37.410 --> 00:02:39.760
of our universe, because
there's certainly no reason
00:02:39.760 --> 00:02:42.870
to trust this mathematical
model all the way back to t
00:02:42.870 --> 00:02:45.820
equals zero, where the
densities become infinite.
00:02:45.820 --> 00:02:47.400
We know a lot
about matter and we
00:02:47.400 --> 00:02:49.300
think we can predict
how matter will behave,
00:02:49.300 --> 00:02:52.260
even the temperatures and
energy densities somewhat
00:02:52.260 --> 00:02:54.230
beyond what we
measure in laboratory.
00:02:54.230 --> 00:02:56.080
But we don't think
we can't necessarily
00:02:56.080 --> 00:02:59.780
extrapolate all the way to
infinite matter density.
00:02:59.780 --> 00:03:01.730
So these equations
do break down when
00:03:01.730 --> 00:03:03.220
you get very close
to t equals zero
00:03:03.220 --> 00:03:06.240
and nobody really knows exactly
what should be said about t
00:03:06.240 --> 00:03:08.170
equals zero.
00:03:08.170 --> 00:03:09.990
When we talk about
the age, we're
00:03:09.990 --> 00:03:12.950
really talking about the age
since the extrapolated time
00:03:12.950 --> 00:03:16.050
at which a would have
been zero in this model,
00:03:16.050 --> 00:03:18.500
but we don't really know that
it ever actually was zero.
00:03:21.350 --> 00:03:25.570
We next discussed the concept
of the horizon distance.
00:03:25.570 --> 00:03:27.850
If the universe, at
least the universe
00:03:27.850 --> 00:03:32.220
as we know it, we want to be
agnostic about what happened
00:03:32.220 --> 00:03:35.435
before t equals zero, but
certainly the universe as we
00:03:35.435 --> 00:03:37.560
know it, really began at
t equals zero in the sense
00:03:37.560 --> 00:03:42.220
that that's when structure
and complicated things
00:03:42.220 --> 00:03:44.330
started to develop.
00:03:44.330 --> 00:03:47.090
So since t equals zero there's
been only a finite amount
00:03:47.090 --> 00:03:48.660
of time elapsed.
00:03:48.660 --> 00:03:51.560
And since light travels
at a finite speed,
00:03:51.560 --> 00:03:53.690
that means that light
could only have traveled
00:03:53.690 --> 00:03:58.380
some finite distance since the
big bang, since t equals zero.
00:03:58.380 --> 00:04:00.850
And that means that
there's some object which
00:04:00.850 --> 00:04:03.490
is the furthest possible
object that we could see,
00:04:03.490 --> 00:04:07.070
and any object further than
that would be in a situation
00:04:07.070 --> 00:04:08.930
where light from that
object would not yet
00:04:08.930 --> 00:04:10.980
have had time to reach us.
00:04:10.980 --> 00:04:14.780
And that leads to this notion
of our horizon distance,
00:04:14.780 --> 00:04:17.500
where the definition
of the horizon distance
00:04:17.500 --> 00:04:21.490
is that it is defined to be the
present distance of the most
00:04:21.490 --> 00:04:25.200
distant objects which we
are capable of seeing,
00:04:25.200 --> 00:04:28.380
limited only by
the speed of light.
00:04:28.380 --> 00:04:32.170
And once able to calculate that,
and in particular for the model
00:04:32.170 --> 00:04:34.660
that we so far
understood at this point,
00:04:34.660 --> 00:04:37.160
the matter-dominated
flat universe,
00:04:37.160 --> 00:04:41.225
the horizon distance turned
out to be three times c t.
00:04:41.225 --> 00:04:42.720
Tt
00:04:42.720 --> 00:04:44.850
Now remember, if the
universe were just static
00:04:44.850 --> 00:04:48.750
and appeared at time t ago,
then the horizon distance
00:04:48.750 --> 00:04:50.500
would just be c t, the
distance that light
00:04:50.500 --> 00:04:52.630
could travel in time t.
00:04:52.630 --> 00:04:55.607
What makes it larger is the fact
that the universe is expanding,
00:04:55.607 --> 00:04:57.690
and that means that
everything was closer together
00:04:57.690 --> 00:04:59.670
in the early time and
light could make more
00:04:59.670 --> 00:05:02.690
progress at the early time, and
then these objects have since
00:05:02.690 --> 00:05:05.380
moved out to much
larger distances.
00:05:05.380 --> 00:05:09.040
So that allows the horizon
distance to be larger than c t,
00:05:09.040 --> 00:05:12.470
and in this particular
model, it's three times c t.
00:05:16.094 --> 00:05:17.510
Next we began a
calculation, which
00:05:17.510 --> 00:05:20.790
is what we're going to
pick up to continue on now.
00:05:20.790 --> 00:05:25.210
We were calculating how to
extend our understanding of a
00:05:25.210 --> 00:05:27.990
of t, the behavior
of the scale factor,
00:05:27.990 --> 00:05:31.560
away from the flat case,
to ultimately discuss
00:05:31.560 --> 00:05:35.000
the two other cases, the
open and closed universe.
00:05:35.000 --> 00:05:37.310
And we decided on
a flip of a coin--
00:05:37.310 --> 00:05:39.504
I promise you I flipped
a coin at some point--
00:05:39.504 --> 00:05:40.920
to start with the
closed universe.
00:05:40.920 --> 00:05:43.761
We could have done
either one first.
00:05:43.761 --> 00:05:46.010
The equation that we start
with is basically the same,
00:05:46.010 --> 00:05:48.160
it's the sine of k that
makes the difference.
00:05:48.160 --> 00:05:51.150
This is the so-called
Friedmann equation,
00:05:51.150 --> 00:05:54.440
and for a closed
universe k is positive.
00:05:54.440 --> 00:05:56.700
This is the evolution
equation, but it
00:05:56.700 --> 00:05:58.600
has to be coupled
with an equation that
00:05:58.600 --> 00:06:01.340
describes how rho
behaves with time.
00:06:01.340 --> 00:06:04.920
And for a
matter-dominated universe,
00:06:04.920 --> 00:06:07.860
rho is just representing
non-relativistic matter,
00:06:07.860 --> 00:06:11.800
which is nothing but spread
as the universe expands.
00:06:11.800 --> 00:06:14.740
And the spreading
gives a factor of one
00:06:14.740 --> 00:06:19.310
over a cubed as the
volume grows as a cubed.
00:06:19.310 --> 00:06:22.430
And that means that rho
times a cubed is a constant,
00:06:22.430 --> 00:06:24.890
and that expresses
everything that there
00:06:24.890 --> 00:06:27.460
is to know about how
rho behaves with time.
00:06:30.820 --> 00:06:33.430
OK, then after writing
these equations,
00:06:33.430 --> 00:06:36.260
we said that things will
simplify a little bit, not
00:06:36.260 --> 00:06:39.756
a lot, but a little bit if we
redefine variables basically
00:06:39.756 --> 00:06:42.380
to incorporate all the constants
that appear in these equations
00:06:42.380 --> 00:06:45.590
into one overall constant.
00:06:45.590 --> 00:06:50.520
And we decided, or I
claimed, that a good way
00:06:50.520 --> 00:06:53.040
to do that, an economical
way to do that,
00:06:53.040 --> 00:06:56.470
is to define things so
that the variables all
00:06:56.470 --> 00:06:59.120
have units which are
easily understood.
00:06:59.120 --> 00:07:01.620
And in this case
the units of length
00:07:01.620 --> 00:07:03.460
can describe everything
that we need,
00:07:03.460 --> 00:07:05.490
so we chose to
express everything
00:07:05.490 --> 00:07:08.890
in terms of variables
that have units of length.
00:07:08.890 --> 00:07:12.450
So the scale factor itself
is units of meters per notch,
00:07:12.450 --> 00:07:13.724
and that's not a length.
00:07:13.724 --> 00:07:15.640
And notches we'd like
to get rid of because we
00:07:15.640 --> 00:07:17.740
know they're un-physical.
00:07:17.740 --> 00:07:23.220
That is, there's no standard
for what the notch should be.
00:07:23.220 --> 00:07:25.930
So if we divide a of t
by the square root of k,
00:07:25.930 --> 00:07:27.336
the notches
disappear, and we get
00:07:27.336 --> 00:07:29.210
something which just
has the units of meters,
00:07:29.210 --> 00:07:36.160
or units of length, and I
call that a twiddle of t.
00:07:36.160 --> 00:07:40.810
Similarly, but more obviously,
t can be turned into a length
00:07:40.810 --> 00:07:43.130
by just multiplying by
c, the speed of light.
00:07:43.130 --> 00:07:46.310
So I defined a variable t
tilde which is just c times t.
00:07:46.310 --> 00:07:49.650
So both of these new
variables, with the tildes,
00:07:49.650 --> 00:07:51.830
have units of length.
00:07:51.830 --> 00:07:53.600
And then the
Friedmann equation can
00:07:53.600 --> 00:07:55.500
be rewritten, just
reshuffling things
00:07:55.500 --> 00:08:00.560
according to these new
definitions, in this way where
00:08:00.560 --> 00:08:05.060
all the constants are lumped
into this variable alpha, where
00:08:05.060 --> 00:08:07.550
alpha is this complicated
expression, which absorbs all
00:08:07.550 --> 00:08:09.910
the constants from
the earlier equation.
00:08:09.910 --> 00:08:13.090
Alpha also has units
of length, and even
00:08:13.090 --> 00:08:15.880
though it has a rho times
a tilde cubed in it,
00:08:15.880 --> 00:08:19.650
and both rho and a tilde
each depend on time,
00:08:19.650 --> 00:08:24.420
the product of rho times a tilde
cubed is a constant because rho
00:08:24.420 --> 00:08:27.930
times a cubed is a constant
and a tilde differs only
00:08:27.930 --> 00:08:31.080
by the square root of k,
which is also a constant.
00:08:31.080 --> 00:08:32.659
So alpha is a constant
even though it
00:08:32.659 --> 00:08:35.140
has time dependent factors.
00:08:35.140 --> 00:08:37.100
The time dependence
of those two factors
00:08:37.100 --> 00:08:38.980
cancel each other
out to give something
00:08:38.980 --> 00:08:42.890
which is time independent and
can be evaluated any old time.
00:08:42.890 --> 00:08:49.510
So this now is our equation, and
we proceeded to manipulate it.
00:08:49.510 --> 00:08:54.010
So the first thing we did
was to take it's square root,
00:08:54.010 --> 00:08:57.670
and to rearrange it so that d
t tilde appeared on one side,
00:08:57.670 --> 00:08:59.670
and everything else
was on the other side.
00:08:59.670 --> 00:09:02.140
And everything else
depends on a tilde
00:09:02.140 --> 00:09:05.330
but not explicitly on time.
00:09:05.330 --> 00:09:07.720
So this completely
separates everything
00:09:07.720 --> 00:09:09.540
that depends on t
tilde on the left,
00:09:09.540 --> 00:09:13.430
and everything that depends
on a tilde on the right.
00:09:13.430 --> 00:09:16.110
And now we can just integrate
both sides of that equation,
00:09:16.110 --> 00:09:19.200
and and they point to that,
that there are basically
00:09:19.200 --> 00:09:22.000
two ways of proceeding
here, one of which
00:09:22.000 --> 00:09:24.810
we already did when
we did the flat case.
00:09:24.810 --> 00:09:27.900
When we did the flat case,
we integrated both sides
00:09:27.900 --> 00:09:29.820
as indefinite integrals.
00:09:29.820 --> 00:09:33.750
And when you carry out an
indefinite integration,
00:09:33.750 --> 00:09:35.600
you get a constant
of integration,
00:09:35.600 --> 00:09:38.960
which then becomes a
constant in your solution.
00:09:38.960 --> 00:09:43.470
And in that case we discovered
that the constant really
00:09:43.470 --> 00:09:45.660
just shifted the origin of time.
00:09:45.660 --> 00:09:48.480
And since we had not said
anything previously that
00:09:48.480 --> 00:09:51.190
in any way determined
the origin of time,
00:09:51.190 --> 00:09:54.260
we used that constant to
arrange the origin of time
00:09:54.260 --> 00:09:56.880
so that a of zero
would equal zero,
00:09:56.880 --> 00:09:59.400
and that eliminated
the constant.
00:09:59.400 --> 00:10:03.950
Just for variety, I am going
to do it another way this time.
00:10:03.950 --> 00:10:05.860
Instead of doing an
indefinite integral,
00:10:05.860 --> 00:10:07.734
I will do a definite integral.
00:10:07.734 --> 00:10:09.150
And if you do a
definite integral,
00:10:09.150 --> 00:10:10.290
you have to make sure
you're integrating
00:10:10.290 --> 00:10:12.540
both sides over the
same range, or at least
00:10:12.540 --> 00:10:13.880
corresponding ranges.
00:10:13.880 --> 00:10:15.960
We have different names
for the variables,
00:10:15.960 --> 00:10:18.520
but the range of
integration of the two sides
00:10:18.520 --> 00:10:21.560
has to match in
order to maintain
00:10:21.560 --> 00:10:24.050
the equality between
the two sides.
00:10:24.050 --> 00:10:26.850
So we're going to integrate
the left side from zero
00:10:26.850 --> 00:10:28.580
to some final time.
00:10:28.580 --> 00:10:34.470
And I'm going to call the
final time t tilde variable sub
00:10:34.470 --> 00:10:36.700
f, for f to stand for final.
00:10:36.700 --> 00:10:38.200
And there's no real
final time here.
00:10:38.200 --> 00:10:40.116
It could be any time,
it's just the final time
00:10:40.116 --> 00:10:42.290
for the integration,
and in the end
00:10:42.290 --> 00:10:46.290
discover then how things
behave at time t tilde sub f.
00:10:46.290 --> 00:10:48.940
And then once we've figured
that out we can drop the effort.
00:10:48.940 --> 00:10:53.790
It will be a formula that
will be valid for any time.
00:10:53.790 --> 00:10:56.740
So the left hand side
is integrated from zero
00:10:56.740 --> 00:10:58.910
to time t tilde f,
the right hand side
00:10:58.910 --> 00:11:02.650
has to be integrated over the
corresponding time interval.
00:11:02.650 --> 00:11:08.040
And we would like to define a
tilde and the origin of time
00:11:08.040 --> 00:11:10.725
so that a tilde equals
zero at time zero,
00:11:10.725 --> 00:11:14.570
the same convention we
used for the other case.
00:11:14.570 --> 00:11:17.110
The standard convention in
cosmology, t equals zero,
00:11:17.110 --> 00:11:18.750
is the instance of the Big Bang.
00:11:18.750 --> 00:11:20.740
And the instance of the
Big Bang is the time
00:11:20.740 --> 00:11:23.600
at which the scale
factor vanished.
00:11:23.600 --> 00:11:28.160
So that means when we have
a lower limit of integration
00:11:28.160 --> 00:11:30.390
of t tilde equals zero
on the left hand side,
00:11:30.390 --> 00:11:32.280
we should have a lower
limit of integration
00:11:32.280 --> 00:11:36.620
of a tilde equals zero
on the right hand side.
00:11:36.620 --> 00:11:39.190
And similarly the upper
limits of integration
00:11:39.190 --> 00:11:41.540
should correspond to each other.
00:11:41.540 --> 00:11:44.120
So the upper limit of
integration on the left hand
00:11:44.120 --> 00:11:48.680
side was t tilde sub f,
really just an arbitrary time
00:11:48.680 --> 00:11:52.120
that we designated
by the subscript f.
00:11:52.120 --> 00:11:54.020
So the right hand
side the limit should
00:11:54.020 --> 00:11:56.760
be the value of a
tilde at that time.
00:11:56.760 --> 00:12:00.300
And I'll call that
a tilde sub f.
00:12:00.300 --> 00:12:02.710
So a tilde sub f is
just defined to be
00:12:02.710 --> 00:12:08.020
the value of a tilde at
the time t tilde sub f.
00:12:08.020 --> 00:12:11.130
And in this way the limits of
integration on the two sides
00:12:11.130 --> 00:12:13.150
correspond and now
we can integrate them
00:12:13.150 --> 00:12:16.630
and we don't need any new
integration constants.
00:12:16.630 --> 00:12:19.100
These definite integrals
have definite values,
00:12:19.100 --> 00:12:21.929
which is why they're called
definite integrals, I suppose.
00:12:21.929 --> 00:12:23.720
OK any questions about
that, because that's
00:12:23.720 --> 00:12:26.590
where we're ready to take off
and start doing new material.
00:12:26.590 --> 00:12:27.090
Yes.
00:12:27.090 --> 00:12:29.230
AUDIENCE: I have a
general question.
00:12:29.230 --> 00:12:33.720
So regarding the issue of
how we're not really sure how
00:12:33.720 --> 00:12:36.054
to extrapolate up
to t equals zero,
00:12:36.054 --> 00:12:37.720
this is just kind of
a general question.
00:12:37.720 --> 00:12:40.220
I was wondering how
we're kind of assuming
00:12:40.220 --> 00:12:42.930
throughout that time
kind of flows uniformly,
00:12:42.930 --> 00:12:44.400
and so I've heard
about something
00:12:44.400 --> 00:12:46.420
like gravitational
time dilation.
00:12:46.420 --> 00:12:48.750
So at the beginning
especially when there's
00:12:48.750 --> 00:12:52.420
such a high density of
matter or radiation,
00:12:52.420 --> 00:12:56.240
then wouldn't that affect
how, I guess, time flows?
00:12:56.240 --> 00:12:58.100
PROFESSOR: OK good
question, good question.
00:12:58.100 --> 00:13:02.300
The question was, does things
like general realistic time
00:13:02.300 --> 00:13:05.970
dilation affect how
time flows, and are we
00:13:05.970 --> 00:13:09.750
perhaps being overly simplistic
and assuming that time just
00:13:09.750 --> 00:13:13.210
flows smoothly from
time zero onward.
00:13:13.210 --> 00:13:16.130
The answer to that is that
general relativity does
00:13:16.130 --> 00:13:18.250
predict an extra
time dilation, which
00:13:18.250 --> 00:13:21.560
in fact is built into the
Doppler shift calculation
00:13:21.560 --> 00:13:23.690
that we already did.
00:13:23.690 --> 00:13:26.760
And there are other instances
where similar things happen.
00:13:26.760 --> 00:13:28.920
If a photon travels from
the floor of this room
00:13:28.920 --> 00:13:30.490
to the ceiling of
this room, there's
00:13:30.490 --> 00:13:35.080
a small Doppler shift, a
small shift in the timing.
00:13:35.080 --> 00:13:37.460
And you could see it in
principle with clocks as well.
00:13:37.460 --> 00:13:39.960
If you had a clock on the floor,
and a clock on the ceiling,
00:13:39.960 --> 00:13:42.260
they would not run at
quite the same rate.
00:13:42.260 --> 00:13:44.690
But to talk about time
dilation, you always
00:13:44.690 --> 00:13:47.340
have to have two
clocks to compare.
00:13:47.340 --> 00:13:50.220
In the case of the
universe, we have this thing
00:13:50.220 --> 00:13:55.130
that we call cosmic time, which
can be measured on any clock.
00:13:55.130 --> 00:13:56.890
The homogeneity
assumption implies
00:13:56.890 --> 00:13:58.680
that all clocks will
do the same thing,
00:13:58.680 --> 00:14:04.580
so so the issue of time
dilation really does not arise.
00:14:04.580 --> 00:14:06.940
Our definition of
cosmic time defines
00:14:06.940 --> 00:14:10.290
time that is the time
variable that we will use.
00:14:10.290 --> 00:14:13.510
And when we say it flows
uniformly from zero up
00:14:13.510 --> 00:14:17.610
to the present time, that word
uniformly sounds sensible.
00:14:17.610 --> 00:14:20.170
But if you think about it,
I don't know what it means.
00:14:20.170 --> 00:14:23.880
So I don't even know how to ask
if it's really uniform or not.
00:14:23.880 --> 00:14:27.080
It's certainly true
that our time variable
00:14:27.080 --> 00:14:29.440
evolves from zero up
to some final time,
00:14:29.440 --> 00:14:33.030
but smoothly or uniformly
is not really a question we
00:14:33.030 --> 00:14:36.685
can ask until one has some other
clock that one can compare it
00:14:36.685 --> 00:14:37.525
to.
00:14:37.525 --> 00:14:38.150
PROFESSOR: Yes.
00:14:38.150 --> 00:14:41.024
AUDIENCE: So you
said at the beginning
00:14:41.024 --> 00:14:43.419
that something like
an infinite density
00:14:43.419 --> 00:14:46.772
is just an effect
of our equation.
00:14:46.772 --> 00:14:50.010
Aren't a lot of theories
like inflation come out
00:14:50.010 --> 00:14:53.300
of assuming that the
universe had gotten
00:14:53.300 --> 00:14:57.310
some infinite,
dense, small area?
00:14:57.310 --> 00:15:02.053
So what effect does this
assumption that may or may not
00:15:02.053 --> 00:15:04.920
be correct have on theories?
00:15:04.920 --> 00:15:08.180
PROFESSOR: OK, the
question is, if we are not
00:15:08.180 --> 00:15:11.180
sure we should believe the
t equals zero singularity,
00:15:11.180 --> 00:15:13.660
how does that affect other
theories like inflation, which
00:15:13.660 --> 00:15:17.740
are based on extrapolating
backwards to very early times.
00:15:17.740 --> 00:15:20.110
And there is an answer to that.
00:15:20.110 --> 00:15:22.275
The answer may or may
not sound sensible,
00:15:22.275 --> 00:15:24.030
and it may or may
not be sensible.
00:15:24.030 --> 00:15:26.240
It's hard to know for sure.
00:15:26.240 --> 00:15:29.440
But one can be more
detailed and ask
00:15:29.440 --> 00:15:32.590
how far back do we think
we can trust our equations?
00:15:32.590 --> 00:15:34.090
And nobody really
knows the answer
00:15:34.090 --> 00:15:37.380
to that of course, that's
part of the uncertainty here.
00:15:37.380 --> 00:15:42.050
But a plausible answer which is
kind of the working hypothesis
00:15:42.050 --> 00:15:45.880
for many of us, is that the
only obstacle to extrapolating
00:15:45.880 --> 00:15:48.320
backwards is our
lack of knowledge
00:15:48.320 --> 00:15:51.125
of the quantum
effects of gravity,
00:15:51.125 --> 00:15:52.500
and therefore the
quantum effects
00:15:52.500 --> 00:15:55.340
of what space time looks like.
00:15:55.340 --> 00:15:57.930
And we can estimate
where that sets in.
00:15:57.930 --> 00:16:01.870
And it's at a time called the
Planck time, which is about 10
00:16:01.870 --> 00:16:06.010
to the minus 43
seconds, and inflation,
00:16:06.010 --> 00:16:08.880
which we'll see later as
sort of a natural timescale
00:16:08.880 --> 00:16:12.640
of about probably 10 to
the minus 37 seconds.
00:16:12.640 --> 00:16:14.976
So although that's
incredibly small,
00:16:14.976 --> 00:16:16.600
it's actually incredibly
large compared
00:16:16.600 --> 00:16:19.390
to 10 to the minus 43 seconds.
00:16:19.390 --> 00:16:23.810
So we think there is at
least a basis for believing
00:16:23.810 --> 00:16:26.450
that things like our discussions
of inflation which we'll
00:16:26.450 --> 00:16:30.050
be talking about later, are
valid even though we don't
00:16:30.050 --> 00:16:32.510
think we can extrapolate
back to t equals zero.
00:16:35.410 --> 00:16:35.910
Yes.
00:16:35.910 --> 00:16:38.368
AUDIENCE: I have a
question about use
00:16:38.368 --> 00:16:39.600
of the definite integrals.
00:16:39.600 --> 00:16:40.225
PROFESSOR: Yes.
00:16:40.225 --> 00:16:42.872
AUDIENCE: So we have
a twiddle defined
00:16:42.872 --> 00:16:44.655
as a over square root of k.
00:16:44.655 --> 00:16:47.075
And we noticed in
our equation that a
00:16:47.075 --> 00:16:53.180
goes to zero as t goes to zero,
so a twiddle also goes to zero.
00:16:53.180 --> 00:16:55.692
How do we know then that
that definite integral
00:16:55.692 --> 00:16:57.150
is convergent,
because then we have
00:16:57.150 --> 00:17:01.640
zero over zero competing
case. [INAUDIBLE]
00:17:01.640 --> 00:17:06.390
PROFESSOR: Let me think.
00:17:09.676 --> 00:17:11.300
Yeah, we'll see, I
think is the answer.
00:17:11.300 --> 00:17:12.349
How do we know it's convergent?
00:17:12.349 --> 00:17:13.940
Well, we're going to
actually do the integral,
00:17:13.940 --> 00:17:15.599
and then the integral
does converge.
00:17:15.599 --> 00:17:16.310
You are right.
00:17:16.310 --> 00:17:20.030
The integrand does become zero
over zero, but that in fact
00:17:20.030 --> 00:17:24.430
means the integrand is some
finite number actually.
00:17:24.430 --> 00:17:27.839
Both numerator and
denominator go to zero,
00:17:27.839 --> 00:17:29.770
I mean let me think about this.
00:17:29.770 --> 00:17:31.570
I guess a tilde squared
becomes negligible,
00:17:31.570 --> 00:17:34.760
so the denominator goes like
one over the square root
00:17:34.760 --> 00:17:35.650
of a tilde.
00:17:35.650 --> 00:17:38.570
So in fact the numerator
divided by the denominator
00:17:38.570 --> 00:17:41.567
goes like the square root of
a tilde as time goes to zero.
00:17:41.567 --> 00:17:43.650
Because you have a square
root in the denominator,
00:17:43.650 --> 00:17:46.130
the a tilde squared
becomes negligible,
00:17:46.130 --> 00:17:48.710
so it's manifestly convergent.
00:17:48.710 --> 00:17:51.210
The integrand does
not even blow up,
00:17:51.210 --> 00:17:54.564
even though a tilde
does go to zero.
00:17:54.564 --> 00:17:56.480
It's certainly worth
looking at, you're right.
00:17:56.480 --> 00:17:58.880
One should always check to make
sure integrals are convergent.
00:17:58.880 --> 00:18:01.379
But since we will actually be
explicitly doing the interval,
00:18:01.379 --> 00:18:03.890
if it was divergent we would
get a divergent answer,
00:18:03.890 --> 00:18:08.357
and we will not as you'll
see in a couple minutes.
00:18:08.357 --> 00:18:09.190
Any other questions?
00:18:11.960 --> 00:18:15.525
OK, so in that case,
to the blackboard.
00:18:36.770 --> 00:18:38.340
OK, so I will write
on the blackboard
00:18:38.340 --> 00:18:42.150
the same equation we have up
there, so we can continue.
00:18:42.150 --> 00:18:48.240
t twiddle f is equal to
the integral from zero
00:18:48.240 --> 00:18:56.940
to a twiddle sub f,
a twiddle d a twiddle
00:18:56.940 --> 00:19:02.870
over the square
root of two alpha
00:19:02.870 --> 00:19:07.655
a twiddle minus a
twiddle squared.
00:19:12.480 --> 00:19:14.290
OK, so what we'd
like to do now is
00:19:14.290 --> 00:19:17.246
to carry out the integral
on the right hand side.
00:19:17.246 --> 00:19:19.620
Ideally what we'd like to do
is to carry out the integral
00:19:19.620 --> 00:19:26.486
on the right hand side and
get some function of a tilde f
00:19:26.486 --> 00:19:28.490
and then we'd like to
convert that function
00:19:28.490 --> 00:19:31.240
to be able to write a tilde
f as a function of time.
00:19:31.240 --> 00:19:33.550
We actually won't quite
be able to do that.
00:19:33.550 --> 00:19:36.130
We'll end up with what's
called the parametric solution,
00:19:36.130 --> 00:19:38.850
and you'll see how that
arises and what that means.
00:19:38.850 --> 00:19:40.790
I don't need to try
to describe exactly
00:19:40.790 --> 00:19:43.580
in advance what that means.
00:19:43.580 --> 00:19:46.980
Doing the integral can
be done by some tricks,
00:19:46.980 --> 00:19:48.480
some substitutions.
00:19:48.480 --> 00:19:51.990
And the first
substitution is based
00:19:51.990 --> 00:19:56.610
on completing the square
in the denominator,
00:19:56.610 --> 00:19:59.180
and that motivates the
substitution that we will make.
00:19:59.180 --> 00:20:03.070
So we can rewrite this
just by doing some algebra
00:20:03.070 --> 00:20:05.210
on the denominator, which
is called completing
00:20:05.210 --> 00:20:06.820
the square for
reasons that you'll
00:20:06.820 --> 00:20:09.270
see when I write
down what it is.
00:20:09.270 --> 00:20:13.840
The numerator will
stay a tilde d a tilde.
00:20:13.840 --> 00:20:18.370
And the denominator
can be written
00:20:18.370 --> 00:20:28.572
as alpha squared minus a tilde
minus alpha quantity squared.
00:20:28.572 --> 00:20:30.030
So completing the
square just means
00:20:30.030 --> 00:20:34.040
to put the a tilde
inside a perfect square.
00:20:34.040 --> 00:20:35.980
And the nice thing about
this is that now we
00:20:35.980 --> 00:20:38.160
can shift our variable
of integration,
00:20:38.160 --> 00:20:40.790
and turn this into just
a single variable instead
00:20:40.790 --> 00:20:42.090
of the sum of the two.
00:20:42.090 --> 00:20:43.673
And then you have
an expression, which
00:20:43.673 --> 00:20:45.900
is clearly simpler looking
than this one, which
00:20:45.900 --> 00:20:48.130
had a tildes in both places.
00:20:48.130 --> 00:20:50.760
Now the variable integration
will appear only there.
00:20:50.760 --> 00:20:54.360
And substitution which does
that obviously enough, is we're
00:20:54.360 --> 00:20:57.240
going to let something, we
can choose whatever we want,
00:20:57.240 --> 00:20:58.830
and then once I call
it x, we're going
00:20:58.830 --> 00:21:03.910
to let x equal a
tilde minus alpha.
00:21:07.200 --> 00:21:11.560
And we're just going to rewrite
that integral in terms of x.
00:21:11.560 --> 00:21:15.070
So what we get when
we do that is t
00:21:15.070 --> 00:21:21.030
sub f tilde is equal to the
rewriting of that integral,
00:21:21.030 --> 00:21:28.110
and just substituting the a
tilde becomes x plus alpha.
00:21:30.900 --> 00:21:34.240
So we get x plus alpha
where a tilde was,
00:21:34.240 --> 00:21:36.795
and d a tilde becomes just d x.
00:21:40.060 --> 00:21:42.520
And the denominator,
which was our motivation
00:21:42.520 --> 00:21:44.850
for making the substitution
in the first place,
00:21:44.850 --> 00:21:47.890
becomes just alpha
squared minus x squared.
00:21:55.950 --> 00:21:57.806
So this is perfectly
straight forward.
00:21:57.806 --> 00:21:59.180
The next step
which is important,
00:21:59.180 --> 00:22:01.200
is to get the limits
of integration right,
00:22:01.200 --> 00:22:03.040
because with this
definite integral method
00:22:03.040 --> 00:22:05.456
we really have to make sure
that our limits of integration
00:22:05.456 --> 00:22:06.250
are correct.
00:22:06.250 --> 00:22:10.570
So straightforward to do that,
the lower limit of integration
00:22:10.570 --> 00:22:16.130
was a tilde equals zero,
and if a tilde equals zero,
00:22:16.130 --> 00:22:18.560
x is equal to minus alpha.
00:22:18.560 --> 00:22:20.180
So the lower limit
of integration
00:22:20.180 --> 00:22:22.770
expressed as a value
of x is minus alpha.
00:22:25.600 --> 00:22:28.770
And the upper limit
expressed as x
00:22:28.770 --> 00:22:31.890
was a tilde f, and
that means x is
00:22:31.890 --> 00:22:35.390
equal to a tilde f minus alpha.
00:22:35.390 --> 00:22:39.167
So the limit here is a
tilde sub f minus alpha
00:22:39.167 --> 00:22:40.625
for the upper limit
of integration.
00:22:47.680 --> 00:22:51.980
OK, now this integral
is still not easy,
00:22:51.980 --> 00:22:56.360
but it can be made easy
by one more substitution.
00:22:56.360 --> 00:22:59.500
And the one more substitution
is a trigonometric substitution
00:22:59.500 --> 00:23:02.620
to simplify the denominator.
00:23:02.620 --> 00:23:09.970
We can let x equal
minus alpha cosine
00:23:09.970 --> 00:23:15.410
of theta, where theta is our
new variable of integration.
00:23:15.410 --> 00:23:17.550
And then alpha squared
minus x squared
00:23:17.550 --> 00:23:19.250
becomes alpha
squared minus alpha
00:23:19.250 --> 00:23:20.890
squared cosine squared theta.
00:23:20.890 --> 00:23:23.348
And the alphas factor out and
you have the square root of 1
00:23:23.348 --> 00:23:24.560
minus cosine squared theta.
00:23:24.560 --> 00:23:27.559
1 minus cosine squared theta
is sine squared theta, which
00:23:27.559 --> 00:23:29.600
is a convenient thing to
take the square root of,
00:23:29.600 --> 00:23:30.860
you just get sine theta.
00:23:30.860 --> 00:23:33.190
And everything else
also simplifies.
00:23:33.190 --> 00:23:36.230
And the bottom line, which
I will just give you,
00:23:36.230 --> 00:23:49.380
is that now we find that t sub s
tilde is just the integral of 1
00:23:49.380 --> 00:23:55.040
minus cosine theta d theta.
00:23:57.890 --> 00:23:58.490
That's it.
00:23:58.490 --> 00:24:01.930
Everything simplifies to that
which is easy to integrate.
00:24:01.930 --> 00:24:05.880
We also have to keep track
of our limits of integration.
00:24:05.880 --> 00:24:09.850
The limit of integration, the
lower limit of integration,
00:24:09.850 --> 00:24:12.790
is where x equals minus alpha.
00:24:12.790 --> 00:24:15.410
And if x equals
minus alpha, that
00:24:15.410 --> 00:24:18.400
means cosine theta equals 1.
00:24:18.400 --> 00:24:22.060
And cosine theta equals
1 means theta equals 0.
00:24:22.060 --> 00:24:27.710
So the limit of integration on
theta is easy, it starts at 0.
00:24:27.710 --> 00:24:30.370
And the final value
is really just a value
00:24:30.370 --> 00:24:35.130
that corresponds to
the final value of x
00:24:35.130 --> 00:24:39.350
which is a twiddle
f minus alpha.
00:24:39.350 --> 00:24:41.730
For now I'm just going
to call it theta sub f.
00:24:49.732 --> 00:24:51.190
Things that we have
just determined
00:24:51.190 --> 00:24:53.560
is the value of theta that goes
with the upper limit there.
00:24:53.560 --> 00:24:56.101
We'll figure out in a second
how to write it more explicitly.
00:24:56.101 --> 00:24:58.810
Let me first do the integral to
just get that out of the way.
00:24:58.810 --> 00:25:06.410
Doing the integral,
you get alpha times 1
00:25:06.410 --> 00:25:09.935
minus cosine theta sub f.
00:25:19.100 --> 00:25:22.560
So this in fact becomes
half of our solution.
00:25:22.560 --> 00:25:27.062
It expresses t sub f in
terms of theta sub f.
00:25:27.062 --> 00:25:29.020
And now that we're done
with the whole problem,
00:25:29.020 --> 00:25:30.880
I'm going to drop
the subscript f.
00:25:30.880 --> 00:25:32.520
There's just some
time that we're
00:25:32.520 --> 00:25:35.852
interested in which is called t
and the value of a at that time
00:25:35.852 --> 00:25:38.060
will be called a, and the
value of theta at that time
00:25:38.060 --> 00:25:39.010
will be called theta.
00:25:39.010 --> 00:25:41.430
So I'm just going to drop
the subscript everywhere
00:25:41.430 --> 00:25:44.740
because we are now in a
situation were the subscript is
00:25:44.740 --> 00:25:46.580
everywhere, so
dropping everywhere
00:25:46.580 --> 00:25:48.950
does not lose any information.
00:25:48.950 --> 00:25:50.640
So one of our equations
is going to be
00:25:50.640 --> 00:25:56.040
simply t is equal to alpha
times 1 minus cosine theta.
00:25:59.850 --> 00:26:02.660
And another equation will
come from figuring out
00:26:02.660 --> 00:26:04.297
what theta sub f
really is, which
00:26:04.297 --> 00:26:06.380
I said comes from making
sure that the upper limit
00:26:06.380 --> 00:26:08.620
of integration here
corresponds to the upper limit
00:26:08.620 --> 00:26:11.970
of integration in the
previous integral.
00:26:11.970 --> 00:26:14.480
So theta sub f, I
had to determine
00:26:14.480 --> 00:26:17.755
this on another blackboard and
then put the final equations
00:26:17.755 --> 00:26:18.255
together.
00:26:18.255 --> 00:26:19.129
AUDIENCE: [INAUDIBLE]
00:26:25.010 --> 00:26:28.460
PROFESSOR: Yes, that's right
I don't want to drop twiddles.
00:26:28.460 --> 00:26:32.490
T twiddle, thank you, is
equal to alpha times 1
00:26:32.490 --> 00:26:44.390
minus cosine theta, and and then
we have the final value of x.
00:26:44.390 --> 00:26:50.790
Xx x is equal to a
tilde minus alpha.
00:26:50.790 --> 00:26:54.540
So the final value of x is
equal to the final value
00:26:54.540 --> 00:26:56.765
of a tilde minus alpha.
00:26:59.720 --> 00:27:04.600
And the final value of x
is also related to theta
00:27:04.600 --> 00:27:06.825
by this equation, which is
what we're trying to get,
00:27:06.825 --> 00:27:10.090
how theta is related
to the other variables.
00:27:10.090 --> 00:27:17.800
So this is equal to minus
alpha cosine of theta sub f.
00:27:23.010 --> 00:27:25.920
And now we might
want to, for example,
00:27:25.920 --> 00:27:30.060
solve this for a tilde sub f,
which just involves looking
00:27:30.060 --> 00:27:32.700
at the right hand
half of the equation,
00:27:32.700 --> 00:27:36.690
bringing this alpha to the
other side making a plus alpha.
00:27:36.690 --> 00:27:42.300
So this implies that a tilde
sub f is equal to alpha times 1
00:27:42.300 --> 00:27:46.442
minus cosine theta sub f.
00:27:46.442 --> 00:27:48.150
So this equation now
just says that theta
00:27:48.150 --> 00:27:50.110
sub f means what it
should mean to give us
00:27:50.110 --> 00:27:52.510
a final limit of
integration that corresponds
00:27:52.510 --> 00:27:53.885
to the final limit
of integration
00:27:53.885 --> 00:27:57.560
on our original integral,
which we called a tilde sub f.
00:27:57.560 --> 00:27:58.060
Yes.
00:27:58.060 --> 00:27:59.976
AUDIENCE: Sorry,
perhaps I missed it.
00:27:59.976 --> 00:28:01.892
But when you are doing
the integral of 1
00:28:01.892 --> 00:28:04.121
minus cosine theta, how come--
00:28:04.121 --> 00:28:05.370
PROFESSOR: Oh, I got it wrong.
00:28:08.110 --> 00:28:08.710
Good point.
00:28:12.920 --> 00:28:14.430
The integral of
cosine theta d theta
00:28:14.430 --> 00:28:16.855
is sine theta, not cosine theta.
00:28:16.855 --> 00:28:17.737
AUDIENCE: [INAUDIBLE]
00:28:22.189 --> 00:28:23.230
PROFESSOR: Wait a minute.
00:28:23.230 --> 00:28:23.730
Hold on.
00:28:23.730 --> 00:28:25.950
Do I still have it wrong?
00:28:25.950 --> 00:28:28.500
AUDIENCE: [INAUDIBLE]
00:28:28.500 --> 00:28:29.977
PROFESSOR: OK hold on.
00:28:29.977 --> 00:28:31.310
There was an alpha missing here.
00:28:31.310 --> 00:28:34.080
That's part of the
problem, coming
00:28:34.080 --> 00:28:38.610
from the original equation here.
00:28:38.610 --> 00:28:40.360
Let's see if I have
this right now.
00:28:55.767 --> 00:28:58.380
This is copying from a
long line altogether.
00:28:58.380 --> 00:29:00.020
So I got both factors wrong.
00:29:03.000 --> 00:29:04.950
And then wait a minute.
00:29:12.640 --> 00:29:15.689
I think this is right now.
00:29:15.689 --> 00:29:17.105
There's still a
wrong [INAUDIBLE]?
00:29:17.105 --> 00:29:17.979
AUDIENCE: [INAUDIBLE]
00:29:27.860 --> 00:29:30.220
PROFESSOR: If I differentiate
sine, I get cosine.
00:29:30.220 --> 00:29:33.200
So if I differentiate minus
sine, I get minus cosine.
00:29:33.200 --> 00:29:34.890
So differentiating
this, I get this.
00:29:34.890 --> 00:29:36.973
That should mean that
integrating this I get that.
00:29:36.973 --> 00:29:38.872
I think I have it right.
00:29:38.872 --> 00:29:40.080
Sometimes I get things right.
00:29:40.080 --> 00:29:41.486
It's a surprise, but--
00:29:41.486 --> 00:29:42.360
AUDIENCE: [INAUDIBLE]
00:29:42.360 --> 00:29:43.026
PROFESSOR: What?
00:29:43.026 --> 00:29:45.100
AUDIENCE: The last
equation you wrote.
00:29:45.100 --> 00:29:47.433
PROFESSOR: The last equation
needs to be changed, right.
00:29:47.433 --> 00:29:49.880
This was copied
from that, right.
00:29:49.880 --> 00:29:52.580
Thanks for reminding me.
00:29:52.580 --> 00:29:59.520
So t tilde is equal to alpha
times theta minus sine theta.
00:30:05.280 --> 00:30:08.290
OK, and now I was working
out the relationship
00:30:08.290 --> 00:30:13.390
between theta and a tilde.
00:30:13.390 --> 00:30:15.177
And let me just remind
you that all I did
00:30:15.177 --> 00:30:17.260
was make sure that the
upper limits of integration
00:30:17.260 --> 00:30:18.770
correspond to each other.
00:30:18.770 --> 00:30:21.380
So I'm just basically rewriting
the upper limit of integration
00:30:21.380 --> 00:30:23.690
in terms of the new
variable each time
00:30:23.690 --> 00:30:26.810
when we change variables
starting from a tilde going
00:30:26.810 --> 00:30:30.460
to a tilde to x,
and from x to theta.
00:30:30.460 --> 00:30:32.265
And then I use the
equality of these two
00:30:32.265 --> 00:30:35.270
to write a tilde
in terms of theta.
00:30:38.750 --> 00:30:43.480
And these equations now hold
with the subscript f present.
00:30:43.480 --> 00:30:45.600
When subscript f's appear
everywhere we can just
00:30:45.600 --> 00:30:48.620
drop them and say that
the time that we called
00:30:48.620 --> 00:30:51.890
t sub f now we're
just going to call t.
00:30:51.890 --> 00:30:54.400
And now we could put together
our two final results
00:30:54.400 --> 00:30:56.830
which are maybe right over here.
00:30:56.830 --> 00:30:58.570
We have ct.
00:30:58.570 --> 00:31:01.780
I'll eliminate my
tildes altogether now.
00:31:01.780 --> 00:31:04.840
ct, which is t tilde, is
equal to alpha times theta
00:31:04.840 --> 00:31:06.000
minus sine theta.
00:31:16.540 --> 00:31:21.380
And a divided by the
square root of k,
00:31:21.380 --> 00:31:27.040
which is a tilde and its sub f,
but we're dropping the sub f,
00:31:27.040 --> 00:31:32.580
is equal to alpha times
1 minus cosine theta.
00:31:43.130 --> 00:31:46.700
And this is as good as it gets.
00:31:46.700 --> 00:31:49.560
Ideally, it would
be nice if we could
00:31:49.560 --> 00:31:53.040
solve the top equation for
theta as a function of t,
00:31:53.040 --> 00:31:55.120
and plug that into
the bottom equation,
00:31:55.120 --> 00:31:57.330
and then we would get
a as a function of t.
00:31:57.330 --> 00:32:00.030
That's what ideally
we would love to have.
00:32:00.030 --> 00:32:02.460
But there's no way to
do that analytically.
00:32:02.460 --> 00:32:05.790
In principle of course, this
equation can be inverted.
00:32:05.790 --> 00:32:07.460
You could do it numerically.
00:32:07.460 --> 00:32:09.432
For any particular
value of t you
00:32:09.432 --> 00:32:11.640
could figure out what value
of theta makes this work,
00:32:11.640 --> 00:32:13.940
and then plug that value
of theta into here.
00:32:13.940 --> 00:32:15.440
But there's no
analytic way that you
00:32:15.440 --> 00:32:16.898
can write theta as
a function of t.
00:32:16.898 --> 00:32:19.470
It's not a soluble equation.
00:32:19.470 --> 00:32:21.760
So this is called a parametric
solution in the sense
00:32:21.760 --> 00:32:24.930
that theta is a parameter.
00:32:24.930 --> 00:32:29.100
And as theta varies, both t and
a vary in just the right way
00:32:29.100 --> 00:32:31.970
so that a is always related
to t in the correct way
00:32:31.970 --> 00:32:35.930
to solve our original
differential equation.
00:32:35.930 --> 00:32:39.190
That's what's meant by
a parametric solution.
00:32:39.190 --> 00:32:40.937
We can also see
from this equation,
00:32:40.937 --> 00:32:43.270
or maybe from thinking back
about how things are defined
00:32:43.270 --> 00:32:45.510
in the first place,
how theta varies
00:32:45.510 --> 00:32:49.230
over the lifetime of
our model universe.
00:32:49.230 --> 00:32:52.190
Theta we discovered starts at 0.
00:32:52.190 --> 00:32:59.000
We discovered that when we
wrote our integral over there.
00:32:59.000 --> 00:33:02.550
And we could also
probably see it from here.
00:33:02.550 --> 00:33:04.900
We start our universe
at a equals 0.
00:33:04.900 --> 00:33:08.560
And at a equals 0, we want
1 minus cosine theta to be 0
00:33:08.560 --> 00:33:10.400
and theta equals 0 does that.
00:33:10.400 --> 00:33:13.650
So theta starts at 0, which
corresponds to a equals 0,
00:33:13.650 --> 00:33:16.470
and it also corresponds
putting theta equals 0 here
00:33:16.470 --> 00:33:17.414
to t equals 0.
00:33:17.414 --> 00:33:18.830
So we have arranged
things the way
00:33:18.830 --> 00:33:21.180
we intended so that
a equals 0 happens
00:33:21.180 --> 00:33:24.260
the same time t
equals 0 happens.
00:33:24.260 --> 00:33:27.090
And then theta starts to grow.
00:33:27.090 --> 00:33:30.700
As theta grows, a
increases, the universe
00:33:30.700 --> 00:33:34.270
gets bigger until
theta reaches pi.
00:33:34.270 --> 00:33:37.930
And when theta reaches pi,
cosine of pi is minus 1,
00:33:37.930 --> 00:33:40.630
you get a factor
of 2 here, 2 alpha.
00:33:40.630 --> 00:33:42.740
That's as big as
our universe gets.
00:33:42.740 --> 00:33:46.360
And then as theta
continues beyond that,
00:33:46.360 --> 00:33:49.620
1 minus cosine theta starts
getting smaller again.
00:33:49.620 --> 00:33:51.330
So our universe
reaches a maximum size
00:33:51.330 --> 00:33:54.470
when theta equals pi, and
then starts to contract.
00:33:54.470 --> 00:33:56.900
And then by the time
theta equals 2 pi,
00:33:56.900 --> 00:33:58.930
you are back to where
you started from,
00:33:58.930 --> 00:34:00.530
a is again equal to 0.
00:34:00.530 --> 00:34:03.760
We have universe which starts at
0 size, goes to a maximum size,
00:34:03.760 --> 00:34:08.179
goes back to 0 size, giving
a Big Crunch at the end.
00:34:08.179 --> 00:34:10.555
And that's the way this
closed universe behaves.
00:34:13.480 --> 00:34:17.739
It turns out that these
equations actually
00:34:17.739 --> 00:34:19.343
correspond to some
simple geometry.
00:34:21.850 --> 00:34:24.929
It corresponds to a cycloid.
00:34:24.929 --> 00:34:30.480
And you may or may not
remember what a cycloid is.
00:34:30.480 --> 00:34:33.510
There are the equations
written on the screen, which
00:34:33.510 --> 00:34:37.050
are hopefully the same equations
that I have on the blackboard.
00:34:37.050 --> 00:34:39.130
Can't always count on
these things it turns out.
00:34:39.130 --> 00:34:42.820
But yeah, they are the same
equations, that's healthy.
00:34:42.820 --> 00:34:45.290
And I have a diagram
here which explains
00:34:45.290 --> 00:34:49.909
this cycloid correspondence.
00:34:49.909 --> 00:34:58.600
A cycloid is defined as what
happens in this picture.
00:34:58.600 --> 00:35:00.000
Let me explain the picture.
00:35:00.000 --> 00:35:03.660
We have a disk shown
in the upper left
00:35:03.660 --> 00:35:07.210
in its original position
with a dot on the disk, which
00:35:07.210 --> 00:35:09.040
is initially at the bottom.
00:35:09.040 --> 00:35:11.060
And initially we're
going to put this disk
00:35:11.060 --> 00:35:12.720
at the origin of our
coordinate system
00:35:12.720 --> 00:35:15.440
to make things as
simple as possible.
00:35:15.440 --> 00:35:18.450
And then we're going to
imagine that this disk rolls
00:35:18.450 --> 00:35:21.440
without slipping to the right.
00:35:21.440 --> 00:35:25.900
And the path that
this dot traces out,
00:35:25.900 --> 00:35:29.780
which is shown along that line,
is a cycloid by definition.
00:35:29.780 --> 00:35:31.850
That's what defines a cycloid.
00:35:31.850 --> 00:35:41.000
It's the path that a point on
a rolling disk evolved through.
00:35:41.000 --> 00:35:42.980
So what I would like
to do is convince you
00:35:42.980 --> 00:35:47.470
that this geometric picture
corresponds to those equations.
00:35:47.470 --> 00:35:49.720
Let me put the equations
higher to make sure everybody
00:35:49.720 --> 00:35:50.261
can see them.
00:35:54.560 --> 00:35:56.580
And it's actually
not so complicated
00:35:56.580 --> 00:36:00.190
once you figure out how
to parse the pictures.
00:36:00.190 --> 00:36:03.960
So what I've drawn in the upper
right is a blow up of this disk
00:36:03.960 --> 00:36:07.010
after it's rolled
through some angle theta.
00:36:07.010 --> 00:36:09.810
And I even made it the same
angle theta in the two cases.
00:36:09.810 --> 00:36:11.830
But this is just
a bigger version
00:36:11.830 --> 00:36:15.405
of what's in the corner there
showing the disk after it
00:36:15.405 --> 00:36:16.800
rolled a little bit.
00:36:16.800 --> 00:36:19.010
So after it's rolled
through an angle theta, what
00:36:19.010 --> 00:36:22.070
we want to verify is that
the horizontal and vertical
00:36:22.070 --> 00:36:24.840
coordinates here correspond
to these two equations.
00:36:24.840 --> 00:36:27.150
And if they do, it means
that we're tracing out
00:36:27.150 --> 00:36:30.250
the behavior of
those two equations.
00:36:30.250 --> 00:36:32.460
So look first at the
horizontal component.
00:36:32.460 --> 00:36:35.270
The horizontal component
is the ct axis.
00:36:35.270 --> 00:36:37.340
So that should correspond
to alpha times theta
00:36:37.340 --> 00:36:40.650
minus alpha times sine theta.
00:36:40.650 --> 00:36:42.150
And you can see
that in the picture.
00:36:42.150 --> 00:36:44.200
We're talking about the
horizontal component
00:36:44.200 --> 00:36:47.550
of the coordinates
of this point P.
00:36:47.550 --> 00:36:49.310
And the point is
that you can get
00:36:49.310 --> 00:36:51.864
to P starting at the origin.
00:36:51.864 --> 00:36:54.030
Now remember we're only
looking at horizontal motion
00:36:54.030 --> 00:36:57.120
so we could start
anywhere on that line.
00:36:57.120 --> 00:37:00.540
We could start at the origin,
go to the right by alpha theta,
00:37:00.540 --> 00:37:03.960
and to the left by alpha times
sine theta, and we get there.
00:37:03.960 --> 00:37:06.920
And that's exactly
what this formula says.
00:37:06.920 --> 00:37:09.740
So if we just understand the
alpha theta and the alpha sine
00:37:09.740 --> 00:37:14.210
theta of those two
lines we have it made.
00:37:14.210 --> 00:37:17.180
So let's look first
at what happens where
00:37:17.180 --> 00:37:20.900
this first arrow comes
from, the alpha theta line.
00:37:20.900 --> 00:37:26.030
That's just the total distance
that the point of contact
00:37:26.030 --> 00:37:29.060
has moved during
the rolling process.
00:37:29.060 --> 00:37:31.690
And the claim is that
as something rolls,
00:37:31.690 --> 00:37:34.770
really the definition of
rolling without slipping,
00:37:34.770 --> 00:37:38.590
is that the arc length that
is swept out by the rolling
00:37:38.590 --> 00:37:41.510
is the same as the length
along the surface on which it's
00:37:41.510 --> 00:37:42.810
rolling.
00:37:42.810 --> 00:37:45.390
You could imagine, if
you like, that as it
00:37:45.390 --> 00:37:47.010
rolls there's a
tape measure that's
00:37:47.010 --> 00:37:50.840
wrapped around it that gets
left on the ground as it rolls.
00:37:50.840 --> 00:37:52.620
And if you could
picture that happening,
00:37:52.620 --> 00:37:54.750
the existence of
that movie really
00:37:54.750 --> 00:37:57.260
guarantees that the
length on the ground
00:37:57.260 --> 00:38:01.170
is the same as what the length
was when the tape measure was
00:38:01.170 --> 00:38:04.410
wrapped around the cylinder.
00:38:04.410 --> 00:38:09.740
So the length that the
point of contact has moved
00:38:09.740 --> 00:38:11.930
is just alpha times
the angle through which
00:38:11.930 --> 00:38:13.470
the disk is rolled.
00:38:13.470 --> 00:38:17.840
So that explains the alpha
theta label on that line.
00:38:17.840 --> 00:38:22.170
To get the alpha sine
theta on the line above,
00:38:22.170 --> 00:38:24.790
that's the distance
between the point P
00:38:24.790 --> 00:38:29.500
and a vertical line that goes
through the center of the disk.
00:38:29.500 --> 00:38:33.820
And that's just simple
trigonometry on this triangle.
00:38:33.820 --> 00:38:35.725
The radius of our
circle is alpha.
00:38:39.250 --> 00:38:42.940
And then by simple
trigonometry, this length
00:38:42.940 --> 00:38:47.650
is alpha times sine theta,
which is what the label says.
00:38:47.650 --> 00:38:49.650
So to summarize
what we've got here
00:38:49.650 --> 00:38:51.690
we can find the x-coordinate
of the horizontal
00:38:51.690 --> 00:38:53.510
according to the
point P by going
00:38:53.510 --> 00:38:55.220
to the right a
distance alpha theta,
00:38:55.220 --> 00:38:59.770
and then back to the left the
distance of minus alpha theta.
00:38:59.770 --> 00:39:02.170
And that gives us
an x-coordinate,
00:39:02.170 --> 00:39:05.430
which is exactly the formula
that appears in the ct formula.
00:39:05.430 --> 00:39:07.190
So ct works.
00:39:07.190 --> 00:39:08.930
The horizontal
component of that dot
00:39:08.930 --> 00:39:12.890
is just where it should be to
trace out the equations that
00:39:12.890 --> 00:39:15.650
describe the evolution
of a closed universe.
00:39:15.650 --> 00:39:18.880
Similarly we can now look
at the vertical components
00:39:18.880 --> 00:39:20.330
of that dot.
00:39:20.330 --> 00:39:22.250
Again it's most easily
seen as the difference
00:39:22.250 --> 00:39:23.830
of two contributions.
00:39:23.830 --> 00:39:27.180
We're trying to reproduce this
formula that says that's alpha
00:39:27.180 --> 00:39:31.100
minus alpha cosine theta.
00:39:31.100 --> 00:39:35.600
So if we start at vertical
coordinate 0, which
00:39:35.600 --> 00:39:38.970
means on the
x-axis, we can begin
00:39:38.970 --> 00:39:41.370
by going up to the
center of the disk.
00:39:41.370 --> 00:39:43.650
And the disk has
radius alpha, so that's
00:39:43.650 --> 00:39:46.490
going up the distance alpha.
00:39:46.490 --> 00:39:51.120
And then we go down the distance
of this piece of the triangle,
00:39:51.120 --> 00:39:53.960
going from the
center of the disk
00:39:53.960 --> 00:39:57.540
to the point which is
parallel to the point P.
00:39:57.540 --> 00:40:00.730
And that again is just
trigonometry on that triangle,
00:40:00.730 --> 00:40:05.160
and it's alpha cosine theta
by simple trigonometry.
00:40:05.160 --> 00:40:07.450
So we can get to the
elevation of point P
00:40:07.450 --> 00:40:11.280
by going up by alpha and down
by alpha times cosine theta.
00:40:11.280 --> 00:40:14.520
And that's exactly that formula.
00:40:14.520 --> 00:40:16.490
So it works.
00:40:16.490 --> 00:40:19.150
The x and y components,
the horizontal
00:40:19.150 --> 00:40:20.720
and vertical
components of that dot,
00:40:20.720 --> 00:40:23.330
are exactly the
two formulas here.
00:40:23.330 --> 00:40:26.223
So one of them can be thought of
as the x-axis, and one of them
00:40:26.223 --> 00:40:27.700
can be thought of as the y-axis.
00:40:27.700 --> 00:40:29.550
And the rolling of the
disk just traces out
00:40:29.550 --> 00:40:33.550
the evolution of
our closed universe.
00:40:33.550 --> 00:40:36.699
So closed universes
evolve like a cycloid.
00:40:36.699 --> 00:40:37.740
Any questions about that?
00:40:41.570 --> 00:40:44.560
OK, great.
00:40:44.560 --> 00:40:50.600
OK, Let me just mention that
this angle theta is sometimes
00:40:50.600 --> 00:40:51.560
given a name.
00:40:51.560 --> 00:40:54.240
It's called the development
angle of the universe
00:40:54.240 --> 00:40:56.280
or of the solution.
00:41:08.620 --> 00:41:11.840
And that is just intended to
have the connotation that theta
00:41:11.840 --> 00:41:15.210
describes how developed
the universe is
00:41:15.210 --> 00:41:17.390
and theta has a fixed scale.
00:41:17.390 --> 00:41:20.340
It always goes from 0 to
2 pi over the lifetime
00:41:20.340 --> 00:41:21.890
of this closed
universe, no matter
00:41:21.890 --> 00:41:25.170
how big the closed
universe might be.
00:41:25.170 --> 00:41:30.070
That brings me to my next
question I want to mention.
00:41:30.070 --> 00:41:34.280
How many parameters do
we have in this solution?
00:41:34.280 --> 00:41:36.190
The way we've
written it, it looks
00:41:36.190 --> 00:41:52.680
like it can depend on both
alpha and k because both of them
00:41:52.680 --> 00:41:54.229
appear in the answer.
00:41:54.229 --> 00:41:56.020
And k is positive for
our closed universes.
00:41:56.020 --> 00:41:58.860
These formulas will not make
sense if k were negative,
00:41:58.860 --> 00:42:00.230
square root of k appears there.
00:42:00.230 --> 00:42:03.200
We don't want anything
to be imaginary.
00:42:03.200 --> 00:42:04.830
But k can have any
value in principle
00:42:04.830 --> 00:42:07.740
and these equations
would still be valid.
00:42:07.740 --> 00:42:10.910
So on the surface it appears
like there's a two parameter
00:42:10.910 --> 00:42:13.710
class of closed
universe solutions.
00:42:13.710 --> 00:42:16.640
But that's actually not true.
00:42:16.640 --> 00:42:19.971
Can somebody tell me
why it's not true?
00:42:19.971 --> 00:42:20.470
Yes.
00:42:20.470 --> 00:42:23.750
AUDIENCE: [INAUDIBLE]
00:42:23.750 --> 00:42:25.490
PROFESSOR: Exactly.
00:42:25.490 --> 00:42:28.770
Yes, since k has units
of 1 over notch squared,
00:42:28.770 --> 00:42:31.370
you can change k to
anything you want
00:42:31.370 --> 00:42:33.610
by changing your
definition of a notch.
00:42:33.610 --> 00:42:37.110
And there's nothing fixed about
the definition of a notch.
00:42:37.110 --> 00:42:39.210
So k is an irrelevant parameter.
00:42:39.210 --> 00:42:42.310
If we change k we're just
rescaling the same solution,
00:42:42.310 --> 00:42:45.170
and not actually
creating a new solution.
00:42:45.170 --> 00:42:47.810
So there's really only one
parameter class of solutions.
00:42:47.810 --> 00:42:50.464
One could, for example,
fix k to always be 1,
00:42:50.464 --> 00:42:51.880
and then we'd have
a one parameter
00:42:51.880 --> 00:42:55.560
class of solutions
indicated by alpha.
00:42:55.560 --> 00:43:02.180
Alpha, unlike k, really does
have a clear, physical meaning
00:43:02.180 --> 00:43:04.135
related to the behavior
of the universe.
00:43:06.890 --> 00:43:21.970
And we could see
what that is if we
00:43:21.970 --> 00:43:25.220
ask, what is the total lifetime
of this universe from beginning
00:43:25.220 --> 00:43:28.330
to end, from Big
Bang to Big Crunch.
00:43:28.330 --> 00:43:32.990
We can answer that by just
looking at the ct equation.
00:43:32.990 --> 00:43:38.370
From Big Bang to Big Crunch, we
know that theta evolves from 0
00:43:38.370 --> 00:43:42.270
to pi back to 2 pi,
which is the same as 0.
00:43:42.270 --> 00:43:44.220
So theta goes through
one cycle of 2 pi
00:43:44.220 --> 00:43:47.780
during a lifetime of
our model universe.
00:43:47.780 --> 00:43:52.210
As theta goes from 0 to 2
pi, sine theta starts at 0
00:43:52.210 --> 00:43:55.450
and eventually comes back
to 0 when theta equals 2 pi.
00:43:55.450 --> 00:44:00.960
But theta increases from
0 to 2 pi over one cycle.
00:44:00.960 --> 00:44:04.060
So over one cycle
of our universe,
00:44:04.060 --> 00:44:07.950
ct increases by
alpha times 2 pi.
00:44:07.950 --> 00:44:10.470
So that tells us what the
total lifetime of the universe
00:44:10.470 --> 00:44:23.035
is, I'll call it t total.
00:44:28.910 --> 00:44:37.930
And we get it by just writing
c times t total equals 2 pi
00:44:37.930 --> 00:44:38.950
times alpha.
00:44:44.380 --> 00:44:49.950
And I think I can do this one
without making a mistake. t
00:44:49.950 --> 00:44:55.190
total is then 2 pi
alpha divided by c.
00:44:59.030 --> 00:45:00.660
And we can even check
our units there.
00:45:00.660 --> 00:45:05.500
Alpha has units of length,
so length divided by c
00:45:05.500 --> 00:45:07.960
becomes a time, c
being a velocity.
00:45:07.960 --> 00:45:09.450
So it has the right units.
00:45:09.450 --> 00:45:10.790
And that's the total
lifetime of the universe.
00:45:10.790 --> 00:45:12.140
It's just determined by alpha.
00:45:12.140 --> 00:45:14.101
So alpha can be viewed
as just the measure
00:45:14.101 --> 00:45:16.600
of the total lifetime of the
universe, which can be anything
00:45:16.600 --> 00:45:18.141
for different sized
closed universes.
00:45:21.260 --> 00:45:25.330
Alpha is also related to the
maximum value of a over root k.
00:45:25.330 --> 00:45:30.680
And a has no fixed meaning,
this is meters per notch.
00:45:30.680 --> 00:45:33.430
But a over root k does
have units of meters.
00:45:33.430 --> 00:45:35.965
We haven't yet really seen
what it means physically,
00:45:35.965 --> 00:45:38.940
because that's related to the
geometry of the closed universe
00:45:38.940 --> 00:45:41.010
which we'll be discussing later.
00:45:41.010 --> 00:45:43.470
But in any case as
a mathematical fact,
00:45:43.470 --> 00:45:47.490
we could always say that the
maximum value of a over root k
00:45:47.490 --> 00:45:48.765
is determined by alpha.
00:45:53.660 --> 00:46:01.070
So a max over root k is
equal to the maximum value
00:46:01.070 --> 00:46:02.576
of this expression.
00:46:02.576 --> 00:46:04.200
And this expression
has a maximum value
00:46:04.200 --> 00:46:06.760
when theta equals pi,
which gives cosine theta
00:46:06.760 --> 00:46:08.530
equals minus 1,
which gives a 2 here.
00:46:08.530 --> 00:46:10.237
And that's as big
as it ever gets,
00:46:10.237 --> 00:46:11.570
so that's just equal to 2 alpha.
00:46:18.430 --> 00:46:22.650
So alpha is related in a very
clear way to the total lifetime
00:46:22.650 --> 00:46:25.890
of our universe, and is
also related to a over root
00:46:25.890 --> 00:46:27.770
k, although we
haven't really given
00:46:27.770 --> 00:46:29.510
a physical meaning
to a over root k yet.
00:46:29.510 --> 00:46:31.280
But we know it has
dimensions of meters.
00:46:42.080 --> 00:46:44.850
OK, the next
calculation I want to do
00:46:44.850 --> 00:46:48.080
is to calculate the
age of the universe
00:46:48.080 --> 00:46:50.740
as a function of
measurable things.
00:46:50.740 --> 00:46:52.815
We learned for the flat
matter dominated case
00:46:52.815 --> 00:46:54.440
that there was a
simple answer to that.
00:46:54.440 --> 00:46:59.170
The age was just 2/3 times
the inverse Hubble parameter.
00:46:59.170 --> 00:47:02.680
So what you do now is get
the analogous formula here,
00:47:02.680 --> 00:47:05.620
it follows in
principle immediately
00:47:05.620 --> 00:47:08.070
from our description
of the evolution.
00:47:08.070 --> 00:47:10.580
But we have to do a fair
number of substitutions
00:47:10.580 --> 00:47:13.360
before we can really see how
to express the age in terms
00:47:13.360 --> 00:47:16.580
of things that
we're interested in.
00:47:16.580 --> 00:47:20.100
The formula here
tells us directly
00:47:20.100 --> 00:47:22.100
how to express the
age of the universe. t
00:47:22.100 --> 00:47:26.720
is the age of the universe as
a function of alpha and theta.
00:47:26.720 --> 00:47:29.670
But if you tell an astronomer
to go out and measure alpha
00:47:29.670 --> 00:47:32.280
and theta so I could
calculate the age,
00:47:32.280 --> 00:47:34.810
he says what in the world
are alpha and theta.
00:47:34.810 --> 00:47:37.330
So what we'd like to do
is to express the age
00:47:37.330 --> 00:47:39.750
in terms of things that
astronomers know about.
00:47:39.750 --> 00:47:42.090
And the characterizations
of the universe
00:47:42.090 --> 00:47:44.090
like this that an
astronomer would know about
00:47:44.090 --> 00:47:46.830
would be the Hubble
expansion rate,
00:47:46.830 --> 00:47:49.540
and some notion of
the mass density.
00:47:49.540 --> 00:47:52.170
And the easiest way to
talk about the mass density
00:47:52.170 --> 00:47:54.490
is in terms of
omega, the fraction
00:47:54.490 --> 00:47:57.430
of the critical density that
the actual mass density has.
00:47:57.430 --> 00:48:00.169
So our goal is going to be to
manipulate these equations.
00:48:00.169 --> 00:48:01.710
All the information
is already there.
00:48:01.710 --> 00:48:05.020
But our goal will be to
manipulate these equations
00:48:05.020 --> 00:48:09.310
to be able to express the age
t in terms of h and omega.
00:48:41.260 --> 00:48:46.900
OK, so first we need to remind
ourselves what omega is.
00:48:46.900 --> 00:48:49.380
The critical density is
defined as that density
00:48:49.380 --> 00:48:51.060
which makes the universe flat.
00:48:51.060 --> 00:48:53.830
And we've calculated that
the critical density is
00:48:53.830 --> 00:48:58.300
equal to 3 h squared
over 8 pi times newtons
00:48:58.300 --> 00:49:09.530
constant, capital G. We can
then write the mass density rho
00:49:09.530 --> 00:49:13.210
as omega times the
critical density,
00:49:13.210 --> 00:49:14.840
which is just the
definition of omega.
00:49:14.840 --> 00:49:19.130
Omega is rho divided by rho c,
the actual mass density divided
00:49:19.130 --> 00:49:20.780
by the critical density.
00:49:20.780 --> 00:49:25.800
And putting what rho c
is, we can express rho
00:49:25.800 --> 00:49:46.650
as 3h squared omega divided by
8 pi G. And being very pedantic,
00:49:46.650 --> 00:49:48.410
I'm just going to
rewrite that in the form
00:49:48.410 --> 00:49:52.010
that we're actually going to
use it by multiplying through.
00:49:52.010 --> 00:49:57.290
8 pi over 3 G rho,
taking these factors
00:49:57.290 --> 00:49:59.470
and bringing them
to the other side,
00:49:59.470 --> 00:50:02.270
becomes just equal to
h squared times omega.
00:50:06.020 --> 00:50:08.650
And you might recognize
this particular combination
00:50:08.650 --> 00:50:11.370
as what appears in the
Friedmann equation.
00:50:11.370 --> 00:50:17.250
The Friedmann equation told
us that a dot over a squared
00:50:17.250 --> 00:50:27.100
is equal to 8 pi over
3 G rho, minus kc
00:50:27.100 --> 00:50:29.500
squared over a squared.
00:50:40.580 --> 00:50:44.410
And in order to get the
substitutions that I want,
00:50:44.410 --> 00:50:47.030
I'm going to just
rewrite this putting
00:50:47.030 --> 00:50:51.120
h squared for a
dot over a squared.
00:50:51.120 --> 00:50:52.900
8 pi over 3 G rho
we said we could
00:50:52.900 --> 00:50:54.310
write as 8 squared times omega.
00:50:58.080 --> 00:51:01.620
And then we have minus kc
squared over a squared.
00:51:01.620 --> 00:51:04.380
Note that we really have
here a tilde squared.
00:51:04.380 --> 00:51:07.660
This is a squared divided
by k if I put them together.
00:51:07.660 --> 00:51:14.740
So this term can be
written as minus c squared
00:51:14.740 --> 00:51:15.760
over a tilde squared.
00:51:22.766 --> 00:51:24.390
And this accomplishes
one of our goals.
00:51:24.390 --> 00:51:27.580
It allows us to express a tilde
in terms of the quantities
00:51:27.580 --> 00:51:30.960
that we want in our
answers, h and omega.
00:51:30.960 --> 00:51:32.840
And if we can do
the same for theta,
00:51:32.840 --> 00:51:36.530
we have everything we
need to express the age.
00:51:36.530 --> 00:51:43.960
So the implication here
is that a tilde squared
00:51:43.960 --> 00:51:53.775
is equal to c squared divided by
h squared times omega minus 1.
00:52:23.062 --> 00:52:24.770
To take the square
root of that equation,
00:52:24.770 --> 00:52:26.310
to find out what a
tilde is, we need
00:52:26.310 --> 00:52:29.970
to think a little bit about
signs and things like that.
00:52:29.970 --> 00:52:33.040
A tilde is always positive.
00:52:33.040 --> 00:52:35.510
This is the scale factor
divided by the square root of k,
00:52:35.510 --> 00:52:38.070
square root of k is
positive, scale factors
00:52:38.070 --> 00:52:40.340
are always positive
the way we defined it.
00:52:40.340 --> 00:52:41.960
So we can take the
square root of that
00:52:41.960 --> 00:52:44.490
taking the positive square
root of the right hand side.
00:52:44.490 --> 00:52:48.430
Omega is bigger than 1 for
our case, so omega minus 1
00:52:48.430 --> 00:52:51.780
is a positive number, 8
squared is a positive number.
00:52:51.780 --> 00:52:56.120
So taking the square root
there offers no real problem.
00:52:56.120 --> 00:53:00.260
We can write a tilde
is equal to c over,
00:53:00.260 --> 00:53:03.155
I guess this point I might
not notice until later.
00:53:03.155 --> 00:53:05.290
h h can be positive or
negative over the course
00:53:05.290 --> 00:53:06.240
of our calculation.
00:53:06.240 --> 00:53:07.823
We're going to talk
about an expanding
00:53:07.823 --> 00:53:10.190
phase and a contracting phase.
00:53:10.190 --> 00:53:13.250
So when we take the
square root of h squared,
00:53:13.250 --> 00:53:16.250
we want the positive number
to give us a positive a tilde.
00:53:16.250 --> 00:53:18.250
So it's the positive
square root that we want,
00:53:18.250 --> 00:53:21.090
which is the magnitude
of h, not necessarily h.
00:53:21.090 --> 00:53:23.237
When h is positive, the
magnitude of h is h.
00:53:23.237 --> 00:53:25.320
h could be negative though,
and the magnitude of h
00:53:25.320 --> 00:53:28.130
is still positive, and then
the square root of omega
00:53:28.130 --> 00:53:32.420
minus 1, which is
always positive.
00:53:32.420 --> 00:53:35.370
So that's our formula for a
tilde in terms of h and omega.
00:53:45.590 --> 00:53:48.790
OK, now we want
to evaluate alpha,
00:53:48.790 --> 00:53:55.750
and I guess I did not keep
the formula for alpha quite as
00:53:55.750 --> 00:53:58.290
long as we needed it.
00:53:58.290 --> 00:54:00.070
When we defined alpha
in the first place,
00:54:00.070 --> 00:54:26.670
let me remind you how it
was defined, 4 pi over 3 G
00:54:26.670 --> 00:54:35.000
rho a tilde cubed
over c squared.
00:55:01.510 --> 00:55:12.040
And that can be evaluated
using our formula for rho
00:55:12.040 --> 00:55:21.950
and we'll put that in
for rho, and what we get
00:55:21.950 --> 00:55:29.490
is c over 2 times
the magnitude of h
00:55:29.490 --> 00:55:31.116
using this formula
for a tilde as well.
00:55:35.220 --> 00:55:43.820
And then omega over omega
minus 1 to the 3/2 power.
00:55:43.820 --> 00:55:45.380
Just using this
formula, and we know
00:55:45.380 --> 00:55:47.670
how to express rho from
the right hand side,
00:55:47.670 --> 00:55:50.550
and we know how to express a
tilde from this formula here.
00:55:50.550 --> 00:55:53.580
So everything's straightforward,
and this is what we get.
00:55:53.580 --> 00:56:08.550
And now I want to use these
to rewrite this equation,
00:56:08.550 --> 00:56:14.320
a over the square root of
k is equal to alpha times 1
00:56:14.320 --> 00:56:15.630
minus cosine theta.
00:56:22.030 --> 00:56:25.040
I'm going to replace a over
root k by this formula.
00:56:25.040 --> 00:56:27.855
I'm going to replace
alpha by that formula.
00:56:32.360 --> 00:56:36.430
So this implies,
rewriting it, that c
00:56:36.430 --> 00:56:41.640
over the magnitude of h times
the square root of omega
00:56:41.640 --> 00:56:51.300
minus 1 is equal to c over twice
the magnitude of h times omega
00:56:51.300 --> 00:56:57.930
minus 1 to the 3/2 power
times 1 minus cosine theta.
00:57:21.420 --> 00:57:23.530
And now we've had to
survive some boring algebra.
00:57:23.530 --> 00:57:26.750
But notice that now most
things cancel away here.
00:57:26.750 --> 00:57:30.160
We get a very simple
relationship between theta
00:57:30.160 --> 00:57:34.840
and h and omega,
actually just omega .
00:57:34.840 --> 00:57:58.840
In particular,
when we solve that,
00:57:58.840 --> 00:58:01.920
we get simply that
cosine theta is
00:58:01.920 --> 00:58:08.770
equal to two minus
omega over omega.
00:58:08.770 --> 00:58:10.800
So theta is directly
linked to omega.
00:58:10.800 --> 00:58:13.008
If you know omega, you know
theta, If you know theta,
00:58:13.008 --> 00:58:15.169
you know omega, by that formula.
00:58:15.169 --> 00:58:16.710
And we can rewrite
this the other way
00:58:16.710 --> 00:58:19.530
around by solving
for omega if we want.
00:58:19.530 --> 00:58:24.745
Omega is equal to 2 over
1 plus cosine theta.
00:58:33.860 --> 00:58:35.600
Now we can look at
this qualitatively
00:58:35.600 --> 00:58:38.420
to understand how omega
is going to behave.
00:58:38.420 --> 00:58:46.460
At the very beginning
cosine theta is equal to 1,
00:58:46.460 --> 00:58:51.650
theta is equal to 0, so omega
is 2 over 1 plus 1, which is 1.
00:58:51.650 --> 00:58:53.810
So at very early times
omega is driven to 1 even
00:58:53.810 --> 00:58:56.630
in a closed universe.
00:58:56.630 --> 00:59:01.710
As theta gets larger, cosine
theta gets less than 1.
00:59:01.710 --> 00:59:04.060
This then becomes more than 1.
00:59:04.060 --> 00:59:09.490
So omega starts to grow as
the universe starts to evolve.
00:59:09.490 --> 00:59:12.215
At the turning point,
when the universe
00:59:12.215 --> 00:59:16.140
has reached its maximum
size, theta is pi,
00:59:16.140 --> 00:59:19.410
cosine theta is
minus 1, omega is
00:59:19.410 --> 00:59:20.850
infinite at the turning point.
00:59:20.850 --> 00:59:23.220
That may or may
not be a surprise.
00:59:23.220 --> 00:59:25.530
But it you think about
it, it's obvious.
00:59:25.530 --> 00:59:28.710
At the turning point h is 0,
therefore the critical density
00:59:28.710 --> 00:59:31.690
is 0, but the actual
density is not 0.
00:59:31.690 --> 00:59:33.920
And the only way the
actual density can be 0
00:59:33.920 --> 00:59:36.500
while the critical
density is 0 is for omega
00:59:36.500 --> 00:59:39.570
to be infinite, so we
should have expected that.
00:59:39.570 --> 00:59:42.620
And then the return trip,
the collapsing phases,
00:59:42.620 --> 00:59:45.940
a mirror image of
the expanding phase,
00:59:45.940 --> 00:59:48.980
omega goes from infinity at
the turning point back to 1
00:59:48.980 --> 00:59:51.360
at the moment of the Big Crunch.
00:59:51.360 --> 00:59:51.860
Yes.
00:59:51.860 --> 00:59:54.320
AUDIENCE: I'm confused with
what the universe would
00:59:54.320 --> 00:59:56.937
look like when it
gets to infinity.
00:59:56.937 --> 00:59:58.270
PROFESSOR: It would look static.
00:59:58.270 --> 00:59:59.294
It's temporarily static.
00:59:59.294 --> 01:00:00.710
It's reached a
maximum size and is
01:00:00.710 --> 01:00:03.245
about to turn around
and collapse, so h is 0.
01:00:03.245 --> 01:00:06.822
AUDIENCE: OK, but
like with the density.
01:00:06.822 --> 01:00:08.780
PROFESSOR: Well we could
calculate the density.
01:00:08.780 --> 01:00:10.446
It's some number which
depends on alpha.
01:00:10.446 --> 01:00:13.394
AUDIENCE: OK, but that
doesn't diverge or anything.
01:00:13.394 --> 01:00:15.310
PROFESSOR: It doesn't
diverge or anything, no.
01:00:15.310 --> 01:00:17.470
It's just some finite density.
01:00:17.470 --> 01:00:20.454
At the turning point,
it's just a finite density
01:00:20.454 --> 01:00:22.120
that can be expressed
in terms of alpha.
01:00:25.371 --> 01:00:26.870
Sounds like a good
homework problem.
01:00:26.870 --> 01:00:29.750
I think maybe I'll do that.
01:00:34.530 --> 01:00:39.530
OK, so this now allows
us to express theta
01:00:39.530 --> 01:00:42.730
as a function of omega,
which is what we wanted.
01:00:42.730 --> 01:00:45.000
If we express theta as
a function of omega,
01:00:45.000 --> 01:00:48.720
and alpha as a function of
omega and h, we have our answer.
01:00:48.720 --> 01:00:52.910
We have t expressed as a
function of h and theta.
01:00:52.910 --> 01:00:59.470
So there are choices about
how exactly to express omega
01:00:59.470 --> 01:01:01.390
in terms of theta
that will involve
01:01:01.390 --> 01:01:03.100
inverse trigonometric functions.
01:01:03.100 --> 01:01:04.620
And anything that
can be expressed
01:01:04.620 --> 01:01:06.590
in terms of an inverse
cosine, can also
01:01:06.590 --> 01:01:08.640
be expressed as an
inverse sine by doing
01:01:08.640 --> 01:01:10.510
a little bit of manipulations.
01:01:10.510 --> 01:01:13.390
We have our choice here
about what we want to do.
01:01:13.390 --> 01:01:15.910
But in any case,
the answer already
01:01:15.910 --> 01:01:19.860
has a factor of
sine theta in it.
01:01:19.860 --> 01:01:22.020
So it's most useful
to express theta
01:01:22.020 --> 01:01:29.710
as the inverse sine of what
we get from that formula,
01:01:29.710 --> 01:01:31.920
to express the answer
in what at least to me
01:01:31.920 --> 01:01:33.507
is the simplest form.
01:01:33.507 --> 01:01:35.340
So I'm going to manipulate
this a little bit
01:01:35.340 --> 01:01:37.570
to find out what sine
theta is in terms of omega,
01:01:37.570 --> 01:01:41.130
and then invert that to express
theta in terms of omega.
01:01:41.130 --> 01:01:49.240
So sine theta is of
course plus or minus
01:01:49.240 --> 01:01:53.330
the square root of 1 minus
cosine squared theta.
01:01:53.330 --> 01:01:55.280
And cosine theta we
know in terms of omega,
01:01:55.280 --> 01:01:58.080
so we can express this
in terms of omega.
01:01:58.080 --> 01:02:02.110
And if you do that it's
straightforward enough algebra.
01:02:02.110 --> 01:02:05.310
It's plus or minus, depending
on which sign of the square root
01:02:05.310 --> 01:02:08.570
is relevant, and we'll talk
about that in a minute.
01:02:08.570 --> 01:02:11.640
It's plus or minus the
square root of 2 times omega
01:02:11.640 --> 01:02:13.395
minus 1 over omega.
01:02:17.880 --> 01:02:20.634
So we can express theta
as the inverse sine
01:02:20.634 --> 01:02:21.425
of this expression.
01:02:24.720 --> 01:02:27.900
And now what I want to
do is to make use of this
01:02:27.900 --> 01:02:32.650
to put into this expression
using the value for alpha
01:02:32.650 --> 01:02:37.220
that we've already calculated
and wrote over there.
01:02:37.220 --> 01:02:41.440
So we get our
final answer, which
01:02:41.440 --> 01:02:48.350
is that t is equal
to omega over twice
01:02:48.350 --> 01:02:54.050
the magnitude of the Hubble
expansion rate times omega
01:02:54.050 --> 01:03:06.990
minus 1 to the 3/2 power
times the arc sine or inverse
01:03:06.990 --> 01:03:17.890
sine of twice the square root
of omega minus 1 over omega.
01:03:17.890 --> 01:03:21.210
That's just the theta that
appears in this formula written
01:03:21.210 --> 01:03:25.320
in terms of sine theta or the
inverse sine of the quantity
01:03:25.320 --> 01:03:28.010
that we determined
was the sine of theta.
01:03:28.010 --> 01:03:33.520
And then I see here I
should have a plus or minus
01:03:33.520 --> 01:03:35.467
because we haven't
figured out our signs yet.
01:03:35.467 --> 01:03:37.050
Either is actually
possible, depending
01:03:37.050 --> 01:03:38.740
on where we are
in the evolution.
01:03:38.740 --> 01:03:45.870
And then minus or plus twice
the square root of omega
01:03:45.870 --> 01:03:49.430
minus 1 over omega.
01:03:54.740 --> 01:03:57.330
That's a plus or minus,
this is a minus or plus.
01:03:57.330 --> 01:04:00.300
And the reason I wrote one
upstairs and one downstairs
01:04:00.300 --> 01:04:05.760
is that we don't yet really
know how to evaluate theta,
01:04:05.760 --> 01:04:08.689
but the sign of
this term is always
01:04:08.689 --> 01:04:10.980
going to be the negative of
the sign of that term, this
01:04:10.980 --> 01:04:11.900
or that minus sign.
01:04:16.190 --> 01:04:18.680
So theta can be the inverse
sine of this expression
01:04:18.680 --> 01:04:22.070
with either sign of the sine.
01:04:22.070 --> 01:04:24.500
Sorry for the puns.
01:04:24.500 --> 01:04:26.380
But whichever it is,
it's the same on there
01:04:26.380 --> 01:04:30.170
as it is here but with a minus
sign in front, that minus sign.
01:04:33.100 --> 01:04:36.200
OK, so this is our final
formula for the age.
01:04:36.200 --> 01:04:39.530
But we still need to think a
little bit about the s-i-g-n
01:04:39.530 --> 01:04:44.950
signs of these inverse
functions that appear here,
01:04:44.950 --> 01:04:48.050
that and that, and
straightforward if you just
01:04:48.050 --> 01:04:50.560
take it case by case.
01:04:50.560 --> 01:04:52.910
And in the notes we
have a table which
01:04:52.910 --> 01:04:54.880
I'll put shortly on the screen.
01:04:54.880 --> 01:04:58.710
But let's start by just talking
about the earliest phase where
01:04:58.710 --> 01:05:01.850
the universe is shortly
after the Big Bang,
01:05:01.850 --> 01:05:05.315
so that the development
angle is a small angle.
01:05:05.315 --> 01:05:06.690
We know that theta
is going to go
01:05:06.690 --> 01:05:09.760
from 0 to 2 pi over the
lifetime of our universe.
01:05:09.760 --> 01:05:12.030
So I now want to think
of theta being small.
01:05:12.030 --> 01:05:15.800
And small means small compared
to any number you think about.
01:05:15.800 --> 01:05:19.450
So when theta is small the
sine theta is nearly theta.
01:05:19.450 --> 01:05:21.640
Both are positive.
01:05:21.640 --> 01:05:28.360
And in that case the
sine theta being positive
01:05:28.360 --> 01:05:31.650
means this is the positive
root in this equation,
01:05:31.650 --> 01:05:34.570
and therefore the positive
root in that equation.
01:05:34.570 --> 01:05:37.870
So for early times this
would be the plus sign,
01:05:37.870 --> 01:05:40.960
and that would mean that
this would be the minus sign.
01:05:40.960 --> 01:05:43.280
Again the minus sign
just coming from there
01:05:43.280 --> 01:05:45.460
and theta being positive.
01:05:45.460 --> 01:05:49.420
So for early times
it's a plus and minus.
01:05:49.420 --> 01:05:52.540
And the arc sine is
itself ambiguous.
01:05:52.540 --> 01:05:55.270
For early times
the angle we know
01:05:55.270 --> 01:05:57.400
is going to be just a
little bit bigger than 0.
01:05:57.400 --> 01:06:03.026
So that's the evaluation that we
make of the arc sine function.
01:06:03.026 --> 01:06:05.154
Pi plus that would also
give us the same sine.
01:06:05.154 --> 01:06:07.320
It would be another possible
value for the arc sine.
01:06:07.320 --> 01:06:09.450
And of course 2 pi
plus that would also
01:06:09.450 --> 01:06:11.010
be another possible root.
01:06:11.010 --> 01:06:13.560
So you have to know
which root to take
01:06:13.560 --> 01:06:18.450
to know the right answer
here because as an angle,
01:06:18.450 --> 01:06:21.737
0 is equivalent to 2 pi,
but as a time, 0 is not
01:06:21.737 --> 01:06:22.820
at all equivalent to 2 pi.
01:06:22.820 --> 01:06:25.450
So you do have to know
the right one to take.
01:06:25.450 --> 01:06:28.955
We'll continue doing that
on a case by case basis.
01:06:28.955 --> 01:06:34.040
Those are the equations,
that's the formula for the age,
01:06:34.040 --> 01:06:38.970
and that's the formula for the
age with a description of which
01:06:38.970 --> 01:06:42.560
roots to take for each
case, which just comes out
01:06:42.560 --> 01:06:45.580
by following the
evolution, we know
01:06:45.580 --> 01:06:47.770
that theta is going
from 0 to 2 pi,
01:06:47.770 --> 01:06:51.800
and this last column, the
inverse sine of the expression
01:06:51.800 --> 01:06:54.520
which means the expression
that appears here.
01:06:58.220 --> 01:07:00.400
For the smallest angle
is 0 to pi over 2.
01:07:00.400 --> 01:07:03.720
We can think of this actually
as defining our columns.
01:07:03.720 --> 01:07:06.240
Theta starts at 0 so the
time lengths between 0
01:07:06.240 --> 01:07:08.960
and pi over 2, a time
length between pi over 2,
01:07:08.960 --> 01:07:12.520
a time length between
pi and 3 pi over 2,
01:07:12.520 --> 01:07:19.384
and a final time length
between 3 pi over 2 and 2 pi.
01:07:19.384 --> 01:07:22.680
And the first two correspond to
the expanding phase, second two
01:07:22.680 --> 01:07:25.570
correspond to the
contracting phase.
01:07:25.570 --> 01:07:27.540
We can easily see
what values of omega
01:07:27.540 --> 01:07:30.190
are relevant in those cases.
01:07:30.190 --> 01:07:31.860
Omega we said
starts at 1 and gets
01:07:31.860 --> 01:07:36.772
larger, the borderline where
the angle is pi over 2 one
01:07:36.772 --> 01:07:38.230
could just plug
into these formulas
01:07:38.230 --> 01:07:41.980
and see amounts
to omega equals 2.
01:07:41.980 --> 01:07:46.390
So that is a division line
between these first two
01:07:46.390 --> 01:07:52.186
quadrants just calculated
from the value of theta.
01:07:52.186 --> 01:07:53.560
Omega then goes
to infinity as we
01:07:53.560 --> 01:07:57.310
said, comes backwards
and back to 1 in the end.
01:07:57.310 --> 01:08:00.340
And in this column
we just figure
01:08:00.340 --> 01:08:02.480
out which sign
choice corresponds
01:08:02.480 --> 01:08:05.870
to getting the right
value for omega
01:08:05.870 --> 01:08:08.400
and the angle that
appears in the arc
01:08:08.400 --> 01:08:12.192
sine of our formula for
the age, the formula there.
01:08:12.192 --> 01:08:14.025
So any one of these I
claim is very obvious.
01:08:14.025 --> 01:08:16.907
Seeing the whole
picture takes time
01:08:16.907 --> 01:08:19.240
because I think you really
have to look at each case one
01:08:19.240 --> 01:08:22.130
at a time to make sure you
understand it in detail.
01:08:22.130 --> 01:08:25.029
But if you understand the
initial expanding phase
01:08:25.029 --> 01:08:27.069
that's what corresponds
to our universe
01:08:27.069 --> 01:08:29.300
if our universe were closed.
01:08:29.300 --> 01:08:31.359
And the others are just as easy.
01:08:31.359 --> 01:08:34.120
You just have to take them
one at a time I think.
01:08:34.120 --> 01:08:35.790
OK, we're going to end there.
01:08:35.790 --> 01:08:39.020
We will continue on Thursday.