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PROFESSOR: OK, in
that case I can
00:00:27.270 --> 00:00:31.270
begin by giving a quick
review of last time.
00:00:31.270 --> 00:00:36.050
We began last time by talking
about the data of measurements
00:00:36.050 --> 00:00:38.700
of the cosmic
microwave background,
00:00:38.700 --> 00:00:43.185
starting with the data as
it existed in 1975 which
00:00:43.185 --> 00:00:46.575
I advertised as being an
incredible mess, which it was.
00:00:46.575 --> 00:00:52.640
You could easily believe that
this data fit this solid line
00:00:52.640 --> 00:00:54.465
curve, which was what
it's supposed to fit.
00:00:54.465 --> 00:00:58.120
But you could equally well
believe that it didn't.
00:00:58.120 --> 00:01:00.720
Things got worse
before they got better.
00:01:00.720 --> 00:01:02.980
There was the famous
Berkeley-Nagoya Rocket Flight
00:01:02.980 --> 00:01:07.820
experiment of 1987 which
had a data point which
00:01:07.820 --> 00:01:11.940
missed the theoretical curve by
16 standard deviations, which
00:01:11.940 --> 00:01:15.410
might seem fairly disappointing.
00:01:15.410 --> 00:01:17.504
It reminds one, by the
way, of a famous quote
00:01:17.504 --> 00:01:19.420
of Arthur Eddington--
which you may or may not
00:01:19.420 --> 00:01:21.990
be familiar with-- but
Eddington pointed out
00:01:21.990 --> 00:01:25.480
that while we always say that
we should not believe theories
00:01:25.480 --> 00:01:28.340
until the confirmed
by experiment,
00:01:28.340 --> 00:01:32.050
it's in fact equally true that
we should not believe data
00:01:32.050 --> 00:01:34.980
that's put forward until
it's confirmed by theory,
00:01:34.980 --> 00:01:37.280
and that certainly
was the case here.
00:01:37.280 --> 00:01:39.110
This data was never
confirmed by theory
00:01:39.110 --> 00:01:41.292
and turned out to be wrong.
00:01:41.292 --> 00:01:44.740
The beautiful data
was achieved in 1990
00:01:44.740 --> 00:01:49.040
by this fabulous COBE
satellite experiment, which
00:01:49.040 --> 00:01:53.370
showed-- unambiguously,
for the first time--
00:01:53.370 --> 00:01:56.180
that the cosmic background
radiation really
00:01:56.180 --> 00:02:00.610
does obey an essentially
perfect blackbody curve, which
00:02:00.610 --> 00:02:03.770
is just gorgeous.
00:02:03.770 --> 00:02:05.970
We then went on, in
our last lecture,
00:02:05.970 --> 00:02:09.759
to begin to talk about
the cosmological constant
00:02:09.759 --> 00:02:12.580
and its effect on the
evolution of the universe--
00:02:12.580 --> 00:02:17.680
completely changing gear
here-- and the key issue
00:02:17.680 --> 00:02:22.580
is the cosmological
effect of pressure.
00:02:22.580 --> 00:02:26.540
Earlier we had
derived this equation.
00:02:26.540 --> 00:02:32.470
The equation shows the
significant role of pressure
00:02:32.470 --> 00:02:35.650
during the
radiation-dominated era,
00:02:35.650 --> 00:02:38.672
but it also shows
that pressure--
00:02:38.672 --> 00:02:40.130
if it were negative--
could perhaps
00:02:40.130 --> 00:02:42.610
have the opposite
effect, causing
00:02:42.610 --> 00:02:46.020
an acceleration of the universe.
00:02:46.020 --> 00:02:48.820
Furthermore, we
learned last time
00:02:48.820 --> 00:02:53.530
that vacuum energy-- first
thing we learned, I guess,
00:02:53.530 --> 00:02:55.990
is just that's synonymous
with Einstein's cosmological
00:02:55.990 --> 00:03:01.060
constant, related to Einstein's
cosmological constant lambda
00:03:01.060 --> 00:03:02.602
by this equation.
00:03:02.602 --> 00:03:04.310
That is, the energy
density of the vacuum
00:03:04.310 --> 00:03:06.309
is equal to the mass
density of the vacuum times
00:03:06.309 --> 00:03:09.720
C squared and is equal to
this expression in terms
00:03:09.720 --> 00:03:13.180
of Einstein's original
cosmological constant.
00:03:13.180 --> 00:03:15.900
And most important,
in terms of physics,
00:03:15.900 --> 00:03:22.010
we learned that if we have a
non-zero vacuum energy-- vacuum
00:03:22.010 --> 00:03:24.440
energy by definition
does not change with time
00:03:24.440 --> 00:03:26.280
because the vacuum
is the vacuum,
00:03:26.280 --> 00:03:29.320
it's simply the lowest
possible energy density allowed
00:03:29.320 --> 00:03:31.660
by the laws of physics,
and the laws of physics,
00:03:31.660 --> 00:03:33.820
as far as we know, do
not change with time,
00:03:33.820 --> 00:03:36.060
and therefore vacuum
energy density does not
00:03:36.060 --> 00:03:39.460
change with time-- and
that is enough to imply
00:03:39.460 --> 00:03:41.650
that the pressure
of the vacuum has
00:03:41.650 --> 00:03:46.080
to be equal to minus the
energy density in a vacuum,
00:03:46.080 --> 00:03:48.970
and therefore minus
that expression in terms
00:03:48.970 --> 00:03:54.080
of the cosmological constant--
which is exactly what will give
00:03:54.080 --> 00:03:57.500
us a repulsive
gravitational effect,
00:03:57.500 --> 00:03:59.725
where we put that into
the Friedmann equation.
00:04:02.460 --> 00:04:05.470
Now I should emphasize
here that the effect
00:04:05.470 --> 00:04:07.730
of the pressure that
we are talking about
00:04:07.730 --> 00:04:12.060
is not the mechanical
effect of the pressure.
00:04:12.060 --> 00:04:13.890
The mechanical effect
of the pressure
00:04:13.890 --> 00:04:17.779
is literally zero because the
pressure that we are discussing
00:04:17.779 --> 00:04:21.579
here is a uniform pressure,
and pressures only
00:04:21.579 --> 00:04:23.810
cause mechanical
forces when there's
00:04:23.810 --> 00:04:25.760
gradients in the
pressure-- more pressure
00:04:25.760 --> 00:04:27.550
on one side than the other.
00:04:27.550 --> 00:04:29.500
And if all this pressure
is always balanced,
00:04:29.500 --> 00:04:32.150
the mechanical force of
the pressure is zero.
00:04:32.150 --> 00:04:36.260
But nonetheless, that pressure
contributes-- according
00:04:36.260 --> 00:04:39.850
to Einstein's equations--
as a contribution
00:04:39.850 --> 00:04:44.870
to the gravitational field,
and a positive pressure
00:04:44.870 --> 00:04:47.740
creates an attractive
gravitational field
00:04:47.740 --> 00:04:51.790
but a negative pressure produces
repulsive gravitational fields.
00:04:51.790 --> 00:04:54.350
And a positive vacuum
energy corresponds
00:04:54.350 --> 00:04:57.140
to a negative pressure which,
in fact, would dominate
00:04:57.140 --> 00:05:00.660
this equation, resulting in
a gravitational repulsion.
00:05:03.530 --> 00:05:08.110
So it's useful to
divide the total energy
00:05:08.110 --> 00:05:13.750
density into a normal
component plus vacuum energy,
00:05:13.750 --> 00:05:15.450
and similarly we can
divide the pressure
00:05:15.450 --> 00:05:19.880
into a normal component
plus the vacuum contribution
00:05:19.880 --> 00:05:21.230
to the pressure.
00:05:21.230 --> 00:05:23.390
The vacuum contribution
to the pressure
00:05:23.390 --> 00:05:25.766
will instantly disappear
from all of our equations
00:05:25.766 --> 00:05:27.140
because we know
how to express it
00:05:27.140 --> 00:05:29.720
in terms of the
vacuum mass density.
00:05:29.720 --> 00:05:33.020
It's just minus
Rho vac C squared.
00:05:33.020 --> 00:05:35.270
So we can then re-write
the Friedmann equations
00:05:35.270 --> 00:05:39.610
making those
substitutions, and we
00:05:39.610 --> 00:05:41.870
conceive-- in the
second-order equation--
00:05:41.870 --> 00:05:45.670
that the vacuum contribution,
negative, negative
00:05:45.670 --> 00:05:49.510
is a positive, produces
a positive acceleration--
00:05:49.510 --> 00:05:53.850
as we've been saying-- and a
positive vacuum energy also
00:05:53.850 --> 00:05:55.850
contributes to the
right-hand side
00:05:55.850 --> 00:05:58.560
of the first-order
Friedmann equation.
00:05:58.560 --> 00:06:03.370
And in many, many situations--
although not quite all--
00:06:03.370 --> 00:06:07.100
this vacuum energy will
dominate at late times.
00:06:07.100 --> 00:06:08.950
It definitely falls
off more slowly
00:06:08.950 --> 00:06:10.780
than any of the
other contributions.
00:06:10.780 --> 00:06:14.120
The vacuum energy is a constant,
and every other contribution
00:06:14.120 --> 00:06:16.450
to that right-hand
side falls off
00:06:16.450 --> 00:06:20.080
with A. The only
way that Rho vac can
00:06:20.080 --> 00:06:22.629
fail to dominate
if it's non-zero
00:06:22.629 --> 00:06:25.170
is if you have a closed universe
that collapses before it has
00:06:25.170 --> 00:06:28.460
a chance to dominate,
which is a possibility.
00:06:28.460 --> 00:06:32.610
But barring that, eventually
the vacuum energy will dominate,
00:06:32.610 --> 00:06:35.870
and once the vacuum
energy dominates,
00:06:35.870 --> 00:06:38.360
we just have H
squared-- A dot over A
00:06:38.360 --> 00:06:41.990
is H. H squared
equals a constant,
00:06:41.990 --> 00:06:46.280
and that just says that
H approaches its vacuum
00:06:46.280 --> 00:06:51.230
value, which is the square
root of 8 pi over 3 G Rho vac,
00:06:51.230 --> 00:06:53.720
and with H being a
constant we can also
00:06:53.720 --> 00:06:56.650
solve for A. The
scale factor itself
00:06:56.650 --> 00:07:00.591
is just proportional to an
exponential of E to the H T,
00:07:00.591 --> 00:07:05.110
where H is the H
associated with the vacuum.
00:07:05.110 --> 00:07:13.252
So this will be a very easy
to obtain asymptotic solution
00:07:13.252 --> 00:07:15.210
to the equations of the
universe, and, in fact,
00:07:15.210 --> 00:07:20.360
we think that our real universe
is approaching an exponentially
00:07:20.360 --> 00:07:22.790
expanding phase of
exactly this sort today.
00:07:22.790 --> 00:07:25.420
We're not there yet, but
we are approaching it.
00:07:25.420 --> 00:07:25.920
Yes?
00:07:25.920 --> 00:07:27.816
AUDIENCE: So does an
exponential A of T
00:07:27.816 --> 00:07:30.090
mean that the universe just
keeps expanding forever
00:07:30.090 --> 00:07:33.911
and just spins out
into nothingness?
00:07:33.911 --> 00:07:35.660
PROFESSOR: Yes, an
exponentially expanding
00:07:35.660 --> 00:07:39.360
A means the universe will
continue expanding forever
00:07:39.360 --> 00:07:41.910
and ordinary matter will
thin out to nothing,
00:07:41.910 --> 00:07:43.530
but this vacuum
energy density will
00:07:43.530 --> 00:07:47.560
remain as a constant
contribution.
00:07:47.560 --> 00:07:50.250
So the universe would go
on expanding exponentially
00:07:50.250 --> 00:07:51.830
forever.
00:07:51.830 --> 00:07:56.000
Now, there is the possibility
that this vacuum that we're
00:07:56.000 --> 00:08:00.630
living in is actually what might
be called a false vacuum-- that
00:08:00.630 --> 00:08:03.310
is, an unstable
vacuum-- a vacuum which
00:08:03.310 --> 00:08:05.220
behaves like a vacuum
for a long time
00:08:05.220 --> 00:08:08.730
but is subjected to the
possibility of decay.
00:08:08.730 --> 00:08:10.600
If that's the case,
it's still true
00:08:10.600 --> 00:08:16.390
that most of our
future space time
00:08:16.390 --> 00:08:19.080
will continue to
exponentially expand exactly
00:08:19.080 --> 00:08:23.400
as this equation shows, but
a kind of a Swiss cheese
00:08:23.400 --> 00:08:26.120
situation will develop
where decays in the vacuum
00:08:26.120 --> 00:08:30.860
would occur in places,
producing spherical holes
00:08:30.860 --> 00:08:34.643
in this otherwise exponentially
expanding background.
00:08:34.643 --> 00:08:36.059
We'll be talking
a little bit more
00:08:36.059 --> 00:08:37.830
about that later in the course.
00:08:40.557 --> 00:08:41.390
Any other questions?
00:08:46.900 --> 00:08:50.130
OK, well what I
want to do now is
00:08:50.130 --> 00:08:53.920
to work on a few
calculations which
00:08:53.920 --> 00:08:55.547
I'd like to present today.
00:08:55.547 --> 00:08:58.130
If all goes well, we might have
as many as three calculations,
00:08:58.130 --> 00:09:00.320
or at least two
calculations that we'll do
00:09:00.320 --> 00:09:03.070
and one that we'll talk
about a little bit.
00:09:03.070 --> 00:09:04.990
The first thing I
want to do-- and I
00:09:04.990 --> 00:09:08.100
guess we just talked about
starting it last time--
00:09:08.100 --> 00:09:12.140
is calculate the age of the
universe in this context.
00:09:12.140 --> 00:09:14.490
How do we express the
age of the universe
00:09:14.490 --> 00:09:17.400
in terms of measurable,
cosmological parameters,
00:09:17.400 --> 00:09:20.770
taking into account the
fact that vacuum energy is
00:09:20.770 --> 00:09:23.080
one of the ingredients
of our universe
00:09:23.080 --> 00:09:25.752
along with radiation and
non-relativistic matter, which
00:09:25.752 --> 00:09:26.835
we have already discussed.
00:09:34.410 --> 00:09:50.227
So to start that calculation
we can write down
00:09:50.227 --> 00:09:51.685
the first-order
Friedmann equation.
00:09:55.680 --> 00:10:06.050
A dot over A squared is equal
to 8 pi over 3 G times Rho
00:10:06.050 --> 00:10:08.960
and now I'm going to divide
Rho into all contributions
00:10:08.960 --> 00:10:10.480
we know about.
00:10:10.480 --> 00:10:17.430
Rho sub M, which represents
non-relativistic matter,
00:10:17.430 --> 00:10:24.070
plus Rho sub radiation,
which represents radiation,
00:10:24.070 --> 00:10:28.930
plus vacuum energy, which is
our new contribution, which
00:10:28.930 --> 00:10:32.280
will not depend on time at all.
00:10:32.280 --> 00:10:34.310
And then to complete
the equation
00:10:34.310 --> 00:10:38.835
there is minus KC
squared over A squared.
00:10:42.640 --> 00:10:44.890
And the strategy
here is really simply
00:10:44.890 --> 00:10:47.570
that because we know the
time evolution of each
00:10:47.570 --> 00:10:49.850
of the terms on the
right-hand side,
00:10:49.850 --> 00:10:52.270
we will be able to start
from wherever we are today
00:10:52.270 --> 00:10:55.650
in the universe-- which
will just take from data,
00:10:55.650 --> 00:10:59.180
the values of these
mass densities--
00:10:59.180 --> 00:11:02.050
and we will be able to integrate
backwards and ask how far back
00:11:02.050 --> 00:11:04.080
do we have to go
before we find the time
00:11:04.080 --> 00:11:07.180
when the scale factor vanished,
which is the instant of the Big
00:11:07.180 --> 00:11:07.680
Bang.
00:11:15.160 --> 00:11:19.660
So what we want to do is
to put into this equation
00:11:19.660 --> 00:11:22.660
the time dependents
that we know.
00:11:22.660 --> 00:11:25.020
So Rho sub M of T,
for example, can
00:11:25.020 --> 00:11:39.370
be written as A of T naught
divided by A of T cubed,
00:11:39.370 --> 00:11:43.020
times Rho sub M zero.
00:11:43.020 --> 00:11:46.720
And all these zeroes, of
course, mean the present time.
00:11:46.720 --> 00:11:48.690
So this formula
says, first of all,
00:11:48.690 --> 00:11:51.440
that the mass density
falls off as 1
00:11:51.440 --> 00:11:53.000
over the cube of
the scale factor.
00:11:53.000 --> 00:11:56.070
A of T is the only factor
on the right-hand here
00:11:56.070 --> 00:12:00.830
that depends on T. The
numerator depends on T sub zero,
00:12:00.830 --> 00:12:04.260
but not T. The
other constant, T is
00:12:04.260 --> 00:12:07.080
zero in the numerator
and Rho sub M zero.
00:12:07.080 --> 00:12:08.990
Rho sub M zero denotes
the present value
00:12:08.990 --> 00:12:11.241
of the mass density.
00:12:11.241 --> 00:12:12.990
And the constants here
are just rearranged
00:12:12.990 --> 00:12:15.280
so that when T equals
T naught, you just
00:12:15.280 --> 00:12:17.220
get Rho is equal to
its present value.
00:12:20.370 --> 00:12:26.040
And we can do the same
thing for radiation,
00:12:26.040 --> 00:12:28.340
and here I won't
write everything out
00:12:28.340 --> 00:12:31.240
because most things
are the same.
00:12:31.240 --> 00:12:33.000
The quantity in brackets
will be the same
00:12:33.000 --> 00:12:35.910
but this time it will
occur to the fourth power
00:12:35.910 --> 00:12:38.080
because radiation falls
off like the fourth power
00:12:38.080 --> 00:12:41.110
of the matter-- fourth
power of the scale factor--
00:12:41.110 --> 00:12:45.550
and then we have
Rho radiation zero.
00:12:45.550 --> 00:12:54.151
And then finally, for
the vacuum energy,
00:12:54.151 --> 00:12:56.900
we will just write on the
blackboard what we already
00:12:56.900 --> 00:12:59.700
know, which is that's
independent of time.
00:13:02.270 --> 00:13:04.640
So this gives us the time
dependents of all three terms
00:13:04.640 --> 00:13:05.140
here.
00:13:12.640 --> 00:13:15.940
Now we could just go from
there, but cosmologists
00:13:15.940 --> 00:13:17.820
like to talk about
mass densities in terms
00:13:17.820 --> 00:13:20.970
of the fraction of the
critical density, omega.
00:13:20.970 --> 00:13:22.720
So we're going to
change the notation just
00:13:22.720 --> 00:13:25.020
to correspond to the
way people usually
00:13:25.020 --> 00:13:27.520
talk about these things.
00:13:27.520 --> 00:13:31.190
So we will recall that the
critical density-- which
00:13:31.190 --> 00:13:33.960
is just that total
mass density that
00:13:33.960 --> 00:13:36.830
makes little K equal
to zero and hence
00:13:36.830 --> 00:13:41.430
the universe geometrically
flat-- so Rho sub C,
00:13:41.430 --> 00:13:45.450
we learned, is 3 H
squared over 8 pi G
00:13:45.450 --> 00:13:51.280
and then we will introduce
different components of omega.
00:13:51.280 --> 00:13:54.450
So I'm going to write omega sub
X here where X really is just
00:13:54.450 --> 00:13:59.460
a stand-in for matter or
radiation or vacuum energy.
00:13:59.460 --> 00:14:01.560
And whichever one of
those we're talking about,
00:14:01.560 --> 00:14:07.310
omega sub X is just a shorthand
for the corresponding mass
00:14:07.310 --> 00:14:10.050
density, but
normalized by dividing
00:14:10.050 --> 00:14:11.235
by this critical density.
00:14:19.210 --> 00:14:23.080
And then I'm just going to
rewrite these three equations
00:14:23.080 --> 00:14:27.150
in terms of omega
instead of Rho.
00:14:27.150 --> 00:14:42.150
So Rho sub M of T becomes then
3 H naught squared over 8 pi G
00:14:42.150 --> 00:14:51.026
times the same A of T
zero over A of T cubed,
00:14:51.026 --> 00:14:57.140
but not I'm multiplying,
omega sub M zero.
00:14:57.140 --> 00:14:59.210
And from the definitions
we've just written,
00:14:59.210 --> 00:15:03.890
this equation is just a
rewriting of that equation.
00:15:03.890 --> 00:15:06.925
And we can do the same thing,
of course, for radiation.
00:15:10.250 --> 00:15:13.610
Rho radiation of T is
equal to the same factor
00:15:13.610 --> 00:15:17.770
out front, the same
quantity in brackets
00:15:17.770 --> 00:15:21.000
but this time to
the fourth power,
00:15:21.000 --> 00:15:26.130
and then times omega
radiation at the present time.
00:15:29.650 --> 00:15:37.020
And finally, Rho vac doesn't
really depend on time--
00:15:37.020 --> 00:15:41.220
but we'll write it as if
it was a function of time--
00:15:41.220 --> 00:15:44.450
it consists of the same factor
of 3 H naught squared over 8 pi
00:15:44.450 --> 00:15:50.670
G, and no powers of the
quantity in brackets,
00:15:50.670 --> 00:15:55.960
but then just multiplying
omega sub vac zero.
00:16:02.304 --> 00:16:03.280
[ELECTRONIC RINGING]
00:16:07.669 --> 00:16:09.960
Everybody should turn off
their cell phone, by the way.
00:16:09.960 --> 00:16:11.852
[LAUGHING]
00:16:18.010 --> 00:16:19.615
OK, sorry about that.
00:16:32.010 --> 00:16:40.160
OK, now to make the
equation look prettier,
00:16:40.160 --> 00:16:43.470
I'm going to rewrite
even this last term
00:16:43.470 --> 00:16:45.480
as if it has something
to do with an omega.
00:16:48.110 --> 00:16:59.970
And we can do that
by defining omega sub
00:16:59.970 --> 00:17:10.319
K zero to be equal to minus
KC squared over A squared of T
00:17:10.319 --> 00:17:21.450
naught times H naught
squared, which is just
00:17:21.450 --> 00:17:23.880
the last term that appears
there [INAUDIBLE] of factor
00:17:23.880 --> 00:17:28.260
of H squared, which we'll
be able to factor out.
00:17:28.260 --> 00:17:32.580
And doing all that, the original
Friedmann equation can now
00:17:32.580 --> 00:17:47.670
be rewritten as H squared-- also
known as A dot over A squared--
00:17:47.670 --> 00:17:50.500
can be written as H naught
squared-- oh, I'm sorry,
00:17:50.500 --> 00:17:54.230
one other definition I
want to introduce here.
00:17:54.230 --> 00:17:58.150
This ratio-- A of T naught
over A-- keeps recurring,
00:17:58.150 --> 00:18:04.400
so it's nice to give
it a name, and I'm
00:18:04.400 --> 00:18:06.230
going to give 1
over that a name.
00:18:06.230 --> 00:18:12.690
I'm going to let X equal
the scale factor normalized
00:18:12.690 --> 00:18:13.960
by the scale factor today.
00:18:17.627 --> 00:18:19.960
And I might point out that
in Barbara Ryden's book, what
00:18:19.960 --> 00:18:22.330
I'm calling X is just what
she calls the scale factor,
00:18:22.330 --> 00:18:25.460
because she chooses to normalize
the scale factor so that it's
00:18:25.460 --> 00:18:27.030
equal to 1 today.
00:18:27.030 --> 00:18:29.630
So we haven't done that yet
but we are effectively doing it
00:18:29.630 --> 00:18:34.640
here by redefining a
new thing X. Having done
00:18:34.640 --> 00:18:39.290
that, the right-hand side
of the Friedmann equation
00:18:39.290 --> 00:18:41.470
can now be written
in a simple way.
00:18:41.470 --> 00:18:45.430
It's H naught squared
over X squared
00:18:45.430 --> 00:18:49.360
times a function F of X--
which is just an abbreviation
00:18:49.360 --> 00:18:53.720
to not have to write something
many times-- this is not,
00:18:53.720 --> 00:18:55.330
by any means, a
standard definition.
00:18:55.330 --> 00:18:57.020
It really is just for today.
00:18:57.020 --> 00:19:01.500
It allows us to save some
writing on the blackboard.
00:19:01.500 --> 00:19:04.920
So I'm going to, for today,
be using the abbreviation
00:19:04.920 --> 00:19:19.140
F of X is equal to omega sub
M zero times X plus omega sub
00:19:19.140 --> 00:19:25.780
radiation zero times no
powers of X plus omega
00:19:25.780 --> 00:19:33.310
sub vac zero times X to
the fourth, and finally
00:19:33.310 --> 00:19:39.260
plus omega sub K
zero times X squared.
00:19:42.450 --> 00:19:44.237
And this just lists
all the things
00:19:44.237 --> 00:19:46.820
that would occur in parentheses
here if we factored everything
00:19:46.820 --> 00:19:49.220
out.
00:19:49.220 --> 00:19:51.300
Notice I factored out
some powers of X squared,
00:19:51.300 --> 00:19:55.610
so the powers of X that appear
here do not look familiar,
00:19:55.610 --> 00:19:57.700
but the relative powers do.
00:19:57.700 --> 00:20:00.320
That is, omega should fall
like-- omega matter should
00:20:00.320 --> 00:20:03.670
fall like 1 over X cubed.
00:20:03.670 --> 00:20:07.830
Omega radiation should fall
off like one power faster
00:20:07.830 --> 00:20:09.660
than that, and it does.
00:20:09.660 --> 00:20:12.400
This is one less power
of X there than there,
00:20:12.400 --> 00:20:15.970
and omega vacuum should fall
off like four powers of X
00:20:15.970 --> 00:20:18.970
different from radiation,
and it does, et cetera.
00:20:18.970 --> 00:20:22.590
But there's no real
offset here that makes
00:20:22.590 --> 00:20:24.410
the factors there
not look familiar.
00:20:33.990 --> 00:20:38.270
OK, all of this was just to
put things in a simple form,
00:20:38.270 --> 00:20:42.790
but there's one other very
useful fact to look at.
00:20:42.790 --> 00:20:46.850
Suppose we now apply
this for T equals
00:20:46.850 --> 00:20:51.132
T zero, which means
for X equals 1.
00:20:51.132 --> 00:20:54.535
OK, it's true at any
time, but in particular
00:20:54.535 --> 00:20:56.285
we can look at what
it says for X equals 1
00:20:56.285 --> 00:20:58.243
and it tells us something
about our definitions
00:20:58.243 --> 00:21:01.526
that we could have
noticed in other ways--
00:21:01.526 --> 00:21:06.510
but didn't notice yet-- which is
that we set X equal to 1 here.
00:21:06.510 --> 00:21:08.090
These just becomes
the sum of omegas.
00:21:08.090 --> 00:21:11.870
The powers of X's
all become just ones.
00:21:11.870 --> 00:21:15.690
And the left-hand side
is just H zero squared,
00:21:15.690 --> 00:21:18.300
which matches the
H zero squared here
00:21:18.300 --> 00:21:21.760
because at T equals T naught
H squared is H zero squared,
00:21:21.760 --> 00:21:23.800
so these H zeros
squareds cancel.
00:21:23.800 --> 00:21:27.820
So applying it to
T equals T naught X
00:21:27.820 --> 00:21:38.800
equal to 1, what
you get is simply
00:21:38.800 --> 00:21:47.520
1 is equal to omega sum M zero
plus omega sub radiation zero
00:21:47.520 --> 00:21:54.020
plus omega sub vac
zero plus omega sub K
00:21:54.020 --> 00:21:58.080
zero, which gives
us the simplest
00:21:58.080 --> 00:22:01.630
way of thinking about what this
omega sub K zero really means.
00:22:01.630 --> 00:22:05.680
We defined it initially up
there in terms of little K,
00:22:05.680 --> 00:22:09.250
but for this equation, we
can see that omega sub K
00:22:09.250 --> 00:22:12.260
zero really is just
another way of writing
00:22:12.260 --> 00:22:15.020
1 minus all the other omegas.
00:22:26.914 --> 00:22:29.330
So you could think of this as
being the definition of what
00:22:29.330 --> 00:22:32.220
I'm calling omega sub k zero.
00:22:32.220 --> 00:22:34.260
So it's a language in
which you essentially
00:22:34.260 --> 00:22:37.130
think that the total
omega has to equal 1,
00:22:37.130 --> 00:22:41.410
and whatever is not contained
in real matter becomes
00:22:41.410 --> 00:22:44.290
a piece of omega sub k, the
curvature or contribution
00:22:44.290 --> 00:22:44.850
to omega.
00:22:52.940 --> 00:22:54.390
OK, now it's really
just a matter
00:22:54.390 --> 00:22:58.322
of simple manipulations
and I-- the main purpose
00:22:58.322 --> 00:23:00.030
of defining F of x is
to be able to write
00:23:00.030 --> 00:23:02.790
these simple manipulations
simpler than they would be
00:23:02.790 --> 00:23:05.484
if you had to write out
what F of x was every time.
00:23:05.484 --> 00:23:07.400
We're first just going
to take the square root
00:23:07.400 --> 00:23:11.390
of the key equation up there--
the Friedmann equation-- and we
00:23:11.390 --> 00:23:18.290
get a dot over a is also
equal to x dot over x.
00:23:18.290 --> 00:23:21.430
Note that the constant of
proportionality there, a
00:23:21.430 --> 00:23:23.160
of t naught-- which
is a constant--
00:23:23.160 --> 00:23:25.260
cancels when you
take a dot over a.
00:23:25.260 --> 00:23:27.960
So a dot over a is the
same as x dot over x.
00:23:30.950 --> 00:23:34.770
And that-- just taking the
square root of that equation--
00:23:34.770 --> 00:23:37.720
is H naught over x.
00:23:40.240 --> 00:23:41.248
Hold on a second.
00:23:54.720 --> 00:23:57.030
Yeah, we're taking the
square root of the equation,
00:23:57.030 --> 00:23:59.130
so yeah, we had H naught
squared over x squared.
00:23:59.130 --> 00:24:01.130
Here we have H naught
over x and then just times
00:24:01.130 --> 00:24:02.170
the square root of f.
00:24:09.630 --> 00:24:12.280
And these x's cancel each other.
00:24:15.620 --> 00:24:16.440
Wait a minute.
00:24:28.390 --> 00:24:29.884
Oh, I'm sorry.
00:24:29.884 --> 00:24:31.550
They're not supposed
to cancel because I
00:24:31.550 --> 00:24:33.357
didn't write this
quite correctly.
00:24:33.357 --> 00:24:34.940
That should have
been x to the fourth.
00:24:34.940 --> 00:24:35.440
Apologies.
00:24:42.020 --> 00:24:43.665
And now here we have x squared.
00:24:51.990 --> 00:24:54.310
And this can just
be-- by manipulating
00:24:54.310 --> 00:25:02.289
where the x's go--
rewritten as x times dx dt.
00:25:02.289 --> 00:25:03.830
So I multiply the
whole equation by x
00:25:03.830 --> 00:25:06.570
squared to get rid of
that factor on the right,
00:25:06.570 --> 00:25:08.770
and now on the
right-hand side we just
00:25:08.770 --> 00:25:23.510
have H zero times
square root of F.
00:25:23.510 --> 00:25:25.940
And now I just want to do
the usual trick of separating
00:25:25.940 --> 00:25:30.270
the dx pieces from dt
pieces in this equation.
00:25:30.270 --> 00:25:34.260
And we can rewrite
that equation as dt
00:25:34.260 --> 00:25:40.540
is equal to 1 over
H naught times
00:25:40.540 --> 00:25:46.380
x dx over the square root of
F. And maybe I'll rewrite it
00:25:46.380 --> 00:25:49.310
as the square root of F
of x to make it explicit
00:25:49.310 --> 00:25:50.990
that F depends on x.
00:25:50.990 --> 00:25:54.190
So this is just the
rewriting of this equation,
00:25:54.190 --> 00:25:57.550
moving factors around,
and in this form
00:25:57.550 --> 00:25:59.080
we can integrate
it and determine
00:25:59.080 --> 00:26:02.510
the age of the universe.
00:26:02.510 --> 00:26:06.280
The present age of the
universe can be obtained just
00:26:06.280 --> 00:26:08.820
by integrating this
expression from the big bang
00:26:08.820 --> 00:26:10.720
up to the present.
00:26:10.720 --> 00:26:14.150
And that will be the integral
dt from the big bang up
00:26:14.150 --> 00:26:15.670
to present, the
sum of all the time
00:26:15.670 --> 00:26:19.150
intervals from the
big bang until now.
00:26:19.150 --> 00:26:23.620
And it's just equal
to 1 over H zero times
00:26:23.620 --> 00:26:32.320
the integral of x dx over
the square root of F of x.
00:26:32.320 --> 00:26:36.400
And now-- just to think about
the limits of integration--
00:26:36.400 --> 00:26:39.810
what should limits
of integration be?
00:26:39.810 --> 00:26:41.630
AUDIENCE: Zero to one.
00:26:41.630 --> 00:26:44.400
PROFESSOR: I hear zero to
one, and that's correct.
00:26:44.400 --> 00:26:48.190
We're integrating from the
big bang up to the present.
00:26:48.190 --> 00:26:50.860
At the big bang,
a is equal to zero
00:26:50.860 --> 00:26:53.790
and therefore x
is equal to zero.
00:26:53.790 --> 00:26:56.665
And, at the present time,
t is equal to t naught
00:26:56.665 --> 00:26:58.750
and therefore x is equal to 1.
00:26:58.750 --> 00:27:00.480
So we integrate up
to one if we want
00:27:00.480 --> 00:27:02.480
the present age of the universe.
00:27:02.480 --> 00:27:04.760
We could also integrate
it up to any other value
00:27:04.760 --> 00:27:08.150
of x that we want, and it will
tell us the age of universe
00:27:08.150 --> 00:27:09.750
when the scale factor
had that value.
00:27:16.850 --> 00:27:19.250
So this is the final,
state of the art
00:27:19.250 --> 00:27:22.370
formula for the age
of the universe,
00:27:22.370 --> 00:27:25.810
expressed in terms of
the matter contribution
00:27:25.810 --> 00:27:28.390
to omega, the radiation
contribution to omega,
00:27:28.390 --> 00:27:30.970
and the vacuum
contribution to omega,
00:27:30.970 --> 00:27:32.501
and the value of H naught.
00:27:32.501 --> 00:27:35.000
Those are the only ingredients
on the right-hand side there.
00:27:35.000 --> 00:27:36.740
And then you can
calculate the age.
00:27:36.740 --> 00:27:39.140
And it's the completely
state of the art formula.
00:27:39.140 --> 00:27:42.320
It's exactly what the Planck
people did when they told you
00:27:42.320 --> 00:27:47.780
that the age of the universe
was 13.9 billion years, using
00:27:47.780 --> 00:27:50.050
that formula.
00:27:50.050 --> 00:27:53.470
Now as far as actually doing the
integral, in the general case,
00:27:53.470 --> 00:27:55.940
the only way to do
it is numerically.
00:27:55.940 --> 00:27:58.060
That's how it's usually done.
00:27:58.060 --> 00:28:02.020
Special cases can be
done analytically.
00:28:02.020 --> 00:28:05.110
We've already talked about
the case where there's
00:28:05.110 --> 00:28:17.900
no cosmological constants, no
vacuum energy, but just matter
00:28:17.900 --> 00:28:20.765
and curvature-- omega,
in this language.
00:28:23.730 --> 00:28:27.060
There's another special
case which can be done,
00:28:27.060 --> 00:28:34.130
which is the case
that involves vacuum
00:28:34.130 --> 00:28:36.845
energy and
nonrelativistic matter.
00:28:45.430 --> 00:28:49.990
And this is the flat case, only,
that can be done analytically.
00:28:49.990 --> 00:28:54.190
So it corresponds
to omega radiation
00:28:54.190 --> 00:29:00.400
equals omega k
equals zero, and that
00:29:00.400 --> 00:29:05.959
means that omega matter plus
omega vac is equal to 1,
00:29:05.959 --> 00:29:08.250
because the sum of all the
omegas is always equal to 1.
00:29:08.250 --> 00:29:12.730
So in this case I can
write an answer for you.
00:29:12.730 --> 00:29:21.147
And I don't intend to try
to derive this answer,
00:29:21.147 --> 00:29:23.480
but it's worth knowing that
can be written analytically.
00:29:23.480 --> 00:29:26.300
That's the main point, I guess.
00:29:26.300 --> 00:29:28.930
So it does get divided
into three cases
00:29:28.930 --> 00:29:32.730
depending on whether omega
matter is larger than, smaller
00:29:32.730 --> 00:29:35.120
than, or equal to 1.
00:29:35.120 --> 00:29:39.830
So the first case will
be if omega matter
00:29:39.830 --> 00:29:43.780
zero is greater than 1.
00:29:43.780 --> 00:29:45.480
And if omega matter
is greater than 1,
00:29:45.480 --> 00:29:49.030
that corresponds to
omega vac less than 1
00:29:49.030 --> 00:29:51.720
because the sum of the two is
equal to 1 in all cases, here.
00:29:55.052 --> 00:29:58.064
So omega vac has
to be less than 0.
00:29:58.064 --> 00:30:00.355
So this is not our real
universe but it's a calculation
00:30:00.355 --> 00:30:06.640
that you can do, and
it's 2 over 3 H naught
00:30:06.640 --> 00:30:14.590
times the inverse tangent of
the square root of omega sub
00:30:14.590 --> 00:30:25.150
m zero 1 over the square root
of omega sub m zero minus 1.
00:30:28.590 --> 00:30:30.945
So if you plug this
integral into mathematica,
00:30:30.945 --> 00:30:33.320
you should get that answer or
something equivalent to it.
00:30:36.340 --> 00:30:40.770
For the case-- the
borderline case, here--
00:30:40.770 --> 00:30:45.140
where omega matter
zero equals 1,
00:30:45.140 --> 00:30:47.950
that's the special case
where omega vac is equal to 0
00:30:47.950 --> 00:30:50.410
because the sum of
these is always one.
00:30:50.410 --> 00:30:52.006
So this special
case in the middle
00:30:52.006 --> 00:30:53.380
is the case we
already knew, it's
00:30:53.380 --> 00:30:55.660
just the matter-dominated
flat universe.
00:30:55.660 --> 00:31:00.230
So that's two thirds H inverse.
00:31:00.230 --> 00:31:05.600
So it's 2 over 3 H naught.
00:31:05.600 --> 00:31:17.310
And then, finally, if omega
matter zero is less than 1
00:31:17.310 --> 00:31:19.690
and then omega vac is positive.
00:31:22.334 --> 00:31:23.750
And this [INAUDIBLE]
approximation
00:31:23.750 --> 00:31:27.780
is our universe, that is, that
our universe has possibly zero
00:31:27.780 --> 00:31:31.080
curvature-- in any case,
unmeasureably small curvature--
00:31:31.080 --> 00:31:35.310
and very, very small radiation
for most of its evolution.
00:31:35.310 --> 00:31:37.360
So this last case
is our universe
00:31:37.360 --> 00:31:43.840
except for cases that are near
the radiation-dominated era,
00:31:43.840 --> 00:31:53.610
and the formula here is
2 over 3 H naught times
00:31:53.610 --> 00:31:57.700
the inverse hyperbolic
tangent of 1
00:31:57.700 --> 00:32:07.390
minus-- square root, excuse
me-- of 1 minus omega sub m
00:32:07.390 --> 00:32:17.470
zero over the square root
of 1 minus omega sub m zero.
00:32:20.592 --> 00:32:23.110
OK, so this is just
a result obtained
00:32:23.110 --> 00:32:27.430
by doing that integral
for the special case
00:32:27.430 --> 00:32:30.570
that we're talking about.
00:32:30.570 --> 00:32:33.430
Now, I don't know
any simpler way
00:32:33.430 --> 00:32:36.764
to write it except
as these three cases.
00:32:36.764 --> 00:32:38.555
It is, however, a single
analytic function,
00:32:38.555 --> 00:32:40.680
and when you graph it--
and I'll show you a graph--
00:32:40.680 --> 00:32:44.080
it is one smooth function right
across the range of these three
00:32:44.080 --> 00:32:47.160
cases, which is similar
to what we found
00:32:47.160 --> 00:32:50.855
the earlier for the flat,
matter-dominated case.
00:32:57.819 --> 00:32:58.360
So let's see.
00:32:58.360 --> 00:33:00.400
This is not that yet.
00:33:00.400 --> 00:33:03.260
This is the case that
we did a long time ago,
00:33:03.260 --> 00:33:09.610
actually, the case of a
matter-dominated universe
00:33:09.610 --> 00:33:11.590
with nothing but
nonrelativistic matter
00:33:11.590 --> 00:33:13.930
and possibly with curvature.
00:33:13.930 --> 00:33:19.830
And I can remind
you, here, that what
00:33:19.830 --> 00:33:22.210
we found for that model is
that we tended to get ages
00:33:22.210 --> 00:33:24.540
there were too young.
00:33:24.540 --> 00:33:28.090
So if we take a
reasonable value for H
00:33:28.090 --> 00:33:33.150
of 67 to 70 kilometers per
second per megaparsec--
00:33:33.150 --> 00:33:36.140
which is in this range--
and take a reasonable value
00:33:36.140 --> 00:33:40.010
for omega-- which is somewhere
between, say, 0.2 and 1
00:33:40.010 --> 00:33:41.940
depending on what you
consider reasonable,
00:33:41.940 --> 00:33:44.530
this model doesn't work
anyway-- but if you
00:33:44.530 --> 00:33:48.820
take omega anywhere
between 0.1 and 1,
00:33:48.820 --> 00:33:52.780
you get numbers
for the age which
00:33:52.780 --> 00:33:56.164
are in the order of 10 billion
years, which is not old enough
00:33:56.164 --> 00:33:57.580
to be consistent
with what we know
00:33:57.580 --> 00:33:59.700
about the ages of
the older stars.
00:33:59.700 --> 00:34:02.000
And especially if you think
that omega should be one,
00:34:02.000 --> 00:34:05.530
you get a very young age of
more like 9 billion years,
00:34:05.530 --> 00:34:06.970
which is what we found earlier.
00:34:06.970 --> 00:34:10.989
This is just a graph of
those same equations.
00:34:10.989 --> 00:34:13.600
But, if we include
vacuum energy,
00:34:13.600 --> 00:34:15.330
it makes all the difference.
00:34:15.330 --> 00:34:20.090
So this now is a graph
of those equations.
00:34:20.090 --> 00:34:25.530
What's shown is the age, T
naught, as a function of H
00:34:25.530 --> 00:34:30.360
and for various values of
omega sub m, the same omega sub
00:34:30.360 --> 00:34:35.570
m that's called omega sub
m zero on the blackboard.
00:34:35.570 --> 00:34:44.110
And shown here are the Barbara
Ryden a benchmark point,
00:34:44.110 --> 00:34:48.429
which is the left-most of these
two almost overlapping points.
00:34:48.429 --> 00:34:52.409
And also shown here
is the favored point
00:34:52.409 --> 00:34:56.420
from the WMAP satellite
seven-year data.
00:34:56.420 --> 00:34:58.530
They lie almost on
top of each other.
00:34:58.530 --> 00:35:01.190
I didn't get a chance to
plot the Planck point, which
00:35:01.190 --> 00:35:03.020
is the one that we
would consider the most
00:35:03.020 --> 00:35:04.940
authoritative these
days, but I'll
00:35:04.940 --> 00:35:08.370
add that before I
post the lectures.
00:35:08.370 --> 00:35:11.020
It lies almost on top of
these, and it corresponds
00:35:11.020 --> 00:35:16.600
to a Hubble expansion
rate of a little under 70,
00:35:16.600 --> 00:35:24.070
and a vacuum energy
contribution of about 0.7,
00:35:24.070 --> 00:35:30.700
and therefore a matter
contribution of about 0.3.
00:35:30.700 --> 00:35:33.010
This curve.
00:35:33.010 --> 00:35:38.530
And it gives an age of 13.7,
13.8 billion years-- perfectly
00:35:38.530 --> 00:35:42.390
consistent with estimates of
the age of the oldest stars.
00:35:42.390 --> 00:35:45.480
So this age problem
which had been,
00:35:45.480 --> 00:35:48.630
until the discovery of the
dark energy, a serious problem
00:35:48.630 --> 00:35:51.210
in cosmology for
many, many years
00:35:51.210 --> 00:35:55.800
goes away completely once
one adds in the dark energy.
00:35:55.800 --> 00:35:57.960
So that's it for
the age calculation.
00:35:57.960 --> 00:36:01.471
Are there any questions about
the age of the universe?
00:36:01.471 --> 00:36:01.970
Yes?
00:36:01.970 --> 00:36:03.329
AUDIENCE: So when
you say dark energy,
00:36:03.329 --> 00:36:05.524
are you using that synonymously
with vacuum energy?
00:36:05.524 --> 00:36:06.440
PROFESSOR: Sorry, yes.
00:36:06.440 --> 00:36:07.840
I used the word
dark energy there
00:36:07.840 --> 00:36:09.904
and I've been talking
about vacuum energy,
00:36:09.904 --> 00:36:11.070
and what's the relationship?
00:36:13.910 --> 00:36:17.240
When I said dark energy I
really meant vacuum energy.
00:36:17.240 --> 00:36:20.100
In general, the way
these words are used
00:36:20.100 --> 00:36:24.230
is that vacuum energy has
a very specific meaning.
00:36:24.230 --> 00:36:27.230
It really does mean the
energy of the vacuum,
00:36:27.230 --> 00:36:29.210
and by definition,
therefore, it does not
00:36:29.210 --> 00:36:32.180
change with time, period.
00:36:32.180 --> 00:36:34.090
We don't know for
sure what this stuff
00:36:34.090 --> 00:36:36.690
is that's driving the
acceleration of the universe,
00:36:36.690 --> 00:36:41.650
and hence the name dark energy,
which is more ambiguous.
00:36:41.650 --> 00:36:44.120
I think the technical
definition of dark energy
00:36:44.120 --> 00:36:47.510
is it's whatever the
stuff is that's driving
00:36:47.510 --> 00:36:49.420
the acceleration
of the universe.
00:36:49.420 --> 00:36:54.480
And the other conceivable
possibility-- and observers
00:36:54.480 --> 00:36:57.800
are hard at work trying to
distinguish, experimentally,
00:36:57.800 --> 00:37:00.890
between these two options-- the
other possibility is that it
00:37:00.890 --> 00:37:03.910
could be a very
slowly evolving scalar
00:37:03.910 --> 00:37:08.080
field of the same type
that drives inflation
00:37:08.080 --> 00:37:10.100
that we'll be
talking about later.
00:37:10.100 --> 00:37:12.000
But this would be a
much lower energy scale
00:37:12.000 --> 00:37:14.550
than the inflation of
the early universe,
00:37:14.550 --> 00:37:17.560
and much more slowly evolving.
00:37:17.560 --> 00:37:22.270
So far, we have not yet
found any time variation
00:37:22.270 --> 00:37:24.829
in the dark energy.
00:37:24.829 --> 00:37:27.370
So, so far, everything we' have
learned about the dark energy
00:37:27.370 --> 00:37:29.210
is consistent with
the possibility
00:37:29.210 --> 00:37:32.910
that it is simply vacuum energy.
00:37:32.910 --> 00:37:33.410
Question.
00:37:33.410 --> 00:37:35.326
AUDIENCE: Is the amount
of dark energy related
00:37:35.326 --> 00:37:36.626
to the amount of dark matter?
00:37:36.626 --> 00:37:38.750
PROFESSOR: Is the amount
of dark energy related to.
00:37:38.750 --> 00:37:40.370
the amount of dark matter?
00:37:40.370 --> 00:37:41.210
No.
00:37:41.210 --> 00:37:43.300
They're both numbers
and they differ
00:37:43.300 --> 00:37:45.990
by a factor of 2 and
1/2 or so, but there's
00:37:45.990 --> 00:37:49.200
no particular relationship
between them that we know of.
00:37:49.200 --> 00:37:50.655
AUDIENCE: But
doesn't dark matter
00:37:50.655 --> 00:37:53.192
imply that they have a
certain attraction to bodies
00:37:53.192 --> 00:37:56.370
around it, which is
a form of energy?
00:37:56.370 --> 00:37:59.706
PROFESSOR: Yeah, well let's
talk about this later.
00:37:59.706 --> 00:38:02.081
AUDIENCE: Do we have any idea
what dark energy is at all?
00:38:02.081 --> 00:38:03.622
PROFESSOR: OK, the
question is, do we
00:38:03.622 --> 00:38:05.760
have any idea what
dark energy is at all?
00:38:05.760 --> 00:38:07.090
And the answer is probably yes.
00:38:07.090 --> 00:38:09.780
That is, I think there's a good
chance it is vacuum energy.
00:38:09.780 --> 00:38:11.950
Now if you ask what is
vacuum energy, what is it
00:38:11.950 --> 00:38:14.236
about the vacuum that gives
it this nonzero energy,
00:38:14.236 --> 00:38:15.610
there we're pretty
much clueless.
00:38:15.610 --> 00:38:17.443
I was going to talk
about that a little more
00:38:17.443 --> 00:38:19.260
at the end of today,
if we get there.
00:38:19.260 --> 00:38:23.080
But whatever property
of the vacuum
00:38:23.080 --> 00:38:26.310
it is that gives it its
energy-- we know of many,
00:38:26.310 --> 00:38:27.810
it's just a matter
of what dominates
00:38:27.810 --> 00:38:31.180
and how they add up--
the end result is pretty
00:38:31.180 --> 00:38:34.680
much the same as far as the
phenomenology of vacuum energy.
00:38:34.680 --> 00:38:37.040
So we understand the
phenomenology of vacuum energy,
00:38:37.040 --> 00:38:39.580
I would say, completely.
00:38:39.580 --> 00:38:42.020
The big issue, which
I'll talk about either
00:38:42.020 --> 00:38:43.920
at the end of
today or next time,
00:38:43.920 --> 00:38:47.164
is trying to estimate the
magnitude of the vacuum energy,
00:38:47.164 --> 00:38:48.830
and there we're really
totally clueless,
00:38:48.830 --> 00:38:50.040
as I will try to describe.
00:38:53.600 --> 00:38:56.700
OK, that's it for my slides.
00:38:56.700 --> 00:38:58.920
OK, I wanted to now talk
about another very important
00:38:58.920 --> 00:39:02.300
calculation, which is
basically the calculation which
00:39:02.300 --> 00:39:08.270
led to the original evidence
that the universe is
00:39:08.270 --> 00:39:10.630
accelerating to begin with.
00:39:14.190 --> 00:39:18.020
OK, discovery that the
universe was accelerating
00:39:18.020 --> 00:39:20.810
was made, as I said earlier,
by two groups of astronomers
00:39:20.810 --> 00:39:28.355
in 1998, and the key observation
was using a type 1a supernovae
00:39:28.355 --> 00:39:31.860
as standard candles to
measure the expansion
00:39:31.860 --> 00:39:33.810
rate of the universe
versus time,
00:39:33.810 --> 00:39:35.580
looking back into the past.
00:39:35.580 --> 00:39:38.530
And basically what they found
is that when they look back
00:39:38.530 --> 00:39:42.180
about 5, 6 billion years,
the expansion rate then
00:39:42.180 --> 00:39:45.530
was actually slower
than expansion rate now,
00:39:45.530 --> 00:39:47.480
meaning that the
universe has accelerated.
00:39:47.480 --> 00:39:49.270
And that was the
key observation.
00:39:49.270 --> 00:39:51.700
So the question
for us to calculate
00:39:51.700 --> 00:39:56.040
is, what do we expect, as a
function of these parameters,
00:39:56.040 --> 00:40:00.940
for redshift versus luminosity?
00:40:00.940 --> 00:40:03.890
These astronomers, by
using type 1a supernovae
00:40:03.890 --> 00:40:06.050
as standard candles,
are basically
00:40:06.050 --> 00:40:09.200
using the luminosity
measurements of these type
00:40:09.200 --> 00:40:13.810
1a supernovae as estimates
of their distance.
00:40:13.810 --> 00:40:16.390
So what they actually measured
was simply luminosity verses
00:40:16.390 --> 00:40:18.960
redshift, and that's what we
will learn how to calculate,
00:40:18.960 --> 00:40:20.700
and the formula
that will get will
00:40:20.700 --> 00:40:22.570
be, again, exactly
the formula that they
00:40:22.570 --> 00:40:27.420
used when they were trying to
fit their data-- to understand
00:40:27.420 --> 00:40:28.920
what their data
was telling them--
00:40:28.920 --> 00:40:31.610
about possible acceleration
of the universe.
00:40:40.550 --> 00:40:42.240
So the calculation
we're about to do
00:40:42.240 --> 00:40:43.990
is really nothing
new to you folks
00:40:43.990 --> 00:40:45.830
because we have
calculated luminosities
00:40:45.830 --> 00:40:47.220
in another contexts.
00:40:47.220 --> 00:40:50.630
Now we will just write down the
equations in their full glory,
00:40:50.630 --> 00:40:55.210
including the contribution
due to vacuum energy.
00:41:13.609 --> 00:41:15.150
So we'd like to do
these calculations
00:41:15.150 --> 00:41:16.925
in a way that allows
for curvature, even
00:41:16.925 --> 00:41:18.800
though-- in the end--
we're going to discover
00:41:18.800 --> 00:41:20.550
that the curvature
of our universe
00:41:20.550 --> 00:41:23.055
is-- as far as anybody
can tell-- negligible.
00:41:23.055 --> 00:41:25.180
But people still look for
it and it still very well
00:41:25.180 --> 00:41:28.077
could be there at the
level of one part in 1,000
00:41:28.077 --> 00:41:29.035
or something like that.
00:41:29.035 --> 00:41:32.770
But at the level of 1 part
in 100, it's not there.
00:41:32.770 --> 00:41:39.380
So we begin by writing down
the Robertson-Walker metric,
00:41:39.380 --> 00:41:43.375
ds squared is equal
to minus c squared
00:41:43.375 --> 00:41:51.850
dt squared plus a
squared of t times dr
00:41:51.850 --> 00:42:02.290
squared over 1 minus little k
times r squared plus r squared
00:42:02.290 --> 00:42:11.320
d theta squared plus sine
squared theta d phi squared,
00:42:11.320 --> 00:42:14.660
end curly brackets.
00:42:14.660 --> 00:42:17.620
OK, so this is the metric
that we're familiar with.
00:42:17.620 --> 00:42:20.280
We're going to be interested,
mainly, in radial motion,
00:42:20.280 --> 00:42:22.630
and if you're interested
mainly in radial motion,
00:42:22.630 --> 00:42:25.540
it helps to simplify the
radial part of this metric
00:42:25.540 --> 00:42:28.030
by using a different
radial variable.
00:42:28.030 --> 00:42:31.200
And we've done
this before, also.
00:42:31.200 --> 00:42:33.310
At this point, we
really need to pick
00:42:33.310 --> 00:42:35.102
whether we're talking
about open or closed.
00:42:35.102 --> 00:42:36.643
If we're talking
about flat, we don't
00:42:36.643 --> 00:42:37.890
need to do anything, really.
00:42:37.890 --> 00:42:39.810
If you eliminate k,
here, the radial part
00:42:39.810 --> 00:42:41.362
is as simple as it gets.
00:42:41.362 --> 00:42:43.070
But if we want talk
about open or closed,
00:42:43.070 --> 00:42:46.070
it pays to use
different variables,
00:42:46.070 --> 00:42:47.650
and the variable
that we'd use would
00:42:47.650 --> 00:42:49.507
be different in two cases.
00:42:49.507 --> 00:42:51.590
So I'm going to consider
the closed-universe case.
00:43:00.010 --> 00:43:07.870
And I'm going to introduce
an angle, sine of psi
00:43:07.870 --> 00:43:10.580
being equal to the
square root of k-- which
00:43:10.580 --> 00:43:14.870
is positive in this
case-- times little r.
00:43:14.870 --> 00:43:18.920
And this psi is, in fact, if
you trace everything back,
00:43:18.920 --> 00:43:22.462
the angle from the
w-axis that we originally
00:43:22.462 --> 00:43:23.920
used when we
constructed the closed
00:43:23.920 --> 00:43:26.320
Robertson-Walker universe
in the first place.
00:43:26.320 --> 00:43:29.160
But now we're essentially
working backwards.
00:43:29.160 --> 00:43:32.426
We've learned to know
and love this expression,
00:43:32.426 --> 00:43:33.800
so we're going to
just rewrite it
00:43:33.800 --> 00:43:36.440
in terms of the new
variable, sin of psi
00:43:36.440 --> 00:43:41.690
equals the square
root of k times r.
00:43:41.690 --> 00:43:46.330
And from this, by
just differentiation,
00:43:46.330 --> 00:43:49.730
you discover that deep psi
is equal to the square root
00:43:49.730 --> 00:43:57.840
of k times dr over cosine psi.
00:43:57.840 --> 00:44:00.100
And that is equal
to the square root
00:44:00.100 --> 00:44:07.430
of k times dr over the square
root of 1 minus kr squared.
00:44:07.430 --> 00:44:11.270
So this, then, fits in very
nicely with the metric itself.
00:44:11.270 --> 00:44:13.489
The metric is just the
square of this factor,
00:44:13.489 --> 00:44:15.530
and therefore it is just
proportional to deep psi
00:44:15.530 --> 00:44:18.910
squared all by itself.
00:44:18.910 --> 00:44:23.670
And rewriting the whole
metric, we can write it
00:44:23.670 --> 00:44:29.640
as ds squared is equal to
minus c squared dt squared
00:44:29.640 --> 00:44:33.560
plus a new scale factor--
which I'll define
00:44:33.560 --> 00:44:42.990
in a second in terms of the old
one-- times deep psi squared
00:44:42.990 --> 00:44:46.560
plus-- now, the
angular term becomes
00:44:46.560 --> 00:44:49.090
nonstandard instead of just
having an r squared here,
00:44:49.090 --> 00:44:52.052
we have sine squared of psi.
00:44:52.052 --> 00:44:54.010
Which is, of course,
proportional to r squared.
00:44:59.780 --> 00:45:05.030
And that multiplies d theta
squared plus sine squared
00:45:05.030 --> 00:45:12.770
theta d phi squared,
end curly brackets.
00:45:12.770 --> 00:45:25.372
And a tilde is just
equal to our original
00:45:25.372 --> 00:45:30.822
a divided by the
square root of k.
00:45:30.822 --> 00:45:34.020
So we scaled it.
00:45:34.020 --> 00:45:37.480
And I should mention that
I'm putting a tilde here
00:45:37.480 --> 00:45:40.250
because we've already written
an a without a tilde there,
00:45:40.250 --> 00:45:41.730
and they're not equal to other.
00:45:41.730 --> 00:45:44.200
If you want to just
start here, you can,
00:45:44.200 --> 00:45:46.290
and then there's no
need for the tilde.
00:45:46.290 --> 00:45:48.245
You could just call
this the scale factor
00:45:48.245 --> 00:45:49.680
and it doesn't need a tilde.
00:45:49.680 --> 00:45:52.600
The tilde is only to distinguish
the two cases from each other.
00:46:17.998 --> 00:46:21.202
AUDIENCE: [INAUDIBLE]
00:46:21.202 --> 00:46:21.910
PROFESSOR: Sorry?
00:46:21.910 --> 00:46:24.200
AUDIENCE: Do a and a tilde
have different units?
00:46:24.200 --> 00:46:25.348
PROFESSOR: Didn't hear you?
00:46:25.348 --> 00:46:27.440
AUDIENCE: Do a and a tilde
have different units?
00:46:27.440 --> 00:46:29.690
PROFESSOR: They do have
different-- yes, a and a tilde
00:46:29.690 --> 00:46:31.660
do have different units.
00:46:31.660 --> 00:46:37.710
That's right, and
that's because in what
00:46:37.710 --> 00:46:40.130
one might call
conventional units here,
00:46:40.130 --> 00:46:43.070
r is some kind of a
coordinate distance.
00:46:43.070 --> 00:46:45.960
So in my language I'd
measure it in notches,
00:46:45.960 --> 00:46:49.810
and then a has units
of meters per notch.
00:46:49.810 --> 00:46:52.250
On the other hand,
here psi is an angle.
00:46:52.250 --> 00:46:54.900
It is naturally dimensionless.
00:46:54.900 --> 00:46:57.360
So one doesn't introduce
notches in this case,
00:46:57.360 --> 00:46:59.470
and therefore a just
has units of length--
00:46:59.470 --> 00:47:01.532
a tilde, rather-- just
has units of length.
00:47:06.730 --> 00:47:11.670
OK, now we want to imagine
that some distant galaxy is
00:47:11.670 --> 00:47:15.410
radiating-- or a distant
supernova, perhaps--
00:47:15.410 --> 00:47:17.815
and we want to ask,
what is the intensity
00:47:17.815 --> 00:47:20.780
of the radiation that
we receive on earth?
00:47:20.780 --> 00:47:22.530
And we'll draw the
same picture that we've
00:47:22.530 --> 00:47:25.390
drawn at least twice
before, if not more.
00:47:25.390 --> 00:47:27.600
We'll put the source
in the middle.
00:47:32.520 --> 00:47:35.740
We'll imagine a sphere
surrounding the source,
00:47:35.740 --> 00:47:38.570
with the source of
the center, and we'll
00:47:38.570 --> 00:47:42.270
imagine that the sphere has been
drawn so that our detector is
00:47:42.270 --> 00:47:44.357
on the surface of the sphere.
00:47:44.357 --> 00:47:45.440
This will be the detector.
00:47:54.650 --> 00:47:59.760
And we'll give a symbol for
the area of the detector.
00:47:59.760 --> 00:48:00.560
It will be a.
00:48:05.240 --> 00:48:07.860
And we'll imagine drawing this
in our co-moving coordinate
00:48:07.860 --> 00:48:11.590
system where psi is
our radial variable.
00:48:11.590 --> 00:48:17.070
So the sphere here
will be at some value,
00:48:17.070 --> 00:48:22.327
psi equals psi sub d-- where d
stands for detector-- and psi
00:48:22.327 --> 00:48:23.410
equals zero at the center.
00:49:07.780 --> 00:49:09.780
OK, I'm going to make the
same kind of arguments
00:49:09.780 --> 00:49:11.260
we've made in the past.
00:49:11.260 --> 00:49:18.690
We say that the
fraction of light
00:49:18.690 --> 00:49:24.480
that hits the sphere--
which hits the detector--
00:49:24.480 --> 00:49:36.359
is just equal to the
area of the detector
00:49:36.359 --> 00:49:37.525
over the area of the sphere.
00:49:40.612 --> 00:49:44.682
Now, the area of detector
is, by definition, a.
00:49:44.682 --> 00:49:46.182
The area of the
sphere we have to be
00:49:46.182 --> 00:49:47.840
a little bit careful
about because we
00:49:47.840 --> 00:49:51.540
have to calculate the area of
the sphere using the metric.
00:49:51.540 --> 00:49:54.000
Now, the metric is
slightly nontrivial,
00:49:54.000 --> 00:49:58.070
but the sphere is just described
by varying theta and phi.
00:49:58.070 --> 00:50:00.580
And if we just vary theta and
phi, this piece of the metric
00:50:00.580 --> 00:50:04.180
is what we're used to-- it's
the standard Euclidean spherical
00:50:04.180 --> 00:50:07.730
element-- and the coefficient
that multiplies is just
00:50:07.730 --> 00:50:10.100
the square of the
radius of that sphere.
00:50:10.100 --> 00:50:16.850
So the radius of our sphere
is a tilde times sine psi.
00:50:16.850 --> 00:50:20.640
That's the important thing
that we get from the metric.
00:50:20.640 --> 00:50:22.930
The thing that multiplies
d theta squared and d phi
00:50:22.930 --> 00:50:24.513
squared, et cetera,
is just the square
00:50:24.513 --> 00:50:26.750
of the radius of the sphere
that determines distances
00:50:26.750 --> 00:50:28.460
on the surface of the sphere.
00:50:28.460 --> 00:50:32.690
So what goes here is 4 pi
times the radius squared.
00:50:32.690 --> 00:50:41.570
So it's 4 pi times a tilde
squared of t naught times
00:50:41.570 --> 00:50:45.860
sine squared of psi sub d.
00:50:45.860 --> 00:50:49.770
It's t naught because
we're interested in what
00:50:49.770 --> 00:50:53.410
happens when we detect
this radiation today.
00:50:53.410 --> 00:50:56.710
Our detector is detecting it
today and has area a today,
00:50:56.710 --> 00:51:01.480
and we want to compare it with
entire sphere that surrounds
00:51:01.480 --> 00:51:04.990
this distant source as
that sphere appears today,
00:51:04.990 --> 00:51:07.960
so that all of the distances
are measured today,
00:51:07.960 --> 00:51:09.900
and therefore can be
properly compared.
00:51:14.580 --> 00:51:18.090
The other thing we
have to remember
00:51:18.090 --> 00:51:19.340
is the effect of the redshift.
00:51:24.040 --> 00:51:25.555
The redshift,
we've said earlier,
00:51:25.555 --> 00:51:29.810
and it's just a repetition,
it reduces the energy
00:51:29.810 --> 00:51:34.790
of each photon by a factor
of 1 plus z, the redshift,
00:51:34.790 --> 00:51:38.480
and similarly it reduces
the rate at which photons
00:51:38.480 --> 00:51:43.570
are arriving at the sphere by
that same factor-- 1 plus z.
00:51:43.570 --> 00:51:46.420
It basically says that
any clock slows down
00:51:46.420 --> 00:51:49.320
by a factor of 1 plus
z, and that clock
00:51:49.320 --> 00:51:51.230
could be the frequency
of the photon-- which
00:51:51.230 --> 00:51:53.870
affects its energy
proportionally--
00:51:53.870 --> 00:51:55.490
or the arrival rate
of the photons.
00:51:55.490 --> 00:52:02.110
That's also a clock that get
time dilated in the same way.
00:52:02.110 --> 00:52:15.440
So we get two factors of 1
plus z sub s, I'll call it.
00:52:15.440 --> 00:52:20.310
s for z of the source,
the z between the time
00:52:20.310 --> 00:52:22.780
of emission at the
source to a time
00:52:22.780 --> 00:52:26.350
where it arrives at us today.
00:52:26.350 --> 00:52:32.580
So 1 plus z is equal to
a of t naught divided
00:52:32.580 --> 00:52:34.885
by a of t emission.
00:52:42.705 --> 00:52:44.900
I'll just put it
here to remind us.
00:52:48.280 --> 00:53:00.240
One from redshift of photons
and one from arrival rate.
00:53:15.310 --> 00:53:17.960
OK, putting that
together we can now
00:53:17.960 --> 00:53:26.200
say that the total
power received is
00:53:26.200 --> 00:53:30.830
equal to the power originally
emitted by the source-- p
00:53:30.830 --> 00:53:37.060
will just be the power emitted
by the source-- divided
00:53:37.060 --> 00:53:42.810
by 1 plus the redshift
z of the source squared,
00:53:42.810 --> 00:53:44.330
and then just
times the fraction.
00:53:44.330 --> 00:53:57.130
A over 4 pi a twiddle
squared sine squared psi d.
00:54:28.630 --> 00:54:30.720
And then, finally, what
we're really interested in
00:54:30.720 --> 00:54:35.370
is J-- the intensity
of the source
00:54:35.370 --> 00:54:39.300
as we measure it-- which
is just the power received
00:54:39.300 --> 00:54:47.430
by our detector
divided by its area.
00:54:47.430 --> 00:54:51.370
So from this formula we
just get rid of the A there.
00:54:51.370 --> 00:54:55.080
We can write it as the power
emitted by the source, capital
00:54:55.080 --> 00:55:02.480
P, divided by 4
pi 1 plus z sub s
00:55:02.480 --> 00:55:10.660
squared a twiddle squared
of t naught times sine
00:55:10.660 --> 00:55:14.350
squared psi sub d.
00:55:25.220 --> 00:55:27.040
Now that effectively
is the answer
00:55:27.040 --> 00:55:31.950
to this question except
that we prefer to rewrite it
00:55:31.950 --> 00:55:35.342
in terms of things
that are more directly
00:55:35.342 --> 00:55:36.425
meaningful to astronomers.
00:55:39.160 --> 00:55:42.050
a twiddle is not particularly
meaningful to the astronomer.
00:55:42.050 --> 00:55:43.970
The redshift is, that's OK.
00:55:43.970 --> 00:55:45.550
But a twiddle is
not particularly
00:55:45.550 --> 00:55:49.899
meaningful to an astronomer, nor
is sine squared of psi sub d.
00:55:49.899 --> 00:55:51.940
Now, many astronomers who
know general relativity
00:55:51.940 --> 00:55:54.030
can figure this out,
of course, but it's
00:55:54.030 --> 00:55:55.197
our job to figure it out.
00:55:55.197 --> 00:55:56.780
We would like to
express this in terms
00:55:56.780 --> 00:56:01.490
of things that are directly
measured by astronomers.
00:56:01.490 --> 00:56:08.460
So to do that, first
of all, a tilde--
00:56:08.460 --> 00:56:10.551
to get a tilde related
to other things,
00:56:10.551 --> 00:56:12.300
it really just goes
back to the definition
00:56:12.300 --> 00:56:15.180
that we gave for
omega sub k sub 0.
00:56:15.180 --> 00:56:19.930
And if you look back
at that definition,
00:56:19.930 --> 00:56:24.730
you'll find that a
of t naught tilde
00:56:24.730 --> 00:56:30.130
is just equal to c times the
inverse of the present Hubble
00:56:30.130 --> 00:56:36.800
expansion rate times the
square root of minus omega
00:56:36.800 --> 00:56:38.280
sub k comma zero.
00:56:41.840 --> 00:56:45.000
And this is for the
close-universe case.
00:56:45.000 --> 00:56:48.260
The closed-universe case,
little k is positive.
00:56:48.260 --> 00:56:51.247
But if you remember the
definition of omega sub k
00:56:51.247 --> 00:56:53.580
naught-- maybe I should write
it back on the blackboard,
00:56:53.580 --> 00:56:54.570
or is it findable?
00:56:54.570 --> 00:56:55.540
It's not findable.
00:57:06.630 --> 00:57:09.840
The original definition big
A for this omega sub k naught
00:57:09.840 --> 00:57:20.120
was just minus Kc squared over
a squared of t naught H naught
00:57:20.120 --> 00:57:21.620
squared.
00:57:21.620 --> 00:57:23.340
So this is just
rewriting of that,
00:57:23.340 --> 00:57:26.230
and for our closed-universe
case, k is positive,
00:57:26.230 --> 00:57:28.395
omega sub k naught is
negative, this is then
00:57:28.395 --> 00:57:30.260
the square root of a positive
number with that extra minus
00:57:30.260 --> 00:57:31.680
sign, so everything
fits together.
00:57:36.600 --> 00:57:38.580
So that takes care
of expressing a tilde
00:57:38.580 --> 00:57:41.060
in terms of measurable things.
00:57:41.060 --> 00:57:42.290
We use this formula.
00:57:42.290 --> 00:57:44.875
Expansion rate is measurable,
omega sub k comma zero
00:57:44.875 --> 00:57:47.500
is measurable.
00:57:47.500 --> 00:58:06.290
And then, in terms of
sine squared psi sub d,
00:58:06.290 --> 00:58:09.750
we obtain that by
reminding ourselves
00:58:09.750 --> 00:58:15.800
that we know how to trace light
rays through this universe.
00:58:15.800 --> 00:58:18.250
Light waves just travel
locally at the speed c,
00:58:18.250 --> 00:58:21.240
they travel locally
on null geodesics.
00:58:21.240 --> 00:58:23.650
So if we're looking
at a radial light ray,
00:58:23.650 --> 00:58:31.550
this metric tells us-- if we
apply it to a radial light ray
00:58:31.550 --> 00:58:36.600
where ds squared equals minus dc
squared-- where ds squared has
00:58:36.600 --> 00:58:44.080
to be zero-- that says
that minus c squared dt
00:58:44.080 --> 00:58:54.502
squared plus a twiddle squared
of t d psi squared equals zero.
00:58:54.502 --> 00:58:55.960
This is just the
equation that says
00:58:55.960 --> 00:59:00.060
we have a null line,
a null radial line.
00:59:00.060 --> 00:59:07.180
That implies that
deep psi dt has
00:59:07.180 --> 00:59:17.354
to equals c over a
twiddle of t, which
00:59:17.354 --> 00:59:18.770
is a formula that,
in other cases,
00:59:18.770 --> 00:59:21.650
we try to motivate just
by using intuition.
00:59:21.650 --> 00:59:23.410
But, in that case,
we were probably not
00:59:23.410 --> 00:59:24.868
talking about curved
universe where
00:59:24.868 --> 00:59:26.670
the intuition is a
little bit less strong.
00:59:26.670 --> 00:59:28.480
But you see it does
follow immediately
00:59:28.480 --> 00:59:31.140
from assuming that we're
talking about a null geodesic
00:59:31.140 --> 00:59:32.610
in the Robertson-Walker metric.
00:59:51.260 --> 00:59:54.130
Now, the point is that the
Friedmann equation, which we've
00:59:54.130 --> 00:59:58.835
been writing and rewriting,
tells us what to do with that.
01:00:07.076 --> 01:00:08.700
The Friedmann equation
basically allows
01:00:08.700 --> 01:00:12.310
us to integrate that
because it allows
01:00:12.310 --> 01:00:19.750
us to express a in
terms of x, and we
01:00:19.750 --> 01:00:21.580
know some things about x.
01:00:21.580 --> 01:00:24.010
So let me try to get that
on the blackboard, here.
01:00:27.910 --> 01:00:35.120
We know that H squared-- which
is x dot over x squared--
01:00:35.120 --> 01:00:40.050
can be written as H zero squared
over x to the fourth times
01:00:40.050 --> 01:00:42.700
this famous function F of x.
01:00:49.440 --> 01:01:00.840
And psi of a given redshift,
according to this equation,
01:01:00.840 --> 01:01:07.380
could just be obtained by
integration of the time
01:01:07.380 --> 01:01:09.960
that the source emits
the radiation up
01:01:09.960 --> 01:01:16.990
to the present time of c
over a twiddle of t times dt.
01:01:26.400 --> 01:01:30.130
And now to rewrite this
in terms of redshift,
01:01:30.130 --> 01:01:35.370
we can use the fact that
1 plus z is equal to 1
01:01:35.370 --> 01:01:38.680
over x because we know
how to relate 1 plus z
01:01:38.680 --> 01:01:40.179
to the scale factor.
01:01:40.179 --> 01:01:41.970
1 plus z is just the
ratio of scale factors
01:01:41.970 --> 01:01:46.200
and it's precisely the ratio
that we called 1 over x.
01:01:46.200 --> 01:01:52.770
And we can then
differentiate this equation
01:01:52.770 --> 01:02:02.560
and find that dz is equal to
minus a twiddle of t naught
01:02:02.560 --> 01:02:11.040
over a twiddle of t squared
times a twiddle dot times
01:02:11.040 --> 01:02:17.460
dt, rewriting x in terms of
a of t naught over a of t.
01:02:20.780 --> 01:02:27.095
And this, then, is equal to
minus a twiddle of t naught
01:02:27.095 --> 01:02:36.420
times H of t times, oops,
times dt over a tilde of t.
01:02:42.640 --> 01:02:52.450
And this allows us to replace
the dt that appears there
01:02:52.450 --> 01:03:00.080
and the final relationship
is that psi of s
01:03:00.080 --> 01:03:06.020
is equal to 1 over a tilde of
t naught times the integral
01:03:06.020 --> 01:03:17.910
from zero up to z sub
s of c over H of z dz.
01:03:25.130 --> 01:03:26.970
Yeah, I think that
looks like it works.
01:03:35.770 --> 01:03:38.260
So it really is just a
matter of changing variables
01:03:38.260 --> 01:03:42.570
to express things in terms
of H and integrating over z
01:03:42.570 --> 01:03:44.340
instead of integrating over t.
01:03:46.781 --> 01:03:48.280
And the usefulness
of that is simply
01:03:48.280 --> 01:03:50.280
that z is the variable
that astronomers
01:03:50.280 --> 01:03:51.300
use to measure time.
01:03:55.600 --> 01:03:58.120
And this then can be
written in more detail,
01:03:58.120 --> 01:04:02.820
and it really finishes
the answer more or less.
01:04:02.820 --> 01:04:07.840
Psi of z sub s can be
written just-- writing
01:04:07.840 --> 01:04:14.060
in what a tilde is according
to our definition here--
01:04:14.060 --> 01:04:18.030
square root of the
magnitude of omega comma k
01:04:18.030 --> 01:04:22.660
zero-- this could also have
been written as minus omega of k
01:04:22.660 --> 01:04:26.380
comma 0 because we know
it's a negative quantity--
01:04:26.380 --> 01:04:39.940
and then times the integral from
0 to z sub s and integral dz.
01:04:39.940 --> 01:04:43.680
Now I'm just writing
H as a function of z.
01:04:43.680 --> 01:04:48.910
Earlier we had written H--
it's no longer on the screen,
01:04:48.910 --> 01:04:52.980
I guess-- earlier we had
written H in terms of F of z,
01:04:52.980 --> 01:04:56.040
oh, excuse me, F of x.
01:04:56.040 --> 01:05:00.890
x is related to z
simply by this formula.
01:05:00.890 --> 01:05:02.340
So since the
integral was written
01:05:02.340 --> 01:05:04.430
with z as the variable
of integration,
01:05:04.430 --> 01:05:06.600
I'm going to rewrite the
integrand in terms of z,
01:05:06.600 --> 01:05:08.835
but it really is just
our old friend F of x.
01:05:13.720 --> 01:05:18.970
So it would be the square
root of omega sub m zero 1
01:05:18.970 --> 01:05:31.420
plus z cubed plus omega sub
radiation zero times 1 plus z
01:05:31.420 --> 01:05:36.090
to the fourth, all
inside the square root,
01:05:36.090 --> 01:05:43.950
here, plus omega sub vac
zero plus omega sub k
01:05:43.950 --> 01:05:47.025
zero times 1 plus z squared.
01:05:54.120 --> 01:05:58.920
And this, then, is the
answer for psi of z.
01:05:58.920 --> 01:06:04.470
And then we put that into here
and replace a twiddle by this,
01:06:04.470 --> 01:06:07.950
and we get a formula for
what we're looking for,
01:06:07.950 --> 01:06:11.360
an expression for the
actual measured intensity
01:06:11.360 --> 01:06:16.190
of the source at the Earth in
terms of the parameters chosen
01:06:16.190 --> 01:06:19.530
here-- the current
values of omega
01:06:19.530 --> 01:06:21.270
and the redshift of the source.
01:06:21.270 --> 01:06:24.010
And that's all that goes
into this final formula.
01:06:24.010 --> 01:06:25.960
So if you know the
current values of omega
01:06:25.960 --> 01:06:27.640
and the redshift
of the source, you
01:06:27.640 --> 01:06:32.450
can calculate what you expect
the measure intensity to be
01:06:32.450 --> 01:06:34.685
in terms of the
intrinsic intensity.
01:06:34.685 --> 01:06:36.600
And that's exactly
what the supernova
01:06:36.600 --> 01:06:39.970
people did in 1998 using
exactly this formula--
01:06:39.970 --> 01:06:43.350
nothing different--
and discovered that,
01:06:43.350 --> 01:06:46.640
in order to fit their data,
they needed a very significant
01:06:46.640 --> 01:06:49.585
contribution from this vacuum
energy, namely a contribution
01:06:49.585 --> 01:06:50.970
in the order of 60 or 70%.
01:06:53.800 --> 01:06:56.460
So we will stop there for today.
01:06:56.460 --> 01:06:59.470
We will continue on Thursday
to talk a little bit more
01:06:59.470 --> 01:07:03.630
about the physics
of vacuum energy.