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PROFESSOR: OK.

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Good morning, everybody.

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Let's get started.

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00:00:30,300 --> 00:00:33,060
Let me just begin by asking
if there are any questions,

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00:00:33,060 --> 00:00:36,960
either about logistical issues
or about physics issues?

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00:00:42,180 --> 00:00:42,860
OK.

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00:00:42,860 --> 00:00:45,650
Today we'll be
finishing our discussion

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of black-body
radiation by talking

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about the actual spectrum of
the cosmic microwave background

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00:00:52,480 --> 00:00:54,480
that we find in our universe.

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And then move on to talk about
the rather exciting discovery

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00:00:59,520 --> 00:01:04,810
in 1998 of the fact that
our universe today appears

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00:01:04,810 --> 00:01:09,480
to have a nonzero
cosmological constant.

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00:01:09,480 --> 00:01:13,490
So I want to begin by reviewing
what we did last time.

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And one of the reasons
why I do this is I

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think it's a good
opportunity for you

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00:01:17,810 --> 00:01:20,690
to ask questions that don't
occur to you the first time we

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00:01:20,690 --> 00:01:21,190
go through.

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And that, from my
point of view, has

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been an extraordinary success.

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00:01:23,950 --> 00:01:26,470
I think you've asked
great questions.

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00:01:26,470 --> 00:01:29,010
So we'll see what
comes up today.

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00:01:29,010 --> 00:01:33,440
We began the last
lecture by recalling,

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I think from the previous
lecture actually,

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00:01:36,080 --> 00:01:39,810
the basic formulas for
black-body radiation, which

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00:01:39,810 --> 00:01:42,280
is just the
radiation of massless

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00:01:42,280 --> 00:01:45,020
particles at a
given temperature.

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00:01:45,020 --> 00:01:48,800
And we have formulas for the
energy density, the pressure,

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00:01:48,800 --> 00:01:52,370
the number density, and the
entropy density, all of which

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00:01:52,370 --> 00:01:55,530
are given in terms of
two constants, little g

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00:01:55,530 --> 00:01:59,580
and little g star, which
is the only place where

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00:01:59,580 --> 00:02:04,230
the actual nature of
the matter comes in.

40
00:02:04,230 --> 00:02:08,150
G and g-star are both
equal to 2 for photons,

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00:02:08,150 --> 00:02:10,310
but these formulas
allow us to talk

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00:02:10,310 --> 00:02:13,590
about other kinds of
black-body radiation

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00:02:13,590 --> 00:02:18,660
as well, black-body radiation
of other kinds of particles.

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00:02:18,660 --> 00:02:21,280
As neutrinos are
also effectively

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00:02:21,280 --> 00:02:23,690
massless, so they contribute.

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00:02:23,690 --> 00:02:26,820
And in addition, e
plus e minus pairs,

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00:02:26,820 --> 00:02:29,020
if the temperature
gets hot enough so

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00:02:29,020 --> 00:02:31,880
that the mass of the
electrons becomes negligible

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00:02:31,880 --> 00:02:35,770
compared to kt, also contribute
to the cosmic background

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00:02:35,770 --> 00:02:36,820
radiation.

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00:02:36,820 --> 00:02:38,504
And if we want the
higher temperatures,

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00:02:38,504 --> 00:02:40,170
other particle will
start to contribute.

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00:02:40,170 --> 00:02:42,750
And at the highest
temperatures all particles

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act like black-body radiation.

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00:02:46,060 --> 00:02:50,430
The general formula
for g and g star

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00:02:50,430 --> 00:02:55,220
is that there is a factor
out front that depends on

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00:02:55,220 --> 00:02:58,210
whether the particle is
a boson or a fermion,

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00:02:58,210 --> 00:03:01,860
a particle which does or does
not obey the Pauli exclusion

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00:03:01,860 --> 00:03:02,870
principle.

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00:03:02,870 --> 00:03:06,980
Fermions do not, bosons-- excuse
me, I said that backwards.

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00:03:06,980 --> 00:03:09,670
Fermions obey the Pauli
exclusion principle,

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00:03:09,670 --> 00:03:13,100
bosons do not.

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00:03:13,100 --> 00:03:16,710
G and g star are
both 1 for bosons.

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00:03:16,710 --> 00:03:20,170
But for fermions there's
a factor of 7/8 for g

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00:03:20,170 --> 00:03:22,570
and 3/4 for g star.

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Yes?

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00:03:23,070 --> 00:03:24,820
AUDIENCE: Would you
mind quickly restating

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why the positron-electron
pairs act

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like radiation above
that temperature?

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00:03:29,657 --> 00:03:30,240
PROFESSOR: OK.

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The question is, why do
electron-positron pairs

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act like radiation at
these high temperatures?

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00:03:35,784 --> 00:03:37,450
And the answer is
that radiation is just

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00:03:37,450 --> 00:03:40,410
characterized by the fact that
the particles are effectively

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00:03:40,410 --> 00:03:41,840
massless.

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00:03:41,840 --> 00:03:44,110
And the effective
energy scale is

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00:03:44,110 --> 00:03:48,450
kt, that's the average thermal
energy in a thermal mix.

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00:03:48,450 --> 00:03:53,470
So as long as m e c squared
is small compared to kt,

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00:03:53,470 --> 00:03:55,220
electrons and positrons
think that they're

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00:03:55,220 --> 00:03:59,140
massless and act like
they're massless.

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And as I said, if you go to
higher temperatures still,

82
00:04:01,680 --> 00:04:03,755
all particles will act
like they're massless.

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00:04:06,580 --> 00:04:10,160
Coming back to the
story of g and g star,

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we have the factor out
front which depends on

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00:04:12,430 --> 00:04:14,350
whether they're
bosons or fermions.

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00:04:14,350 --> 00:04:16,570
And then that just
multiplies the total number

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00:04:16,570 --> 00:04:20,980
of particle types,
whereby a particle type--

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00:04:20,980 --> 00:04:25,410
we made a complete specification
of what kind of a thing it is.

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00:04:25,410 --> 00:04:28,300
And that includes specifying
what species of particle

90
00:04:28,300 --> 00:04:32,370
it is, whether it's a
particle or an anti-particle

91
00:04:32,370 --> 00:04:35,100
if that distinction
exists, and what the spin

92
00:04:35,100 --> 00:04:38,900
state is if the
particle has spin.

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00:04:38,900 --> 00:04:43,890
So we can try this out
now on some examples.

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00:04:43,890 --> 00:04:46,310
First example, will
be neutrinos which

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00:04:46,310 --> 00:04:50,090
play a very important role
in the early universe,

96
00:04:50,090 --> 00:04:55,910
and even in the particle number
balance of today's universe.

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00:04:55,910 --> 00:04:58,320
Neutrinos actually
have a small mass,

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00:04:58,320 --> 00:05:02,090
as we talked about last time and
as I'll review again this time.

99
00:05:02,090 --> 00:05:05,630
But nonetheless, as far
as cosmology is concerned,

100
00:05:05,630 --> 00:05:08,490
they effectively act
like massless particles

101
00:05:08,490 --> 00:05:10,490
although the story
about why they

102
00:05:10,490 --> 00:05:13,775
act like massless particles
is a little complicated.

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00:05:13,775 --> 00:05:15,400
It's more than just
saying that they're

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00:05:15,400 --> 00:05:18,530
mass is small, for
reasons that we'll see.

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00:05:18,530 --> 00:05:20,360
But anyway, I'm
nonetheless going

106
00:05:20,360 --> 00:05:24,410
to start by describing neutrinos
as if they were massless,

107
00:05:24,410 --> 00:05:27,520
as was believed to be the
case really until 15 years

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00:05:27,520 --> 00:05:28,170
ago or so.

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00:05:31,100 --> 00:05:33,390
The massless model
of the neutrino

110
00:05:33,390 --> 00:05:38,330
was a particle which
was always left-handed.

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00:05:38,330 --> 00:05:39,890
And by left-handed
what I mean is

112
00:05:39,890 --> 00:05:42,680
that for neutrinos, if
you took the angular

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00:05:42,680 --> 00:05:46,740
momentum of the neutrino in
the direction of the momentum,

114
00:05:46,740 --> 00:05:49,680
p hat there means dotted
with the unit vector

115
00:05:49,680 --> 00:05:52,360
in the direction of
the spatial momentum,

116
00:05:52,360 --> 00:05:56,650
you'd always get minus
1/2 in units of h bar.

117
00:05:56,650 --> 00:06:00,170
And conversely, all new
bars are right-handed which

118
00:06:00,170 --> 00:06:03,370
just means the same equation
holds with the opposite sign.

119
00:06:03,370 --> 00:06:06,370
So neutrinos always
have spins that oppose

120
00:06:06,370 --> 00:06:10,070
the direction of motion,
and anti-neutrinos always

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00:06:10,070 --> 00:06:13,710
have spins aligned with
the direction of motion.

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00:06:13,710 --> 00:06:16,140
Now, it's not obvious
but, if neutrinos

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00:06:16,140 --> 00:06:22,350
were massless this would be a
Lorentz invariant statement.

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00:06:22,350 --> 00:06:26,030
If neutrinos have a
mass, that statement

125
00:06:26,030 --> 00:06:28,720
is obviously not learn
Lorentz invariant.

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00:06:28,720 --> 00:06:32,340
As you can see by imagining
a neutrino coming by,

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00:06:32,340 --> 00:06:35,170
and you can get
into a rocket ship,

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00:06:35,170 --> 00:06:38,770
chase it, and pass it, and
then see it going the other way

129
00:06:38,770 --> 00:06:41,700
out your window because
you're going faster than it.

130
00:06:41,700 --> 00:06:44,699
You would see the momentum
in the opposite direction

131
00:06:44,699 --> 00:06:46,240
from the way it
looked to begin with.

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00:06:46,240 --> 00:06:48,960
But the spin would look like
it was the same direction as it

133
00:06:48,960 --> 00:06:51,740
did to begin with,
and therefore the spin

134
00:06:51,740 --> 00:06:54,041
would now be aligned
with the momentum

135
00:06:54,041 --> 00:06:55,415
instead of opposite
the momentum.

136
00:06:58,350 --> 00:07:00,310
So this could not
possibly hold universally

137
00:07:00,310 --> 00:07:02,840
if the neutrino has a mass.

138
00:07:02,840 --> 00:07:06,727
But for the time being our
neutrinos are massless.

139
00:07:06,727 --> 00:07:08,560
So we're going to take
this as a given fact.

140
00:07:08,560 --> 00:07:10,930
And it certainly is a fact
for all neutrinos that

141
00:07:10,930 --> 00:07:14,710
have ever been
actually measured.

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00:07:14,710 --> 00:07:20,052
Given this model of the
neutrino, the g for neutrinos

143
00:07:20,052 --> 00:07:22,850
is 7/8 because they're fermions.

144
00:07:22,850 --> 00:07:24,830
Then there's a factor
of 3, because there

145
00:07:24,830 --> 00:07:28,670
are three different species
of neutrinos- electron

146
00:07:28,670 --> 00:07:33,610
neutrinos, muon
neutrinos, tau neutrinos.

147
00:07:33,610 --> 00:07:36,620
Neutrinos come in particles
or anti-particles which

148
00:07:36,620 --> 00:07:39,340
are distinct from
each other, we think.

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00:07:39,340 --> 00:07:42,360
So there's a factor of 2
associated with the particle

150
00:07:42,360 --> 00:07:45,150
anti-particle duality.

151
00:07:45,150 --> 00:07:46,850
And there's only one spin state.

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00:07:46,850 --> 00:07:51,530
The spin that's anti-aligned
with the momentum,

153
00:07:51,530 --> 00:07:53,520
or aligned for the
anti-neutrinos.

154
00:07:53,520 --> 00:07:55,510
But only one spin
state in either case.

155
00:07:55,510 --> 00:07:57,680
So just a factor of
1 from spin states,

156
00:07:57,680 --> 00:08:02,180
and multiplying that
through we get 21/4 for g,

157
00:08:02,180 --> 00:08:04,860
and 9/2 for g star.

158
00:08:04,860 --> 00:08:05,360
Yes?

159
00:08:05,360 --> 00:08:08,044
AUDIENCE: If we found out
that they were Majorana,

160
00:08:08,044 --> 00:08:09,710
that they were their
own anti-particles,

161
00:08:09,710 --> 00:08:13,970
would that change what we expect
the temperature [INAUDIBLE]

162
00:08:13,970 --> 00:08:14,930
to be?

163
00:08:14,930 --> 00:08:16,320
PROFESSOR: No, it would not.

164
00:08:16,320 --> 00:08:16,520
OK.

165
00:08:16,520 --> 00:08:17,740
The question was, if
we find that they're

166
00:08:17,740 --> 00:08:19,044
Majorana particles--
which I'm going

167
00:08:19,044 --> 00:08:20,419
to be talking
about in a minute--

168
00:08:20,419 --> 00:08:23,560
where the particles would be
their own anti-particles, which

169
00:08:23,560 --> 00:08:27,220
would mean that the right-handed
anti-neutrino would really just

170
00:08:27,220 --> 00:08:30,250
be the anti-particle of
the left-handed neutrino,

171
00:08:30,250 --> 00:08:33,490
it would not change these
final numbers at all.

172
00:08:33,490 --> 00:08:37,110
What it would do is, instead
of having the 2 for particle

173
00:08:37,110 --> 00:08:42,090
anti-particle, we would
have a 2 for spin states.

174
00:08:42,090 --> 00:08:45,285
So there would still be
two kinds of neutrinos,

175
00:08:45,285 --> 00:08:47,910
but instead of calling them the
neutrino and the anti-neutrino,

176
00:08:47,910 --> 00:08:50,490
the right words would
be right-handed neutrino

177
00:08:50,490 --> 00:08:51,545
and left-handed neutrino.

178
00:08:55,156 --> 00:08:57,749
But the product would
still be the same.

179
00:08:57,749 --> 00:09:02,050
AUDIENCE: Wait, they have
mass and they are Majorana?

180
00:09:02,050 --> 00:09:03,800
PROFESSOR: If they
have mass and Majorana,

181
00:09:03,800 --> 00:09:05,541
what I just said applies.

182
00:09:05,541 --> 00:09:07,040
The fact that they
have a mass would

183
00:09:07,040 --> 00:09:09,830
mean at the lowest possible
temperatures they would not

184
00:09:09,830 --> 00:09:11,820
act like black-body radiation.

185
00:09:11,820 --> 00:09:15,944
Kt would have to be bigger than
their mass times c squared.

186
00:09:15,944 --> 00:09:18,235
But that's only on the order
of electron volts at most.

187
00:09:22,060 --> 00:09:28,470
So I'll talk later about why
the true model neutrinos which

188
00:09:28,470 --> 00:09:30,260
have masses give the
same result as this.

189
00:09:34,190 --> 00:09:35,160
OK.

190
00:09:35,160 --> 00:09:37,280
Then we can also,
just as an exercise,

191
00:09:37,280 --> 00:09:38,365
calculate g and g star.

192
00:09:38,365 --> 00:09:39,490
It's more than an exercise.

193
00:09:39,490 --> 00:09:41,010
We like to know the results.

194
00:09:41,010 --> 00:09:44,510
We can calculate g and g star
for e plus e minus pairs, which

195
00:09:44,510 --> 00:09:46,800
is relevant for when
kt is large compared

196
00:09:46,800 --> 00:09:50,060
to the rest energy
of an electron.

197
00:09:50,060 --> 00:09:52,880
And again, they
are fermions so we

198
00:09:52,880 --> 00:09:56,240
get a factor of 7/8 appearing
in the expression for g,

199
00:09:56,240 --> 00:09:59,527
and 3/4 appearing in the
expression for g star.

200
00:09:59,527 --> 00:10:01,360
And then we just have
to multiply that times

201
00:10:01,360 --> 00:10:06,642
the total number of types
of electrons that exist.

202
00:10:06,642 --> 00:10:08,475
There's only one species
called an electron,

203
00:10:08,475 --> 00:10:11,280
so we only get a factor
of one in the species

204
00:10:11,280 --> 00:10:13,640
slot of the product.

205
00:10:13,640 --> 00:10:16,470
There are both electrons
and anti-electrons

206
00:10:16,470 --> 00:10:19,870
where the anti-electrons are
usually called positrons.

207
00:10:19,870 --> 00:10:22,620
So we get a factor of 2
in particle anti-particle.

208
00:10:22,620 --> 00:10:25,050
Two spin states because an
electron can be spin up or spin

209
00:10:25,050 --> 00:10:29,260
down, and that
gives us 7/2 and 3.

210
00:10:33,990 --> 00:10:38,200
Given that, we can go
ahead and calculate

211
00:10:38,200 --> 00:10:41,980
what the energy density
and radiation should

212
00:10:41,980 --> 00:10:44,200
be for the present
universe given

213
00:10:44,200 --> 00:10:46,980
the temperature of the
photons, the temperature

214
00:10:46,980 --> 00:10:50,360
of the cosmic
microwave background.

215
00:10:50,360 --> 00:10:53,900
And in doing that there's
an important catch which

216
00:10:53,900 --> 00:10:57,220
is something which is the
subject of a homework problem

217
00:10:57,220 --> 00:11:00,380
that you'll be doing
on problem set seven.

218
00:11:00,380 --> 00:11:03,460
When the electron-positron
pairs disappear

219
00:11:03,460 --> 00:11:06,680
from the thermal
equilibrium mix,

220
00:11:06,680 --> 00:11:10,400
if everything were still
in thermal contact,

221
00:11:10,400 --> 00:11:16,580
its heat would be shared between
the photons and the neutrinos

222
00:11:16,580 --> 00:11:19,770
in a way that would keep
a common temperature.

223
00:11:19,770 --> 00:11:25,220
But in fact, when the e plus
e minus pairs disappear,

224
00:11:25,220 --> 00:11:27,350
things are not in
thermal contact anymore.

225
00:11:27,350 --> 00:11:29,600
And in particular, the
neutrinos have decoupled.

226
00:11:29,600 --> 00:11:32,350
They're effectively not
interacting with anything

227
00:11:32,350 --> 00:11:33,640
anymore.

228
00:11:33,640 --> 00:11:36,320
So the neutrinos keep
their own entropy

229
00:11:36,320 --> 00:11:39,940
and do not absorb any entropy
coming from the e plus

230
00:11:39,940 --> 00:11:41,900
e minus pairs.

231
00:11:41,900 --> 00:11:46,890
So all the entropy of the
e plus e minus pairs gets

232
00:11:46,890 --> 00:11:50,190
transferred only to the photons.

233
00:11:50,190 --> 00:11:54,240
And that heats the photons
relative to the neutrinos

234
00:11:54,240 --> 00:11:57,260
in a calculable
amount, which you

235
00:11:57,260 --> 00:11:59,780
will calculate on
the homework problem.

236
00:11:59,780 --> 00:12:02,090
And the answer is that the
temperature of the neutrinos

237
00:12:02,090 --> 00:12:06,170
ends up being only
4/11 to the 1/3 power,

238
00:12:06,170 --> 00:12:08,907
times the temperature
of the photons.

239
00:12:08,907 --> 00:12:10,490
And that's important
for understanding

240
00:12:10,490 --> 00:12:12,870
what's been happening in the
universe since this time.

241
00:12:12,870 --> 00:12:17,699
That ratio is maintained
forever from that time onward.

242
00:12:17,699 --> 00:12:20,115
So if we want to write down
the formula for energy density

243
00:12:20,115 --> 00:12:24,460
and radiation today it
would have two terms.

244
00:12:24,460 --> 00:12:28,100
The 2 here is the g for the
photons, and this, times

245
00:12:28,100 --> 00:12:32,920
that expression is the
energy density in photons.

246
00:12:32,920 --> 00:12:36,790
The second term is the
energy density in neutrinos.

247
00:12:36,790 --> 00:12:39,370
And it has the
factor of 21/4 which

248
00:12:39,370 --> 00:12:42,210
was the g factor for neutrinos.

249
00:12:42,210 --> 00:12:44,470
But then there's also
a correction factor

250
00:12:44,470 --> 00:12:47,260
for the temperature, because
on the right hand side here

251
00:12:47,260 --> 00:12:49,200
I put t gamma to the fourth.

252
00:12:49,200 --> 00:12:52,420
So this factor corrects it
to make it into t neutrino

253
00:12:52,420 --> 00:12:54,470
to the fourth, which
is what we need there

254
00:12:54,470 --> 00:12:59,580
to give the right energy
density for the neutrinos today.

255
00:12:59,580 --> 00:13:02,850
And this is just that
ratio to the fourth power.

256
00:13:02,850 --> 00:13:04,810
And once you plug in
numbers there it's

257
00:13:04,810 --> 00:13:09,700
7.01 times 10 to the minus 14th
joules per meter cubed, which

258
00:13:09,700 --> 00:13:11,200
is, from the
beginning, what we said

259
00:13:11,200 --> 00:13:15,210
was the energy density in
radiation of the universe

260
00:13:15,210 --> 00:13:15,710
today.

261
00:13:21,581 --> 00:13:22,080
OK.

262
00:13:22,080 --> 00:13:26,120
Finally I'd like to come back
to this real story of neutrinos

263
00:13:26,120 --> 00:13:33,520
and their masses and why, even
though they have small masses,

264
00:13:33,520 --> 00:13:37,550
the answers that we gave for the
massless model of the neutrino

265
00:13:37,550 --> 00:13:41,280
are completely
accurate for cosmology.

266
00:13:41,280 --> 00:13:44,640
We've never actually measured
the mass of a neutrino.

267
00:13:44,640 --> 00:13:48,440
But what we have seen is
that neutrinos of one species

268
00:13:48,440 --> 00:13:53,650
can oscillate into neutrinos
of the other species.

269
00:13:53,650 --> 00:13:55,400
And it turns out,
theoretically, that that

270
00:13:55,400 --> 00:13:57,610
requires them to have a mass.

271
00:13:57,610 --> 00:14:00,100
And by seeing how
fast they oscillate

272
00:14:00,100 --> 00:14:02,320
you can actually
measure the difference

273
00:14:02,320 --> 00:14:06,254
in the mass squareds
between the two species.

274
00:14:06,254 --> 00:14:08,170
So it's still possible
actually, in principle,

275
00:14:08,170 --> 00:14:10,610
that one of the species
could have zero mass.

276
00:14:10,610 --> 00:14:12,380
But they can't
all have zero mass

277
00:14:12,380 --> 00:14:16,670
because we know the differences
in the squares of their masses.

278
00:14:16,670 --> 00:14:19,800
So in particular, delta
m squared 2 1 times c

279
00:14:19,800 --> 00:14:23,560
to the fourth, meaning the
mass expressed as an energy,

280
00:14:23,560 --> 00:14:28,420
is 7.5 times 10 to the
minus 5 eV squared.

281
00:14:28,420 --> 00:14:31,760
And larger values
obtained for 2 3,

282
00:14:31,760 --> 00:14:36,670
which is 2.3 times 10 to
the minus 3 eV squared.

283
00:14:36,670 --> 00:14:41,000
We're still talking
about fractions of an eV.

284
00:14:41,000 --> 00:14:44,670
The other of the three
possible combinations

285
00:14:44,670 --> 00:14:48,340
here are just not known yet.

286
00:14:48,340 --> 00:14:51,030
Now, if neutrinos have a mass,
that does actually change

287
00:14:51,030 --> 00:14:56,790
things rather dramatically
because of what we said about--

288
00:14:56,790 --> 00:14:59,760
the statement that the neutrinos
always align their spins with

289
00:14:59,760 --> 00:15:03,990
their motion just cannot be
true if neutrinos have a mass.

290
00:15:03,990 --> 00:15:06,840
And more generally,
for any particle

291
00:15:06,840 --> 00:15:10,950
with a mass of arbitrary
spin j, the statement

292
00:15:10,950 --> 00:15:14,780
is that, the component of j
along any particular axis--

293
00:15:14,780 --> 00:15:17,460
and we'll call it
the z-axis-- always

294
00:15:17,460 --> 00:15:20,590
takes on the possible
values in terms of h bar

295
00:15:20,590 --> 00:15:24,587
going from minus j up
to j with no emissions.

296
00:15:24,587 --> 00:15:26,170
It's different for
massless particles.

297
00:15:26,170 --> 00:15:29,400
For massless particles
every one of these elements

298
00:15:29,400 --> 00:15:32,550
on the right hand side is
independent and, by itself,

299
00:15:32,550 --> 00:15:36,990
a Lorentz invariant possibility.

300
00:15:36,990 --> 00:15:40,810
But, coming back to
neutrinos-- if the neutrinos

301
00:15:40,810 --> 00:15:45,880
have a mass, in addition to
the left-handed neutrinos there

302
00:15:45,880 --> 00:15:49,470
has to be a
right-handed partner.

303
00:15:49,470 --> 00:15:55,342
And the question then is,
what's the story behind that?

304
00:15:55,342 --> 00:15:57,050
And it turns out we
don't know the story.

305
00:15:57,050 --> 00:16:01,510
But we know two
possible stories.

306
00:16:01,510 --> 00:16:03,470
And one of the
possible stories is

307
00:16:03,470 --> 00:16:07,740
that could be what's
called a Dirac mass.

308
00:16:07,740 --> 00:16:10,300
And for Dirac mass
what it means is that,

309
00:16:10,300 --> 00:16:16,130
the right-handed neutrino is
simply a new type of particle

310
00:16:16,130 --> 00:16:18,470
which just happens to be a
particle that we've never

311
00:16:18,470 --> 00:16:20,970
seen, but a particle
which would have

312
00:16:20,970 --> 00:16:23,960
a perfectly real existence.

313
00:16:23,960 --> 00:16:27,920
It would however, to fit
into theory and observation,

314
00:16:27,920 --> 00:16:31,340
be an extremely weakly
interacting particle.

315
00:16:31,340 --> 00:16:33,170
The interactions of
the right-handed one

316
00:16:33,170 --> 00:16:35,510
do not have to be the
same as the interactions

317
00:16:35,510 --> 00:16:38,300
of the left-handed one.

318
00:16:38,300 --> 00:16:40,580
That is, the interactions
can depend explicitly

319
00:16:40,580 --> 00:16:43,016
on p hat dot j.

320
00:16:43,016 --> 00:16:45,270
So depending its
value, you could

321
00:16:45,270 --> 00:16:48,270
affect what the
interactions are, again,

322
00:16:48,270 --> 00:16:50,750
in a Lorentz invariant way.

323
00:16:50,750 --> 00:16:55,250
And in practice, the
right-handed neutrinos

324
00:16:55,250 --> 00:16:58,400
would indirect so
weakly that we would not

325
00:16:58,400 --> 00:17:00,410
expect to see them
in the laboratory.

326
00:17:00,410 --> 00:17:03,173
And we would not expect
even in the early universe

327
00:17:03,173 --> 00:17:04,589
that they would
have been produced

328
00:17:04,589 --> 00:17:06,640
in any significant number.

329
00:17:06,640 --> 00:17:08,710
So even though it would
be a particle that,

330
00:17:08,710 --> 00:17:12,829
in principle, exists, we
would not expect to see it.

331
00:17:12,829 --> 00:17:18,000
And we would not expected it
to affect the early universe.

332
00:17:18,000 --> 00:17:22,550
Alternatively-- and in some
ways a more subtle idea--

333
00:17:22,550 --> 00:17:24,805
is that the mass of
the neutrino could

334
00:17:24,805 --> 00:17:27,530
be what's called
a Majorana mass.

335
00:17:27,530 --> 00:17:30,400
Where Majorana, like Dirac,
is the name of a person--

336
00:17:30,400 --> 00:17:33,510
perhaps less well
known than Dirac,

337
00:17:33,510 --> 00:17:39,210
but made important contributions
in this context nonetheless.

338
00:17:39,210 --> 00:17:43,830
In this case, it can only
occur if lepton number is not

339
00:17:43,830 --> 00:17:45,650
conserved.

340
00:17:45,650 --> 00:17:47,780
And if lepton number
is not conserved then

341
00:17:47,780 --> 00:17:49,710
there are really
no quantum numbers

342
00:17:49,710 --> 00:17:51,210
that separate the
particle that we

343
00:17:51,210 --> 00:17:52,740
call a neutrino
from the particle

344
00:17:52,740 --> 00:17:56,000
that we call an anti-neutrino.

345
00:17:56,000 --> 00:17:59,190
And if that's the
case, the particle

346
00:17:59,190 --> 00:18:01,750
that we call the anti-neutrino
could, in fact, just

347
00:18:01,750 --> 00:18:06,740
be the right-handed
partner of the neutrino.

348
00:18:06,740 --> 00:18:09,190
So for the Majorana
mass case we don't

349
00:18:09,190 --> 00:18:12,850
need to introduce any new things
that we haven't already seen.

350
00:18:12,850 --> 00:18:15,570
We just have to rename
the thing that we've

351
00:18:15,570 --> 00:18:21,520
been calling the anti-neutrino
the neutrino with helicity

352
00:18:21,520 --> 00:18:27,132
plus 1 instead of minus 1-- it's
with j hat dot p with j dot p

353
00:18:27,132 --> 00:18:32,840
hat, equal to plus 1/2
instead of minus 1/2.

354
00:18:32,840 --> 00:18:34,590
So these would just
be the two spin states

355
00:18:34,590 --> 00:18:37,230
of the neutrino instead of the
neutrino and the anti-neutrino.

356
00:18:37,230 --> 00:18:38,271
And that's a possibility.

357
00:18:40,840 --> 00:18:43,670
And this would also change
nothing as far as the counting

358
00:18:43,670 --> 00:18:44,600
that we did.

359
00:18:44,600 --> 00:18:48,430
It would just change where
the factors go instead

360
00:18:48,430 --> 00:18:50,040
of having a factor
of 2 for particle

361
00:18:50,040 --> 00:18:53,670
anti-particle and
this type counting.

362
00:18:53,670 --> 00:18:56,322
We'd have a factor of 2
in the spin state factor,

363
00:18:56,322 --> 00:18:58,280
and a factor of 1 in the
particle anti-particle

364
00:18:58,280 --> 00:18:58,780
[INAUDIBLE].

365
00:18:58,780 --> 00:19:00,196
The particle and
the anti-particle

366
00:19:00,196 --> 00:19:01,250
would be the same thing.

367
00:19:03,580 --> 00:19:04,080
OK.

368
00:19:04,080 --> 00:19:05,140
Any questions about that?

369
00:19:10,490 --> 00:19:11,700
OK.

370
00:19:11,700 --> 00:19:15,260
Finally-- and I think this is
my last slide of the summary.

371
00:19:15,260 --> 00:19:17,510
At the end of the lecture
we just pointed out

372
00:19:17,510 --> 00:19:21,650
a number of tidbits
of information.

373
00:19:21,650 --> 00:19:24,300
We can calculate the temperature
of the early universe

374
00:19:24,300 --> 00:19:26,550
at any time from the
formulas that we already

375
00:19:26,550 --> 00:19:28,860
had on the slide.

376
00:19:28,860 --> 00:19:31,505
We know how to calculate the
energy density at any time.

377
00:19:31,505 --> 00:19:33,950
And by knowing about
black-body radiation

378
00:19:33,950 --> 00:19:35,970
we can convert that
into a temperature.

379
00:19:35,970 --> 00:19:39,690
And for an important
interval of time,

380
00:19:39,690 --> 00:19:44,000
which is when kt is small enough
so that you don't make muon

381
00:19:44,000 --> 00:19:47,850
anti-muon pairs, but
large enough so that

382
00:19:47,850 --> 00:19:52,470
electron-positron pairs
act like they are massless,

383
00:19:52,470 --> 00:19:59,310
and this very large range
kt is equal to 0.860 m e v,

384
00:19:59,310 --> 00:20:01,940
divided by the square root of
time where time is measured

385
00:20:01,940 --> 00:20:03,980
in seconds.

386
00:20:03,980 --> 00:20:09,296
So in particular, at 1
second kt is 0.86 m e

387
00:20:09,296 --> 00:20:14,410
v. And it does
apply at 1 second.

388
00:20:17,296 --> 00:20:20,070
Because 0.86 m e v
is in this range.

389
00:20:24,790 --> 00:20:26,570
We also then talked
about the implications

390
00:20:26,570 --> 00:20:29,340
of the conservation of entropy.

391
00:20:29,340 --> 00:20:32,670
If total entropy is
conserved, the entropy density

392
00:20:32,670 --> 00:20:37,620
has to just fall off like 1
over the cube of the volume.

393
00:20:37,620 --> 00:20:40,380
Total entropy is conserved
for almost all processes

394
00:20:40,380 --> 00:20:42,370
in the early universe.

395
00:20:42,370 --> 00:20:46,810
So the entropy falls
off like 1 over a cubed.

396
00:20:46,810 --> 00:20:48,560
And that means that,
as long as we're

397
00:20:48,560 --> 00:20:52,920
talking about a period of time
during which little g does not

398
00:20:52,920 --> 00:20:57,040
change-- and little g only
changes when particles freeze

399
00:20:57,040 --> 00:20:59,140
out, like when the
electron-positron pairs

400
00:20:59,140 --> 00:21:02,960
disappear-- but as long as
little g doesn't change,

401
00:21:02,960 --> 00:21:04,980
s [INAUDIBLE] 1 over
a cubed means simply

402
00:21:04,980 --> 00:21:08,670
that the temperature
falls like 1 over a.

403
00:21:08,670 --> 00:21:11,420
And when little g changes you
can even calculate corrections

404
00:21:11,420 --> 00:21:14,140
to this as, effectively
you're doing

405
00:21:14,140 --> 00:21:16,020
when you calculate
this relationship

406
00:21:16,020 --> 00:21:20,600
between the neutrino temperature
and the photon temperature.

407
00:21:20,600 --> 00:21:24,240
And finally, we talked
about the behavior

408
00:21:24,240 --> 00:21:30,220
of the atoms in the universe
as the universe cools.

409
00:21:30,220 --> 00:21:34,020
For temperatures above
about 4,000 degrees

410
00:21:34,020 --> 00:21:36,730
the universe, which
is mainly hydrogen,

411
00:21:36,730 --> 00:21:40,030
is mainly a hydrogen plasma.

412
00:21:40,030 --> 00:21:42,970
Isolated protons and electrons
zipping through space

413
00:21:42,970 --> 00:21:44,890
independently.

414
00:21:44,890 --> 00:21:48,340
At about 4,000 Kelvin-- and
this is a stat [? mac ?]

415
00:21:48,340 --> 00:21:54,830
calculation, which we're not
doing-- but using the answer.

416
00:21:54,830 --> 00:21:57,660
At about 4,000 Kelvin--
which is a number which

417
00:21:57,660 --> 00:21:59,950
depends on the density of
hydrogen in the universe,

418
00:21:59,950 --> 00:22:03,450
it's not a universal
property of hydrogen--

419
00:22:03,450 --> 00:22:05,860
but for the density of
hydrogen in the universe,

420
00:22:05,860 --> 00:22:09,800
at about 4,000 Kelvin
hydrogen recombines.

421
00:22:09,800 --> 00:22:11,105
It becomes neutral atoms.

422
00:22:14,090 --> 00:22:21,210
And slightly colder, at about
3,000, the degree of ionization

423
00:22:21,210 --> 00:22:23,800
becomes small enough
so that the photons

424
00:22:23,800 --> 00:22:25,800
become effectively free.

425
00:22:25,800 --> 00:22:27,630
The photons decouple.

426
00:22:27,630 --> 00:22:31,610
In between 4,000 and 3,000 the
hydrogen is mostly neutral,

427
00:22:31,610 --> 00:22:34,860
but they're still enough ionized
so that the photons are still

428
00:22:34,860 --> 00:22:37,420
interacting.

429
00:22:37,420 --> 00:22:39,170
So the most important
temperatures--

430
00:22:39,170 --> 00:22:41,940
the 3,000 Kelvin,
when the photons

431
00:22:41,940 --> 00:22:45,160
are released, when the
photons are no longer trapped

432
00:22:45,160 --> 00:22:48,640
with the matter of the universe.

433
00:22:48,640 --> 00:22:53,530
And last time we estimated the
time at which that happens.

434
00:22:53,530 --> 00:22:55,320
That should be a small t, sorry.

435
00:22:55,320 --> 00:22:59,510
The time of decoupling
is about 380,000 years.

436
00:22:59,510 --> 00:23:01,260
And that number is
actually very accurate,

437
00:23:01,260 --> 00:23:05,150
even though we didn't
calculate it very accurately.

438
00:23:05,150 --> 00:23:08,230
And that's the
end of my summary.

439
00:23:08,230 --> 00:23:10,420
Any questions about the summary?

440
00:23:16,560 --> 00:23:17,760
OK.

441
00:23:17,760 --> 00:23:22,400
In that case, let's
go on to talk first

442
00:23:22,400 --> 00:23:24,830
about the spectrum of the
cosmic background radiation.

443
00:23:27,280 --> 00:23:28,780
And then we'll move
on to talk about

444
00:23:28,780 --> 00:23:29,863
the cosmological constant.

445
00:23:46,270 --> 00:23:49,680
CMB is cosmic
microwave background.

446
00:23:49,680 --> 00:23:52,370
And that's a very, very standard
abbreviation these days.

447
00:23:55,410 --> 00:23:57,850
So when the cosmic microwave
background was first

448
00:23:57,850 --> 00:24:02,492
discovered by Penzias and
Wilson in 1965-- which,

449
00:24:02,492 --> 00:24:03,950
I might point out,
is going to have

450
00:24:03,950 --> 00:24:09,100
its 50th anniversary
in the coming year--

451
00:24:09,100 --> 00:24:11,840
they only measured
it at one frequency.

452
00:24:11,840 --> 00:24:16,800
It was a real tour de force to
measure it at the one frequency

453
00:24:16,800 --> 00:24:20,860
and to convince themselves that
the buzz that they were hearing

454
00:24:20,860 --> 00:24:23,500
in their detector was
not just some kind

455
00:24:23,500 --> 00:24:27,730
of random electrical noise,
but really was some signal

456
00:24:27,730 --> 00:24:30,090
coming from outer space.

457
00:24:30,090 --> 00:24:33,130
And the main clue that
it was some signal coming

458
00:24:33,130 --> 00:24:37,340
from outer space
was that they were

459
00:24:37,340 --> 00:24:41,702
able to compare it
with a cold load,

460
00:24:41,702 --> 00:24:47,720
a liquid helium-cooled
source, and find

461
00:24:47,720 --> 00:24:50,460
that that comparison worked
the way they expected.

462
00:24:50,460 --> 00:24:52,340
And the main reason
for believing

463
00:24:52,340 --> 00:24:54,599
it was cosmological
rather than local

464
00:24:54,599 --> 00:24:56,390
is that they got the
same reading no matter

465
00:24:56,390 --> 00:24:58,140
what direction they
pointed their antenna.

466
00:25:01,660 --> 00:25:06,770
This just took a lot of
radio technique skill

467
00:25:06,770 --> 00:25:08,520
to convince themselves
that it wasn't just

468
00:25:08,520 --> 00:25:12,674
some radio tube that was
malfunctioning or something.

469
00:25:12,674 --> 00:25:14,090
They even worried
that it may have

470
00:25:14,090 --> 00:25:18,510
been caused by pigeon
droppings in their antenna,

471
00:25:18,510 --> 00:25:21,600
I actually read about
in Weinberg's book.

472
00:25:21,600 --> 00:25:25,850
But they finally convinced
themselves that it was real.

473
00:25:25,850 --> 00:25:27,470
They were still not
convinced really

474
00:25:27,470 --> 00:25:30,510
that it was a sign
for the big bang

475
00:25:30,510 --> 00:25:34,470
and-- you may recall,
again, from reading Weinberg

476
00:25:34,470 --> 00:25:37,960
that there were two papers
published back-to-back.

477
00:25:37,960 --> 00:25:40,110
The experimental paper
by Penzias and Wilson,

478
00:25:40,110 --> 00:25:42,360
which really just
described the experiment,

479
00:25:42,360 --> 00:25:44,660
mentioning that a
possible explanation

480
00:25:44,660 --> 00:25:47,380
was in this other paper
by Dickie, Peebles, Roll,

481
00:25:47,380 --> 00:25:50,820
and Wilkinson which described
the theory that this

482
00:25:50,820 --> 00:25:55,010
was radiation that
originated with the big bang.

483
00:25:55,010 --> 00:25:58,720
But it's all based on one
point at one frequency.

484
00:25:58,720 --> 00:26:02,810
Shortly afterwards, I
guess within the same year,

485
00:26:02,810 --> 00:26:04,843
Roll and Wilkinson
were able to measure it

486
00:26:04,843 --> 00:26:07,160
at a slightly
different frequency.

487
00:26:07,160 --> 00:26:11,440
And when I wrote my
popular-level book

488
00:26:11,440 --> 00:26:18,300
I tabulated all of the data
that was known in 1975.

489
00:26:18,300 --> 00:26:22,350
And this mess is the graph.

490
00:26:22,350 --> 00:26:23,940
This shows sort
of the full range

491
00:26:23,940 --> 00:26:25,820
of interesting frequencies.

492
00:26:25,820 --> 00:26:30,180
The solid line here is the
expected theoretical curve

493
00:26:30,180 --> 00:26:31,850
corresponding to a
modern measurement

494
00:26:31,850 --> 00:26:37,412
of the temperature
2.726 degrees Kelvin.

495
00:26:37,412 --> 00:26:39,860
All of the interesting
historical points

496
00:26:39,860 --> 00:26:42,560
are in this tiny little
corner on the left, which

497
00:26:42,560 --> 00:26:45,610
is magnified above.

498
00:26:45,610 --> 00:26:48,230
The original Penzias
and Wilson point

499
00:26:48,230 --> 00:26:50,620
is way down here at
very low frequencies

500
00:26:50,620 --> 00:26:58,190
by the standards of radiation
at 2.726 degrees Kelvin.

501
00:26:58,190 --> 00:27:00,940
The Roll and Wilkinson
point is there.

502
00:27:00,940 --> 00:27:04,850
These blobs indicate error bars.

503
00:27:04,850 --> 00:27:08,240
The [? cyanogen ?] points
that you read about

504
00:27:08,240 --> 00:27:12,490
in Weinberg are shown
there and there.

505
00:27:12,490 --> 00:27:15,840
The first measurement that
showed that, it didn't only

506
00:27:15,840 --> 00:27:18,810
go up but started to go down
like black-body radiation

507
00:27:18,810 --> 00:27:25,110
should, was a balloon flight--
this 1971 balloon flight

508
00:27:25,110 --> 00:27:29,970
which produce that
blob and that bound.

509
00:27:29,970 --> 00:27:33,222
This was an experiment
by MIT's own Ray Weiss.

510
00:27:33,222 --> 00:27:35,670
And it was very
important in the history

511
00:27:35,670 --> 00:27:39,060
because it was the first
evidence that we weren't just

512
00:27:39,060 --> 00:27:41,070
seeing some straight
line, but we

513
00:27:41,070 --> 00:27:44,230
were seeing something which
did indeed rise and fall

514
00:27:44,230 --> 00:27:46,870
the way black-body
radiation should.

515
00:27:46,870 --> 00:27:52,070
A later balloon flight in
1974 produced error bars

516
00:27:52,070 --> 00:27:54,950
that are shown by
this gray area.

517
00:27:54,950 --> 00:27:56,990
Incredibly broad.

518
00:27:56,990 --> 00:27:59,840
So the bottom line
that this graph

519
00:27:59,840 --> 00:28:04,040
was intended by me to
illustrate is that, in 1975 you

520
00:28:04,040 --> 00:28:07,460
could believe that this was
black-body radiation if you so

521
00:28:07,460 --> 00:28:09,020
wished.

522
00:28:09,020 --> 00:28:12,140
But there was not
really a lot of evidence

523
00:28:12,140 --> 00:28:14,355
that it was
black-body radiation.

524
00:28:17,060 --> 00:28:21,680
The situation did not
get better quickly.

525
00:28:21,680 --> 00:28:25,950
The next significant
measurement came

526
00:28:25,950 --> 00:28:30,760
in 1987 which was a
rocket flight, which

527
00:28:30,760 --> 00:28:33,750
was a collaboration
between a group at Berkeley

528
00:28:33,750 --> 00:28:37,550
and a group at Nagoya, Japan.

529
00:28:37,550 --> 00:28:39,220
I believe it was the
Japanese group that

530
00:28:39,220 --> 00:28:41,510
supplied the rocket and
the American group that

531
00:28:41,510 --> 00:28:45,070
supplied the instrumentation.

532
00:28:45,070 --> 00:28:51,322
And they measured the
radiation at three points.

533
00:28:51,322 --> 00:28:52,780
I can give you the
number that goes

534
00:28:52,780 --> 00:28:54,240
with those graphical points.

535
00:29:05,380 --> 00:29:10,800
I guess what I
have tabulated here

536
00:29:10,800 --> 00:29:13,231
is the effect of temperature
that those points correspond

537
00:29:13,231 --> 00:29:13,730
to.

538
00:29:20,820 --> 00:29:23,260
As you can see from
the graph, those points

539
00:29:23,260 --> 00:29:26,390
are all well above
the black-body curve.

540
00:29:26,390 --> 00:29:28,480
Significantly more
radiation than what

541
00:29:28,480 --> 00:29:30,830
was expected by
people who thought

542
00:29:30,830 --> 00:29:34,000
it should be black-body.

543
00:29:34,000 --> 00:29:42,790
And 0.2 up there
would correspond

544
00:29:42,790 --> 00:30:00,040
to a temperature of 2.955 plus
or minus 0.017 K. The size

545
00:30:00,040 --> 00:30:02,480
of the vertical bars
there are the error

546
00:30:02,480 --> 00:30:06,970
bars that the
experimenters found.

547
00:30:06,970 --> 00:30:31,560
And 0.3 was t equals 3.175
plus or minus 0.027 K.

548
00:30:31,560 --> 00:30:33,780
So these were higher
temperatures then

549
00:30:33,780 --> 00:30:38,490
the 2.7 that fit the lower
part of the spectrum.

550
00:30:38,490 --> 00:30:40,640
And very, very small error bars.

551
00:30:43,250 --> 00:30:45,330
So this data came out in 1987.

552
00:30:45,330 --> 00:30:50,700
And, in truth, nobody knew
what to make out of it.

553
00:30:50,700 --> 00:30:54,730
The experimental group were
well aware that this was not

554
00:30:54,730 --> 00:30:58,530
what people wanted them to find.

555
00:30:58,530 --> 00:31:01,560
And they certainly
examined their data

556
00:31:01,560 --> 00:31:04,880
very carefully to figure at
what could have conceivably gone

557
00:31:04,880 --> 00:31:05,930
wrong.

558
00:31:05,930 --> 00:31:08,390
And they were going
around the country

559
00:31:08,390 --> 00:31:09,650
giving talks about this.

560
00:31:09,650 --> 00:31:12,250
And I heard one or
two of them in which

561
00:31:12,250 --> 00:31:15,180
they described how surprised
they were by the results,

562
00:31:15,180 --> 00:31:19,410
but emphasized that they
analyzed the experiment very,

563
00:31:19,410 --> 00:31:23,420
very carefully and couldn't
find anything wrong with it.

564
00:31:23,420 --> 00:31:26,250
And this was the
situation for awhile.

565
00:31:26,250 --> 00:31:29,600
I should point out that I think
this point number three is

566
00:31:29,600 --> 00:31:34,020
something like 16 standard
deviations off of the theory.

567
00:31:34,020 --> 00:31:37,220
And usually when somebody
makes a measurement that's

568
00:31:37,220 --> 00:31:39,930
three or four standard
deviations off of your theory,

569
00:31:39,930 --> 00:31:41,950
you really start to worry.

570
00:31:41,950 --> 00:31:46,910
16 standard deviations is
certainly a bit extreme.

571
00:31:46,910 --> 00:31:53,095
Nonetheless, nobody had any
good explanation for this.

572
00:31:53,095 --> 00:31:55,220
So, well, different people
had different attitudes.

573
00:31:55,220 --> 00:31:57,470
There were some people who
tried to construct theories

574
00:31:57,470 --> 00:31:59,260
that would account for this.

575
00:31:59,260 --> 00:32:04,870
And there were others who
waited for it to go away.

576
00:32:04,870 --> 00:32:06,290
I'm pretty sure
I was among those

577
00:32:06,290 --> 00:32:11,960
who waited for it to go
away, and we were right.

578
00:32:11,960 --> 00:32:15,240
So the next important
piece of data

579
00:32:15,240 --> 00:32:17,770
came from the first
satellite dedicated

580
00:32:17,770 --> 00:32:21,260
to measuring the cosmic
background radiation.

581
00:32:21,260 --> 00:32:26,654
The famous COBE Satellite--
Cosmic Background Explorer-- I

582
00:32:26,654 --> 00:32:28,320
guess I didn't write
down the name here.

583
00:32:28,320 --> 00:32:29,370
Oh, it's in the title.

584
00:32:29,370 --> 00:32:32,250
Preliminary measurement of the
cosmic microwave background

585
00:32:32,250 --> 00:32:35,440
spectrum by the Cosmic
Background Explorer,

586
00:32:35,440 --> 00:32:37,290
COBE Satellite.

587
00:32:37,290 --> 00:32:39,820
So COBE was the first
satellite dedicated

588
00:32:39,820 --> 00:32:43,605
to measuring the cosmic
background radiation.

589
00:32:43,605 --> 00:32:48,770
It was launched in 1989,
I guess, and released

590
00:32:48,770 --> 00:32:53,980
its first data in
January of 1990.

591
00:32:53,980 --> 00:32:58,700
Back in those days there
was no internet or archive.

592
00:32:58,700 --> 00:33:01,510
So you may or may not know that
the way physics results were

593
00:33:01,510 --> 00:33:03,200
first announced
to the world were

594
00:33:03,200 --> 00:33:06,510
in the form of what were
called pre-prints, which

595
00:33:06,510 --> 00:33:12,130
were essentially xeroxed copies
of the paper that were sent out

596
00:33:12,130 --> 00:33:14,220
to a mailing list.

597
00:33:14,220 --> 00:33:16,870
Typically, I think,
institutions had mailing lists

598
00:33:16,870 --> 00:33:20,260
of maybe 100 other institutions.

599
00:33:20,260 --> 00:33:24,930
And every physics department
had a pre-print library

600
00:33:24,930 --> 00:33:28,510
that people can go to and
find these pre-prints.

601
00:33:28,510 --> 00:33:30,940
So this is the COBE pre-print.

602
00:33:30,940 --> 00:33:35,560
90-01, the first
pre-print from 1990.

603
00:33:35,560 --> 00:33:36,770
And this is the data.

604
00:33:46,080 --> 00:33:49,600
So it is kind of
breathtaking, I think.

605
00:33:49,600 --> 00:33:54,130
It suddenly changed the entire
field, and in some sense

606
00:33:54,130 --> 00:33:55,764
really change cosmology
for the field.

607
00:33:55,764 --> 00:33:57,680
Where we only had
approximate ideas of the way

608
00:33:57,680 --> 00:34:05,530
things worked, to suddenly
having a really precise science

609
00:34:05,530 --> 00:34:08,960
in which precise
measurements could be made,

610
00:34:08,960 --> 00:34:10,849
and cleared up the
issue of the radiation.

611
00:34:13,969 --> 00:34:19,219
It wasn't just a mess like this,
or a terrible fit like that,

612
00:34:19,219 --> 00:34:23,460
but a fantastically good fit.

613
00:34:23,460 --> 00:34:30,320
Really nailing the radiation
as having a thermal spectrum.

614
00:34:30,320 --> 00:34:34,360
So the history is that John
Mather presented this data

615
00:34:34,360 --> 00:34:40,179
at the January 1990 American
Physical Society meeting,

616
00:34:40,179 --> 00:34:43,870
and was given a
standing ovation.

617
00:34:43,870 --> 00:34:47,150
And he later won the
Nobel Prize for this work.

618
00:34:47,150 --> 00:34:49,490
He was the head of the team
that brought this data.

619
00:34:49,490 --> 00:34:53,000
He won the Nobel Prize in
2006 along with George Smoot,

620
00:34:53,000 --> 00:34:55,719
who was responsible for one
of the other experiments

621
00:34:55,719 --> 00:34:57,761
on the COBE satellite.

622
00:34:57,761 --> 00:34:58,260
Yes?

623
00:34:58,260 --> 00:34:59,801
AUDIENCE: So do we
know what happened

624
00:34:59,801 --> 00:35:01,120
with the other measurements?

625
00:35:01,120 --> 00:35:01,856
PROFESSOR: To tell
you the truth,

626
00:35:01,856 --> 00:35:03,390
I don't think the
other measurements

627
00:35:03,390 --> 00:35:05,330
ever-- the other people
ever really published

628
00:35:05,330 --> 00:35:06,371
what they think happened.

629
00:35:06,371 --> 00:35:10,280
But the widespread rumor,
which I imagine is true,

630
00:35:10,280 --> 00:35:14,850
is that they were seeing
their own rocket exhaust.

631
00:35:14,850 --> 00:35:19,000
And there were, I think,
some arguments going on

632
00:35:19,000 --> 00:35:23,260
between the Americans
and the Japanese,

633
00:35:23,260 --> 00:35:27,694
with the Americans more or less
accusing the Japanese of not

634
00:35:27,694 --> 00:35:29,610
really telling them how
the rocket was set up.

635
00:35:33,070 --> 00:35:33,580
Yes?

636
00:35:33,580 --> 00:35:36,256
AUDIENCE: Are the error bars
plugged on those points,

637
00:35:36,256 --> 00:35:37,980
or is it just that good?

638
00:35:37,980 --> 00:35:39,605
PROFESSOR: Those
are the error bars?

639
00:35:39,605 --> 00:35:40,146
AUDIENCE: OK.

640
00:35:47,130 --> 00:35:51,860
PROFESSOR: And even
more spectacular,

641
00:35:51,860 --> 00:35:54,210
a couple years later,
I guess it was-- this

642
00:35:54,210 --> 00:35:56,710
was actually just based on nine
minutes of data or something

643
00:35:56,710 --> 00:35:58,080
like that.

644
00:35:58,080 --> 00:36:01,840
But a couple years later they
published their full data set,

645
00:36:01,840 --> 00:36:08,380
where the size of the error bars
were reduced by a factor of 10.

646
00:36:08,380 --> 00:36:11,649
And still a perfect fit.

647
00:36:11,649 --> 00:36:13,190
They didn't even
know how to plot it,

648
00:36:13,190 --> 00:36:14,814
so I think they
plotted the same graph,

649
00:36:14,814 --> 00:36:16,910
and said the error
bars are a factor of 10

650
00:36:16,910 --> 00:36:19,385
smalled than what's shown.

651
00:36:19,385 --> 00:36:20,295
It was gorgeous.

652
00:36:34,400 --> 00:36:37,520
So I think I forgot to tell you
what the spectrum is supposed

653
00:36:37,520 --> 00:36:39,975
to look like exactly.

654
00:36:39,975 --> 00:36:41,840
And this is just
a formula that I

655
00:36:41,840 --> 00:36:44,420
want you to understand
the meaning of, but not

656
00:36:44,420 --> 00:36:46,830
the derivation of.

657
00:36:46,830 --> 00:36:53,378
We-- as with the other stat mech
results that we're relying on,

658
00:36:53,378 --> 00:36:56,810
we're going to relegate their
derivation to the stat mech

659
00:36:56,810 --> 00:36:59,740
course that you either
have taken or will take.

660
00:36:59,740 --> 00:37:01,610
But the spectrum is
completely determined

661
00:37:01,610 --> 00:37:04,830
because the principle
of thermal equilibrium

662
00:37:04,830 --> 00:37:07,890
is sort of absolute in
statistical mechanics.

663
00:37:07,890 --> 00:37:11,510
And in order for a
black-body radiating

664
00:37:11,510 --> 00:37:15,120
object to be in
thermal equilibrium

665
00:37:15,120 --> 00:37:19,340
with an environment
at that temperature,

666
00:37:19,340 --> 00:37:22,430
it has to have not only the
right emission rate but also

667
00:37:22,430 --> 00:37:23,666
the right spectrum.

668
00:37:23,666 --> 00:37:25,040
If the spectrum
weren't right you

669
00:37:25,040 --> 00:37:26,581
could imagine putting
in filters that

670
00:37:26,581 --> 00:37:30,130
would trap in some frequencies
and let out others.

671
00:37:30,130 --> 00:37:32,860
And then you would move away
from thermal equilibrium

672
00:37:32,860 --> 00:37:35,750
if the spectrum were right
or wrong because you'd

673
00:37:35,750 --> 00:37:38,880
be trapping in more
radiation-- you could arrange

674
00:37:38,880 --> 00:37:44,030
for the filters to trap in more
radiation than they are letting

675
00:37:44,030 --> 00:37:45,970
out.

676
00:37:45,970 --> 00:37:51,400
So the spectrum is calculable.

677
00:37:51,400 --> 00:37:58,995
And in terms of-- I guess
this is energy density.

678
00:37:58,995 --> 00:38:01,210
I have to admit, I usually
call energy density u

679
00:38:01,210 --> 00:38:03,070
and in these notes
here it's called rho.

680
00:38:09,520 --> 00:38:12,440
We'll figure out the units after
I write it down and make sure

681
00:38:12,440 --> 00:38:14,750
that it is energy density.

682
00:38:14,750 --> 00:38:21,890
Rho sub nu of nu d nu,
means-- with this product

683
00:38:21,890 --> 00:38:27,340
it means the total energy
density, energy per volume,

684
00:38:27,340 --> 00:38:30,891
per frequency interval, d
nu-- well, it's times d nu,

685
00:38:30,891 --> 00:38:32,390
so if you multiply
by times nu, this

686
00:38:32,390 --> 00:38:36,050
is the total energy for
frequencies between nu and nu

687
00:38:36,050 --> 00:38:38,220
plus d nu.

688
00:38:38,220 --> 00:38:51,770
And the formula is 16 pi
squared h bar nu cubed, divided

689
00:38:51,770 --> 00:39:07,980
by c cubed times 1 over e to
the 2 pi h bar nu over kt,

690
00:39:07,980 --> 00:39:11,290
minus 1 d nu.

691
00:39:26,480 --> 00:39:26,980
OK.

692
00:39:26,980 --> 00:39:28,896
And actually, the unit's
not that transparent.

693
00:39:28,896 --> 00:39:32,800
I believe this is energy
density and not mass density.

694
00:39:32,800 --> 00:39:37,350
But maybe I'll make sure of
that and let you know next time.

695
00:39:39,880 --> 00:39:41,610
And this is what
produces that curve

696
00:39:41,610 --> 00:39:43,310
that you saw on the slides.

697
00:39:47,890 --> 00:39:49,660
I've included the
subscript nu here

698
00:39:49,660 --> 00:39:52,550
to indicate that it's the number
which, when you multiply it

699
00:39:52,550 --> 00:39:56,640
by d nu, gives you the energy
density between nu and nu

700
00:39:56,640 --> 00:39:58,230
plus d nu.

701
00:39:58,230 --> 00:40:00,110
If instead you wanted
to know the energy

702
00:40:00,110 --> 00:40:04,690
density between lambda
and lambda plus d lambda,

703
00:40:04,690 --> 00:40:06,620
there'd be a kinematic
factor that you'd

704
00:40:06,620 --> 00:40:10,660
have to put in here-- the factor
that relates d lambda to d nu.

705
00:40:10,660 --> 00:40:15,460
And you could imagine
working that out.

706
00:40:15,460 --> 00:40:17,090
I might add that,
in Weinberg's book,

707
00:40:17,090 --> 00:40:22,772
he actually plots
both sub lambda of nu.

708
00:40:22,772 --> 00:40:24,980
So his curve looks somewhat
different than the curves

709
00:40:24,980 --> 00:40:26,492
that I showed you.

710
00:40:26,492 --> 00:40:27,950
This is not exactly
the same thing.

711
00:40:34,910 --> 00:40:39,050
Now, what this extremely
accurately black-body curve

712
00:40:39,050 --> 00:40:42,480
proves is that the
early universe really

713
00:40:42,480 --> 00:40:46,970
was very accurately in
thermal equilibrium.

714
00:40:46,970 --> 00:40:50,754
And that can only happen if the
early universe was very dense.

715
00:40:50,754 --> 00:40:52,420
And of course, our
model of the universe

716
00:40:52,420 --> 00:40:54,090
goes back to infinite density.

717
00:40:54,090 --> 00:40:55,860
So the model predicts
that it should

718
00:40:55,860 --> 00:40:57,880
be in thermal equilibrium.

719
00:40:57,880 --> 00:41:01,530
But in particular, the
numbers that we have here,

720
00:41:01,530 --> 00:41:03,610
if you ask how much could
you change the model

721
00:41:03,610 --> 00:41:08,040
and still expect these
curves the answer

722
00:41:08,040 --> 00:41:13,110
is roughly that, all of the
important energy-releasing

723
00:41:13,110 --> 00:41:16,080
processes have to have
happened before about one

724
00:41:16,080 --> 00:41:18,520
year after the Big Bang.

725
00:41:18,520 --> 00:41:20,540
Anything that happened
after one year

726
00:41:20,540 --> 00:41:22,670
would still show
up as some glitch

727
00:41:22,670 --> 00:41:26,110
in the black-body spectrum.

728
00:41:26,110 --> 00:41:30,030
So the big bag model really
is confirmed back to about one

729
00:41:30,030 --> 00:41:33,350
year on the basis of
this precise measurement

730
00:41:33,350 --> 00:41:35,974
of the spectrum of the
cosmic background radiation.

731
00:41:35,974 --> 00:41:37,890
And the COBE measurement
is still, by the way,

732
00:41:37,890 --> 00:41:39,860
the best measurement
of the spectrum.

733
00:41:39,860 --> 00:41:41,860
We've had other very
important experiments,

734
00:41:41,860 --> 00:41:43,500
that we'll talk
about later, which

735
00:41:43,500 --> 00:41:47,230
measure the non-uniformity
of the black-body radiation.

736
00:41:47,230 --> 00:41:49,600
Which is very small,
but nonetheless very,

737
00:41:49,600 --> 00:41:51,690
very important [? effect. ?]

738
00:41:51,690 --> 00:41:56,460
So we've had WMAP
and now Planck which

739
00:41:56,460 --> 00:41:59,790
have been dedicated to
measuring the anisotropies

740
00:41:59,790 --> 00:42:00,870
of the radiation.

741
00:42:00,870 --> 00:42:02,490
COBE also made
initial measurements

742
00:42:02,490 --> 00:42:04,210
of the anisotropies.

743
00:42:04,210 --> 00:42:06,420
And we'll be talking
about anisotropies later

744
00:42:06,420 --> 00:42:08,380
in the course.

745
00:42:08,380 --> 00:42:09,170
Yes?

746
00:42:09,170 --> 00:42:12,337
AUDIENCE: [INAUDIBLE].

747
00:42:12,337 --> 00:42:13,045
PROFESSOR: Sorry?

748
00:42:13,045 --> 00:42:14,930
AUDIENCE: The units
of the right-hand side

749
00:42:14,930 --> 00:42:16,050
are energy density.

750
00:42:16,050 --> 00:42:16,370
PROFESSOR: Energy density.

751
00:42:16,370 --> 00:42:16,870
OK.

752
00:42:16,870 --> 00:42:17,380
Thanks.

753
00:42:17,380 --> 00:42:17,880
OK.

754
00:42:17,880 --> 00:42:18,380
Good.

755
00:42:18,380 --> 00:42:20,100
So my words were right.

756
00:42:20,100 --> 00:42:22,169
I should have called
it u, I think,

757
00:42:22,169 --> 00:42:23,835
to be consistent with
my usual notation.

758
00:42:31,340 --> 00:42:31,840
Thanks.

759
00:42:36,050 --> 00:42:36,550
OK.

760
00:42:36,550 --> 00:42:37,966
Any other questions
about the CMB?

761
00:42:37,966 --> 00:42:41,230
Because if not, we're going
to change gears completely

762
00:42:41,230 --> 00:42:46,800
and start talking about one of
the other crucially important

763
00:42:46,800 --> 00:42:52,940
observational discoveries in
cosmology in the last 20 years.

764
00:43:01,571 --> 00:43:02,070
OK.

765
00:43:02,070 --> 00:43:03,590
So what I want to
talk about next

766
00:43:03,590 --> 00:43:10,040
is the very important
discovery originally made

767
00:43:10,040 --> 00:43:19,584
in 1998-- also resulting
in a Nobel Prize--

768
00:43:19,584 --> 00:43:21,000
that the universe
is accelerating.

769
00:43:35,110 --> 00:43:36,940
And this was a
discovery that involved

770
00:43:36,940 --> 00:43:40,030
two experimental groups,
and a total of something

771
00:43:40,030 --> 00:43:43,960
like 52 astronomers
between the two groups.

772
00:43:43,960 --> 00:43:46,980
Which actually meant that--
I'm exaggerating slightly,

773
00:43:46,980 --> 00:43:47,560
I suppose.

774
00:43:47,560 --> 00:43:49,544
But it really
involved the majority

775
00:43:49,544 --> 00:43:51,710
of the astronomers of the
world, and therefore there

776
00:43:51,710 --> 00:43:53,918
weren't a lot of astronomers
to argue with them about

777
00:43:53,918 --> 00:43:57,220
whether or not the
result was right.

778
00:43:57,220 --> 00:43:59,110
But there still
was some argument.

779
00:43:59,110 --> 00:44:03,350
The announcement was initially
made at a AAS meeting

780
00:44:03,350 --> 00:44:10,250
in January of 1998 by--
which group was first?

781
00:44:10,250 --> 00:44:18,570
I think that was the High-Z
Supernova-- where are they?

782
00:44:18,570 --> 00:44:19,070
Yeah.

783
00:44:19,070 --> 00:44:22,220
That was the High-Z
Supernova Search Team.

784
00:44:22,220 --> 00:44:26,080
And then there was also a group
largely based at Berkeley.

785
00:44:26,080 --> 00:44:29,670
The High-Z Supernova Search Team
was actually fairly diffused,

786
00:44:29,670 --> 00:44:33,190
although based to some
extent at Harvard.

787
00:44:33,190 --> 00:44:35,710
And the Supernova
Cosmology Project

788
00:44:35,710 --> 00:44:38,600
was based rather
squarely in Berkeley,

789
00:44:38,600 --> 00:44:42,260
headed by Saul Perlmutter.

790
00:44:42,260 --> 00:44:43,700
And they both agreed.

791
00:44:43,700 --> 00:44:47,390
And what they found
was, by looking

792
00:44:47,390 --> 00:44:54,510
at distant supernovae of a
particular type-- type 1a--

793
00:44:54,510 --> 00:45:00,530
they were able to use these
supernovae as standard candles.

794
00:45:00,530 --> 00:45:02,530
And because supernovae
are brighter

795
00:45:02,530 --> 00:45:06,320
than other standard candles
that had been studied earlier

796
00:45:06,320 --> 00:45:10,660
in history, they were able to go
out to much greater distances.

797
00:45:10,660 --> 00:45:14,610
And that means to look
much further back in time

798
00:45:14,610 --> 00:45:17,520
than previous studies.

799
00:45:17,520 --> 00:45:21,420
And what they discovered
was that the expansion rate

800
00:45:21,420 --> 00:45:23,680
of the universe
today was actually

801
00:45:23,680 --> 00:45:27,600
faster, and not slower,
than the expansion rate

802
00:45:27,600 --> 00:45:31,080
about five billion years ago.

803
00:45:31,080 --> 00:45:36,030
And that was a big shock because
until then everybody expected

804
00:45:36,030 --> 00:45:38,850
that gravity would
be slowing down

805
00:45:38,850 --> 00:45:41,430
the expansion of the universe.

806
00:45:41,430 --> 00:45:45,710
And when these guys started
to make these measurements

807
00:45:45,710 --> 00:45:48,730
they were just simply
trying to figure out

808
00:45:48,730 --> 00:45:52,330
how fast the universe
was slowing down.

809
00:45:52,330 --> 00:45:55,730
And they were shocked to
discover that it was not

810
00:45:55,730 --> 00:46:00,400
slowing down, but
instead speeding up.

811
00:46:00,400 --> 00:46:02,420
Initially there was some
controversy about it.

812
00:46:02,420 --> 00:46:06,055
People did try to invent other
explanations for this data.

813
00:46:09,130 --> 00:46:14,240
But the data has, in fact, held
up for the period from 1998

814
00:46:14,240 --> 00:46:15,320
to the present.

815
00:46:15,320 --> 00:46:20,000
And in fact, it has been
strongly supported by evidence

816
00:46:20,000 --> 00:46:23,040
from these anisotropies in the
cosmic microwave background

817
00:46:23,040 --> 00:46:26,070
radiation, which we'll
be talking about later.

818
00:46:26,070 --> 00:46:28,330
But it turns out, you can
get a lot of information

819
00:46:28,330 --> 00:46:33,270
from these anisotropies in the
cosmic background radiation.

820
00:46:33,270 --> 00:46:36,820
So the picture now is
really quite secure,

821
00:46:36,820 --> 00:46:41,670
that the acceleration-- the
expansion of the universe

822
00:46:41,670 --> 00:46:47,790
is actually accelerating,
and not decelerating.

823
00:46:47,790 --> 00:46:49,950
And the simplest
explanation for that,

824
00:46:49,950 --> 00:46:54,290
which is the one that--
well, certainly because it's

825
00:46:54,290 --> 00:46:57,330
the most plausible, and the one
that most of us take seriously,

826
00:46:57,330 --> 00:46:59,913
and it's the only one that fits
the data extraordinarily well.

827
00:46:59,913 --> 00:47:03,410
So we've not seen any reason
not to use this explanation.

828
00:47:03,410 --> 00:47:07,690
The simplest explanation is
that there's a nonzero energy

829
00:47:07,690 --> 00:47:11,360
density to the
vacuum, which is also

830
00:47:11,360 --> 00:47:13,400
what Einstein called the
cosmological constant.

831
00:47:16,140 --> 00:47:17,970
So we should begin
by writing down

832
00:47:17,970 --> 00:47:23,245
the equations that
describe this issue.

833
00:47:44,060 --> 00:47:46,910
So we've learned how to write
down the second order Friedmann

834
00:47:46,910 --> 00:47:55,972
equation, which
describes how the scale

835
00:47:55,972 --> 00:47:57,430
factor of the
universe accelerates.

836
00:48:05,330 --> 00:48:09,730
And on the right-hand side,
once we included materials

837
00:48:09,730 --> 00:48:12,480
with nonzero
pressures, we discover

838
00:48:12,480 --> 00:48:14,860
that we need on the
right-hand side, rho plus 3 p,

839
00:48:14,860 --> 00:48:17,810
over c squared--
excuse me-- times a.

840
00:48:22,860 --> 00:48:25,790
Now when the cosmological
constant was born,

841
00:48:25,790 --> 00:48:29,720
was when Einstein
first turned his theory

842
00:48:29,720 --> 00:48:32,730
of general relativity
to cosmology.

843
00:48:32,730 --> 00:48:36,700
Einstein invented the theory
of general relativity in 1916.

844
00:48:36,700 --> 00:48:40,210
And just one year
later, in 1917,

845
00:48:40,210 --> 00:48:42,370
he was applying it to
the universe as a whole

846
00:48:42,370 --> 00:48:45,800
to see if he could get
a cosmological model

847
00:48:45,800 --> 00:48:49,370
consistent with
general relativity.

848
00:48:49,370 --> 00:48:55,860
Einstein at that point was
under the misconception

849
00:48:55,860 --> 00:48:59,790
that the universe was static,
as Newton had also thought,

850
00:48:59,790 --> 00:49:02,260
and as far as we
know, as everybody

851
00:49:02,260 --> 00:49:06,830
between Newton and
Einstein thought.

852
00:49:06,830 --> 00:49:11,350
If you look up at the stars, the
universe looks pretty static.

853
00:49:11,350 --> 00:49:13,845
And people took
this very seriously.

854
00:49:13,845 --> 00:49:16,220
In hindsight, it's a little
hard to know why they took it

855
00:49:16,220 --> 00:49:19,490
so seriously, but they did.

856
00:49:19,490 --> 00:49:25,820
So when Einstein
discovered this equation

857
00:49:25,820 --> 00:49:27,990
he was assuming that
the universe consisted

858
00:49:27,990 --> 00:49:30,210
of basically
non-relativistic stuff.

859
00:49:30,210 --> 00:49:33,300
Stars are essentially
non-relativistic hunks

860
00:49:33,300 --> 00:49:35,220
of matter.

861
00:49:35,220 --> 00:49:37,300
So he thought that
rho would be positive,

862
00:49:37,300 --> 00:49:41,390
the effective pressure
would be zero.

863
00:49:41,390 --> 00:49:45,400
And he immediately
noticed that this equation

864
00:49:45,400 --> 00:49:49,240
would imply that the
scale factor would

865
00:49:49,240 --> 00:49:51,800
have a negative acceleration.

866
00:49:51,800 --> 00:49:54,300
So that if you tried to
set up a static universe

867
00:49:54,300 --> 00:49:57,170
it would instantly collapse.

868
00:49:57,170 --> 00:50:00,370
And as we talked
about earlier, Newton

869
00:50:00,370 --> 00:50:03,910
had talked himself out
of that conclusion.

870
00:50:03,910 --> 00:50:06,090
And I think the real
difference, as I think we also

871
00:50:06,090 --> 00:50:08,850
talked about earlier, was
that Newton was thinking

872
00:50:08,850 --> 00:50:12,310
of the law of gravity as
an action at a distance,

873
00:50:12,310 --> 00:50:14,390
where you determine the
total force on something

874
00:50:14,390 --> 00:50:18,850
by integrating the forces
caused by all other masses.

875
00:50:18,850 --> 00:50:21,520
And then things get complicated
and divergent, actually,

876
00:50:21,520 --> 00:50:24,220
for an infinite,
static universe.

877
00:50:24,220 --> 00:50:26,500
And Newton managed
to convince himself

878
00:50:26,500 --> 00:50:30,540
that you could have a static
universe of that type,

879
00:50:30,540 --> 00:50:32,540
a statement that we now
consider to be incorrect

880
00:50:32,540 --> 00:50:35,800
even in the context of
Newtonian mechanics.

881
00:50:35,800 --> 00:50:38,279
But this fact that
it's incorrect even

882
00:50:38,279 --> 00:50:39,820
in the context of
Newtonian mechanics

883
00:50:39,820 --> 00:50:42,230
was really not discovered
until Einstein wrote down

884
00:50:42,230 --> 00:50:43,520
this equation.

885
00:50:43,520 --> 00:50:50,090
And then Einstein himself
also gave a Newtonian argument

886
00:50:50,090 --> 00:50:53,860
showing that, at least with
a modern interpretation

887
00:50:53,860 --> 00:50:55,050
of Newtonian mechanics.

888
00:50:55,050 --> 00:50:56,970
It doesn't work in
Newtonian gravity

889
00:50:56,970 --> 00:50:59,940
either to have a
static universe.

890
00:50:59,940 --> 00:51:05,440
But Einstein was still convinced
that the universe was static.

891
00:51:05,440 --> 00:51:12,049
And he realized that he could
modify his field equations--

892
00:51:12,049 --> 00:51:13,840
the equations that we
have not written down

893
00:51:13,840 --> 00:51:17,190
in this course, the equations
that describe how matter create

894
00:51:17,190 --> 00:51:22,280
gravitational fields--
by adding a new term

895
00:51:22,280 --> 00:51:31,195
with a new coefficient in front
of it which he called lambda.

896
00:52:00,580 --> 00:52:05,060
And this extra
term, lambda, could

897
00:52:05,060 --> 00:52:09,320
produce a kind of a universal
gravitational repulsion.

898
00:52:09,320 --> 00:52:12,330
And he realised he had to adjust
the constant to be just right

899
00:52:12,330 --> 00:52:15,050
to balance the amount of
matter in the universe.

900
00:52:15,050 --> 00:52:17,840
But he didn't let
that bother him.

901
00:52:17,840 --> 00:52:19,690
And if he adjusted
it to be just right,

902
00:52:19,690 --> 00:52:21,820
and the universe was
perfectly homogeneous,

903
00:52:21,820 --> 00:52:23,770
he could arrange
for it to balance

904
00:52:23,770 --> 00:52:27,375
the standard force of gravity.

905
00:52:27,375 --> 00:52:29,500
We can understand what
lambda does to the equations

906
00:52:29,500 --> 00:52:31,625
because it does, in fact,
have a simple description

907
00:52:31,625 --> 00:52:34,240
in terms of things that we have
discussed and do understand.

908
00:52:34,240 --> 00:52:36,020
That is, you could
think of lambda

909
00:52:36,020 --> 00:52:41,540
as simply corresponding to
a vacuum energy density.

910
00:52:41,540 --> 00:52:45,470
Einstein did not
make that connection.

911
00:52:45,470 --> 00:52:48,790
And not being an
historian of science,

912
00:52:48,790 --> 00:52:50,800
I can speculate
as much as I want.

913
00:52:50,800 --> 00:52:54,420
So my speculation is that,
the reason this did not

914
00:52:54,420 --> 00:52:57,250
occur to Einstein
is that Einstein

915
00:52:57,250 --> 00:53:00,840
was a fully classical
physicist who was not

916
00:53:00,840 --> 00:53:03,390
at this time or
maybe never accepting

917
00:53:03,390 --> 00:53:04,669
the notions of quantum theory.

918
00:53:04,669 --> 00:53:06,210
And in any case,
quantum field theory

919
00:53:06,210 --> 00:53:08,940
was still far in the future.

920
00:53:08,940 --> 00:53:13,130
So in classical physics the
vacuum is just plain empty.

921
00:53:13,130 --> 00:53:15,540
And if the vacuum
is just plain empty

922
00:53:15,540 --> 00:53:18,760
it shouldn't have
any energy density.

923
00:53:18,760 --> 00:53:21,060
The quantum field theory
picture of the vacuum,

924
00:53:21,060 --> 00:53:24,860
however, is vastly more complex.

925
00:53:24,860 --> 00:53:28,250
So to a modern quantum
field theory-oriented

926
00:53:28,250 --> 00:53:32,090
theoretical physicist
the vacuum has

927
00:53:32,090 --> 00:53:35,260
particle, anti-particle pairs
appearing and disappearing

928
00:53:35,260 --> 00:53:36,890
all the time.

929
00:53:36,890 --> 00:53:38,970
We are now convinced
that there's also

930
00:53:38,970 --> 00:53:43,330
this Higgs field that has
even a nonzero mean value

931
00:53:43,330 --> 00:53:44,850
in the vacuum.

932
00:53:44,850 --> 00:53:47,180
So the vacuum is a
very complicated state

933
00:53:47,180 --> 00:53:48,910
which, if anything
characterizes it,

934
00:53:48,910 --> 00:53:52,790
it's simply the state of
lowest possible energy density.

935
00:53:52,790 --> 00:53:56,120
But because of, basically,
the uncertainty principles

936
00:53:56,120 --> 00:53:58,830
of quantum mechanics, the
lowest possible energy density

937
00:53:58,830 --> 00:54:00,810
does not mean that all
the fields are just zero

938
00:54:00,810 --> 00:54:02,210
and stay zero.

939
00:54:02,210 --> 00:54:05,226
They're constantly
fluctuating as they must

940
00:54:05,226 --> 00:54:07,100
according to the
uncertainty principle, which

941
00:54:07,100 --> 00:54:08,955
applies to fields
as well particles.

942
00:54:11,490 --> 00:54:15,220
So we have no reason
anymore to expect the energy

943
00:54:15,220 --> 00:54:18,010
density of the
vacuum to be zero.

944
00:54:18,010 --> 00:54:20,490
So from a modern perspective
it's very natural

945
00:54:20,490 --> 00:54:24,110
to simply equate the idea
of the cosmological constant

946
00:54:24,110 --> 00:54:27,430
to the idea of a nonzero
vacuum energy density.

947
00:54:27,430 --> 00:54:31,050
And there are some
unit differences-- just

948
00:54:31,050 --> 00:54:34,110
constants related to
the historical way

949
00:54:34,110 --> 00:54:37,930
that Einstein added
this term his equations.

950
00:54:37,930 --> 00:54:43,480
So the energy density of
the vacuum-- which is also

951
00:54:43,480 --> 00:54:45,460
the mass density of
the vacuum times c

952
00:54:45,460 --> 00:54:53,790
squared-- is equal to Einstein's
lambda times c to the fourth,

953
00:54:53,790 --> 00:55:01,050
over 8 pi G.

954
00:55:01,050 --> 00:55:03,680
And this is really just
an historical accident

955
00:55:03,680 --> 00:55:04,840
that it's defined this way.

956
00:55:04,840 --> 00:55:06,673
But this is the way
Einstein defined lambda.

957
00:55:11,780 --> 00:55:15,910
Now, if the vacuum
has an energy density,

958
00:55:15,910 --> 00:55:19,370
as the universe
expands the space

959
00:55:19,370 --> 00:55:20,640
is still filled with vacuum.

960
00:55:20,640 --> 00:55:22,400
At least, if it was
filled with vacuum.

961
00:55:22,400 --> 00:55:24,174
If it was matter
it would thin out.

962
00:55:24,174 --> 00:55:26,590
But we can imagine a region
of space that was just vacuum,

963
00:55:26,590 --> 00:55:29,360
and as it expands it would
have to just stay vacuum.

964
00:55:29,360 --> 00:55:31,850
What else could it become?

965
00:55:31,850 --> 00:55:35,110
And that means that we
know that, for a vacuum,

966
00:55:35,110 --> 00:55:39,250
rho dot should equal zero.

967
00:55:39,250 --> 00:55:43,170
Now we've also learned earlier,
by applying conservation

968
00:55:43,170 --> 00:55:46,270
of energy to the
expanding universe,

969
00:55:46,270 --> 00:55:54,680
that rho dot in an expanding
universe, is equal to minus 3

970
00:55:54,680 --> 00:55:56,280
a dot over a.

971
00:55:56,280 --> 00:56:00,650
Or we could write
this as h times

972
00:56:00,650 --> 00:56:04,065
rho plus p over c squared.

973
00:56:07,820 --> 00:56:10,710
This is basically
a rewriting of d u

974
00:56:10,710 --> 00:56:16,380
equals minus p d v, applying
it to the expanding universe.

975
00:56:16,380 --> 00:56:17,380
So I won't re-derive it.

976
00:56:17,380 --> 00:56:19,629
We already derived it.

977
00:56:19,629 --> 00:56:21,670
Actually, I think you
derived it on the homework,

978
00:56:21,670 --> 00:56:23,070
was the way it actually worked.

979
00:56:23,070 --> 00:56:26,090
But in any case,
this immediately

980
00:56:26,090 --> 00:56:33,680
tells us that if
rho dot is going

981
00:56:33,680 --> 00:56:47,530
to be 0 for vacuum
energy, this has to be 0.

982
00:56:47,530 --> 00:56:55,860
And therefore p vacuum has
to be equal to minus rho

983
00:56:55,860 --> 00:56:58,620
vacuum times c squared.

984
00:57:15,060 --> 00:57:17,260
And if we know the energy
density in the pressure

985
00:57:17,260 --> 00:57:20,174
of this stuff called
vacuum, that's

986
00:57:20,174 --> 00:57:22,590
all we need to know to put it
into the Friedmann equations

987
00:57:22,590 --> 00:57:25,370
and find out how things behave.

988
00:57:25,370 --> 00:57:27,660
Otherwise this vacuum energy
behaves no differently

989
00:57:27,660 --> 00:57:28,640
from anything else.

990
00:57:28,640 --> 00:57:31,090
It just has a
particular relationship

991
00:57:31,090 --> 00:57:34,700
between the pressure
and the energy density,

992
00:57:34,700 --> 00:57:37,455
with a very peculiar feature-
that the pressure is negative.

993
00:57:43,550 --> 00:57:45,300
And that's an important
feature because we

994
00:57:45,300 --> 00:57:47,660
had commented earlier that
a negative pressure can

995
00:57:47,660 --> 00:57:50,280
drive acceleration.

996
00:57:50,280 --> 00:57:55,490
And now we're in a good position
to see exactly how that works.

997
00:58:11,566 --> 00:58:13,690
To sort of keep
things straight I'm

998
00:58:13,690 --> 00:58:20,060
going to divide the mass
density of the universe

999
00:58:20,060 --> 00:58:26,105
into a vacuum piece
and a normal piece,

1000
00:58:26,105 --> 00:58:30,100
where normal represents matter,
or radiation, or anything else,

1001
00:58:30,100 --> 00:58:32,662
if we ever discover
something else.

1002
00:58:32,662 --> 00:58:34,620
But in fact it will just
be matter or radiation

1003
00:58:34,620 --> 00:58:36,190
for anything that we'll
be doing in this course,

1004
00:58:36,190 --> 00:58:38,356
or anything that's really
done in current cosmology.

1005
00:58:42,000 --> 00:58:48,100
And similarly, I'm going to
write pressure as p vac plus p

1006
00:58:48,100 --> 00:58:48,830
normal.

1007
00:58:48,830 --> 00:58:49,690
"N" is for normal.

1008
00:58:55,800 --> 00:58:57,800
But p vac I don't
really need to use,

1009
00:58:57,800 --> 00:59:01,589
because p vac I can rewrite
in terms of rho vac.

1010
00:59:01,589 --> 00:59:03,380
So in the end I can
express everything just

1011
00:59:03,380 --> 00:59:06,290
in terms of rho vac.

1012
00:59:06,290 --> 00:59:09,650
And I can write down the second
order Friedmann equation.

1013
00:59:14,537 --> 00:59:16,620
And it's just a matter of
substituting in that rho

1014
00:59:16,620 --> 00:59:19,640
and that p into the
Friedmann equation

1015
00:59:19,640 --> 00:59:21,430
that we've already written.

1016
00:59:26,130 --> 00:59:34,960
And we get minus 4 pi
over 3 G, times rho

1017
00:59:34,960 --> 00:59:42,730
normal plus 3 p
normal, over c squared.

1018
00:59:42,730 --> 00:59:45,890
And the vacuum pieces-- have two
pieces because there's a vacuum

1019
00:59:45,890 --> 00:59:49,730
piece there and a
vacuum piece there. it

1020
00:59:49,730 --> 00:59:53,290
can all be expressed in terms
of rho vac and collected.

1021
00:59:53,290 --> 01:00:02,650
And what you get is
minus 2 rho vac times a.

1022
01:00:06,834 --> 01:00:08,500
Showing just what we
were talking about.

1023
01:00:08,500 --> 01:00:10,700
That because of that minus
sign, multiplies that

1024
01:00:10,700 --> 01:00:17,790
minus sign, vacuum energy drives
acceleration, not deceleration.

1025
01:00:17,790 --> 01:00:20,640
And that's why vacuum
energy can explain

1026
01:00:20,640 --> 01:00:23,810
these famous results of 1998.

1027
01:00:23,810 --> 01:00:26,110
And we'll see later that,
for the same reason vacuum

1028
01:00:26,110 --> 01:00:28,690
energy or things
like vacuum energy

1029
01:00:28,690 --> 01:00:30,700
can actually drive the
expansion of the universe

1030
01:00:30,700 --> 01:00:33,320
in the first place in
what we call inflation.

1031
01:00:36,320 --> 01:00:37,130
Yes?

1032
01:00:37,130 --> 01:00:39,950
AUDIENCE: So for the equation
without the cosmological

1033
01:00:39,950 --> 01:00:46,110
constant it's, let's say, rho
and p are about the constant,

1034
01:00:46,110 --> 01:00:48,054
then wouldn't that
be the equation

1035
01:00:48,054 --> 01:00:50,000
for a simple harmonic
function [INAUDIBLE]

1036
01:00:50,000 --> 01:00:54,546
or the oscillation
of a [INAUDIBLE]

1037
01:00:54,546 --> 01:00:56,981
is some negative
constant times a?

1038
01:00:59,910 --> 01:01:04,880
PROFESSOR: That's right, except
that you would probably not

1039
01:01:04,880 --> 01:01:06,949
believe the equations
with the bounds.

1040
01:01:06,949 --> 01:01:07,490
AUDIENCE: OK.

1041
01:01:07,490 --> 01:01:09,781
PROFESSOR: And when a went
negative you wouldn't really

1042
01:01:09,781 --> 01:01:11,880
have a cosmological
interpretation anymore,

1043
01:01:11,880 --> 01:01:12,970
I don't think.

1044
01:01:12,970 --> 01:01:17,340
But it is, in fact, true that
if rho and p were constants--

1045
01:01:17,340 --> 01:01:20,360
I'm not sure of any model that
actually does that-- this would

1046
01:01:20,360 --> 01:01:24,640
give you sinusoidal behavior
during the expanding

1047
01:01:24,640 --> 01:01:27,670
and contracting phase.

1048
01:01:27,670 --> 01:01:28,613
Yes?

1049
01:01:28,613 --> 01:01:32,960
AUDIENCE: [INAUDIBLE] the vacuum
energy is constant over time,

1050
01:01:32,960 --> 01:01:36,867
is it also makes
sense [INAUDIBLE]?

1051
01:01:36,867 --> 01:01:38,700
AUDIENCE: Are you asking,
does it make sense

1052
01:01:38,700 --> 01:01:40,970
for maybe the vacuum
energy to change with time?

1053
01:01:48,055 --> 01:01:49,430
I think, if it
changed with time,

1054
01:01:49,430 --> 01:01:51,900
you wouldn't call
it vacuum energy.

1055
01:01:51,900 --> 01:01:54,070
Because the vacuum
is more or less

1056
01:01:54,070 --> 01:01:55,870
defined as the lowest
possible energy

1057
01:01:55,870 --> 01:02:00,130
state allowed by
the laws of physics.

1058
01:02:00,130 --> 01:02:02,210
And the laws of physics,
as far as we know,

1059
01:02:02,210 --> 01:02:03,481
do not change with time.

1060
01:02:03,481 --> 01:02:05,980
It's certainly true that, in a
completely different context,

1061
01:02:05,980 --> 01:02:08,630
you might imagine the laws of
physics might change with time.

1062
01:02:08,630 --> 01:02:10,700
And then thing would
get more complicated.

1063
01:02:10,700 --> 01:02:13,580
But that would really take you
somewhat outside the sphere

1064
01:02:13,580 --> 01:02:16,910
of physics as we know it.

1065
01:02:16,910 --> 01:02:18,660
You could always explore
things like that,

1066
01:02:18,660 --> 01:02:21,170
and it may turn out to be right.

1067
01:02:21,170 --> 01:02:24,940
But at least within
the context of physics

1068
01:02:24,940 --> 01:02:27,500
as we currently envision
it, vacuum energies

1069
01:02:27,500 --> 01:02:29,280
are constant, pretty
much by definition.

1070
01:02:35,920 --> 01:02:39,940
Now I should maybe qualify
that within the context of what

1071
01:02:39,940 --> 01:02:45,650
we understand, there may,
in fact, be multiple vacua.

1072
01:02:45,650 --> 01:02:48,740
For example, if you
have a field theory

1073
01:02:48,740 --> 01:02:51,350
one can have a potential
energy function

1074
01:02:51,350 --> 01:02:53,830
for one or more fields.

1075
01:02:53,830 --> 01:02:55,280
And that potential
energy function

1076
01:02:55,280 --> 01:02:58,680
could have more than
one local minimum.

1077
01:02:58,680 --> 01:03:03,080
And then any one of those local
minima is effectively a vacuum.

1078
01:03:03,080 --> 01:03:06,130
And that could very
likely be the situation

1079
01:03:06,130 --> 01:03:07,970
that describes the real world.

1080
01:03:07,970 --> 01:03:10,820
And then you could
tunnel from one vacuum

1081
01:03:10,820 --> 01:03:13,990
to another, changing
the vacuum energy.

1082
01:03:13,990 --> 01:03:15,890
But that would not be
a smooth evolution.

1083
01:03:15,890 --> 01:03:17,350
That would be a
sudden tunneling.

1084
01:03:33,490 --> 01:03:33,990
OK.

1085
01:03:33,990 --> 01:03:37,120
So this is what happens to
the second order Friedmann

1086
01:03:37,120 --> 01:03:39,576
equation.

1087
01:03:39,576 --> 01:03:42,890
It is also very useful to look
at the first order Friedmann

1088
01:03:42,890 --> 01:04:03,380
equation, which is a dot over
a squared, 8 pi over 3 G.

1089
01:04:03,380 --> 01:04:08,970
And in its native
way of being written

1090
01:04:08,970 --> 01:04:11,990
we would just have
8 pi over 3 G rho,

1091
01:04:11,990 --> 01:04:16,100
minus k over-- kc
squared over a squared.

1092
01:04:16,100 --> 01:04:20,450
And all I want to do now
is replace rho by rho

1093
01:04:20,450 --> 01:04:21,690
vac plus rho n.

1094
01:04:39,770 --> 01:04:41,670
And this is a first
order Friedmann equation.

1095
01:04:47,450 --> 01:04:54,530
And we can expand rho n
if we want more details,

1096
01:04:54,530 --> 01:04:59,070
as rho matter plus
rho radiation.

1097
01:05:01,650 --> 01:05:07,940
And rho matter, we know,
varies with time proportional

1098
01:05:07,940 --> 01:05:09,135
to 1 over a cubed.

1099
01:05:12,710 --> 01:05:18,250
Rho radiation behaves with
time as 1 over a to the fourth.

1100
01:05:21,020 --> 01:05:23,900
So all of the terms
here, except for rho vac,

1101
01:05:23,900 --> 01:05:25,460
fall off as a grows.

1102
01:05:28,200 --> 01:05:33,760
And that implies that
if you're not somehow

1103
01:05:33,760 --> 01:05:35,889
turned around firsts,
which you can be--

1104
01:05:35,889 --> 01:05:37,430
you could have a
closed universe that

1105
01:05:37,430 --> 01:05:41,840
collapses before vacuum
energy can take over.

1106
01:05:41,840 --> 01:05:47,430
But as the universe gets larger,
if it doesn't turn around,

1107
01:05:47,430 --> 01:05:49,974
eventually rho vac will win.

1108
01:05:49,974 --> 01:05:51,640
It will become larger
than anything else

1109
01:05:51,640 --> 01:05:53,660
because everything else is just
getting smaller and smaller.

1110
01:05:53,660 --> 01:05:55,576
And once that starts to
happen everything else

1111
01:05:55,576 --> 01:05:57,930
will get smaller and
smaller, faster and faster,

1112
01:05:57,930 --> 01:06:03,350
because a will start
to grow exponentially.

1113
01:06:03,350 --> 01:06:17,140
If rho vac dominates--
which it will, as I said,

1114
01:06:17,140 --> 01:06:18,890
unless the universe
re-collapses first--

1115
01:06:18,890 --> 01:06:21,620
so for a large class
of solutions rho vac

1116
01:06:21,620 --> 01:06:27,740
will dominate-- then you
can solve that equation.

1117
01:06:27,740 --> 01:06:39,460
And you have h, which is
a dot over a, approaches,

1118
01:06:39,460 --> 01:06:49,255
as a goes to infinity, the
square root of 8 pi over 3 G

1119
01:06:49,255 --> 01:06:49,755
rho vac.

1120
01:07:02,230 --> 01:07:09,250
So h will approach a fixed
value for a universe which

1121
01:07:09,250 --> 01:07:13,540
is ultimately dominated
by rho vacuum.

1122
01:07:13,540 --> 01:07:15,260
And if a dot over
a is a constant,

1123
01:07:15,260 --> 01:07:16,845
that means that a
grows exponentially.

1124
01:08:06,800 --> 01:08:10,840
So we could maybe give
this a name-- h vac.

1125
01:08:10,840 --> 01:08:12,680
The value h has
when it's completely

1126
01:08:12,680 --> 01:08:15,890
dominated by the vacuum energy.

1127
01:08:15,890 --> 01:08:30,485
And then we can
write that a of t

1128
01:08:30,485 --> 01:08:35,090
is ultimately going to be
proportional to e to the h vac

1129
01:08:35,090 --> 01:08:36,557
times t.

1130
01:08:36,557 --> 01:08:38,640
Which is what you get when
you solve the equation,

1131
01:08:38,640 --> 01:08:40,760
a dot over a equals
this constant.

1132
01:08:53,520 --> 01:08:54,479
OK.

1133
01:08:54,479 --> 01:09:00,485
Now one thing which you
can see very quickly--

1134
01:09:00,485 --> 01:09:02,359
let's see how far I
should plan to get today.

1135
01:09:20,220 --> 01:09:20,720
OK.

1136
01:09:20,720 --> 01:09:22,511
I'll probably make one
qualitative argument

1137
01:09:22,511 --> 01:09:25,399
and then start a calculation
that won't get very far.

1138
01:09:25,399 --> 01:09:27,680
I will continue next time.

1139
01:09:27,680 --> 01:09:29,750
One qualitative
point which you can

1140
01:09:29,750 --> 01:09:33,710
see from just glancing
at these equations

1141
01:09:33,710 --> 01:09:37,950
is that the cosmological
constant, when

1142
01:09:37,950 --> 01:09:40,330
added to the other
ingredients that we've already

1143
01:09:40,330 --> 01:09:42,779
put into our model
universe, will

1144
01:09:42,779 --> 01:09:45,550
have the effect of
increasing the age

1145
01:09:45,550 --> 01:09:48,950
of the universe for
a given value of h.

1146
01:09:48,950 --> 01:09:52,410
And that's something that we
said earlier in the course,

1147
01:09:52,410 --> 01:09:53,910
we're looking forward to.

1148
01:09:53,910 --> 01:09:56,460
Because the model of
the universe that we're

1149
01:09:56,460 --> 01:09:58,420
been constructing
so far have always

1150
01:09:58,420 --> 01:10:04,359
turned out to be too young
for the measured value of h.

1151
01:10:04,359 --> 01:10:06,150
That is, the oldest
stars look like they're

1152
01:10:06,150 --> 01:10:07,310
older than the universe.

1153
01:10:07,310 --> 01:10:08,370
And that's not good.

1154
01:10:08,370 --> 01:10:11,030
So we'd like to make
universe look older.

1155
01:10:11,030 --> 01:10:17,460
And one of the beauties of
having this vacuum energy,

1156
01:10:17,460 --> 01:10:19,120
as far as making
things fit together,

1157
01:10:19,120 --> 01:10:21,410
is that it does make
the universe older.

1158
01:10:21,410 --> 01:10:26,050
And the easiest way to
see that-- at least a way

1159
01:10:26,050 --> 01:10:32,530
to see that-- is to imagine
drawing a graph of h versus t.

1160
01:10:32,530 --> 01:10:36,320
Hubble expansion rate versus t.

1161
01:10:36,320 --> 01:10:40,510
And if we look at
the formula for h

1162
01:10:40,510 --> 01:10:44,770
here we see that the
rho vac piece just

1163
01:10:44,770 --> 01:10:49,800
puts in a floor as
h evolves with time,

1164
01:10:49,800 --> 01:10:53,420
instead of going to 0 as it
wood in most models-- at least,

1165
01:10:53,420 --> 01:10:56,250
as it would in
open-universe model.

1166
01:10:56,250 --> 01:10:57,855
It stops at some floor.

1167
01:11:00,560 --> 01:11:11,050
And certainly for the models
that we've been dealing with,

1168
01:11:11,050 --> 01:11:15,370
h just decreases to
some-- this is supposed

1169
01:11:15,370 --> 01:11:18,650
to represent the present time.

1170
01:11:18,650 --> 01:11:24,115
So this is previous models.

1171
01:11:27,852 --> 01:11:30,310
Now as you might say, that what
I'm trying to describe here

1172
01:11:30,310 --> 01:11:35,220
is not quite a theorem if you
considered closed universes

1173
01:11:35,220 --> 01:11:43,471
where this k piece
could be causing

1174
01:11:43,471 --> 01:11:47,010
a positive-- a negative
contribution to h, which

1175
01:11:47,010 --> 01:11:48,640
is then decreasing with time.

1176
01:11:48,640 --> 01:11:50,674
Things can get complicated.

1177
01:11:50,674 --> 01:11:52,840
But for the models that
we've been considering which

1178
01:11:52,840 --> 01:11:57,130
are nearly flat, that
k piece is absent.

1179
01:11:57,130 --> 01:12:00,660
And then we just have pieces
that go like, 1 over a cubed,

1180
01:12:00,660 --> 01:12:03,060
1 over a to the
fourth, and constant.

1181
01:12:03,060 --> 01:12:05,650
All of which are positive.

1182
01:12:05,650 --> 01:12:11,920
Then in the absence of vacuum
energy we would have h falling.

1183
01:12:11,920 --> 01:12:16,270
And with the presence
of vacuum energy

1184
01:12:16,270 --> 01:12:17,820
it would not fall
as fast because we

1185
01:12:17,820 --> 01:12:21,890
have this constant piece
that would not be decreasing.

1186
01:12:21,890 --> 01:12:24,110
So this is previous models.

1187
01:12:24,110 --> 01:12:29,780
This is with rho vac.

1188
01:12:29,780 --> 01:12:32,340
And I'm always talking
about positive rho vac

1189
01:12:32,340 --> 01:12:34,800
because that is what
our universe has.

1190
01:12:38,307 --> 01:12:40,140
So this would be the
two different behaviors

1191
01:12:40,140 --> 01:12:46,110
of h for the model
without vacuum energy

1192
01:12:46,110 --> 01:12:48,140
and the model with
vacuum energy.

1193
01:12:48,140 --> 01:12:51,170
And if we're trying to calculate
the age of the universe

1194
01:12:51,170 --> 01:12:53,900
we would basically be
extrapolating this curve back

1195
01:12:53,900 --> 01:12:57,410
to the point where h was
infinite, at the big bang.

1196
01:12:57,410 --> 01:12:59,900
And we could see that,
since this curve is always

1197
01:12:59,900 --> 01:13:01,750
below this curve,
it will take longer

1198
01:13:01,750 --> 01:13:05,140
before it turns up
and becomes infinite.

1199
01:13:05,140 --> 01:13:10,886
So the age will always increase
by adding vacuum energy.

1200
01:13:22,310 --> 01:13:34,500
With rho vac h equals infinity
is further to the left.

1201
01:13:38,902 --> 01:13:41,110
And notice that I'm comparing
two different theories,

1202
01:13:41,110 --> 01:13:42,651
both of which are
the same age today.

1203
01:13:42,651 --> 01:13:44,920
Because that's what
we're interested in.

1204
01:13:44,920 --> 01:13:46,170
We've measured the value of h.

1205
01:13:46,170 --> 01:13:48,475
We're trying to infer
the age of the universe.

1206
01:14:02,450 --> 01:14:02,950
OK.

1207
01:14:02,950 --> 01:14:05,283
Maybe I'll just say a couple
words about the calculation

1208
01:14:05,283 --> 01:14:07,242
that we'll be starting
with next time.

1209
01:14:07,242 --> 01:14:08,950
We want to be able to
precisely calculate

1210
01:14:08,950 --> 01:14:10,490
things like the age
of the universe,

1211
01:14:10,490 --> 01:14:13,340
including the effect
of this vacuum energy.

1212
01:14:13,340 --> 01:14:16,840
And we'll be able to do that
in a very straightforward way

1213
01:14:16,840 --> 01:14:19,165
by using this first
order Friedmann equation.

1214
01:14:22,180 --> 01:14:25,100
We know how each term in
this Friedmann equation

1215
01:14:25,100 --> 01:14:28,240
varies with a.

1216
01:14:28,240 --> 01:14:30,227
And we can measure
the amount of matter,

1217
01:14:30,227 --> 01:14:32,560
and the amount of radiation,
and in principal the amount

1218
01:14:32,560 --> 01:14:35,030
of curvature-- it's
negligibly small--

1219
01:14:35,030 --> 01:14:37,480
in our current universe.

1220
01:14:37,480 --> 01:14:40,600
And once you have
those parameters

1221
01:14:40,600 --> 01:14:43,070
you can use that
equation to extrapolate,

1222
01:14:43,070 --> 01:14:47,020
to know what h was at
any time in the past.

1223
01:14:47,020 --> 01:14:51,990
And that tells you
how the derivative--

1224
01:14:51,990 --> 01:14:55,820
it tells you the value of a
dot at any time in the past.

1225
01:14:55,820 --> 01:14:58,670
And if you know the value of
a dot at any time in the past,

1226
01:14:58,670 --> 01:15:01,080
it's a principle just
a matter of integration

1227
01:15:01,080 --> 01:15:04,330
to figure out when a was 0.

1228
01:15:04,330 --> 01:15:07,710
And that's the calculation that
we'll begin by doing next time.

1229
01:15:07,710 --> 01:15:11,020
And we'll be able to get
an integral expression

1230
01:15:11,020 --> 01:15:14,980
for the age of the universe
for an arbitrary value.

1231
01:15:14,980 --> 01:15:18,180
We'll, at the end,
express the matter density

1232
01:15:18,180 --> 01:15:20,710
and the radiation
density as fractions

1233
01:15:20,710 --> 01:15:24,470
of omega, fractions of
the critical density.

1234
01:15:24,470 --> 01:15:27,480
And for any value
of omega matter,

1235
01:15:27,480 --> 01:15:30,810
omega radiation, and we'll
even express the curvature

1236
01:15:30,810 --> 01:15:31,920
as an omega curvature.

1237
01:15:31,920 --> 01:15:34,040
The effective fraction
of the critical density

1238
01:15:34,040 --> 01:15:36,120
that this term represents.

1239
01:15:36,120 --> 01:15:38,075
And in terms of those
different omegas,

1240
01:15:38,075 --> 01:15:39,700
we'll be able to
write down an integral

1241
01:15:39,700 --> 01:15:42,080
for the total age
of the universe.

1242
01:15:42,080 --> 01:15:44,100
And that really is going
to be state of the art.

1243
01:15:44,100 --> 01:15:48,160
That is what the
Planck team uses

1244
01:15:48,160 --> 01:15:50,854
when they're analyzing their
data to try to understand what

1245
01:15:50,854 --> 01:15:53,020
the age of universe is
according to the measurements

1246
01:15:53,020 --> 01:15:55,180
that they're making.

1247
01:15:55,180 --> 01:15:59,180
So we will finally come
up to the present as far

1248
01:15:59,180 --> 01:16:05,210
as the actual understanding
of cosmology by the experts.

1249
01:16:05,210 --> 01:16:07,530
So that's all for today.

1250
01:16:07,530 --> 01:16:10,000
see you all next Tuesday.