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PROFESSOR: OK, in that
case, let's take off.

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There's a fair
amount I'd like to do

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before the end of the term.

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First, let me quickly review
what we talked about last time.

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We talked about the
actual supernovae data,

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which gives us brightness as a
function of redshift for very

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distant objects and
produced the first discovery

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that the model is not fit very
well by the standard called

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00:00:49,530 --> 00:00:52,140
dark matter model, which
would be this lower line.

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But it fits much better
by this lambda CDM,

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00:00:55,590 --> 00:01:01,090
a model which involves a
significant fraction-- 0.76 was

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used here-- and significant
fraction of vacuum energy

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along with cold, dark
matter and baryons.

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And that was a model
that fit the data well.

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And this, along
with similar data

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from another group
of astronomers,

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was a big bombshell
of 1998, showing

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that the universe appears
to not be slowing down

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under the influence of gravity
but rather to be accelerating

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due to some kind of
repulsive, presumably,

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gravitational force.

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We talked about the
overall evidence

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for this idea of an
accelerating universe.

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It certainly began with
the supernovae data

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that we just talked about.

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The basic fact
that characterizes

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that data is that the most
distant supernovae are

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00:01:43,050 --> 00:01:46,780
dimmer than you'd
expect by 20% to 30%.

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00:01:46,780 --> 00:01:50,310
And people did cook up
other possible explanations

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for what might have made
supernovae at certain distances

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00:01:53,380 --> 00:01:54,460
look dimmer.

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00:01:54,460 --> 00:01:56,890
None of those really
held up very well.

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00:01:56,890 --> 00:01:59,860
But, in addition, several other
important pieces of evidence

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00:01:59,860 --> 00:02:03,930
came in to support this idea
of an accelerating universe.

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Most important, more
precise measurements

43
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of the cosmic background
radiation anisotropies came in.

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00:02:13,940 --> 00:02:17,030
And this pattern
of antisotropies

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00:02:17,030 --> 00:02:18,980
can be fit to a
theoretical model, which

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00:02:18,980 --> 00:02:22,290
includes all of the parameters
of cosmology, basically,

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00:02:22,290 --> 00:02:25,850
and turns out to give a very
precise setting for essentially

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all the parameters of cosmology.

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They now really all have their
most precise values coming out

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00:02:31,970 --> 00:02:33,680
of these CMB measurements.

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00:02:33,680 --> 00:02:36,240
And the CMB measurements
gave a value

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of omega [? vac ?], which
is very close to what we'll

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get from the supernovae,
which makes it all

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look very convincing.

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And furthermore, the
cosmic background radiation

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shows that omega total is
equal to 1 to about 1/2%

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accuracy, which is very hard
to account for if one doesn't

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assume that there's a
very significant amount

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of dark energy.

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Because there just
does not appear

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to be nearly enough of anything
else to make omega equal to 1.

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And finally, we pointed out
that this vacuum energy also

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improves the age calculations.

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Without vacuum energy,
we tend to define

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that the age of the universe
as calculated from the Big Bang

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theory always ended up
being a little younger

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than the ages of the oldest
stars, which didn't make sense.

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But with the vacuum
energy, that changes

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the cosmological calculation of
the age producing older ages.

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So with vacuum energy of the
sort that we think exists,

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we get ages like 13.7
or 13.8 billion years.

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00:03:36,570 --> 00:03:38,940
And that's completely
consistent with what

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we think about the ages
of the oldest stars.

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So everything fits together.

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So by now, I would say that
with these three arguments

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together, essentially,
everybody is convinced

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that this acceleration is real.

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I do know a few people
who aren't convinced,

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but they're oddballs.

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Most of us are convinced.

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And the simplest explanation
for this dark energy

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is simply vacuum energy.

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And every measurement
that's been made so far

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is consistent with the
idea of vacuum energy.

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00:04:05,300 --> 00:04:07,330
There is still an
alternative possibility

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which is called
quintessence, which

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would be a very slowly
evolving scalar field.

88
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And it would show up, because
you would see some evolution.

89
00:04:15,100 --> 00:04:17,490
And so far nobody has
seen any evolution

90
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of the amount of dark
energy in the universe.

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So that's basically
where things stand as far

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as the observations of
acceleration of dark energy.

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Any questions about that?

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00:04:32,690 --> 00:04:36,230
OK, next, we went on to talk
about the physics of vacuum

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00:04:36,230 --> 00:04:39,680
energy or a
cosmological constant.

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00:04:39,680 --> 00:04:41,970
A cosmological constant
and vacuum energy

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are really synonymous.

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00:04:43,830 --> 00:04:45,900
And they're related
to each other

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by the energy
entering the vacuum

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00:04:47,480 --> 00:04:50,960
being equal to this expression,
where lambda is what Einstein

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00:04:50,960 --> 00:04:52,775
originally called the
cosmological constant

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and what we still call
the cosmological constant.

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We discussed the fact
that there are basically

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three contributions
in a quantum field

105
00:05:00,310 --> 00:05:02,845
theory to the
energy of a vacuum.

106
00:05:02,845 --> 00:05:04,720
We do not expect it to
be zero, because there

107
00:05:04,720 --> 00:05:07,489
are these complicated
contributions.

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00:05:07,489 --> 00:05:09,530
There are, first of all,
the quantum fluctuations

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00:05:09,530 --> 00:05:12,750
of the photon and
other Bosonic fields,

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00:05:12,750 --> 00:05:14,950
Bosonic fields
meaning particles that

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00:05:14,950 --> 00:05:18,050
do not obey the Pauli
exclusion principle.

112
00:05:18,050 --> 00:05:20,380
And that gives us a
positive contribution

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00:05:20,380 --> 00:05:23,790
to the energy, which
is, in fact, divergent.

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00:05:23,790 --> 00:05:27,870
It diverges because every
standing wave contributes.

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00:05:27,870 --> 00:05:31,740
And there's no lower bound to
the wavelength of a standing

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00:05:31,740 --> 00:05:32,610
wave.

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So by considering shorter
and shorter wavelengths,

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00:05:35,280 --> 00:05:37,880
one gets larger and
larger contributions

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to this vacuum energy.

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00:05:39,900 --> 00:05:44,030
And in the quantum field
theory, it's just unbounded.

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00:05:44,030 --> 00:05:46,020
Similarly, there are
quantum fluctuations

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00:05:46,020 --> 00:05:48,160
to other fields like
the electron field

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00:05:48,160 --> 00:05:50,290
which is a Fermionic
field, a field that

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00:05:50,290 --> 00:05:53,450
describes a particle that obeys
the Pauli exclusion principle.

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00:05:53,450 --> 00:05:56,710
And those fields behave
somewhat differently.

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00:05:56,710 --> 00:05:58,840
Like the photon, the
electron is viewed

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00:05:58,840 --> 00:06:02,610
as the quantum
excitation of this field.

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00:06:02,610 --> 00:06:06,550
And that turns out to be by
far, basically, the only way we

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00:06:06,550 --> 00:06:09,270
know to describe
relativistic particles

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00:06:09,270 --> 00:06:12,090
in a totally consistent way.

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00:06:12,090 --> 00:06:13,960
In this case, again,
the contribution

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00:06:13,960 --> 00:06:15,420
to the vacuum
energy is divergent.

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But in this case, it's
negative and divergent,

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00:06:18,020 --> 00:06:20,320
allowing possibilities of
some kind of cancellation,

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but no reason that
we know of why

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they should cancel each other.

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00:06:23,380 --> 00:06:26,350
They seem to just be
totally different objects.

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00:06:26,350 --> 00:06:28,410
And, finally, there
are some fields

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00:06:28,410 --> 00:06:30,220
which have nonzero
values in the vacuum.

140
00:06:30,220 --> 00:06:33,910
And, in particular, the Higgs
field of the standard model

141
00:06:33,910 --> 00:06:37,170
is believed to have a nonzero
value even in the vacuum.

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00:06:40,170 --> 00:06:43,310
So this is the basic story.

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00:06:43,310 --> 00:06:46,110
We commented that if we
cut off these infinities

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00:06:46,110 --> 00:06:48,550
by saying that we don't
understand things at very, very

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00:06:48,550 --> 00:06:53,050
short wavelengths, at
least one plausible cut off

146
00:06:53,050 --> 00:06:56,300
would be the Planck scale,
which is the scale associated

147
00:06:56,300 --> 00:06:59,460
with where we think quantum
gravity becomes important.

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00:06:59,460 --> 00:07:02,050
And if we cut off at
this Planck scale,

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00:07:02,050 --> 00:07:05,880
the energies become
finite but still too large

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00:07:05,880 --> 00:07:09,340
compared to what we observe
by more than 120 orders

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00:07:09,340 --> 00:07:10,470
of magnitude.

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00:07:10,470 --> 00:07:12,650
And on the homework set
that's due next Monday,

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00:07:12,650 --> 00:07:15,010
you'll be calculating
this number for yourself.

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00:07:15,010 --> 00:07:18,720
It's a little bit more than
120 orders of magnitude.

155
00:07:18,720 --> 00:07:21,360
So it's a colossal failure
indicating that we really

156
00:07:21,360 --> 00:07:24,790
don't understand what controls
the value of this vacuum

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00:07:24,790 --> 00:07:27,010
energy.

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00:07:27,010 --> 00:07:28,707
And I think I
mentioned last time,

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00:07:28,707 --> 00:07:30,540
and I'll mention it a
little more explicitly

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00:07:30,540 --> 00:07:32,090
by writing it on
the transparency

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00:07:32,090 --> 00:07:36,490
this time, that the situation
is so desperate in that we've

162
00:07:36,490 --> 00:07:40,150
had so much trouble trying to
find any way of explaining why

163
00:07:40,150 --> 00:07:45,860
the vacuum energy should be so
small that it has become quite

164
00:07:45,860 --> 00:07:50,080
popular to accept the
possibility, at least,

165
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that the vacuum energy is
determined by what is called

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00:07:53,190 --> 00:07:55,720
the anthropic
selection principal

167
00:07:55,720 --> 00:07:58,352
or anthropic selection effect.

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00:07:58,352 --> 00:07:59,810
And Steve Weinberg
was actually one

169
00:07:59,810 --> 00:08:03,062
of the first people who
advocated this point of view.

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00:08:03,062 --> 00:08:06,080
I'm sort of a recent convert
to taking this point of view

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00:08:06,080 --> 00:08:07,790
seriously.

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00:08:07,790 --> 00:08:11,380
But the idea is
that there might be

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00:08:11,380 --> 00:08:13,620
more than one possible
type of vacuum.

174
00:08:13,620 --> 00:08:15,720
And, in fact, string
theory comes in here

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00:08:15,720 --> 00:08:16,949
in an important way.

176
00:08:16,949 --> 00:08:18,490
String theory seems
to really predict

177
00:08:18,490 --> 00:08:21,830
that there's a colossal number
of different types of vacuum,

178
00:08:21,830 --> 00:08:25,680
perhaps 10 to the 500 different
types of vacuum or more.

179
00:08:25,680 --> 00:08:27,840
And each one would have
its own vacuum energy.

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00:08:27,840 --> 00:08:31,560
So with that many, some of them
would have a, by coincidence,

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00:08:31,560 --> 00:08:33,220
near perfect
cancellation between

182
00:08:33,220 --> 00:08:35,730
the positive and negative
contributions producing

183
00:08:35,730 --> 00:08:38,770
a net vacuum energy that
could be very, very small.

184
00:08:38,770 --> 00:08:42,840
But it would be a tiny fraction
of all of the possible vauua,

185
00:08:42,840 --> 00:08:45,970
a fraction like 10
to the minus 120,

186
00:08:45,970 --> 00:08:48,030
since we have 120
orders of magnitude

187
00:08:48,030 --> 00:08:50,730
mismatch of these ranges.

188
00:08:50,730 --> 00:08:54,170
So you would still have to
press yourself to figure out

189
00:08:54,170 --> 00:08:56,550
what would be the explanation
why we should be living

190
00:08:56,550 --> 00:08:59,260
in such an atypical vacuum.

191
00:08:59,260 --> 00:09:03,840
And the proposed answer is that
it's anthropically selected,

192
00:09:03,840 --> 00:09:07,280
where anthropic means
having to do with life.

193
00:09:07,280 --> 00:09:11,110
Whereas, the claim is
made that life only

194
00:09:11,110 --> 00:09:16,840
evolves in vacuua which
have incredibly small vacuum

195
00:09:16,840 --> 00:09:18,090
energies.

196
00:09:18,090 --> 00:09:20,910
Because if the vacuum energy is
much larger, if it's positive,

197
00:09:20,910 --> 00:09:24,520
it blows the universe apart
before structures can form.

198
00:09:24,520 --> 00:09:27,590
And it it's negative,
it implodes the universe

199
00:09:27,590 --> 00:09:30,760
before there's time
for structures to form.

200
00:09:30,760 --> 00:09:34,750
So a long-lived universe
requires a very small vacuum

201
00:09:34,750 --> 00:09:35,664
energy density.

202
00:09:35,664 --> 00:09:37,580
And the claim is that
those are the only kinds

203
00:09:37,580 --> 00:09:39,810
of universes that support life.

204
00:09:39,810 --> 00:09:42,100
So we're here because it's
the only kind of universe

205
00:09:42,100 --> 00:09:44,961
in which life can
exist is the claim.

206
00:09:44,961 --> 00:09:45,460
Yes?

207
00:09:45,460 --> 00:09:46,655
AUDIENCE: So, different
types of energies,

208
00:09:46,655 --> 00:09:48,806
obviously, affect the
acceleration rate and stuff

209
00:09:48,806 --> 00:09:49,472
of the universe.

210
00:09:49,472 --> 00:09:52,106
But do they also affect, in any
way, the fundamental forces,

211
00:09:52,106 --> 00:09:54,064
or would those be the
same in all of the cases?

212
00:09:54,064 --> 00:09:56,510
PROFESSOR: OK, the question
is, would the different kinds

213
00:09:56,510 --> 00:09:59,630
of vacuum affect the kinds
of fundamental forces

214
00:09:59,630 --> 00:10:03,760
that exist besides the force
of the cosmological constant

215
00:10:03,760 --> 00:10:05,839
on the acceleration of
the universe itself?

216
00:10:05,839 --> 00:10:08,130
The answer is, yeah, it would
affect really everything.

217
00:10:08,130 --> 00:10:10,350
These different vacuua would be
very different from each other.

218
00:10:10,350 --> 00:10:12,000
They would each have
their own version

219
00:10:12,000 --> 00:10:14,585
of what we call the standard
model of particle physics.

220
00:10:14,585 --> 00:10:16,960
And that's because the standard
model of particle physics

221
00:10:16,960 --> 00:10:18,890
would be viewed as
what happens when

222
00:10:18,890 --> 00:10:22,380
you have small perturbations
about our particular type

223
00:10:22,380 --> 00:10:23,434
of vacuum.

224
00:10:23,434 --> 00:10:24,850
And with different
types of vacuum

225
00:10:24,850 --> 00:10:27,670
you get different time
types of small perturbations

226
00:10:27,670 --> 00:10:29,530
about those vaccua.

227
00:10:29,530 --> 00:10:32,020
So the physics really could
be completely different in all

228
00:10:32,020 --> 00:10:36,330
the different vacuua that
string theory suggests exist.

229
00:10:40,840 --> 00:10:43,030
So the story here,
basically, is a big mystery.

230
00:10:43,030 --> 00:10:45,880
Not everybody accepts
these anthropic ideas.

231
00:10:45,880 --> 00:10:48,160
They are talked about.

232
00:10:48,160 --> 00:10:49,780
At almost any
cosmology conference,

233
00:10:49,780 --> 00:10:52,446
there will be some session where
people talk about these things.

234
00:10:52,446 --> 00:10:55,920
They are widely
discussed but by no means

235
00:10:55,920 --> 00:10:58,080
completely agreed upon.

236
00:10:58,080 --> 00:11:00,230
And it's very much
an open question,

237
00:11:00,230 --> 00:11:03,840
what it is that explains
the very small vacuum energy

238
00:11:03,840 --> 00:11:07,300
density that we observe.

239
00:11:07,300 --> 00:11:10,370
OK, moving on, in
the last lecture

240
00:11:10,370 --> 00:11:14,779
I also gave a quick historical
overview of the interactions

241
00:11:14,779 --> 00:11:16,320
between Einstein
and Friedmann, which

242
00:11:16,320 --> 00:11:18,680
I found rather interesting.

243
00:11:18,680 --> 00:11:24,460
And just a quick summary here,
in 1922 June 29, to be precise,

244
00:11:24,460 --> 00:11:27,530
Alexander Friedmann's first
paper about the Friedmann

245
00:11:27,530 --> 00:11:31,450
equations and the dynamical
model of the universe

246
00:11:31,450 --> 00:11:35,150
were received at [INAUDIBLE].

247
00:11:35,150 --> 00:11:37,820
Einstein learned about it
and immediately decided

248
00:11:37,820 --> 00:11:42,840
that it had to be wrong and
fired off a refutation claiming

249
00:11:42,840 --> 00:11:44,820
that Friedmann had gotten
his equations wrong.

250
00:11:44,820 --> 00:11:46,140
And if he had gotten
them right, he

251
00:11:46,140 --> 00:11:48,389
would have discovered that
rho dot, the rate of change

252
00:11:48,389 --> 00:11:50,500
of the energy density,
had to be zero

253
00:11:50,500 --> 00:11:53,170
and that there was nothing but
the static solution allowed.

254
00:11:55,840 --> 00:12:00,030
Einstein then met a friend
of Friedmann's Yuri Krutkov

255
00:12:00,030 --> 00:12:05,110
at a retirement lecture
by Lawrence in Leiden

256
00:12:05,110 --> 00:12:07,210
the following spring.

257
00:12:07,210 --> 00:12:09,400
And Krutkov convinced
Einstein that he

258
00:12:09,400 --> 00:12:11,570
was wrong about
this calculation.

259
00:12:11,570 --> 00:12:13,390
Einstein had also
received a letter

260
00:12:13,390 --> 00:12:14,960
from Friedmann,
which he probably

261
00:12:14,960 --> 00:12:17,005
didn't read until this
time, but the letter

262
00:12:17,005 --> 00:12:19,290
was apparently also convincing.

263
00:12:19,290 --> 00:12:21,450
So Einstein did finally retract.

264
00:12:21,450 --> 00:12:26,760
And at the end of May
1923, his refraction

265
00:12:26,760 --> 00:12:28,425
was received at
Zeitschrift fur Physik.

266
00:12:31,240 --> 00:12:33,700
And another interesting
fact about that

267
00:12:33,700 --> 00:12:37,400
is that the original handwritten
draft of that retraction

268
00:12:37,400 --> 00:12:38,620
still exists.

269
00:12:38,620 --> 00:12:42,010
And it had the curious
comment, which was crossed out,

270
00:12:42,010 --> 00:12:46,270
where Einstein suggested that
the Friedmann solutions could

271
00:12:46,270 --> 00:12:48,670
be modified by the phrase,
"a physical significance

272
00:12:48,670 --> 00:12:51,290
can hardly be ascribed to them."

273
00:12:51,290 --> 00:12:53,330
But at the last minute,
apparently, Einstein

274
00:12:53,330 --> 00:12:55,800
decided he didn't really
have a very good foundation

275
00:12:55,800 --> 00:13:00,414
for that statement
and crossed it out.

276
00:13:00,414 --> 00:13:01,830
So I like the
story, first of all,

277
00:13:01,830 --> 00:13:03,884
because it illustrates
that we're not

278
00:13:03,884 --> 00:13:05,300
the only people
who make mistakes.

279
00:13:05,300 --> 00:13:07,140
Even great physicists
like Einstein

280
00:13:07,140 --> 00:13:09,065
make really silly mistakes.

281
00:13:09,065 --> 00:13:12,114
It really was just a
dumb calculational error.

282
00:13:12,114 --> 00:13:14,030
And it also, I think,
points out how important

283
00:13:14,030 --> 00:13:16,620
it is not to allow
yourself to be caught

284
00:13:16,620 --> 00:13:20,066
in the grip of some firm idea
that you cease to question,

285
00:13:20,066 --> 00:13:21,440
which apparently
is what happened

286
00:13:21,440 --> 00:13:24,830
to Einstein with his belief
that the universe was static.

287
00:13:24,830 --> 00:13:26,330
He was so sure that
the universe was

288
00:13:26,330 --> 00:13:29,150
static that he very quickly
looked at Friedmann's paper

289
00:13:29,150 --> 00:13:32,100
and reached the incorrect
conclusion that Friedmann

290
00:13:32,100 --> 00:13:34,310
had gotten his calculus wrong.

291
00:13:34,310 --> 00:13:37,960
In fact, it was Einstein
who got it wrong.

292
00:13:37,960 --> 00:13:39,630
So that summarizes
the last lecture.

293
00:13:39,630 --> 00:13:40,900
Any further questions?

294
00:13:44,280 --> 00:13:47,380
OK, in that case, I think
I am done with that.

295
00:13:47,380 --> 00:13:49,750
Yeah, that comes later.

296
00:13:49,750 --> 00:13:52,890
OK, what I want to
do next is to talk

297
00:13:52,890 --> 00:13:58,800
about two problems associated
with the conventional cosmology

298
00:13:58,800 --> 00:14:01,280
that we've been learning
about and, in particular,

299
00:14:01,280 --> 00:14:02,941
I mean cosmology
without inflation,

300
00:14:02,941 --> 00:14:04,440
which we have not
learned about yet.

301
00:14:04,440 --> 00:14:06,530
So I am talking
about the cosmology

302
00:14:06,530 --> 00:14:09,140
that we've learned about so far.

303
00:14:09,140 --> 00:14:12,340
So there are a total
of three that I

304
00:14:12,340 --> 00:14:16,080
want to discuss
problems associated

305
00:14:16,080 --> 00:14:19,400
with conventional cosmology
which serve as motivations

306
00:14:19,400 --> 00:14:23,570
for the inflationary
modification that, I think,

307
00:14:23,570 --> 00:14:27,710
you'll be learning about
next time from Scott Hughes.

308
00:14:27,710 --> 00:14:30,570
But today I want to
talk about the problems.

309
00:14:30,570 --> 00:14:37,030
So the first of 3 is sometimes
called the homogeneity,

310
00:14:37,030 --> 00:14:38,220
or the horizon problem.

311
00:14:46,280 --> 00:14:49,870
And this is the problem
that the universe

312
00:14:49,870 --> 00:14:53,130
is observed to be
incredibly uniform.

313
00:14:53,130 --> 00:14:56,370
And this uniformity
shows up most clearly

314
00:14:56,370 --> 00:14:59,390
in the cosmic microwave
background radiation,

315
00:14:59,390 --> 00:15:01,960
where astronomers have now made
very sensitive measurements

316
00:15:01,960 --> 00:15:04,390
of the temperature as a
function of angle in the sky.

317
00:15:04,390 --> 00:15:07,770
And it's found that that
radiation in uniform to one

318
00:15:07,770 --> 00:15:10,970
part in 100,000,
part in 10 to the 5.

319
00:15:26,770 --> 00:15:29,260
Now, the CMB is
essentially a snapshot

320
00:15:29,260 --> 00:15:33,610
of what the universe looked
like at about 370,000 years

321
00:15:33,610 --> 00:15:36,160
after the Big Bang at the
time that we call t sub

322
00:15:36,160 --> 00:15:37,590
d, the time of decoupling.

323
00:16:00,088 --> 00:16:00,588
Yes?

324
00:16:00,588 --> 00:16:02,560
AUDIENCE: This measurement,
the 10 to the 5,

325
00:16:02,560 --> 00:16:05,026
it's not a limit
that we've reached

326
00:16:05,026 --> 00:16:06,150
measurement technique-wise?

327
00:16:06,150 --> 00:16:09,186
That's what it actually
is, [INAUDIBLE]?

328
00:16:09,186 --> 00:16:11,630
PROFESSOR: Yes, I was
going to mention that.

329
00:16:11,630 --> 00:16:15,710
We actually do see fluctuations
at the level of one part in 10

330
00:16:15,710 --> 00:16:17,540
to the five.

331
00:16:17,540 --> 00:16:18,730
So it's not just a limit.

332
00:16:18,730 --> 00:16:22,230
It is an actual observation.

333
00:16:22,230 --> 00:16:25,840
And what we interpret is
that the photons that we're

334
00:16:25,840 --> 00:16:30,480
seeing in the CMB have been
flying along on straight lines

335
00:16:30,480 --> 00:16:31,906
since the time of decoupling.

336
00:16:31,906 --> 00:16:33,530
And therefore, what
they show us really

337
00:16:33,530 --> 00:16:35,071
is an image of what
the universe look

338
00:16:35,071 --> 00:16:37,080
like at the time of decoupling.

339
00:16:37,080 --> 00:16:39,200
And that image is an image
of the universe which

340
00:16:39,200 --> 00:16:42,767
is almost a perfectly smooth
mass density and a perfectly

341
00:16:42,767 --> 00:16:44,350
smooth temperature--
it really is just

342
00:16:44,350 --> 00:16:47,620
radiation-- but tiny
ripples superimposed

343
00:16:47,620 --> 00:16:50,210
on top of that uniformity
where the ripples have

344
00:16:50,210 --> 00:16:53,240
an amplitude of order
of 10 to the minus 5.

345
00:16:56,049 --> 00:16:57,840
And those ripples are
important, because we

346
00:16:57,840 --> 00:17:00,110
think that those are the
origin of all structure

347
00:17:00,110 --> 00:17:01,760
in the universe.

348
00:17:01,760 --> 00:17:05,260
The universe is
gravitationally unstable

349
00:17:05,260 --> 00:17:08,069
where there's a positive
ripple making the mass density

350
00:17:08,069 --> 00:17:10,160
slightly higher than average.

351
00:17:10,160 --> 00:17:12,170
That creates a slightly
stronger than average

352
00:17:12,170 --> 00:17:15,609
gravitational field
pulling in extra matter,

353
00:17:15,609 --> 00:17:18,010
creating a still stronger
gravitational field.

354
00:17:18,010 --> 00:17:20,339
And the process cascades
until you ultimately

355
00:17:20,339 --> 00:17:22,660
have galaxies and
clusters of galaxies

356
00:17:22,660 --> 00:17:24,990
and all the complexity
in the universe.

357
00:17:24,990 --> 00:17:28,960
But it starts from these
seeds, these minor fluctuations

358
00:17:28,960 --> 00:17:32,260
at the level of one
part in 10 to the five.

359
00:17:32,260 --> 00:17:33,970
But for now we want
to discuss is simply

360
00:17:33,970 --> 00:17:36,960
the question of how
did we get so uniform.

361
00:17:36,960 --> 00:17:40,210
We'll talk about how the
non-uniformities arise later

362
00:17:40,210 --> 00:17:42,640
in the context of inflation.

363
00:17:42,640 --> 00:17:47,030
The basic picture is
that we are someplace.

364
00:17:47,030 --> 00:17:49,540
I'll put us here in
a little diagram.

365
00:17:49,540 --> 00:17:53,690
We are receiving photons,
say, from opposite directions

366
00:17:53,690 --> 00:17:54,190
in the sky.

367
00:17:57,100 --> 00:17:59,930
Those little arrows represent
the incoming patterns

368
00:17:59,930 --> 00:18:03,310
of two different
CMB photons coming

369
00:18:03,310 --> 00:18:05,020
from opposite directions.

370
00:18:05,020 --> 00:18:09,480
And what I'm interested
in doing to understand

371
00:18:09,480 --> 00:18:12,570
the situation with regard
to this uniformity is I'm

372
00:18:12,570 --> 00:18:15,240
interested in tracing
these photons back

373
00:18:15,240 --> 00:18:19,570
to where they originated
at time t sub d.

374
00:18:19,570 --> 00:18:21,169
And I want to do
that on both sides.

375
00:18:21,169 --> 00:18:22,460
But, of course, it's symmetric.

376
00:18:22,460 --> 00:18:25,052
So I only need to
do one calculation.

377
00:18:25,052 --> 00:18:26,520
And what I want
to know is how far

378
00:18:26,520 --> 00:18:28,260
apart were these two points.

379
00:18:28,260 --> 00:18:30,690
Because I want to explore
the question of whether

380
00:18:30,690 --> 00:18:32,360
or not this uniformity
in temperature

381
00:18:32,360 --> 00:18:34,290
could just be mundane.

382
00:18:34,290 --> 00:18:36,590
If you let any object
sit for a long time,

383
00:18:36,590 --> 00:18:38,580
it will approach a
uniform temperature.

384
00:18:38,580 --> 00:18:42,230
That's why pizza gets cold when
you take it out of the oven.

385
00:18:42,230 --> 00:18:44,730
So could that be responsible
for this uniformity?

386
00:18:44,730 --> 00:18:46,860
And what we'll see
is that it cannot.

387
00:18:46,860 --> 00:18:49,040
Because these two points
are just too far apart

388
00:18:49,040 --> 00:18:51,600
for them to come in
to thermal equilibrium

389
00:18:51,600 --> 00:18:54,830
by ordinary thermal
equilibrium processes

390
00:18:54,830 --> 00:18:59,210
in the context of the
conventional big bang theory.

391
00:18:59,210 --> 00:19:01,560
So we want to calculate how
far apart these points were

392
00:19:01,560 --> 00:19:03,252
at the time of emission.

393
00:19:07,220 --> 00:19:09,180
So what do we know?

394
00:19:09,180 --> 00:19:13,900
We know that the temperature
at the time of decoupling

395
00:19:13,900 --> 00:19:15,920
was about 3,000
Kelvin, which is really

396
00:19:15,920 --> 00:19:18,850
where we started with our
discussion of decoupling.

397
00:19:18,850 --> 00:19:20,540
We did not do the
statistical mechanics

398
00:19:20,540 --> 00:19:22,370
associated with this statement.

399
00:19:22,370 --> 00:19:24,485
But for a given density,
you can calculate

400
00:19:24,485 --> 00:19:27,880
at what temperature
hydrogen ionizes.

401
00:19:27,880 --> 00:19:30,740
And for the density that we
expect for the early universe,

402
00:19:30,740 --> 00:19:34,254
that's the temperature at
which the ionization occurs.

403
00:19:34,254 --> 00:19:35,670
So that's where
decoupling occurs.

404
00:19:35,670 --> 00:19:39,900
That's where it becomes neutral
as the universe expands.

405
00:19:39,900 --> 00:19:44,977
We also know that
during this period, aT,

406
00:19:44,977 --> 00:19:46,560
the scale factor
times the temperature

407
00:19:46,560 --> 00:19:48,950
is about equal to
a constant, which

408
00:19:48,950 --> 00:19:52,370
follows as a consequence
of conservation of entropy,

409
00:19:52,370 --> 00:19:54,879
the idea that the universe
is near thermal equilibrium.

410
00:19:54,879 --> 00:19:56,170
So the entropy does not change.

411
00:19:59,690 --> 00:20:04,260
Then we can calculate
the z for decoupling,

412
00:20:04,260 --> 00:20:09,070
because it would just be the
ratio of the temperatures.

413
00:20:09,070 --> 00:20:11,630
It's defined by the ratio
of the scale factors.

414
00:20:11,630 --> 00:20:15,250
This defines what you mean
by 1 plus z decoupling.

415
00:20:15,250 --> 00:20:17,690
But if aT is about
equal to a constant,

416
00:20:17,690 --> 00:20:20,990
we can relate this to the
temperatures inversely.

417
00:20:20,990 --> 00:20:24,510
So the temperature of decoupling
goes in the numerator.

418
00:20:24,510 --> 00:20:28,810
And the temperature today
goes into the denominator.

419
00:20:28,810 --> 00:20:34,680
And, numerically,
that's about 1,100.

420
00:20:34,680 --> 00:20:37,780
So the z of the cosmic
background radiation

421
00:20:37,780 --> 00:20:42,020
is about 1,100, vastly larger
than the red shifts associated

422
00:20:42,020 --> 00:20:43,930
with observations of
galaxies or supernovae.

423
00:20:49,350 --> 00:20:53,160
From the z, we can calculate
the physical distance today

424
00:20:53,160 --> 00:20:59,870
of these two locations, because
this calculation we already

425
00:20:59,870 --> 00:21:00,370
did.

426
00:21:06,900 --> 00:21:11,770
So I'm going to call l sub p
the physical distance between us

427
00:21:11,770 --> 00:21:14,520
and the source of
this radiation.

428
00:21:14,520 --> 00:21:18,951
And its value today-- I'm
starting with this formula

429
00:21:18,951 --> 00:21:20,450
simply because we
already derived it

430
00:21:20,450 --> 00:21:30,430
on a homework set-- it's 2c h
naught inverse times 1 minus 1

431
00:21:30,430 --> 00:21:35,250
over the square
root of 1 plus z.

432
00:21:35,250 --> 00:21:38,480
And this calculation was done
for a flat matter dominated

433
00:21:38,480 --> 00:21:55,064
universe, flat matter dominated.

434
00:21:55,064 --> 00:21:56,730
Of course, that's
only an approximation,

435
00:21:56,730 --> 00:21:59,930
because we know our
real universe was matter

436
00:21:59,930 --> 00:22:02,380
dominated at the
start of this period.

437
00:22:02,380 --> 00:22:04,830
But it did not remain
matter dominated

438
00:22:04,830 --> 00:22:07,980
through to the present at
about 5,000 or 6,000 years ago.

439
00:22:07,980 --> 00:22:10,990
We switched to a situation where
the dark energy is actually

440
00:22:10,990 --> 00:22:13,330
larger than the
non-relativistic matter.

441
00:22:13,330 --> 00:22:14,830
So we're ignoring
that effect, which

442
00:22:14,830 --> 00:22:16,996
means we're only going to
get an approximation here.

443
00:22:16,996 --> 00:22:19,770
But it will still be easily
good enough to make the point.

444
00:22:22,830 --> 00:22:26,400
For a z this large, this
factor is a small correction.

445
00:22:26,400 --> 00:22:28,390
I think this ends up
being 0.97, or something

446
00:22:28,390 --> 00:22:31,160
like that, very close to 1,
which means what we're getting

447
00:22:31,160 --> 00:22:33,500
is very close to 2c h
naught inverse, which

448
00:22:33,500 --> 00:22:35,610
is the actual horizon.

449
00:22:35,610 --> 00:22:37,894
The horizon corresponds
to z equals infinity.

450
00:22:37,894 --> 00:22:39,310
If you think about
it, that's what

451
00:22:39,310 --> 00:22:41,055
you expect the horizon to be.

452
00:22:41,055 --> 00:22:43,026
It corresponds to
infinite red shift.

453
00:22:43,026 --> 00:22:44,650
And you don't see
anything beyond that.

454
00:22:50,100 --> 00:22:56,000
So if we take the best value
we have for h naught, which I'm

455
00:22:56,000 --> 00:23:00,970
taking from the
Planck satellite, 57.3

456
00:23:00,970 --> 00:23:06,130
kilometers per second
per megaparsec,

457
00:23:06,130 --> 00:23:14,980
and put that and the value
for z into this formula,

458
00:23:14,980 --> 00:23:24,080
we get l sub p of t naught
of 28.2 billion light years

459
00:23:24,080 --> 00:23:27,700
times 10 to the 9 light year.

460
00:23:32,440 --> 00:23:35,020
So it's of course larger
than ct as we expect.

461
00:23:35,020 --> 00:23:38,450
It's basically 3ct for a
matter dominated universe.

462
00:23:38,450 --> 00:23:42,520
And 3ct is the same
as 2ch 0 inverse.

463
00:23:47,930 --> 00:23:50,010
Now, what we want
to know, though,

464
00:23:50,010 --> 00:23:53,510
is how far away was this
when the emission occurred,

465
00:23:53,510 --> 00:23:55,200
not the present distance.

466
00:23:55,200 --> 00:23:56,840
We looked at the
present distance

467
00:23:56,840 --> 00:24:00,030
simply because we had a formula
for it from our homework set.

468
00:24:00,030 --> 00:24:03,290
But we know how to extrapolate
that backwards, l sub

469
00:24:03,290 --> 00:24:06,070
t at time t sub d.

470
00:24:06,070 --> 00:24:08,550
Distances that are fixed
in co-moving space,

471
00:24:08,550 --> 00:24:10,050
which these are,
are just stretched

472
00:24:10,050 --> 00:24:12,420
with the scale factor.

473
00:24:12,420 --> 00:24:16,280
So this will just
be the scale factor

474
00:24:16,280 --> 00:24:19,180
at the time of decoupling
divided by the scale factor

475
00:24:19,180 --> 00:24:21,450
today times the
present distance.

476
00:24:26,290 --> 00:24:30,790
And this is, again, given by
this ratio of temperatures.

477
00:24:30,790 --> 00:24:33,400
So it's 1 over
1,100, the inverse

478
00:24:33,400 --> 00:24:35,170
of what we had over there.

479
00:24:35,170 --> 00:24:37,160
So the separation
at this early time

480
00:24:37,160 --> 00:24:41,070
is just 1,100 times smaller
than the separation today.

481
00:24:41,070 --> 00:24:46,350
And that can be
evaluated numerically.

482
00:24:46,350 --> 00:24:54,900
And it gives us 2.56 times
10 to the seven light years,

483
00:24:54,900 --> 00:24:59,747
so 25 million light years.

484
00:24:59,747 --> 00:25:01,830
Now, the point is that
that's significantly larger

485
00:25:01,830 --> 00:25:05,724
than the horizon
distance at that time.

486
00:25:05,724 --> 00:25:07,140
And remember, the
horizon distance

487
00:25:07,140 --> 00:25:09,770
is the maximum possible
distance that anything

488
00:25:09,770 --> 00:25:12,280
can travel limited
by the speed of light

489
00:25:12,280 --> 00:25:14,710
from the time of the
big bang up to any given

490
00:25:14,710 --> 00:25:17,790
point in cosmic history.

491
00:25:17,790 --> 00:25:23,060
So the horizon at
time t sub d is just

492
00:25:23,060 --> 00:25:26,480
given by the simple formula
that the physical value

493
00:25:26,480 --> 00:25:30,180
of the horizon
distance, l sub h phys,

494
00:25:30,180 --> 00:25:36,570
l sub horizon physical, at
time t sub d is just equal to,

495
00:25:36,570 --> 00:25:41,720
for a matter dominated
universe, 3c times t sub d.

496
00:25:41,720 --> 00:25:44,670
And that we can evaluate,
given what we have.

497
00:25:44,670 --> 00:25:52,050
And it's about 1.1 times 10 to
the sixth light years, which

498
00:25:52,050 --> 00:25:54,840
is significantly less than
2.56 times 10 to the seven

499
00:25:54,840 --> 00:25:56,790
light years.

500
00:25:56,790 --> 00:26:01,920
And, in fact, the ratio of
the two, given these numbers,

501
00:26:01,920 --> 00:26:10,870
is that l sub p of t sub d
over l sub h is also of t sub

502
00:26:10,870 --> 00:26:19,720
d is about equal to 23,
just doing the arithmetic.

503
00:26:19,720 --> 00:26:23,170
And that means if we go
back to this picture,

504
00:26:23,170 --> 00:26:27,820
these two points of emission
were separated from each other

505
00:26:27,820 --> 00:26:29,995
by about 46 horizon distances.

506
00:26:40,000 --> 00:26:42,900
And that's enough to
imply that there's no way

507
00:26:42,900 --> 00:26:45,880
that this point could have known
anything whatever about what

508
00:26:45,880 --> 00:26:48,100
was going on at this point.

509
00:26:48,100 --> 00:26:50,690
Yet somehow they knew to
emit these two photons

510
00:26:50,690 --> 00:26:54,100
at the same time at
the same temperature.

511
00:26:54,100 --> 00:26:56,300
And that's the mystery.

512
00:26:56,300 --> 00:26:58,630
One can get around this
mystery if one simply

513
00:26:58,630 --> 00:27:03,460
assumes that the singularity
that created all of this

514
00:27:03,460 --> 00:27:07,359
produced a perfectly homogeneous
universe from the very start.

515
00:27:07,359 --> 00:27:09,150
Since we don't understand
that singularity,

516
00:27:09,150 --> 00:27:11,630
we're allowed to attribute
anything we want to it.

517
00:27:11,630 --> 00:27:14,330
So in particular, you can
attribute perfect homogeneity

518
00:27:14,330 --> 00:27:15,729
to the singularity.

519
00:27:15,729 --> 00:27:17,270
But that's not really
an explanation.

520
00:27:17,270 --> 00:27:19,260
That's an assumption.

521
00:27:19,260 --> 00:27:23,060
So if one wants to be able
to explain this uniformity,

522
00:27:23,060 --> 00:27:25,760
then one simply cannot do it
in the context of conventional

523
00:27:25,760 --> 00:27:27,010
cosmology.

524
00:27:27,010 --> 00:27:28,880
There's just no
way that causality,

525
00:27:28,880 --> 00:27:31,240
the limit of the speed of
light, allows this point

526
00:27:31,240 --> 00:27:34,150
to know anything about what's
going on at that point.

527
00:27:34,150 --> 00:27:34,650
Yes?

528
00:27:34,650 --> 00:27:36,910
AUDIENCE: How could a
singularity not be uniform?

529
00:27:36,910 --> 00:27:39,520
Because If it had
non-uniform [INAUDIBLE],

530
00:27:39,520 --> 00:27:41,537
then not be singular?

531
00:27:41,537 --> 00:27:42,953
PROFESSOR: OK, the
question is how

532
00:27:42,953 --> 00:27:46,020
can a singularity
not be uniform?

533
00:27:46,020 --> 00:27:47,480
The answer is,
yes, singularities

534
00:27:47,480 --> 00:27:49,560
can not be uniform.

535
00:27:49,560 --> 00:27:53,090
And I think the way one can
show that is a little hard.

536
00:27:53,090 --> 00:27:55,810
But you have to imagine a
non-uniform thing collapsing.

537
00:27:55,810 --> 00:27:58,060
And then it would just be
the time reverse, everything

538
00:27:58,060 --> 00:27:59,750
popping out of the singularity.

539
00:27:59,750 --> 00:28:01,850
So you can ask, does
a non-uniform thing

540
00:28:01,850 --> 00:28:03,534
collapse to a singularity?

541
00:28:03,534 --> 00:28:05,450
And the answer to that
question is not obvious

542
00:28:05,450 --> 00:28:08,510
and really was debated
for a long time.

543
00:28:08,510 --> 00:28:11,830
But there were theorems
proven by Hawking and Penrose

544
00:28:11,830 --> 00:28:16,170
that indeed not only do the
homogeneous solutions that we

545
00:28:16,170 --> 00:28:18,722
look at collapse but in
homogeneous solutions also

546
00:28:18,722 --> 00:28:19,805
collapse to singularities.

547
00:28:23,505 --> 00:28:25,380
So a singularity does
not have to be uniform.

548
00:28:30,190 --> 00:28:34,150
OK, so this is the story
of the horizon problem.

549
00:28:34,150 --> 00:28:35,950
And as I said, you
can get around it

550
00:28:35,950 --> 00:28:38,325
if you're willing to just
assume that the singularity was

551
00:28:38,325 --> 00:28:39,040
homogeneous.

552
00:28:39,040 --> 00:28:41,550
But if you want to have a
dynamical explanation for how

553
00:28:41,550 --> 00:28:44,100
the uniformity of the
universe was established,

554
00:28:44,100 --> 00:28:46,730
then you need some model
other than this conventional

555
00:28:46,730 --> 00:28:49,670
cosmological model that
we've been discussing.

556
00:28:49,670 --> 00:28:51,660
And inflation will
be such a twist

557
00:28:51,660 --> 00:28:55,120
which will allow a
solution to this problem.

558
00:28:55,120 --> 00:28:57,830
OK, so if there
are no questions,

559
00:28:57,830 --> 00:28:59,400
no further questions,
we'll go on

560
00:28:59,400 --> 00:29:01,270
to the next problem
I want to discuss,

561
00:29:01,270 --> 00:29:05,520
which is of a similar nature
in that you can get around it

562
00:29:05,520 --> 00:29:10,670
by making strong assumptions
about the initial singularity.

563
00:29:10,670 --> 00:29:13,020
But if one wants,
again, something

564
00:29:13,020 --> 00:29:15,530
you can put your hands on,
rather than just an assumption

565
00:29:15,530 --> 00:29:19,550
about a singularity, then
inflation will do the job.

566
00:29:19,550 --> 00:29:22,590
But you cannot solve the
problem in the context

567
00:29:22,590 --> 00:29:24,055
of a conventional
big bang theory,

568
00:29:24,055 --> 00:29:26,430
because the mechanics of the
conventional big bang theory

569
00:29:26,430 --> 00:29:28,720
are simply well-defined.

570
00:29:28,720 --> 00:29:31,510
So what I want to
talk here is what

571
00:29:31,510 --> 00:29:36,980
is called the flatness
problem, where flatness

572
00:29:36,980 --> 00:29:40,840
is in the sense of
Omega very near 1.

573
00:29:40,840 --> 00:29:42,470
And this is
basically the problem

574
00:29:42,470 --> 00:29:48,930
of why is Omega today
somewhere near 1?

575
00:29:48,930 --> 00:29:52,360
So Omega naught is the
present value of Omega,

576
00:29:52,360 --> 00:29:55,795
why is it about equal to 1?

577
00:29:55,795 --> 00:29:57,170
Now, what do we
know first of all

578
00:29:57,170 --> 00:29:59,750
about it being about equal to 1?

579
00:29:59,750 --> 00:30:03,900
The best value from
the Planck group,

580
00:30:03,900 --> 00:30:05,890
this famous Planck
satellite that I've

581
00:30:05,890 --> 00:30:08,820
been quoting a lot
of numbers from--

582
00:30:08,820 --> 00:30:10,970
and I think in all cases,
I've been quoting numbers

583
00:30:10,970 --> 00:30:14,310
that they've established
combining their own data

584
00:30:14,310 --> 00:30:16,897
with some other pieces of data.

585
00:30:16,897 --> 00:30:18,480
So it's not quite
the satellite alone.

586
00:30:18,480 --> 00:30:20,730
Although, they do give numbers
for the satellite alone

587
00:30:20,730 --> 00:30:22,800
which are just a little
bit less precise.

588
00:30:22,800 --> 00:30:26,410
But the best number they give
for Omega naught is minus

589
00:30:26,410 --> 00:30:34,560
0.0010 plus or minus 0.0065.

590
00:30:34,560 --> 00:30:36,460
Oops, I didn't put
enough zeroes there.

591
00:30:44,030 --> 00:30:47,657
So it's 0.0065 is the error.

592
00:30:47,657 --> 00:30:49,740
So the error is just a
little bit more than a half

593
00:30:49,740 --> 00:30:52,320
of a percent.

594
00:30:52,320 --> 00:30:56,330
And as you see, consistent
with-- I'm sorry,

595
00:30:56,330 --> 00:30:57,650
I meant this to be 1.

596
00:31:00,910 --> 00:31:01,680
Hold on.

597
00:31:04,480 --> 00:31:09,750
This is Omega naught minus 1
that I'm writing a formula for.

598
00:31:09,750 --> 00:31:12,060
So Omega naught is very
near 1 up to that accuracy.

599
00:31:28,200 --> 00:31:30,620
What I want to emphasize in
terms of this flatness problem

600
00:31:30,620 --> 00:31:33,920
is that you don't need to know
that Omega naught is very, very

601
00:31:33,920 --> 00:31:36,180
close to 1 today,
which we now do know.

602
00:31:36,180 --> 00:31:43,090
But even back when inflation
was first invented around 1980,

603
00:31:43,090 --> 00:31:44,630
in circa 1980 we
certainly didn't

604
00:31:44,630 --> 00:31:47,740
know that Omega was so
incredibly close to 1.

605
00:31:47,740 --> 00:31:50,950
But we did know that
Omega was somewhere

606
00:31:50,950 --> 00:31:57,790
in the range of about
0.1 and 0.2, which is not

607
00:31:57,790 --> 00:32:02,140
nearly as close to 1 as what we
know now, but still close to 1.

608
00:32:02,140 --> 00:32:04,420
I'll argue that the
flatness problem exists

609
00:32:04,420 --> 00:32:06,640
for these numbers
almost as strongly

610
00:32:06,640 --> 00:32:09,000
as it exists for those numbers.

611
00:32:09,000 --> 00:32:11,990
Differ, but this is still a
very, very strong argument

612
00:32:11,990 --> 00:32:15,240
that even a number like
this is amazingly close to 1

613
00:32:15,240 --> 00:32:18,350
considering what
you should expect.

614
00:32:18,350 --> 00:32:22,255
Now, what underlies this is the
expectations, how close should

615
00:32:22,255 --> 00:32:24,070
we expect Omega to be to 1?

616
00:32:24,070 --> 00:32:26,170
And the important
underlying piece

617
00:32:26,170 --> 00:32:31,560
of dynamics that controls this
is the fact that Omega equals 1

618
00:32:31,560 --> 00:32:34,370
is an unstable
equilibrium point.

619
00:32:34,370 --> 00:32:37,290
That means it's like a
pencil balancing on its tip.

620
00:32:37,290 --> 00:32:39,920
If Omega is exactly
equal to 1, that

621
00:32:39,920 --> 00:32:41,650
means you have a flat universe.

622
00:32:41,650 --> 00:32:43,270
And an exactly
flat universe will

623
00:32:43,270 --> 00:32:46,390
remain an exactly
flat universe forever.

624
00:32:46,390 --> 00:32:48,055
So if Omega is
exactly equal to 1,

625
00:32:48,055 --> 00:32:51,580
it will remain exactly
equal to 1 forever.

626
00:32:51,580 --> 00:32:54,420
But if Omega in
the early universe

627
00:32:54,420 --> 00:32:56,390
were just a tiny
bit bigger than 1--

628
00:32:56,390 --> 00:32:58,220
and we're about
to calculate this,

629
00:32:58,220 --> 00:33:05,170
but I'll first qualitatively
describe the result--

630
00:33:05,170 --> 00:33:08,522
it would rise and would
rapidly reach infinity,

631
00:33:08,522 --> 00:33:10,980
which is what it reaches if
you have a closed universe when

632
00:33:10,980 --> 00:33:13,010
a closed universe
reaches its maximum size.

633
00:33:13,010 --> 00:33:17,830
So Omega becomes infinity and
then the universe recollapses.

634
00:33:17,830 --> 00:33:19,395
So if Omega were
bigger than 1, it

635
00:33:19,395 --> 00:33:20,870
would rapidly approach infinity.

636
00:33:20,870 --> 00:33:23,300
If Omega in the early universe
were just a little bit less

637
00:33:23,300 --> 00:33:26,250
than 1, it would rapidly
trail off towards 0

638
00:33:26,250 --> 00:33:29,990
and not stay 1 for
any length of time.

639
00:33:29,990 --> 00:33:34,000
So the only way to
get Omega near 1 today

640
00:33:34,000 --> 00:33:36,310
is like having a pencil
that's almost straight

641
00:33:36,310 --> 00:33:39,470
up after standing there
for 1 billion years.

642
00:33:39,470 --> 00:33:41,195
It'd have to have
started out incredibly

643
00:33:41,195 --> 00:33:42,320
close to being straight up.

644
00:33:42,320 --> 00:33:44,250
It has to have
started out incredibly

645
00:33:44,250 --> 00:33:46,240
close to Omega equals 1.

646
00:33:46,240 --> 00:33:49,080
And we're going to
calculate how close.

647
00:33:49,080 --> 00:33:53,160
So that's the set-up.

648
00:33:53,160 --> 00:33:56,700
So the question we want to
ask is how close did Omega

649
00:33:56,700 --> 00:33:58,940
have to be to 1 in
the early universe

650
00:33:58,940 --> 00:34:02,110
to be in either one of
these allowed ranges today.

651
00:34:02,110 --> 00:34:04,100
And for the early
universe, I'm going

652
00:34:04,100 --> 00:34:07,220
to take t equals one second
as my time at which I'll

653
00:34:07,220 --> 00:34:08,389
do these calculations.

654
00:34:08,389 --> 00:34:10,639
And, historically, that's
where this problem was first

655
00:34:10,639 --> 00:34:17,610
discussed by Dicke and
Peebles back in 1979.

656
00:34:17,610 --> 00:34:20,040
And the reason why one
second was chosen by them,

657
00:34:20,040 --> 00:34:23,150
and why it's a sensible time
for us to talk about as well,

658
00:34:23,150 --> 00:34:25,729
is that one second
is the beginning

659
00:34:25,729 --> 00:34:27,770
of the processes of
nucleosynthesis, which you've

660
00:34:27,770 --> 00:34:30,370
read about in
Weinberg and in Ryden,

661
00:34:30,370 --> 00:34:34,610
and provides a real test of
our understanding of cosmology

662
00:34:34,610 --> 00:34:35,969
at those times.

663
00:34:35,969 --> 00:34:38,721
So we could say that we
have real empirical evidence

664
00:34:38,721 --> 00:34:40,929
in the statement that the
predictions of the chemical

665
00:34:40,929 --> 00:34:42,678
elements work.

666
00:34:42,678 --> 00:34:44,719
We could say that we have
real empirical evidence

667
00:34:44,719 --> 00:34:47,900
that our cosmological model
works back to one second

668
00:34:47,900 --> 00:34:49,900
after the Big Bang.

669
00:34:49,900 --> 00:34:51,745
So we're going to
choose one second

670
00:34:51,745 --> 00:34:54,120
for the time at which we're
going to calculate what Omega

671
00:34:54,120 --> 00:34:56,840
must've been then for it to
be an allowed range today.

672
00:35:00,910 --> 00:35:20,480
How close must Omega have been
to 1 at t equals 1 second?

673
00:35:20,480 --> 00:35:23,530
Question mark.

674
00:35:23,530 --> 00:35:25,030
OK, now, to do this
calculation, you

675
00:35:25,030 --> 00:35:26,930
don't need to know anything
that you don't already know.

676
00:35:26,930 --> 00:35:29,070
It really follows as a
consequence of the Friedmann

677
00:35:29,070 --> 00:35:33,710
equation and how matter and
temperature and so on behave

678
00:35:33,710 --> 00:35:37,902
with time during radiation
in matter dominated eras.

679
00:35:37,902 --> 00:35:40,360
So we're going to start with
just the plain old first order

680
00:35:40,360 --> 00:35:46,680
Fiedmann equation, h squared
is equal to 8 pi over 3 g Rho

681
00:35:46,680 --> 00:35:51,370
minus kc squared
over a squared, which

682
00:35:51,370 --> 00:35:56,640
you have seen many, many
times already in this course.

683
00:35:56,640 --> 00:35:58,320
We can combine that
with other equations

684
00:35:58,320 --> 00:36:00,140
that you've also
seen many times.

685
00:36:00,140 --> 00:36:03,920
The critical density is just
the value of the density

686
00:36:03,920 --> 00:36:05,540
when k equals 0.

687
00:36:05,540 --> 00:36:07,900
So you just solve
this equation for Rho.

688
00:36:07,900 --> 00:36:14,484
And you get 3h
squared over h pi g.

689
00:36:14,484 --> 00:36:15,900
This defines the
critical density.

690
00:36:15,900 --> 00:36:20,650
It's that density which makes
the universe flat, k equals 0.

691
00:36:20,650 --> 00:36:22,130
And then our standard
definition is

692
00:36:22,130 --> 00:36:26,412
that Omega is just defined to be
the actual mass density divided

693
00:36:26,412 --> 00:36:27,620
by the critical mass density.

694
00:36:27,620 --> 00:36:31,990
And Omega will be the quantity
we're trying to trace.

695
00:36:31,990 --> 00:36:34,430
And we're also going
to make use of the fact

696
00:36:34,430 --> 00:36:37,250
that during the era that
we're talking about,

697
00:36:37,250 --> 00:36:40,990
at is essentially
equal to a constant.

698
00:36:40,990 --> 00:36:43,870
It does change a little bit
when electron and positron

699
00:36:43,870 --> 00:36:44,910
pairs freeze out.

700
00:36:44,910 --> 00:36:46,600
It changes by a
factor of something

701
00:36:46,600 --> 00:36:49,470
like 4/11 to the 1/3 power
or something like that.

702
00:36:49,470 --> 00:36:51,760
But that power will be of
order one for our purposes.

703
00:36:51,760 --> 00:36:53,180
But I guess this is
a good reason why

704
00:36:53,180 --> 00:36:54,679
I should put a
squiggle here instead

705
00:36:54,679 --> 00:36:56,910
of an equal sign as an
approximate equality,

706
00:36:56,910 --> 00:36:59,240
but easily good enough
for our purposes,

707
00:36:59,240 --> 00:37:02,565
meaning the corrections
of order one.

708
00:37:02,565 --> 00:37:03,940
We're going to
see the problem is

709
00:37:03,940 --> 00:37:05,170
much, much bigger
than order one.

710
00:37:05,170 --> 00:37:07,010
So a correction of order
one doesn't matter.

711
00:37:10,510 --> 00:37:14,210
Now, I'm going to start by using
the Planck satellite limits.

712
00:37:14,210 --> 00:37:18,230
And at the end, I'll just make
a comment about the circa 1980

713
00:37:18,230 --> 00:37:19,160
situation.

714
00:37:19,160 --> 00:37:25,922
But if we look at the
Planck limits-- I'm sorry.

715
00:37:25,922 --> 00:37:28,380
Since I'm going to write an
equation for a peculiar looking

716
00:37:28,380 --> 00:37:30,560
quantity, I should
motivate the peculiar

717
00:37:30,560 --> 00:37:31,667
looking quantity first.

718
00:37:34,650 --> 00:37:36,700
It turns out to be useful
for these purposes.

719
00:37:39,482 --> 00:37:40,940
And this purpose
means we're trying

720
00:37:40,940 --> 00:37:44,220
to track how Omega
changes with time.

721
00:37:44,220 --> 00:37:46,580
It turns out to be useful
to reshuffle the Friedmann.

722
00:37:46,580 --> 00:37:49,220
And it is just an
algebraic reshuffling

723
00:37:49,220 --> 00:37:52,680
of the Friedmann equation
and the definitions

724
00:37:52,680 --> 00:37:54,200
that we have here.

725
00:37:54,200 --> 00:37:56,360
We can rewrite the
Friedmann equation

726
00:37:56,360 --> 00:38:02,560
as Omega minus 1 over Omega
is equal to a quantity called

727
00:38:02,560 --> 00:38:07,174
a times the temperature
squared over Rho.

728
00:38:07,174 --> 00:38:09,840
Now, the temperature didn't even
occur in the original equation.

729
00:38:09,840 --> 00:38:12,460
So things might look a
little bit suspicious.

730
00:38:12,460 --> 00:38:14,440
I haven't told
you what a is yet.

731
00:38:14,440 --> 00:38:26,925
a is 3k c squared over 8
pi g a squared t squared.

732
00:38:26,925 --> 00:38:29,550
So when you put the a into this
equation, the t squares cancel.

733
00:38:29,550 --> 00:38:32,410
So the equation doesn't
really have any temperature

734
00:38:32,410 --> 00:38:33,660
dependence.

735
00:38:33,660 --> 00:38:36,360
But I factored things
this way, because we

736
00:38:36,360 --> 00:38:38,789
know that at is
approximately a constant.

737
00:38:38,789 --> 00:38:40,830
And that means that this
capital a, which is just

738
00:38:40,830 --> 00:38:42,205
other things which
are definitely

739
00:38:42,205 --> 00:38:45,260
constant times, a square t
square in the denominator,

740
00:38:45,260 --> 00:38:47,420
this a is approximately
a constant.

741
00:38:51,770 --> 00:38:53,330
And you'll have to
check me at home

742
00:38:53,330 --> 00:38:55,705
that this is exactly equivalent
to the original Friedmann

743
00:38:55,705 --> 00:38:58,870
equation, no approximations
whatever, just substitutions

744
00:38:58,870 --> 00:39:03,620
of Omega and the
definition of Rho sub c.

745
00:39:03,620 --> 00:39:06,040
So the nice thing about
this is that we can read off

746
00:39:06,040 --> 00:39:07,790
the time dependence
of the right-hand side

747
00:39:07,790 --> 00:39:09,290
as long as we know the time
dependence in the temperature

748
00:39:09,290 --> 00:39:11,280
and the time dependence
of the energy density.

749
00:39:11,280 --> 00:39:15,350
And we do for matter dominated
and radiation dominated eras.

750
00:39:15,350 --> 00:39:17,870
So this, essentially,
solves the problem for us.

751
00:39:17,870 --> 00:39:20,120
And now it's really
just a question

752
00:39:20,120 --> 00:39:22,500
of looking at the
numerics that follow

753
00:39:22,500 --> 00:39:25,920
as a consequence
of that equation.

754
00:39:25,920 --> 00:39:28,840
And this quantity,
we're really interested

755
00:39:28,840 --> 00:39:30,572
in just Omega minus 1.

756
00:39:30,572 --> 00:39:32,780
The Friedmann equation gave
us the extra complication

757
00:39:32,780 --> 00:39:34,372
of an Omega in the denominator.

758
00:39:34,372 --> 00:39:36,580
But in the end, we're going
to be interested in cases

759
00:39:36,580 --> 00:39:38,121
where Omega is very,
very close to 1.

760
00:39:38,121 --> 00:39:41,200
So the Omega in the denominator
we could just set equal to one.

761
00:39:41,200 --> 00:39:43,200
And it's the Omega minus
1 in the numerator that

762
00:39:43,200 --> 00:39:47,390
controls the value of
the left-hand side.

763
00:39:47,390 --> 00:39:57,080
So if we look at these Planck
limits, we could ask how big

764
00:39:57,080 --> 00:39:59,220
can that be?

765
00:39:59,220 --> 00:40:02,030
And it's biggest if the error
occurs on the negative side.

766
00:40:02,030 --> 00:40:04,120
So it contributes to
this small mean value

767
00:40:04,120 --> 00:40:06,260
which is slightly negative.

768
00:40:06,260 --> 00:40:10,970
And it gives you 0.0075
for Omega minus 1.

769
00:40:10,970 --> 00:40:15,140
And then if you put that in the
numerator and the same thing

770
00:40:15,140 --> 00:40:20,560
in the denominator, you
get something like 0.0076.

771
00:40:20,560 --> 00:40:23,760
But I'm just going
to use the bound

772
00:40:23,760 --> 00:40:32,330
that Omega naught minus 1
over Omega is less than 0.01.

773
00:40:32,330 --> 00:40:34,950
But the more accurate
thing would be 0.076.

774
00:40:34,950 --> 00:40:37,430
But, again, we're not really
interested in small factors

775
00:40:37,430 --> 00:40:37,930
here.

776
00:40:41,170 --> 00:40:42,830
And this is a one signa error.

777
00:40:42,830 --> 00:40:45,010
So the actual error could
be larger than this,

778
00:40:45,010 --> 00:40:46,730
but not too much
larger than this.

779
00:41:12,980 --> 00:41:16,010
So I'm going to divide the time
interval between one second

780
00:41:16,010 --> 00:41:18,940
and now up into two integrals.

781
00:41:18,940 --> 00:41:20,920
From one second to
about 50,000 years,

782
00:41:20,920 --> 00:41:22,940
the universe was
radiation dominated.

783
00:41:22,940 --> 00:41:25,680
We figured out earlier that the
matter to radiation equality

784
00:41:25,680 --> 00:41:27,470
happens at about 50,000 years.

785
00:41:27,470 --> 00:41:29,830
I think we may have gotten
47,000 years or something

786
00:41:29,830 --> 00:41:32,530
like that when we calculated it.

787
00:41:32,530 --> 00:41:41,962
So for t equals 1
second to-- I'm sorry,

788
00:41:41,962 --> 00:41:43,420
I'm going to do it
the other order.

789
00:41:43,420 --> 00:41:45,711
I'm going to start with the
present and work backwards.

790
00:41:50,130 --> 00:42:01,966
So for t equals 50,000
years to the present,

791
00:42:01,966 --> 00:42:03,340
the universe is
matter dominated.

792
00:42:08,514 --> 00:42:09,930
And the next thing
is that we know

793
00:42:09,930 --> 00:42:11,710
how mattered dominated
universe's behave.

794
00:42:11,710 --> 00:42:14,030
We don't need to recalculate it.

795
00:42:14,030 --> 00:42:17,360
We know that the scale
factor for a matter

796
00:42:17,360 --> 00:42:22,340
dominated flat universe goes
like t to the 2/3 power,

797
00:42:22,340 --> 00:42:25,290
I should have a
portionality sign here.

798
00:42:25,290 --> 00:42:29,390
a of t is proportional
to t to 2/3.

799
00:42:29,390 --> 00:42:31,602
And it's fair to assume
flat, because we're always

800
00:42:31,602 --> 00:42:33,935
going to be talking about
universes that are nearly flat

801
00:42:33,935 --> 00:42:36,059
and becoming more and more
flat as we go backwards,

802
00:42:36,059 --> 00:42:37,866
as we'll see.

803
00:42:37,866 --> 00:42:39,990
And again, this isn't an
approximate calculation.

804
00:42:39,990 --> 00:42:41,890
One could do it more
accurately if one wanted to.

805
00:42:41,890 --> 00:42:43,960
But there's really no need
to, because the result

806
00:42:43,960 --> 00:42:46,610
will be so extreme.

807
00:42:46,610 --> 00:42:51,101
The temperature behaves like
one over the scale factor.

808
00:42:51,101 --> 00:42:53,225
And that will be true for
both the matter dominated

809
00:42:53,225 --> 00:42:57,070
and a radiation
dominated universe.

810
00:42:57,070 --> 00:43:00,680
And the energy density
will be proportional to one

811
00:43:00,680 --> 00:43:01,910
over the scale factor cubed.

812
00:43:08,180 --> 00:43:11,040
And then if we put those
together and use the formula

813
00:43:11,040 --> 00:43:14,200
on the other blackboard and ask
how Omega minus 1 over Omega

814
00:43:14,200 --> 00:43:19,600
behaves, it's proportional
to the temperature squared

815
00:43:19,600 --> 00:43:22,160
divided by the energy density.

816
00:43:22,160 --> 00:43:23,820
The temperature
goes like 1 over a.

817
00:43:23,820 --> 00:43:26,990
So temperature squared
goes like 1 over a squared.

818
00:43:26,990 --> 00:43:30,160
But Rho in the denominator
goes like 1 over a cubed.

819
00:43:30,160 --> 00:43:33,310
So you have 1 over a squared
divided by 1 over a cubed.

820
00:43:33,310 --> 00:43:36,990
And that means it just goes
like a, the scale factor itself.

821
00:43:36,990 --> 00:43:41,740
So Omega minus 1 over
Omega is proportional to a.

822
00:43:41,740 --> 00:43:44,310
And that means it's
proportional to t to the 2/3.

823
00:43:47,990 --> 00:43:55,240
So that allows us to
write down an equation,

824
00:43:55,240 --> 00:43:57,810
since we want to
relate everything

825
00:43:57,810 --> 00:44:00,630
to the value of Omega
minus 1 over Omega today,

826
00:44:00,630 --> 00:44:08,410
we can write Omega
minus 1 over Omega

827
00:44:08,410 --> 00:44:21,660
at 50,000 years is about equal
to the ratio of the 2 times

828
00:44:21,660 --> 00:44:37,210
the 50,000 years and today,
which is 13.8 billion years,

829
00:44:37,210 --> 00:44:41,640
to the 2/3 power
since Omega minus 1

830
00:44:41,640 --> 00:44:42,989
grows like t to the 2/3.

831
00:44:42,989 --> 00:44:44,530
I should maybe have
pointed out here,

832
00:44:44,530 --> 00:44:46,800
this telling us that Omega
minus 1 grows with time.

833
00:44:46,800 --> 00:44:48,762
That's the important feature.

834
00:44:48,762 --> 00:44:51,670
It grows t to the 2/3.

835
00:44:51,670 --> 00:44:55,420
So the value at 50,000
years is this ratio

836
00:44:55,420 --> 00:45:00,870
to the 2/3 power times
Omega minus 1 over Omega

837
00:45:00,870 --> 00:45:04,060
today, which I can
indicate just by putting

838
00:45:04,060 --> 00:45:06,280
subscript zeros on my Omegas.

839
00:45:06,280 --> 00:45:07,310
And that makes it today.

840
00:45:10,200 --> 00:45:13,430
And I've written this as
a fraction less than one.

841
00:45:13,430 --> 00:45:15,810
This says that Omega
minus 1 over Omega

842
00:45:15,810 --> 00:45:18,840
was smaller than it
is now by this ratio

843
00:45:18,840 --> 00:45:23,720
to the 2/3 power, which
follows from the fact

844
00:45:23,720 --> 00:45:28,380
that Omega minus 1 over Omega
grows like t to the 2/3.

845
00:45:28,380 --> 00:45:32,230
OK, we're now halfway there.

846
00:45:32,230 --> 00:45:38,010
And the other half is similar,
so it will go quickly.

847
00:45:38,010 --> 00:45:41,020
We now want to go
from 50,000 years

848
00:45:41,020 --> 00:45:46,054
to one second using the
fact that during that era

849
00:45:46,054 --> 00:45:47,595
the universe was
radiation dominated.

850
00:45:53,790 --> 00:46:06,660
So for t equals 1
second to 50,000 years,

851
00:46:06,660 --> 00:46:08,350
the universe is
radiation dominated.

852
00:46:13,490 --> 00:46:18,390
And that implies
that the scale factor

853
00:46:18,390 --> 00:46:20,040
is proportional to t to the 1/2.

854
00:46:23,960 --> 00:46:27,000
The temperature is,
again, proportional to 1

855
00:46:27,000 --> 00:46:28,880
over the scale factor.

856
00:46:28,880 --> 00:46:32,070
That's just
conservation of entropy.

857
00:46:32,070 --> 00:46:36,212
And the energy density goes
one over the scale factor

858
00:46:36,212 --> 00:46:36,795
to the fourth.

859
00:46:40,080 --> 00:46:42,630
So, again, we go
back to this formula

860
00:46:42,630 --> 00:46:45,500
and do the corresponding
arithmetic.

861
00:46:45,500 --> 00:46:46,960
Temperature goes like 1 over a.

862
00:46:46,960 --> 00:46:49,140
Temperature squared goes
like 1 over a squared.

863
00:46:49,140 --> 00:46:50,616
That's our numerator.

864
00:46:50,616 --> 00:46:51,990
This time, in the
denominator, we

865
00:46:51,990 --> 00:46:55,130
have Rho, which goes like
one over a to the fourth.

866
00:46:55,130 --> 00:46:57,650
So we have 1 over a
squared divided by 1

867
00:46:57,650 --> 00:46:58,647
over a to the fourth.

868
00:46:58,647 --> 00:47:00,230
And that means it
goes like a squared.

869
00:47:06,930 --> 00:47:10,130
So we get Omega
minus 1 over Omega

870
00:47:10,130 --> 00:47:13,080
is proportional to a squared.

871
00:47:13,080 --> 00:47:16,630
And since goes like
the square root of t,

872
00:47:16,630 --> 00:47:19,230
a squared goes like t.

873
00:47:19,230 --> 00:47:20,962
So during the
radiation dominated era

874
00:47:20,962 --> 00:47:22,420
this diverges even
a little faster.

875
00:47:22,420 --> 00:47:24,294
PROFESSOR: It goes like
t, rather than like t

876
00:47:24,294 --> 00:47:28,060
to the 2/3, which is a
slightly slower growth.

877
00:47:28,060 --> 00:47:32,170
And using this fact, we
can put it all together now

878
00:47:32,170 --> 00:47:41,970
and say that Omega minus
1 over Omega at 1 second

879
00:47:41,970 --> 00:47:47,520
is about equal to 1
second over 50,000

880
00:47:47,520 --> 00:47:55,020
years to the first
power-- this is going

881
00:47:55,020 --> 00:47:57,950
like the first
power of t-- times

882
00:47:57,950 --> 00:48:12,420
the value of Omega minus 1
over Omega at 50,000 years.

883
00:48:12,420 --> 00:48:16,380
And Omega at 50,000 years,
we can put in that equality

884
00:48:16,380 --> 00:48:18,560
and relate everything
to the present value.

885
00:48:42,640 --> 00:48:46,760
And when you do that,
putting it all together,

886
00:48:46,760 --> 00:48:53,220
you ultimately find that Omega
minus 1 in magnitude at t

887
00:48:53,220 --> 00:48:59,440
equals 1 second is less than
about 10 to the minus 18.

888
00:48:59,440 --> 00:49:01,780
This is just putting
together these inequalities

889
00:49:01,780 --> 00:49:05,121
and using the Planck value for
the present value, the Planck

890
00:49:05,121 --> 00:49:05,620
inequality.

891
00:49:08,597 --> 00:49:10,930
So then 10 to the minus 18
is a powerfully small number.

892
00:49:10,930 --> 00:49:13,810
What we're saying is that to
be in the allowed range today,

893
00:49:13,810 --> 00:49:16,010
at one second
after the Big Bang,

894
00:49:16,010 --> 00:49:17,870
Omega have to have
been equal to 1

895
00:49:17,870 --> 00:49:19,870
in the context of this
conventional cosmology

896
00:49:19,870 --> 00:49:23,680
to an accuracy of
18 decimal places.

897
00:49:23,680 --> 00:49:26,900
And the reason
that's a problem is

898
00:49:26,900 --> 00:49:28,630
that we don't know
any physics whatever

899
00:49:28,630 --> 00:49:31,390
that forces Omega
to be equal to 1.

900
00:49:31,390 --> 00:49:33,680
Yet, somehow Omega
apparently has

901
00:49:33,680 --> 00:49:37,930
chosen to be 1 to an accuracy
of 18 decimal places.

902
00:49:37,930 --> 00:49:41,760
And I mention that the argument
wasn't that different in 1980.

903
00:49:41,760 --> 00:49:45,520
In 1980, we only knew
this instead of that.

904
00:49:45,520 --> 00:49:48,590
And that meant that instead
of having 10 to the minus 2

905
00:49:48,590 --> 00:49:51,840
on the right-hand side
here, we would have had 10

906
00:49:51,840 --> 00:49:54,050
differing by three
orders of magnitude.

907
00:49:54,050 --> 00:49:56,270
So instead of getting
10 to the minus 18 here,

908
00:49:56,270 --> 00:49:58,255
we would have gotten
10 to minus 15.

909
00:49:58,255 --> 00:50:00,880
And 10 to minus 15 is, I
guess, a lot bigger than 10

910
00:50:00,880 --> 00:50:03,620
to minus 18 by a
factor of 1,000.

911
00:50:03,620 --> 00:50:05,530
But still, it's an
incredibly small number.

912
00:50:05,530 --> 00:50:07,975
And the argument really
sounds exactly the same.

913
00:50:07,975 --> 00:50:10,660
The question is, how
did Omega minus 1

914
00:50:10,660 --> 00:50:12,610
get to be so incredibly small?

915
00:50:12,610 --> 00:50:15,100
What mechanism was there?

916
00:50:15,100 --> 00:50:16,830
Now, like the
horizon problem, you

917
00:50:16,830 --> 00:50:20,250
can get around it by
attributing your ignorance

918
00:50:20,250 --> 00:50:21,340
to the singularity.

919
00:50:21,340 --> 00:50:22,840
You can say the
universe started out

920
00:50:22,840 --> 00:50:25,470
with Omega exactly equal
to 1 or Omega equal to 1

921
00:50:25,470 --> 00:50:28,459
to some extraordinary accuracy.

922
00:50:28,459 --> 00:50:30,000
But that's not really
an explanation.

923
00:50:30,000 --> 00:50:32,740
It really is just a
hope for an explanation.

924
00:50:32,740 --> 00:50:34,330
And the point is
that inflation, which

925
00:50:34,330 --> 00:50:35,970
you'll be learning about
in the next few lectures,

926
00:50:35,970 --> 00:50:37,540
provides an actual explanation.

927
00:50:37,540 --> 00:50:39,710
It provides a
mechanism that drives

928
00:50:39,710 --> 00:50:44,280
the early universe towards Omega
equals 1, thereby explaining

929
00:50:44,280 --> 00:50:47,810
why the early universe
had a value of Omega

930
00:50:47,810 --> 00:50:49,640
so incredibly close to 1.

931
00:50:49,640 --> 00:50:51,800
So that's what we're going
to be learning shortly.

932
00:50:51,800 --> 00:50:53,675
But at the present time,
the takeaway message

933
00:50:53,675 --> 00:50:56,540
is simply that for Omega to
be in the allowed range today

934
00:50:56,540 --> 00:51:00,130
it had to start out unbelievably
close to 1 at, for example, t

935
00:51:00,130 --> 00:51:01,800
equals 1 second.

936
00:51:01,800 --> 00:51:04,170
And within
conventional cosmology,

937
00:51:04,170 --> 00:51:07,920
there's no explanation for
why Omega so close to 1

938
00:51:07,920 --> 00:51:10,610
was in any way preferred.

939
00:51:10,610 --> 00:51:13,630
Any questions about that?

940
00:51:13,630 --> 00:51:14,424
Yes?

941
00:51:14,424 --> 00:51:16,844
AUDIENCE: Is there
any heuristic argument

942
00:51:16,844 --> 00:51:21,684
that omega [INAUDIBLE]
universe has total energy zero?

943
00:51:21,684 --> 00:51:25,556
So isn't that, at
least, appealing?

944
00:51:25,556 --> 00:51:28,440
PROFESSOR: OK the question
is, isn't it maybe appealing

945
00:51:28,440 --> 00:51:31,510
that Omega should equal
1 because Omega equals

946
00:51:31,510 --> 00:51:34,525
1 is a flat universe,
which has 0 total energy?

947
00:51:34,525 --> 00:51:36,650
I guess, the point is that
any closed universe also

948
00:51:36,650 --> 00:51:38,430
has zero total energy.

949
00:51:38,430 --> 00:51:42,200
So I don't think Omega
equals 1 is so special.

950
00:51:42,200 --> 00:51:46,590
And furthermore, if you look at
the spectrum of possible values

951
00:51:46,590 --> 00:51:52,270
of Omega, it can be positive--
I'm sorry, not with Omega.

952
00:51:52,270 --> 00:51:55,564
Let me look at the
curvature itself, little k.

953
00:51:55,564 --> 00:51:57,230
Little k can be
positive, in which case,

954
00:51:57,230 --> 00:51:58,035
you have a closed universe.

955
00:51:58,035 --> 00:51:59,451
It can be negative,
in which case,

956
00:51:59,451 --> 00:52:00,800
you have an open universe.

957
00:52:00,800 --> 00:52:02,510
And only for the one
special case of k

958
00:52:02,510 --> 00:52:07,460
equals 0, which really is one
number in the whole real line

959
00:52:07,460 --> 00:52:10,576
of possible numbers, do
you get exactly flat.

960
00:52:10,576 --> 00:52:12,450
So I think from that
point of view flat looks

961
00:52:12,450 --> 00:52:17,420
highly special and not at
all plausible as what you'd

962
00:52:17,420 --> 00:52:20,620
get if you just grabbed
something out of a grab bag.

963
00:52:20,620 --> 00:52:24,372
But, ultimately, I think there's
no way of knowing for sure.

964
00:52:24,372 --> 00:52:26,830
Whether or not Omega equals 1
coming out of the singularity

965
00:52:26,830 --> 00:52:29,080
is plausible really depends
on knowing something about

966
00:52:29,080 --> 00:52:30,740
the singularity, which we don't.

967
00:52:30,740 --> 00:52:32,550
So you're free to speculate.

968
00:52:32,550 --> 00:52:34,430
But the nice thing
about inflation

969
00:52:34,430 --> 00:52:36,220
is that you don't
need to speculate.

970
00:52:36,220 --> 00:52:38,500
Inflation really does
provide a physical mechanism

971
00:52:38,500 --> 00:52:41,070
that we can understand
that drives Omega

972
00:52:41,070 --> 00:52:44,830
to be 1 exactly
like what we see.

973
00:52:44,830 --> 00:52:47,500
Any other questions?

974
00:52:47,500 --> 00:52:49,540
OK, in that case,
what I'd like to do

975
00:52:49,540 --> 00:52:53,120
is to move on to
problem number three,

976
00:52:53,120 --> 00:52:56,010
which is the magnetic monopole
problem, which unfortunately

977
00:52:56,010 --> 00:52:59,410
requires some background
to understand.

978
00:52:59,410 --> 00:53:01,770
And we don't have too much time.

979
00:53:01,770 --> 00:53:04,240
So I'm going to go through
things rather quickly.

980
00:53:06,920 --> 00:53:09,860
This magnetic
monopole problem is

981
00:53:09,860 --> 00:53:14,100
different from the other two
in the first two problems

982
00:53:14,100 --> 00:53:18,270
I discussed are just problems
of basic classical cosmology.

983
00:53:18,270 --> 00:53:19,750
The magnetic
monopole problem only

984
00:53:19,750 --> 00:53:24,200
arises if we believe that
physics at very high energies

985
00:53:24,200 --> 00:53:27,080
is described what are called
grand unified theories, which

986
00:53:27,080 --> 00:53:30,400
then imply that these magnetic
monopoles exist and allow us

987
00:53:30,400 --> 00:53:32,410
a root for estimating
how many of them

988
00:53:32,410 --> 00:53:33,950
would have been produced.

989
00:53:33,950 --> 00:53:35,520
And the point is
that if we assume

990
00:53:35,520 --> 00:53:38,290
that grand unified theories
are the right description

991
00:53:38,290 --> 00:53:41,060
of physics at very
high energies, then

992
00:53:41,060 --> 00:53:43,750
we conclude that far too
many magnetic monopoles would

993
00:53:43,750 --> 00:53:46,970
be produced if we had just the
standard cosmology that we've

994
00:53:46,970 --> 00:53:48,534
been talking about
without inflation.

995
00:53:48,534 --> 00:53:50,575
So that's going to be the
thrust of the argument.

996
00:53:50,575 --> 00:53:52,220
And it will all go
away if you decide

997
00:53:52,220 --> 00:53:54,595
you don't believe in grand
unified theories, which you're

998
00:53:54,595 --> 00:53:55,120
allowed to.

999
00:53:55,120 --> 00:53:57,370
But there is some evidence
for grand unified theories.

1000
00:53:57,370 --> 00:53:59,900
And I'll talk about
that a little bit.

1001
00:53:59,900 --> 00:54:03,230
Now, I'm not going to have
time to completely describe

1002
00:54:03,230 --> 00:54:04,320
grand unified theories.

1003
00:54:04,320 --> 00:54:06,892
But I will try to tell
you enough odd facts

1004
00:54:06,892 --> 00:54:08,100
about grand unified theories.

1005
00:54:08,100 --> 00:54:10,760
So there will be kind of
a consistent picture that

1006
00:54:10,760 --> 00:54:14,314
will hang together, even though
there's no claim that I can

1007
00:54:14,314 --> 00:54:16,730
completely teach you grand
unified theories in the next 10

1008
00:54:16,730 --> 00:54:18,510
minutes and then talk
about the production

1009
00:54:18,510 --> 00:54:20,884
of magnetic monopoles and
those theories in the next five

1010
00:54:20,884 --> 00:54:21,390
minutes.

1011
00:54:21,390 --> 00:54:23,780
But that will be
sort of the goal.

1012
00:54:23,780 --> 00:54:26,342
So to start with, I
mentioned that there's

1013
00:54:26,342 --> 00:54:28,300
something called the
standard model of particle

1014
00:54:28,300 --> 00:54:30,790
physics, which is
enormously successful.

1015
00:54:30,790 --> 00:54:33,720
It's been developed
really since the 1970s

1016
00:54:33,720 --> 00:54:37,710
and has not changed too
much since maybe 1975 or so.

1017
00:54:37,710 --> 00:54:40,950
We have, since 1975, learned
that neutrinos have masses.

1018
00:54:40,950 --> 00:54:43,200
And those can be incorporated
into the standard model.

1019
00:54:43,200 --> 00:54:46,460
And that's a recent addition.

1020
00:54:46,460 --> 00:54:48,380
And, I guess, in
1975 I'm not sure

1021
00:54:48,380 --> 00:54:53,040
if we knew all three
generations that we now know.

1022
00:54:53,040 --> 00:54:54,840
But the matter, the
fundamental particles

1023
00:54:54,840 --> 00:54:57,277
fall into three
generations, these particles

1024
00:54:57,277 --> 00:54:58,110
of a different type.

1025
00:54:58,110 --> 00:54:59,464
And we'll talk about them later.

1026
00:54:59,464 --> 00:55:00,505
But these are the quarks.

1027
00:55:00,505 --> 00:55:02,890
These are the
spin-1/2 particles,

1028
00:55:02,890 --> 00:55:05,620
these three columns on the left.

1029
00:55:05,620 --> 00:55:09,680
On the top, we have the quarks,
up, down, charm, strange, top,

1030
00:55:09,680 --> 00:55:10,180
and bottom.

1031
00:55:10,180 --> 00:55:14,460
There are six different
flavors of quarks.

1032
00:55:14,460 --> 00:55:18,502
Each quark, by the way, comes
in three different colors.

1033
00:55:18,502 --> 00:55:19,960
The different colors
are absolutely

1034
00:55:19,960 --> 00:55:21,510
identical to each other.

1035
00:55:21,510 --> 00:55:23,870
There's a perfect
symmetry among colors.

1036
00:55:23,870 --> 00:55:25,730
There's no perfect
symmetry here.

1037
00:55:25,730 --> 00:55:28,354
Each of these quarks is a little
bit different from the others.

1038
00:55:28,354 --> 00:55:30,630
Although, there are
approximate symmetries.

1039
00:55:30,630 --> 00:55:32,860
And related to each
family of quarks

1040
00:55:32,860 --> 00:55:35,320
is a family of
leptons, particles

1041
00:55:35,320 --> 00:55:37,660
that do not undergo
strong interactions

1042
00:55:37,660 --> 00:55:41,310
in the electron-like
particles and neutrinos.

1043
00:55:41,310 --> 00:55:42,750
This row is the neutrinos.

1044
00:55:42,750 --> 00:55:45,480
There's an electron neutrino,
a muon neutrino, and a tau

1045
00:55:45,480 --> 00:55:47,590
neutrino, like
we've already said.

1046
00:55:47,590 --> 00:55:50,180
And there's an electron,
a muon, and a tao,

1047
00:55:50,180 --> 00:55:52,820
which I guess we've
also already said.

1048
00:55:52,820 --> 00:55:56,690
So the particles on the left are
all of the spin-1/2 particles

1049
00:55:56,690 --> 00:56:00,060
that exist in the standard
model of particle physics.

1050
00:56:00,060 --> 00:56:03,240
And then on the right, we
have the Boson particles,

1051
00:56:03,240 --> 00:56:05,560
the particles of
integer span, starting

1052
00:56:05,560 --> 00:56:07,910
with the photon on the top.

1053
00:56:07,910 --> 00:56:09,870
Under that in this
list-- there's

1054
00:56:09,870 --> 00:56:12,080
no particular order
in here really-- is

1055
00:56:12,080 --> 00:56:14,840
the gluon which is
the particle that's

1056
00:56:14,840 --> 00:56:17,340
like the photon but the
particle which describes

1057
00:56:17,340 --> 00:56:20,530
the strong interactions, which
are somewhat more complicated

1058
00:56:20,530 --> 00:56:24,890
and electromagnetism, but still
described by spin-1 particles

1059
00:56:24,890 --> 00:56:26,940
just like the photon.

1060
00:56:26,940 --> 00:56:30,190
And then two other spin-1
particles, the z0 and the w

1061
00:56:30,190 --> 00:56:33,130
plus and minus, which
are a neutral particle

1062
00:56:33,130 --> 00:56:35,000
and a charged
particle, which are

1063
00:56:35,000 --> 00:56:37,355
the carriers of the
so-called weak interactions.

1064
00:56:37,355 --> 00:56:40,490
The weak interactions
being the only

1065
00:56:40,490 --> 00:56:43,380
non-gravitational interactions
that neutrinos undergo.

1066
00:56:43,380 --> 00:56:45,660
And the weak interactions
are responsible

1067
00:56:45,660 --> 00:56:47,340
for certain particle decays.

1068
00:56:47,340 --> 00:56:50,870
For example, a neutron can
decay into a proton giving off

1069
00:56:50,870 --> 00:56:53,940
also an electron--
producing a proton,

1070
00:56:53,940 --> 00:56:57,810
yeah-- charge has to be
conserved, proton is positive.

1071
00:56:57,810 --> 00:57:00,250
So it's produced with
an electron and then

1072
00:57:00,250 --> 00:57:04,750
an anti-electron neutrino to
balance the so-called electron

1073
00:57:04,750 --> 00:57:06,480
lepton number.

1074
00:57:06,480 --> 00:57:08,045
And that's a weak direction.

1075
00:57:08,045 --> 00:57:09,920
Essentially, anything
that involves neutrinos

1076
00:57:09,920 --> 00:57:12,474
is going to be weak interaction.

1077
00:57:12,474 --> 00:57:13,640
So these are the characters.

1078
00:57:13,640 --> 00:57:15,070
And there's a set
of interactions

1079
00:57:15,070 --> 00:57:17,720
that go with this
set of characters.

1080
00:57:17,720 --> 00:57:21,510
So we have here a
complete model of how

1081
00:57:21,510 --> 00:57:24,020
elementary particles interact.

1082
00:57:24,020 --> 00:57:26,580
And the model has been
totally successful.

1083
00:57:26,580 --> 00:57:29,130
It actually gives
predictions that

1084
00:57:29,130 --> 00:57:32,580
are consistent with every
reliable experiment that

1085
00:57:32,580 --> 00:57:36,930
has been done since the
mid-1970s up to the present.

1086
00:57:36,930 --> 00:57:38,920
So it's made particle
physics a bit dull

1087
00:57:38,920 --> 00:57:42,140
since we have a theory that
seems to predict everything.

1088
00:57:42,140 --> 00:57:44,340
But it's also a
magnificent achievement

1089
00:57:44,340 --> 00:57:46,510
that we have such a theory.

1090
00:57:46,510 --> 00:57:49,570
Now, in spite of the fact that
this theory is so unbelievably

1091
00:57:49,570 --> 00:57:54,660
successful I don't think I know
anybody who really regards this

1092
00:57:54,660 --> 00:57:59,360
as a candidate even or the
ultimate theory of nature.

1093
00:57:59,360 --> 00:58:02,559
And the reason for that
is maybe twofold, first

1094
00:58:02,559 --> 00:58:04,225
is that it does not
incorporate gravity,

1095
00:58:04,225 --> 00:58:07,060
it only incorporates
particle interactions.

1096
00:58:07,060 --> 00:58:10,057
And we know that gravity exists
and has to be put in somehow.

1097
00:58:10,057 --> 00:58:12,640
And there doesn't seem to be any
simple way of putting gravity

1098
00:58:12,640 --> 00:58:15,300
into this theory.

1099
00:58:15,300 --> 00:58:20,152
And, secondly-- maybe there's
three reasons-- second,

1100
00:58:20,152 --> 00:58:22,610
it does not include any good
candidates for the dark matter

1101
00:58:22,610 --> 00:58:25,390
that we know exists
in cosmology.

1102
00:58:25,390 --> 00:58:29,860
And third, excuse me-- and this
is given a lot of importance,

1103
00:58:29,860 --> 00:58:33,600
even though it's an aesthetic
argument-- this model has

1104
00:58:33,600 --> 00:58:37,690
something like 28 free
parameters, quantities that you

1105
00:58:37,690 --> 00:58:40,190
just have to go out and measure
before you can use the model

1106
00:58:40,190 --> 00:58:41,540
to make predictions.

1107
00:58:41,540 --> 00:58:44,110
And a large number
of free parameters

1108
00:58:44,110 --> 00:58:49,500
is associated, by theoretical
physicists, with ugliness.

1109
00:58:49,500 --> 00:58:52,460
So this is considered
a very ugly model.

1110
00:58:52,460 --> 00:58:55,530
And we have no real
way of knowing,

1111
00:58:55,530 --> 00:58:57,650
but almost all
theoretical physicists

1112
00:58:57,650 --> 00:59:00,130
believe that the
correct theory of nature

1113
00:59:00,130 --> 00:59:03,850
is going to be simpler and
involve many fewer, maybe

1114
00:59:03,850 --> 00:59:06,147
none at all, free knobs
that can be turned

1115
00:59:06,147 --> 00:59:07,730
to produce different
kinds of physics.

1116
00:59:11,470 --> 00:59:14,500
OK, what I want to
talk about next,

1117
00:59:14,500 --> 00:59:16,100
leading up to grand
unified theories,

1118
00:59:16,100 --> 00:59:19,730
is the notion of gauge theories.

1119
00:59:19,730 --> 00:59:22,270
And, yes?

1120
00:59:22,270 --> 00:59:26,206
AUDIENCE: I'm sorry, question
real quick from the chart.

1121
00:59:26,206 --> 00:59:28,666
I basically heard
the explanation

1122
00:59:28,666 --> 00:59:31,454
that the reason for the
short range of the weak force

1123
00:59:31,454 --> 00:59:34,324
was the massive mediator that is
the cause of exponential field

1124
00:59:34,324 --> 00:59:35,062
decay.

1125
00:59:35,062 --> 00:59:37,030
But if the [INAUDIBLE]
is massless,

1126
00:59:37,030 --> 00:59:39,490
how do we explain
that to [INAUDIBLE]?

1127
00:59:39,490 --> 00:59:41,724
PROFESSOR: Right,
OK, the question

1128
00:59:41,724 --> 00:59:43,265
is for those of you
who couldn't hear

1129
00:59:43,265 --> 00:59:45,915
it is that the short range
of the weak interactions,

1130
00:59:45,915 --> 00:59:48,500
although I didn't talk about
it, is usually explained

1131
00:59:48,500 --> 00:59:51,970
and is explained by the
fact that the z and w naught

1132
00:59:51,970 --> 00:59:53,170
Bosons are very heavy.

1133
00:59:53,170 --> 00:59:55,270
And heavy particles
have a short range.

1134
00:59:55,270 --> 00:59:57,770
But the strong interactions
seem to also have a short range.

1135
00:59:57,770 --> 01:00:01,290
And yet, the gluon is
effectively massless.

1136
01:00:01,290 --> 01:00:04,210
That's related to a
complicated issue which

1137
01:00:04,210 --> 01:00:05,990
goes by the name of confinement.

1138
01:00:05,990 --> 01:00:08,940
Although the gluon is
massless, it's confined.

1139
01:00:08,940 --> 01:00:11,820
And confined means that
it cannot exist as a free

1140
01:00:11,820 --> 01:00:13,260
particle.

1141
01:00:13,260 --> 01:00:14,960
In some sense, the
strong interactions

1142
01:00:14,960 --> 01:00:18,670
do have a long range in that
if you took a meson, which

1143
01:00:18,670 --> 01:00:20,620
is made out of a quark
and an anti-quark,

1144
01:00:20,620 --> 01:00:23,520
in principle, if
you pulled it apart,

1145
01:00:23,520 --> 01:00:25,720
there'd be a string
of gluon force

1146
01:00:25,720 --> 01:00:27,865
between the quark
and the anti-quark.

1147
01:00:27,865 --> 01:00:29,490
And that would produce
a constant force

1148
01:00:29,490 --> 01:00:32,970
no matter how far
apart you pulled them.

1149
01:00:32,970 --> 01:00:34,680
And the only thing
that intervenes,

1150
01:00:34,680 --> 01:00:36,870
and it is important
that it does intervene,

1151
01:00:36,870 --> 01:00:39,480
is that if you pulled
them too far apart

1152
01:00:39,480 --> 01:00:41,380
it would become
energetically favorable

1153
01:00:41,380 --> 01:00:45,776
for a quark anti-quark pair
to materialize in the middle.

1154
01:00:45,776 --> 01:00:48,150
And then instead of having a
quark here and an anti-quark

1155
01:00:48,150 --> 01:00:50,112
here and a string
between them, you

1156
01:00:50,112 --> 01:00:52,070
would have a quark here
and an anti-quark there

1157
01:00:52,070 --> 01:00:55,140
and a string between them,
and an anti-quark here

1158
01:00:55,140 --> 01:00:57,046
and-- I'm sorry, I
guess it's a quark here

1159
01:00:57,046 --> 01:00:59,170
and an anti-quark there
and a string between those.

1160
01:00:59,170 --> 01:01:01,160
And then they would
just fly apart.

1161
01:01:01,160 --> 01:01:02,890
So the string can
break by producing

1162
01:01:02,890 --> 01:01:04,210
quark anti-quark pairs.

1163
01:01:04,210 --> 01:01:07,110
But the string can never just
end in the middle of nowhere.

1164
01:01:07,110 --> 01:01:08,967
And that's the essence
of confinement.

1165
01:01:08,967 --> 01:01:10,675
And it's due to the
peculiar interactions

1166
01:01:10,675 --> 01:01:14,772
that these gluons
are believed to obey.

1167
01:01:14,772 --> 01:01:16,230
So the gluons behave
in a way which

1168
01:01:16,230 --> 01:01:18,490
is somewhat uncharacteristic
of particles.

1169
01:01:18,490 --> 01:01:20,990
Except at very short distances,
they behave very much

1170
01:01:20,990 --> 01:01:21,990
like ordinary particles.

1171
01:01:21,990 --> 01:01:24,970
But at larger distances,
these effects of confinement

1172
01:01:24,970 --> 01:01:28,570
play a very significant role.

1173
01:01:28,570 --> 01:01:30,570
Any other questions?

1174
01:01:30,570 --> 01:01:33,810
OK, onward, I want talk
about gauge theories,

1175
01:01:33,810 --> 01:01:35,400
because gauge
theories have a lot

1176
01:01:35,400 --> 01:01:38,530
to do with how one gets
into grand unified theories

1177
01:01:38,530 --> 01:01:41,380
from the standard model.

1178
01:01:41,380 --> 01:01:44,520
And, basically, a gauge
theory is a theory

1179
01:01:44,520 --> 01:01:49,500
in which the fundamental fields
are not themselves reality.

1180
01:01:49,500 --> 01:01:51,680
But rather there's a
set of transformations

1181
01:01:51,680 --> 01:01:54,020
that the fields can
undergo which take you

1182
01:01:54,020 --> 01:01:56,320
from one description
to an equivalent

1183
01:01:56,320 --> 01:01:58,370
description of the same reality.

1184
01:01:58,370 --> 01:01:59,940
So there's not a
one to one mapping

1185
01:01:59,940 --> 01:02:01,490
between the fields and reality.

1186
01:02:01,490 --> 01:02:03,364
There are many different
field configurations

1187
01:02:03,364 --> 01:02:05,500
that correspond to
the same reality.

1188
01:02:05,500 --> 01:02:07,560
And that's basically
what characterizes

1189
01:02:07,560 --> 01:02:08,824
what we call a gauge theory.

1190
01:02:08,824 --> 01:02:10,740
And you do know one
example of a gauge theory.

1191
01:02:10,740 --> 01:02:12,896
And that's e and m.

1192
01:02:12,896 --> 01:02:16,990
If e and m is expressed in terms
of the potentials Phi and A,

1193
01:02:16,990 --> 01:02:19,120
you can write e in terms
of the potential that way

1194
01:02:19,120 --> 01:02:23,140
and b as the curl of A, you
could put Phi and A together

1195
01:02:23,140 --> 01:02:25,000
and make a four-vector
if you want

1196
01:02:25,000 --> 01:02:28,052
to do things in a
Lorentz covariant way.

1197
01:02:28,052 --> 01:02:30,260
And the important point,
though, whether you put them

1198
01:02:30,260 --> 01:02:32,440
together or not, is
that you can always

1199
01:02:32,440 --> 01:02:35,380
define a gauge transformation
depending on some arbitrary

1200
01:02:35,380 --> 01:02:38,595
function Lambda, which is a
function of position and time.

1201
01:02:38,595 --> 01:02:40,720
I didn't write in the
arguments, but Lambda is just

1202
01:02:40,720 --> 01:02:43,190
an arbitrary function
of position and time.

1203
01:02:43,190 --> 01:02:46,450
And for any Lambda you
can replace 5 by 5 prime,

1204
01:02:46,450 --> 01:02:50,365
given by this line, and a by
a prime, given by that line.

1205
01:02:50,365 --> 01:02:51,990
And if you go back
and compute e and b,

1206
01:02:51,990 --> 01:02:53,860
you'll find that
they'll be unchanged.

1207
01:02:53,860 --> 01:02:55,610
And therefore, the
physics is not changed,

1208
01:02:55,610 --> 01:02:59,140
because the physics really
is all contained in e and b.

1209
01:02:59,140 --> 01:03:01,590
So this gauge transformation
is a transformation

1210
01:03:01,590 --> 01:03:04,570
on the fields of the theory--
it can be written covariantly

1211
01:03:04,570 --> 01:03:08,065
this way-- which leaves
the physics invariant.

1212
01:03:08,065 --> 01:03:09,940
And it turns out that
all the important field

1213
01:03:09,940 --> 01:03:13,770
theories that we know
of are gauge theories.

1214
01:03:13,770 --> 01:03:15,970
And that's why it's
worth mentioning here.

1215
01:03:15,970 --> 01:03:20,290
Now, for e and m, the gauge
parameter is just this function

1216
01:03:20,290 --> 01:03:23,087
Lambda, which is a function
of position and time.

1217
01:03:23,087 --> 01:03:24,670
And an important
issue is what happens

1218
01:03:24,670 --> 01:03:26,480
when you combine
gauge transformations,

1219
01:03:26,480 --> 01:03:29,850
because the succession
of two transformations

1220
01:03:29,850 --> 01:03:32,150
had better also be a
symmetry transformation.

1221
01:03:32,150 --> 01:03:35,620
So it's worth understanding
that group structure.

1222
01:03:35,620 --> 01:03:37,430
And for the case of e
and m, these Lambdas

1223
01:03:37,430 --> 01:03:40,930
just add if we make
successive transformations.

1224
01:03:40,930 --> 01:03:42,430
And that means the
group is Abelian.

1225
01:03:42,430 --> 01:03:44,910
It's commutative.

1226
01:03:44,910 --> 01:03:47,060
But that's not always the case.

1227
01:03:47,060 --> 01:03:48,530
Let's see, where am I going?

1228
01:03:48,530 --> 01:03:50,646
OK, next slide actually
comes somewhat later.

1229
01:03:50,646 --> 01:03:52,020
Let me go back to
the blackboard.

1230
01:03:57,100 --> 01:04:00,340
It turns out that the important
generalization of gauge

1231
01:04:00,340 --> 01:04:03,710
theories is the generalization
from Abelian gauge

1232
01:04:03,710 --> 01:04:06,680
theories to non-Abelian ones,
which was done originally

1233
01:04:06,680 --> 01:04:16,130
by Yang and Mills
in 1954, I think.

1234
01:04:16,130 --> 01:04:19,140
And when it was first proposed,
nobody knew what to do with it.

1235
01:04:19,140 --> 01:04:21,310
But, ultimately, these
non-Abelian gauge theories

1236
01:04:21,310 --> 01:04:26,170
became the standard model
of particle physics.

1237
01:04:26,170 --> 01:04:28,570
And in non-Abelian
gauge theories

1238
01:04:28,570 --> 01:04:31,720
the parameter that describes the
gauge transformation is a group

1239
01:04:31,720 --> 01:04:35,850
element, not just
something that gets added.

1240
01:04:35,850 --> 01:04:39,030
And group elements
multiply, according

1241
01:04:39,030 --> 01:04:41,040
to the procedures of some group.

1242
01:04:41,040 --> 01:04:43,772
And in particular,
the standard model

1243
01:04:43,772 --> 01:04:44,980
is built out of three groups.

1244
01:04:47,431 --> 01:04:49,430
And the fact that there
are three groups not one

1245
01:04:49,430 --> 01:04:52,880
is just an example
of this ugliness

1246
01:04:52,880 --> 01:04:54,980
that I mentioned and is
responsible for the fact

1247
01:04:54,980 --> 01:04:58,470
that there's some significant
number of parameters

1248
01:04:58,470 --> 01:05:01,120
even if there were no
other complications.

1249
01:05:01,120 --> 01:05:07,460
So the standard model is based
on three gauge groups, SU3,

1250
01:05:07,460 --> 01:05:10,370
SU2, and U1.

1251
01:05:14,459 --> 01:05:16,250
And it won't really be
too important for us

1252
01:05:16,250 --> 01:05:17,920
what exactly these groups are.

1253
01:05:21,460 --> 01:05:25,280
Let me just mention
quickly, SU3 is a group of 3

1254
01:05:25,280 --> 01:05:28,530
by 3 matrices which are
unitary in the sense

1255
01:05:28,530 --> 01:05:33,130
that u adjoint times u is equal
to 1 and special in the sense

1256
01:05:33,130 --> 01:05:35,390
that they have determinant 1.

1257
01:05:35,390 --> 01:05:36,980
And the set of all
3 by 3 matrices

1258
01:05:36,980 --> 01:05:38,438
have those properties
form a group.

1259
01:05:38,438 --> 01:05:39,820
And that group is SU3.

1260
01:05:39,820 --> 01:05:41,430
SU2 is the same
thing but replace

1261
01:05:41,430 --> 01:05:44,400
the 3 in all those
sentences by 2s.

1262
01:05:44,400 --> 01:05:47,306
U1 is just the group of phases.

1263
01:05:47,306 --> 01:05:48,680
That is the group
of real numbers

1264
01:05:48,680 --> 01:05:51,080
that could be written
as either the i phi.

1265
01:05:51,080 --> 01:05:55,380
So it's just a complex number of
magnitude 1 with just a phase.

1266
01:05:55,380 --> 01:05:58,510
And you can multiply those
and they form a group.

1267
01:05:58,510 --> 01:06:00,740
And the standard model
contains these three groups.

1268
01:06:00,740 --> 01:06:03,370
And the three groups
all act independently,

1269
01:06:03,370 --> 01:06:05,952
which means that if you
know about group products,

1270
01:06:05,952 --> 01:06:09,120
one can say that the full
group is the product group.

1271
01:06:12,640 --> 01:06:16,670
And that just means that a full
description of a group element

1272
01:06:16,670 --> 01:06:20,980
is really just a set
of an element of SU3,

1273
01:06:20,980 --> 01:06:24,490
and an element of SU2,
and an element of U1.

1274
01:06:24,490 --> 01:06:28,094
And if you put together three
group elements in each group

1275
01:06:28,094 --> 01:06:29,510
and put them
together with commas,

1276
01:06:29,510 --> 01:06:31,840
that becomes a group element
of the group SU3 cross,

1277
01:06:31,840 --> 01:06:33,660
SU2 cross U1.

1278
01:06:33,660 --> 01:06:37,322
And that is the gauge
group of the standard model

1279
01:06:37,322 --> 01:06:38,155
of particle physics.

1280
01:06:42,010 --> 01:07:08,050
OK, now grand unified theories,
a grand unified theory

1281
01:07:08,050 --> 01:07:10,800
is based on the idea that
this set of three groups

1282
01:07:10,800 --> 01:07:13,220
can all be embedded in
a single simple group.

1283
01:07:37,760 --> 01:07:40,160
Now, simple actually has a
mathematical group theory

1284
01:07:40,160 --> 01:07:40,660
meaning.

1285
01:07:40,660 --> 01:07:42,660
But it also, for
our purposes, just

1286
01:07:42,660 --> 01:07:46,580
means simple, which is
good enough for our brush

1287
01:07:46,580 --> 01:07:49,220
through of these arguments.

1288
01:07:49,220 --> 01:07:52,052
And, for example--
and the example

1289
01:07:52,052 --> 01:07:54,010
is shown a little bit in
the lecture notes that

1290
01:07:54,010 --> 01:07:57,900
will be posted
shortly-- an example

1291
01:07:57,900 --> 01:07:59,570
of a grand unified
theory, and indeed

1292
01:07:59,570 --> 01:08:02,070
the first grand unified
theory that was invented,

1293
01:08:02,070 --> 01:08:06,260
is based on the full gauge group
SU5, which is just a group of 5

1294
01:08:06,260 --> 01:08:10,750
by 5 matrices which are
unitary and have determinant 1.

1295
01:08:10,750 --> 01:08:15,380
And there's an easy way to embed
SU3 and SU2 and U1 into SU5.

1296
01:08:15,380 --> 01:08:17,490
And that's the way that
was used to construct

1297
01:08:17,490 --> 01:08:19,930
this grand unified theory.

1298
01:08:19,930 --> 01:08:29,200
One can take a 5 by 5 matrix--
so this is a 5 by 5 matrix--

1299
01:08:29,200 --> 01:08:36,029
and one can simply take
the upper 3 by 3 block

1300
01:08:36,029 --> 01:08:40,850
and put an SU3 matrix there.

1301
01:08:40,850 --> 01:08:46,080
And one can take the
lower 2 by 2 block

1302
01:08:46,080 --> 01:08:47,830
and put an SU2 matrix there.

1303
01:08:51,840 --> 01:08:53,939
And then the U1 piece--
there's supposed

1304
01:08:53,939 --> 01:08:57,359
to be a U1 left
over-- the U1 piece

1305
01:08:57,359 --> 01:09:01,120
can be obtained by giving
an overall phase to this

1306
01:09:01,120 --> 01:09:03,250
and an overall phase
to that in such a way

1307
01:09:03,250 --> 01:09:05,590
that the product of
the five phases is 0.

1308
01:09:05,590 --> 01:09:08,620
So the determinant
has not changed.

1309
01:09:08,620 --> 01:09:17,660
So one can put an e to the i2
Phi there and a factor of e

1310
01:09:17,660 --> 01:09:22,020
to the minus i3 Phi
there for any Phi.

1311
01:09:24,810 --> 01:09:28,130
And then this Phi
becomes the description

1312
01:09:28,130 --> 01:09:31,950
of the U1 piece of
this construction.

1313
01:09:31,950 --> 01:09:35,020
So we can take an arbitrary
SU3 matrix, and arbitrary SU2

1314
01:09:35,020 --> 01:09:39,240
matrix, and an arbitrary
U1 value expressed by Phi

1315
01:09:39,240 --> 01:09:42,810
and put them together
to make an SU5 matrix.

1316
01:09:42,810 --> 01:09:44,810
And if you think about
it, the SU3 piece

1317
01:09:44,810 --> 01:09:47,634
will commute with the SU2
piece and with the U1 piece.

1318
01:09:47,634 --> 01:09:49,800
These three pieces will all
commute with each other,

1319
01:09:49,800 --> 01:09:51,341
if you think about
how multiplication

1320
01:09:51,341 --> 01:09:52,620
works with this construction.

1321
01:09:52,620 --> 01:09:53,953
So it does exactly what we want.

1322
01:09:53,953 --> 01:09:57,310
It decomposes SU5.

1323
01:09:57,310 --> 01:10:01,710
So it has a subgroup of
SU3 cross SU2 across U1.

1324
01:10:01,710 --> 01:10:06,560
And that's how the simplest
grand unified theory works.

1325
01:10:06,560 --> 01:10:11,060
OK, now, there are important
things that need to be said,

1326
01:10:11,060 --> 01:10:14,540
but we're out of time.

1327
01:10:14,540 --> 01:10:18,590
So I guess what we need
to do is to withhold

1328
01:10:18,590 --> 01:10:21,690
from the next problem set,
the magnetic monopole problem.

1329
01:10:21,690 --> 01:10:25,110
Maybe I was a bit over-ambitious
to put it on the problem set.

1330
01:10:25,110 --> 01:10:27,320
So I'll send an email
announcing that.

1331
01:10:27,320 --> 01:10:29,474
But the one problem
on the problem

1332
01:10:29,474 --> 01:10:31,390
set for next week about
grand unified theories

1333
01:10:31,390 --> 01:10:32,290
will be withheld.

1334
01:10:32,290 --> 01:10:37,250
And Scott Hughes will pick up
this discussion next Tuesday.

1335
01:10:37,250 --> 01:10:40,319
So I will see all of
you-- gee willikers,

1336
01:10:40,319 --> 01:10:42,360
if you come to my office
hour, I'll see you then.

1337
01:10:42,360 --> 01:10:45,110
But otherwise, I may not
see you until the quiz.

1338
01:10:45,110 --> 01:10:49,990
So have a good Thanksgiving
and good luck on the quiz.