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[SQUEAKING]

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[RUSTLING]

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[CLICKING]

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SCOTT HUGHES: All right.

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So at this point we're
going to switch gears.

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Everything that we have done
over the past several lectures

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has been in service
of the approach

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to solving the Einstein
field equations in which we

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assume a small perturbation
around an exact background.

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Most of it was spent looking at
perturbations around flat space

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time.

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A little bit in the
last lecture we touched

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on some of the mathematics
and some of the analysis

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when you curve--

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you expand around some
non-specified curved

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background.

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I didn't tell you where that
curve background comes from.

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Today we'll be--
this lecture will

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be the first one
in which we begin

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thinking about different forms
of different kinds of solutions

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that arise from
different principles.

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We're going to begin this
by studying cosmology.

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It's the large-scale
structure of the universe.

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So from the standpoint of--

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from the standpoint of
the calculational toolkit

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that we will be
using, this is going

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to be the first
example of a spacetime

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that we construct using
a symmetry argument.

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We are not going to
make any assumptions

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that anything is weak or small
or any kind of approximation

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can be--

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any kind of an approximation
can be applied.

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What we're going to do is ask
ourselves, suppose spacetime--

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at least spacetime on some
particular very large scales--

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is restricted by
various symmetries.

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So we will apply various
restrictions to the equations

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and to the spacetime
by the assumption

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that certain symmetric
symmetries must hold.

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Let me reword this.

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By demanding that
certain symmetries hold.

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Doing so will
significantly reduce

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the complicated non-linear
dynamics of the field equations

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of general relativity.

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This will allow us to
reduce those complicated

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generic equations into
something that is tractable.

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So let me describe--
whoops-- let's

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me give a little bit of
background to this discussion.

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Let me get some
better chalk, too.

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So as background,
I'm going to give

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a little bit of a synopsis
of some stuff that

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is described very nicely
in the textbook by Carroll.

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So for background,
part of what we're

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going to consider
as we move into this

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is a notion of what are called
maximally symmetric spaces.

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So I urge you to read
Section 3.9 of Carroll

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for extensive
discussion of this.

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But the key concept of this
is that a maximally symmetric

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space is a space that
has the largest number--

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so let's say MSS,
maximally symmetric space,

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has the largest number of
allowed Killing vectors.

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If your space has
n dimensions, it

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has n times n plus 1 over
2 such Killing vectors.

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And recall, if you
do a lead derivative

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of the metric along the
Killing vector, you get 0, OK?

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So it's that these n
times m plus 1 Killing

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vectors all define
ways in which, as you

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sort of flow along
these vectors,

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spacetime is left unchanged.

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Intuitively, what these
do is define a spacetime

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that is maximally homogeneous--

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I shouldn't say
spacetime yet, we

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haven't specified the
nature of this manifold.

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So this defines a space that
is maximally homogeneous.

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And homogeneous
means that just--

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it has uniform properties
in all locations.

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And it is maximally isotopic.

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Which is a way of
saying essentially

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that it looks the same
in all directions.

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In the kind of spacetimes
that we are familiar with,

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something-- a spacetime
that is highly isotropic

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is one that is
invariant with respect

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to rotations and boosts,
and one that is homogeneous

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is something that is invariant
with respect to translations.

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So let me give two examples.

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In Euclidean space, that
is a maximally symmetric

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three-dimensional space, n
times n plus 1 over 2 equals 6.

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And those 6 Killing
vectors in Euclidean space

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correspond to 3 rotations
and 3 translations.

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Minkowski flat
spacetime: n equals 4.

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n times n plus 1 over
2 is equal to 10.

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I have 3 rotations, 3
translations, and 4 boosts, OK?

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The requirement that your
space satisfy these properties,

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it leads to a condition
that the Riemann tensor

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must be Lorentz-invariant
within the local Lorentz frame.

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So for these two examples
that I talked about,

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the Riemann tensor
actually vanishes,

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and 0 is certainly
Lorentz-invariant,

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so there's no problem there.

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But as I start thinking
about more general classes

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of spacetimes, which I'm going
to consider to be examples

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of massively
symmetric spaces, they

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might not have vanishing
Riemann tensors.

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But the Riemann tensors,
in order to be massively

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symmetric, if I go into
a freely-falling frame,

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that has to look--

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everything has to look
Lorentz-invariant.

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This leads to a condition that
my Riemann tensor must take--

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it is constrained to take
one particular simple form.

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It must be R over n times n
minus 1 times metric like so.

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This is-- so Carroll goes
through this in some detail.

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Essentially what's
going on here is

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this is the only way in
which I am guaranteed

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to create a tensor that
is Lorentz-invariant

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in a local Lorentz frame.

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So I go to my
local Lorentz frame

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and I must have a form
it looks like this,

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and this is a way of putting
all my various quantities

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on my metric tensors
together in such a way

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that I recover the symmetries
of the Riemann tensor,

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and is my number of dimensions.

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Because it will
prove useful, let

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me generate the Ricci tensor
and the Ricci scalar from this.

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So R mu nu is going to be
R over n times m minus 1.

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I'm taking the trace on
indices alpha and beta.

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If I trace on this guy, I get n.

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The trace in the metric
always just gives me back

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the number of dimensions.

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And when I trace on
alpha and beta here,

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I basically just contract
these two indices,

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and so I get the metric
back over n times g mu nu.

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Take a further trace
and you can see

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that that R that
went into this thing

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is indeed nothing more
than the Ricci curvature.

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Excuse me, the scalar
Ricci curvature.

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OK?

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You can construct the
Einstein tensor out of this,

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and what you see is that
the Einstein tensor must

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be proportional to the metric.

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And in fact, there are--

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the only solutions for
the Einstein tensor that--

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the only solution is the
Einstein field equations

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in which the Einstein tensor is
proportional to the metric are

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either flat spacetime or
a cosmological constant.

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So empty space,
empty flat spacetime

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and cosmological constant
are the only maximally

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symmetric four-dimensional
spacetimes.

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That does not necessarily
describe our universe.

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So our universe, how am I
going to tie all this together?

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We begin with the
observation that our universe

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is, in fact,
homogeneous and isotopic

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on large spatial scales.

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I emphasize spatial because
the spacetime of our universe

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is not homogeneous, OK?

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In fact, the past
of our universe

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is very different
from the present.

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Because light travels
at a finite time, when

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we observe two
large distances, we

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are looking back into
the distant past.

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And we see that the universe
is a lot denser in the past

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than it is today.

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It remains the case,
though, that it is still

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homogeneous and
isotropic spatially,

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at least on large scales.

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So we are going
to take advantage

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of these notions of
maximally symmetric spaces

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to define a spacetime that is
maximally symmetric spatially,

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but is not so symmetric
with respect to time, OK?

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So we'll get to that
in just a few moments.

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There is a wiggle word in here.

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I said that our
universe, by observation,

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is homogeneous and isotropic
on large spatial scales.

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What does large mean?

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Well the very largest scales
that we can observe of all--

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so that when we go back and
we probe sort of the largest

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coherent structure that can
be observed in our universe,

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we have to go all the way
back to a time which is

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approximately
13-point-something or another--

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I forget the exact
number, but let's say

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about 13.7 billion years ago,
and we see the cosmic microwave

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background.

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So the cosmic
microwave background

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describes what our universe
looked like 13.7 billion years

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ago or so, and what you
see is that this guy is

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homogeneous and isotropic to
about a part in 100,000, OK?

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With a lot of
interesting physics

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in that deviation from--

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that sort of part in
100,000 deviation,

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but that's a topic
for a different class.

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We then sort of imagine
you move forward in time,

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you look on-- so that
tells you about the largest

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scales in the earliest times.

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Look at the universe
on smaller scales.

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I mean, clearly you
look in this room, OK?

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I'm standing here, there's
a table over there,

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this is not homogeneous
and not isotropic,

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things look quite different.

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What we start to see is things
deviate from homogeneity

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and isotropy on scales that are
on the order of several tens

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of megaparsecs in size, OK?

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Parsec, for those of you
who are not astrophysicists,

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is a unit of measure.

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It's approximately
3.2 light years.

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So once you get
down to boxes that

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are on the orders of 50 million
light years or so on a side,

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you start to see deviations
from homogeneity and isotropy.

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And this is caused by
gravitational clumping.

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These are things
like galaxy clusters.

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So when I talk about
cosmology and I

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want to describe the universe
as large-scale structure,

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I am going to be
working on defining

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a description of spacetime
that averages out

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over small things like
clusters of galaxies, OK?

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00:17:19,369 --> 00:17:23,420
So this is sort of a fun lecture
in that sense, in that anything

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larger than an
agglomeration of a couple

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dozen or a couple
hundred galaxies,

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I'm going to treat
that like a point.

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So here's what I am going to
choose for my spacetime metric.

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This is where you start to see
the power of assuming a given

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symmetry.

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So the line element,
I'm going to write it

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as minus dt squared
plus some function R

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00:18:08,410 --> 00:18:18,310
squared of t gamma ij dx i dx j.

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The function R of t
I've written down here.

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It's one variant of-- there's
a couple functions that

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are going to get this name.

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We call this the scale factor.

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Caution, it's the same capital
R we used for the Ricci scalar,

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it's not the Ricci scalar.

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Just a little bit of
unfortunate notation,

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but it should be clear
from context which is which

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when they come up.

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I have chosen gtt equals
minus 1 and gti equals 0.

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Remember from our discussion
of linearized theory

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around a flat background,
that the spacetime-- the 10

235
00:19:05,110 --> 00:19:10,350
independent functions of my
spacetime metric, of those 10,

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four of them were
things that I could

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specify by choosing a gauge.

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00:19:14,850 --> 00:19:18,870
Well here, I have specified four
functions pretty much by fiat.

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Think of this as defining the
gauge that I am working in, OK?

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00:19:21,800 --> 00:19:24,900
In a very similar way, I have
chosen a coordinate system

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00:19:24,900 --> 00:19:29,340
by specifying gtt to be
minus 1 and gti to be 0.

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00:19:29,340 --> 00:19:31,530
This means that I am
working in what are

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00:19:31,530 --> 00:19:34,260
called co-moving coordinates.

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00:19:39,500 --> 00:19:55,810
So if I am an observer who
is at rest in the spacetime

245
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so that I would define
my four velocity like so,

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I will be essentially co
moving with the spacetime.

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Whatever the spacetime
is doing, I'm

248
00:20:06,367 --> 00:20:08,200
just going to sort of
homologously track it.

249
00:20:31,890 --> 00:20:36,420
It's worth noting-- so those of
you who think about astronomy,

250
00:20:36,420 --> 00:20:39,060
astrophysics, and
observational cosmology,

251
00:20:39,060 --> 00:20:42,420
the earth is not co-moving, OK?

252
00:20:42,420 --> 00:20:44,250
We build our telescopes
on the surface

253
00:20:44,250 --> 00:20:46,110
of the Earth which rotates.

254
00:20:46,110 --> 00:20:49,320
The Earth itself
orbits around the Sun.

255
00:20:49,320 --> 00:20:52,387
The Sun is in a solar system.

256
00:20:52,387 --> 00:20:54,720
Or excuse me-- the Sun at the
center of our solar system

257
00:20:54,720 --> 00:20:57,300
is itself orbiting our galaxy.

258
00:20:57,300 --> 00:21:00,420
And our galaxy is
actually falling

259
00:21:00,420 --> 00:21:05,460
into a large cluster of galaxies
called the Virgo Cluster.

260
00:21:05,460 --> 00:21:09,360
This basically means that when
we are making cosmologically

261
00:21:09,360 --> 00:21:12,390
interesting measurements, we
have to correct for the fact

262
00:21:12,390 --> 00:21:16,140
that we make measurements using
a four velocity that is not

263
00:21:16,140 --> 00:21:18,150
a co-moving four velocity.

264
00:21:18,150 --> 00:21:19,650
This actually shows
up in the fact

265
00:21:19,650 --> 00:21:21,120
that when one
makes measurements,

266
00:21:21,120 --> 00:21:23,078
one of the most impressive
places that shows up

267
00:21:23,078 --> 00:21:25,650
is that when you measure the
cosmic microwave background,

268
00:21:25,650 --> 00:21:28,380
it has what we call
a dipole isotropy.

269
00:21:28,380 --> 00:21:32,730
And that dipole is just
essentially a Doppler shift

270
00:21:32,730 --> 00:21:36,300
that is due to the fact that
when we make our measurements,

271
00:21:36,300 --> 00:21:41,980
we are moving with respect to
the co-moving reference frame.

272
00:21:41,980 --> 00:21:42,670
All right.

273
00:21:42,670 --> 00:21:45,140
So that's the metric that
we're going to use here.

274
00:21:45,140 --> 00:21:47,410
Setting gtt and
gti like so means

275
00:21:47,410 --> 00:21:50,860
I have chosen these
co-moving coordinate systems.

276
00:21:50,860 --> 00:21:58,820
I'm going to take
gamma ij, I'm going

277
00:21:58,820 --> 00:22:12,410
to take this to be
maximally symmetric, OK?

278
00:22:12,410 --> 00:22:16,950
So this is my statement
that at any moment of time,

279
00:22:16,950 --> 00:22:20,720
space is maximally symmetric.

280
00:22:20,720 --> 00:22:23,540
So a few words on the
units, a few things

281
00:22:23,540 --> 00:22:24,810
that I'm going to set up here.

282
00:22:24,810 --> 00:22:34,940
So my coordinate, I'm going to
take my xi to be dimensionless,

283
00:22:34,940 --> 00:22:36,800
and all notions of length--

284
00:22:36,800 --> 00:22:45,950
all length scales
and the problem

285
00:22:45,950 --> 00:22:53,860
are going to be absorbed
into this factor R of t.

286
00:22:56,570 --> 00:23:00,200
We're going to see that the
overall scale of the universe

287
00:23:00,200 --> 00:23:02,330
is going to depend on the
dynamics of that function

288
00:23:02,330 --> 00:23:05,310
R of t.

289
00:23:05,310 --> 00:23:08,660
So let's imagine
that on a given--

290
00:23:08,660 --> 00:23:10,790
at some given
moment of time, you

291
00:23:10,790 --> 00:23:14,030
want to understand the
curvature associated

292
00:23:14,030 --> 00:23:16,070
with that constant time slice.

293
00:23:20,600 --> 00:23:34,530
So the Riemann tensor that we
build from our spatial metric,

294
00:23:34,530 --> 00:23:38,930
I'm going to write this as 3 R--

295
00:23:38,930 --> 00:23:40,600
and I'm doing purely
spatial things,

296
00:23:40,600 --> 00:23:45,120
so I'm going to use Latin
letters for my indices.

297
00:23:45,120 --> 00:23:47,360
It's going to equal
to some number k--

298
00:23:50,860 --> 00:23:52,670
not to be confused
with the index k.

299
00:23:52,670 --> 00:23:55,477
It's unfortunate,
but there's only so

300
00:23:55,477 --> 00:23:56,560
many letters to work with.

301
00:24:01,510 --> 00:24:02,500
That looks like so.

302
00:24:02,500 --> 00:24:19,650
And if I take a trace to make
my Ricci curvature, I get this,

303
00:24:19,650 --> 00:24:24,510
and your Ricci scalar will
turn out to be equal to 6k.

304
00:24:24,510 --> 00:24:26,670
We won't actually need
that, but just so you

305
00:24:26,670 --> 00:24:28,440
can establish what
that k actually means.

306
00:24:28,440 --> 00:24:30,385
It's simply related to
the Ricci curvature.

307
00:24:30,385 --> 00:24:33,170
Oops, that should
have a 3 on it.

308
00:24:33,170 --> 00:24:37,110
Ricci curvature of that
particular instant in time.

309
00:24:41,470 --> 00:24:48,980
Now I'm going to require
my coordinate system

310
00:24:48,980 --> 00:24:53,790
to reflect the fact
that space is isotopic.

311
00:24:53,790 --> 00:24:56,150
So if it's isotropic,
it must look the same

312
00:24:56,150 --> 00:24:59,210
in all directions, and in
a three-dimensional space,

313
00:24:59,210 --> 00:25:01,160
anything that is the
same in all directions

314
00:25:01,160 --> 00:25:03,089
must be spherically symmetric.

315
00:25:19,226 --> 00:25:31,965
And so what this means is that
when I compute gamma ij dx i dx

316
00:25:31,965 --> 00:25:41,050
j, it must be equal to
some function of radius.

317
00:25:41,050 --> 00:25:43,540
The bar on that radius
just reminds you this

318
00:25:43,540 --> 00:25:45,580
is meant to be a dimensionless
notion of radius.

319
00:25:45,580 --> 00:25:46,750
Remember, all length
scales are going

320
00:25:46,750 --> 00:25:48,417
to be absorbed into
the function capital

321
00:25:48,417 --> 00:25:56,226
R. It's R squared d omega.

322
00:26:06,450 --> 00:26:08,580
And my angular sector
is just related

323
00:26:08,580 --> 00:26:14,660
to circle coordinate
angles the usual way.

324
00:26:14,660 --> 00:26:28,600
So it's convenient for
us to put this f of R--

325
00:26:28,600 --> 00:26:35,423
I can rewrite this as an
exponential function, OK?

326
00:26:35,423 --> 00:26:37,590
Just think of this as the
definition of the function

327
00:26:37,590 --> 00:26:39,350
beta.

328
00:26:39,350 --> 00:26:42,920
The reason why this is
handy is that suppose

329
00:26:42,920 --> 00:26:48,650
I now take gamma ij, I
compute the three-dimensional

330
00:26:48,650 --> 00:26:52,160
Christoffel symbols, I compute
my three-dimensional Riemann,

331
00:26:52,160 --> 00:26:54,770
ignoring for a moment
that my Riemann is meant

332
00:26:54,770 --> 00:26:56,480
to be maximally symmetric, OK?

333
00:26:56,480 --> 00:26:58,700
I'm just going to
say, I know the recipe

334
00:26:58,700 --> 00:27:01,340
for how to make
Riemann from a metric.

335
00:27:01,340 --> 00:27:02,407
I will do that.

336
00:27:02,407 --> 00:27:04,115
I will then make Ricci
from that Riemann.

337
00:27:14,540 --> 00:27:20,725
When you do this,
what you find is

338
00:27:20,725 --> 00:27:23,100
that-- let's just look at the
Rr component of this thing.

339
00:27:28,400 --> 00:27:31,220
This turns out to be 2 over
R times a regular derivative

340
00:27:31,220 --> 00:27:32,420
of beta, OK?

341
00:27:32,420 --> 00:27:34,910
It's just a little bit
of tensor manipulation

342
00:27:34,910 --> 00:27:39,400
to do that using some of the
tools, the mathematical tools

343
00:27:39,400 --> 00:27:41,660
I'm going to post in
the 8.962 website,

344
00:27:41,660 --> 00:27:43,210
you can verify this yourself.

345
00:27:48,070 --> 00:28:00,730
If I compute Ricci from the
maximally symmetric assumption,

346
00:28:00,730 --> 00:28:13,180
what I find is that this is
equal to 2k times gamma, which

347
00:28:13,180 --> 00:28:15,460
is itself exponent of 2 beta.

348
00:28:23,460 --> 00:28:25,440
Let's equate these
and solve for beta.

349
00:28:49,160 --> 00:28:58,162
So one side I've got 2
over r bar dr bar of eta.

350
00:28:58,162 --> 00:29:03,200
On the near side I
have 2k and gamma r bar

351
00:29:03,200 --> 00:29:09,320
r bar is itself e to the 2 beta.

352
00:29:09,320 --> 00:29:11,075
Cancel, cancel.

353
00:29:11,075 --> 00:29:13,670
A little bit of algebra.

354
00:29:13,670 --> 00:29:15,240
First let's write it this way.

355
00:29:15,240 --> 00:29:17,570
Let's move this to
the other side...

356
00:29:17,570 --> 00:29:26,240
e to the minus 2 beta
is equal to this.

357
00:29:26,240 --> 00:29:30,590
Let us make the assumption-- so
we have a choice of a boundary

358
00:29:30,590 --> 00:29:31,970
condition.

359
00:29:31,970 --> 00:29:37,250
Let's put beta equals
0 at r bar equals 0.

360
00:29:37,250 --> 00:29:40,190
This is basically
saying that on my--

361
00:29:40,190 --> 00:29:41,780
so r bar is sort of the origin.

362
00:29:41,780 --> 00:29:43,322
We're just sort of
saying that things

363
00:29:43,322 --> 00:29:47,720
look like a flat spacetime
in the vicinity of the origin

364
00:29:47,720 --> 00:29:49,220
of the coordinates
we're using here,

365
00:29:49,220 --> 00:29:51,470
that's a fine
assumption to make.

366
00:29:51,470 --> 00:30:02,250
Doing so, we can easily
integrate this guy up,

367
00:30:02,250 --> 00:30:04,500
and here's what we get.

368
00:30:04,500 --> 00:30:08,540
So with this, we now
have a full line element.

369
00:30:32,690 --> 00:30:36,380
So there's a few unknown
quantities in here.

370
00:30:36,380 --> 00:30:38,240
What is k?

371
00:30:38,240 --> 00:30:41,180
What is R?

372
00:30:41,180 --> 00:30:46,640
So far I have only talked about
the geometry of this spacetime.

373
00:30:46,640 --> 00:30:50,000
We haven't yet connected
this-- any of the dynamics

374
00:30:50,000 --> 00:30:53,120
of the spacetime to a source.

375
00:30:53,120 --> 00:30:54,930
It is when we hook
this up to a source

376
00:30:54,930 --> 00:30:57,097
that we're going to learn
something about these two.

377
00:30:57,097 --> 00:30:58,400
So hold that thought for now.

378
00:30:58,400 --> 00:31:00,380
This essentially has
just said that here

379
00:31:00,380 --> 00:31:05,060
is what my maximally spatially
symmetric spacetime looks like,

380
00:31:05,060 --> 00:31:07,730
allowing for there to be a
difference between the past

381
00:31:07,730 --> 00:31:09,770
and the present.

382
00:31:09,770 --> 00:31:17,640
Before I move on, so I can't
tell you what k is yet,

383
00:31:17,640 --> 00:31:19,920
but I can make the
following observation which

384
00:31:19,920 --> 00:31:24,963
allows me to restrict
what values of k

385
00:31:24,963 --> 00:31:25,880
I need to worry about.

386
00:31:44,220 --> 00:31:58,870
Suppose I take k and I replace
it with k prime equal alpha k.

387
00:32:01,960 --> 00:32:07,870
But in doing so,
I define R tilde

388
00:32:07,870 --> 00:32:10,060
to be square root of alpha--

389
00:32:15,230 --> 00:32:15,730
yeah.

390
00:32:15,730 --> 00:32:21,030
Square root of
alpha times R bar.

391
00:32:21,030 --> 00:32:36,670
And I also require that
my overall scale factor

392
00:32:36,670 --> 00:32:38,650
look like the original
scale factor divided

393
00:32:38,650 --> 00:32:41,170
by square root of alpha.

394
00:32:41,170 --> 00:32:45,850
Rewriting my spacetime, my line
element in terms of k prime

395
00:32:45,850 --> 00:33:20,090
and the tilde R into-- the
two tilde R's, I get this.

396
00:33:20,090 --> 00:33:24,200
Basically, that transformation
leaves the line element

397
00:33:24,200 --> 00:33:28,390
completely irrelevant to me.

398
00:33:28,390 --> 00:33:30,030
That was completely
the wrong word.

399
00:33:30,030 --> 00:33:35,610
That re-prioritization of k
and R and the two different R's

400
00:33:35,610 --> 00:33:39,060
here, that leaves the line
element completely invariant.

401
00:33:39,060 --> 00:33:40,920
It is unchanged when I do this.

402
00:33:46,210 --> 00:33:48,413
So what this tells
me is-- by the way,

403
00:33:48,413 --> 00:33:50,830
alpha has to be a positive
number so that the square roots

404
00:33:50,830 --> 00:33:51,910
make sense there.

405
00:33:51,910 --> 00:33:58,060
It tells me that the
normalization associated with k

406
00:33:58,060 --> 00:34:05,460
can be absorbed into
my scale factor.

407
00:34:10,830 --> 00:34:15,870
And so what it suggests
we ought to do is just--

408
00:34:15,870 --> 00:34:18,210
you don't need to
worry about whether k

409
00:34:18,210 --> 00:34:23,250
is equal to 15 or pi or
negative the 38th root of e

410
00:34:23,250 --> 00:34:24,540
or anything silly like that.

411
00:34:29,620 --> 00:34:34,659
The only three values
of k that matter for us

412
00:34:34,659 --> 00:34:39,190
are whether it is
negative 1, 0, or 1.

413
00:34:39,190 --> 00:34:42,750
This stands for all
negative values of k,

414
00:34:42,750 --> 00:34:49,320
0 as a set onto itself, and
all positive values of k, OK?

415
00:34:49,320 --> 00:34:53,328
So we will use this to
say, great, the thing which

416
00:34:53,328 --> 00:34:54,870
I'm going to care
about, once I start

417
00:34:54,870 --> 00:34:57,360
looking at the physics
associated with this,

418
00:34:57,360 --> 00:34:59,590
is whether--

419
00:34:59,590 --> 00:35:01,800
let's go back over to
this version of it--

420
00:35:01,800 --> 00:35:06,630
I'm going to care about
whether k is negative 1, 0, 1,

421
00:35:06,630 --> 00:35:10,170
and I want to understand
how my scale factor behaves.

422
00:35:19,320 --> 00:35:21,900
So before I start hooking
this up to my source

423
00:35:21,900 --> 00:35:25,650
and doing a little bit
of physics, many of you

424
00:35:25,650 --> 00:35:27,660
are going to do something
involving cosmology

425
00:35:27,660 --> 00:35:30,480
at some point in your
lives, and so it's

426
00:35:30,480 --> 00:35:34,950
useful to introduce a few
other bits of notation that

427
00:35:34,950 --> 00:35:42,928
are commonly used here,
as well as to describe

428
00:35:42,928 --> 00:35:45,220
some important terminology
that comes up at this point.

429
00:36:00,240 --> 00:36:03,650
So here's some common
notation and terminology.

430
00:36:03,650 --> 00:36:21,770
Let us define a
radial coordinate chi

431
00:36:21,770 --> 00:36:25,910
via the following definition--

432
00:36:25,910 --> 00:36:32,330
d chi will be equal to d R
bar over square root of 1

433
00:36:32,330 --> 00:36:37,350
minus kR bar squared.

434
00:36:37,350 --> 00:36:39,500
Now remember, we just
decided that k can only

435
00:36:39,500 --> 00:36:41,510
take on one of three
interesting values.

436
00:36:44,210 --> 00:36:46,130
I can immediately
integrate this up,

437
00:36:46,130 --> 00:36:50,840
and I will find that my
R bar is equal to sine

438
00:36:50,840 --> 00:37:00,250
of chi of k equals plus 1 is
equal to chi of k equal 0.

439
00:37:03,560 --> 00:37:08,160
And it's the sinh of
chi of k equals minus 1.

440
00:37:15,115 --> 00:37:17,800
So let's take a
look at what this

441
00:37:17,800 --> 00:37:23,005
means with these sort of
three possible choices.

442
00:37:23,005 --> 00:37:26,320
The three possible
values that k can take.

443
00:37:26,320 --> 00:37:27,820
What is our line
element looks like?

444
00:37:32,000 --> 00:37:34,050
So let's look at k
equals plus 1 first.

445
00:37:38,920 --> 00:37:47,360
I get minus dt squared R square
root of t d chi squared plus--

446
00:37:47,360 --> 00:37:57,650
I'm going to use
the fact that R bar,

447
00:37:57,650 --> 00:38:04,160
this describes a spacetime in
which every spacial slice is

448
00:38:04,160 --> 00:38:06,020
what is called a 3-sphere, OK?

449
00:38:06,020 --> 00:38:09,020
You're all nicely familiar
with the 2-sphere.

450
00:38:09,020 --> 00:38:12,110
So a 2-sphere is the
three-dimensional surface

451
00:38:12,110 --> 00:38:16,090
in which you pick a point
and every point that is,

452
00:38:16,090 --> 00:38:19,340
let's say, a unit radius
away from that point that,

453
00:38:19,340 --> 00:38:23,550
defines a 2-sphere in
three-dimensional space.

454
00:38:23,550 --> 00:38:26,560
So this defines the space--

455
00:38:26,560 --> 00:38:28,590
the spatial
characteristics of pick

456
00:38:28,590 --> 00:38:31,670
a point in
four-dimensional space

457
00:38:31,670 --> 00:38:35,360
and ask for all of the points
that are a unit distance away

458
00:38:35,360 --> 00:38:37,670
from it in three dimensions,
that is a 3-sphere.

459
00:38:53,430 --> 00:38:56,910
Notice that my 3-sphere
has a maximum--

460
00:38:56,910 --> 00:39:00,330
there's a maximum distance
associated with it, OK?

461
00:39:00,330 --> 00:39:02,970
So there's no bounds on chi, OK?

462
00:39:02,970 --> 00:39:04,950
Chi can go from 0 to infinity.

463
00:39:04,950 --> 00:39:06,840
But this one's
periodic, isn't it?

464
00:39:06,840 --> 00:39:12,690
So as chi reaches pi over 2,
the separation between any two

465
00:39:12,690 --> 00:39:14,310
points on that
single slice, they've

466
00:39:14,310 --> 00:39:17,550
reached their maximum value, and
as chi continues to increase,

467
00:39:17,550 --> 00:39:19,680
the distance gets
smaller again, OK?

468
00:39:19,680 --> 00:39:22,230
And eventually, when
chi gets up to pi,

469
00:39:22,230 --> 00:39:24,120
you come back to
where you started.

470
00:39:24,120 --> 00:39:28,733
We call this a closed universe.

471
00:39:28,733 --> 00:39:30,150
This is something
where if it were

472
00:39:30,150 --> 00:39:33,300
possible to step out of
time and just run around

473
00:39:33,300 --> 00:39:38,050
on a spacial slice, you would
find that it is a finite size.

474
00:39:38,050 --> 00:39:40,650
The best you could
do is run around

475
00:39:40,650 --> 00:39:44,660
on that three-dimensional sphere
in four-dimensional space.

476
00:39:57,530 --> 00:39:59,290
Let's do k equals 0 next.

477
00:40:17,850 --> 00:40:34,580
If I do k equals 0,
there's my line element.

478
00:40:34,580 --> 00:40:38,170
Each spacial slice is
simply Euclidean space.

479
00:40:47,790 --> 00:40:52,660
So this is often
described as flat space.

480
00:40:58,310 --> 00:41:01,670
A significant word of caution.

481
00:41:01,670 --> 00:41:04,730
When you talk to a
cosmologist, they will often

482
00:41:04,730 --> 00:41:06,778
talk about how the best--

483
00:41:06,778 --> 00:41:09,320
we're going to talk about sort
of the observational situation

484
00:41:09,320 --> 00:41:12,410
in the next lecture that
I record a little bit.

485
00:41:12,410 --> 00:41:15,460
Our evidence actually
suggests that this

486
00:41:15,460 --> 00:41:17,210
is what our universe
looks like right now.

487
00:41:17,210 --> 00:41:21,530
We're in a k equal 0
universe in which space--

488
00:41:21,530 --> 00:41:24,120
each spatial slice is flat.

489
00:41:24,120 --> 00:41:27,920
That does not mean
spacetime is flat, OK?

490
00:41:27,920 --> 00:41:29,540
So when they say
that it's flat, that

491
00:41:29,540 --> 00:41:31,280
is referring to
the geometry only

492
00:41:31,280 --> 00:41:34,020
of the spatial slices in this
co-moving coordinate system.

493
00:41:38,940 --> 00:41:59,580
k equals minus 1, you get a
form that looks like this.

494
00:41:59,580 --> 00:42:02,830
This describes the
geometry of a hyperbole.

495
00:42:17,540 --> 00:42:20,120
We call this an
open spatial slice.

496
00:42:20,120 --> 00:42:23,018
So notice for both
choices 2 and 3,

497
00:42:23,018 --> 00:42:24,560
if you could sort
of step out of time

498
00:42:24,560 --> 00:42:28,280
and explore the full geometry
of that spatial slice,

499
00:42:28,280 --> 00:42:29,480
it goes on forever, OK?

500
00:42:29,480 --> 00:42:31,950
Again, there's really
no boundary on that chi

501
00:42:31,950 --> 00:42:34,037
as near as we can tell,
and so that spatial slide

502
00:42:34,037 --> 00:42:35,120
can just kind of go, whee!

503
00:42:35,120 --> 00:42:37,190
And take off forever.

504
00:42:37,190 --> 00:42:41,330
This one sort of goes to large
distances a little slower

505
00:42:41,330 --> 00:42:42,170
than this one does.

506
00:42:42,170 --> 00:42:44,270
This hyperbolic
function means that this

507
00:42:44,270 --> 00:42:48,500
is really bloody large, OK?

508
00:42:48,500 --> 00:42:54,800
So both of these tend
to imply a universe that

509
00:42:54,800 --> 00:42:57,040
is sort of spatially unbounded.

510
00:42:57,040 --> 00:43:04,142
The closed universe, because
each slice is a 3-sphere,

511
00:43:04,142 --> 00:43:05,100
it's a different story.

512
00:43:09,640 --> 00:43:14,070
So another bit of notation
which you should be aware of--

513
00:43:17,040 --> 00:43:19,320
and I unfortunately
am going to want

514
00:43:19,320 --> 00:43:22,230
to sort of flip back and forth
between the notation I've

515
00:43:22,230 --> 00:43:25,157
been using so far and this
one I'm about to introduce.

516
00:43:28,887 --> 00:43:31,470
It can be a little bit annoying
when you're first learning it,

517
00:43:31,470 --> 00:43:34,710
but just keep track of
context, it's not that hard.

518
00:43:34,710 --> 00:43:38,520
So what we're going
to do is let's

519
00:43:38,520 --> 00:43:41,440
choose a particular value
of the scale factor,

520
00:43:41,440 --> 00:43:43,500
and we will normalize
things to that.

521
00:43:58,310 --> 00:44:04,540
So what I'm going to do is
define some particular value

522
00:44:04,540 --> 00:44:10,570
of k such that the scale factor
there I will call it R sub 0.

523
00:44:10,570 --> 00:44:17,800
And as we'll see, a particularly
useful choice for this

524
00:44:17,800 --> 00:44:20,323
is to choose the
value right now, OK?

525
00:44:20,323 --> 00:44:22,740
What we're doing, then, is
we're kind of norm-- what we're

526
00:44:22,740 --> 00:44:25,073
going to see in a moment is
this means we're normalizing

527
00:44:25,073 --> 00:44:27,880
all the scales associated
with our universe

528
00:44:27,880 --> 00:44:30,180
to where they are right now.

529
00:44:30,180 --> 00:44:30,680
OK.

530
00:44:30,680 --> 00:44:37,240
Having done this, I'm
going to define a of t

531
00:44:37,240 --> 00:44:43,923
to be R of t divided
by this special value.

532
00:44:43,923 --> 00:44:45,340
For dimensional
reasons, I'm going

533
00:44:45,340 --> 00:44:49,750
to need to put this into
my radial coordinate.

534
00:44:49,750 --> 00:44:51,180
So notice, what's
going on here is

535
00:44:51,180 --> 00:44:54,300
that my R will now have
dimensions associated with it,

536
00:44:54,300 --> 00:45:00,655
and so essentially everything
is just being scaled by that R0.

537
00:45:00,655 --> 00:45:03,030
And this is the bit where it
gets a tiny bit unfortunate,

538
00:45:03,030 --> 00:45:09,995
you sort of lose the beauty
of k only having three values.

539
00:45:09,995 --> 00:45:11,745
So I'm going to replace
that with a kappa.

540
00:45:14,580 --> 00:45:16,920
This is unfortunately a
little bit hard to read,

541
00:45:16,920 --> 00:45:19,310
so whenever I make it sort
of with my messy cursive,

542
00:45:19,310 --> 00:45:21,870
it will be k; whenever it
looks a little bit more

543
00:45:21,870 --> 00:45:23,600
like a printed thing,
it will be kappa.

544
00:45:27,970 --> 00:45:32,730
And so kappa is k
divided by R0 squared.

545
00:45:32,730 --> 00:45:53,150
And when you do that, your
line element becomes this.

546
00:46:00,800 --> 00:46:01,490
OK?

547
00:46:01,490 --> 00:46:03,750
So that's a form that we're
going to use a little bit.

548
00:46:03,750 --> 00:46:08,177
What's a little bit annoying
about it is just that my--

549
00:46:08,177 --> 00:46:10,010
the kappa that appears
in there doesn't just

550
00:46:10,010 --> 00:46:12,190
come as a set of one of
these parts of three,

551
00:46:12,190 --> 00:46:15,660
but basically if kappa
is a negative number,

552
00:46:15,660 --> 00:46:19,010
then you know k must be minus 1;
kappa equals 0 corresponds to k

553
00:46:19,010 --> 00:46:22,360
equals 0; if kappa is a positive
number, then k equals plus 1.

554
00:46:22,360 --> 00:46:24,860
This form where we're using
this sort of dimensionless scale

555
00:46:24,860 --> 00:46:28,970
factor a is particularly useful.

556
00:46:28,970 --> 00:46:30,680
If you look at this,
this is telling you

557
00:46:30,680 --> 00:46:43,150
that with the choice that R0
defines a scale factor now,

558
00:46:43,150 --> 00:46:47,860
this means a now equals 1.

559
00:46:47,860 --> 00:46:50,470
And so this gives us a
nice dimensionless factor

560
00:46:50,470 --> 00:46:54,760
by which we can compare
all of our spatial scales

561
00:46:54,760 --> 00:46:56,800
at different moments in
the universe to the size

562
00:46:56,800 --> 00:46:57,550
that they are now.

563
00:47:00,800 --> 00:47:01,950
OK.

564
00:47:01,950 --> 00:47:04,890
Everything I have
said so far has really

565
00:47:04,890 --> 00:47:06,990
been just discussing
the geometry

566
00:47:06,990 --> 00:47:08,970
that I'm going to
use to describe

567
00:47:08,970 --> 00:47:13,060
the large-scale
structure of spacetime.

568
00:47:13,060 --> 00:47:16,150
I haven't said
anything about what

569
00:47:16,150 --> 00:47:19,570
happens when I solve the
Einstein field equations

570
00:47:19,570 --> 00:47:22,210
and connect this
geometry to physics.

571
00:47:25,730 --> 00:47:27,680
So what we need to do
is choose a source.

572
00:47:27,680 --> 00:47:29,560
And so what we're
going to do is we

573
00:47:29,560 --> 00:47:33,010
will do what is sort of the
default choice in many analyses

574
00:47:33,010 --> 00:47:34,190
in general relativity.

575
00:47:34,190 --> 00:47:38,170
We will choose our source
to be a perfect fluid.

576
00:47:48,330 --> 00:47:51,180
What's nice about this
is that it automatically

577
00:47:51,180 --> 00:47:57,390
satisfies the requirements
of isotropy and homogeneity.

578
00:47:57,390 --> 00:48:03,930
At least it does so if
the fluid is at rest

579
00:48:03,930 --> 00:48:05,627
in co-moving coordinates.

580
00:48:30,000 --> 00:48:34,560
So let's fill this in:
t mu nu with everything

581
00:48:34,560 --> 00:48:42,850
in the downstairs position looks
like rho plus P mu nu mu nu

582
00:48:42,850 --> 00:48:46,000
plus Pg mu nu.

583
00:48:49,122 --> 00:48:53,430
And this becomes in my
co-moving coordinate system.

584
00:48:58,510 --> 00:49:01,820
So then it looks like this, OK?

585
00:49:01,820 --> 00:49:06,080
A handy fact to
have, this is going

586
00:49:06,080 --> 00:49:09,418
to be quite useful for
a calculation or two

587
00:49:09,418 --> 00:49:10,710
that we do a little bit later--

588
00:49:13,670 --> 00:49:16,180
actually, not just a minute
later, almost right away.

589
00:49:37,610 --> 00:49:43,175
This looks like a diagonal
of all this stuff, OK?

590
00:49:46,640 --> 00:49:47,490
All right.

591
00:49:47,490 --> 00:49:53,110
So what we want to do is use
this stress energy tensor

592
00:49:53,110 --> 00:49:54,260
as the right-hand side.

593
00:49:54,260 --> 00:49:57,670
So we've worked out
our Ricci tensor.

594
00:49:57,670 --> 00:50:01,180
With a little bit of work, we
can make the Einstein tensor,

595
00:50:01,180 --> 00:50:03,490
couple it to this guy, we
can set up our differential

596
00:50:03,490 --> 00:50:07,090
equations, and we can solve
for the free functions that

597
00:50:07,090 --> 00:50:09,190
specify the spacetime.

598
00:50:09,190 --> 00:50:13,420
Before doing this, always a good
sanity check, remind yourself,

599
00:50:13,420 --> 00:50:16,718
your fluid has to satisfy
local energy conservation.

600
00:50:21,020 --> 00:50:22,360
Actually, let's just do the 0.

601
00:50:22,360 --> 00:50:25,120
So this is energy and
momentum conservation,

602
00:50:25,120 --> 00:50:30,339
we set that equal to 0, this
is local energy conservation.

603
00:50:39,930 --> 00:50:51,210
Expanding out these
derivatives, what you

604
00:50:51,210 --> 00:50:53,190
find is that this turns into--

605
00:51:09,050 --> 00:51:12,280
so it looks like this.

606
00:51:12,280 --> 00:51:16,960
And plugging in-- so
using the spacetime--

607
00:51:21,040 --> 00:51:24,660
by the way, I made a
small mistake earlier.

608
00:51:24,660 --> 00:51:34,870
I should have told you
that this spacetime,

609
00:51:34,870 --> 00:51:40,080
this is now called the
Robertson-Walker spacetime.

610
00:51:50,522 --> 00:51:52,730
So this was actually first
written down in the 1920s,

611
00:51:52,730 --> 00:51:54,620
and Robertson and Walker
developed this basically

612
00:51:54,620 --> 00:51:56,328
just as I have done
it here, just arguing

613
00:51:56,328 --> 00:51:58,160
on the basis of looking
for something that

614
00:51:58,160 --> 00:52:03,800
is as symmetric as possible with
respect to space if not time,

615
00:52:03,800 --> 00:52:05,702
and they came out with
that line element.

616
00:52:05,702 --> 00:52:07,660
My apologies, I didn't
mention that beforehand.

617
00:52:07,660 --> 00:52:09,077
This is my third
lecture in a row,

618
00:52:09,077 --> 00:52:10,370
I'm getting a little bit tired.

619
00:52:10,370 --> 00:52:13,130
So if I take that
Robertson-Walker metric,

620
00:52:13,130 --> 00:52:17,610
plug it into here to
evaluate all these,

621
00:52:17,610 --> 00:52:19,860
this gives me a
remarkably simple form.

622
00:52:40,070 --> 00:52:45,195
So rho is the pressure
of my perfect fluid--

623
00:52:45,195 --> 00:52:47,070
excuse me, the density
of my perfect fluid, P

624
00:52:47,070 --> 00:52:50,870
is the pressure of my perfect
fluid, a is my scale factor.

625
00:52:57,170 --> 00:53:01,030
If you like, you can put the
factor of R0 back in there,

626
00:53:01,030 --> 00:53:07,700
and an equivalent way of
writing this, which I think

627
00:53:07,700 --> 00:53:10,910
is very useful for giving some
physical insight as to what

628
00:53:10,910 --> 00:53:11,570
this means--

629
00:53:30,266 --> 00:53:31,980
so put that R back in.

630
00:53:48,708 --> 00:53:51,360
OK, so let's look at
what this is saying.

631
00:53:51,360 --> 00:53:54,990
R cubed is modulo
numerical factor,

632
00:53:54,990 --> 00:53:58,140
that is the volume
of a spacial slice.

633
00:53:58,140 --> 00:54:01,380
And so this is saying,
the rate of change

634
00:54:01,380 --> 00:54:02,670
of energy in a volume--

635
00:54:11,830 --> 00:54:14,740
so a volume describing
my spatial slice

636
00:54:14,740 --> 00:54:20,260
is equal to negative pressure
times the rate of change

637
00:54:20,260 --> 00:54:21,340
of that volume.

638
00:54:33,330 --> 00:54:36,790
I hope this looks familiar.

639
00:54:36,790 --> 00:54:41,080
This, in somewhat more
convoluted notation,

640
00:54:41,080 --> 00:54:43,330
is negative dp--

641
00:54:43,330 --> 00:54:47,450
du equals negative P dv.

642
00:54:47,450 --> 00:54:49,686
It's just the first
law of thermodynamics.

643
00:54:52,910 --> 00:54:53,410
All right.

644
00:54:53,410 --> 00:54:56,980
So this relationship,
whether written in this form

645
00:54:56,980 --> 00:54:58,810
or in that form, is
something that we

646
00:54:58,810 --> 00:55:00,310
will exploit moving forward.

647
00:55:05,360 --> 00:55:10,047
Let's now solve the
Einstein field equations.

648
00:55:25,970 --> 00:55:34,640
So we'll begin with g mu
nu equals 8 pi g t mu nu.

649
00:55:34,640 --> 00:55:37,040
The equations that
are traditionally

650
00:55:37,040 --> 00:55:42,300
used to describe cosmology are
a little bit more naturally

651
00:55:42,300 --> 00:55:42,800
written.

652
00:55:42,800 --> 00:55:47,330
If I change this into the form
that uses the Ricci tensor--

653
00:55:47,330 --> 00:55:54,240
so let me rewrite this as R
mu nu equals 8 pi g t mu nu.

654
00:55:57,720 --> 00:55:58,220
OK?

655
00:55:58,220 --> 00:56:03,850
So this is equivalent where
t is just the usual trace

656
00:56:03,850 --> 00:56:06,890
of the stress energy tensor.

657
00:56:06,890 --> 00:56:10,330
And what you find,
there are two--

658
00:56:10,330 --> 00:56:14,890
if you just look at the 0, 0
components of this equation,

659
00:56:14,890 --> 00:56:30,890
it tells you the acceleration of
the scale factor a divided by a

660
00:56:30,890 --> 00:56:31,490
is--

661
00:56:31,490 --> 00:56:38,930
it is simply related to
the density and 3 times

662
00:56:38,930 --> 00:56:41,100
the pressure.

663
00:56:41,100 --> 00:56:44,850
If you evaluate Rii--

664
00:56:44,850 --> 00:56:45,907
in other words, any--

665
00:56:45,907 --> 00:56:47,490
this is-- there's
no sum implied here,

666
00:56:47,490 --> 00:56:54,070
just take any spatial
component of this guy,

667
00:56:54,070 --> 00:56:56,570
and add on R0,0 because
it's a valid equation,

668
00:56:56,570 --> 00:57:11,820
it helps you to
clear out some stuff,

669
00:57:11,820 --> 00:57:15,570
you get the following
relationship between the rate

670
00:57:15,570 --> 00:57:20,100
of change to the scale factor,
the density, and remember,

671
00:57:20,100 --> 00:57:24,010
kappa is your rescaled k.

672
00:57:24,010 --> 00:57:28,540
So I'm going to call
this equation F1,

673
00:57:28,540 --> 00:57:32,020
I'm going to call this one F2.

674
00:57:32,020 --> 00:57:34,680
These are known as the
Friedmann equations.

675
00:57:41,180 --> 00:57:47,250
When one uses them to solve
to describe your line element,

676
00:57:47,250 --> 00:57:59,480
you get
Friedmann-Robertson-Walker

677
00:57:59,480 --> 00:57:59,980
metrics.

678
00:58:17,408 --> 00:58:18,950
So just a little
bit of nomenclature.

679
00:58:18,950 --> 00:58:21,860
Robertson-Walker tells
you about the geometry,

680
00:58:21,860 --> 00:58:25,640
you then equate these
guys to a source,

681
00:58:25,640 --> 00:58:29,640
and that gives you
Friedmann-Robertson-Walker line

682
00:58:29,640 --> 00:58:30,140
elements.

683
00:58:37,340 --> 00:58:39,678
One other bit of
information-- so

684
00:58:39,678 --> 00:58:41,720
let's introduce a little
bit of terminology here.

685
00:58:54,300 --> 00:59:02,580
So a dot over a, this tells
me how the overall length

686
00:59:02,580 --> 00:59:05,370
scale associated with
my spatial slices

687
00:59:05,370 --> 00:59:07,020
is evolving as a
function of time.

688
00:59:10,480 --> 00:59:16,542
This is denoted H and it's
known as the Hubble parameter.

689
00:59:24,740 --> 00:59:30,950
H0 is the value of
H that we measure

690
00:59:30,950 --> 00:59:35,180
in our universe corresponding
to its expansion right now, OK?

691
00:59:35,180 --> 00:59:39,050
And the notes that I have
scanned and placed online

692
00:59:39,050 --> 00:59:42,020
claim a best value
for this of 73

693
00:59:42,020 --> 00:59:45,650
plus or minus 3 kilometers
per second per megaparsec.

694
00:59:45,650 --> 00:59:47,360
These notes were
originally hand written

695
00:59:47,360 --> 00:59:50,390
about 11 or 12 years ago, that
number is already out of date,

696
00:59:50,390 --> 00:59:51,750
OK?

697
00:59:51,750 --> 00:59:55,790
If you went to Adam
Reese's colloquium

698
00:59:55,790 --> 00:59:59,223
shortly before MIT went
into its COVID shutdown,

699
00:59:59,223 --> 01:00:00,890
you will have seen
that there's actually

700
01:00:00,890 --> 01:00:02,848
little bit of controversy
about this right now.

701
01:00:02,848 --> 01:00:04,760
So our best measurements
of this thing,

702
01:00:04,760 --> 01:00:08,578
indeed, they are clustering
around 72 or 73 in these units,

703
01:00:08,578 --> 01:00:11,120
but they're inconsistent with
some other measures by which we

704
01:00:11,120 --> 01:00:15,200
can infer to be the-- what the
Hubble parameter should be.

705
01:00:15,200 --> 01:00:16,310
And it's a very--

706
01:00:18,767 --> 01:00:19,850
very interesting problems.

707
01:00:19,850 --> 01:00:21,530
Unclear sort of--
it sort of smells

708
01:00:21,530 --> 01:00:22,970
like something might
be a little bit off

709
01:00:22,970 --> 01:00:25,387
in our cosmological models,
but we're not quite there yet.

710
01:00:25,387 --> 01:00:27,470
Let's consider-- let's
proceed with sort

711
01:00:27,470 --> 01:00:29,780
of the standard picture,
and just bear in mind

712
01:00:29,780 --> 01:00:32,300
that this is an evolving field.

713
01:00:32,300 --> 01:00:41,600
The one thing I will note
is that H has the dimensions

714
01:00:41,600 --> 01:00:43,910
of inverse time, OK?

715
01:00:43,910 --> 01:00:45,680
The way one actually
measures it.

716
01:00:45,680 --> 01:00:48,770
So the dimensions in which most
astronomers quote its value,

717
01:00:48,770 --> 01:00:51,590
it looks like a velocity over
a length, which is, of course,

718
01:00:51,590 --> 01:00:53,360
also an inverse time.

719
01:00:53,360 --> 01:00:57,830
And that is because objects
that are at rest with respect

720
01:00:57,830 --> 01:01:00,860
to the-- that are at rest in
these co-moving coordinates,

721
01:01:00,860 --> 01:01:06,050
as this fluid is meant to be,
if the universe is expanding,

722
01:01:06,050 --> 01:01:08,810
we see them moving away from us.

723
01:01:08,810 --> 01:01:10,790
OK, I'm going to make
a few definitions.

724
01:01:14,640 --> 01:01:29,390
Let us define rho crit to
be 3H squared over 8 pi j.

725
01:01:29,390 --> 01:01:38,520
So the way I got
that was take F1--

726
01:01:38,520 --> 01:01:42,900
imagine kappa is equal to 0,
just ignore kappa for a second.

727
01:01:42,900 --> 01:01:46,109
Left-hand side as H
squared, solve for rho, OK?

728
01:02:10,590 --> 01:02:15,098
Notice, rho quit-- rho crit is a
parameter that you can measure.

729
01:02:15,098 --> 01:02:16,640
You can measure the
Hubble parameter,

730
01:02:16,640 --> 01:02:19,015
I'll describe to you how that
is done in my next lecture,

731
01:02:19,015 --> 01:02:21,390
but it's a number
that can be measured.

732
01:02:21,390 --> 01:02:25,070
And then 3 and 8 pi
are just exact numbers,

733
01:02:25,070 --> 01:02:26,962
g is a fundamental constant.

734
01:02:26,962 --> 01:02:28,670
So that's something
that can be measured.

735
01:02:31,480 --> 01:02:40,750
Let's define omega to be any
density divided by rho crit.

736
01:02:40,750 --> 01:02:43,805
Putting all these together, I
can rewrite the first Friedmann

737
01:02:43,805 --> 01:02:44,305
equation.

738
01:02:49,190 --> 01:02:59,840
This guy can be written as omega
minus 1 equals kappa over H

739
01:02:59,840 --> 01:03:04,230
squared a squared.

740
01:03:04,230 --> 01:03:08,950
Now notice, H and a,
they are real numbers.

741
01:03:08,950 --> 01:03:14,060
H squared and a squared
are positive definite.

742
01:03:14,060 --> 01:03:19,580
We at last can now see how
the large-scale distribution

743
01:03:19,580 --> 01:03:21,920
of matter in our
universe allows us

744
01:03:21,920 --> 01:03:26,150
to constrain one of
the parameters that

745
01:03:26,150 --> 01:03:29,420
sets our Robertson-Walker
line element.

746
01:03:29,420 --> 01:03:35,150
If omega is less than 1--

747
01:03:35,150 --> 01:03:41,350
in other words, if rho
is less than rho crit,

748
01:03:41,350 --> 01:03:46,960
then it must be the case
that kappa is negative,

749
01:03:46,960 --> 01:03:51,243
k equals minus 1, and we
have an open universe.

750
01:03:56,920 --> 01:04:00,090
If a omega equals 1
such that rho is exactly

751
01:04:00,090 --> 01:04:07,350
equal to rho crit, kappa
must equal 0, k must equal 0,

752
01:04:07,350 --> 01:04:10,432
and we have a Euclidean
spatially flat universe.

753
01:04:17,040 --> 01:04:23,250
If omega is greater than
1, kappa is greater than 1,

754
01:04:23,250 --> 01:04:28,496
k equals 1, and we
have a closed universe.

755
01:04:36,473 --> 01:04:38,223
Clean up my handwriting
a little bit here.

756
01:04:52,027 --> 01:04:54,220
OK, this is really interesting.

757
01:04:54,220 --> 01:04:59,760
This is telling us
if we can determine

758
01:04:59,760 --> 01:05:04,710
whether the density of
stuff in our universe

759
01:05:04,710 --> 01:05:11,170
exceeds, is equal to, or is
less than that critical value,

760
01:05:11,170 --> 01:05:12,990
we know something
pretty profound

761
01:05:12,990 --> 01:05:16,500
about the spatial
geometry of our universe.

762
01:05:16,500 --> 01:05:20,620
Either it's finite,
sort of simply infinite,

763
01:05:20,620 --> 01:05:23,025
or ridiculously infinite.

764
01:05:25,382 --> 01:05:27,840
Let me do a few more things
before I conclude this lecture.

765
01:05:44,150 --> 01:05:47,000
First, this isn't that
important for our purposes,

766
01:05:47,000 --> 01:05:50,305
but it's something that some
of you students will see.

767
01:05:50,305 --> 01:05:51,805
A little bit of
notational trickery.

768
01:05:56,160 --> 01:05:58,980
It's not uncommon
in the literature

769
01:05:58,980 --> 01:06:01,440
to see people define what's
called a curvature density.

770
01:06:06,240 --> 01:06:11,180
And what this is,
is you just combine

771
01:06:11,180 --> 01:06:16,880
factors of kappa, g, and the
scale factor in such a way

772
01:06:16,880 --> 01:06:20,060
that this has the
dimensions of density.

773
01:06:20,060 --> 01:06:26,690
You can then find an omega
associated with curvature

774
01:06:26,690 --> 01:06:32,150
to be rho with curvature
over the critical density.

775
01:06:32,150 --> 01:06:37,650
And when you do this, F1,
the first Friedmann equation,

776
01:06:37,650 --> 01:06:45,540
becomes simply omega plus
omega curvature equals 1, OK?

777
01:06:45,540 --> 01:06:47,910
Just bear in mind, that
is not really a density,

778
01:06:47,910 --> 01:06:49,390
it's just a concept--

779
01:06:49,390 --> 01:06:52,110
it's a useful auxiliary concept.

780
01:06:52,110 --> 01:06:55,180
This is often for certain
kinds of calculations,

781
01:06:55,180 --> 01:06:58,750
a nice constraint
to bear in mind, OK?

782
01:06:58,750 --> 01:07:01,290
People are very interested
in understanding

783
01:07:01,290 --> 01:07:04,620
the geometry of our
universe, and this

784
01:07:04,620 --> 01:07:12,090
is a way of formulating it that
sort of puts the term involving

785
01:07:12,090 --> 01:07:15,600
the k parameter or the kappa
parameter on the same footing

786
01:07:15,600 --> 01:07:18,180
as other densities that
contribute to the energy

787
01:07:18,180 --> 01:07:19,555
budget of our universe.

788
01:07:28,110 --> 01:07:28,650
OK.

789
01:07:28,650 --> 01:07:32,910
So the equations that
we are working with here

790
01:07:32,910 --> 01:07:34,380
involve these--

791
01:07:37,790 --> 01:07:41,300
it involves the pressure
and density here.

792
01:07:41,300 --> 01:07:44,350
I haven't said too
much about them so far.

793
01:07:44,350 --> 01:07:48,070
If I want to make
further progress,

794
01:07:48,070 --> 01:07:50,210
I've got to know a little
bit about the matter that

795
01:07:50,210 --> 01:07:51,639
fills my universe.

796
01:08:02,630 --> 01:08:09,480
So to make more progress,
I need to choose

797
01:08:09,480 --> 01:08:12,355
what is called an
equation of state that

798
01:08:12,355 --> 01:08:14,480
relates the pressure and
the density to each other.

799
01:08:23,649 --> 01:08:27,819
What I really need is to
know that my pressure is

800
01:08:27,819 --> 01:08:30,550
some function of
the density, OK?

801
01:08:30,550 --> 01:08:33,279
This can be written down for
just about all kinds of matter

802
01:08:33,279 --> 01:08:35,170
that we care about.

803
01:08:35,170 --> 01:08:49,970
In cosmology, one
usually take and assumes

804
01:08:49,970 --> 01:08:55,029
that the pressure is a linear
function-- it's just linearly

805
01:08:55,029 --> 01:08:57,340
related to the energy density.

806
01:08:57,340 --> 01:09:00,899
Let me emphasize that as
a very restrictive form.

807
01:09:00,899 --> 01:09:02,899
When we finish cosmology,
one of the next things

808
01:09:02,899 --> 01:09:05,080
we're going to talk about
are spherically symmetric

809
01:09:05,080 --> 01:09:07,420
compact objects--

810
01:09:07,420 --> 01:09:10,580
stars-- and we want to
describe them as a fluid,

811
01:09:10,580 --> 01:09:13,149
and we'll need an equation of
state to make progress there,

812
01:09:13,149 --> 01:09:15,565
we do not use a form
like that for stars.

813
01:09:18,250 --> 01:09:22,330
As we'll see, though, for
the kind of matter that

814
01:09:22,330 --> 01:09:25,823
dominates the behavior of our
universe on the largest scales,

815
01:09:25,823 --> 01:09:27,490
this is actually a
very reasonable form.

816
01:09:31,359 --> 01:09:33,819
So if I were to write
down my thoughts on that,

817
01:09:33,819 --> 01:09:39,399
I would say
restrictive but useful

818
01:09:39,399 --> 01:09:41,859
on the large scales
appropriate to cosmology.

819
01:09:54,570 --> 01:09:57,950
Pardon me just one moment.

820
01:09:57,950 --> 01:10:08,668
Let's imagine
that-- yeah, sorry.

821
01:10:08,668 --> 01:10:10,460
Let's imagine that I
have a universe that's

822
01:10:10,460 --> 01:10:14,660
dominated by a single species
of some kind of stuff, OK?

823
01:10:14,660 --> 01:10:16,490
So in reality, what
you will generally

824
01:10:16,490 --> 01:10:19,168
have is a universe
in which there

825
01:10:19,168 --> 01:10:20,710
are several different
things present.

826
01:10:20,710 --> 01:10:22,730
So you might have
a W corresponding

827
01:10:22,730 --> 01:10:26,742
to one form of matter, another W
for a different form of matter,

828
01:10:26,742 --> 01:10:28,700
and you'll sort of have
a superposition of them

829
01:10:28,700 --> 01:10:32,400
all present at one given moment.

830
01:10:32,400 --> 01:10:47,110
So to start, start by
imagining a universe dominated

831
01:10:47,110 --> 01:11:01,520
by a particular what I
will call a species rho i,

832
01:11:01,520 --> 01:11:04,970
and the pressure will
be related to this

833
01:11:04,970 --> 01:11:08,770
by a particular W for whatever
that rho happens to be.

834
01:11:13,380 --> 01:11:21,620
So before I even hook this up
to the Friedmann equations,

835
01:11:21,620 --> 01:11:26,540
let's require that this
form of matter respects

836
01:11:26,540 --> 01:11:29,250
stress energy conservation.

837
01:11:29,250 --> 01:11:31,028
OK, so the equation
I just wrote,

838
01:11:31,028 --> 01:11:33,070
let me rewrite that in a
slightly different form.

839
01:11:33,070 --> 01:11:35,305
I can divide both
sides by R0 cubed.

840
01:11:51,664 --> 01:11:53,780
OK, that looks like so.

841
01:11:53,780 --> 01:11:59,040
Now it's not too hard to show
using this assumed form here--

842
01:11:59,040 --> 01:12:07,000
so if I plug in that
my p is Wi rho i,

843
01:12:07,000 --> 01:12:09,790
in a line or two of algebra,
you can turn this into--

844
01:12:22,350 --> 01:12:28,625
and using-- well, I
didn't even really do

845
01:12:28,625 --> 01:12:30,250
anything that
sophisticated, I can just

846
01:12:30,250 --> 01:12:32,560
integrate up both sides.

847
01:12:32,560 --> 01:12:36,550
And what you see is
that rho normalized

848
01:12:36,550 --> 01:12:38,770
to some initial time,
some initial value.

849
01:12:47,820 --> 01:12:51,360
It is very simply related to the
behavior of the scale factor.

850
01:12:54,198 --> 01:12:55,150
OK?

851
01:12:55,150 --> 01:12:57,330
But if you like, you
can set a0 equal to 1,

852
01:12:57,330 --> 01:13:00,330
if you make that your stuff
now, and this gives you

853
01:13:00,330 --> 01:13:02,640
a simple relationship
that allows

854
01:13:02,640 --> 01:13:06,390
you to see how matter behaves
as the large-scale structure

855
01:13:06,390 --> 01:13:08,010
of the universe changes.

856
01:13:08,010 --> 01:13:10,810
Let's look at a couple
examples of how this behaves.

857
01:13:44,890 --> 01:13:48,655
So I'm going to call my
first category matter.

858
01:13:51,874 --> 01:13:53,970
OK, that's pretty broad.

859
01:13:53,970 --> 01:13:57,100
When a cosmologist speaks
of matter, generally what

860
01:13:57,100 --> 01:14:02,260
they are thinking of, this is
stuff for which W equals 0.

861
01:14:02,260 --> 01:14:06,690
So this is something
that is pressureless.

862
01:14:06,690 --> 01:14:10,320
And we talked about pressureless
stuff very early in this class.

863
01:14:10,320 --> 01:14:11,945
This is what we call dust.

864
01:14:15,423 --> 01:14:17,340
So what we're talking
about here is a universe

865
01:14:17,340 --> 01:14:21,330
that is filled with dust,
which seems kind of stupid

866
01:14:21,330 --> 01:14:22,560
at first approximation, OK?

867
01:14:22,560 --> 01:14:25,290
Our universe sure as hell
doesn't look like dust.

868
01:14:25,290 --> 01:14:27,982
But bear in mind, what we
really mean about this is,

869
01:14:27,982 --> 01:14:29,190
go back to this pressureless.

870
01:14:29,190 --> 01:14:30,960
We're just referring
to something

871
01:14:30,960 --> 01:14:36,480
that is sufficiently
non-interactive, that when

872
01:14:36,480 --> 01:14:40,750
particles basically do not
interact with each other.

873
01:14:40,750 --> 01:14:43,330
Our typical dust
particle is going

874
01:14:43,330 --> 01:14:48,250
to actually be something
on the scale of a galaxy.

875
01:14:48,250 --> 01:14:54,660
On cosmological scales,
matter-matter interactions

876
01:14:54,660 --> 01:14:58,120
are, in fact, quite weak.

877
01:14:58,120 --> 01:15:00,663
So this is a very,
very good description.

878
01:15:09,970 --> 01:15:11,470
Sort of imagine the
universe is kind

879
01:15:11,470 --> 01:15:13,870
of a gas of galaxies
and galaxy clusters,

880
01:15:13,870 --> 01:15:17,140
it's a pressureless gas of
galaxy and galaxy clusters.

881
01:15:17,140 --> 01:15:19,270
When you put all this
together-- so let's

882
01:15:19,270 --> 01:15:22,900
take a look at this form here.

883
01:15:22,900 --> 01:15:29,830
The density of matter looks like
I'm going to set a0 equal to 1,

884
01:15:29,830 --> 01:15:38,600
it looks like the density
now times a to the minus 3.

885
01:15:38,600 --> 01:15:39,100
OK?

886
01:15:39,100 --> 01:15:40,475
What I've done is
I've just taken

887
01:15:40,475 --> 01:15:45,070
that evolution law there and
I have plugged in Wi equals 0.

888
01:15:49,550 --> 01:15:55,580
What this is basically saying is
that the conservation of stress

889
01:15:55,580 --> 01:15:59,760
energy demands that the--

890
01:15:59,760 --> 01:16:01,970
excuse me-- that the
density of this matter

891
01:16:01,970 --> 01:16:11,170
changes in such a way that
the number of dust particles

892
01:16:11,170 --> 01:16:23,600
is constant, but their density
varies as a to the minus 3.

893
01:16:23,600 --> 01:16:25,490
a sets all of my length scales.

894
01:16:25,490 --> 01:16:27,050
If I make the
universe twice as big,

895
01:16:27,050 --> 01:16:29,180
the density will
be 1/8 as large.

896
01:16:33,500 --> 01:16:36,310
Your second species of matter
that your cosmologist often

897
01:16:36,310 --> 01:16:39,190
likes to worry-- or
second species of stuff

898
01:16:39,190 --> 01:16:42,560
that your cosmologist likes
to worry about is radiation.

899
01:16:42,560 --> 01:16:44,680
Here, just go back to Stat Mech.

900
01:16:44,680 --> 01:16:50,350
If you have a gas of photons,
it exerts photon pressure,

901
01:16:50,350 --> 01:16:51,880
and that is of the form--

902
01:16:54,990 --> 01:17:01,730
the radiation pressure is
1/3 of the energy density.

903
01:17:01,730 --> 01:17:04,170
Factors of speed of light
are being omitted here.

904
01:17:04,170 --> 01:17:07,796
So this corresponds
to a law in which--

905
01:17:07,796 --> 01:17:09,210
here, let me put it this way.

906
01:17:09,210 --> 01:17:10,400
I should've make this an m.

907
01:17:10,400 --> 01:17:13,460
So this is my i
equals m for matter.

908
01:17:13,460 --> 01:17:17,190
So my W for radiation is 1/3.

909
01:17:17,190 --> 01:17:21,950
And what you find in
this case is that rho

910
01:17:21,950 --> 01:17:25,430
of radiation scales
with the scale

911
01:17:25,430 --> 01:17:29,210
factor to the fourth power.

912
01:17:29,210 --> 01:17:31,070
What's going on here?

913
01:17:31,070 --> 01:17:33,590
Well let's imagine that
the scale factor increases

914
01:17:33,590 --> 01:17:36,500
by a factor of 2, OK?

915
01:17:36,500 --> 01:17:39,720
Imagine that the number of
photons is not changing.

916
01:17:39,720 --> 01:17:41,720
So what this is
basically saying is,

917
01:17:41,720 --> 01:17:44,270
OK, I get my factor
of 8 corresponding

918
01:17:44,270 --> 01:17:47,110
to the volume increasing
by a factor of 8,

919
01:17:47,110 --> 01:17:50,210
but I have an additional factor
of 2, where's that coming from?

920
01:17:50,210 --> 01:17:52,140
Well remember, that's
an energy density.

921
01:17:52,140 --> 01:17:55,310
So this is saying that not only
is the density being diluted

922
01:17:55,310 --> 01:17:58,670
by the volume growing,
but each packet of energy

923
01:17:58,670 --> 01:18:03,380
is also getting smaller as the
universe increases in size.

924
01:18:06,060 --> 01:18:09,230
Each quantum of
radiation is redshifting.

925
01:18:29,740 --> 01:18:31,690
It's redshifting with
a scale factor a.

926
01:18:34,610 --> 01:18:37,690
We are going to revisit
that in the next lecture.

927
01:18:37,690 --> 01:18:39,150
That's an important
point and we're

928
01:18:39,150 --> 01:18:49,060
going to re-derive that result
somewhat more rigorously as we

929
01:18:49,060 --> 01:18:51,970
began exploring how it is that
we can observational it probe

930
01:18:51,970 --> 01:18:54,118
the properties of our universe.

931
01:18:57,190 --> 01:19:00,760
Just for fun, there's
another form of--

932
01:19:00,760 --> 01:19:02,620
there's another kind
of perfect fluid

933
01:19:02,620 --> 01:19:06,650
that cosmologists
worry about, and that's

934
01:19:06,650 --> 01:19:07,760
the cosmological constant.

935
01:19:17,013 --> 01:19:21,710
So a cosmological
constant has pressure

936
01:19:21,710 --> 01:19:26,880
equal to minus the density.

937
01:19:26,880 --> 01:19:28,890
This corresponds to
an equation of state

938
01:19:28,890 --> 01:19:33,540
parameter W equal to minus 1.

939
01:19:38,980 --> 01:19:43,100
If I go to my form here, I
plug in W equals minus 1,

940
01:19:43,100 --> 01:19:48,855
rho goes as a to the
0th power, a constant.

941
01:19:53,232 --> 01:19:55,190
Well, it is a cosmological
constant, after all,

942
01:19:55,190 --> 01:19:57,970
so that shouldn't
be too surprising.

943
01:19:57,970 --> 01:20:01,170
This is a very interesting
one because it is basically

944
01:20:01,170 --> 01:20:07,430
telling us that the amount
of energy in spacial slices--

945
01:20:07,430 --> 01:20:08,970
the energy density
does not change.

946
01:20:08,970 --> 01:20:12,288
The amount of energy
appears to be Increasing.

947
01:20:12,288 --> 01:20:14,580
Now bear in mind, it's hard
to define the total energy,

948
01:20:14,580 --> 01:20:17,670
we cannot really in a covariant
way add up energy at various

949
01:20:17,670 --> 01:20:19,440
different kinds of points.

950
01:20:19,440 --> 01:20:21,967
Local energy is still
being conserved,

951
01:20:21,967 --> 01:20:24,300
but there's no question that
this guy is a little weird.

952
01:20:28,610 --> 01:20:32,440
So one of the next
things that we want to do

953
01:20:32,440 --> 01:20:36,970
is take this stuff, run it
through Einstein's equations.

954
01:20:36,970 --> 01:20:38,680
Einstein's equations,
of course, give us

955
01:20:38,680 --> 01:20:40,540
the Friedmann equations.

956
01:20:40,540 --> 01:20:43,420
And solve to see what the
expansion the universe

957
01:20:43,420 --> 01:20:43,960
looks like.

958
01:20:43,960 --> 01:20:48,040
We saw already that if the
density of the universe

959
01:20:48,040 --> 01:20:51,640
relative the critical density
is either higher, the same,

960
01:20:51,640 --> 01:20:54,880
or lower, that tells us about
the value of this k parameter,

961
01:20:54,880 --> 01:20:56,650
or rather, the kappa parameter.

962
01:20:56,650 --> 01:21:01,480
We haven't yet seen how to
solve for the scale factor a.

963
01:21:05,030 --> 01:21:07,430
However, we have the
two Friedmann equations,

964
01:21:07,430 --> 01:21:12,860
and if nothing else, write them
down, write out your stuff,

965
01:21:12,860 --> 01:21:15,620
you got yourself a
system of equations,

966
01:21:15,620 --> 01:21:19,370
Odin gave us
mathematica-- attack.

967
01:21:19,370 --> 01:21:21,590
To give you some intuition
as to what you end up

968
01:21:21,590 --> 01:21:24,970
seeing when you look at
these kind of solutions,

969
01:21:24,970 --> 01:21:26,845
let me look at the
simplest kind of universes

970
01:21:26,845 --> 01:21:28,820
that we can solve this way.

971
01:21:28,820 --> 01:21:33,710
So let's examine what I will
call a monospecies universe--

972
01:21:33,710 --> 01:21:36,650
in other words, a universe
that only contains

973
01:21:36,650 --> 01:21:40,060
one of these forms of matter
that I have described here,

974
01:21:40,060 --> 01:21:42,560
one of these sources of stress
energy that I described here.

975
01:21:46,180 --> 01:21:51,370
And for simplicity, I'm going
to take it to be spatially flat.

976
01:21:58,860 --> 01:22:02,340
Neither of these two
conditions are true in general,

977
01:22:02,340 --> 01:22:03,700
but they are--

978
01:22:03,700 --> 01:22:05,760
they're fine for us
to wrap our heads

979
01:22:05,760 --> 01:22:09,310
around what the characteristics
of the solutions look like.

980
01:22:09,310 --> 01:22:20,240
So in this limit, Friedmann 1
becomes a dot over a squared

981
01:22:20,240 --> 01:22:26,930
equals 8 pi g rho over 3.

982
01:22:26,930 --> 01:22:33,800
I can borrow this form that
I've got here to write this as 8

983
01:22:33,800 --> 01:22:40,280
pi over 3 rho at some
particular moment times scale

984
01:22:40,280 --> 01:22:42,600
factor to the minus n.

985
01:22:42,600 --> 01:22:45,230
So what I'm doing here
is I'm assuming a0 now.

986
01:22:45,230 --> 01:22:48,260
My a right now is 1,
and I'm defining n

987
01:22:48,260 --> 01:23:01,265
to be 3 times 1 plus W.

988
01:23:01,265 --> 01:23:02,640
This is easy enough
to solve for.

989
01:23:18,590 --> 01:23:19,530
OK.

990
01:23:19,530 --> 01:23:20,450
Take the square root.

991
01:23:26,630 --> 01:23:28,382
What you find is--

992
01:23:28,382 --> 01:23:29,840
I'm going to just
sort of-- there's

993
01:23:29,840 --> 01:23:32,510
a constant you guys can work
out on your own if you like.

994
01:23:32,510 --> 01:23:42,960
a dot must be proportional
to a 1 minus n over 2,

995
01:23:42,960 --> 01:23:50,190
or a is proportional
to t to the 2 over n.

996
01:23:57,360 --> 01:23:59,250
n equals 0 is a
special case that we'll

997
01:23:59,250 --> 01:24:01,092
talk about in just a moment.

998
01:24:01,092 --> 01:24:02,550
If we are dealing
with what we call

999
01:24:02,550 --> 01:24:13,620
a matter-dominated
universe, well,

1000
01:24:13,620 --> 01:24:17,820
in this sort of monospecies,
spatially flat form,

1001
01:24:17,820 --> 01:24:21,070
this would have W
equal 0, n equals

1002
01:24:21,070 --> 01:24:28,110
3, and a scale factor that
grows as t to the 2/3 power.

1003
01:24:28,110 --> 01:24:31,530
A matter-dominated universe
is one that expands,

1004
01:24:31,530 --> 01:24:34,050
but it expands with
this kind of a loss.

1005
01:24:34,050 --> 01:24:37,120
So it slows down with time.

1006
01:24:37,120 --> 01:24:50,140
A radiation-dominated
universe, W equals 1/3, n

1007
01:24:50,140 --> 01:25:02,720
equals 4, that is a
universe that expands

1008
01:25:02,720 --> 01:25:03,720
as the square root of t.

1009
01:25:09,030 --> 01:25:10,760
What about my
cosmological constant?

1010
01:25:10,760 --> 01:25:11,260
Ah.

1011
01:25:11,260 --> 01:25:12,385
OK, well that's a problem--

1012
01:25:15,030 --> 01:25:19,050
n equals 0, and my solution
doesn't work for that one.

1013
01:25:36,820 --> 01:25:38,350
So what you do is just--

1014
01:25:38,350 --> 01:25:44,762
let's just go back to
our F or W equations.

1015
01:25:44,762 --> 01:25:46,220
Or rather, our
Friedmann equations.

1016
01:25:46,220 --> 01:25:48,980
Let's write F1 down again.

1017
01:25:48,980 --> 01:25:54,410
I have a dot over
a equals 8 pi--

1018
01:25:54,410 --> 01:26:02,075
whoops-- 8 pi g rho over 3,
and this is now a constant.

1019
01:26:08,460 --> 01:26:11,310
Rho equals rho 0
because it's a constant.

1020
01:26:11,310 --> 01:26:13,980
I can rewrite this in terms
of the cosmological constant

1021
01:26:13,980 --> 01:26:16,070
lambda.

1022
01:26:16,070 --> 01:26:20,020
And so another way
to write this is--

1023
01:26:20,020 --> 01:26:22,840
sorry about that-- a dot
a-- a dot over a squared

1024
01:26:22,840 --> 01:26:30,420
equals this, which
equals lambda over 3.

1025
01:26:33,800 --> 01:26:43,715
So this leads to an
exponential solution.

1026
01:26:48,270 --> 01:26:52,590
Now our real universe is
not as simple as these three

1027
01:26:52,590 --> 01:26:55,440
illustrative cases that
I have put in here just

1028
01:26:55,440 --> 01:26:57,660
to illustrate what the
extremes look like, OK?

1029
01:26:57,660 --> 01:27:01,110
We are not a
monospecies universe,

1030
01:27:01,110 --> 01:27:05,370
we have a mixture of matter,
we have a mixture of radiation,

1031
01:27:05,370 --> 01:27:07,770
we appear to have
something that smells

1032
01:27:07,770 --> 01:27:09,660
a lot like a
cosmological constant,

1033
01:27:09,660 --> 01:27:14,580
although the jury is still out
if one is being perfectly fair.

1034
01:27:14,580 --> 01:27:17,410
Work is ongoing.

1035
01:27:17,410 --> 01:27:20,830
What you need to do in general
is sort of model things.

1036
01:27:20,830 --> 01:27:24,790
You try to make models of
the universe that correspond

1037
01:27:24,790 --> 01:27:28,540
to different mixtures of
things that can go into it,

1038
01:27:28,540 --> 01:27:32,770
and then you go through
and ask yourself,

1039
01:27:32,770 --> 01:27:36,280
do the observables that emerge
in this universe match what

1040
01:27:36,280 --> 01:27:38,860
we see?

1041
01:27:38,860 --> 01:27:40,660
Generally you will
see sort of trends

1042
01:27:40,660 --> 01:27:42,580
that are similar to
this that emerge, right?

1043
01:27:42,580 --> 01:27:44,500
There might be a particular
epoch where matter

1044
01:27:44,500 --> 01:27:45,730
is more important,
there might be

1045
01:27:45,730 --> 01:27:47,500
an epoch where radiation
is more important,

1046
01:27:47,500 --> 01:27:49,750
there might be an epoch where
cosmological constant is

1047
01:27:49,750 --> 01:27:50,540
more important.

1048
01:27:50,540 --> 01:27:52,810
And so you might see sort
of you know transitions

1049
01:27:52,810 --> 01:27:55,210
between these things where
it's mostly a square root t

1050
01:27:55,210 --> 01:27:58,240
expansion, and then
something happens

1051
01:27:58,240 --> 01:28:00,598
and the radiation
becomes less important,

1052
01:28:00,598 --> 01:28:03,140
there's an intermediate regime
where both are playing a role,

1053
01:28:03,140 --> 01:28:04,180
and then matter
becomes important

1054
01:28:04,180 --> 01:28:06,550
and it kicks over to a t to
the 2/3 kind of expansion

1055
01:28:06,550 --> 01:28:09,550
when it becomes
matter-dominated.

1056
01:28:09,550 --> 01:28:11,830
You don't want to assume
the universe is flat,

1057
01:28:11,830 --> 01:28:14,530
you need to do your analysis,
including a non-zero flatness

1058
01:28:14,530 --> 01:28:16,655
parameter in there, which
makes things a little bit

1059
01:28:16,655 --> 01:28:17,560
complicated.

1060
01:28:17,560 --> 01:28:20,560
So in the next
lecture, I am going

1061
01:28:20,560 --> 01:28:23,470
to talk a little bit about
how one extracts observables

1062
01:28:23,470 --> 01:28:24,670
from these spacetimes.

1063
01:28:24,670 --> 01:28:31,450
How is it that we are able
to actually go into an FRW

1064
01:28:31,450 --> 01:28:35,145
universe and measure
things-- what can we measure,

1065
01:28:35,145 --> 01:28:36,520
how can we use
those measurements

1066
01:28:36,520 --> 01:28:39,280
to learn about the energy
budget of our universe

1067
01:28:39,280 --> 01:28:42,880
and formulate cosmology as
an observational and physical

1068
01:28:42,880 --> 01:28:44,200
science?

1069
01:28:44,200 --> 01:28:47,640
And with that, I will
end this lecture.