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SCOTT HUGHES: All right.
So at this point we're
going to switch gears.
Everything that we have done
over the past several lectures
has been in service
of the approach
to solving the Einstein
field equations in which we
assume a small perturbation
around an exact background.
Most of it was spent looking at
perturbations around flat space
time.
A little bit in the
last lecture we touched
on some of the mathematics
and some of the analysis
when you curve--
you expand around some
non-specified curved
background.
I didn't tell you where that
curve background comes from.
Today we'll be--
this lecture will
be the first one
in which we begin
thinking about different forms
of different kinds of solutions
that arise from
different principles.
We're going to begin this
by studying cosmology.
It's the large-scale
structure of the universe.
So from the standpoint of--
from the standpoint of
the calculational toolkit
that we will be
using, this is going
to be the first
example of a spacetime
that we construct using
a symmetry argument.
We are not going to
make any assumptions
that anything is weak or small
or any kind of approximation
can be--
any kind of an approximation
can be applied.
What we're going to do is ask
ourselves, suppose spacetime--
at least spacetime on some
particular very large scales--
is restricted by
various symmetries.
So we will apply various
restrictions to the equations
and to the spacetime
by the assumption
that certain symmetric
symmetries must hold.
Let me reword this.
By demanding that
certain symmetries hold.
Doing so will
significantly reduce
the complicated non-linear
dynamics of the field equations
of general relativity.
This will allow us to
reduce those complicated
generic equations into
something that is tractable.
So let me describe--
whoops-- let's
me give a little bit of
background to this discussion.
Let me get some
better chalk, too.
So as background,
I'm going to give
a little bit of a synopsis
of some stuff that
is described very nicely
in the textbook by Carroll.
So for background,
part of what we're
going to consider
as we move into this
is a notion of what are called
maximally symmetric spaces.
So I urge you to read
Section 3.9 of Carroll
for extensive
discussion of this.
But the key concept of this
is that a maximally symmetric
space is a space that
has the largest number--
so let's say MSS,
maximally symmetric space,
has the largest number of
allowed Killing vectors.
If your space has
n dimensions, it
has n times n plus 1 over
2 such Killing vectors.
And recall, if you
do a lead derivative
of the metric along the
Killing vector, you get 0, OK?
So it's that these n
times m plus 1 Killing
vectors all define
ways in which, as you
sort of flow along
these vectors,
spacetime is left unchanged.
Intuitively, what these
do is define a spacetime
that is maximally homogeneous--
I shouldn't say
spacetime yet, we
haven't specified the
nature of this manifold.
So this defines a space that
is maximally homogeneous.
And homogeneous
means that just--
it has uniform properties
in all locations.
And it is maximally isotopic.
Which is a way of
saying essentially
that it looks the same
in all directions.
In the kind of spacetimes
that we are familiar with,
something-- a spacetime
that is highly isotropic
is one that is
invariant with respect
to rotations and boosts,
and one that is homogeneous
is something that is invariant
with respect to translations.
So let me give two examples.
In Euclidean space, that
is a maximally symmetric
three-dimensional space, n
times n plus 1 over 2 equals 6.
And those 6 Killing
vectors in Euclidean space
correspond to 3 rotations
and 3 translations.
Minkowski flat
spacetime: n equals 4.
n times n plus 1 over
2 is equal to 10.
I have 3 rotations, 3
translations, and 4 boosts, OK?
The requirement that your
space satisfy these properties,
it leads to a condition
that the Riemann tensor
must be Lorentz-invariant
within the local Lorentz frame.
So for these two examples
that I talked about,
the Riemann tensor
actually vanishes,
and 0 is certainly
Lorentz-invariant,
so there's no problem there.
But as I start thinking
about more general classes
of spacetimes, which I'm going
to consider to be examples
of massively
symmetric spaces, they
might not have vanishing
Riemann tensors.
But the Riemann tensors,
in order to be massively
symmetric, if I go into
a freely-falling frame,
that has to look--
everything has to look
Lorentz-invariant.
This leads to a condition that
my Riemann tensor must take--
it is constrained to take
one particular simple form.
It must be R over n times n
minus 1 times metric like so.
This is-- so Carroll goes
through this in some detail.
Essentially what's
going on here is
this is the only way in
which I am guaranteed
to create a tensor that
is Lorentz-invariant
in a local Lorentz frame.
So I go to my
local Lorentz frame
and I must have a form
it looks like this,
and this is a way of putting
all my various quantities
on my metric tensors
together in such a way
that I recover the symmetries
of the Riemann tensor,
and is my number of dimensions.
Because it will
prove useful, let
me generate the Ricci tensor
and the Ricci scalar from this.
So R mu nu is going to be
R over n times m minus 1.
I'm taking the trace on
indices alpha and beta.
If I trace on this guy, I get n.
The trace in the metric
always just gives me back
the number of dimensions.
And when I trace on
alpha and beta here,
I basically just contract
these two indices,
and so I get the metric
back over n times g mu nu.
Take a further trace
and you can see
that that R that
went into this thing
is indeed nothing more
than the Ricci curvature.
Excuse me, the scalar
Ricci curvature.
OK?
You can construct the
Einstein tensor out of this,
and what you see is that
the Einstein tensor must
be proportional to the metric.
And in fact, there are--
the only solutions for
the Einstein tensor that--
the only solution is the
Einstein field equations
in which the Einstein tensor is
proportional to the metric are
either flat spacetime or
a cosmological constant.
So empty space,
empty flat spacetime
and cosmological constant
are the only maximally
symmetric four-dimensional
spacetimes.
That does not necessarily
describe our universe.
So our universe, how am I
going to tie all this together?
We begin with the
observation that our universe
is, in fact,
homogeneous and isotopic
on large spatial scales.
I emphasize spatial because
the spacetime of our universe
is not homogeneous, OK?
In fact, the past
of our universe
is very different
from the present.
Because light travels
at a finite time, when
we observe two
large distances, we
are looking back into
the distant past.
And we see that the universe
is a lot denser in the past
than it is today.
It remains the case,
though, that it is still
homogeneous and
isotropic spatially,
at least on large scales.
So we are going
to take advantage
of these notions of
maximally symmetric spaces
to define a spacetime that is
maximally symmetric spatially,
but is not so symmetric
with respect to time, OK?
So we'll get to that
in just a few moments.
There is a wiggle word in here.
I said that our
universe, by observation,
is homogeneous and isotropic
on large spatial scales.
What does large mean?
Well the very largest scales
that we can observe of all--
so that when we go back and
we probe sort of the largest
coherent structure that can
be observed in our universe,
we have to go all the way
back to a time which is
approximately
13-point-something or another--
I forget the exact
number, but let's say
about 13.7 billion years ago,
and we see the cosmic microwave
background.
So the cosmic
microwave background
describes what our universe
looked like 13.7 billion years
ago or so, and what you
see is that this guy is
homogeneous and isotropic to
about a part in 100,000, OK?
With a lot of
interesting physics
in that deviation from--
that sort of part in
100,000 deviation,
but that's a topic
for a different class.
We then sort of imagine
you move forward in time,
you look on-- so that
tells you about the largest
scales in the earliest times.
Look at the universe
on smaller scales.
I mean, clearly you
look in this room, OK?
I'm standing here, there's
a table over there,
this is not homogeneous
and not isotropic,
things look quite different.
What we start to see is things
deviate from homogeneity
and isotropy on scales that are
on the order of several tens
of megaparsecs in size, OK?
Parsec, for those of you
who are not astrophysicists,
is a unit of measure.
It's approximately
3.2 light years.
So once you get
down to boxes that
are on the orders of 50 million
light years or so on a side,
you start to see deviations
from homogeneity and isotropy.
And this is caused by
gravitational clumping.
These are things
like galaxy clusters.
So when I talk about
cosmology and I
want to describe the universe
as large-scale structure,
I am going to be
working on defining
a description of spacetime
that averages out
over small things like
clusters of galaxies, OK?
So this is sort of a fun lecture
in that sense, in that anything
larger than an
agglomeration of a couple
dozen or a couple
hundred galaxies,
I'm going to treat
that like a point.
So here's what I am going to
choose for my spacetime metric.
This is where you start to see
the power of assuming a given
symmetry.
So the line element,
I'm going to write it
as minus dt squared
plus some function R
squared of t gamma ij dx i dx j.
The function R of t
I've written down here.
It's one variant of-- there's
a couple functions that
are going to get this name.
We call this the scale factor.
Caution, it's the same capital
R we used for the Ricci scalar,
it's not the Ricci scalar.
Just a little bit of
unfortunate notation,
but it should be clear
from context which is which
when they come up.
I have chosen gtt equals
minus 1 and gti equals 0.
Remember from our discussion
of linearized theory
around a flat background,
that the spacetime-- the 10
independent functions of my
spacetime metric, of those 10,
four of them were
things that I could
specify by choosing a gauge.
Well here, I have specified four
functions pretty much by fiat.
Think of this as defining the
gauge that I am working in, OK?
In a very similar way, I have
chosen a coordinate system
by specifying gtt to be
minus 1 and gti to be 0.
This means that I am
working in what are
called co-moving coordinates.
So if I am an observer who
is at rest in the spacetime
so that I would define
my four velocity like so,
I will be essentially co
moving with the spacetime.
Whatever the spacetime
is doing, I'm
just going to sort of
homologously track it.
It's worth noting-- so those of
you who think about astronomy,
astrophysics, and
observational cosmology,
the earth is not co-moving, OK?
We build our telescopes
on the surface
of the Earth which rotates.
The Earth itself
orbits around the Sun.
The Sun is in a solar system.
Or excuse me-- the Sun at the
center of our solar system
is itself orbiting our galaxy.
And our galaxy is
actually falling
into a large cluster of galaxies
called the Virgo Cluster.
This basically means that when
we are making cosmologically
interesting measurements, we
have to correct for the fact
that we make measurements using
a four velocity that is not
a co-moving four velocity.
This actually shows
up in the fact
that when one
makes measurements,
one of the most impressive
places that shows up
is that when you measure the
cosmic microwave background,
it has what we call
a dipole isotropy.
And that dipole is just
essentially a Doppler shift
that is due to the fact that
when we make our measurements,
we are moving with respect to
the co-moving reference frame.
All right.
So that's the metric that
we're going to use here.
Setting gtt and
gti like so means
I have chosen these
co-moving coordinate systems.
I'm going to take
gamma ij, I'm going
to take this to be
maximally symmetric, OK?
So this is my statement
that at any moment of time,
space is maximally symmetric.
So a few words on the
units, a few things
that I'm going to set up here.
So my coordinate, I'm going to
take my xi to be dimensionless,
and all notions of length--
all length scales
and the problem
are going to be absorbed
into this factor R of t.
We're going to see that the
overall scale of the universe
is going to depend on the
dynamics of that function
R of t.
So let's imagine
that on a given--
at some given
moment of time, you
want to understand the
curvature associated
with that constant time slice.
So the Riemann tensor that we
build from our spatial metric,
I'm going to write this as 3 R--
and I'm doing purely
spatial things,
so I'm going to use Latin
letters for my indices.
It's going to equal
to some number k--
not to be confused
with the index k.
It's unfortunate,
but there's only so
many letters to work with.
That looks like so.
And if I take a trace to make
my Ricci curvature, I get this,
and your Ricci scalar will
turn out to be equal to 6k.
We won't actually need
that, but just so you
can establish what
that k actually means.
It's simply related to
the Ricci curvature.
Oops, that should
have a 3 on it.
Ricci curvature of that
particular instant in time.
Now I'm going to require
my coordinate system
to reflect the fact
that space is isotopic.
So if it's isotropic,
it must look the same
in all directions, and in
a three-dimensional space,
anything that is the
same in all directions
must be spherically symmetric.
And so what this means is that
when I compute gamma ij dx i dx
j, it must be equal to
some function of radius.
The bar on that radius
just reminds you this
is meant to be a dimensionless
notion of radius.
Remember, all length
scales are going
to be absorbed into
the function capital
R. It's R squared d omega.
And my angular sector
is just related
to circle coordinate
angles the usual way.
So it's convenient for
us to put this f of R--
I can rewrite this as an
exponential function, OK?
Just think of this as the
definition of the function
beta.
The reason why this is
handy is that suppose
I now take gamma ij, I
compute the three-dimensional
Christoffel symbols, I compute
my three-dimensional Riemann,
ignoring for a moment
that my Riemann is meant
to be maximally symmetric, OK?
I'm just going to
say, I know the recipe
for how to make
Riemann from a metric.
I will do that.
I will then make Ricci
from that Riemann.
When you do this,
what you find is
that-- let's just look at the
Rr component of this thing.
This turns out to be 2 over
R times a regular derivative
of beta, OK?
It's just a little bit
of tensor manipulation
to do that using some of the
tools, the mathematical tools
I'm going to post in
the 8.962 website,
you can verify this yourself.
If I compute Ricci from the
maximally symmetric assumption,
what I find is that this is
equal to 2k times gamma, which
is itself exponent of 2 beta.
Let's equate these
and solve for beta.
So one side I've got 2
over r bar dr bar of eta.
On the near side I
have 2k and gamma r bar
r bar is itself e to the 2 beta.
Cancel, cancel.
A little bit of algebra.
First let's write it this way.
Let's move this to
the other side...
e to the minus 2 beta
is equal to this.
Let us make the assumption-- so
we have a choice of a boundary
condition.
Let's put beta equals
0 at r bar equals 0.
This is basically
saying that on my--
so r bar is sort of the origin.
We're just sort of
saying that things
look like a flat spacetime
in the vicinity of the origin
of the coordinates
we're using here,
that's a fine
assumption to make.
Doing so, we can easily
integrate this guy up,
and here's what we get.
So with this, we now
have a full line element.
So there's a few unknown
quantities in here.
What is k?
What is R?
So far I have only talked about
the geometry of this spacetime.
We haven't yet connected
this-- any of the dynamics
of the spacetime to a source.
It is when we hook
this up to a source
that we're going to learn
something about these two.
So hold that thought for now.
This essentially has
just said that here
is what my maximally spatially
symmetric spacetime looks like,
allowing for there to be a
difference between the past
and the present.
Before I move on, so I can't
tell you what k is yet,
but I can make the
following observation which
allows me to restrict
what values of k
I need to worry about.
Suppose I take k and I replace
it with k prime equal alpha k.
But in doing so,
I define R tilde
to be square root of alpha--
yeah.
Square root of
alpha times R bar.
And I also require that
my overall scale factor
look like the original
scale factor divided
by square root of alpha.
Rewriting my spacetime, my line
element in terms of k prime
and the tilde R into-- the
two tilde R's, I get this.
Basically, that transformation
leaves the line element
completely irrelevant to me.
That was completely
the wrong word.
That re-prioritization of k
and R and the two different R's
here, that leaves the line
element completely invariant.
It is unchanged when I do this.
So what this tells
me is-- by the way,
alpha has to be a positive
number so that the square roots
make sense there.
It tells me that the
normalization associated with k
can be absorbed into
my scale factor.
And so what it suggests
we ought to do is just--
you don't need to
worry about whether k
is equal to 15 or pi or
negative the 38th root of e
or anything silly like that.
The only three values
of k that matter for us
are whether it is
negative 1, 0, or 1.
This stands for all
negative values of k,
0 as a set onto itself, and
all positive values of k, OK?
So we will use this to
say, great, the thing which
I'm going to care
about, once I start
looking at the physics
associated with this,
is whether--
let's go back over to
this version of it--
I'm going to care about
whether k is negative 1, 0, 1,
and I want to understand
how my scale factor behaves.
So before I start hooking
this up to my source
and doing a little bit
of physics, many of you
are going to do something
involving cosmology
at some point in your
lives, and so it's
useful to introduce a few
other bits of notation that
are commonly used here,
as well as to describe
some important terminology
that comes up at this point.
So here's some common
notation and terminology.
Let us define a
radial coordinate chi
via the following definition--
d chi will be equal to d R
bar over square root of 1
minus kR bar squared.
Now remember, we just
decided that k can only
take on one of three
interesting values.
I can immediately
integrate this up,
and I will find that my
R bar is equal to sine
of chi of k equals plus 1 is
equal to chi of k equal 0.
And it's the sinh of
chi of k equals minus 1.
So let's take a
look at what this
means with these sort of
three possible choices.
The three possible
values that k can take.
What is our line
element looks like?
So let's look at k
equals plus 1 first.
I get minus dt squared R square
root of t d chi squared plus--
I'm going to use
the fact that R bar,
this describes a spacetime in
which every spacial slice is
what is called a 3-sphere, OK?
You're all nicely familiar
with the 2-sphere.
So a 2-sphere is the
three-dimensional surface
in which you pick a point
and every point that is,
let's say, a unit radius
away from that point that,
defines a 2-sphere in
three-dimensional space.
So this defines the space--
the spatial
characteristics of pick
a point in
four-dimensional space
and ask for all of the points
that are a unit distance away
from it in three dimensions,
that is a 3-sphere.
Notice that my 3-sphere
has a maximum--
there's a maximum distance
associated with it, OK?
So there's no bounds on chi, OK?
Chi can go from 0 to infinity.
But this one's
periodic, isn't it?
So as chi reaches pi over 2,
the separation between any two
points on that
single slice, they've
reached their maximum value, and
as chi continues to increase,
the distance gets
smaller again, OK?
And eventually, when
chi gets up to pi,
you come back to
where you started.
We call this a closed universe.
This is something
where if it were
possible to step out of
time and just run around
on a spacial slice, you would
find that it is a finite size.
The best you could
do is run around
on that three-dimensional sphere
in four-dimensional space.
Let's do k equals 0 next.
If I do k equals 0,
there's my line element.
Each spacial slice is
simply Euclidean space.
So this is often
described as flat space.
A significant word of caution.
When you talk to a
cosmologist, they will often
talk about how the best--
we're going to talk about sort
of the observational situation
in the next lecture that
I record a little bit.
Our evidence actually
suggests that this
is what our universe
looks like right now.
We're in a k equal 0
universe in which space--
each spatial slice is flat.
That does not mean
spacetime is flat, OK?
So when they say
that it's flat, that
is referring to
the geometry only
of the spatial slices in this
co-moving coordinate system.
k equals minus 1, you get a
form that looks like this.
This describes the
geometry of a hyperbole.
We call this an
open spatial slice.
So notice for both
choices 2 and 3,
if you could sort
of step out of time
and explore the full geometry
of that spatial slice,
it goes on forever, OK?
Again, there's really
no boundary on that chi
as near as we can tell,
and so that spatial slide
can just kind of go, whee!
And take off forever.
This one sort of goes to large
distances a little slower
than this one does.
This hyperbolic
function means that this
is really bloody large, OK?
So both of these tend
to imply a universe that
is sort of spatially unbounded.
The closed universe, because
each slice is a 3-sphere,
it's a different story.
So another bit of notation
which you should be aware of--
and I unfortunately
am going to want
to sort of flip back and forth
between the notation I've
been using so far and this
one I'm about to introduce.
It can be a little bit annoying
when you're first learning it,
but just keep track of
context, it's not that hard.
So what we're going
to do is let's
choose a particular value
of the scale factor,
and we will normalize
things to that.
So what I'm going to do is
define some particular value
of k such that the scale factor
there I will call it R sub 0.
And as we'll see, a particularly
useful choice for this
is to choose the
value right now, OK?
What we're doing, then, is
we're kind of norm-- what we're
going to see in a moment is
this means we're normalizing
all the scales associated
with our universe
to where they are right now.
OK.
Having done this, I'm
going to define a of t
to be R of t divided
by this special value.
For dimensional
reasons, I'm going
to need to put this into
my radial coordinate.
So notice, what's
going on here is
that my R will now have
dimensions associated with it,
and so essentially everything
is just being scaled by that R0.
And this is the bit where it
gets a tiny bit unfortunate,
you sort of lose the beauty
of k only having three values.
So I'm going to replace
that with a kappa.
This is unfortunately a
little bit hard to read,
so whenever I make it sort
of with my messy cursive,
it will be k; whenever it
looks a little bit more
like a printed thing,
it will be kappa.
And so kappa is k
divided by R0 squared.
And when you do that, your
line element becomes this.
OK?
So that's a form that we're
going to use a little bit.
What's a little bit annoying
about it is just that my--
the kappa that appears
in there doesn't just
come as a set of one of
these parts of three,
but basically if kappa
is a negative number,
then you know k must be minus 1;
kappa equals 0 corresponds to k
equals 0; if kappa is a positive
number, then k equals plus 1.
This form where we're using
this sort of dimensionless scale
factor a is particularly useful.
If you look at this,
this is telling you
that with the choice that R0
defines a scale factor now,
this means a now equals 1.
And so this gives us a
nice dimensionless factor
by which we can compare
all of our spatial scales
at different moments in
the universe to the size
that they are now.
OK.
Everything I have
said so far has really
been just discussing
the geometry
that I'm going to
use to describe
the large-scale
structure of spacetime.
I haven't said
anything about what
happens when I solve the
Einstein field equations
and connect this
geometry to physics.
So what we need to do
is choose a source.
And so what we're
going to do is we
will do what is sort of the
default choice in many analyses
in general relativity.
We will choose our source
to be a perfect fluid.
What's nice about this
is that it automatically
satisfies the requirements
of isotropy and homogeneity.
At least it does so if
the fluid is at rest
in co-moving coordinates.
So let's fill this in:
t mu nu with everything
in the downstairs position looks
like rho plus P mu nu mu nu
plus Pg mu nu.
And this becomes in my
co-moving coordinate system.
So then it looks like this, OK?
A handy fact to
have, this is going
to be quite useful for
a calculation or two
that we do a little bit later--
actually, not just a minute
later, almost right away.
This looks like a diagonal
of all this stuff, OK?
All right.
So what we want to do is use
this stress energy tensor
as the right-hand side.
So we've worked out
our Ricci tensor.
With a little bit of work, we
can make the Einstein tensor,
couple it to this guy, we
can set up our differential
equations, and we can solve
for the free functions that
specify the spacetime.
Before doing this, always a good
sanity check, remind yourself,
your fluid has to satisfy
local energy conservation.
Actually, let's just do the 0.
So this is energy and
momentum conservation,
we set that equal to 0, this
is local energy conservation.
Expanding out these
derivatives, what you
find is that this turns into--
so it looks like this.
And plugging in-- so
using the spacetime--
by the way, I made a
small mistake earlier.
I should have told you
that this spacetime,
this is now called the
Robertson-Walker spacetime.
So this was actually first
written down in the 1920s,
and Robertson and Walker
developed this basically
just as I have done
it here, just arguing
on the basis of looking
for something that
is as symmetric as possible with
respect to space if not time,
and they came out with
that line element.
My apologies, I didn't
mention that beforehand.
This is my third
lecture in a row,
I'm getting a little bit tired.
So if I take that
Robertson-Walker metric,
plug it into here to
evaluate all these,
this gives me a
remarkably simple form.
So rho is the pressure
of my perfect fluid--
excuse me, the density
of my perfect fluid, P
is the pressure of my perfect
fluid, a is my scale factor.
If you like, you can put the
factor of R0 back in there,
and an equivalent way of
writing this, which I think
is very useful for giving some
physical insight as to what
this means--
so put that R back in.
OK, so let's look at
what this is saying.
R cubed is modulo
numerical factor,
that is the volume
of a spacial slice.
And so this is saying,
the rate of change
of energy in a volume--
so a volume describing
my spatial slice
is equal to negative pressure
times the rate of change
of that volume.
I hope this looks familiar.
This, in somewhat more
convoluted notation,
is negative dp--
du equals negative P dv.
It's just the first
law of thermodynamics.
All right.
So this relationship,
whether written in this form
or in that form, is
something that we
will exploit moving forward.
Let's now solve the
Einstein field equations.
So we'll begin with g mu
nu equals 8 pi g t mu nu.
The equations that
are traditionally
used to describe cosmology are
a little bit more naturally
written.
If I change this into the form
that uses the Ricci tensor--
so let me rewrite this as R
mu nu equals 8 pi g t mu nu.
OK?
So this is equivalent where
t is just the usual trace
of the stress energy tensor.
And what you find,
there are two--
if you just look at the 0, 0
components of this equation,
it tells you the acceleration of
the scale factor a divided by a
is--
it is simply related to
the density and 3 times
the pressure.
If you evaluate Rii--
in other words, any--
this is-- there's
no sum implied here,
just take any spatial
component of this guy,
and add on R0,0 because
it's a valid equation,
it helps you to
clear out some stuff,
you get the following
relationship between the rate
of change to the scale factor,
the density, and remember,
kappa is your rescaled k.
So I'm going to call
this equation F1,
I'm going to call this one F2.
These are known as the
Friedmann equations.
When one uses them to solve
to describe your line element,
you get
Friedmann-Robertson-Walker
metrics.
So just a little
bit of nomenclature.
Robertson-Walker tells
you about the geometry,
you then equate these
guys to a source,
and that gives you
Friedmann-Robertson-Walker line
elements.
One other bit of
information-- so
let's introduce a little
bit of terminology here.
So a dot over a, this tells
me how the overall length
scale associated with
my spatial slices
is evolving as a
function of time.
This is denoted H and it's
known as the Hubble parameter.
H0 is the value of
H that we measure
in our universe corresponding
to its expansion right now, OK?
And the notes that I have
scanned and placed online
claim a best value
for this of 73
plus or minus 3 kilometers
per second per megaparsec.
These notes were
originally hand written
about 11 or 12 years ago, that
number is already out of date,
OK?
If you went to Adam
Reese's colloquium
shortly before MIT went
into its COVID shutdown,
you will have seen
that there's actually
little bit of controversy
about this right now.
So our best measurements
of this thing,
indeed, they are clustering
around 72 or 73 in these units,
but they're inconsistent with
some other measures by which we
can infer to be the-- what the
Hubble parameter should be.
And it's a very--
very interesting problems.
Unclear sort of--
it sort of smells
like something might
be a little bit off
in our cosmological models,
but we're not quite there yet.
Let's consider-- let's
proceed with sort
of the standard picture,
and just bear in mind
that this is an evolving field.
The one thing I will note
is that H has the dimensions
of inverse time, OK?
The way one actually
measures it.
So the dimensions in which most
astronomers quote its value,
it looks like a velocity over
a length, which is, of course,
also an inverse time.
And that is because objects
that are at rest with respect
to the-- that are at rest in
these co-moving coordinates,
as this fluid is meant to be,
if the universe is expanding,
we see them moving away from us.
OK, I'm going to make
a few definitions.
Let us define rho crit to
be 3H squared over 8 pi j.
So the way I got
that was take F1--
imagine kappa is equal to 0,
just ignore kappa for a second.
Left-hand side as H
squared, solve for rho, OK?
Notice, rho quit-- rho crit is a
parameter that you can measure.
You can measure the
Hubble parameter,
I'll describe to you how that
is done in my next lecture,
but it's a number
that can be measured.
And then 3 and 8 pi
are just exact numbers,
g is a fundamental constant.
So that's something
that can be measured.
Let's define omega to be any
density divided by rho crit.
Putting all these together, I
can rewrite the first Friedmann
equation.
This guy can be written as omega
minus 1 equals kappa over H
squared a squared.
Now notice, H and a,
they are real numbers.
H squared and a squared
are positive definite.
We at last can now see how
the large-scale distribution
of matter in our
universe allows us
to constrain one of
the parameters that
sets our Robertson-Walker
line element.
If omega is less than 1--
in other words, if rho
is less than rho crit,
then it must be the case
that kappa is negative,
k equals minus 1, and we
have an open universe.
If a omega equals 1
such that rho is exactly
equal to rho crit, kappa
must equal 0, k must equal 0,
and we have a Euclidean
spatially flat universe.
If omega is greater than
1, kappa is greater than 1,
k equals 1, and we
have a closed universe.
Clean up my handwriting
a little bit here.
OK, this is really interesting.
This is telling us
if we can determine
whether the density of
stuff in our universe
exceeds, is equal to, or is
less than that critical value,
we know something
pretty profound
about the spatial
geometry of our universe.
Either it's finite,
sort of simply infinite,
or ridiculously infinite.
Let me do a few more things
before I conclude this lecture.
First, this isn't that
important for our purposes,
but it's something that some
of you students will see.
A little bit of
notational trickery.
It's not uncommon
in the literature
to see people define what's
called a curvature density.
And what this is,
is you just combine
factors of kappa, g, and the
scale factor in such a way
that this has the
dimensions of density.
You can then find an omega
associated with curvature
to be rho with curvature
over the critical density.
And when you do this, F1,
the first Friedmann equation,
becomes simply omega plus
omega curvature equals 1, OK?
Just bear in mind, that
is not really a density,
it's just a concept--
it's a useful auxiliary concept.
This is often for certain
kinds of calculations,
a nice constraint
to bear in mind, OK?
People are very interested
in understanding
the geometry of our
universe, and this
is a way of formulating it that
sort of puts the term involving
the k parameter or the kappa
parameter on the same footing
as other densities that
contribute to the energy
budget of our universe.
OK.
So the equations that
we are working with here
involve these--
it involves the pressure
and density here.
I haven't said too
much about them so far.
If I want to make
further progress,
I've got to know a little
bit about the matter that
fills my universe.
So to make more progress,
I need to choose
what is called an
equation of state that
relates the pressure and
the density to each other.
What I really need is to
know that my pressure is
some function of
the density, OK?
This can be written down for
just about all kinds of matter
that we care about.
In cosmology, one
usually take and assumes
that the pressure is a linear
function-- it's just linearly
related to the energy density.
Let me emphasize that as
a very restrictive form.
When we finish cosmology,
one of the next things
we're going to talk about
are spherically symmetric
compact objects--
stars-- and we want to
describe them as a fluid,
and we'll need an equation of
state to make progress there,
we do not use a form
like that for stars.
As we'll see, though, for
the kind of matter that
dominates the behavior of our
universe on the largest scales,
this is actually a
very reasonable form.
So if I were to write
down my thoughts on that,
I would say
restrictive but useful
on the large scales
appropriate to cosmology.
Pardon me just one moment.
Let's imagine
that-- yeah, sorry.
Let's imagine that I
have a universe that's
dominated by a single species
of some kind of stuff, OK?
So in reality, what
you will generally
have is a universe
in which there
are several different
things present.
So you might have
a W corresponding
to one form of matter, another W
for a different form of matter,
and you'll sort of have
a superposition of them
all present at one given moment.
So to start, start by
imagining a universe dominated
by a particular what I
will call a species rho i,
and the pressure will
be related to this
by a particular W for whatever
that rho happens to be.
So before I even hook this up
to the Friedmann equations,
let's require that this
form of matter respects
stress energy conservation.
OK, so the equation
I just wrote,
let me rewrite that in a
slightly different form.
I can divide both
sides by R0 cubed.
OK, that looks like so.
Now it's not too hard to show
using this assumed form here--
so if I plug in that
my p is Wi rho i,
in a line or two of algebra,
you can turn this into--
and using-- well, I
didn't even really do
anything that
sophisticated, I can just
integrate up both sides.
And what you see is
that rho normalized
to some initial time,
some initial value.
It is very simply related to the
behavior of the scale factor.
OK?
But if you like, you
can set a0 equal to 1,
if you make that your stuff
now, and this gives you
a simple relationship
that allows
you to see how matter behaves
as the large-scale structure
of the universe changes.
Let's look at a couple
examples of how this behaves.
So I'm going to call my
first category matter.
OK, that's pretty broad.
When a cosmologist speaks
of matter, generally what
they are thinking of, this is
stuff for which W equals 0.
So this is something
that is pressureless.
And we talked about pressureless
stuff very early in this class.
This is what we call dust.
So what we're talking
about here is a universe
that is filled with dust,
which seems kind of stupid
at first approximation, OK?
Our universe sure as hell
doesn't look like dust.
But bear in mind, what we
really mean about this is,
go back to this pressureless.
We're just referring
to something
that is sufficiently
non-interactive, that when
particles basically do not
interact with each other.
Our typical dust
particle is going
to actually be something
on the scale of a galaxy.
On cosmological scales,
matter-matter interactions
are, in fact, quite weak.
So this is a very,
very good description.
Sort of imagine the
universe is kind
of a gas of galaxies
and galaxy clusters,
it's a pressureless gas of
galaxy and galaxy clusters.
When you put all this
together-- so let's
take a look at this form here.
The density of matter looks like
I'm going to set a0 equal to 1,
it looks like the density
now times a to the minus 3.
OK?
What I've done is
I've just taken
that evolution law there and
I have plugged in Wi equals 0.
What this is basically saying is
that the conservation of stress
energy demands that the--
excuse me-- that the
density of this matter
changes in such a way that
the number of dust particles
is constant, but their density
varies as a to the minus 3.
a sets all of my length scales.
If I make the
universe twice as big,
the density will
be 1/8 as large.
Your second species of matter
that your cosmologist often
likes to worry-- or
second species of stuff
that your cosmologist likes
to worry about is radiation.
Here, just go back to Stat Mech.
If you have a gas of photons,
it exerts photon pressure,
and that is of the form--
the radiation pressure is
1/3 of the energy density.
Factors of speed of light
are being omitted here.
So this corresponds
to a law in which--
here, let me put it this way.
I should've make this an m.
So this is my i
equals m for matter.
So my W for radiation is 1/3.
And what you find in
this case is that rho
of radiation scales
with the scale
factor to the fourth power.
What's going on here?
Well let's imagine that
the scale factor increases
by a factor of 2, OK?
Imagine that the number of
photons is not changing.
So what this is
basically saying is,
OK, I get my factor
of 8 corresponding
to the volume increasing
by a factor of 8,
but I have an additional factor
of 2, where's that coming from?
Well remember, that's
an energy density.
So this is saying that not only
is the density being diluted
by the volume growing,
but each packet of energy
is also getting smaller as the
universe increases in size.
Each quantum of
radiation is redshifting.
It's redshifting with
a scale factor a.
We are going to revisit
that in the next lecture.
That's an important
point and we're
going to re-derive that result
somewhat more rigorously as we
began exploring how it is that
we can observational it probe
the properties of our universe.
Just for fun, there's
another form of--
there's another kind
of perfect fluid
that cosmologists
worry about, and that's
the cosmological constant.
So a cosmological
constant has pressure
equal to minus the density.
This corresponds to
an equation of state
parameter W equal to minus 1.
If I go to my form here, I
plug in W equals minus 1,
rho goes as a to the
0th power, a constant.
Well, it is a cosmological
constant, after all,
so that shouldn't
be too surprising.
This is a very interesting
one because it is basically
telling us that the amount
of energy in spacial slices--
the energy density
does not change.
The amount of energy
appears to be Increasing.
Now bear in mind, it's hard
to define the total energy,
we cannot really in a covariant
way add up energy at various
different kinds of points.
Local energy is still
being conserved,
but there's no question that
this guy is a little weird.
So one of the next
things that we want to do
is take this stuff, run it
through Einstein's equations.
Einstein's equations,
of course, give us
the Friedmann equations.
And solve to see what the
expansion the universe
looks like.
We saw already that if the
density of the universe
relative the critical density
is either higher, the same,
or lower, that tells us about
the value of this k parameter,
or rather, the kappa parameter.
We haven't yet seen how to
solve for the scale factor a.
However, we have the
two Friedmann equations,
and if nothing else, write them
down, write out your stuff,
you got yourself a
system of equations,
Odin gave us
mathematica-- attack.
To give you some intuition
as to what you end up
seeing when you look at
these kind of solutions,
let me look at the
simplest kind of universes
that we can solve this way.
So let's examine what I will
call a monospecies universe--
in other words, a universe
that only contains
one of these forms of matter
that I have described here,
one of these sources of stress
energy that I described here.
And for simplicity, I'm going
to take it to be spatially flat.
Neither of these two
conditions are true in general,
but they are--
they're fine for us
to wrap our heads
around what the characteristics
of the solutions look like.
So in this limit, Friedmann 1
becomes a dot over a squared
equals 8 pi g rho over 3.
I can borrow this form that
I've got here to write this as 8
pi over 3 rho at some
particular moment times scale
factor to the minus n.
So what I'm doing here
is I'm assuming a0 now.
My a right now is 1,
and I'm defining n
to be 3 times 1 plus W.
This is easy enough
to solve for.
OK.
Take the square root.
What you find is--
I'm going to just
sort of-- there's
a constant you guys can work
out on your own if you like.
a dot must be proportional
to a 1 minus n over 2,
or a is proportional
to t to the 2 over n.
n equals 0 is a
special case that we'll
talk about in just a moment.
If we are dealing
with what we call
a matter-dominated
universe, well,
in this sort of monospecies,
spatially flat form,
this would have W
equal 0, n equals
3, and a scale factor that
grows as t to the 2/3 power.
A matter-dominated universe
is one that expands,
but it expands with
this kind of a loss.
So it slows down with time.
A radiation-dominated
universe, W equals 1/3, n
equals 4, that is a
universe that expands
as the square root of t.
What about my
cosmological constant?
Ah.
OK, well that's a problem--
n equals 0, and my solution
doesn't work for that one.
So what you do is just--
let's just go back to
our F or W equations.
Or rather, our
Friedmann equations.
Let's write F1 down again.
I have a dot over
a equals 8 pi--
whoops-- 8 pi g rho over 3,
and this is now a constant.
Rho equals rho 0
because it's a constant.
I can rewrite this in terms
of the cosmological constant
lambda.
And so another way
to write this is--
sorry about that-- a dot
a-- a dot over a squared
equals this, which
equals lambda over 3.
So this leads to an
exponential solution.
Now our real universe is
not as simple as these three
illustrative cases that
I have put in here just
to illustrate what the
extremes look like, OK?
We are not a
monospecies universe,
we have a mixture of matter,
we have a mixture of radiation,
we appear to have
something that smells
a lot like a
cosmological constant,
although the jury is still out
if one is being perfectly fair.
Work is ongoing.
What you need to do in general
is sort of model things.
You try to make models of
the universe that correspond
to different mixtures of
things that can go into it,
and then you go through
and ask yourself,
do the observables that emerge
in this universe match what
we see?
Generally you will
see sort of trends
that are similar to
this that emerge, right?
There might be a particular
epoch where matter
is more important,
there might be
an epoch where radiation
is more important,
there might be an epoch where
cosmological constant is
more important.
And so you might see sort
of you know transitions
between these things where
it's mostly a square root t
expansion, and then
something happens
and the radiation
becomes less important,
there's an intermediate regime
where both are playing a role,
and then matter
becomes important
and it kicks over to a t to
the 2/3 kind of expansion
when it becomes
matter-dominated.
You don't want to assume
the universe is flat,
you need to do your analysis,
including a non-zero flatness
parameter in there, which
makes things a little bit
complicated.
So in the next
lecture, I am going
to talk a little bit about
how one extracts observables
from these spacetimes.
How is it that we are able
to actually go into an FRW
universe and measure
things-- what can we measure,
how can we use
those measurements
to learn about the energy
budget of our universe
and formulate cosmology as
an observational and physical
science?
And with that, I will
end this lecture.
