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SCOTT HUGHES: So in the lectures
I'm going to record today,
we're going to
conclude cosmology.
And then we're going to begin
talking about a another system
in which we saw the
Einstein field equations
using asymmetry.
So this is one
where we will again
consider systems that
are spherically symmetric
but we're going to consider
them to be compact.
In other words,
things that are sort
of, rather than filling
the whole universe,
the spacetime that arises
from a source that's
localized in some
particular region in space.
So let me just do a
quick recap of what
we did in the previous
lecture that I recorded.
So by arguing solely
from the basis of how
to make a spacetime that is as
symmetric as possible in space
but has a time asymmetry,
so the past and the future
look different, we came up with
the Robertson-Walker metric,
which I will call the RW metric,
which has the form I've written
there on the top one line.
There were actually
several forms for this.
I've actually written down two
variants of it on the board
here.
Key thing which I
want to highlight
is that there is, in
the way I've written it
on the first line, there's a
hidden scale in there, r zero,
which you can think
of as, essentially,
setting an overall length
scale to everything
that could be measured.
There is a parameter,
either k or kappa.
k is either minus 1, 0, or 1.
Kappa is just k with a factor
of that length scale squared
thrown in there.
We define the overall
scale factor to be--
it's a number.
And we define it so that
its value is 1 right now.
And then everything is scaled
to the way distances look
at the present time.
A second form of that--
excuse me-- which is essentially
the change of variables
is you change your
radial coordinates
to this parameter chi.
And depending on
the value of k, what
you find then is
that the relationship
between the radius r and the
coordinate chi, if k equals 1,
you have what we call
a closed universe.
And the radius is equal to
the sign of the parameter chi.
If you have an open
universe, k equals minus 1,
then the radius is the
sinh of that parameter chi.
And for a flat
universe, k equals 0,
they are simply 1 is just the
other modulo, a choice of units
with the R of 0 there.
All this is just geometry.
OK.
So when you write
this down, you are
agnostic about the form of a
and you have no information
about the value of k.
So to get more insight
into what's going on,
you need to couple
this to your source.
And so we take these things.
And using the Einstein
field equations,
you equate these to a perfect
fluid stress energy tensor.
What pops out of that
are a pair of equations
that arise in the
Einstein field equations.
I call these F1 and F2.
These are the two
Friedmann equations.
F1 tells me about the velocity
associated with that expansion
premise.
There's an a dot
divided by a squared.
We call that the Hubble
parameter, h of a squared.
And it's related to the density
of the source of your universe
as well as this kappa term.
OK.
And as we saw last time,
we can get some information
about kappa from this equation.
The second Friedmann equation
relates the acceleration
of this expansion
term, a double dot--
dot, by the way, d by dy.
A double dot divided
by a is simply
related to a quantity
that is the density plus 3
times the pressure of this
perfect fluid that makes up
our universe.
We also find by requiring that
local energy conservation be
held, in other words that
your stress energy tensor be
divergence free, we
have a constraint that
relates the amount of energy--
the rate of change of energy
in a fiducial volume--
to the negative pressure
times the rate of change
of that fiducial volume.
And this, as I discussed
in the last lecture,
is essentially nothing
more than the first law
of thermodynamics.
It's written up
in fancy language
appropriate to a
cosmological spacetime.
As we move forward, we
find it useful to make
a couple of definitions.
So if you divide the
Hubble parameter squared
by Newton's
gravitational constant,
that's got the
dimensions of density.
And so we're going to define
a critical density to be 3h
squared over 8 pi g.
And we're going to define
density parameters, omega,
as the actual physical
density is normalized
to that critical density.
And when you do this, you
find that the critical--
the first Friedmann equation
can be written as omega--
oh, that's a typo--
omega plus omega
curvature equals
1 where omega curvature--
pardon the typo here.
Omega curvature is not
actually related to identity
but it sort of plays
one in this equation.
It is just a parameter that
has the proper dimensions
to do what is necessary
to fit this equation.
And it only depends on what
the curvature parameter is.
So remember that this
kappa is essentially
minus 1, 0, or 1 module of
factor of my overall scale.
So this is either a
positive number, 0,
or a negative number.
All right.
So let's carry things
forward from here.
Mute my computer so I'm not
distracted by things coming in.
We now have everything we
need using this framework
to build a universe.
Let's write down the
recipe to build a universe,
or I should say to build
a model of the universe.
So first thing you do is
pick your spatial curvature.
So pick the parameter k
to be minus 1, 0, or 1.
Pick a mixture of
species that contribute
to the energy density budget
of your model universe.
So what you would say is that
the total density of things
in your universe is a sum
over whatever mixture of stuff
is in your universe.
You will find it helpful to
specify an equation of state
for each species.
Cosmologists typically
choose equation of state
that has the following form.
OK.
So you require your species--
if you follow this model
that most cosmologists use,
each species will
have a pressure
that is linear in the density.
If you do choose
that form, then when
you enforce local conservation
of energy, what you will then
find is that for every
one of your species,
there is a simple relationship.
There's a simple
differential equation
that governs how
that species evolves
as the scale factor changes.
This can be immediately
integrated up
to find that the amount of,
at some particular moment,
the density of species i
depends on how it looks.
So 0 again denotes now.
It is simply proportional to
some power of the scale factor
where the power that enters
here can be simply calculated
given that equation
of state parameter.
Once you have these things
together, you're ready to roll.
You've got your
Friedmann equations.
You've got all the
constraints and information
you need to dig into
these equations.
Sit down, make your
models, have a party.
Now what we really want to do--
OK.
We are physicists.
And our goal in
doing these models
is to come up with some kind of
a description of the universe
and compare it with
our data so that we
can see what is the
nature of the universe
that we actually live in.
OK.
So what is of interest to
us is how does varying all
these different terms change--
well, we're going to talk
about various observables.
But really, if you
think about this model,
how does it change the
evolution of the scale factor?
Everything is bound up in that.
That is the key thing.
OK.
So if I can make a
mathematical model that
describes a universe with all
these various different kinds
of ingredients, yeah, I
can sit down and make--
if I make a really
complicated thing,
I probably can't do this
analytically so that's fine.
I will make a
differential equation
integrator that solves all these
coupled differential equations.
And I will then make predictions
for how AFT evolves depending
upon what the different
mixtures of species are,
what the curvature
term is equal to,
all those together,
and make my model.
But my goal as a physicist is
then to compare this to data.
And so what I need to do is to
come up with some kind of a way
this will then be useful.
We then need some kind of
an observational surrogate
for a of t.
What I would like to
be able to do is say,
great, model A predicts
the following evolution
of the scale factor.
Model B predicts this
evolution of the scale factor.
Can I look at the
universe and deduce
whether we are closer to
model A or closer to Model B?
And in order to do
that, I need to know
how do I measure a of t.
And as we're going
to see, this really
boils down to two things.
I need to be able
to deduce if I look
at some event in the universe,
if I look at something,
I want to know what scale
factor to associate with that.
I need to measure a.
And I need to know what
t to label that a with.
So this kind of sounds
like I'm reading
from the journal of duh.
But if I want to do this, what
it basically boils down to is
I need to know how to associate
an a with things that I measure
and how to associate the t
with the things that I measure.
That's what we're going
to talk about today.
What are the actual
observational surrogates,
the ways in which we can
go out, point telescopes
and other instruments
at things in the sky,
and deduce what a of t is
for the kind of events,
the kind of things that
we are going to measure?
Let's talk about how I can
measure the scale factor first.
OK.
So let's ignore the fact
there's a t on here.
How can I measure a?
We saw a hint of this
in the previous lecture
that I recorded.
So recall-- pardon me.
I've got a bit of extra
junk here in my notes.
Recall that in my
previous lecture,
we looked at the way different
kinds of densities behaved.
But I've got the
results right here.
For matter which had an
equation state parameter of 0,
what we found was that
the density associated
with that matter, it just
fell as the scale factor
to the inverse third power.
That's essentially saying
that the number of particles
of stuff is constant.
And so as the universe
expands, it just
goes with the volume
of the universe.
If it was radiation, we found
it went as a scale factor
to the inverse fourth power.
And that's consistent
with diluting the density
as the volume gets larger
provided we also decrease
the energy per particle
of radiation per photon,
per graviton, per whateveron.
If I require that the energy
per quantum of radiation
is redshifted with this thing,
that explains the density flaw
that we found for radiation.
And so that sort of suggests
that what we're going to find
is that the scale
factor is directly
tied to a redshift measure.
OK.
I just realized what
this page of notes was.
My apologies.
I'm getting myself
organized here.
Let's make that a little
bit more rigorous now.
OK.
So that argument on the basis
of how the density of radiation
behaves.
It's not a bad one
as a first pass.
It is quite indicative.
But let's come at it from
another point of view.
And this allows me to
introduce in a brief
aside a topic that
is quite useful here.
So we talked about Killing
vectors a couple of weeks ago.
Let's now talk about a
generalization of this
known as Killing tensors.
So recall that a
Killing vector was
defined as a particular
vector in my space time
manifold such that
if I Lie transport
the metric along that
Killing vector, I get 0.
This then leads to the
statement that if I put together
the symmetrized covariant
gradient of the Killing vector,
I get 0.
Another way to write this
is to use this notation.
Whoops.
OK.
So these are equations
that tell us about the way
that Killing vectors behave.
A Killing tensor is
just a generalization
of this idea to an object
that has more than one index
line, to a higher
rank tensorial object.
So we consider this to be
a rank one Killing tensor.
A rank n Killing
tensor satisfies--
so let's say k is
my Killing tensor.
Imagine I have n
indices here if I
take the covariant gradient
of that Killing tensor
and I symmetrize
over all n indices.
That gives me 0.
This defines a Killing tensor.
Starting with this
definition, it's not at all
hard to show that if I
define a parameter, k,
which is what I get when
I contract the Killing
tensor, every one of its
indices with the four-velocity
of a geodesic.
If my u satisfies the geodesic
equation-- or this could be--
let's write this as a momentum.
Which you say is the
tangent to a world line.
Could be either a
velocity or a momentum.
So if I define the scalar
k by contracting my Killing
tensor with n copies of
tangent to the world line,
and that thing satisfies
the geodesic equation,
then the following is true.
You guys did this on
a homework exercise
for when we thought about a
spacetime-- you did something
similar to this, I should say,
for a spacetime containing
an electromagnetic field.
We talked about how this works
for the case of a Killing
vector.
Hopefully you can
kind of see the way
you would do this
calculation at this point.
Now the reason I'm
doing this aside
is that if you have a
Friedmann-Robertson-Walker
spacetime, search
spacetimes actually
have a very useful
Killing tensor.
So let's define k with
two indices, mu nu.
And this is just given
by the scale factor.
Multiplying the metric
u mu, u mu, u nu.
Where this u comes
from the four-velocity
of a co-moving fluid element.
So this is the
four-velocity that we
use to construct
the stress energy
tensor that is the source
of our Friedmann equations.
So here's how we're
going to use this.
Let's look at what we get
for this Killing vector.
Excuse me, this Killing
tensor when I consider
it's a long a null geodesic.
We're going to want to think
about null geodesics a lot,
because the way that we are
going to probe our universe
is with radiation.
We're going to look at it
with things like telescopes.
These days people are
starting to probe it
with things like
gravitational wave detectors.
All things that involve
radiation that moves on null
geodesics.
So let's examine the
associated conserved quantity
that is associated
with a null geodesic.
So let's say v--
let's make it a p, actually.
So it's going to
be a null geodesic,
so we're going to imagine it's
radiation that is following.
It has a four-momentum, pu.
And let's define k, case of
ng, from my null geodesic.
That is going to be
k mu nu, p mu, p nu.
Let's plug-in the definition
of my Killing tensor.
So this is a square root
of t, g mu nu, p nu, p nu.
This is zero.
It's a null geodesic.
Then I get u mu p nu, u nu p mu.
Now remind you of something.
Go back to a nice
little Easter egg,
an exercise you guys
did a long time ago.
If I look at the dot
product of a four-momentum
and a four-velocity, what I
get is the energy associated
with that four-momentum as
measured by the observer whose
four-velocity is u.
So what we get
here is two copies
of the energy of that
null geodesic measured
by the observer
who is co-moving.
So what this null geodesic--
what this quantity associated
with this null geodesic
is two powers of the scale
factor times the energy that
would be measured by
someone who is co-moving
with the fluid that
fills my universe.
Energy of p mu, as
measured by u mu.
And remember, this
is a constant.
So as this radiation travels
across the universe--
as this radiation travels
across the universe,
the product of the scale
factor and the energy
associated with that
radiation as measured
by co-moving observers
is a constant.
So this is telling
us that the energy,
as measured by a
co-moving observer,
let's say it is emitted at some
time with a scale factor is a.
When it propagates to us, we
define our scale factor as 1,
the energy will have fallen
down by a factor of 1 over a.
So this makes it a little bit
more rigorous, this intuitive
argument that we
saw from considering
how the density of
radiation fell off.
What we see is the energy
is indeed redshifting
with the scale factor.
So if I use the fact that
the energy that I observe--
if I'm measuring light, light
has a frequency of omega--
what I see is the omega
that I observe at my scale
factor, which I define to
be 1, normalized to that
when it was emitted, it
looks like the scale factor
when it was admitted.
Divided by a now, a observed.
I call this 1.
I can flip this
over, another way
of saying this is
that, if I write it
in terms of wavelengths of
the radiation, the wavelength
of the radiation and when it was
emitted versus the wavelength
that we observe it tells
me about the scale factor
when the radiation was emitted.
Astronomers like to
work with redshift.
They like to work
with wavelength
when they study things
like the spectra
of distant astronomical objects.
And they use it to define
a notion of redshift.
So we define the redshift
z to be the wavelength
that we observe, minus the
wavelength at the radiation
that when it is emitted
divided by the wavelength when
it was emitted.
Put all of these
definitions together,
and what this tells
me is that the scale
factor at which the
radiation was emitted
is simply related to the
redshift that we observe.
So this at last gives us
a direct and not terribly
difficult to use
observational proxy
that directly encodes the
scale factor of our universe.
Suppose we measure the spectrum
of radiation from some source,
and we see the
distinct fingerprint
associated with emission
from a particular set
of atomic transitions.
What we generally find is
some well-known fingerprints
of well-characterized
transitions,
but in general they are
stretched by some factor
that we call the redshift z.
Actually, you usually stretch
by-- when you go through this,
you'll find that
what you measure
is actually stretched
by 1 plus z.
You measure that,
you have measured
the scale factor at which
this radiation was measured--
was emitted.
So this is beautiful.
This is a way in which
the universe hands
us the tool by which we can
directly characterize some
of the geometry of the
universe at which light
has been emitted.
This is actually
one of the reasons
why a lot of people who do
observational cosmology also
happen to be expert
atomic spectroscopists.
Because you want to know
to very high precision what
is the characteristics of
the hydrogen Balmer lines.
Some of the most important
sources for doing these
tend to be galaxies
in which there's
a lot of matter falling onto
black holes, some of the topics
we'll be talking about
in an upcoming video.
As that material
falls in, it gets hot,
it generates a lot of
radiation, and you'll
see things like transition
lines associated
with carbon and iron.
But often all reddened
by a factor of several.
You sort of go,
oh, look at that,
carbon falling onto a
black hole at redshift 4.8.
That is happening at
a time when the scale
factor of the universe
was 1 over 4.8--
or 1 over 5.8, forgot my
factor of 1 plus there.
So you measure the
redshift, and you
have measured the scale factor.
But you don't know when
that light was emitted.
We need to connect
the scale factor
that we can measure so
directly and so beautifully
to the time at which
it was emitted.
We now have a way
of determining a,
but we need a as
a function of t.
And in truth, we do this
kind of via a surrogate.
Because we are using
radiation as our tool
for probing the scale
factor, we really
don't measure t directly.
When we look at light
and it's coming to us,
it doesn't say I
was emitted on March
27th of the year
negative 6.8 billion
BC, or something like that.
We do know, though,
that it traveled
towards us at the speed of
light on a null geodesic.
And because it's
a null geodesic,
there's a very simple--
simple's a little bit
of an overstatement,
but there is at least a
calculable connection.
Because it's moving at the
speed of light, we can simply--
I should stop using that word--
we can connect time to space.
And so rather than
directly determining
the time at which the
radiation was emitted,
we want to calculate the
distance of the source from us
that emitted it.
So rather than directly
building up a of t,
we're going to build
up an a of d, where d
is the distance of the source.
And if you're used to working
in Euclidean geometry,
you sort of go, ah, OK, great.
I know that light travels
at the speed of light,
so all I need to do is
divide the distance by c,
and I've got the time,
and I build a of t.
Conceptually, that
is roughly right,
and that gives at least a
cartoon of the idea that's
going on here.
But we have to be a
little bit careful.
Because it turns out when
you are making measurements
in a curved space time, the
notion of distance that you use
depends on how you make
the distance measurement.
So this leads us now to our
discussion of distance measures
in cosmological spacetime.
So just to give a
little bit of intuition
as to what's the
kind of calculation
we're going to need to do,
let me describe one distance
measure that is observation,
which is about useless,
but not a bad thing
to at least begin
to get a handle on, the
way different parameters
of the spacetime
come in and influence
what the distance measure is.
So let's just think about
the proper distance from us
to a source.
So let's imagine that--
well, let's just begin by
first, let's write down
my line element.
And here's the form
that I'm going to use.
OK, so here's my line element.
This is my
differential connection
between two events spaced
between one another by dt,
d chi, d theta, d phi, all
hidden in that angular element,
the omega.
Let's imagine that
we want to consider
two sources that are separated
purely in the radial direction.
So my angular displacement
between the two events
is going to be 0.
So the only thing I need
to care about is d chi,
and let's imagine
that I determine
the distance between these
two at some instant--
So then you just get
ds squared equals
a squared or zero squared d chi
squared, and you can integrate
this up and you get our
first distance measure,
d sub p equals scale factor.
Your overall distance
scale are zero and chi.
So Carroll's textbook calls
this the instantaneous physical
distance.
Let's think about what
this means if you do this.
This is basically
the distance you
would get if you
took a yardstick,
you put one end at
yourself, you're
going to call
yourself chi equals 0,
you put the other
end of the yardstick
at the object in your
universe at some distance
chi in these coordinates,
and that's the distance.
d sub p is the distance
that you measure.
It is done, you're
sort of imagining
that both of the events at
the end of this yardstick.
You're sort of ascertaining
their position at exactly
the same instant, hence
the term instantaneous,
and you get something
out of it that
encodes some important
aspects of how we think
about distances in cosmology.
So notice everything scales
with the overall length scale
that we associated
with our spatial slices
with our spatial sector
of this metric, the r0.
Notice that whatever is going
on your scale factor, your a,
your distance is
going to track that.
As a consequence of
this, two objects
that are sitting in
what we call the Hubble
flow, in other words,
two objects that
are co-moving with the
fluid that makes up
the source of our universe.
They have an apparent motion
with respect to each other.
If I take the time
derivative of this,
the apparent is just
a dot, or 0 chi,
which is equal to the
Hubble parameter times dp.
Recall that the Hubble
parameter is a dot over a,
and if I'm doing this
right now, that's
the value of the
Hubble parameter now.
So this is the Hubble Expansion
Law, the very famous Hubble
Expansion Law.
So we can see it
hidden in this--
not even really
hidden, it's quite
apparent in this notion of
an instantaneous physical
distance.
Let me just finally
emphasize, though,
that the instantanaeity
that is part
of this object's name,
instantaneous measurements,
are not done.
As I said-- I mean,
this is it sounds
like I'm being slightly
facetious, but it's not.
The meaning of this
distance measure
is, like I said, it's
a yardstick where
I have an event at me, the
other end of my yardstick
is at my cosmological event.
Those are typically
separated by millions,
billions of light years.
Even if you could--
OK, the facetious bit was
imagining it as a yardstick.
But the non-facetious
point I want to make
is we do not make instantaneous
measurements with that.
When I measure an event that is
billions of light years away,
I am of course
measuring it using light
and I'm seeing light that was
emitted billions of years ago.
So we need to think
a little bit more
carefully about how
to define distance
in terms of quantities
that really correspond
to measurements we can make.
And to get a little
intuition, here
are three ways where, if you
were living in Euclidean space
and you were looking at
light from distant objects,
here are three ways that
you could define distance.
So if spacetime were that
of special relativity--
well, let's just say if
space were purely Euclidean.
Let's just leave it like that.
Here are three notions
that we could use.
One, imagine there was
some source of radiation
in the universe that you
understood so well that you
knew its intrinsic luminosity.
What you could do is compare
the intrinsic luminosity
of a source to its
apparent brightness.
So let's let f be the flux
we measure from the source.
This will be related to
l, the luminosity, which--
suspend disbelief
for a moment, we
want to imagine that we
know it for some reason.
And if we imagine this
is an isotropic emitter,
this will be related
by a factor of 4 pi,
and the distance between
us and that source.
Let's call this d sub l.
This is a luminosity distance.
It is a distance that
we measure by inferring
the behavior of luminosity
of a distant object.
Now it turns out, and this is a
subject for a different class,
but nature actually
gives us some objects
whose luminosity is known or
at least can be calibrated.
In the case of much of what
is done in cosmology today,
we can take advantage of the
behavior of certain stars
whose luminosity is strongly
correlated to the way that--
these are stars whose
luminosity is variable,
and we can use the fact that
their variability is correlated
to their luminosity to infer
what their absolute luminosity
actually is.
There are other supernova
events whose luminosity
likewise appears to
follow a universal law.
It's related to the
fact that the properties
of those explosions are
actually set by the microphysics
of the stars that set them.
More recently, we've been
able to exploit the fact
that gravitational wave sources
have an intrinsic luminosity
in gravitational waves,
the dedt associated
with the gravitational
waves that they emit,
which depends on the source
gravitational physics in a very
simple and predictable
way that doesn't
depend on very many parameters.
I actually did a
little bit of work
on that over the
course of my career,
and it's a very
exciting development
that we can now use
these as a way of setting
the intrinsic luminosity
of certain sources.
At any rate, if you can take
advantage of these objects that
have a known luminosity
and you can then
measure the flux of
radiation in your detector
from these things, you
have learned the distance.
At least you have learned
this particular measure
of the distance.
That's measure one.
Measure two is imagine
you have some object
in the sky that has a particular
intrinsic size associated
with it.
You can sort of think of the
objects whose luminosity you
know about as standard candles.
Imagine if nature builds
standard yardsticks,
there's some object whose
size you always know.
Well, let's compare that
physical size to the angular
size that you measure.
The angle that the object
sub tends in the sky
is going to be that intrinsic
size, delta l divided
by the distance.
We'll call this distance d sub
a, for the angular diameter
distance.
Believe it or not,
nature, in fact,
provides standard
yardsticks type
so that we can actually do this.
Finally, at least as
a matter of principle,
imagine you had
some object that's
moving across the sky with
a speed that you know.
You could compare
that transverse speed
to an apparent angular speed.
So the theta dot,
the angular speed
that you measure, that
would be the velocity
perpendicular to your line
of sight divided by distance.
We'll call this d sub m,
the proper motion distance.
So if our universe
were Euclidean,
not only would it be easy
for us to use these three
measures of distance,
all three of them
would give the same
result. Because this is all
just geometry.
Turns out when you study
these notions of distance,
in an FRW spacetime, there's
some variation that enters.
Let me just emphasize here that
there is an excellent summary
on this stuff.
I can't remember if I linked
this to the course website
or not.
I should and I shall.
An excellent
summary of all this,
really emphasizing
observationally significant
aspects of these things
come from the article that
is on the archive called
Distance Measures in Cosmology
by David Hogg, a colleague
at New York University.
You can find this on the
Astro PH archive, 9905116.
It's hard for me to believe
this is almost 21 years old now.
This is a gem of a paper.
Hogg never submitted
it to any journal,
just posted it on the
archive so that the community
could take advantage of it.
So the textbook by Carroll goes
through the calculation of d
sub l.
On a problem set
you will do d sub m.
We're going to go
through d sub a,
just so you can see some
of the way that this works.
Let me emphasize one thing,
all of these measures
use the first
Friedmann equation.
So writing your Friedmann
equation like so.
i is a sum of all
the different species
of things that can contribute
including curvature.
So recall that even though
curvature isn't really
a density, you can
combine enough factors
to make it act as though
it were a density.
You assume a power law.
So this n sub i is related to
the equation of state parameter
for each one of these species.
And let's now take
advantage of the fact
that we know the scale factor
directly ties to redshift.
I can rewrite this as
how the density evolves
as a function of redshift.
So when you put all
of this together,
this allows us to
write h of a as h of z.
This is given by the
Hubble parameter now
times some function e of z.
And that is simply--
you divide everything
out to normalize
to the critical
density, and that
is a sum over all these
densities with the redshift
waiting like so.
Let's really do an
example, like I said.
So if you read Carroll, you
will see the calculation
of the luminosity distance.
If you do the cosmology
problem set that will be--
I believe its p set 8, you
will explore the proper motion
distance.
So let's do the angular
diameter distance.
Seeing someone work
through this, I think,
is probably useful for
helping to solidify
the way in which these
distance measures work
and how it is that one can tie
together important observables.
So let's consider
some source that we
observe that, according to us,
subtends an angle, delta phi.
Every spacial sector is
spherically symmetric,
and so we can orient
our coordinate system
so that this thing, it would
be sort of a standard ruler--
what we're going to do is
orient our coordinate system
so that object lies in the
theta equals pi over 2 plane.
The proper size of that source--
so the thing is just
sitting in the sky there--
the proper size
of the source, you
can get this in
the line element.
The delta l of
the source will be
the scale factor at the time
at which it is emitted, r0.
So this is using one of the
forms of the FRW line elements
I wrote down at the
beginning of this lecture.
And so the angular
diameter distance,
that's a quantity that
I've defined over here,
it's just the ratio of
this length to that angle.
Let's rewrite this
using the redshift.
Redshift is something that
I actually directly observe,
so there we go.
This is not wrong,
but it's flawed.
So this is true.
This is absolutely true.
Here's the problem, I don't
know the overall scale
of my universe and this
coordinate chi doesn't really
have an observable
meaning to it.
It's how I label events, but I
look at some quasar in the sky
and I'm like, what's your chi?
So what we need to do is
reformulate the numerator
of this expression in such a
way as to get rid of that chi
and then see what happens,
see if we have a way
to get rid of that r0.
Let's worry about chi first.
We're going to eliminate it by
taking advantage of the fact
that the radiation
I am measuring
comes to me on a null path.
Not just any null path.
I'm going imagine
it's a radial one.
We are allowed to be somewhat
self-centered in defining
FRW cosmology, we put
ourselves at the origin.
So any light that
reaches us moves
on a purely radial trajectory
from its source to us.
So looking at how the time
and the radial coordinate chi
are related for a radial null
path we go into our FRW metric.
I get this.
So I can integrate this up
to figure out what chi is.
So this is right.
Let's massage it a
little bit to put it
in a form that's a little
bit more useful to us.
Let's change our variable--
change our variable of
integration from time to a.
So this will be an integral
from the scale factor at which
the radiation is emitted
to the scale factor
which we observe it, i.e. now.
And when you do that
change of variables
your integral changes like so.
Let's rewrite this once more
to insert my Hubble parameter.
Now let's change
variables once more.
We're going to use the fact
that our direct measurable is
redshift.
And so if we use a
equals 1 over 1 plus z,
I can further write this as an
integral over redshift like so.
And that h0 can come
out of my integral.
So this is in a form that
is now finally formulated
in terms of an
observable redshift
and my model
dependent parameters.
The various omegas that, when
I construct my universe model,
I am free to set.
Or if I am a
phenomenologist, that
are going to be
knobs that I turn
to try to design a model
universe that matches
the data that I am measuring.
r0, though, is still
kind of annoying.
We don't know what
this guy is, so what
we do is eliminate r0 in
favor of a curvature density
parameter.
So using the fact
that omega curvature--
go back to how this was
originally defined--
it was negative kappa
over h0 squared.
That's negative k over
r0 squared h0 squared.
That tells me that r0 is
the Hubble constant now,
divided by the square
root of the absolute value
of the curvature, at least
if k equals plus or minus 1.
What happens when it's not plus
or minus 1, if it's equal to 0?
Well, hold that thought.
So let's put all
these pieces together.
So assembling all
the ingredients
I have here, what we find is
the angular diameter distance.
There's a factor
of 1 over 1 plus z,
1 over the Hubble constant.
Remember, Hubble has
units of 1 over length--
excuse me, 1 over time,
and with the factor
of the speed of
light that is a 1
over length, so one over
the Hubble parameter now
is essentially a kind of
fiducial overall distance
scale.
And then our solution breaks
up into three branches,
depending upon whether k
equals minus 1, 0, or 1.
So you get one term where it
involves minus square root
the absolute value of the
curvature parameter times sine.
That same absolute value of
the curvature parameters square
root.
So here's your k
equals plus 1 branch.
For your k equals 0
branch, basically what
you'll find when you
plug-in your s of k
is that r0 cancels out.
So that ends up
being a parameter
you do not need to worry
about, and I suggest you just
work through the
algebra and you'll
find that for k equals 0,
it simply looks like this.
And then finally, if you
are in an open universe--
that is supposed
to be curvature--
what we get is this.
So this is a
distance measure that
tells me how angular
diameter distance depends
on observable parameters.
Hubble is something
that we can measure.
Redshift is something
we can measure.
And it depends on
model parameters,
the different densities
that go into e of z, and--
which I have on the
board right here--
the different densities
that go into e of z.
My apologies, I left
out that h0 there--
and your choice
of the curvature.
When you analyze
these three distances
here is what you find.
You find that the
luminosity distance
is related to the proper motion
distance by a factor of 1
plus z, and that's related to
the angular diameter distance
by a factor of 1 plus z squared.
So when you read Carroll, you
will find that 1 plus z factor
there--
excuse me, 1 plus z to the minus
1 power turns into a 1 plus z--
is the camera not looking at me?
Hello?
There we go.
So that 1 over 1 plus z
turns into a 1 plus z.
When you do it on the p set, you
do the proper motion distance,
so it will just be no 1 plus z
factor in front of everything.
So the name of the game
when one is doing cosmology
as a physicist is
to find quantities
that you can measure that
allow you to determine
luminosity distances,
angular diameter distances,
proper motion distances.
Now it turns out that the
proper motion distance
is not a very practical
one for basically
any cosmologically
interesting source.
They are simply so far
away that even for a source
moving essentially at
the speed of light,
the amount of
angular motion that
can be seen over essentially a
human lifetime is negligible.
So this turns into
something that--
hi.
MIT POLICE: [INAUDIBLE]
SCOTT HUGHES: That's OK.
Yeah, I'm doing some
pre-recording of lectures.
[LAUGHS] I was warned
you guys might come by.
I have my ID with me and
things like that, so.
MIT POLICE: That's fine.
Take care.
MIT POLICE: You look official.
SCOTT HUGHES: [LAUGHS]
I appreciate it.
So those of you watching the
video at home, as you can see,
the MIT police is
keeping us safe.
Scared the crap out
of me for a second
there, but it's all good.
All right, so let's go
back to this for a second.
So the proper motion
distance is something
that is not
particularly practical,
because as I said, even if you
have an object that is moving
close to the speed of
light this is not something
that even over the course
of a human lifetime
you are likely to see
significant angular motion.
So this is generally not used.
But luminosity distances and
angular diameter distances,
that is, in fact,
extremely important,
and a lot of cosmology is based
on looking for well understood
objects where we can
calibrate the physical size
and infer the angular
diameter distance,
or we know the
intrinsic brightness
and we can determine
the luminosity distance.
So let me just give
a quick snapshot
of where the measurements come
from in modern cosmology that
are driving our
cosmological model.
One of the most important is
the cosmic microwave background.
So, vastly oversimplifying, when
we look at the cosmic microwave
background after removing
things like the flow
of our solar system with respect
to the rest frame of-- excuse
me, the co-moving
reference frame
of the cause of the
fluid that makes up
our universe, the size
of hot and cold spots
is a standard ruler.
By looking at the
distribution of sizes
that we see from
these things, we
can determine the
angular diameter distance
to the cosmic
microwave background
with very high precision.
This ends up being one of the
most important constraints
on determining what the
curvature parameter actually
is.
And it is largely thanks to
the cosmic microwave background
that current prejudice, I would
say, the current best wisdom--
choose your descriptor
as you wish--
is that k equals
0 and our universe
is in fact spatially flat.
Second one is what are
called type 1a supernova.
These are essentially the
thermonuclear detonations
of white dwarfs.
Not even really
thermonuclear, it's just--
my apologies, I'm
confusing a different event
that involves white dwarfs.
The type 1a's are
not the thermonuclear
explosions of these things.
This is actually the core
collapse of a white dwarf star.
So this is what happens when
a white dwarf it creates
or in some way
accumulates enough mass
such that electron
degeneracy pressure is
no longer sufficient to hold it
against gravitational collapse,
and the whole thing basically
collapses into a neutron star.
That happens at a defined
mass, the Chandrasekhar mass,
named after one of my
scientific heroes, Subramanian
Chandrasekhar.
And because it has a defined
mass associated with it,
every event basically
has the same amount
of matter participating.
This is a standard candle.
So there's a couple
others that I'm not
going to talk about in
too much detail here.
While I'm erasing the board
I will just mention them.
So by looking at things like
the clustering of galaxies
we can measure the
distribution of mass
in the universe that allows
us to determine the omega m
parameter.
That's one of the
bits of information
that tells us that
much of the universe
is made of matter that
apparently does not
participate in standard
model processes
as we know them today--
the so-called dark
matter problem.
We can look at chemical
abundances, which tells us
about the behavior of
nuclear processes in the very
early universe.
And the last one which
I will mention here
is we can look at
nearby standard candles.
And nearby standard
candles allow
us to probe the local
Hubble law and determine h0.
"Probble," that's not a word.
If you combine "probe" and
"Hubble" you get "probble."
And when I say
nearby, that usually
means events that are merely
a few tens of millions
of light years away.
It's worth noting that all
of these various techniques,
all of these different things,
you can kind of even see it
when you think about the
mathematical form of everything
that went into our
distance measures,
they're all highly
entangled with each other.
And so to do this kind
of thing properly,
you need to take just
a crap load of data,
combine all of your
data sets, and do
a joint analysis of
everything, looking at the way
varying the parameters and
all the different models
affects the outcome
of your observables.
You also have to carefully
take into account the fact
that when you measure something,
you measure it with errors.
And so many of these
things are not known.
Turns out you can usually
measure redshift quite
precisely, but these
distances always
come with some error bar
associated with them.
And so that means that
the distance you associate
with a particular
redshift, which
is equivalent to associating
a time with a redshift,
there's some error bar on that.
And that can lead to
significant skew in what you
determine from things.
There's a lot more we could
say, but time is finite
and we need to
change topic, so I'm
going to conclude this lecture
by talking about two mysteries
in the cosmological
model that have been
the focus of a lot
of research attention
over the past several decades.
Two mysteries.
One, why is it that
our universe appears
to be flat, spatially flat?
So to frame why this
is a bit of a mystery,
you did just sort
of go, eh, come on.
You've got three choices for
the parameter, you got 0.
Let's begin by thinking about
the first Friedmann equation.
I can write this
like so, or I can
use this form, where I say
omega plus omega curvature--
I'm going to call
that omega c for now--
that equals 1.
The expectation had long
been that our universe would
basically be dominated by
various species of matter
and radiation for
much of its history,
especially in the
early universe.
If it was radiation
dominated, you'd
expect the density to
go as a to the minus 4.
If it's matter
dominated, you expect
it to go as a to the minus 3.
Now, your curvature density
goes as a to the minus 2.
And so what this
means is that if you
look at the ratio of
omega curvature to omega,
this will be proportional
to a, for matter,
a squared for radiation.
If your universe--
in some sense,
looking at these parameter
k of the minus 1, 0, 1,
that's a little bit misleading.
It's probably a little
bit more useful to think
about things in terms
of the kappa parameter.
And when you look at
that, your flat universe
is a set of measure 0 in the
set of all possible curvature
parameters that you could have.
And physicists tend
to get suspicious
when something that
could take on any range
of possible random values
between minus infinity
and infinity picks out zero.
That tends to tell
us that there may
be some principle at play
that actually derives things
to being 0.
Looking at it this
way, imagine you
have a universe
that at early times
is very close to being
flat, but not quite.
Any slight deviation
from flatness
grows as the universe expands.
That's mystery one.
Mystery two, why is the
cosmic microwave background
so homogeneous?
So when we look at the
cosmic microwave background,
we see that it has
the same properties.
The light has the
same brightness,
it has the same temperature
associated with it,
to within in a part in 100,000.
Now the standard
model of our universe
tells us that at
very early times
the universe was essentially
a dense hot plasma.
This thing cooled as
the universe expanded,
much the same way that
if you have a bag of gas,
you squeeze it very
rapidly, it will get hot,
you stretch it very
rapidly, it will cool.
There's a few more details
of this in my notes,
but when we look at this
one the things that we
see is that in a universe that
is only driven by matter or by
radiation--
so the matter dominated and
radiation dominated picture--
it shows us that points on
opposite sides of the sky
were actually out of causal
contact with each other
in the earliest moments
of the universe.
In other words, I
look at the sky,
and the patch of sky over
here was out of causal contact
with the patch of sky
over here in the earliest
days of the universe.
And yet they had the
same temperature,
which suggests that they
were in thermal equilibrium.
How can two disparate,
unconnected patch of the sky
have the same temperature
if they cannot exchange
information?
You could imagine it
being a coincidence
if one little bit of the
sky has the same temperature
as a bit of another
piece, but in fact,
when you do this calculation,
you find huge patch of the sky
could not communicate
with any other.
And so how then is it that the
entire sky that we can observe
has the same temperature
at the earliest
times within a part in 100,000.
You guys will explore this on--
I believe it's
problem set eight.
The solution to both of these
problems that has been proposed
is cosmic inflation.
So what you do is imagine
that at some earlier
moment in the
universe, our universe
was filled with
some strange field,
and I'll describe the
properties of that field
in just a moment, such
that it acted like it
had a cosmological constant.
In such an epic, the scale
factor of the universe
goes as exponentially with
the square root of the size
of the cosmological constant.
What you find when you look at
this is that this-- it still,
of course, goes as
a to the minus 2--
but a to the minus--
well, sorry.
Let me back up for a second.
So my scale factor in
this case you'll find
goes are a to the
minus 2, and that's
because the density associated
with the cosmological constant
remains constant.
So even if you start
in the early universe
with some random value
for the curvature,
if you are in this epic
of exponential inflation,
just for-- you know, you have
to worry about timescales
a little bit, but if
you do it long enough
you can drive this
very, very close to 0.
So much so that when
you then move forward,
let's say you come out of this
period of cosmic inflation
and you enter a universe that
is radiation dominated or matter
dominated, it will
then begin to grow,
but if you drive it sufficiently
close to 0 it doesn't matter.
You're never going to catch up
with what inflation did to you.
On the P set you will
also show that if you
have a period of
inflation like this, then
that also cures the problem
of piece of the sky being
out of causal contact.
So when you do
that, what you find
is that essentially everything
is in causal contact early on.
It may sort of come
out of causal contact
after inflation has ended,
more on that in just a moment,
and then things sort of
change as the universe
continues to evolve.
OK so it looks like
recording is back on.
My apologies, everyone.
So as I was in the
middle of talk,
I talked a little bit too long
in this particular lecture
so we're going to spill
over into a little bit
of an addendum, just a
five-ish minute piece that
goes a bit beyond this.
Doing this by myself is a
little bit weird, I'm tired,
and I will confess,
I got a little bit
rattled when the police
came in to check in on me.
Let's back up for a second.
So I was talking
about two mysteries
of the modern
cosmological model.
One of them is this question
of why the universe is
so apparently flat.
The spatial sector of the
universe appears to be flat.
And we had this expectation
that the universe
is either radiation dominated
or matter dominated,
which would give us the density
associated with radiation.
If it was radiation
dominated, then the density
of stuff in our universe would
fall off as the scale factor
to the fourth power.
If it's matter dominant,
it's scale factor
to the third power.
When you define a density
associated with the curvature,
it falls off as scale
factor to the second power.
And so if we look at the
ratio of the curvature density
to any other kind of density, it
grows as the universe expands.
So any slight deviation
from flatness we
would expect to grow.
And that's just confusing.
Why is it when we make the
various measurements that we
have been making for the
past several decades,
all the evidence is
pointing to a universe that
has a flatness of 0?
If you sort of imagine that
the parameter kappa can
be any number between
minus infinity to infinity,
why is nature picking out 0?
Another mystery is why is the
cosmic microwave background
so homogeneous?
We believe that the universe
was a very hot dense plasma
at very early times.
It cooled as the
universe expanded,
and when we measured the
radiation from that cooling
expanding ball of
plasma, what we find
is that it has the
same temperature
at every point in the sky
to within a part in 100,000.
But when one looks at the
behavior of how light moves
around in the early universe,
if the universe is matter
dominated or
radiation dominated,
what you find is that a piece
of the sky over here cannot
communicate with a
piece of sky over here.
Or over here, or over here.
You actually find that
the size of the sky
that, if I look at
a piece of sky over
here, how much of the
universe could it talk to,
it's surprisingly small.
So how is it that
the entire universe
has the same temperature?
How is that they are apparently
in thermal equilibrium,
even if they cannot
exchange information?
So I spent some while
talking to myself
after the cameras went out.
So I'll just sketch
what I wrote down.
A proposed solution to this,
to both of these mysteries,
is what is known as cosmic
inflation, something
that our own Alan Guth shares
a lot of the credit for helping
to develop.
So recall, if we have a
cosmological constant,
then the scale factor
grows exponentially
and the density of
stuff associated
with that cosmological
constant is constant.
As the universe expands,
the energy density
associated with that
cosmological constant
does not change.
If our universe is dominated
by such a constant,
then the ratio of density is
associated with curvature,
the density associated with
cosmological constant, actually
falls off inversely with
the curvature scale squared,
and the curvature scale
is growing exponentially,
it means that omega
c is being driven
to zero relative to the density
in cosmological constant, as e
to a factor like this.
It's being exponentially
driven close to 0.
You'd do do a little bit
of work to figure out
what the timescales
associated with this are,
but this suggests that if you
can put the universe in a state
where it looks like a
cosmological constant,
you can drive the density
associated with curvature as
close to 0 as you want.
Recall that a cosmological
constant is actually
equivalent to there
being a vacuum energy.
If we think about the
universe being filled
with some kind of a scalar
field at early times,
it can play the role of
such a vacuum energy.
Without going into
the details, one
finds that in an
expanding universe
there is an equation of
motion for that scalar field.
How the field itself behaves
is driven by a differential
equation, looks like this.
Here's your Hubble
parameters, so this
has to do with a
scale factor in here.
v is a potential for this
scalar field, which I'm not
going to say too much about.
Take this guy, couple it to
your Friedmann equations,
and the one that's
most important
is the first Friedmann equation.
What you see is that v of
phi is playing the role
of a cosmological constant.
It's playing the
role of a density.
So if we can put the
universe into a state where
it is in fact being dominated
by this scalar field,
by the potential associated
with the scalar field,
it will inflate.
A lot of the research in
early universe physics
that has gone on over
the past couple decades
has gone into understanding what
are the consequences of such a
such a potential.
Can we make such a potential?
Do the laws of physics permit
something like this to exist?
If the universe is in this
state early on, what changed?
How is it that
this thing evolves?
Is there is an
equation of motion
here so that scalar
field is presumably
evolving in some kind of a way?
Is there a potential,
does nature
permit us to have a field of the
sort with a potential that sort
of goes away after some time?
Once it goes away, what
happens to that field?
Is there any smoking gun
associated with this?
If we look at the
universe and this
is a plausible explanation
for why the universe appears
to be spatially flat and
why the cosmic microwave
background is so homogeneous.
Is there anything
else that we can
look at that would basically say
yes, the universe did in fact
have this kind of an expansion?
Without getting into
the weeds too much,
it turns out that if the
universe expanded like this,
we would expect there to
be a primordial background
of gravitational waves, very low
frequency gravitational waves.
Sort of a moaning background
filling the universe.
And so there's a
lot of experiments
looking for the imprints
of such gravitational waves
on our universe.
If it is measured,
it would allow
us to directly probe what this
inflationary potential actually
is.
I'm going to conclude our
discussion of cosmology here.
So some of the
quantitative details,
the way in which inflation
can cure the flatness problem
and cure the homogeneity problem
you will explore on problem
set seven.
Going into the weeds of how
one makes it a scalar field,
designs a potential and
does things like this
beyond the scope of 8.962--
there are courses like
this, and I would not
be surprised if some of
the people in this class
spent a lot more time
studying this in their futures
than I have in my life.
All right, so that is
where we will conclude
our discussion of cosmology.
