>> I planned to spend the next few
lectures talking about cosmology.
Cosmology, of course, is
the study of the universe
as a whole on the largest length scales.
And the starting point for this study is the
observation that the universe is isotropic.
This means when we look out beyond the
nearby planets beyond the Milky Way galaxy,
beyond our local cluster of galaxies
to the largest length scales,
we find that the universe looks
the same in all directions.
So this observed isotropy occurs at
length scales of clusters of galaxies.
So, greater than about 10 mega parsecs.
And we observe this isotropy whether
we look at the visible light or x-rays
or any other wavelengths, we also
see that the universe is isotropic
when we measure the microwave
background radiation.
The microwave background radiation is uniform in
all directions within about 1/1000 of a percent.
And those tiny variations are understood to be
a result of the fluctuations in the universe
in the early universe that led to
the local formation of structure,
local meaning on smaller length
scales the formation of galaxies and,
of course, stars and planets and so forth.
Of course, the observed isotropy of the
universe is only from our own vantage point.
So this is us, here on Earth.
Isotropy at a single point like this
doesn't preclude the possibility
that the universe consists of concentric
shells, different layers of different density,
different chemical composition, and so forth.
So isotropy at a single point does not
imply that the universe is homogeneous.
It does not imply.
But in fact, when we look out in any
direction, we don't see different shells,
different layers of different
density and composition.
What we see is pretty much the same density
and same chemical composition
as the universe around us.
So we observed the universe to be not
only isotropic but also homogeneous.
So we observe universe is homogeneous.
Now this is a fairly nontrivial assertion
because keep in mind, as we look outward
to the distant universe, we're
also looking back in time.
So if we look far enough out, we see the
universe at an earlier stage of development.
We see earlier generations of
stars and galaxies and so forth.
So we do see a difference.
But those differences become really significant,
at length scales of more like 100 mega parsecs.
And at any rate, that can be understood
as a consequence of fact that when we look
out to those distance scales, we're observing
the universe at an earlier stage of development.
So the starting point for
cosmology is the assumption
that the universe is isotropic and homogeneous.
Now, from the point of view
of general relativity,
these assumptions are really
very rich and powerful.
Notice that when we say the universe
is isotropic and homogeneous,
what we mean is that space
is isotropic and homogeneous.
In other words, we're assuming that
spacetime can be split into slices,
which are space at different times.
And each one of these slices,
these three-dimensional slices
of spacetime, is isotropic and homogeneous.
So this is a diagram of spacetime.
And here we are.
This is us now.
And each of these slices is
a three-dimensional space.
So these are all isotropic and homogeneous.
But these spaces don't have to be identical.
So the space at the now time,
so this is the now time,
can be different from the
space at some past time.
So, for example, the density on
this past slice could be different
from the density on this now slice.
Now, from a general relativity point of view, we
have to ask what's special about these slices?
Why not slice the spacetime say this way?
And what's special about these
slices is these slices are the slices
in which the matter content of the
universe is on average at rest.
So let me erase these alternative slices
and draw some world lines of matter.
So here's our world line.
Here's another cluster of galaxies,
another cluster of galaxies.
So, these clusters of galaxies are
at rest with respect to these slices.
In other words, the four-velocity of any
of these world lines is orthogonal to any
of the basis vectors within
three-dimensional space.
And that must be true because
if there were some velocity,
if this cluster of galaxies had some velocity
with respect to space is defined this way,
then there would be preferred
direction at this point.
But we have a preferred direction
that breaks isotropy.
OK. We know that the universe is isotropic
at this point, because we know it's isotropic
about us and we're assuming it's homogenous.
So the universe must be isotropic
about any point.
So in particular, the cluster of galaxies
here can't have any motion on average.
So, each of these world lines is at rest
with respect to each of these spaces.
And that's what defines his
preferred slicing of spacetime.
It's the slicing such that the matter which
follows these world lines is on average at rest.
So our first task will be to determine how
this structure restricts the spacetime metric.
And as usual, in general relativity, our
life will be easiest if we choose coordinates
that are adapted to whatever structure we have.
So let's use these slices to define
constant values of coordinate t.
And we'll use spatial coordinates that
are constant along these world lines.
So let's denote the spatial coordinates as xi.
So these remain constant
along these world lines.
So now, these world lines are the integral
curves of the vector field d by dt.
And t itself we can choose to be
proper time between the slices.
After all, the proper time along this world
line between these slices must be the same
as this proper time, the same as
this proper time due to homogeneity.
So we can use proper time as
the label for these slices.
So using these coordinates, the
metric must have the following form.
It's ds squared equals minus dt
squared plus gij dxidxj where this part
of the spacetime metric is a spatial metric.
It's the metric for three-dimensional
isotropic homogeneous space.
And it in general is going to be t dependent.
So it's going to change as
the proper time t changes.
Now the reason the gtt component of the metric
is minus 1 is because if we look along a curve
of constant spatial coordinates, if xi is
constant and dx squared is minus dt squared,
or in other words, d tau
squared is equal to dt squared.
So this is just saying that t is within an
additive constant, just the proper time.
And that's what we expect
from the way we defined t.
So this explains why gtt is minus 1.
Now, next question.
Why is gti equal to zero?
And the answer is quite simple.
If you recall, these world lines
are orthogonal to the slices.
So that means the basis vector
d by dt is orthogonal to any
of the basis vectors d by
dxi that lie in the slices.
So we have d by dt is orthogonal
to any of the d by dxi's.
But that is just gti, the ti
component with the metric.
Now let's turn to this part of
the metric, the spatial metric,
and ask what is a 3D isotropic
homogeneous space?
From a mathematical point of view,
this is a maximally symmetric space.
Which means it has the maximum possible
number of killing vector fields.
I should say that number of independent.
And in three dimensions, that maximum
number of killing vector fields is 6.
Loosely speaking, three translations
and three rotations.
So let's see if we can deduce
what the metric looks like.
So let's pick a point in
the three-dimensional space.
And let's send out geodesics from that point
in every direction in all three dimensions.
So these are all geodesics.
And now let's measure along each geodesic
out to one unit of proper distance.
Let's say, there's one unit of proper
distance along each of these geodesics.
And now we can connect all of these
points to form a spherical shell.
So this is in three dimensions, remember, and
here's the spherical shell at a proper distance
of two units away from our central point.
So these are spherical shells
And they are in fact spheres.
Each one has the geometry
of a two-dimensional sphere.
And that's due to isotropy.
We've traveled out along the geodesic
along the same distance in each direction.
And so the curvature of the shell
must be the same in any direction.
So, since the curvature is constant all
across this surface, it must be a sphere.
So it's a 2D sphere, each one of these shells.
Now on each one of these spherical shells,
we can introduce the usual
spherical coordinates, theta and phi.
And we'll also choose these spherical
coordinates on each shell in such a way
that they're lined up along
the radial directions.
So in other words, whatever the value of theta
and phi is here, we'll use the same value here.
And now let's choose a radial
coordinate for our space,
which is just the proper
distance along the geodesics.
Let's call it r. It's the proper
distance along the geodesics.
And of course, [inaudible] tells
us that the proper distance
between two spherical shells is the same
along any of these radial directions.
So now let's write down the metric
for this isotropic homogeneous space
using the coordinates r, theta, and phi.
It's ds squared equals dr squared plus
some function of r, which I'll call g of r,
times d omega squared, where as
usual d omega squared is shorthand
for d theta squared plus sine
squared theta d phi squared.
So it's the metric on a unit sphere.
So why is the rr component
of the metric equal to 1?
Well, that's because we've
chosen the coordinates
such the r measures proper
distance along the geodesics,
which are curves of constant theta
and phi, r is proper distance.
And why are the off-diagonal terms, the
r, theta, and our phi terms in the metric?
So gr theta and gr phi equal to zero.
Well, if you look at one of our
spherical shells and one of our geodesics,
so this is one of the geodesics,
and this is a spherical shell.
The geodesic must be orthogonal to the shell.
In other words, the geodesic must
point along the normal to the shell.
If it didn't point along
the normal to the shell,
then there would be a preferred direction.
That would violate isotropy.
So, the geodesic must be orthogonal to the
shell, but the geodesics are integral curves
of d by dr. The shell has basis
vectors d by d theta and d by d phi.
So, these must be zero.
The inner product for d by dr with
either d by d theta or d by d phi,
but this is just gr theta and this is gr phi.
So, what we've now shown is the metric for
three-dimensional space that's isotropic
and homogeneous must have this form.
But actually, if you think about it, all
we've used is isotropy about a single point.
We haven't imposed that the
space should be homogeneous yet.
So we can impose homogeneity by demanding
that the curvature is a constant
at each point in the space.
Now by curvature here, I mean really any
measure of curvature that's independent
of any arbitrariness that the
coordinates might introduce.
So, any geometrically defined
measure of curvature.
In other words, we need a curvature invariant.
Now, the simplest curvature
invariant is the curvature scalar.
But that turns out to be a
little bit of a messy choice.
There's a simpler choice and that is
the rr component of the Ricci tensor.
So the Ricci tensor is Rij, and we're
going to look at the rr component.
Now, normally when you look at a component
of a tensor, you have to be concerned
that that is a coordinate dependent quantity,
it depends on the way the
coordinates have been defined.
But remember, in our case,
we defined the r coordinate
to be proper distance along a geodesic.
So it's actually a geometric quantity.
So the rr component is actually the
type 02 tensor Rij acting on two copies
of the vector field d by dr. And these
vector fields are defined geometrically
as proper distance along geodesics.
So in particular, homogeneity tells
us this quantity should be the same
at each point in the space.
So homogeneity implies rrr is equal to a
constant, let's call that constant k. So now
if we compute the Ricci tensor
for this metric of this form,
that just has this one unspecified function g of
r, the rr component of the Ricci tensor is equal
to g prime squared minus 2g times g double
prime, or prime denotes differentiation
with respect to r divided by 2g squared.
And that must equal some constant k.
Now just a little bit of
playing around with this,
shows that the left-hand side can be
written as minus 2 over the square root
of g times the square root of g double prime.
So that tells us that square
root of g double prime is equal
to minus 1/2k times the square root of g. So we
know how to solve this differential equation.
For example, if k is positive, then the
solution is just a simple harmonic oscillator.
Square root of g is equal to some
constant times sine of square root of k
over 2 times r minus another constant.
Let's call that B. And if k is equal to
zero, this just tells us that square root
of g is equal to a linear function of r. Let's
write it as A times r minus B. And finally,
if k is negative, we have the solution
square root of g is equal to instead
of sign we have a hyperbolic sine sinh.
And here we'll put the absolute value of k
over 2 square rooted times r minus
B. Now let me remind you the form
of the metric we're dealing with.
It's ds squared equals dr squared
plus g of r times d omega squared.
Now in constructing these coordinates,
we've introduced a coordinate singularity
at r equals zero, where the
radial geodesics converge.
So we need to be careful
about the point r equals zero.
And one thing we know about that
point is just like any geometry,
it should be smooth at r equals zero.
And if we look in a close enough region it
should be approximately flat near r equals zero.
So, that tells us that close to
r equals zero the geometry --
the metric should look like dr squared
plus r squared d omega squared.
So in particular, the square root of g should
be approximately equal to r near r equals zero.
Now if we look at these expressions for square
root of g, you'll see that this will be true,
if in this case B is equal to zero and A is
equal to square root of 2 over k. In this case,
we need B equals zero and A equals 1,
and in this case, we need B equals zero
and A equals square root of 2 over
absolute value of k. So with these values
with constants A and B, the metric
for the isotropic homogeneous space,
ds squared equals dr squared plus 2 over k
times sine squared of the square root of k
over 2 times r, d omega squared,
this is the case k greater than zero.
For k equal to zero, we have dr
squared plus r squared d omega squared.
And finally for k less than zero, we have
dr squared plus 2 over absolute value
of k times hyperbolic sine squared,
the square root of absolute value
of k over 2 times r d omega squared.
So each of these three metrics
represents a homogeneous isotropic space,
and of course this k equals
zero case is just flat space.
The case with positive k is
actually a three-dimensional sphere,
and this k less than zero case is
a three-dimensional hyperboloid.
Let's take a closer look at that.
Let's start with this k positive case.
So when k is greater than
zero, this is a three sphere --
three-dimensional sphere of
radius square root of 2 over k.
To see this, let's change coordinates.
Let chi equals square root of k over 2 times
r and d chi squared of k over 2 times dr.
And it's easy to see then the metric becomes
too over k times d chi squared plus 2
over k times sine squared
of chi d omega squared.
And now let's make a slight change of notation.
Let's let the radius of this
three-sphere, call it a,
then the metric becomes a squared times
d chi squared plus sine squared chi.
And then d omega squared is, of course, d theta
squared plus sine squared theta, d phi squared.
And now you can show that this
metric is a three-dimensional sphere
by considering a flat four-dimensional space.
So this is not physical.
This has nothing to do with physics.
It's just mathematics.
Consider the space described by
flat metric in four dimensions,
dx squared plus dy squared plus dz
squared plus let's say dw squared.
And now to find a surface in this space by
x equals a sine chi sine theta cosine phi,
y equals a sine chi sine theta sine phi,
z equals a sine chi cosine
theta, and w equals a cosine chi.
If you use this flat metric
to define the geometry
of this surface, you find precisely this metric.
And furthermore, the surface satisfies x squared
plus y squared plus z squared plus w squared
equals a squared, which is means it's
a three-dimensional sphere of radius a.
So the three-dimensional
isotropic homogeneous space
with k positive is a three-dimensional sphere.
The case with k negative is a
three-dimensional hyperboloid.
And we can show that in much the same way.
So let's consider k less than zero case.
This is a 3D hyperboloid with radius -- what
we'll call radius square root of 2 over k
over absolute value k. To see this,
we make a change of variables.
We let chi equal square root of
absolute value of k over 2 times r.
And we'll also define this radius as little
a. And then the metric, it's not hard to see,
becomes a squared times d chi
squared plus hyperbolic sine squared
of chi times d omega squared.
That's d theta squared plus sine
squared theta d phi squared.
And now to show that this is a hyperboloid,
we define a four-dimensional flat spacetime
and unphysical spacetime by minus dt squared
plus dx squared plus dy squared plus dz squared.
And then we consider the
surface in this spacetime defined
by x equals a times hyperbolic sine
of chi times sine theta cosine phi,
y equals a hyperbolic sine chi sine theta sine
phi, z is a hyperbolic sine chi cosine theta,
and t is a times hyperbolic cosine of chi.
So when you plug this surface into this line
element, you find this metric precisely.
And furthermore, this surface satisfies t
squared minus x squared minus y squared minus z
squared equals a squared.
This is a hyperboloid.
We could draw it if we suppress one
of the dimensions say the z direction.
So this is the x axis, y axis, and the t axis.
On this diagram, the surface looks like a bow.
But remember, this surface is
not a surface in Euclidean space.
It's a surface in Minkowski spacetime.
So now let me summarize our results for the
three-dimensional isotropic homogeneous spaces.
We have three cases.
First, the sphere, the case
of a three-dimensional sphere,
the line element is a squared times d chi
squared plus sine squared chi d omega squared.
And that's the k greater than zero case.
And for the k less than zero case,
we have a squared times d chi
squared plus hyperbolic sine squared
of chi times d omega squared.
And for that k equals zero case, you'll recall
the metric is just the flat metric dr squared
plus r squared d omega squared.
And we can write this in a way that
looks similar to these line elements
by making a change of variables, r
equals a times chi or some constant a,
and then the metric in this case becomes a
squared times d chi squared plus chi squared d
omega squared.
Now let's recall the meaning of this constant
k. K was the rr component of the Ricci tensor
where we were using coordinates in which are
measured proper distance along radial geodesics.
And for these three metrics, it turns out that
this is proportional to the curvature scalar.
We could write the curvature scalar in
terms of k. But notice that we replaced k
in these line elements with this new
parameter a so it's more convenient
to write the curvature scalar in terms of
a. So when we do that for the k greater
than zero case, we find curvature
scalar is equal
to 2 divided by a squared, which is positive.
The curvature scalar for the k equals
zero case is, of course, R equals zero.
This is just flat space.
And curvature scalar for the k less than
zero case is negative 2 over a squared.
So these three cases of three-dimensional
isotropic homogeneous spaces are characterized
by their scalar curvature being
positive, zero, or negative.
Now the coordinates used in writing these three
metrics have the advantage of being closely tied
to the symmetries of these spaces.
But there's another set of coordinates that
are very convenient for use in cosmology.
And those coordinates are defined by
replacing chi with a new coordinate r,
which is defined as sine chi for the R positive
case is just chi for the R equals zero case,
and hyperbolic sine chi for the R negative case.
Now this new coordinate r is not the same as the
coordinate r that I used earlier in the lecture.
So this is a new definition of r. When
we make this coordinate transformation,
it's easy to see that the metric
becomes a squared times dr squared
over 1 minus r squared plus r squared d omega
squared for the scalar curvature positive case.
For scalar curvature equals zero, we just have
a squared times dr squared plus r squared d
omega squared.
And for the scalar curvature negative
case, we have a squared times dr squared
over 1 plus r squared plus
r squared d omega squared.
Now these three line elements
can be written all together,
if we introduce a new parameter that's
usually called k. Now this is a new definition
of k. It's not the same constant
that I called k before.
And this is defined as the
sign of the scalar curvature.
So it's k -- this k is equal
to plus 1 in this case,
minus 1 in this case, and zero in this case.
So it's either plus or minus 1 or zero.
And this is a new definition of k. Although
it is true that the sign of R is the same
as the sign of the old k, so our new k is the
sign of the old K. But with this definition,
we can now write the metric as a
squared times dr squared divided
by 1 minus k times r squared
plus r squared d omega squared.
And this accounts for all three cases.
When k is equal to plus 1, that's
the positive curvature case.
If k is equal to zero, that's the flat case.
And k equals minus 1 corresponds
to the negative curvature case.
And now we're finally ready to put all this
together and write down the spacetime metric.
Recall, show that spacetime metric has
for minus dt squared plus the metric
for homogeneous isotropic space.
That's of course what we have written here.
But remember, this is in general t dependent.
So spacetime can be viewed as a stack of slices,
each one of these slices is a three-dimensional
isotropic homogeneous space but they don't have
to all be the same isotropic homogeneous space.
So space can evolve in time t. Now,
if you look at this line element
for an isotropic homogeneous space, you
see that it depends on two quantities,
continuous parameter a and a discrete parameter
k. So this continuous parameter a could change
and evolve with t. So it can
depend on t. On the other hand,
this discrete parameter k
can't change because if it's --
for example, if it's equal to 1 on
one slice, it can't suddenly jump
to equal minus 1 on the next slice.
That would be a discontinuity
in the spacetime metric.
So k is fixed in spacetime
metric but a is a function of t.
So the spacetime metric has the form ds
squared equals minus dt squared plus a
of t squared times dr squared over 1 minus k
times r squared plus r squared d omega squared.
And this is referred to as
the Robertson-Walker metric.
Or the Robertson-Walker spacetime
or the Robertson-Walker cosmology.
And there are 3 cases.
The case with k equal plus 1, or space has
the geometry of a three-dimensional sphere,
this is often referred to as a closed cosmology.
The case k equals zero space
has a flat geometry.
This is often referred to as a flat cosmology.
In the case k equal minus 1
or space is a hyperboloid,
and this is often referred
to as an open cosmology.
The term closed comes from the fact
that a sphere is a closed geometry.
It has a finite volume.
Open refers to the fact that a
hyperboloid is an open geometry.
It extends to infinity in all
directions and has infinite volume.
I think it's useful at this point to
pause and consider how far we've come.
The only input to our analysis so far is that
the universe is isotropic and homogeneous.
And from those assumptions
alone, we're able to deduce
that the metric has this very limited form.
Notice that the only freedom here in the
Robertson-Walker metric is a single function
of a single coordinate, this a
of t, along with this parameter k
that can only take three
values, plus or minus 1 or zero.
So considering that the metric in general is 10
functions of four variables or four coordinates,
this is quite significant
restriction on the form of the metric.
At this point, you might be wondering,
how do we determine this function a of t
which by the way is referred to as the scale
factor in parameter k, how are they determined?
And they're determined when we
impose the Einstein equation.
And in turn, the Einstein
equation depends on our choice
for the matter content in the universe.
I planned to go through that analysis
in detail in the next lecture.
But for now, I just want to focus on some of
the properties of this Robertson-Walker metric.
Let me begin by drawing a tr diagram or t
and r are the Robertson-Walker
time and radial coordinates.
So this is the r axis.
This is the t axis.
And we'll place ourselves our own cluster
of galaxies at the origin at r equals zero,
we can place it anywhere we
like because of homogeneity.
But we'll just choose ourselves
to be at r equals zero.
So here's our world line.
This is us.
And this is a world line of some
distant cluster of galaxies.
And remember, all of the matter on average
is at rest in these space-like slices,
these isotropic homogeneous spaces.
And they -- the world lines
are at constant values
of the Robertson-Walker spatial
coordinates, so constant r theta and phi.
So this is r equals zero and this
is r equals some constant value.
Let's call it big R, not to be
confused with the curvature scalar.
And what I'd like to do now is
compute the proper distance between us
and this distant cluster of galaxies.
So we'll call this distance d. And this distance
will change in time so it's a function of time.
So let me write down the
Robertson-Walker metric for reference.
It's ds squared equals minus dt squared plus
a of t squared times quantity dr squared
over 1 minus k times r squared
plus r squared d omega squared.
Now this distance is being computed
along a constant angular direction.
So theta equal constant and phi equal constant.
So in the line element, d omega squared is zero.
And we're also computing this distance
along t equal constant surface.
So dt is zero.
So, the proper distance squared
is just equal to a
of t squared times dr squared
divided by 1 minus kr squared.
And from this, we find, of
course, ds is equal to a times --
that's a of t times dr divided by the
square root of 1 minus kr squared.
Now we can integrate both sides.
The left side is just what we're calling the
total distance d of t. And on the right side,
let's pull out this factor a of t since
it's independent of the coordinate little r.
So it's a of t times the integral
from zero to big R of dr divided
by the square root of 1 minus kr squared.
Now this factor, this integral, just
depends on capital R. In other words,
it depends on which cluster
of galaxies we're looking at.
And it depends on this parameter
little k, but it's just a constant,
in particular it's a constant in t.
It doesn't depend on t. So we see here
that the total distance is proportional
to this scale factor a. So for example,
if the scale factor is increasing
in time, then the proper distance
between clusters of galaxies
increases with time.
And now we can take this relation that says
d of t is equal to a of t times a constant.
And differentiate this with respect to
time, let's use dot for the derivative.
So we have d dot is equal to
a dot times the same constant.
Now we can divide these two equations
to obtain the following result.
We have d dot is equal to a dot divided
by a times d. This combination a dot
over a appears a lot in cosmology, so we give
it a special symbol H. And H depends on time t
because a depends on t. So H is defined
as a dot of t divided by a of t.
And this is the Hubble constant or the Hubble
parameter, sometimes called the Hubble constant.
Of course, it's not constant, it depends
on t but it is a constant in the sense
that it's independent of the spatial coordinate.
It's the same value everywhere in space.
So it's a spatial constant.
We can now write this result for the proper
distance between clusters of galaxies
as d dot equals H of t times d. So the rate
of separation d dot is proportional
to d, the separation.
So, if we double the separation,
then we double the relative velocity.
And the proportionality between distance
and velocity is the Hubble parameter H. Now,
the current value of the Hubble parameter --
-- sometimes called H naught,
this is H at time t0
or we often use the notation
t0 to denote the current time.
This is approximately 70 kilometers
per second per mega parsec.
So, two clusters of galaxies that are separated
by mega parsec are receding from each other
at about 70 kilometers per second.
Now, this number, which comes from observations
of our universe at the present time,
is positive, which tells
us that a dot is positive.
So the scale factor is increasing in time.
So the universe is expanding.
Now let's consider a graph of the
scale factor as a function of t.
So along the horizontal axis is the t axis.
And vertically I'll plot a of t. And we haven't
really talked about what a of t looks like yet.
That's for the Einstein equation to determine.
But let's say it looks something like this.
And this is t naught.
So it's now it's the current, the present time.
If you look at the slope of
this graph at this point,
this slope is a dot at the
present time t naught.
And now let's form a little triangle here.
This is the distance a at
the present time t naught.
And this integral we refer to as t
sub H which is called the Hubble time.
And now you see from this diagram that
slope at the present time can be written
as the scale factor at the present
time divided by the Hubble time.
So this tells us that t sub
H is the present value
of the scale factor divided
by a dot at the present time.
But this is just the inverse of the
Hubble parameter at the present time.
So, we have t sub H is equal to 1 over H0,
the present value of the Hubble parameter.
And you can see from this graph that t sub H is
the time that the scale factor would go to zero
if we assume the scale factor
has a constant slope,
constant equal to the present value of a dot.
So, the Hubble time is the age of the universe,
assuming a dot is approximately
constant equal to the present value.
Of course, a dot will not
turn out to be a constant,
so t sub H is only an approximation
to the age of the universe.
But nevertheless, t sub H is
approximately 14 billion years.
When we observe the universe, we
observe distant clusters of galaxies.
We don't see those galaxies at
the present cosmological time.
Instead, what we have observe is the
light that was emitted from those galaxies
in the distant past that light travels
along null geodesics is just now reaching
our location.
So let me once again draw a tr diagram.
This is the r axis.
This is the t axis.
And we'll place our cells
at the origin r equals zero.
So this is our world line
for our cluster of galaxies.
So this is us and r equals zero.
And here's some distant cluster of galaxies.
At some coordinate value say capital R. And when
we observe this distant cluster of galaxies,
we observe the light that's
been emitted from the past
and this light travels along in all geodesic.
So this light was admitted at
some earlier cosmological time.
Let's call that t sub e for the time emitted.
And the time observed is
the present time t sub zero.
Now the Robertson-Walker metric is ds
squared equals minus dt squared plus a
of t squared times dr squared over 1 minus
kr squared plus r squared d omega squared.
And for a null geodesic, ds squared is zero.
So zero equals minus dt squared plus a
of t squared dr squared divided
by 1 minus kr squared.
And because we placed ourselves
at the origin of coordinates,
this null geodesics along a radial
direction, so d omega squared is zero.
So from here we find dt over a of t
equals dr divided by 1 minus kr squared.
And we can integrate both sides.
So this is integral from te to to, the
admission time to the observation time.
And this is integral from zero to capital
R. Actually, this should be the square root
of 1 minus kr squared in the denominator.
So now let's consider a second null ray that's
emitted just after this first one was emitted.
So here's a second null ray
that's emitted at some time.
It's called emission time te plus delta te.
And the observation time in is to plus delta to.
So we can do the same calculation
for this second null ray.
And of course what we'll find is a similar
result, but with te replaced by te plus delta te
and to replaced by to plus delta to.
And on the right-hand side we'll have the same
integral, integral from zero to R dr divided
by the square root of 1 minus kr squared.
And now let's subtract this first
equation from the second equation.
And of course, the right-hand sides cancel.
And what we're left with is the integral from
te plus delta te to te plus the integral from to
to to plus delta to equals
zero, or in other words,
integral from te to te plus delta te is equal
to the integral from to to to plus delta to.
And now if we take the limit as delta te
and delta to become infinitesimally small,
the integral on the left just becomes delta
te divided by a of t. And the integral
on the right is delta to divided by a of to.
Now let's imagine that these two
null geodesics are the world lines
of successive peaks of an electromagnetic wave.
Then delta te is the period of the wave
as viewed when it's emitted and delta
to is the period of the wave as it's observed.
And so this is equal to 1 over
the frequency, the emitted wave,
and this is 1 over the frequency
of the observed wave.
So this equation becomes 1 over the emission
frequency times the scale factor at the time
of emission equals 1 over the observed
frequency times the scale factor
at the time of observation.
So here I'm defining ae as a at the emission
time and ao as a at the present time.
So from here we have the observed frequency
divided by the emitted frequency is equal to ae
over ao or equivalently, we remember
that frequency is 1 over wavelength.
This tells us the observed wavelength divided
by the emitted wavelength is ao divided by ae.
And this just describes the
cosmological redshift.
So we have since the universe is
expanding, ao is larger than ae.
So that tells us that the observed wavelength
is larger than the emitted wavelength,
or equivalently the observed frequency
is less than the emitted frequency.
Now, the cosmological redshift is commonly used
as a distance measure in cosmology because it's
so closely tied with observations.
So we defined what's often called
the redshift z as a distance measure.
So z is defined as the change in wavelength
divided by the emitted wavelength.
So that's lambda o minus lambda e
divided by lambda e. Or in other words,
lambda o divided by lambda e minus 1.
And from this result, we have lambda o
divided by lambda e is just ao divided by ae.
So there are lots of different measures of
distance that are introduced in cosmology.
Besides the redshift distance, we have
things like the luminosity distance,
and these are measures that are closely tied
to observations that we can actually make.
And, of course, the reason we do this is
because we can't directly observe the distance
between us and some other cluster of
galaxies at a single cosmological time.
So this is us, this is another
cluster of galaxies,
this is the present cosmological time t0.
The distance, the proper distance between
us in this cluster we'll call it d0.
We can't measure directly.
We can only observe photons that were
emitted at a past cosmological time te.
Now, the relationship between these
different measures of distance can be worked
out in detail with little effort.
For example, we can relate the redshift
distance z to the proper distance d0.
This is a very rough calculation.
So we write z this way.
This is a zero minus ae divided by ae.
And if we assume that this cluster
of galaxies is very fairly close,
so the scale factor a hasn't changed very much,
then this is approximately a zero minus ae is
approximately a0 dot times this time integral
delta t. That's this time integral to minus te.
And the denominator is since a hasn't
changed very much, we can just call this a0.
So this is a0 dot over a0,
which is the present value
of the Hubble parameter times delta t. Delta t,
again, if this is a cluster of galaxies is close
and we just assume this is approximately flat
spacetime, then delta t is approximately D0.
So this is H0D0.
And now turning this around, we can write
the proper distance as approximately 1
over the present value of the Hubble constant
times z. Now more careful analysis shows
that this is just a leading
order term in a series expansion
that looks like z plus 1/2 H naught dot.
That's the time derivative of the
Hubble parameter at the present time,
divided by H naught zero
times z squared, and so forth.
If you're interested in how this
calculation is done, I'll include it along
with some guidance as one
of the practice problems.
