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PROFESSOR: OK.
I think it's time for us to start.
Last time we talked about the Doppler shift and a little bit
of special relativity.
Today we'll be going on to talk more about cosmological topics.
We'll be talking about kinematically
how one describes a homogeneously-expanding
universe like the one that we think
we're living to a very good approximation.
In that case, let's get started.
What I want to do today is talk about some
of the basic descriptive properties of the universe
as we will describe it.
The universe is, of course, a very complicated place.
It includes you and me, for example,
and we're pretty complicated structures.
But cosmology is not really the study of all that.
Cosmology is the study of the universe in the large,
and we'll begin by discussing the universe on its largest
scales in which you view approximated by a very
simple model, which we'll be learning about.
So in particular on very large scales,
the universe is pretty well described by threes properties,
which we will talk about one by one.
The first is isotropy, and that just
comes from some Greek root, which
means the same in all directions.
Now, of course, as we look around say the room here,
the room doesn't look the same in all directions.
The front of the room looks different from the back
of the room.
And looking towards Mass Ave looks
different from looking towards the river,
and looking further out into space,
looking towards the Virgo cluster, which
is the center of our local super cluster,
looks rather different from looking
in the opposite direction.
But when one gets out to looking at things
on the very large scale where in this case very large
means on the scale of a few hundred million light years,
things begin to look very isotropic.
That is no matter what direction you look,
as long as you're averaging over these very large scales,
you find that you see pretty much the same thing.
This becomes most emphatic when one looks
at the cosmic background radiation, which is really
the furthest object that we can look at.
It's radiation that was admitted shortly after the Big Bang.
The history of the cosmic background radiation
in a nutshell is worth keeping in mind here.
I'll refer to it as the CMB for cosmic microwave background.
And in a nutshell, the things to keep in mind
in thinking about this history is
that until about 400,000 years after the beginning,
the universe was a plasma, or maybe I
should say more accurately that the universe was
filled with plasma.
And within a plasma, photons essentially go nowhere.
They're constantly moving at the speed of light,
but they have a very large cross section for scattering off
of the free electrons that fill the plasma.
And that means that the photons are constantly
changing directions and the net progress in any one direction
is negligible.
So the photons are frozen with the matter,
I'll say frozen inside the matter, which
means that the net velocity relative to this plasma
is essentially zero.
But according to our calculations,
and we'll learn later how to do these calculations,
at about 400,000 years after the Big Bang,
the universe cooled enough so that it neutralized
and then it became a neutral gas like the air in this room.
And the air in this room you know
this is very transparent to photons,
and that means that light travels from my face
to your eyes on straight lines and allows
you to see an image of what my face looks like and vice versa,
by the way.
And it's a little dicey to extrapolate something
from the room to the universe.
The orders of magnitude are very different.
But in this case, the physics actually
ends up being exactly the same.
Once the universe becomes filled with a neutral gas,
it really does become transparent to the photons
of the cosmic microwave background.
So these photons have for the most part been travelling
on perfectly-straight lines since 400,000 years after
the Big Bang.
And that means that when we see them today,
we are essentially seeing an image of what
the universe looks like at 400,000 years after the Big
Bang.
So at 400,000 years, gas neutralized
and became transparent.
This by the way has a name, which is universally
what is called in cosmology, nobody actually
understands why it's called this, by the way,
but the name is recombination.
And the mystery is what the re is doing there
because as far as we know, the gas is combining
for the first time in the history of universe,
but that's otherwise what everybody calls it.
I did actually once ask Jim Peebles who
might be the person who first called it this why it was
called this, and he told me that this
is what the plasma physicists called it,
so it was natural to just pick up the same word
when he was doing cosmology, so maybe
that's how the word originated.
But coming from the point of cosmology,
it is a misnomer in that for the theory
that we're discussing the prefix re here
has absolutely no business being there.
So what do we see when we look at the cosmic microwave
background?
We see that it is unbelievably isotropic.
What we find is that there are deviations
in the temperature of the radiation.
The intensity is measured as an effective temperature.
There are deviations in the temperature of the radiation
of a fractional amount of about 10 to the minus 3,
which is a very small number, but it's
even stronger than that.
This deviation of one part in 10 to the 3
has a particular angular pattern,
and it's not the angular pattern that you
would expect if the source system were moving
through the cosmic microwave background,
and that's how we interpret this 10 to the minus 3 effect.
Motion of solar system through the CMB.
And after removing the effect of the motion.
Now actually when we move it, it's
not like we have an independent way of measuring it.
We don't really, not to enough accuracy.
So we're really just fitting it to the data and removing it.
But when we do the split to the data,
it's a three-parameter fit, that is,
we have three components of a velocity to fit.
We have a whole angular pattern on the sky,
and we only have three numbers to play with.
So it's strongly constrained even though we're
using the data itself to determine
what we think our velocity is relative
to the cosmic microwave background.
And after removing it, then what we find
is that the residual deviations, delta t over t,
are only at the level of about 10 to the minus 5,
1 part in 100,000, which is really unbelievably
isotropic, unbelievably uniform.
One time I decided to think about how
round that is, how much the same in all directions
it is by asking myself the question,
is it possible to grind a marble that
would be spherical to an accuracy of 10 to the minus 5.
And you can think about that yourself.
The answer I came up with was that yes it is, but it really
strains the limits of our technology.
It correspond to sort of the best technology
we have for building highly-precise lenses
basically fractions of a wavelength of light.
So to round to 1 part in 10 to the 5
is really being unbelievable round, unbelievably isotropic.
And that's the way the universe looks.
Next item in our description of the universe is homogeneity.
Homogeneity is harder to test with precision
because it means looking out into space
and trying to see, for example, if the density of galaxies
is uniform as a function of distance.
We always talked about as a function of angle, that's
isotropy, and it's very uniform where
one could make very precise statements
about the cosmic microwave background.
But to talk about homogeneity, one
has to be able to talk about how the galaxy distribution varies
with distance, and distances are very
hard to measure cosmologically.
So as far as we could tell, the universe
is perfectly compatible with being homogeneous, again,
on length scales of a few hundred million light years,
but it's hard to make any very precise statement.
There is, of course, relationships
between isotropy and homogeneity.
Homogeneity, by the way I didn't define that.
I assumed you know what it meant,
but I should definitely define it.
Isotropy means the same in all directions.
Homogeneity means the same at all places.
So sometimes these are just put together and called uniformity
because they are very similar concepts.
They are, however, distinct concepts logically,
and it is worth spending a little time understanding
how they connect to each other, in particular
how you can have one without the other is the best way
to understand what they individually mean.
So suppose, for example, we had a universe
that was homogeneous but not isotropic.
Is that possible, and if so, what
would be an example of a feature that
would be described that way?
Let me throw it out to you.
We want to be homogeneous, but not isotropic.
Yes.
AUDIENCE: It would be parallel universes like a cylinder
pointing in a z direction, and I mean,
matter is all homogeneous with a cylinder
but there is preferred directions for isotropic.
PROFESSOR: A preferred direction fixed
by the direction of the periodicity?
That is an example.
That's right.
Let me ask if there are other examples people could think of.
Yes.
AUDIENCE: There are galaxies everywhere
with constant density, but they're all
aligned in a particular direction.
PROFESSOR: That's right.
That's right.
Galaxies have a shape, in particular
they have an angular momentum.
The angular momentum could be a line,
and that would be an example of a universe that
would be homogeneous but not isotropic.
Very good.
Another example that I'll just throw out, which I think maybe
is simple to think about is the universe is filled
with this cosmic microwave background radiation.
suppose all the photons going in the z direction
were more energetic than the ones
going in the x and y direction.
That would be a possible situation
that could be completely homogeneous,
but would be an example of something that would not
be isotropic.
So there are many examples you can come up with.
I'm very glad you came up with the ones you did.
That's great.
Going the other way it's harder.
Suppose we try to think of the universe that's
isotropic but not homogeneous.
Isotropic, by the way, does depend on the observer, so
let's first talk about isotropic relative to us.
I was just going to say imagining a universe that
would be isotropic relevant to us,
but would not be homogeneous.
Yes.
AUDIENCE: Could it be like if we lived in some shell.
PROFESSOR: That's right.
A shell structure.
AUDIENCE: In all direction, the shell would be there.
PROFESSOR: That's right.
That's right.
I think I'll even draw that on the blackboard.
Example of isotropy without homogeneity.
So we would be here, and the matter
could be distributed in a perfectly spherically-symmetric
distribution with us at the center.
And that would be an example of something
that would be isotopic to us but not homogeneous.
Now, things like that, of course,
are considered weird because we don't think of ourselves
as living in any special place in the universe,
and that's basically what the Copernican Revolution was
all about.
And the Copernican Revolution is sunken very deeply
into the psychology of scientists.
So I think scientists would be very loathed to imagine
the universe that look like this,
but it does help to understand what these words mean.
If a universe is going to be isotropic to all observers,
then it does have to be homogeneous,
and that's part of the reason why
we're pretty confident that our universe is basically
homogeneous, because we just decided that's isotropic to us,
and we decide we're not special then it
has to be isotropic to everybody and then it
has to be homogeneous.
If the universe is isotropic to all observers,
it is homogeneous
Now, a thought which I will leave
for you to think about between now and the next lecture
is whether or not really knowing that a universe is
isotropic with respect to two observers
is enough to prove that it's homogeneous.
That turned out to be a more subtle question
than it might sound.
I don't know if it sounds subtle or not.
I should maybe just tell you basically what the answer is
and then you can try to think if you can understand the answer.
In the Euclidean space, isotropy about two distinct observers
is enough to make it homogeneous,
which is kind of what you visualize.
But if you can allow yourself to think
about non-Euclidean spaces, and I
know we haven't talked about non-Euclidean spaces
yet so you might not have in the way of tools to think about it.
But think, for example, about surfaces in three dimensions.
Surfaces are very good examples of
non-Euclidean two-dimensional geometries.
And see if you can invent a two-dimensional geometry that
would be isotropic about two points,
but would not be homogeneous.
So that's your thought assignment for next time,
not to be handed in just to be talked about in the lecture
next time.
So isotropy and homogeneity are two of the key properties that
define the simplicity of our universe on very large scales.
The next thing I want to talk about
is the expansion of the universe, which is basically
characterized by Hubble's law.
Last time I think I said I was going
to call it the Lemaitre-Hubble law.
I decided I'll probably call it Hubble's law.
Now, Hubble, I think, really does
deserve credit for demonstrating observations
that the law is true, and that's really
what he is getting credit for and that was not
believed until he discovered it.
So it really did have a tremendous effect
on the course of cosmology.
So Hubble's law says that on average all galaxies
are receding from us with a velocity which
is equal to a constant, H, called the Hubble constant--
Hubble called it K, by the way, capital
K-- times the distance to the Galaxy, r.
And so it's not true exactly for our universe,
but it's true in some average sense,
just as isotropy and homogeneity are,
we're only true on an average sense.
I want to tell you about the units in which it's measured
and that leads me to the parsec.
Let me write this on the board.
But astronomers always measure the Hubble constant
or I will sometimes call it the Hubble expansion
rate in kilometers per second per megaparsec.
And it's a relationship between a velocity and distance,
so kilometers per second is velocity
and velocity per megaparsec is the velocity
per distance, which is what it should be.
Notice, however, that I wrote that.
A kilometer and a megaparsec are both units of distance.
So they actually just have some fixed ratio.
So in the end, this Hubble constant really
is just an inverse time, and obviously,
if you multiply an inverse time times the distance
you get a distance per time, which is the velocity,
so that works.
But it's very seldom quoted as simply an inverse time,
instead it's quoted by the units that astronomers like to use.
They measure velocities as a normal person would
in kilometers per second, but they
measure distances in megaparsecs,
where a megaparsec is a million parsecs,
and a parsec is defined by that diagram.
The base of this triangle is one astronomical unit,
the mean distance between the Earth the sun.
And the distance at which the angles attended
by one astronomical unit is one second of arc
is what's called a parsec and abbreviated pc.
And a parsec is about three light years.
I'll write these things on the board.
One parsec equals 3.2616 light years,
and a megaparsec is a million of those.
Another useful number to keep in mind for converting,
if you want to think of H as inverse years,
then the useful equality is that 1 over 10 to the 10 years
is equal to 97.8, and it's suitable to remember this
as being 100-- you can look up the exact number when you need
it-- and these funny things kilometers per second
per megaparsec.
So what is the value of Hubble's constant?
It actually has a very interesting and
historically-significant history.
It was first measured in this paper by George Lemaitre
and in 1927, published only in French
and ignored by the rest of world, at the time at least.
It got discovered later.
And Lemaitre was not an astronomer.
He was a theoretical cosmologist.
I mentioned a few times I think he had a PhD from MIT
in theoretical cosmology in physics, in principle.
And the value that he got based on looking at other people's
data, in 1927, had the value of-- I guess actually,
I'll give you the range.
He gave two different methods of calculating it.
We've got two slightly different answers.
So we had 575 to 625 of these [INAUDIBLE] units kilometers
per second per megaparsec.
And two years later in his famous paper
"Hubble," got the value of 500 kilometers
per second per megaparsec.
I have a picture of Hubble too.
Yes.
AUDIENCE: That last in the board right there
where you have 1 over 10 to the 10 amperes, is that H?
PROFESSOR: That's just an equality of units.
AUDIENCE: Quality of units.
PROFESSOR: That's just the unit equality.
It's relevant to H, because H is measured in those units.
But it really is just an equality of units.
1 over 10 to the 10th years has units of inverse time,
and kilometers per second per megaparsec
has units of inverse time also because kilometers
is distance and megaparsec is inverse distance.
So both sides have the same units and the same dimensions,
I should say, and it's just two different ways
of measuring the same thing, inverse times.
So in 1929, Hubble published his famous paper
which he got the value of 500, and there's
an important difference really between the papers by Lemaitre
and Hubble.
First of all, Hubble was using largely his own data.
Lemaitre was using other people's data mostly
Hubble's actually.
And furthermore, Hubble made the claim
that the data justified the relationship
that v is equal to a constant times r.
Lemaitre knew that relation theoretically
for a uniformly-expanding universe,
which we'll be talking about shortly.
But he did not claim to be able to get it from the data.
The data he had he decided was not strong enough
to reach that conclusion, but he was
still able to get a value for H by taking the average velocity
dividing it by the average distance and got a number.
I think I have Hubble's data next.
Yeah, here's Hubble's data.
The data obviously was not very good.
It only goes up to about 1,000.
One curiosity of this graph that you might notice
is that the vertical axis is a velocity meaning
it should be measured in kilometers per second,
but nonetheless Hubble wrote it as kilometers.
Not getting his units right, so minus 10 or something
like that on the graded sheet.
But somehow it did not stop the paper
from getting published in the proceedings of the National
Academy of Sciences and had become,
of course, a monumentally-famous paper.
But you can see that the data is scattered,
and it has those nice lines drawn through which
guide your eye, but if you imagine taking away the lines,
it's not that clear on the data itself
that it really is a linear relationship.
But it's suggested, at least, and Hubble
thought it was pretty convincing and later Hubble
gathered more data for this project,
and it did become quite convincing that there
is a linear relationship, and today there's
no doubt that there is a linear relationship between velocity
and distance.
At very large distances there's deviations,
which we can understand and we'll
be talking about later, but basically, at least
for moderate distances, one has this linear relationship.
I should mention that the velocity of the solar system
through the CMB is also the velocity of the solar system
through this pattern of Hubble expansion,
and both Hubble and Lemaitre had to make estimates
of the velocity of the solar system
relative to these galaxies and subtract that out
to get things that resemble a straight line.
Lemaitre estimated the velocity of our solar system
as 300 kilometers per second, and Hubble estimated it
as 280 kilometers per second.
So it was a relevant feature because remember,
the maximum velocity there is only 1,000 kilometers
per second, so the correction that he's putting in
is about a third of the maximum velocity seen.
So it's a very important and not that it was easy to determine.
AUDIENCE: What were they using to determine
the [INAUDIBLE] CMB?
PROFESSOR: I think they were just
looking for what they could assume
that would make the average expansion
in all directions about the same.
To be honest, I'm not sure about that.
But that's the only thing I can see that they would have,
so I think that must be what they were using.
Now since these ancient times, there
have been many measurements of the Hubble expansion rate,
and they changed a great deal.
So in the '40s through '60s, there
was a whole series of measurements dominated
by people like Walter Baade and Allan Sandage.
And generally speaking, the values
came down steadily from the high values
that were measured by Hubble and Lemaitre
in the very early days.
When I was a graduate student, if you asked anybody
what the Hubble constant was, you always got the same answer.
It was somewhere between 50 and 100,
still uncertain by a factor of 2,
but much lower by a factor of 5 or 10 from the values
that Hubble was talk about, and was still
a major source of uncertainty in talking about cosmology.
Values started to become more precise around 2001.
So in 2001, there was the Hubble Key Project
that released these results.
The word Hubble here refers to the Hubble satellite,
which was named after Hubble-- Hubble, Edwin.
And they were able to use the Hubble telescope to see
Cepheid variables and galaxies, that was significantly further
than Cepheid variables can ever be seen before
and thereby make a much better calibration of the distance
scale.
As you'll learn about when you do your reading,
Cepheid variables are crucial to determining
the cosmological distance scale.
So the value that they got was much more precise
than anything previous, 72 plus or minus 8 of these [? quad ?]
units kilometers per second per megaparsec.
Meanwhile things were still controversial.
I should have added that when people said it was 50 to 100
when I was a graduate student, it wasn't that people really
understood the error bars to be that large.
The real situation is that there were a group of astronomers
that claimed adamantly it was 50,
and there were other groups of astronomers that
claimed adamantly that it was 100.
Anyway On person is shouting in your ear saying it's 100,
another person is shouting in your ear
saying it's 50 the conclusion is that it's 50 to 100,
and that's the situation when I was a graduate student.
So this was a somewhat high value relative to the argument.
The people who are arguing on the low side
were still in business at this time and still in fact also
using Hubble telescope data.
So Tammann and Sandage, the same year
using the same instrument-- let me put the year here,
and it's 2001-- Tammann and Sandage were estimating
60 plus or minus they said less than 10%.
so these didn't quite mesh.
Coming to more modern times, in 2003, WMAP,
the satellite called the Wilkinson Microwave Anisotropy
Probe, a satellite dedicate to measuring
these minute variations of the cosmic microwave background
at the level of 1 part in 100,000,
it turns out that those measurements
are estimated at the Hubble expansion rate
by fitting the data to a theoretical model.
And their initial number was 72 plus or minus 5.
And that was based on one year of data.
And in 2011, the same WMAP satellite team
was based on seven years of data,
came up with a number of 70.2 plus or minus
1.4, so to very precise.
And the most recent number comes from a similar satellite
to WMAP but more recent and more powerful
satellite called Planck, which just
released its data last March.
And it came up with a somewhat surprisingly low number 67.3
plus or minus 1.2.
Yes.
AUDIENCE: The other measurements there
are kind of inconsistent with one another
and then with one measurement sort of 20th century
make this big jump down suggesting
those early guys were making the same kind of mistake.
What was it?
PROFESSOR: Good question.
The early guys were making a big mistake
in estimating the distance scale,
and I'm not sure I understand the details of that,
but I think it had something to do with misidentifying Cepheid
variables, equating two different types that should not
have been compared with each other.
But I'm not altogether sure of the details,
but it was definitely the distance scale they had wrong.
The velocities are pretty easy to measure accurately,
and they were very wrong.
Yes.
AUDIENCE: There's two types of Cepheids,
one has a certain period of velocity relation
that would give it, and it's like a completely different
type of star, and so we got mixed up,
and we got completely different absolute magnitudes, which
will give you two completely different distance estimates.
So I don't know how far, but measuring
Cepheids and Andromeda was way off the distance scale
because we thought they were Type 1,
but they were actually Type 2.
AUDIENCE: I think the difference between Type 1
and Type 2 are a factor of 4, so that would make sense.
AUDIENCE: Yeah.
It's like two completely different linear relations.
PROFESSOR: An intensity goes like 1
over the distance squared, so I think that, I mean,
a factor of 4 in intensity I think would mean a factor of 16
in distance estimates.
Yes.
AUDIENCE: I'm following so much that these
are like, they both have error bars,
but they're not within error of each other.
PROFESSOR: Right.
AUDIENCE: Well, this is like current data.
PROFESSOR: So what's going on?
Nobody knows for sure.
One thing I should mention though
is that these are what are called 1 sigma error
bars, which means that you don't expect
them to necessarily agree.
You expect the right answer to be within one sigma error bar
2/3 of the, time but 1/3 of the time
it should be outside the error bar.
The questions is, the error bar is on both.
But the comparison of this, and this
is usually viewed as something like 2
and 1/2 sigma effect, which naively, I think,
means the probability of something
on the order of 1% or something like that
of getting errors that large at random.
And it's debated whether or not it's significant or not.
It's the abstract of the Planck paper use
words something like there's a tension between their value
and other recent values.
When somebody does see things like that
happen more frequently than the probabilities indicate,
which I think it proves a theorem
that experimenters always underestimate their error bars.
But there's no absolute proof of that theorem.
So these thing were early debatable.
People don't know-- there are many things that turn up
in experimental physics and especially
in cosmology that turn up regularly where people have
different opinions about whether or not it's pointing
to something very important or something
that's going to go away.
So very often they go away, that's a fact.
But you never know in any one case,
whether it's something important that
will become more definite as for the measurements are made
or whether it's just a spurious effect that
will disappear in a few years.
Yes.
AUDIENCE: So I imagine in the 1940s
when people started saying yes, that Hubble,
for whom the constant is named, was off by a factor of 10.
That's very controversial.
Was there any kind of sloping trend
where people may have changed their data to make it seem,
oh, we're not that far off the Hubble standards.
Has this happened a couple of times before?
PROFESSOR: Question is did people perhaps try to fudge
their data during a period in the middle to make it look more
like Hubble's.
I think, I don't know, and there were,
as I said, pretty much through the middle of the 20th century
two groups, one of which was getting a high value,
and one of which was getting a low value.
The high value is where, in fact, disciples of Hubble,
rather directly-- wait a minute.
That's not right.
The most direct disciple of Hubble was Allan Sandage,
and he was, in fact, abdicating the low value.
So the sociological trends are not that clear.
What is clear is that they were way off.
I was going to add concerning the way offness,
that it really does have or did have a very significant effect
on the history of cosmology because when one looks at a Big
Bang model and tries to use that model to estimate when it all
started, what you're doing is you're trying to extrapolate
backwards, ask when was everything on top
of each other given that things are moving at the speed now.
There is more that goes into the calculation
then just H. It depends on your model, the matter, and things
like that.
But nonetheless H is obviously a crucial ingredient there.
The faster things are moving now outward
when you extrapolate backwards, the faster they're
moving inward and the younger the universe is.
And to a very good degree of reliability,
any age estimate-- and we'll make age calculations later--
but any age estimate is proportional to 1
over the Hubble parameter, 1 over the Hubble expansion rate.
So if you're off by now we would say a factor of 7
between Hubble's value and the current value, 70 versus 500,
if you're off by a factor of 7, you
get ages for the universe, which are factors of 7
smaller than what you should be getting.
And this was noticed early on.
People were calculating ages of the universe and Big Bang
models and getting numbers like 2 billion years
instead of 14 billion years, a factor of 7.
And even back in the '20s and '30s,
there was significant geological evidence
that the Earth was much older than 2 billion years,
and people understood something about the evolution of stars,
and it would seem pretty clear that the stars were older
than 2 billion years, so you couldn't tolerate
a universe that was only 2 billion years old.
And it led to very significant problems
with the development of the Big Bang theory,
and in particular, it certainly gave extra credence
to what was called the steady state theory, which
you may have heard of, which held that the universe was
infinitely old and as it expanded,
more matter was created in the steady state theory
to fill in the gaps so the density of matter
would be constant.
And Lemaitre himself in his 1927 paper,
built a very complicated, from my standards,
theory in order to get the age to be compatible.
Instead of having a Big Bang model,
his 1927 model was not a Big Bang model.
His 1927 model started out in a static equilibrium
where he had a positive cosmological constant which
produces a repulsive gravity, like what
we talked about in my opening lecture,
balancing against the normal attractive gravity
of ordinary matter, producing what was almost
a static universe of exactly the type
that Einstein had been advocating.
But Lemaitre's universe started out with just a slightly less
mass density than Einstein would have had,
so it gradually started to get bigger and bigger.
The force of ordinary gravity wasn't quite enough
to hold it together, but when it did
that, it starts to get bigger and bigger very slowly
initially and then picks up speed
and allows you to have universities
that are much older than what you would get
in a straightforward Big Bang model.
Let's go on.
What I want to talk about next is what this Hubble expansion
is telling us about the universe,
and I want to go through this a little bit
carefully because it's a very important point even
though it's possible you've already figured it out
from the reading.
I don't know for sure.
Naively, Hubble's law makes it sound
like we're saying that we are the center of the universe
after all.
Copernicus was really wrong.
Everything is moving away from us, so we must be the center.
But that's actually not the case.
It turns out that when you look at things a little bit
carefully, and that's what we'll do in this diagram,
if Hubble's law looks like it holds to one observer, it
in fact, also looks like it holds to any other observer
as long as you recognize that there's no way
to measure absolute velocity.
So we think that we're at rest, but that's
really just our definition of the rest frame.
If we lived on some other galaxy,
we would equally well attribute the state
of being at rest to that other galaxy.
And that's what's being shown in this picture, which I Xeroxed
from Steve Weinberg's book so this might seem familiar to you
if you've read that chapter yet.
It shows just expansion in one direction,
but that's enough to illustrate the point.
And the top diagram we imagine that we
are living on the galaxy labeled A.
The other galaxies are moving away
from us with velocities proportional to the distance,
and we've spaced these galaxies from the diagram evenly,
so the other galaxies are moving away at v,
and then the next one is moving away at 2v.
And if we continue, it would be 3v, 4v, et cetera,
all the way out to infinity.
And what we want to do in going from A to B
is to ask suppose we were living in exactly this universe
as described on line A. But suppose we were living
in galaxy B and considered galaxy B to be at rest.
So we'd describe everything from the rest frame of galaxy B.
Then galaxy B would have no velocity,
because that would defined the rest frame.
When you change frames, this was all
done in the context of Galilean transformations.
We'll build more relativistic models later.
Then the context of the Galilean transformation,
if you go from one frame to a frame moving
at a constant velocity, the only thing you have
to do to transform velocity is you add to each velocity
a fixed velocity, that velocity difference
between the two frames.
So to go from the top to the bottom picture, what
we do in all cases is just add a velocity, v, to the left
to each velocity, and that takes the velocity of v
here when we move it with v to the right.
When we add a v to the left, we get 0.
It does the right thing there, which
is what defines the transformation we're
trying to make.
We're trying to make the transformation that
brings B to rest.
And that means that when we add v
to the left to the velocity of z,
where we already had a v to the left, we get 2v to the left.
When we add v to the left to y, which had 2v to the left,
we get 3v to the left.
Going the other direction, when we add v
to the left to c, which had a velocity 2v to the right,
we're left with the velocity of 1v to the right,
and that gives us what we have on the second row.
And if we look from the second row
from the point of view of v, the galaxy
is one way or each moving away from us with a velocity,
v. The velocity is two way and moving away
from us with velocities 2v, et cetera.
That's exactly the same.
So even though the Hubble expansion pattern
is phrased in a way that makes it look like you're talking
about yourself as the center of the universe, in fact,
it does describe a completely homogeneous picture.
And it's a picture that, in fact,
has a very simple description.
It's a picture of just uniform expansion,
and I think I have my favorite, at least the best picture I've
every drawn of uniform expansion on the next slide here.
The idea is that if you look at some region of the universe,
the claim-- and the claim is just
called homogeneous expansion-- is
that each picture at successive times would look identical,
but it would look like a photographic blowup.
Each picture would just be a bigger image
of the same picture with one important exception,
and I did try to draw this correctly,
the positions of the galaxies-- this little
lob there is supposed to be a galaxy,
by the way, in case you can't tell my great artistry.
The positions of each galaxy is just expand uniformly,
the pattern of positions, but each individual galaxy
does not expand.
The individual of the galaxies maintain their size
as the universe undergoes this public expansion.
Now if we're talking about the very early universe
before there were any galaxies, you would just have basically
a uniform distribution of matter of gas,
and that would just uniformly expand every molecule
and move away from every other molecule on average.
So this is the picture of Hubble expansion.
And now what I'd like to do is provide a description
of how we're going to treat this mathematically.
If we have this uniformly-- I'm sorry.
Yes.
AUDIENCE: I'm still getting confused
whether like the expansion is the galaxies expanding
into the universe or if the universe itself is expanding.
PROFESSOR: Yeah.
The question in case you didn't hear it was there's
some confusion here about whether we should think
of the galaxies as moving through space
or whether we should think of space itself as expanding.
And the answer really is that both points of view
should be right.
If space were like water, then you
could imagine putting little dust
in the water, little grains of salt or something
you can see and see if they are carried by the water or apart.
But there's no way to mark space.
It's intrinsic to the principle of relativity
that you can't tell if you're moving relative to space
or not.
There's no meaning to moving relative to space.
And there's no meaning for you to move relative to space.
There is also no meaning for space to move relative to you.
they do the same thing.
So you can't really tell, and both points of view
should be correct.
There are cases where you can tell, however,
which is not locally but if, for example, you
had a closed universe, which we'll talk about later how that
works exactly, then you could ask
does the volume of a closed universe get bigger with time
as this Hubble expansion takes place.
And the answer there is yes.
AUDIENCE: That would mean actually,
the universe is expanding.
PROFESSOR: The actual universe is expanding.
So we will normally think of it, globally at least,
as the actual universe expanding.
That is how we will think about it.
But locally, there's not really any distinction between that
and saying that these galaxies are just moving through space.
Yes.
AUDIENCE: So given that the galaxies are actually
two points, why can it be claimed
that the galaxies themselves are not expanding?
PROFESSOR: OK.
How do we understand the fact that the galaxies themselves
are not expanding is what you're asking.
And I'll give you a nutshell answer,
and we might be talking about it more later.
One should imagine that this starts out shortly
after the Big Bang as an almost perfectly uniform gas, which
is just uniformly expanding, everything moving away
from everything else.
But the gas is not completely uniform.
It has tiny ripples in the matter density, which
are the same ripples that we see in the cosmic microwave
background radiation today or at least the cosmic ripples
that we see in the cosmic background radiation are caused
by the ripples in the mass density of the early universe.
These ripples eventually form galaxies
because they're gravitationally unstable.
Wherever there's a slight excess of mass,
that will create a slightly stronger gravitational field
in that region pulling in more mass creating
a still stronger gravitational field pulling in more mass,
and eventually instead of having this nice uniform distribution
with just ripples at 1 part in 100,000,
you eventually have huge clumps of matter which are galaxies.
And as you go from this transition of things
being almost completely uniform and uniformly expanding
to these lumps that form galaxies,
the ones that are being formed by extra gravity pulling
in the matter.
And what happens is that extra gravity
that forms the galaxy overcomes the Hubble expansion.
The matter that makes up the galaxy
had been expanding in the early days,
but the gravitational pull of the matter that
forms the galaxy pulls it back in.
So the galaxy actually reaches a maximum size and then, in fact,
starts to get smaller and then reaches
equilibrium, an equilibrium where
the rotational motion keeps a finite size.
Yes.
AUDIENCE: So the diagrams that you're
applying up here that all the distance relations have been
galaxies that are being preserved.
Is that a potential or is it exactly [INAUDIBLE]?
PROFESSOR: Well, yeah.
It's supposed to be just a photographic blow
up as far as where the relocations of dots are.
Yeah.
I mean is that what you're asking?
AUDIENCE: Well, like will there be
equal distance between galaxies as a sub [INAUDIBLE] member?
PROFESSOR: Yeah.
I think the picture shows that, doesn't it?
AUDIENCE: Well, the notches basically
are spaces between [INAUDIBLE].
PROFESSOR: Oh, that's right.
That's right.
I haven't talked about the notches yet.
The diagrams are supposed to show actual physical distances.
So the physical distance between this galaxy and this galaxy
is a little bit there and much more there.
So that's how you're supposed to interpret that picture.
But what I was about to get to and you've got there
so I'll continue, is that the best
way to describe this uniformly-expanding system
is to introduce a coordinate system that expands with it,
and that's what these notches are.
The notches are artificial things that we create,
and we could think of them as just being labels on a map.
Once we know that the expansion is uniform this way,
we could take any one of these pictures
and think of it as a map of our region of the universe.
And we can then get to any other picture on the slide
simply by converting units on a map to physical distances
with a different scale factor.
So if Massachusetts was forever getting bigger and bigger
and we had a map of Massachusetts,
we would not have to throw away that map every day
and buy a new map.
We could handle the expansion of Massachusetts keeping
the same map just crossing out the place in the corner
of the map where it says 1 inch equals 7 miles,
and the next day 1 inch equals 8 miles,
and 1 inch equals 9 miles, and 1 inch equals 10 and 1/2 miles.
So by changing the scale factor on the map and the use
of the word scale factor here is exactly the same meaning as it
would have for a map, you can allow
yourself to describe an expanding system
without ever throwing away the original map.
And that's the kind of coordinate system
that we will be using, and these are
called comoving coordinates.
And the idea here is that galaxies
are at-- I'm sticking in the word
approximately here because none of this is exact,
but we'll be thinking in a toy model as it was exact.
So galaxies are at approximately constant values
of the coordinates, and the scale factor, which
means the physical distance per coordinate distance increases
with time.
So that describes this all-important comoving
coordinate system that we'll be using
for the rest of the course to describe the expanding
universe.
Yes.
AUDIENCE: Do we have to do anything funny to the Lorentz
transform to come to the fact that coordinates are now
not moving at the same velocity relative to each other?
PROFESSOR: It depends on what questions you ask.
There are questions where you do have to think carefully,
and we'll have one of those shortly as probably
an extra credit problem.
But for most things, it actually makes things very simple,
and you can ignore most of the complications
of special relativity.
And we'll try to think as we go along where we need to worry
about special relativity and where we don't, and usually we
don't.
So the key relationship is that the physical distance
between any two points on the map, by physical distance,
I mean what it is in the real world,
miles if we're talking about Massachusetts,
and this is miles between the real physical points,
is equal to a time-dependent scale
factor times the coordinate distance.
Now, here I'm going to use conventions
for defining things that are slightly different from what
are often used.
A common procedure where I think it's done in most of the books,
is to think of both the coordinate distance
and the physical distances being measured in normal distance
units, meters, and then the scale factor is dimensionless,
and it just tells you how much you
have to blow up the map to be able to make
it the physical map.
I find it significantly easier to know
what you're doing as things go along
to label the map in units that are not centimeters, but are
what shown on the picture as notches.
One advantage of that logically is it
means that if you have different copies of the map
that you've made on a Xerox machine
with different scalings, you have a big copy of the map
and a little copy of the map, that they're
marked off with notches.
The notches grow with the physical size of the page,
the scale factor is the same no matter
which copy of the map you're using.
But most importantly, it allows you to, I think,
do dimensional tests.
The size of the map is clearly something
that's unrelated to the actual length of a meter,
it's just how many units you put on your map.
So there's a clear and logical separation
between what is meant by a certain number of units
and distance here and a real meter by any standards.
So you can keep that straight by just imagining
that your map is calibrated in some new arbitrary unit which
is just special to the map, and I'm going to call that a notch.
So notches are just arbitrary units
that you use to mark off your map.
And then the physical distance is, of course,
measured in meters or any other standard unit of distance.
And then the scale factor is measured
in units of meters per notch instead of being dimensionless.
And the basic advantage of this is that when you're all done,
nothing had better have any notches in it
as you're calculating something physical, that is physical
why you should not depend on the size of your map.
So you have a nice dimensional check
to make sure that the notches drop out
of any physical calculation that you try to do.
What I want to do next is to show you
that this relationship leads me to Hubble's law
and furthermore will allow us to figure out what this Hubble
expansion rate is in terms of what the scale factor is doing.
So it's an easy enough calculation
if we're looking at some object out there
and it's physical distance l sub p is given by that formula,
and we want to know what its velocity is.
It's velocity is by definition, just
a time derivative of that quantity.
So the velocity of some object, some distance out in space,
is just equal to d dt of l sub p of t,
and that will be da dt times l sub c,
because l sub c is constant.
On average, all our galaxies are at rest
in this coordinate system, in this expanding coordinate
system.
Now this could be written in a way that ends up
being more useful by dividing and multiplying by a.
So I could write it as 1 over a times
da dt times a of t times lc.
And the advantage of multiplying and dividing by a this way
is that this quantity is again just l
sub p, the physical distance.
So now we say that the velocity of any distant object
is equal to 1 over a da dt times the distance to that object.
And that is Hubble's law, and it tells us
that the Hubble expansion rate, which is itself
going to be a function of time, is
equal to 1 over a times da dt.
And this allows us to illustrate the unit check
that I talked about for the first time.
Notice that a is measured in meters per notch,
so the meters per notch here cancels the meters per notch
here, and you just get inverse time,
and the really important thing is
that the notches have gone away.
And again, notches have to go away
in any calculation of any physical,
and that makes a nice check.
And once you know how a of t is behaving,
you know exactly how the Hubble expansion rate behaves.
It's determined by a of t.
You might mention one notational item.
Nowadays almost everybody calls this scale factor little a.
In the early days, it was very first
introduced by Alexander Friedmann who first invented
the equation describing expanding universe
in the early 1920s.
He used the letter R, capital R. Lemaitre also
used the letter capital R, and I guess Einstein probably
used the capital R. I'm not sure.
And going into more modern times,
Steve Weinberg wrote a book on gravitation and cosmology
which still used the letter capital R,
but that was sort of the last major work that
used the capital R for the scale factor.
The disadvantage of it is at the same time
this capital R means something else in general relativity.
It's the standard symbol for what's
called the scalar curvature in general relativity.
So to avoid confusion between those two quantities, nowadays
almost everybody calls the scale factor little a.
If you look at old A286 notes, I used
to follow Steve Weinberg's textbook on gravitation
and cosmology and call it capital R,
but now it's hopefully all switched to little a.
OK.
Next item.
If we're going to understand what
it would look like to live in a universe like this,
we're going to need to know how to trace
light rays through our expanding universe.
And that turned out to be easy.
If I let x be equal to a coordinate, that
means it's measured in notches, and if I imagine
I have a light ray moving in the x direction,
I can describe how that light ray is going to move.
If I can write down a formula for the dx dt.
Tells me how fast in the coordinate system
the light ray travels.
Well the basic principle that we're going to use
here is that light, in fact, always
travels at the speed of light at some fixed value c,
but c is the physical velocity of light, the velocity measured
in meters per second.
But dx dt is the velocity measured in notches per second
because our coordinate system is marked off
not in meters, but in notches.
And that really is very important
because meters are going to be constantly changing lengths
relative to our notches, and we want
to keep track of things in notches
so we have a nice picture within our coordinate description
of the universe that we can think about.
So we're going to want to know what dx dt is,
but it's just a unit conversion problem.
dx dt is the speed of light in notches per second.
We know the speed of light in meters per second, c.
So to convert is just a matter of multiplying by the scale
factor to convert the units of meters to notches.
And here again it helps to have this meters versus notches
because it guarantees that you can't get it wrong
if you just check your units.
So this is not really a question mark.
It is just c divided by a of t, the scale factor.
And we can make sure we got it right by checking our units.
I'm going to use brackets to indicate units of.
So we're going to work out what the units are
of c over a of t, trivial problem, of course,
but we'll make sure we got the right answer.
The units of c are, of course, meters per second.
a of t we said is meters per notch.
So the meters cancel, and we get notches per second.
Now, I told you that you should never get notches
in the physical answer because this is not a physical answer.
This is a coordinate velocity of light,
so it does depend on our coordinates
depending on what coordinates we've chosen.
So it should certainly be notches per second
because x is measured in notches and t is measured in seconds.
So we put in the a of t in the right place.
It does belong in the denominator
and not the numerator.
Yes.
AUDIENCE: Why aren't we worrying about the fact
that as the universe expands, there's
also a velocity component with a light rate from its position
moving according the Hubble expansion?
PROFESSOR: Right.
The reason we don't worry about that
is that special relativity tells us
that all inertial observers are equivalent
and that the speed of light does not depend on the cannon
that the light beam was shot out of.
So if I'm at rest in this expanding coordinate system,
I'm not really an inertial observer
because there is gravity in this whole system,
but we're going to ignore that.
If we're really being rigorous here,
we have to do the full general relatively thing.
But I think the intuitive explanation is pretty obviously
valid.
It is, in fact, rigorously valid.
If I'm standing still in this expanding coordinate system,
I am an inertial observer.
And if a light beam comes by me, I
should measure it's B and C, no matter where it started,
no matter what's happened in the past.
So the conversion between my units of distance
and physical units of distance, my coordinate distances
and physical distances, is just a of t.
So that's the only factor that appears
and this is completely rigorous.
One can drive this in a more general context
in general relativity.
Here we're starting out with a premise
that the light pulse travels at speed
say if one had the full theory of general relativity
coupled to Maxwell's equations we
could really derive the fact of exactly how rays
travel and would give us this.
So we have a very simple equation
for how light rays travel.
Now I want to spend a little bit of time, and this, I guess,
will be the last topic I'll talk about today discussing
the synchronization of clocks in this world system,
in this cosmological coordinate system.
In special relativity, you know that it's
hard to talk about synchronizing clocks at large distances.
The synchronization of clocks depends
on the velocity of the observer.
That was one of the principles we learned
about when I wrote down those three
kinematic properties of special relativity.
So in general, in the context of special relativity,
there is no universal way of synchronize the clocks.
You could synchronize clocks with respect to one observer
but then they would not be synchronized with respect
to another observer moving with respect to that observer.
In this case, we have perhaps even a further complication
although in the end everything is simple,
but we have the further complication that the different
clocks that we're talking about, which are clocks carried
by these observers that are stationary in our comoving
coordinate system-- clocks carried essentially by galaxies
that are uniformly expanding-- all these clocks
are moving relative to each other.
because the Hubble expansion tells us that.
So the notion of trying to synchronize clocks
seems a bit formidable.
Turns out however that we can synchronize clocks
and one can develop a notion of what we call cosmic time, which
is the time that would be right on all these clocks
where all these clocks, I mean all the clocks
that are stationary with respect to the local matter,
in other words stationary with respect to this comoving
but expanding coordinate system.
So why can't we synchronize clocks?
What we're using as our core assumption, which
is what makes everything simple, is
that the model universe that we're building this homogeneous
and that means that what I would see if I was living
in this universe would not depend on where I was.
So if I were living on galaxy number one
and took out my stopwatch and timed
how long it took before say the Hubble parameter changed
from Hubble's value to the current value
say, as an example, any two numbers,
if I were living any place else and timed the same thing,
how long it took for the Hubble expansion rate
to change from A to B, I would have to get exactly
the same time interval, otherwise, it
would not be homogeneous.
Homogeneous means everybody sees exactly the same thing.
So we all have, no matter where we
live in this universe, a common history,
and that means that the only thing we don't know
is how to synchronize our clocks what time on my watch
might correspond to what time on your watch.
But if we imagine that we could send signals
or that we're some global observer watching
the whole thing, then we could just tell each other let's
all set our clocks to noon when the Hubble expansion
rate is 500 kilometers per second per megaparsec.
And then we would have a well-defined synchronization.
And once we synchronized our watch that way,
if we each looked at how the Hubble expansion rate changed
with time, we would get exactly the same function of time
by this principle of homogeneity.
None of us could see anything different
as long as we're measuring time intervals,
and we've fixed it so now we're measuring nothing
but time intervals because we've arranged to all
set our clocks to the same time at some particular value
to the Hubble parameter.
So to synchronize, we can ask what are the options.
I mentioned the Hubble parameter.
That's certainly one parameter that
can be used in principle to synchronize clocks
throughout our model universe.
You might wonder if we can use the scale factor
itself to synchronize times.
And the answer there I would say is
no because of this ambiguity of the notch.
I have no way of comparing my notch to your notch.
We can compare physical distances
because they're related to physical properties
as the size of a hydrogen atom is a certain physical size,
no matter where it is in this universe.
So we could use hydrogen atoms to measure meters,
and we would all be talking about the same meters.
And we could use those meters to define seconds
by how long it takes light to travel through a meter,
and so on.
So meters and seconds, we can all
agree on because they're related to physical phenomena
that we can all see and that will
be the same everywhere in our homogeneous model universe.
But notches, not so, everybody gets to make up his own notch.
It's just the size of the map he happens to draw.
So we cannot compare scale factors and say,
we'll set our clocks to a certain time when both
of our scale factors have a certain value.
We would get different synchronizations
depending on what choices we've made
about having to find a notch.
So the scale factor does not work as a synchronization
mechanism, Hubble expansion rates does.
Also we haven't really talked about an ideal cosmic microwave
background, but we certainly talked about it,
the cosmic microwave background has a temperature which
is falling as the universe expands,
so that could be used to synchronize clocks as well.
Might mention in the last 30 seconds
one interesting phenomenon.
For our universe, the Hubble expansion rate
is changing with time, the temperature
is changing with time.
There's no problem talking about this synchronization.
But if you're talking about different kinds
of mathematical models of the universes,
you can imagine a universe where the Hubble expansion
rate is just constant, and in fact
that is a space that was studied very
early in the history of general relativity.
It's called de Sitter space.
And it's approximately what happens during inflation,
so we'll even be talking about de Sitter space
later in the course.
In de Sitter space, the Hubble constant
is absolutely constant, so at least one
of the mechanisms I mentioned to synchronize the clocks
goes away.
There's also, in fact, no cosmic microwave background radiation
in pure de Sitter space, so that goes away.
You could ask, is there anything else,
it turns out there is not, so you really
can construct a well-defined model of the universe,
the so-called de Sitter space, where there really
is no way of synthesizing clocks.
And you could really show that you could make transformations
so that if you synchronize the clocks one way,
you could make a symmetry transformation if we take
all those clocks out of synchronization
and otherwise the space would be just as
good as what you started with.
So the synchronization is subtle,
and it depends on having something which actually
changes with time, but that will be
the case where our real universe and for the model
universes that we'll be talking about.
So I'll stop there.
See you folks on Thursday.