The following
content is provided
under a Creative
Commons license.
Your support will help MIT
OpenCourseWare continue
to offer high quality
educational resources for free.
To make a donation or to
view additional materials
from hundreds of MIT courses,
visit MIT OpenCourseWare
at ocw.mit.edu.
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.
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.
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.
