Professor Charles
Bailyn: Welcome back for
more cosmology on a rainy day.
You'll recall,
maybe, what we were trying to
do.
We're trying to determine the
past and future of the
Universe--which,
it turns out,
can be summarized by a single
number, namely,
the scale factor of the
Universe.
It's actually not a single
number.
It's actually a function. Right?
It's a function of time.
The scale factor changes with
time.
So, when I make this grandiose
statement that I'm going to talk
about the past and future of the
Universe,
what I really mean is,
I want to determine what the
scale factor is as a function of
time.
And so, that's just a standard
kind of a graph.
Here's the scale factor,
usually given the letter
a.
Function of t.
Here's time.
The present is somewhere along
this graph.
Call that now.
And so, here's--and the
Universe has some scale factor
at the present time.
And what we're going to do is
we're going to call that--define
the scale factor at the present
time to be 1.
We're just going to--that's
called normalizing:
when you set--when you multiply
something by some convenient
number so that it comes out to
be 1 at a convenient moment.
So, that's just a definition.
We define the current scale
factor to be 1,
to be unity.
And so, there's a point here
at--right now,
the scale factor is 1.
We also know what the slope of
the scale factor is,
because we measure that from
the Hubble Constant.
And so, in an appropriate set
of units, the scale factor is
increasing at a rate of the
Hubble Constant,
which is, I think,
2 x 10^(-18) per second.
So, if the scale factor is one
right now, then one second from
now, the scale factor is going
to be 1 + 2 x 10^(-18).
So, every second,
it gets 2 x 10^(-18) bigger.
That's not a real large
increase and,
in fact, over the entire
history of humanity,
it won't increase by that much.
Let's see, in a million years,
there is something like--okay,
in 10^(6) years,
there are 3 x 10^(7) x 10^(6)
seconds.
That's 3 x 10^(13) seconds.
Multiply that by 2 x 10^(-18)
and the scale factor increases
by 3 x 10^(13) x 2 x 10^(-18).
That's 6 x 10^(-5).
So, it still hasn't made a
whole lot difference,
even a million years from now.
But, keep at it for a while,
and the scale factor is going
to pile up and will increase.
So, we know that it's
increasing.
And then, the question is,
"Is this increase changing with
time?"
So, let's see.
If it doesn't change--if it's
just a straight linear increase,
this is the situation in which
there's no matter in the
Universe.
So, the Universe is not slowing
down because of gravity,
because there's no matter in
it.
And that gets labeled with this
factor Omega [Ω]
being equal to 0.
Let me remind you:
Ω is defined as the density of
the Universe divided by the
critical density.
The critical density is how
dense it has to be to turn
around.
So, another possible track for
the Universe:
we know it's expanding at a
particular rate now,
but it could slow down in the
future, turn around and fall
back--could look something like
this.
This is the case where Ω is
greater than 1.
And then, the intermediate case
where Ω is less than 1,
but not equal to 0.
It sort of does this.
It falls off a little bit,
continues to expand,
something like that.
And this Ω less than 1,
but still greater than 0.
Okay.
So, here's the goal.
We've got to figure out which
track we're on.
At the moment,
we sit here,
and we measure the Hubble
Constant.
So, all we know is where we are
and where we're going in the
short run.
And what we want to do is to
figure out which of these
possibilities the Universe is
actually following.
And in so doing,
decide whether we're going to
end up falling back together in
this Big Crunch or expanding
forever.
And we talked a little bit last
time about a kind of direct way
of determining this,
namely, measuring Ω.
This Ω factor is the density
of the Universe divided by the
critical density.
You could, in principle,
go out and measure the density
of the Universe.
You can compute the critical
density.
You divide one by another,
and you know which of these
possibilities turns out to be
true.
And that ran into trouble,
as you'll recall.
So, direct measurement of Ω.
This leads to trouble in the
form of dark matter.
So, the Universe turns out to
be full of this dark matter.
If you add up all the
galaxies--the mass of all the
galaxies, including dark
matter--so,
mass in galaxies gives you an
Ω that's less than 1,
that's around 1/3 or 1/4.
But there might be--you know,
now that we've already
discovered that the vast
majority of matter is something
we know nothing about,
there might be more of it
between galaxies that we don't
know about.
So, we want a different
approach.
So, here's something you can do
that's really quite a different
approach to this problem.
Let's draw that plot up again.
And so, here's now,
and then, there are these
various possibilities.
And it might go like this,
or it might,
you know, fall all the way back
down like that.
So, here's something you can do.
Remember that you can look back
into the past.
You can look into the past by
looking at very distant things.
And you see the light coming to
you from the time in the past.
So, you look into the past
because of the light travel
time.
And so, if now is here,
we can actually look at objects
which were existing at some time
in the past,
back, say, here,
I don't know--by looking at
things that are three billion
light years away,
or something like that.
And the question is,
if you can do that--you can
figure out how far in the past
it is, by the way,
that's easy.
Measure distance and time in
the past is just distance.
Miles per hour.
Which way up is it going to be?
Divided by the speed of light.
So, if you can measure the
distance to something you know
how long--how far in the past it
was.
So, that tells you where it is
on this axis.
So, then, all you need to do is
measure the scale factor at that
time.
And you can see that if you
could do this,
you would be able to determine
whether at some point in the
past you are here,
if the scale factor was low,
or here, or here.
And therefore,
if you made accurate
measurements of the scale factor
in the past, you could figure
out which one of these tracks
you're on.
And if you could figure that
out, then you can infer,
if that track continues without
some other form of physics
kicking in,
you could figure out where it
will go in the future.
So, you've got to be able to
determine what the scale factor
was, which doesn't seem like
it's actually all that easy,
but it turns out that this can
be done.
And the way to think about this
is you have to think about the
redshift, the cosmological
redshift,
in different terms from what
we've been thinking about it so
far.
So, a different view of
redshift.
You remember this is where we
started, right?
We measured the distance to
things.
We measured their velocity.
We got Hubble's Law;
therefore, the Universe is
expanding.
And as we were doing that,
we had the idea that the
redshift is telling you
something about velocity.
So, one view is that the
redshift tells you something
about velocity,
and the velocities are all
moving away from us.
And therefore,
the Universe is expanding.
But you remember that the
cosmological redshift caused by
the expansion of the
Universe--that velocity is a
little different in kind from
the way we ordinarily think of
velocity.
It's not--objects are not
moving through space.
It's just, the space itself is
expanding and that leads to some
differences.
For example,
in principle,
the cosmological redshift can
cause things that look like the
velocity is potentially greater
than the speed of light,
and things like that.
So, the cosmological redshift
is not the same as an ordinary
velocity.
And one way to think about what
the redshift is,
is the following.
So, another view.
The Universe is expanding.
But a lot of the things in the
Universe are not expanding.
You're not expanding.
The galaxy's not expanding.
We talked about this a little.
That's because there are forces
holding it together.
The Earth isn't expanding
because its gravitational force
holds it together.
I'm not expanding because the
chemical bonds between the atoms
in my body hold it together.
The galaxy is not expanding
because gravity holds it
together.
But a wavelength of light
doesn't have such a force to
hold it together.
So, as the Universe expands,
wavelengths of light expand
along with it.
So, this is another view of
where the redshift comes from:
that wavelengths of light
expand along with the Universe.
That's an interesting thought.
So, the question is,
"Why do we observe light from
distant galaxies to be
redshifted?"
In this interpretation,
it's not because these things
are moving away from us.
It's because,
during the time between when
that wavelength was emitted and
when we see it,
the Universe has expanded and
the wavelength has expanded
along with it.
And, therefore,
we see a longer wavelength than
we would have from a photon that
hadn't been traveling as long,
because the Universe,
in that case,
wouldn't have expanded as much.
Wavelengths of light expand
along with the Universe.
And so, when we observe distant
objects, the wavelength we
observe is longer than when it
was emitted,
because the Universe has been
expanding during that time.
This is--yes, go ahead.
Student: So,
if the Universe is expanding as
the light is traveling,
by the time the light reaches
this Universe,
will be somewhat larger than
when--than when the light was
emitted?
Professor Charles
Bailyn: Yeah.
Student: So,
is the time--is the time that
we, you know,
that we even have [inaudible]
in the past,
is that the distance divided
by--I mean, is that the original
distance divided by the speed of
light or the-- Professor
Charles Bailyn: Okay.
So, this is this complicated
thing about, what do we mean by
distance in an expanding
Universe?
And what you mean--this is a
good question.
What you mean by distance is
how much--okay.
So, the first thing to say is
that cosmologists worry about
this a lot, and there are
different kinds of distances
that can be defined.
There's the coordinate
distance, which doesn't change
as the Universe expands.
There are other kinds of
distances.
In the particular case of what
we're doing right now,
the key way you measure
distance, remember,
is the standard candle.
So, you define what's called
the luminosity distance,
which is how much fainter
something looks.
And so, basically,
distance is defined--so let
me--this is kind of a
parenthesis.
What do I mean by distance?
Distance is defined by this
equation.
So, what you mean by distance,
in this case,
is how much fainter are the
objects you're looking at than
otherwise.
And so, something that's twice
as far away--what does that mean
for something to be twice as far
away?
It means the light from a
standard candle is four times
fainter.
And so, you define distance as
the dimming of light,
because that's the way you
measure the thing.
And then, you have to be
careful when you're doing this
out in strict mathematical
terms.
You have to be careful that
that kind--that the way you've
defined, you don't get confused
about which kind of distance
you're working on.
So, that defines distance.
Then, time, the lookback time,
turns out to be,
more or less,
certainly for short distances,
correctly defined as Distance
/ c.
But you do have to worry about
this when you get into large
cosmological distances and I
don't want to go into that,
but the different kinds of
distances are related to each
other in complicated ways.
Okay.
So, this is basically back to
the question of how you get the
x-axis on that graph.
I'm now talking about the
y-axis,
the scale factor.
So, then, how do you measure
the scale factor?
The scale factor--so,
the scale factor now,
of the Universe now,
divided by the scale factor
then,
when you emitted that
wavelength, is going to be equal
to the observed wavelength,
divided by the wavelength of
the light when it was emitted.
Because that's how much larger
the Universe has gotten,
and the wavelengths have gotten
big along with it.
So that's equal to
λ_emit plus Δλ,
as we've defined it before,
over λ_emit,
which is equal to 1 + Z.
Because the redshift,
Z, remember,
is the change in the wavelength
divided by the wavelength when
it was emitted.
So, by measuring redshifts,
you can tell what the scale of
the Universe was relative to the
scale of the Universe now,
when that light was emitted.
Now, remember,
I defined the scale factor now
to be 1.
So, that means 1 / a is
equal to 1 + Z, or the
scale factor of the Universe in
units where it's currently equal
to 1,
is equal to 1 / (1 + Z).
And Z, you measure.
And so, in principle,
you can measure both axes of
that graph.
And so, if I now go back to my
plot, you could measure some
points on this plot.
Now, all of them are in the
past, not the future,
because you can't look into the
future by this light travel time
trick.
But you could go out there and
you could discover,
if you made,
you know, really excellent
measurements,
you could discover that there's
a point right here.
That a particular galaxy,
or whatever it is you're
looking at, is far away such
that the time back is this.
And then, you measure the
redshift.
You get a scale factor.
And you could measure that at
that time the Universe had such
and such a scale factor.
And if you measured a whole
bunch of these things and they
sort of lined up in this nice
way,
then you might conclude,
well, the Universe is doing
this, and you might then
extrapolate to determine what
the Universe is going to do in
the future.
And so, this is just the Hubble
Diagram interpreted in a
slightly different way and being
looked at at a distance
sufficiently far away that the
rate of expansion would have
been different back then,
than it is now.
So, I'm going to draw it all
again.
Here's a.
Here's t.
There's this.
Here's one of these.
And here's this.
But, you know,
that's not usually how we plot
the Hubble Diagram.
What we're usually plotting,
as you'll recall,
is distance versus velocity,
because those are the things we
actually measure.
Velocity is redshift, Z.
And remember that 1 / (1 +
Z) = a,
or--let's see.
1 / a = 1 + Z
a / a 1 / a
1 – a / a,
yeah.
1 – a / a =
Z.
So, this represents the scale
factor, this axis,
whereas this--what we actually,
usually, plot is m -
M, which,
you'll recall,
is equal to 5 log (D /
10 parsecs).
And so, this scale represents
time, because distance can be
converted to time,
because we know the speed of
light.
So, this is a time axis in
really weird units.
And this is a scale factor axis
in really weird units.
But, you can,
therefore, transform each of
these lines, which is scale
factor versus time,
into an equivalent line on this
plot.
So, the green line is the plot
where the Universe just expands
and there's no mass.
Nothing slows down.
So, it kind of looks like this.
All right.
Now, what are the points?
This is now in the time factor,
and up here is then.
Okay, so, you're going from now
to then.
So, you're kind of sitting on
this point, looking back this
way.
And here is scale factor
a = 1,
and here's scale factor
a = small.
Okay?
Because the Z is getting
bigger, therefore a is
getting smaller.
So, what would the red curve
look like on this plot of the
things we actually measure?
Well, when it's close,
the red curve sits right on top
of the green curve.
So, it kind of starts out like
this, with no distinction.
Then, at a certain point in the
past, let's see.
How do I want to do this?
Let's pick a particular time.
So, that's a particular
distance.
At that time,
the scale factor is smaller on
the red curve than it is on the
green curve.
If the scale factor is smaller,
the redshift has to be bigger.
So, what's going to be
happening, here,
is the red line is going to
gradually diverge from the green
line in just the same way it
does here,
in such a way that at a given
distance--that is to say,
a given time in the past,
the redshift is bigger,
which means the scale factor is
smaller.
And then, the brown curve will
kind of do the same thing,
only more so.
So, this is what we expect to
see.
This is Ω greater than 1.
Ω less than 1,
but still greater than 0.
Ω = 0.
So, all the Hubble Diagrams
we've looked at so far have been
down at this end,
here, where they're all lying
on top of each other.
But you can imagine that if you
get out to big redshifts,
redshift of 1.
At a redshift of 1,
the scale factor is half of
what it is today.
You're halfway back to the Big
Bang in terms of scale factor.
You can imagine that these
things would really look quite
different, as indeed they do in
this little plot I've drawn,
when the scale factor is half
of what it is today.
If the scale factor is 1% less
than it is today,
you'll have a tough time
telling these things apart.
So, what do you need to do?
What you want to do is measure
redshifts and distances.
Plot them on this plot,
and see which of these lines
turns out to be true.
What do you need to do that?
What you need is a really
bright standard candle.
Why does it need to be bright?
Because you need to be able to
see it at large distances.
And the thing that happened in
the 1990s, and that has
transformed cosmology,
is that we are now able to make
these kinds of measurements all
the way out to a redshift of 1,
and in some cases, further.
And the particular standard
candle we're looking at are the
so-called Type 1a Supernovae.
I'll discuss what these are
later on, but for the moment
let's just take the point that
they're standard candles,
therefore we know how bright
they are.
And they're really,
really bright--absolute
magnitude of something like -19.
So, they're as bright as whole
galaxies.
In fact, they often outshine
the galaxy they're in.
These are exploding stars and,
as I say, we'll talk about the
details of exactly how these
things work later on.
So, Type 1a Supernovae turn out
to be these really bright
standard candles,
and you can make this kind of
measurement on them.
I have the data here,
but it's--I have to talk about
one more technical thing about
the way the data's presented.
Let's see.
Ordinarily, the way you tend to
plot these things,
as I said, is you plot redshift
versus apparent minus absolute
magnitude.
This is a quantity known as the
distance modulus,
and it's related to distance by
the equation we had before.
And so, then,
what you get is,
you get this,
and you get this,
and you get this,
as we've written it down
before.
And the problem is that it's
hard to tell the differences
between these lines because this
is now being measured in
magnitude.
The separation out here at a
redshift of 1.
This is about--this interval
here, in magnitude--let me write
it that way--is about--oh,
I don't know,
a third of a magnitude or
something like that.
But this whole plot is,
you know, if you go all the way
back to time 0,
this whole plot is,
I don't know,
can be 10 to 20 magnitudes.
And so, it's just kind of hard
to see these little differences
between the things.
So, they plot it differently so
that you can bring out these
slight differences.
And what they do is,
they do this.
They plot a quantity called Δ
(m – M).
What's that?
That's the difference between
the measured distance modulus
and the distance modulus that
you would expect at that
particular redshift in an empty
Universe.
So, m - M at a
given red shift in an empty
Universe.
So, what does the green curve
look like?
The green curve is Ω = 0.
That is to say,
an empty Universe.
Student: [Inaudible]
Professor Charles
Bailyn: Yeah,
yeah, yeah.
It's just a straight line at 0,
because it's always true that
the empty Universe,
minus what you expect for the
empty Universe,
is 0.
So, what you've done,
basically, is you've taken this
plot and tipped it 45º so that
this green line is straight at
0.
And then the brown line will
look like this--sorry,
the red line will look like
this.
The brown line will look like
this.
This, again,
Ω greater than 1,
Ω less than 1.
Ω, 0.
So, that's just convenient,
because then,
you can have your scale here go
from, I don't know,
-0.3 to +0.3,
or something like this.
And so, you can see the whole
thing.
Actually, I've got the signs
wrong, haven't I?
Because bright things are a
minus.
So, let's just put it as bright
and faint.
So, we've done our very best to
confuse you, right?
First of all,
we started with a magnitude
system that's upside down and
logarithmic.
Second of all,
we took the intuitively simple
plot of scale factor versus
time,
which actually tells you what
you need to know about the
Universe: is it getting bigger?
Is it getting smaller?
Is it going to have a Big
Crunch?
And we transformed it into this
rather stranger thing,
where the things we are
plotting are the things we
measure--but it's not at all
obvious how this translates into
what the Universe is doing.
Then, not content with that,
we take that whole thing and
subtract something you don't
understand from it,
so as to flatten out one of the
lines.
Okay?
So, it's a three-part exercise
in confusion.
And I would say that the number
one conceptually difficult thing
in this entire course is to
understand how this plot relates
to this plot,
and more difficultly,
how this plot relates back to
the scale of the Universe plot.
And if you can understand that,
you can understand twenty-first
century cosmology.
And so, we will work on this a
little bit, so,
if you're not quite getting it
yet, don't freak out.
You'll have plenty more
opportunities to discuss it,
I assure you.
In any case,
even if you're a little puzzled
at the moment,
let's look at this plot because
this is how the data show up.
So, there's an Ω = 0 line,
which is straight.
And then, lines below it are
increasingly large values of Ω.
And so, if you go out and
measure a bunch of points,
you expect to see them kind of
lying in a straight line here.
And then, you expect to see
them falling below that straight
line, and depending on how far
below the straight line they
fall,
you can determine whether the
Universe is going to re-collapse
or not.
So, if they end up,
you know, down here,
it's going to re-collapse.
If they end up here,
it's going to expand forever.
Okay?
How are we doing?
Ready for some data?
Okay.
So, in the 1990s,
as I mentioned,
they actually made this
measurement.
In fact, two different groups
made the measurement,
and they were rivals.
They did things slightly
differently.
They're measuring the same
kinds of objects.
Here's what they found.
Okay.
So, let's see.
This is this Δ (m –
M).
So, 0 on this plot is the empty
Universe.
And then, this dotted line is
Ω.
So, let me color code this the
way I had color-coded everything
earlier.
So, I'm going to just take this
and draw the relevant lines.
So, here's the green line at
exactly Ω = 0.
And then, this thing is the
brown line--sorry,
the red line.
Got to get my colors straight.
Red line.
This is Ω less than 1.
In fact, this particular line
is for Ω = 1 / 4.
And then, the brown line is
this thing here,
which is Ω = 1.
So, if you fall below the brown
line, the Universe is going to
re-collapse.
Right.
What else do I want to say
about that?
Nothing, because I've now put
some points on there.
Yes?
Student: Why are they
lines [inaudible]
whereas the ones that you used
before they were curved?
Professor Charles
Bailyn: Oh,
these are actually curved.
It's just, I exaggerated the
curvature.
The reason that it comes out
less curved on this is it's only
going up to a redshift of 1.
If you continued this out much
further, the curvature would
become more obvious.
But these are not,
in fact, straight lines.
The green one's a straight line
because it's just at everything
equals 0.
But there is curvature on these
lines.
Okay.
So, the problem's obvious.
Here are the observed points,
and they don't go through any
of these lines.
And you can first of--okay.
So, there are many things to
say about this.
First, the error bars.
Let's talk about the error bars.
Redshifts are relatively easy
to observe.
You can do that to high
accuracy.
This distance modulus is more
complicated.
There are two kinds of errors
incorporated into these error
bars.
One is measurement errors--that
is to say, how accurately can
you actually measure the
brightness of these things?
That's an error in m,
you know, how bright--how
accurately can you measure these
things?
They're pretty faint by the
time they're out at a redshift
of .6.
But the other is how good a
standard candle is.
How well do you know exactly
what the absolute magnitude is
supposed to be?
So, accuracy of the standard
candle.
And that's an error in big
M, because you don't
know--these things turn out to
have a little bit of a spread in
brightness, and you don't know
that.
And so, that's why the nearby
ones, these ones down on this
end, which you can measure,
really, quite precisely,
still have substantial errors,
because you don't know how
bright the standard candle is
supposed to be to perfect
precision.
And so, you can see that down
below a redshift of .2,
down in this region here,
you're going to have a really
tough time telling the
difference between these lines.
Because the lines are separated
by a tiny fraction of a
magnitude and the errors on
these things,
on any individual object,
is several tenths of a
magnitude.
So, you know,
pretty much all of these points
go through all of the lines,
and so, that isn't going to
tell you a great deal.
By the time you're out in here,
you're starting to get several
tenths of a magnitude difference
in the prediction.
Now, the errors are still
several tenths of a magnitude.
So, observing one such object
isn't going to tell you anything
because, supposing I observe any
one of these things,
I don't care.
Here's an error bar.
And while it doesn't go through
this line, that's only,
you know, two times the error.
Two Sigma error for those of
you have studied statistics,
isn't such a crazy thing.
And so, one of these points
would not tell you all that
much.
But on this plot,
which was published in 2003 and
contains data up until about
2001, there are a couple dozen
points.
Now, this line here,
the red line with Ω = 1 / 4.
We already know that Ω has to
be equal to at least 1 / 4,
because we've measured all that
dark matter out there.
We added it up,
and it comes out to about Ω =
1 / 4 or 1 / 3,
or something like that.
And so, if there's nothing else
in the Universe beyond what we
already know about,
you expect it to be on this
line.
If there is matter in the
Universe that we don't know
about, you expect it to be below
that line.
So, if you take the points at
redshifts greater than .2,
the interesting fact is,
all of them lie above this
line, and none of them lie below
that line.
Now, each of them has a
different error.
But nevertheless,
the points are all above the
line.
So, if the Universe truly were
following this red line here,
then what would you say?
You would say that half of them
should lie above and half of
them--and points should lie
above--below just by random
chance.
So, this is throwing two dozen
coins and having them all come
up heads because they've all,
by random chance,
if the Universe is following
this line, landed above the
line.
So, you can figure out what the
probability of that is.
It's--you know,
if there are 20 points and
they're all heads,
that's 2^(20).
That's about--a probability of
one in a million.
There was another team doing
the same experiment.
They measured about the same
number of points,
got the same result.
So, that's two incidences of a
one in a million chance.
That's one in a trillion.
So, random errors are not going
to get you back down onto the
redline, much less there.
Now, look at the green line.
This is an empty Universe.
Nothing in it at all.
Maybe we're just totally wrong
and we've screwed up all the
dark matter, and we've got only
two points below that.
That's almost equally
improbable.
And so, what we think is going
on is that this line actually
goes above the green line.
And one current hypothesis for
what such a line should be is
this solid line up here,
which nicely goes through the
points in such a way that,
kind of, half of them are above
and half of them are below--the
way points ought to be.
So, this is quite strong
evidence that,
in fact, what the Universe is
doing is not going below that
line, but going above it.
So, what does that tell us?
All right.
Let's do this whole exercise
backwards.
Δ (m – M)
versus Z.
Here's the empty Universe,
and here's what we see.
So, how does that go onto the
next plot?
This is now m - M
versus redshift again.
Here's the empty Universe.
And so, what appears to be
happening is this.
How does that,
then, map onto the scale factor
versus the redshift?
Scale factor versus time, sorry.
Here's now.
Here's 1.
Here's this.
This is now still the empty
Universe.
It's a straight line in this
case, and so,
what is happening here?
If you go back and you look at
a particular distance--that is
to say, a particular time
ago--so, you're looking here.
The purple line here has a
redshift that is smaller and
therefore the scale factor is
larger.
So, you ought to be here, right?
So, that's what we're
observing--the implication of
which is that the future is
going to look like this.
So, the Universe in this
is--Universe is expanding.
We already knew that.
And the expansion is
accelerating.
Something is pushing the
Universe outwards.
So, we had a thought all along.
What's going to happen?
Gravity is going to pull it
back together again.
It's going to be slowing down.
No, not true.
The Universe is being pushed
outward by some kind of force,
by some kind of energy,
which has the characteristics
of a repulsive gravity.
So, it's anti-gravity,
if you like.
This is what the science
fiction writers would certainly
call it.
It's something that pushes
things away from each other.
And this pervades the Universe
in such a way that it actually
dominates the behavior of the
Universe as a whole.
We give this the name dark
energy, and we have absolutely
no idea what it is.
But, of course,
as my colleague,
Charlie Baltay,
likes to point out,
we never actually admit to
people that we don't know what
we're talking about.
So, you observe this kind of
thing.
You don't say:
we have no idea what's going on
in the Universe.
No.
You say: we discovered dark
energy.
Big triumph.
And so, we have discovered dark
energy, and furthermore,
we know something about how
much of it there is--namely,
because we can actually measure
this curve.
And so, you can ask:
what is the density of the dark
energy compared to matter?
Now, you know how to compare
energy to matter,
right?
You have to use E =
mc^(2).
So, if you have energy--some
amount of energy per meter
cubed, you can transform that
into energy / c^(2) per
meter cubed,
and that now is in the same
units as a matter density.
And so, you can have Ω of the
matter, which we think is about
1 / 3, or 1 / 4,
or something like that.
And you can ask:
how much energy is there.
You can ask:
how much Ω is there in energy,
in this dark energy?
Not photons.
Not ordinary kinds of energy,
but this repulsive--this
mysterious, repulsive dark
energy.
It's about three times the
amount of matter--where Ω,
remember, is equal to density
over the critical density,
and, you know,
it doesn't matter whether it's
an energy density divided by
c^(2),
or some amount of matter.
Okay.
That's what we think the
Universe consists of.
So, here is the famous pie
chart of the Universe,
which tells you what's going
on.
So, you know how pie charts
work?
You start with a circle.
That's everything,
and here's us.
So, this is all ordinary
matter, and all ordinary kinds
of energy, photons,
what have you,
that we know about.
And it's something like 4% of
the Universe.
Then, there's a big chunk
that's dark matter.
And that, we discussed last
time.
The dark matter is this stuff
we can't see.
It might be WIMPs.
It might be MACHOs,
we don't know.
We can't see it,
but we know it's there.
Why do we know it's there?
Because it exerts gravitational
force on galaxies and stars in
galaxies, and we can measure how
much of them there is by looking
at stars--galaxies and clusters
of galaxies,
and inferring the mass of them
by observing the orbits of
things around them,
just as we did with black holes;
just as we did with planets.
This is a fairly
straightforward astronomical
technique.
The rest of it--this region,
here, is dark energy,
which not only can we not
detect directly,
but it has a physical effect
unlike anything else we know
anything about at all--namely,
it pushes out.
It's a purely repulsive force
of some kind and that purely
repulsive force provides 3/4 of
the mass energy of the Universe.
Well, that's bad,
because we have even less
idea--oh, and I should say--So,
we've been counting up
frontiers and controversies with
time.
This is Frontiers and
Controversies in 2007.
What the heck is going on with
this pie chart?
What's this?
What's this and why?
That's what we need to find out.
Now, oddly enough,
it turns out,
Einstein predicted this.
Einstein was a genius.
Only, he then retracted it and
decided it was the greatest
mistake he had ever made.
And then, because he said it
was a great mistake,
everybody ignored it for
seventy years,
until we discovered the dark
energy, and then,
somebody said:
oh, you know,
Einstein told us about this
already.
And here's how--why he did it.
Remember, Einstein wanted a
static Universe.
This is before Hubble
discovered that the Universe was
expanding.
And so, Einstein was looking at
the implications of his
equations of general relativity,
and he thought it would be nice
if the Universe was static.
And the problem is,
there's all this matter in the
Universe, so it attracts itself.
So, it can't be static,
because if you try and hold it
in one place,
it'll fall down.
It has to either be expanding
up or expanding down,
just like the pen is either
going up or going down.
But that's bad,
thought Einstein,
because it ought to be static.
So, he invented an additional
term for his equation,
which he wrote down as a
capital lambda [Λ]
and called the cosmological
constant.
And this had generated a
repulsive force that would
balance gravity and lead to a
static Universe.
And it was allowed in the
equations.
It fit fine into the equations.
There was no problem there.
Those of you who have done
calculus, this shows up as a
constant of integration,
so, it can be any value you
like.
If you don't know calculus,
ignore that last sentence.
Okay.
And so, he invented this
repulsive force,
this cosmological constant,
to balance gravity,
so that you could have a static
Universe.
Then, Hubble discovers that the
Universe isn't static--discovers
expansion.
And Einstein says:
damn, I could have predicted
that, and then,
I would have been famous.
So, Einstein says:
Λ was my biggest mistake.
Because if he hadn't been so
insistent that the Universe
needed to turn out to be static,
he could have said,
you know, my equations predict
that the Universe either has to
be expanding or contracting.
It has to be moving around.
And then, Hubble would have
verified Einstein's prediction
and Einstein might have won the
Nobel Prize, or something.
So, Λ was my biggest mistake.
Except, it turns out that it
might be true.
So, this is one of those
fables, right?
The fable of Einstein's biggest
mistake.
And the moral is:
interesting ideas can turn up
in a different context and turn
out to be even more important
than you thought they were--can
turn up in other contexts.
So, for seventy years,
you know, the idea of this Λ
term, this cosmological
constant, was kind of buried in
the textbooks.
It would be,
you know, Problem 53 at the end
of chapter 7.
"Supposing there is a
cosmological constant,
then, what would happen?"
And generations of grad
students thought about this for
fifteen minutes and then forgot.
And then, all of a sudden,
it turns out to be an
explanation for this very
mysterious force.
Now, just because you stick it
in an equation doesn't tell you
what it is.
So, what might it be?
The particle physicists,
it turns out,
have a little bit of an
explanation for this,
called the vacuum energy.
I'll talk more about that later.
Never mind what it is.
I just wanted to point out that
if you calculate what the vacuum
energy is, what you discover is
that Ω associated with this Λ
is equal to 10^(120).
That's a particle physics
calculation.
The observed Ω is equal to
about 3/4, okay?
So, this is the wrongest
calculation in the history of
science.
It's off by 120 orders of
magnitude.
And in fact,
it would be very bad if this
was true, because if there was
that much repulsive energy,
no galaxies would have formed.
No planets would have formed.
We wouldn't be here.
All our stuff would have been
sprayed out into the far reaches
of the Universe.
So, we knew already that that
couldn't really be true.
And so, what the particle
physicists said was,
you know, obviously,
this can't be true,
so it must somehow cancel out.
So, it must be 0,
because otherwise we have this
embarrassing problem,
which we know can't be true.
The fact that it's small,
but not 0, is completely
mysterious, and that's where
we'll start next time.
 
