We started the class by talking about classical Astronomy.
We moved on to slowly expanding our point of view on the universe starting with the
solar system, proceeding to stars and star clusters and galaxy's last week and
that seems only fitting to end our journey by talking about the [UNKNOWN]
scene. What is it that we can say about the
universe as a whole? And principle that guides us and being able to say anything
and all about the universe is the cosmological principle, the assertion
that the universe is homogenous and isotropic.
Homogeneous, recall, means that they're in no preferred locations in the
universe. Everywhere is the same as everywhere
else. In particular, this implies there's no
point that is the center of the universe nor is there an edge of the universe,
those will be distinguished points and there are none.
Furthermore, the universe is isotropic. It's the same in all directions.
It's the same in all directions no matter where you are looking from, and so the
existence of a homogeneous isotropic universe, all of this symmetry, is what's
going to allow is to say something useful about the universe.
Now clearly, the universe is neither homogeneous nor isotropic at small
distances, so the best we can hope to imagine is that, in fact, at large
distances, when you average out little perturbations like stars and galaxies and
clusters and super clusters at really large distances, there is no leftover
structure. And we saw some evidence for that last
week in the decrease of the correlation function and galaxy counts at distances
of over a 100 Mpc or more. we will see a lot more evidence this week
and I want to say that we already saw another indirect piece of evidence, which
is the Hubble Flow that we discussed last week.
So, at the end of last week we mentioned Hubble's result that galaxies are
receding from us and the average recessional velocity is proportional to
the distance of galaxies, the constant of proportionality being the Hubble
Constant. And so, here's a cartoon of what that could look like, here's a
one-dimensional universe. Of course, I've only drawn a piece of it
these arrows are meant to indicate that the universe goes on in all directions.
This is the universe, let's say, t=0. There are three galaxies here.
These are the same three galaxies over here at some later time t and we can
label them, we'll label this galaxy number 1, 0, 1,
-1, and if I wasn't so stingy, I would have added galaxy number 2 over here and
galaxy number -2 over here. And what we see is that the father the
galaxy is from galaxy 0, the faster it's motion, the velocities of these galaxies
are attempting to represent the Hubble Law and in particular, I'm going to erase
all my figures when I move on but here we go if the I label these galaxies, i.
So, i counts the name, [UNKNOWN] is the name of a galaxy and the position of a
galaxy at time t is its position at time 0 times 1 plus H0t.
This, if you look at the speed that this, or the velocity that this position
implies, then this is just a constant shift.
This is xi(0)+xi(0)*H0t, and that tells you that the velocity of the ith galaxy
is just H0*xi(0). The farther a galaxy is, the faster it's
moving. This is the Hubble Expansion Law and this
is all from the point of view of galaxy number 0.
In particular, xi is the distance if the ith galaxy from galaxy number 0,
vi is the velocity as drawn in this graph relative to galaxy number 0.
What if you tried to represent the same thing but from a point of view of galaxy
number 2? Well, in that case, what goes on, of course, is that the velocity you
measure, if you said j=2 here., the relative velocity, this is all
non-relativistic of galaxy i, is the difference of velocity between i and j,
plugging the values from the Hubble Law in.
I see that indeed, this is the Hubble Constant times the relative position of
galaxy i relative to galaxy j. In other words, Hubble's Law looks the
same as observed from galaxy j as it did observed from galaxy 0.
Now, let me draw, erase all my scribbling so we can see what we're doing and make a
point. this is true for Hubble's Law because it
is a linear dependence of velocity on distance.
Imagine, for example, that Hubble had discovered a quadratic dependence of
velocity on distance. In other words, galaxies twice as far
would be moving four times as fast rather than twice as fast.
if this were true, then imagine that that's fine.
Each of these galaxies is moving with a velocity proportional to the square of
its distance, however, if you now observe the relative velocity of the ith galaxy
relative to the j. So, if you're now observing with from the
point of view of galaxy j, then the relative velocity is the difference of
their velocity and that is not the same as the quadratic velocity law you would
have obtained by simply imagining that this law just as it applies viewed from
galaxy 0, applies viewed from galaxy j. In other words a quadratic velocity law
would not be consistent with the homogeneous and isotropic universe, but
the Hubble Flow is. So, the Hubble Flow is another subtle
hint about, that we may consider the universe isotropic and homogeneous.
We'll get many more of those, but for now, we're just going to assert it.
Now, the Hubble Flow tells us a few other things that I want to emphasize.
One is at every location, at every event, at every location at every given time in
the universe, the Hubble Flow selects, in fact, a preferred rest frame.
The reason is that this is the frame in which the Hubble Flow looks isotropic.
Of course, if I see an isotropic Hubble Flow and you are moving relative to me
and at my same position at the same time, at the same event, you are moving with
some large velocity relative to me than in the direction of your motion, galaxies
will appear to be receding at a smaller velocity than what I observe.
Whereas, behind you, galaxies will be observed to be receding with larger
velocities, all this completely non-relativistic and it is not a
violation of special relativity. This is not a preferred reference frame
with respect to the laws of Physics, it's a preferred reference frame with respect
to the positions of the motions of the objects in the universe.
And so, the observation is, that by and large, galaxies follow the Hubble Flow.
They have peculiar velocities, they move relative to the local Hubble Flow.
But on average, in any region of space at a given time, the average peculiar
velocities of the galaxies average to 0 if the region of spaces large enough, by
and large, galaxies follow the Hubble flow and then relative to that, they have
their peculiar velocities. And so at every point in space and time,
and at every event, there is a unique frame, unique reference frame in which
the Hubble Flow looks isotropic. Now, we use the Hubble Flow or we
discover the Hubble Flow by measuring the redshift.
Again, everything here is non-relativistic.
you remember that the redshift z is given by 1+z is lambda over lambda 0 and this
is equal in turn to 1+V/C and V/C in turn is 1+H0D/C.
So, the redshift was given by H0/C times D, so we have a linear redshift distance
relationship. When is this valid? Well, this is valid
because we used a non-relativistic version of the Doppler formula.
We expect it to be valid for small non-relativistic velocities, and this
corresponds to distances small compared to C times the inverse of the Hubble
constant. In other words there's this distance, a
typical size, characteristic size of the universe, that is and the Hubble Law, we
expect and we found, holds for distances small to this, small relative to this.
What happens if you want to understand larger z's?
Well, you could try these the special relativistic Doppler Law formula
replacing this by that square root. As we saw in the homework, it doesn't
quite work and, in fact, in the context of an expand of dynamical expanding
universe corrections to this beyond small z are going to be sensitive on the one
hand to relativistic corrections because the velocities
that you observe when z is not much less than 1 are going to be relativistic.
On the other hand the light will have been traveling a significant fraction at
z not much smaller than 1 of the history of the universe and so, you will be
sensitive not only to the state of the universe now but to the entire history of
the universe. And, in particular, if you try to extend
an expression like this to distances D that are not small relative to the amount
of the, the distance that light would have traveled in the age of the universe,
then when you're talking about objects that are that far away the definition of
distance becomes ambiguous. What do you mean by distance? You can't
go around laying a ruler from galaxy at one side of the universe to a galaxy at
the other side of the universe because if those galaxies are moving, then by the
time you finish your ruler, they will not be ready, they were, when you started.
So, you can understand what you mean by distance will become ambiguous and
because of all of these, you would be surprised to find that the special
relativistic expression for Doppler shift inserted into this would give an actual
correct interpretation and understanding the redshift distance relation will tell
us, in fact, a lot about what goes on in the universe.
Now, naively, just extending from this, the statement that distances between
galaxies at time t are given in terms of their distances at time zero times 1 plus
H0t, then it's clear that going into the past
things were closer. And you predict that when t is equal to the inverse Hubble
constant distances were all 0, so you predict a singularity or a Big Bang with
a characteristic time of 13.8 billion years given by the inverse of the Hubble
Constant. Now, this assumes some sort of constant
ongoing expansion extrapolated into the past.
We don't really expect that. Hubble certainly knew not to expect that.
what are we describing? We're describing a bunch of massive objects.
Galaxies flying away from each other. These objects interact principally
gravitationally. And what we expect is gravitation is a
long range force. these galaxies are distant but there's a
lot of them. And we expect that these galaxies flying
apart will be slowed down by gravitational interactions.
Sure, I can make something fly away from a massive object by giving it the correct
initial conditions like I can throw a rubber ball in the air.
But eventually, gravity will slow it down.
And the question is, if I throw it fast enough, it has the escape velocity, it
will slow down but keep receding. If I throw with less than the escape
velocity, it will slow down, stop, and return.
But Hubble certainly did not expect his law to, to extend to the entire history
of the universe because what one expects is that expansion is slowing down.
This is a very reasonable expectation given our understanding of gravity.
Now, just this bit of the picture as a first tidbit from this week is going to
give us the resolution to Olbers paradox that has followed us for a few weeks.
And the resolution is precisely the one that Edgar Allen Poe intuited so many
years ago. in a universe whose age is finite the sky
is dark even if the universe is infinite and even if it's filled with a
homogeneous distribution of stars. there are stars that are so far away that
light from them has not yet had time to reach us and, in fact, if the universe is
infinite, almost all of it is too far away for us to have seen it.
Yet, in fact, we only see a finite distance out into the universe because
light has only had a finite amount of time to travel from there to here.
This means that most of the infinite sum of equal terms that made the divergence
that caused the paradox in Olbers most of that infinite sum is not yet visible and
so at any given time, you see out to a fit, finite distance.
Within a finite distance, the stars take up a small fraction of the sky and the
sky appears dark. Of course, we expect by now that it's not
that you see all the stars up to a certain distance and then beyond that,
it's dark. That kind of sudden transition is not
what happens. Stars closer and closer to the maximal
distance will be increasingly redshifted so that they will be dimmed.
And eventually there will be sort of an infinite dimming and we will not see any
stars past a particular distance. We'll talk about that.
We haven't seen the entire universe yet, as we'll see by the end of this week.
We never will, in fact. We've seen most of what we ever will see
and understanding the meaning of that statement will be part of the joys of
studying cosmology this week. And so, what is it that we're going to
do? Well, we're going to start with the intellectual framework that allows us to
discuss relativistic corrections to Hubble,
which, of course, will lead us to general, general relativistic point of
view. We'll apply general relativity to a
homogeneous isotropic universe. We will understand the kinds of
cosmological models that we construct. We will understand the parameters of the
model and how they are determined. And in the process, understand this a
very famous pie chart over here on the right that tells you what it is that
comprises the total energy density of the university today.
about 5% of that is in the form of atoms, baryons, protons, neutrons.
24% is that dark matter of which we've had some indications and about the nature
of which we've made some reasonable discussions.
And then, 71.4% of the energy density of the universe is in the form of this even
more mysterious dark energy about which we have not spoken but we will get to it,
and about which, in fact, we know very little.