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.