welcome in the last lecture we learned about
hubble's law of expansion
hubble's law which tells us that the galaxies
are all moving away from us with the speed
that is proportional to the distance this
is what hubble observe this is an observe
fact which was first noticed by hubble that
the galaxies are moving away from us with
the speed that is proportional to the distance
and the constant of proportionality h 0 is
called the hubble parameters
and it is useful and customary to parameterised
the hubble parameter h 0 as 100 h km/s/mpc
that is how it is usually parameterised so
current observations seem to indicate that
the hubble parameter has a value 70 which
you would represent by saying that the parameter
h so h 0=70 you would say that h=07 so this
small h is a dimension less number that parameterises
the hubble parameter
in the last lecture we also learned about
the cosmological principle that the universe
is homogeneous and isotropic there is no preferred
position there is the preferred direction
in the universe
and then i told you that if you combine these
two things you are led to a model which has
no boundaries so the model is in the simplest
situation the universe is infinite the universe
can have no boundaries so the simplest model
is that the universe is infinite and it is
filled with galaxies
because the universe around us is also filled
with galaxies and since the universe is homogeneous
the entire universe is filled with galaxies
and the galaxies are all moving away from
one another this was the basic pictureso if
you combine these two things the observational
fact that the galaxies are all moving away
from us and the theoretical input that we
expect the universe to be homogeneous and
isotropic we are led to a model where the
universe has no boundaries
it is filled with galaxies uniformly filled
with galaxies and the galaxies are all moving
away from one another now this is the model
of the expanding universe and today we are
going to learn a little bit about the nature
of this expanding universe so the first thing
that i should mention is that it is convenient
to introduce a coordinate system called the
comoving coordinates
so if you are going to describe the expanding
universe it is convenient to work in a coordinate
system called the comoving coordinates so
this is a coordinate system that is fixed
to the galaxies 
the galaxies i have told you are all moving
away from one another so you can think of
it like this
imagine a galaxy sitting at each point of
grid point in this graph paper that fills
this part of the universe it is a part of
the universe and there is a galaxy sitting
at every grid point and at these galaxies
are all moving away from one another so at
a later time the same part of the universe
will look like this a gird spacing has become
larger
and the separation between the galaxies has
increased this is the basic idea of the expanding
universe so the comoving coordinate system
is a coordinate system that is attached to
these galaxies so let me draw picture and
explain this to you
so let me draw a picture so these are let
us say this is a small part of that picture
and these are four galaxies and this is my
comoving coordinate x i denote my comoving
coordinate as x and in this comoving coordinate
system this has a coordinate 0 0 this has
1 0 this is 0 1 and 1 1 now at a later instant
of time what will happen these four things
are the four galaxies over here
they are more or less uniformly distributed
for convenience i have represented them as
being equally distributed at equal intervals
now just imagine at a later instant of time
what will happen is that the separation between
these galaxies has increased so what it looks
like is like this 
the comoving coordinates of these four galaxies
continued to remain the same
so this still remains 0 0 this is 1 0 and
0 1 and 1 1 the comoving coordinates of these
four galaxies continues to remain the same
but you can see that the physical coordinate
r so the physical coordinate r has increased
right the physical coordinate is the actual
separation between these galaxies so we write
the physical separation as a of t into x 
okay
here x is the x refers to the comoving coordinate
system r is the actual physical coordinate
a is something called the scale factor which
actually tells you that whether that the separation
is increasing so these are two galaxies their
comoving coordinate remain same so this is
a still at x=0 0 x=1 0 it continues to be
that there
the x coordinate of these galaxies remain
fixed as the universe expect what increases
with time is the scale factor a of t okay
and the expansion of the universe the fact
that the galaxies are moving apart has gone
into this factor function a of t which is
the scale okay so let me write these things
here so we have a three things
we have the comoving coordinates x we have
the scale factor a of t 
and we have the physical coordinate r 
physical coordinator or distance 
and the relation again let me as write it
here is r=a d into x so the galaxies in this
coordinate system sit at fixed values of x
they are uniformly distributed in x and as
the universe expands this a of t is keeps
on increasing essentially okay so this is
the way you this is a convenient coordinate
system to describe the expanding universe
(sir cannot the universe all like this comoving
coordinates the scale factor is changing and
that coordinate is not changing that means
they are following a linear path) well no
who says they are following a linear path
(in this we have a picture) the question is
that x is fixed and a is changing and so it
follows a linear path that is not so a could
be some arbitrary function of time
we do not know what time it is right what
the function of time is the basic thing is
this is you have a universe which is homogeneous
and isotropic right that is what we believe
and you want to also have the universe follow
you have also incorporate hubble's law into
this picture so the way to do that is to assume
that the galaxies are all moving apart from
one another okay
so if since the universe is homogenous the
galaxies are uniformly distributed okay so
this is a picture this picture in this picture
just imagine this is just a picture okay do
not take it literally it is a picture so it
is a kind of cartoon so in this picture there
are galaxies uniformly distributed and this
fills the entire space okay it is isotropic
because the galaxies are all moving away from
each other without reference to any special
direction okay
so this is a model which is consistent with
hubble’s public law and also consistent
with homogeneity and isotropy where the entire
universe is filled with galaxies like this
and they all moving away from each other that
is the picture okay so now you can have a
coordinate system that is fixed to the galaxies
and the fact that the intergalactic separation
is increasing is there in this function a
of t okay
now let us ask the question so now we could
ask a question that there is an observer sitting
over here and this is a galaxy g at a distance
r physical distance r a corresponding comoving
coordinate x what is the speed with which
this observer will see this galaxy moving
to determine that we have to just differentiate
r now as the universe expands the galaxies
sit at fixed x coordinate there comoving coordinate
is fixed
so the velocity of the galaxy is going to
be a dot where a dot refers to the time derivative
dadt a dot into x the x the comoving coordinate
does not change with time and this can be
written as a dot/a into r right so you see
this is the hubble's law and a is a function
of time so you can identify the hubble parameter
from this so the hubble parameter h which
actually now here the function of time
so you can evaluate it at any time instant
it is related to a dot it is a dot/a both
of these are functions of time okay so in
this picture in the comoving if you go over
to the comoving coordinate system the entire
expansion has gone into the scale factor and
the hubble parameter is basically with the
rate of change of the scale factor divided
by the scale factor itself okay
and the present value of the hubble parameter
h 0 t=to a dot/a evaluated at t=t 0 t 0 refers
to 
the present epoch the value of t at present
okay so h 0 is the present value of the hubble
parameter it is the rate of the change of
the scale factor with divided by a at the
present epoch okay so let me again recapitulate
what we have done we have
so i told you in the last class that we are
led to a model where the entire universe is
filled with galaxies and they are all moving
away from one another so now we go over to
coordinate system which is also moving expanding
with the galaxy which is fixed to the galaxies
and the fact that the separation between the
galaxies is increasing is gone into this function
called the scale factor
now we write the hubble we look at the velocity
in terms of this model so the comoving coordinate
of each galaxy does not change as the universe
expands as the galaxies move away from one
another all that changes is the scale factor
so we are led to a relation where the velocity
of the galaxy is proportional to the distance
into some function of time this function of
time is a dot/a
and comparing this with the hubble law we
see that we can identify this with observe
fact see by hubble we can identify this with
the hubble parameter and the present value
of the hubble parameter is the value of the
derivative at the present epoch okay so all
that we have done till now is that we have
just changed the coordinate system nothing
else okay
we have gone over to a coordinate system which
is fixed to the galaxies the galaxies are
homogeneously distributed so i can imagine
them to be equally spaced along different
direction in the universe so i have a coordinate
system attached to that and they are all moving
away so the scale factor is increasing with
time and we saw them that we write the hubble
parameter is a dot/a
let us now determine what this a of t is what
is the functional form of this a of t we still
do not know what that is so far this let us
take a very simple model the simplest model
that one could adopt is that we have free
expansion there are no forces okay so that
is the simplest model which we shall take
up now
so this is the model of free expansion 
so in this model there are no forces acting
on this galaxy it is moving freely okay the
whole universe is expanding like a set of
free particles all the galaxies are moving
away from us and their motion is just like
free particles moving away from us okay we
have ignored all any effect influence of enforce
so let us write down the equation of the motion
for this galaxy i am the observer sitting
here let us write down the equation of motion
for this galaxy
the equation of motion we all know is d square
r bt square=0 in the absence of any force
and the physical coordinate as been written
in terms of the comoving coordinate and as
the galaxy moves away from us the comoving
coordinate remains fixed because the coordinate
system moves with the galaxy so all that changes
is the scale factor so this essentially tells
us that the second derivative of the scale
factor 
is 0
so we can straightaway write down the solution
the solution is a of t=k1t+k2 these k1 and
k2 are constants of which you obtain constants
of this they are basically determined by the
boundary conditions they are constants of
motion okay now this k2 is just an additive
constant you see the behaviour of a of t let
us plot a of t so this is a this is t and
the solution is a straight line
that is free motion particle moves with a
fixed velocity which is basically what we
have done so it is a straight line and i am
free to place the k2 decides where the straight
line will cut the x access so it decides where
the scale factor because 0 and it is most
convenient to choose k2 so that the scale
factor becomes 0 at t=0 so a=0 we choose the
scale factor so that a=0 at t=0 which implies
that k2 is 0
so this thing is not there okay so that the
time t=0 the scale factor collapses is becomes
0 now you see what is the significance of
the scale factor becoming 0 the scale factor
becoming 0 means that the separation between
different galaxies whatever be the comoving
coordinate they all are at 0 separation the
entire universe is filled with galaxies and
different galaxies have different coordinate
comoving coordinate so not only i have galaxy
here with comoving coordinate 2 0
and one beyond etc but when the scale factor
becomes 0 the distance between all of these
galaxies is 0 and the entire infinite universe
and the separation between any two galaxies
in the infinite universe becomes 0 this is
a singularity right so this is singularity
so this is a singularity 
and the singularity was interestingly given
a name big bang by fred hoyle who do not believe
in this evolving universe model he you wanted
to make fun of it but somehow the name stuck
so this is also called the big bang
it was given by a person who did who did who
did not wanted to basically make fun of this
theory but anyway the name stuck so the singularity
is also called the big bang so is essentially
a singularity so we do not know what happens
over there the density becomes infinite and
we can assume for the time being that the
universe started at the beginning of the universe
that is the time starts from there so t is
essentially the time t is essentially the
time that elapsed since the big bang
that is the time that elapsed since the big
bang since the epoch when a became 0 this
is also called the age of the universe this
is the cosmic age cosmic time or the age of
the universe if you believe that the universe
started at the big bang and the singularity
then this is the age of the universe okay
t so the time t here it is called the cosmic
time so t is also the cosmic time and it is
the age of the universe okay
now in this model so we have worked out this
model we have looked out everything in this
model this is a simple model where the universe
is expanding freely for this model let us
see what the hubble parameter is what is the
hubble parameter at any instant of time so
how does one calculate the hubble parameter
to calculate the hubble parameter what one
has to do is one has to calculate a dot/a
so let us calculate a dot/a remember a is
constant k1 into t so we want to calculate
a dot/a
so a dot/a in this freely expanding model
is k1/k1t so we see that this is one by time
this is the hubble parameter at any epoch
so if you can measure the hubble you see that
at any epoch the age of the universe is inverse
of the hubble parameter okay and the present
age so if you can determine the present the
value of the hubble parameter you can determine
the present age of the universe
this gives us the age of the universe one
by the hubble parameter gives us the age of
the universe in this simple model where the
universe is expanding freely okay and just
tell you that this makes sense let us look
at the hubble parameter the hubble parameter
can be parameterised as 100 h km/sec/mpc what
is the dimension of h 0
so in the numerator you see we have length
by time divided by length again so the length
cancels out so the hubble parameter has dimension
inverse of time right the hubble parameter
has dimension inverse of time and this is
called the hubble time so this time 1/h 0
is referred to as the hubble time
and i leave it to you as an assignment to
work out what this is in years so the assignment
is determine the hubble time assuming that
h 0 100 h km/sec/mpc determine this in years
so determine this in years how many what is
the hubble time what is the age of the universe
if it were expanding free but we know that
the universe is not expanding freely
so this gives you just an estimate an order
of magnitude estimate of the h it does not
give you a real estimate of what you expect
the age of the universe to be okay as we shall
see shortly this the assumption that the universe
is freely expanding is not a correct it is
not correct so we have to also take into account
forces acting on the galaxy okay but it does
give us some idea of the nature of the expansion
and the big bang and the hubble time scale
okay
so the hubble time scale gives us an order
of magnitude estimate of the age of the universe
let us now incorporate the effect of gravity
and strictly speaking one should be using
the general theory of relativity but in this
course we shall not use the general theory
relativity we shall use simple newtonian physics
and let we tell you that the results that
we will get will be correct so the results
match okay
so there is no difference in the results if
you were to use the general theory of relativity
with this let me get down to calculating the
dynamics of the expanding universe so we are
going to basically study now the dynamics
of the expanding universe so let us right
down the same thing again so we have an observer
over here
this is the observer and there is a galaxy
here g and we would like to write down the
equation of motion for this galaxy let us
say that the galaxy has mass m now since the
universe is homogenous and isotropic the mass
distribution the density of the universe is
everywhere the same so you can think of it
as a spherical mass distribution around the
observer because it is isotropic
so the observer around this observer the mass
has a spherically symmetric distribution and
it is well known that if you want to calculate
the force on a mass which is inside a spherically
symmetric mass distribution then only the
mass inside sphere will contribute to the
force the mass outside will not contribute
okay so we can write down the equation of
motion of this galaxy it is very simple the
only force here is gravity and at these large
scales the only force that acts is gravity
only force which is known is gravity
the universe we assume is charge neutral on
large scales so there are no net charges so
it is neutral on the average so we can then
write down the equation of motion for this
galaxy what you have to do is we have to just
consider the force due to the mass inside
a sphere of radius r so this is my galaxy
and the force on this galaxy is due to the
mass inside a sphere of radius r
and so the equation of motion is quite straightforward
m d square r dt square is equal to so i can
write it as a vector if you wish or you could
write it as a scalar it does not matter because
it the forces are all radial the acceleration
2 is radial this is = g into the mass within
this sphere of radius r into the mass of the
galaxy divided by r square or i can write
it as divided by r cube into r
so this is my equation of motion for the galaxy
in the previous example of free expansion
we had ignored the effect of gravity so that
gravity here is going to slow down the expansion
of the universe that is the main thing okay
now we can write this in terms of the density
of the universe remember that the density
is the same throughout because we have assumed
that the universe is homogeneous
so if i write this in terms of the density
we know that the mass inside a sphere of radius
r is 4/3 pi r cube into rho 
okay so we have these two equations and then
if we substitute this here this r cube cancels
out from the denominator and the other point
to note is that the mass of the galaxy also
does not figure anyway this is the equivalent’s
principle
the gravitational acceleration is independent
of the mass of the galaxy that cancels out
from the inertial mass and the gravitational
mass are the same so this cancels out and
it does not depend on the mass of this particular
galaxy so we have then the equation further
what we could do is we could write in terms
of the comoving coordinate system and the
scale factor okay we will do that in step
so let us write down this equation first the
question then becomes d square r dt square=
4/3 pi g rho into r
so that is the equation and that we have 
and this equation can further simplified and
we can write this is for a particular galaxy
now we could write it in terms of the scale
factor so let us replace the physical coordinate
in terms of the scale factor into the comoving
coordinate the comoving coordinate of the
galaxy is fixed it does not change so i can
take it outside the derivative and you see
that it cancels from both the left and the
right hand side
so we are left to an equation for just the
scale factor and the question is d square
a dt square= 4/3 pi g rho into a so this is
the equation that governs the evolution of
the scale factor so the point here is once
you solve this equation and get the scale
factor you know how the entire universe is
expanding so by just studying the equation
of motion of a single galaxy you can work
out how the entire universe is expanding because
the scale factor is the thing that has the
encodes information about the expansion of
the universe
once i know how i can use this to determine
the motion of any galaxy different galaxies
correspond to difficult comoving coordinates
okay the other point to note here is that
the density of the universe does not vary
from place to place but it does vary with
time why does it vary with time basically
the mass is conserved so as the universe expands
you can see as the universe expands the mass
inside any volume physical volume has to be
fixed so the length the volume of the universe
goes up as a cube
but the mass has to be constant mass is density
into length cube so we can see that the density
into the scale factor cube has to be a constant
right as the universe expands the volume increases
as a cube the mass is conserved mass within
any volume any region is conserved mass is
neither produced not destroyed so in addition
to this we also have rho a cube is constant
as the universe expands the density goes down
as 1/a cube and we can write this as the present
so this refers to the present a subscript
0 always refers to the present so this is
the present density into the present value
of the scale factor cube and we will use a
comoving coordinate system so that it matches
with the physical coordinates at present so
the value of the scale factor at present is
assumed to be 1 that is how we shall work
so we shall assume that the present value
of the scale factor is 1 okay
so now we have to simultaneously solve these
two equations one equation being the equation
over here and the equation over here so we
have to simultaneously solve these two questions
so what we could do is you see there are two
functions of time in this equation one function
of time being the scale factor other is the
density so what we could do is we could just
eliminate density from this whole thing by
replacing density by the present value of
the density divided by a cube okay
so let us do that so what this equation then
becomes is a double dot= 4/3 pi g and we are
replacing the density with rho 0/a cube a
0 is 1 so this will be replaced by rho 0 divided
by a cube so i will have a 1/a square here
okay so what i get here is rho 0 a 2 and the
way to 
solve this equation is as follows what you
do is you multiply the left hand side and
the right hand side of this equation with
a dot
so this you can see is a time derivative d/dt
of a dot square/2 if i differentiate a dot
square i will have two terms one of these
terms will be a double dot the other term
will be a dot so i would get a double dot
into a dot and there will be two such terms
so this two will be cancelled out okay and
so this same equation can be written like
this
now if you look at the right hand side a to
the power 2 into a dot this can be written
as 4/3 pi g rho 0 into d/dt of 1/a because
if i differentiate 1/a i will get a dot into
a to the power 2 okay so this same equation
can be written like this and now i can integrate
both the left hand side and the right hand
side of this integration and in doing this
in integration i will then have to introduce
if i do the integral i have to introduce a
constant of integration
so the solution to this equation is a dot
square/2=4/3 pi g rho 0 into 1/a+ a constant
of integration e this is the solution to this
equation where e is a constant of integration
that we introduce when we integrate this okay
so we worked out the solution the first integral
of the equation that governs the expansion
of the universe
and you have to do one more integral in order
to solve this now let us just spend a little
time and look at this equation 
okay this equation if you notice is essentially
the left hand side the term on the left hand
side of this equation is essentially the kinetic
energy of the galaxy consider a galaxy at
unit comoving coordinate let us consider a
galaxy at the comoving coordinate x=1
then its velocity will be a dot and its kinetic
energy let us say its mass is 1 so its velocity
is a dot its kinetic energy is half a dot
square so if i have a galaxy at unit comoving
distance and unit mass then its velocity its
kinetic energy is half a dot square similarly
let us look at the potential energy of that
same galaxy so the potential energy of a galaxy
here is going to be –gm/r
and for a galaxy at unit comoving coordinate
the mass inside is 4/3 pi into the volume
into the density the density inside is rho
which can be written as so the density inside
so the mass inside here is pi 4/3 so let us
just look at the mass inside this the mass
inside this is here 4/3 pi r cube rho r cube
for a unit comoving coordinate is basically
a cube into rho a cube into rho can be written
as rho 0 okay
so the mass inside this sphere is 4/3 pi a
cube rho 0 basically 4/3 pi rho 0 and the
potential energy is the mass divided by the
distance – mass times the distance so this
is the potential energy okay so this is essentially
the kinetic energy t+the potential energy
this equation essentially same as this okay
and this e is a constant okay so this e can
be thought of has been the energy of the galaxy
which we are discussing which is conserved
in the motion
here it appears as a constant of integration
when i do the first integral okay now the
question is how to determine the value of
this e from observations i have a universe
i am living in a universe and how will i determine
to integrate this equation once more and determine
the solution for the scale factor i have to
determine the value of e the whole thing determines
depends on the value of e
how to determine the value of this e okay
to determine the value so let us discuss that
so to determine the value of e let me write
down the same equation in a slightly different
way let us put back the value of the density
over here so rho 0 is rho into aq if i put
back the value of the density over here this
becomes rho into a cube/a so rho into a square
so if i put back the value of the density
over here this equation then reads a dot square
8/3 pi g rho a square=2e so i have multiplied
that equation by 2 and i have replaced rho
0 as rho into a cube now let me divide this
equation by a square throughout so if i divide
this equation by a square throughout what
i get is 1/a square here this a square is
gone and i have a square here okay
so this is the equation that we have now look
at this equation carefully this equation essentially
reads that the value of the hubble parameter
at any instant of time because a dot/a is
the hubble parameter this is the square of
the hubble parameter= 8/3 pi g rho is equal
to this constant 2e/a square okay this is
the general equation if you apply this equation
at present so we want to determine the value
of e from present day observations
to do that what do you do then you apply this
equation at present t=t 0 at t=t 0 this becomes
h 0 square so at present what we get is h
0 square and i can take h 0 square common
over here on the left hand side then i have
1 – rho 0 divided by 3 h 0 square by 8 pi
g=2 e
so if i apply this equation to the present
epoch present epoch means t=t 0 then this
becomes the present value of the hubble parameter
this becomes the present density and this
becomes a 0 which is 1 which we have chosen
to be 1 so we have this equation
now the point to note here is that in this
equation we have this quantity over here in
the ratio of these two quantities which has
to be compared to 1 okay so the first thing
that you can see here is that this combination
3 h 0 square/8 pi g is something that is of
dimension of density okay and this is called
the critical density so the present value
of the critical density
so this is rho c 0 this is called a critical
density 
so another homework determine the value of
the critical density right you know the value
of h 0 and the others are all known so determine
the value of the critical density of the universe
okay the critical density is a very crucial
density scale in the universe so the ratio
of the if you look at this expression it is
h 0 square into 1 the ration of the actual
density now rho 0 that is the actual density
of the universe now to the critical density
now
this is what determines the constant e okay
constant of the integration e so 2e and this
ratio is given a name it is very important
in cosmology there is a special name for this
so this is the ratio that occurs over here
it is denoted by omega 0 it is called the
density parameter 
and the equation the constant e is now can
be written in the following way the constant
e 
okay
so let me again remind you want these things
that we have discussed are we see that the
hubble parameter defines a density scale in
the universe so the present value of the hubble
parameter defines the present value of the
critical density which is rho c 0 and the
critical density is three times the hubble
parameter square by 8 pi g
so this is at present at any instant of time
you can write down the critical density rho
c as 3 h square/8 pi g this is true at any
time instant not only at the present but at
any instant of time okay so the hubble parameter
define the density scale and the ratio of
the actual density of the universe to the
critical density is called the density parameter
of the universe omega and its present value
so this omega can also be defined at any instant
of time so mega at any instant of time=rho
these are all actually functions of time possibly
function because h also is the function of
time so the critical density is the function
of time and density parameter also is a function
of time
we are interested here in the present day
values so the present day value of the density
parameter omega 0 is the value of the actual
density of the universe to the critical density
of the universe okay now what is the specialty
of this ratio density parameter the value
of the density parameter decides the value
of the constant of integration e
if the density parameter has value 1 this
implies that the constant of integration is
0 okay this is the critical value it divides
between two different kinds of cosmological
models okay if omega 0 is  0 if omega
0 > than 1 it implies that the constant of
integration e < 0 and this constant of integration
e i told you can be thought as being the energy
total energy of this galaxy
let me end today's lecture over here in tomorrow's
lecture we shall interpret what the significance
of these and then look at some of the solutions
yeah it could be right it is a function of
time why not that is the present value of
the critical density h 0 square but h is the
function of time so the critical density is
also function of time okay
