welcome let me remind about you that in the
last class we had embarked upon trying to
understand the nature of the space time in
cosmology
and i had started off by first briefly recapitulating
the nature of space-time without the effect
of gravity and the space-time properties are
can be characterized by this quantity ds square
which we refer to as the line element or the
interval and this formula tells us how to
calculate the interval between two events
events can be represented in the space time
diagram like this so each event has a corresponding
position and time when it occurred
so you can represent events on a space-time
diagram and if a and b are two events you
can calculate the interval if these two events
correspond to the same point same co-ordinate
at different time so imagine a person sitting
at one place then you will get positive value
for the interval and these two events are
said to be time like what was very important
i had pointed out is if you look at the propagation
of light light will propagate along a curve
so if this is the curve which along which
the light is propagating we are in considering
only the x axis let us assume that the y and
z direction remain fixed then x is delta x=c
delta t for light so delta x=c delta t and
if you put that in here and calculate the
interval you will find that the interval turns
out to be 0 so for two events which are along
the trajectory of a photon the interval is
0
the expression for the interval or the line
element can also be written like this where
we have written this spatial distance separation
in spherical polar coordinates now in einsteins
theory of relativity the gravitational field
manifest itself as the curvature of space-time
and here in cosmology the cosmological space-time
has the property that it has to be space rather
the cosmological space time has the property
that the space has to be homogeneous and isotropic
so the line element or the metric corresponding
the metric allows us to calculate distance
between points the line element the cosmological
line element where this space is homogeneous
and isotropic has been written over here this
is called the friedman robertson walker metric
it was independently written down by friedman
robertson and walker and possibly by other
scientist also at the same time
and i was trying to explain to you what the
various terms in this mean now time here is
the cosmic time c is the speed of light a
is the scale factor which is already familiar
to us it is the it quantifies the expansion
of the universe the x is the comoving radius
we are working in spherical polar coordinates
so x is the comoving radius we are working
in comoving coordinates because you can see
that there is the scale factor over here
so x is the comoving radius and you have this
extra factor if this factor k were 0 if the
factor k were 0 then this line element would
be exactly the same as this line element except
for a scale factor squared over here but we
have this extra term k into x square/l square
also appearing and k this presence of this
non-0 k indicates that there is spatial curvature
my spaces is curved my space time is going
to be curved because of this scale factor
okay
but the presence of this k indicates that
my space is curved and l is the radius of
the curvature in comoving so this is the comoving
radius of curvature of that space
so k=0 if i have k=0 this is spatially flat
and here the line element is ds square is=c
square dt square-a square [dx square+x square
(d theta square+sin square theta d phi square)]
so this is how you calculate the length in
the usual flat curved space polar coordinates
and the lengths these are comoving lengths
get multiplied by a square lengths square
gets multiplied by a square because universe
is expanding and the line element is just
given by this
so i would think that this should be quiet
comprehensible to all of you comprehendible
to all of you you should be able to understand
this quite easily okay now the question is
what happens when there is a k and l which
are non 0 l there is no the radius of curvature
there is no radius of curvature infinite it
is infinite in this when k is 0 okay
now i have told you that we had introduced
when we solved the einstein when we solve
the equation for the expansion of universe
we had encountered a constant of integration
2e and this appeared in my equation for the
hubble parameter in this combination if you
remember and this we had replaced we had parametrized
this by a fictitious density omega k which
whose contribution for the a square as universe
expands
and the contribution from this fictitious
thing which i have told you corresponds to
curvature was quantified using omega k0 so
if you now take you see in einsteins theory
of gravity you do not use the laws of gravity
or modified they are given by einstein equation
and if you write down these equations you
have something some relations between the
metric so they relate the curvature of space
time to the matter which produces gravity
and they give you relations between the scale
factor and k and l so these things are related
to matter density and these equations turn
out to be exactly the same equation that we
derived from newtonian considerations with
the modification that rho gets replaced by
rho+3p/c square okay so they turn out to be
exactly the same so if you now workout einsteins
equation in those equations you will have
k and l appearing
if you relate those if you identify that equation
with the equation that we have derived you
will find that you can relate omega curvature
or 2e/a square to this k and l which i have
introduced here it follow some einstein equation
and the identification is let me just write
it down i will not work it out here the identification
is that omega curvature 0=-k c square/h0 square
into l square 
okay
let me remind you again this is something
that we have already encountered we have introduced
to this to essentially parameterized contribution
from this constant of integration this is
the value of k and l are related to that constant
of integration which we had encountered that
relation comes if you look at the einstein
equation i will not do the derivation here
let me just place it before you that relation
comes out to be this okay
so you are led to this relation if you really
work through the einstein equation so it follows
that if the constant of integration is 0 then
so if 
e or omega k0=0 it implies that k=0 the space
is specially flat this part of the line element
refers to the space it is spatially flat it
the usual flat space okay now the question
is what happens when this constant of integration
is not 0 what is the curvature? what is the
value of k?
so it is quite obvious that if omega curvature
0 if the constant of integration is positive
these are the models if i have only matter
and this constant of integration these are
the models that will keep on expanding forever
this is the critical model
if i have omega curvature 0 > 0 then k=-1
k is -1 and if omega curvature is <0 then
k is 1 and this is called so the previous
one was specially flat this k=0 is specially
flat this has the positive this is space of
positive constant positive curvature 
and this is the space of constant negative
curvature 
okay and the radius of curvature l is 
this is the radius of curvature comoving radius
of curvature 
and the curvature is 1/the radius of curvature
squared
so if you want the physical radius of curvature
then it is a into l and the curvature which
is denoted by r is the order of is actually=1/a
square l square the curvature of space not
of space time or space 
okay so this is a brief gist of the properties
of space time let me know explain to you try
to explain to you what we mean by a space
of constant let us say positive curvature
okay so let us look at this case a little
closer take a closer look at this case where
k=+1
so the situation where k is =+1 and we shall
only deal with the spatial part of this which
allows us to calculate the comoving distance
so let us try to interpret what this space
looks like so the line the distance between
the two points in that space is given 
by dl square=dx square 1-x square/l square+x
square (d theta square+sin square theta d
phi square) let us try to get the picture
of what this space looks like what is this
space look like?
for this it is convenient to introduce new
coordinates x=l sin sorry not x chi or sorry
x=sin chi/l so this is chi/l x=l sin (chi/l)
before we embark up on this let me point out
that the range of x you see this space the
radius x is the distance from the origin x
is the radius and in the usual flat space
the radius can go from 0 to infinity there
is no restriction you see in the usual flats
space look at this there is no restriction
on the radius it can go from 0 to infinity
but in the space there is a limit the value
of x cannot exceed l so x=the range x has
to vary in the range beyond which this thing
will become negative obviously distance is
cannot be negative this allows us to calculate
the distances between different points and
the square of the distance that obviously
cannot be negative so l has to be in the range
0 to l x has to be in the range 0 to l or
equivalently chi has to be in the range 0
to pi
the variable chi that we had been introduced
has to be in the range 0 to pi okay x is the
distance from the origin and x is restricted
0 to l so you cannot go distance further than
l away from the origin it is a different kind
of space all together from the usual space
that we are normally used to
“professor-student conversation starts”
yes is there a question? sir why cannot x
lead from – l to l? well x is the distance
right it is a radial distance how can the
radial distance be- so the x is the radial
distance from the origin where in the polar
coordinates system so x goes from 0 to l not
from –l to l okay fine “professor-student
conversation ends”
so let us now replace the variable x with
chi so to do that all that you have to do
is you have to replace it here and this becomes
sin square sorry let me remove this chi/l
here let us make chi is dimensionless okay
sorry otherwise i have to put extra factor
of pi l so x=l sin chi and the range of chi
is 0 to pi okay so with this this term it
becomes d chi square because the numerator
becomes l square cos square chi d chi square
chi the denominator also cos square
so the line element over here if i write it
in terms of this variable chi it takes on
the form dl square=l square 
d chi square+chi square 
sorry it will not be + sin square chi d theta
square+sin square theta d phi square this
is how we calculate length all that i have
done is i just changed variables and gone
from x to chi where chi is sin well x is l
into sin chi
“professor-student conversation starts”
sir where do you get this x=l into sin chi?
i am making a substitution that is all i am
given this formula for calculating the distance
between points every i am replacing x with
a new variable chi i can always change my
coordinates right so that is all i am doing
for my convenience the convenience will be
cleared just now “professor-student conversation
ends”
so you see the way you calculate the distance
in this co-ordinate system it is a d chi square
sin square chi d theta square+sin square theta
d phi square okay so this is like a spherical
polar co-ordinate system except that you have
this extra coordinate chi appearing okay so
this is the thing that we would try to interpret
the interpretation of this is quite easy to
interpret this let us fix the value of chi
some let us look at some fixed value chi okay
so if a look at a fixed value of chi then
the if i look at a fixed value of chi let
me write this down here if i look at a fixed
value of chi then the distance between two
points the chi does not change becomes l square
sin squared chi [d square theta+sin square
theta d phi square] so i am looking at a fixed
value of chi let me now look at a fixed value
of theta phi is the only variable
so for a fixed value of theta phi is the only
variable so now you see that you can think
of this as being a it is basically a circle
if phi is the only variable we have a circle
right phi is the only variable so we have
a circle and the distance between two points
and that circle is d phi into l sin chi sin
theta let us just ignore this part
so this is the circle and that radius of the
circle just increases with theta it is 0 when
theta=0 it is 0 when theta =phi and the radius
of the circle this is the radius of the this
whole thing is the radius of the circle we
will just focus only on this part so the radius
of the circle keeps on increasing with the
theta so let me draw it here this is the this
you can see is the surface in 2 dimension
the 2 dimensional surface 2-d surface it is
basically 2d sphere right nothing more than
that
the radius of the sphere is l square sin square
chi sorry the radius is l into sign chi and
let me draw it so this is my xy z and this
is my sphere this is the angle theta for a
fixed theta this is my angle phi various in
this direction in the x y plane this is my
circle so essentially what this tells us only
this part what it tells us is that i can think
of a circle as a collection of as a sphere
collection of circles
the radius of the circle is 0 at the north
pole the radius of the circle is also 0 at
the south pole where theta is 0 and phi respectively
the radius of the circle increases as it comes
near the equator and it is maximum at the
equator and again it become 0 at the two poles
okay so 2d sphere can we thought as the collection
of 1d spheres a circle is a 1d sphere okay
circle basically circle and they act up like
this okay
i hope it is clear so 2 dimensional sphere
this is a 2 dimensional sphere this surface
is 2 dimensional can be thought of as a collection
of 1 dimensional sphere at different values
of theta as the radius of the sphere varies
from 0 to the maximum value which is given
here that is the maximum value into this that
is the maximum value and then again it goes
to 0 at the other pole okay so this is the
2d sphere
now look at this this you can see is a collection
again so this is the 1d sphere 1 dimensional
sphere collection of which gives me a 2 dimensional
sphere this is a collection of 2 dimensional
spheres along the variable chi the radius
of the sphere is l sin chi the range of chi
is 0 to pi this is chi so for chi=0 i have
a sphere of radius 0 okay it is a point and
then for chi=pi this sphere has the maximum
size and again it become 0 here
so you see this is a collection of 2d spheres
this is what is called whose radius keeps
on varying as i go towards the equator the
equator is at chi=pi pi/2 and then again it
decreases as i go to chi=pi this is what is
called 3d sphere it is a 3-dimensional volume
which is a sphere it is a generalization of
a 2 dimensional sphere it is a 3 dimensional
sphere okay
and this is a collection of i can think of
it as a collection of 2d spheres which i have
been all stuck together okay or i can generalize
this i can have 4 dimensional sphere 5 dimensional
sphere 6 dimensional sphere all i have to
do is more variables theta sin square theta
etc okay so the radius of this 3 dimensional
sphere is l just like the radius of a 2 dimensional
sphere here is l square forget about this
is l right this represents the 2d sphere with
this=1
this represents the 3 dimensional sphere with
radius l okay so k=+1 is this space is essentially
3d sphere so 3 dimensional sphere the surface
of the earth for example is a 2 dimensional
sphere okay this is a 3 dimensional sphere
and the way you can imagine this is that there
is a space in that space 2 dimensional sphere
for example on the surface of the earth if
i go forward on the surface of the earth after
sometime i will circum navigate the earth
and come back from the back
if i go this way i will after sometime go
around the entire earth and come back from
this side similarly in a 3 dimensional sphere
if i go up given adequate time i will go around
the entire sphere and come back from the bottom
it is a 2 dimensional sphere it is a finite
volume it has no boundaries though okay there
are no boundaries but the volume is finite
just like the surface of the earth it is a
finite area but the boundary there are no
boundaries okay
so this is what we have when k=+1 it is a
space of constant curvature just like 2 dimensional
sphere and we have some idea now what it looks
like similarly k=-1 is a space of negative
curvature okay k=+1 is a space of positive
curvature and k=constant positive curvature
k=-1 is a space of constant negative curvature
okay
and it is convenient to rewrite this cosmological
metric sorry not this it is convenient no
it is convenient to rewrite the cosmological
metric in the following form let me just write
it down for you so it is convenient to rewrite
the cosmological metric in the following form
we will write as ds square=c square dt square-a
square and then we have [dr square+s (r) square
d theta square+sin square theta d phi square]
okay so i have written the same line element
which i have written down right at the start
of todays class it is exactly the same line
element where we have now introduce so this
is the line element i was talking about i
have just written it in terms of new variable
r r is also a comoving distance i written
it like this
where remember that we had x square here we
now have a function of r over here this function
of r s (r)=r if k=0 that is very familiar
flat space okay d square r+r square d theta
square+sin square theta etc okay now this
function s (r) is l sin (r/l) if k =+1 we
just saw this except that we use the variable
chi which was dimensionless
now we will use the variable r/l so the range
of r is from 0 all the way to pi into l okay
and this is if k =+1 positive curvature if
k is =-1 then we will represent this a sine
hyperbolic that function s becomes sine hyperbolic
we have to use the exactly same thing as you
did for k=+1 you will then find that the function
s is of the form sine hyperbolic l into (r/l)
if k=-1 and there is no restrictions for both
these cases r 
can take any value with more than 0 okay
this space r infinite these two in these two
cases negative curvature and no curvature
but if it is positively curved then the space
is finite okay so this is the general cosmological
metric that line element that we shall we
using so the cosmological space time in general
can be represented like this the form of this
function s (r) depends on the value of the
omega curvature 0 how much curvature basically
is parameterized by that okay
so this is the form of the cosmological space
time now let us use this to calculate certain
interesting things right after all the space
time the form of a space-time is important
only if it is useful so the question is what
use is this in cosmology we have already seen
that we have determined the evolution of the
in principle one would take this and put it
in a einsteins equation determine the equations
governing a
but we have already done this from newtonian
considerations just extending it a little
bit okay so now we will see what other utility
this thing has and the place where you cannot
do without relativity is essentially if you
look at a propagation of light propagation
of massive particles can be very well studied
without the aid of relativity in the most
situations if they move slowly but things
are moved at the speed of light you cannot
study without relativity
because the galileos laws of transformation
do not hold okay so let us look at the propagation
of light in this space time so the situation
that we are going to consider is as follows
we have an observer 
sitting over here and there is a galaxy over
here and the observer receives some light
coming from the galaxy that is what we normally
have in astrophysics right if you have observation
observer is receiving light coming from a
galaxy
and let us say that the galaxy is at comoving
distance r with respect to the observer now
and later say that the light was emitted here
at a time t emitted and it was received here
at a time t observed this is t observed cannot
be necessarily t present t o and t e so we
would like to see the propagation study the
propagation of this light i have already told
you that the most important thing about this
interval is that it is 0 along the trajectory
of a photon along the trajectory of light
so the propagation of light from this galaxy
to this observer is along trajectory which
satisfies c dt=a (t) dr the light we are assuming
propagates entirely along the radial direction
so these terms are not going to be there d
theta and d phi and this is nothing very special
because we can always choose the coordinates
system so that one of the point is observer
is at the origin and galaxy is somewhere else
okay
so this is the situation that we are dealing
with light propagates along a curve which
satisfies this so we would like to now to
determine the trajectory you have to now integrate
from t emitted to t observed right so what
we can say from this is that see the integral
t emitted and there will be a-sign here because
as time increases the distance r decreases
well you would have taken a positive sign
if the lights were going outwards okay
so we have to integrate this and we have to
integrated from t emitted to t observed and
we have to integrate dt/a (t) and we have
to integrate the right hand side from r to
0 this will give us r now light we know is
a wave okay so there is a wave actually which
is coming from here and there is a wave which
is being received over here we would like
to now calculate the redshift of the light
we address the question does the frequency
of the light change when it propagates from
the galaxy to the observer that is the question
that we would like to address so we know that
light is wave so let us just draw a wave which
is emitted so the wave which is emitted at
the emitter will look something like this
this is t emitted and this is the oscillation
that is being transmitted that is being emitted
so this is t emitted
this is the amplitude of light let us call
this some capital a amplitude of the wave
that is being transmitted and let us say that
we are this is the time instant t e when 1
crust is emitted this is the time instant
t e+delta t e when the next crust was emitted
so there are two crusts in this wave which
are emitted at the time interval of delta
t e and we know that the frequency of the
light mu emitted=1/delta t e and this is also=c/lambda
emitted okay
so basically delta t e is proportional to
lambda emitted similarly this is the emitter
what the emitter emits the observer will also
receive some thing is exactly identical except
that maybe the time period will be different
so the observer will receive the first crust
at a time t observed and he will receive the
second crust at the time t observed+delta
t observed and the observed frequency are
1/delta t observed=c/lambda observed which
is okay
now so 1 there is 1 crust which is emitted
at t emitted and received here t observed
and that will follow this trajectory it will
travel at the speed of light so we have written
down the equation that governs it let us now
write down the equation that governs the propagation
of the second crust the second crust will
follow exactly you see in the comoving co-ordinate
system the galaxy remains at the fixed coordinate
the observer will also remains at the fixed
co-ordinate
so this right hand side will not change the
right hand side is the comoving separation
from the galaxy to the observer which remains
fixed okay the limits of the right hand side
integral the left hand side integral will
be different so for the next crust for this
crust and this one the limits of the integral
will be t emitted+delta t emitted that is
when it was emitted and the time when it reaches
here will be t observed+delta t observed this
dt/a (t) is also=r right
the second crust has to follow exactly the
same equation so now we can equate these two
so let me equate these two and i can expand
out this integral so let me expand out the
integral first if i expand out the integral
what i have is t emitted+delta t emitted to
t emitted 
so this is what i get if i expand out the
second integral i have just taken the thing
the integral limits of the limits are from
t emitted+delta t emitted to t observed+delta
t observed
i have written it out broken it up into these
three parts and you can see that this part
this exactly cancels with this so we are led
to the relation that this integral should
be equal to this now you see this integral
delta t emitted is extremely small so we can
replace the integral by just delta t emitted
by the value a (t) emitted we may assume that
the function a (t) does not change in this
interval okay similarly we may assume that
the function a (t) does not change in this
interval
so we are led to the relation that this=delta
t observed/a (t) observed okay and since the
time interval and the wavelength are exactly
proportional we see that the wavelength satisfied
this relation so the wavelength actually changed
because the time intervals change okay so
the wavelength of the light which is emitted
is different from the wavelength of the light
which is received okay and this is occurring
because essentially because of the expansion
of the universe
the first peak over here the first crust has
to travel a smaller physical distance a (t)
is an increasing function of time so the physical
distance at the first crust has to travel
is less compared to physical distance the
second crust has to travel so the time the
first crust takes to cover the separation
is less the time that the second one takes
is more as a consequence delta t observed
is more than delta t emitted okay
and this implies that the ratio of the wavelength
also sales according to the scale factor okay
so essentially you can think of it as follows
that the wavelength gets stretched with the
expansion of the universe because the ratio
of the wavelength to the scale factor remains
a constant as the universe expense if i have
a wavelength light of wavelength of 1 meter
the size of the universe is becomes doubled
the wavelength also becomes 2 meters okay
and then we can now calculate the redshift
z remember that if i look at the ratio of
the observed wavelength to the emitted wavelength
this ratio is what i call 1+the redshift so
light is emitted at the wavelength lambda
emitted it is received at the wavelength lambda
observed this i called 1+the redshift okay
and this here is then a emitted/a observed
okay
the ratio of the scale factors and if i assume
that the observer is sitting at present then
we have the 1+the redshift is this should
be a observed by a emitted because they are
proportional so this is observer in the top
emitter in the denominator this=1/a emitted
if i assume that the observer is sitting at
present okay in general it is the ratio of
the scale factors the observer is sitting
at present
so let us assume that the observer at present
see is light coming from a galaxy faraway
the value of the scale factor we have assume
to be 1 at present so i can put a0=1 1+the
redshift is essentially 1/the scale factor
when the light was emitted okay so this now
allows us to interpret the redshift straight
away in terms of the scale factor of the universe
when the light was emitted so let me ask you
a simple question suppose i observe a galaxy
at redshift 1
if i observe a galaxy at a redshift 1 i know
then that the scale factor when the light
was emitted the scale factor had a value which
is how much is it it is half the universe
was half the present size and suppose we also
assume that the universe is matter dominated
completely matter dominated then we know that
the scale factor is proportional to t to the
power 2/3 this is something very important
that you should remember
as a universe is matter dominated the scale
factor is proportional to t to the power 2/3
okay if this is very important because this
gives an with approximation to the real universe
for most of its evolution except when it was
radiation dominated in the past but we do
not see objects in that epoch okay and if
it is if there is a cosmological constant
then it will be different in the future may
be somewhere near the present but most of
the evolution it is like this
we can use it for most purposes we can use
it to make estimates so if i know this then
i can ask you the question what was the age
of universe compared to the present age when
the light was emitted so what was the age
of the universe please work out this problem
it is a very simple exercise what was age
of how far back in the universe are we seeing
when the light was emitted so how far back
in the universe are we seeing when we see
a galaxy at redshift 1
and you can determine this from the fact that
the scale factor is proportional to t to the
power 2/3 so we know the value of this scale
factor now is 1 it is half at the time when
the light was emitted so i determine the age
of the how far determine the age of the universe
at the instant when the light was emitted
or how far back in the time compared to the
present epoch is the light coming from okay
right
so we have learnt how the redshift of a source
we have learnt that due to the expansion of
the universe the light from distant sources
light from sources which we see from the past
of the universe get redshifted light from
distant sources get redshifted and the next
thing that we would like to calculate is that
we would like to okay
before we go into anymore calculations let
me now introduce something called the conformal
time which is very useful if we are studying
the propagation of light in an expanding universe
so the conformal time is introduced as follows
the same cosmological metric can also be written
in the following way so let me introduce something
called conformal time
so i have not written out the length element
in full
i have just represented this entire thing
by dl square it is essentially the length
distance between two points okay so the same
interval expression for the interval can also
be written in this way where we have introduced
a conformal time d eta which is dt/a (t) you
see all that we have done is that we have
taken the factor a (t) which occurs over here
common outside okay so to do that i will get
dt/a (t) here so i have call that d eta the
conformal time
and line element now looks like this okay
this is called conformal time because this
is very much like the metric and the absence
of gravity and the absence of gravity that
is correct right in the absence of expansion
and the expansion of the universe as gone
in as conformal factor outside so if i have
an element line element and i multiplied with
the function this is called a conformal factor
okay so that is the reason why this is called
conformal time
now if you look at this the advantage of this
using the conformal time is as follows let
me look at the propagation of light again
in a space-time diagram in my space time diagram
if i plot r and here if a plot c eta then
the propagation of light is along 45 degree
lines because light will propagate along curves
where ds is 0 so c d eta=dl i am looking at
only the radial direction so r will be=c eta
so the light propagates along 45 degree lines
in this space time diagram
if you go back to the original cosmological
metric if i were to draw the space-time diagram
in terms of t and r the comoving coordinate
the light would not propagate along 45 degree
lines because of this factor a (t) here okay
so the conformal time is very useful if we
are looking at the propagation of light the
light covers equal comoving distance in equal
intervals of the conformal time that is the
important property okay
so let me time is nearly over so let me bring
todays lecture to end over here let me recapitulate
before what we have learnt today we learned
first how to interpret the cosmological space-time
which is represented through the cosmological
line interval or what is called a metric this
is essentially allows us to calculate distances
and intervals between evens that is the crux
of the whole matter and today we learned how
to interpret it
we saw that you can the space can be of three
different kinds it could be flat or the cosmological
space the universe at we are leaving could
be spatially flat or it could be space of
constant positive curvature or of constant
negative curvature we then took a closer look
at we mean by a space of constant positive
curvature it is a three dimensional sphere
and finally i explained to you how to calculate
the redshift of light from a distance source
how that is related to the scale factor and
we saw that the wavelength essentially scales
with the expansions of the universe which
gives rise to the observed redshift of distant
sources let me end todays lecture here
