The Friedmann model describes how the universe expands or contracts based on Einstein's field equations and the cosmological principle of homogeneity and isotropy. It assumes the universe looks the same from all positions and in all directions on large scales. The Friedmann equations relate the expansion rate and density of the universe. The horizon problem arises because regions emitting the microwave background radiation were too far apart to be in causal contact, yet the radiation is isotropic. The flatness problem is that the density of the universe must have been extremely close to the critical density shortly after the Big Bang to remain flat today within observational limits.

2. Problems in
Freidmann Model
Horizon Problem
&
Flatness Problem

3. What is Friedmann
Model???

4. The "Friedmann model" is a model of the Universe
governed by the Friedmann equations, which
describes how the Universe expands or contracts.
These equations are a solution to Einstein's field
equations, and with two very important assumptions
they form the basis for our understanding of the
evolution and structure of our Universe. These
assumptions, together called "the cosmological
principle", are that the Universe is homogeneous,
and that it's isotropic.

5. ! Friedmann Models consist of :
 Principle of general relativity
 Cosmological principles

6. Principle Of General Relativity:
The equations describing the laws of physics have the
same form irrespective of the coordinate system
fulfilled by Einstein’s Field Equations
Rμν −1/2 Rgμν −Λgμν = 8πGTμν

7. Cosmological Principles:
 Homogeneity
 Isotropy

8. Homogeneity:
The Universe is homogeneous means that its "the
same" everywhere. Obviously, it's not, really. For
instance, under your feet is a dense, rocky planet,
whereas above your head there's thin air. We live in a
galaxy full of stars and molecular clouds and what not,
while 100,000 lightyears from the Milky Way, there is
virtually nothing. But on very large scales, say above
half a billion lightyears, the Universe actually looks
the same all over.

9. Isotropy:
Isotropy means the universe looks the same in all directions.
Again, obviously it doesn't on small scales, but on large scales,
it does. If it didn't, it would mean that we occupied a special
place in the Universe, and we don't think we do.
So, neither of these assumptions have to be true, but
observations tell us that apparently, they are.

10. What is Friedmann
Equation???

11. The time-time component of the Einstein equations is called
the Friedmann equation
Where H=a’/a is the Hubble parameter and ρ=Σρa is the total
energy density of the Universe. We have included the
curvature contribution k, to highlight the fact that in a flat
universe (k = 0) the total density always equals the critical
density ρcrit defined as
2
2
8
3
G k
H
a

 
2
3
8
crit
H
G




12. The Horizon Problem
The Flatness Problem

13. The horizon problem arises from the observed uniformity of
the microwave background radiation. The temperature of this
radiation is isotropic to better than 1 part in 104 , after
accounting for a small dipole variation due to the peculiar
motion of the earth and the galaxy. Yet at the time of emission,
regions in opposite directions in the sky from which the
radiation emanated were well outside of each other's horizons.
Thus, no physical process propagating with velocity at or
below the speed of light could have brought these regions into
thermal equilibrium. The horizon problem is the inability of the
standard model to account for this state of homogeneity.
The Horizon Problem:

14. Light propagates along world lines for which the spacetime
interval is zero. Thus,
We consider light emitted by one comoving observer at the
coordinate re and cosmic time te and received by another
comoving observer at coordinate ro and time to. For
convenience, choose ro, the coordinate of the receiving
observer as the origin.
 
2
2 2 2 2 2 2 2 2
2
0 (t) sin
1
dr
ds c dt R r d d
kr
  
 
     
 

15. By the cosmological principle, any point can be chosen as the
origin without loss of generality. With this choice of
coordinates, light will propagate along lines of constant θ, φ.
Then, from the above metric, the time of propagation is
related to the coordinate distance transversed by the following
expression,
This expression can be used to derive a relationship between
the observed redshift of light emitted from distant galaxies
and the scale factor.
0
2
0
(t) 1
e
e
t r
t
c dr
dt
R kr


 

16. If light with frequency ve and period Δte = 1/ve, is being
emitted, then the first wavefront will follow a trajectory
described by the above equation. The crest of the second
wavefront will be emitted from the same comoving coordinate
at a time Δte later, and will follow a trajectory described by the
following equation.
The right hand side of both equations are the same.
Subtracting the first equation from the second and noting that
Δte and Δt0 are small enough that for both R(t + Δ t) ≈ R(t).
0 0
2
0
(t) 1
e
e e
t t r
t t
c dr
dt
R kr




 

17. Then
or equivalently,
We define the redshift parameter z as the fractional change
in wavelength of the emitted light.
0
0
( )
( )
e
e
t R t
t R t



0
0
( )
( )
e
e
R t
R t



0 0
0
0
1
( )
1
( )
e
e e
e
e
z
R t
z
R t
  
 



  
  

18. Given an explicit expression for R(t), measurements of the
redshift would then allow us to calculate the time at which the
light was emitted. More generally, the redshift parameter z can
be used as an implicit expression for time. Note also that z
expresses the amount of expansion of the universe since the
light was emitted. If z = 1, then the scale factor and thus the
size of the universe, has doubled between emission and
reception of the light.
The proper distance of the emitting source at the time the light
is received is the coordinate distance of the emitting region
multiplied by the scale factor at the time of reception.

19. 0
0 02
0
( ) ( )
(t)1
e
e
r t
prop
t
dr c
d R t R t dt
Rkr
 

 
The finite speed for the propagation of light sets a limit on the
distance from which signals can be received by an observer
located at r = 0. For any given time of emission te, there is a
region of comoving space defined by r < re such that signals
outside of this region could not have reached r = 0 by time to. If
the integral
0
( )
(t )
t
c
R t dt
R


converges, then there exists what is known as a particle
horizon, which represents the farthest distance over which
regions could have communicated.

20. For Friedmann models, the integral does converge. We will call
dH the proper or physical distance of the horizon and rH its
radial comoving coordinate.
For the case of k = 0, the horizon distance can be easily
calculated. In the radiation dominated era, R(t)∝ t1/2 and
In the radiation dominated era, R(t)∝ t2/3,
2
0 0
(t) R(t) (t)
(t ) 1
Hrt
H
c dr
d dt R
R kr
 
 
 
1
2
1
'
0 2
(t) t 2
t
H
cdt
d ct
t

 
2
3
2
'
0 3
(t) t 3
t
H
cdt
d ct
t

 

21. Today we detect microwave radiation coming uniformly from
every direction of the sky.
The physical distance of the source of that radiation at the
time of emission was
Thus at the time of emission, the sources of radiation coming
from opposite directions of the sky would have been
separated by a physical distance of approximately
0t
t
(t ) R(t )
(t)e
CBR e e
cdt
d
R
 
2 (t )sep CBR ed d

22. The size of the horizon at the time of emission was
Now we can ask whether regions emitting the microwave
radiation would have been in causal contact with each other.
To do this, we calculate the ratio of the separation distance to
the horizon size. If this ratio is greater than 1, the two regions
are outside each other's horizon; if the ratio is greater than 2,
no signal emitted from an intermediate point could have
reached both regions.
0t
0
(t ) R(t )
(t)
H e e
cdt
d
R
 
0
0
t
0
t
t
2
(t ) (t)
(t )
(t)e
sep e
CBR e
cdt
d R
d cdt
R




23. To calculate this ratio, consider the simple case of a matter
dominated k =0 universe.
This expression can be written directly in terms of the redshift
parameter z, eliminating the dependence on time.
0
1
t 3
0
0 0
1
t 3
0
0 00
1
3
0
6 t (t ) t
(t ) 2R(t ) 1
(t) (t ) t
3 t (t ) t
(t ) R(t )
(t) (t ) t
(t ) t
2 1
(t ) t
e
e
e e
sep e e
t
e e
H e e
sep e e
H e
c Rcdt
d
R R
c Rcdt
d
R R
d
d
 
      
  
 
 
   
 
 
     
  
 



24.  
2
3
0
0
1
2
(t ) t
1
(t ) t
(t )
2 1 1
(t )
e
e
sep e
H e
R
z
R
d
z
d
 
    
 
 
   
 
The microwave background was emitted at about z = 1500,
which corresponds to a temperature of T≈4000° K. Thus,
the separation distance was approximately 80 times the
horizon distance at the time of emission.

25. The flatness problem arises from the observational constraints
on omega (Q), a dimensionless parameter that describes the
ratio of the actual mass density of the universe to the critical
mass density. A universe with critical density (Q = 1) is
globally flat, at the borderline between a closed, finite universe
and an open, infinite universe. Observational constraints put
omega well between between 0.01 and 2 today, and most likely
within an order of magnitude of 1.
However, Q = 1 is an unstable equilibrium point. In order for
omega to fall within the given constraints today, it had to be
equal to 1 to within one part in 1015 at the time of
nucleosynthesis, approximately one second after the big bang.
The Flatness Problem:

26. Why is the universe density so
nearly at the critical density or
put another way, why is the
universe so flat????

27. The Flatness Problem:
The universe is so well-balanced between the positively-curved
closed universe and the negatively-curved open universe that
astronomers have a hard time figuring out which model to
choose. Of all the possibilities from very positively-curved
(very high density) to very negatively-curved (very low
density), the current nearly flat condition is definitely a special
case. The balance would need to have been even finer nearer
the time of the Big Bang because any deviation from perfect
balance gets magnified over time.

28. If the universe density was slightly greater than the
critical density a billion years after the Big Bang, the
universe would have re-collapsed by now.
For example:

29. Consider the analogy of the difficulty of shooting an arrow at a
small target from a distance away. If your angle of shooting is a
little off, the arrow misses the target. The permitted range of
deviation from the true direction gets narrower and narrower as
you move farther and farther away from the target. The earlier
in time the universe's curvature became fixed, the more finely
tuned the density must have been to make the universe's current
density be so near the critical density. If the curvature of the
universe was just a few percent off from perfect flatness within
a few seconds after the Big Bang, the universe would have
either re-collapsed before fusion ever began or the universe
would expanded so much that it would seem to be devoid of
matter. It appears that the density/curvature was very finely
tuned.