The Cosmological Parameters: H, Omega(Lambda), Omega(M), Omega(K) The Age of the Universe versus the CMBR


The purpose of this document is to discuss the problems involved in order to calculate these Cosmological Parameters completely and solely from the Cosmic Microwave Background Radiation.
This raises four issues:
  1. Which are the Cosmological Parameters. What is the definition. What does each physical describe.
  2. What is this radiation.
  3. What is measured and What is calculated.
  4. The program CAMB. This program calculates the Power Spectrum as a function of 6 parameters.
To answer those questions in general is very difficult and speculative. Lets starts with the Big Bang.

The Big Bang.

To understand the Big Bang is physics. That means to describe the physical process that happened during that epoch. This is different from mathematics and the mathematical laws that describe these processes.

IMO the evolution of the Universe can de divided into four periods.

  1. The first period we can call the Quark Soup. The basic building blocks of that period are: the elements of the standard model.
  2. The second period we call the Baryon soup. The basic building blocks of that period are : protons, neutrons, electrons and photons.
  3. The third period. The basic building blocks of that period are: atoms, molecules. The evolution of both went "slowly"
  4. This is the period when the first physical objects started to appear. This is the starting point of Newtons Law between masses. More complex atoms and molecules also started to form. There are now 4 fundamental forces of nature.
The transition between the different periods went "slowly".

One way to learn something about the Big Bang is to compare it with three diffent objects:

a Black Hole, a Supernova and the Sun.

What is the Background radiation

The Cosmic Background radiation are photons. The origin of those photons is very close after the Big Bang. The origin is much earlier than the youngest stars (galaxies) we see.

One parameter to describe the background radiation is temperature. But this is wrong because temperature is a human parameter. The only two parameters to describe the background radiation is by intensity and frequency.
The same concept that can also not be used is cooling. Again this is a human concept. What happened is that the wave length of the photons increased. The reason is the expanding Universe.

The background radiation (soup) that we measure is not every where the same but is shows a pattern of fluctuations in intensity. The following picture shows this pattern of intensity by + and - signs in a rather abstract way. You can compare this pattern with an interference patern of two waves under an angle of 60 degrees.

  +         +         +         +

  -         -         -         -
       +         +         +         +

       -         -         -         - 
  +         +         +         +

  -         -         -         -
       +         +         +

    Figure 1    BackGround Radiation  

Document Literature (25) from 2010 at page 8 shows that there are roughly 21000 hotspots
With a radius of observed background of R, the distance between two hotspots is d and the distance between a hot and cold spot is a we get:

Surface of sphere = 4*pi*R^2
The surface of the space between three hotspots is (length * height): d * (1/2*d*Sqr(3))
This results in the equation: 4*pi*R^2=1/2*d^2*sqr(3) * n with n = 21000
With d^2= 3 *a^2 we get: 4*pi*R^2=3/2*a^2*sqr(3) * n
That means: a/R = sqr(4*pi/(3/2*sqr(3)*n) = tan (alpha) or
or alpha = atn(sqr(4*pi/(3/2*sqr(3)*n)) * 180 / pi degrees = 0,79 degrees

100/0,79 = 115. This number explains the position of the first peak in the microwave Background radiation power spectrum.

One question to answer is what is the path of the oldest photons which is oberved now as the Microwave BackGround Radiation.
When you study the document: The path of a light ray you can see that the path from a supernova follows the blue line in the x direction as described by Friedmann's Equation. That means in an expanding Universe the path of the photons from oldest supernovae start very close near the origin, but near the rim of the Universe were space expansion is very high. First they move outward but always to a region where space expansion is less. Finally they move inward towards the origin.
In an ideal empty universe the number of photons that reach the observer is a function of L/d^2. With d the distance that the lightray has travelled. This function describes for example from a lighthouse the amount of light that reaches the observer at a distance d from a lighthouse. However this function is not totally correct. When there is fog you cannot see the light. The reason is dispersion by the free floating water molecules.
The same problem exists for light from a supernova, but is even more severe from photons emitted immediate after the Big Bang.

Figure 2 and Figure 3 shows the issues infolved in more detail. The most important event is the * in Figure 3. The horizontal axis is the time since the Big Bang. The vertical axis is the line of sight from an observer at t = 14 billion years after the Big Bang.

|                                     a
|                                a
|                          a
|                    a
|               a
|           a        b  b               
|        a     b             b
|      a  b         g g          b       
|    a b    g               g       b      
|   b  g                       g      b
| b  g                           g      b 
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 
|         time --->                  g
|                                      g
|                                        g
        Figure 2      No Dispersion
|                                a--------            
|                          a                  
|                    a
|               a
|           a        b  b               
|        a     b             b
|      a  b         g g          b--------      
|    a b    g               g             
|   b  g                       g------*---      
| b  g                                  
0  1  2  3  4  5  6  7  8  9 10 11 12 13 14 
|         time --->                            
        Figure 3       With Dispersion
Figure 2 shows the path of three light rays without dispersion.
  1. The line marked with the letters "b" (blue), shows the path of a light ray which starts for example 500 million years after the Big Bang and which reaches the observer 14 Billion years after the Big Bang. This is point 14 at the time axis.
  2. The line marked with the letters "a", shows the path of a light ray which started 2 billion years after the Big Bang but closer to the rim of the Universe as the blue line at that moment. This light ray will never pass the path of the Observer (since the Big Bang)
  3. The line marked with the letters "g" green, shows the path of a light ray which started 2 billion years after the Big Bang but closer towards the Observer as the blue line at that moment. This light ray crossed the path of the Observer 11.5 years after the Big Bang.
Figure 3 shows what happened if a light ray is reflected perpendicular. If the light ray follows the x axis reflection in this case means that the reflected path lies somewhere in the y,z plane.
After the reflection the distance towards the observer does does not change any more. In figure 3 this is depicted as a horizontal line.
  1. This has no consequences from the observer point of view for line "a". The path stays outside the blue region.
  2. For the blue light ray "b" this means that, in case of such an event, after 14 billion years the original photons cannot be observed. Even if a second reflection occurs.
  3. For the green light ray "g" this is different. The green light ray after this event will pass the undisturbed path of the blue line. This is very important because if at that moment there is again a reflection it is possible that the path starts to follow the blue line ! (and will be observed after 14 billion year.
The above sketched situation in figure 3 (of a single photon) is in reality highly unlikely. But that is not the point. The importance is that in the past many single photons (from all directions) could have crossed the path of the blue line and at the same time could be reflected in the direction of the blue line. This reflection is more probable in the past towards the Big Bang. Towards the present we observe this reflection as gravitational lensing around galaxies. It is important to remark that the Andromeda Galaxy is a source of such reflection. Even our own Galaxy the Milky Way. Even our Sun.

Those reflected single photons we observe now as the Back Ground Radiation. The wave length is increased being caused by an expanding Universe.

The following url: Cosmic Inflation explains the origin of the CMB radiation. IMO this picture is misleading. We assume that our Universe is homogeneous. That means for example that is has the same density in all directions. Recombination happend 300.000 years after the Big Bang. That is the origin of the CMB radiation we observe today. Also during that period the Universe was homogeneous, implying that everywhere this recombination happened at the same time. However that is not what is observed. What we observe are the events happening on the surface of a sphere almost at the rim of the universe at that epoch. As such we do not observe what has happened at the center of the sphere, nor, and that is very important, that we can claim by observing the CMB radiation that the Universe is homogeneous.

Which are the Cosmological Parameters: H, Omega(Lambda), Omega(M), Omega(K) and The Age of the Universe

The Cosmological Parameters are H, Omega(lambda), Omega(M) and Omega(k) are calculated from the Friedmann equation. Besides those parameters there are two more: Omega(Baryon) and Omega(CDM). Together those two are equal to Omega(M).
None of those parameters are constants. They are all a function of time.
When we discuss H0 we mean the value of the Hubble parameter now, at present. The same is true for almost all the Cosmological Parameters. That means we have a Omega0(Lambda), Omega0(M), Omega0(K), Omega0(Baryon) and a Omaga0(CDM).
It has to be mentioned that in case k=0 that both Omaga(K) and Omega0(k) are zero.

When you go to the documents mentioned in Literature (2), specific to: the Intermediate Level CMB tutorial, Third Peak, Summary "Tab" you will read that the first peak of the power spectrum is explained by the parameter Omega(K), the second peak by the parameter Omega(Baryon) and the third peak by the parameter Omega(CDM). That means (my impression) the CMB radiation can not be used to calculate the parameter H, Omega(Lambda) and the Age of the Universe.

However this also raises a serious question: At which epoch are we speaking ?

Are this the Cosmological Parameters: Omega0(K), Omega(Baryon) and Omega0(CDM) at present ?
When you go to the document: Intermediate Level CMB tutorial, First Peak, Spatial Curvature "Tab" you will read:
As advertised, the position of that first peak in the power spectrum of the anisotropies, and indeed all of the peaks, depend sensitively on the spatial curvature of the universe.
That may be the case, but how do you know this dependency between curvature versus powerspectrum ?
The first peak moves from 200 to 1000. That means the angle decreases with a factor 5. Besides, the concept of curvature is a mathematical issue. There are three versions of Friedmann equation and the issue is to select the one which is closest to observation. This is the parameter k which has the values 1, 0 and -1.

When you go to the document: Intermediate Level CMB tutorial, Angular Peaks, Streaming "Tab" you will read:

After recombination, the photons stream unimpeded.
As time progresses, you see photons from more distant regions.
What the simulation shows is that photons, originating during the Big Bang, follow a straight line. This picture to a large extend gives the wrong impression. Photons only follow straight lines under undisturbed idealized conditions. In reality individual photons follow curved lines, influenced by matter, during their path to the observer. Gravitational lensing is an example of such a behavior.
If the path of photons is straight than the Law Luminosity/ (2 * Pi * d^2) = Magtitude (with d being the length of the light path) should hold for any distance d. However that is not true. In reality dispersion is at stake, meaning that less light reaches the observer.
                    Figure 4
The above sketch shows a light cone starting at position 0. Observer at the positions 1, 2 3 and 4 should measure the same number of photons per second going through the surface indicated. This is not happening. The Observer at position 4 measures the least. The Law that describes this behavour is Luminosity/ (2 * pi * (d * (1 + a* d + b * d^2))^2) with b smaller than a

Document Literature (25) Paragraph 3.2.4 Luminosity Distances page 13 is written:

There is an indication that the constraints on dark energy parameters are different when different methods are used to fit the light curves of Type Ia supernovae (Hicken et al. 2009b; Kessler et al. 2009).
This conclusion is as expected because the results depend very much about the Luminosity versus Magnitude function used. At the same time this result identifies how difficult it is to calculate the Cosmological Constants: Lambda, C, k and the age of the Universe.

Document Literature (23) only mentions two cases where Super Novae SN Data is used. (At page 17).

Reflection first peak and the parameter k

When you go to the documents mentioned in Literature (2), specific to: the Intermediate Level CMB tutorial, First Peak, Curvature "Tab" the simualtion shows that the Omaga(K) peak moves to the right.
This simulation is highly misleading, because it gives the impression that the relation between curvature and the Microwave BackGround is properly understood and that the equations that describe this performance are correct. That is not true. There is no proof.
The outcome of the simulation is that k=0 and that the Universe is flat because the CPS matches the OPS. This result is what we wish because it makes the issue involved disapear, like the snow in the sun. but there is no prove that it is correct. The issue is curvature, and that is IMO purely a mathematical concept.
Mathematical flat means that the distance between parallel lines always is the same. closed means that the distance decreases. Open means that the distance increases. How you can demonstrate that the BackGround Radiation agrees with one concept and not with the others, I do not know.

The fact that the Universe is mathematical flat still means that the Universe can be either physical Open or Closed as a function of the parameter Lambda. Lambda < 0 means closed. Lambda > or equal zero means "Open"

The following table shows certain combinations of k=-1, k=0 and k=1 with the objective that omega(m)*h2 = Constant = 0.1366 The program used is Friedmann's Equation.xls. The parameter optimised is Lambda.

Lambda 0.00908 0.00643 0,0042140,0154130,00994 0,0224630,016357
k -1 -1 -1 0 0 1 1
age 13 13,5 14 13 14 13 14
H0 71,162 64,931 59,219 79,359 67,429 88,097 76,207
omega m0,269750,324000,38951 0,2169 0,300430,176010,23521
omega L0,573770,488050,38453 0,7831 9,699570,926080,90122
omega k0,156470,187950,22596 0 0 -0,1021-0,1364
o(m)*h20,1366 0,1366 0,1366 0,1366 0,1366 0,1366 0,1366
Table 1

The purpose of table 2 is to adjust the parameters Omaga(k), by try and error, such that the age of the universe is either 13 or 14. The program used is CAMB (See next paragraph), The boundary condition is that omega(m)*h2 = Constant = 0.1366.
The purpose of table 3 is to adjust both the parameter k and Lambda when the age of the Universe is either 13 or 14. The program used is: Friedmann's Equation.xls The boundary condition is that omega(m)*h2 = Constant = 0.1366.

Omega_K -0.055 0 0.178
Omega_L 0.7762 0.72120.5432
Omega_m 0.2788 0.27880.2788
age 14.000 13.73813.000
H0 70 70 70
1 Peak l 202,1 221.6 278.7
3 Peak l 747,1 816,3 1030,9
5 Peak l 1308,8 1429,2 1806,7
1 Peak H 5733,7 5734,1 5747,5
3 Peak H 2513,3 2514,7 2514,7
5 Peak H 819,1 819,1,5 819,5
Table 2
Lambda 0.012062 0.00642 0,00836
k 0,344 0 -1,11
age 14 13,738 13
H0 70,378 70,355 70,291
omega m0,275790,275970,27647
omega L0,780110,724030,54552
omega k-0,055 0 0,17801
o(m)*h20,1366 0,1366 0,1366
Table 3

Table 2 shows that in case the age of the Universe is 13 Omaga(k) = -0.55 and when the age = 14 that Omega(k) is 0.178.

The program CAMB

The program CAMB is the program to calcualte the Power Spectrum as a function of 7 parameters: H, Omega(CMB), Omega(Baryon), Omega(Lamba), Omega(K), Tcmb (Temperature CMB) and Helium Factor. Omega(K) is considered zero here, in this discussion. The Calculated Power Spectrum is Called the CPS
The Power Spectrum calculated from the Cosmic Micro Background Radiation is called the OPS. This is Test 3 in Table 4. Table 4 shows the results of 9 different CPS
For a copy of the program CAMB go to: CAMB
To study the listing of the Fortran program go to: CAMB listing
1 Peak l
3 Peak l
5 Peak l
1 Peak H
66 67 70 73 74 78 79
- - - - - - -
223.7 222,9 221.6 219.5 219.5 216,0 216,0
826,5 822,2 816,3 809,7 807,0 799,0 796,2
1447,3 1441,7 1429,2 1416,7 1413,9 1400,0 1394,4
5737 5734 5731 5736 5734 5737 5734
14.100 14.008 13.738 13.481 13.398 13.076 12.999
0.6864 0.6957 0.7212 0.7437 0.7505 0.7755 0.7811
14.8459 14.624 13.9976 13.4224 13.241 12.562 12.403
1 2 3 4 5 6 7
Om_C T cmb HF
0.12 3 0.54
218,1 220,9 222,3
796,2 820,5 824,0
1394,4 1434,1 1443,8
5705 6085 6138
13.738 13.736 13.738
0.7212 0.7212 0.7212
13.9976 13.9976 13.9976
8 9 10
Table 4
The tests 8,9 and 10 also use H0=70
The only difference between the tests 1-7 in the middle is the parameter H0. Omega(baryon)*h2 = 0.0226, Omega(CDM)*h2 = 0.114 that means Omega(m)*h2 = 0.1366.
The row with 1 Peak l shows the parameter l value of the first peak in the horizotal direction. 3 Peak l shows the third peak and 5 Peak l the Fifth peak. The overall pattern is that the distance decreases for higher values of H0
The row with 1 Peak H shows the height of the first peak in the vertical direction. The overall pattern is that the height is independent of H0
The calculated Age of the Universe for H0 = 67 is close to 14 billion years and the calculated Age of the Universe for H0 = 79 is close to 13 billion years.
Using the Excel program "Friedmann equations.xls" with age = 14 and Lambda = 0.009716 we get:
H0=66,998 Omega(L)=0,69262 0mega(M)=0,30738 om*h2=0,13798 1/H0=14,625
Using the Excel program "Friedmann equations.xls" with age = 13 and Lambda = 0.0152 we get:
H0=78,998 Omega(L)=0,77935 0mega(M)=0,22065 om*h2=0,1377 1/H0=12,403
That means both programs are in agreement with each other.
Using the Excel program "Friedmann equations.xls" with age = 13.74 and Lambda = 0.01155 we get:
H0=71,000 Omega(L)=0,73316 0mega(M)=0,26684 om*h2=0,13451 1/H0=13,801
This example demonstrates the coincidence problem (age of the Universe is close to Hubble time)

The tests 8-10 also use H0=70 with different values for the parameters: Omega(CDM)*h^2, T CMB and Helium Factor. The three values are respectivily: 0.120 , 3 and 0.54 The "standard" values for H0=70 (in Test 3) are: 0.114 , 2.725 and 0.24
In all the three cases the shape of the power spectrum changes, specific for the higher peaks.
It is important to mention that in these three tests the parameters H0, Omega(Lambda) and Omega(M) are identical as in the case of H0=70.
In the first case the parameter Omega(CDM)*h^2 is modified and is equal to 0.114. But so is Omega(Baryon)*h^2 and is set equal to 0.0166. The sum of those values is 0.1366 ( = Omega(m)*h^2) which is the same as for the other two cases. The result is that in all these three cases the value of Omega(Lambda) is the same and equal to the Omega(Lambda) of H0=70.

The question could be asked: are the calculations involved to calculate the power spectra correct. That means is the program CAMB correct. The problem is there is no way to test that.
The Power Spectra in the three tests 8 to 10 are different compared to the case of H0=70 but that does not mean that the corresponding parameters for H0=70 are correct for t=0 at present ?

The Power Spectra of Test 8 is clearly different than the OPS but there is noway to decide that the difference is correct, nor that either Power Spectra is correct. There is no way to test that, because it is almost mainly a mathematical problem based on underlying physical assumptions.
There is no way to test if in the present Universe the relation between Cold Dark Matter and Baryonic matter is actual 5 to 1.


Supernovae 1A are very distict "point" objects. Their main problem is that the magtitude observed decreases more than pure based on distance. The reason is that a certain amount of the photons is reflected caused by interstellar "dust" or interference of other objects. The further away the more.
For the MicroWave BackGround Radiation, which are single photons, emitted right after the Big Bang IMO this problem is more severe. The result is that IMO: Discussion Usenet sci.astro.research:

Created: 19 December 2012

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