Answer question 7 - Visible matter, Dark matter and Dark energy

For Visible matter and Dark matter the relation is indirect. For Dark energy the relation is a maybe.
Friedmann's equation describes the relation between objects. In the book "Introducing Einstein's Relativity" by Ray d'Inverno he writes in paragraph 22.3: "it is assumed that the Universe consists of a finite number of Galaxies, each having a mass m(i) etc". The final result is Equation 22.6 which is a function of all the masses m(i) and which consists of three energy terms including one which contains the cosmological constant Lambda. This term is considered to describe dark energy.
This means:

• that Equation 22.6 does not make a distinction between visible and invisible matter
• and that dark energy is a property of all the matter in the Universe.

In fact this raises a serious problem. First you have estimate all Visible matter directly in the Universe including the matter near the outer rim of the Universe i.e. what I call the 100% line. This is difficult because IMO a lot of that mass is very small and difficult to observe.
Next you have to calculate the Total matter in the Universe using Equation 22.6. using different observations (redshift). This is very difficult or maybe impossible because the parameter Lambda is not known.
The difference between the Total matter and the Visible matter is the Dark matter.
What makes this whole exercise even more difficult is that you have to calculate first all the matter (including Dark matter) in all the galaxies (Using galaxy rotation curve) before you can start with all the Dark matter in the space outside the galaxies.

Interesting reading is the following document:Lecture 06: Age of the Universe. In the text near equation 147 is written: Consider a mixture of matter and dark energy rho = rho_m + rho_de etc. IMO such a mixture does not exist because the Cosmological Constant and the Gravitational Constant are qualities of the same objects i.e. galaxies. That means it is very difficult to make a disctinction how much each contributes to actual observations. In that light equation 154, 155 and Figure 7 At page 30 are difficult to understand and to accept.

Answer question 8 - Calculation C, Lambda and k based on observations.

The reason why Supernova can be used to calculate distance is because it is assumed that there intrinsic brightness (L) is the same. A supernova is an exploding star. One class is of specific importance and that is the type 1A. They are important because SN of type 1A all explode physical the same i.e. when they explode they all emit the same amount of light (energy).

The most up to date document which shows the relation between magnitude and z based on observations is the document from 2011:

Supernova Constraints and systematic uncertainties from the first 3 years of the Supernova Legacy Survey specific Figure 5 at page 32.

See also: Supernova Legacy Survey: using spectral signatures to improve Type Ia supernovae as distance indicators
Go to paragraph 5.3.3. and select Figure 11.

What the document shows is the relation between magnitude and z. What we want is the relation between distance and z based on observations. Using that information as a mask and comparing it with different simulations of the Friedmann equation it is possible to find the optimum fit between the two. With different simulations is meant different combinations of the parameters C, Lambda and k.

It should be emphasized that the object of the SNLS document is to calculate the cosmological parameters omaga(m) and the equation of state parameter w. See Table 7. The object here is to calculate the parameters C, Lambda and k of the Friedmann equation. Using those parameters it is possible to calculate omega(Lamba) and omega(M).

In the next 6 paragragh six different FL relations between magnitude and distance (d) are discussed.

1. F = L/(4*pi*d^2)

   This information can be used to calculate distance.
   The following two sets of calculations are performed:
   1. To go from magnitude (m) to distance (d):
      n = 0 : K = (m-n)/-2.5 : F = 10^K : d = SQR(L/4*pi*F)
      To go from distance (d) to magnitude (m)

      n = 0 : F = L/(4*pi*d*d) : K = 10 Log(F) : m = -2.5*K+n

   Figure 13a z versus magnitude

   Figure 13b z versus distance

   Figure 13a shows three theoretical curves and one observed curve (black line)
   The magnitude values of the three theoretical curves are calculated as a function of x (distance) by using set 2 of the calculations mentioned above.
   
   • The horizontal axis shows z between 0 and 3
     
   • The vertical axis shows magnitude between 13 and 30
   The black line or SNLS line is copied from Figure 5 in the above mentioned document.

   Figure 13b shows three theoretical curves and one observed curve. The distance of the observed curve is calculated as a function of magnitude by using set 1 of the calculations mentioned above.
   

   • The horizontal axis shows z between 0 and 3
     
   • The vertical axis shows distance between 0 and 20 billion years

   Figure 13b shows:

   • For small values of z that there is a linear relation between z and distance.
   • For larger values of z that :
     • The z values of the SNLS line as a function of distance decrease
     • The z values of the theoretical curves (based on the Friedmann equation) as a function of distance increase
   This difference in behaviour is quite remarkable. The implications are that based on the observations (assuming the calculations are correct) it is almost impossible to calculate the parameters C, Lambda and k of the observed curve i.e. the SNLS line. The best matches are with Lambda<0 and k=1 for small values of z. A different implication is that the Friedmann equations are incorrect or that the assumed relation of magnitude versus distance is wrong.

   The same curve as Figure 13b is also shown in the document from 2003: Measuring cosmology with Supernovae by Saul Perlmutter and Brian P. Schmidt. See page 17 Figure 4. The range of the z value is shorter as in the 2011 document but the conclusions are the same.

   At page 16 of that document is written:

   Fig 4 shows the Hubble diagram for both teams. Both samples show that SNe are, on average, fainter than would be expected, even for an empty Universe, indicating that the Universe is accelerating.
   In order to measure acceleration you should observe the same galaxy over a certain period and establish that its radial speed (z value) is not constant but increasing. This is a very difficult exercise.
   It would be interesting to know exactly what is meant with fainter than would be expected

   Figure 13a shows the calculated magnitude versus z function for three different values of Lambda (0, 0.05 and 0.1) and the observed magnitude versus z function (this is the black line). What we want is to find the optimum Lambda value (Lambda Fitting) which closest matches the observations i.e. the black line. This also involves Luminosity Fitting.
   For an explanation about Luminosity Fitting and Lambda Fitting go to:

   Luminosity Fitting, Lambda Fitting and minimum error calculation

   The following table shows the Luminosity Fitting values and the corresponding error values as a function of Lambda.
   

   Figure 13c Lambda between 0 and 1

   Nr Lambda L * 10^9 Error 1/H0 1 0 136,596 0,01034775 20,89459 - 0,0824 33,8855 0,00519802 3 0,1 28,016 0,00520296 5,4729 - 0,1143 24,6194 0,00518310 5 0,2 14,164 0,00530294 3,87382 9 0,4 7,180 0,00506521 2,73961 13 0,6 4,784 0,00505857 2,23706 17 0,8 3,589 0,00505636 1,93749 19 0,9 3,195 0,00505650 1,82674 - 0,94 3,055 0,00505435 1,78747 21 1 2,870 0,00505605 1,73305 Table 7.1

   Figure 13c Detail Lambda between 0 and 0.1

   Nr Lambda L * 10^9 Error 1 0 136,596 0,01034775 5 0,02 92,071 0,00689313 9 0,04 62,591 0,00567553 13 0,06 45,356 0,00531722 17 0,08 34,823 0,0052039 - 0,0824 33,8855 0,00519802 21 0,1 28,016 0,00520296 - 0,1143 24,6194 0,00518310 25 0,12 23,46 0,00519191 29 0,14 20,065 0,0052504 Table 7.1 Detail

   The table shows that there are 3 optimum values for Lambda (smallest error)

   1. One at Lambda = 0.0824
   2. One at Lambda = 0.1143
   3. One at Lambda = 0.94. When you select Figure 13c you will discover that starting from Lambda = 0.7 the error value is almost constant. That means it is very difficult to calculate Lambda accurately.

   IMO the most important issue is the relation between Luminosity (L) and Flux (F).
   

   In Figure 13 the Flux is calculated as: L/4*pi*d^2
   However assume that over larger distances this relation is not correct.
   In Figure 14 the Flux is calculated as: L/4*pi*d^3
   That means the measured Flux values are smaller. The K value will also decrease, but the magnitude m will increase.
   What that means consider Figure 13a and Figure 14a and study the three coloured lines. Those lines are the simulated conditions for different values of Lambda and are a function of x (distance) and z. Take the right hand side and compare the average distance. In figure 4 the lines at the right side are higher as would be expected.
   The black line in both Figure 13a and Figure 14a is identical because that line is a function of m and z.
   In Figure 13b and Figure 14b the reverse situation exists. The coloured lines are fixed and the black line moves. The Flux values on the right hand side are identical. If you call the distance in Figure 13b d and in Figure 14b r than you get the following relation:
   F = L/4*pi*d^2 = L/4*pi*r^3 = Constant
   You can rewrite that as : d^2 = r^3 = Constant.
   That means r (Figure 14b) is smaller than d (Figure 13b). Which is what is observed for the black line.

   F = L/(4*pi*d^2*d)

   Figure 14a z versus magnitude

   Figure 14b z versus distance

   Figure 14 uses the following calculations

   1. To go from magnitude (m) to distance (d):
      n = 0 : K = (m-n)/-2.5 : F = 10^K : d = (L/4*pi*F)^(1/3)
      To go from distance (d) to magnitude (m)

      n = 0 : F = L/(4*pi*d^2*d) : K = 10 Log(F) : m = -2.5*K+n

   The agreement between the theoretical line of Lambda=0.05 and the observed line (black line) is quite remarkable.

   Figure 14c Lambda between 0 and 1

   Nr Lambda L * 10^9 Error 1/H0 1 0 1932,062 0,00408746 20,89459 - 0,019375 944,219 0,00180396 2 0,05 359,521 0,0021122 7,67691 3 0,1 135,766 0,0022281 5,47290 5 0,2 48,808 0,00229701 3,87382 9 0,4 17,555 0,00237469 2,73961 13 0,6 9,557 0,00239219 2,23706 17 0,8 6,209 0,00240729 1,93749 21 1 4,445 0,00242096 1,73305 Table 7.2

   Figure 14c Detail Lambda between 0 and 0.1

   Nr Lambda L * 10^9 Error 1 0 1932,062 0,00408746 5 0,01 1339,143 0,00230221 8 0,0175 1008,304 0,00183421 - 0,019375 944,219 0,00180396 9 0,02 922,184 0,00180465 13 0,03 649,237 0,00187891 17 0,04 474,492 0,0020199 21 0,05 359,521 0,0021122 Table 7.2 Detail

   The table shows that the optimum value for Lambda = 0.019375

   F = L/(4*pi*d^2*(1+z))

   Figure 15a z versus magnitude

   Figure 15b z versus distance

   Figure 15 uses the following calculations

   1. To go from magnitude (m) to distance (d):
      n = 0 : K = (m-n)/-2.5 : F = 10^K : d = SQR(L/4*pi*F*(1+z))
      To go from distance (d) to magnitude (m)

      n = 0 : F = L/(4*pi*d^2*(1+z)) : K = 10 Log(F) : m = -2.5*K+n

   For higher values of z there is still a discrapency between the theoretical line of Lambda=0.05 and the observed line (black line)

   Figure 15c Lambda between 0 and 1

   Nr Lambda L * 10^9 Error 1/H0 1 0 344,384 0,00566396 20,89459 2 0,05 115,393 0,00193277 7,67691 - 0,096903 62,430361 0,00178093 3 0,1 60,531 0,00178222 5,47290 4 0,15 40,495 0,00181968 4,47235 5 0,2 30,307 0,00191678 3,87382 9 0,4 15,486 0,00165174 2,73961 13 0,6 10,324 0,00164418 2,23706 17 0,8 7,745 0,00164158 1,93749 19 0,9 6,887 0,00164188 1,82674 0,91 6,808 0,00164070 1,81668 21 1 6,200 0,00164190 1,73305 Table 7.3

   Figure 15c Detail Lambda between 0 and 0.1

   Nr Lambda L * 10^9 Error 1 0 344,384 0,00566396 5 0,02 211,376 0,00295582 9 0,04 137,829 0,00210554 13 0,06 98,1 0,00185863 17 0,08 75,038 0,00181168 19 0,09 66,966 0,00179083 - 0,096903 62,430361 0,00178093 21 0,1 60,531 0,00178222 25 0,12 50,437 0,00184101 Table 7.3 Detail

   The table shows that there are 2 optimum values for Lambda (smallest error)

   1. One at Lambda = 0.096903
   2. One at Lambda = 0.91. When you select Figure 13c you will discover that starting from Lambda = 0.65 the error value is almost constant. That means it is very difficult to calculate Lambda accurately.

   F = L/(4*pi*d^2*(1+d))

   Figure 16a z versus magnitude

   Figure 16b z versus distance

   Figure 16 uses the following calculations

   1. To go from magnitude (m) to distance (d):
      n = 0 : K = (m-n)/-2.5 : F = 10^K : d^2 + d^3 = L/4*pi*F
      To go from distance (d) to magnitude (m)

      n = 0 : F = L/(4*pi*d*d*(1+d)) : K = 10 Log(F) : m = -2.5*K+n

   The above equations are in fact a combination of the equations mentioned in section 1 and section 2.

   1. For small distances the equation for F = L/(4*pi*d^2)
   2. For larger distances the equation for F = L/(4*pi*d^3)

   The result give the best match between the observed curve and the calculated curve when Lambda = 0.05

   Figure 16c

   Nr Lambda L * 10^9 Error 1/H0 1 0 2093,943 0,00453812 20,89459 2 0,05 424,277 0,00086602 7,67691 - 0,097469 176,330 0,000681958 3 0,1 170,187 0,00068625 5,4729 - 0,14625 100,7138384 0,000664015 4 0,15 97,251 0,00066556 4,47235 5 0,2 65,315 0,00085365 3,87382 6 0,25 48,700 0,00072810 3,46506 0,2841 41,242 0,000620371 3,25015 7 0,3 38,327 0,00063818 3,16326 9 0,4 25,981 0,00075753 2,73961 13 0,6 15,116 0,00098621 2,23706 17 0,8 10,349 0,00117552 1,93749 21 1 7,739 0,00133413 1,73305 Table 7.4

   Figure 16c Detail

   Lambda L * 10^9 Error 0 2093,943 0,00453812 0,02 1034,244 0,00146251 0,04 550,546 0,00092830 0,06 337,615 0,00080924 0,08 230,480 0,00074579 0,097469 176,330 0,000681958 0,1 170,187 0,00068625 0,12 132,000 0,00070700 0,14 106,906 0,00067107 0,14625 100,7138384 0,000664015 0,16 88,931 0,00070427 0,18 75,404 0,00080758 0,2 65,315 0,00085365 Table 7.4 Detail

   The table shows that there are 3 optimum values for Lambda (smallest error)

   1. One at Lamba = 0.097469
   2. One at Lamba = 0.14625
   3. One at Lamba = 0.2841 This one shows the smallest error value.

   F = L/(4*pi*d^2*(1+z)^2)

   Figure 17a z versus magnitude

   Figure 17b z versus distance

   Figure 18 uses the following calculations

   1. To go from magnitude (m) to distance (d):
      n = 0 : K = (m-n)/-2.5 : F = 10^K : d = SQR(L/4*pi*F*(1+z)^2)
      To go from distance (d) to magnitude (m)

      n = 0 : F = L/(4*pi*d^2*(1+z)^2) : K = 10 Log(F) : m = -2.5*K+n

   Figure 17c

   Nr Lambda L * 10^9 Error 1/H0 1 0 743,242 0,0021986 20,89459 - 0,019654 435,7513 0,00014195 2 0,05 227,557 0,0008190 7,67691 3 0,1 118,45 0,00090210 5,47290 4 0,15 79,226 0,00084130 4,47235 0,16875 70.595 0,00078566 4,21694 5 0,2 60,095 0,00085676 3,87382 9 0,4 30,390 0,00104048 2,73961 13 0,6 20,259 0,00104978 2,23706 17 0,8 15,202 0,00105885 1,93749 21 1 12,163 0,00106720 1,73305 Table 7.5

   Figure 17c Detail

   Lambda L * 10^9 Error 0 743,242 0,0021986 0,01 563,414 0,0008097 0,019654 435,751 0,000141954 0,02 431,906 0,00014315 0,03 339,227 0,00047669 0,04 274,285 0,00071430 0,05 227,557 0,00081900 0,06 193,353 0,00089824 0,07 167,358 0,00090472 0,08 147,595 0,00093738 0,09 131,506 0,00093987 0,1 118,450 0,00090210 Table 7.5 Detail

   The table shows that there are 2 optimum values for Lambda (smallest error)

   1. One at Lamba = 0.019654 This one shows the smallest error value.
   2. One at Lamba = 0.16875

   F = L/(4*pi*d^2*(1+alpha*d)^2)

   The following table shows the parameter alpha as a function of error.
   In order to calculate alpha both Lambda Fitting and Luminosity Fitting are used. The purpose of this exercise is to calculate alpha which gives the smallest error between theory and observation. The result is alpha = 0,145.

   Figure 18a

   alpha Lambda L * 10^9 Error 0,13 0,0455 222,097 0,0002713 0,14 0,0609 164,444 0,0002223 0,145 0,06344 161,09917 0,00020676 0,15 0,0656 158,749 0,0002161 0,16 0,0833 121,156 0,0002481 0,17 0,0921 111,212 0,0002272 0,18 0,0979 107,317 0,0002538 Table 7.6.1

   The following table shows Lambda as a function of error.
   In order to calculate Lambda Luminosity Fitting is used.
   

   The parameter Alpha = 0.145

   Figure 18b

   Nr Lambda L * 10^9 Error 1/H0 1 0 1260,645 0,0032479 20,89459 - 0,0634 161,0986 0,00020676 3 0,1 86,921 0,00057687 5,47290 5 0,2 33,901 0,00152054 3,87382 7 0,3 20,357 0,00170848 3,16326 9 0,4 14,087 0,00201326 2,73961 11 0,5 10,634 0,00223821 2,45048 13 0,6 8,485 0,00241394 2,23706 15 0,7 7,020 0,00255671 2,07119 17 0,8 5,971 0,00267751 1,93749 19 0,9 5,181 0,00278079 1,82674 21 1,00 4,566 0,00287027 1,73305 Table 7.6.2

   Figure 18b Detail

   Nr Lambda L * 10^9 Error 1 0 1260,645 0,0032479 5 0,02 564,269 0,00071551 9 0,04 287,664 0,00038335 13 0,06 173,391 0,00022008 - 0,0634 161,0986 0,00020676 15 0,07 141,091 0,00025897 17 0,08 117,739 0,00036151 21 0,1 86,921 0,00057687 Table 7.6.2 Detail

   The table shows that the optimum value for Lambda = 0.0634

   F = L/(4*pi*d^2*(1+alpha*d+beta*d^2)

   In order to calculate alpha and beta both Lambda Fitting and Luminosity Fitting are used. The purpose of this exercise is to calculate the parameters alpha and beta which give the smallest error between theory and observation. There are two solutions:

   1. The first result is alpha = 0,163 and beta = 0.2197
   2. The second result is alpha = 0,204 and beta = 0.766

   The following table shows the parameter alpha as a function of error.
   The parameter Alpha = 0.163 The parameter Beta = 0.2197

   Figure 19a

   alpha Lambda L * 10^9 Error 0,140 0,0423 205,36 0,0001699 0,150 0,0429 206,084 0,0001637 0,160 0,044 204,357 0,0001625 0,163 0,0445 202,583 0,0001622 0,170 0,0447 204,54 0,0001634 0,180 0,0449 207,814 0,0001691 Table 7.7.1

   The following table shows beta as a function of error.
   In order to calculate Lambda Luminosity Fitting is used.
   The parameter Alpha = 0.204 The parameter Beta = 0.766

   Figure 19c

   Nr Beta Lambda L * 10^9 Error 1 0,016 0,0429 211,151 0,0003592 9 0,024 0,0596 155,550 0,0001775 10 0,025 0,0617 149,959 0,0001782 20 0,035 0,0890 99,704 0,0002342 30 0,045 0,1051 84,542 0,0002607 40 0,055 0,1352 62,770 0,0002823 50 0,065 0,1558 53,769 0,0001921 60 0,075 0,1660 51,417 0,0001139 0,0766 0,1680 50,863 0,0001127 70 0,085 0,1790 48,137 0,0001564 80 0,095 0,1987 43,152 0,0002394 Table 7.7.3

   The following table shows Lambda as a function of error.
   In order to calculate Lambda Luminosity Fitting is used.
   The parameter Alpha = 0.204 The parameter Beta = 0.766

   Figure 19e

   Nr Lambda L * 10^9 Error 1/H0 1 0 2440,011 0,00169228 20,89459 0,005 1920,304 0,00132201 3 0,1 110,766 0,00101839 5,4729 0,16815 50,84027 0,0001127 4.22446 0,17 50,037 0,00011621 5 0,2 39,632 0,00035848 3,87382 7 0,3 22,652 0,00079111 3,16326 9 0,4 15,248 0,00124387 2,73961 13 0,6 8,892 0,00184551 2,23706 17 0,8 6,139 0,00223959 1,93749 21 1 4,639 0,00252427 1,73305 Table 7.7.5

   Figure 19e Detail

   Nr Lambda L * 10^9 Error 1 0 2440,011 0,00169228 2 0,005 1920,304 0,00132201 3 0,01 1498,567 0,00154418 21 0,10 110,729 0,00101838 33 0,16 54,693 0,00017016 0,16815 50,84027 0,0001127 35 0,17 50,037 0,00011621 37 0,18 46,054 0,00019386 Table 7.7.5 Detail

   The table shows that there are 2 optimum values for Lambda (smallest error)

   1. One at Lamba = 0.005
   2. One at Lamba = 0.16815 This one shows the smallest error value for F/L relation #7 and also the smallest error value among all seven tested F/L relations.

• Summary: Flux Luminosity Relation - Age 14

  The results of the seven FL relations are summarised in the following table:

  Nr Description Lambda L * 10^9 1/H0 Error Omega(M) 1.1 F = L/(4*pi*d^2) 0.11432 24,619 5.12044 0.0051831 0,00109 1.2 F = L/(4*pi*d^2) 0.94 3,055 1.78747 0.00505435 0 2 F = L/(4*pi*d^2*d) 0.01937 944,219 11.60739 0.0018039 0,12786 3.1 F = L/(4*pi*d^2*(1+z)) 0.09690 62,432 5.55897 0.0017809 0,00210 3.2 F = L/(4*pi*d^2*(1+z)) 0.91 6,887 1.82674 0.0016418 0 4.1 F = L/(4*pi*d^2*(1+d)) 0.14625 100,713 4.52922 0.0006640 0,00037 4.2 0,2841 41,242 3,25015 0,000620371 0,00001 5.1 F = L/(4*pi*d^2*(1+z)^2) 0.01965 435,752 11.54382 0.0001419 0,12498 5.2 0.012 533,602 13.739 0.000607 0,24502 6.1 F = L/(4*pi*d^2*(1+alpha*d)^2) 0,06344 161,098 6.84648 0.0002067 0,00885 6.2 F = L/(4*pi*d^2*(1+alpha*d)^2) 0,012 1068,957 13.739 0.001058 0,24502 7.1 F = L/(4*pi*d^2*(1+alpha*d+beta*d^2) 0,04457 202,583 8.10498 0.0001622 0,02361 7.2 0,16815 50,84027 4.22446 0.0001127 0,00019 7.3 0,012 435,454 13.739 0.000418 0,24502

  Table 8 FL relation 1 shows two minimum conditions.
  FL relation 3 shows two minimum conditions.
  FL relation 4 ex1 and ex 2 both show minimum conditions.
  In FL relation 6 ex 1 the value of alpha = 0.145
  In FL relation 6 ex 2 the value of alpha = 0.1886
  In FL relation 7 ex 1 the value of alpha = 0.163 beta = 0.02197
  In FL relation 7 ex 2 the value of alpha = 0.204 beta = 0.0766
  In FL relation 7 ex 3 the value of alpha = -0,0579 beta = 0.0287
  

  FL relation 7 shows the smallest error between observations and theory (Friedmann equation)

  Of specific importance are FL relation Ex 2, FL relation 6 ex 2 and FL relation 7 Ex3. Each of those show the same 1/H0 value 0f 13,739 billion years. This is equivalent with the Hubble constant value of 71,316 km/sec/Mpc. The importance becomes obvious if you compare this with the same values of Table 13 for an universe of 28 billion years.

  Lambda versus error for F/L 1 to 7 Figure 20 A

  Figure 20 A shows Lambda values for six F/L relations for Lambda between 0 and 1.
  
  • F/L relation #1 and #3 are similar in the sense that in both cases it is very difficult to calculate Lambda. Starting from Lambda = 0.7 the error value is almost constant.
    
  • F/L relation #7 shows the smallest error between theory and observation

• Summary: curvature constant k values

  Using Labda Fitting and Luminosity Fitting it also possible to calculate error values using different values of k.
  Only the values k = -1, k = 0 and k = 1 are possible. See: Lecture 5: Solutions of Friedmann Equations.
  The book "Introducing Einstein's Relativity" by d'Inverno in chapter 23 "Cosmological Models" does the same.
  The document The Friedmann Equation considers the three cases: k=0, k<0 and k>0 (Select: curvature Parameter)

  Figure 19 shows the error values for k between -2 and 2 for three FL relations.

  1. The red line is FL relation 5 (method 4)
  2. The green line is FL relation 6 (method 5) alpha = 0,145
  3. The blue line is FL relation 7 (method 6) alpha = 0,163 beta=0,02197

  k between 2 and -2 Figure 20 B

  1. FL relation 5 - method 4

     nr k Lambda L * 10^9 Error 1 2 0,0211 491,241 0,0001608 11 1 0,0199 462,574 0,0001345 12 0,9 0,0199 462,574 0,00013427 21 0 0,0196 435,734 0,0001419 31 -1 0,0196 414,48 0,0001455 41 -2 0,0194 401,157 0,0001481 Table 9.4

  2. FL Relation 6 - method 5

     nr k Lambda L * 10^9 Error 1 2 0,0688 151,65 0,0002051 8 1,3 0,06724 153,655 0,00020162 11 1 0,0666 154,445 0,0002022 21 0 0,0634 161,1 0,0002067 31 -1 0,0621 162,135 0,0002176 41 -2 0,0595 168,361 0,0002316 Table 9.5

  3. FL Relation 7 - method 6

     nr k Lambda L * 10^9 Error 1 2 0,0555 165,475 0,0001972 11 1 0,0482 191,132 0,0001778 21 0 0,0445 202,59 0,0001622 31 -1 0,0416 211,475 0,0001753 41 -2 0,0403 212,176 0,0002085 Table 9.6

  4. FL relation 2-7

     The following table shows the curvature constant k values for 6 FL relations with the smallest error value between theory and observation.
     
     • Column 1 shows the Flux Luminosity relation.
     • Column 2 shows the Table Nr which gives more detail.
     • Column 3 shows the measurement point in the corresponding figure.
     • Column 4 shows the curvature constant k value with the smallest error value

     FL Table nr k Lambda L * 10^9 Error 2 9.1 18 -11 0,0193 664,766 0,0017059 3 9.2 1 7 0,0663 181,602 0,0004725 4 9.3 1 8 0,158 96,626 0,0006094 5 9.4 12 0,9 0,0199 462,574 0,00013427 6 9.5 8 1,3 0,06724 153,655 0,00020162 7 9.6.1 21 0 0,0445 202,59 0,0001622 7 9.6.2 37 -14 0,1649 51,738 0,00011231 Table 9.7

• Summary: age calculation

  Figure 21 A shows the error value as a function of age, with age going from 14 to 28 billion years. The error value is the difference between theory and observation. This calculation requires both the Friedmann equation and the Flow Luminosity relation.
  In Figure 21 A this is relation # 4.
  In Figure 21 B FL relation #5 is used
  In Figure 21 C FL relation #6 is used
  In Figure 21 D FL relation #7 is used
  

  Figure 21 A F/L relation 4	Figure 21 B F/L relation 5
  Figure 21 C F/L relation 6	Figure 21 D F/L relation 7

• Summary: C calculation

  C between 5 and 100 Figure 22 A

  K = 0 Nr C Lambda L * 10^9 Error 1 5 0,0196 435,674 0,000142 4 20 0,0196 435,719 0,0001419 8 40 0,0196 435,740 0,0001419 12 60 0,0196 435,749 0,0001419 16 80 0,0196 435,755 0,0001419 20 100 0,0196 435,772 0,0001419

  C between 5 and 100 Figure 22 B

  Nr C Lambda L * 10^9 Error 1 5 0,1681 50,842 0,00011275350 4 20 0,1681 50,842 0,00011275361 8 40 0,1681 50,843 0,00011275369 12 60 0,1681 50,841 0,00011275370 16 80 0,1681 50,840 0,00011275373 20 100 0,1681 50,841 0,00011275377

  What Figure 22 A clearly shows is that for FL relation 5 (method 4) with k =0 (the blue line) the parameter C is independent of the error value (Difference between theory and observation). That means it is not possible calculating the parameter C.
  Figure 22 B For FL relation 7 (method 6) shows the same behaviour.

Answer question 9 - Age = 28 and Age = 42

• Figure 23a (23b) shows the same information as Figure 14a (14b) except over a period of 28 billion years

  Figure 23a z versus magnitude

  Figure 23b z versus distance

  4. F = L/(4*pi*d^2*(1+d))

  Lambda L * 10^9 Error 0 16338.4 0.010149 0.01 4161.731 0.001992 0.011 3721.976 0.001962 0.011022 3716.043 0.001962 0.012 3356.378 0.001992 0.013 3038.983 0.002010 0.02 1708.5 0.002164 0.03 959.7 0.002225 0.05 461.7 0.00225 0.1 169.1 0.00228

  Table 10.1

  5. F = L/(4*pi*d^2*(1+z)^2)

  Lambda L * 10^9 Error 0 2942.5 0.003961 0.005 1750.7 0.000598 0.005624 1623.848 0.000419 0.006 1571.4 0.000439 0.007 1423.8 0.000666 0.008 1314.8 0.000909 0.01 1106.763 0.001258 0.02 602.3 0.002145 0.03 402.3 0.002650 0.05 237.2 0.002763 0.1 116.4 0.003361

  Table 10.2

• Figure 24a (24b) show the same information as Figure 14a (14b) except over a period of 42 billion years

  Figure 24a z versus magnitude

  Figure 24b z versus distance

  4. F = L/(4*pi*d^2*(1+d))

  Lambda L * 10^9 Error 0 52912,178 0.006829 0.003 20483,102 0.001936 0.004257 14436,282 0.001734 0.01 4763,150 0.002085 0.02 1747,485 0.002211 0.03 966,907 0.002272 0.04 635.669 0.002244 0.05 460.181 0.002252 0.1 168,463 0.002275

  Table 10.3

  5. F = L/(4*pi*d^2*(1+z)^2)

  Lambda L * 10^9 Error 0 6732,037 0.003278 0.002466 3688,605 0.000315 0.003 3291,119 0.000449 0.01 1208,331 0.001811 0.02 606,647 0.002177 0.03 400,415 0.002718 0.04 303,918 0.002850 0.05 237,432 0.003240 0.1 117.352 0.004041

  Table 10.4

What Figure 14, 23 and 24 (a and b) demonstrate is that it is not possible to calculate the age of the universe based on the information supplied in the "3 years SNLS document"

The following table shows the redshift values or z values immediate after the Big Bang in billions of years for different combinations of the parameter Lambda. The age of the Universe is respectively: 14 or 28 or 42 billion years.
The left column shows the age of the Universe after the Big Bang in billion of light years.
Age 0 is not the moment of the Big Bang but 100 million years after the Big Bang.

t BB 0 1 2 3 4 5 6 7 8 9 10 1/H0 q0 - -

L=0 0.01 0.02 0.03 z z z z 25,120 29,53 34,249 39,29 4,781 5,75 6,79 7,89 2,651 3,25 3,89 4,575 1,788 2,23 2,71 3,206 1.303 1,65 2,02 2,414 0,985 1,27 1,57 1,879 0,758 0,99 1,24 1,484 0,587 0,78 0,97 1,174 0,452 0,61 0,76 0,921 0,342 0,46 0,59 0,710 0,251 0,35 0,44 0,528 21 14,51 11,481 9,70 0,5 -0,552 -0,817 -0,913 age = 14 Table 11a

L=0 0.01 0.02 0.03 z z z z 40,457 71,49 111,92 163,9 8,175 15,03 23,95 35,4 4,794 9,10 14,67 21,8 3,426 6,68 10,87 16,2 2,655 5,30 8,69 12,98 2,151 4,39 7,23 10,8 1,790 3,73 6,16 9,17 1,518 3,23 5,33 7,9 1,304 2,82 4,65 6,8 1,130 2,48 4,09 6,0 0,986 2,20 3,61 5,2 42,01 17,054 12,224 9,998 0,500 -0,954 -0,994 -0,998 age=28 Table 11b

L=0 0.01 0.02 0.03 z z z z 53,322 162,45 353,77 1163 11,023 35,14 77,30 255 6,593 21,77 48,20 159 4,799 16,32 36,27 119 3,789 13,22 29,42 96 3,128 11,16 24,84 80 2,656 9,68 21,49 68 2,300 8,53 18,87 59 2,019 7,61 16,75 51 1,791 6,86 14,98 45 1,602 6,21 13,47 40 63 17,299 12,25 8,662 0,5 -0,996 -0,999 -1,000 age=42 Table 11c

In the following document Super-solar metal abundances in two Galaxies at z=3.57 revealed by the GRB090323 afterglow spectrum the chemical contents of two galaxies is discussed which are less then 2 billion years old.
In that document at page 9 the relation age versus redshift or z is discussed. At present z = 0 and near the Big Bang z = 7. Roughly 2 billion years z is approximate 3.5 which coincide with the age of the two galaxies or slightly earlier. The question is if that argumentation is correct.

Table 11a first column shows the evolution of z for Lambda=0 over a period of 14 billion years. At 1,4 billion years after the Big Bang z=3.630. At 1.5 billion years z=3.423. The measured value of z=3.57 is somewhere in between.

The question is does Table 11a show the right condition of the Universe. When you consider that the age of the Universe is 28 billion years old than the value of z=5.7 is between 2.8 and 2.9 billion years after the big bang. When the age of the Universe is 42 billion years between z=3.565 after 4.3 billion years. What that means that there is a clear linear relation between the age of the Universe and the age of an event for the same redshift value.

The 4 columns of Table 11a shows Lambda values of: 0 0.01 0.02 and 0.03 Table 11a and Table 11b show that for the same event (same z value) when you increase the age of the universe with a factor 2 that than also the age of the event increases. When Lambda is considered non zero the age of the event increases more. For example with L=0.03 and z=3.57 and age=14 the event happened at approximate 2.75 billion years
With L=0.01 and age=28 the event happened at approximate 6.5 billion years, with L=0.02 at approximate 11 billion years and with L=0.03 at approx 13 billion years

• When Lambda=0 and k=1 for an assumed age of 14 billion years z=3,536 after 1.6 billion years.
• When Lambda=0 and k=-1 for an assumed age of 14 billion years z=3,687 and 3.450 after 1.2 and 1.3 billion years.

The results of the seven FL relations assuming an age of the universe of 28 billion years are summarised in the following table:

Nr Description Lambda L * 10^9 Error 1/H0 rho c rho Omega M 1 F = L/(4*pi*d^2) 0.03744 76,055 0.0050636 8,950 0,00149 0 0,00034 2 F = L/(4*pi*d^2*d) 0.00548 6936,51 0.0018405 22,142 0,000243 0,000025 0,1045 3 F = L/(4*pi*d^2*(1+z)) 0.09440 64,530 0.0017468 5,6383 0,00375 0 0 4 F = L/(4*pi*d^2*(1+d)) 0.09862 175,310 0.0006804 5,5164 0,00392 0 0 5.1 F = L/(4*pi*d^2*(1+z)^2) 0.004904 1760,50 0.0001404 23,132 0,000223 0,000028 0,1254 5.2 0.0158 744,998 0.000941 13,718 0,00063 0,0000056 0,00896 6.1 F = L/(4*pi*d^2*(1+alpha*d)^2) 0,06445 156,098 0.0001932 6,8235 0,00256 0 0,00002 6.2 0.012865 759,067 0.000291 15.1472 0,00052 0,0001 0,01619 6.3 0.0158 642,811 0.000195 13,718 0,00063 0,0000056 0,00896 7.1 F = L/(4*pi*d^2*(1+alpha*d+beta*d^2) 0,03840 254,583 0.0001433 8,83852 0,00153 0 0,00030 7.2 0,0158 564,313 0.000131 13,718 0,00063 0,0000056 0,00896

Table 12 In FL relation 6.1 the value of alpha = 0.14
In FL relation 6.2 the value of alpha = 0.70193
In FL relation 6.3 the value of alpha = 0.0711
In FL relation 7.1 the value of alpha = 0.195 beta = 0.0114
In FL relation 7.2 the value of alpha = 0.09046 beta = 0.00629

Of specific importance are FL relation 5 Ex 2, FL relation 6 ex 3 and FL relation 7 Ex 2. Each of those show the same 1/H0 value 0f 13,718 billion years. This is equivalent with the Hubble constant value of 71,318 km/sec/Mpc. The importance becomes obvious if you compare this with the same error values of Table 8 for an universe of 14 billion years. The errors in Table 12 are smaller. This means the Universe is older than 14 billion years.

The following table shows FL relations for an Universe of 21 billion years.

Nr Description Lambda L * 10^9 Error 1/H0 rho c rho Omega M 5.1 F = L/(4*pi*d^2*(1+z)^2) 0.0151 703,586 0.000598 13,777 0,00063 0,000028 0,0474 6.1 F = L/(4*pi*d^2*(1+alpha*d)^2) 0.0151 717,027 0.000396 13,777 0,00063 0,000028 0,0474 7.1 F = L/(4*pi*d^2*(1+alpha*d+beta*d^2) 0.0151 551,323 0.000169 13,777 0,00063 0,000028 0,0474

Table 12.1 In FL relation 6.1 the value of alpha = 0.0885
In FL relation 7.1 the value of alpha = 0.0680 beta = 0.0099

Answer question 10: The parameter q

Accordingly to feedback received the best document to study to calculate the parameters of the Friedmann equations is: Numerical calculations on relativistic cosmological models Authors: S. Refsdal; R. Stabell; and F.G. de Lange in 1967
In that document the parameter q is used, called deceleration parameter.

q is defined as: -R*a(R)/v(R) with V(R)=dR/dt and a(R) as dv(R)/dt

See Table 4 the right column, for q values from different combinations of Lambda and k

The reason why the parameter q is used in the document to calculate the parameters of the Friedmann equation is not clear because the parameter q can not be directly observed. The calculation of q requires the three values R, v(R) and a(R) and those values are easy to calculate using the Friedmann equation. But R is a global parameter and impossible to observe.
To calculate q based on observations you need:

1. The distance of a galaxy.
2. The speed of a galaxy
3. The acceleration of a galaxy

The first two are relative simple because you can use the redshift (z) parameter.
The third one is very difficult because you must follow the galaxy over a long period of time and measure the change in z value.

Answer question 11: The parameters omega(M), omega(Lambda) and omega(K)

The equations and pages mentioned in this paragraph are from the book: "Introducing Einstein's Relativity" by Ray d'Inverno.
The next steps define omega(Matter), omega(Lambda) = omega dark energy and omega(K).

1. starting point is the equation 23.1 used above:
   (dR/dt)^2= C/R + 1/3 * Lambda*R^2 - k
   using equation 22.57 C= (8*pi/3)*R^3*rho we get:

   (dR/dt)^2= (8*pi/3)*R^2*rho + 1/3 * Lambda*R^2 - k

   dividing all components by R^2 and substituting H=(dR/dt)/R we get:

   H^2= (8*pi/3)*rho + Lambda/3 - k/R^2

   rhoc (rho crital) is defined in equation (23.36) as: (3/8*pi)*H0^2
   

   substituting 8*pi/3 with H0^2/rhoc we get:

   H^2= H0^2*(rho/rhoc) + Lambda/3 - k/R^2

   and next:

   H^2= H0^2 * (rho/rhoc + Lambda/(3*H0^2) - k/(R^2*H0^2))

   With is equal to:

   H^2=H0^2 * (omega(M) + omega(Lambda) +omaga(K))

   omega(Matter) is defined at page 355 as: rho/rhoc

   with H2 = H0 at t = 0 this leads to:

   omega(M) + omega(Lambda) +omaga(K) = 1

The following table shows the results.

Nr Lambda C k L Error 1/H omega(M) rho c rho omega(L) omega(k) 1 0 60 0 742,601 0.0021988 21,00127 0,99995 0.0002705 0.0002705 0 0 2 0 60 -1 688,218 0,0018941 19,38226 0,64757 0,0003177 0,0002057 0 0,35239 3 0 60 +1 827,009 0,0026405 23,91846 1,80202 0,0002086 0,0003760 0 -0,80211 4 0 10 0 743,174 0,0021985 21,00485 0,99995 0,0002705 0,0002705 0 0 5 0 10 -1 612,028 0,0015821 17,45902 0,32068 0,0003915 0,0001255 0 0,6793 6 0 10 +1 1213,289 0,0061800 115,93216 137,37738 0,0000088 0,0122101 0 -136,37858 7 0.01 60 0 563,414 0,0008097 14,50704 0,29856 0,0005672 0,0001693 0,70144 0 8 0.01 60 -1 528,691 0,0007038 13,95589 0,21508 0,0006128 0,0001318 0,64916 0,13576 9 0.01 60 +1 615,291 0,0009574 15,36127 0,45043 0,0005058 0,0002278 0,78646 -0,23689 10 0.01 10 0 563,320 0,0008099 14,50637 0,29862 0,0005672 0,0001693 0,70144 0 11 0.01 10 -1 480,246 0,0005968 13,21788 0,12178 0,0006832 0,000083 0,58237 0,29591 12 0.01 10 +1 881,684 0,0024968 21,98605 2,51915 0,0002469 0,0006220 1,61128 -3,13042 13 0.01211 60 0 531,996 0,0005964 13,70001 0,24244 0,0006359 0,0001541 0,75764 0 14 0.01211 10 0 531,984 0,0005966 13,69951 0,24250 0,0006360 0,0001542 0,75758 0 15 0.01211 2 0 531,878 0,0005967 13,69922 0,24253 0,0006360 0,0001542 0,75755 0 16 0.010724 60 -1 518,883 0,0006356 13,70016 0,20096 0,0006359 0,0001278 0,67094 0,12817 17 0.008468 10 -1 497,871 0,0007123 13,70019 0,13910 0,0006360 0,0000884 0,52980 0,33111 18 0.005389 2 -1 468,779 0,0008690 13,70014 0,06713 0,0006360 0,0000426 0,33716 0,59575

Table 13 - method = 4

• In line #1 with k=0 the parameter 1/H = 21 and Omega(M) = 1, Omega(Lambda)= 0. As expected.
  In line #4 with k=0 the parameters 1/H and omega are the same as in line #1. Also as expected.
  In both lines the parameter rho shows rhoc (Rho critical)
• In line #2 with k=-1 Omega(M) = 0.76, Omega(Lambda)= 0. This means that Omega(k) = 0.24
  
• In the lines #7 to #9 the parameter Lambda = 0.02. The error value is much lower compared with the lines #1 to #3 with Lambda = 0.
  That means Lambda = 0.02 seems to be a better fit.
• In the lines #13 to #15 the parameter Lambda = 0.02. The error value is much lower compared with the lines #10 to #12 with Lambda = 0.
• The difference between the lines #7 to #9 and #13 to 15 is the method used. In the first combination this is F/L #5 and in the second combination this is F/L #1.

The most important consideration of the above table is that the calculated values: 1/H, omega and rho are independent of the F/L relation selected.

The values most often mentioned in the Literature are Omega(M) = 0.28, Omega(Lamba)=0.72, Omega(K)=0
For example See:

• Improved Cosmological Constraints from New, Old, and Combined Supernova Data Sets page 758 and page 771 Figure 15.
  The object of this document is to calculate the parameters omega(Lambda), omega(M) and the equation of state parameter w. The basic assumption is that the universe is flat i.e. that k = 0
  The same document is also: Improved Cosmological constraints etc page 11 and page 23 Figure 15. This version of the document contains the "raw" data values.
• Also see for example Measurements of Omega and Lambda from 42 High-REDSHIFT SUPERNOVAE This is a document from 1999 and shows the values omega(M)=0.28 and omega(Lamba)=0.72
• Cosmological constraints from supernova data set with corrected redshift. This "arxiv.org" document gives for omega(M) (See Table 1 and Figure 1) a value of 0.7. That means omage(L) = 0.3. Observing Table 8 this is the value close for F/L relation 2 and 5. The same document also mentions omega(M) of 1. That means omega(L) = 0. That is the value close to the other F/L relations.

What this means that it is difficult to calculate the parameters of the Friedmann equation and other cosmological parameters.

The following table shows the parameters omega(M) and omega(Lambda) as a function of Lambda for C = 60 and k=0.

Lambda L Error 1/H omega(M) omega(L) 0 743,242 0,0021986 21,00709 0,99995 0 0,002 703,181 0,0018497 19,16382 0,75515 0,24484 0,004 665,971 0,0015294 17,67046 0,58368 0,41631 0,006 630,534 0,0012515 16,43500 0,45981 0,54018 0,008 595,884 0,0010242 15,39509 0,36803 0,63196 0,01 563,414 0,0008097 14,50704 0,29856 0,70143 0,011 548,238 0,0007064 14,10974 0,27010 0,72989 0,012 533,601 0,0006070 13,73928 0,24502 0,75497 0,013 519,457 0,0005125 13,39297 0,22282 0,77717 0,014 505,833 0,0004239 13,06845 0,20311 0,79688 0,015 492,691 0,0003425 12,76370 0,18555 0,81444 0,016 479,892 0,0002704 12,47690 0,16986 0,83013 0,018 455,014 0,0001731 11,95101 0,14317 0,85682 0,019 443,200 0,0001477 11,70927 0,13179 0,86820 0,02 431,906 0,0001431 11,48014 0,12151 0,87848 0,022 410,708 0,0001904 11,05585 0,10378 0,89621 0,024 391,052 0,0002711 10,67132 0,08914 0,91085

Table 14 What the table shows is that for Lambda = 0.011 omega(M) = 0.27 omega(Lambda) = 0.73. This is the current most accepted value for omega(Lamba).
It is important to remark that this particular value has nothing to do with the F/L relation selection. In the above table this is #5 (method = 4).
The above table shows that the smallest error value is for Lambda = 0.02 with method = 4 (F/L relation = 5)

1. An important document to study is: The Hubble Space Telescope Cluster Supernova Survey: V. Improving the Dark Energy Constraints Above z>1 and Building an Early-Type-Hosted Supernova Sample Submitted on 17 May 2011.
   Specific study the text at the bottom of page 19:
   
   To examine constraints on the existence of dark energy at different epochs, we study rho(z), which is the density of the dark energy and allowed to have different values in fixed redshift bins. Within each bin, rho is constant. (Note that the discontinuities in rho(z) at the bin boundaries introduce discontinuities in H(z).) We choose the same binning as above, but note that binned rho and binned w models give different expansion histories. Our results are shown in Figure 9 and Table 8
   Table 8 at page 22 shows the parameter rho(DE)/rho(0) for 4 bins:

   z <0.5 0.5 < z < 1.0 1.0 < z<1.6 z > 1.6 0.731 0.85 0.23(0.33) 0.9(0.7)

   The parameter rho(DE)/rho(0) is the same as omega(Lambda)
   Specific the third value is interesting because this means that also much smaller values for omega(Lambda) are possible.
   It should be remarked that in this study all the cosmological parameters are calculated based on the total curve ( z between 0.5 and 1.4) and not based on bins.

   Also important is the following document:SPECTRA AND HUBBLE SPACE TELESCOPE LIGHT CURVES OF SIX TYPE Ia SUPERNOVAE AT 0.511 < z < 1.12 AND THE UNION2 COMPILATION
   This document is important to study the best fitted cosmological parameters: Z, Omega(m), Omega(w) and w.
   See paragraph 7.2 "Fitting Cosmology" (page 724):

   The blind technique is implemented by adjusting the magnitudes of the SNe until they match a fiducial cosmology (Omega(M) = 0.25, w = -1). This procedure leaves the residuals only slightly changed, so that the performance of the analysis framework can be studied. This seems to me that this document is biased towards the values of Omega(L) = 0.75 and Omega(M) = 0.25

Technical Information.

First consider the green line in figure 8 above.
This line starts at t0 = 2 and the initial distance is the 80% line.
The calculation of the green line involves 3 parameters:

• The parameter dist1. This is the distance of the 100% line.
• The parameter dist2. This distance of light ray. In this case the green line
• The parameter fac. This is the relative distance. fac = dist2/dist1. In this case fac = 0.8 at t0 = 2.

The time in all calculations is subdivided in 14 segments of 1 billion years each. Each of those segments is subdivided in 1000 increments of 1000 million years.

1. The calculation of the parameter dist1. (the 100% line) requires the friedmann equation:
   delta_t = 1/1000
   t = t + delta_t
   R = dist1
   v = sqr(C/R + Lambda * R * R /3)
   R = R + v * delta_t
   dist1 = R
   
   This calculation is performed in a loop for t going from 0 to 14.
   At regular intervals (when t is an integer) the value dist1 is stored in an array of the Excel spread sheet.

   The initial value of R = dist1 at t=0 is defined as: R = v0*delta_t with v0 = 3

   In parallel with this calculation the parameter dist2 of the green line is calculated
   This calculation starts when t = t0 (with t0 = 2) and proceeds in the following steps as part of the above mentioned loop.

   First space expansion is calculated :

   dist2 = fac * dist1 with fac = 0.8 at t0=2

   This calculation is only done once.

   Next the movement of the light ray towards the observer is calculated:

   speed of light c = 1

   dist2 = dist2 - c * delta_t

   Next a new parameter fac is calculated:

   fac = dist2/dist1

This terminates dist2 calculation. This whole process is repeated until t = 14. At regular intervals (when t is an integer) the value dist2 is stored in an array of the Excel spread sheet. This terminates the calculation of the green line.

In summary the calculation of one light ray involves a starting distance (the parameter fac) at an initial moment and a final distance (dist2) at t = 14.

The calculate of the blueline starts with an initial parameter fac=0.8 at t0. The calculation of dist2 is identical as for the green line.

The two parameters are called fach and dist2h. The letter h stands for high.

• It is important that dist2h should be greater than zero else you repeat the same process for fac = fac * 2
  
• Next you repeat the same process for fac = 0.4. The result is a final position dist2.
  Those two parameters are called now facl and dist2l. The letter l stands for low.
  It is important that dist2l should be less than zero else you repeat the same process for fac = fac / 2

  This means you have two initial distances: fach and facl, with the property:

  • That one ends above (after) the observer at t=14 This is fach at a distance dist2h.
  • That one ends below (before) the observer at t=14 This is facl at a distance dist2l.

  This means that the initial distance that reaches the observer at t=14 (distance=0) should be inbetween fach and facl.

  Next you repeat the same process for facm = (fach + facl) / 2. The result is a final position dist2m.

  When dist2m is greater than zero you replace fach by facm.

  When dist2m is less than zero you replace facl by facm.

  You repeat this whole calculation: 25 times. The last time the result is the final blue line. The final value of the parameter dist2 will be close to zero. The final value of of the parameter fac gives the initial position at t = t0.

For the blue it is important that t0 is as small as possible. That means it should be equal to delta_t.

Technical Information: redshift, z and delta l/l

In order to calculate the redshift parameter z as part of the 25 cycle the fac values are stored in an array facx at 14 regular intervals of 1 billion years.
The calculation of z goes in four steps: For example if you want to calculate z at position 7 than t0 = 7. fac = facx(7).

1. First you calculate the present position of the light ray at t0. That means you need dist1 at t0. dist2 becomes: dist2 = fac * dist1. This distance is called d10.
2. Next you calculate the position at t = 14 for that same light ray. This requires the same calculations as out lined for the green line. The result is dist2 for t = 14. This distance is called d20.
3. Next you repeat the same calculation for the light ray at t0 as above but instead of fac = facx(7) you use: facx(7) * (1+1/100). That means you change the initial position just a little. The distance dist2 calculated is called d11 at t0
4. You also repeat the same calculation for t = 14. The distance is called d21.

That means you get two distances at t0 (d10 and d11) and two distances at t=14 (d20 and d21)
The calculation of z is now straight forward: z = (d21-d20)/(d11-d10) - 1.
Or in words: "Present length at t=14"/"Initial length at t=t0" - 1. Simple comme bonjour.
The parameter z is the observed redshift or frequency shift towards the observer.

--------- 100% ----- --- - - - - - d11 ^***** - d10 ^.... ***** - | ... ***^ d21 - | ... | - | .. | - | .^ d20 ------------------------------- t=0 t=7 t=14 Figure 25

At t=0 is the Big Bang. At t=7 there is an event, a supernovae. The present moment is at t=14 The dotted line from d10 to d20 represents the blue line. The line starts at the event at t=7 and ends near the observer at t=14. The distance d20 is zero. The line with the "stars" *** going from d11 to d21 represents a light ray which starts very close to the event at t=7. This line ends at t = 14 at a distance d21 from the observer. The line segment d11-d10 represents the distance (length) between those two lines at t=7. The line segment d21-d20 represents the distance (length) between those two lines at t=14. z is defined as : (d21-d20)/(d11-d10) - 1

Reflection Light rays

• Light rays which originate near the Big Bang and that reach an observer now are the blue lines in Figure 3, Figure 5, Figure 8 and Figure 10.
  In each of these figures the horizontal axis represents the time in years (billion of years).
  The vertical axis (the x axis) the distance x from the observer (Earth/Sun) in light years (billion of ly).
  At first inspection the shape of those light rays which originate at the Big Bang look rather strange:
  They all start at the left side at t = 0 at the moment of the Big Bang at x = 0 and they end at the right side at t = 14 (the present) also at x = 0 near the Observer. In between the shape looks like a hill.
  

  To explain this behaviour let us consider what happens to the light of a star:

  in our galaxy, in Andromeda Galaxy, at 10 billion ly, at 1 billion ly, at 2 billion ly and at 10 billion ly.
  • If you consider a star in our galaxy at 1 light year than it takes 1 year for a light ray to reach the observer.
    That means the light ray (The blue line) starts at a position 1 year left from the Observer (at t=14) and 1 light year in the x direction and moves in a straight line with an angle of 45 degrees towards the observer at t=14 and x = 0.
    This is identical as the brown line but the length and time involved will be extremely short.
  • If you consider a star in the Andromeda Galaxy than the x axis is again the line connecting that star with our present position (Earth/Sun). The blue line is the same but longer because the Andromeda Galaxy is at 2.5 million light years.
    
  • Consider there is no star at that position but one at 10 million light years. The only consequence is that the blue line becomes 4 times as long but the angle stays at 45 degrees.
    
  • Consider there is no star at that position but one at 1 billion light years. The situation will slightly change. The reason is space expansion in the direction away from the observer. The result is when a light ray is transmitted 1 billion years ago the local speed is c but the global speed is slightly smaller because space expands. That in turn means that the distance at the moment of transmission was less than 1 billion light years. For the blue lines this means that its length becomes 100 times as long and starts below the brown line. The initial angle of the blue line will be less than 45 degrees and increases slowly when the light ray moves towards the observer. The maximum angle will be 45 degrees, that means the blue line is slightly curved.
    

    Consider what happens if you divide the time in 1000 increments of 1 million years. In the first increment because of space expansion the source (the star) will move a small distance dx away from the observer but the light ray will travel a distance of 1 million light years towards the observer. That means the global distance travelled by the light ray will be less than 1 million light years and the angle will be less than 45 degrees. In the second increment the source (the light ray) will again move a small distance dx away from the observer but less because space expansion will be less. This is because the distance towards the observer will be smaller but what is more important that that distance compared with the total size of the Universe at that moment (which increases) will be less. The global distance travelled by the light ray will increase slightly and the angle will also grow a little. This process will repeat its self at each increment. Both the global distance travelled and the angle will increase. At the final increment the global distance travelled will be 1 million light years and the angle will be 45 degrees.

  • Consider there is no star at that position but one at 2 billion light years. Again we have to take space expansion into account, but more. For the blue lines this means that its length becomes 2 times as long and the distance between the blue line and the brown line increases. The initial angle of the blue line decreases.
    
  • Consider there is no star at that position but one at 10 billion light years. Again we have to take space expansion into account. For the blue lines this means that its length becomes 5 times as long and the distance between the blue line and the brown line increases still more.

    Consider the case that space expansion at 10 billion light years is exactly equal to c. That means in the first increment of 1 million years the distance away from the observer will be 1 million light years (space expansion) but towards the observer also. That means the global distance will be zero, the angle = 0 and the blue line will be horizontal In the second increment the distance away will become less than 1 million, but towards the observer will stay the same. That means the light ray will start slowly to move towards the observer. You are at the top of the hill.

  • Consider there is no star at 10 billion light years but one at 12 billion light years. In that region space expansion should be larger than 1 to offset the distance travelled towards the observer.
  • Let us see what happens at t = 0, at the moment of the Big Bang. At that moment the size of space is zero. For the blue line this means its distance is zero at t = 0. For the blue line in total this means its shape becomes a hill.

  However this is not the complete picture. In order that the blue line starts from a distance zero at t=0 and reaches the observer at distance zero at t = 14, space expansion should be much larger than 1 at least a factor 1000. That means you need some form of inflation.

  • Figure 5 and Figure 10 (C = 2) show what happens if there is not enough space expansion. The point where the blue line crosses the 100% line is after roughly 6 billion years. That means you cannot observe anything at present that is younger than 2.5 billion years.
  • In Figure 3 and Figure 8 (C = 60) space expansion is also not enough to observe the Big Bang.
  • In order to see the Big Bang C has to be equal to roughly 400. The problem is the higher the value of C the more difficult it becomes to estimate the influence of the cosmological constant Lambda.

  In order to study space expansion distances and speeds are involved.
  Two types of speeds are considered: Global and Local
  

  • The Global speed is the speed at end of the Universe i.e. at the 100% line.
  • The Local speed is the speed of the light ray studied i.e. the blue line.

  The following tables shows the detail of each configuration:

Figure 26 C = 2 Lambda = 0 v0 = 3 t r max r v max v 0 0 0 2.38 8.40 1 1.65 4.24 1.10 2.82 2 2.62 5.48 0.87 1.82 3 3.44 6.04 0.76 1.34 4 4.16 6.22 0.69 1.04 5 4.83 6.14 0.64 0.82 6 5.45 5.87 0.61 0.65 Figure 26

Figure 27 C = 60 Lambda = 0 v0 = 3 t r max r v max v 0 0 0 7.30 8.28 1 5.15 4.25 3.42 2.81 2 8.16 5.49 2.71 1.82 3 10.69 6.05 2.37 1.34 4 12.94 6.22 2.15 1.04 5 15.01 6.14 2.00 0.82 6 16.95 5.88 1.88 0.65 Figure 27

Figure 28 C = 400 Lambda = 0 v0 = 3 t r max r v max v 0 0 0 13.54 8.14 1 9.73 4.26 6.42 2.80 2 15.39 5.49 5.10 1.82 3 20.14 6.05 4.46 1.34 4 24.38 6.23 4.05 1.03 5 28.28 6.15 3.76 0.82 6 31.92 5.88 3.54 0.65 Figure 28

• The r max shows the distance at the 100% line.
• The r shows the distance at the blue line.
• The v max shows the speed at the 100% line. This is the global speed.
• The v shows the speed at the blue line. This is the local speed
  • This speed is close to zero near the observer at present.
  • This speed is much larger than zero near the Big Bang

The importance of Figure 26, 27 and 28 is:

• That both the local distance (the column r) and the local speed (the column v) are the same and independent of C
• That means the path of the light ray is independent of the parameter C.
• The parameter C is important how far we can see in the past. For small values C=1 the maximum you can see is roughly 7 billion years ago. For large values C=200 you can observe the Big Bang (or close).
• That the local speed increases as a function of distance also for Lambda= 0. This means that based on friedmann's equation the expansion of space locally includes acceleration. "Locally" implies what is measured.

Figure 29 C = 60 Lambda = -0.04 v0 = 3 t r max r v max v 0 0 0 7.30 9.68 1 5.15 5.22 3.37 3.42 2 8.18 6.95 2.57 2.21 3 10.37 7.82 2.09 1.57 4 12.26 8.16 1.70 1.13 5 13.78 8.12 1.35 0.80 6 14.96 7.77 1.01 0.53 Figure 29

Figure 30 C = 60 Lambda = 0.03 v0 = 3 t r max r v max v 0 0 0 7.30 7.68 1 5.17 3.83 3.45 2.56 2 8.24 4.87 2.82 1.67 3 10.93 5.31 2.59 1.26 4 13.47 5.44 2.50 1.01 5 15.97 5.36 2.51 0.84 6 18.51 5.14 2.58 0.72 Figure 30

Figure 27, Figure 29 and Figure 30 have in common that the parameter C is identical. The difference is in the parameter Lambda.

• In Figure 29, with Lambda < 0, the Universe is closed and collapses into one point.
• In Figure 30, with Lambda > 0, the Universe is more open compared to Lambda = 0. Space expansion globally (100% line) involves acceleration.

Reflection part 2

What the simulations all have in common is a large period of roughly 4 billion years where the speed of expansion locally is larger than the speed of light c in order to observe close to the moment of the Big Bang at the present epoch.
This is in conflict with the inflation theory which assumes a large expansion of a split second after the Big Bang.

There are two more questions:

• The first question to answer is what was the temperature of the Universe 300000 years of the Big Bang ?
  Accordingly to the book "The Big Bang" at the decoupling era when the Universe was 300000 years old this was 4000 degrees. Presently this temperature is 2.7 degrees above absolute zero because the Universe has expanded.
• The second question to answer is does the Cosmic Microwave Background Radiation clearly shows an image of the state 300000 years after the Big Bang or much later ?
  The fact that it is cold means that the state that the radiation shows is more local and recent. That does mean that the origin is not 300000 years after the Big Bang.

If this assumption is correct than the conclusion that the earliest Universe we can observe is much older becomes more logical.
Light travels in a straight line (almost). It is the question if this is also true for radiation. Specific if the origin is more local than more mixing can take place because space expansion is limited.

Reflection part 3 - Question 7

The purpose of Question 7 is to establish the parameters of "the Friedmann equation" based on the SNLS data. The method used is to find the closest fit between theory and observations.
The theory used is the Friedmann equation which gives the relation between z and distance and which requires a set of parameters. The object is to calculate those parameters.
The observations are the SNLS data which gives the relation between z and magnitude.
To make the connection between distance and magnitude you need the Flux Luminosity relation. This is maybe also the weakest link. In the above document 7 different versions are discussed (also called method's)

F/L relation 7 gives the smallest error between theory and observation.
The problem with F/L relation 7 is that the smallest curvature constant k value obtained is -14 which is outside the three possible physical values (-1,0 and 1). It is also not possible to calculate the parameter C ambiguous.

What is wisdom.

Feedback

Discussion Usenet sci.physics.research:

• Is the Universe a closed system ?
• Friedmann's equation and the path of a light ray
• Inverse square vs inverse cube

Discussion Usenet sci.astro.research:

• Luminosity Fitting Friedmann equation and SNLS data
  Latest entry 13 April 2012. While studying this thread it is important to consider that there are two versions of the Friedmann equation.
  See also: Friedmann equations
  1. One versions using the parameters: Lambda, k, rho and p=0.
     This is the version used in this document. The parameter C = (8/3)*pi*R^3 * rho.
  2. One version using the parameters: omega(Lambda), omega(k) and omega(m). The sum of those parameters is 1.
  In posting #4 Phillip Helbig writes:
  
  "It is possible to get a GOOD fit with the standard Friedmann-Lemaître equation, with Omega of about 0.27 and lambda of about 0.73."
  Reinvestigation of this posting shows that he meant Omega(M) of 0.27 and omega(Lambda) of 0.73.
  See also Question 11 omega

  A related subject is: The dark matter crisis

Back to my home page Contents of This Document