Question 1  What is the path of a light ray when the Cosmological Constant (Lambda) = 0 (Open Universe ) 
Question 2  What is the path of a light ray when Lambda < 0 (Closed Universe) 
Question 3  What is the path of a light ray when Lambda > 0 (Open with acceleration) 
Question 4  What is the path of a light ray when Lambda = 0 and k=1 or k=1 
Question 5  For the above three conditions is it possible to validate Hubble's Law? Lemaître's Law? 
Question 6  What is the relation of the value 1/H versus the age of the Universe. 
Question 7  What is the relation between Friedmann's Equation and ordinary matter, Dark matter and Dark energy ? 
Question 8  What is the current state to calculate the parameters C, Lambda and k, including the age of the Universe based on observations. 
Question 9  What are the implications when the age of the Universe is 28 or 42 billion years. 
Question 10  How important is the deceleration parameter q ? 
Question 11  What are the present values for omega(M), omega(Lambda) and omega(K) ? 
Question 12  What is the Cosmic Coincidence Problem related to 1/H0 and the Age of the Universe ? 
Question 13  Is it possible to calculate the paramaters H0, omega(Lambda), Omega(M) alone using the CMB radiation ? 
Question 14  What is the influence of the parameter V0 (Which defines the first step size at t=0) in relation to the inflation theory. 
The age of the Universe is considered 14 billion years. This is also the value of parameter "age".
In order to simulate space expansion the Friedmann equation is used:
(See "Introducing Einstein's Relativity" by Ray d'Inverno. Equation 22.58 and 23.1
It should be understood that the parameter Lambda used in this document and the parameter Omega(Lambda) used in the Questions 11 and 13 are different parameters.
To understand the above equation better start from the first equation after : Friedmann equations  Wikipedia and replace both c and G with 1. You get than:
In addition to R(t) also two additional functions are implemented: a(t) and b(t)




The following table shows the most important results of the simulations for three different values of C: 2, 60 and 400.
The First Column shows the time in billion of years, or the distance along the x axis in billion of light years.








What is necessary is the ralation between z and r.
The following table shows the most important results of the simulations with C= 60 for two different values of Lambda: 0,03 and 0,06.
The First Column shows the time in billion of years, or the distance along the x axis in billion of light years.





Table 2d shows the results for C = 60 and different values of Lambda



In order to study the most recent values study this link: Friedmann L=0.01155 Table 2E shows the results for Lambda = 0.01155 and different values of C














Table 5 shows the Hubble constant H0 (Hubble parameter H at t=0) calculated over a distance of 100 million light year near the observer for different combinations of the parameters Lambda and k. The parameter C = 60.
The parameter 1/H0 is also shown.















It is also interesting to study the evolution of the redshift or parameter z.


The following table shows the result of a simulation. The maximum age of the Universe is 18. The parameters C, Lambda and k are respectivily: 60, 0.011288 and 0.

The current accepted position is that the age of the Universe is equal to the Hubble time. See Reflection part 4  Question 6 for more detail.
The following table is used to test the relation: E(z) = H(z)/H(0) = sqr(rho c(z)/rho c(0))
"rho c" is the critical density

Interesting reading is the following document: Literature (14) 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.
What Figure 5 shows is the relation between magnitude and z based on SNLS observations. What we want is the relation between distance and z (or reverse). To calulate the distance based on magtitude you need the Flux/Luminosity relation. The assumption is that the Luminosity (L) is the same for all supernova. The details are discussed below
Using Figure 5 as a mask and using the F/L relation it is possible to calculate the distance based on observations of z. Using the Friedmann equation and based on a certain combination of the parameters parameters C, Lambda and k also a distance is calculated as a function of z. Comparing those two curves it is possible to calculate an error.
With different combinations of the parameters C, Lambda and k it is possible to find the smallest error (the closest fit between the two) and the best values for C, Lamba 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(M), omega(Lambda) and omega(k). See Answer: Question 11 
In the next 7 paragragh seven different FL relations between magnitude and distance (d) are discussed.


  0,94  3,055  0,00505435  1,78747 
21  1  2,870  0,00505605  1,73305 



0,91  6,808  0,00164070  1,81668  
21  1  6,200  0,00164190  1,73305 




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 




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 








The table shows that the optimum value for Lambda = 0.0634
The following table shows the parameter alpha as a function of error.
The parameter Alpha = 0.163 The parameter Beta = 0.2197









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/H value of 13,739 billion years. This is equivalent with the Hubble parameter 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.






















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.




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

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/H value of 13,718 billion years. This is equivalent with the Hubble parameter 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.


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




Literature (19) "Solves" the coincidence problem by introducing a variable Constants. In fact what they do is to change the friedmann equation.
To study the Cosmic Microwave BackGround Radiation a similar problem exists.
The object is to calculate the three parameters omega(m), omega(Lambda) and the Age of the Universe as a function of the Back Ground Radiation. In order to do that the program CAMB is used. For a copy select: CAMB. To study the listing select: CAMB_all_html
The program CAMB calculates the power spectrum as a function of H0, omega(k), omega(baryon)*h^2, omega(CDM)*h^2 at t=0 and other parameters like Tcmb. with h = H0/100.
Using omega(m)*h^2 = omega(baryon)*h^2 + omega(CDM)*h^2 , the program calculates omega(m).
Using omega(m) + omega(Lambda) = 1 (assuming omega(k) = 0 ) the program calculates omega(Lambda)
The final step is to calculate the Age of the Universe with the Friedmann equation.
The strategy is to try different values for H0, omega(baryon) and omega(CDM) and to compare the result with the power spectrum calculated from the Background radiation. The result is also a power spectrum. The optimum values are the one with the smallest error value between the two spectra
A simpler strategy is to use H0 and omega(m) only.
What ever the strategy the most important problem is the calculation of the power spectrum as a function of the three parameters
H, omega(baryon)*h^2 and omega(CDM)*h^2 at present. How do we know that those functions (equations) are correct. The problem is that this is very difficult, even impossible . The equations are partly a description of processes (conditions) which should have happened in 3D in the very early universe as a function of parameters we can measure now. What we measure is the Cosmic MicroWave Back Ground Radiation, photons released from that early period. The problem is that we have no way to demonstrate that those equations are a correct description meaning that the parmeters at t=0 are correct.
A typical case is the parameter Tcmb (Temperature CMB).
The standard value is 2.725 Using that value and with H0 = 70 the program calculates that the age of the Universe is 13.738 billion years. With Tcmb = 3 and with the same value for H0 and omega(m)*h2 the program calculates that the age of the Universe is 13.736. The two power spectra's with Tcmb=2.725 and with Tcmb=3 are clearly different. It is also clear that the first closely matches the power spectrum calculated from the observed CMB radiation (Observed Power Spectrum = OPS) but that does not mean that the two calculations involved are correct. If we know that Tcmb=2.725 is the measured value of the CMB radiation by some other means than we can calibrate the equations such that the calculated power spectrum with Tcmb=2.725 matches the OPS. But that still does not mean that the power spectrum calculated with T=3 is correct. There is no way how to test that, because the only spectrum we have is the OPS. And that the Observed and calculated Power Spectra are different.
For more detail information about the problems involved cheque the above link.
The following table shows the result of simulations with the following standard parameters:
C = 60, Lambda=0,01155, k = 0, age = 13,74
For a general explanation of these parameters see: Friedmann Lambda=0.01155


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


Consider what happens if you divide the time in 1000 increments of 1 million years. 
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. 





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.
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.
The table on the right shows the name of the Galaxy, the distance, z and H0 for 30 measurements. Many of the Galaxies show two measurements: a minimum and a maximum The two columns at the right side shows the minimum and the maximum values. The bottom line shows the average values. The results are 65,8 and 78,4 

Accordingly to Wikipedia Age of the Universe  History
and to Nasa: How old is the Universe it is possible to calculate the age of the Universe solely on WMAP data.
Both documents do not make a clear distinction that 1/H0 and the age of Universe are two different parameters.
Literature (23) at page 10 Table 1 shows 7 parameters (Omaga(b)*h^2, Omega(CDM)*h^2, Omega(Lambda), Delta R^2,nz, tau and Asz) which can be calculated at high accuracy using WMAP data. The two parameters t0 and H0 are indicated as derived parameters in that document.
See also Question 12 Coincidence problem.