Author 
Comment/Response 
Yukterez

06/04/13 8:55pm
In Response To 'Re: Re: NSolve / Solve Problem with Integral'  Ok it really might help if I tell what the calculation is about. It also might be that I didn't get the answear right, so I'll try to figure it out.
Maybe it was the confusion with a and A  both are the same variable, a of t is here exactly A of t because a is in the integral and runs from 0 to A.
So let's futher call the variable A[t] instead of a[t] (Originally, the notation is Integrate[f[a'], {a',0,a}] but Mathematica does not take ' so i take Capitals.)
Anyway, the calculation is about the size of the universe by time. The variable "a" is the dimensionless scalefactor, "t" is the cosmological time since the big bang.
The expansion rate of the universe, "H[a]", a funtion of the scalefactor, is
H0*Sqrt[ΩR*a^4+ΩM*a^3+(1ΩRΩMΩΛ)*a^2+ΩΛ]
where H0 ist the expansion rate today in m/sek/m and the Omegas are the density parameters for radiation, matter, curvature and dark energy and are dimensionless.
With that you can solve for the time "t" by the scalefactor "a", or in another notation, for t[a], which is evaluated by the formula
t = 1/H0 NIntegrate[1/(a ea), {a, 0, A},
Method > {"GlobalAdaptive", "MaxErrorIncreases" > 10000,
Method > "GaussKronrodRule"}, MaxRecursion > 10000]
which I can plot over A (also the scalefactor, but integrationvariable of it) with
Plot[{tA}, {A, 0, 2}}}]
Now I have the specific age of the universe by its size.
If I want to get the size of the universe by its age, it is not that easy. I tried to make a diagramm like the old WMAP diagramm at
http://imageshack.us/a/img842/1913/expdiag1750.jpg
which I finally did, but with an analytical simplification that can only handle two Omegas. That is still fine since ΩR is very small (the plot would need to have more than 1000 px height to have 1 pixel difference) and ΩK is 0 is most modells.
The result is now at
http://yukterez.enabled.io/horizonte_nach_zeit.png
and some other, more special modells, at
http://yukterez.enabled.io/forum/phpBB3/viewtopic.php?t=43&f=7
Now I try to make a plot like the horizondiagramm above with modells that have other forms of dark energy (ΩΛ) than constant. Therefore I need to solve for a[t] and apply the formulas I collected in the forum link.
I try if I can do that with your advice, but until now I could not Plot[]
URL: , 
