# easy(?) cosmology question: what to integrate to get R(t)

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### Phillip Helbig (undress to reply)

Jan 19, 2020, 6:05:06 PM1/19/20
to
When discussing Friedmann models, the usual approach is to start with
the Friedmann equation then express dr/dt as a function of R, Omega,
lambda, and so on, then re-arranging it to get an expression for dt as a
function of R (scale factor), Omega, and lambda, which can be integrated
to give the lookback time (or time since the big bang) as a function of
the scale factor. This can be re-arranged to express it as a function
of redshift, then one can compute distances as a function of redshift
and so on.

This is standard stuff.

But what about an expression which one can integrate to get R(t), the
scale factor as a function of time?

For special case, one can invert t(R) to get R(t). Of course, if I
numerically calculate t(R), I can invert it to get R(t). But what I
would like is an expression for dR as a function of t which I can
integrate from 0 to t to get R(t), just like I have an expression for dt
as a function of R which I can integrate from 0 to R to get t(R).

Does such a thing exist?

Formally, dR/dt = f(R;Omega,lambda) (with H as a scale factor). The
usual approach is then dt = dr/f, which I can integrate from 0 to R to
get t, which is fine because f = f(R). Algebraically, I can also write
dR = f(R)dt, but I can't integrate it from 0 to t to get R, since f is a
function of R, not of t.

Another way to ask the question: I give you arbitrary Omega
(Omega_matter) and lambda (Omega_Lambda) and ask you to make a plot of
R(t). How would you do it?

I'm pretty sure that one can do it with Jacobian elliptic functions
(essentially the inverse functions of Legendre elliptic integrals).
This is more elegant, and faster, but more difficult to implement.
However, I want something simpler: a function which I can integrate
numerically to get R(t) for arbitrary lambda and Omega. Any ideas?

### hapor...@gmail.com

Apr 30, 2020, 1:56:08 PM4/30/20
to
Analytic solution comes in parametric form which contains elliptic
integral of first kind and other part simple trigonometric function.
Analytic solution can be found in my postings in sci.physics.relativity.
I don't remember date when I posted it, it was couple years ago. (my old
email was h*******@luukku.com).

Best Regards, Hannu Poropudas