### momentum of particles in FLRW universe

The expression for the line element in a FLRW universe is

The geodesic equation, used to describe the kinematics of particles or photons in the manifold is

Let’s look at $\frac{dt}{ds}$ specifically,

say $u^{\theta}$ and $u^{\phi}$ are zero

And if you compute all of the relevant christoffel symbols i.e $\Gamma$s, you will see that $\Gamma^{t}_{rr}$ is the only non-zero component.

and since we know that $u^r = \frac{dr}{ds}$

If we substitute the value of $\Gamma^t_rr$, we get

Now, we also know that $u_{\mu}u^{\mu} = 1$ for particles.

Substituting the above expression in the expression below, we get

say

we wrote earlier that $u^{\mu}u_{\mu} = 1$

where $i$ represents the spatial components.

where $|p|^2 = \sigma_{ij}p^i p^j$.

Note that we are working in units of $c=1$. If I’m missing factors anywhere, I apologize.

We basically get that the spatial momentum of any free particle in an FLRW universe is inversely proportional to the scale factor $a(t)$ and since $a(t)$ is always positive and increasing, the momentum keeps decreasing with time.

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### Animation using GNUPlot

Animation using GNUPlotI've been trying to create an animation depicting a quasar spectrum moving across the 5 SDSS pass bands with respect to redshift. It is important to visualise what emission lines are moving in and out of bands to be able to understand the color-redshift plots and the changes in it.
I've tried doing this using the animate function in matplotlib, python but i wasn't able to make it work - meaning i worked on it for a couple of days and then i gave up, not having found solutions for my problems on the internet.
And then i came across this site, where the gunn-peterson trough and the lyman alpha forest have been depicted - in a beautiful manner. And this got me interested in using js and d3 to do the animations and make it dynamic - using sliders etc.
In the meanwhile, i thought i'd look up and see if there was a way to create animations in gnuplot and whoopdedoo, what do i find but nirvana!

In the image, you see 5 static curves and one dynam…