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 specifically,
say and are zero
And if you compute all of the relevant christoffel symbols i.e s, you will see that is the only non-zero component.
and since we know that
If we substitute the value of , we get
Now, we also know that for particles.
Substituting the above expression in the expression below, we get
we wrote earlier that
where represents the spatial components.
Note that we are working in units of . 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 and since is always positive and increasing, the momentum keeps decreasing with time.
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