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On my lecturer's notes on Dark Matter I am told:

"It is customary to define densities normalised by the time dependent volume $V (t) = a(t)^3$. The reason for this is that, in the absence of number changing processes, the comoving number density remains constant with time evolution."

I don't understand what is meant by a number changing process.

The image is given as an example of the preservation of the comoving number density.

Initially I thought a number changing process was one in which the number of the particles changed as the process evolved but I believe the image is trying to emphasise that the number of particles is kept constant. enter image description here

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You are right. The number of particles does not change, but the volume increases as $V(t) = V_0 a^3(t)$, because of expansion. Naively one can compute the number density as $$ n(t) = \frac{N(t)}{V(t)}=\frac{N_0}{V_0 a^3(t)}, $$ where I used the fact that $N(t)=N_0$ in the absence of process that change the number of particles.

However, if we spice up things by normalizing the volume by the expansion factor $a^3(t)$ in the previous expression, we have $$ \hat{n}(t) = \frac{N_0}{V_0 a^3(t)/a^3(t)}= \frac{N_0}{V_0}, $$ which is indeed the definition of comoving number density and it is manifestly time-independent.

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