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


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.

| cite | improve this answer | |

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.