I am reading a text which talks about the WIMP speed distribution in the galactic halo in the frame of the Sun and Earth. The point where I am stuck it is trying to explain the concept of Gravitational Focusing of WIMPs at the location of the Earth due to the gravitational well of the Sun.

While talking about the speed distribution, it then moves to take the full phase space density and attempts to write everything in those terms. The relation it uses looks like this, $$\text{Phase space density} = \tilde{f}(v)=n_v f(v)$$ Here, $f(v)$ is the speed distribution function and $n_v$ is said to be the phase space number density.

From what I understand about these terms, and what I have been able to gather: $f(v)dv$ is the probability that a certain particle, out of $N$, has a speed between $v$ and $v +dv$. Moreover, $\tilde{f}(v)d^Nv$ is the probability that the state of the system, the speed of all N particles, lies in a cube of $d^Nv$. What is referred here as the "phase space number density $n_v$" is still unclear to me and I am not able to find it's definition.

From what I understand, if I try to write the phase space density in terms of the speed distribution I would write it something like this: $f(v_1)dv$ is the probability of finding a particle in the speed range $v_1$ and $v_1 +dv$ and same goes for $f(v_2)dv$ and so forth for all $N$ particles. The probability of finding one particle in speed range $v_1$ and $v_1 +dv$ and then particle 2 in speed range $v_2$ and $v_2 +dv$ and then so forth should be then $f(v_1)f(v_2)f(v_3)...f(v_N)d^Nv$ which should be just $\tilde{f}(v)d^Nv$. But I can't understand how it is reduced to the relationship given in the literature i.e. $\tilde{f}(v)=n_v f(v)$, if I am doing it right.

So am I understanding the terms "speed distribution" and "phase space density" correctly? And if yes then is my calculation correct? And finally, what is the relationship $\tilde{f}(v)=n_v f(v)$ and what is phase space number density $n_v$?


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