Annihilation Signal Scaling

For edits on my annihilation paper, I have a problem with how I’m rescaling my profiles, so I’m going to record my calculation here.

Rescale profiles

In particular, my annihilation signal is in units $M_{\rm unit}^2 / r_{\rm unit}^3$, where $r_{\rm unit}=r_s$ and $M_{\rm unit}=M(<10 r_s)$ (assuming an NFW profile).

Therefore

$$\dfrac {M_{\rm unit}^2} {r_{\rm unit}^3} = \dfrac {M (10 r_s)^2 } {r_s^3} = \dfrac {M (10 r_s)^2 c^3 } {r_{\rm vir}^3}$$

And then, using $200 \rho_c = M_{\rm vir} / (4/3 \pi r_{\rm vir}^3)$,

$$\dfrac {M_{\rm unit}^2} {r_{\rm unit}^3} = \dfrac {4\pi \times 200 \rho_c M (10 r_s)^2 c^3 } {3 M_{\rm vir}} = \dfrac{4}{3}\pi \times 200 \rho_c M_{\rm vir} c^3 \left( \dfrac {M(10 r_s)} {M_{\rm vir}} \right)^2$$

Concentration–mass relation

Now I need a way to choose a concentration for a halo of a given mass. I’m going to just assume it lies on the concentration–mass relation. I’ll use one of the Colossus functions. here. I chose the Ishiyama 2021 model, with the $200\rho_c$ definition.

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