Wednesday, April 3, 2013

limitations of the NFW-profile shape

the NFW-profile is an approximation to the density distribution inside cold dark matter structures. where does the NFW-approximation break down and for what reason? if there was annihilation of dark matter, what could you say about the central brightness?

bonus question: where is the first minimum of $x^{\sin(x)}$, $x>0$?


  1. the NFW-profile shape needs to break down at small radii because the density diverges $\propto 1/r$ and it needs to be truncated at large radii because the total mass diverges logarithmically. if there was CDM annihilation with a luminosity density $\propto\rho^2$ you'd get infinite integrated luminosity if the profile were just a bit steeper than $1/r$ in the centre.

  2. The bonus question:
    $$=> f'(x)=\frac{sin(x)*cos(x)}{x}*f(x)$$
    $$f(x)> 0 for$$ $$x > 0$$
    $$f'(x)= 0 => \frac{sin(x)*cos(x)}{x}=0$$
    $$=> sin(x)*cos(x)=0$$
    so there first minimum is at $$x = \frac{\pi}{4}$$