## Wednesday, June 20, 2012

### haloes of all sizes

what are the largest and smallest objects (a.k.a. dark matter haloes) that can form in a given cosmology? which processes determine their sizes?

bonus question: what's $\log_i(\mathrm{e})$ with $i=\sqrt{-1}$

1. Since the $\log$ can also be defined for complex numbers, we can use
$$\log_i{e}= \frac{\log_e{e}}{\log_e{i}} = \frac{1}{\log_e{e^{i\pi/2}}} = \frac{2}{i \pi} = -i \frac{2}{\pi}$$
2. the size of the smallest dark matter haloes that can form is determined by the free-streaming length of the CDM particles, which causes the CDM spectrum to cut off exponentially. the largest objects need to collapse within a Hubble-time... at least in SCDM. the exponential expansion caused by dark energy or $\Lambda$ stops structure formation altogether, so one should rather say that the largest objects need to collapse before the expansion dynamics is dominated by dark energy.