Parameters

sigma_min:

The standard deviatiation of density fluctuations on the scale corresponding to the minimum mass for a halo to be counted.

delta_crit:

The critical (over)density of collapse.

sigma_mass:

The standard deviation of density fluctuations on the scale corresponding to the mass of the region under consideration. Use zero to consider the entire universe.

delta:

The overdensity of the region under consideration. Zero corresponds to the mean density of the universe.

Notes

The fraction of the mass in a region of mass m that has already collapsed into halos above mass m_min is:

f_\mathrm{col} = \mathrm{erfc} \left[ \frac{\delta_c - \delta(m)}
{ \sqrt {2 [\sigma^2(m_\mathrm{min}) - \sigma^2(m)]}} \right]

The answer isn't real if sigma_mass > sigma_min.

Note that there is a slight inconsistency in the mass used to calculate sigma in the above formula, since the region deviates from the average density.

Examples

>>> import numpy
>>> import perturbation as cp
>>> cosmo = {'omega_M_0' : 0.27,
...  'omega_lambda_0' : 1-0.27,
...  'omega_b_0' : 0.045,
...  'omega_n_0' : 0.0,
...  'N_nu' : 0,
...  'h' : 0.72,
...  'n' : 1.0,
...  'sigma_8' : 0.9,
...  'baryonic_effects' : False
...  }
>>> fc = cp.collapse_fraction(*cp.sig_del(1e4, 0, **cosmo))
>>> print round(fc, 4)
0.7328
>>> fc = cp.collapse_fraction(*cp.sig_del(1e2, 0, **cosmo))
>>> print round(fc, 4)
0.8571