# AssembiasMBK10Sats¶

class halotools.empirical_models.AssembiasMBK10Sats(**kwargs)[source]

Bases: AssembiasZheng07Sats

Power law model for the occupation statistics of satellite galaxies, introduced in Kravtsov et al. 2004, arXiv:0308519. This implementation uses Zheng et al. 2007, arXiv:0703457, to assign fiducial parameter values, with a second moment defined by a negative binomial distribution, as in Boylan-Kolchin et al. 2010, arXiv:0911.4484, and uses the Decorated HOD to incorporate galaxy assembly bias.

In contrast to MBK10Sats, here the non-Poissonian fluctuations are implemented at fixed primary and secondary halo properties. Note that even in the parent class AssembiasZheng07Sats, fluctuations in satellite occupation statistics are non-Poissonian at fixed primary halo property, which is a generic feature of assembly bias.

The behavior of a negative binomial distribution is controlled by two parameters, n and p. The relationship between these parameters and the mean and variance is as follows:

$$p = \frac{\mu}{\sigma^2}$$

$$n = \frac{\mu^2}{\sigma^2 - \mu}$$

In MBK10Sats, the behavior of the deviations from Poisson behavior is controlled by the parameter nsat_up0, which can take on any value on the real line. The parameter nsat_up0 is related to the negative binomial distribution parameter p by a simple sigmoid transformation, which enforces the mathematical constraint $$0<p<1$$

For any Poisson distribution, the following relationship between the first and second moments holds:

$$\sigma^2 = \mu$$

In MBK10Sats, changing the parameter nsat_up0 has no effect on the first moment, but the second moment is altered as shown in the figure below.

In terms of two-point correlation functions, deviations from Poisson fluctuations only influence satellite-satellite pair counts, and so the parameter nsat_up0 will strictly influence clustering in the 1-halo term, and the effects will be larger in galaxy samples with larger satellite fractions.

Parameters:
thresholdfloat, optional

Luminosity threshold of the mock galaxy sample. If specified, input value must agree with one of the thresholds used in Zheng07 to fit HODs: [-18, -18.5, -19, -19.5, -20, -20.5, -21, -21.5, -22]. Default value is specified in the model_defaults module.

prim_haloprop_keystring, optional

String giving the column name of the primary halo property governing the occupation statistics of gal_type galaxies. Default value is specified in the model_defaults module.

Methods Summary

 mc_occupation([seed]) Method to generate Monte Carlo realizations of satellite abundance non_poissonian_p(**kwargs) Parameter controlling the variance of the negative binomial distribution std_occupation(**kwargs) Standard deviation of the occupation statistics (square root of the second moment)

Methods Documentation

mc_occupation(seed=None, **kwargs)[source]

Method to generate Monte Carlo realizations of satellite abundance

Parameters:
prim_haloproparray, optional

Array of mass-like variable upon which occupation statistics are based. If prim_haloprop is not passed, then table keyword argument must be passed.

tableobject, optional

Data table storing halo catalog. If table is not passed, then prim_haloprop keyword argument must be passed.

seedint, optional

Random number seed used to generate the Monte Carlo realization. Default is None.

Returns:
mc_abundancearray

Integer array giving the number of galaxies in the input halos

non_poissonian_p(**kwargs)[source]

Parameter controlling the variance of the negative binomial distribution

Parameters:
prim_haloproparray, optional

Array storing a mass-like variable that governs the occupation statistics. If prim_haloprop is not passed, then table keyword arguments must be passed.

tableobject, optional

Data table storing halo catalog. If table is not passed, then prim_haloprop keyword arguments must be passed.

Returns:
pfloat or array
std_occupation(**kwargs)[source]

Standard deviation of the occupation statistics (square root of the second moment)

Parameters:
prim_haloproparray, optional

Array of mass-like variable upon which occupation statistics are based. If prim_haloprop is not passed, then table keyword argument must be passed.

tableobject, optional

Data table storing halo catalog. If table is not passed, then prim_haloprop keyword argument must be passed.

Returns:
std_nsatarray

Standard deviation of the number of satellites in the input halos.

Notes

At fixed value of the non_poissonian_p parameter nsat_up0, the quantity x = std_occupation**2 / mean_occupation is a constant. Increasing up0 will decrease x with no change to mean_occupation. In the limit of infinite up0, x approaches the Poisson value of unity