One of the chalenges of cosmological parameter estimation is marginalizing out the nuisance parameters such as the parameters that model the connection between galaxies and dark matter. For such marginalizations we are required to evaluate this conditional probability p(galaxy properties | dark matter halo properties). The list of galaxy properties consists of stellar mass, star formation rate, etc and the list of dark matter halo properties consists of mass, maximum circular velocity, etc.

In order to evaluate these we can either use parametrized empirical models or expensive hydrodynamic simulations. The parametrized models are not physically motivated by they can be inserted in a cosmological inference setup. The hydro simulations are realistic and physically motivated but it is not tractable to include these inside a cosmological inference framework.

We propose a solution for this problem by modeling the conditional probability p(galaxy properties | dark matter halo properties) with a mixture density network.

Mixture Density Network

A mixture denisty network models a conditional probability p(y|x) by a Gaussian Mixture Model where the parameters of the mixture model are some nonlinear funcitons of x. This nonlinear function is given by a deep neural network:

At the bottom of the graphical model shown above we have the dark matter halo properties and at the top of the model we have the astrophysical parameters.

Application to a complex hydrodynamic simulation

We applied this model to a realistic simulation of the universe where a list of dark matter properties were passed as input to the model and a list of galaxy properties were passed as output to the model. The modern learns the probability distribution of galaxy properties conditioned on dark matter halo properties. Here is an example:

In the contour plot shown here, we see the joint probability distribution of the stellar mass of galaxies and the maximum circular velocity of dark matter halos. The empty(filled) contours correspond to the network prediction(simulation).

We see that the mixture denisty networks provide a fast way of evaluating the hard-to-evaluate conditional probability inside our cosmological inferences.