In this tutorial we'll use Galacticus to compute and output solutions to the excursion set problem, which is an important ingredient in Press-Schechter-like halo mass function calculations. Briefly, the problem is to consider a set of random walk trajectories in overdensity, δ, as a function of variance (of the cosmological density field), S, and the compute the fraction, f(S)d_S_, of these trajectories which make their first upcrossing through a barrier, B(S), between S and S+d_S_.

This tutorial assumes that you've previously completed some of the earlier tutorials, so are familiar with the basics of Galacticus usage. If you haven't a good place to start is with the halo mass function tutorial.

A more mathematical and detailed discussion of the excursion set problem, and how it is implemented in Galacticus can be found here.

To run this tutorial:

$ ./Galacticus.exe parameters/tutorials/excursionSets.xml

If everything is working you should see output which looks something like:

              ##                                     
   ####        #                  #                  
  #   #        #             #                       
 #       ###   #  ###   ### ###  ##   ### ## ##   ## 
 #       #  #  #  #  # #  #  #    #  #  #  #  #  #   
 #   ###  ###  #   ### #     #    #  #     #  #   #  
  #   #  #  #  #  #  # #     #    #  #     #  #    # 
   ####  #### ### ####  ###   ## ###  ###   #### ##  

 © 2009, 2010, 2011, 2012, 2013, 2014, 2015, 2016,
   2017, 2018, 2019, 2020
   - Andrew Benson

M: Memory:         code +      nodes +       misc =      total
M:            22.246Mib +   1.000  b +   9.000  b =  22.246Mib
M: -> Begin task: excursion sets
M: <- Done task: excursion sets

and a file excursionSets.hdf5 will have been created.

You can look at the entire parameter file for this tutorial here. Below we'll explore just those sections which are specific to this tutorial.

  <!-- Specify tasks to perform -->
  <task value="excursionSets"/>

This block tells Galacticus what task we want it to perform - here we're just asking it to solve the excursion set problem.

  <excursionSetBarrier       value="criticalOverdensity"/>
  <excursionSetFirstCrossing value="linearBarrier"      />

The first option specifies the form of the barrier function, B(S). Here we choose B(S)=δ_c where δ_c is the critical overdensity for collapse of a dark matter halo.

The second option selects the solver to use to compute the distribution of first crossings of the barrier. Here we choose a solver which utilizes the analytic solution for a linear barrier, i.e. a barrier of the form B(S) = a + b S.

The output file excursionSets.hdf5 has the excursion set problem solutions in the excursionSets group:

$ h5ls excursionSets.hdf5/excursionSets
barrier                  Dataset {15, 51}
firstCrossingProbability Dataset {15, 51}
firstCrossingRate        Dataset {15, 51, 51}
mass                     Dataset {51}
massFunction             Dataset {15, 51}
powerSpectrum            Dataset {15, 51}
time                     Dataset {15}
variance                 Dataset {15, 51}
wavenumber               Dataset {51}

These datasets contain the following information:

• mass: Halo mass [M_☉];
• time: Cosmic time [Gyr];
• wavenumber: Wavenumber corresponding to this halo mass [Mpc^-1];
• powerSpectrum: Power spectrum at this wavenumber [Mpc³];
• variance: The variance, S(M) = σ²(M), at this halo mass;
• barrier: The excursion set barrier, B(S);
• firstCrossingProbability: The probability of first crossing this barrier between S and S+d S;
• firstCrossingRate: The rate of first crossing of the barrier per unit time [Gyr^-1] for all pairs of halo mass;
• massFunction: The halo mass function [M^-1_☉ Mpc^-3].