# Presentation: 488476

## Dark Matter Halo Mergers & Smoluchowski's Equation

*Andrew Benson (Caltech)**‏*

## Overview

- Press-Schechter theory and extensions
- Progenitor mass functions
- Merger trees
- Problems
- Smoluchowski's coagulation equation
- What is it?
- Solutions in simple cases
- Simple analytic forms
- Fits
- Mass function evolution

- Dark Matter Halo Merging

- March 11, 2008

## What is Press-Schechter?

- An analytical model for the distribution of dark matter halo masses in hierarchical Universes
- Based on the statistics of peaks in Gaussian fields
- First derived by Press & Schechter (1974), extended in early 90's by Lacey & Cole, BCEK, Bower etc.
- Predicts halo mass function:
- Rate of change of mass function easily determined

- Dark Matter Halo Merging

- March 11, 2008

## Extended Press-Schechter (ePS)‏

- Theory can be manipulated to predict progenitor mass functions
- Agree reasonably well with N-body simulations
- Leads to predictions for merger rates of halos...

- Somerville et al. (2001)‏

- Prediction for halo merger rate: Γ = n(M1)n(M2)QePS(M1,M2;t)‏
- Obvious problem with this:

- Dark Matter Halo Merging

- March 11, 2008

## Merger Trees

- Halos form through merging of sub-units
- Process of merging described by merger trees
- Extended Press-Schechter can be used to construct these statistically

- Dark Matter Halo Merging

- March 11, 2008

## Uses of Press-Schechter

- Press-Schechter (and in particular ePS) is used extensively in studies of cosmology and galaxy formation
- Galaxy evolution
- Galaxy morphology
- AGN activity
- Lyman-break galaxies
- Abundance of binary SMBHs...
- ....and resulting event rate for LISA
- Formation of the first stars
- Substructure in Galactic halos
- Reionization of the Universe
- Halo angular momenta and concentrations
- Particle acceleration in clusters
- Formation redshifts of clusters

- Dark Matter Halo Merging

- March 11, 2008

## The Problem with ePS

- Problem is that ePS predicts *two different* merger rates!
- Rates are similar for equal mass mergers, but very different for larger mass ratios
- Can affect many calculations using ePS

- Dark Matter Halo Merging

- March 11, 2008

## Why is There a Problem?

- Press-Schechter doesn't deal with discrete halos
- Halo mass is a continuously varying function of position
- So, it doesn't really incorporate halo mergers at all
- (Also, filtering is not spatially localized)‏

- Dark Matter Halo Merging

- March 11, 2008

## Smoluchowski's Coagulation Equation

- Smoluchowski (1916) writes a simple equation governing coagulation processes, e.g.:
- Formation of polymers, growth of planetisimals, hierarchical formation of halos
- Halos are created by mergers of lower mass halos...
- ...and destroyed by mergers with other halos
- Net rate of change of abundance of halos is:

- Dark Matter Halo Merging

- March 11, 2008

## Results: Merger Rate Functions

- Merger rate functions are symmetric and smooth
- Similar form to symmetrized ePS merger rates
- Can be parameterized by a simple fitting formula

- Dark Matter Halo Merging

- March 11, 2008

## CDM Power Spectra.............

- Apply same techniques to CDM power spectra
- Used same regularization conditions
- Improved solver with improved dynamic range
- Can find solutions to Smoluchowski's equation......
- ........but do not agree with N-body simulations!
- The smoothest solution is not the correct one
- Tried other regularization conditions, but no real success

- Dark Matter Halo Merging

- March 11, 2008

## Modified ePS Merger Rates

- Recently proposed empirical modifications to ePS
- Tuned to match N-body progenitor mass functions
- Parkinson, Cole & Helly (2007) method:

- In Smoluchowski terms, just multiplies the merger kernel

- Dark Matter Halo Merging

- March 11, 2008

## Fits to Conditional Mass Functions

- Dark Matter Halo Merging

- March 11, 2008

## Constraints from Smoluchowski

- Additional constraint on 3 parameters from Smoluchowski equation
- Slice through parameter space at fixed 2
- Highly degenerate
- Consistent constraints from N-body and Smoluchowski

- Conditional MF (N-body)‏
- Smoluchowski

- Dark Matter Halo Merging

- March 11, 2008

## Solutions....

- Difference in rate of change of mass function per halo
- G() functional form does not admit an exact solution
- We weight towards high masses (fastest change)‏
- Our fits to conditional mass functions...

- Dark Matter Halo Merging

- March 11, 2008

## Constraints from Smoluchowski

- Evolve Sheth-Tormen mass function over large redshift intervals
- New fits work extremely well, even for very low abundance halos
- Other functional forms for G() could work better still

- Dark Matter Halo Merging

- March 11, 2008

## Merger Tree Factory: Orbits

- What are orbital properties of just merged halos?
- How to find them
*N-body simulation*
*Find all the halos*
*Find those about to merge*
- Distribution of orbital velocities
- Measured to very good precision

- Dark Matter Halo Merging

- March 11, 2008

## Halo Mass-ConcentrationDan Grin & AJB

-  Halo concentrations well measured
*Millennium Simulation (Neto et al., arXiv:0706.2919)**‏*
*C(M,z)**‏*
-  Scatter larger than current analytic models predict (<30%)‏
-  Can we rectify this by considering complete merging history?
-  Currently testing this hypothesis using Millennium Simulation data

- Dark Matter Halo Merging

- March 11, 2008

## Summary

- Merger trees ubiquitous in structure/galaxy formation work
*Need to be accurate*
*N-body trees good, but still very limited*
- Extended Press-Schechter doesn't work
*Inconsistencies*
*Inaccurate*
- Coagulation equation
*Can solve directly, but what regularization to use?*
*Using simple analytic forms a better/easier approach*
- High-accuracy merger trees possible
*Can construct trees back to z**≥**4 maintaining mass function*

- Dark Matter Halo Merging

- March 11, 2008