by F. van den Bosch, slides here.

- Press - Schechter theory:
- filters of density fields: Gaussian, top-hat, sharp k-space filter (same, as top-hat, except we end up having a nice box in Fourier space)
- mass variance
- halo mass function: how can we assign halo mass to a collapsed region?
- PS: only half of the mass of the Universe can become collapsed (overdense), but due to smoothing, this is not entirely correct. fudge factor of 2
- halo mass function: many more small haloes than higher, exponential suppression

- Excursion set (EPS) formalism (Bond et al, ~1991, statistics of peak heights):
- excursion: locations of space
- sharp k-space filter: Markovian trajectory
- http://ned.ipac.caltech.edu/level5/Sept06/Loeb/Loeb4.html
- all mass elements will eventually sit in a halo of arbitrarily low mass
- works in a statistical sense, not necessarily correct on a per-halo basis
- the sharp k-space filter looks ugly in real space (sinc-like)
- the Spherical Cow
- simulations: underpredicts number of high-mass haloes, over-predicts low-mass haloes
- halo finding: how we define the haloes? It changes the halo mass function!

- Ellipsoidal collapse EPS theory:
- moving barrier up-crossing, mcmc simulations
- much better agreement w/ simulations

- Halo merger trees:
- accurately sampling progenitor mass f-n
- mass conservation
- impossible to do properly: conditional probabilities, many ways to draw from the distribution
- tests on the accuracy of merger trees: consistency test, comparison w/ numerical simulation

- mass assembly history:
- main central galaxy accretes lower mass haloes (subhaloes, which may survive or not)
- satellite
- assembly time: main progenitor reaches 1/2 the final mass (more massive haloes assemble later)

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