fleshed out the introductory sections

halo_mass_history.typ

@@ -1 +1,121 @@
-= Halo mass history
+#import "importer/main.typ": *
+#import "helpers.typ": *
+
+= Halo mass history <halo_mass_history>
+
+This section delves into the central role of the halo mass evolution for the results of the simulation.
+
+== Modelling mass accretion
+
+Generalized mass accretion rate and its simplification in the exponential growth model.
+
+As described in @hmreio the fundamental assumption of #beorn is the halo model of reionization by @schneider2023cosmologicalforecast21cmpower.
+// no need to recite?
+It describes how observables of reionization can be parametrized in terms of the halo mass and more specifically its rate of change since they are derived from the star formation rate expressed in @eq:star_formation_rate.
+In this simplified model, for a given star formation efficiency
+#footnote[
+  Note that the assumption of a fixed star formation efficiency or even an analytic expression as a function of halo mass is a simplification.
+  // Citation
+  The investigation of stochasticity has been subject to separate research (e.g. "missing").
+],
+the halo mass history is the single most impactful property besides the mass itself.
+
+
+// . In particular we express the star formation rate $dot(M)_star$ through the star formation efficiency $f_star$ and the halo mass $M_"h"$ as follows:
+// $
+//   dot(M)_star = f_star (M_h) dot dot(M_h)
+// $
+
+#beorn's goal is to provide simulations of the map-level contributions to the $21 "cm"$ signal, meaning that we can not rely on a distribution of halo masses and accretion rates alone. Instead #beorn leverages large scale N-body simulations to provide a spatial distribution of halos. In its introduction #cite(<Schaeffer_2023>, form: "normal") #beorn used the PkdGrav3 suite as a generator of the halo distribution. Halo growth was then modelled through an exponential growth model
+$
+  M_"h" (z) = M_"h" (z_0) dot exp[-alpha (z - z_0)]
+$
+where $alpha = - dot(M_"h") / M_"h"$ is a free parameter describing the specific mass accretion rate. Following `@???` a value of $alpha = 0.79$ was used as a fiducial value for all halos, independent of their mass or redshift.
+
+In the following we will motivate the need for a more precise treatment of the halo mass history and show how we can leverage the data provided by the THESAN simulation suite to obtain a more precise model.
+
+// In a purely formal investigation where a qualitative prediction is derived from a well-defined halo mass distribution, the mass history is simply obtained as a direct derivation from the mass distribution. The simulations made by #beorn aim to provide 3D data that allows for quantitative conclusions. To this end a spatial distribution of the halo mass history is required, as provided by large scale simulations
+// #footnote[
+//   As described previously the halo model allows us to restrict the simulation to dark matter only, allowing for a more efficient simulation of the large scale structure.
+// ]
+// // Cite pkdgrav, Illustris, THESAN
+// .
+
+
+== Effect on radiation profiles
+#let notebook = json("../workdir/11_visualization/alpha_dependence_of_profiles.ipynb")
+
+In order to illustrate the necessity of a refined mass history model, we first investigate the effect of different mass accretion rates on the resulting radiation profiles. To this end we consider halos at fixed masses and vary their accretion rates around the fiducial value of $alpha = 0.79$.
+
+
+@Kannan_2021 also shows that reionization history is different for different gas densitites, i.e. halo masses. We also show from a profile perspective that treating halo accretion as a free parameter can lead to significant differences in the resulting profiles.
+
+
+== The THESAN simulation
+
+In order to generate precise map-level predictions of the 21cm signal, #beorn combines the halo model of reionization with large scale N-body simulations which provide realistic snapshots of the dark matter distribution.  They constitute the fundamental input to the halo model amd give a spatial context to the generated profiles.
+
+Past iterations @Schaeffer_2023 #beorn have used different
+#cite(<Schaeffer_2023>, form: "normal")
+means to generate these snapshots, including the 21cmfast emulator as a validation and the PkdGrav3 N-body code as a large signal generator.
+
+For the purposes of this thesis we don't aim to run the largest possible simulation, but rather to refine the underlying model. To this end, we use the publicly available data from the THESAN simulation suite
+#cite(<Kannan_2021>, form: "normal")
+#cite(<Garaldi_2022>, form: "normal")
+#cite(<Smith_2022>, form: "normal")
+. The #smallcaps[Thesan-Dark] simulation in particular provides a dark matter only simulation and provides halo catalogs and merger trees.
+
+With a box length of $95.5 "cMpc"$ it provides a sufficient volume to avoid box size effects
+// CITATION
+while still allowing us to refine the underlying model without excessive computational cost. The simulation has two variants with different mass resolutions:
+... // TODO
+which allow us to perform convergence tests as described in @validation.
+
+
+
+// The simulation has two main limitations: First, the mass resolution of $3.12 * 10^7 "M_⊙"$ means that halos below a mass of $10^9 "M_⊙"$ are not resolved. This is particularly relevant as these low mass halos are expected to contribute significantly to the ionizing photon budget at high redshifts #cite(<Kannan_2021>, form: "normal"). To account for this, we use boosted models of star formation efficiency as described in section <sf_efficiency>. Second, the simulation only provides snapshots down to a redshift of $z=5.5$. As reionization is expected to be completed by this time, this does not impact our results.
+
+Thesan halo catalog and the motivation to increase the cutoff.
+
+At the same time THESAN low mass halos seem overabundant which is why we use boosted models of star formation efficiency.
+
+
+
+
+@Kannan_2021 describes the nuance of using thesan 1 vs thesan 2 for the halo mass:
+the lowest mass halos which are not resolved by thesan 2 form small bubbles quickly and as early as z=10 and contribute to the ionization budget at early times
+
+
+
+=== Merger trees
+#let notebook = json("../workdir/11_visualization/show_trees.ipynb")
+
+The central representation of halo mass evolution is given by merger trees. These tree-like structures describe the halo history in terms of the mergers of its smaller progenitors. A merger tree is constructed by linking halos in consecutive snapshots of the simulation where each halo as a single descendant but potentially multiple progenitors. As described in ... THESAN
+
+The main progenitor can be used as a tracer of the halo mass history if we assume that the halo mass growth is dominated by mergers.
+// Has this been shown to be true somewhere?
+Beyond that, the main progenitor will be the main contributor in terms of stellar mass which is the main quantity of interest for the reionization model. Utilizing the trees provided by
+
+#figure(
+  image_cell(notebook, cell_id: "merger_tree_and_fitting"),
+  caption: "Example of a merger tree and the fitting of its main progenitor's mass history.",
+
+) <fig:merger_tree_and_fitting>
+
+
+- How we treat incomplete trees
+- how we treat invalid trees
+
+
+
+== Resulting distribution
+#let notebook = json("../workdir/11_visualization/evolution_of_alphas.ipynb")
+
+
+#figure(
+  image_cell(notebook, cell_id: "alpha_evolution_vs_redshift"),
+  caption: "??",
+
+
+) <fig:alpha_evolution_vs_redshift>
+