#import "importer/main.typ": *
#import "helpers.typ": *

= Results <results>
This section presents the results of the different simulation runs. We compare the effect of different accretion models on the global signal, map-level differences and statistical properties of the 21 cm brightness temperature field. We focus on three different implementations:
- The fiducial model where the accretion rate is kept fixed independently of the halo and the redshift. This corresponds to the original implementation of #beorn where $alpha = 0.79$.
- A model where the accretion rate is computed individually for each halo based on its mass growth history and is considered during the painting of each halo.
- A model where the accretion rate is computed individually for each halo but the considered value during the painting is set to the mean accretion rate of all halos at the respective redshift (effectively reducing the dynamic range of accretion rates).



== Effect on the global signal
#let notebook = json("../workdir/11_visualization/simulation_signals.ipynb")
#figure(
  image_cell(notebook, cell_id: "signal_comparison"),
  caption: [
    Signal comparison between full runs with the different accretion models: Single value of $alpha$ for all halos according to the mean accretion rate (blue), individual accretion rates for each halo allowing a range from $alpha = 0$ to $alpha = 5$ (green), from $alpha = 0$ to $alpha = 2$ (yellow), and the previously model fixing $alpha = 0.79$ (purble, dashed).
    From _left_ to _right_:
    Evolution of the value of the coupling coefficient $x_alpha$.
    Evolution of the mean kinetic temperature $T_k$.
    Mean ionization fraction history $x_"HII"$.
    Global evolution of the differential brightness temperature $d T_"b"$.
    The bottom row shows the difference to the reference model - in this case we chose the model following the mean accretion rate. The comparison of the original model with fixed $alpha = 0.79$ is omitted for clarity.
  ],
) <fig:global_signal_combined>

We first compare the effect of the different accretion models on the global i.e. averaged quantities that consititute the 21 cm signal. @fig:global_signal_combined shows the evolution of the coupling coefficient $x_alpha$, the kinetic temperature $T_k$, the ionization fraction $x_"HII"$, and their combined effect on the differential brightness temperature $d T_"b"$. Moving away from the initial model where $alpha = 0.79$ for all halos, we see a clear delay in the evolution of all quantities - since the overall star formation rate is reduced,


== Map-level investigation
#let notebook = json("../workdir/11_visualization/simulation_maps.ipynb")

#figure(
  caption: [
    Map slices of the core profiles applied onto the simulation grid for the different accretion models plotted at a fixed ionization fraction of $x_"HII" = 0.5$. From _top_ to _bottom_:
    Map of the $x_alpha$ coupling coefficient. Difference to the fiducial model.
    Map of the kinetic temperature $T_k$. Difference to the fiducial model.
    Map of the ionization fraction $x_"HII"$. Difference to the fiducial model.
    In the difference plots blue regions correspond to values lower than the fiducial model while red regions are higher than the fiducial model.
  ]
)[
  #set image(height: 90%)
  #image_cell(notebook, cell_id: "grids_and_diffs"),
] <fig:grids_and_diffs>


@fig:grids_and_diffs shows slices through the simulation box for the different accretion models. Instead of fixing the redshift, we show slices at a fixed ionization fraction of $x_"HII" = 0.5$. This helps to remove the effect of different timing of reionization and focus on the spatial differences and to compare the morphology of the ionized regions. Since the slices we pick are identical the maps resemble each other closely so we focus on the difference plots that highlight specific deviations produced when changing the accretion model.

The coupling coefficient maps sees a decrease in most regions which is explained by an overall lower star formation rate compared to the fiducial case where $alpha = 0.79$. Only a select few halos with high mass accretion rates produce a positive difference. Comparing directly between the individually modelled halos and the case where only the mean accretion rate is used, we see that the bulk of the halos are similar but that both positive and negative deviations occur. This shows that simply using the mean accretion rate is not sufficient to capture map level differences.

This observation is repeated in the kinetic temperature maps. Many regions are colder than in the fiducial case due to the lower heating by fewer stars. Still some regions clearly stand out as being hotter than in the fiducial case. Again, the mean accretion rate model is not able to capture these differences. The background temperature is higher in both alternative accretion models since fixing the ionization fraction means that we show earlier redshifts where the universe has not yet adiabatically cooled as much.

Finally the ionization maps show the clearest differences due to the modelling using a step function. The ionized regions are more compact in the alternative accretion models and the strands of ionized gas connecting the larger bubbles are less pronounced. This hints towards a more individual ionization history where large structures of contiguous ionized gas are less common. When we consider the difference to the fiducial model we see that using the mean accretion rate already captures this distinction well. There are however multiple bubbles where the detailed mass accretion history generates a clear contrast compared to the mean model. Capturing the diversity of halo histories is therefore important to generate maps with the realistic dynamic range.
// not a fan of "dynamic" here.


#figure(
  image_cell(notebook, cell_id: "results_lightcones"),
  caption: [
    // Lightcone images showing brightness temperature slices as they evolve with time.
    // Blue regions absorb the background radiation while orange regions correspond to IGM that has been heated above the background temperature. Black regions are ionized and do not affect the brightness temperature.
    Map slices of the brightness temperature $d T_"b"$ for the different accretion models plotted at a fixed ionization fraction of $x_"HII" = 0.5$.
    // of the 21 cm brightness temperature $d T_"b"$ at
  ]
) <fig:results_lightcones>


// TODO - show a sense of scale

We give special attention
// reformulate
to the derived brightness temperature map. As a reminder, these are not a direct output of the simulation but the spatial distribution can be obtained from the local values of the simulated quantities via `@eq:???`. @fig:results_lightcones shows slices and their comparison to the fiducial model, as previously done for the individual fields. Our observations are compounded here since the contrasts of the fields are combined.
// A little sentence describing the changes explicitly


== Effect on statistic properties

We also compare summary statistics of the $d T_b$ field. The time evolution of the power spectrum describes
// what exactly?

// #figure(
//   image_cell(notebook, cell_id: "power_spectra_comparison"),
//   caption: [
//     ]

// ) <fig:power_spectra_comparison>


#lorem(50)

Importance of RSD for the 21 cm signal
https://arxiv.org/abs/2011.03558