filling a few things in

introduction.typ

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+#import "importer/main.typ": *
+#import "@preview/physica:0.9.5": *
+#import "@preview/touying:0.6.1": *
+
+
+= Simulating the Epoch of Reionization
+== The Epoch of Reionization
+- Marks the universe’s last major phase transition: from neutral to ionized hydrogen.
+
+- Shapes the large-scale structure of the intergalactic medium (IGM).
+
+- Is strongly linked to the formation and growth of the first galaxies and black holes.
+
+- Sets the stage for many observables: CMB secondary anisotropies, 21-cm signal, high-z galaxy surveys.
+// reformulate
+
+== The 21cm signal
+
+The brigthtness temperature describes the difference between the CMB temperature and the spin temperature of neutral hydrogen
+#grid[
+  #image("assets/cmb_black_body_spectrum.svg", height: 1fr, fit: "contain") @cmb_spectrum
+][
+  Removing the contribution from the black body spectrum of the CMB yields the explicit 21-cm signal:
+  #image("assets/cmb_black_body_spectrum.svg", height: 1fr, fit: "contain")
+
+]
+
+
+
+#align(center)[
+  #image("assets/evolution_of_dtb.png", fit: "contain")
+]
+@Pritchard2012
+
+#pagebreak()
+
+$
+  d T_"b" (bold(x), z) tilde.eq T_0 (z) dot x_"HI" (bold(x), z)  dot (1 + delta_b (bold(x), z)) dot (x_alpha (bold(x), z)) / (1 + x_alpha (bold(x), z) ) dot ((1 - T_"CMB" (z)) / (T_"gas" (bold(x), z)))
+$ <eq:dTb>
+(e.g @Pritchard2012)
+
+
+
+== The halo model of reionization
+
+
+
+Following @Schneider_2021 @schneider2023cosmologicalforecast21cmpower:
+
+$
+  rho_alpha (r bar M, z) = (1 + z)^2 / (4 pi r^2) dot sum_(n=2)^(n_m)f_n dot epsilon_alpha (nu prime) dot f_star dot dot(M)(z prime bar M, z)
+$
+#line(
+  length: 100%
+)
+$
+  3/2 dot derivative(rho_h (r bar M, z), z) = (3 rho_h (r bar M, z)) / (1 + z) - (rho_"xray" (r bar M, z)) /(k_B (1 + z) H(z))
+$
+#line(
+  length: 100%
+)
+$
+  x_("HII")(r bar M, z) = theta_"H" lr([R_b (M, z) - r], size: #150%)
+$
+
+
+
+== The current state of simulations
+#lorem(20)
+
+- slow and big
+- radiative transfer // what are the downsides
+
+
+$==>$ semi-numerical approaches such as `BEoRN` @Schaeffer_2023
+
+
+#link(<backup_validation>, "validation")
+
+== `BEoRN`