nearly fully fleshed out now

backup.typ

@@ -1,9 +1,50 @@
 #import "globals.typ": *
 
 
+== Full profiles <backup_full_profiles>
+$
+  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)
+$
+with the lookback redshift $z prime$ so $nu prime = nu dot (1+z prime) slash (1+z)$
+
+$=>$ coupling coefficient
+$
+  x_alpha (r bar M, z) = (1.81 dot 10^11) / (1 + z) dot S_alpha (z) dot rho_alpha (r bar M, z)
+$
+with a suppression factor $S_alpha (z)$
+
+
+#pagebreak()
+$
+  rho_"xray" (r bar M, z) = 1 / r^2 sum_i f_i f_(X,h) dot integral_(nu^i_"th")^oo d nu (nu - nu^i_"th") h_P sigma_i (nu) e^(-tau_nu) f_star dot(M) (z prime bar M, z)
+$
+$
+  ==> 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))
+$
+with the Boltzmann constant $k_B$ and $H(z)$ is the Hubble parameter
+// where i={H, He} and νi thhP ={13.6, 26.5} eV.
+// Contribution from both He and H since
+// No defined sum but integral due to all radiations being xray
+
+// Lookback comes from the definition of comoving r:
+// $r = integral_(z)^(z prime) c dz dot (1+z) / H(z)$
+
+#pagebreak()
+
+The comoving ionized volume around a source of ionizing photons satisfies the differential equation
+$
+  derivative(V, t) = dot(N)_"ion"(t) / overline(n)_H^0 - alpha_B dot C / a^3 dot overline(n)_H^0 dot V
+$
+bubble radius $R_b = root(3, 3/ (4pi) V(M,z))$ and using the Heaviside step function $theta_H$:
+$
+  x_("HII")(r bar M, z) = theta_"H" lr([R_b (M, z) - r], size: #150%)
+$
+
+
 == Validation <backup_validation>
-#[
-  #set image(height: 45%, fit: "contain")
+#line(length: 100%, stroke: 0pt)
+#align(center)[
+  #set image(height: 43%, fit: "contain")
   #let notebook = json("../workdir/11_visualization/validation_simple_run.ipynb")
   #image_cell(notebook, cell_id: "validation_signal_comparison_old_v_new")
 
@@ -13,11 +54,11 @@
 
 
 
-== Backup slides
-This is a backup slide
-#lorem(50)
-
-== Another backup slide
-#lorem(50)
+== Thesan resolution issues
+#let notebook = json("../workdir/11_visualization/halo_mass_function_thesan_1_2.ipynb")
 
+#line(length: 100%, stroke: 0pt)
+#align(center)[
+  #image_cell(notebook, cell_id: "halo_mass_functions")
+]