#import "globals.typ": *


== The halo model of reionization
Following @Schneider_2021 @schneider2023cosmologicalforecast21cmpower, the halo model describes (#link(<backup_full_profiles>, "derivation")):
#line(length: 100%, stroke: color.white.transparentize(100%))
#pause

$
  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)
$

#pause

$
  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))
$
// From the xray emission
// primordial + heating term
// expansion + deposition by xrays
// => xrays are assumed to be the only source of heating
//
#pause
$
  derivative(V_b, t) = dot(N)_"ion"(t) / overline(n)_H^0 - alpha_B dot C / a^3 dot overline(n)_H^0 dot V_b
$
// $
//   x_("HII")(r bar M, z) = theta_"H" lr([R_b (M, z) - r], size: #150%)
// $

#pagebreak()
Visually:

#image("assets/profiles_demo.png", height: 70%)
(from @Schaeffer_2023)
// COMMENTS:
// - contribution from the lyman lines
// - 1/r^2 decrease from spreading photons
// - more steep outwards + sharp drop due to redshifting out of line

#pagebreak()


== Revisiting the 21cm signal
$
  d T_"b" (bold(x), z) tilde.eq T_0 (z) dot
  #pin(1) x_"HI" (bold(x), z) #pin(2) dot
  (1 + delta_b (bold(x), z)) dot
  (x_alpha (bold(x), z)) / (#pin(3)  1 + x_alpha (bold(x), z) #pin(4) ) dot
  ((1 - T_"CMB" (z)) / (#pin(5) T_"gas" (bold(x), z) #pin(6)))
$ <eq:dTb>

#pause
// #pinit-highlight(1, 20)
#pinit-point-from((1, 2))[from $x_"HII"$]
#pause

#pinit-point-from((3, 4))[from $rho_alpha$]
#pause

#pinit-point-from((5, 6))[from $rho_"h"$]

#pagebreak()