64 lines
1.6 KiB
Typst
64 lines
1.6 KiB
Typst
#import "globals.typ": *
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== The halo model of reionization
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Following @Schneider_2021 @schneider2023cosmologicalforecast21cmpower, the halo model describes (#link(<backup_full_profiles>, "derivation")):
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#line(length: 100%, stroke: color.white.transparentize(100%))
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#pause
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$
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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)
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$
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#pause
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$
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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))
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$
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// From the xray emission
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// primordial + heating term
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// expansion + deposition by xrays
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// => xrays are assumed to be the only source of heating
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//
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#pause
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$
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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
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$
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// $
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// x_("HII")(r bar M, z) = theta_"H" lr([R_b (M, z) - r], size: #150%)
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// $
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#pagebreak()
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Visually:
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#image("assets/profiles.png")
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// COMMENTS:
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// - contribution from the lyman lines
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// - 1/r^2 decrease from spreading photons
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// - more steep outwards + sharp drop due to redshifting out of line
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#pagebreak()
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== Revisiting the 21cm signal
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$
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d T_"b" (bold(x), z) tilde.eq T_0 (z) dot
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#pin(1) x_"HI" (bold(x), z) #pin(2) dot
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(1 + delta_b (bold(x), z)) dot
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(x_alpha (bold(x), z)) / (#pin(3) 1 + x_alpha (bold(x), z) #pin(4) ) dot
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((1 - T_"CMB" (z)) / (#pin(5) T_"gas" (bold(x), z) #pin(6)))
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$ <eq:dTb>
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#pause
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// #pinit-highlight(1, 20)
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#pinit-point-from((1, 2))[from $x_"HII"$]
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#pause
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#pinit-point-from((3, 4))[From $rho_alpha$]
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#pause
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#pinit-point-from((5, 6))[From $rho_"h"$]
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#pagebreak()
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