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final adjustments before the hand in

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Remy Moll
2025-09-15 11:50:00 +00:00
parent 221bdcda07
commit 04598e6bb1
8 changed files with 48 additions and 27 deletions
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procedure.typ
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@@ -13,7 +13,7 @@ This section presents the model describing the sources of radiation that drive r
The distinguishing feature
// don't like that word
of #beorn is the parametrization of sources of radiation through the properties of their host dark matter halos. This approach is based on the model presented by @schneider2023cosmologicalforecast21cmpower and gives a description
of #beorn is the parametrization of sources of radiation through the properties of their host dark matter halo. This approach is based on the model presented by @schneider2023cosmologicalforecast21cmpower and gives a description
// bad word
of the 21-cm signal through the treatment of flux profiles around sources. Using these profiles and allowing them to overlap enables efficient computations of the ionization state of the IGM without the need for detailed radiative transfer simulations. The following description follows the derivation made by @schneider2023cosmologicalforecast21cmpower @Schneider_2021 and we refer to these works for components that are not defined here.
@@ -35,7 +35,7 @@ with $M_"p"$ the pivot mass where the efficiency peaks, and $gamma_1$, $gamma_2$
=== Expression of the profiles
Derived from the star formation rate the halo model predicts the production and distribution of photons in three distinct energy bands:
Lyman-$alpha$ photons, ionizing UV photons, and X-ray photons. Each of these bands has a different effect on the IGM and is treated separately.
Lyman-$alpha$ photons, x-ray photons, and ionizing UV photons. Each of these bands has a different effect on the IGM and is treated separately.
// Not really sure that's true
==== Lyman-$alpha$ flux profile
Lyman-$alpha$ photons induce a coupling between the spin temperature and the kinetic temperature of the gas. This effect, known as the Wouthuysen-Field effect
@@ -69,9 +69,9 @@ The comoving ionized volume around a source of ionizing photons satisfies the di
$
derivative(V, t) = dot(N)_"ion"(t) / overline(n)_H^0 - alpha_B dot C / a^3 dot overline(n)_H^0 dot V
$
where $alpha_B$ is the recombination coefficient, $C$ is the clumping factor, $a$ is the scale factor, and $overline(n)_H^0$ is the mean density of hydrogen.
where $alpha_B$ is the recombination coefficient, $C$ is the clumping factor, $a$ is the scale factor, and $overline(n)_H^0$ is the mean density of hydrogen. We expressed this volume in terms of $dot(N)_"ion"$ the rate of change of the total number of ionizing photons per baryon. This description is not dependent on the frequency since we simply consider the contribution of all photons above the ionization threshold.
Ionizing photons, i.e. photons with energies above $13.6 "eV"$ ,experience a large optical depth, which justifies the sharp cutoff of the ionization profile at the bubble radius $R_b = root(3, 3/ (4pi) V(M,z))$. The radial dependence of the ionized fraction is expressed through the Heaviside step function $theta_"H"$ and reads
Ionizing photons, i.e. photons with energies above $13.6 "eV"$, experience a large optical depth, which justifies the sharp cutoff of the ionization profile at the bubble radius $R_b = root(3, 3/ (4pi) V(M,z))$. The radial dependence of the ionized fraction is expressed through the Heaviside step function $theta_"H"$ and reads
$
x_("HII")(r bar M, z) = theta_"H" lr([R_b (M, z) - r], size: #150%)
$
@@ -80,14 +80,14 @@ $
The above profiles express the local effect of radiation around a single halo as a 1-d simplification. A representation of a typical profile can be seen in @fig:alpha_evolution_vs_redshift. Using an estimate of the spatial distribution of halos, these profiles can be applied to generate a full 3-d map if we assume spherical symmetry.
The observable signal of the 21-cm line is obtained from a combination of the mapped quantities. It is expressed as the differential brightness temperature $d T_"b"$ which describes the contrast between the foreground and the CMB background.
The observable signal of the 21-cm line is now a spatially dependent quantity obtained from a combination of the mapped quantities. It is expressed as the differential brightness temperature $d T_"b"$ which describes the contrast between the foreground and the background radiation emitted by the CMB.
// might want to rephrase that
Following e.g. @Pritchard2012 an expression for $d T_"b"$ is given by
$
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>
// where $T_0 (z) = 27 "mK" sqrt((1 + z)/10 (0.15 / (Omega_m h^2))) (Omega_b h^2 / 0.023)$, $x_"HI"$ is the neutral hydrogen fraction, $delta_b$ is the baryonic overdensity, $x_alpha$ is the coupling coefficient introduced by the Wouthuysen-Field effect, $T_"CMB"$ is the temperature of the CMB, and $T_"gas"$ is the kinetic temperature of the gas. The expression assumes that the spin temperature is closely coupled to the kinetic temperature which is valid in most regimes of interest during reionization.
where the background radiation originates from the CMB. The above considerations give us the values of $x_"HI"$, $x_alpha$, and $T_"gas"$. The baryonic overdensity $delta_b$ is assumed to trace the dark matter overdensity $delta_"dm"$ which is obtained from the underlying #nbody simulation. The amplitude of the signal is given by
where the background radiation originates from the CMB. The previous considerations allow us to infer the values of $x_"HI"$, $x_alpha$, and $T_"gas"$ from $x_"HII"$, $rho_alpha$, and $rho_h$ respectively. The baryonic overdensity $delta_b$ is assumed to trace the dark matter overdensity $delta_"dm"$ which is obtained from the underlying #nbody simulation. The amplitude of the signal is given by
$
T_0 (z) = 27 dot (Omega_b h^2) / 0.023 dot sqrt((1 + z)/10 0.15 / (Omega_m h^2)) "mK"
$
@@ -112,10 +112,11 @@ In accordance with the astrophysical parameters set by the user, radiation profi
The last step consists of applying the ionization and temperature distributions defined by the profiles onto a 3-d grid. This is done by iterating over the halos in the catalog and using their corresponding profile. For a given profile a 3-d kernel is generated and mapped onto the grid via convolution using the fast fourier transform implemented by the `astropy` #cite(<astropy:2022>, form: "normal") package. We refer to this procedure as "painting" since the addition of the contributions of each halo allows us to sequentially build up the final map. In general, contributions from multiple halos can overlap without any restrictions. The ionization map is treated specially: In order to conserve the overall number of ionizing photons, we ensure that each cell is only ionized once.
// If multiple halos contribute to the ionization of a cell, the excess photons are redistributed to neighboring cells until they are either used up or reach the edge of the simulation box.
This final step ensures consistent painting: While the contributions to the temperature and coupling maps can be simply added, the ionization map requires a binary treatment. Cells cannot be ionized past a value of $x_"HII" = 1$. Discarding the excesses would violate photon conservation. A redistribution of the excess photons to neighboring cells is performed to ensure that the total number of ionizations matches the total number of emitted photons.
The usage of precomputed profiles is crucial to the efficiency of the simulation but it introduces a discretization in halo mass since each halo is assigned the profile of the closest mass bin. The effect of this simplification has been shown to converge for a sufficient number of mass bins by @Schaeffer_2023.
One final step is required to ensure consistent painting: While the contributions to the temperature and coupling maps can be simply added, the ionization map requires a binary treatment. Cells cannot be ionized past a value of $x_"HII" = 1$. Discarding the excesses would violate photon conservation. A redistribution of the excess photons to neighboring cells is performed to ensure that the total number of ionizations matches the total number of emitted photons.
=== Derivation of global quantities
The global signal as well as the power spectrum are derived from the map data and compared to other models or observations. Being derived from a full 3-d #nbody simulation, the results are sensitive to the underlying cosmology and the detailed profile modeling. This means that the results depend on the underlying astrophysical model.
The global signal as well as the power spectrum are derived from the map data to be compared to other models or observations. Being derived from a full 3-d #nbody simulation, the results are sensitive to the underlying cosmology and the detailed profile modeling. This means that the results also depend on the underlying astrophysical model.