Module greens

Green’s function for the cosmological thermalization problem.

Provides a fast, approximate method for computing spectral distortions from arbitrary energy release histories. The distortion from a delta-function energy injection at redshift z_h is decomposed into μ, y, and temperature shift components using visibility/branching functions.

The Green’s function approach is “quasi-exact” for small distortions (Δρ/ρ << 1) and much faster than solving the full PDE.

Also includes the photon injection Green’s function (Chluba 2015), which handles injection of photons at a specific frequency x_inj. Unlike pure energy injection, photon injection changes both energy and number, producing negative μ when x_inj < x₀ ≈ 3.60.

Energy non-conservation in the three-component ansatz

The temperature-shift coefficient uses J_T = 1 − J_bb* following Chluba (2013). The y-component uses the independently fitted J_y of Chluba (2013) Eq. 5, which is NOT simply (1 − J_μ) × J_bb*. As a result, the three branching ratios do not sum to unity:

J_μ × J_bb* + J_y + (1 − J_bb*) ≠ 1

Chluba (2013) §3 notes that the “missing” energy in this ansatz stays within the residual and never exceeds ~16–17%, maximised in the μ-y transition region (z ~ 7–8×10⁴). Using the independent J_y fit matches PDE results more closely than the strictly energy-conserving choice J_y = (1 − J_μ) · J_bb*. Callers that need strict energy conservation must use the full PDE solver.

References:

• Chluba (2013), MNRAS 436, 2232 [arXiv:1304.6120], Eqs. 5–6 for J_μ, J_y, G_th
• Chluba (2015), MNRAS 454, 4182 [arXiv:1506.06582], Eq. 13 for J_bb*
• Chluba & Jeong (2014), MNRAS 438, 2065

Functions

distortion_from_heating

Compute the spectral distortion from an arbitrary energy release history.

distortion_from_photon_injection

Compute spectral distortion from an arbitrary photon injection history.

greens_function

Compute the Green’s function G_th(x, z_h) for a delta-function energy injection at redshift z_h, observed at z = 0.

greens_function_photon

Green’s function for monochromatic photon injection.

integration_crosses_photon_gf_gap

Returns true iff the GF integration window crosses the μ-y transition band; callers can use this to emit a warning at function entry.

mu_from_heating

Extract μ parameter from the Green’s function approximation.

mu_from_photon_injection

Compute μ from monochromatic photon injection at frequency x_inj.

mu_y_from_heating

Compute both μ and y from an arbitrary energy release history in a single pass.

photon_survival_probability

Photon survival probability P_s(x, z).

visibility_j_bb

Thermalization visibility: probability that energy injection at z is fully thermalized into a blackbody (temperature shift).

visibility_j_bb_star

Improved thermalization visibility with correction factor.

visibility_j_mu

μ-distortion branching ratio: fraction of energy going into μ-type distortion.

visibility_j_t

Temperature shift branching: fraction of energy going into temperature shift.

visibility_j_y

y-distortion branching ratio: fraction of energy going into y-type distortion.

x_c

Combined critical frequency for photon absorption.

x_c_br

Critical frequency for bremsstrahlung absorption.

x_c_dc

Critical frequency for double Compton absorption.

y_from_heating

Extract y parameter from the Green’s function approximation.