Improved frequency hierarchy treatment for anisotropic spectral distortions

TL;DR

This paper exacts a refined frequency-hierarchy framework for CMB spectral distortions, embedding full thermodynamic equilibrium into the evolution and adding stimulated scattering, kinematic corrections, and photon sources. By separating distortion variables from standard perturbations and recasting scattering terms in a spectral-basis formalism, it achieves improved numerical stability and physically transparent equilibration, including a detailed treatment of Compton, double Compton, and Bremsstrahlung processes. A new worked example demonstrates how changes in the temperature-redshift relation can induce distortion anisotropies in the pre-recombination era, while consistency checks show that previous results remain robust under the enhanced formulation. The methodology paves the way for consistent predictions of CMB spectral-distortion anisotropies in photon-dark photon and photon-axion conversion scenarios, broadening the scope of early-universe probes accessible with CMB data.

Abstract

Spectral distortion anisotropies of the cosmic microwave background (CMB) provide a new probe of the early Universe that can be accessed using traditional CMB imaging techniques. It is possible to compute the creation and evolution of anisotropic signals for various scenarios using the frequency hierarchy method recently developed for CosmoTherm. However, the current treatment is not perfect and some approximations had to be made. Here, we carefully construct a modified form for the evolution equations that has the full equilibrium solutions built into the formulation. We improve the formalism to account for i) additional stimulated scattering effects, ii) kinematic corrections to the thermalization terms, iii) corrections to the standard perturbation variables and iv) direct photon sources. These effect could not be captured with the original formulation of the frequency hierarchy method but are indeed important for cleanly separating real distortions from temperature signals. However, we show that previous results are not altered significantly when compared to the improved formulation presented here. As a new worked example, which could indeed not be treated before, we also illustrate how possible changes in the temperature-redshift relation would create spectral distortion anisotropies in the pre-recombination era. The theoretical methods presented here are also an important step towards being able to consistently predict the CMB spectral distortion anisotropies in photon-dark photon and photon-axion conversion scenarios.

Figures (6)

• Figure 1: Transfer functions for the monopole perturbations as a function of scale factor $a=1/(1+z)$ and assuming a single injection of a temperature shift at $z_{\rm s}=z$ as annotated. The left column shows the evolution of the background temperature (no background distortions are sourced by construction) and temperature perturbations. The right column shows the induced temperature correction and $y$ and $\mu$-parameters. For comparison we show the scaled solutions $\delta\Theta^{(1)}_0 \approx 3 \bar{\Theta} \Theta^{(1)}_0$ and $y^{(1)}_{0,0} \approx \bar{\Theta} \Theta^{(1)}_0$. We note the changing scale on the $y$-axis.
• Figure 2: Transfer functions for the monopole perturbations assuming a temperature source term with parameters as annotated. The left column shows that evolution of the background temperature (no background distortions are sourced by construction) and temperature perturbations. The right column shows the induced temperature correction and distortion $y$ and $\mu$-parameter. For comparison we show the expected equilibrium solutions $\delta\Theta^{(1)}_0 \approx 3 \bar{\Theta} \Theta^{(1)}_0$ and $y^{(1)}_{0,0} \approx \bar{\Theta} \Theta^{(1)}_0$, which match the solutions at late times very well for $k=10^{-1}\,{\rm Mpc}^{-1}$, while departures are visible for $k=10^{-2}\,{\rm Mpc}^{-1}$ due to the lower Thomson scattering rate when the mode is inside the horizon. We note the varying change of scale on the $y$-axis.
• Figure 3: Transfer functions for the monopole perturbations assuming a temperature source term with parameters as annotated. The left column shows that evolution of the background temperature (no background distortions are sourced by construction) and temperature perturbations. The right column shows the induced temperature correction and distortion $y$ and $\mu$-parameter. For comparison we show the expected equilibrium solutions $\delta\Theta^{(1)}_0 \approx 3 \bar{\Theta} \Theta^{(1)}_0$ and $y^{(1)}_{0,0} \approx \bar{\Theta} \Theta^{(1)}_0$, which match the solutions at late times very well for $k=10^{-1}\,{\rm Mpc}^{-1}$, while departures are visible for $k=10^{-2}\,{\rm Mpc}^{-1}$ due to the lower Thomson scattering rate when the mode is inside the horizon. We note the varying change of scale on the $y$-axis.
• Figure 4: CMB power spectra for the settings of Fig. \ref{['fig:T-shift-I']} [observational basis with up to $Y_5$.]. The solid black line is the $\Lambda$CDM case (which has no distortion contributions), while the blue-dashed lines are computed for $\beta=10^{-2}$ and $z_{\rm s}=10^4$, showing the total $TT$, $yT$ and $\mu T$ power spectra. Around the maximum of the Thomson visibility function at $z\simeq 1080$ one has $\bar{\Theta}\approx 0.02$. The scaled models (solid red lines) use the $\Lambda$CDM case to obtain $C_\ell^{TT-{\rm scaled}}=(1+3 \bar{\Theta})^2 \,C_\ell^{TT}$ and $C_\ell^{yY-{\rm scaled}}=(1+3 \bar{\Theta})\times \bar{\Theta} \,C_\ell^{TT}$. For the $\mu T$ signal, negative branches are shown with a thinner line.
• Figure 5: CMB power spectra for the settings of Fig. \ref{['fig:T-shift-II']} [observational basis with $Y_k$ up to $Y_5$.]. The solid black line is the $\Lambda$CDM case (which has no distortion contributions), while the blue-dashed lines are computed for $\beta=10^{-2}$ and $z_{\rm s}=10^5$, showing the total $TT$, $yT$ and $\mu T$ power spectra. Around the maximum of the Thomson visibility function one has $\bar{\Theta}\approx 0.041$, but we used $\bar{\Theta}\approx 0.038$ in the scaling, which matches the result better. The scaled models (solid red lines) use the $\Lambda$CDM case to obtain $C_\ell^{TT-{\rm scaled}}=(1+3 \bar{\Theta})^2 \,C_\ell^{TT}$ and $C_\ell^{yY-{\rm scaled}}=(1+3 \bar{\Theta})\times \bar{\Theta} \,C_\ell^{TT}$. For the $\mu T$ signal, negative branches are shown with a thinner line.