Parameterization of temperature and spectral distortions in future CMB experiments

Abstract

CMB spectral distortions are induced by Compton collisions with electrons. We review the various schemes to characterize the anisotropic CMB with a non-Planckian spectrum. We advocate using logarithmically averaged temperature moments as the preferred language to describe these spectral distortions, both for theoretical modeling and observations. Numerical modeling is simpler, the moments are frame-independent, and in terms of scattering the mode truncation is exact.

Appendix

Distribution functions with a chemical potential. At redshifts higher than $z \simeq 10^4$ and smaller than $z \simeq 3\,\times \, 10^{5}$, the thermalization of photons with an excess of energy with respect to a Planck spectrum results in a Bose–Einstein distribution with a chemical potential [29]. However, at low energies, processes such as double Compton emission and Bremsstrahlung are enough to modify the spectrum and remove the effect of the chemical potential so that the low energy limit of the distribution is still $\propto 1/E$. It is indeed approximately described by a Bose–Einstein distribution with an energy dependent chemical potential. At high energies, this chemical potential converges to a constant, but at low energies it is suppressed, and the transition from one regime to the other is governed by a cut-off $x_c=E_c/T$ [29]. The distribution function of this ansatz is approximately of the form

$$\begin{aligned} n(x) = \frac{1}{e^{x+\mu (x)} -1},\quad \mu (x) = \mu _{\infty } e^{-x_c/x}. \end{aligned}$$

(17)

with $x\equiv E/T$. A typical cut-off is $x_c=0.01$ and one can check that when $\mu _\infty \rightarrow 0$, this distribution approaches a BBR as the various moments decrease, as expected. In Fig. 1 we plot the first moments as a function the chemical potential $\mu _\infty$ to illustrate this convergence.

Fig. 1

The main moments for a pseudo Bose–Einstein distribution (17) with cut-off $x_c=0.01$. In black $\log _{10} \bar{T}$. In colors $-u_k/k!$ for $k=2,3,4,5$ (from bottom to top) (color figure online)