Fifty shades of grayness: parameterizations of spectral distortions and applications in cosmology

TL;DR

The work addresses how to efficiently parameterize and constrain spectral distortions of cosmological backgrounds beyond ideal blackbodies. It introduces three temperature-transform formalisms (TT, LTT, NLTT) and an Orthonormal Polynomial Expansion (OPE) to capture distortions with a small set of coefficients, linking them directly to observable moments and densities. The authors demonstrate the practicality of these methods by applying them to primordial neutrino spectra and CMB distortions, deriving constraints on distortion parameters from BBN, $N_{\rm eff}$, and FIRAS data, and by extending the approach to the massive case. The OPE framework emerges as a robust, model-independent, and instrument-agnostic tool for describing a wide range of distortions in cosmology and beyond, with clear pathways for future high-precision probes and nonstandard physics scenarios.

Abstract

Thermal distribution functions can only be of the Fermi-Dirac or Bose-Einstein types, whereas distorted spectra encompass any possible deviations from these shapes. It is fruitful to devise parameterizations of these distortions with only a few parameters which depend on the physical system considered. A method proposed by Stebbins consists in describing a distorted spectrum as a sum of thermalized spectra with a distribution of temperatures, the moments of which are the parameters of interest. After revisiting and extending this approach by working at the level of the number density distribution instead of the standard spectrum, we build another method which consists in describing the distorted spectrum by a polynomial modulating a reference thermalized spectrum. The distortion parameters are then the coefficients of a decomposition on a suitable orthonormal polynomial basis. We advocate that the latter is computationally easier and allows to describe a wide range of distortions. With this formalism, we efficiently describe the standard distortions of the cosmological backgrounds of neutrinos and photons, and we obtain model-independent constraints on nonstandard distortions of these cosmological relics.

Figures (14)

• Figure 1: Orthonormal polynomials with respect to the weights \ref{['ScalarProduct_def']} with plus sign for fermions (continuous lines) and minus sign for bosons (dashed lines). The dotted lines are the normalized Laguerre polynomials \ref{['Laguerre']}. The black lines are the corresponding weight functions (rescaled for better visibility).
• Figure 2: Illustration of the determination of the various $T_{\rm ref}$ for a given thermalized spectrum (solid point) of fermions. Solid lines correspond to lines of thermal spectra. When $\mu$ is specified, $T$ is varied along the line (black and gray lines), and conversely when $T$ is specified to the three $T_{\rm ref}$ defined in the text, $\mu$ is varied (colored lines). Dashed lines correspond to the directions of projection, with a vertical projection for constant number density, an horizontal projection for constant energy density, or a line of fixed ratio $\rho/n$ for constant average energy per particle.
• Figure 3: Left: Reconstruction of $\tilde{q}(\mathcal{T})$ for the spectrum \ref{['example_spectrum']} of either bosons or fermions, using $\alpha = 10^{-25}$ in the regularization \ref{['Wiener_regularization']}. The Dirac functions of the analytical result \ref{['example_spectrum_analytical_transform']} are marked by the vertical dashed lines. Right: Moments of the logarithmic temperature transform of the spectrum \ref{['example_spectrum']}, calculated for the bosonic case. Our method and Stebbins' method are very close the analytic result \ref{['eq:centered_moments_twoT']} at low $n$, but our implementation of Stebbins' method breaks down around $n \sim 7$ whereas our method works for larger $n$.
• Figure 4: Left: Reconstructions of the number density distribution of the spectrum \ref{['example_spectrum']} in the bosonic case using the LTT and Eq. \ref{['eq:fnu_from_moments']} (dotted lines), or the NLTT and Eq. \ref{['RecconstructionFromTilde']} (dashed lines), truncated to various orders. The order 0 truncation is labeled as $g, \, \overline{\mathcal{T}}$ (resp. $\tilde{g}, \, \widetilde{\overline{\mathcal{T}}}$) to highlight that it is the graybody spectrum with the corresponding grayness and temperature. Right: OPE of the spectrum \ref{['example_spectrum']} in the bosonic case for various orders. The reference temperature corresponds to the graybody choice \ref{['Tref_equal_average_energy']}. We see that the approximations quickly converge towards the true spectrum.
• Figure 5: Left: Reconstructions of the number density distribution of a $y$ distortion spectrum (i.e. the true spectrum minus the undistorted blackbody spectrum) of bosons with $y=0.1$, using the LTT and Eq. \ref{['eq:fnu_from_moments']} (dotted lines), or the NLTT and Eq. \ref{['RecconstructionFromTilde']} (dashed lines), truncated to various orders. The order 0 truncation is labeled as $g, \, \overline{\mathcal{T}}$ (and $\tilde{g}, \, \widetilde{\overline{\mathcal{T}}}$) to highlight that it is the graybody spectrum with the associated grayness and temperature. We see that both methods work well, but the NLTT method converges faster. Right: OPE of the same bosonic $y$ distortion for various truncation orders. The reference temperature corresponds to the number density choice \ref{['Trefnrelation']}, which is especially natural for $y$ distortions because they do not alter the number density. We see that the approximations quickly converge towards the true spectrum.