The degeneracy between the dust colour temperature and the spectral index. The problem of multiple chi^2 minima

TL;DR

This paper investigates how noise and model assumptions cause multiple minima in chi^2 fits of dust spectra, leading to potential misinterpretations of temperature and spectral index relations in interstellar clouds.

Contribution

It demonstrates the conditions under which chi^2 minimization yields non-Gaussian, multi-modal solutions, highlighting the importance of proper weighting in spectral analysis.

Findings

Multiple chi^2 minima occur at low signal-to-noise ratios.

Noise can produce artificial populations with low temperature and high spectral index.

Model choice and wavelength range influence the presence of local minima.

Abstract

Because of the Herschel and Planck satellite missions, there is strong interest in the interpretation the sub-millimetre dust spectra from interstellar clouds. Much work has been done to understand the dependence between the spectral index beta_Obs and the colour temperature T_C that is partly caused by the noise. The (T_C, beta_Obs) confidence regions are elongated, banana-shaped structures. We studied under which conditions these exhibit anomalous, strongly non-Gaussian behaviour that could affect the interpretation of the observed (T_C, beta_Obs) relations. We examined modified black body spectra and spectra calculated from radiative transfer models of filamentary clouds at wavelengths 100um-850um. We performed modified black body fits and examined the structure of the chi^2(T_, beta_Obs) function. We show cases where, when the signal-to-noise ratio is low, the chi^2 has multiple local minima in the (T_C, beta_Obs) plane. A small change in the weighting of the data points can cause the solution to jump to completely different values. In particular, noise can lead to the appearance of a separate population of solutions with low colour temperatures and high spectral indices. The anomalies are caused by the noise but the tendency to show multiple chi^2 minima depends on the model and the wavelengths analysed. Deviations from the assumed single modified black body spectrum are not important. The presence of local minima implies that the results obtained from the chi^2 minimisation depend on the starting point of the optimisation and may correspond to non-global minima. The (T_C,beta_Obs) distributions may be contaminated by a few solutions with unrealistically low colour temperatures and high spectral indices. Proper weighting must be applied to avoid the determination of the beta_Obs(T_C) relation to be unduly affected by these measurements.