The degeneracy between the dust colour temperature and the spectral index. The problem of multiple χ2 minima

Context. With the current Herschel and Planck satellite missions, there is strong interest in the interpretation of the details of the sub-millimetre dust emission spectra. The data contain information on the properties of the interstellar clouds and the physics of the dust grains. A lot of work has been done to understand the negative correlation observed between the spectral index β_Obs and the colour temperature T_C that in the χ² fits is partly caused by the observational noise.
Aims: In the (T_C, β_Obs) plane, the confidence regions are elongated, banana-shaped structures. Previous studies have indicated that the errors may exhibit strongly asymmetric features that have important consequences for the investigation of individual objects and the interpretation of the relation between the T_C and β_Obs parameters. We study under which conditions the confidence regions exhibit such anomalous, strongly non-Gaussian behaviour that could affect the interpretation of the observed (T_C, β_Obs) relations.
Methods: We examined a set of modified black body spectra and spectra calculated from radiative transfer models of filamentary interstellar clouds. We analysed simulated observations at discrete wavelengths between 100 μm and 850 μm. We performed modified black body fits and examined the structure of the χ²(T_C, β_Obs) function of the fits.
Results: We demonstrate cases where, when the signal-to-noise ratio is low, the χ² has multiple local minima in the (T_C, β_Obs) plane. A small change in the weighting of the data points can cause the solution to jump to completely different values. In particular, if there is noise, the analysis of spectra with T > 10 K and β_Obs ≲ 2 can lead to a separate population of solutions with much lower colour temperature and higher spectral indices. The anomalies are caused by the noise. However, the tendency to show multiple χ² minima depends on the model (in part via the influence on the signal-to-noise ratios) and on the set of wavelengths included in the analysis. Deviations from the underlying assumption of a single modified black body spectrum are not significant.
Conclusions: The presence of several local minima implies that the results obtained from the χ² minimisation depend on the starting point of the optimisation and may correspond to non-global minima. Because of the strongly non-Gaussian nature of the errors, the obtained (T_C, β_Obs) distribution 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 β_Obs(T_C) relation to be unduly affected by these measurements.