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{We explored the {softness parameter} in the infrared, whose main purpose isthe characterisation of the hardness of the incident ionising radiation inemission-line nebulae. This parameter is obtained from the combination ofmid-infrared wavelength range transitions corresponding to consecutiveionisation stages in star-forming regions. We compiled observational data froma sample of star-forming galaxies (SFGs), including luminous and ultraluminousinfrared galaxies (LIRGs and ULIRGs), to study the softness parameter and itsequivalent expression in two dimensions, the softness diagram. We compared themwith predictions from photoionisation models to determine the shape of theionising continuum energy distribution in each case. We also used the measuredemission-line ratios as input for HCmistry-Teff-IR, a code that performs aBayesian-like comparison with photoionisation model predictions in order toquantify the equivalent effective temperature (T*) and the ionisationparameter. We found similar average values within the errors of the softnessparameter in (U)LIRGs (-0.57) in the rest of the SFGs (-0.51), which could beinterpreted as indicative of a similar incident radiation field. This result isconfirmed from the analysis using HCm-Teff-IR, which simultaneously points to aslightly lower, although similar within the errors, T* scale for (U)LIRGs, evenwhen a higher dust-to-gas mass ratio is considered in the models for theseobjects. These derived T* values are compatible with the ionisation frommassive stars, without any need of harder ionising sources, both for (U)LIRGsand the rest of the SFGs. However, the derived T* in (U)LIRGs do not show anycorrelation with metallicity. This could be interpreted as a sign that theirsimilar average T* values are due to the attenuation of the energetic incidentflux from massive stars by the heated dust mixed with the gas.

Exploring the hardness of the ionising radiation with the infrared  softness diagram. I. Similar effective temperature scales for starbursts and  (ultra)luminous infrared galaxies