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

Aims. We explored the softness parameter in the infrared, eta(')(IR), whose main purpose is the characterisation of the hardness of the incident ionising radiation in emission-line nebulae. This parameter is obtained from the combination of mid-infrared wavelength range transitions corresponding to consecutive ionisation stages in star-forming regions. Methods. We compiled observational data from a sample of star-forming galaxies (SFGs), including luminous and ultraluminous infrared galaxies (LIRGs and ULIRGs), to study the eta(')(IR) and its equivalent expression in two dimensions, the softness diagram. We compared them with predictions from photoionisation models to determine the shape of the ionising continuum energy distribution in each case. We also used the measured emission-line ratios as input for HCMISTRY-TEFF-IR, a code that performs a Bayesian-like comparison with photoionisation model predictions in order to quantify the equivalent effective temperature (T-*) and the ionisation parameter. Results. We found similar average values within the errors of eta(')(IR) in (U)LIRGs (-0.57) in the rest of the SFGs (-0.51), which could be interpreted as indicative of a similar incident radiation field. This result is confirmed from the analysis using HCM-TEFF-IR, which simultaneously points to a slightly lower, although similar within the errors, T-* scale for (U)LIRGs, even when a higher dust-to-gas mass ratio is considered in the models for these objects. These derived T-* values are compatible with the ionisation from massive stars, without any need of harder ionising sources, both for (U)LIRGs and the rest of the SFGs. However, the derived T-* in (U)LIRGs do not show any correlation with metallicity. This could be interpreted as a sign that their similar average T-* values are due to the attenuation of the energetic incident flux from massive stars by the heated dust mixed with the gas. This is supported by the known very large amounts of small grains associated with the very high star formation rates measured in galaxies of this type.