  We study the narrowest spectra expected from gamma-ray bursts. We present an
analytical function for the spectrum that is emitted from the photosphere of a
radiation-dominated flow that is under acceleration. This is the narrowest
possible spectrum and it differs from a Planck function. We also present
numerical spectra from photospheres occurring during the transition into the
coasting phase of the flow. Using these spectral models, we reanalyse Fermi
observations of GRB100507 and GRB101219, which both have been reported to have
very narrow spectra. The bursts can be fitted by the spectral models: For
GRB101219 the spectrum is consistent with the photosphere occurring below or
close to the saturation radius, while for GRB100507 the photosphere position
relative to the saturation radius can be determined as a function of time. In
the latter case, we find that the photosphere initially occurs in the
acceleration phase and thereafter transitions into the coasting phase. We also
find that this transition occurs at the same time as the change in observed
cooling behaviour: the temperature is close to constant before the break and
decays after. We argue that such a transition can be explained by an increasing
mass outflow rate. Both analysed bursts thus give strong evidence that the jets
are (initially) radiation dominated.
