  We investigate the non-equilibrium electronic structure and characteristic
time scales in a candidate excitonic insulator, Ta$_2$NiSe$_5$, using time- and
angle-resolved photoemission spectroscopy with a temporal resolution of 50 fs.
Following a strong photoexcitation, the band gap closes transiently within 100
fs, i.e., on a time scale faster than the typical lattice vibrational period.
Furthermore, we find that the characteristic time associated with the rise of
the photoemission intensity above the Fermi energy decreases with increasing
excitation strength, while the relaxation time of the electron population
towards equilibrium shows an opposite behaviour. We argue that these
experimental observations can be consistently explained by an excitonic origin
of the band gap in the material. The excitonic picture is supported by
microscopic calculations based on the non-equilibrium Green's function
formalism for an interacting two-band system. We interpret the speedup of the
rise time with fluence in terms of an enhanced scattering probability between
photo-excited electrons and excitons, leading to an initially faster decay of
the order parameter. We show that the inclusion of electron-phonon coupling at
a semi-classical level changes only the quantitative aspects of the proposed
dynamics, while the qualitative features remain the same. The experimental
observations and microscopic calculations allow us to develop a simple and
intuitive phenomenological model that captures the main dynamics after
photoexcitation in Ta$_2$NiSe$_5$.
