  The TRAPPIST-1 system is unique in that it has a chain of seven terrestrial
Earth-like planets located close to or in its habitable zone. In this paper, we
study the effect of potential cometary impacts on the TRAPPIST-1 planets and
how they would affect the primordial atmospheres of these planets. We consider
both atmospheric mass loss and volatile delivery with a view to assessing
whether any sort of life has a chance to develop. We ran N-body simulations to
investigate the orbital evolution of potential impacting comets, to determine
which planets are more likely to be impacted and the distributions of impact
velocities. We consider three scenarios that could potentially throw comets
into the inner region (i.e within 0.1au where the seven planets are located)
from an (as yet undetected) outer belt similar to the Kuiper belt or an Oort
cloud: Planet scattering, the Kozai-Lidov mechanism and Galactic tides. For the
different scenarios, we quantify, for each planet, how much atmospheric mass is
lost and what mass of volatiles can be delivered over the age of the system
depending on the mass scattered out of the outer belt. We find that the
resulting high velocity impacts can easily destroy the primordial atmospheres
of all seven planets, even if the mass scattered from the outer belt is as low
as that of the Kuiper belt. However, we find that the atmospheres of the
outermost planets f, g and h can also easily be replenished with cometary
volatiles (e.g. $\sim$ an Earth ocean mass of water could be delivered). These
scenarios would thus imply that the atmospheres of these outermost planets
could be more massive than those of the innermost planets, and have
volatiles-enriched composition.
