  The presence of neutral hydrogen in the shock proximity changes the structure
of the shock and affects the spectra of particles accelerated through the
first-order Fermi mechanism. This phenomenon has profound implications for the
interpretation of the multifrequency spectra of radiation from supernova
remnants. Neutrals that undergo charge exchange with hot ions downstream of the
shock may result in fast neutrals moving towards the upstream gas, where they
can suffer additional charge exchange or ionisation reactions, thereby
depositing energy and momentum upstream. Here we discuss the implications of
this neutral return flux, which was already predicted in our previous work on
neutral mediated supernova shocks, and show how the spectra of accelerated
particles turn out to be appreciably steeper than $p^{-4}$, thereby affecting
the gamma ray spectra from supernova remnants in general and from Tycho
specifically. The theory that describes non-linear diffusive shock acceleration
in the presence of neutral hydrogen has been developed in recent years. Here we
use a semi-analytical theory developed in previous work and specialise our
predictions to the case of the Tycho supernova shock, where there is evidence
that the spectrum of the accelerated cosmic rays is steeper than expected from
the traditional theory of diffusive shock acceleration. We show that, if the
fraction of neutral hydrogen in the vicinity of the Tycho supernova shock is,
as suggested by observations, ~70-90, then spectra of accelerated protons
steeper than $p^{-4}$ may be a natural consequence of charge exchange reactions
and the associated neutral return flux. The spectral shape is affected by this
phenomenon for particles with energies below ~100-1000 GeV, for which the
diffusion length is less than or at most comparable to the pathlength of charge
exchange and ionisation upstream of the shock.
