  In this paper we study the behaviour of gravitational wave background (GWB)
generated during inflation in the environment of the noncommutative field
approach. From this approach we derive out one additive term, and then we find
that the dispersion relation of the gravitational wave would be modified and
the primordial gravitational wave would obtain an effective mass. Therefore it
breaks lorentz symmetry in local. Moreover, this additive term would a little
raise up the energy spectrum of GWB in low frequency and then greatly suppress
the spectrum at even lower energy scale of which the wave length may be near
the current horizon. Therefore, a sharp peak is formed on the energy spectrum
in the range of low frequencies. This peak should be a key criterion to detect
the possible existence of noncommutativity of space-time in the background of
our universe and a critical test for breaking lorentz symmetry in local field
theory. Adding all possible effects on the evolution of GWB, we give some new
information of the tensor power spectrum and its energy spectrum which may be
probed in the future cosmological observations.
