  We propose a variation of spacetime noncommutative field theory to realize
the stringy spacetime uncertainty relation without breaking any of the global
symmetries of the homogeneous isotropic universe. We study the spectrum of
metric perturbations in this model for a wide class of accelerating background
cosmologies. Spacetime noncommutativity leads to a coupling between the
fluctuation modes and the background cosmology which is nonlocal in time. For
each mode, there is a critical time at which the spacetime uncertainty relation
is saturated. This is the time when the mode is generated. These effects lead
to a spectrum of fluctuations whose spectral index is different from what is
obtained for commutative spacetime in the infrared region, but is unchanged in
the ultraviolet region. In the special case of an exponentially expanding
background, we find a scale-invariant spectrum. but with a different magnitude
than in the context of commutative spacetime if the Hubble constant is above
the string scale.
