  We study the primordial gravitational wave background produced in models of
single field inflation. Using the inflationary flow approach, we investigate
the amplitude of gravitational wave spectrum in the frequency range 1 mHz - 1
Hz pertinent to future space-based laser interferometers. For models that
satisfy the current observational constraint on the tensor-to-scalar ratio,
r<0.36, we derive a strict upper bound of omega_{gw}<1.6 x 10^{-15},
independent of the form of the inflationary potential. Applying, in addition,
the observational constraints on the spectral index and its running, omega_{gw}
is expected to be considerably lower than this bound unless the shape of the
potential is finely tuned. We contrast our numerical results with those based
on simple power-law extrapolation of the tensor power spectrum from CMB scales.
In addition to single field inflation, we summarise a number of other possible
cosmological sources of primordial gravitational waves and assess what might be
learnt from direct detection experiments such as LISA, Big Bang Observer and
beyond.
