  We study the implications of recent indications for a red spectrum of
primordial density perturbations for the detection of inflationary
gravitational waves (IGWs) with forthcoming cosmic microwave background
experiments. We find that if inflation occurs with a single field with an
inflaton potential minimized at V=0, then Planck will be able to detect IGWs at
better than 2$\sigma$ confidence level, unless the inflaton potential is a
power law with a very weak power. The proposed satellite missions of the Cosmic
Vision and Inflation Probe programs will be able to detect IGWs from all the
models we have surveyed at better than 5$\sigma$ confidence level. We provide
an example of what is required if the IGW background is to remain undetected
even by these latter experiments.
