content_image/349086.jpg content_image/349087.jpg content_image/349088.jpg
As any massless field, the gravitational waves also feel the quantum fluctuations in an expanding background. The spectrum mimics that of Eq. (16) 

<img src='content_image/349086.jpg'>

Note that the spectrum has a Planck mass suppression, which suggests that the ampli- tude of the gravitational waves is smaller compared to that of the adiabatic perturbations. Therefore it is usually assumed that their contribution to the CMB anisotropy is small. The corresponding spectral index can be expanded in terms of the slow-roll parameters at first order as 10 

<img src='content_image/349087.jpg'>

Note that the tensor spectral index is negative. It is expected that PLANCK could detect gravity waves if r > ∼ 0 . 1, however the spectral index will be hard to measure in forthcoming experiments. The primordial gravity waves can be generated for large field value inflationary models. Using the definition of the number of e-foldings it is possible to derive the range of Δ φ (see for instance [210–212]) 

## C. Multi-field perturbations 

<img src='content_image/349088.jpg'>

In multi-field inflation models contributions to the density perturbations come from all the fields. However unlike in a single scalar case, in the multi-field case there might not be a unique late time trajectory corresponding to all the fields. In a very few cases it is possible to obtain a late time attractor behavior of all the fields; an example is assisted inflation [157]. If there is no late time attractor then different trajectories inherit difference in perturbations, known as entropy perturbations, which opens new set of constraints which we will discuss below [10, 183, 213–215]. 

