content_image/363629.jpg content_image/363630.jpg content_image/363631.jpg content_image/363632.jpg content_image/363633.jpg
negligible during slow-roll inflation. They argue that this mechanism could only be con- sidered as a phenomenological idea that still lacks some theoretical support and an explicit regime where this can be realized. 

## 4. Assisted inflation 

There could be many more light fields during inflation, they could collectively assist inflation by increasing the effective friction term for all the individual fields [157, 285, 286, 548–551]. This idea can be illustrated with the help of ′ m ′ identical scalar fields with an exponential potentials [157]: 

For a particular solution; where all the scalar fields are equal: φ$_{1}$ = φ$_{2}$ = · · · = φ$_{m}$ . 

<img src='content_image/363629.jpg'>

<img src='content_image/363630.jpg'>

<img src='content_image/363631.jpg'>

These can be mapped to the equations of a model with a single scalar field ˜ φ by the redefi- nitions 

<img src='content_image/363632.jpg'>

so the expansion rate is a ∝ t ˜ $^{p}$, provided that ˜ p > 1 / 3. The expansion becomes quicker the more scalar fields there are. In particular, potentials with p < 1, which for a single field are unable to support inflation, can do so as long as there are enough scalar fields to make mp > 1. 

In order to calculate the density perturbation produced in multi-scalar field models, we recall the results from [213]: 

<img src='content_image/363633.jpg'>

where P R is the spectrum of the curvature perturbation R , N is the number of e -foldings of inflationary expansion remaining, and there is a summation over i and j . Since N = − ∫ H dt , we have ∂N ∂φ$_{i}$ ˙ φ$_{i}$ = − H , (193) 

