content_image/356467.jpg content_image/356468.jpg content_image/356469.jpg content_image/356470.jpg
where in our case each term in the sum is the same, yielding 

<img src='content_image/356467.jpg'>

Note that this last expression only contains one of the scalar fields, chosen arbitrarily to be φ$_{1}$ . This estimation for the spectral tilt is given by [213]: 

where there is a summation over repeated indices and the commas indicate derivatives with respect to the corresponding field component. Under our assumptions, the complicated second term on the right-hand side of the above equation cancels out, and Eq. (195) reduces to the simple form 2 

<img src='content_image/356468.jpg'>

<img src='content_image/356469.jpg'>

This result shows that the spectral index also matches that produced by a single scalar field with ˜ p = mp . The more scalar fields there are, the closer to scale-invariance is the spectrum that they produce. Note however that if the fields have such steep potentials as to be individually non-inflationary, p < 1, then many fields are needed before the spectrum is flat enough. The above calculation can be repeated for arbitrary slopes, p$_{i}$ in Eq. (188). In which case the spectral tilt would have been given by n = 1 − 2 / ˜ p , where ˜ p = ∑ p$_{i}$ . The above scenario has been generalized to study arbitrary exponential potentials with couplings, V = ∑ $^{n}$z$_{s}$ exp( ∑ $^{m}$α$_{sj}$ φ$_{j}$ ) in Ref. [548], see also [551]. Such potentials are expected to arise in dimensionally reduced SUGRA models [552]. 

One particular nice observation for m scalar potentials of chaotic type, V ∼ ∑ $_{i}$f ( φ n $_{i}$/M n − 4 P ) (for n ≥ 4), is that inflation can now be driven at VEVs below the Planck scale [285, 286, 549, 550] $^{52}$. The effective slow-roll parameters are given by: ϵ$_{eff}$ = ϵ/m ≪ 1 and | η$_{eff}$ | = | η | /m ≪ 1, where ϵ, η are the slow-roll parameters for the individual fields. Inflation can now occur for field VEVs [286]: 

<img src='content_image/356470.jpg'>

