where in our case each term in the sum is the same, yielding 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 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]: 89 52 The double inflation model has been studied extensively with two such fields, V = m 2 $_{1}$φ 2 1 + m$_{2}$φ 2 $_{2}$, in Refs. [214, 217, 553–556]. In general one could expect: V ∼ ∑ $_{i}$m 2 $_{i}$φ 2 i [285, 549, 550], or V ∼ λ$_{i}$ ( φ n i /M n − 4 P ), where n ≥ 4 [286], where φ$_{i}$ ≪ M$_{P}$ . P R = ( H 2 π ) 2 1 m H 2 ˙ φ 2 1 . (194) n − 1 = 2 ˙ H H 2 − 2 ∂N ∂φ$_{i}$ ( ˙ φ$_{i}$ ˙ φ$_{j}$ M 2 $_{P}$H 2 − M 2 $_{P}$V$_{,i,j}$ V ) ∂N ∂φ$_{j}$ δ$_{ij}$ ∂N ∂φ$_{i}$ ∂N ∂φ$_{j}$ , (195) 1 − n = − 2 ˙ H H 2 = m 2 Pl 8 π [ ∂V ( φ 1 ) ∂φ 1 V ( φ$_{1}$ ) ] 2 = 2 mp . (196) Δ φ M$_{P}$ ∼ ( 600 m ) ( N$_{Q}$ 60 ) ( ϵ$_{eff}$ 2 ) 1 / 2 ≪ 1 , (197)