Degree of scale invariance of inflationary perturbations

Most inflationary models predict that the power-law index of the spectrum of density perturbations is close to 1, though not precisely equal to 1, $|n-1|\sim \mathcal{O}\mathrm{}\left(0.1\right),$ implying that the spectrum of density perturbations is nearly, but not exactly, scale invariant. However, there are models where n is significantly less than 1 $(n\sim 0.7);$ a spectral index significantly greater than 1 is more difficult to achieve. Without recourse to specific models, we show very generally and very explicitly that $n\approx 1$ is a consequence of the slow-roll conditions for inflation and “naturalness,” and thus, that near scale invariance is a generic prediction of inflation and a test of the inflationary framework. We derive the conditions needed to deviate significantly from scale invariance, and then show, by explicit construction, the existence of smooth potentials that satisfy all the conditions for successful inflation and give n as large as 2.