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I've been reading that the results from the Planck satellite constrain a number called the "scalar spectral index" to be 0.96 rather than 1 at the 5-sigma level.

This is supposed to be big news, but I don't understand why (my ignorance, of course).

For instance, with an index of 0.96, can we say how many e-foldings occured during inflation? What else does it tell us?

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The scalar spectral index (usually denoted '$n_s$') describes how density fluctuations vary with scale. An index of unity means that the variations are the same on all scales. $n_s$ is an input parameter to $\Lambda$CDM, and influences the characteristic size-scales of structure formation (where this small of an adjustment has little effect). Inflationary models, on the other hand, do generally suggest particular values (often ranges) of $n_s$, and $n_s = 0.96$ is still very compatible with current models of inflation.

The plots I've seen definitely state that $n_s$ effects the number of e-folds, but I'm no expert on the subject, and will refrain from conjecturing wildly. I'd recommend taking a look here, and the references there-in.

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The scalar spectral index, like the above answer states, describes how the density fluctuations vary with scale. As the size of these fluctuations depends upon the inflaton's motion when these quantum fluctuations are becoming super-horizon sized, different inflationary potentials predict different spectral indices. These depend upon the slow roll parameters, in particular the gradient and curvature of the potential. In models where the curvature is large and positive $n_s>1$ on the other hand models such as monomial potentials predict a red spectral index $n_s<1$.

The fact that $n_s$ is significantly different from 1 adds support to the inflationary paradigm. As inflation can produce an almost-but-not-quite scale invariant spectrum of density fluctuations which are largely gaussian.

The number of e-folds are unknown. Largely because of the unknown mechanism and duration of the reheating period (and any late time entropy production can make this more uncertain). Typical canonical values are $N_e \sim50,60$. Again different inflationary models will predict different numbers of e-folds. For the monomial potential $n_s = f(N_e)$ so to get the observed value of the spectral index requires a certain number of e-folds. However the tensor-to-scalar ratio, $r$, (which is a measure of the relative power in tensor perturbations compared to scalar perturbations) is also $r\sim f(N_e)$. Bounds on $r$ then constrain $N_e$ and can help to rule out inflationary models. For example inflationary potentials which are quadratic or quartic in the field value are disfavoured as the value of $r$ they predict is too large for the observed value of $n_s$. (Note this is not necessarily true in the warm inflation scenario).

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