Editing Primordial fluctuations (section)

===Scalar modes===
Scalar modes have the power spectrum defined as the mean squared density fluctuation for a specific wavenumber <math>k</math>, i.e., the average fluctuation amplitude at a given scale:
:<math>\mathcal{P}_\mathrm{s}(k) = \langle\delta_k\rangle^2.</math>

Many inflationary models predict that the scalar component of the fluctuations obeys a [[power law]]{{why|date=March 2017}} in which
:<math>\mathcal{P}_\mathrm{s}(k) \propto k^{n_\mathrm{s}}.</math>
For scalar fluctuations, <math>n_\mathrm{s}</math> is referred to as the scalar [[spectral index]], with <math>n_\mathrm{s} = 1</math> corresponding to [[scale invariance|scale invariant]] fluctuations (not scale invariant in <math>\delta</math> but in the comoving curvature perturbation <math>\zeta</math> for which the power <math>\mathcal{P}_{\zeta}(k) \propto k^{n_s-1}</math> is indeed invariant with <math>k</math> when <math>n_s=1</math>).<ref>{{cite book|author=Liddle & Lyth |title=Cosmological inflation and large-scale structure |page=75}}</ref>

The scalar ''spectral index'' 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 <math>n_s > 1</math>.  On the other hand, models such as monomial potentials predict a red spectral index <math>n_s < 1</math>.  Planck provides a value of <math>n_s = 0.968 \pm 0.006</math>.<ref name=":0" />