Primordial Fluctuations - Formalism

Formalism

Primordial fluctuations are typically quantified by a power spectrum which gives the power of the variations as a function of spatial scale. Within this formalism, one usually considers the fractional energy density of the fluctuations, given by:

\delta(\vec{x}) \ \stackrel{\mathrm{def}}{=}\ \frac{\rho(\vec{x})}{\bar{\rho}} - 1 = \int \text{d}k \; \delta_k \, e^{i\vec{k} \cdot \vec{x}},

where is the energy density, its average and the wavenumber of the fluctuations. The power spectrum can then be defined via the ensemble average of the Fourier components:

Many inflationary models predict that the scalar component of the fluctuations obeys a power law in which

For scalar fluctuations, is referred to as the scalar spectral index, with corresponding to scale invariant fluctuations.

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