# Primordial fluctuations

Primordial fluctuations are density variations in the early universe which are considered the seeds of all structure in the universe. Currently, the most widely accepted explanation for their origin is in the context of cosmic inflation. According to the inflationary paradigm, the exponential growth of the scale factor during inflation caused quantum fluctuations of the inflaton field to be stretched to macroscopic scales, and, upon leaving the horizon, to "freeze in". At the later stages of radiation- and matter-domination, these fluctuations re-entered the horizon, and thus set the initial conditions for structure formation.

The statistical properties of the primordial fluctuations can be inferred from observations of anisotropies in the cosmic microwave background and from measurements of the distribution of matter, e.g., galaxy redshift surveys. Since the fluctuations are believed to arise from inflation, such measurements can also set constraints on parameters within inflationary theory.

## 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 $\rho$ is the energy density, $\bar{\rho}$ its average and $k$ the wavenumber of the fluctuations. The power spectrum $\mathcal{P}(k)$ can then be defined via the ensemble average of the Fourier components:

$\langle \delta_k \delta_{k'} \rangle = \frac{2 \pi^2}{k^3} \, \delta(k-k') \, \mathcal{P}(k).$

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

$\mathcal{P}_\mathrm{s}(k) \propto k^{n_\mathrm{s} - 1}.$

For scalar fluctuations, $n_\mathrm{s}$ is referred to as the scalar spectral index, with $n_\mathrm{s} = 1$ corresponding to scale invariant fluctuations.