Primordial fluctuations

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:

${\displaystyle \delta (\overrightarrow{x})\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\rho (\overrightarrow{x})}{\overline{\rho }}-1=\int \text{d}k{\delta }_k{e}^{i\overrightarrow{k}\cdot \overrightarrow{x}},}$

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

${\displaystyle ⟨{\delta }_k{\delta }_{k^{\prime }}⟩=\frac{2{\pi }^2}{k^3}{\delta }_D(k-k^{\prime })\mathcal{P}(k),}$

where ${\displaystyle {\delta }_D}$ is the Dirac delta function and angle brackets denote an ensemble average.

There are both scalar and tensor modes of fluctuations.

Scalar modes

Scalar modes have the power spectrum defined as the mean squared density fluctuation for a specific wavenumber ${\displaystyle k}$, i.e., the average fluctuation amplitude at a given scale:

${\displaystyle {\mathcal{P}}_{\mathrm{s}}(k)=⟨{\delta }_k{⟩}^2.}$

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

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

For scalar fluctuations, ${\displaystyle n_{\mathrm{s}}}$ is referred to as the scalar spectral index, with ${\displaystyle n_{\mathrm{s}}=1}$ corresponding to scale invariant fluctuations (not scale invariant in ${\displaystyle \delta }$ but in the comoving curvature perturbation ${\displaystyle \zeta }$ for which the power ${\displaystyle {\mathcal{P}}_{\zeta }(k)\propto k^{n_s-1}}$ is indeed invariant with ${\displaystyle k}$ when ${\displaystyle n_s=1}$).

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 ${\displaystyle n_s>1}$. On the other hand, models such as monomial potentials predict a red spectral index ${\displaystyle n_s<1}$. Planck provides a value of ${\displaystyle n_s=0.968\pm 0.006}$.

Tensor modes

The presence of primordial tensor fluctuations is predicted by many inflationary models. As with scalar fluctuations, tensor fluctuations are expected to follow a power law and are parameterized by the tensor index (the tensor version of the scalar index). The ratio of the tensor to scalar power spectra is given by

${\displaystyle r=\frac{2|{\delta }_h{|}^2}{|{\delta }_R{|}^2},}$

where the 2 arises due to the two polarizations of the tensor modes. 2015 CMB data from the Planck satellite gives a constraint of ${\displaystyle r<0.11}$.