CosmicMicrowaveBackground

The Gaussian process is a generalization of the Gaussian probability distribution and is specified by its mean and covariance functions. A Gaussian process can be used as a prior probability distribution over functions in Bayesian inference. These properties make it a useful tool in a variety of fields, including physics and cosmology.

In the context of the cosmic microwave background (CMB) and inflation theory, the Gaussian process hypothesis is linked to the idea that the fluctuations in the CMB are Gaussian-distributed. Inflation theory posits that the universe underwent a period of rapid expansion shortly after the Big Bang, and this led to quantum fluctuations being 'stretched' into macroscopic variations in the energy density of the universe. These variations were then imprinted in the CMB, a snapshot of the universe roughly 380,000 years after the Big Bang.

If these fluctuations are indeed Gaussian-distributed, the power spectrum of the CMB (i.e., the magnitude of the fluctuations as a function of angular scale) should be a straightforward prediction of inflation theory. Specifically, the power spectrum $P(k)$ should be a power-law function of the wavenumber $k$, or equivalently the multipole moment $l$ in the spherical harmonic decomposition of the CMB:

$$ P(k) \sim A_s \left(\frac{k}{k_{\text{pivot}}}\right)^{n_s - 1} $$

where $A_s$ is the amplitude of the fluctuations, $k_{\text{pivot}}$ is a reference wavenumber, and $n_s$ is the spectral index, which quantifies the slight deviation from a pure power law predicted by simplest inflation models ($n_s=1$ corresponds to a scale-invariant spectrum). This equation is typically probed using measurements of the CMB anisotropy $C_l$, where $l$ is the multipole moment:

$$C_l \sim \frac{2}{\pi} \int k^2 P(k) |\Theta_l(k)|^2dk$$

where $\Theta_l(k)$ is the radiation transfer function that depends on the cosmological model and the physics of recombination.

The CMB power spectrum has been measured by experiments like COBE, WMAP, Planck, and more recent experiments with great precision. The results are in excellent agreement with the inflationary prediction of a nearly scale-invariant spectrum ($n_s$ slightly less than 1), providing strong evidence for inflation and the Gaussian process hypothesis.

If the primordial fluctuations were not Gaussian, additional non-Gaussian features (higher order correlations, referred to as bispectrum, trispectrum, etc.) would be present in the CMB. Such non-Gaussianities could give clues about the physics of inflation, and are the subject of ongoing research. The non-Gaussian features can be quantified by parameters $f_{NL}$, $g_{NL}$, etc., that parameterize the deviations from Gaussianity in the primordial fluctuations. So far, no significant non-Gaussianities have been detected in the CMB, consistent with the simplest models of inflation.