Statistical Entropy of Cosmic Microwave Background Fluctuation Fields

Povzetek

The cosmic microwave background is close to a Gaussian random field, but small departures from Gaussianity,
seeded by inflationary dynamics and reprocessed by late-time effects such as lensing and the integrated
Sachs-Wolfe effect, are of direct cosmological interest, and Shannon-type configuration entropy offers a
model-independent way to quantify them beyond the standard bispectrum-based statistics. We computed
pixel-space and wavelet-domain statistical entropy for 36 full-sky temperature fluctuation maps, comprising 6
realizations each of a pure Gaussian random field, an equilateral-type non-Gaussian field, a local-type
non-Gaussian field, and three field realizations built to approximate Planck-like foreground-cleaned temperature
maps, all generated at HEALPix resolution N_side = 512 with matched angular power spectra. Mean pixel-space
entropy for the pure Gaussian realizations was 7.842 ± 0.019 bits, statistically indistinguishable from the theoretical
maximum-entropy value of 7.851 bits for a Gaussian field with matched variance (difference 0.11%, within 1 sigma),
whereas local-type non-Gaussian realizations showed a mean entropy deficit of 0.086 ± 0.014 bits relative to the
Gaussian case, a 1.1% reduction significant at p < 0.001 by a two-sample t-test across the 6 realizations of each
type. Equilateral-type non-Gaussianity produced a smaller but still detectable deficit of 0.031 ± 0.011 bits (0.4%
reduction, p = 0.008), consistent with its comparatively weaker imprint on the pixel-amplitude distribution used here.
Wavelet-domain entropy, computed across 5 scale bands spanning angular scales of 2 to 20 degrees, showed the
local-type entropy deficit concentrated at the two largest scale bands, accounting for 68% of the total deficit, while
the equilateral-type deficit was more evenly distributed across scales, differing by less than 15 percentage points
between the largest and smallest bands. Applying the same pipeline to the Planck-like simulated maps yielded a
pixel-space entropy of 7.798 ± 0.024 bits, a deficit of 0.044 bits relative to pure Gaussian, roughly half the
magnitude of the local-type non-Gaussian deficit and attributable in the simulation to a combination of realistic noise
inhomogeneity and residual foreground structure rather than primordial non-Gaussianity alone. These results
demonstrate that pixel- and wavelet-domain statistical entropy can detect non-Gaussian signatures at percent-level
precision using sample sizes an order of magnitude smaller than typically required for robust bispectrum estimation,
and that scale-resolved entropy decomposition helps distinguish candidate non-Gaussianity types by their
characteristic scale dependence.