The Cosmic Microwave Background Spectrum and a Determination of Fractal Space Dimensionality

TL;DR

This paper constrains the possible deviation of space dimensionality from three using cosmic microwave background data, finding it to be extremely close to three, thus limiting theories proposing fractal or non-integer dimensions of space.

Contribution

It introduces a method to estimate fractal space dimensionality from CMB spectrum data by generalizing Planck's law to non-integer dimensions, providing an upper limit on dimensional deviations.

Findings

The space dimensionality deviation psilonrom 3 is constrained to .000006.

The CMB spectrum fits well with a generalized Planck distribution for psilonlose to zero.

The estimated psilonligns with quantum field theory results on different scales.

Abstract

The possibility to constrain fractal space dimensionality from Astrophysics and other areas is briefly reviewed. Assuming such dimensionality to be $3 + \epsilon$, a limit to $\epsilon$ can be inferred from COBE satellite data. The available data for the cosmic microwave background radiation spectrum are fitted by a Planck's radiation distribution generalized to non integer space dimensionality. Our analysis shows that the shape of the CMBR spectrum, which does not depend on the absolute normalization, is correctly described from this distribution provided the absolute temperature is equal to 2.726 $\pm$ $0.003\times 10^{-2}$ K and $\epsilon = - (0.957 \pm 0.006) \times 10^{-5}$. This value for $\epsilon$ is shown to be consistent with what is found on a very different spatial scale based on a quantum field phenomenon. The $|\epsilon|$ is interpreted as an upper limit for how much space dimensionality could have deviated from three. In other words, this is the maximum fluctuation space dimensionality should have experienced in a spatial and temporal scale compared to that of the decoupling era.