The Heisenberg limit at cosmological scales

Abstract: For an observation time {equal to} the universe age, the Heisenberg principle fixes the value of the smallest measurable mass at $m_{\rm H}=1.35 \times 10^{-69}$ kg and prevents to probe the masslessness for any particle using a balance. The corresponding reduced Compton length to $m_{\rm H}$ is $\lambdabar_{\rm H}$, and represents the length limit beyond which masslessness cannot be proved using a metre ruler. In turns, $\lambdabar_{\rm H}$ is equated to the luminosity distance $d_{\rm H}$ which corresponds to a red shift $z_{\rm H}$. When using the Concordance-Model parameters, we get $d_{\rm H} = 8.4$ Gpc and $z_{\rm H}=1.3$. Remarkably, $d_{\rm H}$ falls quite short to the radius of the {\it observable} universe. According to this result, tensions in cosmological parameters could be nothing else but due to comparing data inside and beyond $z_{\rm H}$. Finally, in terms of quantum quantities, the expansion constant $H_0$ reveals to be one order of magnitude above the smallest measurable energy, divided by the Planck constant