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 $$m_\mathrm{H}=1.35 \times 10^{-69}$$ kg and prevents to probe the masslessness for any particle using a balance. The corresponding reduced Compton length to $$m_\mathrm{H}$$ is , and represents the length limit beyond which masslessness cannot be proved using a metre ruler. In turns, is equated to the luminosity distance $$d_\mathrm{H}$$ which corresponds to a red shift $$z_\mathrm{H}$$ . When using the Concordance-Model parameters, we get $$d_\mathrm{H} = 8.4$$ Gpc and $$z_\mathrm{H}=1.3$$ . Remarkably, $$d_\mathrm{H}$$ falls quite short to the radius of the observable universe. According to this result, tensions in cosmological parameters could be nothing else but due to comparing data inside and beyond $$z_\mathrm{H}$$ . 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.