2019/07/31 by Foteini Skara, F. Skara, Leandros Perivolaropoulos +1
Mathematics · Physics and Astronomy · #Black Holes and Theoretical Physics #Cosmology and Gravitation Theories #Mathematical physics #Mathematics #Noncommutative and Quantum Gravity Theories #Physics #Quantum #Quantum mechanics #Spectral density #Spectral line #Uncertainty principle #astro-ph.CO #gr-qc #hep-ph #hep-th
paper · pdf · doi:10.1103/physrevd.100.123527
published as Phys. Rev. D 100, 123527 (2019) · 12 pages, 3 figures. Published version. Comments added, figure added, improved statistical analysis. The Mathematica file that was used for the production of the figures may be downloaded from http://leandros.physics.uoi.gr/GUP-GFC/
openalex created_date 2019/08/13 · openalex publication_date 2019/12/17 · arxiv created 2019/12/19 · arxiv updated 2019/12/25 · openalex updated_date 2026/08/05
The existence of the cosmological particle horizon as the maximum measurable length lmax in the Universe leads to a generalization of the quantum uncertainty principle (GUP) to the form \mathrm\ensuremathΔx\mathrm\ensuremathΔp\ensuremath≥\phantom\rule0ex0ex\frac\ensuremathℏ2\frac11\ensuremath-\ensuremathα\mathrm\ensuremathΔx2, where \ensuremathα\ensuremath≡lmax^\ensuremath-2. The implication of this GUP and the corresponding generalized commutation relation [x,p]=i\ensuremathℏ\frac11\ensuremath-\ensuremathαx2 on simple quantum mechanical systems has been discussed recently by one of the authors [Cosmological horizons, uncertainty principle and maximum length quantum mechanics, Phys. Rev. D 95, 103523 (2017).] and shown to have extremely small (beyond current measurements) effects on the energy spectra of these systems due to the extremely large scale of the current particle horizon. This may not be the case in the early Universe during the quantum generation of the inflationary primordial fluctuation spectrum. Here we estimate the effects of such a GUP on the primordial fluctuation spectrum and on the corresponding spectral index. In particular, motivated by the above GUP we generalize the field commutation (GFC) relation to [\ensuremathφ(k),\ensuremathπ_\ensuremathφ(k^\ensuremath')]=i\ensuremathδ(k\ensuremath-k^\ensuremath')\frac11\ensuremath-\ensuremathμ\ensuremathφ2(k), where \ensuremathμ\ensuremath≃\ensuremathα2\ensuremath≡lmax^\ensuremath-4 is a GFC parameter, \ensuremathφ denotes a scalar field, and \ensuremathπ_\ensuremathφ denotes its canonical conjugate momentum. In the context of this GFC we use standard methods to obtain the primordial scalar perturbation spectrum and show that it is of the form PS(k)=PS(0)(k)(1+\frac\ensuremathμk), where \ensuremathμ\ensuremath≡\ensuremathμV*\ensuremath≃√\ensuremathα=lmax^\ensuremath-1 (here V*\ensuremath≃lmax3 is the volume corresponding to the maximum measurable scale lmax) and PS(0)(k) is the standard primordial spectrum obtained in the context of the Heisenberg uncertainty principle (HUP \ensuremathμ=0). We show that the scalar spectral index predicted by the model, defined from PS(k)=ASk^ns\ensuremath-1, is running and may be written as ns=1\ensuremath-\ensuremathλ\ensuremath-\frac\ensuremathμk with \ensuremathλ=6\ensuremathε\ensuremath-2\ensuremathη (where \ensuremathε and \ensuremathη are the slow-roll parameters). Using observational constraints on the scale dependence of the spectral index ns, a cosmological constraint may be imposed on \ensuremathμ as \ensuremathμ=(0.9\ifmmode±\else\textpm\fi7.6)\ifmmode×\else\texttimes\fi10^\ensuremath-6 h/Mpc. Using this result we estimate the GUP parameter \ensuremathα\ensuremath\lesssim10^\ensuremath-54 m^\ensuremath-2 at 1\ensuremathσ and \ensuremathα\ensuremath\lesssim10^\ensuremath-52 m^\ensuremath-2 at 2\ensuremathσ. The 2\ensuremathσ range of \ensuremathα corresponds to lmax\ensuremath\gtrsim1026 m, which is of the same order as the current particle horizon. Thus the assumption that a maximum measurable length could emerge as a result of the presence of the cosmological particle horizon remains a viable assumption at the 2\ensuremathσ level.