Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length quantum mechanics
Abstract
The existence of the cosmological particle horizon as the maximum measurable length l_{max} in the Universe leads to a generalization of the quantum uncertainty principle (GUP) to the form Δ x Δ p ≥ℏ/2 1/1 α Δ x^{2} , where α ≡l_{max}^{2}. The implication of this GUP and the corresponding generalized commutation relation [x ,p ]=i ℏ 1/1 α x^{2} 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)., 10.1103/PhysRevD.95.103523] 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 [φ (k ),π_{φ}(k^{'})]=i δ (k k^{'})1/1 μ φ^{2}(k ) , where μ ≃α^{2}≡l_{max}^{4} is a GFC parameter, φ denotes a scalar field, and π_{φ} 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 P_{S}(k )=P_{S}^{(0 )}(k )(1 +μ/¯k ) , where μ ¯≡μ V_{*}≃√{α }=l_{max}^{1} (here V_{*}≃l_{max}^{3} is the volume corresponding to the maximum measurable scale l_{max}) and P_{S}^{(0 )}(k ) is the standard primordial spectrum obtained in the context of the Heisenberg uncertainty principle (HUP μ =0 ). We show that the scalar spectral index predicted by the model, defined from P_{S}(k )=A_{S}k^{ns1}, is running and may be written as n_{s}=1 λ μ/¯ k with λ =6 ɛ 2 η (where ɛ and η are the slowroll parameters). Using observational constraints on the scale dependence of the spectral index n_{s}, a cosmological constraint may be imposed on μ ¯ as μ ¯ =(0.9 ±7.6 )×10^{6} h /Mpc . Using this result we estimate the GUP parameter α ≲10^{54} m^{2} at 1 σ and α ≲10^{52} m^{2} at 2 σ . The 2 σ range of α corresponds to l_{max}≳10^{26} 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 σ level.
 Publication:

Physical Review D
 Pub Date:
 December 2019
 DOI:
 10.1103/PhysRevD.100.123527
 arXiv:
 arXiv:1907.12594
 Bibcode:
 2019PhRvD.100l3527S
 Keywords:

 General Relativity and Quantum Cosmology;
 Astrophysics  Cosmology and Nongalactic Astrophysics;
 High Energy Physics  Phenomenology;
 High Energy Physics  Theory
 EPrint:
 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/GUPGFC/