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- W2965743411 abstract "The existence of the cosmological particle horizon as the maximum measurable length ${l}_{mathrm{max}}$ in the Universe leads to a generalization of the quantum uncertainty principle (GUP) to the form $mathrm{ensuremath{Delta}}xmathrm{ensuremath{Delta}}pensuremath{ge}phantom{rule{0ex}{0ex}}frac{ensuremath{hbar}}{2}frac{1}{1ensuremath{-}ensuremath{alpha}mathrm{ensuremath{Delta}}{x}^{2}}$, where $ensuremath{alpha}ensuremath{equiv}{l}_{mathrm{max}}^{ensuremath{-}2}$. The implication of this GUP and the corresponding generalized commutation relation $[x,p]=iensuremath{hbar}frac{1}{1ensuremath{-}ensuremath{alpha}{x}^{2}}$ 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).] 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 $[ensuremath{varphi}(mathbf{k}),{ensuremath{pi}}_{ensuremath{varphi}}({mathbf{k}}^{ensuremath{'}})]=iensuremath{delta}(mathbf{k}ensuremath{-}{mathbf{k}}^{ensuremath{'}})frac{1}{1ensuremath{-}ensuremath{mu}{ensuremath{varphi}}^{2}(mathbf{k})}$, where $ensuremath{mu}ensuremath{simeq}{ensuremath{alpha}}^{2}ensuremath{equiv}{l}_{mathrm{max}}^{ensuremath{-}4}$ is a GFC parameter, $ensuremath{varphi}$ denotes a scalar field, and ${ensuremath{pi}}_{ensuremath{varphi}}$ 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 ${P}_{S}(k)={P}_{S}^{(0)}(k)(1+frac{overline{ensuremath{mu}}}{k})$, where $overline{ensuremath{mu}}ensuremath{equiv}ensuremath{mu}{V}_{*}ensuremath{simeq}sqrt{ensuremath{alpha}}={l}_{mathrm{max}}^{ensuremath{-}1}$ (here ${V}_{*}ensuremath{simeq}{l}_{mathrm{max}}^{3}$ is the volume corresponding to the maximum measurable scale ${l}_{mathrm{max}}$) and ${P}_{S}^{(0)}(k)$ is the standard primordial spectrum obtained in the context of the Heisenberg uncertainty principle (HUP $ensuremath{mu}=0$). We show that the scalar spectral index predicted by the model, defined from ${P}_{S}(k)={A}_{S}{k}^{{n}_{s}ensuremath{-}1}$, is running and may be written as ${n}_{s}=1ensuremath{-}ensuremath{lambda}ensuremath{-}frac{overline{ensuremath{mu}}}{k}$ with $ensuremath{lambda}=6ensuremath{epsilon}ensuremath{-}2ensuremath{eta}$ (where $ensuremath{epsilon}$ and $ensuremath{eta}$ are the slow-roll parameters). Using observational constraints on the scale dependence of the spectral index ${n}_{s}$, a cosmological constraint may be imposed on $overline{ensuremath{mu}}$ as $overline{ensuremath{mu}}=(0.9ifmmodepmelsetextpmfi{}7.6)ifmmodetimeselsetexttimesfi{}{10}^{ensuremath{-}6}text{ }text{ }h/mathrm{Mpc}$. Using this result we estimate the GUP parameter $ensuremath{alpha}ensuremath{lesssim}{10}^{ensuremath{-}54}text{ }text{ }{mathrm{m}}^{ensuremath{-}2}$ at $1ensuremath{sigma}$ and $ensuremath{alpha}ensuremath{lesssim}{10}^{ensuremath{-}52}text{ }text{ }{mathrm{m}}^{ensuremath{-}2}$ at $2ensuremath{sigma}$. The $2ensuremath{sigma}$ range of $ensuremath{alpha}$ corresponds to ${l}_{mathrm{max}}ensuremath{gtrsim}{10}^{26}text{ }text{ }mathrm{m}$, 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 $2ensuremath{sigma}$ level." @default.
- W2965743411 created "2019-08-13" @default.
- W2965743411 creator A5051749552 @default.
- W2965743411 creator A5076121373 @default.
- W2965743411 date "2019-12-17" @default.
- W2965743411 modified "2023-10-03" @default.
- W2965743411 title "Primordial power spectra of cosmological fluctuations with generalized uncertainty principle and maximum length quantum mechanics" @default.
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- W2965743411 doi "https://doi.org/10.1103/physrevd.100.123527" @default.
- W2965743411 hasPublicationYear "2019" @default.