Primordial-Power-Spectra-of-Cosmological-Fluctuations-with-GUP

This repository contains the Mathematica files of arXiv:1907.12594

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 $\Delta x \Delta p \geq \frac{\hbar}{2}\frac{1}{1-\alpha\Delta x^2} $, where $\alpha\equiv l_{max}^{-2}$. The implication of this GUP and the corresponding generalized commutation relation $[x,p] =i\hbar \frac{1}{1-\alpha x^2}$ on simple quantum mechanical systems has been discussed recently \cite{Perivolaropoulos:2017rgq} by one of the authors and shown to have extremely small (beyond current measurements) effects of the energy spectra of these systems due to the extremely large scale of the current particle horizon. This may not the case in the Early Universe during the quantum generation of the inflationary primordial fluctuation spectrum. Here we estimate the effects of such 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 $[\varphi(\bold{k}),\pi_{\varphi}(\bold{k'})]=i\delta(\bold{k}-\bold{k'})\frac{1}{1-\mu\varphi^2(\bold{k})}$, where $\mu\simeq\alpha^2\equiv l_{max}^{-4}$ is a GFC parameter, $\varphi$ denotes a scalar field and $\pi_{\varphi}$ denotes its canonical conjugate momentum. In the context of this GFC we use standard methods to obtain the primordial scalar perturbations spectrum and show that it is of the form $P_S(k)=P_S^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)$ where $\bar{\mu}\equiv\mu V_* \simeq \sqrt{\alpha}= l_{max}^{-1}$ (here $V_*\simeq l_{max}^3$ 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 $\mu=0$). We show that the scalar spectral index predicted by the model, defined from $P_S(k)=A_Sk^{n_s -1}$ is running and may be written as $n_s=1-\lambda-\frac{\bar{\mu}}{k}$ with $\lambda=6\epsilon-2\eta$ (where $\epsilon$ and $\eta$ 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 $\bar{\mu}$ as $\bar{\mu}=(0.9\pm 7.6)\cdot 10^{-6} h/Mpc$. Using this result we estimate the GUP parameter $\alpha\lesssim 10^{-54} m^{-2}$ at $1\sigma$ and $\alpha\lesssim 10^{-52} m^{-2}$ at $2\sigma$. The $2\sigma$ range of $\alpha$ corresponds to $l_{max}\gtrsim 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 presence of the cosmological particle horizon remains a viable assumption at the $2\sigma$ level.

Citing the paper

If you use any of the above codes or the figures in a published work please cite the following paper:

Primordial Power Spectra of Cosmological Fluctuations with Generalized Uncertainty Principle and Maximum Length Quantum Mechanics

Leandros Perivolaropoulos and Foteini Skara arXiv:1907.12594

Any further questions/comments are welcome.

Authors List

Leandros Perivolaropoulos - leandros@uoi.gr

Foteini Skara - f.skara@uoi.gr