Primordial Power Spectra of Cosmological Fluctuations with Generalized
  Uncertainty Principle and Maximum Length Quantum Mechanics
  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 effects of this GUP on simple quantum mechanical systems has
been shown recently by one of the authors\cite{Perivolaropoulos:2017rgq} to be
extremely small (beyond current measurements) due to the extremely large scale
of the current particle horizon. This is not the case in the Early Universe
during the quantum generation of the inflationary primordial fluctuation
spectrum. We estimate the effects of such GUP on the primordial fluctuation
spectrum and on the corresponding spectral index. We generalize the field
commutation (GFC) relation to
$[\varphi(k),\pi_{\varphi}(k')]$=$i\delta(k-k')\frac{1}{1-\mu\varphi^2(k)}$,
where $\mu\sim \alpha^2\equiv l_{max}^{-4}$ is a GFC parameter, $\varphi$
denotes a scalar field and $\pi_{\varphi}$ denotes its canonical conjugate
momentum. We obtain the predicted primordial perturbation spectrum as
$P_S(k)=P_S^{(0)}(k)\left(1+\frac{\bar{\mu}}{k}\right)$ where
$\bar{\mu}\equiv\mu V_* \simeq \sqrt{\alpha}= l_{max}^{-1}$ (here $V_*\simeq
l_{max}^3$ is the volume corresponding to $l_{max}$) and $P_S^{(0)}(k)$ is the
standard primordial spectrum obtained in the context of the Heisenberg
uncertainty principle ($\mu=0$). We show that the predicted scalar spectral
index is $n_s=1-\lambda-\frac{\bar{\mu}}{k}$ where $\lambda$ is a slow-roll
parameter. Using observational constraints on the scale dependence of the
spectral index $n_s$ we show that 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.
