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The existence of the cosmological particle horizon as the maximum measurablelength $l_{max}$ in the universe leads to a generalization of the quantumuncertainty principle (GUP) to the form $\Delta x \Delta p \geq\frac{\hbar}{2}\frac{1}{1-\alpha\Delta x^2} $, where $\alpha\equivl_{max}^{-2}$. The effects of this GUP on simple quantum mechanical systems hasbeen shown recently by one of the authors\cite{Perivolaropoulos:2017rgq} to beextremely small (beyond current measurements) due to the extremely large scaleof the current particle horizon. This is not the case in the Early Universeduring the quantum generation of the inflationary primordial fluctuationspectrum. We estimate the effects of such GUP on the primordial fluctuationspectrum and on the corresponding spectral index. We generalize the fieldcommutation (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 conjugatemomentum. 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 $V_*\simeql_{max}^3$ is the volume corresponding to $l_{max}$) and $P_S^{(0)}(k)$ is thestandard primordial spectrum obtained in the context of the Heisenberguncertainty principle ($\mu=0$). We show that the predicted scalar spectralindex is $n_s=1-\lambda-\frac{\bar{\mu}}{k}$ where $\lambda$ is a slow-rollparameter. Using observational constraints on the scale dependence of thespectral index $n_s$ we show that the $2\sigma$ range of $\alpha$ correspondsto $l_{max}\gtrsim 10^{26} m $ which is of the same order as the currentparticle horizon.

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