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 $[\phi (k),{\pi }_{\phi }(k^{\mathrm{\prime }})]$=$i\delta (k-k^{\mathrm{\prime }})\frac{1}{1-\mu {\phi }^2(k)}$, where $\mu \sim {\alpha }^2\equiv l_{max}^{-4}$ is a GFC parameter, $\phi$ denotes a scalar field and ${\pi }_{\phi }$ denotes its canonical conjugate momentum. We obtain the predicted primordial perturbation spectrum as $P_S(k)=P_S^{(0)}(k)(1+\frac{\bar{\mu }}{k})$ where $\bar{\mu }\equiv \mu V_{\ast }\simeq \sqrt{\alpha }=l_{max}^{-1}$ (here $V_*\simeq l_{max}^3$isthevolumecorrespondingto$l_{max}$)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.