Natural inflation from polymer quantization

We study the polymer quantization of a homogeneous massive scalar field in the early Universe using a prescription inequivalent to those previously appearing in the literature. Specifically, we assume a Hilbert space for which the scalar field momentum is well defined but its amplitude is not. This is closer in spirit to the quantization scheme of loop quantum gravity, in which no unique configuration operator exists. We show that in the semiclassical approximation, the main effect of this polymer quantization scheme is to compactify the phase space of chaotic inflation in the field amplitude direction. This gives rise to an effective scalar potential closely resembling that of hybrid natural inflation. Unlike polymer schemes in which the scalar field amplitude is well defined, the semiclassical dynamics involves a past cosmological singularity; i.e., this approach does not mitigate the big bang.

Figure 1

Phase portraits for the $l=0$ case. We have selected $\beta =1.0$. The dark circles are the attractive fixed points ${\mathrm{\Gamma }}_{+}$ and the light circles are the repulsive fixed points ${\mathrm{\Gamma }}_{-}$. Note that the periodicity of the orbits in the $\mathrm{\Phi }$ direction. The shaded regions indicate when the condition for inflation ${\epsilon }_{\mathrm{H}}<1$ is satisfied.

Figure 2

Phase portraits for the $l=0$ case drawn on a cylinder. We have selected $\beta =1.0$ and $Z_0=1.0$.

Figure 3

Time evolution of the ${\epsilon }_{\mathrm{H}}$ slow roll parameter (top) and phase portrait (bottom) for the $(l,\beta ,Z_0)=(0,1,0.8)$. The shaded regions indicate when the condition for “slow-roll” inflation is satisfied: ${\epsilon }_{\mathrm{H}}<0.5$. Trajectories colored in red exit the slow roll inflationary phase at finite time and later reenter.

Figure 4

These figures show how the duration of the initial slow-roll inflationary phase (as measured in $e$-folds $N$) varies with $\beta$, $Z_0$, and the initial field amplitude ${\mathrm{\Phi }}_0$ when $l=0$. We chose initial data with $d\mathrm{\Phi }/d\tau =2$. Note that some initial data leads to an infinite amount of initial inflation, as indicated by the dotted lines. The trajectories with finite $N$ are colored red in Fig. 3.