Oscillatory universes in loop quantum cosmology and initial conditions for inflation

Positively curved oscillatory universes are studied within the context of loop quantum cosmology subject to a consistent semiclassical treatment. The semiclassical effects are reformulated in terms of an effective phantom fluid with a variable equation of state. In cosmologies sourced by a massless scalar field, these effects lead to a universe that undergoes ever-repeating cycles of expansion and contraction. The presence of a self-interaction potential for the field breaks the symmetry of the cycles and can enable the oscillations to establish the initial conditions for successful slow-roll inflation, even when the field is initially at the minimum of its potential with a small kinetic energy. The displacement of the field from its minimum is enhanced for lower and more natural values of the parameter that sets the effective quantum gravity scale. For sufficiently small values of this parameter, the universe can enter a stage of eternal self-reproduction.

Figure 1

Time evolution of the scalar field velocity $\dot{\varphi }$, the logarithmic scale factor and the Hubble parameter when the potential $V=0$, $j=100$, $a_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}/a_{*}=0.9$ and $H_i=0$. Axes are labeled in Planck units.

Figure 2

(a) Schematically illustrating the logarithmic variation of the effective energy density of a massless scalar field (solid line) and the curvature term in the Friedmann equation (dashed line). The universe oscillates indefinitely between the intersection points of the two lines. (b) Schematically illustrating the effects of introducing a self-interaction potential for the field. The cycles are eventually broken as the potential becomes dynamically significant, thereby resulting in slow-roll inflation. In both figures, the slope of the trajectories is given by $d\ln {\rho }_{\mathrm{e}\mathrm{f}\mathrm{f}}/d\ln a=-3[1+w(a)]$.

Figure 3

Illustrating the time evolution of the scalar field $\varphi$ (top panel) and the logarithmic scale factor (middle panel) for a quadratic potential $V=m^2{\varphi }^2/2$. The bottom panel illustrates how the effective kinetic energy ${\dot{\varphi }}^2/2D$ (solid line) and potential energy (dashed line) of the field vary compared to the curvature term (dot-dashed line). The cycles end and slow-roll inflation commences at the point when the potential begins to dominate the kinetic energy. Initial conditions are chosen such that ${\varphi }_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=H_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}=0$ and $a_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}/a_{*}=0.9$, with $m={10}^{-6}{l}_{\mathrm{P}\mathrm{l}}^{-1}$ and $j=5\times {10}^{11}$ (see the text for details). The axes are labeled in Planck units.

Figure 4

Constraints on the initial velocity of the field $|\dot{\varphi }{|}_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}$ for a given initial value of the scale factor, $a_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}/a_{*}$, for different values of the quantization parameter, $j$. The axes are labeled in Planck units and the shaded areas represent the regions where constraints (11, 12) are satisfied. For all values of $j$, the area of the shaded region is finite, and the points of intersection occur farther from $a_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}/a_{*}\approx 1$ as $j$ is increased. Note that for $a_{\mathrm{i}\mathrm{n}\mathrm{i}\mathrm{t}}>a_{*}$, the universe is initially in a contracting phase and subsequently bounces, after which the behavior discussed in the text is followed.