Inflation: A graceful entrance from loop quantum cosmology

The evolution of a scalar field is explored taking into account the presence of a background fluid in a positively curved universe in the framework of loop quantum cosmology. Though the mechanism that provides the initial conditions for inflation extensively studied in the literature is still available in this setup, it demands that the initial kinetic energy of the field be comparable to the energy density of the background fluid if the field is initially situated at the minimum of the potential. It is found, however, that for potentials with a minimum such as the chaotic inflation model, there is an additional mechanism that can provide the correct initial conditions for successful inflation even if initially the kinetic energy of the field is subdominant by many orders of magnitude. In this latter mechanism the field switches direction when the Universe is still in the expanding phase. The kinetic energy gained while the field rolls down the potential is subsequently enhanced when the Universe enters the collapsing phase pushing the field one step up the potential. This behavior is repeated on every cycle of contraction and expansion of the Universe until the field becomes dominant and inflation follows.

Figure 1

Dependence of $v$ on the critical value $q_c$. From top to bottom the curves represent $r=0$, 0.3, 0.6, 0.9, and 1.2. Solid lines represent stable critical points and dashed lines represent saddle points. Dotted lines correspond to the position of the critical points if condition (25) were satisfied.

Figure 2

Areas of the parameter space $(v,r)$ where there exists one region of exclusion (unshaded area) and where there are two exclusion regions (shaded area). Regions of exclusion are regions in the phase space where the kinetic energy of the field is negative.

Figure 3

Illustrating the phase space trajectories (solid lines) and regions of exclusion (shaded areas) for $r=0.6$ and varying $v$. The trajectories evolve counterclockwise.

Figure 4

Evolution of $q$ and the potential $V$, with time. The initial conditions are ${\varphi }_{\mathrm{init}}=0$, $H_{\mathrm{init}}=0$, $q_{\mathrm{init}}=0.866$, $j=1.94\times {10}^{11}$, and $r=0.6$. The $V$ axis is labeled in Planck units.

Figure 5

Evolution of the energy densities with the scale factor. The scalar field is represented by the solid line, the background fluid by the dashed line, and the curvature term by the dash-dotted line. We have used $\alpha ={10}^{-20}$, ${\varphi }_{\mathrm{init}}=0$, $q_{\mathrm{init}}=4\times {10}^{-4}$, and $j=9\times {10}^7$ for a quadratic potential $V=m^2{\varphi }^2/2$ with $m={10}^{-6}{m}_{\mathrm{Pl}}$. The vertical axis is labeled in Planck units.

Figure 6

The dashed line represents the lower limit on $j$ resulting from requiring that the field freezes before the bounce at $q_2$. This is estimated using Eqs. (32, 33). The solid line represents the borderline between an evolution that pushes the field up the potential in successive steps (below the curve) and an evolution where the field is displaced from where the initial position stops before the bounce, turns around and is displaced again in the opposite direction in the collapsing phase (above the curve). This curve was obtained numerically and fits to $q_f=q_2^{0.82}$ accurately. The dash-dotted line gives a lower bound on the values of $j$ that satisfy the consistency condition $H{a}_i<1$.

Figure 7

Maximum achieved value of the field (in Planck units) with respect to the quantization parameter $j$ for three different sets of initial conditions. Upper panel: $q_{\mathrm{init}}=4\times {10}^{-4}$ and $\alpha ={10}^{-20}$; middle panel: $q_{\mathrm{init}}=4\times {10}^{-4}$ and $\alpha =2\times {10}^{-15}$; lower panel: $q_{\mathrm{init}}=2.5\times {10}^{-3}$ and $\alpha =2\times {10}^{-15}$. Dashed lines represent $\varphi =3m_{\mathrm{Pl}}$. The vertical axes are labeled in Planck units.

Figure 8

Maximum achieved value of the Hubble ratio (in Planck units) with respect to the quantization parameter $j$ for three different sets of initial conditions. Upper panel: $q_{\mathrm{init}}=4\times {10}^{-4}$ and $\alpha ={10}^{-20}$; middle panel: $q_{\mathrm{init}}=4\times {10}^{-4}$ and $\alpha =2\times {10}^{-15}$; lower panel: $q_{\mathrm{init}}=2.5\times {10}^{-3}$ and $\alpha =2\times {10}^{-15}$. Dashed lines represent the consistency constraint $H{a}_i<1$. The vertical axes are labeled in Planck units.