Inflation in loop quantum cosmology: Dynamics and spectrum of gravitational waves

Loop quantum cosmology provides an efficient framework to study the evolution of the Universe beyond the classical Big Bang paradigm. Because of holonomy corrections, the singularity is replaced by a “bounce.” The dynamics of the background is investigated into the details, as a function of the parameters of the model. In particular, the conditions required for inflation to occur are carefully considered and are shown to be generically met. The propagation of gravitational waves is then investigated in this framework. By both numerical and analytical approaches, the primordial tensor power spectrum is computed for a wide range of parameters. Several interesting features could be observationally probed.

Figure 1

Shark fin-type evolution of a scalar field for $m={10}^{-3}{m}_{\mathrm{Pl}}$. The (red) dot represents the point where the initial conditions in classical cosmology are usually set.

Figure 2

Exemplary phase trajectories of the scalar field with $m=m_{\mathrm{Pl}}$.

Figure 3

Time evolution of the scalar field. Different evolutionary scenarios leading to a slow-roll inflation phase are displayed. The bottom (solid) line represents the symmetric case. The middle (dotted) line represents the shark fin-type evolution mostly investigated in this paper. The top (dashed) line corresponds to a larger fraction of potential energy. For all curves $m=0.01m_{\mathrm{Pl}}$.

Figure 4

Time evolution of the scale factor. Different evolutionary scenarios leading to a slow-roll inflation phase are displayed. The bottom (solid) line represents the symmetric case. The middle (dotted) line represents the shark fin-type evolution. The top (dashed) line corresponds to a larger fraction of potential energy. For all curves $m=0.01m_{\mathrm{Pl}}$.

Figure 5

The shark fin-type evolution of the scalar field for $m={10}^{-3}{m}_{\mathrm{Pl}}$. Curves from bottom to top were computed for ${\rho }_{\mathrm{c}}={10}^{-6}$, ${10}^{-4}$, ${10}^{-2}$, 1, and $100[m_{\mathrm{Pl}}^4]$, respectively.

Figure 6

Evolution of the effective mass used in the analytical approximation [Eq. (19)]. On this plot, $ϵ$ is set to zero as an example. The dashed line represents the case without a bounce. The points where the transfer matrices are computed in our model are also indicated.

Figure 7

Analytical tensor power spectra, normalized to the non-LQC-corrected spectrum, for three different values of $k_0$ in the $ϵ=0$ case. The parameters are: $k_0=0$ (solid line), $k_0=1.5$ (dashed line), $k_0=3$ (dotted line), ${\eta }_i=-1$, and $\Delta \eta =1$.

Figure 8

Analytical tensor power spectra, normalized to the non-LQC-corrected spectrum, for two different values of $\Delta \eta$ in the $ϵ=0$ case. The parameters are: $\Delta \eta =0$ (solid line), $\Delta \eta =10$ (dashed line), $k_0=1$, and ${\eta }_i=-1$.

Figure 9

Comparison of numerical and analytical spectra [Eq. (38)] for $m={10}^{-2}{m}_{\mathrm{Pl}}$. In the IR region the spectra behave as ${\mathcal{P}}_{\mathrm{T}}\propto k^2$ while in the UV region they behave as ${\mathcal{P}}_{\mathrm{T}}\propto k^{-2ϵ}$, where $ϵ\ll 1$ is the slow-roll parameter. Here: $H=0.037m_{\mathrm{Pl}}$, $ϵ=0.0246$, $k_0=0.037m_{\mathrm{Pl}}$, ${\eta }_i=-750$, and $\Delta \eta =10$.

Figure 10

Numerically computed power spectra for $m={10}^{-4}$, ${10}^{-3}$, ${10}^{-2}{m}_{\mathrm{Pl}}$ (from bottom to top in the UV range).

Figure 11

Numerically computed power spectra for ${\rho }_{\mathrm{c}}={10}^{-4}$, ${10}^{-2}$, $10m_{\mathrm{Pl}}^4$ (from bottom to top in the UV range) with $m={10}^{-3}{m}_{\mathrm{Pl}}$.

Figure 12

Ratio defined by Eq. (47) as a function of inflaton mass in Planck units. Dots are values obtained from the numerical computations. The straight line is the fit given by Eq. (48).

Figure 13

Comparison of the numerical spectrum for $m={10}^{-2}{m}_{\mathrm{Pl}}$ with formulas (49, 50). The solid (blue) line corresponds to (49) while the dashed (red) line corresponds to (50).

Figure 14

Schematic picture of evolution of the Hubble radius (solid line) and of the different length scales (dashed lines) for the considered model of the universe. Different times are distinguished: $t_1$ time when the initial conditions are set; $t_2$ bounce ($H=0$); $t_3$ beginning of inflation; $t_4$ end of inflation; $t_5$-present epoch of dark energy domination.