Coherent semiclassical states for loop quantum cosmology

The spatially flat Friedmann-Robertson-Walker cosmological model with a massless scalar field in loop quantum cosmology admits a description in terms of a completely solvable model. This has been used to prove that: (i) the quantum bounce that replaces the big bang singularity is generic; (ii) there is an upper bound on the energy density for all states, and (iii) semiclassical states at late times had to be semiclassical before the bounce. Here we consider a family of exact solutions to the theory, corresponding to generalized coherent Gaussian and squeezed states. We analyze the behavior of basic physical observables and impose restrictions on the states based on physical considerations. These turn out to be enough to select, from all the generalized coherent states, those that behave semiclassical at late times. We study then the properties of such states near the bounce where the most “quantum behavior” is expected. As it turns out, the states remain sharply peaked and semiclassical at the bounce and the dynamics is very well approximated by the “effective theory” throughout the time evolution. We compare the semiclassicality properties of squeezed states to those of the Gaussian semiclassical states and conclude that the Gaussians are better behaved. In particular, the asymmetry in the relative fluctuations before and after the bounce are negligible, thus ruling out claims of so-called “cosmic forgetfulness.”

Figure 1

The asymptotic relative fluctuation ${\Delta }_{\pm }$ is plotted as a function of ${\sigma }^2$ with $k_0=30000$ in Planck units, where $\alpha \approx 6.14$. Here we have chosen $n=0$ and $p=0$. Two features characterize the behavior of the function near the minimum $\tilde{\sigma }$. The first one is that for ${\sigma }^2<{\tilde{\sigma }}^2$ the function is exponential; the second one is that for ${\sigma }^2>{\tilde{\sigma }}^2$ the function has a polynomial behavior.

Figure 2

The relative fluctuation $(\Delta \widehat{V}{)}^2/⟨\widehat{V}{⟩}^2$ is plotted as a function of internal time $\varphi$ with $k_0=30000$, in Planck units where $\alpha \approx 6.14$. Note that the minimum of this quantity is at the bounce (here corresponding to $\varphi =0$). It is found that the value at the bounce is approximately $1/2$ of its asymptotic value for the generalized Gaussian states.

Figure 3

The values $\tilde{\varphi }$ of $\varphi$ for which one reaches the classical region are plotted as a function of $k_0$, where $n=0$, $\sigma =\tilde{\sigma }$ and $p=0$ at $\varphi =0$. In the plot, $\delta =1$ dotted, $\delta =0.5$ dashed, $\delta =0.25$ dot-dashed, $\delta =0.125$ down line, $\delta =0.0625$ middle line, and $\delta =0.03125$ the top line. Note that for high values of $k_0$, the time $\tilde{\varphi }$ remains a constant.

Figure 4

Values of density are plotted for different values of $\delta (\tilde{\varphi })$, as function of $k_0$, where $n=0$, $\sigma =\tilde{\sigma }$ and $p=0$ at $\varphi =0$. Here, ${\rho }_{1,2,3,4}/{\rho }_{\mathrm{crit}}$ for which $\delta =1$ dotted, $\delta =0.5$ dashed, $\delta =0.25$ dotdashed, $\delta =0.125$ up line, $\delta =0.0625$ middle line, and $\delta =0.03125$ the bottom line.

Figure 5

Density and asymptotic relative fluctuation ${\Delta }_{+}$. In the plot $1/{\eta }_R={\sigma }^2\mathrm{\text{:}}(0,100000)$, ${\eta }_I\mathrm{\text{:}}(-0.001,0.001)$, ${\beta }_R=k_0=30000$, ${\beta }_I=0$ and $n=0$. The maximal value of the density correspond to the Gaussian states. Is clear that the squeezed states have a maximal density close to ${\rho }_{\mathrm{crit}}$ only if the state is near to the Gaussian states.

Figure 6

Volume at the Bounce. In the plot $1/{\eta }_R={\sigma }^2\mathrm{\text{:}}(0,8\times {10}^6)$, ${\eta }_I\mathrm{\text{:}}(-0.0001,0.0001)$, ${\beta }_R=k_0=30000$, ${\beta }_I=0$ and $n=0$. The global minimum is over the Gaussian states and has an exponential behavior if we move far away from the Gaussian state or if ${\sigma }^2$ is smaller that ${\tilde{\sigma }}^2=2\alpha k_0$.

Figure 7

Relative error of the difference in the asymptotic values of the relative fluctuation $\frac{{\Delta }_{+}-{\Delta }_{-}}{{\Delta }_{-}}$. In the plot $1/{\eta }_R={\sigma }^2\mathrm{\text{:}}(0,{10}^6)$, ${\eta }_I\mathrm{\text{:}}(-0.0003,0.0003)$, ${\beta }_R=k_0=30000$, ${\beta }_I=0$ and $n=0$. Outside of the vertical one the squeezed state have an asymptotic relative fluctuations (${\Delta }_{+}$ and ${\Delta }_{-}$) bigger that 1. One line mark the Gaussian states (${\eta }_I=0$) that is just when the curve is zero and the other line is for states with ${\tilde{\sigma }}^2=2\alpha k_0$.