Quantum analysis of the recent cosmological bounce in the comoving Hubble length

We formulate the transition from decelerated to accelerated expansion as a bounce in connection space and study its quantum cosmology, knowing that reflections are notorious for bringing quantum effects to the fore. We use a formalism for obtaining a time variable via the demotion of the constants of nature to integration constants, and focus on a toy universe containing only radiation and a cosmological constant $\mathrm{\Lambda }$ for simplicity. We find that, beside the usual factor-ordering ambiguities, there is an ambiguity in the order of the quantum equation, leading to two distinct theories: one second order, and one first order. In both cases two time variables may be defined, conjugate to $\mathrm{\Lambda }$ and the radiation constant of motion. We make little headway with the second-order theory, but are able to produce solutions to the first-order theory. They exhibit the well-known “ringing” whereby incident and reflected waves interfere, leading to oscillations in the probability distribution even for well-peaked wave packets. We also examine in detail the probability measure within the semiclassical approximation. Close to the bounce, the probability distribution becomes double peaked, with one peak following a trajectory close to the classical limit but with a Hubble parameter slightly shifted downwards, and the other with a value of $b$ stuck at its minimum. An examination of the effects still closer to the bounce, and within a more realistic model involving matter and $\mathrm{\Lambda }$, is left to future work.

Figure 1

Cosmological solutions with initial data (set at $t=0$) $a=1$, $b=2$, and $m=1.2$, (top: $k=0,\mathrm{\Lambda }=8.4$; bottom: $k=1,\mathrm{\Lambda }=11.4$). These follow the radiation-dominated (orange dotted) branch at small $a^2$ and the $\mathrm{\Lambda }$-dominated (green dashed) branch at large $a^2$. The time coordinate is defined by setting $N=1/a$.

Figure 2

Solutions with $m=1.2$, $k=0$, $\mathrm{\Lambda }=8.4$, and in units $\mathfrak{h}=1$ (real part: blue; imaginary part: orange). The top panel shows a solution given in terms of a triconfluent Heun function that diverges at $b\approx 6.8$. In the bottom panel we match this to the approximate solution (32) by matching the wave function and its derivative at $b=6$, which leads to a solution defined at arbitrarily large $b$. The classically allowed region is $b>b_0\approx 1.91$. This solution agrees with a numerical solution constructed from the same initial data $\psi (0)=1,{\psi }^{\prime }(0)=0$ (black, dashed and dotted).

Figure 3

Imaginary part of the wave functions ${\psi }_{s\pm }$ [$\mathrm{\Lambda }$ branch ($+$) in blue, radiation branch ($-$) in orange], with $\mathfrak{h}=1$, $m=1$, and $\varphi ={10}^6$, defined with the lower limit $b_0$ (here and in the following plots $b_0\approx 0.0447$). Notice how the oscillation frequency increases/decreases with $b$ for the $\mathrm{\Lambda }$-/radiation-dominated branches.

Figure 4

Imaginary part of the wave functions ${\psi }_{s-}$ with $\mathfrak{h}=1$, $m=1$, and $\varphi ={10}^6$, defined with the lower limit of integration $b_0$ (orange) and with lower limit infinity (green), compared with the asymptotic radiation-dominated wave function (blue).

Figure 5

Imaginary part of the evanescent wave function ${\psi }_{s-}$ valid for $b<b_0$ with $\mathfrak{h}=1$, $m=1$, and $\varphi ={10}^6$, defined with the lower limit $b_0$. (Strictly speaking, the integrals used are only valid for $b>0$ but we ignore the region $b<0$).

Figure 6

Imaginary part of the full wave function ${\psi }_s$ normalized so as to match the asymptotic radiation-dominated expression, with parameters $\mathfrak{h}=1$, $m=1$, and $\varphi ={10}^6$. The incident (orange) and reflected (blue) waves, when superposed, match the evanescent wave (green) up to second derivatives in this plot.

Figure 7

Snapshots of the wave function in the classically allowed region $b\ge b_0$ for a wave packet with ${\sigma }_{Tm}=4$ at times $T_m=8,0,-8$. Note that on shell $T_m=-(\eta -{\eta }_0)$, where $\eta$ and ${\eta }_0$ are the conformal time and conformal time at the bounce, respectively. The envelope picks the right portion of the ${\psi }_s$, $+$ or $-$, away from the bounce. Close to the bounce, however, the $+$ and $-$ waves interfere.

Figure 8

Plot of $|\psi {|}^2$ for the same situation as in Fig. 7 at $T_m=12$, 8, 4, 0. (For the particular case of $T_m$—but not for a generic time—this function is symmetric, so for clarity we have refrained from plotting the equivalent $T_m<0$).

Figure 9

Probability with the semiclassical measure, for the same situation as in Fig. 7, at the various times $T_m=16$, 12, 10, 8, 0. We have explicitly verified that this probability density, unlike the function plotted in Fig. 8, always integrates to unity.