Arrows of time in the bouncing universes of the no-boundary quantum state

We derive the arrows of time of our universe that follow from the no-boundary theory of its quantum state (NBWF) in a minisuperspace model. Arrows of time are viewed four-dimensionally as properties of the four-dimensional Lorentzian histories of the universe. Probabilities for these histories are predicted by the NBWF. For histories with a regular “bounce” at a minimum radius fluctuations are small at the bounce and grow in the direction of expansion on either side. For recollapsing classical histories with big bang and big crunch singularities the fluctuations are small near one singularity and grow through the expansion and recontraction to the other singularity. The arrow of time defined by the growth in fluctuations thus points in one direction over the whole of a recollapsing spacetime but is bidirectional in a bouncing spacetime. We argue that the electromagnetic, thermodynamic, and psychological arrows of time are aligned with the fluctuation arrow. The implications of a bidirectional arrow of time for causality are discussed.

Figure 1

A bouncing universe. This figure shows the scale factor $\widehat{a}$ (top) and the scalar field $\widehat{\varphi }$ (bottom) for a homogeneous, isotropic Lorentzian history in the NBWF classical ensemble. The example shown has ${\varphi }_0=4$ for $m=.05$. The quantities $\widehat{a}$, $\widehat{\varphi }$ and $t$ are all in Planck units. The solution was found by extrapolating the asymptotically real behavior of the ${\varphi }_0=4$ fuzzy instanton, where the semiclassical approximation applies, backward in time using the classical equations of motion. The universe inflates on both sides of the bounce.

Figure 2

Fluctuations in a bouncing universe. This figure shows the behavior of the gauge-invariant fluctuation mode $z_{(n)}$ with wave number $n=20$, as a function of time in Planck units in the bouncing background of Fig. 1. The fluctuation history was calculated by starting with the behavior of the $n=20$ perturbation mode around the fuzzy instanton at large $t$ and extrapolating that backwards in time using the Lorentzian perturbation equations. This solution determines the wave function of the perturbation to leading order in $\hslash$. The fluctuation oscillates in its ground state in the regime of three-surfaces near the bounce at $t=0$ when the size of the universe is small. The fluctuation amplitude freezes when the physical wavelength of the mode becomes larger than the horizon size. This happens for the mode shown around $t=\pm 4$ on either side of the bounce. The fluctuation modes will enter the horizon again much further away from the bounce, on both sides, as classical perturbations with an expected amplitude $Q^2\sim {\mathcal{V}}^3(\widehat{\varphi })/{\mathcal{V}}_{,\widehat{\varphi }}^2$ evaluated at horizon exit during inflation. During matter domination perturbations continue to grow in the two directions away from the bounce (on time scales much larger than illustrated here), leading to a bidirectional fluctuation arrow of time in the bouncing histories predicted by the NBWF.

Figure 3

A contour in the complex $\tau$-plane for representing both a complex saddle point and the Lorentzian history it predicts. The figure shows the contour $C_B(X)$ that starts along the real axis from the SP at $\tau =0$, breaks at $X$, and extends upward in the imaginary $\tau$ direction. The complex saddle points start at the SP with $a(0)=0$, $\varphi (0)\equiv {\varphi }_0\exp (i\theta )$ and conditions of regularity. By tuning $\theta$ and $X$ for each ${\varphi }_0$ a vertical contour can be found along which the imaginary parts of $a(X+iy),\varphi (X+iy)$ become negligible beyond some value $y_c$, which decreases for increasing ${\varphi }_0$. The variation in $I_R$ also becomes negligible so that the classicality conditions are satisfied. For large ${\varphi }_0$ one has $y_c\ll 1$. The approximately real values of $a$ and $\varphi$ for $y>y_c$ provide Cauchy data for the classical Lorentzian “background” history $(\widehat{a}(t),\widehat{\varphi }(t))$ predicted by the saddle point, which can therefore be labeled by ${\varphi }_0$. The Lorentzian histories can be extrapolated classically to values lower than $y_c$ as indicated by the dotted line. For low values of ${\varphi }_0\sim \mathcal{O}(1)$ the histories are singular in the past. However for larger values of ${\varphi }_0$ the histories exhibit a bounce in the past at a finite radius $b$ at $y\approx 0$. In the latter case we assume one can reliably extrapolate along the vertical part of the contour to negative values of $y$ where we find the history enters another inflationary regime. Regularity at the SP of the fuzzy instantons requires that the saddle point fluctuations vanish there. Saddle point fluctuations will therefore be small in a region around the SP indicated by the circle. We find this region includes the part of the contour up to $y_c$. The predicted Lorentzian histories of fluctuations coincide with the saddle point fluctuations along the solid part of the contour. The wave function of the fluctuations can be extrapolated along the $x=X$ line using the perturbation equations [cf. EI(A2)]. We find the Lorentzian fluctuations are small in the regime near $t\sim y_c$ where the universe is also small. For bouncing universes fluctuations oscillate in their ground state near the bounce as illustrated in Fig. 2 and eventually increase in both directions away from there indicated by the arrows. This gives rise to large-scale structures on both sides of the bounce. Hence the bouncing universes predicted by the NBWF exhibit a bidirectional fluctuation arrow of time.