The Classical Universes of the No-Boundary Quantum State

TL;DR

The paper investigates how the no-boundary wave function (NBWF) of the universe selects a quasiclassical, inflationary cosmos within simple minisuperspace models containing a cosmological constant and a quadratic scalar potential. By applying a leading semiclassical (steepest descents) analysis, complex extremal geometries—fuzzy instantons—generate a conserved classical-history measure on a bounded region of phase space, with individual histories weighted by $e^{-2 I_R}$. Inflation emerges as a generic feature of histories that are classical at late times, while the NBWF also reveals regimes with bounces or initial singularities, and a nuanced dependence of inflation on model parameters — notably the parameter $\mu = (3/\Lambda)^{1/2} m$. Volume weighting pushes probabilities toward histories with many e-foldings, potentially aligning with our observation of a long inflationary epoch, and the framework offers a route to connecting the quantum state to late-time observables, including the thermodynamic arrow of time and the fate of the universe. Overall, the NBWF provides a concrete, predictive link between quantum initial conditions and the emergence of a classical, inflationary cosmos within the studied minisuperspace setting.

Abstract

We analyze the origin of the quasiclassical realm from the no-boundary proposal for the universe's quantum state in a class of minisuperspace models. The models assume homogeneous, isotropic, closed spacetime geometries, a single scalar field moving in a quadratic potential, and a fundamental cosmological constant. The allowed classical histories and their probabilities are calculated to leading semiclassical order. We find that for the most realistic range of parameters analyzed a minimum amount of scalar field is required, if there is any at all, in order for the universe to behave classically at late times. If the classical late time histories are extended back, they may be singular or bounce at a finite radius. The ensemble of classical histories is time symmetric although individual histories are generally not. The no-boundary proposal selects inflationary histories, but the measure on the classical solutions it provides is heavily biased towards small amounts of inflation. However, the probability for a large number of efoldings is enhanced by the volume factor needed to obtain the probability for what we observe in our past light cone, given our present age. Our results emphasize that it is the quantum state of the universe that determines whether or not it exhibits a quasiclassical realm and what histories are possible or probable within that realm.

Figures (24)

• Figure 1: The complex solutions that provide the steepest descents approximation to the NBWF are found by integrating the field equations along a broken contour $C_B(X)$ in the complex $\tau$-plane. In order for the solutions to behave classically at late times one ought to tune the tangent $\gamma$ of the phase of $\phi$ at the South Pole and the turning point $X$ of the contour. We show the tangent $\gamma$ (left) and the turning point $X$ (right) here as a function of the absolute value $\phi_0$ of $\phi$ at the South Pole. There is a qualitative difference between $\mu<3/2$ models on the one hand where $\gamma$ remains finite for all $\phi_0$ (dotted curve), and $\mu >3/2$ models on the other hand where $\gamma$ diverges at a critical value $\phi_0^c$ (remaining curves). In the latter case there is no combination $(X,\gamma)$ for which the classicality conditions at large scale factor hold when $\phi_0 < \phi_0^c$: the ensemble of possible classical histories is restricted to a bounded surface in phase space. The right panel shows the critical value $\phi_0^c$ increases slightly with $\mu$, for fixed $m$, and tends to $1.27$ as $\Lambda \rightarrow 0$, independently of the value of $m^2$. From top to bottom, the different curves show $\gamma$ and $X$ for $\mu=3/4,\ 33/20,\ 9/4, 3$ and (in the left panel) for a scalar field model with $m^2=.05$ and $\Lambda=0$.
• Figure 2: The real and imaginary part of the scalar field $\phi$ for a typical complex solution that provides the semiclassical approximation to the NBWF. This solution has $\mu=3/4$ and $\phi_0=2$. It is shown along a broken contour $C_B(X)$ in the complex $\tau=(x,y)$ plane that runs first along the $x$-axis from the South Pole at $x=0$ to a value $x=X$, and then vertically in the $y$-direction. The turning point $X$ is the largest value of $x$ plotted in the left hand two figures. It and the imaginary part of $\phi$ at the SP are determined by the requirement that the imaginary part of the action becomes constant with increasing $y$ (cf. Figure \ref{['angle']}) . This is necessary for classicality at late times and it implies that the imaginary part of $\phi$ decays rapidly to zero with increasing $y$.
• Figure 3: The real and imaginary part of the scale factor for the same complex solution as in Figure \ref{['ex']}, along the same broken contour $C_B(X)$ in the complex $\tau$-plane. The imaginary part of $a(\tau)$ rapidly decays to zero along the $y$-axis as a consequence of the classicality conditions, whereas the real part grows exponentially for some time.
• Figure 4: Left panel: The ratio of the gradient squared of the real to the imaginary part of the action plotted along the $y$-axis, for the complex solution that behaves classically at large scale factor, with $\mu=3/4$ and $\phi_0 =2$. For $y <2$ the ratio still significantly deviates from zero, not because $I_R$ varies in the $y$-direction (as can be seen in the right panel) but because the gradient of $I_R$ in the $X$-direction is not small. Right panel: The real part of the action of this complex solution rapidly stabilizes along the $y$-axis.
• Figure 5: The asymptotic value of the real part of the action of the complex solutions that behave classically at large scale factor plotted as a function of $\phi_0$ and for $\mu=3/4$. This determines the relative probabilities predicted by the NBWF for the corresponding classical Lorentzian histories. The upper curve shows the prediction of the perturbation theory for small $\phi$ around the empty de Sitter space with cosmological constant $\Lambda$.