Quantum propagation across cosmological singularities

The initial singularity is the most troubling feature of the standard cosmology, which quantum effects are hoped to resolve. In this paper, we study quantum cosmology with conformal (Weyl) invariant matter. We show that it is natural to extend the scale factor to negative values, allowing a large, collapsing universe to evolve across a quantum “bounce” into an expanding universe like ours. We compute the Feynman propagator for Friedmann-Robertson-Walker backgrounds exactly, identifying curious pathologies in the case of curved (open or closed) universes. We then include anisotropies, fixing the operator ordering of the quantum Hamiltonian by imposing covariance under field redefinitions and again finding exact solutions. We show how complex classical solutions allow one to circumvent the singularity while maintaining the validity of the semiclassical approximation. The simplest isotropic universes sit on a critical boundary, beyond which there is qualitatively different behavior, with potential for instability. Additional scalars improve the theory’s stability. Finally, we study the semiclassical propagation of inhomogeneous perturbations about the flat, isotropic case, at linear and nonlinear order, showing that, at least at this level, there is no particle production across the bounce. These results form the basis for a promising new approach to quantum cosmology and the resolution of the big bang singularity.

Figure 1

Associating positive, negative and imaginary scale factor $a$ to different regions in field space. The light cone corresponds to the singularity $a=0$.

Figure 2

The points of constant (real) $a$ form a hyperboloid parametrized by $v^{\alpha }$.

Figure 3

Integration contours in the complex $\tau$ plane, for the relativistic massive propagator, defined in Eq. (31). As $\sigma \equiv -(x-x^{\prime }{)}^2$ is varied, from timelike values $\sigma =T^2$ with $T$ real, to spacelike values $\sigma =-R^2$, with $R$ real, by passing beneath the origin in the complex $\sigma$ plane, the saddle point in $\tau$, shown by the black point, moves correspondingly. In each case, the integral is defined by the associated steepest descent contour running from 0 to $\infty$, also shown.

Figure 4

Feynman propagator for flat universes as a function of $T$, for $m=7$, showing real part (blue), imaginary part (dashed) and absolute value (black) which is constant here.

Figure 5

Integration contours in the complex $\tau$ plane for $\kappa \le 0$ (solid line) and for $\kappa >0$ (dashed line), for the case where $x=(T,\overrightarrow{0})$ and $x^{\prime }=(-T,\overrightarrow{0})$.

Figure 6

Feynman propagator for open universes as a function of $T$, for $m=7$ and $\kappa =-1$, showing real part (blue), imaginary part (dashed) and absolute value (black).

Figure 7

Feynman propagator for closed universes as a function of $T$, for $m=7$ and $\kappa =1$, showing real part (blue), imaginary part (dashed) and absolute value (black). As discussed in the text, for $T\ge 1$ the semiclassical interpretation fails.