Black hole pair creation in de Sitter space beyond leading semiclassical order: constrained instantons, prefactors, and spherical dominance

The Lorentzian condition of Bousso and Hawking (Phys. Rev. D 59, 103501 (1999)) lets one extract expanding Lorentzian histories from the Euclidean no-boundary path integral by working in the momentum representation of the wave function of the universe, where a purely imaginary argument signals real-time evolution. Applied to Schwarzschild-de Sitter spacetime, it yielded the leading semiclassical nucleation rate $\Gamma=\exp(S_{\rm SdS}-S_{\rm dS})=\exp(-\pi bc)$ for black hole pairs of arbitrary (sub-maximal) mass in a de Sitter background, resting on a constrained instanton (a saddle that solves the field equations only after the black hole mass is fixed) with a conical singularity on the black hole horizon. The source paper itself flagged three uncontrolled steps: the required regular horizon slicings were never exhibited, the saddlepoint prefactor was dropped, and spherical dominance of the path integral was assumed. Later work has confirmed the leading exponent and given the constrained instanton a modern path-integral foundation (Draper and Farkas 2022; Morvan, van der Schaar and Visser 2023), and the one-loop prefactor is known for the smooth Nariai instanton (Volkov and Wipf 2000). The open problem audited here is the controlled completion of this computation: the rigorous geometric status of the conical constrained saddle together with an explicit regular slicing construction, the one-loop fluctuation determinant, zero-mode measure and negative modes that make up the multiplicative prefactor of the sub-maximal rate, and a justification (or refutation) of the spherical-symmetry reduction.

Background

In quantum cosmology the universe is described by a wave function $\Psi$, a functional of the three-dimensional geometry on a spacelike surface $\Sigma$. Two rival boundary conditions define it through a path integral over four-geometries: the no-boundary proposal of Hartle and Hawking sums over compact Euclidean (positive-definite signature) geometries whose only boundary is $\Sigma$, while the tunneling proposal of Vilenkin selects outgoing solutions of the Wheeler-DeWitt equation (the canonical evolution equation of quantum gravity). In the semiclassical approximation the integral is dominated by saddle points, called instantons, which are solutions of the Euclidean Einstein equations; probabilities of competing histories are ratios $\Psi^*\Psi$ that reduce to exponentials of differences of the Euclidean action $I$.

A subtlety is that the observed universe is Lorentzian (it has a time direction) while the computation is Euclidean, so one must identify which part of the wave function describes real-time evolution. The classical signal is the second fundamental form (extrinsic curvature) $K_{ij}=\nabla_i n_j$ of $\Sigma$, the rate of change of the induced metric as $\Sigma$ moves along its unit normal $n$. Bousso and Hawking (1999) noted that $K_{ij}$ is conjugate to the three-metric through the canonical momentum $\pi_{ij}$, so one can use a momentum representation of the wave function (related to the metric representation by a Laplace transform): then $\Sigma$ belongs to a Lorentzian region precisely when the argument of $\Psi$ is purely imaginary. This Lorentzian condition is a sharp criterion for extracting expanding Lorentzian histories, and it makes contact with the tunneling proposal by giving a precise meaning to outgoing.

The application is gravity with a positive cosmological constant $\Lambda$ and no matter. The Lorentzian solution with spatial topology $S^1\times S^2$ is Schwarzschild-de Sitter (Kottler) spacetime,

$$ds^2 = -V(r)\,dt^2 + V(r)^{-1} dr^2 + r^2 d\Omega_2^2,\qquad V(r)=1-\frac{2\mu}{r}-\frac{\Lambda r^2}{3},$$

which describes a pair of black holes in a de Sitter background and has a black-hole horizon and a cosmological horizon of different surface gravities (the redshifted acceleration of the horizon). After the Euclidean continuation $t\to -i\tau$, Euclidean time must be made periodic to avoid a conical singularity at each horizon, a point where the manifold looks locally like a cone rather than smooth space; the two horizons demand different periods, so at most one can be made smooth. Only in the maximal-mass limit do the two horizons match and the solution degenerates to the Nariai geometry, two-dimensional de Sitter space times a sphere, whose smooth $S^2\times S^2$ instanton (Ginsparg and Perry 1983; Bousso and Hawking 1995, 1996) gives the pair creation rate of maximal black holes. It had been thought that lighter black holes could not nucleate semiclassically; Z. C. Wu (1997) had proposed constrained instantons, saddles that solve the field equations only after a parameter such as the mass is fixed, for the same process.

Bousso and Hawking showed that sub-maximal (arbitrary-mass) black holes do nucleate: keep the conical singularity on the black hole horizon but place it on the measurement surface $\Sigma$, where the metric is prescribed so the action stays stationary under the remaining variations, impose the Lorentzian condition (purely imaginary second fundamental form) elsewhere on $\Sigma$, and normalize the probability against empty de Sitter. The resulting rate is

$$\Gamma_{\rm SdS} = \exp\big(S_{\rm SdS}-S_{\rm dS}\big)=\exp(-\pi b c),$$

where $b$ and $c$ are the black-hole and cosmological horizon radii and $S=(A_b+A_c)/4G$ is the total horizon entropy, one quarter of the total horizon area, as in black hole thermodynamics. This action-entropy identity for Schwarzschild-de Sitter is now standard and has been rederived in several ways.

The paper itself flagged three uncontrolled steps, each stated in its text. First, the family of spacelike slicings of Schwarzschild-de Sitter used to build $\Sigma$, which must pass through the conical singularity and whose neighboring slices must approach each other in proper time at least quadratically in the proper distance from the black hole horizon so that every Lorentzian slice is regular there, was drawn only schematically: the authors write that they shall not give any such slicing explicitly. Second, the multiplicative prefactor $A$ of the saddlepoint approximation is dropped throughout, so the quantitative correction to the rate is unknown. Third, the path integral was restricted to spherically symmetric configurations to make the problem tractable, with the dominance of spherical configurations asserted as reasonable but not proven; the computation is set up for general relativity with a cosmological constant and without matter fields.

Problem Statement

Close the three gaps that the Bousso-Hawking (1999) computation of the nucleation rate of Schwarzschild-de Sitter black hole pairs in de Sitter space left open, so that the rate is a controlled prediction rather than a leading exponent. The common objective uniting the three components is a quantitatively reliable, first-principles rate for the spontaneous nucleation of neutral black hole pairs of arbitrary (sub-maximal) mass in a de Sitter background, derived from the no-boundary (or equivalently constrained) Euclidean path integral, with the validity of the saddle used established rather than assumed. The theory is four-dimensional Einstein gravity with a positive cosmological constant and no classical matter, and the rate is normalized against empty de Sitter, exactly as in the source.

(1) Geometry and status of the saddle. Exhibit explicitly a family of spacelike slicings of Schwarzschild-de Sitter that contains the conical singularity (equivalently, the black hole horizon in the Euclidean sector) and satisfies the regularity condition of the source: the proper time between points on neighboring slices vanishes at least quadratically as a function of proper distance from the black hole horizon, so that all Lorentzian spacelike slices are regular on the horizon. Establish the precise sense in which the conical, constrained instanton is a legitimate saddle of the gravitational path integral.

(2) Beyond-leading-order accuracy. Compute the multiplicative prefactor of the sub-maximal nucleation rate: the one-loop fluctuation determinant (the Gaussian integral over small quadratic fluctuations around the saddle) on the conical constrained instanton; the zero-mode and measure factors (zero modes are fluctuation directions that leave the action unchanged, generated by symmetries and by the constraint parameter, and their normalization enters the prefactor), including the measure over the mass or constraint parameter, which the most recent dedicated computation (Morvan, van der Schaar and Visser 2023) assumes flat and parameter-independent; and the treatment of the negative modes (unstable fluctuations that decrease the Euclidean action), whose near-Nariai coefficients are in open disagreement in the literature (Ginsparg and Perry 1983 versus Bousso-Hawking 1996 versus Morvan et al. 2023). Quantify how these corrections modify $\Gamma_{\rm SdS}=\exp(-\pi bc)$; the complete one-loop result of Volkov and Wipf for the smooth Nariai instanton is the existing computation in the maximal-mass limit and provides a comparison point.

(3) Validity of the symmetry reduction. Establish, rather than assume, that spherically symmetric configurations dominate the gravitational path integral for this process: analyze the non-spherical fluctuation modes around the constrained instanton and show that they neither destabilize the saddle nor change the leading rate, or else identify the dominant non-spherical channel and recompute the rate.

The known limiting results (the leading exponent $\exp(S_{\rm SdS}-S_{\rm dS})$ obtained in the source and the smooth-Nariai one-loop rate of Volkov and Wipf) are reference points against which a proposed prefactor can be compared.

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