Survival of black holes pair-created during inflation: the initial nondegeneracy distribution and quantum-corrected evolution

In 1996 Bousso and Hawking computed the instanton-mediated rate at which black holes are pair-created during inflation from the no-boundary proposal for the wave function of the universe, and traced the nucleated black holes' classical and quantum evolution. They found that essentially all neutral pair-created black holes evaporate before inflation ends, survival being possible only if the nucleated near-degenerate (Nariai-type) geometry starts with nondegeneracy $\epsilon_0$ below an exponentially tiny threshold $\sim\exp(-m^{-2}/720\pi)$. The paper itself left the load-bearing inputs uncontrolled: the probability distribution of $\epsilon_0$ was guessed to be uniform over a few Planck lengths, quantum radiation was treated only in a thermal Stefan–Boltzmann approximation, and the charged analysis was confined to Einstein–Maxwell theory. Later work resolved the fixed-cosmological-constant counterparts of these threads — dilatonic pair-creation rates, exact one-loop pair-creation rates, and the static-patch quantum evolution including the removal of the anti-evaporation artifact — but the actual inflationary, time-dependent question remains open: the distribution of the initial nondegeneracy fixed by the no-boundary complex saddle for general values of the effective cosmological constant, and the one-loop quantum-corrected evolution that determines whether any pair-created black holes survive to the end of inflation.

Background

Inflation is a period of quasi-exponential expansion of the very early universe, driven by the vacuum energy of a slowly rolling scalar field (the inflaton). In chaotic inflation with a massive inflaton, the potential is $V(\phi)=m^2\phi^2/2$ with $m\approx 10^{-6}$ in Planck units, so the effective cosmological constant $\Lambda_{\rm eff}=8\pi G V(\phi)$ starts near the Planck density and decreases slowly until inflation ends. De Sitter space is the maximally symmetric solution for a positive cosmological constant, and a black hole embedded in it is described by the Schwarzschild–de Sitter spacetime, which has two horizons: a black hole horizon and a cosmological horizon. In the degenerate (Nariai) limit the two horizons coincide and the near-horizon geometry becomes a direct product $dS_2\times S^2$, where $dS_2$ is two-dimensional de Sitter space.

Quantum-mechanically, black hole pairs can nucleate in a de Sitter background by direct analogy with Schwinger pair production — the spontaneous creation of charged particle–antiparticle pairs out of the vacuum by a strong electric field. In the Euclidean path integral formulation of gravity, such processes are mediated by gravitational instantons: smooth solutions of the Euclidean field equations that interpolate between the vacuum and the nucleated configuration, with a rate of order $\exp(-\Delta S_E)$, the difference of Euclidean actions. Only the degenerate Nariai solution (Euclidean topology $S^2\times S^2$) has a smooth Euclidean section within the neutral Schwarzschild–de Sitter family, so it dominates the neutral channel; away from exact degeneracy, and on a slow-roll background with time-dependent $\Lambda_{\rm eff}$, the corresponding stationary points are complex saddles — solutions of the field equations continued to complexified field values, matched to real Lorentzian initial data where the geometry becomes real. The relevant framework is quantum cosmology: the no-boundary proposal defines the wave function of the universe by a path integral over compact Euclidean geometries with no boundary, in competition with the tunnelling proposal, which instead nucleates the universe from nothing via an outgoing wave at zero scale factor; Bousso and Hawking found in 1996 that only the no-boundary proposal gives physically sensible black hole pair creation rates during inflation.

After nucleation, the pair-created black hole evolves. Classically, its horizon grows with the cosmological horizon during slow roll, which the 1996 authors showed is exactly the flux of inflaton field energy across the horizon as required by the First Law of black hole mechanics ($dM=(\kappa/8\pi G)\,dA$ plus work terms, with $\kappa$ the surface gravity, i.e. the redshifted acceleration at the horizon). Quantum mechanically, each horizon radiates thermally at temperature $T=\kappa/2\pi$. In the exactly degenerate Nariai state the two horizons have equal temperature and are in equilibrium; any small nondegeneracy, parameterized by $\epsilon$ via horizon radii $r_b=b(1-\epsilon)$ and $r_c=b(1+\epsilon)$ with initial value $\epsilon_0$, makes the smaller black hole horizon hotter, so the black hole evaporates. Bousso and Hawking evolved $\epsilon(t)$ using a thermal (Stefan–Boltzmann flux) approximation for the radiation and found that a Planck-size nucleated black hole survives only until the end of inflation if $\epsilon_0$ lies below a threshold of order $\exp(-m^{-2}/720\pi)$ — absurdly small for $m\approx 10^{-6}$.

The paper then declared the load-bearing inputs open. First, the probability $P(\epsilon_0\le\epsilon_0^{\max})$ depends on the distribution of the initial horizon sizes, which was simply assumed uniform between zero and a few Planck lengths; a controlled semiclassical treatment of the quantum fluctuations that shift the nucleated geometry away from degeneracy, for general values of $\Lambda_{\rm eff}$, was called 'an interesting problem by itself, and beyond the scope of this paper'. Second, the thermal approximation stood in for the one-loop effective action of the low-mass fields (the leading quantum correction to the classical action), which the authors deemed 'impossibly difficult' to compute except in very simple metrics. Third, the analysis of magnetically charged (Reissner–Nordström-type, i.e. charged) black holes was confined to Einstein–Maxwell theory, whereas in dilatonic theories — extended gravity models with an extra scalar field coupled to the gauge and gravitational sectors — both the effective charge and Newton's constant can vary in time, potentially removing the exponential suppression of charged black hole production. Fourth, nucleation near the Planck regime lies beyond semiclassical control altogether.

Subsequent literature brought the fixed-cosmological-constant version of this problem largely under control: dilatonic pair creation rates were computed by Bousso in 1997; Volkov and Wipf computed the exact one-loop pair-creation rate in de Sitter space in 2000; and in 2026 Shi, Turiaci and Wu showed that, in the static patch of de Sitter (the causally connected region inside the cosmological horizon) at fixed $\Lambda$, the nucleated near-Nariai black hole simply evaporates thermally (the once-suspected anti-evaporation channel being an artifact of a singular quantum state), with no large quantum-gravity corrections. What has remained without a controlled calculation is the original setting itself: a slow-roll background with time-dependent $\Lambda_{\rm eff}$, where the initial-data distribution at the complex saddle and the competition between classical horizon growth and quantum evaporation determine the answer.

Problem Statement

Consider a slow-roll inflationary universe (for definiteness, chaotic inflation with a massive inflaton, or an equivalent model in which the effective cosmological constant $\Lambda_{\rm eff}$ starts near the Planck density and decreases slowly). Black holes nucleate in pairs during inflation through the no-boundary instantons constructed by Bousso and Hawking in 1996; away from exact degeneracy the relevant saddle points are complex, i.e. continued into the complexified fields. Determine, within a controlled semiclassical treatment of quantum gravity:

1. the probability distribution of the initial nondegeneracy parameter $\epsilon_0$ of the nucleated pair — the parameter in the horizon radii $r_b=b(1-\epsilon_0)$ and $r_c=b(1+\epsilon_0)$ at nucleation, where $b$ is the common horizon radius, so that $\epsilon_0=0$ is the exactly degenerate (Nariai) geometry and $\epsilon_0\ll 1$ is near-degenerate — as fixed by the no-boundary condition through the relevant complex saddle, for general values of $\Lambda_{\rm eff}$ during slow roll, replacing the assumed uniform distribution of initial horizon sizes (Eq. (5.13) of the 1996 paper);

2. the subsequent quantum-corrected evolution of the two horizon radii beyond the thermal Stefan–Boltzmann approximation — that is, from the one-loop effective action (or equivalently the renormalized stress-energy tensor) of the matter fields in the time-dependent near-Nariai background, together with the classical inflaton-energy flux implied by the First Law of black hole mechanics — including whether a physical, horizon-smooth (Hadamard, i.e. regular at both horizons) quantum state is well defined in this non-stationary setting.

3. the analogue of items 1 and 2 for the magnetically charged and dilatonic variants that cannot evaporate completely — the no-boundary distribution of the initial nondegeneracy and the quantum-corrected evolution of the horizon radii, evaluated on their own charged (Reissner–Nordström–de Sitter) and dilatonic near-Nariai backgrounds rather than the neutral one — and the resulting relic abundance with which those variants survive to the present universe.

These components jointly answer one scientific question: with what probability do black holes pair-created during inflation survive until the end of inflation — for the neutral pair through items 1 and 2, and for the magnetically charged and dilatonic variants that cannot evaporate completely, through their relic abundance in the present universe (item 3)? The neutral components (items 1 and 2) do not by themselves determine item 3, which requires the same analysis carried out on the charged and dilatonic backgrounds. Nucleation in the near-Planckian regime lies outside the domain of validity of this controlled semiclassical treatment.

References

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