Pair Creation of Black Holes During Inflation

Cambridge); Raphael Bousso; Stephen W. Hawking (DAMTP

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Pair Creation of Black Holes During Inflation

Raphael Bousso, Stephen W. Hawking (DAMTP, Cambridge)

General Relativity and Quantum Cosmology · Submitted 18 Jun 1996 · 23 references

Pith's verdict: The no-boundary derivation and the First Law check are solid and citable; the charged-black-hole exponent in Sec. 6 does not follow from the paper's own equations.

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CitationCambridge), Raphael Bousso, Stephen W. Hawking (DAMTP. "Pair Creation of Black Holes During Inflation." arXiv:gr-qc/9606052 (1996). https://pith.science/paper/6SYWTJRR

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Editor's note

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Worth your time for Secs. 3 and 4. Sec. 3 compares the no-boundary measures exp(3π/Λ) for the de Sitter four-sphere and exp(2π/Λ) for the S²×S² Nariai instanton to get Γ = exp(−π/Λ_eff), then shows in Eq. 3.16 that this is exactly the standard instanton formula exp[−(I_obj − I_bg)]. That is a clean saddle-point computation and it earns the claim that the two prescriptions agree. Sec. 4 also holds up: I checked Eq. 4.8 against Eq. 4.13 and the First Law bookkeeping works — κȦ/8π = 1/φ² = A T_ab l^a l^b with b = 1/(mφ). The Appendix is explicit enough to redo by hand; the coordinate change (A.11), the perturbed Nariai metric (A.14), the surface gravities (A.16) and the negative mode (A.18) all hang together. The picture of each Hubble volume nucleating through its own instanton is a heuristic, not a derived statement, but it is presented as one and does no harm. Note also that the instantons themselves are quoted from [15], [18] and [11]; the new content is the measure comparison, the First Law correspondence, and the abundance estimate, and the abstract oversells the instanton part slightly. That is a framing quibble, not a flaw — the cited results are published and checkable. The soft spot is Sec. 6, and it is real. The text there switches to Λ_eff = mφ while Eq. 4.2 says Λ_eff = m²φ². Eq. 6.10 is consistent with the former; Eq. 6.12's φ_max ∝ (αm²)^(−1/6) follows from neither; and Eq. 6.15's exponent (9π²/(16αm²))^(1/3) ≈ 9×10⁴ comes from balancing against 3 ln φ with the m²φ² form and dropping the (3/2)φ² dilution term. Redo it consistently and the exponent is of order 10⁶–10⁷, not 10⁵. The qualitative conclusion — no magnetic black holes in the observable universe — is untouched, since the exponent is huge and negative under every reading. But the quoted number and Eqs. 6.12/6.15 are not reproducible from the paper's own inputs, and the '−10⁵' remark inherits that. Sec. 5's survival estimate rests on a flat prior for ǫ₀, which the authors flag themselves as beyond the semi-classical treatment; take their word and do not press it. Eq. 5.9 does mix m²b with 1/b³, which differ by m² in dimension — worth a look, though these are order-of-magnitude estimates. The 'tunnelling proposal cannot be maintained' line is a plausibility argument from one process, not a derivation; defensible, but stated harder than the evidence. Bottom line: the derivation and the First Law correspondence are the durable parts and are worth citing. The charged-black-hole numerology needs redoing before anyone quotes a number from it.

Referee report

Significance

If the central derivation holds, the paper supplies a first-principles derivation of the standard pair-creation formula from the no-boundary measure rather than a restatement of it, and that is a genuine result. The Appendix is the strongest part: the coordinate transformation (A.11), the perturbed Nariai metric (A.14), the surface gravities (A.16) and the negative mode (A.18) are explicit and checkable line by line, and the correction to Ginsparg–Perry is well motivated. The First Law check of Eqs. (4.8)–(4.13) is a parameter-free consistency relation, and it verifies: with φ̇ = −m and A = 4π/(m²φ²) one recovers Ṁ = 1/φ² from both sides. The magnetic-monopole abundance bound is a falsifiable prediction, and the qualitative conclusion is robust to the arithmetic problems noted below.

Major objections (4)

[Section 6, Eqs. (6.10)–(6.15)] The quoted exponent is not reproducible from the paper's own equations, and the source is a convention mismatch. Eq. (4.2) defines Λ_eff = m²φ², but the text opening Section 6 states 'Λ_eff = mφ', and Eq. (6.10) is written as exp[−(√3/α)(π/(mφ_pc))], which corresponds to Λ_eff = mφ, not m²φ². With the paper's Eq. (4.2), Eq. (6.9) gives I_luke − I_dS = π√3/(αm²φ²), so the exponent is −A/φ² with A = π√3/(αm²). Maximising −A/φ² + 3 ln φ − (3/2)φ² (Eq. 6.11) gives φ_max ∝ (αm²)^{−1/4}, not the (αm²)^{−1/6} of Eq. (6.12), and an exponent of order −6.7×10⁷ for α = 1/137, m = 10⁻⁶, rather than the −10⁵ quoted after Eq. (6.15). Using instead the Section 6 convention Λ_eff = mφ gives φ_max ∝ (αm)^{−1/3} and an exponent of order −1.8×10⁶. Under either reading the exponent is wrong by one to three orders of magnitude. The qualitative conclusion of [abstract:s7] survives and is in fact strengthened, but Eq. (6.15) is a displayed result a reader would quote, and it should be recomputed with a single, explicitly stated Λ_eff convention used consistently in Sections 4 and 6.
[Section 5, Eq. (5.13), and abstract:s5] The conclusion that most neutral black holes evaporate before inflation ends rests on the assumption that the initial horizon radii are 'distributed roughly uniformly between zero and a few Planck lengths'. The authors acknowledge that the semi-classical treatment of the fluctuations that make the geometry non-degenerate is 'beyond the scope of this paper' and that the approximation breaks down at the Planck scale, and they note in passing that the alternative route (Planck-size holes growing to the safe size) is more suppressed, by exp(−m^{−2}/720π) versus exp(−π/m). That robustness argument is the right one, but it is made only in the body, while the abstract and Section 7 state the evaporation result unconditionally. Please either carry the caveat into the abstract and summary, or state explicitly there that the surviving fraction is exponentially small under any distribution of ǫ₀, since the near-degenerate route is itself more suppressed than the large-hole route.
[Section 3.3] The conclusion that the tunnelling proposal 'cannot be maintained' follows from flipping the sign in Eq. (2.3) to Ψ = e^{+I}, giving Γ_TP = 1/Γ_NBP. This is a one-line argument applied to a proposal whose saddle-point contour and weighting are themselves part of the disputed content, and the paper does not address the possibility that a different contour, or a different choice of fluctuating direction, is intended. The claim appears in the abstract [abstract:s2] as a stated result. I would ask the authors either to soften the language to 'the tunnelling proposal, read this way, gives the inverse rate and is hard to reconcile with Eq. (1.1)', or to add a paragraph specifying which saddle points and which contour the tunnelling proposal is being taken to select.
[Appendix, Eq. (A.18)] The calculation finds a negative mode, I = −2π/Λ − 17πǫ²/(9Λ) + O(ǫ⁴), in the direction of decreasing black hole mass, but the manuscript never states what this implies for the saddle-point approximation used in Section 3. A negative mode makes the Gaussian integral over that direction imaginary, so the one-loop prefactor is imaginary; the paper neglects prefactors throughout ('where we neglect a prefactor', after Eq. 2.3), so the rate itself is unaffected, but a reader is entitled to know whether the Nariai instanton is being treated as a genuine saddle point of the no-boundary path integral or as a complex saddle whose real part alone enters Eq. (2.4). One or two sentences in the Appendix or in Section 3.1.2 would settle this.

Minor objections (7)

[Section 5, Eqs. (5.10) and (5.12)] Eq. (5.10) gives ǫ(t) = ǫ₀(φ/φ₀)^{2/3} exp[(1/(720πm²φ₀⁴))(1 − (φ/φ₀)⁴)], while Eq. (5.12) gives ǫ₀^max = m^{1/3} exp[−(m^{−2}/720π)(1 − m²)]. The exponential coefficients are not the same quantity (one is 1/(720πm²φ₀⁴), the other m^{−2}/720π), and the power-law prefactors differ by m^{2/3} at φ = m^{−1/2}. Please check the algebra; the coefficient in Eq. (5.12) is the one that follows from integrating Eq. (5.9) with b = 1/(mφ).
[Section 4, Eq. (4.8)] The step from A T_ab l^a l^b to 1/φ² is not shown. With φ̇ = −m, A = 4π/(m²φ²) and l^a l^b T_ab = m²/(4π), the result follows in one line and should be displayed, since the First Law check of Eq. (4.13) is one of the paper's cleanest results.
[Section 6, Eq. (6.9)] The Maxwell contribution π√3/(αΛ) should be written in terms of q₀², using q₀ = 1/(2e₀) and α = e₀², so that its dimensions and its relation to Eq. (6.5) are transparent to a reader who has not worked through Mann and Ross.
[Section 1] The statement that 'by the de Sitter no hair theorem, we can regard each of these 20 Hubble volumes as having been nucleated independently through gravitational instantons' is an interpretive heuristic rather than a derivation, and a sentence marking it as such would help, since the formal derivation of Eq. (3.16) does not use it.
[Reference [17]] Please verify the publisher and year for Gibbons and Hawking (eds.), Euclidean Quantum Gravity; the standard edition is World Scientific (1993), not Cambridge University Press (1973).
[Section 6, Eq. (6.14) and following text] The dilution factor quoted as 10^{−91}m^{−3/2} is dimensionally odd for a number density; please check whether this should be 10^{−91}m^{−3} or whether a different normalisation is intended.
[Figures 2 and 3] The captions should state explicitly what the axes are in physical units and identify the points N, D, C, U, U₁, U₂ and E in the caption text rather than only in the body, so that the figures can be read independently.

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The paper

The paper claims

The same expansion that drove inflation could have pulled pairs of black holes out of the vacuum, and the no-boundary wave function says how often.

The reading

Inflation is driven by a large effective cosmological constant, and this paper asks whether that same expansion can pull a pair of black holes out of the vacuum the way an electric field pulls apart an electron-positron pair. Working within the no-boundary proposal, the authors compare the probability of the ordinary de Sitter instanton, a four-sphere, with that of the $S^2$ x $S^2$ Nariai instanton that carries a black hole pair, and read off a creation rate $Gamma = exp(-pi/Lambda_{eff})$. The rate approaches one only when the effective cosmological constant is near the Planck value, so Planck-mass black holes are produced freely in the earliest stages of inflation while larger ones are exponentially rare. The authors then follow the pairs through inflation: classically they grow in step with the horizon, and that growth is accounted for exactly by the flux of inflaton energy across the horizon through the first law of black hole mechanics, while quantum mechanically most of them evaporate before inflation ends. Finally, magnetically charged black holes cannot evaporate, but the same instanton calculation puts their number in the presently observable universe at roughly $exp(-10^5)$.

What carries the argument

The machinery is the pair of saddle-point instantons and the ratio of their probability measures. The background is half of a round four-sphere of radius $sqrt{3/Lambda}$, continued onto a Lorentzian de Sitter hyperboloid, with real action $-3pi/(2Lambda)$. The black hole pair is carried by the Nariai solution, the topological product $S^2$ x $S^2$ with both two-spheres of radius $1/sqrt{Lambda}$, whose real action is $-pi/Lambda$. The load-bearing technical point is that regularity of the Euclidean section forces the two horizons to have equal radius, so the instanton approach fixes the mass of the created pair rather than leaving it free; the black holes are born degenerate, at the maximum Schwarzschild-de Sitter size. Dividing the two measures gives $Gamma = exp(-pi/Lambda_{eff})$, and doubling the real parts of the actions recovers the conventional $I_{obj} - I_{bg}$ formula.

Extended reading notes

Core claim

The central claim is that the no-boundary wave function, applied to the two competing gravitational instantons, yields a pair creation rate for black holes on an inflationary background that agrees with the standard instanton prescription $Gamma = exp[-(I_{obj} - I_{bg})]$ and behaves sensibly: unsuppressed near the Planck scale, exponentially suppressed at low energies. The rate comes out as $exp(-pi/Lambda_{eff})$, the ratio of the $S^2$ x $S^2$ probability measure $exp(2pi/Lambda)$ to the de Sitter measure $exp(3pi/Lambda)$. The authors argue that the rival tunnelling proposal, which flips the sign of the exponent, would give the reciprocal rate, make de Sitter space catastrophically unstable to black hole formation, and favour larger black holes over smaller ones at low energy, so it cannot be maintained. Two further results ride on the same machinery: the classical growth of the pair during slow roll is shown to satisfy the first law of black hole mechanics with the surface gravity of the degenerate Schwarzschild-de Sitter solution, and the extension to Reissner-Nordstrom-de Sitter instantons gives an exponentially small abundance for magnetically charged primordial black holes in ordinary Einstein-Maxwell theory.

Load-bearing premise

The claim that most neutral black holes evaporate before inflation ends depends on assuming the initial mismatch between the two horizon radii is spread uniformly from zero up to a few Planck lengths, a distribution the authors explicitly say is beyond the scope of the paper and lies where the semi-classical approximation breaks down anyway.

Editorial extensions

If this is right

Black hole pair creation during inflation is a real, computable effect whose rate is fixed by the effective cosmological constant at the moment of nucleation, so the primordial black hole mass spectrum is tied directly to the inflationary trajectory.
The tunnelling proposal is excluded on physical grounds, since it would make de Sitter space violently unstable to black hole formation and would preferentially create the largest black holes at the lowest energies.
Neutral black holes created during inflation grow with the cosmological horizon, and this growth is not an independent assumption but a consequence of energy flux across the horizon obeying the first law of black hole mechanics.
Almost all neutral primordial black holes evaporate before inflation ends, so they leave no relics; only pairs born late and large, or born extremely close to degenerate, survive.
Magnetically charged primordial black holes survive to the present but are so rare in Einstein-Maxwell theory, roughly $exp[-(9pi^2/(16 alpha m^2))^{1/3}]$, that the observable universe should contain none.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Because the creation rate depends on $Lambda_{eff}$ at nucleation, the resulting black hole abundance is a probe of the inflaton potential at energies far above what the microwave background constrains, so a detection or a firm null result would constrain the shape of $V(phi)$ rather than just its slope.
The same probability-ratio argument applies to any instanton whose Euclidean action can be computed, which suggests a general recipe: any topology-changing fluctuation of the inflationary vacuum is suppressed by the difference of real Euclidean actions, and the no-boundary versus tunnelling sign question can be settled case by case.
The dilaton remark at the end points to a concrete programme: if the effective fine structure constant or Newton constant was larger in the early universe, the exponent suppressing magnetic black holes shrinks, and the resulting abundance could become a constraint on dilatonic couplings rather than a null prediction.
The evaporation conclusion rests on an assumed flat distribution of initial horizon splittings at the Planck scale, so a first-principles calculation of that distribution from the same path integral would either rescue a population of surviving Planck-mass relics or close the question.

Original abstract

Black holes came into existence together with the universe through the quantum process of pair creation in the inflationary era. We present the instantons responsible for this process and calculate the pair creation rate from the no boundary proposal for the wave function of the universe. We find that this proposal leads to physically sensible results, which fit in with other descriptions of pair creation, while the tunnelling proposal makes unphysical predictions. We then describe how the pair created black holes evolve during inflation. In the classical solution, they grow with the horizon scale during the slow roll-down of the inflaton field; this is shown to correspond to the flux of field energy across the horizon according to the First Law of black hole mechanics. When quantum effects are taken into account, however, it is found that most black holes evaporate before the end of inflation. Finally, we consider the pair creation of magnetically charged black holes, which cannot evaporate. In standard Einstein-Maxwell theory we find that their number in the presently observable universe is exponentially small. We speculate how this conclusion may change if dilatonic theories are applied.

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The abstract, sentence by sentence

arXiv listing abstract v2. Results and the main claim are underlined; show every role.

Black holes came into existence together with the universe through the quantum process of pair creation in the inflationary era. We present the instantons responsible for this process and calculate the pair creation rate from the no boundary proposal for the wave function of the universe. We find that this proposal leads to physically sensible results, which fit in with other descriptions of pair creation, while the tunnelling proposal makes unphysical predictions. We then describe how the pair created black holes evolve during inflation. In the classical solution, they grow with the horizon scale during the slow roll-down of the inflaton field; this is shown to correspond to the flux of field energy across the horizon according to the First Law of black hole mechanics. When quantum effects are taken into account, however, it is found that most black holes evaporate before the end of inflation. Finally, we consider the pair creation of magnetically charged black holes, which cannot evaporate. In standard Einstein-Maxwell theory we find that their number in the presently observable universe is exponentially small. We speculate how this conclusion may change if dilatonic theories are applied.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	Black holes came into existence together with the universe through the quantum process of pair creation in the inflationary era.	background 0.63	0.17	0.48	asserted 1.00
s1	We present the instantons responsible for this process and calculate the pair creation rate from the no boundary proposal for the wave function of the universe.	contribution 0.58	0.95	0.07	asserted 0.98
s2	We find that this proposal leads to physically sensible results, which fit in with other descriptions of pair creation, while the tunnelling proposal makes unphysical predictions.	result 0.78	0.95	0.26	asserted 1.00
s3	We then describe how the pair created black holes evolve during inflation.	contribution 0.97	0.85	0.04	asserted 0.69
s4	In the classical solution, they grow with the horizon scale during the slow roll-down of the inflaton field; this is shown to correspond to the flux of field energy across the horizon according to the First Law of black hole mechanics.	result 0.87	0.85	0.07	asserted 1.00
s5	When quantum effects are taken into account, however, it is found that most black holes evaporate before the end of inflation.	result 1.00	0.94	0.22	asserted 1.00
s6	Finally, we consider the pair creation of magnetically charged black holes, which cannot evaporate.	contribution 0.98	0.56	0.04	asserted 0.58
s7	In standard Einstein-Maxwell theory we find that their number in the presently observable universe is exponentially small.	result 1.00	0.93	0.19	asserted 1.00
s8	We speculate how this conclusion may change if dilatonic theories are applied.	future work 0.88	0.18	0.04	speculative 1.00

What kind of paper the classifier reports this is: theoretical result 1.00 · evidence from proof or derivation 1.00.

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sources read

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not examined

full text
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** The creation of a de Sitter universe. The lower region is half of a Euclidean

**Figure 2.** The hypersurfaces in the parameter space of the Reissner{Nordstrom{

**Figure 3.** The action of the di erent types of Reissner{Nordstrom{de Sitter instan

Figures reproduced from arXiv:gr-qc/9606052 by the authors.

References

Works cited (23)

[15]

P. Ginsparg and M. J. Perry: Semiclassical perdurance of de Sitter space . Nucl. Phys. B222, 245 (1983)

View on Pith 1983
[18]

R. Bousso and S. W. Hawking: The probability for primordial black holes . Phys. Rev. D 52, 5659 (1995), gr–qc/9506047

View on Pith 1995
[11]

R. B. Mann and S. F. Ross: Cosmological production of charged black hole pairs . Phys. Rev. D 52, 2254 (1995), gr–qc/9504015

View on Pith 1995
[1]

A. D. Linde: Chaotic inﬂation . Phys. Lett. 129B, 177 (1983)

View on Pith 1983
[2]

S. W. Hawking: The quantum state of the Universe . Nucl. Phys. B239, 257 (1984)

View on Pith 1984
[3]

A. D. Linde: Eternally existing selfreproducing chaotic inﬂationary u niverse. Phys. Lett. B175, 395 (1986)

View on Pith 1986
[4]

A. D. Linde: Eternal chaotic inﬂation . Mod. Phys. Lett. A1, 81 (1986)

View on Pith 1986
[5]

S. W. Hawking and S. F. Ross: Pair production of black holes on cosmic strings . Phys. Rev. Lett. 75, 3382 (1995), gr–qc/9506020

View on Pith 1995
[6]

G. W. Gibbons: Quantized ﬂux tubes in Einstein–Maxwell theory and noncom- pact internal spaces . In Fields and Geometry 1986: Proceedings of the 22nd Karpacz Winter School of Theoretical Physics , edited by A. Jadczyk, World Scientiﬁc, Singapore (1986)

View on Pith 1986
[7]

D. Garﬁnkle, S. B. Giddings and A. Strominger: Entropy in black hole pair production. Phys. Rev. D 49, 958 (1994), gr–qc/9306023

View on Pith 1994
[8]

F. Dowker, J. P. Gauntlett, D. A. Kastor and J. Traschen: Pair creation of dilaton black holes . Phys. Rev. D 49, 2909 (1994), hep–th/9309075. 27

View on Pith 1994
[9]

F. Dowker, J. P. Gauntlett, S. B. Giddings and G. T. Horowi tz: On pair creation of extremal black holes and Kaluza–Klein monopoles . Phys. Rev. D 50, 2662 (1994), hep–th/9312172

View on Pith 1994
[10]

S. W. Hawking, G. T. Horowitz and S. F. Ross: Entropy, area, and black hole pairs. Phys. Rev. D 51, 4302 (1995), gr–qc/9409013

View on Pith 1995
[12]

S. W. Hawking and S. F. Ross: Duality between electric and magnetic black holes. Phys. Rev. D 52, 5865 (1995), hep–th/9504019

View on Pith 1995
[13]

R. R. Caldwell, A. Chamblin and G. W. Gibbons: Pair creation of black holes by domain walls . Phys. Rev. D 53, 7103 (1996), hep–th/9602126

View on Pith 1996
[14]

J. B. Hartle and S. W. Hawking: Wave function of the Universe . Phys. Rev. D 28, 2960 (1983)

View on Pith 1983
[16]

A. Vilenkin: Boundary conditions in quantum cosmology . Phys. Rev. D 33, 3560 (1986)

View on Pith 1986
[17]

G. W. Gibbons and S. W. Hawking (eds.): Euclidean Quantum Gravity . Cam- bridge University Press, Cambridge, England (1973)

View on Pith 1973
[19]

G. F. Smoot et al.: Structure in the COBE DMR ﬁrst year maps . Astrophys. J. 396, L1 (1992)

View on Pith 1992
[20]

J. J. Halliwell and S. W. Hawking: The origin of structure in the universe . Phys. Rev. D 31, 1777 (1985)

View on Pith 1985
[21]

J. M. Bardeen, B. Carter and S. W. Hawking: The four laws of black hole mechanics. Commun. Math. Phys. 31, 161 (1973). 28

View on Pith 1973
[22]

G. W. Gibbons and S. W. Hawking: Action integrals and partition functions in quantum gravity . Phys. Rev. D 15, 2752 (1977)

View on Pith 1977
[23]

F. Mellor and I. Moss: Black holes and gravitational instantons . Class. Quant. Grav. 6, 1379 (1989). 29

View on Pith 1989

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The paper shows that small AdS4 black holes reproduce flat-space black hole partition functions in the limit of vanishing cosmological constant, and constructs a Carrollian-like limit of ABJM that yields the entropy o...
Semiclassical decay of de Sitter space into black holes with vortex-deformed horizons
hep-th 2026-06 unverdicted novelty 5.0 of 10

De Sitter space decays into vortex-dressed black holes via a regular Euclidean instanton that generalizes the Nariai solution, with rates organized by topological charge of the vortices.
The fate of Reissner--Nordstr\"om--de Sitter black holes: nonequilibrium discharge and evaporation
hep-th 2026-05 unverdicted novelty 5.0 of 10

Semiclassical RN-dS evaporation via 2D dilaton gravity and anomaly flux yields monotonic neutral mass loss and rapid discharge, making classical equilibrium loci non-attractors and leading to empty de Sitter space.
Dynamical black holes in the inflationary epoch
gr-qc 2026-03 unverdicted novelty 5.0 of 10

Only black holes with initial masses in a narrow range formed during inflation survive to the present day, reaching a maximum mass of approximately 1.043 times 10 to the minus 3 solar masses.
Further remarks on de Sitter space, extremal surfaces and time entanglement
hep-th 2023-09 conditional novelty 5.0 of 10

The complex areas of de Sitter extremal surfaces, and the pseudo-entanglement wedge they bound, follow from analytic continuation of anti-de Sitter Ryu-Takayanagi surfaces and satisfy entropy relations that continue t...
On the Euclidean Action of de Sitter Black Holes and Constrained Instantons
hep-th 2022-03 conditional novelty 5.0 of 10

The Euclidean action of a Schwarzschild-de Sitter black hole equals minus the sum of its two horizon entropies and is independent of the Euclidean time period, which yields a pair-creation rate for black holes in de S...
Quasi-local energy and microcanonical entropy in two-dimensional nearly de Sitter gravity
hep-th 2022-03 accept novelty 5.0 of 10

The on-shell Euclidean microcanonical action of a causal diamond in semi-classical JT gravity equals minus the generalized entropy, so minimizing the action extremizes the entropy and yields the island rule in dS2 wit...
Quantum Corrected Black Holes from String T-Duality
gr-qc 2019-02 conditional novelty 5.0 of 10

String T-duality yields a regular black hole metric identical in form to Bardeen's, with the ultraviolet regulator fixed by the string scale.
Baryogenesis from Hawking Radiation
hep-ph 2014-04 conditional novelty 5.0 of 10

Black hole evaporation in a chemical potential generated by derivative couplings to the B-L current produces a baryon asymmetry, and the same mechanism can set the dark matter abundance.
TASI Lectures on Holographic Space-Time, SUSY and Gravitational Effective Field Theory
hep-th 2010-07 conditional novelty 5.0 of 10

Quantum gravity has no field-theoretic vacuum states; instead each asymptotic geometry defines its own finite-dimensional quantum system, with de Sitter space a finite-state system whose gravitino mass scales as the q...
Extremal limits and black hole entropy
hep-th 2009-01 conditional novelty 5.0 of 10

The extremal limit of a Reissner-Nordstrom black hole is discontinuous: the interior region survives as a disconnected AdS2 x S2 space, which the authors propose carries the entropy that semiclassical methods assign to zero.
Scattering and Hawking Radiation from Einstein--Euler--Heisenberg--de Sitter Black Holes
gr-qc 2026-06 unverdicted novelty 4.0 of 10

Numerical scattering calculations for neutral scalar and Dirac fields on Einstein-Euler-Heisenberg-de Sitter black holes show that Euler-Heisenberg coupling raises barriers and shifts transmission frequencies upward w...
Grey-Body Factors and Thermodynamics of Asymptotically de Sitter Black Holes in Generalized Proca Theory
gr-qc 2026-05 unverdicted novelty 4.0 of 10

Applies sixth-order WKB to compute grey-body factors for test fields on generalized Proca black holes with effective de Sitter scale, finding that scalar mass and couplings raise barriers, suppress low-frequency trans...
de Sitter space, extremal surfaces and "time-entanglement"
hep-th 2022-10 conditional novelty 4.0 of 10

De Sitter extremal surfaces anchored at the future boundary either stretch timelike to the past boundary or join a Euclidean hemisphere, and the resulting areas motivate toy models of entanglement between timelike-sep...
Review of the No-Boundary Wave Function
hep-th 2023-03 accept novelty 3.0 of 10

The no-boundary wave function, implemented via a regular momentum condition at the South Pole rather than a literal sum over compact metrics, yields stable semiclassical saddle points that prefer low inflationary ener...
De Sitter Musings
hep-th 2012-05 accept novelty 3.0 of 10

A survey of de Sitter physics arguing that the static-patch observer and the far-future-boundary metaobserver are genuinely distinct problems, and that most open questions in the field come from conflating them.

How later papers used it

75 later papers cite it

Evidence: substantial. 74 of the citing papers share no author with it, and 14 of those carry a confident label from Pith's citation classifier.

Pith read 21 of the 143 citing sentences, choosing only those whose citation markers point at this paper alone, and wrote the paraphrases from them. The summary, the counts, the evidence level, the links and the quotes come from code; the summary counts the papers listed below. A later paper is listed under a label when Pith's citation classifier gives one of its sentences that label at probability 0.8 or more (0.95 for support), Pith's reading finds that the citing paper itself did it, and a second check confirms the paraphrase against the sentence. This section lists uses only. It does not list criticism or disagreement, so their absence here says nothing about how the paper has been received. Papers that share an author with this one are listed apart, under "By the same authors". A quote is the citing authors' own sentence, shown when their paper's licence is CC BY, CC BY-SA or CC0. JSON

4 later papers by other authors are listed below: 3 used its method or tool and 1 compared its own results with it.

Uses its method or tool 3 of 4 papers with this label

Following Bousso and Hawking [ref] , we fix the normalization using the unique static geodesic observer at $r_O$, defined by $f'(r_O)=0$.

The Fate of Nucleated Black Holes in de Sitter Quantum Gravity (2026) · The near-Nariai throat and two-dimensional dilaton gravity · 0.92
Followed the gravitational instanton approach proposed by another work and by this paper to estimate the PBH nuclear rate.

Primordial black holes in a dimensionally reduced universe (2018) · 1.00 · paraphrased: the licence does not allow quoting
Used the normalization proposed by this paper so that the norm of the Killing vector is one where gravitational attraction and cosmological repulsion cancel.

Some aspects of entropic gravity in the presence of a noncommutative Schwarzschild-deSitter black hole (2013) · 0.95 · paraphrased: the licence does not allow quoting

Compares with it 1 of 8 papers with this label

Said that the process it described resembles this paper's cosmological pair production of four-dimensional Schwarzschild-de Sitter black holes in the Nariai limit.

Holography For a De Sitter-Esque Geometry (2011) · 0.99 · paraphrased: the licence does not allow quoting

How the literature uses it

61 arXiv papers cite this in 126 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 89 sentences without a confident label

Contests it 1 paper

To be clear, we agree with all the equations in the Appendix of [ref] , except for the final result (A.18) for the action.

On the Euclidean Action of de Sitter Black Holes and Constrained Instantons · Near-Nariai expansion · contest 0.81

Builds on it 3 papers

In previous work [ref] we provided a general and covariant derivation of the Euclidean on-shell action for neutral, nonrotating black holes in de Sitter space, generalizing classic results for the Nariai instanton [ref] , and showed it is equal to minus the sum of the two horizon entropies for arbitrary mass and in any number of dimensions (as was previously done for four dimensions in [ref] ).

Action, entropy and pair creation rate of charged black holes in de Sitter space · Introduction · 0.84
This process is a generalization of the pair creation of black holes described in Refs. [ref] , when a $CP^1$ field is turned on.

Semiclassical decay of de Sitter space into black holes with vortex-deformed horizons · The decay rate · 1.00
Our result generalizes a previous computation [ref] of the probability of nucleating a Nariai black hole to arbitrary SdS black holes [ref] in general dimensions.

On the Euclidean Action of de Sitter Black Holes and Constrained Instantons · Conclusion · 0.99

Uses its method or tool 4 papers

In order to estimate the PBH nuclear rate, we follow the gravitational instanton approach proposed by Mann & Ross [ref] and Bousso & Hawking [ref] .

Primordial black holes in a dimensionally reduced universe · Introduction · 1.00
Following Bousso and Hawking [ref] , we fix the normalization using the unique static geodesic observer at $r_O$, defined by $f'(r_O)=0$.

The Fate of Nucleated Black Holes in de Sitter Quantum Gravity · The near-Nariai throat and two-dimensional dilaton gravity · 0.92
In order to prevent a complication in taking the normalization of Killing vector, we use the normalization proposed by Bousso and Hawing [ref] so that the norm of the Killing vector is one at the area where the gravitational attraction and the cosmological repulsion cancel each other and thus the force vanishes.

Some aspects of entropic gravity in the presence of a noncommutative Schwarzschild-deSitter black hole · Verlinde's Entropic Formalism · 0.95

Compares with it 7 papers

This is precisely what [ref] found for the extremal case.

Festina Lente: EFT Constraints from Charged Black Hole Evaporation in de Sitter · Quasistatic · 0.94
The process we have described resembles the cosmological pair production of four-dimensional Schwarzschild-de Sitter black holes in the Nariai limit (where the black hole and cosmological horizons coincide) [ref] .

Holography For a De Sitter-Esque Geometry · Instantons · 0.99
For $d=4$ we recover the result by Bousso and Hawking [ref] , cf. their Eq. (A.16), $κ̃_b,c = √( Λ) (1 ±2/3ϵ̃_0 + 11/18 ϵ̃_0^2 ) + 𝒪(ϵ̃_0^3)$ (see also [ref] ).

On the Euclidean Action of de Sitter Black Holes and Constrained Instantons · SdS geometry and its non-equilibrium thermodynamics · 0.99

Cites it as support 1 paper

Producing them during inflation is unlikely [ref] .

Comments on magnetic black holes · Production in the early universe? · 0.98

Cites it as background 15 papers

In this regard, it is well known that dS space is semi-classically unstable to the thermal nucleation of black holes [ref] .

Semiclassical decay of de Sitter space into black holes with vortex-deformed horizons · Introduction · 0.94
The nucleation process was studied by Bousso and Hawking in the context of primordial black holes (PBHs) created during inflation [ref] .

The Fate of Nucleated Black Holes in de Sitter Quantum Gravity · Introduction · 0.94
A special role in this paper is played by the Nariai limit [ref] of the wavefunction of a $S^1×S^2$ universe.

The wavefunction of a quantum $S^1 \times S^2$ universe · The Nariai limit of the $S^1 ×S^2$ wavefunction · 0.92

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

[1] [15]

P. Ginsparg and M. J. Perry: Semiclassical perdurance of de Sitter space . Nucl. Phys. B222, 245 (1983)

View on Pith 1983

[2] [18]

R. Bousso and S. W. Hawking: The probability for primordial black holes . Phys. Rev. D 52, 5659 (1995), gr–qc/9506047

View on Pith 1995

[3] [11]

R. B. Mann and S. F. Ross: Cosmological production of charged black hole pairs . Phys. Rev. D 52, 2254 (1995), gr–qc/9504015

View on Pith 1995

[4] [1]

A. D. Linde: Chaotic inﬂation . Phys. Lett. 129B, 177 (1983)

View on Pith 1983

[5] [2]

S. W. Hawking: The quantum state of the Universe . Nucl. Phys. B239, 257 (1984)

View on Pith 1984

[6] [3]

A. D. Linde: Eternally existing selfreproducing chaotic inﬂationary u niverse. Phys. Lett. B175, 395 (1986)

View on Pith 1986

[7] [4]

A. D. Linde: Eternal chaotic inﬂation . Mod. Phys. Lett. A1, 81 (1986)

View on Pith 1986

[8] [5]

S. W. Hawking and S. F. Ross: Pair production of black holes on cosmic strings . Phys. Rev. Lett. 75, 3382 (1995), gr–qc/9506020

View on Pith 1995

[9] [6]

G. W. Gibbons: Quantized ﬂux tubes in Einstein–Maxwell theory and noncom- pact internal spaces . In Fields and Geometry 1986: Proceedings of the 22nd Karpacz Winter School of Theoretical Physics , edited by A. Jadczyk, World Scientiﬁc, Singapore (1986)

View on Pith 1986

[10] [7]

D. Garﬁnkle, S. B. Giddings and A. Strominger: Entropy in black hole pair production. Phys. Rev. D 49, 958 (1994), gr–qc/9306023

View on Pith 1994

[11] [8]

F. Dowker, J. P. Gauntlett, D. A. Kastor and J. Traschen: Pair creation of dilaton black holes . Phys. Rev. D 49, 2909 (1994), hep–th/9309075. 27

View on Pith 1994

[12] [9]

F. Dowker, J. P. Gauntlett, S. B. Giddings and G. T. Horowi tz: On pair creation of extremal black holes and Kaluza–Klein monopoles . Phys. Rev. D 50, 2662 (1994), hep–th/9312172

View on Pith 1994

[13] [10]

S. W. Hawking, G. T. Horowitz and S. F. Ross: Entropy, area, and black hole pairs. Phys. Rev. D 51, 4302 (1995), gr–qc/9409013

View on Pith 1995

[14] [12]

S. W. Hawking and S. F. Ross: Duality between electric and magnetic black holes. Phys. Rev. D 52, 5865 (1995), hep–th/9504019

View on Pith 1995

[15] [13]

R. R. Caldwell, A. Chamblin and G. W. Gibbons: Pair creation of black holes by domain walls . Phys. Rev. D 53, 7103 (1996), hep–th/9602126

View on Pith 1996

[16] [14]

J. B. Hartle and S. W. Hawking: Wave function of the Universe . Phys. Rev. D 28, 2960 (1983)

View on Pith 1983

[17] [16]

A. Vilenkin: Boundary conditions in quantum cosmology . Phys. Rev. D 33, 3560 (1986)

View on Pith 1986

[18] [17]

G. W. Gibbons and S. W. Hawking (eds.): Euclidean Quantum Gravity . Cam- bridge University Press, Cambridge, England (1973)

View on Pith 1973

[19] [19]

G. F. Smoot et al.: Structure in the COBE DMR ﬁrst year maps . Astrophys. J. 396, L1 (1992)

View on Pith 1992

[20] [20]

J. J. Halliwell and S. W. Hawking: The origin of structure in the universe . Phys. Rev. D 31, 1777 (1985)

View on Pith 1985

[21] [21]

J. M. Bardeen, B. Carter and S. W. Hawking: The four laws of black hole mechanics. Commun. Math. Phys. 31, 161 (1973). 28

View on Pith 1973

[22] [22]

G. W. Gibbons and S. W. Hawking: Action integrals and partition functions in quantum gravity . Phys. Rev. D 15, 2752 (1977)

View on Pith 1977

[23] [23]

F. Mellor and I. Moss: Black holes and gravitational instantons . Class. Quant. Grav. 6, 1379 (1989). 29

View on Pith 1989