arxiv: 2603.04590 · v2 · submitted 2026-03-04 · 🌀 gr-qc

Recognition: 2 theorem links

· Lean Theorem

Dynamical black holes in the inflationary epoch

Ertola Urtubey Milos , Daniela P\'erez

Authors on Pith no claims yet

Pith reviewed 2026-05-15 16:00 UTC · model grok-4.3

classification 🌀 gr-qc

keywords black holesinflationMcVittie geometryprimordial black holesHawking evaporationcosmological evolutionStarobinsky inflation

0 comments

The pith

Black holes formed during inflation survive to today only within a narrow initial mass range and reach at most about 0.001 solar masses.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper studies black holes that existed during inflation and tracks how their masses change as the universe expands through radiation, matter, and dark-energy eras. It assumes the holes stay coupled to the background so their gravitational mass grows with the cosmic scale factor, then adds the opposing effects of radiation accretion and Hawking evaporation. The requirement that the event horizon never exceeds the particle horizon, together with bounds from evaporation during inflation and runaway accretion later, restricts the allowed starting masses to a narrow window. Only those holes reach the present day, and even then their final mass cannot exceed roughly 1.043 times 10 to the minus three solar masses.

Core claim

Only black holes formed with initial masses inside a narrow window during inflation persist through all later eras; their masses evolve with the scale factor in a generalized McVittie geometry, radiation accretion is limited to prevent runaway growth, and Hawking evaporation sets a minimum starting mass, yielding a present-day upper limit of M(t_0) ≃ 1.043×10^{-3} M_⊙.

What carries the argument

Generalized McVittie geometry in which black-hole gravitational mass scales proportionally with the cosmic scale factor, combined with explicit integration of Hawking evaporation and radiation accretion across the inflationary, radiation, matter, and dark-energy epochs.

If this is right

Black holes with initial masses below the lower bound evaporate completely during inflation.
Masses above the upper bound cause the event horizon to exceed the particle horizon or trigger runaway accretion in the radiation era.
The maximum mass any such surviving hole can reach today is fixed at approximately 1.043×10^{-3} solar masses.
No population of much heavier black holes can be traced back to formation during inflation under this coupling.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Any primordial black holes detected today in the sub-solar-mass range would have to have formed inside this narrow window if they followed the same dynamical coupling.
The result supplies a concrete upper mass cutoff that could be compared with microlensing or gravitational-wave surveys of low-mass compact objects.
Changing the assumed geometry or the strength of accretion would shift or remove the allowed mass window, offering a direct test of the coupling assumption.

Load-bearing premise

Black holes remain dynamically coupled to the expanding universe through a generalized McVittie geometry so that their gravitational mass grows in proportion to the cosmic scale factor.

What would settle it

A calculation or observation showing a black hole that formed during inflation with a mass outside the narrow window yet still satisfies the horizon condition and reaches the present day without complete evaporation or runaway growth.

read the original abstract

We investigate the evolution of black holes present during the inflationary epoch, assuming they are dynamically coupled to the cosmological background through a generalized McVittie geometry, such that their gravitational mass scales with the cosmic scale factor. Adopting Starobinsky's $\mathcal{R}^2$ inflation model, we analyse the combined effects of cosmological coupling, Hawking evaporation and radiation accretion during the subsequent cosmic eras: inflation, radiation, matter, and dark energy. Requiring the black hole event horizon to remain smaller than the particle horizon at all times yields an upper bound on the mass parameter. Radiation accretion during the radiation era further constrains the parameter space to prevent runaway growth. Hawking evaporation sets a lower bound on the initial mass to ensure survival through inflation. We find that only black holes formed within a narrow initial mass range during inflation can persist to the present day, reaching a maximum mass of $M(t_0) \simeq 1.043\times10^{-3} M_\odot$.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper finds a narrow initial mass range for black holes that survive inflation to today, but this hinges on an assumed mass scaling with the scale factor that is not derived from the Starobinsky model.

read the letter

The one thing to know is that this paper concludes only black holes with initial masses in a narrow range during inflation can make it to the present day, with a maximum mass today of about 1.043 times 10^{-3} solar masses. Everything else follows from enforcing the horizon condition while adding up cosmological growth, evaporation, and accretion in the Starobinsky model. They apply the generalized McVittie framework across all eras and produce a concrete numerical result that tightens the allowed parameter space. That specific bound and the narrow window look new compared to prior work on the topic. The modeling is clear in outline, but the load-bearing piece is the assumption that the black hole mass scales with the scale factor. The paper adopts this from the generalized McVittie geometry without showing a derivation from the Starobinsky R^2 inflation or the junction conditions in the effective theory. The abstract mentions the final value but gives no equations, integration details, or checks on how the result changes if the scaling is modified. If that assumption does not hold once the full dynamics are solved, the integrated bounds on initial mass would move by orders of magnitude and the narrow survival range would disappear. The lower bound from Hawking evaporation during inflation and the upper bound from radiation accretion both rest on it. This is relevant for anyone working on primordial black holes and early universe constraints. It engages honestly with the literature on dynamical black holes even if the central assumption needs more support. I would send it to peer review so that the derivations can be checked properly.

Referee Report

2 major / 2 minor

Summary. The paper investigates the evolution of black holes during the inflationary epoch in Starobinsky's R² inflation model. It assumes dynamical coupling to the cosmological background via a generalized McVittie geometry in which gravitational mass scales with the scale factor a(t). The analysis integrates cosmological mass growth, Hawking evaporation, and radiation accretion across inflation, radiation, matter, and dark-energy eras, imposing the condition that the black-hole horizon remains inside the particle horizon at all times. This yields the claim that only black holes formed in a narrow initial-mass window survive to the present day, reaching a maximum mass M(t₀) ≃ 1.043 × 10^{-3} M⊙.

Significance. If the mass-scaling assumption and horizon condition can be placed on a firmer footing, the result supplies a concrete, falsifiable upper bound on primordial black-hole masses that persist through inflation, with possible relevance to dark-matter constraints and gravitational-wave backgrounds. The work also illustrates how the interplay of accretion, evaporation, and expansion can carve out a narrow survival window in a specific inflationary model.

major comments (2)

[modeling approach (abstract and § on generalized McVittie geometry)] The headline result (narrow initial-mass window and M(t₀) ≃ 1.043 × 10^{-3} M⊙) is obtained by integrating M ∝ a(t) growth, Hawking evaporation, and radiation accretion under the horizon condition. The scaling M ∝ a(t) is stated as an assumption of the generalized McVittie geometry rather than derived from the junction conditions or the effective stress-energy tensor of the Starobinsky f(R) model; if this scaling does not hold once R² corrections are included, the integrated bounds shift by orders of magnitude and the narrow-range conclusion is lost.
[evolution and numerical results] The abstract reports a specific present-day mass but supplies no explicit evolution equations, numerical integration scheme, or error analysis. Without these, it is impossible to verify that the reported bounds follow from the stated assumptions or that the narrow window is independent of the choice of initial-mass fitting procedure.

minor comments (2)

[abstract] The abstract states the final mass value without quoting the precise initial-mass window or the functional form of the scale-factor coupling used in the integration.
[introduction/references] Ensure that earlier literature on McVittie metrics in f(R) gravity and on dynamical black holes in inflation is cited to clarify how the generalized geometry differs from standard treatments.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful and constructive review of our manuscript. We address each major comment in detail below and have revised the paper to improve transparency and rigor where appropriate.

read point-by-point responses

Referee: [modeling approach (abstract and § on generalized McVittie geometry)] The headline result (narrow initial-mass window and M(t₀) ≃ 1.043 × 10^{-3} M⊙) is obtained by integrating M ∝ a(t) growth, Hawking evaporation, and radiation accretion under the horizon condition. The scaling M ∝ a(t) is stated as an assumption of the generalized McVittie geometry rather than derived from the junction conditions or the effective stress-energy tensor of the Starobinsky f(R) model; if this scaling does not hold once R² corrections are included, the integrated bounds shift by orders of magnitude and the narrow-range conclusion is lost.

Authors: We acknowledge that the M ∝ a(t) scaling is introduced as an assumption within the generalized McVittie ansatz rather than derived ab initio from the junction conditions of the Starobinsky effective stress-energy tensor. This choice follows the standard phenomenological approach used in the literature for embedding black holes in FLRW backgrounds. In the revised manuscript we have added an extended discussion in the generalized McVittie section that (i) recalls the effective-fluid interpretation of the R² corrections, (ii) shows that the leading-order cosmological coupling remains compatible with the scaling to within the slow-roll regime, and (iii) explicitly states the limitation that a full numerical-relativity treatment would be required to quantify possible deviations. The headline result is therefore presented as conditional on this assumption, and we have added a brief sensitivity estimate illustrating how order-of-magnitude changes in the scaling exponent would alter the final mass window. revision: partial
Referee: [evolution and numerical results] The abstract reports a specific present-day mass but supplies no explicit evolution equations, numerical integration scheme, or error analysis. Without these, it is impossible to verify that the reported bounds follow from the stated assumptions or that the narrow window is independent of the choice of initial-mass fitting procedure.

Authors: We apologize for the insufficient detail in the original presentation. The evolution equations (including the separate terms for cosmological mass growth, Hawking evaporation, and radiation accretion) are now written explicitly in a new subsection of Section 3, together with the horizon condition. The numerical integration employs a fourth-order Runge-Kutta scheme with adaptive step-size control; we have added a description of the integrator, the convergence criterion, and an error analysis based on step-size halving. Additional figures demonstrate that the narrow survival window remains stable under modest variations in the initial-mass fitting procedure. The abstract has been updated to direct readers to these sections. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained under explicit assumptions

full rationale

The paper states the key mass scaling explicitly as an assumption ('assuming they are dynamically coupled to the cosmological background through a generalized McVittie geometry, such that their gravitational mass scales with the cosmic scale factor'). The claimed result (narrow initial-mass window and M(t0) ≃ 1.043×10^{-3} M_⊙) follows from integrating the stated effects (cosmological coupling, Hawking evaporation, radiation accretion) subject to the independent horizon condition. This is a standard forward calculation from inputs to consequences, not a reduction of the output to the inputs by construction. No self-citations, uniqueness theorems, or smuggled ansatzes are load-bearing in the provided text. The specific numerical bound is a model output, not tautological.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

The central claim rests on the generalized McVittie geometry and the Starobinsky inflation model as background assumptions; the initial mass serves as the primary free parameter whose allowed range is determined by the horizon and accretion conditions.

free parameters (1)

initial black hole mass
The starting mass is varied to find the narrow window that satisfies the horizon condition and avoids runaway accretion or complete evaporation.

axioms (2)

domain assumption Generalized McVittie geometry couples black hole mass to cosmic scale factor
Invoked to make the black hole dynamical within the inflationary background.
domain assumption Starobinsky R^2 inflation model governs the early expansion history
Adopted to describe the inflationary epoch and subsequent eras.

pith-pipeline@v0.9.0 · 5464 in / 1401 out tokens · 69397 ms · 2026-05-15T16:00:55.774491+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

we assume that these black holes are coupled to the background cosmological dynamics and that, as a result, their properties are modified... M(t)=m0 a(t) (Eq. 2.9, 3.8)
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the gravitational mass of the black hole... is encoded in the Weyl part... M(t)=m0 a(t)

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

36 extracted references · 36 canonical work pages · 14 internal anchors

[1]

McVittie,The mass-particle in an expanding universe,Monthly Notices of the Royal Academic Society93(1933) 325

G.C. McVittie,The mass-particle in an expanding universe,Monthly Notices of the Royal Academic Society93(1933) 325

work page 1933
[2]

Influence of global cosmological expansion on local dynamics and kinematics

M. Carrera and D. Giulini,Influence of global cosmological expansion on local dynamics and kinematics,Reviews of Modern Physics82(2010) 169 [0810.2712]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[3]

On the generalization of McVittie's model for an inhomogeneity in a cosmological spacetime

M. Carrera and D. Giulini,Generalization of McVittie’s model for an inhomogeneity in a cosmological spacetime,Physical Review D81(2010) 043521 [0908.3101]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[4]

The effect of an expanding universe on massive objects

R. Nandra, A.N. Lasenby and M.P. Hobson,The effect of an expanding universe on massive objects,Mon. Not. R. Astron. Soc.422(2012) 2945 [1104.4458]. – 14 –

work page internal anchor Pith review Pith/arXiv arXiv 2012
[5]

The effect of a massive object on an expanding universe

R. Nandra, A.N. Lasenby and M.P. Hobson,The effect of a massive object on an expanding universe,Mon. Not. R. Astron. Soc.422(2012) 2931 [1104.4447]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[6]

Faraoni,Cosmological and Black Hole Apparent Horizons, vol

V. Faraoni,Cosmological and Black Hole Apparent Horizons, vol. 907 (2015), 10.1007/978-3-319-19240-6

work page doi:10.1007/978-3-319-19240-6 2015
[7]

Electromagnetic fields and charges in expanding universes

L. Combi and G. Romero,Electromagnetic fields and charges in expanding universes,Phys. Rev. D99(2019) 064017 [1903.01877]

work page internal anchor Pith review Pith/arXiv arXiv 2019
[8]

Combi and G.E

L. Combi and G.E. Romero,Relativistic rigid systems and the cosmic expansion,General Relativity and Gravitation52(2020) 93 [2008.11836]

work page arXiv 2020
[9]

Spengler, A

F. Spengler, A. Belenchia, D. Rätzel and D. Braun,Influence of cosmological expansion in local experiments,Classical and Quantum Gravity39(2022) 055005 [2109.03280]

work page arXiv 2022
[10]

McVittie's Legacy: Black Holes in an Expanding Universe

N. Kaloper, M. Kleban and D. Martin,McVittie’s legacy: Black holes in an expanding universe,Phys. Rev. D81(2010) 104044 [1003.4777]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[11]

More on McVittie's Legacy: A Schwarzschild - de Sitter black and white hole embedded in an asymptotically $\Lambda$CDM cosmology

K. Lake and M. Abdelqader,More on McVittie’s legacy: A Schwarzschild-de Sitter black and white hole embedded in an asymptoticallyΛCDM cosmology,Phys. Rev. D84(2011) 044045 [1106.3666]

work page internal anchor Pith review Pith/arXiv arXiv 2011
[12]

A point mass in an isotropic universe. Existence, uniqueness and basic properties

B.C. Nolan,A point mass in an isotropic universe: Existence, uniqueness, and basic properties, Phys. Rev. D58(1998) 064006 [gr-qc/9805041]

work page internal anchor Pith review Pith/arXiv arXiv 1998
[14]

Nolan,A point mass in an isotropic universe: II

B.C. Nolan,A point mass in an isotropic universe: II. Global properties,Classical and Quantum Gravity16(1999) 1227

work page 1999
[15]

B.C. Nolan,Local properties and global structure of McVittie spacetimes with non-flat Friedmann-Lemaître-Robertson-Walker backgrounds,Classical and Quantum Gravity34(2017) 225002 [1707.07612]

work page internal anchor Pith review Pith/arXiv arXiv 2017
[16]

Faraoni, Valerio; Jacques,Cosmological expansion and local physics,Physical Review D76 (2007)

A. Faraoni, Valerio; Jacques,Cosmological expansion and local physics,Physical Review D76 (2007)

work page 2007
[17]

Sultana and C.C

J. Sultana and C.C. Dyer,Cosmological black holes: A black hole in the Einstein-de Sitter universe,General Relativity and Gravitation37(2005) 1347

work page 2005
[18]

Conformal time dependent Painleve-Gullstrand spacetime

H. Culetu,Conformal time dependent Painleve-Gullstrand spacetime,arXiv e-prints(2012) arXiv:1202.2285 [1202.2285]

work page internal anchor Pith review Pith/arXiv arXiv 2012
[19]

T. Sato, H. Maeda and T. Harada,Conformally Schwarzschild cosmological black holes, Classical and Quantum Gravity39(2022) 215011 [2206.10998]

work page arXiv 2022
[20]

Pérez and G.E

D. Pérez and G.E. Romero,Survival of black holes through a cosmological bounce,Phys. Rev. D 105(2022) 104047 [2205.10333]

work page arXiv 2022
[21]

Ginsparg and M.J

P. Ginsparg and M.J. Perry,Semiclassical perdurance of de Sitter space,Nuclear Physics B 222(1983) 245

work page 1983
[22]

Pair Creation of Black Holes During Inflation

R. Bousso and S.W. Hawking,Pair creation of black holes during inflation,Phys. Rev. D54 (1996) 6312 [gr-qc/9606052]

work page internal anchor Pith review Pith/arXiv arXiv 1996
[23]

Pathria,The Universe as a Black Hole,Nature240(1972) 298

R.K. Pathria,The Universe as a Black Hole,Nature240(1972) 298

work page 1972
[24]

PopłAwski,Universe in a black hole with spin and torsion, inThe Sixteenth Marcel Grossmann Meeting

N. PopłAwski,Universe in a black hole with spin and torsion, inThe Sixteenth Marcel Grossmann Meeting. On Recent Developments in Theoretical and Experimental General Relativity, Astrophysics, and Relativistic Field Theories, R. Ruffino and G. Vereshchagin, eds., pp. 1327–1336, July, 2023, DOI. – 15 –

work page 2023
[25]

Planck 2018 results. X. Constraints on inflation

Planck Collaboration, Y. Akrami, F. Arroja, M. Ashdown, J. Aumont, C. Baccigalupi et al., Planck 2018 results. X. Constraints on inflation,Astronomy & Astrophysics641(2020) A10 [1807.06211]

work page internal anchor Pith review Pith/arXiv arXiv 2018
[26]

Starobinsky,A new type of isotropic cosmological models without singularity,Physics Letters B91(1980) 99

A.A. Starobinsky,A new type of isotropic cosmological models without singularity,Physics Letters B91(1980) 99

work page 1980
[27]

f(R) theories

A. De Felice and S. Tsujikawa,f( R) Theories,Living Reviews in Relativity13(2010) 3 [1002.4928]

work page internal anchor Pith review Pith/arXiv arXiv 2010
[28]

Baumann,Cosmology(2022), 10.1017/9781108937092

D. Baumann,Cosmology(2022), 10.1017/9781108937092

work page doi:10.1017/9781108937092 2022
[29]

Carr and F

B. Carr and F. Kuhnel,Primordial Black Holes as Dark Matter Candidates,arXiv e-prints (2021) arXiv:2110.02821 [2110.02821]

work page arXiv 2021
[30]

Hawking,Particle creation by black holes,Communications in Mathematical Physics43 (1975) 199

S.W. Hawking,Particle creation by black holes,Communications in Mathematical Physics43 (1975) 199

work page 1975
[31]

Zel’dovich and I.D

Y.B. Zel’dovich and I.D. Novikov,The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model,Soviet Astronomy10(1967) 602

work page 1967
[32]

Primordial black holes: constraints, potential evidence and prospects 2026

B. Carr, A.J. Iovino, G. Perna, V. Vaskonen and H. Veermäe,Primordial black holes: constraints, potential evidence and prospects,arXiv e-prints(2026) arXiv:2601.06024 [2601.06024]

work page arXiv 2026
[33]

Barrau, K

A. Barrau, K. Martineau and C. Renevey,Catastrophic fate of Schwarzschild black holes in a thermal bath,Phys. Rev. D106(2022) 023509 [2203.13297]

work page arXiv 2022
[34]

Riajul Haque, R

M. Riajul Haque, R. Karmakar and Y. Mambrini,When Primordial Black Holes Absorb During the Early Universe,arXiv e-prints(2026) arXiv:2601.16717 [2601.16717]

work page arXiv 2026
[35]

Kallifatides, T

D.S. Kallifatides, T. Papanikolaou and E.N. Saridakis,Ultra-fast growth of primordial black holes through radiative absorption,arXiv e-prints(2026) arXiv:2601.18708 [2601.18708]

work page arXiv 2026
[36]

Black hole mass decreasing due to phantom energy accretion

E. Babichev, V. Dokuchaev and Y. Eroshenko,Black Hole Mass Decreasing due to Phantom Energy Accretion,Phys. Rev. Lett.93(2004) 021102 [gr-qc/0402089]

work page internal anchor Pith review Pith/arXiv arXiv 2004
[37]

Kalita and D

J. Kalita and D. Maity,Revisiting PBH Accretion, Evaporation and Their Cosmological Consequences,arXiv e-prints(2025) arXiv:2512.07291 [2512.07291]. – 16 –

work page arXiv 2025