Formation of Black Holes from Collapsed Cosmic String Loops

R. R. Caldwell & Paul Casper

REVIEW

Formation of Black Holes from Collapsed Cosmic String Loops

R. R. Caldwell & Paul Casper

General Relativity and Quantum Cosmology · Submitted 6 Sep 1995

Pith's verdict: The loop-fragmentation numerics are solid and the power law is a real result; the final Gμ bound leans on a slope fitted over 0.7 decades and a tail of roughly a dozen loops.

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arXiv IDgr-qc/9509012 (version 1)

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CitationR. R. Caldwell & Paul Casper. "Formation of Black Holes from Collapsed Cosmic String Loops." arXiv:gr-qc/9509012 (1995). https://pith.science/paper/Z7IM6B7Z

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Editor's note

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The thing worth knowing up front: the new content here is the empirical power law, not the bound. Caldwell and Casper push 25,576 fragmentation-generated loops through a minimum-bounding-sphere search over a full oscillation period and get f_BH = 10^{4.9±0.2}(Gμ)^{4.1±0.1}. That is a real step past the order-of-magnitude estimates in Hawking 1989 and Polnarev–Zembowicz 1991. The machinery is described well enough to reimplement: N log N bounding spheres with N = 20,000, radius error under 10^-4, four very different parent ensembles giving near-identical distributions, and an angular-momentum check that raises 11% of the μ_min values. The heuristic in §IV — P(ε) ∝ ε^3 giving f ∝ δ^4 — is a plausibility argument, but it is the right kind, and it independently supports a slope near 4 rather than, say, 2. That is worth something when the fit window is thin. Speaking of which: [abstract:s3] advertises validity over 10^-3 ≲ Gμ ≲ 3×10^-2, but §IV fits over x ∈ [-2.0, -1.3], which is 0.7 decades. At the lower edge f_BH ≈ 5×10^-4, about thirteen loops out of 25,576. The authors note that noise dominates the lowest bins in Figs. 2–5 but never show how the slope moves when the window is widened or the tail resampled. That matters more than it looks, because the final bound is a crossing of two power laws. Setting 10^4.9(Gμ)^4.1 equal to the Caldwell–Gates f_obs ∝ (Gμ)^{-2} gives (Gμ)^{6.1} ≈ 2×10^-33.9, so the slope enters with weight about 6. The quoted ±0.7 is the formal propagation of ±0.1 in a; it does not carry fit-window or tail systematics. A slope of 3.9 gives ~2×10^-6, 4.3 gives ~4.6×10^-6. The qualitative conclusion — steep power law, bound near 10^-6 — is probably safe; the coefficient 3.1 is softer than advertised. One more: the loop ensemble, which is the entire evidence base, comes from Casper–Allen [8], an unpublished preprint. That is a real dependency, not a formality. Send it out. The power law belongs in the literature whatever happens to the bound, and a referee should ask for the fit-window sensitivity study before the 3.1 is quoted as a number.

Referee report

Significance

If it holds, the result is a genuinely new and independent constraint on the one free parameter of the cosmic string scenario, competitive with (and consistent with) the nucleosynthesis, pulsar-timing and CMB bounds quoted in Eq. (1.1). The numerical work is substantial and, on the evidence presented, carefully done: four parent ensembles with very different mode content and daughter multiplicities give nearly identical f_BH(Gμ), the bounding-sphere algorithm is described in enough detail to be reimplemented, the time-refinement error is quoted (below 0.0001%), and the angular-momentum barrier is applied rather than ignored. The paper is also explicit about its two main idealizations — the weak-field/negligible-back-reaction approximation (§II.B) and the assumption that the fitted power law continues to Gμ ~ 10^-6 (§IV) — which is the right way to present an extrapolation of this kind. The heuristic ε^3 → δ^4 argument, while not a derivation, gives the empirical slope a plausible structural origin. The main things I would want before the quoted precision is taken at face value are a quantified fit-window/tail systematic on the slope and an unambiguous statement of the normalization

Major objections (4)

[§IV, Eq. (4.2) and Figs. 6–7, versus abstract sentence s3] The abstract advertises the power law as valid over 10^-3 ≲ Gμ ≲ 3×10^-2, but §IV states that the linear fit is performed over x = log10(Gμ) ∈ [-2.0, -1.3], i.e. Gμ ∈ [10^-2, 5×10^-2]. The advertised lower limit is a full decade below the fitted window, and it is precisely the decade in which the authors themselves say statistical noise dominates (Figs. 2–5). At the lower edge of the fit, f_BH ≈ 5×10^-4, so only of order ten of the 25,576 loops lie below it. Please state the actual fitted interval in the abstract and either extend the fit downward with an explicit noise treatment or restrict the quoted range to the interval actually fitted.
[§IV, Eq. (4.2) and §V, final bound] The final bound is the crossing of f_BH ∝ (Gμ)^a with f_BH,obs ∝ (Gμ)^{-3/2}, so the fitted slope enters the answer with weight a + 3/2 ≈ 5.6–6.1: a = 3.9 gives ≈2.0×10^-6 and a = 4.3 gives ≈4.6×10^-6. The quoted ±0.7 tracks only the formal ±0.1 on a and does not include fit-window or tail-resampling systematics, even though the fit spans only 0.7 decades and its lower edge is populated by roughly a dozen loops. Please report the fitted slope for at least one alternative window (for example [-1.8, -1.3] and [-2.0, -1.5]) and a bootstrap or jackknife resampling of the low-f_BH tail, and propagate the resulting spread into the bound on Gμ. I expect the central number to survive at the factor-of-two level; the request is that the quoted uncertainty reflect the analysis actually performed.
[§V, Eqs. (5.1)–(5.2)] The two observational expressions must agree at Gμ = 10^-6, since Eq. (5.1) is quoted as the value of the bound at that point, but as printed they do not. Reading the last factor of Eq. (5.2) as (κGμ)^{-1/2} and substituting κ = γGμ gives (γG^2μ^2)^{-1/2} ≈ 1.4×10^5 at Gμ = 10^-6, so Eq. (5.2) would return ≈10^-12 rather than 10^-17. The factor is presumably (κ/(γGμ))^{-1/2}, which is O(1) over the stated range 0.1 ≲ κ/(γGμ) ≲ 1 and which also reproduces the μ^{3/2} scaling of the cosmic-ray emission quoted in the text. Please write the factor unambiguously, state the value of γ used, and show explicitly that Eq. (5.2) reduces to Eq. (5.1) at Gμ = 10^-6, since the final number depends on this normalization.
[§IV, angular-momentum constraint, Eq. (2.6)] The hoop condition (2.4) and the extremality condition (2.6) must be satisfied at the same instant τ, but the text does not say at which time J is evaluated when μ_min is raised. If J is evaluated at τ_min, then a loop that fails (2.6) there might satisfy both conditions at a different τ where its angular momentum is smaller, and the procedure of raising μ_min at fixed τ_min would then overstate the threshold. Please state the evaluation time explicitly; if it is τ_min, say so and note that the resulting f_BH is a lower bound. The effect on the final bound is modest because it enters only through the normalization b (weight 1/(a + 3/2)), but the 11% incidence and 30% mean fractional change reported in §IV show it is not negligible.

Minor objections (8)

[§II.B and abstract] The upper end of the quoted range, Gμ ~ 3×10^-2, lies well outside the weak-field regime that §II.B invokes when it says the fitted function may only be used where the weak-field approximation is valid. The abstract should distinguish the empirical fit range from the physically applicable range, as §II.B already does in the body.
[§IV, Eqs. (4.4)–(4.5)] The heuristic derivation assumes that contraction to within a fraction ε of L in each of the three spatial directions occurs independently, with probability ε per direction. For an oscillating loop these contractions are correlated, so the agreement with the fitted slope 4.1 should not be presented as independent confirmation. The text already calls the argument heuristic; one sentence making the independence assumption explicit would be enough.
[§III.A, Ref. [8]] The loop-fragmentation code and its validation are documented in Ref. [8], which is cited as an unpublished preprint (WISC-MILW-95-TH-11). Since the entire loop ensemble rests on that code, a short summary of the validation performed there — for example, the comparison of radiation-rate distributions across parent types — would help a reader assess the ensemble without consulting the preprint.
[§III.A and §IV] Because each loop is followed through only one oscillation period, loops that would collapse on a later period are not counted, so f_BH as defined is a lower bound and the resulting limit on Gμ is conservative. This is worth stating explicitly, since it strengthens the bound's interpretation.
[Fig. 1] The caption explains the kink-crossing discontinuity in R(τ); marking τ_min and R(τ_min) directly on the plot would make the definition of μ_min easier to read off.
[Figs. 2–5] The number of loops contributing to each bin is not shown, so the reader cannot judge where the noise that the text mentions actually sets in. Adding counts per bin, or a shaded band, would make the onset of the noisy region quantitative.
[§V] The value of γ used in evaluating Eq. (5.2) should be stated where the bound is computed, not only as 'typically ~50' in §II.B, since the final number depends on it through the κ factor.
[Text] There are a few typographical errors, for example 'Similarily' in §V and 'earler' in the discussion of Fig. 7.

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The paper

The paper claims

Cosmic string loops collapse into black holes at a rate that falls as a steep power of the string's mass-per-unit-length, and comparing that rate with the observed gamma-ray background caps the string tension at $Gμ ≤ 3.1(±0.7)×10^{-6}$.

The reading

The paper asks a concrete question: out of all the loops a cosmic string network produces, what fraction shrink inside their own Schwarzschild radius and become black holes? The authors answer it by brute force, taking 25,576 loop trajectories produced by a loop-fragmentation code, watching each one through a full oscillation, and finding the smallest sphere that ever encloses it. Equating that sphere's radius with $2GμL$ gives, for each loop, the minimum mass-per-unit-length it would need in order to satisfy the hoop conjecture. The resulting fraction is a clean power law, $f_{BH} = 10^{4.9}(Gμ)^{4.1}$, over the range the simulation actually covers. The payoff is cosmological: extrapolated to $Gμ ∼ 10^{-6}$ the collapse fraction is about $2×10^{-20}$, small enough to sit under the bound on evaporating primordial black holes set by the diffuse gamma-ray background, which converts into an upper limit on the string tension.

What carries the argument

Two pieces of machinery carry the argument. The first is the hoop conjecture in its operational form: a loop of invariant length $L$ that at some moment fits inside a sphere of radius $R$ collapses to a black hole only if $R ≤ 2GμL$, so each loop can be assigned a threshold $μ_{min} = R/(2GL)$. The second is the numerical procedure for finding the smallest enclosing sphere of a discretised loop (20,000 points plus exact kink positions): start from five random points, brute-force their minimal sphere, replace the least useful point with the point lying farthest outside, and iterate — an $O(N log N)$ scheme that converges in order $log N$ iterations. The global minimum of $R(τ)$ over one oscillation period is then located by four successive refinements of the time grid, giving radii accurate to about $10^{-5}$ relative.

Extended reading notes

Core claim

The central claim is that the fraction of cosmic string loops which collapse to black holes is a steep power of the string mass-per-unit length, $f_{BH}(Gμ) = 10^{4.9±0.2}(Gμ)^{4.1±0.1}$, measured over $10^{-3} ≲ Gμ ≲ 3×10^{-2}$ from 25,576 non-self-intersecting child loops descended from four very different families of parent loops. The same power law emerges for every parent family, and it survives the addition of the angular-momentum barrier: about 11% of the loops needed their $μ_{min}$ raised so that $|J| ≤ GM^2$, and the fitted slope barely moved (from $a = 4.0$ to $a = 4.1$). Extrapolating to $Gμ ∼ 10^{-6}$ gives $f_{BH} ≈ 2.0(±1.5)×10^{-20}$, which is compared with the Caldwell–Gates bound from evaporating black hole cosmic rays, $f_{BH,obs} ≲ 10^{-17}$ at that tension. The mismatch yields the paper's headline result, $Gμ ≤ 3.1(±0.7)×10^{-6}$, consistent with the nucleosynthesis, pulsar-timing and microwave-background-anisotropy limits quoted in the introduction.

Load-bearing premise

The power law measured between $Gμ ≈ 10^{-2}$ and $10^{-1.3}$ is assumed to hold unchanged four orders of magnitude lower, at $Gμ ≈ 10^{-6}$, where no simulated loop exists.

Editorial extensions

If this is right

Any cosmic string scenario with $Gμ$ above roughly $3×10^{-6}$ overproduces evaporating black holes relative to the observed diffuse gamma-ray background, so that region of parameter space is closed.
The collapse fraction is a steep power law, so the bound is only weakly sensitive to order-of-magnitude uncertainties in the cosmological loop-formation model: the paper estimates that loosening the loop-size parameter $α$ weakens the limit by only about 20%.
Because the four parent-loop families give nearly identical $f_{BH}(Gμ)$, the result is presented as generic to loops formed by fragmentation rather than an artefact of initial conditions.
The heuristic argument that a loop must shrink to a fraction $ε$ of its length in all three directions, giving $P(ε) ∝ ε^3$ and hence $f ∝ δ^4$, matches the measured slope and supplies a normalisation for the collapse fraction.

Reading between the lines

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

The power law is measured over roughly $10^{-2}$ to $10^{-1.3}$ in $Gμ$ and then extrapolated four orders of magnitude; a direct simulation at smaller $Gμ$ — or a reweighting argument showing the minimal-radius distribution stays scale-free — is the natural next test.
If the slope steepened at small $Gμ$, the bound would tighten; if it flattened, the bound would loosen substantially, so the slope's stability is the single most leveraged number in the chain.
The same minimal-bounding-sphere machinery could be applied to other collapsing extended objects — monopole-antimonopole pairs, domain walls, or non-Abelian string networks — wherever a hoop-conjecture-style criterion sets the collapse threshold.
Because the bound sits within a factor of a few of the microwave-anisotropy bound, an independent improvement in either the gamma-ray background data or the black-hole evaporation modelling would make the two constraints genuinely competitive rather than merely consistent.

Original abstract

The fraction of cosmic string loops which collapse to form black holes is estimated using a set of realistic loops generated by loop fragmentation. The smallest radius sphere into which each cosmic string loop may fit is obtained by monitoring the loop through one period of oscillation. For a loop with invariant length $L$ which contracts to within a sphere of radius $R$, the minimum mass-per-unit length $\mu_{\rm min}$ necessary for the cosmic string loop to form a black hole according to the hoop conjecture is $\mu_{\rm min} = R /(2 G L)$. Analyzing $25,576$ loops, we obtain the empirical estimate $f_{\rm BH} = 10^{4.9\pm 0.2} (G\mu)^{4.1 \pm 0.1}$ for the fraction of cosmic string loops which collapse to form black holes as a function of the mass-per-unit length $\mu$ in the range $10^{-3} \lesssim G\mu \lesssim 3 \times 10^{-2}$. We use this power law to extrapolate to $G\mu \sim 10^{-6}$, obtaining the fraction $f_{\rm BH}$ of physically interesting cosmic string loops which collapse to form black holes within one oscillation period of formation. Comparing this fraction with the observational bounds on a population of evaporating black holes, we obtain the limit $G\mu \le 3.1 (\pm 0.7) \times 10^{-6}$ on the cosmic string mass-per-unit-length. This limit is consistent with all other observational bounds.

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The fraction of cosmic string loops which collapse to form black holes is estimated using a set of realistic loops generated by loop fragmentation. The smallest radius sphere into which each cosmic string loop may fit is obtained by monitoring the loop through one period of oscillation. For a loop with invariant length $L$ which contracts to within a sphere of radius $R$, the minimum mass-per-unit length $\mu_{\rm min}$ necessary for the cosmic string loop to form a black hole according to the hoop conjecture is $\mu_{\rm min} = R /(2 G L)$. Analyzing $25,576$ loops, we obtain the empirical estimate $f_{\rm BH} = 10^{4.9\pm 0.2} (G\mu)^{4.1 \pm 0.1}$ for the fraction of cosmic string loops which collapse to form black holes as a function of the mass-per-unit length $\mu$ in the range $10^{-3} \lesssim G\mu \lesssim 3 \times 10^{-2}$. We use this power law to extrapolate to $G\mu \sim 10^{-6}$, obtaining the fraction $f_{\rm BH}$ of physically interesting cosmic string loops which collapse to form black holes within one oscillation period of formation. Comparing this fraction with the observational bounds on a population of evaporating black holes, we obtain the limit $G\mu \le 3.1 (\pm 0.7) \times 10^{-6}$ on the cosmic string mass-per-unit-length. This limit is consistent with all other observational bounds.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
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s0	The fraction of cosmic string loops which collapse to form black holes is estimated using a set of realistic loops generated by loop fragmentation.	contribution 0.79	0.88	0.06	asserted 0.99
s1	The smallest radius sphere into which each cosmic string loop may fit is obtained by monitoring the loop through one period of oscillation.	method 1.00	0.89	0.02	asserted 0.88
s2	For a loop with invariant length $L$ which contracts to within a sphere of radius $R$, the minimum mass-per-unit length $\mu_{\rm min}$ necessary for the cosmic string loop to form a black hole according to the hoop conjecture is $\mu_{\rm min} = R /(2 G L)$.	method 0.77	0.25	0.03	asserted 0.99
s3	Analyzing $25,576$ loops, we obtain the empirical estimate $f_{\rm BH} = 10^{4.9\pm 0.2} (G\mu)^{4.1 \pm 0.1}$ for the fraction of cosmic string loops which collapse to form black holes as a function of the mass-per-unit length $\mu$ in the range $10^{-3} \lesssim G\mu \lesssim 3 \times 10^{-2}$.	result 1.00	0.97	0.28	asserted 1.00
s4	We use this power law to extrapolate to $G\mu \sim 10^{-6}$, obtaining the fraction $f_{\rm BH}$ of physically interesting cosmic string loops which collapse to form black holes within one oscillation period of formation.	method 0.72	0.95	0.07	asserted 0.99
s5	Comparing this fraction with the observational bounds on a population of evaporating black holes, we obtain the limit $G\mu \le 3.1 (\pm 0.7) \times 10^{-6}$ on the cosmic string mass-per-unit-length.	result 0.97	0.93	0.72	asserted 1.00
s6	This limit is consistent with all other observational bounds.	interpretation 0.96	0.59	0.09	asserted 0.65

What kind of paper the classifier reports this is: simulation study 0.98 · evidence from simulation or computation 1.00.

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sources read

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full text
figures and tables
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References

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How the literature uses it

37 arXiv papers cite this in 43 sentences

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Papers citing it · 12 sentences without a confident label

Uses its data or values 1 paper

Following earlier analytic calculations [ref] , numerical simulations by Caldwell & Casper found that $f = 10^4.9 ±0.2 (G μ/c^2)^4.1 ±0.1$ for $10^-3 ≲(G μ/c^2) ≲3 ×10^-2$ [ref] .

Constraints on the cosmic string loop collapse fraction from Primordial Black Holes · Cosmic string loop formation and collapse · 0.85

Cites it as background 26 papers

However, we are interested in macroscopically-large loops, which lie well above this dispersion regime for any value of $Gμ$ consistent with observational constraints.] and PBH formation from circular loop collapse has remained an active research topic ever since [ref] .

Primordial black holes from cusp collapse on cosmic strings · Introduction · 0.83
There are currently many proposed mechanisms for the formation of PBHs, such as: First-order phase transition (FOPT) [ref] , inflaton ultraslow-roll [ref] , the collapse of cosmic strings [ref] , domain walls [ref] , scalar condensates [ref] , or processes in a dissipative dark sector [ref] .

Numerical simulations of density perturbation and gravitational wave production from cosmological first-order phase transition · Introduction · 0.99
The most commonly considered scenario involves the collapse of large density perturbations generated during inflation [ref] , but they may be produced also in cosmological phase transitions [ref] , by the collapse of cosmic strings [ref] , false vacuum bubbles [ref] or domain walls [ref] , or post-inflationary scalar field fragmentation [ref] .

Anatomy of single-field inflationary models for primordial black holes · Introduction · 0.97

Discussion

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