An upper limit on the frequency of short-period black hole companions to Sun-like stars

Dan Maoz; George Hume; Hans-Walter Rix; Ikram Hamoudy; James Munday; Kareem El-Badry; Marco C. Lam; Matthew J. Green; Paige Yarker; Simchon Faigler

REVIEW 3 major objections 5 minor 79 references

An upper limit on the frequency of short-period black hole companions to Sun-like stars

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read Short-period black holes around Sun-like stars are rarer than optimistic models predict, the paper claims.

desk verdict Careful upper-limit measurement; the headline 'one in 10^5' needs a systematic inflation before it is robust. read the letter →

arxiv 2412.02082 v2 pith:RGQZADC6 submitted 2024-12-03 astro-ph.SR astro-ph.HE

Matthew J. Green , Yoav Ziv , Hans-Walter Rix , Dan Maoz , Ikram Hamoudy , Tsevi Mazeh , Simchon Faigler , Marco C. Lam

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Kareem El-Badry George Hume James Munday Paige Yarker

This is my paper · ORCID

classification astro-ph.SRastro-ph.HE

keywords blackholecompanionsellipsoidalvariabilityTESSsolar-typestarsupperlimitsstellarbinariesdormantholespopulationsynthesis

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper tries to answer a census question: how often do Sun-like stars have a black hole companion on a very short orbit, under about three days? It argues that the answer is almost never: after hunting through TESS light curves of 4.7 million AFGK-type stars for the tidal 'ellipsoidal' distortion a dark massive companion would imprint, and following up 250 of the 457 most promising candidates with spectroscopy, none is consistent with a black hole. From this non-detection the paper derives a 2-$\sigma$ upper limit of $9.5\times10^{-6}$ on the fraction of solar-type stars with such a companion, tightening to about $1\times10^{-6}$ near one-day orbits. If correct, this rules out the most optimistic population models and shows that short-period dormant black holes are not hiding in large numbers around Sun-like stars.

What carries the argument

The engine is the ellipsoidal amplitude $A_\mathrm{ell}$: the photometric variation at twice the orbital period caused by tidal distortion. Under the assumption that all light comes from the primary star, the amplitude formula links $A_\mathrm{ell}$ to the mass ratio $q=M_2/M_1$, and the ellipsoidal mass function $\mathcal{M}_\mathrm{ell}=\sin^2 i\, q/(q+1)$ yields a lower limit $q_\mathrm{min}$ that can exceed unity only for a dark, high-mass companion. Candidate selection used $q_\mathrm{min}>1$ for periods above one day and the alternative MMMR statistic for shorter periods, with by-eye eclipse checks to remove contact binaries. The calculation is carried by injection-recovery simulations that add synthetic black-hole binary light curves to real TESS light curves, measuring an average selection efficiency $\bar{S}'=0.18$; the upper limit then follows from inverting the expected number and applying a Wilson score interval.

What would settle it

A single confirmed short-period black-hole companion among the 207 unobserved candidates, or in a re-run of the same selection on the full 4.7-million-star TESS sample, would directly violate the limit at the claimed confidence. Conversely, an injection-recovery calculation that includes O'Connell-effect star spots and finds the selection efficiency drops by more than a factor of two would invalidate the $9.5\times10^{-6}$ number.

Watch

Extended reading notes

Core claim

The central discovery is a null result with a number attached. For orbital periods below three days, black hole companions to AFGK main-sequence stars exist in at most $9.5\times10^{-6}$ of such stars at 2-$\sigma$ confidence, and at most $1\times10^{-6}$ when the period is close to one day, under the paper's fiducial priors on companion mass and period. The claim is established by combining the photometric selection of ellipsoidal binaries (the tidal deformation of the visible star where a compact companion's gravity modulates its projected area) with a measured selection efficiency from injection-recovery simulations, and then using the absence of any innocent explanation among 250 spectroscopically observed candidates to invert the expected count. The paper presents this as the strongest direct constraint yet in this period range, and notes it is in tension with predictions as high as one in $10^{4-5}$ stars.

Load-bearing premise

The quoted upper limit assumes that unmodeled star-spot variability does not reduce the measured selection efficiency by more than about a factor of two, and that the 250 candidates actually followed up fairly represent the 457 selected candidates; if either assumption fails, the limit is too strong.

Editorial extensions

If this is right

The most optimistic published population models, predicting $10^{-4}$ to $10^{-5}$ short-period black-hole companions per solar-type star, are excluded at the 2-3 sigma level.
More pessimistic recent models at $10^{-7}$ to $10^{-8}$ remain consistent but untestable with current data; reaching them would need a survey 30 to 100 times larger than the 4.7 million stars processed here.
The space density of non-accreting short-period binaries cannot exceed the space density of accreting low-mass X-ray binaries by more than about two orders of magnitude.
At periods near one day the limit tightens to $\lesssim 10^{-6}$, so the absence is most severe exactly where the ellipsoidal signal is strongest.
Future searches should focus on the $q_\mathrm{min}$ method and on less strict harmonic cuts, because the MMMR method has very low completeness for black-hole companions.

Reading between the lines

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

Editorial inference: if this limit survives, binary population synthesis must be tuned so that the vast majority of massive stars with low-mass companions either merge, are disrupted, or avoid shrinking to $P<3$ days; the bottleneck is the survival fraction, not the survey volume.
Editorial inference: the star-spot caveat cuts both ways; a future injection-recovery calculation that includes explicit O'Connell-effect spot signals could find that the true sensitivity is lower than $\bar{S}'=0.18$, which would push the upper limit upward.
Editorial inference: the same TESS dataset could be re-mined with the $q_\mathrm{min}$ method extended below one day and with less aggressive amplitude cuts; any recovered candidates would directly test the limit rather than assuming it.
Editorial inference: the comparison to X-ray binaries implies that the detached, pre-mass-transfer phase of black-hole low-mass binaries is not enormously longer than the accreting phase, otherwise more non-accreting systems would have been found.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

This paper delivers the first statistically grounded upper limit on short-period black hole companions to solar-type stars, and the core work is solid. From 4.7 million TESS targets the authors select 457 ellipsoidal candidates, follow up 250 with RVs (NTT/INT plus Gaia), find none consistent with a black hole, and then use injection-recovery through their actual BEER pipeline to turn that null into a completeness-corrected limit. The selection-function estimation is the real contribution: adding synthetic BH light curves to real TESS data and measuring how many survive each cut is the right way to make such a limit credible. The comparison to population-synthesis predictions in Fig. 14, with explicit caveats about converting literature predictions to a common metric, is fairly done. The authors also deserve credit for flagging their own main weakness in Section 7.3 rather than burying it.

The soft spots are real but not disqualifying. The star-spot systematic is the load-bearing one. The paper admits that a factor-of-2 reduction in selection efficiency S' would qualitatively change the results, yet the quoted 2-sigma limit of 9.5e-6 does not include any systematic inflation. A factor of exactly 2 moves that limit to 1.9e-5, which changes the "one in 10^5" headline to about one in 50,000. That is not a fatal flaw for the paper's central conclusion: even with a factor-2 penalty, the most optimistic models (Masuda & Hotokezaka, Mashian & Loeb) are still ruled out, and intermediate models are challenged at roughly the 1-sigma level rather than 1-2 sigma. But the headline number should be restated with this uncertainty built in, or the systematic should be quantified. The second issue is the follow-up completeness factor fobserved: 250 of 457 candidates were observed, with priority given to brightness and mmmr membership, and the paper does not demonstrate that the observed subset is representative. This is probably a minor effect, but it deserves a sentence or two more than it gets.

I do not see circularity here. The priors on Porb and q are explicit modeling choices, and the limit is inverted from an observed null via a measured selection function. That is the standard logic. The paper is honest, clearly written, and the data products (RV tables, light-curve injections) are reproducible in principle.

Who is this for? Anyone working on BH binary populations, TESS variability selection, or survey completeness methodology. It deserves a serious referee. My recommendation: accept after major revision, with the authors either quantifying the star-spot effect on S' or presenting the headline limit with an explicit systematic term. The science is good; the presentation of the central number needs to catch up with the caveats the authors already know about.

Referee Report

3 major / 5 minor

Summary. This paper searches TESS light curves of 4.7 million AFGK-type stars from Paper I for ellipsoidal variability indicative of short-period (P_orb < 3 days) dark companions. Two selection methods (a qmin-based mass-function method and the MMMR method of Gomel et al.) produce 457 candidate BH companions. Spectroscopic follow-up of 250 candidates with NTT/INT RVs and Gaia RV data firmly excludes a high-mass dark companion in every case. Using injection-recovery tests with synthetic BH+MS light curves added to real TESS photometry, the authors measure an average selection efficiency S'_Overall = 0.18 and convert the zero detections into f_BH < 2.4e-6 (1σ), 9.5e-6 (2σ), and 21e-6 (3σ), under stated uniform priors on P_orb and q. They compare this limit to population-synthesis predictions and find that the most optimistic models are ruled out, while more recent, pessimistic models remain consistent with the limit.

Significance. If the headline limit is robust, it is a valuable new empirical constraint on the frequency of short-period dormant black hole companions to Sun-like stars, a quantity that population-synthesis models predict to span several orders of magnitude. The paper's strengths include the use of injection-recovery into real TESS light curves to measure selection efficiency, the cross-check of measured K amplitudes against Gaia RV data, and the presentation of a two-dimensional upper-limit map in M2 and P_orb that avoids some of the assumptions needed for the marginalized headline number. The main limitation is that the quoted one-in-10^5 upper limit is sensitive to an unquantified star-spot systematic that the authors themselves identify in Section 7.3, as well as to the assumed priors on P_orb and q; these issues do not undermine the method or the 2D map, but they require the headline claim to be conditioned or revised.

major comments (3)

[§7.3 (and §8)] The paper itself states that the a1 > a2 cut removes genuine binaries exhibiting the O'Connell star-spot effect, that three published active K-dwarf plus white dwarf binaries would have been selected with qmin > 1 in the absence of this cut, and that a reduction of the selection efficiency by a factor of roughly 2 would 'qualitatively change' the results. Nevertheless, the central upper limits in Section 6.1 and the abstract (f_BH < 9.5e-6 at 2σ) do not include any allowance for this systematic. Since the headline claim rests directly on S'_Overall, the limit should be either inflated by a conservative spot-related systematic, presented as a band, or explicitly restated as conditional on the assumption that star spots reduce S' by less than a factor of 2.
[§6.1 (Eq. 14)] The factor fobserved is introduced in Eq. (14) but no value is ever stated; from Table 1 the reader must infer fobserved = 250/457. More importantly, Section 3 describes follow-up priority by MMMR membership and by brightness, so the 250 observed candidates are not demonstrated to be a random subset of the 457 candidates. A single scalar fobserved applied to the combined qmin and MMMR samples may therefore not equal the true observed fraction per selection method or per brightness bin. Please report fobserved explicitly, justify the representativeness assumption (for example by computing S' for the actual observed subset), or marginalize over the unknown fobserved.
[§6.1 (Eq. 15)] The headline upper limit marginalizes over p(P_orb) uniform in 0-3 days and p(q) uniform in 1-30, but S_phys varies by more than an order of magnitude across the period range (Fig. 13 and Table 5). No sensitivity test to these prior choices is presented, so the abstract's 'fewer than one in 10^5' claim is conditioned on untested distributional assumptions. Please add a robustness check (for example a log-uniform P_orb prior or a period distribution matched to the known short-period binary population) and quote the resulting range of f_BH.

minor comments (5)

[§3 / Table 2] The text says 60 targets were observed on NTT and INT, but the NTT and INT rows in Table 2 sum to 61; please correct the count or clarify overlaps.
[§5.2] The Gaia variable name 'rv_ampltidue_robust' is a typo for 'rv_amplitude_robust'.
[§7.1 / Fig. 14] The figure label 'Masuda (2019)' is inconsistent with the text citation 'Masuda & Hotokezaka (2019)'.
[References] The reference entry 'Mazeh, T., Faigler, S., Mazeh, T., & Faigler, S. 2010' duplicates author names; it should be corrected to cite the actual author list of the 2010 A&A paper.
[Fig. 11] The colorbar is labeled 'log(1σ upper limit)' while the text and caption describe two-dimensional upper limits more generally; please specify which confidence level is plotted.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fBH upper limit is derived from independent non-detections and injection-recovery efficiencies, not from the claimed bound itself.

full rationale

The derivation of the upper limit is a standard survey-limit calculation and does not reduce to its inputs. The paper defines fBH as a constant fraction of AFGK stars (Section 6.1, Eqs. 11-17), converts non-detections among 250 spectroscopically followed candidates into a Poisson/Wilson upper limit, and calibrates the selection efficiency S' by injecting synthetic BH-LC light curves into real TESS light curves and rerunning the same selection pipeline (Section 6.2, Table 5). The assumed uniform priors on Porb and q are explicit modeling choices, not fitted parameters, and no equation sets fBH equal to the injected population fraction. Citations to Paper I and to BEER/MMMR work are infrastructure for candidate selection, not an unverified premise used to forbid alternatives. The acknowledged possibility that star spots reduce S' by a factor of about 2 or more (Section 7.3) is an unquantified systematic robustness concern, and the fobserved scaling is a representativeness assumption, but neither is a circular step in the derivation. The result is self-contained against external benchmarks: the upper limit is compared with, not derived from, literature population predictions.

Assumptions & free parameters 3 free parameters · 7 assumptions · 0 invented entities

The central limit rests on the BEER selection efficiency measured by injection-recovery, the assumption that the observed subset is representative, and the unquantified star-spot systematic. The main free choices are the priors p(Porb) and p(q) used for marginalization, plus the implicit follow-up completeness factor fobserved. No new physical entities are introduced.

free parameters (3)

p(Porb) prior for marginalization = Uniform 0-3 days
Section 6.2 assumes a uniform distribution of orbital periods between 0 and 3 days to compute the overall selection efficiency S'_Overall. The 2D upper limits in Fig. 11 do not depend on this prior, but the headline marginal fBH does.
p(q) prior for companion mass ratio = Uniform 1-30, then M2 > 3 Msun filter
Section 6.2 draws q uniformly from 1 to 30, computes M2 from q and M1, and discards simulated binaries with M2 < 3 Msun. This weighting enters the overall selection efficiency, though the paper shows the result is weakly dependent on mass for M2 > 5 Msun.
fobserved (follow-up completeness factor) = Not explicitly given; inferred near 0.5 from quoted limits and 250/457 candidates
Equation 13-14 introduces fobserved to account for candidates without follow-up, but its value is never stated. The 1-sigma limit of 2.4e-6 with N=4.7e6 and S'=0.18 implies fobserved around 0.5. The upper limit is inversely proportional to this factor, and non-random follow-up selection could make it period- and mass-dependent.

assumptions (7)

domain assumption Primary star dominates the light: f1=1, f2=0 in the ellipsoidal amplitude model
Used in Eq. 1 and 2 for qmin and MMMR. This is valid for an invisible BH companion, but it is also the source of contamination from MS-MS binaries and inflated primaries.
domain assumption All orbits are circular
Assumed in Section 6.1 and for RV fitting, justified by tidal circularization at P<3 days (Zahn 1977; Bashi et al. 2023).
domain assumption Host star masses and radii follow main-sequence interpolation from TIC temperatures (Pecaut and Mamajek 2013)
Used in qmin selection and in simulations. The paper notes the ellipsoidal selection favors inflated primaries, making this assumption imperfect and a source of false positives.
ad hoc to paper Selection efficiency measured by injection-recovery is not degraded by unmodeled star spots by more than about a factor of 2
Section 7.3 concedes that the O'Connell effect can hide ellipsoidal signals and removed three known active K-dwarf WD binaries. The final limit assumes this does not reduce S' enough to change the qualitative result.
standard math Uniform-in-cos i orbital inclination distribution
Assumed for geometric marginalization in Eq. 10; standard for randomly oriented orbits.
standard math Wilson score interval provides the 1, 2, 3 sigma event-count thresholds (1.0, 3.9, 8.7)
Used to convert zero detected BH candidates into upper limits on fBH in Section 6.1.
ad hoc to paper The observed 250 candidates are a representative subset of the 457 candidates, so a single fobserved applies across period and mass
Section 3 prioritizes mmmr-selected and brighter targets; Eq. 13-14 nevertheless use one overall fobserved. The text only asserts independence from qmin, not from period or companion mass.

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Cite this review

Pith. "Pith review of An upper limit on the frequency of short-period black hole companions to Sun-like stars." pith.science (2026). https://pith.science/paper/RGQZADC6

@misc{pith2026241202082,
  author       = {Pith},
  title        = {Pith review of: An upper limit on the frequency of short-period black hole companions to Sun-like stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/RGQZADC6}},
  note         = {Machine review of arXiv:2412.02082}
}

abstract

Stellar-mass black holes descend from high-mass stars, most of which had stellar binary companions. However, the number of those binary systems that survive the binary evolution and black hole formation is uncertain by multiple orders of magnitude. The survival rate is particularly uncertain for massive stars with low-mass companions, which are thought to be the progenitors of most black hole X-ray binaries. We present a search for close black hole companions (separations less than 20 solar radii) to AFGK-type stars in TESS, i.e. the non-accreting counterparts to and progenitors of low-mass X-ray binaries. Such black holes can be detected by the tidally induced ellipsoidal deformation of the visible star, and the ensuing photometric light-curve variations. From an initial sample of 4.7 million TESS stars, we have selected 457 candidates for such variations. However, spectroscopic followup of 250 of them shows that none are consistent with a close black hole companion. On the basis of this non-detection, we determine (2 $\sigma$ confidence) that fewer than one in $10^5$ Solar-type stars in the Solar neighbourhood host a short-period black hole companion. This upper limit is in tension with a number of ``optimistic'' population models in the literature that predict short-period black hole companions around one in $10^{4-5}$ stars. Our limits are still consistent with other models that predict only a few in $10^{7-8}$.

Figures

Figures reproduced from arXiv: 2412.02082 by the authors.

**Figure 1.** Left: Expected Aell for a 1 M⊙ MS star with an 8M⊙ dark companion (i.e. mass ratio q = 8) at a range of orbital periods (denoted in days in the legend) as a function of orbital inclination cosi (0 is edge-on, 1 is face-on). Because the probability distribution of cosi is uniform for a randomly oriented orbit, every y value plotted here is equally likely. We note that this binary system will overflow its Roche lobe a… view at source ↗

**Figure 2.** Values of the maximum fraction (derived qmin / true q) by which qmin may be overestimated due to the unknown value of C. We assume here a Sun-like primary star and an edge-on inclination. For Porb ≳ 1 day, the overestimation is relatively minor unless the donor is unusually massive. and period. Approximately 10% of detached binaries and 20% of contact binaries with Porb < 1 day become contaminants using the qmin met… view at source ↗

**Figure 4.** Several example light curves that were identified as showing eclipses. Grey points show the individual TESS data, black points are phasebinned data, and the red line shows the best-fit ellipsoidal model. Eclipse features may be shallow, but can be identified as a departure from the ellipsoidal model at orbital phases 0 and 0.5. 0.0 0.5 1.0 1.5 BP − RP 0 1 2 3 4 5 6 7 MG 0.0 0.5 1.0 1.5 2.0 log( qmin) or log( qMMMR)… view at source ↗

Figures from the paper (11 more)

**Figure 5.** Figure 5: Colour-magnitude diagram of our targets (coloured circles) compared to a magnitude-limited sample of stars (grey two-dimensional histogram). Our targets are coloured according to their ellipsoidal-implied qmin for targets with Porb > 1 day (Section 2.2), or by MMMR f…

**Figure 6.** Figure 6: Values of MMMR for a range of secondary masses, assuming a 1 M⊙ primary star and an edge-on inclination. Tracks are plotted for the period range in which neither star fills its Roche lobe. Even for unusually high-mass secondaries, MMMR> 1 is only possible for a Sun-li…

**Figure 7.** Figure 7: Orbital solutions for several example targets from our target list, all observed on the 2.5 m INT. TIC ID numbers are given in the figure panels, while fM is the spectroscopic mass function. Black points are measured RV epochs, phase-folded on the photometric orbital p…

**Figure 8.** Figure 8: Verification of the Gaia SB1 orbital solutions. Bashi et al. (2022) previously noted that a number of Gaia orbital solutions at short periods are unreliable due to aliasing issues. Here we show that, of our photometrically selected targets that have Gaia SB1 solutions…

**Figure 9.** Figure 9: The K-amplitudes derived via two methods from Gaia data, against those measured from orbital solutions, for all targets with both measurements. There is generally reasonable agreement, with some outlying points, as is discussed in the text. < 3.5σ, while the other fou…

**Figure 10.** Figure 10: Measured K-amplitudes (top) and the implied lower limits on M2 (middle) and q (bottom) for our observed targets. Also plotted in the top panel are the expected K-amplitudes for a 1M⊙ star with an 8M⊙ companion at the median orbital inclination (solid black line) and t…

**Figure 11.** Figure 11: Two-dimensional upper limits on the frequency of black hole companions to solar-type stars as a function of orbital period and black hole mass, fBH(M2, Porb). The existence of orbital periods close to 1 day is more tightly constrained than the existence of periods clo…

**Figure 12.** Figure 12: Properties of the simulated BH-LC population, with coloured highlights showing the subsets of systems that were selected using the mmmr and qmin methods. The panels show the input distributions of Porb, q, M1 and M2, the distributions of the measured properties Aell a…

**Figure 11.** Figure 11 [PITH_FULL_IMAGE:figures/full_fig_p012_11.png]

**Figure 14.** Figure 14: Upper limits on the frequency of short-period BH companions to solar-type stars derived from this work (horizontal dashed lines), compared to various theoretical predictions from the literature, with error bars showing the range of theoretical predictions when multi…

**Figure 13.** Figure 13: Probability for simulated binary systems of various types to be accepted into the beer sample (top), qmin sample (middle), and mmmr sample (bottom), as a function of period. Other variables (cosi, M1, and M2) have been marginalised over [PITH_FULL_IMAGE:figures/full_…

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