Read arXiv:2601.06024v2 from Sec. VI and Figs. 4–6 outwards. Fig. 4 compiles single-mass limits; Sec. VI explains why a mass function cannot be read off it. Then Sec. II.A.2 shows how non-Gaussian statistics move the abundance, Secs. III and IV.E cover evaporation and accretion, Sec. IV.C the gravitational-wave limits and Sec. V the asteroid-mass window. Keep four labels apart while reading: measured limits, the review’s own synthesis, model results and open claims.
Which version, and where to start
This guide uses version 2 on arXiv, dated 1 April 2026, by Carr, Iovino, Perna, Vaskonen and Veermäe. arXiv lists the journal reference Riv. Nuovo Cim. (2026); the guide was checked against the arXiv text, not the journal typesetting, so equation or figure numbers may differ there.
Start with the caption of Fig. 4. It compiles limits for a monochromatic population: every black hole at one mass. Bold curves are limits; dashed curves mark potential evidence, labelled SNe, OGLE and LVK in the caption. The figure maps what each probe excludes at a single mass; testing a formation model takes the integral of Sec. VI.
Sec. VI gives the rule for linear probes in Eqs. 25–28 and the quadrature combination in Eq. 29, the same relations derived in the L3 chapter from Carr et al. 2017. Read the paragraph on limitations next to them: mergers, Poisson clustering, Lyman-alpha, dynamical limits, mass segregation and accretion each make the rule approximate or inapplicable, and the authors suggest treating the result as an effective mass function.
Fig. 5 has four panels: a critical-collapse mass function, a lognormal with σ=1 and two cut power laws with opposite slopes. Solid curves are the extended-mass-function limits and dashed curves the monochromatic ones. The text refers to the colour-coded map of the lognormal width against the mean mass as the “right panel of Fig. 5”; in v2 that map is the left panel of Fig. 6. The review states that the asteroid-mass window closes for widths above about σ≈2.5.
The abundance comes from a far tail, so anything that changes the tail changes the answer by orders of magnitude. The review separates two channels. The relation between curvature and density is non-linear, which matters even when the curvature perturbation is Gaussian. The curvature perturbation itself can also be non-Gaussian, as in curvaton or ultra-slow-roll models.
The local expansion of Eq. 16, with coefficients fNL and gNL, is the familiar starting point, but the review notes that it fails for the closed forms of the curvaton and ultra-slow-roll cases. A truncated series can converge for a narrow spectrum; a broad spectrum needs a non-perturbative treatment.
Fig. 2 shows a mass function near one solar mass changing with the curvaton decay parameter rdec. Fig. 3 shows the spectrum amplitude needed for f=1, with a threshold of 0.55, for the curvaton and an ultra-slow-roll-like case. Keeping only the quadratic term overestimates the abundance for the curvaton and underestimates it for the ultra-slow-roll-like case. These are model calculations: they show how large the effect can be, not which model is realised.
ζ=ζG+53fNLζG2+259gNLζG3+…
The review’s Eq. 16. The Lab sets every non-Gaussian term to zero.
Sec. III gives the Hawking temperature (Eq. 21) and the lifetime (Eq. 23), about 0.4s(M/109g)3. A non-rotating black hole born with about 5×1014 grams is finishing its evaporation today. Lighter ones are gone and constrained through their effect on nucleosynthesis and the photon backgrounds, the limits summarised by Carr et al. 2010.
Sec. IV.E covers black holes heavier than about ten solar masses, whose accretion would heat gas and change recombination, the CMB, X-ray counts and the 21-cm signal. The review stresses that these limits depend on the accretion model, such as disk versus spherical inflow and the Bondi rate. Accretion can also grow the black holes and weaken limits, while clustering can strengthen them. Serpico et al. 2020 is one such CMB analysis; its abstract says the result depends on accretion physics.
Black holes that pair up in the early universe would merge today. With hundreds of LIGO-Virgo-KAGRA mergers now observed, the review quotes a merger limit of about f≲0.1% from an O3 analysis; the abstract of Andrés-Carcasona et al. 2024 gives the same order, about 10−3 between 1 and 200 solar masses, for several mass functions. The stochastic background gives a weaker limit of order 10−2. Limits from scalar-induced gravitational waves are discussed separately and are not drawn in Fig. 4.
The same section lists potential evidence: events in the mass gaps at about 60–120 and 2–5 solar masses, and a sub-solar-mass candidate from the GstLAL search with a false-alarm rate of one in five years. The review reports them as possibilities; this guide relies on the review’s account of them.
Between about 10−17 and 10−10 solar masses, black holes could still be all of the dark matter. The review explains why several proposed limits there no longer hold: hydrodynamic simulations removed the white-dwarf ignition limit, femtolensing limits are invalid, limits from neutron stars in globular clusters are uncertain, and main-sequence lifetimes are unaffected.
Its upper edge is set by microlensing. The Subaru limit is weaker than first published once finite source size is included, as the Smyth et al. 2020 abstract states. Future tests the review names include X-ray microlensing with instruments such as NICER and STROBE-X, reaching below about 10−13 solar masses, and solar-system ephemerides.
Published limits from data: OGLE microlensing (Mróz et al., Nature 2024), the revised Subaru limit (Smyth et al. 2020), the O3 merger analysis (Andrés-Carcasona et al. 2024) and the CMB accretion limit (Serpico et al. 2020). This series read their abstracts only, so it relies on their stated results, not on a check of their methods.
The review’s synthesis: the Fig. 4 compilation, the extended-mass-function treatment in Sec. VI and the judgement about which old limits are invalid. The guide read these sections in full in arXiv v2.
Model results: the non-Gaussian amplitudes of Figs. 2–3, the accretion-model dependence and the Lab’s own Gaussian, linear formation model. They show how a prediction responds to an assumption.
Open claims: the mass-gap and sub-solar candidates, the correlation in source-subtracted infrared and X-ray backgrounds named in Sec. IV.E, and whether black holes make up any of the dark matter at all. Primordial black holes remain candidates; no population has been confirmed.
The default preset puts all the dark matter in the asteroid-mass window. Check that the tightest ratio, Subaru at about 0.08, stays far below one.
Choose “Ruled out by evaporation” and read the evaporation ratio of about 50; locate the matching part of Fig. 4.
Choose “Stellar-mass black holes”. The gravitational-wave ratio is greyed out; explain why, using Sec. VI and Sec. IV.C.
Exercises with explanations
A talk shows a lognormal mass function drawn on top of Fig. 4 and says it is allowed because it never crosses a curve. What two questions should you ask?
Read the explanation
First, was it tested with the integral of Sec. VI, Eqs. 25–28, rather than by eye? A density per lnM cannot be compared with a curve that caps a total. Second, are the probes in the relevant mass range linear in ψ? If the decisive limit is a merger rate, the linear rule does not apply and the answer needs the merger analysis itself.
Why does the review say the asteroid window closes for widths above about 2.5, when the monochromatic curves leave it open?
Read the explanation
A wide lognormal centred in the window puts part of its mass below about 10−17 solar masses, where evaporation limits are strong, and part above 10−10, where microlensing is strong. The weighted ratio adds those contributions, so a spread that is safe at its centre can fail through its tails.
Fig. 3 uses a threshold of 0.55, while the Lab defaults to 0.45. Can you compare its amplitudes with the Lab’s?
Read the explanation
Not directly. The required amplitude depends steeply on the threshold, the window function and the definition of the density contrast. Compare trends within one paper’s conventions, such as how the curvaton and ultra-slow-roll cases depart from the quadratic approximation.
Review; arXiv lists the journal reference Riv. Nuovo Cim. (2026). Read in full: Sec. II.A.2 on non-Gaussianity, Sec. IV.C (gravitational waves), IV.E (accretion), V (asteroid-mass window), VI (Eqs. 25–29) and the captions of Figs. 2–6; parts of Sec. II and III.
Abstract only. Twenty years of OGLE data on nearly 80 million LMC stars: no events longer than one year, so compact objects from 1.8×10−4 to 6.3 solar masses make up at most 1% of the dark matter.
Abstract only. Accretion changes the ionization history seen in the CMB; the bound reaches a fraction of 3×10−9 near 104 solar masses and depends on accretion physics.
Abstract only. A hierarchical analysis of O3 mergers with several mass functions limits PBHs to at most about 10−3 of the dark matter between 1 and 200 solar masses.
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