Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves

Kazunori Kohri; Takahiro Terada

REVIEW

Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves

Kazunori Kohri, Takahiro Terada

Cosmology and Nongalactic Astrophysics · Submitted 24 Sep 2020 · 147 references

Pith's verdict: A transparent, well-written consistency fit, but the central fPBH claim hinges on an uncalibrated delta_c choice and the title overreaches.

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arXiv ID2009.11853 (version 4)

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CitationKazunori Kohri, Takahiro Terada. "Solar-Mass Primordial Black Holes Explain NANOGrav Hint of Gravitational Waves." arXiv:2009.11853 (2020). https://pith.science/paper/LVMBU2YC

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Editor's note

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Quick read: competent, clearly written interpretation of the NANOGrav 12.5-yr excess as induced GWs from solar-mass PBH formation, but the headline claim is conditional on an unvalidated choice in the PBH abundance calculation. Read it as a consistency fit, it's fine; read the title literally, it overreaches.

What the paper does well: it is unusually transparent about assumptions. The discussion of the prior-dependence of the tension between NANOGrav 12.5 and older PTA bounds is careful, the mapping from scalar amplitude to fPBH is explicit, and the forecast for the merger background is a genuinely useful testable prediction. The authors also list the important systematics--window function, transfer function, critical collapse, QCD softening, non-Gaussianity--rather than burying them, and the future tests via SKA and interferometers are sensible.

The soft spots are real, and one is load-bearing. The conversion from As to fPBH uses the modified Gaussian window w(x)=exp(-x^2/4) and delta_c=0.42. Footnote 4 admits the value calibrated for that window is delta_c=0.18, and they choose 0.42 based on a 'naive expectation.' Because beta is an Erfc of delta_c/sqrt(2 sigma_2^2), that choice changes fPBH by orders of magnitude at fixed As. If the calibrated threshold is right, the NANOGrav-favored As would push fPBH above the EROS/MACHO, caustic, and CMB accretion bounds shown in Fig. 4. If the conversion is suppressed instead, the abundance drops below the 'substantial amount' needed for the merger background forecasts in Fig. 5. The Section 5 discussion concedes the window function is 'the largest difference' from Refs [36,37], and the Note Added claim of consistency with Ref [37] is not shown quantitatively. That is not a minor caveat; it is the hinge of the paper.

There is also the fitting issue: As and k* are scanned so the resulting GW amplitude lands in the NANOGrav region, so the agreement is a consistency fit, not a parameter-free derivation. No code or data are provided, so independent reproduction requires reimplementing the pipeline from formulas and digitized contours.

Who is this for? People working on PBH interpretations of PTA signals and on PBH formation systematics. It is a useful scenario paper with a clear set of assumptions and a testable future signal, but it is not a detection and the title 'explain' is stronger than the evidence. I would send it to peer review with the expectation that referees press on the delta_c/window choice and require a robustness scan across thresholds and window functions. The paper deserves serious referee time, but only if the referee treats the central abundance claim as the main issue to resolve.

Referee report

Significance

If the central claim holds, the paper would establish a concrete, testable link between the NANOGrav signal and solar-mass PBHs, with forecasts for future PTA, interferometer, microlensing, and CMB observations. The induced-GW computation uses the authors' own published semianalytic formula, the merger-background estimate is based on standard PBH binary calculations, and the paper is explicit about many of its assumptions. The quantitative force of the claim, however, rests entirely on the conversion from As to fPBH, which is exponentially sensitive to the threshold δc and the window function; the paper itself identifies this choice as the largest difference from competing analyses. The agreement with NANOGrav is a consistency fit rather than a parameter-free prediction, so the significance of the result is conditional on the robustness of that conversion.

Major objections (4)

[Section 3, Eqs. (5)-(7) and footnote 4] The conversion from the scalar amplitude As to the PBH abundance fPBH is exponentially sensitive to the critical threshold δc, because β involves Erfc(δc/√(2σ2^2)) and σ2^2 is proportional to As. The paper adopts δc=0.42 while noting in footnote 4 that Ref. [76] calibrates δc=0.18 for the modified Gaussian window w(x)=exp(-x^2/4), justifying the choice by a 'naive expectation' rather than a calculation. At fixed As, changing δc from 0.18 to 0.42 changes the argument of the error function by a factor of about 2.3, which changes fPBH by orders of magnitude. Since the fPBH values in Fig. 3 and the NANOGrav contours mapped to the (MPBH, fPBH) plane in Fig. 4 depend directly on this choice, the paper's quantitative conclusion--that fPBH ~ 0.1 is allowed and that the merger background is detectable--is not robust. The authors should provide a quantitative sensitivity analysis showing fPBH(As) for δc=0.18 with the same window, and for a standard Gaussian window, to demonstrate whether the central claim survives.
[Section 5, first and second paragraphs] The paper concedes that the modified Gaussian window 'boosts the value of fPBH for a given value of As' and is 'the largest difference compared to Refs. [36,37]', and that omission of the transfer function reduces σ2^2 by several percent and 'reduces fPBH non-negligibly'. These concessions, combined with the δc ambiguity in footnote 4, mean that the claimed abundance window and the resulting detectability forecasts in Section 4 are not stable under plausible variations of the PBH formation criterion. The Note Added asserts consistency with Ref. [37] but does not quantify this consistency; a direct comparison of fPBH(As) or the (MPBH, fPBH) contours under the different choices is needed before the central claim can be accepted.
[Sections 2-3, parameter scan] Because As is scanned freely so that the induced GW amplitude lands in the NANOGrav region, the overlap between the model band and the NANOGrav contours in Fig. 2 and Fig. 4 is a consistency fit rather than an independent prediction. The physically meaningful content of the paper is therefore the implied PBH abundance for the NANOGrav-favored As, which brings the δc/window sensitivity in the previous comments to the fore. The authors should state more explicitly that no inflationary model is being tested, and should present the abundance predictions for the NANOGrav-favored As values under the alternative threshold/window choices.
[Section 4 and Fig. 5] The merger-background forecasts in Fig. 5 assume fPBH values derived from the chosen δc and window function. If a lower threshold δc=0.18 or a standard window is used, the NANOGrav-favored As could push fPBH above the excluded regions in Fig. 4, while if the conversion is suppressed by the transfer function the fPBH values would drop and the merger background would fall below the plotted sensitivity curves. The statement that 'a large part of the parameter space can be probed by the future GW observations' is therefore conditional on the same uncalibrated abundance conversion and should be re-derived for the alternative choices.

Minor objections (5)

[Footnote 4] The term 'naive expectation' should be quantified or replaced with a reference to a derivation of δc for the modified Gaussian window; as written, the footnote leaves the threshold choice unexplained.
[Section 2, Eq. (3)] The quantities kηc and I^2(t,s,kηc) are used without a self-contained definition; a sentence defining the conformal time ηc and referencing the explicit kernel formula would improve readability.
[Fig. 4 caption and text] The sentence that the contour-extraction uncertainty is 'very roughly an order of magnitude' is vague; the authors should either quantify this uncertainty or state how it was estimated.
[Note Added] Reference [146] is a private communication; the claim that the results are 'largely consistent' with Ref. [37] should be supported by a shown quantitative comparison rather than a footnote reference.
[Section 5, third paragraph] The discussion of critical collapse correctly notes that it is neglected, but the subsequent sentence reporting a 'preliminary study' is not reproducible; either give the details or remove the claim.

How Pith checks papers → · Full audit of this paper →

The paper

The paper claims

The nanohertz pulsar-timing excess can be explained by gravitational waves from solar-mass primordial black hole formation.

The reading

This paper argues that the excess in the 12.5-year pulsar timing array data, reported as a common-spectrum stochastic process, is gravitational waves generated at second order by the enhanced density fluctuations that formed black holes of roughly one solar mass in the early universe. If that interpretation is right, the same curvature perturbations that seeded those black holes also produced a gravitational-wave background in the nanohertz band, and the black holes themselves can be as abundant as roughly ten percent of dark matter in part of the allowed parameter space. The authors scan the curvature amplitude and the peak black-hole mass and show that a large fraction of the scanned region fits the reported signal while avoiding current constraints. They then predict that the same population produces a stochastic gravitational-wave background from binary mergers that future interferometers can detect, making the scenario testable.

What carries the argument

The load-bearing object is the second-order induced gravitational-wave spectrum: scalar curvature perturbations with power spectrum P_zeta source tensor perturbations through a kernel $I^{2}$(t,s,k eta), giving the gravitational-wave energy density as a double convolution of P_zeta with itself. The paper models P_zeta as a log-normal peak of width $\sigma$ = 1, centers it at a wave number related to a horizon mass of about 0.31 solar masses, and converts the same power spectrum to a primordial black hole abundance through a Press-Schechter integral with a coarse-grained variance computed using a modified Gaussian window w(x) = exp(-$x^{2}$/4) and threshold delta_c = 0.42. The same black hole population is then used to compute a merger gravitational-wave background through the comoving merger rate and the inspiral-merger-ringdown spectrum.

Extended reading notes

Core claim

At its core, the paper claims that the reported nanohertz common-spectrum excess can be quantitatively reproduced by the gravitational-wave background induced at second order in cosmological perturbation theory by a log-normal peak in the primordial curvature power spectrum, with the same peak producing solar-mass primordial black holes via Press-Schechter collapse. For curvature amplitudes between about 0.015 and 0.040 and average black-hole masses between 0.2 and 5 solar masses, a large part of the parameter plane falls inside the 1-sigma and 2-sigma signal contours. The implied black-hole abundance can be as high as about 0.1 in part of the allowed mass range, with denser regions excluded by microlensing, caustic-crossing, LIGO, and cosmic-microwave-background accretion bounds. The distinguishing prediction is a stochastic gravitational-wave background from solar-mass primordial black hole mergers at higher frequencies, at a level that future interferometers should be able to reach.

Load-bearing premise

The paper's case rests on the adopted conversion from curvature amplitude to primordial black hole abundance: a smoothing window that is twice as wide as the standard Gaussian and a threshold delta_c = 0.42, with the authors themselves stating that this window boosts the inferred abundance and is the largest difference from prior work.

Editorial extensions

If this is right

If correct, the nanohertz common-spectrum excess is a gravitational-wave background from the same curvature perturbations that formed solar-mass primordial black holes, with an inferred abundance reaching about 0.1 in part of the allowed mass window.
The scenario implies a second, higher-frequency stochastic background from binary primordial black hole mergers whose amplitude scales with the black-hole abundance, so future interferometers can probe most of the allowed curvature-amplitude and mass region.
The induced spectrum is not a pure power law, so continued pulsar timing observations over more pulsars can distinguish it from cosmic-string or supermassive-black-hole-binary explanations by its shape and by the Hellings-Downs correlation.
Because the induced gravitational-wave peak scales as M^{-1/2} while the merger cutoff scales as M^{-1}, measuring both backgrounds would pin down the primordial black hole mass and abundance simultaneously.

Reading between the lines

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

Beyond the paper: the window-function sensitivity implies that an independent numerical-relativity calibration of the black-hole formation criterion could either confirm or overturn the scenario without changing the gravitational-wave computation itself.
Beyond the paper: if the excess is confirmed to be gravitational waves through the Hellings-Downs correlation, the same timing data can be used over time to place upper limits on the curvature power spectrum at primordial-black-hole scales, complementing abundance-based estimates.
Beyond the paper: the method generalizes to other black-hole masses; applying it to the tens-of-solar-masses range inferred from interferometric merger events would predict whether pulsar timing arrays should see a signal from that population, providing a consistency test.

Original abstract

The NANOGrav collaboration for the pulsar timing array (PTA) observation recently announced evidence of an isotropic stochastic process, which may be the first detection of the stochastic gravitational-wave (GW) background. We discuss the possibility that the signal is caused by the second-order GWs associated with the formation of solar-mass primordial black holes (PBHs). This possibility can be tested by future interferometer-type GW observations targeting the stochastic GWs from merger events of solar-mass PBHs as well as by updates of PTA observations.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-09-23: 1 request, 2,997 input tokens.

What the abstract itself states, as probabilities: claims first 0.10 · states limitation 0.33 · human participants 0.02 · controlled comparison 0.03 · randomized assignment 0.01 · states quantitative result 0.05 · compares with prior results 0.18 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v4. Results and the main claim are underlined; show every role.

The NANOGrav collaboration for the pulsar timing array (PTA) observation recently announced evidence of an isotropic stochastic process, which may be the first detection of the stochastic gravitational-wave (GW) background. We discuss the possibility that the signal is caused by the second-order GWs associated with the formation of solar-mass primordial black holes (PBHs). This possibility can be tested by future interferometer-type GW observations targeting the stochastic GWs from merger events of solar-mass PBHs as well as by updates of PTA observations.

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	The NANOGrav collaboration for the pulsar timing array (PTA) observation recently announced evidence of an isotropic stochastic process, which may be the first detection of the stochastic gravitational-wave (GW) background.	background 1.00	0.05	0.05	hedged possibility 0.80
s1	We discuss the possibility that the signal is caused by the second-order GWs associated with the formation of solar-mass primordial black holes (PBHs).	contribution 0.91	0.29	0.33	hedged possibility 0.89
s2	This possibility can be tested by future interferometer-type GW observations targeting the stochastic GWs from merger events of solar-mass PBHs as well as by updates of PTA observations.	future work 1.00	0.10	0.16	hedged possibility 0.76

What kind of paper the classifier reports this is: position or commentary 0.84 · evidence from argument 0.71.

What was examined, and what was not

sources read

arXiv listing abstract v4

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

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.** Example of the spectrum of the second-order GWs induced by the curvature

**Figure 2.** Parameter scan in the range 0.015 ≤ As ≤ 0.040 and 0.2 ≤ MPBH/M ≤ 5 shown as the red shaded region. A larger As corresponds to a larger AGWB, and a larger MPBH corresponds to a larger γ. The thin red lines correspond to fPBH = 10−1 , 10−4 , 10−7 , and 10−10 from top to bottom. The 1σ and 2σ NANOGrav contours are also shown. The scanned parameter range for As corresponds to that of the PBH abundance fPBH as shown in

**Figure 3.** Relation between the scalar amplitude As and the PBH abundance fPBH for MPBH/M = 0.2 (top, solid), 1 (middle, dashed), and 5 (bottom, dotted). signal. Also, it shows that fPBH close to unity is disfavored, but fPBH ∼ 0.1 is within the 2σ contour depending on the value of MPBH. A part of such regions is excluded by existing constraints shown by shaded regions at the top of the figure. These include the microlensing constraints by EROS/MACHO collaborations [88, 89], the caustic crossing constraint [90], Advanced LIGO constraints on the subsolar mass range (individual events [91] and…

**Figure 4.** NANOGrav contours (green) on the plane of the average PBH mass

**Figure 5.** GW spectrum from the superposition of binary PBH merger events (thin black)

Figures reproduced from arXiv:2009.11853 by the authors.

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The angular power spectrum of scalar-induced gravitational waves is derived including all local-type non-Gaussian contributions, showing a 1/[ell(ell+1)] multipole dependence and a frequency-dependent term that breaks...
Search for scalar induced gravitational waves in the International Pulsar Timing Array Data Release 2 and NANOgrav 12.5 years datasets
astro-ph.CO 2023-02 accept novelty 6.0 of 10

The IPTA DR2 common-spectrum signal is decisively better fit by supermassive black hole binaries than by scalar-induced gravitational waves once primordial black hole overproduction caps the curvature power spectrum; ...
From inflation to black hole mergers and back again: Gravitational-wave data-driven constraints on inflationary scenarios with a first-principle model of primordial black holes across the QCD epoch
astro-ph.CO 2022-09 conditional novelty 6.0 of 10

Joint inference on GWTC-3 with a QCD-shaped, power-spectrum-derived primordial black hole mass function limits stellar-mass PBHs to about 10^-3 of dark matter while permitting a small primordial fraction of the detect...
Anatomy of single-field inflationary models for primordial black holes
astro-ph.CO 2022-05 conditional novelty 6.0 of 10

A two-parabola inflaton potential yields a closed-form curvature power spectrum whose peak position and height map onto PBH mass and abundance, and the resulting spectral-distortion bound excludes single-field PBHs he...
Gravitational wave signal from primordial magnetic fields in the Pulsar Timing Array frequency band
astro-ph.CO 2022-01 conditional novelty 6.0 of 10

MHD simulations and an analytical template show that a primordial magnetic field at the QCD scale can produce a stochastic gravitational wave background matching the pulsar timing array common-spectrum signal for gene...
Primordial black holes and stochastic gravitational wave background from inflation with a noncanonical spectator field
astro-ph.CO 2021-08 conditional novelty 6.0 of 10

A kinetically coupled spectator field resonantly amplifies curvature perturbations during inflation, yielding primordial black holes that could be dark matter or LIGO-Virgo progenitors, plus a two-component stochastic...
Non-Gaussianity and the induced gravitational wave background
astro-ph.CO 2021-05 accept novelty 6.0 of 10

The connected part of the local-type primordial trispectrum makes non-negligible, sometimes dominant, contributions to the gravitational wave background induced by scalar perturbations, contrary to earlier claims that...
Gravitational waves from colliding vacuum bubbles in gauge theories
astro-ph.CO 2020-12 accept novelty 6.0 of 10

In gauged first-order phase transitions the gravitational-wave source decays as the cube of the bubble radius after collision, giving a spectrum that rises as frequency to the 2.3 and falls as frequency to the minus 2...
The Threshold for Primordial Black Hole Formation: a Simple Analytic Prescription
astro-ph.CO 2020-11 conditional novelty 6.0 of 10

Once nonlinear growth at horizon crossing is included, the threshold density contrast for primordial black hole formation is about twice the standard superhorizon value, and can be computed from the power spectrum alone.
Black Holes and WIMPs: All or Nothing or Something Else
astro-ph.CO 2020-11 conditional novelty 6.0 of 10

Gamma rays from WIMPs annihilating in halos around primordial black holes cap the black-hole dark-matter fraction at about 2e-9 (WIMP mass/TeV)^1.1 across roughly 24 decades of black-hole mass, and cap the WIMP fracti...
NANOGrav Hints on Planet-Mass Primordial Black Holes
astro-ph.CO 2020-10 conditional novelty 6.0 of 10

A dust-like pre-BBN expansion with -0.091 < w < 0.048 reproduces the NANOGrav 12.5-year GW amplitude and slope and, from the same curvature peak, yields planet-mass primordial black holes at the abundance OGLE microle...
On the gravitational wave sources from the NANOGrav 12.5-yr data
astro-ph.CO 2020-09 conditional novelty 6.0 of 10

In the NANOGrav 12.5-yr common-spectrum data, Bayesian model comparison favors cosmic strings over supermassive black hole binaries and other early Universe gravitational wave sources.
Gravitational wave complementarity and impact of NANOGrav data on gravitational leptogenesis: cosmic strings
hep-ph 2020-09 conditional novelty 6.0 of 10

Cosmic-string gravitational waves can probe gravitational leptogenesis, and if NANOGrav's signal is gravitational waves, it disfavors lightest neutrino masses above about 25 meV within this model.
Cosmic Conundra Explained by Thermal History and Primordial Black Holes
astro-ph.CO 2019-06 conditional novelty 6.0 of 10

Known drops in the early-Universe equation of state imprint four peaks on the primordial black hole mass function, and one tuned spectrum is claimed to explain dark matter, microlensing, LIGO mergers and supermassive ...
Purely quadratic non-Gaussianity from tachyonic instability: Primordial black holes and scalar-induced gravitational waves
astro-ph.CO 2026-04 unverdicted novelty 5.0 of 10

Purely quadratic non-Gaussianity from tachyonic instability allows narrow curvature spectra to exponentially suppress primordial black hole overproduction via correlation coefficient ρ approaching -1 while retaining s...
Implications for Pulsar Timing Arrays of Sub-solar Black Hole Detections: From LVK to Einstein Telescope and Cosmic Explorer
astro-ph.CO 2025-08 conditional novelty 5.0 of 10

A Bayesian analysis shows that a future sub-solar PBH detection would make the primordial SIGW interpretation of PTA data favored over the SMBH interpretation, but this preference is driven by the detection prior.
Induced Gravitational Waves with Kination Era for Recent Pulsar Timing Array Signals
astro-ph.CO 2023-09 conditional novelty 5.0 of 10

Gravitational waves induced by curvature perturbations during a stiff (w=1) kination epoch fit the NANOGrav 15-year spectrum and suppress primordial black hole production by tens of orders of magnitude.
What is the source of the PTA GW signal?
astro-ph.CO 2023-08 conditional novelty 5.0 of 10

A common-likelihood comparison of seven proposed sources of the nanohertz gravitational wave background finds that phase transitions, domain walls, scalar-induced and first-order gravitational waves fit the NANOGrav 1...
Cosmological Background Interpretation of Pulsar Timing Array Data
astro-ph.CO 2023-07 conditional novelty 5.0 of 10

Combined NANOGrav and EPTA pulsar timing data prefer scalar-induced and superstring gravitational-wave backgrounds over a simple supermassive-black-hole-binary power law, and constrain primordial non-Gaussianity to |f...
Axion-Gauge Dynamics During Inflation as the Origin of Pulsar Timing Array Signals and Primordial Black Holes
astro-ph.CO 2023-07 conditional novelty 5.0 of 10

A spectator axion coupled to a U(1) gauge field during inflation can reproduce the NANOGrav nanohertz amplitude, but only inside a narrow window of the particle-production parameter, and that same window implies sub-s...
Implications of Pulsar Timing Array Data for Scalar-Induced Gravitational Waves and Primordial Black Holes: Primordial Non-Gaussianity $f_{\mathrm{NL}}$ Considered
astro-ph.CO 2023-07 conditional novelty 5.0 of 10

A four-parameter scalar-induced gravitational-wave fit to NANOGrav 15-year data bounds local non-Gaussianity to |fNL| < 4.1 and places primordial black holes in the 1e-5 to 1e-2 solar-mass window.
The Detected Stochastic Gravitational Waves and Subsolar-Mass Primordial Black Holes
astro-ph.CO 2023-06 conditional novelty 5.0 of 10

Mapping the NANOGrav 15-year stochastic gravitational-wave contours through standard scalar-induced gravitational-wave and Press-Schechter machinery puts the implied primordial black holes near 10^-4 solar masses, not...
Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays
astro-ph.CO 2023-06 conditional novelty 5.0 of 10

The QCD crossover imprints a calculable, non-power-law shape on the low-frequency tail of a primordial gravitational-wave background, and this shape fits current pulsar-timing data better than the usual cubic power law.
Primordial magnetic field as a common solution of nanohertz gravitational waves and the Hubble tension
astro-ph.HE 2023-06 conditional novelty 5.0 of 10

Primordial magnetic fields with microgauss strength and parsec coherence at the QCD epoch can fit the nanohertz gravitational-wave background and, evolving as a power law with index near 1, also reduce the Hubble and ...
Gravitational Waves from SMBH Binaries in Light of the NANOGrav 15-Year Data
astro-ph.CO 2023-06 conditional novelty 5.0 of 10

Environmental drag on supermassive black-hole binaries, not gravitational-wave emission alone, better explains the spectral tilt of the NANOGrav 15-year gravitational-wave background.
Self-interacting dark matter implied by nano-Hertz gravitational waves
hep-ph 2023-06 conditional novelty 5.0 of 10

A single MeV-scale first-order phase transition in a dark U(1)' sector can set the self-interacting dark matter mediator mass and produce the nano-Hertz gravitational waves seen by pulsar timing arrays.
Primordial gravitational waves in the nano-Hertz regime and PTA data -- towards solving the GW inverse problem
hep-ph 2023-06 accept novelty 5.0 of 10

Concrete particle-physics models for a primordial nano-Hertz gravitational-wave background are fitted directly to pulsar timing data, and most survive cosmological constraints only in narrow parameter windows.
Gravitational waves from bubble collisions and fluid motion in strongly supercooled phase transitions
astro-ph.CO 2022-08 conditional novelty 5.0 of 10

In strongly supercooled phase transitions the radius-dependent efficiency factor that splits energy between bubble walls and fluid rescales the gravitational wave spectrum by a single factor evaluated at R_eff approxi...
NANOGrav Signal from the End of Inflation and the LIGO Mass and Heavier Primordial Black Holes
astro-ph.CO 2022-02 conditional novelty 5.0 of 10

A low-scale double-field inflation model with a first-order phase transition can be tuned to match the NANOGrav gravitational-wave spectrum, and the same transition would produce LIGO-mass and heavier primordial black holes.
Primordial black holes and scalar induced gravitational waves from Higgs inflation with non-canonical kinetic term
gr-qc 2021-11 conditional novelty 5.0 of 10

Higgs inflation with a peaked noncanonical kinetic term and a nonminimal coupling to gravity can enhance curvature perturbations enough to produce primordial black holes and scalar-induced gravitational waves.
Gravitational Waves from Global Cosmic Strings and Cosmic Archaeology
hep-ph 2021-06 conditional novelty 5.0 of 10

A calibrated analytic model of global cosmic string networks predicts a gravitational wave spectrum that falls as a log-cubed power law, and whose shape maps observed frequencies onto cosmic temperatures as high as 10^8 GeV.
Decoding the phases of early and late time reheating through imprints on primordial gravitational waves
astro-ph.CO 2021-05 conditional novelty 5.0 of 10

Perturbative reheating leaves an oscillatory feature in the primordial gravitational wave spectrum at the frequency where the Hubble rate equals the inflaton decay width, and a secondary reheating phase with equation ...
Exploring the Early Universe with Gaia and THEIA
hep-ph 2021-04 conditional novelty 5.0 of 10

Gaia and its proposed successor THEIA can detect gravitational waves in the frequency gap between pulsar timing arrays and LISA, a band where dark-sector confinement transitions and the QCD transition are expected to radiate.
Primordial Black Holes Arise When The Inflaton Falls
astro-ph.CO 2021-04 conditional novelty 5.0 of 10

A downward step in the inflaton potential, crossed in less than an e-fold, amplifies curvature perturbations by the ratio of kinetic energies across the step, enough to seed primordial black holes.
Probing non-Gaussianities with the high frequency tail of induced gravitational waves
astro-ph.CO 2021-03 conditional novelty 5.0 of 10

In single-field inflation, the ultraviolet tilt of the induced gravitational wave spectrum measures the primordial non-Gaussianity parameter f_NL, with non-Gaussian corrections to the spectrum negligible given no prim...
Probing Primordial Features with the Stochastic Gravitational Wave Background
astro-ph.CO 2020-12 conditional novelty 5.0 of 10

A two-field inflation model tuned to produce primordial features in the LISA band yields four distinguishable classes of stochastic gravitational wave spectra, for which the authors give analytic template fits.
Constraining the primordial black hole scenario with Bayesian inference and machine learning: the GWTC-2 gravitational wave catalog
gr-qc 2020-11 conditional novelty 5.0 of 10

Assuming every GWTC-2 black hole merger is primordial, the authors infer that primordial black holes make up about 0.3 percent of dark matter and that accretion, not formation, supplies the observed spins.
NANOGrav results and LIGO-Virgo primordial black holes in axion-like curvaton model
astro-ph.CO 2020-11 conditional novelty 5.0 of 10

Primordial non-Gaussianity in the axion-like curvaton model suppresses induced gravitational waves enough that stellar-mass primordial black holes and the NANOGrav nanohertz signal can be explained by the same early-u...
Testing Stochastic Gravitational Wave Signals from Primordial Black Holes with Optical Telescopes
astro-ph.CO 2020-10 conditional novelty 5.0 of 10

Existing Subaru HSC microlensing data are consistent with, and four additional hours of observation could exclude at 95% CL, the broad primordial-black-hole mass function proposed to explain the NANOGrav gravitational...
Implications of Gravitational-wave Production from Dark Photon Resonance to Pulsar-timing Observations and Effective Number of Relativistic Species
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

An oscillating axion-like field resonantly producing dark photons can generate a gravitational-wave background matching the tentative NANOGrav signal only for a tuned parameter set (m_phi ~ 10^-13 eV, phi_osc ~ 0.1 M_...
Implications of the NANOGrav results for inflation
astro-ph.CO 2020-09 accept novelty 5.0 of 10

A very blue primordial tensor spectrum with r around 1e-6 to 0.07 and a low reheating temperature can make inflation a viable explanation of the NANOGrav common-spectrum signal.
Gauge Invariant Second Order Gravitational Waves
gr-qc 2020-09 conditional novelty 5.0 of 10

This paper builds gauge-invariant synchronous-frame tensor variables and demonstrates that the induced gravitational wave energy spectrum they yield equals the Newtonian gauge spectrum.
Did NANOGrav see a signal from primordial black hole formation?
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

The NANOGrav common-spectrum signal is consistent with gravitational waves produced when large curvature perturbations collapse into primordial black holes, at an amplitude that also matches supermassive black hole se...
Cosmic String Interpretation of NANOGrav Pulsar Timing Data
astro-ph.CO 2020-09 conditional novelty 5.0 of 10

A cosmic string network with tension Gμ between 4×10^-11 and 10^-10 reproduces the NANOGrav 12.5-year common-spectrum signal at 68% confidence, and fits it better than a supermassive-black-hole slope.
NANOGrav 15-year gravitational-wave signals from binary supermassive black-holes seeded by primordial black holes, and implications for the origins of Little Red Dots
astro-ph.CO 2025-11 conditional novelty 4.0 of 10

A PBH abundance of 10^-14–10^-12 of CDM, with seed masses 1–10^3 M_sun, is fitted to the NANOGrav 15-year background via SMBH mergers, consistent with 21-cm limits.
Cosmological constraints on small-scale primordial non-Gaussianity
astro-ph.CO 2025-05 conditional novelty 4.0 of 10

Current pulsar-timing, CMB, BAO and PBH data constrain the small-scale local f_NL to -10.0 < f_NL < 1.2 for a monochromatic primordial power spectrum, with that constraint conditional on the spectral amplitude A_zeta = 10^-2.
PBHs and GWs from $\mathbb{T}^2$-inflation and NANOGrav 15-year data
astro-ph.CO 2023-07 conditional novelty 4.0 of 10

A step-like coupling between the inflaton and the energy-momentum-squared term amplifies small-scale curvature perturbations, seeding primordial black holes and scalar-induced gravitational waves that can be tuned to ...
Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments
astro-ph.CO 2023-06 conditional novelty 4.0 of 10

An inflationary origin for the nanohertz gravitational wave background reported by pulsar timing arrays would require a tensor spectral index near 1.8 and reheating below about 10 GeV, which makes it hard to sustain.
Primordial gravitational waves from NANOGrav: a broken power-law approach
astro-ph.CO 2021-11 conditional novelty 4.0 of 10

A broken power-law primordial gravitational-wave spectrum can fit the NANOGrav signal while satisfying CMB, BBN, and LIGO/Virgo bounds, with a blue tilt n_T about 0.97 below a break frequency under 0.2 Hz.
Electroweak bubble wall expansion: gravitational waves and baryogenesis in Standard Model-like thermal plasma
astro-ph.CO 2021-11 conditional novelty 4.0 of 10

In two Standard Model extensions, the semi-classical fluid treatment of electroweak bubble walls yields solutions only for weak transitions, whose gravitational wave signals fall below future detector sensitivity.
Primordial blackholes from Gauss-Bonnet-corrected single field inflation
astro-ph.CO 2021-08 conditional novelty 4.0 of 10

Coupling the inflaton to the Gauss-Bonnet term creates a balance point where ultra-slow-roll inflation amplifies curvature perturbations enough to seed primordial black holes, with hand-tuned parameters giving either ...
Blue-tilted inflationary tensor spectrum and reheating in the light of NANOGrav results
astro-ph.CO 2020-11 accept novelty 4.0 of 10

A blue-tilted inflationary tensor spectrum can match the NANOGrav nanohertz signal without violating BBN or LIGO-Virgo bounds, provided the reheating temperature is below about 10^3 GeV, or a late-time entropy source ...
Has NANOGrav found first evidence for cosmic strings?
astro-ph.CO 2020-09 conditional novelty 4.0 of 10

Cosmic-string networks with tension between roughly 10^-11 and 10^-7 and loop sizes from 10^-8 to 0.1 can reproduce the NANOGrav 12.5-year common-spectrum signal.
Gravitational wave sources for Pulsar Timing Arrays
astro-ph.HE 2023-06 conditional novelty 3.0 of 10

Bayesian model comparison of five stochastic gravitational-wave background sources against the 2023 NANOGrav, EPTA and PPTA datasets finds no strong preference for any one source.
Observational Evidence for Primordial Black Holes: A Positivist Perspective
astro-ph.CO 2023-06 conditional novelty 3.0 of 10

The authors assemble lensing, dynamical, accretion and gravitational-wave observations into a case that primordial black holes near one solar mass could supply much of the dark matter, while conceding that no single a...

See all 64 Pith citations

How the literature uses it

125 arXiv papers cite this in 153 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 · 93 sentences without a confident label

Uses its method or tool 1 paper

The spectral density of the cumulus of inspiral and mergers is calculated according to [ref] [equation] where $f$ is the measured gravitational wave frequency, $z$ is the redshift, $H(z)$ is the Hubble parameter or expansion rate, $H_0=H(0)$ is the Hubble parameter today, $M_pl^2=(8πG)^-1$, $R(z)$ is the merger rate of primordial black holes in terms of $z$ which can be found in [ref] , and $dE_GW/df_s$ is the energy emitted per binary per frequency where $f_s=(1+z)f$ is the frequency at the source.

Were recently reported MHz events planet mass primordial black hole mergers? · Stochastic gravitational wave background · 0.97

Uses its data or values 1 paper

The analysis result of NANOGrav 12.5-yr data [ref] , if regarded as the stochastic GWB (see inspired studies e.g. [ref] ), suggests $Ω_GW∼10^-9$ with the tilt $-1.5≲n_T≲0.5$, where Ω_GW(τ_0)=k^2/12 a_0^2H^2_0P_T(k)[3 Ω_mj_1(kτ_0)/kτ_0√(1.0+1.36k/k_eq +2.50( k/k_eq) ^2)]^2,

Intermittent null energy condition violations during inflation and primordial gravitational waves · Primordial GWB at low-frequency band · 0.98

Compares with it 5 papers

Although Model WH can not produce significant PBHs using the Gaussian formulate Eq.[ref] , the energy density of SIGWs lies within the $2σ$ region of the NANOGrav signal [ref] .

Primordial black holes and scalar induced gravitational waves from Higgs inflation with non-canonical kinetic term · PBHs and SIGWs from Higgs inflation · 0.81
In contrast, Ref. [ref] consider variations of log-normal power-spectrum and do not address DM abundance with PBHs.

Testing Stochastic Gravitational Wave Signals from Primordial Black Holes with Optical Telescopes · Introduction · 0.93
Compared to Fig. 5 in Ref. [ref] , the peaks of the merger-based GWs are shifted to the high-frequency side.

The Detected Stochastic Gravitational Waves and Subsolar-Mass Primordial Black Holes · Conclusion and Discussion · 1.00

Cites it as support 2 papers

This allows to naturally associate the reported NANOGrav observations with PBH formation [ref] .

Testing Stochastic Gravitational Wave Signals from Primordial Black Holes with Optical Telescopes · Introduction · 0.99
It should be noted that the LIGO-Virgo gravitational wave detection [ref] and recent NANOGrav results [ref] may indeed suggest the existence of a small fraction of DM in PBHs in the $𝒪(1-10)M_⊙$ mass range.

Radio signatures from encounters between Neutron Stars and QCD-Axion Minihalos around Primordial Black Holes · Introduction · 0.97

Cites it as background 48 papers

The observation of a common spectrum process in the NANOGrav 12.5-year data [ref] sparked significant scientific interest and led to numerous interpretations of the signal as potential a stochastic gravitational wave background (SGWB) from cosmological sources, such as first order phase transitions [ref] , cosmic strings and domain walls [ref] , or scalar-induced gravitational waves (SIGWs) generated from primordial fluctuations [ref] (see also [ref] ).

The recent gravitational wave observation by pulsar timing arrays and primordial black holes: the importance of non-gaussianities · Introduction · 0.89
These include mergers of super-massive black holes [ref] or scenarios involving cosmic string [ref] , primordial black holes [ref] , phase transitions [ref] and magneto-hydrodynamics turbulence during the QCD phase transition [ref] and others [ref] .

Constraining models of Inflationary Magnetogenesis with NANOGrav · Introduction · 0.90
Cosmological sources that have been investigated in the context of the tentative NANOGrav signal include GW emission due to inflation [ref] , first-order phase transitions [ref] , cosmic strings [ref] or the formation of primordial black holes [ref] .

Does NANOGrav observe a dark sector phase transition? · Introduction · 0.95

What it cites · 91 sentences without a confident label

Uses the method or tool of 5 papers

In the case of the log-normal power spectrum, the full (approximate) analytic formula of $Ω_GW,c(f)$ is available [ref] although we compute it numerically with the aid of extrapolation into the IR tail using the formula of Ref. [ref] .

Gravitational Waves Induced by Scalar Perturbations with a Lognormal Peak · NANOGrav signals and second-order GWs · 0.98
In the case of the log-normal power spectrum, the full (approximate) analytic formula of $Ω_GW,c(f)$ is available [ref] although we compute it numerically with the aid of extrapolation into the IR tail using the formula of Ref. [ref] .

Log-dependent slope of scalar induced gravitational waves in the infrared regions · NANOGrav signals and second-order GWs · 0.98
This is given by [equation] where $a H$ is the conformal Hubble parameter evaluated at the conformal time $η_c$, $𝒫_ζ(k)$ is the dimensionless power spectrum of the primordial curvature perturbations, and $I^2 (t,s,kη_c)$ is the oscillation average of the kernel function, whose analytic formula has been derived in Refs. [ref] .

Semianalytic Calculation of Gravitational Wave Spectrum Nonlinearly Induced from Primordial Curvature Perturbations · NANOGrav signals and second-order GWs · 0.82

Uses data or values from 3 papers

The top end of the figure is the upper bound $Ω_GWh^2 < 1.8 ×10^-6$ [ref] from the fact that the GWs contribute to the effective number of neutrinos $N_eff$ and affect the big-bang nucleosynthesis.

Semianalytic Calculation of Gravitational Wave Spectrum Nonlinearly Induced from Primordial Curvature Perturbations · Testing the scenario with the GWs from mergers · 0.83
For such PBHs, it turns out that $f ≳f_yr$ does not contribute significantly, and so we consider the frequency range $2.5 ×10^-9 Hz ≤f ≤1.2 ×10^-8 Hz$ [ref] , which corresponds to the orange contour of figure 1 of Ref. [ref] .

Cosmic String Interpretation of NANOGrav Pulsar Timing Data · NANOGrav signals and second-order GWs · 0.85
For such PBHs, it turns out that $f ≳f_yr$ does not contribute significantly, and so we consider the frequency range $2.5 ×10^-9 Hz ≤f ≤1.2 ×10^-8 Hz$ [ref] , which corresponds to the orange contour of figure 1 of Ref. [ref] .

The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background · NANOGrav signals and second-order GWs · 0.85

Compares with 8 papers

Ref. [ref] concluded that the solar-mass PBHs abundance must be negligible and also that the supermassive black holes may be responsible for the NANOGrav signal.

Did NANOGrav see a signal from primordial black hole formation? · Introduction · 0.91
Taking into account the uncertainties of PBH abundance calculations, i.e., the different choices of the window function, the value of $δ_c$ (see footnote [ref]), etc., our results are largely consistent with those of Ref. [ref] [ref] .

NANOGrav Hints to Primordial Black Holes as Dark Matter · Note Added · 1.00
It should be noted that the NANOGrav 12.5-yr signal strength is greater than the upper bound derived in their previous 11-yr result [ref] as well as that in Parkes PTA (PPTA) [ref] (see Ref. [ref] for the NANOGrav 11-yr constraints on PBHs and also Ref. [ref] related particularly to European PTA (EPTA) constraints [ref] ).

The NANOGrav 11-year Data Set: Pulsar-timing Constraints On The Stochastic Gravitational-wave Background · Introduction · 0.91

Cites as background 43 papers

Assuming the observed stochastic process is due to the detection of stochastic GW background, the NANOGrav paper [ref] studied the possibility that the GWs are produced from supermassive black hole merger events (e.g., see Ref. [ref] ).

The NANOGrav 12.5-year Data Set: Search For An Isotropic Stochastic Gravitational-Wave Background · Introduction · 0.87
Assuming the observed stochastic process is due to the detection of stochastic GW background, the NANOGrav paper [ref] studied the possibility that the GWs are produced from supermassive black hole merger events (e.g., see Ref. [ref] ).

Low-frequency gravitational radiation from coalescing massive black hole binaries in hierarchical cosmologies · Introduction · 0.87
More detailed explanation for PBH formation and parameter dependencies can be found, e.g., in Refs. [ref] and in reviews [ref] .

New cosmological constraints on primordial black holes · Implications for the PBH mass and its abundance · 1.00

Discussion

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