Probing Dark Matter Spike with Gravitational Waves from Early EMRIs in the Milky Way Center

Chen Feng; Yong Tang; Yue-Liang Wu

REVIEW 3 major objections 6 minor 2 cited by

Probing Dark Matter Spike with Gravitational Waves from Early EMRIs in the Milky Way Center

T0 review · 3 major / 6 minor · reviewed 2026-08-07 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read A dark matter spike around Sgr A* would leave a measurable fingerprint on the gravitational-wave background from early extreme mass-ratio inspirals, suppressing low frequencies and boosting high frequencies.

desk verdict The paper's headline signal is driven by dark-matter spike profiles that violate the S2 enclosed-mass bound the authors themselves cite; the underlying idea is worth a look but the quantitative claim needs major revision. read the letter →

arxiv 2506.02937 v1 pith:XDCQL4PA submitted 2025-06-03 astro-ph.GA astro-ph.COgr-qchep-ph

Chen Feng , Yong Tang , Yue-Liang Wu This is my paper

classification astro-ph.GAastro-ph.COgr-qchep-ph

keywords darkmatterspikeearlyEMRIsdynamicalfrictiongravitationalwavebackgroundSgrA*GalacticCenterLISATaiji

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

Dense dark matter is expected to pile up around supermassive black holes, but its density near Sgr A* has never been measured directly. This paper argues that such a dark matter spike can be probed through gravitational waves from early extreme mass-ratio inspirals (E-EMRIs), compact objects slowly spiraling into the black hole at the Galactic center. Dark matter drag circularizes these orbits and accelerates the inspiral, so fewer sources linger at low frequencies while more power accumulates at high frequencies. The authors compute the unresolved Galactic-center background and find that the resulting distortion — suppression at low frequencies, enhancement at high frequencies — grows with the spike's steepness and lies within reach of the space-based detectors LISA and Taiji. If the claim is right, future observations would turn the E-EMRI background into a direct readout of the dark matter density profile around Sgr A*.

What carries the argument

The load-bearing object is the dark matter spike, a power-law density profile $\rho(r) = \rho_c (r_c/r)^\gamma$ formed by the adiabatic growth of the black hole in an initially power-law halo, with index $\gamma = (9-2\beta)/(4-\beta)$ and normalization set by the S2-star bound on enclosed mass at $r_c = 0.01$ pc. The argument runs through the competition of three timescales: the relaxation time $t_{\rm rlx}$, which pumps up orbital eccentricity and feeds the loss cone; the gravitational-wave time $t_{\rm gw}$, which drives decay; and the circularization time $t_r$, which includes dark matter dynamical friction computed from a drag acceleration with dark matter velocity distribution obtained from Eddington's formula. The formation condition $t_{\rm gw} < t_{\rm rlx}$ fixes the critical semi-major axis $a_{\rm cri}$, which dark matter does not shift, while the steady-state source count $dN/da = K/\dot{a}(a,e)$ converts the faster orbital evolution into a smaller in-band population. The final spectrum superposes the Peters–Mathews harmonic emission of each sampled source through the Phinney background integral, producing the low-frequency suppression and high-frequency enhancement.

What would settle it

An observed Galactic-center E-EMRI background whose low-frequency end (roughly $10^{-5}$ to $10^{-4}$ Hz) matches the no-dark-matter prediction, with no suppression, would rule out the spike-driven distortion at the assumed densities. Equivalently, an independent measurement of the enclosed mass within 0.01 pc of Sgr A* — from improved stellar orbits or pulsar timing — that falls far below the assumed spike normalizations would remove the premise, and the predicted suppression would vanish.

Watch

Extended reading notes

Core claim

The central claim is that dynamical friction from a dark matter spike around Sgr A* measurably changes both the population and the gravitational-wave spectrum of early EMRIs. Modeling the spike as $\rho(r) = \rho_c (r_c/r)^\gamma$ with $\gamma = 2, 2.5, 3, 3.5$ and normalizations capped by the S2-star upper limit on enclosed mass, the paper shows that dark matter drag accelerates the inspiral and promotes circularization. The formation rate is essentially unchanged — the critical semi-major axis $a_{\rm cri}$ and the relaxation timescale barely move — but the inspiral spends less time in the detector band, so the resident source count falls: for $40\,M_\odot$ black holes the in-band number $N_2$ drops from 133 without dark matter to 16 for $\gamma = 3.5$. The resulting characteristic strain of the unresolved background is suppressed in the low-frequency part of the LISA/Taiji band and enhanced at high frequencies, with the deviation growing with $\gamma$. The paper concludes that unresolved E-EMRIs in the Milky Way center can serve as a probe of the dark matter spike, aided by the known sky location of the signal.

Load-bearing premise

The argument rests on a dense dark matter spike actually surviving around Sgr A*, meaning a density between roughly $1.5\times10^7$ and $1.1\times10^8$ solar masses per cubic parsec at 0.01 pc from the black hole, falling off as a power law with index between 2 and 3.5; if the spike is depleted by annihilation, scattering, or disruption, the dynamical friction becomes negligible and the predicted spectral distortion disappears.

Editorial extensions

If this is right

The unresolved E-EMRI background from the Milky Way center becomes a dark matter probe: a steeper, denser spike (larger $\gamma$) produces a stronger low-frequency suppression and high-frequency enhancement in the characteristic strain.
For a steep spike with $\gamma = 3.5$, the number of $40\,M_\odot$ early EMRIs in the LISA/Taiji band falls from about 133 to 16, while for $10\,M_\odot$ systems it drops from 2.3 to 0.6 — a population change large enough to distort the spectrum.
Dark matter leaves the EMRI formation rate essentially intact — the critical semi-major axis and the relaxation timescale are hardly affected — so the observable signature is the accelerated, circularized inspiral and the spectral distortion it produces, not the event rate.
Because the signal comes from a fixed sky location, the Galactic center, it can be separated from cosmological and other stochastic backgrounds, and loud individual events could subsequently be used for parameter estimation of the spike.

Reading between the lines

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

A null observation would be informative rather than empty: if LISA or Taiji sees no low-frequency suppression in the Galactic-center background, the dark matter density within about 0.01 pc of Sgr A* would have to lie below the spike normalizations assumed here, tightening constraints on spike depletion by annihilation, scattering, or disruption.
The mechanism is generic: any steady drag that circularizes highly eccentric orbits — a stellar cusp, an accretion flow, or self-interacting dark matter — would imprint a similar spectral tilt, so the shape of the distortion, rather than its absolute amplitude, is the more diagnostic signature.
The depth and turnover frequency of the spectral distortion map onto the spike parameters $\gamma$ and $\rho_c$, so inverting a measured background could in principle reconstruct the density profile; the paper does not prove such an inversion is unique, making that a natural target for the parameter-estimation study the authors defer to future work.

Share X Bluesky LinkedIn Reddit HN

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

Colleague,

The genuinely new piece here is the combined calculation: dark matter dynamical friction acting on early EMRIs in the Galactic center, feeding into the unresolved GW background and producing spectral suppression at low frequencies and enhancement at high frequencies. That is a legitimate extension of earlier E-EMRI background work, and the paper's structure is clear. The physics used is standard (Peters equations plus the Dosopoulou dynamical friction term), and the authors are explicit about their assumptions. They also correctly note that the Milky Way center background is not isotropic, which matters for separating it from other backgrounds.

The soft spots are serious, though. The big one is the normalization. The paper cites the S2 orbital constraint of about 1200 solar masses enclosed, then adopts rho_c values for gamma=3 and 3.5 that are inconsistent with that bound. Integrating their power law from their own a_min (8.23e-7 pc) out to the S2 orbit (~0.005 pc) gives about 3.7e3 solar masses for gamma=3 and 4.1e4 for gamma=3.5, both well above the 1200 solar mass limit. Those two cases are precisely the ones that produce the large N2 reductions (133 to 87 and 133 to 16) and the visible spectral deviations in Fig. 5. At an S2-consistent normalization, the gamma=2 and 2.5 cases change N2 by only a few percent, which does not support the 'sizable and detectable' claim in the abstract. This is a load-bearing inconsistency, not a nitpick.

There are other gaps. The assertion that a_cri and the event rate are unaffected by dynamical friction is stated but not derived; the paper just refers to Fig. 3. The detectability claim is not quantified: Fig. 5 plots characteristic strain against LISA/Taiji sensitivity but provides no SNR calculation for the background, no error bars, and no discussion of shot noise from the small numbers of sources (N1 and N2 are sometimes below unity). The sampling scheme uses a single initial orbital template, which is a strong simplification.

The idea is worth a serious referee, and the paper should not be desk-rejected. But in its current form, the central quantitative conclusion is not supported. A revision that recalibrates the spike normalization against S2, derives the event-rate invariance, and adds an SNR estimate could make this a useful contribution. I would send it to review with heavy revision expected.

Referee Report

3 major / 6 minor

Summary. The paper models the dynamical friction exerted by a dark matter spike around Sgr A* on the inspiral of stellar-mass black holes, focusing on early extreme mass-ratio inspirals (E-EMRIs). It computes the resulting E-EMRI population in the Galactic Center under several power-law spike profiles, evolves representative orbits, and constructs the unresolved gravitational-wave background from these sources. The central claim is that dark matter dynamical friction reduces the E-EMRI population and reshapes the background spectrum, suppressing low frequencies and enhancing high frequencies, so that future space-based detectors such as LISA and Taiji could probe the dark matter spike. The analysis is a forward model: spectral deviations are computed from assumed spike parameters rather than fitted to a gravitational-wave signal, and no detection significance is calculated.

Significance. If the claimed effect is real and observable, the paper would provide a new observational window onto the dark matter density profile in the immediate vicinity of Sgr A*, complementing stellar-orbital constraints. The paper is also useful in explicitly connecting early-EMRI populations with the dark matter environment. The modeling is straightforward and reproducible in structure, and the use of standard Peters-Mathews emission plus a dynamical-friction prescription is transparent. However, the significance of the result is conditional on the assumed spike normalizations being consistent with the S2 enclosed-mass upper limit, and the paper does not currently quantify the detectability of the predicted spectral deformation.

major comments (3)

[Sec. 2.2, Eq. (4); Table 2; Fig. 5] The gamma=3 and gamma=3.5 spike profiles used for the headline results are inconsistent with the S2 enclosed-mass upper limit cited in the same section. Integrating Eq. (4) with the adopted rho_c values from a_min=8.23e-7 pc to the S2 orbital radius (about 0.005 pc) gives roughly 3.7e3 M_sun for gamma=3 and 4.1e4 M_sun for gamma=3.5, both well above the quoted limit of about 1200 M_sun. A pure power law with gamma>=3 has a divergent enclosed mass as r goes to 0; imposing the S2 bound would require an inner cutoff near 3e-4 pc (gamma=3) or 5e-4 pc (gamma=3.5), far above a_min, or a substantially lower rho_c. These steep profiles drive the large reductions in N2 in Table 2 (from 133 to 87 for gamma=3 and to 16 for gamma=3.5) and the prominent spectral deviations in Fig. 5. The paper should be recomputed with S2-consistent, truncated spike profiles, and the conclusion should be based on those models rather than on unphysically divergent power laws.
[Sec. 3, Eqs. (17)-(18); Fig. 3] The statement that dark matter dynamical friction does not affect a_cri, and therefore does not alter the event rate, is asserted rather than derived. The text says Fig. 3 demonstrates that a_cri remains the same across all cases, but the definition of a_cri in Fig. 3 (the intersection of the plunge orbit with the trlx=tr curves) is not the same as the EMRI formation condition tgw<trlx given in Eq. (3). Since tr includes the dynamical-friction contribution, it is not self-evident that the intersection is independent of gamma; a quantitative derivation or a numerical check is needed. If a_cri or the loss-cone flux changes with gamma, then the changes in N2 in Table 2 are not purely an evolution-time effect, and the interpretation of the source counts changes.
[Sec. 4.2, Fig. 5] The abstract and conclusions describe the spectral deviations as 'detectable,' but the paper does not compute a detection significance, signal-to-noise ratio, or any other statistical measure for the difference between the with-dark-matter and without-dark-matter spectra. The curves in Fig. 5 are compared visually with LISA and Taiji sensitivity curves, and Fig. 4 reports the SNR of individual E-EMRIs, but neither quantifies whether the distortion of the background is resolvable given noise and source-count variance. An SNR calculation for the model difference, or a clear statement of what would be required to detect the deviation, should be added before claiming detectability.

minor comments (6)

[Sec. 2.2, heading] There is a typo in the section heading: 'Envrionment' should be 'Environment.'
[Sec. 2.2, Eq. (4) and surrounding text] The derivation of the rho_c upper limits from the S2 bound is not shown. The text should specify the integration limits and whether an inner cutoff is used, especially because the gamma>=3 cases are divergent.
[Sec. 3, Eq. (17)] The quantity N_iso(a) is called a number density, but the integration over da indicates it should be a number per unit semi-major axis. Please clarify the notation.
[Fig. 2] The axis labels and some formulas in Fig. 2 are garbled by font-encoding artifacts (e.g., the /uni000000... strings); they need to be regenerated with proper math fonts.
[Sec. 2.4, Fig. 3 and Table 1] The statement that a_band is unaffected by dark matter is based on values rounded to three significant figures. Given that a_band is defined by SNR=10 and the evolution changes with gamma, it would be useful to show more precision or explain why the differences are negligible.
[Sec. 2.2, S2 constraint] The adopted value of the S2 orbital radius (about 0.005 pc) should be stated explicitly when comparing with the enclosed-mass limit, since the normalization of the spike depends on this choice.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the GW spectral deviations are computed from assumed dark-matter spike inputs, not fitted to the GW result; self-citations are contextual and not load-bearing.

full rationale

The paper's derivation chain is: adopt a phenomenological dark-matter spike rho(r) = rho_c (r_c/r)^gamma with r_c = 0.01 pc and rho_c set by an upper limit from S2 orbital dynamics (Sec. 2.2, Eq. 4); compute dynamical friction from that spike (Sec. 2.3); evolve the orbit with Peters GW decay plus dynamical friction (Sec. 2.4); estimate source numbers using an event rate from Seoane et al. 2024 (Sec. 3, Eq. 18); and finally assemble the E-EMRI gravitational-wave background from those evolved orbits (Sec. 4.2, Eqs. 35-36). At no point in this chain is any gravitational-wave observable, source count, or spectral feature used to set a model parameter. The spike normalization and slope are external inputs, explicitly labeled as 'representative' and 'upper limits'; the central prediction of low-frequency suppression and high-frequency enhancement is a derived consequence, not a tautology. The paper does cite several works by its own authors (Zhang & Tang 2024; Zhang et al. 2025; Xie & Tang 2025; Chen & Tang 2025; Zhou et al. 2024; Guo et al. 2024/2025), but those citations support contextual statements about spike-property uncertainty, circularization behavior, or future directionality prospects, not the load-bearing equations. The core formulas come from standard or external sources: Gondolo & Silk for the spike profile, Eddington's formula for the distribution function, Peters for GW-driven orbital decay, Phinney for the background strain, and Seoane et al. for the E-EMRI event rate. The skeptical concern that gamma=3 and 3.5 profiles may exceed the S2 enclosed-mass bound is an internal-consistency or correctness risk, not a circularity: even if those steep cases are excluded, the remaining gamma<=2.5 cases are computed from the same assumed inputs rather than fitted to the GW result. Therefore the derivation is self-contained against its stated assumptions and no prediction reduces by construction to its inputs.

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

The main load-bearing assumptions are the existence and density of the dark matter spike, the S2-based normalization, and the use of an externally derived event-rate formula. The paper's quantitative predictions inherit these assumptions; removing the spike or reducing rho_c would wash out the predicted spectral distortion. No new particles, forces, or fields are introduced.

free parameters (7)

gamma (spike power-law index) = 2.0, 2.5, 3.0, 3.5
Phenomenological values spanning possible spike slopes; the central spectral distortion grows with gamma, so the result is conditional on this choice (Sec. 2.2, Eq. 4).
rho_c (reference density at r_c = 0.01 pc) = <= 1.1e8, 6.5e7, 3.4e7, 1.5e7 M_sun/pc^3 for gamma = 2, 2.5, 3, 3.5
Set to upper limits from S2 star enclosed mass; this normalization governs the strength of dynamical friction and the magnitude of the predicted effect (Sec. 2.2).
r_c (characteristic radius) = 0.01 pc
Chosen as the radius where EMRIs approximately form; it fixes the reference point for densities and affects the mapping of S2 constraints to rho_c (Sec. 2.2).
W (plunge radius factor) = 0.26
Taken from Amaro-Seoane et al. 2013 to define the loss cone boundary; affects a_cri and hence the event rate (Sec. 2.1, Eq. 3).
N0 (number of stellar-mass black holes in influence radius) = 2e4
Adopted from Amaro-Seoane 2018; directly scales the event rate and the source numbers in Tables 1 and 2 (Sec. 3).
Initial orbital parameters (a, e) = a = 0.01 pc, e = 0.9995 for 40 M_sun and e = 0.9997 for 10 M_sun
Chosen as representative high-eccentricity E-EMRI initial conditions; they set the time to merger and the shape of the GW spectrum (Sec. 4.1-4.2).
Stellar-mass black hole masses = 10 and 40 M_sun
Two illustrative populations; the spectral amplitude and source counts scale with mass (Sec. 3 and Sec. 4).

assumptions (7)

domain assumption A cold dark matter spike with power-law profile rho(r) = rho_c (r_c/r)^gamma forms via adiabatic growth of the SMBH.
The spike profile and its existence are taken from Gondolo and Silk 1999 and related work; this is not independently verified within the paper (Sec. 2.2, Eq. 4).
domain assumption The total enclosed mass within the S2 orbit is bounded near 1200 M_sun, and this is used to set the upper limit on rho_c.
Relies on S2 astrometric observations; if the enclosed mass is distributed differently, the density at 0.01 pc and the dynamical friction strength change (Sec. 2.2).
domain assumption Dynamical friction on a compact object from a collisionless dark matter background follows Dosopoulou 2024.
Assumes the dark matter spike can be treated as a smooth, isotropic distribution with an Eddington-derived velocity distribution f(v) (Sec. 2.3, Eqs. 5-11).
domain assumption The event rate of relaxation-driven EMRIs is given by the Hopman and Alexander 2005 form and is unaffected by dark matter dynamical friction.
The paper asserts that a_cri and t_rlx are unchanged without a detailed derivation; this underpins the population numbers in Sec. 3 (Eqs. 17-18).
standard math Gravitational-wave emission follows the Peters 1964 orbit-averaged equations.
Standard quadrupole radiation reaction, appropriate for the separations considered (Sec. 2.4, Eqs. 14-15).
domain assumption Accretion flow is negligible compared to dark matter in the region of interest.
The paper compares densities in Fig. 1 and ignores gas; if the accretion flow is denser near the inner edge, the environmental drag could differ (Sec. 2.2).
domain assumption All stellar-mass black holes have equal mass m, with N0 = 2e4 within R_h = 1 pc.
Simplification used in the event rate formula; a realistic mass spectrum and spatial distribution would change the source counts (Sec. 3).

how reviews work

0 comments

Cite this review

Pith. "Pith review of Probing Dark Matter Spike with Gravitational Waves from Early EMRIs in the Milky Way Center." pith.science (2026). https://pith.science/paper/XDCQL4PA

@misc{pith2026250602937,
  author       = {Pith},
  title        = {Pith review of: Probing Dark Matter Spike with Gravitational Waves from Early EMRIs in the Milky Way Center},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XDCQL4PA}},
  note         = {Machine review of arXiv:2506.02937}
}

read the original abstract

Cold dark matter may form dense structures around supermassive black holes (SMBHs), significantly influencing their local environments. These dense regions are ideal sites for the formation of extreme mass-ratio inspirals (EMRIs), in which stellar-mass compact objects gradually spiral into SMBH, emitting gravitational waves (GWs). Space-based gravitational-wave (GW) observatories such as LISA and Taiji will be sensitive to these signals, including early-stage EMRIs (E-EMRIs) that persist in the low-frequency band for extended periods. In this work, we investigate the impact of dark matter-induced dynamical friction on E-EMRIs in the Milky Way Center, model its effect on the trajectory, and calculate the resulting modifications to the GW spectrum. Our analysis suggests that this influence might be sizable and lead to detectable deviations in the spectrum, namely suppression at low frequencies and enhancement at high frequencies, therefore providing a potential probe for dark matter with future GW detectors in space, such as LISA and Taiji.

Figures

Figures reproduced from arXiv: 2506.02937 by the authors.

**Figure 1.** The comparison of the density profiles between dark matter and the accretion flow. Here, the dark matter power-law index is γ = 2. In contrast, the accretion flow follows a power-law index of α = 3.2, with a number density of 1.87 × 109 cm−3 at the periapse of S2. matter can exert a dynamical friction on its orbital motion. Numerous investigations have examined dynamical friction across diverse dark matter scenario… view at source ↗

**Figure 2.** The timescales of various dynamical effects are shown. As the semi-major axis decreases, circularization dominates the orbital evolution. When a < 0.01 pc, the relaxation effect becomes negligible. When trlx < tr, the relaxation process plays a dominant role in the evolution of the system. This leads to an increase in orbital eccentricity, with the semi-major axis remaining approximately constant. Conversely, when… view at source ↗

**Figure 3.** EMRIs formation in the a − (1 − e) plane for the inspiral of a black hole with mass m = 40M⊙ (Upper) and m = 10M⊙ (Lower) into an SMBH of mass 4.3×106M⊙. The colored area at bottom is the loss cone region, where no stable orbits exist. The boundary line of this region represents the plunge orbit; Objects in this orbit will be swallowed by the SMBH. The black line represents trlx = tgw. The green, blue, orange, and r… view at source ↗

Figures from the paper (2 more)

**Figure 4.** Figure 4: SNR of one-year observation of an E-EMRI, as a function of the remaining time until merger, where the compact object has a mass of m = 40M⊙ (Upper) and m = 10M⊙ (Lower). The black curve illustrates the case without dark matter. The dashed curves in green, blue, orange…

**Figure 5.** Figure 5: The GW background of E-EMRIs at the Galactic Center, where the compact objects have a mass of m = 40M⊙ (Upper) and m = 10M⊙ (Lower). The black solid curve represents the case without dark matter, while the green, blue, orange, and red dashed curves for dark matter spi…

Discussion (0). Sign in to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Distinguishing Monochromatic Signals in LISA and Taiji: Ultralight Dark Matter versus Gravitational Waves
hep-ph 2025-06 conditional novelty 7.0 of 10

Null-response interferometric channels can separate monochromatic ultralight dark matter signals from gravitational waves in LISA and Taiji, most effectively at high frequencies.
Beyond general relativity: gravitational waves in non-minimally coupled theories
gr-qc 2025-10 conditional novelty 5.0 of 10

A generalized propagation parameterization for gravitational-wave strains is extended to O(H²) and O(H′), then mapped to Kalb-Ramond, axion-dilaton–Chern-Simons–Gauss-Bonnet, and U(1) dark-photon models.

Reference graph

Works this paper leans on

88 extracted references · 30 canonical work pages · cited by 2 Pith papers

[1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...
[2]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...
[3]

I<54 ( 1H T@ (- *zT &v P SЇ PX

thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

arXiv 2021
[4]

2024, Astron

Abd El Dayem, K., et al. 2024, Astron. Astrophys., 692, A242, 10.1051/0004-6361/202452274

doi:10.1051/0004-6361/202452274 2024
[5]

2020, Astron

Aghanim, N., et al. 2020, Astron. Astrophys., 641, A6, 10.1051/0004-6361/201833910

doi:10.1051/0004-6361/201833910 2020
[6]

2018, Living Reviews in Relativity, 21, 10.1007/s41114-018-0013-8

Amaro-Seoane, P. 2018, Living Reviews in Relativity, 21, 10.1007/s41114-018-0013-8

doi:10.1007/s41114-018-0013-8 2018
[7]

2019, Physical Review D, 99, 10.1103/physrevd.99.123025

---. 2019, Physical Review D, 99, 10.1103/physrevd.99.123025

work page doi:10.1103/physrevd.99.123025 2019
[8]

R., Freitag, M., et al

Amaro-Seoane, P., Gair, J. R., Freitag, M., et al. 2007, Classical and Quantum Gravity, 24, R113–R169, 10.1088/0264-9381/24/17/r01

doi:10.1088/0264-9381/24/17/r01 2007

Show all 88 references

[9]

R., Pound, A., Hughes, S

Amaro-Seoane, P., Gair, J. R., Pound, A., Hughes, S. A., & Sopuerta, C. F. 2015, Journal of Physics: Conference Series, 610, 012002, 10.1088/1742-6596/610/1/012002

2015 doi
[10]

F., & Freitag, M

Amaro-Seoane, P., Sopuerta, C. F., & Freitag, M. D. 2013, Monthly Notices of the Royal Astronomical Society, 429, 3155–3165, 10.1093/mnras/sts572

2013 doi
[11]

2017, Laser Interferometer Space Antenna

Amaro-Seoane, P., et al. 2017, Laser Interferometer Space Antenna . 1702.00786

2017 arXiv
[12]

2013, Mon

An, J., & Zhao, H. 2013, Mon. Not. Roy. Astron. Soc., 428, 2805, 10.1093/mnras/sts175

2013 doi
[13]

C., Clough, K., Bamber, J., & Ferreira, P

Aurrekoetxea, J. C., Clough, K., Bamber, J., & Ferreira, P. G. 2024, Phys. Rev. Lett., 132, 211401, 10.1103/PhysRevLett.132.211401

2024 doi
[14]

2017, Physical Review D, 95, 10.1103/physrevd.95.103012

Babak, S., Gair, J., Sesana, A., et al. 2017, Physical Review D, 95, 10.1103/physrevd.95.103012

2017 doi
[15]

Babcock , H. W. 1939, Lick Observatory Bulletin, 498, 41, 10.5479/ADS/bib/1939LicOB.19.41B

1939 doi
[16]

Bambi, C., Katsanevas, S., & Kokkotas, K. D., eds. 2022, Handbook of Gravitational Wave Astronomy (Springer), 10.1007/978-981-15-4702-7

2022 doi
[17]

2004, Phys

Barack, L., & Cutler, C. 2004, Phys. Rev. D, 69, 082005, 10.1103/PhysRevD.69.082005

2004 doi
[18]

2007, Phys

---. 2007, Phys. Rev. D, 75, 042003, 10.1103/PhysRevD.75.042003

2007 doi
[19]

2022, in Snowmass 2021

Baryakhtar, M., et al. 2022, in Snowmass 2021 . 2203.07984

2022 arXiv
[20]

2024, JCAP, 06, 024, 10.1088/1475-7516/2024/06/024

Berezhiani, L., Cintia, G., De Luca, V., & Khoury, J. 2024, JCAP, 06, 024, 10.1088/1475-7516/2024/06/024

2024 doi
[21]

Bian, L., et al. 2025. 2505.19747

2025
[22]

2024, Phys

Boey, R., Wang, Y., Kendall, E., & Easther, R. 2024, Phys. Rev. D, 109, 103526, 10.1103/PhysRevD.109.103526

2024 doi
[23]

2020, Physical Review D, 102, 10.1103/physrevd.102.103023

Bonetti, M., & Sesana, A. 2020, Physical Review D, 102, 10.1103/physrevd.102.103023

2020 doi
[24]

2023, Phys

Boudon, A., Brax, P., & Valageas, P. 2023, Phys. Rev. D, 108, 103517, 10.1103/PhysRevD.108.103517

2023 doi
[25]

2024, Channels of Stellar-mass Black Hole Formation

Burrows, A., Wang, T., & Vartanyan, D. 2024, Channels of Stellar-mass Black Hole Formation. 2412.07831

2024 arXiv
[26]

H., & Lee, C

Chan, M. H., & Lee, C. M. 2023, Astrophys. J. Lett., 943, L11, 10.3847/2041-8213/acaafa

2023 doi
[27]

2025, Probing Self-Interacting Dark Matter via Gravitational-Wave Background from Eccentric Supermassive Black Hole Mergers

Chen, M.-C., & Tang, Y. 2025, Probing Self-Interacting Dark Matter via Gravitational-Wave Background from Eccentric Supermassive Black Hole Mergers. 2505.09219

2025 arXiv
[28]

2024, Dark Matter

Cirelli, M., Strumia, A., & Zupan, J. 2024, Dark Matter . 2406.01705

2024 arXiv
[29]

H., et al

Clowe, D., Bradac, M., Gonzalez, A. H., et al. 2006, Astrophys. J. Lett., 648, L109, 10.1086/508162

2006 doi
[30]

2024, LISA Definition Study Report

Colpi, M., et al. 2024, LISA Definition Study Report . 2402.07571

2024 arXiv
[31]

2025, Phys

Cui, Q., Han, W.-B., & Pan, Z. 2025, Phys. Rev. D, 111, 103044, 10.1103/PhysRevD.111.103044

2025 doi
[32]

P., & Tyson, J

Dell'Antonio, I. P., & Tyson, J. A. 1996, Astrophys. J. Lett., 473, L17, 10.1086/310378

1996 doi
[33]

2024, Phys

Dosopoulou, F. 2024, Phys. Rev. D, 110, 083027, 10.1103/PhysRevD.110.083027

2024 doi
[34]

Duque, F., Macedo, C. F. B., Vicente, R., & Cardoso, V. 2024, Phys. Rev. Lett., 133, 121404, 10.1103/PhysRevLett.133.121404

2024 doi
[35]

Feng, W.-X., Bird, S., & Yu, H.-B. 2024. 2411.05065

2024 arXiv
[36]

Ferrer, F., Medeiros da Rosa, A., & Will, C. M. 2017, Physical Review D, 96, 10.1103/physrevd.96.083014

2017 doi
[37]

S., & Thorne, K

Finn, L. S., & Thorne, K. S. 2000, Phys. Rev. D, 62, 124021, 10.1103/PhysRevD.62.124021

2000 doi
[38]

S., & Sagunski, L

Fischer, M. S., & Sagunski, L. 2024, Astron. Astrophys., 690, A299, 10.1051/0004-6361/202451304

2024 doi
[39]

E., & Hughes, S

Flanagan, E. E., & Hughes, S. A. 1998, Phys. Rev. D, 57, 4535, 10.1103/PhysRevD.57.4535

1998 doi
[40]

2025, Phys

Fu, G., Liu, Y., Wang, B., Wu, J.-P., & Zhang, C. 2025, Phys. Rev. D, 111, 084066, 10.1103/PhysRevD.111.084066

2025 doi
[41]

O., Prescod-Weinstein, C., & Wechsler, R

Glennon, N., Musoke, N., Nadler, E. O., Prescod-Weinstein, C., & Wechsler, R. H. 2024, Phys. Rev. D, 109, 063501, 10.1103/PhysRevD.109.063501

2024 doi
[42]

1999, Phys

Gondolo, P., & Silk, J. 1999, Phys. Rev. Lett., 83, 1719, 10.1103/PhysRevLett.83.1719

1999 doi
[43]

M., Barabash, O

Gorkavenko, V. M., Barabash, O. V., Gorkavenko, T. V., et al. 2024, Class. Quant. Grav., 41, 235013, 10.1088/1361-6382/ad8a11

2024 doi
[44]

2024, Results Phys., 60, 107607, 10.1016/j.rinp.2024.107607

Guo, P., Jin, H.-B., Qiao, C.-F., & Wu, Y.-L. 2024, Results Phys., 60, 107607, 10.1016/j.rinp.2024.107607

2024
[45]

2025, Sci

---. 2025, Sci. China Phys. Mech. Astron., 68, 210414, 10.1007/s11433-024-2505-1

2025 doi
[46]

2005, Astrophys

Hopman, C., & Alexander, T. 2005, Astrophys. J., 629, 362, 10.1086/431475

2005 doi
[47]

Elaheh , Zajaček, Michal , Eckart, Andreas , Sabha, Nadeen B

Hosseini, S. Elaheh , Zajaček, Michal , Eckart, Andreas , Sabha, Nadeen B. , & Labadie, Lucas . 2020, Astronomy & Astrophysics, 644, A105, 10.1051/0004-6361/202037724

2020 doi
[48]

2017, Natl

Hu, W.-R., & Wu, Y.-L. 2017, Natl. Sci. Rev., 4, 685, 10.1093/nsr/nwx116

2017 doi
[49]

K., Kavanagh, B

Karydas, T. K., Kavanagh, B. J., & Bertone, G. 2025, Phys. Rev. D, 111, 063070, 10.1103/PhysRevD.111.063070

2025 doi
[50]

2016, Physical Review D, 93, 10.1103/physrevd.93.024003

Klein, A., Barausse, E., Sesana, A., et al. 2016, Physical Review D, 93, 10.1103/physrevd.93.024003

2016 doi
[51]

Koo, H., & Lee, J.-W. 2025. 2504.19219

2025
[52]

2014, Phys

Lacroix, T., Boehm, C., & Silk, J. 2014, Phys. Rev. D, 89, 063534, 10.1103/PhysRevD.89.063534

2014 doi
[53]

2022, Sci

Li, G.-L., Tang, Y., & Wu, Y.-L. 2022, Sci. China Phys. Mech. Astron., 65, 100412, 10.1007/s11433-022-1930-9

2022 doi
[54]

2023, Phys

Liang, D., Xu, R., Mai, Z.-F., & Shao, L. 2023, Phys. Rev. D, 107, 044053, 10.1103/PhysRevD.107.044053

2023 doi
[55]

2023, Physical Review D, 107, 10.1103/physrevd.107.064021

Liu, C., Ruan, W.-H., & Guo, Z.-K. 2023, Physical Review D, 107, 10.1103/physrevd.107.064021

2023 doi
[56]

Mitra, S., Speeney, N., Chakraborty, S., & Berti, E. 2025. 2505.04697

2025 arXiv
[57]

J., Cole, R

Moore, C. J., Cole, R. H., & Berry, C. P. L. 2014, Classical and Quantum Gravity, 32, 015014, 10.1088/0264-9381/32/1/015014

2014 doi
[58]

F., Frenk, C

Navarro, J. F., Frenk, C. S., & White, S. D. M. 1996, Astrophys. J., 462, 563, 10.1086/177173

1996 doi
[59]

Peters, P. C. 1964, Phys. Rev., 136, B1224, 10.1103/PhysRev.136.B1224

1964 doi
[60]

C., & Mathews, J

Peters, P. C., & Mathews, J. 1963, Phys. Rev., 131, 435, 10.1103/PhysRev.131.435

1963 doi
[61]

Phinney, E. S. 2001, A Practical Theorem on Gravitational Wave Backgrounds. astro-ph/0108028

2001 arXiv
[62]

D., Hernquist, L., & Sigurdsson, S

Quinlan, G. D., Hernquist, L., & Sigurdsson, S. 1995, The Astrophysical Journal, 440, 554, 10.1086/175295

1995 doi
[63]

J., & Liu, C

Robson, T., Cornish, N. J., & Liu, C. 2019, Classical and Quantum Gravity, 36, 105011, 10.1088/1361-6382/ab1101

2019 doi
[64]

M., Br\"uggen, M., Schmidt-Hoberg, K., & Fischer, M

Sabarish, V. M., Br\"uggen, M., Schmidt-Hoberg, K., & Fischer, M. S. 2025. 2505.14779

2025
[65]

Sadeghian, L., Ferrer, F., & Will, C. M. 2013, Phys. Rev. D, 88, 063522, 10.1103/PhysRevD.88.063522

2013 doi
[66]

A., Lin, Y., & Tzanavaris, K

Seoane, P. A., Lin, Y., & Tzanavaris, K. 2024, Phys. Rev. D, 110, 064011, 10.1103/PhysRevD.110.064011

2024 doi
[67]

A., & Zhao, S.-D

Seoane, P. A., & Zhao, S.-D. 2025. 2504.20147

2025 arXiv
[68]

Sesana, A., Vecchio, A., & Colacino, C. N. 2008, Monthly Notices of the Royal Astronomical Society, 390, 192–209, 10.1111/j.1365-2966.2008.13682.x

2008
[69]

2025, Phys

Shen, P., Cui, Q., & Han, W.-B. 2025, Phys. Rev. D, 111, 024004, 10.1103/PhysRevD.111.024004

2025 doi
[70]

2023, Mon

Shen, Z.-Q., Yuan, G.-W., Jiang, C.-Z., et al. 2023, Mon. Not. Roy. Astron. Soc., 527, 3196, 10.1093/mnras/stad3282

2023 doi
[71]

Spitzer, L. S. 1987, Dynamical evolution of globular clusters (Princeton, NJ: Princeton University Press)

1987
[72]

2021, Phys

Traykova, D., Clough, K., Helfer, T., et al. 2021, Phys. Rev. D, 104, 103014, 10.1103/PhysRevD.104.103014

2021 doi
[73]

2001, Physical Review D, 64, 10.1103/physrevd.64.043504

Ullio, P., Zhao, H., & Kamionkowski, M. 2001, Physical Review D, 64, 10.1103/physrevd.64.043504

2001 doi
[74]

2022, Phys

Vicente, R., & Cardoso, V. 2022, Phys. Rev. D, 105, 083008, 10.1103/PhysRevD.105.083008

2022 doi
[75]

Wagg, T., Breivik, K., & de Mink, S. E. 2022, The Astrophysical Journal Supplement Series, 260, 52, 10.3847/1538-4365/ac5c52

2022 doi
[76]

D., et al

Wang, Q. D., et al. 2013, Science, 341, 981, 10.1126/science.1240755

2013 doi
[77]

2022, Phys

Wang, Y., & Easther, R. 2022, Phys. Rev. D, 105, 063523, 10.1103/PhysRevD.105.063523

2022 doi
[78]

Will, C. M. 2012, Classical and Quantum Gravity, 29, 217001, 10.1088/0264-9381/29/21/217001

2012 doi
[79]

2025, On Equation of State of Dark Matter around Massive Black Holes

Xie, Z.-M., & Tang, Y. 2025, On Equation of State of Dark Matter around Massive Black Holes. 2501.12574

2025 arXiv
[80]

Yang, S., Zhang, Y.-P., Zhu, T., Zhao, L., & Liu, Y.-X. 2024. 2412.04302

2024 arXiv
[81]

2024, Class

Yue, X.-J., & Cao, Z. 2024, Class. Quant. Grav., 41, 095011, 10.1088/1361-6382/ad38fb

2024 doi
[82]

2019, Astrophys

Yue, X.-J., Han, W.-B., & Chen, X. 2019, Astrophys. J., 874, 34, 10.3847/1538-4357/ab06f6

2019 doi
[83]

2024, JCAP, 04, 088, 10.1088/1475-7516/2024/04/088

Zhang, C., Fu, G., & Dai, N. 2024, JCAP, 04, 088, 10.1088/1475-7516/2024/04/088

2024 doi
[84]

2024, Phys

Zhang, C., & Gong, Y. 2024, Phys. Rev. D, 110, 104052, 10.1103/PhysRevD.110.104052

2024 doi
[85]

2023, JCAP, 06, 054, 10.1088/1475-7516/2023/06/054

Zhang, C., Gong, Y., Liang, D., & Wang, B. 2023, JCAP, 06, 054, 10.1088/1475-7516/2023/06/054

2023 doi
[86]

2024, Phys

Zhang, Z.-C., & Tang, Y. 2024, Phys. Rev. D, 110, 103008, 10.1103/PhysRevD.110.103008

2024 doi
[87]

2025, Universal Density and Velocity Distributions of Dark Matter around Massive Black Holes

Zhang, Z.-C., Yuan, H.-C., & Tang, Y. 2025, Universal Density and Velocity Distributions of Dark Matter around Massive Black Holes. 2503.02573

2025 arXiv
[88]

Zhou, Y.-C., Jin, H.-B., Qiao, C.-F., & Wu, Y.-L. 2024. 2405.19240

2024

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.

[1] [1]

, " * write output.state after.block = add.period write newline

ENTRY address archivePrefix author booktitle chapter doi edition editor eprint howpublished institution journal key month number organization pages publisher school series title misctitle type volume year version url label extra.label sort.label short.list INTEGERS output.state before.all mid.sentence after.sentence after.block FUNCTION init.state.consts ...

[2] [2]

write newline

" write newline "" before.all 'output.state := FUNCTION format.url url empty "" new.block "" url * "" * if FUNCTION format.eprint eprint empty "" archivePrefix empty "" archivePrefix "arXiv" = new.block " " eprint * " " * new.block " " eprint * " " * if if if FUNCTION format.doi doi empty "" " " doi * " " * if FUNCTION format.pid doi empty eprint empty ur...

[3] [3]

I<54 ( 1H T@ (- *zT &v P SЇ PX

thebibliography [1] 20pt to REFERENCES 6pt =0pt -12pt 10pt plus 3pt =0pt =0pt =1pt plus 1pt =0pt =0pt -12pt =13pt plus 1pt =20pt =13pt plus 1pt \@M =10000 =-1.0em =0pt =0pt 0pt =0pt =1.0em @enumiv\@empty 10000 10000 `\.\@m \@noitemerr \@latex@warning Empty `thebibliography' environment \@ifnextchar \@reference \@latexerr Missing key on reference command E...

arXiv 2021

[4] [4]

2024, Astron

Abd El Dayem, K., et al. 2024, Astron. Astrophys., 692, A242, 10.1051/0004-6361/202452274

doi:10.1051/0004-6361/202452274 2024

[5] [5]

2020, Astron

Aghanim, N., et al. 2020, Astron. Astrophys., 641, A6, 10.1051/0004-6361/201833910

doi:10.1051/0004-6361/201833910 2020

[6] [6]

2018, Living Reviews in Relativity, 21, 10.1007/s41114-018-0013-8

Amaro-Seoane, P. 2018, Living Reviews in Relativity, 21, 10.1007/s41114-018-0013-8

doi:10.1007/s41114-018-0013-8 2018

[7] [7]

2019, Physical Review D, 99, 10.1103/physrevd.99.123025

---. 2019, Physical Review D, 99, 10.1103/physrevd.99.123025

work page doi:10.1103/physrevd.99.123025 2019

[8] [8]

R., Freitag, M., et al

Amaro-Seoane, P., Gair, J. R., Freitag, M., et al. 2007, Classical and Quantum Gravity, 24, R113–R169, 10.1088/0264-9381/24/17/r01

doi:10.1088/0264-9381/24/17/r01 2007