From eccentric binaries to nonstationary gravitational wave backgrounds

Alberto Sesana; Hippolyte Quelquejay Leclere; Mikel Falxa

REVIEW 4 major objections 6 minor 1 cited by

From eccentric binaries to nonstationary gravitational wave backgrounds

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Pith's one-line read Eccentric supermassive black hole binaries can make the gravitational-wave background detectably flicker.

desk verdict Solid new analytic result on luminosity variance for eccentric binaries, but the PTA detectability claim compares the wrong quantities and needs a real detection study before it can stand. read the letter →

arxiv 2412.01899 v1 pith:4QRCNUHG submitted 2024-12-02 gr-qc astro-ph.COastro-ph.GAastro-ph.HE

Mikel Falxa , Hippolyte Quelquejay Leclere , Alberto Sesana This is my paper · ORCID

classification gr-qcastro-ph.COastro-ph.GAastro-ph.HE PACS 04.30.-w95.85.Sz04.80.Nn

keywords gravitationalwavebackgroundeccentricbinariessupermassiveblackholepulsartimingarraysnonstationarityspectralfluctuationsGWluminosityvariancenanohertzwaves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

The paper asks whether the stochastic gravitational wave background (GWB) produced by a population of massive black hole binaries is stationary, and shows that it need not be. Because an eccentric binary radiates most strongly near periastron, its GW power depends on its instantaneous orbital phase, so a population of such binaries has a spectrum that fluctuates in time. The authors quantify these fluctuations by $\Delta\Omega^2$, the variance of the spectrum divided by its mean squared, and find that for a homogeneous population of equal binaries the fluctuations are tiny, scaling as $1/N$ with the number of sources. For astrophysically realistic populations, where a small number of massive, highly eccentric binaries dominate the signal, the fluctuations can reach $\Delta\Omega^2 \sim 10^2$, far above the relative measurement uncertainty $\sigma_S^2 \approx 0.9\text{--}8\times10^{-3}$ of current and planned pulsar timing arrays. Even a single eccentric binary with signal-to-noise ratio near unity, invisible to standard deterministic searches, can produce order-unity fluctuations, so detecting nonstationarity would reveal bright eccentric sources in the universe.

What carries the argument

The central object is the variance–covariance structure of the harmonic decomposition of the GW luminosity. Writing each harmonic as $P_n=\langle P_n\rangle f_n(t,e)$, the covariance between two harmonics of the same binary is $\mathrm{cov}\{P_n,P_m\}=P_0^2 g(n,e)g(m,e)(\langle f_n f_m\rangle-1)$, which encodes the fact that harmonics fluctuate together as the binary accelerates through periastron. From this the paper forms the time-fluctuation statistic $\Delta\Omega^2(f)=\mathrm{var}\{h_c^2(f)\}/\langle h_c^2(f)\rangle^2$, and for a single source derives the practical approximation $\Delta\Omega^2 \approx \rho^4[(G(e)/F(e))^2-1]$, where $F(e) = (1 + 73e^2/24 + 37e^4/96)/(1-e^2)^{7/2}$ amplifies the mean luminosity and $G(e)$ is the analogous factor for the squared luminosity. The machinery carries the argument by turning the burst-like radiation of eccentric binaries into the relative variance of the spectrum, a number that can be compared directly with pulsar timing array measurement uncertainties.

What would settle it

Compare the measured variance of the GWB spectrum across independent time segments of a high-cadence pulsar timing array dataset with the variance expected from stationary noise: a source with signal-to-noise ratio $\rho \approx 1.4$ and eccentricity $e=0.5$ should produce an excess fluctuation $\Delta\Omega^2$ of order one localized at its harmonic frequencies, and a realistic massive eccentric population should produce $\Delta\Omega^2$ of order $10^2$; the absence of such excess variance at the predicted level would contradict the paper's central claim.

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Extended reading notes

Core claim

The central claim is that the time variability of the gravitational wave background from supermassive black hole binaries is controlled by the variance of the GW luminosity of an eccentric binary within one orbital period, and that in realistic populations this variance is not averaged away. The paper computes the luminosity variance from the harmonic decomposition of the quadrupole moment and finds it scales as $P_0^2[G^2(e)-F^2(e)]$, where $F(e)$ amplifies the mean luminosity and $G(e)$ is the corresponding factor for the squared luminosity; this quantity vanishes for circular binaries and grows steeply with eccentricity. For a population of identical binaries the resulting spectrum fluctuation $\Delta\Omega^2(f) = \mathrm{var}\{h_c^2(f)\}/\langle h_c^2(f)\rangle^2$ scales as $1/N$ and is at most $\sim10^{-3}$ in the PTA band, too small to detect. When the population is instead sampled from astrophysical merger models with a wide spread in mass and eccentricity, a few bright eccentric binaries dominate and the fluctuations can reach $\Delta\Omega^2 \sim 10^2$, many orders of magnitude above the relative uncertainty $\sigma_S^2$ of current and planned PTAs. For a single source on a stationary background, the total fluctuation obeys approximately $\Delta\Omega^2 \approx \rho^4[(G(e)/F(e))^2-1]$, so a binary with $\rho \approx 1.4$ and $e=0.5$ gives $\Delta\Omega^2$ of order one; the paper argues that such nonstationarity could be the only detectable signature of an otherwise faint eccentric source.

Load-bearing premise

The calculation replaces the time average of the luminosity over one orbital period with an ensemble average over random orbital phases at the observation time, and it neglects the secular shrinking and speeding-up of the orbit (writing $\Phi(t)\approx 2\pi f_p(t-t_0)$); if binaries evolve appreciably over the observing span, the harmonic correlations on which the variance estimate rests would not be stationary.

Editorial extensions

If this is right

A detection of nonstationarity in the nanohertz gravitational wave background would be evidence that a small number of massive, eccentric supermassive black hole binaries, rather than a smooth uniform population, dominate the signal.
For a homogeneous population of equal binaries the fluctuations are predicted to be undetectable with current pulsar timing arrays, so any observed nonstationarity points to a heterogeneous population or to individual bright sources.
A single eccentric binary with signal-to-noise ratio as low as $\rho \approx 1$ can leave a detectable nonstationary imprint even when standard deterministic search methods cannot confidently claim it.
Future pulsar timing arrays with relative uncertainty $\sigma_S^2 \approx 8\times10^{-3}$ should be sensitive to the fluctuations predicted for a massive, highly eccentric population.

Reading between the lines

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

Extension: the same variance mechanism should apply to other gravitational wave backgrounds, such as galactic white-dwarf binaries for a space-based detector and extreme-mass-ratio inspirals, where a few bright eccentric sources could produce analogous detectable nonstationarity.
Extension: the paper's single-source approximation $\Delta\Omega^2 \approx \rho^4[(G(e)/F(e))^2-1]$ implies a sharp dependence on eccentricity; a targeted search for spectral variance in pulsar timing array data could therefore be turned into a measurement of $e$ for the dominant source, which the paper does not develop into a full estimator.
Extension: the neglect of secular frequency evolution over the observation span is the most fragile step; a natural test is to re-run the variance calculation with an evolving orbital frequency over a 10–20 year observation and check whether the harmonic covariance, and hence $\Delta\Omega^2$, changes materially.
Extension: if nonstationarity is detected in current or next-generation pulsar timing array data, it would break the degeneracy between a gravitational wave background of circular binaries and an eccentric-binary interpretation, since circular populations produce no such fluctuations.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

The genuinely new thing here is the closed-form variance of the GW luminosity for an eccentric binary, Equation 4, plus the covariance structure between harmonics. That derivation is clean, self-contained, and they check it against the harmonic decomposition to relative error 10^-14. That part is good, real work. The application to nonstationarity of the GWB—showing that fluctuations scale as 1/N and that a realistic population or a single low-SNR eccentric binary can produce large ΔΩ²—is also new and worth taking seriously. The internal arithmetic and the citation pattern look solid; the population models are drawn from established sources, and the authors are honest about the leading-order post-Newtonian limitation.

The soft spot is the detectability argument in Section VI. They compare ΔΩ²(f) to σ²S, the Fisher-forecast relative uncertainty on the GWB amplitude A², and conclude that if ΔΩ² > σ²S the nonstationarity is detectable. But σ²S is a broadband constraint on a power-law amplitude under stationary Gaussian noise; it is not the per-frequency variance of a spectral estimator, and it is not a detection statistic for time-varying variance. The authors themselves say 'A more in-depth analysis would require defining detection statistics,' which is exactly right, but it means the central claim—that current and future PTAs can detect these fluctuations—is not yet supported. That is an addressable gap, not a fatal one: the paper provides the physical amplitude of the effect, and a proper likelihood or simulation-based study could settle it. I would not desk-reject on this; I would send it to a referee with the explicit request that the detectability section be rewritten around a defined statistic.

The phase-evolution concern raised in the report is minor. Over a 10-year observation, binaries in the PTA band have chirp timescales far longer than the observation span, so Φ(t) ≈ 2πfp(t−t0) is accurate and the time-average-to-ensemble-average mapping is fine. The real burden is the missing detection study.

Who benefits: PTA data analysts thinking about non-Gaussian or nonstationary backgrounds, and people modeling eccentric SMBHB populations. It deserves a serious referee, but the verdict should be conditional on a proper detection analysis and on sharing the population simulation parameters. I would bring it to reading group to discuss exactly where the detectability argument breaks, and I would probably cite Equation 4 in my own work on eccentric backgrounds.

Referee Report

4 major / 6 minor

Summary. This paper studies the time-dependent (nonstationary) character of the gravitational wave background (GWB) produced by eccentric supermassive black hole binaries (SMBHBs) in the pulsar timing array (PTA) band. The authors first derive closed-form expressions for the variance of the GW luminosity of a single eccentric binary within one orbital period, and for the covariance between the harmonic components of the luminosity, starting from the Peters-Mathews quadrupole formulas and a Fourier decomposition of the Keplerian orbit. They verify the consistency of these expressions numerically. They then compute the normalized variance ΔΩ²(f) of the GWB power spectral density for three scenarios: a toy population of equal, homogeneously distributed binaries; two realistic astrophysical population models (POP A and POP B) built from merger-rate and SMBH-host prescriptions; and a single eccentric binary superposed on a stationary GWB. Finally, they compare ΔΩ² with the Fisher-forecast relative uncertainty σ²_S on the GWB amplitude for three PTA configurations (EPTA10, IPTA20, SKA10) and conclude that realistic massive eccentric populations, or even individual binaries with SNR ≈ 1, could produce detectable nonstationary fluctuations.

Significance. The analytical derivation of the luminosity variance and harmonic covariance (Eqs. 4, 10, and Appendix A) is a useful and self-contained contribution, with the internal consistency check in Fig. 10 providing strong support for the algebra. The finding that a single low-SNR eccentric binary can produce relative power fluctuations of order unity, while deterministic searches would struggle to detect it, is interesting and could motivate new data-analysis strategies. The neglect of binary frequency evolution (Sec. II A 3 and Appendix A 2 a) is well justified for SMBHBs in the PTA band, since the chirp timescale vastly exceeds any realistic observation span; I do not regard that assumption as a substantive weakness. However, the paper's central detectability claim is currently supported only by a heuristic comparison between ΔΩ² and a broadband Fisher uncertainty, not by an actual detection statistic or end-to-end simulation. The authors explicitly acknowledge this gap, but the abstract and conclusions present the nonstationarity as 'detectable' without qualification.

major comments (4)

[Sec. VI, Eqs. (38) and Table I] The comparison ΔΩ²(f) > σ²_S is not a valid detection criterion. σ²_S is the Fisher-forecast relative uncertainty on the broadband GWB amplitude A² under the assumption of a stationary, Gaussian GWB with fixed spectral index; it is not the per-frequency variance of the PSD estimator S(f). Table I lists single numbers for each PTA, whereas ΔΩ²(f) varies strongly with frequency. The paper does not construct a likelihood, a model comparison, or a statistic that would actually measure time variations of the PSD, and the authors acknowledge this in Sec. VI A ('A more in-depth analysis would require defining detection statistics'). As it stands, the abstract's statement that the nonstationarity 'might become very large and detectable' and the conclusion's 'easily detectable' are not supported by the evidence presented. This is the load-bearing weakness: either a proper detection study (or at least a per-frequency variance estimate of the spectrum estimator) must be provided, or the detectability claims must be substantially softened.
[Sec. IV B and Fig. 6] The population-simulation estimates of ΔΩ²(f) are obtained from a finite number of sampled binaries ('hundreds of thousands'), but the paper does not state the total number of SMBHBs represented by the sampling nor demonstrate convergence of ΔΩ² with respect to that total number. Since ΔΩ² scales as 1/N for a homogeneous population (Eq. 30), and since the bright, massive binaries dominate the variance, the median and percentile bands in Fig. 6 could depend on the sampling cutoff or the realization size. Please specify the total number of binaries in each simulated universe, the effective number of sources contributing to each frequency bin, and show that the results are converged with respect to the sample size.
[Appendix A 3, Eqs. (A18)-(A23)] The approximation cov{Pn,Pm} ≈ ⟨Pn⟩⟨Pm⟩[(G/F)²-1] does not follow from the equality of weighted sums in Eq. (A18). From Σ_{n,m} g(n,e)g(m,e)[(G/F)²-1] = Σ_{n,m} g(n,e)g(m,e)(⟨fn fm⟩-1), one can only conclude that the weighted average of (⟨fn fm⟩-1) equals (G/F)²-1, not that every individual term satisfies the relation. The derivation presented in Eqs. (A19)-(A23) therefore overstates the rigor of the approximation. Since the main single-source results (Figs. 7 and 8) use the exact covariance from Eq. (10) and only Fig. 9 uses the approximate formula, this issue is not fatal, but the approximation should be presented as a heuristic and its range of validity tested against the exact calculation.
[Sec. III A 1, Eq. (23)] Equation (23) contains an unclear summation structure: the expression 'X_{i=1}' has no upper limit, and the subsequent sum over harmonics 'X_n' is not clearly nested. As written, it is not possible to determine whether the sum over i runs over the N(z)p(f_p,k) sources in the frequency bin or is a typographical artifact. Please clarify the indexing and the definition of Θ(f) so that Eqs. (26)-(27) follow unambiguously.

minor comments (6)

[Eq. (38) and Table I] σ²_S(f) is written as a function of frequency, but the Fisher forecast in [9] yields a single relative uncertainty on A². Please clarify the frequency dependence or define σ²_S as a scalar.
[Fig. 4 caption] The caption reads 'Cross-correlated fluctuations ΔΩ²(f, f′' with a missing closing parenthesis; it should read ΔΩ²(f, f′).
[Sec. III A 1, just above Eq. (23)] The notation 'NP' is used inconsistently: the text writes 'N (z)p(fp,k)' but the equation uses 'N (z)p(fp,k)' with an implicit product. The summation limits and the role of Θ(f) should be stated more explicitly.
[Appendix A, first paragraph and Eq. (A1)] The name 'Peter and Mathews' appears in the appendix text while the reference is 'Peters and Mathews'; please correct the spelling.
[Sec. VI B] The labels 'EPT A10', 'IPT A20', 'SKA10' appear with an extra space in 'EPT A10'; please standardize the notation.
[Sec. II A 4, Eq. (11)] The sentence 'The contribution of consecutive higher harmonics increases 'in phase' as the binary accelerates' is somewhat vague; a more precise statement would be that the fn(t,e) are strongly correlated for nearby n, as shown in Fig. 2.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the fluctuation calculation is self-contained, and the only self-citations are background inputs, not load-bearing reductions.

full rationale

The paper's central chain — luminosity variance (Eq. 4), harmonic covariance (Eq. 10), GWB mean/variance (Eqs. 26–27), and single-source fluctuations (Eqs. 35–36 and A23) — is derived analytically from the Peters–Mathews quadrupole luminosity and the Keplerian Fourier decomposition, with the identity sum_{n,m} cov{Pn,Pm} = var(LGW) checked numerically to epsilon ≈ 10^{-14}. No model parameter is fitted to the predicted DeltaOmega^2; the astrophysical populations POP A/B are sampled from externally published merger-rate and SMBH-host relations ([21], [22]), and the PTA uncertainty sigma^2_S is taken from a published Fisher-forecast algorithm ([9]) that is not constructed from the paper's fluctuation quantity. The paper's own caveats — no detection statistic is defined (Sec. VI A: 'A more in-depth analysis would require defining detection statistics') and realistic simulations are deferred (Sec. VI B) — weaken the strength of the detectability claim but are limitations of scope, not circular reductions. The time-average/ensemble-average identification and the neglect of secular frequency evolution are standard approximations explicitly stated, not hidden redefinitions. Self-citations by the authors appear only as background population/GWB results and sensitivity estimates; they are not used to force the paper's conclusion.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No free parameters are fitted to data. The paper introduces no new physical entities. The axioms are standard domain assumptions: ergodicity, no frequency evolution during observation, the validity of the adopted population models, and the use of a Fisher-based threshold as a proxy for detectability. The load-bearing ones are the phase-uniformity assumption and the detectability threshold assumption.

assumptions (4)

domain assumption The orbital phase of each binary is uniformly random at the time of observation, and the time average over one orbital period equals the ensemble average over phases (ergodic hypothesis).
Section II A 4 and Section III A 1: the mapping from true anomaly to random variable is used to treat Pn as random. If phases are correlated across binaries or evolve significantly during observation, the variance calculation changes.
domain assumption The binary orbital frequency is constant during the observation (Φ(t) ≈ 2πfp(t−t0)), and the secular evolution due to GW emission is neglected for the purpose of computing fluctuations.
Section II A 3 and Appendix A 2 a explicitly state this. This is valid for short observation times relative to the GW inspiral timescale, but it is an assumption that could break for high-eccentricity binaries near merger.
ad hoc to paper The population models used for POP A and POP B accurately represent the astrophysical distribution of SMBHBs.
Section IV A: the populations are drawn from observation-based models in Refs [21,22] by the authors' group. The paper does not validate the models against independent predictions, and the conclusions about detectability depend on the assumed mass and eccentricity distributions.
domain assumption The PTA sensitivity estimates from Babak et al. [9] apply to time-dependent fluctuations, and the relative uncertainty σS2 with fixed spectral index is a valid detectability threshold for ΔΩ2.
Section VI: the paper equates detectability with ΔΩ2 > σS2, but does not construct a detection statistic. This is acknowledged as a brief analysis, but it is load-bearing for the claim that fluctuations are 'detectable by current PTAs.'

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Pith. "Pith review of From eccentric binaries to nonstationary gravitational wave backgrounds." pith.science (2026). https://pith.science/paper/4QRCNUHG

@misc{pith2026241201899,
  author       = {Pith},
  title        = {Pith review of: From eccentric binaries to nonstationary gravitational wave backgrounds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4QRCNUHG}},
  note         = {Machine review of arXiv:2412.01899}
}

abstract

A large population of binary systems in the Universe emitting gravitational waves (GW) would produce a stochastic noise, known as the gravitational wave background (GWB). The properties of the GWB directly depend on the attributes of its constituents. If the binary systems are in eccentric orbits, it is well established that the GW power they radiate strongly depends on their instantaneous orbital phase. Consequently, their power spectrum varies over time, and the resulting GWB can appear nonstationary. In this work, we estimate the amplitude of time-dependent fluctuations in the GWB power spectrum as a function of the eccentricity of the binaries. Specifically, we focus on the GWB produced by a population of supermassive black hole binaries (SMBHB) that should be observable by pulsar timing arrays (PTA). We show that a large population of homogeneously distributed equal SMBHBs produces nonstationary features that are undetectable by current PTA datasets. However, using more realistic and astrophysically motivated populations of SMBHBs, we show that the nonstationarity might become very large and detectable, especially in the case of more massive and eccentric populations. In particular, when one binary is slightly brighter than the GWB, we demonstrate that time fluctuations can become significant. This is also true for individual binary systems with a low signal-to-noise ratio (SNR) relative to the GWB (SNR $\approx$ 1), which standard data analysis methods would struggle to detect. The detection of nonstationary features in the GWB could indicate the presence of some relatively bright GW sources in eccentric orbits, offering new insights into the origins of the signal.

Figures

Figures reproduced from arXiv: 2412.01899 by the authors.

**Figure 2.** FIG. 2. Covariance and variance for the first 60 harmonics of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗

**Figure 3.** FIG. 3. Time-dependent fluctuations ∆Ω [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗

Figures from the paper (6 more)

**Figure 4.** Figure 4: FIG. 4. Cross-correlated fluctuations ∆Ω [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]

**Figure 5.** Figure 5: FIG. 5. The distribution of eccentricity [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]

**Figure 7.** Figure 7: FIG. 7. Time-dependent fluctuations of the GWB spectrum [PITH_FULL_IMAGE:figures/full_fig_p009_7.png]

**Figure 8.** Figure 8: FIG. 8. Total time-dependent fluctuations of the GWB spec [PITH_FULL_IMAGE:figures/full_fig_p009_8.png]

**Figure 9.** Figure 9: FIG. 9. Total time-dependent fluctuations of the GWB spec [PITH_FULL_IMAGE:figures/full_fig_p013_9.png]

**Figure 10.** Figure 10: FIG. 10. Relative error [PITH_FULL_IMAGE:figures/full_fig_p013_10.png]

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

Population statistics of nanohertz gravitational wave sources
astro-ph.HE 2026-07 conditional novelty 6.0 of 10

A hierarchical Bayesian inference framework combining free-spectrum reconstruction with population-level likelihoods distinguishes finite SMBHB populations from Gaussian primordial GWB using mock PTA data.

Reference graph

Works this paper leans on

47 extracted references · 32 canonical work pages · cited by 1 Pith paper

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The luminosity depends greatly on the eccentricity of the binary through the amplification factor F (e), given by F (e) = 1 + 73 24 e2 + 37 96 e4 (1 − e2)7/2

Power as a function of true anomaly Peters and Mathews [1] showed that the average lumi- nosity over one orbital period 1/fp of an eccentric binary with chirp mass M and eccentricity e is given by ⟨LGW ⟩ = P0F (e), (2) with ⟨x⟩ denoting the average over one orbital period and P0 = (32/5c5)G7/3(M2πfp)10/3 the power emitted if the binary was circular. The l...

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(6) This decomposition allows us to write the total lumi- nosity as a sum of harmonic contributions Pn LGW = X n Pn

Fourier analysis of Keplerian orbit The Fourier analysis of the Keplerian orbit provides us with a harmonic decomposition of the quadrupole mo- ment, as defined in Equation 1 I ij = X n I ij n . (6) This decomposition allows us to write the total lumi- nosity as a sum of harmonic contributions Pn LGW = X n Pn. (7) In the following, we estimate the varianc...

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Following [1], we use Φ( t) ≈ 2πfp(t − t0) and neglect the frequency evolution due to the GW radiation of the binary

First moment Consider a binary with orbital period Tp and orbital frequency fp = 1 /Tp. Following [1], we use Φ( t) ≈ 2πfp(t − t0) and neglect the frequency evolution due to the GW radiation of the binary. Using Equation 6, we can split the luminosity on a basis fn(t, e) constructed from Equation A11 as 3 Pn = ⟨Pn⟩fn(t, e), (8) with ⟨fn(t, e)⟩ = 1, and th...

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We apply the formula for the covariance cov{Pn, Pm} ⟨PnPm⟩ − ⟨Pn⟩⟨Pm⟩ = P 2 0 g(n, e)g(m, e) × [⟨fn(t, e)fm(t, e)⟩ −1]

Second moment Now that we have an expression for Pn, it is straight- forward to calculate the covariance between two different components Pn and Pm. We apply the formula for the covariance cov{Pn, Pm} ⟨PnPm⟩ − ⟨Pn⟩⟨Pm⟩ = P 2 0 g(n, e)g(m, e) × [⟨fn(t, e)fm(t, e)⟩ −1]. (10) Note that for n = m, cov{Pn, Pm} is the variance of Pn, var{Pn}. The fn(t, e) do no...

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GWB spectrum A population of binary systems individually emitting GW signals results in a stochastic GW background noise [11]. The properties of this gravitational wave back- ground (GWB) depend on the characteristics of its con- stituents [7, 12, 13]. For a population of GW-driven binary systems, we deduce the power spectral density (PSD) of the GWB by i...

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Using Equation 13, we have ∆Ω2(f ) = |⟨h2 c(f )2⟩ − ⟨h2 c(f )⟩2| ⟨h2c(f )⟩2 = var{h2 c(f )} ⟨h2c(f )⟩2

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T rue anomaly The GW luminosity of a circular binary ( e = 0) is P0 = 32 5 G7/3 c5 (M2πfp)10/3, (A1) with fp the orbital frequency and M the chirp mass. Fol- lowing [1] and using Equation 1, we have the luminosity as a function of true anomaly ψ LGW = 1 12 P0 (1 − e2)5 (1 + e ...
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Pith tools

Reviewed August 12, 2026 · model on record in the stance chip above.

[1] [1]

The luminosity depends greatly on the eccentricity of the binary through the amplification factor F (e), given by F (e) = 1 + 73 24 e2 + 37 96 e4 (1 − e2)7/2

Power as a function of true anomaly Peters and Mathews [1] showed that the average lumi- nosity over one orbital period 1/fp of an eccentric binary with chirp mass M and eccentricity e is given by ⟨LGW ⟩ = P0F (e), (2) with ⟨x⟩ denoting the average over one orbital period and P0 = (32/5c5)G7/3(M2πfp)10/3 the power emitted if the binary was circular. The l...

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[2] [2]

(6) This decomposition allows us to write the total lumi- nosity as a sum of harmonic contributions Pn LGW = X n Pn

Fourier analysis of Keplerian orbit The Fourier analysis of the Keplerian orbit provides us with a harmonic decomposition of the quadrupole mo- ment, as defined in Equation 1 I ij = X n I ij n . (6) This decomposition allows us to write the total lumi- nosity as a sum of harmonic contributions Pn LGW = X n Pn. (7) In the following, we estimate the varianc...

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[3] [3]

Following [1], we use Φ( t) ≈ 2πfp(t − t0) and neglect the frequency evolution due to the GW radiation of the binary

First moment Consider a binary with orbital period Tp and orbital frequency fp = 1 /Tp. Following [1], we use Φ( t) ≈ 2πfp(t − t0) and neglect the frequency evolution due to the GW radiation of the binary. Using Equation 6, we can split the luminosity on a basis fn(t, e) constructed from Equation A11 as 3 Pn = ⟨Pn⟩fn(t, e), (8) with ⟨fn(t, e)⟩ = 1, and th...

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[4] [4]

We apply the formula for the covariance cov{Pn, Pm} ⟨PnPm⟩ − ⟨Pn⟩⟨Pm⟩ = P 2 0 g(n, e)g(m, e) × [⟨fn(t, e)fm(t, e)⟩ −1]

Second moment Now that we have an expression for Pn, it is straight- forward to calculate the covariance between two different components Pn and Pm. We apply the formula for the covariance cov{Pn, Pm} ⟨PnPm⟩ − ⟨Pn⟩⟨Pm⟩ = P 2 0 g(n, e)g(m, e) × [⟨fn(t, e)fm(t, e)⟩ −1]. (10) Note that for n = m, cov{Pn, Pm} is the variance of Pn, var{Pn}. The fn(t, e) do no...

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The properties of this gravitational wave back- ground (GWB) depend on the characteristics of its con- stituents [7, 12, 13]

GWB spectrum A population of binary systems individually emitting GW signals results in a stochastic GW background noise [11]. The properties of this gravitational wave back- ground (GWB) depend on the characteristics of its con- stituents [7, 12, 13]. For a population of GW-driven binary systems, we deduce the power spectral density (PSD) of the GWB by i...

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[6] [6]

Using Equation 13, we have ∆Ω2(f ) = |⟨h2 c(f )2⟩ − ⟨h2 c(f )⟩2| ⟨h2c(f )⟩2 = var{h2 c(f )} ⟨h2c(f )⟩2

Time fluctuations due to eccentricity The characteristic strain hc is related to the noise spec- trum as h2 c(f ) = f S(f ) [11]. Using Equation 13, we have ∆Ω2(f ) = |⟨h2 c(f )2⟩ − ⟨h2 c(f )⟩2| ⟨h2c(f )⟩2 = var{h2 c(f )} ⟨h2c(f )⟩2 . (25) The characteristic strain hc is a random quantity since the luminosity Pn is random. The average ⟨x⟩ is equiv- alent ...

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Enoki and M

M. Enoki and M. Nagashima, Progress of Theoretical Physics 117, 241–256 (2007)

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Binary Massive Black Hole Astrophysics

that characterizing these correlations provides better constraints and inference of the GWB spectrum proper- ties. A realistic GWB will have brighter individual sources that stand out from the background spectrum. In the next section, we investigate whether such sources could produce significant fluctuations. IV. EFFECT OF THE ASTROPHYSICAL DISTRIBUTION O...

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