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\texttt{GWBird}: a toolkit for the characterization of the Stochastic Gravitational Wave Background for Ground, Space, and Pulsar Timing Array detectors

T0 review · 3 major / 6 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper presents GWBird, a public Python toolkit that computes, within one framework, the overlap reduction functions, power-law integrated sensitivity curves, angular response functions, and angular sensitivity curves needed to…

desk verdict A solid, useful SGWB sensitivity toolkit that mostly delivers what it promises; the empirical vector-PTA ORF fit is the one load-bearing soft spot, and it is fixable. read the letter →

arxiv 2507.15791 v1 pith:VFHP7I3G submitted 2025-07-21 astro-ph.IM astro-ph.COgr-qc

Ilaria Caporali , Angelo Ricciardone This is my paper
classification astro-ph.IMastro-ph.COgr-qc PACS 04.80.Nn95.55.Ym
keywords stochasticgravitationalwavebackgroundoverlapreductionfunctionpower-lawintegratedsensitivitycurvesangularresponsefunctionspolarizationspulsartimingarraysLISAEinsteinTelescope
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

GWBird is a single Python framework that computes everything needed to characterize a stochastic gravitational-wave background: the overlap reduction function, the power-law integrated sensitivity curve, the angular response function, and the angular sensitivity curve, for ground-based interferometers, the space-based LISA mission, and pulsar timing arrays. The paper's central claim is that one code covers the full range of gravitational-wave polarizations — tensor, vector, scalar, and circular — and handles both isotropic and anisotropic backgrounds, so the same formalism serves tests of general relativity, searches for parity-violating signals, and anisotropy mapping. A sympathetic reader would care because the toolkit is public, its outputs reproduce known benchmarks such as the Hellings-Downs curve, and it can be applied to third-generation detector designs whose final layouts are still being decided.

What carries the argument

The machinery that carries the argument is the family of overlap reduction function integrals — sky averages of products of the detector or pulsar angular pattern functions $F^\lambda(\theta,\varphi)$, weighted by the phase factor $e^{-2\pi i f \hat{k}\cdot\Delta\vec{x}/c}$ — together with their spherical-harmonic extensions $\gamma_{\ell m,ij}(f)$, which are combined in quadrature into the rotation-invariant angular response $R_\ell(f)$. These integrals feed two sensitivity constructs: the effective noise energy-density spectrum that produces the power-law integrated sensitivity curve, and the multipole-weighted signal-to-noise ratio that produces the angular PLS. Three further objects carry specific weight: the AET transformation that diagonalizes the LISA response, the three-detector combination $\Pi(f)$ of tensor, vector, and scalar overlap reduction functions that isolates one polarization at a time, and, for pulsar timing arrays, the pulsar-term factors together with the fitted vector-mode correlation function of Eq. (2.67).

What would settle it

Recompute the PTA vector-mode overlap reduction function by direct numerical integration, without the fitted form of Eq. (2.67), for pulsar pairs with separation $\alpha_{ij}<\pi/10$ and at the high-frequency end of the PTA band, where the pulsar terms in Eqs. (2.60)–(2.61) are no longer negligible, and check whether GWBird's curve still agrees; independently, recompute the LIGO Livingston-Hanford tensor ORF and the LISA AET responses with a different quadrature scheme, such as spherical-harmonic expansion or a much finer pixel grid, and require pointwise agreement across the full band.

Watch

Extended reading notes

Core claim

On its own terms, the paper establishes that a single modular numerical implementation of the standard cross-correlation formalism works uniformly across detector classes. Starting from the angular pattern functions for each polarization, GWBird evaluates the overlap reduction functions of Eqs. (2.21)–(2.23) and their chiral generalization for circular polarization, then builds power-law integrated sensitivity curves from the signal-to-noise integrals of Section 2.5, and extends the same machinery to anisotropy by decomposing the response into the multipole quantities $\gamma_{\ell m}(f)$ and summing them into $R_\ell(f)$. For pulsar timing arrays the code keeps the frequency-dependent pulsar terms in the response and uses the NANOGrav 15-year catalog; for LISA it works in the AET basis, and for a three-detector network it isolates individual polarizations through the combination $\Pi(f)$ of tensor, vector, and scalar overlap reduction functions. The results section shows tensor, vector, scalar, and circular-polarization channels giving distinct sensitivity curves for LIGO-Virgo, Einstein Telescope (triangular and two-L configurations), Cosmic Explorer, LISA, and the pulsar array, and confirms that the PTA tensor output reduces to the Hellings-Downs curve while interferometer outputs match published results.

Load-bearing premise

The toolkit's correctness rests on numerical sky integrals that are benchmarked against published results for only a subset of outputs; the PTA vector-mode curve additionally rests on a fitting formula checked only against the code's own data, and the PTA forecasts assume identical white timing noise for every pulsar.

Editorial extensions

If this is right

  • A single public code now provides the isotropic and anisotropic sensitivity quantities for all three detector classes, so the same pipeline can serve LIGO-Virgo-KAGRA, ET, LISA, and pulsar-array searches without re-deriving response functions.
  • For Einstein Telescope, the code indicates the 2L aligned layout outperforms the triangular layout for isotropic tensor, vector, and scalar backgrounds, while the 2L misaligned layout is strongly disfavored across all polarizations — concrete input for the ongoing configuration debate.
  • For LISA, the isotropic circular-polarization channel is effectively null in the AET basis, so a parity-violating isotropic background is invisible to LISA and must be probed with moving-frame effects or other channels.
  • For the pulsar array, vector modes are the most sensitive channel for both isotropic and anisotropic backgrounds, so a background seen in the vector channel would immediately point beyond general relativity.
  • The APLS results give concrete monopole thresholds per multipole — for instance $C_\ell^{GW}=10^{-3}$ at $\ell=1,2,3$ — that a next-generation network must reach to resolve anisotropies of a given amplitude.

Reading between the lines

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

  • An editorial extension: because GWBird accepts custom detector geometries, noise curves, and orientations, the same machinery could be pointed at detector classes the paper does not exercise, such as atomic-interferometer or lunar gravitational-wave observatories, provided their pattern functions and noise models are supplied.
  • An editorial caution: the fitted PTA vector-mode curve (Eq. 2.67) is only claimed valid for separations above $\pi/10$; extending the catalog to close pulsar pairs would require direct numerical integration, since the paper itself notes that pulsar terms grow at small separations.
  • An editorial reading of the forecasts: the identical-white-noise assumption for pulsars makes the reported minimum amplitudes ($h^2\Omega_{GW}\sim 10^{-14}$ for tensor modes) optimistic bounds; heterogeneous per-pulsar noise would likely degrade them.
  • An editorial observation: the rotation invariance of $R_\ell(f)$ means the code's agreement between cosmic-rest-frame and computational-frame evaluations doubles as a portable cross-check for any independent anisotropy pipeline.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 6 minor

Summary. The paper presents GWBird, a public Python toolkit for characterizing stochastic gravitational-wave backgrounds. The code computes detector response functions, overlap reduction functions, power-law integrated sensitivity curves, angular response functions, and angular PLS for ground-based (LIGO/ET), space-based (LISA), and pulsar-timing-array detectors, for tensor, vector, scalar, and circular polarizations, including anisotropic and parity-violating backgrounds. The authors validate many outputs against published results (LIGO ORFs, the Hellings–Downs curve, the scalar PTA ORF, and LISA angular responses) and apply the code to ET configurations, LISA, and a NANOGrav-like 60-pulsar array, with reproducibility claimed through example notebooks.

Significance. If the reported implementations are correct, GWBird is a useful community tool and fills a genuine gap: a single framework covering interferometers and PTAs, isotropic and anisotropic components, and non-GR polarizations. The paper's strengths are its explicit benchmark comparisons for tensor/scalar channels and the public, documented code. The main open risk is the vector-mode PTA overlap reduction function, which is validated only against the code's own numerics and which feeds the headline vector PTA sensitivity forecast; this requires independent confirmation before the central claim of correctness can be accepted.

major comments (3)
  1. [§2.7, Eq. (2.67), Figure 6] The vector-mode PTA overlap reduction function is supplied as a five-parameter fit (A=-π, B=0.674, C=-3, D=0.813, E=0.2) that is calibrated against, and tested only against, GWBird's own numerical integration of Eq. (2.61). Because the same code generates both the truth data and the validation curve, any systematic error in the integrator, the normalization, or the pulsar-catalog geometry is invisible to this check. The paper also states the fit is trustworthy only for α>π/10, yet the code's PTA module ingests the full NANOGrav catalog and the text does not say whether the fit is clipped to its validity domain. Since the vector-mode PTA PLS in Table 8 (the headline PTA sensitivity claim, with vector modes beating tensor modes by almost an order of magnitude) rests on this fit, this is load-bearing. Please provide an independent validation (e.g., comparison with Ref. [69] or [70] using a separate code), a residual or error panel for Figure 6, and an explicit statement of how the code handles α≤π/10, or restrict the vector PTA forecasts accordingly.
  2. [§4.3.4, Eq. (2.69), Table 14] The vector-mode PTA angular response and the resulting APLS values in Table 14 are not cross-checked against an external reference. The text states agreement with Refs. [83,85] for tensor and circular polarization only, while the vector-mode angular integrals are frequency-dependent and the paper itself warns that small-separation pulsar pairs require special care. To make the vector-mode anisotropy results credible, please add a direct validation of γ_{lm}^v or R_ℓ^v against Refs. [70] and [85] (or a clearly described independent quadrature), and report the minimum angular separation actually present in the catalog for the ℓ=1,2,3 computations.
  3. [§2.7.1 and §4.2.4] The PTA sensitivity model uses identical white timing noise (σ=100 ns, cadence=20 yr^{-1}) for every pulsar, and the comparison with Ref. [74] is presented as a consistency check even though it differs in observation time, pulsar number, and noise treatment. This is not itself an error, but the text should clearly label that comparison as an order-of-magnitude cross-check rather than a benchmark, and should state how the chosen noise parameters are derived from the NANOGrav catalog (or justify them as assumptions).
minor comments (6)
  1. [Throughout] There are numerous typographical issues, including the repeated 'T able' formatting in Tables 1-14, 'with with M' in Section 4.2.2, and '39..8 Hz' in Table 6; these should be corrected.
  2. [§4.2.1] The text refers to 'left panel of Figure 4.2.1' and 'right panel of Figure 4.2.1'; these should refer to Figure 8.
  3. [§3] The sentence 'GWBird is publicly available at this URL:' is followed by no URL in the body text; the GitHub link appears only in the abstract and should be repeated in Section 3.
  4. [§2.7, Eq. (2.67)] The parenthetical 'As pointed out by [55]' for the frequency dependence of the vector-mode PTA ORF appears to be a mis-citation; the relevant discussion is in Refs. [69,73], and the reference should be corrected.
  5. [Figure 6] The agreement between the fitted curve and the numerical points is described visually; adding a residual panel or a reduced-chi-squared value would make the fit accuracy quantitative.
  6. [References] Reference [52] is incomplete: it should include the full author list or collaboration name, the data release version, and the DOI (10.5281/zenodo.14773896) in the bibliography entry.

Circularity Check

1 steps flagged · score 3.0 of 10

Vector-mode PTA overlap reduction function is validated only by a fit interpolating GWBird's own numerical data; the remaining derivation is standard and externally benchmarked.

  1. fitted input called prediction [Section 2.7, Eq. (2.67); Section 4.1.4, Figure 6; footnote 15]
    "However, despite being true that for vector modes a closed form cannot be found, we provide an expression of an approximated fitting function that interpolates the behavior of the overlap reduction function for vector modes as a function of the angular separation αij between the pulsars γv_ij(αij) = A log (Bαij) + C cos (Dαij + E), where A = −π, B= 0.674, C= −3, D= 0.813, E= 0.2. ... In the case of vector modes instead, we can see that the pulsar pairs match the fitting function in eq. (2.67)."

    The fit in Eq. (2.67) is an explicit interpolation of the γ^v_ij values computed numerically via Eq. (2.61), and Figure 6 validates the fit by plotting the same numerical pulsar-pair points against it. The parameters A–E are chosen to match GWBird's own integration output, so the agreement in Figure 6 is guaranteed by construction and cannot validate the numerical ORF against any independent standard. No external analytic expression or literature value for the vector PTA ORF is provided, so any error in normalization, sky grid, or pulsar-term treatment of the integral is invisible to this check yet propagates into the vector PTA PLS (Table 8) and APLS (Table 14).

full rationale

GWBird's central computation chain is largely non-circular. The ORF, angular response, PLS, and APLS formulas in Sections 2.3–2.6 are standard integrals (Eqs. 2.20–2.35 and 2.58–2.72) built from detector tensors and polarization projectors, and the code outputs are checked against independent published benchmarks for LIGO tensor/vector/scalar/chiral ORFs, ET configurations, LISA AET responses, and PTA tensor (Hellings-Downs) and scalar ORFs. The only load-bearing step that reduces to its own input is the vector-mode PTA ORF: Eq. (2.67) is an empirical five-parameter fit interpolating γ^v_ij computed by GWBird's own integration of Eq. (2.61), and Figure 6 validates the fit against those same numerical points. Because no independent analytic value for the vector PTA ORF is supplied, the fit cannot certify the numerical integral (normalization, grid resolution, pulsar-term handling); any systematic error propagates into the vector PTA PLS and APLS (Tables 8 and 14). The footnote restricting the fit to αij > π/10 further limits the claim, and the text does not state that the code enforces this domain. This is a real but localized circular validation; it does not affect the tensor/scalar PTA results or the interferometer results, which are externally benchmarked. Score 3 reflects partial, localized circularity rather than a central derivation that reduces by definition.

Assumptions & free parameters 4 free parameters · 5 assumptions · 0 invented entities

The sensitivity forecasts rest on standard SGWB cross-correlation formalism borrowed from the literature and on several user-chosen parameters (scalar-mode mixing K, PTA noise model, SNR thresholds), plus a five-parameter empirical fit for the vector-mode PTA ORF. No new physical entities are introduced.

free parameters (4)
  • K (scalar longitudinal/breathing energy ratio) = 0 (chosen by hand)
    In Eq. (2.23) the scalar ORF is defined with a parameter K; the paper sets K=0 throughout, assuming no longitudinal scalar contribution, which affects all scalar-mode sensitivity results.
  • A, B, C, D, E (vector PTA ORF fit) = A=-π, B=0.674, C=-3, D=0.813, E=0.2
    These five constants in Eq. (2.67) are fitted to GWBird's numerical vector-mode overlap reduction function; no uncertainty or derivation is given. They are used in the PLS/APLS computations for PTA vector modes.
  • PTA white noise parameters (σ, Δt, T) = σ=100 ns, Δt=20 yr^-1, T=15 yr
    Section 2.7.1 sets identical white noise for all pulsars using these values. They determine the PTA sensitivity curves and are chosen rather than derived from the catalog.
  • SNR thresholds and observation times = SNRth=1 (ground/PTA), 10 (LISA); T=1 yr, 3 yr, 15 yr
    The PLS and APLS curves depend on these user-set choices (Section 4). They are not fitted, but they set the quoted minimum amplitudes.
assumptions (5)
  • standard math Standard SGWB cross-correlation formalism (Eqs. 2.2, 2.19, 2.38) holds.
    The paper relies on the established relationship between detector strain correlation, overlap reduction functions, and SNR as in Thrane & Romano (2013) and Allen & Romano (1999), without re-deriving it.
  • domain assumption The SGWB is stationary, Gaussian, unpolarized, and isotropic for PLS/APLS definitions.
    Stated in Section 2.5 and assumed in the definition of PLS. Anisotropic cases are treated via the angular power spectrum Cℓ without relaxing the Gaussian assumption.
  • domain assumption All pulsars have identical white timing noise with the PSD in Eq. (2.74).
    Section 2.7.1. This simplification omits red noise, spin-down, and pulsar-specific cadence, as acknowledged by the authors, and directly sets PTA sensitivity forecasts.
  • standard math The XYZ-to-AET rotation matrix in Eq. (2.24) is valid for equal arm lengths and identical noise.
    The paper applies this transformation for LISA and triangular ET, citing Flauger et al. (2021). Real configurations with unequal arms would require correction, as cited in [61,62].
  • domain assumption The detector coordinates listed in Table 15 and the NANOGrav catalog [52] are correct and representative.
    All ground-based results depend on the assumed positions/orientations in Appendix A; PTA results depend on the specific 60-pulsar NANOGrav catalog. Errors in these inputs would propagate to all outputs.

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Cite this review

Pith. "Pith review of \texttt{GWBird}: a toolkit for the characterization of the Stochastic Gravitational Wave Background for Ground, Space, and Pulsar Timing Array detectors." pith.science (2026). https://pith.science/paper/VFHP7I3G

@misc{pith2026250715791,
  author       = {Pith},
  title        = {Pith review of: \textttGWBird: a toolkit for the characterization of the Stochastic Gravitational Wave Background for Ground, Space, and Pulsar Timing Array detectors},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/VFHP7I3G}},
  note         = {Machine review of arXiv:2507.15791}
}
read the original abstract

The detection of the Stochastic Gravitational Wave Background (SGWB) is one of the most challenging tasks for both current and next-generation detectors. Successfully distinguishing the SGWB from instrumental noise and environmental effects requires accurate and flexible analysis tools capable of detecting the signal and determining its origin. In this paper, we introduce a unified framework and a user-friendly tool for SGWB characterization: \texttt{GWBird} (Gravitational Wave Background Inventory of Response functions for Detectors). This code enables the computation of overlap reduction functions (ORFs), power-law integrated sensitivity curves (PLS), angular response functions, and angular PLS (APLS). It supports the full range of gravitational wave polarization modes (tensor, scalar, and vector), allowing for the characterization of both isotropic and anisotropic SGWB components for all the polarizations. Additionally, the code includes functions for circular polarization characterization, which is particularly relevant for probing parity-violating signals. The framework integrates analyses for ground-based, space-based, and Pulsar Timing Array (PTA) detectors, offering a versatile framework for SGWB analysis. The \texttt{GWBird} code is publicly available at:~\github{https://github.com/ilariacaporali/GWBird}

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Forward citations

Cited by 3 Pith papers

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Reference graph

Works this paper leans on

100 extracted references · 7 canonical work pages · cited by 3 Pith papers

  1. [69]

    Chamberlin and X

    S.J. Chamberlin and X. Siemens,Stochastic backgrounds in alternative theories of gravity: overlap reduction functions for pulsar timing arrays, Phys. Rev. D85 (2012) 082001 [1111.5661]

    arXiv 2012

  2. [70]

    da Silva Alves and M

    M.E. da Silva Alves and M. Tinto,Pulsar Timing Sensitivities to Gravitational Waves from Relativistic Metric Theories of Gravity, Phys. Rev. D83 (2011) 123529 [1102.4824]

    arXiv 2011

  3. [85]

    Kato and J

    R. Kato and J. Soda,Probing circular polarization in stochastic gravitational wave background with pulsar timing arrays, Phys. Rev. D93 (2016) 062003 [1512.09139]

    arXiv 2016

  4. [74]

    Babak, M

    S. Babak, M. Falxa, G. Franciolini and M. Pieroni,Forecasting the sensitivity of pulsar timing arrays to gravitational wave backgrounds, Phys. Rev. D110 (2024) 063022 [2404.02864]. – 52 –

    arXiv 2024

  5. [1]

    LIGO Scientific, Virgo collaboration, Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett.116 (2016) 061102 [1602.03837]

    arXiv 2016

  6. [2]

    LIGO Scientific, Virgo collaboration, Tests of general relativity with GW150914, Phys. Rev. Lett.116 (2016) 221101 [1602.03841]

    arXiv 2016

  7. [3]

    LIGO Scientific, Virgo collaboration, GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence, Phys. Rev. Lett.116 (2016) 241103 [1606.04855]

    arXiv 2016

  8. [4]

    LIGO Scientific, VIRGO collaboration, GW170104: Observation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2, Phys. Rev. Lett.118 (2017) 221101 [1706.01812]

    arXiv 2017

Show all 100 references
  1. [5]

    LIGO Scientific, Virgo collaboration, GW170814: A Three-Detector Observation of Gravitational Waves from a Binary Black Hole Coalescence, Phys. Rev. Lett.119 (2017) 141101 [1709.09660]

    2017 arXiv

  2. [6]

    LIGO Scientific, Virgo collaboration, GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett.119 (2017) 161101 [1710.05832]

    2017 arXiv

  3. [7]

    LIGO Scientific, Virgo collaboration, GW190521: A Binary Black Hole Merger with a Total Mass of150M⊙, Phys. Rev. Lett.125 (2020) 101102 [2009.01075]

    2020

  4. [8]

    NANOGra vcollaboration, The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys. J. Lett.951 (2023) L8 [2306.16213]

    2023 arXiv

  5. [9]

    Search for gravitational wave signals, Astron

    EPTA, InPTA: collaboration, The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals, Astron. Astrophys.678 (2023) A50 [2306.16214]

    2023 arXiv

  6. [10]

    Reardon et al.,Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array, Astrophys

    D.J. Reardon et al.,Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array, Astrophys. J. Lett.951 (2023) L6 [2306.16215]

    2023 arXiv

  7. [11]

    Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res

    H. Xu et al.,Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I, Res. Astron. Astrophys.23 (2023) 075024 [2306.16216]

    2023 arXiv

  8. [12]

    LISA collaboration, Laser Interferometer Space Antenna, arXiv e-prints (2017) [1702.00786]

    2017 arXiv

  9. [13]

    LISA collaboration, LISA Definition Study Report, 2402.07571

    arXiv

  10. [14]

    Maggiore et al.,Science Case for the Einstein Telescope, JCAP 03 (2020) 050 [1912.02622]

    M. Maggiore et al.,Science Case for the Einstein Telescope, JCAP 03 (2020) 050 [1912.02622]

    2020 arXiv

  11. [15]

    Reitze et al.,Cosmic Explorer: The U.S

    D. Reitze et al.,Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO, Bull. Am. Astron. Soc.51 (2019) 035 [1907.04833]

    2019 arXiv

  12. [16]

    Evans et al.,A Horizon Study for Cosmic Explorer: Science, Observatories, and Community, 2109.09882

    M. Evans et al.,A Horizon Study for Cosmic Explorer: Science, Observatories, and Community, 2109.09882

    arXiv

  13. [17]

    Branchesi et al.,Science with the Einstein Telescope: a comparison of different designs, JCAP 07 (2023) 068 [2303.15923]

    M. Branchesi et al.,Science with the Einstein Telescope: a comparison of different designs, JCAP 07 (2023) 068 [2303.15923]

    2023 arXiv

  14. [18]

    Abac et al.,The Science of the Einstein Telescope, 2503.12263

    A. Abac et al.,The Science of the Einstein Telescope, 2503.12263

    arXiv

  15. [19]

    Caporali, G

    I. Caporali, G. Capurri, W. Del Pozzo, A. Ricciardone and L. Valbusa Dall’Armi,Impact of correlated noise on the reconstruction of the stochastic gravitational wave background with Einstein Telescope, 2501.09057

    arXiv

  16. [20]

    Begnoni, S

    A. Begnoni, S. Anselmi, M. Pieroni, A. Renzi and A. Ricciardone,Detectability and Parameter Estimation for Einstein Telescope Configurations with GWJulia, 2506.21530. – 49 –

    arXiv

  17. [21]

    Littenberg, N

    T. Littenberg, N. Cornish, K. Lackeos and T. Robson,Global Analysis of the Gravitational Wave Signal from Galactic Binaries, Phys. Rev. D101 (2020) 123021 [2004.08464]

    2020 arXiv

  18. [22]

    Littenberg and N.J

    T.B. Littenberg and N.J. Cornish,Prototype global analysis of LISA data with multiple source types, Phys. Rev. D107 (2023) 063004 [2301.03673]

    2023 arXiv

  19. [23]

    M.L. Katz, N. Karnesis, N. Korsakova, J.R. Gair and N. Stergioulas,Efficient GPU-accelerated multisource global fit pipeline for LISA data analysis, Phys. Rev. D111 (2025) 024060 [2405.04690]

    2025 arXiv

  20. [24]

    S. Deng, S. Babak, M. Le Jeune, S. Marsat, E. Plagnol and A. Sartirana,Modular global-fit pipeline for LISA data analysis, 2501.10277

    arXiv

  21. [25]

    Strub, L

    S.H. Strub, L. Ferraioli, C. Schmelzbach, S.C. Stähler and D. Giardini,Global analysis of LISA data with Galactic binaries and massive black hole binaries, Phys. Rev. D110 (2024) 024005 [2403.15318]

    2024 arXiv

  22. [26]

    LISA Cosmology Working Group collaboration, Testing modified gravity at cosmological distances with LISA standard sirens, JCAP 07 (2019) 024 [1906.01593]

    2019 arXiv

  23. [27]

    Nishizawa, A

    A. Nishizawa, A. Taruya, K. Hayama, S. Kawamura and M.-a. Sakagami,Probing non-tensorial polarizations of stochastic gravitational-wave backgrounds with ground-based laser interferometers, Phys. Rev. D79 (2009) 082002 [0903.0528]

    2009 arXiv

  24. [28]

    Callister, A.S

    T. Callister, A.S. Biscoveanu, N. Christensen, M. Isi, A. Matas, O. Minazzoli et al., Polarization-based Tests of Gravity with the Stochastic Gravitational-Wave Background, Phys. Rev. X 7 (2017) 041058 [1704.08373]

    2017 arXiv

  25. [29]

    Amalberti, N

    L. Amalberti, N. Bartolo and A. Ricciardone,Sensitivity of third-generation interferometers to extra polarizations in the stochastic gravitational wave background, Phys. Rev. D105 (2022) 064033 [2105.13197]

    2022 arXiv

  26. [30]

    Hu, P.-P

    Y. Hu, P.-P. Wang, Y.-J. Tan and C.-G. Shao,Testing the polarization of gravitational wave background with LISA-TianQin network, 2209.07049

    arXiv

  27. [31]

    LIGO Scientific, Virgo collaboration, Search for Tensor, Vector, and Scalar Polarizations in the Stochastic Gravitational-Wave Background, Phys. Rev. Lett.120 (2018) 201102 [1802.10194]

    2018 arXiv

  28. [32]

    Valbusa Dall’Armi, A

    L. Valbusa Dall’Armi, A. Nishizawa, A. Ricciardone and S. Matarrese,Circular Polarization of the Astrophysical Gravitational Wave Background, Phys. Rev. Lett.131 (2023) 041401 [2301.08205]

    2023 arXiv

  29. [33]

    Alonso, C.R

    D. Alonso, C.R. Contaldi, G. Cusin, P.G. Ferreira and A.I. Renzini,Noise angular power spectrum of gravitational wave background experiments, Phys. Rev. D101 (2020) 124048 [2005.03001]

    2020 arXiv

  30. [34]

    Mentasti and M

    G. Mentasti and M. Peloso,ET sensitivity to the anisotropic Stochastic Gravitational Wave Background, JCAP 03 (2021) 080 [2010.00486]

    2021 arXiv

  31. [35]

    LISA Cosmology Working Group collaboration, Probing anisotropies of the Stochastic Gravitational Wave Background with LISA, JCAP 11 (2022) 009 [2201.08782]

    2022 arXiv

  32. [36]

    Schulze, L

    F. Schulze, L. Valbusa Dall’Armi, J. Lesgourgues, A. Ricciardone, N. Bartolo, D. Bertacca et al.,GW_CLASS: Cosmological Gravitational Wave Background in the cosmic linear anisotropy solving system, JCAP 10 (2023) 025 [2305.01602]

    2023 arXiv

  33. [37]

    Mentasti, C

    G. Mentasti, C. Contaldi and M. Peloso,Prospects for detecting anisotropies and polarization of the stochastic gravitational wave background with ground-based detectors, JCAP 08 (2023) 053 [2304.06640]

    2023 arXiv

  34. [38]

    Cusin, C

    G. Cusin, C. Pitrou and J.-P. Uzan,Anisotropy of the astrophysical gravitational wave – 50 – background: Analytic expression of the angular power spectrum and correlation with cosmological observations, Phys. Rev. D96 (2017) 103019 [1704.06184]

    2017 arXiv

  35. [39]

    Jenkins and M

    A.C. Jenkins and M. Sakellariadou,Anisotropies in the stochastic gravitational-wave background: Formalism and the cosmic string case, Phys. Rev. D98 (2018) 063509 [1802.06046]

    2018 arXiv

  36. [40]

    Bertacca, A

    D. Bertacca, A. Ricciardone, N. Bellomo, A.C. Jenkins, S. Matarrese, A. Raccanelli et al., Projection effects on the observed angular spectrum of the astrophysical stochastic gravitational wave background, Phys. Rev. D101 (2020) 103513 [1909.11627]

    2020 arXiv

  37. [41]

    Bellomo, D

    N. Bellomo, D. Bertacca, A.C. Jenkins, S. Matarrese, A. Raccanelli, T. Regimbau et al., CLASS_GWB: robust modeling of the astrophysical gravitational wave background anisotropies, JCAP 06 (2022) 030 [2110.15059]

    2022 arXiv

  38. [42]

    Capurri, A

    G. Capurri, A. Lapi, L. Boco and C. Baccigalupi,Searching for Anisotropic Stochastic Gravitational-wave Backgrounds with Constellations of Space-based Interferometers, Astrophys. J. 943 (2023) 72 [2212.06162]

    2023 arXiv

  39. [43]

    Capurri, A

    G. Capurri, A. Lapi, C. Baccigalupi, L. Boco, G. Scelfo and T. Ronconi,Intensity and anisotropies of the stochastic gravitational wave background from merging compact binaries in galaxies, JCAP 11 (2021) 032 [2103.12037]

    2021 arXiv

  40. [44]

    Allen, D

    B. Allen, D. Agarwal, J.D. Romano and S. Valtolina,Source anisotropies and pulsar timing arrays, Phys. Rev. D110 (2024) 123507 [2406.16031]

    2024 arXiv

  41. [45]

    Valbusa Dall’Armi, A

    L. Valbusa Dall’Armi, A. Mierna, S. Matarrese and A. Ricciardone,Adiabatic or Non-Adiabatic? Unraveling the Nature of Initial Conditions in the Cosmological Gravitational Wave Background, 2307.11043

  42. [46]

    Valbusa Dall’Armi, A

    L. Valbusa Dall’Armi, A. Mierna, S. Matarrese and A. Ricciardone,Inflationary initial conditions for the cosmological gravitational wave background, JCAP 07 (2024) 043 [2407.09405]

    2024 arXiv

  43. [47]

    Mierna, S

    A. Mierna, S. Matarrese, N. Bartolo and A. Ricciardone,Non-linear effects on the Cosmological Gravitational Wave Background anisotropies, 2412.15654

    arXiv

  44. [48]

    Ricciardone, L.V

    A. Ricciardone, L.V. Dall’Armi, N. Bartolo, D. Bertacca, M. Liguori and S. Matarrese, Cross-Correlating Astrophysical and Cosmological Gravitational Wave Backgrounds with the Cosmic Microwave Background, Phys. Rev. Lett.127 (2021) 271301 [2106.02591]

    2021 arXiv

  45. [49]

    Valbusa Dall’Armi,Gravitational Wave Background Anisotropies: Probing Early and Late Universe Cosmology with Interferometers, Ph.D

    L. Valbusa Dall’Armi,Gravitational Wave Background Anisotropies: Probing Early and Late Universe Cosmology with Interferometers, Ph.D. thesis, Università degli Studi di Padova, Italy, Padua U., 2024

    2024

  46. [50]

    Capurri, A

    G. Capurri, A. Lapi and C. Baccigalupi,Detectability of the Cross-Correlation between CMB Lensing and Stochastic GW Background from Compact Object Mergers, Universe 8 (2022) 160 [2111.04757]

    2022 arXiv

  47. [51]

    Thrane and J.D

    E. Thrane and J.D. Romano,Sensitivity curves for searches for gravitational-wave backgrounds, Phys. Rev. D88 (2013) 124032 [1310.5300]

    2013 arXiv

  48. [52]

    10.5281/zenodo.14773896

    The NANOGrav Collaboration,The nanograv 15-year data set, 2025. 10.5281/zenodo.14773896

    2025 doi

  49. [53]

    Depta, V

    P.F. Depta, V. Domcke, G. Franciolini and M. Pieroni,Pulsar timing array sensitivity to anisotropies in the gravitational wave background, 2407.14460

    arXiv

  50. [54]

    Allen and J.D

    B. Allen and J.D. Romano,Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities, Phys. Rev. D59 (1999) 102001 [gr-qc/9710117]

    1999 arXiv

  51. [55]

    Romano and N.J

    J.D. Romano and N.J. Cornish,Detection methods for stochastic gravitational-wave backgrounds: a unified treatment, Living Rev. Rel.20 (2017) 2 [1608.06889]. – 51 –

    2017 arXiv

  52. [56]

    Maggiore,Gravitational Waves

    M. Maggiore,Gravitational Waves. Vol. 1: Theory and Experiments, Oxford University Press (2007), 10.1093/acprof:oso/9780198570745.001.0001

    2007

  53. [57]

    Christensen,Measuring the stochastic gravitational radiation background with laser interferometric antennas, Phys

    N. Christensen,Measuring the stochastic gravitational radiation background with laser interferometric antennas, Phys. Rev. D46 (1992) 5250

    1992

  54. [58]

    Finn, S.L

    L.S. Finn, S.L. Larson and J.D. Romano,Detecting a Stochastic Gravitational-Wave Background: The Overlap Reduction Function, Phys. Rev. D79 (2009) 062003 [0811.3582]

    2009 arXiv

  55. [59]

    Isi, A.J

    M. Isi, A.J. Weinstein, C. Mead and M. Pitkin,Detecting Beyond-Einstein Polarizations of Continuous Gravitational Waves, Phys. Rev. D91 (2015) 082002 [1502.00333]

    2015 arXiv

  56. [60]

    Chatziioannou, M

    K. Chatziioannou, M. Isi, C.-J. Haster and T.B. Littenberg,Morphology-independent test of the mixed polarization content of transient gravitational wave signals, Phys. Rev. D104 (2021) 044005 [2105.01521]

    2021 arXiv

  57. [61]

    Hartwig, M

    O. Hartwig, M. Lilley, M. Muratore and M. Pieroni,Stochastic gravitational wave background reconstruction for a nonequilateral and unequal-noise LISA constellation, Phys. Rev. D107 (2023) 123531 [2303.15929]

    2023 arXiv

  58. [62]

    J. Kume, M. Peloso, M. Pieroni and A. Ricciardone,Assessing the Impact of Unequal Noises and Foreground Modeling on SGWB Reconstruction with LISA, 2410.10342

    arXiv

  59. [63]

    Flauger, N

    R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ricciardone and J. Torrado,Improved reconstruction of a stochastic gravitational wave background with LISA, JCAP 01 (2021) 059 [2009.11845]

    2021 arXiv

  60. [64]

    Seto and A

    N. Seto and A. Taruya,Measuring a Parity Violation Signature in the Early Universe via Ground-based Laser Interferometers, Phys. Rev. Lett.99 (2007) 121101 [0707.0535]

    2007 arXiv

  61. [65]

    Crowder, R

    S.G. Crowder, R. Namba, V. Mandic, S. Mukohyama and M. Peloso,Measurement of Parity Violation in the Early Universe using Gravitational-wave Detectors, Phys. Lett. B726 (2013) 66 [1212.4165]

    2013 arXiv

  62. [66]

    Belgacem,Analytical results for the power-law sensitivity curve of stochastic gravitational-wave backgrounds, 2503.06356

    E. Belgacem,Analytical results for the power-law sensitivity curve of stochastic gravitational-wave backgrounds, 2503.06356

    arXiv

  63. [67]

    Omiya and N

    H. Omiya and N. Seto,Searching for anomalous polarization modes of the stochastic gravitational wave background with LISA and Taiji, Phys. Rev. D102 (2020) 084053 [2010.00771]

    2020 arXiv

  64. [68]

    Hazboun, J.D

    J.S. Hazboun, J.D. Romano and T.L. Smith,Realistic sensitivity curves for pulsar timing arrays, Phys. Rev. D100 (2019) 104028 [1907.04341]

    2019 arXiv

  65. [71]

    Bernardo and K.-W

    R.C. Bernardo and K.-W. Ng,Pulsar and cosmic variances of pulsar timing-array correlation measurements of the stochastic gravitational wave background, JCAP 11 (2022) 046 [2209.14834]

    2022 arXiv

  66. [72]

    Hellings and G.s

    R.w. Hellings and G.s. Downs,UPPER LIMITS ON THE ISOTROPIC GRAVITATIONAL RADIATION BACKGROUND FROM PULSAR TIMING ANALYSIS, Astrophys. J. Lett. 265 (1983) L39

    1983

  67. [73]

    Lee, F.A

    K.J. Lee, F.A. Jenet and R.H. Price,Pulsar timing as a probe of non-einsteinian polarizations of gravitational waves, The Astrophysical Journal685 (2008) 1304

    2008

  68. [75]

    LISA collaboration, Astrophysics with the Laser Interferometer Space Antenna, Living Rev. Rel. 26 (2023) 2 [2203.06016]

    2023 arXiv

  69. [76]

    Buscicchio, J

    R. Buscicchio, J. Torrado, C. Caprini, G. Nardini, N. Karnesis, M. Pieroni et al.,Stellar-mass black-hole binaries in LISA: characteristics and complementarity with current-generation interferometers, JCAP 01 (2025) 084 [2410.18171]

    2025 arXiv

  70. [77]

    Pozzoli, R

    F. Pozzoli, R. Buscicchio, A. Klein, V. Korol, A. Sesana and F. Haardt,Cyclostationary signals in LISA: A practical application to Milky Way satellites, Phys. Rev. D111 (2025) 063005 [2410.08274]

    2025 arXiv

  71. [78]

    Rel.26 (2023) 5 [2204.05434]

    LISA Cosmology Working Group collaboration, Cosmology with the Laser Interferometer Space Antenna, Living Rev. Rel.26 (2023) 5 [2204.05434]

    2023 arXiv

  72. [79]

    Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update, JCAP 03 (2020) 024 [1910.13125]

    C. Caprini et al.,Detecting gravitational waves from cosmological phase transitions with LISA: an update, JCAP 03 (2020) 024 [1910.13125]

    2020 arXiv

  73. [80]

    LISA Cosmology Working Group collaboration, Gravitational waves from first-order phase transitions in LISA: reconstruction pipeline and physics interpretation, JCAP 10 (2024) 020 [2403.03723]

    2024 arXiv

  74. [81]

    LISA Cosmology Working Group collaboration, Gravitational waves from inflation in LISA: reconstruction pipeline and physics interpretation, JCAP 11 (2024) 032 [2407.04356]

    2024 arXiv

  75. [82]

    Domcke, J

    V. Domcke, J. Garcia-Bellido, M. Peloso, M. Pieroni, A. Ricciardone, L. Sorbo et al., Measuring the net circular polarization of the stochastic gravitational wave background with interferometers, JCAP 05 (2020) 028 [1910.08052]

    2020 arXiv

  76. [83]

    Mingarelli, T

    C.M.F. Mingarelli, T. Sidery, I. Mandel and A. Vecchio,Characterizing gravitational wave stochastic background anisotropy with pulsar timing arrays, Phys. Rev. D88 (2013) 062005 [1306.5394]

    2013 arXiv

  77. [84]

    Gair, J.D

    J. Gair, J.D. Romano, S. Taylor and C.M.F. Mingarelli,Mapping gravitational-wave backgrounds using methods from CMB analysis: Application to pulsar timing arrays, Phys. Rev. D 90 (2014) 082001 [1406.4664]

    2014 arXiv

  78. [86]

    Sato-Polito and M

    G. Sato-Polito and M. Kamionkowski,Pulsar-timing measurement of the circular polarization of the stochastic gravitational-wave background, Phys. Rev. D106 (2022) 023004 [2111.05867]

    2022 arXiv

  79. [87]

    N.M.J. Cruz, A. Malhotra, G. Tasinato and I. Zavala,Measuring the circular polarization of gravitational waves with pulsar timing arrays, Phys. Rev. D110 (2024) 103505 [2406.04957]

    2024 arXiv

  80. [88]

    Janssen et al.,Gravitational wave astronomy with the SKA, PoS AASKA14 (2015) 037 [1501.00127]

    G. Janssen et al.,Gravitational wave astronomy with the SKA, PoS AASKA14 (2015) 037 [1501.00127]

    2015 arXiv

  81. [89]

    Van Rossum and F.L

    G. Van Rossum and F.L. Drake,Python 3 Reference Manual, CreateSpace, Scotts Valley, CA (2009)

    2009

  82. [90]

    Harris et al.,Array programming with NumPy, Nature 585 (2020) 357

    C.R. Harris et al.,Array programming with NumPy, Nature 585 (2020) 357

    2020

  83. [91]

    Zonca, L

    A. Zonca, L. Singer, D. Lenz, M. Reinecke, C. Rosset, E. Hivon et al.,Healpy: equal area pixelization and spherical harmonics transforms for data on the sphere in python, Journal of Open Source Software4 (2019) 1298

    2019

  84. [92]

    Górski, E

    K.M. Górski, E. Hivon, A.J. Banday, B.D. Wandelt, F.K. Hansen, M. Reinecke et al., HEALPix: A Framework for High-Resolution Discretization and Fast Analysis of Data Distributed on the Sphere, The Astrophysical Journal622 (2005) 759 [arXiv:astro-ph/0409513]. – 53 –

    2005 arXiv

  85. [93]

    Hunter,Matplotlib: A 2d graphics environment, Computing in Science & Engineering9 (2007) 90

    J.D. Hunter,Matplotlib: A 2d graphics environment, Computing in Science & Engineering9 (2007) 90

    2007

  86. [94]

    Astropy Collaboration,Astropy: A community python package for astronomy, Astronomy & Astrophysics 558 (2013) A33 [1307.6212]

    2013 arXiv

  87. [95]

    Astropy Collaboration,The Astropy Project: Building an Open-science Project and Status of the v2.0 Core Package, The Astrophysical Journal156 (2018) 123 [1801.02634]

    2018 arXiv

  88. [96]

    Astropy Collaboration,The Astropy Project: Sustaining and Growing a Community-oriented Open-source Project and the Latest Major Release (v5.0) of the Core Package, The Astrophysical Journal 935 (2022) 167 [2206.14220]

    2022 arXiv

  89. [97]

    The mpmath development team,mpmath: a Python library for arbitrary-precision floating-point arithmetic (version 1.3.0), 2023

    2023

  90. [98]

    Virtanen et al.,SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17 (2020) 261

    P. Virtanen et al.,SciPy 1.0: Fundamental Algorithms for Scientific Computing in Python, Nature Methods 17 (2020) 261

    2020

  91. [99]

    Luo et al.,PINT: A Modern Software Package for Pulsar Timing, The Astrophysical Journal 911 (2021) 45 [2012.00074]

    J. Luo et al.,PINT: A Modern Software Package for Pulsar Timing, The Astrophysical Journal 911 (2021) 45 [2012.00074]

    2021 arXiv

  92. [100]

    Susobhanan et al.,PINT: Maximum-likelihood Estimation of Pulsar Timing Noise Parameters, The Astrophysical Journal971 (2024) 150 [2405.01977]

    A. Susobhanan et al.,PINT: Maximum-likelihood Estimation of Pulsar Timing Noise Parameters, The Astrophysical Journal971 (2024) 150 [2405.01977]. – 54 –

    2024 arXiv

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