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Pulsar Timing Array Sensitivity to Anisotropy: Empirical Sensitivity Curves, Scaling Relations, and the Multi-Resolution Pixel Basis

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read A pulsar timing array's sensitivity to directional structure in the gravitational wave background scales roughly as the number of pulsars to the 0.8 power, barely moves with timing precision, and gains little from extending a 16-year run…

desk verdict Solid empirical scaling laws for PTA anisotropy sensitivity, but the headline exponents are fit to the Fisher diagonal while the paper's own condition-number analysis shows off-diagonal terms dominate. read the letter →

arxiv 2608.09781 v1 pith:N7WRPHFA submitted 2026-08-10 astro-ph.IM astro-ph.HE

Taha T. Moursy , Nihan S. Pol , Gabriella Agazie , Nikita Agarwal , Akash Anumarlapudi , Anne M. Archibald , Zaven Arzoumanian , Anjana Ashok
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Jeremy G. Baier Paul T. Baker Bence Bécsy Laura Blecha Adam Brazier Paul R. Brook Sarah Burke-Spolaor Rand Burnette Robin Case J. Andrew Casey-Clyde Maria Charisi Shami Chatterjee Tyler Cohen James M. Cordes Neil J. Cornish Fronefield Crawford H. Thankful Cromartie Kathryn Crowter Megan E. DeCesar Paul B. Demorest Heling Deng Lankeswar Dey Timothy Dolch Graham M. Doskoch Elizabeth C. Ferrara William Fiore Emmanuel Fonseca Gabriel E. Freedman Emiko C. Gardiner Nate Garver-Daniels Peter A. Gentile Kyle A. Gersbach Joseph Glaser Deborah C. Good Kayhan Gültekin Aiden Gundersen C. J. Harris Jeffrey S. Hazboun Ross J. Jennings Aaron D. Johnson Megan L. Jones David L. Kaplan Anala K. Sreekumar Luke Zoltan Kelley Matthew Kerr Joey S. Key Nima Laal Michael T. Lam William G. Lamb Bjorn Larsen T. Joseph W. Lazio Natalia Lewandowska Tingting Liu Duncan R. Lorimer Jing Luo Ryan S. Lynch Chung-Pei Ma Dustin R. Madison Ashley Martsen Cayenne Matt Alexander McEwen James W. McKee Maura A. McLaughlin Natasha McMann Bradley W. Meyers Patrick M. Meyers Matthew T. Miles Chiara M. F. Mingarelli Andrea Mitridate Cherry Ng David J. Nice Shania A. Nichols Stella Koch Ocker Daniel J. Oliver Ken D. Olum Timothy T. Pennucci Benetge B. P. Perera Polina Petrov Henri A. Radovan Scott M. Ransom Paul S. Ray Joseph D. Romano Jessie C. Runnoe Alexander Saffer Shashwat C. Sardesai Ann Schmiedekamp Carl Schmiedekamp Kai Schmitz Levi Schult Brent J. Shapiro-Albert Xavier Siemens Joseph Simon Sophia V. Sosa Fiscella Ingrid H. Stairs Daniel R. Stinebring Kevin Stovall Abhimanyu Susobhanan Joseph K. Swiggum Jacob Taylor Stephen R. Taylor Mercedes S. Thompson Jacob E. Turner Michele Vallisneri Rutger van Haasteren Sarah J. Vigeland Haley M. Wahl Kevin P. Wilson Caitlin A. Witt David Wright Olivia Young
This is my paper · ORCID
classification astro-ph.IMastro-ph.HE
keywords gravitationalwavebackgroundpulsartimingarraysanisotropyFisherinformationmatrixsensitivityscalingHEALPixmulti-resolutionpixelbasis
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

This paper asks how a pulsar timing array's ability to map direction-dependent structure in the gravitational wave background depends on the array's size, timing precision, observing frequency, and angular resolution. Using simulations built from a real 15-year dataset, it finds that sensitivity to anisotropy scales roughly as the number of pulsars to the 0.8 power, only weakly with per-pulse timing error, and steeply with the angular size of the anisotropy. It also forecasts that extending observations from 16 to 30 years, by itself, barely improves anisotropy sensitivity except at the lowest frequencies, where newly matured pulsar baselines add weight. These scaling laws give array designers a concrete target: adding pulsars and filling sky coverage buys far more anisotropy sensitivity than waiting longer. The paper closes by introducing a multi-resolution pixel basis that places smaller pixels where the array is more sensitive, and demonstrates that injected sky maps are recovered accurately with it.

What carries the argument

The central object is the cross-correlation Fisher information matrix $\mathbf{M}=\mathbf{R}^T\mathbf{C}^{-1}\mathbf{R}$, where $\mathbf{R}$ maps sky power to pulsar-pair correlations and $\mathbf{C}$ is the covariance matrix of those correlations, including, in the pair-covariant per-frequency optimal statistic, the covariance induced by an isotropic gravitational wave background. The sensitivity measure is $\sqrt{\mathrm{diag}(\mathbf{M})}$, i.e. the reciprocal of the diagonal-approximated uncertainty $\sqrt{1/\mathrm{diag}(\mathbf{M})}$, which deliberately neglects covariances between pixels or spherical harmonics. The angular-scale scaling is driven by the effective number of pulsars that fall inside the characteristic beam of each spherical harmonic, which is why the exponent depends on orientation $m$ and on sky location. The multi-resolution basis replaces the single-resolution HEALPix normalization $1/N_\mathrm{pix}$ with $\Delta\Omega_k/4\pi$, so pixel area can vary across the sky and resolution can be spent where pulsar density is high.

What would settle it

Recompute the scaling-law fits for $N_\mathrm{psr}$, $\delta t$, and $\Delta\Omega$ using the full inverse Fisher matrix $\mathbf{M}^{-1}$ (with SVD regularization, since the paper reports condition numbers $\sim10^{14}$ to $10^{18}$) instead of diagonal-only inversion; if the fitted exponents move by more than the quoted uncertainties, the reported scaling relations are artifacts of the diagonal approximation.

Watch

Extended reading notes

Core claim

The paper's central claim is that, for cross-correlation-based anisotropy searches, PTA sensitivity—quantified by $\sqrt{\mathrm{diag}(\mathbf{M})}$, the reciprocal of the diagonal-approximated per-pixel uncertainty—follows empirical power laws: $\sqrt{\mathrm{diag}(\mathbf{M})}\propto N_\mathrm{psr}^{0.8\pm0.1}$, $\delta t^{-0.08\pm0.01}$, and $f_\mathrm{GW}^{-0.12\pm0.04}$, with spherical-harmonic sensitivity scaling as $\Delta\Omega^{1.58\pm0.03}$ for angular scales corresponding to $8\le\ell\le20$ at $m=0$ and roughly $\Delta\Omega^{1.5}$ to $\Delta\Omega^{2.1}$ depending on orientation. The paper also claims that increasing observation duration from 16 to 30 years has negligible median effect on anisotropy sensitivity except in the lowest frequency bins, because the dominant growth there comes from the number of pulsars whose baselines are long enough to reach those bins. Finally, it claims that a multi-resolution HEALPix pixel basis, with response matrix $R_{ab}^k=(\Delta\Omega_k/4\pi)R_{ab}(\hat{\Omega}_k)$, recovers injected isotropic and anisotropic backgrounds to numerical precision in analytic tests, allowing resolution to be concentrated where pulsar density is high.

Load-bearing premise

The load-bearing premise is that a pulsar array's sensitivity to anisotropy can be quoted from the diagonal of its Fisher information matrix alone, ignoring correlations between neighboring directions on the sky; the paper itself reports near-singular matrices from those correlations, so including them could lower the absolute sensitivities and shift the fitted scaling exponents.

Editorial extensions

If this is right

  • Adding pulsars is the strongest lever: doubling $N_\mathrm{psr}$ raises anisotropy sensitivity by roughly $2^{0.8}\approx1.7$.
  • Reducing per-pulse TOA errors by a factor of 10 improves anisotropy sensitivity by only about 20%, so spending on more pulsars usually beats spending on pushing $\delta t$ down.
  • Extending from 16 to 30 years without adding pulsars gives negligible median sensitivity gain except at the lowest frequency bins, where pulsars whose baselines have matured first enter the analysis.
  • Sensitivity to an anisotropy of angular scale $\Delta\Omega$ grows roughly as $\Delta\Omega^{1.6}$ to $\Delta\Omega^{2.1}$, so smaller-scale features are substantially harder to detect and detections will be sky-position dependent by a factor of 2–3.
  • At the lowest frequencies, neglecting the covariance between pulsar-pair correlations overestimates sensitivity by about a factor of two, so forecasts built on the covariance-free approximation are optimistic there.

Reading between the lines

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

  • Because the paper's sensitivity measure drops off-diagonal Fisher terms, and because the paper itself reports condition numbers $\sim10^{14}$ to $10^{18}$, a direct follow-up would be to repeat the scaling-law fits with a regularized full inverse Fisher matrix; the absolute sensitivities and possibly the fitted exponents could shift.
  • The factor-of-2–3 sky-location dependence suggests a concrete design rule: new pulsars placed in low-density sky regions improve angular resolution at that location more than anywhere else, so sky coverage and pulsar count should be optimized jointly rather than separately.
  • The multi-resolution basis's ability to trade localization for uncertainty points toward a data-driven mesh-selection scheme in which pixel sizes are chosen from the local sensitivity map, extending the paper's planned reversible-jump MCMC approach.
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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. This paper develops empirical scaling relations for PTA sensitivity to anisotropy in the gravitational-wave background, using the per-frequency optimal statistic (PFOS) to build Fisher information matrices in both a HEALPix pixel basis and a spherical harmonic basis. The authors simulate datasets based on the NANOGrav 15-year data and report that their sensitivity measure, sqrt(diag(M)), scales approximately as Npsr^0.8, delta-t^-0.08, f_GW^-0.12, and Delta-Omega^1.58-2.1 depending on the ranges of ell and m. They also simulate a 30-year NANOGrav-like array and find that increasing the observing baseline alone has negligible effect on sensitivity except at the lowest frequencies. Finally, they introduce a multi-resolution HEALPix pixel basis and demonstrate it with analytical injection-recovery tests.

Significance. If the reported scaling relations are robust, they provide practical guidance for PTA array design and for forecasting sensitivity to anisotropic backgrounds, complementing earlier work by Siemens et al. (2013), Pol et al. (2022), and Depta et al. (2024). The paper's strengths are its realistic simulation setup, explicit comparison with hasasia in the isotropic limit, discussion of pair covariance, and public availability of the analysis code as a fork of Defiant. The multi-resolution pixel basis is a promising idea, though it is presented as a proof of concept rather than a complete method. The main limitation is that the headline exponents are derived from a diagonal-only approximation of the Fisher matrix, which the paper's own condition-number analysis shows to be a very strong simplification; this needs to be addressed before the scaling laws can be taken as general statements about PTA anisotropy sensitivity.

major comments (3)
  1. [Section 2.1 / Section 5] The central scaling laws in Eqs. (13)-(15) and Figure 4 are based on sqrt(diag(M)) (equivalently 1/sqrt(diag(M)) under the diagonal approximation), which neglects covariances between basis functions. Section 5 reports that the full pixel-basis Fisher matrix has condition numbers O(10^14) for uniform pulsars and O(10^18) for the NANOGrav distribution, and that zeroing the off-diagonal elements reduces this to O(1). This shows that the off-diagonal structure dominates the Fisher matrix, so the true marginal uncertainty sqrt(diag(M^{-1})) can be very different from 1/sqrt(diag(M)). The authors should recompute the scaling exponents using sqrt(diag(M^{-1})) with an appropriate regularization or pseudo-inverse, or otherwise demonstrate that the exponents are robust to including the off-diagonal covariances. If the off-diagonal structure changes with Npsr, delta-t, or ell, the reported scaling may describe only the diagonal radiometer statistic and not the sensitivity of a full anisotropy search. At minimum, the abstract and conclusions should explicitly qualify that the scaling laws apply to the diagonal-only approximation.
  2. [Section 3.2 / Figure 4] The angular scaling exponent is derived from a non-representative subset of spherical harmonic coefficients: the headline slope 1.58 uses only m=0 with 8<=ell<=20, while the right panel of Figure 4 shows a strong dependence on m, with exponents ranging roughly from 1.5 to 2.2. Because m=0 corresponds to spherical harmonics concentrated near the poles, where the NANOGrav pulsar density is low, a single power law fitted to m=0 may not be a meaningful summary of the angular scaling. The authors should provide an explicit aggregation rule over m and ell, and report the goodness of fit of the power-law model over the chosen ell range. The abstract's range "Delta-Omega^1.6-Delta-Omega^2.1" should be tied directly to the chosen statistic and to the scatter shown in Figure 4.
  3. [Section 2.2 / Section 3.2] There is an inconsistency in the text about which quantity is used for sensitivity curves versus scaling-law fits. Section 2.2 states that uncertainties are "the square root of the diagonal elements of M^{-1}", while Section 3.2 says the scaling laws are fit to sqrt(diag(M)). These two quantities are equal only if M is diagonal, which Section 5 argues is not the case. The authors should clarify the exact quantity used for each figure and analysis, and ensure that the sensitivity curves (e.g., Figure 1) and the scaling-law fits are based on the same definition. This clarification is important for interpreting the reported exponents and for comparing them with prior scaling relations.
minor comments (6)
  1. [Section 3.1 / Figure 1] Please state the HEALPix Nside used for the pixel-basis sensitivity curves and for the scaling-law fits in Section 3.2; this information is needed to reproduce the results and to assess the condition-number discussion in Section 5.
  2. [Abstract / Section 3.2] The abstract reports "Delta-Omega^1.6-Delta-Omega^2.1" while the text reports 1.58 +/- 0.03 for m=0 and a range of roughly 1.5-2.2 for fixed m in Figure 4; please reconcile these numbers and state the precise range and selection criteria in the abstract.
  3. [Section 3.2] The text says the uncertainty on the scaling exponent is "1 standard deviation" but the fits use asymmetric error bars based on the 25th-75th percentiles; please clarify how the standard deviation is obtained from the percentile-based errors or use consistent error reporting throughout.
  4. [Figure 5] The y-axis of Figure 5 is restricted and the text notes that some changes extend beyond the range; please consider showing the full distribution or stating explicitly the fraction of cases that fall outside the plotted range.
  5. [Table 1] The final row labeled "All (GWB)" should be explicitly identified in the table caption as the injected GWB parameters rather than a pulsar-specific noise parameter.
  6. [Section 2.1] The notation "p 1/diag(M)" is nonstandard and could be confused with the power vector P; please write 1/sqrt(diag(M)) or sqrt(diag(M^{-1})) explicitly, depending on which quantity is intended.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the scaling laws are empirical fits to simulated Fisher-matrix diagonals, cross-checked against independent hasasia and prior-work scalings; self-cited PFOS/MAPS are independently published tools, and the diag(M) approximation is an explicitly flagged limitation, not an equation-level identity.

full rationale

The paper's central claims (Eqs. 13-15 and the angular-scale exponent in Fig. 4) are empirical power-law fits to sqrt(diag(M)) computed from simulated datasets (Sec. 3.2). The input to the fit is the Fisher matrix M = R^T C^{-1} R, built from simulated cross-correlations and their covariance via PFOS; the fitted exponents are not present in the inputs by construction, so there is no fitted-input-called-prediction. The diag(M) rather than diag(M^{-1}) choice (Sec. 2.1) is an approximation, and the paper itself flags its consequences: Sec. 5 reports condition numbers O(10^14)-O(10^18) for full pixel-basis Fisher matrices and notes that zeroing off-diagonal elements reduces the condition number to O(1). That is a validity/robustness limitation, not circularity: the exponents describe the stated proxy, and the paper does not claim they are exact inverse-Fisher uncertainties. Self-citations (PFOS Gersbach et al. 2025; MAPS Pol et al. 2022) are tool and method citations with independently published derivations and code, and the results are externally benchmarked against hasasia (Fig. 1) and against Siemens et al. 2013, Pol et al. 2022, and Depta et al. 2024 in Sec. 4. No uniqueness theorem or ansatz is imported from the authors' prior work to force the choice of basis or scaling. The multi-resolution pixel basis is a new construction demonstrated by injection-recovery, not a renaming of a known result. The manuscript explicitly lists its own limitations (near-singular Fisher matrices, the need for further development of the basis), which supports a non-circular reading. The score of 1 reflects the presence of minor self-citations, not load-bearing circularity.

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

The central claims are empirical fits to simulations. The only numbers fitted are the power-law exponents; all other inputs, including pulsar positions, intrinsic red noise amplitudes, and the injected GWB amplitude, come from the NANOGrav 15-year dataset and prior literature. The axioms concern the representativeness of the simulations and the diagonal Fisher approximation. No new physical entities are introduced.

free parameters (4)
  • Npsr scaling exponent = 0.8 ± 0.1
    Fitted by log-log regression of sqrt(diag(M)) medians versus pulsar count across 17, 34, 51, and 67 pulsars.
  • TOA error scaling exponent = -0.08 ± 0.01
    Fitted across 100 ns, 500 ns, and 1 microsecond TOA uncertainties.
  • GW frequency scaling exponent = -0.12 ± 0.04
    Fitted across the lowest six frequency bins of the Fisher matrix values.
  • Angular scale scaling exponent = 1.58 ± 0.03 for m=0 and l=8-20; 1.6-2.1 across m values
    Fitted from spherical harmonic Fisher diagonals; only l=8-20 and m=0 used for the headline line, with m-dependent values shown in Figure 4.
assumptions (4)
  • domain assumption The per-frequency optimal statistic cross-correlations and their covariance matrix correctly describe PTA data.
    Used throughout Section 2.1 to build the Fisher matrix; based on Gersbach et al. 2025.
  • domain assumption Simulated arrays with NANOGrav 15-year pulsar positions and noise parameters are representative enough to yield universal scaling laws.
    Scaling relations are fit to simulations anchored to one array; Section 2.2 describes the injections.
  • ad hoc to paper Sensitivity can be characterized by sqrt(1/diag(M)) even though off-diagonal Fisher elements are large.
    Stated in Section 2.1; condition numbers in Section 5 show off-diagonal terms dominate the full matrix.
  • ad hoc to paper The angular scaling relation is adequately described by a single power law over l=8-20 with m=0, excluding low-l coefficients.
    Section 3.2 and Figure 4; lower l values are excluded because they follow a different, shallower evolution.

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Cite this review

Pith. "Pith review of Pulsar Timing Array Sensitivity to Anisotropy: Empirical Sensitivity Curves, Scaling Relations, and the Multi-Resolution Pixel Basis." pith.science (2026). https://pith.science/paper/N7WRPHFA

@misc{pith2026260809781,
  author       = {Pith},
  title        = {Pith review of: Pulsar Timing Array Sensitivity to Anisotropy: Empirical Sensitivity Curves, Scaling Relations, and the Multi-Resolution Pixel Basis},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/N7WRPHFA}},
  note         = {Machine review of arXiv:2608.09781}
}
abstract

We quantify pulsar timing array (PTA) sensitivity to anisotropy in the gravitational wave background using the cross-correlation based Fisher information matrix in the pixel and spherical harmonic bases. We use a set of simulations to empirically determine scaling relations of a PTA's sensitivity to anisotropy with the number of pulsars $N_\mathrm{psr}$ in the array, the error $\delta t$ on the times of arrival, the frequency $f_\mathrm{GW}$ of the gravitational waves, and the angular scale $\Delta\Omega$ of the anisotropy. The sensitivity scales approximately as $N_\mathrm{psr}^{0.8}$, $\delta t^{-0.08}$, and $\Delta\Omega^{1.6}-\Delta\Omega^{2.1}$ (depending on the ranges of $\ell$ and $m$ under consideration). In addition, we use realistic simulations to project the NANOGrav PTA sensitivity to a 30-year baseline and quantify the growth in sensitivity at several timeslices. Except at the lowest frequencies, we find negligible effect on sensitivity through increasing the observation duration only. Finally, we introduce a multi-resolution pixel basis motivated by the large dependence of the sensitivity on sky location, and demonstrate the operation of the basis through a set of injections and recoveries.

Figures

Figures reproduced from arXiv: 2608.09781 by the authors.

Figure 1
Figure 1. Pixel basis anisotropy sensitivity curves using a simulated dataset with 67 pulsars, 100 ns TOA uncertainty, and pulsars with baselines of identical length. These curves were generated by choosing a sky location for the left and right panels and averaging over all sky locations for the middle panel. The sky location of the left panel is a region of low pulsar density and correspondingly low sensitivity, whereas the … view at source ↗
Figure 2
Figure 2. Pixel basis power law scaling exponents as a function of sky location and (left) number of pulsars, (center) TOA uncertainty, and (right) GW frequency. The Fisher matrix diagonal elements increase with number of pulsars and decrease with TOA uncertainty and GW frequency. 0 5 10 15 20 ` 0.6 0.8 1.0 1.2 Power law exponent 0 5 10 15 20 ` −0.150 −0.125 −0.100 −0.075 −0.050 −0.025 0 5 10 15 20 ` −0.4 −0.3 −0.2 −0.1 0.0 … view at source ↗
Figure 3
Figure 3. Spherical harmonic basis power law scaling exponents as a function of (left) number of pulsars, (center ) TOA uncertainty, and (right) GW frequency. The exponents are grouped by the ℓ value of the clm coefficient, and the spread at each ℓ value corresponds to different m values. p diag(M) scales with a positive exponent (i.e., increases) with number of pulsars and a negative exponent (i.e., decreases) with TOA uncer… view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: Sensitivity scaling in the spherical harmonic basis as a function of angular scale. Here we define angular scale as 180◦ /ℓ for ℓ > 0 and 360◦ for ℓ = 0. The sensitivity increases with the angular scale of the anisotropy. Left. The markers and error bars correspond to …
Figure 5
Figure 5. Figure 5: Percent change in sensitivity as a function of Tspan. All percentages are measured relative to the 16 yr baseline. The distributions contain the medians of each simulation, and the markers and lines represent the medians of the distributions. The rows represent differe…
Figure 6
Figure 6. Figure 6: Sensitivity curves for simulated PTAs with realistic TOA uncertainty distributions and including pulsar pair covari￾ance. The sky location was fixed to a pixel of relatively low (high) sensitivity in the left (right) panel. The curves represent the median sensitivity a…
Figure 7
Figure 7. Figure 7: The difference between injection and recov￾ery are O(10−14), much smaller than the O(1) GWB, indicating successful recoveries. In [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: Uncertainty maps for (left) a single-resolution Nside = 2 parameterization, (center ) a multi-resolution parame￾terization, and (right) a single-resolution Nside = 1 parameterization. The three colorbars have the same limits for easier comparison. Although we have intr…

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Works this paper leans on

69 extracted references · 7 canonical work pages

  1. [1]

    2014, CQGra, 32, 024001, doi: 10.1088/0264-9381/32/2/024001

    Acernese, F., Agathos, M., Agatsuma, K., et al. 2014, CQGra, 32, 024001, doi: 10.1088/0264-9381/32/2/024001

    doi:10.1088/0264-9381/32/2/024001 2014

  2. [2]

    2023, ApJL, 951, L11, doi: 10.3847/2041-8213/acdc91

    Afzal, A., Agazie, G., Anumarlapudi, A., et al. 2023, ApJL, 951, L11, doi: 10.3847/2041-8213/acdc91

    doi:10.3847/2041-8213/acdc91 2023

  3. [3]

    D., Ali-Ha ¨ ımoud, Y., & Smith, T

    Agarwal, D., Romano, J. D., Ali-Ha ¨ ımoud, Y., & Smith, T. L. 2026, Addressing leakage and mode suppression in angular power spectrum estimation for gravitational-wave backgrounds using pulsar timing arrays, https://arxiv.org/abs/2602.20075

    arXiv 2026

  4. [4]

    M., et al

    Agazie, G., Anumarlapudi, A., Archibald, A. M., et al. 2023a, ApJL, 951, L8, doi: 10.3847/2041-8213/acdac6

    doi:10.3847/2041-8213/acdac6

  5. [5]

    M., et al

    Agazie, G., Anumarlapudi, A., Archibald, A. M., et al. 2023b, ApJL, 952, L37, doi: 10.3847/2041-8213/ace18b

    doi:10.3847/2041-8213/ace18b

  6. [6]

    F., Anumarlapudi, A., et al

    Agazie, G., Alam, M. F., Anumarlapudi, A., et al. 2023c, ApJL, 951, L9, doi: 10.3847/2041-8213/acda9a

    doi:10.3847/2041-8213/acda9a

  7. [7]

    2020, Overview of KAGRA: Detector design and construction history, https://arxiv.org/abs/2005.05574 Ali-Ha ¨ ımoud, Y., Smith, T

    Akutsu, T., Ando, M., Arai, K., et al. 2020, Overview of KAGRA: Detector design and construction history, https://arxiv.org/abs/2005.05574 Ali-Ha ¨ ımoud, Y., Smith, T. L., & Mingarelli, C. M. F. 2020, PhRvD, 102, doi: 10.1103/physrevd.102.122005 Ali-Ha ¨ ımoud, Y., Smith, T. L., & Mingarelli, C. M. F. 2021, PhRvD, 103, doi: 10.1103/physrevd.103.042009

    arXiv 2020

  8. [8]

    Allen, B., & Romano, J. D. 2023, PhRvD, 108, doi: 10.1103/physrevd.108.043026

    doi:10.1103/physrevd.108.043026 2023

Show all 69 references
  1. [9]

    Anholm, M., Ballmer, S., Creighton, J. D. E., Price, L. R., & Siemens, X. 2009, PhRvD, 79, doi: 10.1103/physrevd.79.084030 16

    2009 doi

  2. [10]

    2024, Forecasting the sensitivity of Pulsar Timing Arrays to gravitational wave backgrounds, https://arxiv.org/abs/2404.02864

    Babak, S., Falxa, M., Franciolini, G., & Pieroni, M. 2024, Forecasting the sensitivity of Pulsar Timing Arrays to gravitational wave backgrounds, https://arxiv.org/abs/2404.02864

    2024 arXiv

  3. [11]

    Goss, W. M. 1982, Nature, 300, 615, doi: 10.1038/300615a0

    1982 doi

  4. [12]

    Ballmer, S. W. 2006, CQGra, 23, S179, doi: 10.1088/0264-9381/23/8/S23

    2006 doi

  5. [13]

    C., Blandford, R

    Begelman, M. C., Blandford, R. D., & Rees, M. J. 1980, Nature, 287, 307, doi: 10.1038/287307a0

    1980 doi

  6. [14]

    2018, JAX: composable transformations of Python+NumPy programs, 0.3.13 http://github.com/jax-ml/jax

    Bradbury, J., Frostig, R., Hawkins, P., et al. 2018, JAX: composable transformations of Python+NumPy programs, 0.3.13 http://github.com/jax-ml/jax

    2018

  7. [15]

    Caprini, C., & Figueroa, D. G. 2018, CQGra, 35, 163001, doi: 10.1088/1361-6382/aac608

    2018 doi

  8. [16]

    J., Creighton, J

    Chamberlin, S. J., Creighton, J. D., Siemens, X., et al. 2015, PhRvD, 91, doi: 10.1103/physrevd.91.044048

    2015 doi

  9. [17]

    Collaboration, T. L. S., Aasi, J., Abbott, B. P., et al. 2015, CQGra, 32, 074001, doi: 10.1088/0264-9381/32/7/074001

    2015 doi

  10. [18]

    J., & van Haasteren, R

    Cornish, N. J., & van Haasteren, R. 2014, arXiv e-prints, arXiv:1406.4511. https://arxiv.org/abs/1406.4511

    2014 arXiv

  11. [19]

    B., Ferdman, R

    Demorest, P. B., Ferdman, R. D., Gonzalez, M. E., et al. 2012, ApJ, 762, 94, doi: 10.1088/0004-637x/762/2/94

    2012 doi

  12. [20]

    F., Domcke, V., Franciolini, G., & Pieroni, M

    Depta, P. F., Domcke, V., Franciolini, G., & Pieroni, M. 2024, Pulsar timing array sensitivity to anisotropies in the gravitational wave background, https://arxiv.org/abs/2407.14460

    2024 arXiv

  13. [21]

    1979, ApJ, 234, 1100, doi: 10.1086/157593

    Detweiler, S. 1979, ApJ, 234, 1100, doi: 10.1086/157593

    1979 doi

  14. [22]

    2025, Cosmic Variance in Anisotropy Searches at Pulsar Timing Arrays, https://arxiv.org/abs/2508.21131

    Domcke, V., Franciolini, G., & Pieroni, M. 2025, Cosmic Variance in Anisotropy Searches at Pulsar Timing Arrays, https://arxiv.org/abs/2508.21131

    2025 arXiv

  15. [23]

    2017, jellis18/PTMCMCSampler: Official Release, doi: 10.5281/zenodo.1037579

    Ellis, J., & van Haasteren, R. 2017, jellis18/PTMCMCSampler: Official Release, doi: 10.5281/zenodo.1037579

    2017 doi

  16. [24]

    A., Vallisneri, M., Taylor, S

    Ellis, J. A., Vallisneri, M., Taylor, S. R., & Baker, P. T. 2020, ENTERPRISE: Enhanced Numerical Toolbox Enabling a Robust PulsaR Inference SuitE,, Zenodo doi: 10.5281/zenodo.4059815 EPTA Collaboration, InPTA Collaboration, Antoniadis, J., et al. 2023, A&A, 678, A50, doi: 10.1...

    2020 doi

  17. [25]

    2014, MOC - HEALPix Multi-Order Coverage map Version 1.0,, IVOA Recommendation 02 June 2014 doi: 10.5479/ADS/bib/2014ivoa.spec.0602F

    Fernique, P., Boch, T., Donaldson, T., et al. 2014, MOC - HEALPix Multi-Order Coverage map Version 1.0,, IVOA Recommendation 02 June 2014 doi: 10.5479/ADS/bib/2014ivoa.spec.0602F

    2014 doi

  18. [26]

    Flanagan, E. E. 1993, PhRvD, 48, 2389–2407, doi: 10.1103/physrevd.48.2389

    1993 doi

  19. [27]

    A., Taylor, S

    Gersbach, K. A., Taylor, S. R., B´ ecsy, B., et al. 2025, Mapping the Gravitational-wave Background Across the Spectrum with a Next-Generation Anisotropic Per-frequency Optimal Statistic, https://arxiv.org/abs/2509.07090

    2025 arXiv

  20. [28]

    A., Taylor, S

    Gersbach, K. A., Taylor, S. R., Meyers, P. M., & Romano, J. D. 2025, PhRvD, 111, 023027, doi: 10.1103/PhysRevD.111.023027

    2025 doi

  21. [29]

    M., Hivon, E., Banday, A

    Gorski, K. M., Hivon, E., Banday, A. J., et al. 2005, ApJ, 622, 759

    2005

  22. [30]

    Green, P. J. 1995, Biometrika, 82, 711, doi: 10.1093/biomet/82.4.711

    1995 doi

  23. [31]

    J., Thrane, E., et al

    Grunthal, K., Champion, D. J., Thrane, E., et al. 2026, Optimising gravitational-wave sky maps for pulsar timing arrays, https://arxiv.org/abs/2601.13957

    2026

  24. [32]

    R., Millman, K

    Harris, C. R., Millman, K. J., van der Walt, S. J., et al. 2020, Nature, 585, 357, doi: 10.1038/s41586-020-2649-2

    2020 doi

  25. [33]

    2019, Journal of Open Source Software, 4, 1775, doi: 10.21105/joss.01775

    Hazboun, J., Romano, J., & Smith, T. 2019, Journal of Open Source Software, 4, 1775, doi: 10.21105/joss.01775

    2019 doi

  26. [34]

    Hazboun, J. S. 2020, La Forge, 0.3.0 Zenodo, doi: 10.5281/zenodo.4152550

    2020 doi

  27. [35]

    S., Romano, J

    Hazboun, J. S., Romano, J. D., & Smith, T. L. 2019, PhRvD, 100, 104028, doi: 10.1103/PhysRevD.100.104028

    2019 doi

  28. [36]

    W., & Downs, G

    Hellings, R. W., & Downs, G. S. 1983, ApJL, 265, L39, doi: 10.1086/183954

    1983 doi

  29. [37]

    2010, CQGra, 27, 084013, doi: 10.1088/0264-9381/27/8/084013

    Hobbs, G., Archibald, A., Arzoumanian, Z., et al. 2010, CQGra, 27, 084013, doi: 10.1088/0264-9381/27/8/084013

    2010 doi

  30. [38]

    N., et al

    Hobbs, G., Coles, W., Manchester, R. N., et al. 2012, MNRAS, 427, 2780, doi: 10.1111/j.1365-2966.2012.21946.x

    2012

  31. [39]

    N., et al

    Hobbs, G., Guo, L., Caballero, R. N., et al. 2019, MNRAS, 491, 5951, doi: 10.1093/mnras/stz3071

    2019 doi

  32. [40]

    Hunter, J. D. 2007, Computing in Science & Engineering, 9, 90, doi: 10.1109/MCSE.2007.55

    2007 doi

  33. [41]

    R., Pol, N

    Kaiser, A. R., Pol, N. S., McLaughlin, M. A., et al. 2022, ApJ, 938, 115, doi: 10.3847/1538-4357/ac86cc

    2022 doi

  34. [42]

    P., et al

    Lentati, L., Alexander, P., Hobson, M. P., et al. 2013, PhRvD, 87, 104021

    2013

  35. [43]

    P., Burns, E., et al

    Martinez-Castellanos, I., Singer, L. P., Burns, E., et al. 2022, AJ, 163, 259, doi: 10.3847/1538-3881/ac6260

    2022 doi

  36. [44]

    P., Burns, E., et al

    Martinez-Castellanos, I., Singer, L. P., Burns, E., et al. 2024, mhealpy: Object-oriented healpy wrapper with support for multi-resolution maps,, Astrophysics Source Code Library, record ascl:2404.023 http://ascl.net/2404.023

    2024

  37. [45]

    T., Shannon, R

    Miles, M. T., Shannon, R. M., Reardon, D. J., et al. 2024, MNRAS, 536, 1467, doi: 10.1093/mnras/stae2572

    2024 doi

  38. [46]

    Mingarelli, C. M. F., Sidery, T., Mandel, I., & Vecchio, A. 2013, PhRvD, 88, doi: 10.1103/physrevd.88.062005 17

    2013 doi

  39. [47]

    Mingarelli, C. M. F., Lazio, T. J. W., Sesana, A., et al. 2017, Nature Astronomy, 1, 886–892, doi: 10.1038/s41550-017-0299-6

    2017 doi

  40. [48]

    Mingarelli, C. M. F., Casey-Clyde, J. A., Chang, Y. T., et al. 2026, Pulsar timing arrays: the emerging gravitational-wave landscape, https://arxiv.org/abs/2603.13643

    2026

  41. [49]

    2008, PhRvD, 77, 042002, doi: 10.1103/PhysRevD.77.042002

    Mitra, S., Dhurandhar, S., Souradeep, T., et al. 2008, PhRvD, 77, 042002, doi: 10.1103/PhysRevD.77.042002

    2008 doi

  42. [50]

    2025, LMFIT: Non-Linear Least-Squares Minimization and Curve-Fitting for Python, 1.3.4 Zenodo, doi: 10.5281/zenodo.16175987

    Newville, M., Otten, R., Nelson, A., et al. 2025, LMFIT: Non-Linear Least-Squares Minimization and Curve-Fitting for Python, 1.3.4 Zenodo, doi: 10.5281/zenodo.16175987

    2025 doi

  43. [51]

    R., & Romano, J

    Pol, N., Taylor, S. R., & Romano, J. D. 2022, ApJ, 940, 173, doi: 10.3847/1538-4357/ac9836

    2022 doi

  44. [52]

    S., Taylor, S

    Pol, N. S., Taylor, S. R., Kelley, L. Z., et al. 2021, ApJL, 911, L34, doi: 10.3847/2041-8213/abf2c9

    2021 doi

  45. [53]

    J., et al

    Reardon, D. J., et al. 2023, ApJL, 951, L6, doi: 10.3847/2041-8213/acdd02

    2023 doi

  46. [54]

    2015, A&A, 580, A132, doi: 10.1051/0004-6361/201526549

    Reinecke, M., & Hivon, E. 2015, A&A, 580, A132, doi: 10.1051/0004-6361/201526549

    2015 doi

  47. [55]

    A., Bender, R., et al

    Richstone, D., Ajhar, E. A., Bender, R., et al. 1998, Nature, 385, A14, doi: 10.48550/arXiv.astro-ph/9810378

    1998 doi arXiv

  48. [56]

    D., & Cornish, N

    Romano, J. D., & Cornish, N. J. 2017, LRR, 20, 2

    2017

  49. [57]

    A., Sesana, A., & Gair, J

    Rosado, P. A., Sesana, A., & Gair, J. 2015, Monthly Notices of the Royal Astronomical Society, 451, 2417, doi: 10.1093/mnras/stv1098

    2015 doi

  50. [58]

    Sazhin, M. V. 1978, Soviet Ast., 22, 36

    1978

  51. [59]

    Semenzato, F., Bellomo, N., Raccanelli, A., & Mingarelli, C. M. F. 2025, Bias from small-scale leakage in Pulsar Timing Array maps, https://arxiv.org/abs/2510.24857

    2025 arXiv

  52. [60]

    Siemens, X., Ellis, J., Jenet, F., & Romano, J. D. 2013, CQGra, 30, 224015, doi: 10.1088/0264-9381/30/22/224015

    2013 doi

  53. [61]

    P., & Price, L

    Singer, L. P., & Price, L. R. 2016, PhRvD, 93, doi: 10.1103/physrevd.93.024013

    2016 doi

  54. [62]

    Vigeland, S. J. 2021, enterprise extensions, https://github.com/nanograv/enterprise extensions

    2021

  55. [63]

    R., & Gair, J

    Taylor, S. R., & Gair, J. R. 2013, PhRvD, 88, doi: 10.1103/physrevd.88.084001

    2013 doi

  56. [64]

    R., van Haasteren, R., & Sesana, A

    Taylor, S. R., van Haasteren, R., & Sesana, A. 2020, PhRvD, 102, doi: 10.1103/physrevd.102.084039

    2020 doi

  57. [65]

    J., Islo, K., Taylor, S

    Vigeland, S. J., Islo, K., Taylor, S. R., & Ellis, J. A. 2018, PhRvD, 98, doi: 10.1103/physrevd.98.044003

    2018 doi

  58. [66]

    E., et al

    Virtanen, P., Gommers, R., Oliphant, T. E., et al. 2020, Nature Methods, 17, 261, doi: 10.1038/s41592-019-0686-2

    2020 doi

  59. [67]

    2023, RAA, 23, 075024, doi: 10.1088/1674-4527/acdfa5

    Xu, H., Chen, S., Guo, Y., et al. 2023, RAA, 23, 075024, doi: 10.1088/1674-4527/acdfa5

    2023 doi

  60. [68]

    W., & Petty, M

    Youngren, R. W., & Petty, M. D. 2017, Heliy, 3, doi: 10.1016/j.heliyon.2017.e00332

    2017 doi

  61. [69]

    Zonca, A., Singer, L., Lenz, D., et al. 2019, JOSS, 4, 1298, doi: 10.21105/joss.01298 18 APPENDIX A.INJECTED NOISE PARAMETERS Pulsar Amplitude Spectral Index Pulsar Amplitude Spectral Index B1855+09 -13.900 3.613 J1730-2304 -12.974 5×10 −4 B1937+21 -13.530 3.853 J1738+0333 -14...

    2019 doi

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