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Recovering Pulsar Braking Index from a Population of Millisecond Pulsars

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

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Pith's one-line read The paper claims that pulsar braking-index distributions are recoverable from a population, but only after 20 years of timing at 10^-5 ms noise.

desk verdict Novel pipeline, honest feasibility study, but the 20-year 'population recovery' is really a single pulsar. read the letter →

arxiv 2412.10169 v2 pith:47X2OGEN submitted 2024-12-13 astro-ph.HE astro-ph.GAastro-ph.SR

A. L. Hewitt , M. Pitkin , I. M. Hook This is my paper · ORCID
classification astro-ph.HEastro-ph.GAastro-ph.SR
keywords pulsarbrakingindexmillisecondpulsarsspin-downmechanismsgravitational-wavehierarchicalBayesianinferencetimingarraysnoisepopulation
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 braking index of a pulsar population—the exponent $n$ in $\dot{f} = -K f^n$ that distinguishes magnetic-dipole spin-down ($n=3$), gravitational-wave spin-down from a mass quadrupole ($n=5$), and r-mode emission ($n=7$)—can be inferred collectively rather than pulsar by pulsar. Millisecond pulsars have such tiny second frequency derivatives that individual braking-index measurements are usually out of reach. The authors simulate arrival-time data for 47 millisecond pulsars with $n=5$ injected, fit each pulsar's $n$ under a broad prior, and stack the posteriors with a hierarchical Bayesian model that assumes the population values are Gaussian. Their result is that the population-level $n$ is recoverable in principle, but only with observation times above 20 years and RMS timing noise near $10^{-5}$ ms—roughly thirty times better than the mean of the 12.5-year wideband sample that supplies the pulsar parameters. If this holds, future multi-decade timing arrays could determine which spin-down mechanism dominates a whole millisecond-pulsar population without ever measuring an individual $\ddot{f}$.

What carries the argument

The load-bearing quantity is the braking index $n$, defined by $\dot{f} \propto -f^n$ and observable in principle as $n = f\ddot{f}/\dot{f}^2$; it labels the dominant spin-down mechanism. The argument runs on a two-stage statistical machine. First, each pulsar's arrival times are fit to produce a posterior on $f$, $\dot{f}$, and $n$ directly, with $n$ given a uniform prior on $[0,10]$. Second, those posteriors are stacked by a hierarchical Bayesian model that assumes the population's $n$ values are drawn from a Gaussian parent $N(n|\mu,\sigma)$ and approximates each marginal likelihood by the mean of that Gaussian evaluated over the pulsar's posterior samples: $$\int P(D_i|n)\,N(n|\mu,\$\sigma$)\,dn \approx \frac{1}{m_i}\sum_{j=1}^{m_i} N(n_{ij}|\mu,\$\sigma$).$$ The output is summarised by the odds ratio $\mathrm{OR}_{4-6}$, the probability that a random population $n$ lies between 4 and 6 divided by the probability it lies outside that range; $\mathrm{OR}=1$ means the signal is as likely as not.

What would settle it

Repeat the 47-pulsar analysis with red timing noise injected at the amplitude and spectral slope measured in the 12.5-year wideband sample, holding observation length at 20 years and RMS noise at $10^{-5}$ ms; the central claim fails if $\mathrm{OR}_{4-6}$ drops below 1.

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Extended reading notes

Core claim

The central claim is that hierarchical Bayesian stacking can recover the distribution of braking indices of a millisecond-pulsar population even when no individual pulsar's $\ddot{f}$ is measurable, but only under conditions that current data do not meet. With all 47 simulated pulsars spinning down at $n=5$, the method returns the correct mean with odds ratio $\mathrm{OR}_{4-6} \approx 1.0$ at 20 years and $10^{-5}$ ms RMS noise—a coin flip between $n$ lying in $[4,6]$ and outside it. Extending the baseline to 50 years raises $\mathrm{OR}_{4-6}$ to 9.65, whereas reducing noise from $10^{-5}$ to $10^{-6}$ ms only reaches 1.76. The paper concludes that observation time is the parameter with the largest impact on feasibility, that current 12.5-year, $3\times10^{-4}$ ms data cannot support the measurement, and that the study is a proof of principle for future arrays rather than an immediate observational result.

Load-bearing premise

The feasibility thresholds assume the simulated arrival times contain only white noise—red timing noise is discussed but never simulated—and that every pulsar has the same injected braking index; if red noise matters over 20-to-50-year baselines, the required observation times and noise levels would be worse.

Editorial extensions

If this is right

  • At the current mean RMS of $3\times10^{-4}$ ms and a 12.5-year baseline, the stacking method does not confidently recover the injected $n=5$ population; $\mathrm{OR}_{4-6}$ falls below 1.
  • Observation length is the dominant lever: holding noise at $10^{-5}$ ms, $\mathrm{OR}_{4-6}$ rises from 1.00 at 20 years to 1.46 at 30 years and 9.65 at 50 years.
  • Improving timing precision matters less: a factor-of-10 noise reduction from $10^{-5}$ to $10^{-6}$ ms only increases $\mathrm{OR}_{4-6}$ from 1.00 to 1.76.
  • Growing the sample helps: increasing from 10 to 47 pulsars roughly doubles the odds ratio, and one exceptionally well-constrained pulsar contributes disproportionately, so targeted high-precision timing could accelerate the measurement.
  • The method is not biased toward the injected value: separate runs with $n=3$, $n=5$, and $n=7$ are all recovered with similar confidence at 20 years and $10^{-6}$ ms, so the approach can in principle distinguish spin-down mechanisms.

Reading between the lines

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

  • The simulations use a delta-function population in which all 47 pulsars share $n=5$, the most favorable case for a Gaussian parent model; a real population mixing $n=3$ and $n=5$ would blur the recovered mean and width, so the quoted thresholds are likely optimistic.
  • Red timing noise is acknowledged as the main observational obstacle to measuring $\ddot{f}$ but is never injected; because such noise is correlated in time, longer baselines may accumulate it rather than average it away, so the 50-year projection could degrade faster than the white-noise curves suggest.
  • A natural test of the method's reach is to inject a two-component mixture (for example 70% $n=3$ and 30% $n=5$) and ask whether the Gaussian parent model can detect the mixture; if not, a mixture or histogram parent distribution would be needed before applying the method to real data.
  • The same hierarchical stacking recipe could be transported to other weakly measured pulsar population parameters, such as ellipticity or magnetic inclination, though the paper does not make that extension.
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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 presents a simulation-based feasibility study of measuring the distribution of pulsar braking indices from a population of millisecond pulsars. The authors generate simulated TOAs for 47 MSPs drawn from the NANOGrav 12.5-year sample, fixing the second frequency derivative to correspond to an injected braking index of n=5, and add white noise only. Per-pulsar posteriors on n are obtained by sampling directly over the braking index with a uniform prior on [0,10], and the posteriors are combined with a hierarchical Gaussian parent model using the posteriorstacker package. The success metric is an odds-like ratio OR4-6, the posterior probability that a draw from the recovered parent distribution lies between n=4 and n=6 divided by the complementary probability. The main results are that at a mean RMS of 1e-5 ms, a 20-year baseline gives OR4-6=1.00, a 50-year baseline gives OR4-6=9.65, and increasing the number of pulsars improves the recovery. The paper concludes that population-level braking-index recovery is possible but requires observation times over 20 years and very low timing noise.

Significance. If the method were validated, it would offer a way to infer the dominant spin-down mechanism of MSP populations without precise individual measurements of the second frequency derivative, connecting observable timing data to the n=5 gravitational-wave spin-down scenario. The pipeline is coherent and uses appropriate machinery: sampling directly over n is a sensible reparameterization, equation (8) correctly exploits the uniform prior to treat posteriors as likelihoods, and the closed-box injection-recovery tests are the right calibration procedure. The authors also provide a useful technical contribution in the modified enterprise_warp code and document numerical-precision changes. However, the central quantitative claim is weakened by two structural issues: the default 20-year result is effectively driven by a single pulsar, and the simulations exclude red timing noise. The paper is an honest proof-of-principle, but the population-level interpretation at the headline thresholds is not yet demonstrated.

major comments (3)
  1. [§3.3, Table 3] The population-recovery claim for the default 20-year, 1e-5 ms run is not supported by the per-pulsar results in Table 3. For 46 of the 47 pulsars the posterior mean and standard deviation are close to the imposed Uniform(0,10) prior values (mean 5, std 2.89), while only B1937+21 is tightly constrained (n = 5.01 ± 0.06). Section 3.3 further states that this pulsar dominates and is force-included in every subset run. Consequently the reported OR4-6 = 1.00 for the 47-pulsar run is effectively a single-object measurement, and the improvement with pulsar number in Figure 4 does not by itself demonstrate that weak information is being accumulated across the population. Please report the recovered hyperparameters and OR4-6 with B1937+21 removed, and quantify per-pulsar information content (for example, the Kullback-Leibler divergence of each posterior from its prior) to show that the stacking step genuinely combines information from multiple pulsars.
  2. [§2.1, §3.1, Abstract] The headline feasibility thresholds (observation times over 20 years and RMS noise of order 1e-5 ms) are derived from simulated TOAs containing white noise only, as stated in Section 2.1. Red timing noise is cited as a major obstacle to measuring the second frequency derivative (Liu et al. 2019) but is never injected into the simulations, and Section 4 acknowledges it only qualitatively. Because red noise is correlated over the 20-50 year baselines considered here, it could mimic or contaminate the f-dotdot signal and bias the recovered braking index. The authors should either add red-noise injections (even a simple power-law component) to test the thresholds, or explicitly qualify the abstract's claims as conditional on red noise being negligible; as written, the quantitative feasibility statement is overstrong.
  3. [§2.3, Appendix C] The recovery tests use a population in which every pulsar has the same injected n=5, a delta-function parent distribution, and Appendix C tests n=3, 5, and 7 only in separate runs. A realistic MSP population is more likely to contain a mixture of spin-down mechanisms, and the Gaussian parent model's ability to identify the dominant mechanism should be tested on a mixed population (for example, a fraction with n=3 and the remainder with n=5). Without such a test, the claim that the method can 'allow the underlying energy loss process to be inferred' is not fully established, because the favorable delta-function case sidesteps the question of whether the hierarchy can separate a narrow signal from a broad or multi-component distribution.
minor comments (6)
  1. [Abstract, §3.2] The phrase 'observation times of over 20 years' is ambiguous: the 20-year run gives OR4-6 = 1.00, which is an equivocal result, while significant confidence appears only for the 50-year run (OR4-6 = 9.65). Please rephrase to make clear that a confident population-level measurement requires substantially more than 20 years.
  2. [§2.4, Eq. (13)] The quantity OR4-6 is called an 'odds ratio', but it is a ratio of posterior probabilities, not a Bayes factor or a classical odds ratio. Consider renaming it 'probability ratio' or 'odds' to avoid confusion with standard statistical terminology.
  3. [§2.3, Eq. (8)] The proportionality P(Di|n) ∝ P(n|Di) relies on the prior for n being uniform over the full support of the posterior; this is stated in the text but should also be stated immediately after the equation for clarity.
  4. [Table 3] The combined column heading 'Mean n * Std n*' is confusing; please use separate columns for the posterior mean and posterior standard deviation, and explain in the caption that these are for the default 20-year, 1e-5 ms run.
  5. [§2.1, Eq. (6)] The sqrt(N) cadence-to-noise equivalence is valid only for white noise; since the paper later discusses red-noise concerns, this assumption should be flagged at the point where equation (6) is introduced.
  6. [Figure 4] The OR4-6 values are given only in the caption and are easy to miss; consider annotating the curves or adding a small table with the numerical values.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the study is an injection-recovery calibration and its reported limitations are stated honestly.

full rationale

The paper is an injection-recovery calibration rather than a claimed first-principles prediction. The value n=5 is intentionally placed into simulated TOAs by fixing F2 via equation (4), and the enterprise/posteriorstacker pipeline is then tasked with recovering it; this is the standard and non-circular design of a sensitivity study. No output quantity in the derivation chain is defined in terms of an input quantity, and no fitted parameter is renamed as a prediction. The uniform prior on n between 0 and 10 does have midpoint 5, and Table 3 shows that at the headline 20-year, 10^-5 ms configuration most pulsars return posterior standard deviations near the prior value of about 2.887, with only B1937+21 well constrained (n = 5.01 ± 0.06). The paper itself states in Section 3.3 that this pulsar dominates the results, and it honestly reports OR4-6 = 1.00 at that threshold. That is a limitation of the population-level demonstration, but it is not a circular reduction: the paper does not claim the weakly informative pulsars independently constrain n, and Appendix C shows recovery of n=3 and n=7 under the same uniform prior, indicating the method is not simply forced to the prior midpoint. The acknowledged simplifications (white-noise-only TOAs, no red noise, delta-function injected population) are scope limitations and are flagged in Sections 2.1 and 4 as an optimistic proof-of-principle scenario. Self-citations with overlapping authorship (Woan et al. 2018; Goncharov et al. 2024) are motivational or software-related and are not load-bearing for the central simulation results. The derivation chain is self-contained with respect to the stated feasibility claim.

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

The central claim rests on two inputs the authors control: the injected delta-function n=5 population and the white-noise-only TOA model. Neither is independently evidenced, so the feasibility thresholds inherit both assumptions. No new physical entities are introduced; the only novel artifact is software (enterprise_warp braking-index sampling fork, Zenodo 10.5281/zenodo.13274448).

free parameters (4)
  • Injected braking index for all simulated TOAs = 5 (chosen injection)
    F2 is set from n=5 via equation (4) for every pulsar, so the population is a delta function at 5; all feasibility thresholds are conditional on this most favorable case.
  • Target mean RMS noise (mu_desired) = 1e-6 to 2.9e-4 ms
    Equation (5) rescales per-pulsar NANOGrav noise to chosen sample means; the '10^-5 ms needed' headline is one grid point of this hand-chosen set.
  • Hyper-priors on the parent Gaussian = uniform on mu, log-uniform on sigma
    Chosen in posteriorstacker; the log-uniform sigma prior shapes how broad the recovered distribution is when individual posteriors are uninformative.
  • Cadence-to-noise equivalence factor sqrt(N) = N from 4 to 30
    Equation (6), RMSeqv = RMS/sqrt(N), is used instead of simulating denser observation schedules; it assumes purely white noise, which is the same assumption the feasibility thresholds depend on.
assumptions (5)
  • domain assumption White-noise-only TOAs adequately represent MSP timing for f-dotdot inference
    Section 2.1 generates TOAs with white noise alone; red noise is excluded despite the paper citing Liu et al. (2019) on red noise degrading f-dotdot measurability. This is the load-bearing premise for the quantitative thresholds.
  • domain assumption Parent braking-index distribution is Gaussian
    Section 2.3 assumes N(n|mu,sigma) for the population; a mixed population (n=3 and n=5) would be misrepresented, and only delta-function injections are tested (Appendix C).
  • domain assumption Measured F1 is intrinsic spin-down, uncontaminated by acceleration or Shklovskii effects
    Stated in Section 4; binary information is stripped in Section 2.1 so 30 binary pulsars are treated as isolated.
  • ad hoc to paper Uniform prior on n in [0,10] justifies posterior-as-likelihood
    Section 2.3 equation (8) requires a uniform prior on n; this is stated and used consistently.
  • standard math Nested sampling (dynesty) and the Monte Carlo replacement of the integral in equation (9) converge
    The hierarchical likelihood is approximated by sample means of the parent Gaussian over posterior samples; the paper uses about 20,000 samples per pulsar.

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Cite this review

Pith. "Pith review of Recovering Pulsar Braking Index from a Population of Millisecond Pulsars." pith.science (2026). https://pith.science/paper/47X2OGEN

@misc{pith2026241210169,
  author       = {Pith},
  title        = {Pith review of: Recovering Pulsar Braking Index from a Population of Millisecond Pulsars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/47X2OGEN}},
  note         = {Machine review of arXiv:2412.10169}
}
abstract

The braking index, $n$, of a pulsar is a measure of its angular momentum loss and the value it takes corresponds to different spin-down mechanisms. For a pulsar spinning down due to gravitational wave emission from the principal mass quadrupole mode alone, the braking index would equal exactly 5. Unfortunately, for millisecond pulsars, it can be hard to measure observationally due to the extremely small second time derivative of the rotation frequency, $\ddot{f}$. This paper aims to examine whether it could be possible to extract the distribution of $n$ for a whole population of pulsars rather than measuring the values individually. We use simulated data with an injected $n=5$ signal for 47 millisecond pulsars and extract the distribution using hierarchical Bayesian inference methods. We find that while possible, observation times of over 20 years and RMS noise of the order of $10^{-5}$ ms are needed, which can be compared to the mean noise value of $3\times10^{-4}$ ms for the recent wideband 12.5-year NANOGrav sample, which provided the pulsar timing data used in this paper.

Figures

Figures reproduced from arXiv: 2412.10169 by the authors.

Figure 2
Figure 2. Analyses with identical parameters aside from varying the mean RMS noise, calculated using equation (5), in the simulated TOAs. The lines represent the median val￾ues while the shaded regions are bounded by the 5th and 95th percentiles. The OR4−6 for these runs, in order from highest rms to lowest, are: 0.62, 0.88, 0.98, 1.00, 1.06, 1.76. observation length could cost over 262 GB of memory and take on the order of a… view at source ↗
Figure 1
Figure 1. A normalised histogram of the equally weighted posterior samples binned into 100 bins for an observation length of 20 years and an RMS noise value of 1×10−5 ms using all 47 pulsars. Using equation (11), probability that n lies in the bins between 4 and 6 (shaded in red) and the probability that it lies outside that region is calculated to find OR4−6. 3. ANALYSIS RESULTS Following the methods described in Section 2, … view at source ↗
Figure 3
Figure 3. Analyses with identical parameters aside from varying observation time. The solid lines represent the me￾dian values while the shaded regions are bounded by the 5th and 95th percentiles. The OR4−6 for these runs, in order from shortest to longest observation length, are: 0.69, 0.69, 0.92, 1.00, 1.46, 9.65. It can be seen that the confidence increases with length of observation. For 10 or fewer years, the cor￾rect br… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Analyses with identical parameters aside from varying the number of pulsars. For each set, additional pul￾sars are picked at random with the exception of the pulsar with the strongest signal, which was included in all runs. The solid lines represent the median values w…
Figure 5
Figure 5. Figure 5: The result of identical analyses performed on three sets of TOAs produced by Tempo2 given identical parameters. It demonstrates the variation introduced during the TOA generation step and posterior analysis. The solid lines represent the median values while the shaded …
Figure 6
Figure 6. Figure 6: Analyses with identical parameters aside from varying the braking index used to set ¨f when generating the simulate TOAs. The solid lines represent the median values while the shaded regions are bounded by the 5th and 95th percentiles. REFERENCES Abbott, R., Abbott, T.…

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Reference graph

Works this paper leans on

51 extracted references · 13 canonical work pages

  1. [1]

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

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

  2. [2]

    write newline

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

  3. [3]

    6m F_)mKk> ۈQR0 FH) ܞ

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

    2021

  4. [4]

    D., Abraham , S., et al

    Abbott , R., Abbott , T. D., Abraham , S., et al. 2021 a , PhRvX, 11, 021053, 10.1103/PhysRevX.11.021053

    doi:10.1103/physrevx.11.021053 2021

  5. [5]

    2021 b , , 915, L5, 10.3847/2041-8213/ac082e

    ---. 2021 b , , 915, L5, 10.3847/2041-8213/ac082e

    doi:10.3847/2041-8213/ac082e 2021

  6. [6]

    2021 c , , 922, 71, 10.3847/1538-4357/ac0d52

    ---. 2021 c , , 922, 71, 10.3847/1538-4357/ac0d52

    doi:10.3847/1538-4357/ac0d52 2021

  7. [7]

    D., Acernese , F., et al

    Abbott , R., Abbott , T. D., Acernese , F., et al. 2023, Physical Review X, 13, 041039, 10.1103/PhysRevX.13.041039

    doi:10.1103/physrevx.13.041039 2023

  8. [8]

    M., et al

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

    doi:10.3847/2041-8213/acdac6 2023

Show all 51 references
  1. [9]

    F., Arzoumanian , Z., Baker , P

    Alam , M. F., Arzoumanian , Z., Baker , P. T., et al. 2021 a , , 252, 5, 10.3847/1538-4365/abc6a1

    2021 doi

  2. [10]

    2021 b , , 252, 4, 10.3847/1538-4365/abc6a0

    ---. 2021 b , , 252, 4, 10.3847/1538-4365/abc6a0

    2021 doi

  3. [11]

    1998, , 502, 708, 10.1086/305919

    Andersson , N. 1998, , 502, 708, 10.1086/305919

    1998 doi

  4. [12]

    2021, , 507, 2037, 10.1093/mnras/stab2236

    Ashton , G., & Talbot , C. 2021, , 507, 2037, 10.1093/mnras/stab2236

    2021 doi

  5. [13]

    D., et al

    Ashton , G., H \"u bner , M., Lasky , P. D., et al. 2019, , 241, 27, 10.3847/1538-4365/ab06fc

    2019 doi

  6. [14]

    2020, , 498, 5284, 10.1093/mnras/staa2684

    Baronchelli , L., Nandra , K., & Buchner , J. 2020, , 498, 5284, 10.1093/mnras/staa2684

    2020 doi

  7. [15]

    1996, , 312, 675

    Bonazzola , S., & Gourgoulhon , E. 1996, , 312, 675

    1996

  8. [16]

    Buchner , J. 2021. https://github.com/JohannesBuchner/PosteriorStacker

    2021

  9. [17]

    N., Guo , Y

    Chen , S., Caballero , R. N., Guo , Y. J., et al. 2021, , 508, 4970, 10.1093/mnras/stab2833

    2021 doi

  10. [18]

    2002, , 66, 084025, 10.1103/PhysRevD.66.084025

    Cutler , C. 2002, , 66, 084025, 10.1103/PhysRevD.66.084025

    2002 doi

  11. [19]

    T., Hobbs , G

    Edwards , R. T., Hobbs , G. B., & Manchester , R. N. 2006, , 372, 1549, 10.1111/j.1365-2966.2006.10870.x

    2006

  12. [20]

    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, 10.5281/zenodo.4059815

    2020 doi

  13. [21]

    , Arumugam, P

    EPTA Collaboration and InPTA Collaboration , Antoniadis, J. , Arumugam, P. , et al. 2023, A&A, 678, A50, 10.1051/0004-6361/202346844

    2023 doi

  14. [22]

    L., & Morsink , S

    Friedman , J. L., & Morsink , S. M. 1998, , 502, 714, 10.1086/305920

    1998 doi

  15. [23]

    2021, , 507, 116, 10.1093/mnras/stab2048

    Gittins , F., & Andersson , N. 2021, , 507, 116, 10.1093/mnras/stab2048

    2021 doi

  16. [24]

    Goncharov , B. 2021. https://github.com/bvgoncharov/enterprise_warp

    2021

  17. [25]

    2024, mattpitkin/enterprise\_warp: Braking index sampling, braking\_index\_sampling, Zenodo, 10.5281/zenodo.13274448

    Goncharov, B., Zic, A., Reardon, D., & Pitkin, M. 2024, mattpitkin/enterprise\_warp: Braking index sampling, braking\_index\_sampling, Zenodo, 10.5281/zenodo.13274448

    2024 doi

  18. [26]

    2015, Physical Review D, 91, 10.1103/physrevd.91.063007

    Hamil, O., Stone, J., Urbanec, M., & Urbancová, G. 2015, Physical Review D, 91, 10.1103/physrevd.91.063007

    2015 doi

  19. [27]

    R., Millman , K

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

    2020 doi

  20. [28]

    2006, Chinese Journal of Astronomy and Astrophysics Supplement, 6, 189

    Hobbs , G., Edwards , R., & Manchester , R. 2006, Chinese Journal of Astronomy and Astrophysics Supplement, 6, 189

    2006

  21. [29]

    J., et al

    Hobbs , G., Jenet , F., Lee , K. J., et al. 2009, , 394, 1945, 10.1111/j.1365-2966.2009.14391.x

    2009

  22. [30]

    Hunter , J. D. 2007, CSE, 9, 90, 10.1109/MCSE.2007.55

    2007 doi

  23. [31]

    Lam , M. T. 2018, , 868, 33, 10.3847/1538-4357/aae533

    2018 doi

  24. [32]

    Liu , K., Verbiest , J. P. W., Kramer , M., et al. 2011, , 417, 2916, 10.1111/j.1365-2966.2011.19452.x

    2011

  25. [33]

    J., Keith , M

    Liu , X. J., Keith , M. J., Bassa , C. G., & Stappers , B. W. 2019, , 488, 2190, 10.1093/mnras/stz1801

    2019 doi

  26. [34]

    E., Johnston , S., Dunn , L., et al

    Lower , M. E., Johnston , S., Dunn , L., et al. 2021, , 508, 3251, 10.1093/mnras/stab2678

    2021 doi

  27. [35]

    P., & Gunn , J

    Ostriker , J. P., & Gunn , J. E. 1969, , 157, 1395, 10.1086/150160

    1969 doi

  28. [36]

    2000, , 354, 163

    Palomba , C. 2000, , 354, 163

    2000

  29. [37]

    2005, , 359, 1150, 10.1111/j.1365-2966.2005.08975.x

    ---. 2005, , 359, 1150, 10.1111/j.1365-2966.2005.08975.x

    2005

  30. [38]

    M., et al

    Parthasarathy , A., Johnston , S., Shannon , R. M., et al. 2020, , 494, 2012, 10.1093/mnras/staa882

    2020 doi

  31. [39]

    M., et al

    Parthasarathy , A., Bailes , M., Shannon , R. M., et al. 2021, , 502, 407, 10.1093/mnras/stab037

    2021 doi

  32. [40]

    2013, arXiv e-prints, arXiv:1311.3693

    Perrodin , D., Jenet , F., Lommen , A., et al. 2013, arXiv e-prints, arXiv:1311.3693. 1311.3693

    2013 arXiv

  33. [41]

    M., Sip o cz , B

    Price-Whelan , A. M., Sip o cz , B. M., G \"u nther , H. M., et al. 2018, , 156, 123, 10.3847/1538-3881/aabc4f

    2018 doi

  34. [42]

    J., Zic , A., Shannon , R

    Reardon , D. J., Zic , A., Shannon , R. M., et al. 2023, , 951, L6, 10.3847/2041-8213/acdd02

    2023 doi

  35. [43]

    P., Tollerud , E

    Robitaille , T. P., Tollerud , E. J., Greenfield , P., et al. 2013, , 558, A33, 10.1051/0004-6361/201322068

    2013 doi

  36. [44]

    M., Talbot , C., Biscoveanu , S., et al

    Romero-Shaw , I. M., Talbot , C., Biscoveanu , S., et al. 2020, , 499, 3295, 10.1093/mnras/staa2850

    2020 doi

  37. [45]

    Speagle , J. S. 2020, , 493, 3132, 10.1093/mnras/staa278

    2020 doi

  38. [46]

    W., Keane , E

    Stappers , B. W., Keane , E. F., Kramer , M., Possenti , A., & Stairs , I. H. 2018, Philosophical Transactions of the Royal Society of London Series A, 376, 20170293, 10.1098/rsta.2017.0293

    2018

  39. [47]

    R., Baker, P

    Taylor, S. R., Baker, P. T., Hazboun, J. S., Simon, J., & Vigeland, S. J. 2021, enterprise\_extensions. https://github.com/nanograv/enterprise_extensions

    2021

  40. [48]

    R., Baker, P

    Taylor, S. R., Baker, P. T., Hazboun, J. S., et al. 2024, mattpitkin/enterprise\_extensions: Keep float128 precision , float128\_precision, Zenodo, 10.5281/zenodo.13274450

    2024 doi

  41. [49]

    2000, , 319, 902, 10.1046/j.1365-8711.2000.03938.x

    Ushomirsky , G., Cutler , C., & Bildsten , L. 2000, , 319, 902, 10.1046/j.1365-8711.2000.03938.x

    2000

  42. [50]

    D., Haskell , B., Jones , D

    Woan , G., Pitkin , M. D., Haskell , B., Jones , D. I., & Lasky , P. D. 2018, , 863, L40, 10.3847/2041-8213/aad86a

    2018 doi

  43. [51]

    2023, Research in Astronomy and Astrophysics, 23, 075024, 10.1088/1674-4527/acdfa5

    Xu, H., Chen, S., Guo, Y., et al. 2023, Research in Astronomy and Astrophysics, 23, 075024, 10.1088/1674-4527/acdfa5

    2023 doi

Pith tools

Reviewed August 11, 2026 · model on record in the stance chip above.