Pulsar timing in the Galactic Center

Ivan de Martino; Riccardo della Monica

REVIEW 3 major objections 4 minor 3 cited by

Pulsar timing in the Galactic Center

T0 review · 3 major / 4 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read First-order post-Newtonian timing models fail by seconds for pulsars around supermassive black holes, so fully geodesic photon propagation is needed to keep phase connection.

desk verdict Useful proof-of-concept for fully geodesic GC pulsar timing, but the claim that current codes fail is overreach: the 1PN comparison fixes parameters, never fits, and doesn't test real TEMPO2/PINT implementations. read the letter →

arxiv 2501.03912 v2 pith:IGLOAWZP submitted 2025-01-07 gr-qc

Riccardo Della Monica , Ivan de Martino This is my paper · ORCID

classification gr-qc MSC 83C1083C2583C57 PACS 04.20.-q04.25.Nx97.60.Gb

keywords pulsartimingGalacticCentersupermassiveblackholepost-NewtonianapproximationphotonpropagationarrivaltimeresidualsSchwarzschildspacetimegravitationallensing

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 argues that standard pulsar timing models, which compute photon travel time at first post-Newtonian order and assume straight-line light paths, fail when the pulsar orbits a supermassive black hole like Sgr A* at the Galactic Center: the error in predicted pulse arrival time can reach about two seconds, far above the 100-microsecond accuracy goal of future telescopes. The authors propose replacing the approximate delay formulas with a fully relativistic computation that solves the photon geodesic numerically for each emitted pulse and inverts the resulting emission-time-to-arrival-time relation. They demonstrate the model on toy pulsar orbits of increasing compactness and show that misestimating parameters such as black hole mass, pulsar period, or orbital elements produces phase-dependent residuals detectable within months to a few years. If correct, future Galactic Center pulsar timing will require geodesic timing models rather than the post-Newtonian formulas now used in standard codes.

What carries the argument

The load-bearing object is the numerical solution of the emitter-observer problem for null geodesics in a static, spherically symmetric spacetime. For a chosen emission event, the impact parameter b is found by solving the angular integral that links the emitter and observer positions, and the coordinate travel time is the integral of dt/dr along the direct or indirect photon path; the paper composes this with a geodesically integrated pulsar orbit, samples emission times in proper time, computes the relativistic propagation time at each sample, and inverts the interpolated TOA(τ) function to define τ(TOA). This replaces the 1PN delay sum of Eq. (27), whose Rømer and Shapiro terms assume straight-line propagation and whose geometric delay is only a weak-lensing correction.

What would settle it

Take the Toy 0 or S2-like orbit, generate noise-free TOAs from the geodesic pipeline, and fit them with the 1PN timing model of Eq. (27): a residual structure peaking near superior conjunction at the predicted 1-second-to-2-second level would confirm the paper's claim, whereas residuals below 100 microseconds across the full orbit would falsify it.

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Extended reading notes

Core claim

The paper's central claim is that the 1PN timing model—straight-line Rømer delay, Shapiro delay, and weak-lensing geometric delay—cannot reproduce the relativistic photon propagation time for a pulsar orbiting a supermassive black hole. For a circular orbit at 100 gravitational radii, the discrepancy reaches about 2 seconds at superior conjunction, and for an S2-like orbit at the same semimajor axis as Sgr A* the discrepancy is about 0.1 seconds, three orders of magnitude above the nominal 100-microsecond sensitivity. The authors build a timing pipeline in which the pulsar orbit is integrated geodesically in Schwarzschild spacetime in harmonic coordinates, the emitter-observer problem is solved numerically for the primary photon image, and the monotonic TOA(τ) map is inverted by spline interpolation to generate phase-connected residuals. They then use this pipeline to show how misestimating intrinsic and orbital parameters imprints detectable, growing residuals, implying that a single pulsar could constrain the black hole mass far better than current stellar-orbit measurements.

Load-bearing premise

The paper's conclusion that current timing codes fail at the Galactic Center rests on the assumption that the simplified 1PN delay formula of Eq. (27), with straight-line Rømer and Shapiro delays plus a weak-lensing geometric term, is what existing timing codes actually implement for a test particle around a point mass; if real codes use more complete post-Keplerian models, the claimed failure is not established.

Editorial extensions

If this is right

If the central claim is correct, 1PN timing residuals for a pulsar-SMBH system can exceed the pulsar period, so phase connection is lost and standard TOA fitting breaks down.
For an S2-like orbit around Sgr A*, the 1PN-versus-geodesic discrepancy is about 0.1 seconds, still three orders of magnitude above the 100-microsecond accuracy goal, so even mildly relativistic orbits require geodesic modeling.
Misestimating the black hole mass or the semimajor axis by about one part in 10^10 produces residuals above the detection threshold within months to a few years, depending on orbital compactness, which would sharply improve current Sgr A* mass measurements.
Extending the geodesic timing model to a rotating black hole spacetime is the stated next step, and in spherical symmetry the method already applies to any spacetime, not only Schwarzschild.
Indirect photon paths that dip toward the photon sphere are included in the propagation-time calculation, so the model captures strong lensing effects that 1PN formulas omit entirely.

Reading between the lines

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

Beyond the paper's claims: the same 1PN delay formulas would also corrupt timing of pulsars orbiting intermediate-mass black holes or dense cluster centers, so the geodesic pipeline likely has applications well outside the Galactic Center.
Beyond the paper's claims: the single-parameter sensitivity study hints at strong degeneracies between orbital parameters and the black hole mass; a full Bayesian fit on simulated TOAs would show how tightly these parameters can actually be separated rather than independently bounded.
Beyond the paper's claims: the claim that current codes fail depends on Eq. (27) faithfully representing them; testing the same orbit with a complete post-Keplerian timing model would either confirm the failure or localize it to the simplified straight-line delay model the paper adopts.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

Quick take: this is a useful proof-of-concept for building a fully geodesic pulsar timing model around Sgr A*, and the core numerical result—that simple 1PN delay formulas miss by seconds for tight orbits—is solid. The headline claim that current timing codes fail is not established by the comparison as written.

What's genuinely new: they take their earlier emitter-observer solver [37] and turn it into a phase-connected timing pipeline, including interpolation of the TOA(τ) map and residual generation. The validation against the Hackmann–Dhani exact circular-orbit solution (Fig. 2, agreement at 1e-7 s) is the right kind of check and gives real weight to the 1PN-vs-geodesic discrepancy. The sensitivity maps in Fig. 5 and Table II are a reasonable qualitative tool for thinking about what such pulsars could measure; the order-of-magnitude claims about mass constraints are plausible if the pipeline is right.

The soft spots are real and concentrated in Section IV B. The 'current timing codes' comparison uses Eq. (27), a fixed-parameter sum of straight-line Rømer, Shapiro, and geometric delays. That is not what TEMPO2/PINT compute: they use Damour–Deruelle post-Keplerian orbital dynamics, Einstein delay, aberration, and a different Shapiro parametrization—and, critically, they fit the parameters to minimize residuals. The authors fix the parameters at input values and never generate geodesic TOAs and then fit them with a standard code. A real fit could absorb part of the ~2 s discrepancy by shifting mass or orbital elements, so the phase-connection loss is overclaimed. Also, the sensitivity analysis varies one parameter at a time, so the 'tremendous constraining power' phrasing is ahead of an actual covariance or Bayesian inference demonstration. No code or data is released, and the pipeline depends on the authors' own solver from [37], which is not public; that limits reproducibility but is not a fatal flaw given the external validation.

The paper knows it's a work in progress—the conclusion says so, which I take as honesty rather than an excuse.

Who it's for: people planning GC pulsar searches or building timing codes for extreme-mass-ratio binaries. They will get a clear picture of the issues and a pipeline to build on. It deserves a serious referee, but the referee should ask for a realistic comparison to existing codes (or a fitted version of the 1PN model), a proper covariance/full-inference study, and public code/data before the strong claims are accepted.

Referee Report

3 major / 4 minor

Summary. The paper proposes a fully geodesic timing pipeline for future pulsars orbiting Sgr A*. For a given set of orbital and intrinsic parameters, it integrates the pulsar geodesic, solves the emitter-observer problem for each emitted photon, and inverts the TOA-versus-proper-time relation by spline interpolation to build a timing model. The authors compare this model with a fixed-parameter 1PN delay model composed of Rømer, Shapiro, and geometric delays (Eq. 27), reporting discrepancies of order seconds for a 100M circular orbit and much larger discrepancies for tighter and more eccentric orbits (Figure 6). They also compute residuals caused by one-at-a-time misestimations of parameters such as the pulsar period, spin-down rate, SMBH mass, semi-major axis, eccentricity, and inclination, and use these to argue for extreme future constraining power on the SMBH mass and orbital parameters.

Significance. If the comparison were made against the actual timing models used by TEMPO2/PINT and included a full parameter fit, this would be an important proof-of-concept for strong-field pulsar timing around Sgr A*. The numerical solver is a clear strength: it is validated against the independent Hackmann-Dhani exact solution for a circular-orbit case, and the framework is general enough to be extended beyond Schwarzschild to other spherically symmetric spacetimes. However, the paper's headline claim that 'current timing codes' fail is not yet established, because the 1PN comparison uses a simplified fixed-parameter delay sum rather than a fitted post-Keplerian model, and the sensitivity analysis is based on one-at-a-time parameter offsets rather than a correlated fit. These issues are load-bearing for the abstract and conclusions, so the paper needs a revision that either sharpens the claims or adds the missing comparison.

major comments (3)

[Section IV B, Eq. (27), Figure 6] The comparison labeled 'current timing codes' tests a fixed-parameter sum ΔtPN = ΔtR + ΔtSh + Δtgeo, not the timing model implemented in TEMPO2/PINT. Standard codes use the Damour-Deruelle post-Keplerian model, which includes relativistic orbital dynamics, the Einstein delay γ sin u, aberration, and a different Shapiro delay parametrization, and they fit all parameters globally to minimize the residuals. A fixed-parameter comparison cannot determine whether a real timing code loses phase connection, because the best-fit mass, semi-major axis, inclination, and pulsar period can absorb part of the delay discrepancy. Please either reframe the claim as applying to the simplified 1PN model only, or generate geodesic TOAs and fit them with TEMPO2/PINT to test the 'current codes fail' conclusion directly.
[Section IV A, Figure 5, Table II] The claimed constraining power is derived from varying one parameter at a time while keeping all other parameters fixed, and no noise is included in the simulated residuals. In a real fit, parameter covariances and TOA uncertainties strongly affect the posterior, so the percentages reported in Table II should not be presented as projected measurement precisions. Please either run a full correlated fit (e.g., a Fisher-matrix or Markov-chain Monte Carlo forecast) or explicitly label these numbers as qualitative one-parameter sensitivities.
[Section II, Eq. (20), Figure 2] The text states that the exact Hackmann-Dhani solution is for a non-inclined circular orbit, but Figure 2 describes the toy model as having inclination i = 60°. Please clarify whether Eq. (20) applies to an inclined orbit; if it is restricted to zero inclination, the validation should be repeated for the inclined case or the caption corrected. This is the main external validation of the numerical pipeline, so the configuration must be unambiguous.

minor comments (4)

[Section IV B] The sentence 'The results of this analysis are shown in Figure ??' contains an unresolved placeholder; the actual figure appears as Figure 6 and should be cited explicitly in the text.
[Throughout] There are several typographical errors, including 'Galctic' in the introduction to Section III A, 'Earth-bsaed' in the Conclusions, and 'Seciton' in the Conclusions; these should be corrected in a final proofreading pass.
[Figure 2] The inset axis label '5|Num-Ex| (10^-7 s)' is unclear; please label the quantity explicitly, for example '|Numerical - Exact| (10^-7 s)', and ensure the units are unambiguous.
[Section III B] The statement that Ndense roughly 50 times Ndata provides a robust reconstruction of TOA(τ) is not accompanied by a quantitative convergence test; the paper notes that increasing Ndense gives improvements below sensitivity, but showing the interpolation error as a function of Ndense would make the choice more defensible.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the geodesic-vs-1PN comparison is a direct fixed-parameter computation validated against an independent exact solution, and the sensitivity curves are parametric derivatives rather than fitted predictions.

full rationale

The central derivation is self-contained and does not reduce to its inputs. The paper compares the fully relativistic photon propagation time from Eqs. (10)-(16) with the 1PN delay formula in Eq. (27), using fixed toy-model parameters (Table I) and no fitting to data. The 1PN formula is an independent input taken from the cited timing literature, not derived from the geodesic result, so the large residuals reported in Figure 2 are a computed discrepancy rather than a fitted or assumed one. The numerical emitter-observer solver, which is the one load-bearing element inherited from the authors' prior work [37], is validated in this paper against the independent closed-form solution of Hackmann and Dhani [41]; Figure 2 reports agreement within the adopted numerical tolerance of 10^-6 s. That external exact solution provides an anchor outside the present paper's own pipeline, so the self-citation does not carry the argument by itself. The parameter-sensitivity residuals in Figure 5 and Table II are explicitly qualitative parametric derivatives: they show how the same timing model responds when one parameter is varied while the others are held fixed, which is a Fisher-type sensitivity statement rather than a prediction fitted to observations. No self-definitional reduction, fitted-input-called-prediction, imported uniqueness theorem, or ansatz-smuggled-via-citation pattern is present. One non-circular concern should be noted: Section IV B identifies Eq. (27) with what is 'implemented in all current codes devoted to TOA analysis' [43, 50, 55], whereas standard codes include post-Keplerian orbital dynamics, the Einstein delay, aberration, and parameter fitting; this is a possible overreach in the 'current codes fail' wording, but it concerns the adequacy of the comparison, not the logical circularity of the derivation. Therefore the circularity score is 0.

Assumptions & free parameters 6 free parameters · 5 assumptions · 0 invented entities

The central claim rests on the numerical geodesic solver (validated against an exact solution) and on the assumption that Eq. (27) represents current timing codes, which is not established. The scenario parameters (M, r_o, P, toy orbits, 100 microsecond threshold) are inputs chosen by hand, not fitted. No new entities are introduced.

free parameters (6)

Sgr A* mass M = 4e6 M_sun (assumed standard value)
Central mass for all toy models and TOA computations. Sensitivity analysis perturbs it; not fitted in this paper.
Observer distance r_o = 1e9 M
Assumed distance of the Earth-based observer. TOAs and delays scale with this choice; the paper uses it throughout.
Pulsar spin period P = 2 s (exemplary value)
Used in the phase model and in the pulse-count calculation (Npulses). The failure-of-1PN claim depends on comparing discrepancies to P.
Spin-down rate Pdot = Assumed small; perturbed by 0.1% in Figure 5
Included in the phase model; the sensitivity to Pdot is a standard timing effect.
Toy model orbital parameters = Table I: a = 1025, 175.4, 43.8, 5 AU; e = 0.88, 0.80, 0.80, 0.786
Hand-selected to scan increasing strong-field strength Gamma. Quantitative discrepancies depend on these values; the qualitative failure of 1PN does not.
TOA accuracy threshold = 100 microseconds
SKA nominal timing accuracy from Eatough et al. 2015 [39]. All detection-time estimates in Table II depend on this threshold.

assumptions (5)

domain assumption The pulsar is a test particle following a timelike geodesic in the Schwarzschild spacetime of the SMBH.
Used in Sections II and III for orbital motion. Neglects the pulsar mass, the SMBH spin (Kerr), and all other masses; the paper acknowledges in Section V that rotating black holes are future work.
standard math Photon propagation can be restricted to the equatorial plane without loss of generality.
Section II A: spherical symmetry allows this for Schwarzschild, but it breaks for a spinning Sgr A*.
domain assumption The Earth-based observer is static at a coordinate distance r_o = 1e9 M and measures coordinate time.
Section II A. Solar System and interstellar delays are switched off and are assumed to be addable linearly a posteriori (Section III B).
ad hoc to paper The 1PN photon-delay model of Eq. (27) represents what current pulsar timing codes implement.
Load-bearing for the 'failure of current timing codes' claim in Sections I and IV B. Standard codes use the Damour-Deruelle post-Keplerian model, not this simplified sum.
ad hoc to paper Spline interpolation of TOA(tau) with N_dense about 50 times N_data preserves all timing information.
Section III B step (d). The density factor is an empirical choice without a formal error bound; the sensitivity results depend on it.

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Cite this review

Pith. "Pith review of Pulsar timing in the Galactic Center." pith.science (2026). https://pith.science/paper/IGLOAWZP

@misc{pith2026250103912,
  author       = {Pith},
  title        = {Pith review of: Pulsar timing in the Galactic Center},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/IGLOAWZP}},
  note         = {Machine review of arXiv:2501.03912}
}

read the original abstract

We propose a novel approach which implements the relativistic calculations of the photon travel time into a robust timing model for pulsars orbiting supermassive black holes. We demonstrate that timing models relying on the lowest-order (1PN) post-Newtonian approximation can produce significant discrepancies in strong-field configurations, affecting the predicted relativistic times of arrival at an Earth-based observatory. We also show how a misestimation of the pulsar parameters can lead to the appearance of phase-dependent residual, which hints at a tremendous constraining power of the binary and intrinsic parameters for timing observations of potential pulsars at the Galactic Center.

Figures

Figures reproduced from arXiv: 2501.03912 by the authors.

**Figure 2.** Residuals (in seconds) between the post-Newtonian [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗

**Figure 3.** A schematic representation of the timing technique used in popular tools like [PITH_FULL_IMAGE:figures/full_fig_p007_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: A schematic overview of the timing technique that we propose in this work. For a given set of intrinsic, astrometric, [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]

**Figure 5.** Figure 5: Timing residuals over 10 orbits for the Toy 2 model (see Table I), generated by slightly modifying each intrinsic or [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]

**Figure 6.** Figure 6: Maximum discrepancy over one orbital period between the post-Newtonian approximation and the fully relativistic [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 3 Pith papers

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A 1PN numerical timing model shows that a pulsar orbiting Sgr A* produces large, distinctive post-fit residuals from an IMBH companion, enabling constraints that fill gaps left by existing S-star and proper-motion bounds.
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astro-ph.HE 2026-02 conditional novelty 6.0 of 10

A 2PN-accurate pulsar timing model for a pulsar orbiting Sgr A* includes proper motion, aberration, and red noise, and forecasts that proper motion will not break the SMBH spin degeneracy for short-period orbits.
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Pulsar time delays are computed for Kerr, deformed Kerr (Johannsen-Psaltis), and rotating Janis-Newman-Winicour spacetimes, producing model-dependent signatures of up to about 15 seconds.

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