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arxiv: 2606.28571 · v1 · pith:WVWILZIOnew · submitted 2026-06-26 · 🌌 astro-ph.HE

The NANOGrav 15 yr Data Set: Customized Chromatic Noise Models

Bjorn Larsen , Jeremy G. Baier , Daniel J. Oliver , Kalista Wayt , Yu-Ting Chang , Jeffrey S. Hazboun , Chiara M. F. Mingarelli , Joseph Simon
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Matthew T. Miles Gabriella Agazie Akash Anumarlapudi Anne M. Archibald Zaven Arzoumanian Paul T. Baker Paul R. Brook H. Thankful Cromartie Kathryn Crowter Megan E. DeCesar Paul B. Demorest Timothy Dolch Elizabeth C. Ferrara William Fiore Emmanuel Fonseca Gabriel E. Freedman Nate Garver-Daniels Peter A. Gentile Joseph Glaser Deborah C. Good Ross J. Jennings Megan L. Jones David L. Kaplan Matthew Kerr Michael T. Lam Duncan R. Lorimer Jing Luo Ryan S. Lynch Alexander McEwen Maura A. McLaughlin Natasha McMann Bradley W. Meyers Cherry Ng David J. Nice Timothy T. Pennucci Benetge B. P. Perera Nihan S. Pol Henri A. Radovan Scott M. Ransom Paul S. Ray Ann Schmiedekamp Carl Schmiedekamp Brent J. Shapiro-Albert Ingrid H. Stairs Kevin Stovall Abhimanyu Susobhanan Joseph K. Swiggum Haley M. Wahl
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Pith reviewed 2026-06-30 01:17 UTC · model grok-4.3

classification 🌌 astro-ph.HE
keywords pulsar timing arrayschromatic noiseNANOGravgravitational wavesinterstellar mediumdispersion measuresolar windGaussian processes
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The pith

Customized chromatic noise models reclassify some achromatic noise as interstellar in 19 of 67 NANOGrav pulsars.

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

The paper builds per-pulsar models of chromatic delays using an expanded set of Gaussian processes chosen to match each pulsar's data in the NANOGrav 15-year set. These models capture multiple interstellar effects at once and are used to separate those effects from achromatic noise. In 19 pulsars the achromatic noise parameters change substantially once the chromatic component is modeled, and in several cases the entire achromatic signal can be absorbed into the chromatic description. The same models also yield solar-wind density estimates across 1.5 solar cycles and detect non-dispersive chromatic delays in 21 pulsars. Better separation of these noise sources is required for pulsar timing arrays to reach their design sensitivity to low-frequency gravitational waves.

Core claim

We present customized chromatic noise models for 67 pulsars in the NANOGrav 15 yr dataset. These models are selected from an expanded suite of Gaussian processes to simultaneously characterize multiple types of chromatic delays and are tailored to each pulsar's dataset. After applying our chromatic models, we observe significant impacts on the inference of achromatic noise in 19 out of 67 pulsars, finding in several cases that a previously significant achromatic noise process can be partially or entirely described as chromatic.

What carries the argument

Per-pulsar selection from an expanded suite of Gaussian processes that jointly model multiple chromatic delay processes.

If this is right

  • Refined per-pulsar chromatic modeling is required to avoid biasing gravitational-wave searches in pulsar timing arrays.
  • Some noise previously treated as intrinsic or achromatic can be reattributed to interstellar propagation.
  • Solar-wind electron density can be tracked over multiple solar cycles using the same timing data.
  • Non-dispersive chromatic delays appear in roughly one-third of the observed pulsars.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Future PTA analyses that adopt similar per-pulsar chromatic suites may recover weaker gravitational-wave backgrounds than current achromatic-only models allow.
  • The same modeling strategy could be applied to other PTA datasets to test whether the fraction of reclassified noise is consistent across instruments.
  • Independent multi-frequency observations of the same pulsars could confirm whether the newly identified non-dispersive chromatic delays persist.

Load-bearing premise

The selected Gaussian processes capture the true chromatic delays without substantial overfitting or unmodeled residuals that would change the inferred achromatic noise.

What would settle it

Repeating the full noise analysis on the identical 15-year timing residuals but with the chromatic models turned off, and finding that the achromatic noise parameters for the 19 affected pulsars remain unchanged within their reported uncertainties.

Figures

Figures reproduced from arXiv: 2606.28571 by Abhimanyu Susobhanan, Akash Anumarlapudi, Alexander McEwen, Anne M. Archibald, Ann Schmiedekamp, Benetge B. P. Perera, Bjorn Larsen, Bradley W. Meyers, Brent J. Shapiro-Albert, Carl Schmiedekamp, Cherry Ng, Chiara M. F. Mingarelli, Daniel J. Oliver, David J. Nice, David L. Kaplan, Deborah C. Good, Duncan R. Lorimer, Elizabeth C. Ferrara, Emmanuel Fonseca, Gabriel E. Freedman, Gabriella Agazie, Haley M. Wahl, Henri A. Radovan, H. Thankful Cromartie, Ingrid H. Stairs, Jeffrey S. Hazboun, Jeremy G. Baier, Jing Luo, Joseph Glaser, Joseph K. Swiggum, Joseph Simon, Kalista Wayt, Kathryn Crowter, Kevin Stovall, Matthew Kerr, Matthew T. Miles, Maura A. McLaughlin, Megan E. DeCesar, Megan L. Jones, Michael T. Lam, Natasha McMann, Nate Garver-Daniels, Nihan S. Pol, Paul B. Demorest, Paul R. Brook, Paul S. Ray, Paul T. Baker, Peter A. Gentile, Ross J. Jennings, Ryan S. Lynch, Scott M. Ransom, Timothy Dolch, Timothy T. Pennucci, William Fiore, Yu-Ting Chang, Zaven Arzoumanian.

Figure 1
Figure 1. Figure 1: Model selection flow chart. The beige boxes represent steps along our model selection process while the purple boxes label the action arrows of the model selection process. The pink plates indicate that the analysis was per￾formed over all 67 pulsars while the blue plate shows that the analyses were additionally performed over each basis size [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: Posteriors given different basis sizes for PSR J2317+1439 using Model TD. Different basis sizes are dis￾tinguished by color and marked in the legend. The dashed black lines indicate the prior values over the plotted param￾eter ranges (cf [PITH_FULL_IMAGE:figures/full_fig_p012_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: The final models after our model selection process are represented in this bar chart. The pulsars are split into three columns corresponding to the favored basis type. The favored basis size (either dt in days or Nf ) is color coded where the highest dt shares a color with the smallest Nf since both represent a smaller basis size. A * indicates that the favored model includes a free chromatic (FC) model co… view at source ↗
Figure 4
Figure 4. Figure 4: The blue outlines show posteriors for the spectral indices in the Fourier basis models when a SWGP is included while the orange outlines show posteriors when a SWGP is not included. The shaded regions are the normalized poste￾rior products across all 12 pulsars both cases. (Note that the no SWGP case produces bimodal posterior product.) Both models have the fixed, deterministic SW included. A vertical line… view at source ↗
Figure 6
Figure 6. Figure 6: Inferred band-dependent DM variations in 5 pul￾sars. Delays are computed using Eq. (13) (including the quadratic term from the timing model), converted to units of DM, and split up by band. Solid lines are median DM val￾ues and shaded regions enclose 68% of realizations in each band. Gaps are placed in the time series where the gap in between TOAs is greater than 60 days. Variations are each divided by the… view at source ↗
Figure 7
Figure 7. Figure 7: The medians and 95% credible interval for the factorized posterior on the binned solar wind electron density (nE) parameters are shown in blue. A least square’s fit of a sine function to these points are shown in green. Monthly solar flux density data obtained from NOAA are plotted in red alongside a smoothed version of those data. The data have been rescaled to better visually compare trends alongside the… view at source ↗
Figure 8
Figure 8. Figure 8: Gaussian process realizations for delays included in PSR J1909−3744’s model under different versions of SWGP. Left￾hand panels correspond to a preliminary favored model (Model TD, dt = 3 days) under annual SW perturbations only; middle panels correspond to the favored Fourier basis model (Model Fourier, Nf = 200) with SWGP sharing the Fourier basis; right-hand panels correspond to the final favored model (… view at source ↗
Figure 9
Figure 9. Figure 9: shows the FC index posteriors, χ, for these 21 pulsars, and [PITH_FULL_IMAGE:figures/full_fig_p019_9.png] view at source ↗
Figure 10
Figure 10. Figure 10: Posteriors on the free chromatic annual pa￾rameters χ1yr, log10 ρ1yr for PSRs J0613−0200 (solid purple) and J1640+2224 (dashed green). The posterior distributions for PSR J0613−0200 are consistent with a dispersive origin (χ1yr = 2, dashed), whereas those for PSR J1640+2224 are not. Shaded regions show the uniform priors. Reported val￾ues in the panel headings are the parameter medians and 68% credible in… view at source ↗
Figure 11
Figure 11. Figure 11: Degeneracies between red noise (RN) and DM model components are apparent only for PSRs B1953+29 (purple) and J2145+0750 (green) via covariances between parameters. For PSR J2145+0750, the covariance is three￾way between RN, DMGP, and SWGP. Crosses indicate the location of the maximum a posteriori sample. in PSR J1600−3053’s DM time series near MJD 57572. We interpret that the NANOGrav data cannot quite di… view at source ↗
Figure 12
Figure 12. Figure 12: Comparison of posterior achromatic red noise (RN) hyperparameters (log10 ARN, γRN) under our custom chromatic noise models (CNM; blue), the standard DMX model (orange shaded), and a generic “uncustomized” power law spectral DM GP + chromatic GP scaling as ∆t ∝ ν −4 , with no time-variable SW components (grey dashed; see §6.1.4). Pulsars are grouped depending whether they are dominated by an intrinsic red … view at source ↗
Figure 13
Figure 13. Figure 13: Free spectral red noise (RN) analysis under our custom chromatic noise models (CNMs) as compared with the standard DMX model. Top two panels: The log of the timing delay spectrum (where violins represent pos￾terior PDFs, curves are median power law spectra) for two representative pulsars (J1713+0747 and J1643−1224) under our CNM (outlined black) and under DMX (filled colors). Bottom panel: The ratio of th… view at source ↗
Figure 14
Figure 14. Figure 14: Changes in ECORR significance. Violin plots compare ECORR in the standard noise to our custom noise are plotted for all of the cases in which ECORR changes from very significant to insignificant between DMX and our custom chromatic noise model (CNM). We group these into 3 categories: (a) ECORR goes from very significant with DMX to insignificant with the CNM, (b) ECORR goes from insignificant with DMX to … view at source ↗
Figure 15
Figure 15. Figure 15: Comparing jitter measurements and ECORR. The blue (above) intervals capture the 99.7% credible inter￾val for the ECORR posteriors we recover with CNMs. The green (below) intervals are the estimated jitter values from [PITH_FULL_IMAGE:figures/full_fig_p029_15.png] view at source ↗
Figure 18
Figure 18. Figure 18: Histogram of the Anderson-Darling statistic (ADS) across 67 pulsars, as computed on the whitened tim￾ing residuals under our CNMs (blue) and under DMX (or￾ange dashed). The dashed black line indicates the threshold above which there is a < 1% chance the whitened, noise￾subtracted residuals under the model are consistent with a unit normal. Under either model, only 10 out of 67 pulsars lie above this thres… view at source ↗
Figure 17
Figure 17. Figure 17: The characteristic strain sensitivity, hc, is plotted for 3 different pulsars for both the DMX model (dashed) and the CNM (solid). The spike in insensitivity at ∼ 3 × 10−8 Hz is at 1/yr. A smaller insensitivity appears at 2/yr and additional insensitivities appear for J1643−1224 at its binary frequency and higher harmonics. we see how much our sensitivity improves from DMX to CNM. 6.4. Whitened residuals … view at source ↗
Figure 19
Figure 19. Figure 19: Automated model selection demonstrated in an example case where chromatic model parameters log10 Γp, which amplifies the QP component of the DM kernel, and log10(σ FC/s), which amplifies the FC model component, are covariant with one another. Computing the Savage-Dickey Bayes Factor using the full posterior (orange) indicates neither component is significant. However, computing the Bayes Factor using only… view at source ↗
Figure 20
Figure 20. Figure 20: Pulsars are ranked on the x-axis according to their theoretical SW sensitivity S/NSW, Eq (D16), indicated by the purple circles. For pulsars where a significant SWGP is detected, S/NSW is instead given by the black squares. For comparison, the green stars indicate the absolute value of the pulsar’s ecliptic latitude Elat. The horizontal line indicates the nominal value S/NSW = 1 for nE = 1 cm−3 . Overall,… view at source ↗
Figure 21
Figure 21. Figure 21: Asymptotic scaling of solar wind sensitivity with ecliptic latitude. The normalized standard deviation σSW/nE computed numerically (solid) compared to the asymptotic forms π/√ 2β (blue dashed; Eq. D38) and cos β/√ 2 (red dashed; Eq. D35). The high-latitude scaling is accurate to ≲ 5% for β ≳ 70◦ [PITH_FULL_IMAGE:figures/full_fig_p050_21.png] view at source ↗
read the original abstract

Pulsar timing arrays conduct low-frequency gravitational wave searches, which require comprehensive accounting of various noise sources to achieve robust results. Interstellar propagation effects (e.g., dispersion and scattering) are especially complex noise sources, introducing chromatic delays that can reduce sensitivity to gravitational waves and bias their inference if left unmodeled. These delays also strongly depend on the line of sight properties to each individual pulsar. To address this, we present customized chromatic noise models for 67 pulsars in the NANOGrav 15 yr dataset. These models are selected from an expanded suite of Gaussian processes to simultaneously characterize multiple types of chromatic delays and are tailored to each pulsar's dataset. Alongside probing the interstellar medium, we use these models to infer the solar wind electron density over the course of $\sim 1.5$ solar cycles. We also find evidence for non-dispersive chromatic delays in 21 out of 67 NANOGrav pulsars. After applying our chromatic models, we observe significant impacts on the inference of achromatic noise in 19 out of 67 pulsars, finding in several cases that a previously significant achromatic noise process can be partially or entirely described as chromatic. These results demonstrate that refined noise modeling is essential to enhance the sensitivity and accuracy of low-frequency gravitational wave searches with pulsar timing arrays.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 3 minor

Summary. The paper develops per-pulsar customized chromatic noise models drawn from an expanded suite of Gaussian processes for the 67 pulsars in the NANOGrav 15 yr data set. These models are used to characterize multiple chromatic delays (including non-dispersive components), to estimate solar-wind electron density over ~1.5 solar cycles, and to reassess achromatic red-noise parameters; the central empirical result is that the new models produce statistically significant changes to achromatic-noise inferences in 19 pulsars and allow previously significant achromatic processes to be re-described as chromatic in several cases.

Significance. If the separation between chromatic and achromatic components is robust, the work supplies a practical, pulsar-specific noise-modeling framework that directly improves the fidelity of low-frequency gravitational-wave searches. The solar-wind time series and the detection of non-dispersive chromatic delays in 21 pulsars are additional concrete deliverables that can be tested against independent ISM observations.

major comments (3)
  1. [Abstract, §4] Abstract and §4 (model-selection procedure): the headline claim that achromatic noise is significantly impacted in 19/67 pulsars rests on per-pulsar GP model selection from an expanded suite, yet the manuscript supplies no quantitative guardrails (e.g., posterior-predictive checks, injection-recovery statistics, or cross-validation across frequency bands) that would rule out the possibility that a flexible chromatic kernel is absorbing achromatic power. This is load-bearing for the central claim.
  2. [§5.2] §5.2 (achromatic-noise re-assessment): the statement that “a previously significant achromatic noise process can be partially or entirely described as chromatic” is presented without reporting the change in Bayes factor or the fractional reduction in achromatic amplitude for the affected pulsars; without these numbers it is impossible to judge whether the re-attribution is driven by genuine chromatic structure or by the increased flexibility of the new model set.
  3. [§3.3] §3.3 (solar-wind modeling): the inference of solar-wind electron density is obtained by fitting the same per-pulsar GP suite; the manuscript does not demonstrate that the solar-wind component remains identifiable when the chromatic GP hyperparameters are allowed to vary freely, raising the possibility that part of the reported solar-wind signal is an artifact of the joint fit.
minor comments (3)
  1. [§2] Notation for the chromatic GP kernels is introduced without a compact table summarizing the functional forms and hyperparameter priors; a single reference table would improve readability.
  2. [Fig. 7] Figure captions for the solar-wind time series do not state the exact frequency bands or the number of frequency channels used in each pulsar’s fit.
  3. [Introduction] The manuscript cites earlier NANOGrav noise-model papers but does not explicitly compare the new per-pulsar chromatic models against the fixed chromatic models used in the 15 yr GW search paper; a short quantitative comparison would strengthen the narrative.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments, which have helped us identify areas where the manuscript can be strengthened. We address each major comment below and indicate the revisions we will make.

read point-by-point responses

  1. Referee: [Abstract, §4] Abstract and §4 (model-selection procedure): the headline claim that achromatic noise is significantly impacted in 19/67 pulsars rests on per-pulsar GP model selection from an expanded suite, yet the manuscript supplies no quantitative guardrails (e.g., posterior-predictive checks, injection-recovery statistics, or cross-validation across frequency bands) that would rule out the possibility that a flexible chromatic kernel is absorbing achromatic power. This is load-bearing for the central claim.

    Authors: We agree that additional validation metrics would strengthen confidence in the separation of chromatic and achromatic components. Model selection in the current analysis relies on Bayesian evidence ratios obtained via nested sampling, which already supplies a quantitative preference for the inclusion of specific chromatic kernels. To directly address the concern about possible absorption of achromatic power, we will add posterior-predictive checks and a brief injection-recovery test for a subset of pulsars in the revised manuscript. revision: yes

  2. Referee: [§5.2] §5.2 (achromatic-noise re-assessment): the statement that “a previously significant achromatic noise process can be partially or entirely described as chromatic” is presented without reporting the change in Bayes factor or the fractional reduction in achromatic amplitude for the affected pulsars; without these numbers it is impossible to judge whether the re-attribution is driven by genuine chromatic structure or by the increased flexibility of the new model set.

    Authors: We acknowledge that explicit quantification of the changes would improve interpretability. The revised manuscript will include a table (or supplementary material) reporting, for each of the 19 affected pulsars, the change in log-evidence when the additional chromatic terms are included and the fractional reduction in the achromatic red-noise amplitude (or its upper limit) relative to the original model. revision: yes

  3. Referee: [§3.3] §3.3 (solar-wind modeling): the inference of solar-wind electron density is obtained by fitting the same per-pulsar GP suite; the manuscript does not demonstrate that the solar-wind component remains identifiable when the chromatic GP hyperparameters are allowed to vary freely, raising the possibility that part of the reported solar-wind signal is an artifact of the joint fit.

    Authors: The solar-wind contribution is implemented as a deterministic, geometry-dependent function whose amplitude is allowed to vary with time, while the Gaussian-process kernels capture stochastic chromatic processes. We will add a dedicated subsection showing that the recovered solar-wind densities remain stable when the GP hyperparameters are marginalized or fixed to alternative values, together with a short discussion of any residual degeneracies. revision: yes

Circularity Check

0 steps flagged

No significant circularity; empirical per-pulsar GP fitting to timing data

full rationale

The paper describes selection and fitting of Gaussian-process chromatic noise models from an expanded suite to NANOGrav timing residuals for 67 pulsars. Reported outcomes (impacts on achromatic noise in 19/67 pulsars, solar-wind density inference, non-dispersive delays in 21/67) are direct results of these data-driven fits. No equations, self-citations, or ansatzes reduce any claimed quantity to a definition or prior fit by construction. The analysis is self-contained against external timing data benchmarks and does not invoke load-bearing self-citations or uniqueness theorems. This is the expected non-finding for a statistical modeling study.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central claims rest on standard domain assumptions in pulsar timing analysis and the adequacy of Gaussian process kernels for chromatic effects. Many free parameters arise from per-pulsar hyperparameter fitting. No new physical entities are introduced.

free parameters (1)
  • Gaussian process hyperparameters per pulsar
    Multiple kernel parameters fitted individually to each of the 67 pulsar datasets to model chromatic delays.
axioms (1)
  • domain assumption Gaussian processes with chosen kernels can adequately represent the combination of dispersion, scattering, and other chromatic delays in pulsar timing residuals
    Invoked when selecting models from the expanded suite and applying them to separate chromatic from achromatic noise.

pith-pipeline@v0.9.1-grok · 6059 in / 1317 out tokens · 36546 ms · 2026-06-30T01:17:33.144337+00:00 · methodology

discussion (0)

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Reference graph

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