MeerKAT PTA Timing & Noise Analysis

• Pulsar Timing Array (PTA) data is a collection of high-precision pulsar arrival times used to detect gravitational waves via noise characterization and modeling.
• Advanced noise processes—including white noise, red noise, DM noise, chromatic variations, and solar wind effects—are modeled to distinguish astrophysical signals from intrinsic pulsar variability.
• Bayesian inference and composite array likelihoods reveal a significant common red noise process, indicating a strong candidate for a nanohertz gravitational wave background.

4.5 years, from February 2019 to August 2023 (MJD 58526–60157). The typical cadence for each pulsar is about every 14 days.

• Timing Precision:
  • Median uncertainty for sub-band arrival times is 3.1 μs, equivalent to a band-averaged median uncertainty of ~0.5 μs per observation (due to 32 frequency sub-bands: $3.1/\sqrt{32} \approx 0.5\,\mu$s).
  • Observations target high precision, using integration times individually tailored to reach 1 μs precision where possible.
• Data Set Scale:
  • 245,907 arrival time measurements are included.
  • Data is released in standard PTA formats, including pulse profiles, templates, and timing ephemerides.
  • Observations are at L-band (856–1712 MHz) with full-bandwidth and high frequency resolution.

2. Noise Analysis: Types and Astrophysical Motivation

The analysis constructs detailed, custom per-pulsar noise models, recognizing that accurate noise characterization is crucial to avoid spurious GW background claims.

Processes considered:

• Deterministic Model:
  • Includes spin frequency, astrometry, solar system barycentric corrections (DE440 ephemeris), DM, and binary parameters.
• Stochastic Noise Processes:
  • White Noise (uncorrelated in time):
  • EFAC ($\mathrm{E_{F}$): scales formal TOA uncertainties; tests for systematics in time-tagging.
  • EQUAD ($\mathrm{E_{Q}$): adds uncertainty in quadrature; reflects additional systematics.
  • ECORR ($\mathrm{E_{C}$): models epoch-correlated "jitter", i.e., pulse-to-pulse variability and mode-changing.
  • Achromatic Red Noise:
  • Represents intrinsic pulsar "spin noise" or other sources of long-timescale correlated noise, modeled as a power-law in the Fourier domain:

  $\mathrm{P_{Red}(f; \mathrm{A_{Red}, \gamma_\mathrm{Red}) = \frac{\mathrm{A^{2}_{Red}{12\pi^{2} \left (\frac{f}{f_\mathrm{c} \right)^{-\gamma_\mathrm{Red} \;\mathrm{yr}^3$ - Dispersion Measure (DM) Noise: - Stochastic, time-variable electron column density introduces frequency-dependent delays.

  $\mathrm{P_{DM}(f; \mathrm{A_{DM}, \gamma_\mathrm{DM}) = \frac{\mathrm{A^{2}_{DM}{12\pi^{2} \left (\frac{f}{f_\mathrm{c} \right)^{-\gamma_\mathrm{DM} \left( \frac{\nu}{\nu_\mathrm{ref} \right )^{-4} \mathrm{yr}^3$ - Scattering (Chromatic) Noise: - From multi-path propagation in the interstellar medium, modeled with a chromatic index $\beta$ (not always fixed to 4):

  $\mathrm{P_{Chrom}(f; \mathrm{A_{Chrom}, \gamma_\mathrm{Chrom}, \beta) = \frac{\mathrm{A^{2}_{Chrom}{12\pi^{2} \left (\frac{f}{f_\mathrm{c} \right)^{-\gamma_\mathrm{Chrom} \left( \frac{\nu}{\nu_\mathrm{ref} \right )^{-2\beta} \mathrm{yr}^3$ - Solar Wind (Chromatic, Annual) Noise: - Both mean electron density (deterministic) and stochastic solar wind fluctuations modeled, to account for time- and ecliptic latitude-dependent delays.

• Other Deterministic Noise:

  • Gaussian-shaped chromatic events: Short-term, discrete ISM/propagation events.
  • Annual chromatic variations: E.g., annually repeating ISM structures.
• Common Uncorrelated Red Noise (CURN) [Candidate GW Signal]:
  • Search for a common, achromatic, red noise process across the array:

  $\mathrm{P_{CURN}(f; \mathrm{A_{CURN}, \gamma_\mathrm{CURN}) = \frac{\mathrm{A^{2}_{CURN}{12\pi^{2} \left (\frac{f}{f_\mathrm{c} \right)^{-\gamma_\mathrm{CURN} \mathrm{yr}^3$

Astrophysical Motivation:

• Intrinsic pulsar variability (spin noise); ISM-induced delays (DM, scattering); solar wind; and jitter/mode changing.

• CURN models a stochastic nanohertz GW background, but care is taken to avoid confusing noise with a real GW signal.

3. Statistical & Methodological Framework

• Bayesian Inference:

  • Model comparison is performed using Bayesian evidence ($\mathcal{Z}$), with posterior samples for all relevant noise parameters.
  • Noise model selection: Per-pulsar, with nested sampling (parallel-bilby) and MCMC (PTMCMC), using the Enterprise software.
  • Model selection is guided by the log Bayes factor:

  $$\ln(\mathcal{B}) = \ln(\mathcal{Z}_A) - \ln(\mathcal{Z}_B)$$

• Goodness-of-fit:

  • Residuals tested for whiteness and Gaussianity (Anderson-Darling test, reduced $\chi^2$).
  • Noise is modeled in the Fourier domain to accurately distinguish between different processes.
• Composite Array Likelihoods:
  • Search for CURN uses both factorized (per-pulsar) and full PTA likelihoods, including all identified noise processes for fidelity.

4. Results: Common Signal, Amplitude, and Spectral Index

• Common Red Noise (CURN) Detection:
  • Amplitude:

  $\log_{10}\mathrm{A_{CURN} = -14.28^{+0.21}_{-0.21}$

  (For fixed spectral index $\gamma_{CURN}=13/3$). - When allowing spectral index to vary (Full PTA analysis):

  $\log_{10}\mathrm{A_{CURN} = -14.25^{+0.21}_{-0.36},\quad \gamma_\mathrm{CURN} = 3.60^{+1.31}_{-0.89}$

  (Figure: Corner plot showing joint posterior distribution—see Fig. 6 in the manuscript.)

• Significance:

  • Bayes factor (factorized analysis):

  $\ln(\mathcal{B}) = 4.46$ - Bayes factor (full PTA likelihood, spectral index free):

  $\ln(\mathcal{B}) = 3.17$ - Both are considered significant detections (log Bayes factor > 3); factorized likelihood gives slightly higher significance due to better constraint on spectral index.

• Comparison with Other PTAs:

  • MeerKAT amplitude ($-14.28$) is larger than those reported by other major PTAs:
  • EPTA: $-14.60^{+0.11}_{-0.14}$
  • PPTA: $-14.69^{+0.05}_{-0.05}$
  • NANOGrav: $-14.62^{+0.11}_{-0.12}$
  • Difference is $\gtrsim 1.4\sigma$, possibly hinting at growth in the signal amplitude over time or residual unmodeled noise, given MeerKAT's more recent data.
• Sensitivity Forecast:
  • The forecasted sensitivity curve (computed with hasasia) shows that, for a GW background with the measured properties, MeerKAT PTA should recover an optimal S/N of ~4.5 in spatial correlations—a strong forecast for future SGWB detection with longer baselines.
  • For the measured amplitude, “if an SGWB is responsible for the CURN, it should also be detected in spatial correlations at an optimal statistic S/N of $\sim 4.5$.”

5. Summary of Data Release Impact

• Largest and most sensitive sample of MSPs to date.
• High-cadence, high-precision, and wideband observations allow for advanced noise modelling—including ISM and solar wind effects—critical to robust GW searches.
• Rigorous noise analysis and model selection ensures that the PTA noise budget is understood and that claims of common-spectrum processes (potential GW backgrounds) are robust and not due to artifacts.
• Detected CURN is stronger than in other arrays, and MeerKAT forecasts strong spatial-correlation sensitivity in coming years if trends hold.

Key LaTeX Equations

Power Spectrum of Gravitational Wave Background

$S_\mathrm{GW}(f) = \frac{A_{\mathrm{GW}^2}{12\pi^2} \left( \frac{f}{1\,\mathrm{yr}^{-1} \right)^{-13/3}\, \mathrm{yr}^3$

Bayesian Evidence

$$\mathcal{Z} = \int \mathcal{L}(d|\theta)\pi(\theta) d\theta$$

$$\ln(\mathcal{B}) = \ln(\mathcal{Z}_A) - \ln(\mathcal{Z}_B)$$

In summary: The 4.5-year MeerKAT PTA data set is the largest and one of the most precise in pulsar timing to date, with an unprecedented noise analysis that enables robust characterization of intrinsic and propagation-induced noise in each pulsar. The first detailed search for a stochastic GW background in this data finds a common-spectrum red noise process with amplitude higher than in other PTAs and strong statistical support ($\ln(\mathcal{B}) \approx 4.5$). The rigorous noise budget and predicted sensitivity position the MeerKAT PTA to make major contributions to GW astrophysics in the coming years.