Pulsar Noise and Solar-System Ephemerides

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Pulsar timing arrays search for correlations whose amplitude and spectrum resemble several instrumental, propagation, and reference-frame effects. Intrinsic spin noise can imitate red power in one pulsar; clock error is common and monopolar; Solar-System ephemeris error is common and largely dipolar; dispersion and scattering are chromatic. A gravitational-wave interpretation is credible only after these processes are fitted in the same likelihood and the result survives computational and model checks.

Helpful background. Pulsar Timing Residuals defines design-matrix projection and single-pulsar processes. Hellings–Downs Correlations supplies the tensor spatial alternative.

A covariance model for the array

For pulsars $a,b$ , write the residual covariance schematically as

C_{ab}=\delta_{ab} \left(N_a+C_a^{\rm spin}+C_a^{\rm DM}+C_a^{\rm chrom} +C_a^{\rm sys}\right) +C_{ab}^{\rm clock} +C_{ab}^{\rm eph} +C_{ab}^{\rm gw}.

$N_a$ contains radiometer, jitter, and epoch-correlated white noise. Intrinsic red spin noise is achromatic but uncorrelated between pulsars. Dispersion variations scale approximately as $\nu^{-2}$ in radio frequency, scattering often more steeply, and receiver/backend processes can be band or system specific.

A time-standard error adds nearly the same residual to every pulsar. Define $\delta\boldsymbol r(t)$ so that its projection is the error added to the barycentric light-travel correction. Then

r_a^{\rm eph}(t) =\delta\boldsymbol r(t)\cdot\boldsymbol p_a,

with $c=1$ . The same vector projected onto different sky directions produces dipolar correlations. Planetary-mass and orbital-element uncertainty can concentrate power near orbital periods and leak into very-low-frequency bins.

Restoring units gives $r_a^{\rm eph}=\delta\boldsymbol r\cdot\boldsymbol p_a/c$ : a $30\,\mathrm m$ displacement corresponds to at most about $100\,\mathrm{ns}$ . Its pair covariance is generally $p_a^iK_{ij}(t,t')p_b^j/c^2$ , where $K_{ij}$ is the displacement covariance. It reduces to a pure $\cos\zeta$ curve only for isotropic $K_{ij}\propto\delta_{ij}$ . A planetary-orbit error has preferred directions and time dependence, so a dipole control cannot replace physical ephemeris modeling.

Ephemeris modeling

Comparing several deterministic ephemerides is informative but samples only a discrete set of solutions. A perturbative model can vary outer-planet masses, Jupiter’s orbit, and a rotation of the ecliptic frame around a baseline ephemeris. Priors should come from independent planetary constraints where appropriate.

The model is an approximation: omitted objects, correlated orbital elements, or a basis too rigid to represent the true error can leave common residuals. Conversely, an excessively flexible ephemeris process can absorb tensor correlations. Injection tests with known ephemeris perturbations calibrate both failures. Vallisneri and collaborators develop this marginalization and its effect on PTA background inference Vallisneri et al. 2020, §§2–5.

Noise-model uncertainty

Selecting a single best noise model for each pulsar and freezing it before the common-process analysis can understate uncertainty. Fitting every imaginable term can also overfit. Useful approaches include product-space model averaging, hypermodels, dropout parameters, and targeted model expansion justified by residual diagnostics.

Chromatic processes are especially vulnerable to changing receiver coverage. An apparent long-timescale trend may reflect observing-frequency evolution rather than interstellar physics. Wideband and narrowband pipelines can encode profile and dispersion information differently; agreement is a valuable cross-check.

The NANOGrav 15-year detector-characterization analysis is the worked noise-budget reference used through the evidence cutoff 2026-08-18, including pulsar, propagation, receiver, clock, and ephemeris considerations Agazie et al. 2023, §§3–7 and appendices. Its public timing/noise products, not a transcribed plot, are the required inputs.

Reproducible computational path

A clean rerun should record:

exact NANOGrav 15-year .tim/.par files and hashes, TOA flags, clock corrections, and baseline ephemeris;
timing/noise software and commits—such as TEMPO/TEMPO2, enterprise, extensions, and sampler—inside a locked environment;
timing-model marginalization, Fourier frequencies and mode count, chromatic indices, priors, ephemeris basis, and common-process alternatives;
sampler temperatures, proposals, chains, seeds, burn-in, autocorrelation times, effective sample size, and evidence estimator;
posterior products plus monopole, dipole, Hellings–Downs, and uncorrelated model comparisons.

The benchmark is a suite on the real cadence: noise-only, injected clock, injected ephemeris, and injected tensor-background datasets. Each should recover the correct spatial family without falsely preferring another at the advertised calibration rate.

Convergence is tested by independent chains, split-chain summaries, effective sample size for amplitudes and spectral indices, stable model odds across chain halves, and reproduction after increasing the Fourier basis. Thermodynamic-integration or product-space odds require their own temperature-ladder and occupancy checks.

Failure diagnostics

One-pulsar dominance: leave-one-pulsar-out odds change qualitatively.
One-backend dominance: removing a receiver epoch erases common power.
Ephemeris fragility: tensor odds change dramatically among reasonable ephemerides or perturbation priors.
Fourier truncation: amplitude/slope shift as modes are added.
Prior boundary: noise amplitudes or spectral indices pile up at limits.
Poor mixing: label or model-index chains do not traverse alternatives.
Chromatic leakage: a nominally achromatic process changes under radio-frequency subsets.

These are analysis failures or uncertainty signals, not evidence against gravitational waves by themselves.

Exercises

1. Dipolar ephemeris pattern. If $\delta\boldsymbol r$ is an isotropic random vector with covariance $\langle\delta r_i\delta r_j\rangle=\sigma^2\delta_{ij}$ , find $\langle r_a^{\rm eph}r_b^{\rm eph}\rangle$ .

Solution

\langle r_a r_b\rangle =p_a^ip_b^j\sigma^2\delta_{ij} =\sigma^2\boldsymbol p_a\cdot\boldsymbol p_b =\sigma^2\cos\zeta.

This is a dipole in pulsar separation, distinct from the Hellings–Downs curve.

2. More Fourier modes. A common-process amplitude rises steadily as the number of red-noise Fourier modes increases. What should be concluded?