Earth Terms, Pulsar Terms, and Antenna Response

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A plane gravitational wave perturbs the light-travel time along the entire pulsar–Earth path. For a plane wave, that integrated effect reduces to a difference between the metric at reception near Earth and at emission near the pulsar. The common Earth term correlates pulsars; the distant pulsar terms carry source evolution and distance-dependent phase but usually lose coherence across distinct pulsars.

Helpful background. Pulsar Timing Residuals explains the timing-model projection. Polarizations and Observable Wave Content defines the tensor polarization basis.

One-way Doppler response

Let $\boldsymbol p_a$ point from the solar-system barycenter toward pulsar $a$ , at distance $L_a$ . Let $\boldsymbol\Omega$ point in the gravitational-wave propagation direction, so a plane wave depends on $t-\boldsymbol\Omega\cdot\boldsymbol x$ . Use an observed-minus-predicted arrival-time residual: a positive residual means a late pulse. Its derivative is the fractional period increase, or redshift,

z_a=\dot r_a=-\frac{\delta\nu_a}{\nu_a}

to first order. Thus $z_a$ has the opposite sign to a received fractional-frequency fluctuation. For transverse-traceless strain $h_{ij}$ , the response is

z_a(t,\boldsymbol\Omega) =\frac12\frac{p_a^ip_a^j} {1+\boldsymbol\Omega\cdot\boldsymbol p_a} \left[ h_{ij}(t,\boldsymbol 0) -h_{ij}(t-L_a,L_a\boldsymbol p_a) \right].

The first bracketed contribution is the Earth term; the second is the pulsar term at the pulse-emission event. Because the plane wave depends on $t-\boldsymbol\Omega\cdot\boldsymbol x$ , its scalar argument at that event is $t-L_a(1+\boldsymbol\Omega\cdot\boldsymbol p_a)$ . The timing residual is the frequency shift’s time integral,

r_a(t)=\int^t z_a(t')\,dt'.

Writing $h_{ij}=\sum_A h_Ae^A_{ij}$ defines antenna factors

F_a^A(\boldsymbol\Omega) =\frac12 \frac{p_a^ip_a^je^A_{ij}(\boldsymbol\Omega)} {1+\boldsymbol\Omega\cdot\boldsymbol p_a}.

The denominator does not make the tensor antenna factor diverge: the transverse numerator already cancels its zero. As $\boldsymbol\Omega\to-\boldsymbol p_a$ , the Earth and pulsar wave phases also coincide, making the full response vanish. Dropping the pulsar term loses this cancellation and gives an incorrect nonzero limiting response, even though the antenna factor remains bounded.

Why only the endpoints remain

Parameterize the unperturbed ray backward from Earth by $0\leq u\leq L_a$ , so its events are $(t-u,u\boldsymbol p_a)$ . The first-order light-travel delay is

r_a(t)=\frac12p_a^ip_a^j \int_0^{L_a}h_{ij}\!\left(t-u,u\boldsymbol p_a\right)du.

Along a plane wave, its argument is $t-u(1+\boldsymbol\Omega\cdot\boldsymbol p_a)$ . Differentiating with respect to $t$ and using the chain rule to integrate with respect to $u$ gives the Earth-minus-pulsar expression above. This fixes both the denominator and the observable’s sign Romano and Allen 2024, §2.2.

Why Earth terms correlate

At a given barycentric time, every pulsar’s Earth term contains the same gravitational field $h_{ij}(t,\boldsymbol0)$ projected by a different antenna tensor. For an isotropic stochastic tensor background, averaging over propagation direction and polarization yields an angular cross-correlation depending only on the separation between pulsars.

Independent pulsar terms sample distinct field phases when pulsars are separated by many gravitational wavelengths and their distances are uncertain. They then contribute to each pulsar’s auto-covariance but average out of cross-covariances. This is the approximation behind the usual Hellings–Downs curve for distinct pulsar pairs.

The approximation has limits. Physically nearby pulsars, very low frequencies, anisotropic sources, or a deterministic binary can retain pulsar-term correlation. The autocorrelation also includes both terms and needs its separate normalization.

Pulsar terms as long baselines

For a monochromatic source with angular frequency $\omega$ , define the link delay and the magnitude of the Earth–pulsar phase separation by

\tau_a=L_a(1+\boldsymbol\Omega\cdot\boldsymbol p_a), \qquad \Delta\Phi_a=\omega\tau_a.

With the site convention $e^{-i\omega t}$ , the pulsar phasor differs from the Earth phasor by $e^{+i\Delta\Phi_a}$ . Describing the sky by the direction toward the source, $\boldsymbol n=-\boldsymbol\Omega$ , replaces $1+\boldsymbol\Omega\cdot\boldsymbol p_a$ by $1-\boldsymbol n\cdot\boldsymbol p_a$ ; it does not change the physical delay. Changing the Fourier time-sign convention instead complex-conjugates the phasor. A distance error much smaller than the distance can still span many phase cycles. Treating $L_a$ as fixed at an imprecise electromagnetic estimate can therefore bias a continuous-wave search.

For the corresponding transform $\widetilde h(f)=\int h(t)e^{+2\pi ift}dt$ , the complete residual response at $f\ne0$ is

\widetilde r_a(f) =\sum_A F_a^A \frac{1-e^{+2\pi if\tau_a}}{-2\pi if} \widetilde h_A(f).

The factor $1/f$ converts dimensionless redshift into a time delay. For example, a sinusoidal redshift of amplitude $10^{-15}$ at $10\,\mathrm{nHz}$ gives a residual amplitude $10^{-15}/(2\pi\,10^{-8}\,\mathrm{Hz})\simeq16\,\mathrm{ns}$ before timing-model fitting. This is a scale illustration, not a source measurement.

The advantage is a millennial time lever arm. A supermassive-black-hole binary can have a measurably different frequency at the pulsar epoch, encoding chirp mass and environmental evolution. Coherently exploiting that information requires distance priors, phase multimodality, and a source evolution model.

Conventions and independent checks

Some references define $\widehat\Omega$ toward the source rather than along propagation, changing $1+\boldsymbol\Omega\cdot\boldsymbol p$ to $1-\widehat\Omega\cdot\boldsymbol p$ . The physical result is unchanged only if phase, polarization tensors, and retarded pulsar time are changed together.

Anholm and collaborators derive the response and stochastic overlap reduction with explicit Fourier conventions Anholm et al. 2009, §§II–III. Their displayed $\Delta h=h_{\rm pulsar}-h_{\rm Earth}$ in Eqs. (8)–(9) is opposite to the response convention used here; their positive-time Fourier expansion also reverses the frequency-domain phase sign. Translate both before importing a deterministic template. The sign cancels in same-convention pair correlations, but matters for a waveform or memory ramp. Independent checks should verify:

invariance under polarization-basis rotation;
a finite collinear limit;
dimensions: $z$ is dimensionless and $r$ has units of time;
zero response to a constant pure gauge perturbation;
recovery of injected source sky location and pulsar phases;
consistency after marginalizing pulsar distances rather than fixing them.

The concrete public interface through 2026-08-18 is the NANOGrav 15-year timing and noise product family. Any rerun pins TOAs, timing solutions, ephemeris, distance priors, response convention, source/background model, and software version.

Exercises

1. Residual amplitude. If one Fourier component gives $z(t)=z_0\cos\omega t$ , what is the oscillatory timing residual?

Solution

Integrating gives

r(t)=\frac{z_0}{\omega}\sin\omega t+C.

The constant is absorbed by the timing model. At fixed fractional frequency shift, lower-frequency waves produce larger residual amplitudes.

2. Why distance precision matters. A pulsar distance uncertainty changes $\omega L(1+\boldsymbol\Omega\cdot\boldsymbol p)$ by $20\pi$ . Is the pulsar phase localized?