Continuous-Wave Phase Models

A continuous gravitational wave is long-lived enough that a small timing error accumulates into a large phase error. Its useful model therefore separates intrinsic source rotation from propagation delays between the detector, Solar-System barycenter, and—when applicable—a binary barycenter. This page derives that deterministic phase map in declared time standards. Search pipelines, upper limits, and current detections are not included.

Helpful background. Einstein Equations supplies tensor radiation. Waves, Green Functions, and Geometric Optics supplies phase and Doppler propagation. Gravitational Radiation and Waveforms supplies polarization conventions. Relativistic Matter and Compact Objects supplies rotating-star physics. Relativistic Cosmology supplies frame and redshift distinctions for extragalactic sources.

Strain and phase are separate layers

For a nearly triaxial star emitting at one harmonic, an ideal detector response can be written

$$\begin{aligned} s(t)=&\;F_+(t;\boldsymbol n,\psi) \frac{h_0(1+\cos^2\iota)}{2}\cos\Phi(t)\\ &+F_\times(t;\boldsymbol n,\psi) h_0\cos\iota\sin\Phi(t). \end{aligned}$$

$h_0$ and inclination $\iota$ set field amplitudes; $F_A$ sets the rotating detector projection; $\Phi$ carries source and propagation timing. Other emission mechanisms can change the amplitude relation or add harmonics without changing the logic of barycentric phase bookkeeping. The standard source and phase constructions are reviewed in Riles 2023, §§2–3, PDF.

Intrinsic phase at the source

Let $\tau$ be a source-frame emission time referred to epoch $\tau_0$. Expand the gravitational-wave frequency, not necessarily the stellar spin frequency, as

$$f(\tau)=\sum_{k=0}^{s} \frac{f^{(k)}_0}{k!}(\tau-\tau_0)^k.$$

Integrating $d\Phi/d\tau=2\pi f$ gives

$$\Phi(\tau)=\phi_0+2\pi \sum_{k=0}^{s} \frac{f^{(k)}_0}{(k+1)!} (\tau-\tau_0)^{k+1}.$$

The dimensions are $[f^{(k)}_0]=\mathrm{s}^{-(k+1)}$. Differentiating the phase recovers the frequency polynomial term by term, providing a factorial and units check. Timing noise or glitches require an enlarged model; forcing them into a low-order Taylor series can bias every derivative.

From detector time to barycentric time

Let $t$ be a specified detector time standard and $\boldsymbol n$ point from the Solar-System barycenter toward the source. Define barycentric arrival time

$$T=t+\Delta_{\rm R}(t)+\Delta_{\rm E}(t)+\Delta_{\rm S}(t),$$

where the Römer delay is

$$\Delta_{\rm R}=\boldsymbol r_{\rm det}(t)\mathbin{\cdot}\boldsymbol n,$$

with $c=1$. $\Delta_{\rm E}$ accounts for relativistic clock-rate and gravitational-redshift conversion; $\Delta_{\rm S}$ is the Solar-System Shapiro delay in the declared ephemeris. The signs above define $T$: a detector displaced toward the source receives a wave early, so its corresponding barycentric arrival is later by $\boldsymbol r\cdot\boldsymbol n$.

For a slowly varying intrinsic frequency,

$$f_{\rm det}\simeq f(T) \left[1+\boldsymbol v_{\rm det}\mathbin{\cdot}\boldsymbol n\right],$$

before the smaller derivatives of the Einstein and Shapiro delays are added. Sky position, proper motion, parallax, the planetary ephemeris, and the chosen TCB/TDB convention must remain consistent over long baselines.

Binary modulation

If the source orbits a companion, define a positive binary delay $\Delta_{\rm bin}(T)$ as the extra propagation time from the emitter relative to its binary barycenter. Then

$$\tau=T-\Delta_{\rm bin}(T).$$

For a circular orbit one useful convention is

$$\Delta_{\rm bin}(T) =a_p\sin\!\left[\Omega_b(T-T_{\rm asc})\right],$$

where $a_p=a\sin i_b$ is a projected semimajor axis expressed in light-seconds, $\Omega_b=2\pi/P_b$, and $T_{\rm asc}$ is ascending-node time. Eccentric models replace this sinusoid with eccentric, Römer, Einstein, and companion-Shapiro terms. Taking $a_p\to0$ recovers the isolated-source model.

Applying the chain rule exposes the observable Doppler sign:

$$f_{\rm det}(t) =f[\tau(T)]\left(1-\frac{d\Delta_{\rm bin}}{dT}\right) \frac{dT}{dt}.$$

When the source moves farther from its binary barycenter along the line of sight, the propagation delay increases and its received frequency decreases. For the circular model the fractional binary modulation has amplitude $a_p\Omega_b$; this is the line-of-sight orbital speed in units of $c$. The sinusoid as written is the leading timing approximation for a nonrelativistic orbit. A precision timing model must consistently solve the emission-time relation if its delays are supplied as functions of emission time rather than $T$.

Coherence requirements

A timing error $\delta\tau$ produces

$$\delta\Phi\simeq2\pi f\,\delta\tau.$$

Keeping $\lvert\delta\Phi\rvert\lt1$ rad at $1\,\mathrm{kHz}$ requires timing better than about $0.16\,\mathrm{ms}$. A sky error also makes the annual Römer correction wrong; because Earth’s orbital light-travel scale is hundreds of seconds, even a small angular offset matters over a year.

If the first omitted intrinsic derivative is $f_0^{(s+1)}$, its phase contribution after duration $T_{\rm obs}$ is approximately

$$\delta\Phi_{s+1}= \frac{2\pi f_0^{(s+1)}T_{\rm obs}^{s+2}}{(s+2)!}.$$

This is a model-truncation diagnostic, not a search threshold. Phase coherence can also fail through glitches, stochastic timing noise, orbital-parameter drift, ephemeris error, or inconsistent clock standards.

Domain of validity and further applications

The model predicts a deterministic field phase under declared source dynamics and timing corrections. Deformation and torque mechanisms are treated in Relativistic Matter and Compact Objects; template banks, semicoherent combinations, noise, upper limits, and current detections in Gravitational-Wave Astronomy. A good waveform record therefore states the reference epoch, time standard, ephemeris, sky-motion model, binary-delay convention, harmonic relation to stellar rotation, and derivative order.

A minimal phase record

A reproducible continuous-wave phase model should give more than a list of fitted numbers. Record $\phi_0$ and every $f_0^{(k)}$ at the same epoch; say whether frequency is the stellar rotation frequency or a particular radiation harmonic; specify the sky-coordinate frame and epoch; name the barycentric time scale and planetary ephemeris; and supply the complete binary convention when present. Also state the interval over which the Taylor expansion is intended to remain coherent.

Two records with numerically equal $f_0$ and $\dot f_0$ are not equivalent if one uses TCB and the other TDB, or if their epochs differ. Translate the coefficients by differentiating the same physical phase with respect to the new time variable. Similarly, changing $T_{\rm asc}$ by an integer orbital period is a harmless reparameterization, whereas reversing the sign of $\Delta_{\rm bin}$ changes the predicted modulation. These bookkeeping checks precede any statistical use of the waveform.

Exercises

1. Differentiate the model. Retain $f_0$ and $\dot f_0$ only. Verify that the instantaneous source frequency is $f_0+\dot f_0(\tau-\tau_0)$.

Solution

The phase is

$$\Phi=\phi_0+2\pi\left[ f_0\Delta\tau+\frac12\dot f_0\Delta\tau^2 \right].$$

Differentiating and dividing by $2\pi$ gives $f_0+\dot f_0\Delta\tau$. Since $[\dot f_0]=\mathrm{s}^{-2}$, both terms have units of hertz. Omitting the factor $1/2$ in phase would incorrectly double the spindown contribution.

2. Orbital-delay scale. A circular binary has $a_p=2\,\mathrm{s}$ and emits at $500\,\mathrm{Hz}$. What is the maximum phase excursion caused by the binary Römer delay?

Solution

The delay ranges from $-a_p$ to $+a_p$, so relative to the barycenter the phase modulation has amplitude

$$2\pi f a_p=2\pi(500)(2)=2000\pi\ \mathrm{rad}.$$

That is one thousand cycles. The zero-binary limit is smooth, but ignoring a nonzero orbit would destroy long-duration phase coherence rather than create a small perturbation.

References

• Riles, Keith. “Searches for Continuous-Wave Gravitational Radiation.” Living Reviews in Relativity 26 (2023): 3. DOI. Open PDF.