Source-Frame and Detector-Frame Parameters

A compact-binary waveform is generated in a local source frame but recorded after propagation through an expanding universe. Cosmological expansion stretches every source-frame time and wavelength by the same factor $1+z$. Consequently, detectors measure redshifted dimensional masses and frequencies, while mass ratios, dimensionless spins, and geometric angles are unchanged. Distance is subtler: the strain amplitude contains the luminosity distance $D_L$, but converting $D_L$ into $z$ requires a cosmological model or external information.

Helpful background. Einstein Equations supplies local source dynamics and cosmological geometry; Waves, Green Functions, and Geometric Optics supplies phase and amplitude transport; Relativistic Two-Body Dynamics supplies the intrinsic binary parameters; Gravitational Radiation and Waveforms supplies strain modes and flux conventions; and Computational Relativity supplies source-frame strong-field waveforms.

Redshift from FLRW propagation

On scales large compared with the source, write a Friedmann–Lemaître–Robertson–Walker metric in conformal time $\eta$ as

$$ds^2=a^2(\eta) \left[-d\eta^2+d\chi^2+S_K^2(\chi)d\Omega^2\right],$$

where $S_K(\chi)$ encodes the spatial curvature. A radial geometric-optics ray obeys $d\chi=\pm d\eta$. Its comoving wave number is conserved, whereas a comoving observer measures physical angular frequency $\omega\propto a^{-1}$. If the wave is emitted at scale factor $a_{\mathrm e}$ and detected at $a_0$, then

$$1+z=\frac{a_0}{a_{\mathrm e}}, \qquad f_{\rm det}=\frac{f_{\rm src}}{1+z}, \qquad dt_{\rm det}=(1+z)dt_{\rm src}.$$

The last two relations are the same phase statement: $d\Phi=2\pi f_{\rm src}dt_{\rm src}=2\pi f_{\rm det}dt_{\rm det}$. Cosmological propagation of tensor waves and its redshift scaling are developed in Caprini and Figueroa 2018, §§2–3.

These are source-to-observer relations, not Doppler corrections inside the binary model. A source-frame reference frequency must therefore be divided by $1+z$ before it is compared with a detector-frame reference frequency.

The usual binary-waveform scaling treats $z$ as constant during the observation and assumes homogeneous geometric-optics propagation between comoving source and observer frames. Redshift drift, peculiar accelerations, lensing, and observer aberration require additional mappings. In particular, the statement that angles are unchanged concerns cosmological dilation itself; it does not say that lensing or a change of observer velocity cannot change an apparent direction.

Why the measured masses are redshifted

Local vacuum binary dynamics have no independent absolute scale. Let $M_{\mathrm{src}}$ be a source-frame mass scale, $\lambda$ any dimensionless intrinsic parameters, and $\mathcal H$ a dimensionless waveform shape. The observed complex strain can be organized as

$$h_+(t_{\rm det})-ih_\times(t_{\rm det}) =\frac{GM_z}{D_L} \mathcal H\!\left( \frac{t_{\rm det}-t_c}{GM_z}; \lambda,\text{angles} \right), \qquad M_z\equiv(1+z)M_{\mathrm{src}}.$$

Equivalently, the source constructs $H=r(h_+-ih_\times)$ from unit-normalized spin-weight $-2$ modes with the $e^{-i\omega t}$ convention; cosmological propagation replaces the local distance scaling by $D_L$ and the source mass scale by $M_z$. The post-Newtonian source waveform and its mass-frequency scaling are organized explicitly in Blanchet 2024, §§3.4.3–3.4.5, pp. 206–219.

The leading chirp makes the degeneracy transparent. In the source frame,

$$\frac{df_{\rm src}}{dt_{\rm src}} =\frac{96}{5}\pi^{8/3} (G\mathcal M_{\mathrm{src}})^{5/3}f_{\rm src}^{11/3},$$

where $\mathcal M=M\nu^{3/5}$ is the chirp mass. Since $df_{\rm det}/dt_{\rm det}=(1+z)^{-2}df_{\rm src}/dt_{\rm src}$, substitution of $f_{\rm src}=(1+z)f_{\rm det}$ gives

$$\frac{df_{\rm det}}{dt_{\rm det}} =\frac{96}{5}\pi^{8/3} (G\mathcal M_z)^{5/3}f_{\rm det}^{11/3}, \qquad \mathcal M_z=(1+z)\mathcal M_{\mathrm{src}}.$$

Thus an otherwise scale-free gravitational waveform measures $\mathcal M_z$, not $\mathcal M_{\mathrm{src}}$ and $z$ separately. Every dimensional mass transforms in the same way: $m_{A,z}=(1+z)m_{A,\rm src}$. Ratios such as $q=m_1/m_2$ and $\nu=m_1m_2/M^2$, and dimensionless spins $\boldsymbol\chi_A=\boldsymbol S_A/(Gm_A^2)$, are invariant.

Luminosity distance and cosmological assumptions

Set $a_0=1$. In FLRW geometry,

$$D_L(z)=(1+z)S_K[\chi(z)], \qquad \chi(z)=\int_0^z\frac{dz'}{H(z')}.$$

This formula already shows the division of responsibilities. Wave propagation identifies $D_L$ as the amplitude distance. Turning a measured $D_L$ into a redshift requires a choice of $H(z)$ and spatial curvature. The standard relations among comoving, angular-diameter, and luminosity distances, including $D_L=(1+z)^2D_A$, are summarized in Hogg 1999, §§2–7.

Even after choosing a background cosmology, line-of-sight structure perturbs the simple map. For one resolved geometric-optics image, a positive absolute magnification $\lvert\mu_{\rm lens}\rvert$ rescales the amplitude as if $D_L$ were $D_L/\sqrt{\lvert\mu_{\rm lens}\rvert}$. Its arrival-time and Morse-phase shifts must be included separately. Unresolved images or wave-optics diffraction can produce frequency-dependent interference, so they cannot in general be absorbed into one distance. See Gravitational Lensing for that transfer function. Peculiar velocities alter the relation between cosmological redshift and an observed host redshift, especially nearby. These are propagation or environment effects, not changes to the source’s mass ratio or spin.

Geometric parameters and waveform frames

Redshift is a dilation, not a rotation. Sky direction, binary inclination, polarization angle, orbital phase, precession angles, and dimensionless eccentricity do not acquire factors of $1+z$. Their numerical definitions still require declared frames:

• source modes are referred to a source inertial or coprecessing triad;
• the line of sight rotates those modes into a transverse polarization basis;
• detector antenna patterns project that basis onto an instrument.

The mode rotation and cosmological dilation commute because they act on different structures. However, a dimensionless parameter quoted at a reference frequency inherits a frame label indirectly. For example, eccentricity at $20\,\mathrm{Hz}$ detector-frame refers to the source when its frequency was $20(1+z)\,\mathrm{Hz}$, so it is not interchangeable with eccentricity at $20\,\mathrm{Hz}$ source-frame.

The mapping can be kept compact:

Quantity	Detector/source relation	Cosmology needed?
Dimensional mass	$m_z=(1+z)m_{\rm src}$	$z$ needed to recover $m_{\rm src}$
Frequency	$f_{\rm det}=f_{\rm src}/(1+z)$	$z$ needed for source frequency
Time interval	$\Delta t_{\rm det}=(1+z)\Delta t_{\rm src}$	$z$ needed for source time
Mass ratio, $\nu$, dimensionless spin	unchanged	no
Angles and dimensionless eccentricity	unchanged at corresponding events	no, but reference frequency must be mapped
Amplitude distance	waveform contains $D_L$	yes, to infer $z$ from $D_L$

Consistency checks and interpretive limits

Three checks catch most frame mistakes. First, at $z=0$ all source and detector times, frequencies, and masses agree, while $D_L$ reduces to the local distance. Second, the phase product $f\,dt$ is invariant. Third, dimensionless combinations such as $G M f$ are preserved:

$$GM_z f_{\rm det}=GM_{\rm src}f_{\rm src}.$$

A catalog source waveform can therefore be reused at any total mass and redshift by rescaling time and amplitude, provided matter scales or other dimensional source physics have not introduced extra structure. In the stationary-remnant limit, every quasinormal frequency is redshifted by the same factor and every damping time is stretched by it.

This page provides a parameter dictionary, not an inference procedure. A measured luminosity distance has an inclination correlation, and a host association or population model can supply information about $z$; those empirical questions belong to gravitational-wave astronomy. The local orbital equations remain the responsibility of relativistic two-body dynamics.

Exercise: diagnose a frame mix-up

A waveform analysis reports detector-frame component masses $(m_{1,z},m_{2,z})=(45,30)M_\odot$, a detector frequency $f_{\rm det}=80\,\mathrm{Hz}$, and a detector-frame interval $\Delta t_{\rm det}=15\,\mathrm{ms}$. An identified host has $z=0.50$. Find the source-frame masses, mass ratio, corresponding source frequency, and source interval. Verify one invariant dimensionless product.

Solution

Here $1+z=1.5$, so

$$(m_{1,\rm src},m_{2,\rm src})=(30,20)M_\odot, \qquad q=\frac{45}{30}=\frac{30}{20}=1.5.$$

Frequency and time transform oppositely:

$$f_{\rm src}=1.5(80\,\mathrm{Hz})=120\,\mathrm{Hz}, \qquad \Delta t_{\rm src}=\frac{15\,\mathrm{ms}}{1.5}=10\,\mathrm{ms}.$$

Using the total masses, $M_z=75M_\odot$ and $M_{\rm src}=50M_\odot$, the dimensionless frequency check is

$$\frac{GM_z}{c^3}f_{\rm det} =\frac{G(75M_\odot)}{c^3}(80\,\mathrm{Hz}) =\frac{G(50M_\odot)}{c^3}(120\,\mathrm{Hz}) =\frac{GM_{\rm src}}{c^3}f_{\rm src} \simeq 0.02955.$$

The factor $G/c^3$ turns a mass into a time, so every displayed product is genuinely dimensionless. If one instead paired $75M_\odot$ with $120\,\mathrm{Hz}$, the product would fail by the redshift factor and reveal a source/detector-frame mix-up.

References

• Blanchet, Luc. “Post-Newtonian Theory for Gravitational Waves.” Living Reviews in Relativity 27 (2024) 4. DOI. Open PDF
• Caprini, Chiara, and Daniel G. Figueroa. “Cosmological Backgrounds of Gravitational Waves.” Classical and Quantum Gravity 35 (2018) 163001. DOI. Open PDF
• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116 (1999). Abstract and PDF