Cosmological Propagation and Redshift

When a compact source’s size and dynamical time are much smaller than the cosmological curvature radius and expansion time, its local strong-field dynamics can be treated independently of cosmic expansion. Expansion nevertheless changes the relation between source proper time and detector proper time. Frequencies are redshifted, durations and every dimensionful source time scale are dilated, and spherical-wave amplitudes are organized by the luminosity distance. These statements follow from the same null eikonal and amplitude transport used in geometric optics. Here the source and detector frames are declared explicitly, and the derivation is restricted to freely propagating, subhorizon tensor modes in general relativity, with negligible redshift drift during the signal.

Required background. Waves, Green Functions, and Geometric Optics supplies phase and amplitude transport; Curvature supplies null propagation on the FLRW geometry.

Helpful background. Einstein Equations supplies the tensor perturbation equation; Relativistic Cosmology supplies scale factor, redshift, and cosmological distances.

Tensor modes on an FLRW background

Use conformal time $\eta$ and a spatially flat FLRW metric,

$$ds^2=a^2(\eta) \left[-d\eta^2+(\delta_{ij}+h_{ij})dx^i dx^j\right], \qquad \partial_i h_{ij}=0, \qquad h_{ii}=0.$$

After the source and transverse anisotropic stress have become negligible, either tensor polarization mode satisfies

$$h_k''+2\mathcal H h_k'+k^2h_k=0, \qquad \mathcal H=\frac{a'}a.$$

Define the canonically rescaled amplitude $\mu_k=a h_k$. Direct substitution gives

$$\mu_k''+\left(k^2-\frac{a''}{a}\right)\mu_k=0.$$

For a subhorizon, adiabatic mode,

$$k^2\gg\left\lvert\frac{a''}{a}\right\rvert, \qquad k\gg\mathcal H,$$

the positive-frequency solution has

$$\mu_k\simeq C_ke^{-ik\eta}, \qquad h_k\simeq\frac{C_k}{a(\eta)}e^{-ik\eta}.$$

Thus the plane-wave amplitude decays as $1/a$. Caprini and Figueroa derive the free tensor equation, the $\mu_k=a h_k$ rescaling, and the subhorizon solution in Caprini and Figueroa 2018, §2.4, pp. 19–22, eqs. (51)–(70), PDF. This conclusion does not apply unchanged to a superhorizon perturbation, for which the constant and decaying solutions must be treated separately.

Phase and frequency in declared frames

For a local WKB wave take

$$\theta=-k\eta+\boldsymbol{k}\cdot\boldsymbol{x}, \qquad k_\mu=\nabla_\mu\theta.$$

Its raised vector is future directed and follows an affinely parametrized null ray. A comoving cosmological observer has $u^\mu=a^{-1}(1,\boldsymbol{0})$, so the locally measured angular frequency is

$$\omega=-k_\mu u^\mu=\frac{k}{a}.$$

Let $u_s^\mu$ be the source’s comoving cosmological frame at emission and $u_o^\mu$ the detector’s comoving cosmological frame at observation. Then

$$1+z=\frac{a_o}{a_s} =\frac{\omega_s}{\omega_o}, \qquad f_{\mathrm{det}}=f_o=\frac{f_s}{1+z}.$$

Equality of accumulated phase gives the corresponding time map,

$$dt_{\mathrm{det}}=(1+z)\,dt_{\mathrm{src}}.$$

Peculiar velocities add endpoint Doppler factors, and gravitational potentials add gravitational redshifts. They are absent from the comoving relation above; silently folding them into “cosmological redshift” would make the frame declaration incomplete.

Spherical spreading and luminosity distance

The $1/a$ plane-wave behavior is only part of the amplitude law for a localized source. A spherical wave also spreads over a comoving sphere. In a spatially flat background its physical area at observation is $4\pi a_o^2\chi^2$, so geometric-optics flux conservation gives a strain factor proportional to

$$\frac{1}{a_o\chi}.$$

The angular-diameter and luminosity distances are

$$D_A=a_s\chi, \qquad D_L=(1+z)^2D_A=(1+z)a_o\chi.$$

Consequently a fixed source-frame radiative coefficient is multiplied by $(1+z)/D_L$. When the waveform is re-expressed in detector-frame time and frequency, that extra redshift is absorbed into redshifted dimensionful source parameters, leaving the familiar inverse-$D_L$ amplitude law. This combines the $1/r$ ray-bundle behavior developed by Flanagan and Hughes 2005, §5, pp. 23–29, PDF with the FLRW $1/a$ transport above.

For example, a leading circular-binary amplitude has the scaling

$$h_{\mathrm{det}} \propto \frac{[G\mathcal M_{\mathrm{det}}]^{5/3} (\pi f_{\mathrm{det}})^{2/3}}{D_L}, \qquad \mathcal M_{\mathrm{det}}=(1+z)\mathcal M_{\mathrm{src}}.$$

The local source calculation still uses $\mathcal M_{\mathrm{src}}$ and source proper time. Cosmological propagation maps all dimensionful masses and times to detector-frame values by $1+z$; dimensionless mass ratios and dimensionless spins do not acquire such a factor.

Connections and limits

If the wave is lensed at redshift $z_L$, the wave-optics parameter uses the detector frequency and the redshifted lens time scale,

$$w=8\pi G M_L(1+z_L)f_{\mathrm{det}}.$$

This is the same source/lens/detector frame translation used in the cosmological thin-lens integral Takahashi and Nakamura 2003, p. 6, eq. (17), PDF. It should not be confused with the source redshift $z$ unless lens and source coincide.

The derivation assumes general relativity, luminal propagation, wavelengths well inside the Hubble radius, and negligible anisotropic stress along the free-propagation segment. Matter inhomogeneity can focus, lens, and scatter the wave; those effects must be added consistently rather than hidden inside a homogeneous $D_L$. Theory-specific friction, massive dispersion, or birefringence would change the equations and belongs to tests of modified gravity. Instrument selection and likelihood construction likewise lie outside this propagation page.

When $a$ is constant, $\mathcal H=a'/a=0$ and $a''/a=0$, so the tensor equation becomes the flat wave equation. Then $z=0$, $f_{\mathrm{det}}=f_{\mathrm{src}}$, $D_L=a\chi$ is the physical radius, and the result reduces to constant plane-wave amplitude or $1/R$ spherical spreading.

Exercise

A source at redshift $z=2$ emits $N=300$ cycles at an approximately constant source-frame frequency $f_s=150\,\mathrm{Hz}$. Find the detector-frame frequency and duration, and verify that the cycle count is unchanged. How is a source-frame mass scale $M_s$ represented in a waveform written in detector-frame variables?

Solution

Here $1+z=3$, so

$$f_{\mathrm{det}}=\frac{150\,\mathrm{Hz}}3=50\,\mathrm{Hz}.$$

The source duration is $\Delta t_s=N/f_s=2\,\mathrm{s}$, while

$$\Delta t_{\mathrm{det}} =3\Delta t_s=6\,\mathrm{s}.$$

Therefore $f_{\mathrm{det}}\Delta t_{\mathrm{det}}=50\times6=300=N$: redshift changes frequency and duration inversely and preserves phase cycles. A dimensionful mass appears as $M_{\mathrm{det}}=(1+z)M_s=3M_s$ when the same locally generated waveform is parametrized using detector time and frequency. Its propagation amplitude is organized by $D_L(z)$, not by the Euclidean coordinate distance.

References

• Chiara Caprini and Daniel G. Figueroa, “Cosmological Backgrounds of Gravitational Waves,” Classical and Quantum Gravity 35 (2018) 163001, arXiv:1801.04268v3, DOI, Open PDF.
• Éanna É. Flanagan and Scott A. Hughes, “The Basics of Gravitational Wave Theory,” New Journal of Physics 7 (2005) 204, arXiv:gr-qc/0501041v3, DOI, Open PDF.
• Ryuichi Takahashi and Takashi Nakamura, “Wave Effects in the Gravitational Lensing of Gravitational Waves from Chirping Binaries,” The Astrophysical Journal 595 (2003) 1039–1051, arXiv:astro-ph/0305055v2, DOI, Open PDF.