Cosmological Redshift

Cosmological redshift compares a photon’s frequency at emission with its frequency at observation through $1+z=\omega_e/\omega_o$. In an unperturbed FLRW spacetime, with both endpoints comoving, null propagation gives $1+z=a(t_o)/a(t_e)$. The result is geometric: it follows from the metric and observer congruence without using Einstein’s equation. Peculiar motion, gravitational potentials, or propagation through perturbations add observable corrections and must not be hidden inside the background formula.

Required background. Pseudo-Riemannian Metrics and Signature supplies Lorentzian metrics and causal cones; Affine and Variational Geodesics supplies the Geodesics and Jacobi Fields background; this page uses its affine null-geodesic and observer-contraction material for photon frequency, not Jacobi-field theory.

Helpful background. The Einstein Equation supplies the Einstein equations used later to calculate $a(t)$, while Relativistic Matter and Compact Objects supplies the fluid rest frame and peculiar-motion interpretation.

Frequency measured by an observer

Let $k^a$ be the future-directed tangent to an affinely parametrized null ray and $U^a$ the unit four-velocity of any observer at an intersection event. With the site signature, the measured angular frequency is

$$\omega=-U_a k^a\gt0.$$

For an emitter and observer joined by the ray, redshift is defined invariantly by

$$1+z\equiv\frac{\omega_e}{\omega_o} =\frac{(-U_a k^a)_e}{(-U_a k^a)_o}.$$

This equation is more general than any cosmological model. It states exactly which observer frames enter and remains valid with peculiar velocities, gravitational fields, and inhomogeneity. A coordinate component such as $k^0$ becomes a measured frequency only after the observer four-velocity has been specified. Ellis and van Elst develop this observer decomposition in Ellis and van Elst 1999, §§2.1.1–2.1.3, pp. 6–8 (PDF). The symbol $u^a$ below is reserved for the fundamental comoving congruence; $U^a$ remains the four-velocity of a general emitter or observer.

Null propagation in FLRW

Use the dimensionless-scale-factor package from the preceding pages:

$$ds^2=-dt^2+a^2(t)d\Sigma_K^2, \qquad a(t_0)=1,$$

where $d\Sigma_K^2$ carries units of length squared and has constant curvature $K$. Thus $a$ is dimensionless and $K$ has units of inverse length squared. For this cosmic-time metric, take the comoving congruence $u^a=(\partial_t)^a$. Decompose a photon tangent as

$$k^a=\omega(u^a+e^a), \qquad u_a e^a=0, \qquad e_a e^a=1.$$

FLRW kinematics gives $\nabla_a u_b=H(g_{ab}+u_a u_b)$. Along an affine null geodesic,

$$\frac{d\omega}{d\lambda} =-k^a\nabla_a(u_bk^b) =-k^a k^b\nabla_a u_b =-H\omega^2.$$

Since $dt/d\lambda=k^a\nabla_a t=\omega$, it follows that

$$\frac{d\ln\omega}{dt}=-H=-\frac{d\ln a}{dt}, \qquad a\omega=\text{constant}.$$

Therefore two comoving endpoint measurements obey

$$1+z=\frac{\omega_e}{\omega_o} =\frac{a(t_o)}{a(t_e)}.$$

No field equation was used; an arbitrary positive scale factor produces the same kinematical relation.

The successive-crest derivation

The same result follows without solving explicitly for $k^a$. For a radial ray in conformal coordinates,

$$d\chi=\pm\frac{dt}{a(t)}.$$

Suppose one crest is emitted at $t_e$ and received at $t_o$, while the next is emitted at $t_e+\delta t_e$ and received at $t_o+\delta t_o$. Both traverse the same comoving interval, so

$$\int_{t_e}^{t_o}\frac{dt}{a(t)} =\int_{t_e+\delta t_e}^{t_o+\delta t_o}\frac{dt}{a(t)}.$$

To first order in the wave periods,

$$\frac{\delta t_o}{a(t_o)} =\frac{\delta t_e}{a(t_e)}.$$

Because comoving proper time equals $t$ and frequency is the inverse period, $\omega_e/\omega_o=a_o/a_e$. This derivation makes two assumptions especially visible: the source and observer are comoving, and the background does not change appreciably during one wave period. The geometric-optics version above is the appropriate limiting statement. Tong presents this successive-crest argument in Tong 2019, §1.1.3, pp. 13–15 (PDF).

Redshift as a time label

Set $a(t_o)=a_0=1$ at the present epoch. Then background emission time and redshift are related by

$$a(t_e)=\frac{1}{1+z}.$$

Differentiating along the background relation gives

$$\frac{dt_e}{dz}=-\frac{1}{(1+z)H(z)}.$$

This identity is a valid change of variable only on a branch where $a(t)$ is monotonic and $H\ne0$. At a turnaround or bounce, $dt/dz$ diverges and the same redshift need not select a unique emission time. Where the branch conditions hold, the identity underlies lookback-time and distance integrals; redshift is observable whereas $t_e$ is model inferred, and obtaining $H(z)$ requires the cosmic dynamics and chosen matter content. Hogg separates redshift, $H(z)$, and lookback time in Hogg 2000, §3, pp. 2–3; §4, p. 4; §10, p. 8 (PDF).

For a nearby comoving source, expand $a(t_e)$ about $t_o$ and use light-travel time $D$ to obtain $z\simeq H_oD$ in units with $c=1$. This is a local, first-order limit. The exact finite-distance relation is the scale-factor ratio, not the linear Hubble law.

Corrections and limiting checks

If an endpoint observer has $U^a=\gamma(u^a+v^a)$, with $u\cdot v=0$, then the photon decomposition above gives the exact local Doppler factor

$$\omega_U=-U\cdot k=\gamma\omega(1-v\cdot e).$$

Here $e^a$ points along the photon’s propagation direction in the local comoving frame, and $\gamma=(1-v^2)^{-1/2}$. For freely propagating light in the unperturbed FLRW background, arbitrary endpoint velocities therefore give

$$1+z_{\rm obs} =\frac{a_o}{a_e} \frac{\gamma_e(1-v_e\cdot e_e)} {\gamma_o(1-v_o\cdot e_o)}.$$

As a sign check, an emitter moving away from a comoving receiver has $v_e\cdot e_e\lt0$, increasing the redshift. At small velocities, these factors multiply the background $1+z$; adding a peculiar redshift directly to $z$ misses the factor $1+z$ outside the nearby-source limit. Scalar metric perturbations add endpoint gravitational terms and an integrated contribution from evolving potentials; vector and tensor perturbations can also shift the frequency. These effects are derived in the observed-light-cone chapter rather than absorbed into a redefined background scale factor.

Several checks prevent common misinterpretations:

• If $a$ is constant, comoving endpoints have $z=0$.
• On an expanding branch, earlier emission has $a_e\lt a_o$ and hence $z\gt0$. Contraction gives a background blueshift.
• In the dimensionless package, Milne coordinates have $a=t/t_0$ and $K=-t_0^{-2}$ and still describe flat spacetime. The formula agrees with the accumulated special-relativistic Doppler relation between different inertial Milne observers; “cosmological” and “Doppler” are not competing invariant labels in that case.
• Photon frequency decreases as $a^{-1}$, while the number density of freely propagating photons decreases as $a^{-3}$. The radiation energy density therefore scales as $a^{-4}$ once number conservation is supplied.

It is harmless pedagogically to say that expansion stretches wavelengths, but the precise content is $a\omega=\text{constant}$ for the declared ray and comoving observers. A story about photons continuously losing energy to a medium is neither required nor generally meaningful in a spacetime without a global timelike energy symmetry.

Exercises

A photon is emitted when $a_e=0.4$ by a galaxy with negligible peculiar velocity and observed today with $a_o=1$. Find its redshift and the ratio of observed to emitted wavelength.

Solution

The scale-factor law gives

$$1+z=\frac{a_o}{a_e}=\frac{1}{0.4}=2.5, \qquad z=1.5.$$

Since wavelength is inversely proportional to measured frequency, $\lambda_o/\lambda_e=\omega_e/\omega_o=1+z=2.5$. A peculiar velocity or gravitational potential at either endpoint would multiply this background ratio by the corresponding correction and must be specified separately.

References

• Ellis, George F. R., and Henk van Elst. “Cosmological Models.” In Theoretical and Observational Cosmology, edited by Marc Lachièze-Rey, 1–116. Dordrecht: Springer, 1999. DOI; Open PDF.
• Hogg, David W. “Distance Measures in Cosmology.” arXiv:astro-ph/9905116v4, 16 December 2000. Open PDF.
• Tong, David. Cosmology. University of Cambridge Part II Mathematical Tripos lecture notes, Michaelmas Term 2019. Official PDF.