Cosmological Redshift from the Scale Factor

Derivation

Cosmological Redshift from the Scale Factor

D-340 Home PU-308 Threads light · waves · symmetry Depends on The FRW Metric from Homogeneity and Isotropy, null-geodesics

Statement

For a photon travelling on a radial null geodesic through the spatially homogeneous and isotropic Friedmann–Robertson–Walker (FRW) spacetime, the wavelength measured by a comoving observer scales in direct proportion to the cosmic scale factor \(a(t)\). If the light is emitted at cosmic time \(t_e\) with wavelength \(\lambda_e\) and received at \(t_0\) with wavelength \(\lambda_0\), then \(\lambda_0/\lambda_e = a(t_0)/a(t_e)\), so the redshift \(z \equiv (\lambda_0-\lambda_e)/\lambda_e\) obeys \(1+z = a(t_0)/a(t_e)\).

Why it matters

This single relation converts an observable — the fractional shift of spectral lines — into a direct measurement of how much the Universe has expanded since the light left its source. It is the operational meaning of "redshift" in cosmology: \(z\) is not a Doppler velocity but a ratio of scale factors, and it is the coordinate along which nearly all of observational cosmology is organised.

Because \(1+z\) depends only on \(a(t_e)\) and \(a(t_0)\), and not on the expansion history in between, redshift is a clean, model-independent label for the epoch of emission. Combined with a dynamical model for \(a(t)\) (the Friedmann equations) it becomes a distance and a look-back time, underpinning the Hubble diagram, the interpretation of the cosmic microwave background at \(z\approx1100\), and the entire distance ladder.

Assumptions

The spacetime is FRW (homogeneous and isotropic).Without the maximally symmetric spatial slices the metric is not \(ds^2=-c^2dt^2+a^2(t)\,d\Sigma^2\), the notion of a single global scale factor \(a(t)\) is lost, and redshift can no longer be written as a ratio of two scale factors — it becomes path-dependent.

Source and observer are comoving.If either has a peculiar velocity relative to the Hubble flow, an extra special-relativistic Doppler factor multiplies the cosmological factor, and \(1+z=a(t_0)/a(t_e)\) holds only for the pure cosmological part.

Light propagates on null geodesics with \(ds^2=0\).Drop masslessness and the trajectory is timelike; the phase-tracking argument that ties frequency to \(1/a\) no longer applies and there is no clean redshift relation.

Geometric-optics (eikonal) limit: wavelength \(\ll\) curvature and expansion scales.If the wave is not locally plane on the scale over which \(a(t)\) changes, the phase is not well defined along a single ray and the wavecrest-counting argument breaks; one must solve the full wave equation on the curved background.

The metric is torsion-free with the Levi-Civita connection (standard GR).In theories with additional structure (torsion, non-metricity, or a varying fundamental constant) the parallel transport of the photon 4-momentum changes and the simple \(a^{-1}\) frequency scaling can acquire corrections.

Derivation

\[ ds^2 = -c^2\,dt^2 + a^2(t)\left[\frac{dr^2}{1-kr^2} + r^2\,d\Omega^2\right] \]

Start from the FRW line element (assumed prior result), with comoving radial coordinate \(r\), curvature \(k\), and \(d\Omega^2=d\theta^2+\sin^2\theta\,d\varphi^2\). A

\[ 0 = -c^2\,dt^2 + a^2(t)\,\frac{dr^2}{1-kr^2} \]

Place the observer at the origin. By isotropy the photon travels radially, so \(d\theta=d\varphi=0\); light is null, so \(ds^2=0\). A

\[ \frac{c\,dt}{a(t)} = \frac{dr}{\sqrt{1-kr^2}} \]

Take the square root and choose the incoming branch (radial coordinate decreasing toward the observer as \(t\) increases); rearrange to separate \(t\) and \(r\). A

\[ \int_{t_e}^{t_0} \frac{c\,dt}{a(t)} = \int_{0}^{r_e} \frac{dr}{\sqrt{1-kr^2}} \equiv \chi(r_e) \]

Integrate a single wavecrest from emission event \((t_e,r_e)\) to reception \((t_0,0)\). The right side is a fixed comoving distance set by the source, independent of when the crest is emitted. A

\[ \int_{t_e+\delta t_e}^{t_0+\delta t_0} \frac{c\,dt}{a(t)} = \chi(r_e) = \int_{t_e}^{t_0} \frac{c\,dt}{a(t)} \]

The very next wavecrest leaves at \(t_e+\delta t_e\) and arrives at \(t_0+\delta t_0\). Since the source is comoving, \(r_e\) is unchanged, so the comoving distance integral \(\chi(r_e)\) is identical for both crests. B

\[ \int_{t_e+\delta t_e}^{t_0+\delta t_0}\!\frac{c\,dt}{a} - \int_{t_e}^{t_0}\!\frac{c\,dt}{a} = \int_{t_0}^{t_0+\delta t_0}\!\frac{c\,dt}{a} - \int_{t_e}^{t_e+\delta t_e}\!\frac{c\,dt}{a} = 0 \]

Subtract the two equal integrals of Step 5. The overlapping middle interval \([t_e+\delta t_e,\,t_0]\) cancels, leaving only the two short end intervals. B

\[ \frac{c\,\delta t_0}{a(t_0)} = \frac{c\,\delta t_e}{a(t_e)} \]

The wave periods \(\delta t\) are tiny compared with the Hubble time over which \(a\) changes, so \(a(t)\) is constant across each short integral and pulls out: \(\int_t^{t+\delta t} c\,dt/a \approx c\,\delta t/a(t)\). This is the eikonal step. B

\[ \frac{\delta t_0}{\delta t_e} = \frac{a(t_0)}{a(t_e)} \]

Rearrange. The received period is stretched relative to the emitted period by exactly the ratio of scale factors. A

\[ \lambda = c\,\delta t \quad\Rightarrow\quad \frac{\lambda_0}{\lambda_e} = \frac{\delta t_0}{\delta t_e} = \frac{a(t_0)}{a(t_e)} \]

The proper wavelength measured locally by each comoving observer is \(c\) times the locally measured period between crests. Substitute Step 8. A

\[ 1+z \equiv 1 + \frac{\lambda_0-\lambda_e}{\lambda_e} = \frac{\lambda_0}{\lambda_e} = \frac{a(t_0)}{a(t_e)} \]

Apply the definition of redshift \(z=(\lambda_0-\lambda_e)/\lambda_e\) and substitute Step 9. A

Result

\[ 1+z = \frac{a(t_0)}{a(t_e)} \]

Reading. The factor by which a spectral line is shifted to longer wavelength equals the factor by which the Universe has expanded between emission and reception. A galaxy at \(z=1\) emitted its light when the Universe was half its present size (\(a(t_e)=a(t_0)/2\)); the surface of last scattering at \(z\approx1100\) dates from when everything was about \(1100\) times more compressed. Wavelength is dragged along with the expanding metric, not shifted by motion through space.

Units check. \(z\) is defined as a ratio of wavelengths, hence dimensionless; \(a(t)\) is dimensionless (normalised so \(a(t_0)=1\)), so \(a(t_0)/a(t_e)\) is dimensionless. Both sides carry no units. If instead \(a\) is given the dimension of length (physical curvature radius), it cancels in the ratio, preserving dimensionlessness.

Limiting cases

No expansion, \(a=\text{const}\): \(a(t_0)=a(t_e)\Rightarrow z=0\). Static spacetime gives no cosmological shift, as it must.
Low redshift, \(t_0-t_e\) small: Expand \(a(t_e)\approx a(t_0)[1-H_0(t_0-t_e)]\), giving \(z\approx H_0(t_0-t_e)\approx H_0 d/c\) — the Hubble law \(cz\approx H_0 d\) emerges as the \(z\ll1\) limit.
High redshift, \(a(t_e)\to0\): \(1+z\to\infty\); light from arbitrarily early epochs is shifted without bound. The CMB (\(z\approx1100\)) sits far in this regime.
Blueshift, \(a(t_0)<a(t_e)\): A contracting patch would give \(z<0\); formally consistent, realised only in a collapsing model or by peculiar-velocity Doppler on top of the cosmological term.

Breaks when

Peculiar velocities are significant. Cluster members and infalling galaxies carry Doppler shifts of order \(v/c\sim10^{-3}\) that add to (and near \(z\lesssim0.01\) can dominate over) the cosmological term; the observed \(z\) is then \((1+z_{\text{cos}})(1+z_{\text{pec}})-1\), not \(a_0/a_e\) alone.
Strong local gravitational fields intervene. Light climbing out of a deep potential (near a compact source, or through evolving large-scale potentials — the integrated Sachs–Wolfe effect) picks up a gravitational shift not captured by the smooth \(a(t)\); the background must be strictly FRW along the whole path.
Inhomogeneity on the light path (Rees–Sciama, lensing shear). If the metric departs from FRW along the geodesic, redshift becomes path-dependent and no single global \(a\) suffices.
Wavelength approaches the horizon/curvature scale. The eikonal approximation of Step 7 fails; extremely long-wavelength (super-horizon) modes are not described by simple crest-counting.

Failure modes

The "recession-velocity Doppler" error: reading \(z\) as an ordinary special-relativistic Doppler shift and inverting \(1+z=\sqrt{(1+\beta)/(1-\beta)}\) to get a velocity. This gives superluminal "velocities" for \(z>1\) that are meaningless; cosmological redshift is metric expansion, not motion through space.
Confusing \(a(t_e)\) with \(a(t_0)\): writing \(1+z=a(t_e)/a(t_0)\) upside down. Check with a limit: expansion (\(a_0>a_e\)) must give \(z>0\), so the larger scale factor is on top.
Forgetting to normalise: assuming \(a\) is a physical length and quoting \(z\) with units, or setting \(a(t_0)\ne1\) inconsistently between formulas.
Treating \(\delta t\) as varying across the short integral: trying to keep \(a(t)\) inside the wave-period integral of Step 7 and getting an incorrect logarithmic factor, instead of using the eikonal constancy of \(a\) over one period.
Assuming redshift measures distance directly: quoting "\(z=2\) means twice as far as \(z=1\)". The map from \(z\) to distance requires integrating \(1/a\) via the Friedmann equations and is nonlinear.

Discussion

The result reframes redshift as kinematics of the metric itself. Between two comoving points the proper distance grows as \(a(t)\); a light wave threaded through that expanding fabric has its crests carried apart at the same rate, so its wavelength inherits the factor \(a(t_0)/a(t_e)\). Nothing is "moving through" space in the special-relativistic sense — the space between crests is being manufactured. This is why the relation involves only the endpoint scale factors and is blind to the detailed expansion history in between: the total stretch is all that survives.

A complementary derivation uses the geodesic equation for the photon 4-momentum \(p^\mu\). Because the FRW metric has no explicit time-translation symmetry, energy is not conserved; instead the spatial homogeneity gives a conserved comoving momentum, and one finds the physical frequency obeys \(\nu\propto1/a\) directly — the same \(a^{-1}\) scaling, now read off from parallel transport rather than crest-counting. The two routes agree because both encode the same fact: the redshift is the ratio of the photon frequency measured by comoving observers at the two ends of the geodesic.

More deeply, the conserved quantity is a projection of \(p^\mu\) onto a symmetry of the geometry. For each Killing vector \(\xi^\mu\), \(\xi_\mu p^\mu\) is constant along the geodesic. The comoving spatial translations of the maximally symmetric slices supply Killing vectors whose contraction with \(p^\mu\) is the comoving momentum \(a(t)\,p_{\text{phys}}\); its constancy is exactly \(\nu\,a=\text{const}\). Redshift is thus a conservation law dressed by the loss of time-translation invariance — the symmetry thread and the light thread meeting in one line. In the geometric-optics limit the photon phase \(S\) satisfies the eikonal equation \(g^{\mu\nu}\partial_\mu S\,\partial_\nu S=0\), and \(p_\mu=\partial_\mu S\); the crest-counting of Steps 5–7 is precisely the statement that surfaces of constant \(S\) are dragged with the expansion.

Common misconceptions. (i) Cosmological redshift is not the Doppler effect and not gravitational redshift — though at low \(z\) it is numerically indistinguishable from a recession Doppler shift, which is why Hubble's original interpretation as velocity works locally. (ii) Photons do not "lose energy to the expansion" in any globally conserved sense; energy is simply not a conserved quantity in a non-static spacetime, so there is no bookkeeping violation. (iii) The expansion stretches wavelengths of freely propagating radiation, but bound systems (atoms, the Solar System, galaxies) do not expand — they are held by forces far stronger than the cosmological tidal field, so their spectral lines have fixed rest wavelengths that serve as the ruler.

Worked examples

\[ 1+z=\frac{a(t_0)}{a(t_e)}=\frac{\lambda_0}{\lambda_e} \]

Example A — Scale factor at a measured redshift. The \(\text{Ly}\alpha\) line has rest wavelength \(\lambda_e=121.6\ \text{nm}\) and is observed in a quasar at \(\lambda_0=486.4\ \text{nm}\). Find \(z\) and the scale factor \(a(t_e)\) with the convention \(a(t_0)=1\). A

\[ 1+z=\frac{486.4}{121.6}=4.000 \quad\Rightarrow\quad z=3.000 \]

Insert the wavelengths (nm cancels, leaving a pure number). A

\[ a(t_e)=\frac{a(t_0)}{1+z}=\frac{1}{4.000}=0.250 \]

Invert with \(a(t_0)=1\). A

\[ z=3.00,\qquad a(t_e)=0.250 \]

Reading. When this light left the quasar the Universe was one-quarter its present linear size, and every proper wavelength has since been stretched fourfold.

\[ \nu_0=\frac{\nu_e}{1+z}=\frac{a(t_e)}{a(t_0)}\,\nu_e \]

Example B — Redshifting the 21 cm line. Neutral-hydrogen emission has rest frequency \(\nu_e=1420.4\ \text{MHz}\). It is emitted at scale factor \(a(t_e)=0.100\) (with \(a(t_0)=1\)). Find the observed frequency and wavelength. A

\[ 1+z=\frac{a(t_0)}{a(t_e)}=\frac{1}{0.100}=10.0 \]

Compute the redshift from the scale factors. A

\[ \nu_0=\frac{1420.4\ \text{MHz}}{10.0}=142.0\ \text{MHz} \]

Frequency scales as \(1/(1+z)\); MHz carried through. A

\[ \lambda_0=\frac{c}{\nu_0}=\frac{2.998\times10^8\ \text{m s}^{-1}}{1.420\times10^8\ \text{s}^{-1}}=2.11\ \text{m} \]

Convert to wavelength; \((\text{m s}^{-1})/(\text{s}^{-1})=\text{m}\). Rest wavelength was \(0.211\ \text{m}\), stretched by \(1+z=10\). A

\[ z=9.00,\qquad \nu_0=142.0\ \text{MHz},\qquad \lambda_0=2.11\ \text{m} \]

Reading. The 21 cm line from the epoch of reionisation arrives in the low-VHF radio band, an order of magnitude longer in wavelength — the physical basis of 21 cm cosmology.

Problems

A galaxy's \(\text{H}\alpha\) line (rest \(656.3\ \text{nm}\)) is observed at \(787.6\ \text{nm}\). Find \(z\) and \(a(t_e)\) (take \(a(t_0)=1\)).
Solution
\(1+z=787.6/656.3=1.200\Rightarrow z=0.200\). Then \(a(t_e)=1/1.200=0.833\). The Universe was about 83% of its present size when this light was emitted.
The cosmic microwave background is a blackbody at \(T_0=2.725\ \text{K}\) today and was emitted at \(z=1100\). Using \(T\propto1/a\), find the temperature at last scattering.
Solution
Since wavelengths scale as \(a\), a blackbody stays a blackbody with \(T\propto1/a\propto(1+z)\). Thus \(T_e=T_0(1+z)=2.725\times1101=3.00\times10^3\ \text{K}\) (about \(2980\ \text{K}\)) — the temperature at which hydrogen recombines and the Universe becomes transparent.
Show that for small look-back time the exact relation \(1+z=a(t_0)/a(t_e)\) reduces to \(z\approx H_0(t_0-t_e)\), where \(H_0=\dot a(t_0)/a(t_0)\).
Solution
Taylor-expand \(a(t_e)=a(t_0)+\dot a(t_0)(t_e-t_0)+\dots=a(t_0)[1-H_0(t_0-t_e)]\). Then \(1+z=a(t_0)/a(t_e)=[1-H_0(t_0-t_e)]^{-1}\approx1+H_0(t_0-t_e)\), so \(z\approx H_0(t_0-t_e)\). For photons \(t_0-t_e\approx d/c\), giving \(cz\approx H_0 d\), the Hubble law.
A quasar is observed at \(z=6.00\). By what factor has a proper volume comoving with the Hubble flow (\(\propto a^3\)) expanded since the light was emitted, and by what factor has the mean matter density (\(\propto a^{-3}\)) dropped?
Solution
\(a(t_0)/a(t_e)=1+z=7.00\). Volume scales as \(a^3\), so it has grown by \(7.00^3=343\). Matter density scales as \(a^{-3}\), so it has fallen by the same factor \(343\): the Universe was \(343\) times denser in matter at \(z=6\).
Two spectral lines from the same source, rest wavelengths \(\lambda_{1,e}=393.4\ \text{nm}\) (Ca K) and \(\lambda_{2,e}=396.8\ \text{nm}\) (Ca H), are observed. If the K line is seen at \(590.1\ \text{nm}\), predict the observed H-line wavelength, and show the observed separation is stretched by \(1+z\).
Solution
\(1+z=590.1/393.4=1.500\Rightarrow z=0.500\). Every wavelength scales by the same factor, so \(\lambda_{2,0}=1.500\times396.8=595.2\ \text{nm}\). Rest separation \(\Delta\lambda_e=396.8-393.4=3.4\ \text{nm}\); observed \(\Delta\lambda_0=595.2-590.1=5.1\ \text{nm}=1.500\times3.4\ \text{nm}\). The line spacing is stretched by exactly \(1+z\), which is how redshift is confirmed to be multiplicative rather than an additive offset.