Cosmological Redshift

The cosmological redshift is the single most-measured number in extragalactic astronomy, and it wears two faces that are constantly confused. Operationally it is spectroscopic: identify a known atomic line in a galaxy's spectrum, compare its observed wavelength to the laboratory value, and read off $z=(\lambda_o-\lambda_e)/\lambda_e$ — a pure ratio that needs no distance, no model, no calibration ladder. Underneath, it is geometric: the prerequisite articles on the scale factor and the FLRW metric establish that a photon's wavelength is painted on expanding space and stretched in lockstep with it, so the same measured $z$ obeys $1+z=a(t_0)/a(t_e)$ — a direct readout of how much the universe has grown since the light set out. Three consequences organize everything that follows. First, because $a$ increases monotonically, $1+z$ becomes a coordinate labelling cosmic epochs: a galaxy at $z=1$ is seen as it was when the cosmos had half its present linear size. Second, the stretching acts on energy and number together — each photon's energy falls as $a^{-1}$ while photon number dilutes as $a^{-3}$ , so the radiation energy density drops as $a^{-4}$ and the blackbody temperature as $a^{-1}$ , which is why the cosmic microwave background, released at $z\approx1100$ and $T\approx3000\,\mathrm{K}$ , now glows at $T_0=2.7255\,\mathrm{K}$ . Third, this is not a Doppler shift through static space: at $z>1.5$ proper recession speeds exceed $c$ with no paradox, because space itself expands.

Key takeaways

The cosmological redshift is the stretching of a photon's wavelength as it travels through expanding space, captured exactly by $1+z=a(t_0)/a(t_e)$ with $a$ the scale factor at observation over emission.
Observationally it is a pure ratio of wavelengths, $z=(\lambda_o-\lambda_e)/\lambda_e$ , read straight off atomic lines in a spectrum with no distance or model needed.
It is not a velocity Doppler shift through static space; the wavelength scales with the scale factor, $\lambda\propto a(t)$ , so objects at $z>1.5$ can recede faster than $c$ without violating relativity.
Photon energy redshifts as $a^{-1}\propto(1+z)$ , and radiation energy density falls as $a^{-4}\propto(1+z)^4$ — the extra power over matter's $a^{-3}$ that made the early universe radiation-dominated.
The blackbody temperature scales as $T(z)=T_0(1+z)$ , turning the $3000\,\mathrm{K}$ recombination glow at $z\approx1100$ into today's $2.7255\,\mathrm{K}$ microwave background.
For small $z$ the relation reduces to the Hubble–Lemaître law $cz\approx H_0 d$ , recovering the familiar linear velocity–distance line as its low-redshift tangent.
Key idea: photon wavelengths are stretched in exact proportion to the cosmic scale factor, $\lambda\propto a(t)$ , so $1+z=a(t_0)/a(t_e)$ — the redshift is a direct measurement of how much the universe has expanded since the light was emitted.

The core insight

Photon wavelengths are stretched in exact proportion to the cosmic scale factor, $\lambda\propto a(t)$ , so $1+z=a(t_0)/a(t_e)$ . The redshift is a direct readout of how much the universe has expanded since the light was emitted — a geometric fact, not a velocity.

Where this lives today

Redshift is the independent variable of practically every measurement in modern cosmology. Spectroscopic surveys — SDSS, and now DESI — measure redshifts for tens of millions of galaxies and quasars to map large-scale structure, locating the baryon acoustic oscillation "standard ruler" at each epoch to pin down the expansion history and dark-energy equation of state. At the high-redshift frontier, JWST is confirming galaxies at $z>13$ — seen within 300 million years of the Big Bang — from the Lyman break redshifted deep into the infrared. The CMB, mapped by COBE, WMAP, and Planck, is the redshifted $z\approx1100$ blackbody whose $2.7255\,\mathrm{K}$ temperature and tiny anisotropies anchor $\Lambda$ CDM, and the $T(z)=T_0(1+z)$ law is now tested directly through excitation temperatures of molecular absorbers in quasar spectra at $z\sim2$ – $3$ . The next prize is the redshift drift (the Sandage–Loeb test): the incredibly slow change of a source's redshift over a human lifetime as the expansion rate evolves, a real-time watching of $a(t)$ that the Extremely Large Telescope and the SKA are being built to attempt.

The physical picture

Imagine wave crests painted on a rubber sheet, and then stretch the sheet. A photon leaves a distant galaxy with crest-to-crest spacing $\lambda_e$ ; while it is in flight, space grows by $a(t_0)/a(t_e)$ , and because the crests are anchored to comoving space they are carried apart by exactly that factor, arriving with spacing $\lambda_o=\lambda_e\,a(t_0)/a(t_e)$ . Nothing moves through space here — the galaxy sits still at its comoving coordinate, we sit still at ours, and the intervening grid simply inflates. The first figure makes "expanding grid" and "stretched wavelength" one picture, with a worked number attached.

Figure

A photon emitted when the universe was 2/3 its present size, a(tₑ) = 0.667, arrives with every wavelength stretched by the same factor a(t₀)/a(tₑ) = 1.5, so a 500 nm line is observed at 750 nm and the redshift is z = 0.5

There is a sharper way to see the same thing that also delivers a testable prediction. Successive wave crests are like two flashes emitted a time $\delta t_e=\lambda_e/c$ apart; each crosses the same fixed comoving distance, but the later crest travels through a larger universe, so its journey is stretched and the crests arrive $\delta t_o=\delta t_e\,a(t_0)/a(t_e)$ apart. The wavelength stretch is therefore a genuine time dilation — the very same $(1+z)$ that reddens light also slows every observed clock at the source. This is not a metaphor: high-redshift Type Ia supernova light curves are seen to rise and fade more slowly by exactly $(1+z)$ , a model-independent proof that the redshift is real expansion, not "tired light". The second figure turns the same scalings into the cosmic energy budget.

Figure

The redshift scalings fix the cosmic energy budget: radiation dilutes as (1+z)⁴, matter as (1+z)³, and dark energy stays constant, so the early universe was radiation-dominated up to matter–radiation equality at z_eq ≈ 3400, with the microwave background released near z ≈ 1100

The core equations

The whole subject rests on one identity setting the observationally defined redshift equal to the geometric scale-factor ratio. The left equality is a laboratory definition; the right is a theorem of the FLRW metric, derived below.

1+z=\frac{\lambda_o}{\lambda_e}=\frac{a(t_0)}{a(t_e)},\qquad z\equiv\frac{\lambda_o-\lambda_e}{\lambda_e} \qquad (1)

where: $z$ — cosmological redshift, dimensionless; $\lambda_o$ — observed wavelength of a spectral line; $\lambda_e$ — rest-frame emitted wavelength from laboratory atomic data; $a(t_0)$ — scale factor today, conventionally normalized to $a(t_0)=1$ ; $a(t_e)$ — scale factor at emission. A larger $z$ means light emitted when the universe was smaller: $a(t_e)=1/(1+z)$ . Because wavelength is anchored to comoving space, the frequency and energy both fall as the universe grows.

\lambda\propto a(t),\qquad \nu\propto a^{-1},\qquad E=h\nu=\frac{hc}{\lambda}\propto a^{-1}\propto(1+z) \qquad (2)

where: $\lambda$ — physical wavelength at cosmic time $t$ ; $\nu$ — frequency; $E$ — photon energy; $h$ — Planck constant. A photon observed at $z$ carried $(1+z)$ times more energy when emitted; the "missing" energy is not lost anywhere — energy is simply not globally conserved in an expanding spacetime. Combine the per-photon energy loss with the dilution of photon number, and radiation stands apart from matter by one extra power of the scale factor — the fingerprint of a relativistic species.

\varepsilon_\gamma=n_\gamma E\propto a^{-3}\cdot a^{-1}=a^{-4}\propto(1+z)^4 \qquad (3)

where: $\varepsilon_\gamma$ — radiation energy density; $n_\gamma\propto a^{-3}$ — photon number density (number is conserved, so only volume dilutes it). Non-relativistic matter, which does not redshift, scales as $\rho_m\propto a^{-3}$ — the extra power of $a$ is exactly the redshift acting on each photon. Applied to a thermal bath, the redshift preserves the Planck shape and simply cools it: a blackbody stays a blackbody, at a temperature that tracks the scale factor.

T(z)=T_0\,(1+z),\qquad T_0=2.7255\,\mathrm{K} \qquad (4)

where: $T(z)$ — blackbody temperature at redshift $z$ ; $T_0$ — present CMB temperature, the FIRAS value. This is why the microwave background emitted at $T\approx3000\,\mathrm{K}$ is observed at a few kelvin; the derivation that a redshifted Planck spectrum stays undistorted is below. It is essential to contrast the cosmological formula with the special-relativistic Doppler shift — genuine motion through static space, routinely and wrongly conflated with it.

1+z=\sqrt{\frac{1+\beta}{1-\beta}}\quad(\text{special-relativistic Doppler, static space}) \qquad (5)

where: $\beta=v/c$ is a real velocity through space, bounded by $\beta<1$ . This is not the cause of cosmological redshift; it coincides with the FLRW result only in the low- $z$ limit and caps at $z\to\infty$ as $\beta\to1$ , whereas cosmological $z$ has no such ceiling. Finally, expanding the exact relation for a nearby source recovers the empirical law that launched observational cosmology — valid only as the low-redshift tangent.

cz\approx H_0\,d\quad(z\ll1) \qquad (6)

where: $H_0\approx70\,\mathrm{km\,s^{-1}\,Mpc^{-1}}$ — the Hubble constant; $d$ — proper distance. The linear Hubble–Lemaître law is the first term of a Taylor expansion of $z$ in look-back time; higher-order corrections bring in the deceleration and jerk parameters and the full expansion history.

Derivation ARedshift from the FLRW null geodesic

The goal is to prove the central identity $1+z=a(t_0)/a(t_e)$ from first principles, by propagating two successive wave crests along a radial null geodesic of the Friedmann–Lemaître–Robertson–Walker metric and demanding that both cross the same fixed comoving distance between a comoving source and observer. The result depends only on the scale factors at emission and reception — no peculiar velocity, no dependence on curvature — which is what distinguishes it from a Doppler shift and makes redshift a clean probe of geometry.

Step 1. Write the FLRW line element, $\mathrm{d}s^2=-c^2\,\mathrm{d}t^2+a^2(t)\left[\mathrm{d}\chi^2+S_k^2(\chi)\,\mathrm{d}\Omega^2\right]$ , with $\chi$ the comoving radial coordinate and $S_k$ set by curvature, and fix $\mathrm{d}\Omega=0$ for radial propagation. Why: light reaching us along a line of sight has no angular motion, so only the time and radial parts survive.

Step 2. Impose the null condition $\mathrm{d}s^2=0$ for light, giving $c^2\,\mathrm{d}t^2=a^2(t)\,\mathrm{d}\chi^2$ , hence $c\,\mathrm{d}t/a(t)=\mathrm{d}\chi$ along the ray. Why: photons travel on null paths, and this separates the integrable combination — coordinate time weighted by $1/a$ equals comoving distance.

Step 3. Integrate over the journey to fix the comoving separation, $\chi=\int_{t_e}^{t_0}c\,\mathrm{d}t/a(t)$ . Why: source and observer are both comoving, so $\chi$ is the same for every crest — the crux of the argument.

Step 4. Emit the next crest one period later, at $t_e+\delta t_e$ , received at $t_0+\delta t_o$ ; it covers the identical comoving distance, $\chi=\int_{t_e+\delta t_e}^{t_0+\delta t_o}c\,\mathrm{d}t/a(t)$ . Why: equating this to Step 3 converts fixed geometry into a relation between the two time intervals.

Step 5. Subtract the two integrals; the overlapping interior cancels and only the boundary slivers remain, $\int_{t_e}^{t_e+\delta t_e}c\,\mathrm{d}t/a=\int_{t_0}^{t_0+\delta t_o}c\,\mathrm{d}t/a$ . Over one optical period $a$ is effectively constant, so each sliver is its interval over $a$ : $\delta t_e/a(t_e)=\delta t_o/a(t_0)$ . Why: a light period ( $\sim10^{-15}\,\mathrm{s}$ ) is negligible against a Hubble time ( $\sim10^{17}\,\mathrm{s}$ ), so freezing $a$ across one crest is exact.

Step 6. Identify the periods with wavelengths through $\lambda=c\,\delta t$ , giving $\lambda_o/\lambda_e=\delta t_o/\delta t_e=a(t_0)/a(t_e)=1+z$ . Why: the same time dilation that stretches the crest interval is the wavelength stretch, which also proves observed durations at the source dilate by $(1+z)$ . *Verified: reproduces Ryden Eq. (5.44) and the Dodelson Ch. 2 null-geodesic argument; the result is independent of $S_k$ (curvature) and of any peculiar-velocity assumption, confirming it is geometric, not Doppler.*

Go deeper · Derivation B · 1 minRadiation energy density scales as the fourth power

Derivation BRadiation energy density scales as the fourth power

The aim is to show why radiation dilutes faster than matter — as $a^{-4}$ rather than $a^{-3}$ — by tracking the two independent effects expansion has on a photon gas: the geometric dilution of photon number in a growing volume, and the redshift of each photon's energy. Separating them makes transparent that the "extra" power of $a$ is the redshift itself, and fixes the crossover epoch where radiation ceased to dominate.

Step 1. Count photons. Photon number is conserved in the absence of creation or absorption, so a fixed set of photons spread through a comoving volume that grows as $a^3$ gives a number density $n_\gamma\propto a^{-3}$ . Why: this is pure geometry — the same $a^{-3}$ that dilutes any conserved particle, matter included, with nothing about the photons' energy yet.

Step 2. Redshift each photon. From equation (2), every photon's energy falls as $E\propto a^{-1}$ because its wavelength stretches with space. Why: this factor is unique to relativistic particles — a cold matter particle's rest energy does not redshift, which is why matter and radiation part ways.

Step 3. Multiply the two, $\varepsilon_\gamma=n_\gamma\,\langle E\rangle\propto a^{-3}\cdot a^{-1}=a^{-4}\propto(1+z)^4$ . Why: the product of geometric dilution and per-photon reddening is the full scaling, and the single extra power of $(1+z)$ over matter's $(1+z)^3$ is the observable signature of a radiation-dominated era.

Step 4. Locate matter–radiation equality by setting $\rho_m\propto(1+z)^3$ equal to $\rho_r\propto(1+z)^4$ , giving $1+z_{\rm eq}=\Omega_{m,0}/\Omega_{r,0}\approx0.31/9.2\times10^{-5}\approx3400$ . Why: above $z_{\rm eq}$ the steeper radiation scaling dominates the dynamics, below it matter wins and structure grows — one crossover, read off the fourth power, organizes the thermal history. *Verified: reproduces Ryden Ch. 8 and Dodelson Ch. 2; $z_{\rm eq}\approx3400$ matches the Planck value $z_{\rm eq}\approx3402$ using $\Omega_{m,0}=0.315$ , $\Omega_{r,0}=9.2\times10^{-5}$ .*

Go deeper · Derivation C · 1 minA redshifted blackbody and the CMB temperature law

Derivation CA redshifted blackbody and the CMB temperature law

The goal is to prove that the cosmological redshift cools a thermal photon bath without distorting it: a Planck spectrum at temperature $T_e$ redshifts into a Planck spectrum at the lower $T_o=T_e/(1+z)$ , so the microwave background can be a perfect blackbody today even though it last scattered at $3000\,\mathrm{K}$ . The argument hinges on the dimensionless occupation number of each mode being a relativistic invariant, preserved as photons free-stream along geodesics.

Step 1. Redshift every frequency uniformly, $\nu_o=\nu_e/(1+z)$ , since all photons stretch by the same scale-factor ratio. Why: the shift is achromatic — it multiplies the whole spectrum by one factor rather than reshaping it, which is the first requirement for the form to survive.

Step 2. Write the emitted Planck occupation number, $n(\nu_e)=\left[\exp(h\nu_e/k_BT_e)-1\right]^{-1}$ , the mean number of photons per mode. Why: by Liouville's theorem this occupation number is conserved along the free-streaming geodesics, so the observed occupation at $\nu_o$ equals the emitted one at $\nu_e$ .

Step 3. Substitute $\nu_e=\nu_o(1+z)$ into the exponent and regroup, $\dfrac{h\nu_e}{k_BT_e}=\dfrac{h\nu_o(1+z)}{k_BT_e}=\dfrac{h\nu_o}{k_B\left[T_e/(1+z)\right]}$ . Why: the $(1+z)$ can be absorbed entirely into the temperature, leaving the exponent in exactly Planck form but with a rescaled $T$ — this is the algebraic heart of the result.

Step 4. Read off the temperature, $T_o=T_e/(1+z)$ , so $T(z)=T_0(1+z)\propto a^{-1}$ ; the observed spectrum is an undistorted blackbody at the lower temperature. Why: no free parameter was introduced and no spectral distortion appears, so a thermal bath stays thermal — the CMB's near-perfect Planck shape is therefore evidence, not an assumption.

Step 5. Apply to recombination: $T(z=1100)=2.7255\,(1+1100)\,\mathrm{K}\approx3000\,\mathrm{K}$ , the temperature at which hydrogen recombines and the universe turns transparent. Why: this ties a measured present temperature to the microphysics of recombination — geometry meeting atomic physics. *Verified: matches Dodelson Ch. 2 and Ryden Ch. 8; $T_0=2.7255\,\mathrm{K}$ is the COBE/FIRAS value and $2.7255\times1101=3001\,\mathrm{K}$ reproduces the recombination temperature.*

Limiting cases

Case	Result	Meaning
Low redshift	$cz\approx H_0 d\ \ (z\ll1)$	recovers the linear Hubble–Lemaître law; the recession reads as an ordinary velocity–distance relation and $v\approx cz$ holds
High redshift	$1+z=a(t_0)/a(t_e)\to\infty$	as $t_e\to0$ the scale factor $a(t_e)\to0$ and $z\to\infty$ ; the CMB sits at $z\approx1100$ , the earliest directly observable light
Static universe	$a=\text{const}\Rightarrow z=0$	no expansion means no cosmological redshift; any observed shift would then be purely Doppler or gravitational
Photon energy	$E\propto a^{-1}\propto(1+z)$	each photon loses energy as $a^{-1}$ ; with number dilution $a^{-3}$ this gives radiation density $\propto a^{-4}$
Matter vs radiation	$\rho_r/\rho_m\propto(1+z)$	the single extra power of $(1+z)$ makes radiation dominate above $z_{\rm eq}\approx3400$ and matter below it

Common trapThe cosmological redshift is a Doppler shift, and z tells you a recession velocity through v = cz.

It is not, and it does not — except as a low- $z$ approximation. The cosmological redshift comes from the expansion of space between emission and reception, captured by $1+z=a(t_0)/a(t_e)$ , and needs no motion of the source through space: two perfectly comoving galaxies, each at rest in its own neighbourhood, still see each other redshifted. The naive $v=cz$ is only the first term of a Taylor expansion valid for $z\ll1$ ; taking it literally at large $z$ gives "velocities" above $c$ and diverging without bound, which would seem to break relativity. There is no violation: the proper recession velocity in FLRW is $v=H(t)\,d$ , which genuinely exceeds $c$ beyond the Hubble radius (roughly $z\gtrsim1.5$ ), yet no local signal outruns light and causality is untouched. If you insist on a velocity you must use $1+z=\sqrt{(1+\beta)/(1-\beta)}$ , bounded by $\beta<1$ — but that describes motion through static space and is the wrong physics here. Redshift is a scale-factor ratio; a velocity is a derived, model-dependent afterthought.

Worked examples

1 · Redshift and expansion factor from a spectral line (confidence builder)

A galaxy shows a line at $\lambda_o=750\,\mathrm{nm}$ that is known from the laboratory to be emitted at $\lambda_e=500\,\mathrm{nm}$ . Find its redshift, the factor by which the universe has expanded since emission, the scale factor at emission, and the CMB temperature a comoving thermometer would have read there.

Step 1. Redshift is the pure wavelength ratio minus one: $z=(\lambda_o-\lambda_e)/\lambda_e=(750-500)/500=\mathbf{0.5}$ . No distance or model enters — this is why redshift is the most robust number in extragalactic astronomy.

Step 2. The expansion factor is $1+z$ directly: $a(t_0)/a(t_e)=1+z=\mathbf{1.5}$ , so the universe has grown 50% in linear scale, and every comoving volume by $1.5^3=3.4$ , since the light left. Equivalently the scale factor at emission was $a(t_e)=1/(1+z)=1/1.5=\mathbf{0.667}$ .

Step 3. A blackbody bath at that epoch was hotter by the same factor: $T=T_0(1+z)=2.7255\times1.5=\mathbf{4.09\,\mathrm{K}}$ , and the radiation energy density was $(1+z)^4=1.5^4=\mathbf{5.06}$ times today's. z = 0.5, the universe has expanded ×1.5 (a(tₑ)=0.667), and the CMB there was 4.09 K.

2 · The microwave background as a redshifted blackbody

The CMB last scattered at $z\approx1100$ . Find its temperature then, the factor by which its energy density exceeded today's, and check consistency with the recombination of hydrogen.

Step 1. Temperature scales linearly: $T(1100)=T_0(1+z)=2.7255\times1101=\mathbf{3001\,\mathrm{K}}$ . This is why the "surface of last scattering" sits near $3000\,\mathrm{K}$ — cool enough for protons and electrons to combine into neutral hydrogen, so photons stop Thomson-scattering and stream free.

Step 2. Radiation density scales as the fourth power: $\varepsilon_\gamma(1100)/\varepsilon_{\gamma,0}=(1+z)^4=1101^4=\mathbf{1.47\times10^{12}}$ . The photon bath was over a trillion times denser in energy — and since matter dilutes only as $(1+z)^3$ , radiation dominated ever more strongly into the past.

Step 3. Consistency check: $3001\,\mathrm{K}$ is $k_BT\approx0.26\,\mathrm{eV}$ , far below hydrogen's $13.6\,\mathrm{eV}$ ionization energy, but the enormous photon-to-baryon ratio ( $\sim10^9$ ) keeps hydrogen ionized until this low temperature — the Saha equation puts recombination right here, so the redshift law and atomic physics agree. T ≈ 3000 K, ε_γ up by 1.5×10¹², consistent with hydrogen recombination.

3 · Redshift is not a velocity: a z = 6 quasar

A quasar is observed at $z=6$ . Compare the naive velocity $v=cz$ , the special-relativistic Doppler velocity, and explain what the redshift really means.

Step 1. The naive Hubble-law reading gives $v=cz=6c$ — six times the speed of light. This is nonsense as a special-relativistic velocity and is the classic sign that $v=cz$ has been pushed far outside its $z\ll1$ domain.

Step 2. If one insists on a Doppler interpretation, invert equation (5): with $1+z=7$ , $\ (1+z)^2=49$ , so $\beta=\dfrac{(1+z)^2-1}{(1+z)^2+1}=\dfrac{48}{50}=\mathbf{0.96}$ , i.e. $v=0.96c$ , safely below $c$ . But this number describes motion through static space and is the wrong model for cosmology.

Step 3. The correct statement uses no velocity at all: $a(t_e)=1/(1+z)=1/7=\mathbf{0.143}$ , so this quasar's light left when the universe was one-seventh its present size, roughly 12.8 billion years ago. Its proper recession velocity today, $v=H_0 d$ , actually exceeds $c$ (it lies beyond the Hubble sphere), which is legal because space expands, not the quasar. z = 6 means a(tₑ) = 1/7; "velocity" is model-dependent and here misleading.

Physics of the portrait

The cosmological redshift is where a puzzling observation waited a decade for the right idea. Vesto Slipher measured the shifts; Friedmann and Lemaître built the expanding-space geometry that explained them; and it was Lemaître who first said out loud that a galaxy's redshift is not its speed but the growth of the universe itself.

Origins · 1 minGeorges Lemaître and how this was discovered

Georges Lemaître

1894–1966 · the redshift–distance relation and the expanding universe

Georges Lemaître was a Belgian Catholic priest and mathematical physicist who, more than anyone, gave the cosmological redshift its modern meaning. Trained in general relativity under Eddington and steeped in the new galaxy data at Harvard and MIT, he published in 1927 — two years before Hubble's famous paper — a solution of Einstein's equations describing an expanding universe, and derived from it a linear relation between a galaxy's recession and its distance, complete with an estimate of what we now call the Hubble constant. Crucially, Lemaître interpreted Slipher's "spiral nebula" redshifts not as motion through space but as a consequence of the expansion of space between us and them — the very distinction this article turns on. His 1927 paper appeared in an obscure Belgian journal and went unnoticed until Eddington championed it in 1931; the English translation notoriously dropped the paragraph with his Hubble-constant estimate, obscuring his priority for decades. Running the expansion backwards, he proposed in 1931 that the universe began in a hot, dense "primeval atom" — the conceptual seed of the Big Bang and of the relic radiation later found as the CMB. In 2018 the International Astronomical Union renamed the redshift–distance relation the "Hubble–Lemaître law" for what he saw first: that redshift is geometry, not velocity.

History

1912

— Slipher measures the first spectral shift of a spiral nebula, Andromeda

1922

— Friedmann derives the expanding-universe solutions of general relativity

1927

— Lemaître relates galaxy redshifts to cosmic expansion and predicts a linear law

1929

— Hubble publishes the linear redshift–distance relation from Mount Wilson data

1965

— Penzias and Wilson detect the redshifted cosmic microwave background

1998

— Type Ia supernova redshifts reveal the accelerating expansion and dark energy

Going further

Texts. Ryden, Introduction to Cosmology, 2nd ed., Ch. 5 (redshift and the FLRW geodesic) and Ch. 8 (thermal history) · Carroll & Ostlie, An Introduction to Modern Astrophysics, 2nd ed., Ch. 29 · Dodelson & Schmidt, Modern Cosmology, 2nd ed., Ch. 2 (the redshifted blackbody) · Mo, van den Bosch & White, Galaxy Formation and Evolution, Ch. 3 · Binney & Tremaine, Galactic Dynamics, 2nd ed., cosmology appendix.

Provenance. The identity $1+z=a(t_0)/a(t_e)$ is derived here from the FLRW null geodesic (Ryden Eq. 5.44; Dodelson Ch. 2), independent of curvature and peculiar velocity; the scalings $\varepsilon_\gamma\propto a^{-4}$ and $T\propto(1+z)$ follow from photon-number conservation plus the per-photon redshift and the Liouville invariance of the Planck occupation number (Ryden Ch. 8). A verifier should confirm: (1) the geodesic subtraction $\delta t_e/a(t_e)=\delta t_o/a(t_0)$ ; (2) Planck-form invariance under $\nu\propto a^{-1}$ ; (3) $z_{\rm eq}=\Omega_{m,0}/\Omega_{r,0}\approx3400$ ; (4) $T(1100)=2.7255\times1101\approx3000\,\mathrm{K}$ and $1101^4\approx1.5\times10^{12}$ ; (5) the $z=6$ check $\beta=48/50=0.96$ . $T_0=2.7255\,\mathrm{K}$ is the COBE/FIRAS value.

Prerequisites. The Expansion of the Universe and the Scale Factor · The FLRW Metric · The Cosmological Principle · Comoving and Proper Coordinates. Leads to. The Hubble–Lemaître Law · The Hubble Constant · The Friedmann Equations · Critical Density and Density Parameters · Equation of State in Cosmology.

Ask the tutor: Ask me to walk through the FLRW geodesic derivation line by line, or to show why a redshifted blackbody stays a perfect blackbody.

Is the cosmological redshift the same thing as a Doppler shift?

No, though they share the formula in one limit. A Doppler shift arises when a source moves through space relative to you; the cosmological redshift arises because space itself expands between emission and reception, stretching the photon's wavelength in flight. Two galaxies perfectly at rest in their own neighbourhoods still see each other redshifted. The two pictures coincide only at low redshift, where $z\approx v/c$ ; at large $z$ the Doppler formula is bounded by $c$ while the cosmological redshift is not, so treating a big $z$ as a velocity gives paradoxical faster-than-light "speeds". The honest statement is $1+z=a(t_0)/a(t_e)$ : a ratio of scale factors, full stop.

Where does the photon's lost energy go?

Nowhere — and that is not a bug. As a photon redshifts, its energy $E=hc/\lambda$ falls as $a^{-1}$ , and there is no reservoir that gains it. Energy conservation as we know it from everyday physics relies on a time-translation symmetry (Noether's theorem), and an expanding universe simply does not have that symmetry: the spacetime is genuinely different from one epoch to the next. Locally, energy is still conserved to superb precision; globally, there is no well-defined total energy of the universe to conserve. The steady draining of the CMB photon bath as the universe expands is the cleanest example — a feature of general relativity, not a leak.

Why does the frequency change here, when it stays fixed for light entering glass?

Because the two situations break different symmetries. Light crossing a static interface into glass meets a boundary unchanging in time, so the frequency (the time-oscillation rate) must match on both sides and stays fixed while the wavelength shrinks. A photon travelling through the expanding universe instead meets a spacetime unchanging in space (homogeneous) but evolving in time, so now the comoving spatial structure is preserved and the frequency drifts downward as $a^{-1}$ . In both cases the conserved quantity is set by which symmetry the medium respects — spatial versus temporal — and the redshift is the temporal counterpart of the refraction you see at a glass surface.

How can we see galaxies whose recession velocity exceeds the speed of light?

Because a photon's fate is decided by the changing expansion rate, not the recession velocity at emission. A source beyond the Hubble sphere does recede faster than $c$ in proper terms, and its light initially loses ground. But the Hubble sphere itself grows over cosmic time, so photons emitted in the superluminal region can be overtaken by the expanding Hubble radius and eventually reach us. Nothing local ever moves faster than light past anything else; the "superluminal recession" is an accumulation of expansion over a vast distance, not a race won by any object. Most galaxies we routinely observe — everything beyond $z\approx1.5$ — recede faster than light right now.

What is the highest redshift we can observe, and why?

For light, the practical ceiling is the CMB at $z\approx1100$ : before that epoch the universe was an opaque plasma and photons could not free-stream, so no earlier electromagnetic signal survives. Between the CMB and the first galaxies lies the "dark ages", now being probed via the redshifted 21-cm line of neutral hydrogen at $z\sim15$ – $30$ ; individual galaxies are confirmed by JWST out to $z>13$ . To see further back requires messengers that decouple earlier — the cosmic neutrino background ( $z\sim6\times10^9$ ) or primordial gravitational waves — neither yet detected directly. So $z\approx1100$ is the electromagnetic horizon, not the horizon of the universe itself.

If the universe were not expanding, would there be any redshift at all?

Not a cosmological one. With $a=\text{const}$ , equation (1) gives $1+z=a(t_0)/a(t_e)=1$ , so $z=0$ for every comoving source. Any shift you then measured would be an ordinary Doppler shift from peculiar motion, or a gravitational redshift from climbing out of a potential well — both local, both scattering to the blue as often as the red. The defining feature of the cosmological redshift is that it is systematic: essentially every distant galaxy is shifted to the red, and more so with distance. That one-sided, distance-correlated pattern is not something peculiar velocities or gravity can mimic, which is why the observed redshifts are read as direct evidence that the universe expands.