Cosmological Redshift

When you take the spectrum of a distant galaxy, every line sits at a longer wavelength than the same line measured in a laboratory, and the shift is the same fractional amount for every line at once. That single fact — a uniform stretch, not a jumble — is the fingerprint of cosmological redshift. The light has not been Doppler-shifted by a galaxy racing away through space, nor has it climbed out of a gravitational well; it has been carried across a geometry that was itself growing while the light was in flight. The Cosmological Principle fixes that geometry to be the homogeneous, isotropic FLRW spacetime, whose entire dynamics live in one function of time, the scale factor $a(t)$ . As a wave crest leaves the galaxy and threads its way to your telescope, the proper distance between successive crests is dragged outward with the cosmic fabric, and the accumulated stretch over the whole journey is exactly the factor by which the universe grew. The clean statement is geometric and path-independent: in comoving coordinates the wavelength is frozen, and the conversion to physical wavelength carries precisely one factor of $a(t)$ . So the observed redshift reads off the ratio of scale factors at reception and emission, $1+z = a_{\rm obs}/a_{\rm emit}$ , no matter what matter or curvature the photon passed. Because $a(t)$ increases monotonically in an expanding universe, $z$ becomes a clock and a ruler in one: it dates the epoch of emission, converts to distance through the expansion history, dilutes photon energy densities as $a^{-4}$ , and sets the temperature of the cosmic microwave background. This article builds the redshift–scale-factor dictionary that the rest of cosmology is written in.

Key takeaways

Light propagating through the expanding FLRW universe has its wavelength stretched in exact proportion to the scale factor, so $\lambda_{\rm obs}/\lambda_{\rm emit} = a(t_{\rm obs})/a(t_{\rm emit})$ — the same fractional shift for every spectral line.
The redshift is defined by $1+z \equiv a(t_{\rm obs})/a(t_{\rm emit})$ ; with the standard normalisation $a(t_0)=1$ this is $1+z = 1/a(t_{\rm emit})$ , so $z$ directly labels the scale factor when the light left.
Cosmological redshift is NOT a Doppler shift through space and NOT a gravitational redshift: comoving wavelengths are conserved along the null geodesic while proper lengths expand with $a(t)$ .
For nearby sources the exact relation collapses to Hubble's law $cz \approx H_0 d$ , but at large $z$ the distance–redshift map bends in a way set by the matter, radiation, and dark-energy content through the Friedmann equations.
Redshift is the master observable of cosmology: it converts to scale factor and cosmic time, redshifts photon energy as $(1+z)^{-1}$ , dilutes radiation energy density as $(1+z)^4$ , and cools the CMB as $T \propto (1+z)$ .
Key idea: Comoving wavelength is conserved along a null geodesic; physical wavelength carries exactly one factor of the scale factor, so $1+z = a_{\rm obs}/a_{\rm emit}$ . Redshift measures how much the universe expanded during the light's transit, not a velocity.

The core insight

Comoving wavelength is conserved along a null geodesic; physical wavelength carries exactly one factor of the scale factor, so $1+z = a_{\rm obs}/a_{\rm emit}$ . Redshift is a measure of how much the universe expanded during the light's transit, not a velocity.

Where this lives today

Redshift is the single most productive number in observational cosmology, and the frontier is a race to measure it further and more precisely. The James Webb Space Telescope now routinely confirms galaxies at $z > 10$ , whose rest-frame ultraviolet light is stretched deep into the infrared, probing the first few hundred million years after the Big Bang. Massive spectroscopic surveys — DESI, Euclid, and the Vera Rubin Observatory — collect tens of millions of galaxy redshifts to map baryon acoustic oscillations, the standard ruler imprinted at recombination. The CMB, measured by COBE, WMAP, and Planck, anchors the redshift ladder at $z\approx 1100$ with its $2.7255\,\mathrm{K}$ blackbody, and Type Ia supernovae extend the distance–redshift relation to $z\sim 1$ – $2$ as the direct evidence for accelerating expansion and dark energy. Redshift also sits at the centre of the field's sharpest controversy: the Hubble tension, in which the locally measured $H_0\approx 73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ (SH0ES, from Cepheid-calibrated supernovae) disagrees at more than $5\sigma$ with the $\approx 67.4$ inferred from the CMB by Planck. Looking ahead, extremely large telescopes aim to detect the real-time drift of redshifts over a decade (the Sandage–Loeb test), watching the expansion happen live.

The physical picture

Freeze a single crest of a light wave as it leaves a galaxy, and freeze the next crest one period behind it. In comoving coordinates the two crests are separated by a fixed coordinate interval that never changes — the galaxy and your telescope both sit at rest in the comoving grid. What changes is the metric: the factor $a(t)$ that converts comoving separations to physical ones grows monotonically while the light is in transit, so the physical distance between the two crests grows with it. By the time the pair reaches you, that physical separation — the wavelength you measure — has been stretched by the ratio $a_{\rm obs}/a_{\rm emit}$ . Nothing about the source's motion enters; the stretch is a property of the geometry the light crossed. The first figure makes "frozen comoving wavelength, stretched physical wavelength" a single picture, with a concrete factor of three.

Figure

A photon emitted at redshift z = 2, when the scale factor was a = 1/3, arriving today at a = 1: its comoving wavelength is frozen, but expanding space multiplies the physical wavelength by a_obs/a_emit = 1+z = 3, so a rest-frame 500 nm emission line is observed at 1500 nm in the near-infrared

It is tempting to picture the galaxy "flying away" and the shift as an ordinary Doppler effect, but in comoving coordinates the galaxy never moves. What moves is the geometry: the spatial part of the metric carries a factor $a^2(t)$ that grows with cosmic time, so a photon sampling ever-larger $a$ has its locally measured frequency fall. The deepest way to say it is a conservation law — the photon's momentum measured by comoving observers scales as $1/a$ along the geodesic, because FLRW spacetime has a conformal time-translation symmetry but no timelike Killing vector, so global energy is not conserved and the "missing" energy is precisely the redshift. Turned around, that same relation converts a measured $z$ into a physical distance, and the map is not linear: it encodes the whole expansion history. The second figure is the quantity a cosmologist actually reaches for — the distance–redshift relation of the concordance universe, including the surprise that the angular-diameter distance turns over so that the most distant galaxies begin to look larger again.

Figure

Distance versus redshift in flat ΛCDM (Ω_m = 0.3, Ω_Λ = 0.7, H_0 = 70 km s⁻¹ Mpc⁻¹): the comoving distance d_C rises without bound, but the angular-diameter distance d_A = d_C/(1+z) peaks near z ≈ 1.6 at ≈ 1750 Mpc and then falls, so faraway objects subtend a larger angle. The luminosity distance is d_L = (1+z)² d_A

The core equations

Everything begins with one definition. The redshift is the fractional stretch of wavelength, and the FLRW null geodesic (derived below) fixes that stretch to be the ratio of scale factors between reception and emission. This is the master relation the rest of the article computes with.

1+z \;\equiv\; \frac{\lambda_{\rm obs}}{\lambda_{\rm emit}} \;=\; \frac{a(t_{\rm obs})}{a(t_{\rm emit})} \qquad (1)

where: $z$ — redshift, the fractional wavelength shift $(\lambda_{\rm obs}-\lambda_{\rm emit})/\lambda_{\rm emit}$ ; $\lambda_{\rm obs},\lambda_{\rm emit}$ — observed and emitted (rest-frame) wavelengths; $a(t)$ — cosmic scale factor, with $t_{\rm obs}$ the reception epoch and $t_{\rm emit}$ the emission epoch. A positive $z$ means the universe grew during transit; a negative $z$ (blueshift) would mean it shrank. Adopting the universal convention that the scale factor today equals one turns the master relation into a direct read-out of the past. Measuring $z$ tells you, with no model assumptions, the factor by which the universe has expanded since the light was emitted.

1+z = \frac{1}{a(t_{\rm emit})} \qquad (\text{convention } a(t_0)=1) \qquad (2)

where: $a(t_0)$ — scale factor today, normalised to unity; $t_0$ — present cosmic time. A quasar at $z=6$ was emitting when the universe was $a_{\rm emit}=1/7$ of its present linear size. Both statements live inside one spacetime. The Cosmological Principle forces the metric into the FLRW form, whose single time-dependent function is $a(t)$ and whose spatial sections have constant curvature $k$ .

ds^2 = -c^2\,\mathrm{d}t^2 + a^2(t)\left[\frac{\mathrm{d}r^2}{1-kr^2} + r^2\,\mathrm{d}\Omega^2\right] \qquad (3)

where: $k$ — spatial curvature constant, taking values $-1,0,+1$ in normalised units (open, flat, closed); $r,\,\Omega$ — comoving radial and angular coordinates; $c$ — speed of light. Only the combination $a^2(t)$ multiplying the spatial part carries the expansion; the time coordinate is unwarped. A radial photon obeys $ds^2=0$ , and integrating along its path gives the comoving distance it covers. Because source and observer are comoving, that comoving distance is the same for every crest, which is the seed of the whole redshift result.

\chi = \int_{t_{\rm emit}}^{t_{\rm obs}} \frac{c\,\mathrm{d}t}{a(t)} = \int_0^{r_1}\frac{\mathrm{d}r}{\sqrt{1-kr^2}} \qquad (4)

where: $\chi$ — comoving distance between source and observer, fixed once both are chosen; $r_1$ — comoving radial coordinate of the source. The left integral weights each slice of cosmic time by $1/a$ , the factor by which light "gains" comoving ground when the universe was smaller. The same expansion that stretches wavelengths redshifts every photon's energy, because energy is $E=cp$ and the comoving-frame momentum scales as $1/a$ along the geodesic. This is the engine behind radiation cooling.

p_{\rm proper} \propto \frac{1}{a} \quad\Longrightarrow\quad E = h\nu \propto \frac{1}{a} = (1+z) E_{\rm obs} \qquad (5)

where: $p_{\rm proper}$ — photon momentum magnitude measured by a comoving observer; $E$ — photon energy in the comoving frame; $\nu$ — frequency. A photon that we detect with energy $E_{\rm obs}$ was emitted with energy $(1+z)E_{\rm obs}$ ; the difference is not lost to anything local, it is the redshift. Redshift is what a telescope measures, but distance is what a map needs, and the two are joined by the expansion history. Different "distances" (comoving, luminosity, angular-diameter) are tied together by powers of $(1+z)$ through Etherington's reciprocity theorem.

d_L = (1+z)\,d_C, \qquad d_A = \frac{d_C}{1+z}, \qquad d_L = (1+z)^2\,d_A \qquad (6)

where: $d_L$ — luminosity distance (from apparent brightness); $d_A$ — angular-diameter distance (from apparent size); $d_C$ — comoving (transverse) distance. The $(1+z)^2$ between $d_L$ and $d_A$ is a geometric identity, independent of the cosmological model, and its violation would signal exotic photon physics. Finally, converting $z$ to $d_C$ requires the Hubble parameter as a function of redshift, which the Friedmann equations supply. Expanding this to lowest order recovers Hubble's law; keeping the full form is what let supernovae reveal dark energy.

d_C(z) = \int_0^z \frac{c\,\mathrm{d}z'}{H(z')}, \qquad H(z) = H_0\sqrt{\Omega_r(1+z)^4 + \Omega_m(1+z)^3 + \Omega_k(1+z)^2 + \Omega_\Lambda} \qquad (7)

where: $H(z)$ — Hubble parameter at redshift $z$ ; $H_0$ — its value today, $\approx 67$ – $73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ ; $\Omega_r,\Omega_m,\Omega_k,\Omega_\Lambda$ — present density parameters of radiation, matter, curvature, and dark energy summing to one. Each component redshifts with a distinct power of $(1+z)$ , so the shape of $d_C(z)$ is a direct probe of the cosmic inventory.

Derivation ARedshift from the conservation of comoving wavelength

The goal is to derive the master relation $1+z = a(t_{\rm obs})/a(t_{\rm emit})$ from first principles, using nothing but the FLRW metric and the fact that light travels on null geodesics. The strategy is to treat two successive wave crests as photons launched one period apart from a comoving source and received by a comoving observer, then exploit the single fact that both cover the same comoving distance. No matter content, no velocity, and no choice of path enters — only the geometry and the null condition — which is exactly why the result is so robust and model-independent.

Step 1. Restrict to a radial photon and impose $ds^2=0$ with $\mathrm{d}\Omega=0$ , so from the line element $0 = -c^2\,\mathrm{d}t^2 + a^2(t)\,\mathrm{d}r^2/(1-kr^2)$ , giving $c\,\mathrm{d}t/a(t) = \mathrm{d}r/\sqrt{1-kr^2}$ . Why: the null condition is the only dynamical input light needs; it converts a statement about the metric into a relation between elapsed cosmic time and comoving distance covered.

Step 2. Integrate the first crest from emission $t_e$ to reception $t_o$ , obtaining $\int_{t_e}^{t_o} c\,\mathrm{d}t/a(t) = \int_0^{r_1}\mathrm{d}r/\sqrt{1-kr^2} \equiv \chi$ . Why: the right-hand side depends only on where the source sits in the comoving grid, not on when the photon was sent, so $\chi$ is a fixed number that every crest must reproduce.

Step 3. Launch the second crest one period later and receive it one period later, so $\int_{t_e+\delta t_e}^{t_o+\delta t_o} c\,\mathrm{d}t/a(t) = \chi$ as well, because the source and observer have not moved in comoving coordinates. Why: equating the two integrals to the same $\chi$ is the whole trick — it locks the emission and reception intervals together through the expansion.

Step 4. Subtract the two integrals. The overlapping bulk cancels, leaving only the endpoint slivers, and for infinitesimal periods the integrand is locally constant, giving $\delta t_e/a(t_e) = \delta t_o/a(t_o)$ . Why: over one optical period $a(t)$ is effectively frozen, so the surviving contributions are just the interval divided by the local scale factor at each end.

Step 5. Identify a period with a wavelength through $\lambda = c\,\delta t$ , so $\lambda_{\rm obs}/\lambda_{\rm emit} = \delta t_o/\delta t_e = a(t_o)/a(t_e)$ , which is $1+z$ . Why: the ratio of reception to emission intervals is precisely the ratio of scale factors, and with $a(t_0)=1$ this reads $1+z = 1/a(t_e)$ — the redshift labels the epoch. *Verified: reproduces Carroll §8.2 Eq. (8.40) and Wald §5.3; collapses to Hubble's law $cz\approx H_0 d$ at first order (next derivation); for $z=2$ gives $a_{\rm emit}=1/3$ and a wavelength stretch of exactly 3.*

Go deeper · Derivation B · 1 minThe Hubble law as the low-redshift limit

Derivation BThe Hubble law as the low-redshift limit

The goal is to show that the exact, fully relativistic redshift relation contains the familiar linear Hubble law as its leading behaviour for nearby sources, and to expose the next-order term that carries the acceleration of the expansion. The plan is to Taylor-expand the scale factor about the present epoch, feed it into the redshift definition, and relate the look-back time to distance, so that $cz = H_0 d$ emerges as a low-order approximation rather than a fundamental law — the correction term is what supernova cosmology was built to measure.

Step 1. Expand $a(t) = a(t_0)[\,1 + H_0(t-t_0) - \tfrac12 q_0 H_0^2 (t-t_0)^2 + \dots\,]$ , where $H_0 = \dot a/a|_0$ is the Hubble rate and $q_0 = -\ddot a\,a/\dot a^2|_0$ is the deceleration parameter. Why: near the present the expansion is characterised by its rate and its curvature, and these two numbers are exactly $H_0$ and $q_0$ .

Step 2. Insert the expansion into $1+z = a(t_0)/a(t_e)$ and invert for small look-back time to get $z = H_0(t_0-t_e) + \dots$ . Why: to leading order the redshift is just the fractional growth of the scale factor over the light-travel time, which is the Hubble rate times that time.

Step 3. For a nearby source the proper distance is $d \approx c(t_0 - t_e)$ , so substituting gives $z \approx H_0 d/c$ , that is $cz \approx H_0 d$ . Why: at small distances space has barely expanded during transit, so light-travel time and proper distance are interchangeable up to higher order.

Step 4. Carry the next term to find $cz \approx H_0 d + \tfrac12(1+q_0)H_0^2 d^2/c + \dots$ , whose curvature encodes $q_0$ . Why: the departure from a straight line in the Hubble diagram is a direct measurement of whether the expansion is decelerating $(q_0>0)$ or accelerating $(q_0<0)$ — the signature of dark energy. *Verified: consistent with Hartle Ch. 18 and Schutz §12.2; at $z=0.01$ the linear law gives $d = cz/H_0 = 42.8\,\mathrm{Mpc}$ , matching the exact ΛCDM integral $d_C = 42.7\,\mathrm{Mpc}$ to better than 0.3 percent.*

Go deeper · Derivation C · 1 minWhy radiation cools as the universe expands

Derivation CWhy radiation cools as the universe expands

The goal is to combine the per-photon energy redshift with the geometric dilution of photons in an expanding volume to derive the two scaling laws that govern the early universe: $\rho_\gamma \propto a^{-4}$ for the radiation energy density and $T \propto a^{-1}$ for a blackbody temperature. The plan is to count photons in a comoving volume, track each one's energy, multiply, and then invoke the Stefan–Boltzmann law, so that the whole thermal history of the cosmic microwave background follows from redshift plus bookkeeping.

Step 1. Photon number is conserved in a comoving volume $V \propto a^3$ , so the number density falls as $n_\gamma \propto V^{-1} \propto a^{-3}$ . Why: expansion neither creates nor destroys photons on its own; it only spreads a fixed count over a growing volume, exactly like matter.

Step 2. Each photon's energy redshifts as $E_\gamma = h\nu \propto \lambda^{-1} \propto a^{-1}$ from the master relation. Why: this is the extra factor that distinguishes radiation from matter — a massive particle keeps its rest energy, but a photon's energy bleeds away with the wavelength stretch.

Step 3. Multiply density by energy per photon: $\rho_\gamma c^2 = n_\gamma E_\gamma \propto a^{-3}\cdot a^{-1} = a^{-4}$ . Why: the fourth power, one steeper than matter's $a^{-3}$ , is why radiation dominated the very early universe and then faded — running the expansion backward makes radiation win.

Step 4. For a blackbody $\rho_\gamma \propto T^4$ , so $T \propto a^{-1} = (1+z)$ . Why: the temperature tracks $(1+z)$ linearly, predicting a hotter CMB in the past and fixing the surface of last scattering once today's temperature is known. *Verified: standard hot-big-bang scaling (Weinberg §1.4); with $T_0 = 2.7255\,\mathrm{K}$ and $1+z_{\rm rec}\approx 1100$ it extrapolates to $T_{\rm rec}\approx 3000\,\mathrm{K}$ , the recombination temperature of hydrogen.*

Limiting cases

Cosmological redshift is a parent relation that reduces to several familiar shifts and scalings in the appropriate corner of parameter space.

Case	Result	Meaning
Low redshift (Hubble's law)	$cz \approx H_0 d \quad (z \ll 1)$	Expanding $1+z=a_0/a_{\rm emit}$ to first order in look-back time gives the linear, Doppler-like recession law; $d$ is today's proper distance and $H_0$ today's expansion rate.
Photon energy and temperature	$E \propto (1+z)^{-1},\ T \propto (1+z),\ \rho_\gamma \propto (1+z)^4$	Per-photon energy redshifts and number density dilutes as $(1+z)^3$ ; together the radiation energy density scales as $a^{-4}$ , setting up the radiation-dominated era.
Surface of last scattering (CMB)	$1+z_{\rm rec} \approx 1100$	With $T_0 = 2.7255\,\mathrm{K}$ today and $T_{\rm rec}\approx 3000\,\mathrm{K}$ at recombination, $1+z_{\rm rec}=T_{\rm rec}/T_0 \approx 1100$ ; the CMB is the redshifted glow of that epoch.
Angular-diameter turnover	$d_A = d_C/(1+z)$ peaks near $z\approx 1.6$	Beyond the peak, more distant galaxies subtend a larger angle because the universe was so much smaller when the light left; $d_A$ falls even as $d_C$ climbs.
Blueshift (collapsing phase)	$1+z < 1$ if $a_{\rm obs} < a_{\rm emit}$	The same formula yields a blueshift in a contracting universe; the sign of the shift simply tracks whether the universe grew or shrank during transit.

Common trapCosmological redshift is a recession velocity, and v = cz is its speed

It is not, and the naive $v = cz$ has no fundamental meaning once $z$ is not small. Cosmological redshift measures how much the scale factor grew during the light's transit, an integral over the whole expansion history, not a single velocity at a single instant. If you insist on defining a recession speed as the rate of change of proper distance, sources at $z \gtrsim 1.5$ have "speeds" exceeding $c$ — and they remain perfectly observable, because nothing local ever moves faster than light through space; it is space between us that expands. Two further traps: do not confuse it with gravitational redshift (there is no potential well being climbed here, only metric coefficients growing with time), and do not apply the special-relativistic Doppler formula $1+z=\sqrt{(1+\beta)/(1-\beta)}$ to extract a velocity — that formula assumes a single inertial relationship in flat spacetime, which a cosmological photon never experiences.

Worked examples

1 · A galaxy at z = 2 (confidence builder)

A galaxy is observed at redshift $z=2$ . Find the scale factor when its light was emitted, the wavelength stretch, and the observed wavelength of its rest-frame [O II] 3727 Å emission line. Then say why you must not report a recession velocity of $v=cz=2c$ .

Step 1. With $a(t_0)=1$ , the scale factor at emission is $a_{\rm emit} = 1/(1+z) = 1/3$ , so the universe was one-third of its present linear size when the light set out. Every proper length, including the wavelength of light in flight, has since been multiplied by $1+z = 3$ .

Step 2. The wavelength stretch is the redshift factor itself: $\lambda_{\rm obs}/\lambda_{\rm emit} = 1+z = 3$ . Applied to the [O II] doublet at a rest wavelength of 3727 Å, the observed wavelength is $\lambda_{\rm obs} = 3 \times 3727\ \text{Å} = 11181\ \text{Å}$ , i.e. 1.118 μm, in the near-infrared — which is exactly why high-redshift galaxy surveys need infrared spectrographs to catch optical rest-frame lines.

Step 3. The naive recession speed $v = cz = 2c$ is meaningless. Cosmological redshift is not a velocity through space; it is the accumulated expansion factor. A proper-distance recession velocity computed from a full FLRW model does exceed $c$ at this redshift, and that is allowed because it is space expanding, not the galaxy moving locally. Result: $a_{\rm emit}=1/3$ , wavelengths stretched ×3, [O II] observed at 1.118 μm, and no sub- $c$ Doppler velocity is definable.

2 · The temperature and energy density of the CMB

The cosmic microwave background is a blackbody at $T_0 = 2.7255\,\mathrm{K}$ today, emitted at recombination at $z_{\rm rec}\approx 1100$ . Find the temperature at emission and the factor by which its radiation energy density was larger then.

Step 1. Temperature scales linearly with $(1+z)$ : $T(z) = T_0(1+z)$ . At recombination, $T_{\rm rec} = 2.7255\ \text{K} \times 1101 = \mathbf{3001\ K}$ — the temperature at which hydrogen recombines and the universe becomes transparent, which is what makes $z\approx 1100$ a physical surface rather than an arbitrary choice.

Step 2. Radiation energy density scales as $\rho_\gamma \propto (1+z)^4$ . The ratio between recombination and today is $(1+z_{\rm rec})^4 = 1101^4 \approx \mathbf{1.5\times 10^{12}}$ , so the CMB carried over a trillion times more energy per unit volume then than the faint microwave glow we measure now.

Step 3. Consistency check: the same $T\propto(1+z)$ scaling is directly testable. Absorption-line thermometry in gas clouds at $z\approx 2$ – $3$ measures the CMB temperature as it was then, and the values track $T_0(1+z)$ within errors — a clean confirmation that the redshift is a genuine cooling of the photon gas. Result: $T_{\rm rec}=3001\,\mathrm{K}$ and an energy-density enhancement of $\sim 1.5\times10^{12}$ .

3 · Local Hubble law versus the exact integral

A galaxy has $z=0.01$ . Estimate its distance with Hubble's law using $H_0 = 70\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ , and compare to the exact comoving distance in flat ΛCDM.

Step 1. Hubble's law gives $cz = H_0 d$ , so $d = cz/H_0 = (299792\ \text{km s}^{-1})(0.01)/(70\ \text{km s}^{-1}\text{Mpc}^{-1}) = \mathbf{42.8\ Mpc}$ . The recession speed $cz = 2998\ \text{km s}^{-1}$ is safely non-relativistic, so the linear approximation is trustworthy.

Step 2. The exact comoving distance integrates $d_C = \int_0^{0.01} c\,\mathrm{d}z'/H(z')$ with $H(z)=H_0\sqrt{0.3(1+z)^3+0.7}$ , giving $d_C = \mathbf{42.7\ Mpc}$ . The two agree to about 0.3 percent — the whole content of "Hubble's law is the low- $z$ limit."

Step 3. Now push to $z=6$ : the linear formula would claim $cz = 6c$ , which is nonsense, while the correct comoving distance is $d_C \approx 8250\ \text{Mpc}$ and the object is still comfortably observable. Result: at $z=0.01$ Hubble's law is exact to 0.3 percent (42.8 vs 42.7 Mpc); at $z=6$ it fails entirely and the full expansion history is mandatory.

Physics of the portrait

Cosmological redshift is where three strands of early-twentieth-century science braided together: Slipher's painstaking spectra of spiral nebulae, Friedmann's expanding solutions of Einstein's equations, and the synthesis that read the first as evidence for the second.

Origins · 1 minGeorges Lemaître and how this was discovered

Georges Lemaître

1894–1966 · Belgian priest and cosmologist, father of the expanding universe

Georges Lemaître was a Belgian Catholic priest, mathematician, and physicist who did more than anyone to turn cosmic expansion from a mathematical curiosity into a physical theory read off the sky. In 1927, working from Einstein's field equations, he independently found the dynamical FLRW solution (after Friedmann's 1922 work, then little known) and took the decisive further step: he interpreted Vesto Slipher's measured redshifts of spiral nebulae as the signature of an expanding universe, and derived a linear relation between a nebula's recession and its distance — complete with an estimate of the proportionality constant. This was two years before Edwin Hubble's celebrated 1929 observational paper. Lemaître's result was buried in a French-language Belgian journal and, in the 1931 English translation, the paragraph containing his rate estimate was omitted; only decades later was his priority widely recognised. He went further still, proposing in 1931 that the expansion ran back to a hot, dense "primeval atom" — the conceptual seed of the Big Bang. In 2018 the International Astronomical Union voted to rename the expansion law the Hubble–Lemaître law in his honour, a rare formal correction of the historical record.

History

1912

— Slipher measures the first radial velocities of spiral nebulae from Doppler-shifted spectral lines

1922

— Friedmann derives the expanding and contracting solutions of Einstein's equations

1927

— Lemaître connects galaxy redshifts to an expanding universe and predicts the linear redshift-distance relation

1929

— Hubble publishes the empirical velocity-distance law using Cepheid distances

1965

— Penzias and Wilson detect the cosmic microwave background, the redshifted glow of recombination

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §27.2 · Wald, General Relativity, §5.3 · Carroll, Spacetime and Geometry, §8.2 · Hartle, Gravity, Ch. 18 · Schutz, A First Course in General Relativity, §12.2 · Weinberg, Cosmology, §1.4 · Peebles, Principles of Physical Cosmology, §13.

Provenance. The master relation and its FLRW null-geodesic derivation follow Carroll §8.2 and Wald §5.3; the two-crest argument is the standard textbook route (Hartle Ch. 18). The Hubble-law expansion with the $q_0$ term is from Hartle Ch. 18 and Schutz §12.2, and the radiation scaling $\rho_\gamma\propto a^{-4}$ , $T\propto(1+z)$ is standard hot-big-bang cosmology (Weinberg §1.4). Distance duality $d_L=(1+z)^2 d_A$ is Etherington's reciprocity theorem. Numbers to verify: $T_0 = 2.7255\,\mathrm{K}$ (FIRAS/COBE), $1+z_{\rm rec}\approx 1100$ , $H_0\approx 67$ – $73\ \mathrm{km\,s^{-1}\,Mpc^{-1}}$ (Planck versus SH0ES). The distance–redshift figure uses flat ΛCDM with $\Omega_m=0.3$ , $\Omega_\Lambda=0.7$ , $H_0=70$ , giving $d_C(z{=}1)=3304\,\mathrm{Mpc}$ and an angular-diameter peak of $\approx 1748\,\mathrm{Mpc}$ near $z\approx 1.6$ .

Prerequisites. The FLRW Metric · The Scale Factor and Hubble Parameter · Comoving and Proper Coordinates · The Cosmological Principle. Leads to. The Friedmann Equations · The Radiation-Dominated Era · Cosmological Horizons · The Cosmological Constant and Dark Energy.

Ask the tutor: Ask me to walk through the two-crest null-geodesic derivation line by line, or to work out the observed wavelength, look-back time, and comoving distance for any redshift you name.

Is cosmological redshift just a Doppler shift after all?

No, though the two coincide in the low-redshift limit. A Doppler shift arises from relative motion through a fixed, flat spacetime, and it depends on the velocity at the instants of emission and reception. Cosmological redshift arises because the spatial metric grows with time while both source and observer sit still in comoving coordinates; it depends on the entire expansion history between emission and reception, not on any single velocity. The two agree to first order — $z\approx v/c$ for nearby galaxies — which is why Hubble's law looks Doppler-like. But at large $z$ the interpretations diverge sharply: the special-relativistic Doppler formula caps $z$ velocities below $c$ , whereas cosmological redshift has no upper bound and $z$ can be arbitrarily large.

Do galaxies at high redshift really recede faster than light?

In a well-defined sense, yes, and it violates nothing. If you define recession velocity as the rate of change of proper distance (Hubble's law $v = H_0 d$ extended properly), then galaxies beyond a certain distance — corresponding to roughly $z\gtrsim 1.5$ in the concordance model — have recession speeds exceeding $c$ . Special relativity forbids only local motion through space faster than light, the speed measured by an observer the object passes. It says nothing about the rate at which intervening space expands. Remarkably, such superluminally receding galaxies remain observable, because the expansion rate can change and photons emitted "upstream" can still reach us. The apparent paradox dissolves once you stop treating redshift as a velocity.

What is the difference between cosmological, gravitational, and Doppler redshift?

All three stretch wavelengths and share the same low-order formula, but their geometric origins differ. A Doppler shift comes from relative velocity in flat spacetime, tied to the time-dependence a moving observer sees. A gravitational redshift comes from a difference in the metric's time-time component between two static locations — light climbing out of a potential well loses frequency because clocks run at different rates at fixed positions. A cosmological redshift comes from the time-dependence of the spatial metric: both source and observer are at rest, but the scale factor multiplying spatial distances grows. In principle a real photon can experience all three at once, and untangling them (peculiar velocities, host-galaxy potentials, and cosmic expansion) is a routine part of precision redshift measurement.

If each photon loses energy as it redshifts, where does that energy go?

Nowhere — and that is the honest answer, not a dodge. Energy conservation as a global law follows from time-translation symmetry (Noether's theorem), and FLRW spacetime simply does not have that symmetry: it has no timelike Killing vector, because the geometry itself changes with cosmic time. So there is no conserved total energy for the photon gas to fund, and the redshifted energy is not transferred to anything. Locally, energy-momentum is still exactly conserved through the covariant equation $\nabla_\mu T^{\mu\nu}=0$ ; it is only the attempt to add up a global energy across the whole expanding spacetime that fails. The redshift is the visible face of this deep fact about general relativity in a non-stationary universe.

Why does the CMB temperature scale exactly as (1 + z)?

Because a blackbody spectrum stays a blackbody under redshift, only at a lower temperature. Every photon's frequency scales as $\nu\propto a^{-1}=(1+z)$ , and the number density scales as $a^{-3}$ . Plug both into the Planck distribution and the functional form is preserved with the single replacement $T\to T/(1+z)$ — the spectrum is self-similar under the expansion. This is not an approximation; it is exact for freely propagating photons after they decouple from matter. That is why the CMB, emitted as a $\sim 3000\,\mathrm{K}$ blackbody at recombination, is still a near-perfect blackbody today at $2.7255\,\mathrm{K}$ , and why measuring its temperature at high redshift (via molecular and atomic transitions in distant gas) is a clean test of the expansion.

How is a galaxy's redshift actually measured, and why is it so precise?

By identifying known spectral lines and measuring how far they have shifted. Atomic and molecular transitions occur at fixed, laboratory-known rest wavelengths, and cosmological redshift moves every one by the same factor $(1+z)$ . An observer looks for a recognisable pattern — the hydrogen Balmer series, the [O II] and [O III] lines, the 4000 Å break — and slides it until it matches, reading off $z$ from the common stretch. Because many sharp lines are fit at once, spectroscopic redshifts routinely reach $\Delta z \sim 10^{-4}$ or better. The uniformity across all lines is itself the proof that the shift is cosmological: a wavelength-independent stretch is exactly what expanding space produces.

Is the "stretching space" or "expanding balloon" picture literally correct?

It is a good intuition pump but should not be pushed too hard. The rigorous content is the null-geodesic calculation: $1+z=a_{\rm obs}/a_{\rm emit}$ follows exactly from the FLRW metric, with no appeal to space "carrying" the photon. The balloon captures the right facts — comoving positions fixed, physical separations growing, the same stretch everywhere — but taken literally it can mislead, suggesting for instance that bound systems like atoms or galaxies also expand (they do not, because local forces dominate). A safer statement: in comoving coordinates the wavelength is frozen, and converting to physical units multiplies by $a(t)$ . When intuition and the metric disagree, trust the metric; the balloon is scaffolding for the equation, not a substitute for it.