Redshift

Full definition

You're on a sidewalk when a fire truck roars past. The siren emits the same note, but you hear it high-pitched as the truck approaches, then low-pitched as it recedes. That's the Doppler effect. Redshift is exactly that, applied to light instead of sound. When a light source recedes, all its wavelengths are stretched — blue spectral lines slide toward green, green toward orange, red toward infrared. Light 'reddens'. Conversely, an approaching source blueshifts.

The measured quantity is dimensionless:

z = (λ_observed - λ_emitted) / λ_emitted

A galaxy at z = 0.01 is shifted by 1 % — it recedes at ~3,000 km/s (Virgo Cluster galaxy). A galaxy at z = 1 has its light wavelength doubled — its ultraviolet emission reaches us in the visible. A galaxy at z = 6 has its wavelength multiplied by 7 — emitted visible becomes far-infrared, the realm of James Webb.

Central pedagogical point. Redshift comes in three distinct flavours, all producing the same kind of shifted spectrum. First: Doppler redshift, due to peculiar motion of the source through space (binary star, exoplanet wobbling its host, galaxy infalling into a cluster). Second: cosmological redshift, due to stretching of space itself during the light's journey. Third: gravitational redshift, predicted by general relativity, when light climbs out of a gravitational potential well.

In real life, all three can overlap. For a galaxy at z = 0.05, cosmological dominates. For a binary star, Doppler dominates. For a spectral line from a white dwarf surface, gravitational contributes significantly (~100 km/s apparent). Disentangling the components is the astrophysicist's job.

Formula and scale

The basic formula is elegantly simple:

z = Δλ / λ = (λ_obs - λ_em) / λ_em

For small redshifts (z ≪ 1), the relation with velocity is linear:

v ≈ c · z

But for relativistic redshifts, the correct (relativistic Doppler) formula becomes:

1 + z = √((1 + β)/(1 - β))

with β = v/c. At z = 1, v ≈ 0.6 c. At z = 3, v ≈ 0.88 c.

For cosmological redshift, the formula relates 1 + z to the scale factor of the Universe:

1 + z = a(t_obs) / a(t_em)

where a(t) is the relative size of the Universe at time t. At z = 1, the Universe was half its current size. At z = 10, a tenth. At z = 1,090 (CMB emission), the Universe was 1,091 times smaller — about a thousandth.

Observational records.

• Stars: proper motion in the Milky Way → z ≪ 10⁻³. • Nearby galaxies: Virgo at z = 0.004, Coma at z = 0.023. • Typical bright quasars: z ≈ 0.5-3. • Most distant known quasar: J0313-1806 at z = 7.64 (2021). • JWST galaxies: JADES-GS-z14-0 at z = 14.3 (2024) → light emitted when the Universe was ~300 million years old. • CMB: z = 1,090 → cosmic microwave background, oldest observable photon.

Redshift is the fundamental cosmological variable. The entire distant Universe is measured in z.

The three redshift regimes

Understanding the difference between the three types is essential.

Doppler redshift. The source moves through space relative to the observer. Formula: z ≈ v/c (non-relativistic) or relativistic Doppler at high speed. Examples: spectroscopic binary stars, pulsating variable stars, radial-velocity exoplanet detections (the method behind 51 Pegasi b in 1995, Nobel 2019 to Mayor and Queloz), Milky Way stellar proper motions. Typical shifts from a few km/s to a few hundred km/s.

Cosmological redshift. Caused by the stretching of space between source and observer during light's travel. This is not a strict Doppler effect: the source is at rest in its own frame; space is what grows. Exact formula: 1 + z = a(t_obs)/a(t_em). Linked to cosmological parameters (H₀, Ω_m, Ω_Λ) via Friedmann's equation. Governs virtually all galactic redshifts beyond 10 Mpc.

Gravitational redshift. Predicted by general relativity in 1915: a light wave loses energy as it climbs out of a gravitational potential well. Weak-field approximation:

z_grav ≈ GM / (Rc²)

First verified in 1959 by the Pound-Rebka experiment (Harvard, over 22.5 m in a tower, 1 % precision). Concrete examples:

• Sun's surface: z ≈ 2 × 10⁻⁶ (tiny). • White dwarf surface: z ≈ 3 × 10⁻⁴ (~100 km/s apparent). • Neutron star surface: z ≈ 0.3 (significant). • Close to a black hole's horizon: z → ∞.

GPS is the most familiar practical case: atomic clocks on satellites run faster than ours (lower gravity), and the gravitational correction must be applied continuously to maintain metre-level precision.

How do we measure it?

The technique: spectroscopy. Spread the light from a source into its component wavelengths (prism or diffraction grating) and identify characteristic emission or absorption lines of chemical elements (hydrogen Hα at 656.3 nm at rest, calcium H and K lines at 397 and 393 nm, Lyman-α at 121.6 nm, iron lines, magnesium, etc.). If those lines appear shifted, z is measured.

Instruments. The largest ground telescopes (VLT, Keck, GTC) with high-resolution spectrographs (ESPRESSO on VLT, HIRES on Keck). Space missions: Hubble (STIS, COS), James Webb (NIRSpec, MIRI) for very-high-z infrared sources. Massive spectroscopic galaxy surveys: SDSS (2000-present, 5 million spectra), eBOSS, DESI (Dark Energy Spectroscopic Instrument, 40 million spectra planned by 2026).

Photometric redshift. For sources too faint for a spectrum, z is estimated from multi-band photometry (6 to 10 filters). Less precise (~5-10 % vs 10⁻⁴ spectroscopy), but applicable to millions of galaxies. Key to weak-lensing surveys (Euclid, LSST).

Astrophysical applications. Stellar radial velocities (RV exoplanets), galactic kinematics (rotation curves revealing dark matter), 3D mapping of the Universe (z as radial coordinate), cosmic clock (paradox: higher z = deeper in the past).

An amateur example. With a modest spectroscope (Alpy 600 + 80 mm refractor), one can measure the redshift of bright quasars like 3C 273 (z = 0.158) from one's garden. The Hβ line shifted by 97 nm is clearly visible on the recorded spectrum. Guaranteed awe: it's the light of a quasar that left 2.4 billion years ago finishing its journey on your sensor.

Not to be confused with

Several classic confusions about redshift deserve to be cleared up.

Interstellar reddening. Completely different. Interstellar dust scatters short wavelengths (blue) more than long ones (red), so light passing through a dusty medium appears redder. But spectral lines don't move — only the continuum envelope is altered. Redshift, by contrast, shifts all lines proportionally to their wavelength.

Doppler effect. Doppler redshift is a special case of redshift. But 'redshift' in the general sense also encompasses cosmological and gravitational effects, which are not Doppler effects. For a distant galaxy at z = 5, talking about 'recession velocity' via v = c·z gives values exceeding c and has no strict physical meaning — it's not Doppler, it's space stretching.

Cosmological vs. Doppler redshift. For small z (z ≪ 0.1), both give the same numerical result and can be treated interchangeably. For large z, they diverge: cosmological is not motion through space.

Blueshift. Reversed redshift: an approaching source emits light shifted toward shorter wavelengths. The Andromeda Galaxy is the sky's most famous blueshifted source (v ≈ -110 km/s toward us).

Redshift and apparent age. Reading 'galaxy at z = 10' doesn't mean it is 10 billion light-years away — the z → distance relation is non-linear and depends on cosmology. At z = 10, the galaxy is observed as it was ~13 billion years in the past, but its current comoving distance is ~30 billion light-years.