Look-back Time

The age of the light, not the age of the galaxy

Astronomers talk about distances to galaxies, but every measurement actually returns a photon — and photons carry an age. Light from the Andromeda galaxy left 2.5 million years ago. Light from a z = 1 galaxy left 7.9 billion years ago. Light from the cosmic microwave background left 13.787 billion years ago. The look-back time is precisely this age: the elapsed cosmic time between the moment a photon was emitted and the moment it strikes the detector now.

The distinction matters because the galaxy that emitted a photon at lookback 7.9 Gyr has had 7.9 Gyr to evolve before you see it. The light you record at z = 1 shows the galaxy at half its current age; the galaxy may since have merged, quenched, or restructured. Lookback time is the timestamp on the photon, not the galaxy's modern age.

Definition and derivation

In an FLRW universe with scale factor a(t), the cosmological redshift of a photon emitted at time t and received at t₀ is 1 + z = a(t₀)/a(t). Differentiating with respect to z (using ȧ = aH):

  dt/dz = − 1 / [ (1 + z) · H(z) ]

The minus sign reflects that increasing z corresponds to earlier emission times. Integrating from z = 0 (now) up to z (emission) and flipping the sign so t_L(z) is positive:

  t_L(z) = ∫₀^z dz' / [ (1 + z') · H(z') ]

In flat ΛCDM, H(z) is built from the densities:

  H(z) = H₀ · √( Ω_r (1+z)⁴ + Ω_m (1+z)³ + ΩΛ )

There is no closed-form antiderivative; the integral is evaluated numerically with Simpson's rule, Gauss-Legendre quadrature, or any standard ODE solver. In matter-only (Einstein-de Sitter) cosmology a closed form does exist — t_L(z) = (2/3H₀) · (1 − (1+z)^(−3/2)) — but it is wrong for our universe by several Gyr at z ~ 1.

Look-back values across cosmic history

Plugging Planck 2018 best-fit values (H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, ΩΛ = 0.685) into the integrand and integrating numerically:

Redshift z	Cosmic time at emission (Gyr)	Look-back time t_L (Gyr)	Fraction of age	Example object
0.01	13.65	0.14	1.0%	Nearby supernovae
0.1	12.47	1.32	9.6%	SDSS main sample
0.5	8.65	5.13	37.2%	BOSS LOWZ galaxies
1.0	5.92	7.87	57.1%	BOSS CMASS, peak quasar era
2.0	3.32	10.47	75.9%	Cosmic Noon, peak star formation
3.0	2.16	11.62	84.3%	Lyman-break galaxies
6.0	0.94	12.85	93.2%	End of reionization
10	0.47	13.31	96.6%	JWST GN-z11 era
14	0.27	13.52	98.0%	JADES-GS-z14-0
1100 (CMB)	0.00038	13.787	99.997%	Last scattering surface
∞	0	13.787	100%	Big Bang

The first thing to notice is how steeply lookback rises early and how flat it goes late. Half the lifespan of the universe is contained between z = 0 and z ≈ 0.8 — the last few billion years of cosmic time. Everything else, from z = 0.8 to the singularity, fits into the second half. JWST's z = 10 sources are at 96.6% of the age, but z = 1100 is only 99.997% — radiation domination is geometrically tiny in the lookback budget.

Lookback time vs every other cosmic distance

Look-back time and comoving distance are linked but not equal. The relation is

  t_L(z) = ∫₀^z dz' / [ (1+z') · H(z') ]
  D_C(z) = c · ∫₀^z dz' / H(z')

Notice the extra factor of (1+z') in the look-back integrand. At low redshift, (1+z') ≈ 1 and the two integrals coincide up to factors of c — that is why "light-travel distance" and comoving distance look the same for nearby galaxies. At high redshift the (1+z') factor grows large and the lookback integrand shrinks faster than the D_C integrand. The result: lookback saturates at t₀, while D_C saturates at the particle horizon ~14 Gpc.

The press-release distance c · t_L(z) is sometimes called "light-travel distance" and is the natural measure of "how long the light has been in flight." It is not the proper distance, comoving distance, angular-diameter distance, or luminosity distance to the galaxy. For a z = 1 source: c · t_L = 7.9 Gly, but the comoving distance is 10.8 Gly (3.3 Gpc), and the proper distance today is also 10.8 Gly (because a(t₀) ≡ 1). The galaxy is now further away than the light has travelled, because space has been expanding while the photon was in flight.

Worked example: lookback to z = 1

For flat ΛCDM with H₀ = 67.4 km/s/Mpc, Ω_m = 0.315, ΩΛ = 0.685, convert H₀ to inverse Hubble time first:

  1/H₀ = (3.0857 × 10¹⁹ km/Mpc) / (67.4 km/s/Mpc)
       = 4.578 × 10¹⁷ s
       = 1.45 × 10¹⁰ yr
       = 14.51 Gyr     (the Hubble time)

  t_L(1) = (1/H₀) · ∫₀^1 dz' / [(1+z') · √(Ω_m (1+z')³ + ΩΛ)]

Simpson's rule with 8 intervals:
   z'=0.000  integrand = 1/(1.0·√1.0)             = 1.0000
   z'=0.125  integrand = 1/(1.125·√(0.45+0.685))  = 0.8345
   z'=0.250  integrand = 1/(1.25 ·√(0.62+0.685))  = 0.7001
   z'=0.375  integrand = 1/(1.375·√(0.82+0.685))  = 0.5928
   z'=0.500  integrand = 1/(1.50 ·√(1.06+0.685))  = 0.5048
   z'=0.625  integrand = 1/(1.625·√(1.35+0.685))  = 0.4313
   z'=0.750  integrand = 1/(1.75 ·√(1.69+0.685))  = 0.3705
   z'=0.875  integrand = 1/(1.875·√(2.08+0.685))  = 0.3206
   z'=1.000  integrand = 1/(2.00 ·√(2.52+0.685))  = 0.2790

  Integral ≈ 0.5418
  t_L(1)   ≈ 14.51 Gyr · 0.5418  ≈  7.86 Gyr

So light from a z = 1 galaxy reached us after 7.86 Gyr — half the age of the universe. The galaxy was younger than the Earth-Sun system is today when the light was emitted. By the time the photon arrived, the original galaxy may well have evolved into something else: merged, brightened, quenched. Lookback time is the timestamp on the snapshot.

A short history of cosmic ages

The concept of a cosmic age dates to 1922, when Alexander Friedmann derived the expanding-universe solutions of Einstein's equations and gave the first finite age estimate. Edwin Hubble's 1929 distance-velocity relation H₀ ≈ 500 km/s/Mpc implied an age of ~1.8 Gyr — younger than the Earth, an obvious contradiction that prompted decades of "age crisis" cosmology. Walter Baade's 1952 re-calibration of Cepheids halved H₀; the discovery of dark energy in 1998 raised the age by a further ~1.5 Gyr at fixed H₀, since the universe was decelerating in the past. The modern Planck 2018 value t₀ = 13.787 ± 0.020 Gyr is consistent with the oldest globular clusters (~12.5 Gyr) and the iron-line dating of the universe's first stars.

For individual photon ages, the modern era began with the 1948 prediction of the CMB by Gamow, Alpher and Herman, who computed its present temperature and emission redshift. The 1965 discovery of the CMB by Penzias and Wilson at 3 K validated a 13.8 Gyr light-travel time — the longest baseline in observational science. JWST's 2022–2025 confirmed z > 10 galaxies extend the lookback by another factor of ~3 from previous Hubble records, sampling cosmic time 270–500 Myr after the Big Bang directly.

Why the same redshift gives different lookback in different cosmologies

The look-back integrand depends entirely on H(z), which is shaped by the matter, radiation, and dark-energy content. Comparing four cosmologies at z = 1:

Model	(Ω_m, ΩΛ, w)	t_L(z=1) (Gyr)	Δ vs Planck
Planck flat ΛCDM	(0.315, 0.685, −1)	7.87	—
SH0ES (high-H₀)	(0.315, 0.685, −1) with H₀=73	7.27	−0.60
Einstein-de Sitter	(1.0, 0, n/a)	6.36	−1.51
Open matter (1980s)	(0.3, 0, n/a)	7.34	−0.53
w = −0.9 quintessence	(0.315, 0.685, −0.9)	7.74	−0.13
w = −1.1 phantom	(0.315, 0.685, −1.1)	7.99	+0.12

Hubble tension shows up here in the H₀ row: a 9% disagreement in H₀ between Planck and SH0ES produces a ~8% disagreement in look-back time at every redshift. The dark-energy equation-of-state shows up in the bottom two rows; current observational bounds w = −1.03 ± 0.03 already constrain the lookback at z = 1 to within a few tens of Myr — better than any traditional astronomical age determination.

Why the integral converges at infinite redshift

Pushing z to infinity might naïvely give a divergent integral, but it does not. At large z (z >> 1) the integrand behaves as 1/[(1+z')·H₀·√(Ω_m(1+z')³ + Ω_r(1+z')⁴)] ~ 1/[(1+z')·(1+z')²] = 1/(1+z')³ in matter domination, or 1/(1+z')⁴ during radiation domination. Both fall off fast enough for the integral to converge. The integral above z = 1000 contributes only ~0.4 million years to the total — a rounding error in the 13.787 Gyr age budget.

The number t_L(∞) = t₀ is then exactly the cosmic age in flat ΛCDM. It is finite because the early-universe contributions are weighted down by (1+z')H(z'), which grows rapidly with z. The CMB at z = 1100 already buys you 99.997% of the answer; pushing to z = ∞ adds 0.4 Myr — entirely from cosmic time before recombination.

Numerical pitfalls and conventions

• Sign of dt/dz. The physical sign of dt/dz is negative (earlier times have higher z). When writing t_L(z) we flip the sign so the lookback is reported as a positive number. Software conventions vary; double-check that astropy's cosmo.lookback_time(z) returns positive values for z > 0.
• Light-travel distance. c · t_L is not D_C, D_p, D_A or D_L. It is shorter than D_C at all z > 0 because the integrand has the extra (1+z') factor. Press releases that quote a "light-years away" figure usually mean c · t_L. Cosmologists never use it for physics — only for popular outreach.
• Galaxy age ≠ photon age. A galaxy observed at lookback 7.9 Gyr was already several Gyr old at emission. Its stellar population can be aged from spectra; the galaxy may now be 13 Gyr old, far older than the 7.9 Gyr lookback. Press headlines like "earliest galaxy ever seen" refer to photon age, not stellar age.
• Radiation contribution. Ω_r ≈ 9.2 × 10⁻⁵ is tiny today, but (1+z)⁴ scaling makes radiation dominate above z ≈ 3400. Ignoring radiation gives t_L errors of ~0.4 Myr — negligible above the recombination scale but a real (1-2%) error if you compute the radiation-dominated era explicitly.
• Time at emission. Some references quote t_emit (cosmic time at the moment of emission) rather than t_L. They are related by t_emit = t₀ − t_L. For JWST's z = 14 source: t_L = 13.52 Gyr, t_emit = 0.27 Gyr — the universe was 270 Myr old when the photon left.
• Hubble tension. Cepheid-SN measurements give H₀ ≈ 73; Planck gives H₀ ≈ 67.4. Switching between them changes t_L(z) by ~8% at every z. Tension in H₀ is tension in cosmic age.

Where lookback time appears

• Star formation history. The cosmic star-formation rate density ρ_SFR(z) is plotted against z or against cosmic time t = t₀ − t_L. The peak at z ≈ 2 (Cosmic Noon) corresponds to t ≈ 3.3 Gyr — roughly 10.5 Gyr ago.
• JWST press releases. "JADES-GS-z14-0 was emitted 13.5 billion years ago" is a lookback time, computed from the spectroscopic redshift z = 14.32.
• Stellar nucleosynthesis chronology. The light from globular clusters (lookback ≈ 13 Gyr stellar ages) and the CMB (lookback 13.787 Gyr) must agree on the total elapsed cosmic time — a precision consistency check for ΛCDM.
• Reionization. Reionization occurred between z = 6 (t_L = 12.85 Gyr) and z = 10 (t_L = 13.31 Gyr) — a 460 Myr window when neutral hydrogen was ionized by the first generations of stars and AGN.
• Look-back vs Hubble flow. Surveys plotting m vs z (the supernova Hubble diagram) implicitly probe d_L(z); but plotting against t_L converts to a "cosmic chronology" view that emphasises temporal evolution rather than distance.
• Cosmological clocks (chronometers). The "cosmic chronometer" method dates ancient galaxies and uses ΔdAge/Δz = −1/[(1+z)·H(z)] directly to measure H(z) without a distance ladder.

Lookback time vs the cosmic event horizon

Lookback time tells you how long ago the light we receive now was emitted. The cosmic event horizon tells you the maximum cosmic time from which light emitted now will ever reach us in the infinite future. They are complementary: one looks into the past, the other into the future. In flat ΛCDM with Λ > 0, the future is bounded — light emitted now from a galaxy currently 16 Gly away will never arrive, because dark energy carries it out of reach. The past, by contrast, is also bounded: t_L can never exceed t₀ ≈ 13.8 Gyr, because the universe had no earlier history before the Big Bang singularity.