Cosmological Lookback Time Calculator

Calculate how far back in cosmic history you're looking when you observe an object at redshift z, via numerical integration of t_L(z) = (1/H0)∫dz'/[(1+z')E(z')] in a flat ΛCDM cosmology — with standard cosmology presets, a cosmic timeline, and an integral-area chart, validated against Astropy.

Cosmological lookback time calculator

Unlike a plain magnitude or distance conversion, lookback time is a genuine integral over the universe's expansion history: t_L(z) = (1/H₀) ∫₀^z dz′ / [(1+z′)E(z′)], with E(z) = √(Ωm(1+z)³ + Ωk(1+z)² + ΩΛ). It's evaluated here by numerical integration (Simpson's rule) and checked against Astropy — see the note below the calculator. The answer depends explicitly on the cosmology you assume, not just on z.

Redshift (z)

H₀ (km/s/Mpc)

Ωm (matter density)

ΩΛ (dark energy density)

0.685 (flat: 1 − Ωm)

Lookback time: ~7.95 Gyr

You are seeing this object as it was ~5.85 Gyr after the Big Bang (universe is 13.8 Gyr old today).

Cosmic timeline from the Big Bang (left) to today (right). The marked point is where this object's light was emitted; the bracket is exactly the lookback time.

The integrand 1/[(1+z′)E(z′)] versus z′. The shaded area from 0 to z, multiplied by the Hubble time 1/H₀, is the lookback time — an area, not an algebraic formula.

Validated against Astropy's FlatLambdaCDM (matter+Λ, no radiation) across a range of z and cosmologies — lookback times and ages agree to better than 1 part in 10⁶.

When you observe a galaxy at redshift z = 1, you’re not seeing it as it is now — you’re seeing light that left it long before Earth existed. “How long ago” is the lookback time, and unlike most of the conversions on this site, it isn’t a formula you can rearrange with algebra. It’s an integral over the universe’s entire expansion history between then and now, and the answer depends on which cosmology you plug in.

The integral

In a flat universe with matter and a cosmological constant, the expansion rate at redshift z relative to today is

E(z) = \sqrt{\Omega_m (1+z)^3 + \Omega_\Lambda}

and the lookback time to that redshift is

t_L(z) = \frac{1}{H_0} \int_0^z \frac{dz'}{(1+z')E(z')}

There’s no closed-form antiderivative for this in general — it has to be evaluated numerically. This calculator does that with Simpson’s rule at a resolution far finer than the answer’s precision requires, so the numerical method itself isn’t the source of any meaningful error; what you get back is limited by how well your chosen Ωm, ΩΛ, and H₀ actually describe the real universe, not by the integration.

Why the cosmology matters

Change H₀, Ωm, or ΩΛ, and every lookback time changes with them — this is the calculator’s central point, not an edge case. The clearest way to see it: pick z = 1 under Planck 2018 parameters (H₀ = 67.4, Ωm = 0.315) and you get a lookback time of about 7.95 billion years, meaning the object’s light left it about 5.85 billion years after the Big Bang. Keep z = 1 but switch to an Einstein–de Sitter universe (Ωm = 1, no dark energy at all — the standard cosmology before the late-1990s supernova observations) and the lookback time drops to about 6.0 billion years, because a matter-only universe has a very different expansion history. Same redshift, different physical answer — which is exactly why every result here is reported next to the parameters that produced it, and why serious work always states its assumed cosmology explicitly.

Reading the two charts

The cosmic timeline places the emission time directly on a bar running from the Big Bang to today, next to a few landmark epochs (the CMB, the first galaxies, when the Solar System formed) — turning “5.85 Gyr after the Big Bang” into a point on a story whose overall shape you likely already know.
The integrand chart plots 1/[(1+z′)E(z′)] itself and shades the region from 0 to z. That shaded area, multiplied by the Hubble time 1/H₀, is the lookback time — a direct, visual answer to “why isn’t this just algebra.”

Validated against Astropy

Because this result is entirely dependent on the numerical integration and the assumed cosmological model, its outputs were checked directly against Astropy’s cosmology module before publication — specifically FlatLambdaCDM with radiation switched off (Tcmb0=0), matching this calculator’s matter+Λ-only model exactly. Across redshifts from 0.1 to 10 and four different cosmologies (Planck 2018, WMAP9, a simple H₀=70/Ωm=0.3 model, and Einstein–de Sitter), both the lookback time and the age of the universe agreed with Astropy’s values to better than 1 part in 10⁶ — comfortably beyond the precision either tool displays. Radiation is omitted from both, so treat results at very high redshift (z ≳ 1000, near recombination) as approximate; at the redshifts most real observational targets sit at, that omission is negligible.