A quasar at z = 3 is receding at 1.46 times the speed of light
The same calculator page gives you 0.88c and 1.46c for the same quasar. Neither is a bug. They answer different questions, and only one of them is about how fast the galaxy is going away.
Set the redshift calculator to z = 3, the quasar preset, with H₀ = 70 km/s/Mpc. It prints a recession velocity of 264,523 km/s — 88.24% of the speed of light — and, two rows below it, a comoving distance of 6,264.4 Mpc.
Take that second number and apply Hubble's law to it, v = H₀ · D. Seventy kilometres per second per megaparsec, times 6,264.4 megaparsecs, is 438,508 km/s. That is 1.46 times the speed of light.
Two velocities for one object, from two numbers on the same screen, differing by 66%. The larger one is the rate at which the distance to that quasar is increasing. The smaller one answers a different question.
The formula that caps at c
The velocity row uses the relativistic Doppler relation,
v = c · [(1+z)² − 1] / [(1+z)² + 1]
which is the correct way to convert a wavelength shift into a speed when the shift is caused by motion through space. It is the formula for a receding siren, a binary star, a police radar: the physics in the Doppler tool, corrected for time dilation.
Look at its structure. The numerator is always smaller than the denominator, so the result is always below c, for every possible z. Push the slider to the cosmic microwave background at z = 1089 and the readout displays 299,792 km/s and 100.00% of c, the display rounding up to a value the formula can approach and never reach.
That ceiling is the tell. Special relativity forbids anything from moving through space at c, so a formula built on motion through space cannot return more than c no matter what you feed it. If the true recession rate exceeds c, this formula cannot express it, and will quietly report something below the limit instead.
What cosmological redshift is instead
Light from a distant galaxy is not stretched because the galaxy is running away. It is stretched because the space the light is crossing expanded while it was in transit.
The calculator prints the cleanest statement of this as its scale factor: a = 1/(1+z). At z = 3 the scale factor is 0.2500, meaning the universe was exactly a quarter of its present size when that light set out. Every wavelength in flight was stretched by the same factor of four — not by 3.5 or by 4.2 depending on the galaxy's own motion, but by four, because the stretching is a property of the interval of time the light spent travelling. Redshift is a measurement of how much the universe has grown since emission. Read that way, z is not a speedometer at all; it is a ruler laid against cosmic history.
The recession is real, and it follows from the expansion rather than from motion. Distant galaxies are very nearly stationary in the local space around them: their own drift through it, called peculiar velocity, is a few hundred km/s, the same order as the Sun's motion around the galaxy. What increases is the amount of space between us. Nothing crosses space faster than light; the space itself is under no such constraint, because "space expanding at a speed" is not a velocity anything possesses.
Where it stops being academic
At small z the distinction does not matter, and this is why the wrong formula survives in textbooks.
Take z = 0.1, where the tool opens. The Doppler row reads 28,487 km/s. The comoving distance is 418.0 Mpc, so Hubble's law gives 29,260 km/s. A 2.7% disagreement — smaller than the uncertainty in H₀ itself, and invisible in any practical use.
Now walk it up, computing H₀ · D from each comoving distance the tool reports:
- z = 0.1 — D = 418.0 Mpc, recession 0.098c. The Doppler row says 0.095c. Agreement.
- z = 1 — D = 3,274.9 Mpc, recession 0.765c. The Doppler row says 0.600c. A 27% gap.
- z = 1.5 — D = 4,315.6 Mpc, recession 1.008c. The Doppler row says 0.724c.
- z = 3 — D = 6,264.4 Mpc, recession 1.463c. The Doppler row says 0.882c.
- z = 10 — D = 9,275.9 Mpc, recession 2.166c. The Doppler row says 0.984c.
The tool stops integrating at z = 10, so the distance and lookback rows do not move above that: the CMB setting can be read for velocity and scale factor, not for distance.
The crossing sits just below z = 1.5, and it is not a coincidence: the recession rate reaches c exactly where the comoving distance reaches c/H₀ = 4,282.7 Mpc, the Hubble radius. Anything further away than that is receding faster than light. The great majority of the galaxies we can see are past it.
If that quasar's distance is growing at 1.46c, how did its light reach us at all?
Because the expansion rate at a given place is not fixed forever. A photon emitted toward us from beyond the Hubble radius does initially lose ground: the space between us and it grows faster than it can cross. But as it travels it enters regions closer to us, where the recession rate is lower, and in a decelerating or slowly-accelerating universe the Hubble radius itself grows over time. The photon can be overtaken by the expanding sphere within which it makes progress, and from then on it closes. The light we now see from z > 1.5 spent its early life falling behind. Davis and Lineweaver work this through carefully, and identify a run of textbook claims that get it wrong in exactly the way the velocity row here does.
The lookback time makes the scale concrete. The z = 3 quasar's light has been travelling 11.221 billion years, and the object is now 6,264 Mpc away, about 20 billion light years, comfortably more than the distance light could have covered in the age of the universe. It is not a contradiction. The gap opened up behind the photon while it was in flight.
References (2)
- the paper this article is a summary of; it names the error in textbooks Davis & Lineweaver (2004). Expanding Confusion: Common Misconceptions of Cosmological Horizons and the Superluminal Expansion of the Universe. Publications of the Astronomical Society of Australia 21(1).
- the original v = H₀D, with a slope roughly seven times too steep Hubble (1929). A relation between distance and radial velocity among extra-galactic nebulae. PNAS 15(3).