Adjust the Hubble constant and galaxy distance to compute recession velocity, redshift and the age of the universe in real time. Explore the expansion of the universe and the puzzle of recession velocities faster than light.
Solid line: velocity–distance relation for H₀. Dashed: speed of light c. Yellow dot: selected galaxy.
Dark energy (the cosmological constant Λ) makes the expansion of the universe accelerate. The present universe consists of about 68% dark energy, 27% dark matter and 5% ordinary matter (Ω_Λ≈0.68). Because of the acceleration, H₀ has changed over cosmic history, and its present value differs from past values.
1 pc (parsec) ≈ 3.26 light-years. 1 Mpc = 10⁶ pc ≈ 3.26 million light-years ≈ 3.09×10²² m. The Andromeda galaxy is about 0.78 Mpc away, the Local Group is about 5 Mpc across, and the radius of the observable universe is about 14,000 Mpc.
The radius of the observable universe is about 46 billion light-years (comoving distance, allowing for expansion), but the universe as a whole may be far larger, or infinite. It is not an "edge" but an "observation limit": the distance light could travel in 13.8 billion years marks the boundary of the observable universe. Inflation theory holds that the universe is vastly larger than the observable part.
In the low-velocity approximation (v ≪ c), z ≈ v/c = H₀d/c. Relativistically, z = √((1+β)/(1−β)) − 1 with β=v/c. For cosmological redshift the stretching of space a(t) gives z+1 = a(now)/a(emission). A galaxy at z=1 is about 10 billion light-years away (depending on the observations).
In the physical model of the Hubble's law simulator, the observed recession velocity \(v\) of a galaxy and its distance \(d\) from Earth obey the linear relation \(v = H_0 d\), where \(H_0\) is the Hubble constant, the present expansion rate of the universe. The relation implies that the universe expands homogeneously and isotropically. In the simulator, adjusting \(H_0\) recomputes the recession velocity of a galaxy at any distance in real time. The redshift \(z\) is obtained with the low-velocity approximation \(z \approx v/c\). Beyond the distance \(d = c/H_0\) (the Hubble radius), where the recession velocity reaches the speed of light, the formula gives \(z>1\), but the approximation no longer holds; the correct cosmological redshift is defined from the expansion history \(a(t)\) as \(1+z = a_{\text{now}}/a_{\text{emit}}\). The approximate age of the universe is \(t \approx 1/H_0\), the basis for tracing the expansion history backwards. Through these formulas you can grasp intuitively the geometry of cosmic expansion and the limits of observation.
Distance estimation in galaxy surveys
Obtaining v≈cz from the spectroscopic redshift z and converting it to distance with Hubble's law, d=v/H₀, is the basic procedure of large surveys that handle hundreds of thousands to millions of galaxies. Nearby, the peculiar motions of galaxies (a few hundred km/s) become an error source; far away the low-velocity approximation fails and the conversion switches to a cosmological model. The linear d–v relation in this simulator is the first step of that chain.
Determining cosmological parameters
H₀ directly affects estimates of the age of the universe, the size of the observable universe and the dark-energy density. The gap between the H₀ from the distance ladder of Cepheids and type Ia supernovae and the H₀ from the cosmic microwave background (the Hubble tension) is a central issue of modern cosmology; the "Comparison H₀ (2)" input in this tool lets you see side by side how much the velocity–distance relation and the age change between two values.
Education and outreach
In high-school and university astronomy classes and planetarium shows, moving H₀ and the distance while watching recession velocity, redshift and the 1/H₀ age change together quickly conveys the intuitions that "expansion has no centre" and "the farther, the faster". Watching the Hα line move to longer wavelengths in the redshift tab connects the idea to real spectroscopic observation.
It is tempting to think that "if a galaxy recedes faster than light, its light cannot reach Earth". In fact the cause is the stretching of space itself through the expansion of the whole universe; the galaxy is not moving at light speed. Light emitted early in the expansion by galaxies now receding faster than light therefore keeps arriving today.
It is also tempting to think that "the Hubble constant directly gives the age of the universe". In fact the reciprocal 1/H₀ is only an approximation; in the present accelerating universe, the exact age requires a more involved calculation that accounts for dark energy. Note that simply computing 1/H₀ in the simulator does not reproduce the actual age of the universe.
It is easy to misread "the farther the galaxy, the larger its recession velocity" as "distant galaxies move fast". In fact, because the whole universe expands uniformly, a more distant galaxy simply has more space between it and the observer, and the recession velocity only appears large as the cumulative effect of that expanding space. It must be interpreted separately from the galaxy's own peculiar motion (motion driven by local gravity).
At the defaults d=200 Mpc and H₀=70 km/s/Mpc the recession velocity is v=14,000 km/s (4.7% of the speed of light), the redshift z≈0.0467, the age 1/H₀≈14.0 Gyr and the Hubble radius c/H₀≈4,283 Mpc. Setting H₀ to 73 gives v=14,600 km/s (+4.3%) and shortens the age to 13.4 Gyr. Extending d to 4,000 Mpc gives v=280,000 km/s (93% of c), and with H₀=100 it reaches 400,000 km/s (133%), exceeding the speed of light – an apparent velocity caused by the expansion of space.