2The Idea, Step by Step
Start with rising bread
Picture raisins scattered through a lump of bread dough, or dots inked onto a balloon. As the dough puffs up or the balloon inflates, every raisin drifts away from every other one — and the raisins that began farther apart pull apart the fastest. That is exactly what the universe does. Space itself is swelling, carrying the galaxies along for the ride. There is no special raisin at the "centre": pick any one and it looks like everybody else is fleeing from you.
Hubble's simple rule
In 1929 Edwin Hubble pinned this down with numbers. Each galaxy needs just two: its distance $d$ from us — measured in megaparsecs (1 Mpc $\approx$ 3.26 million light-years) — and how fast it is receding, its velocity $v$ in km/s. Hubble found a strikingly clean pattern: the farther a galaxy sits, the faster it runs away, in direct proportion. The simplest form is
Hubble's Law
$$v = H_0\,d$$
where the slope $H_0$, the Hubble constant, is about 70 km/s per Mpc. Worked number: a galaxy 100 Mpc away recedes at $v = 70 \times 100 = 7000$ km/s. Put another galaxy twice as far out and it flees twice as fast.
Why it is a straight line
The proportionality is not a coincidence — it falls out of uniform stretching. If the distance between two "comoving" points is $d(t)=a(t)\,d_0$, where $a(t)$ is the cosmic scale factor, then differentiating gives $v=\dot a\,d_0=(\dot a/a)\,d=H\,d$. Speed proportional to distance, with no centre required. That same stretch reddens the light on its way to us: $1+z=\lambda_{\text{obs}}/\lambda_{\text{emit}}=a_{\text{obs}}/a_{\text{emit}}$. And the slope secretly carries a clock — if the expansion rate had never changed, the age of the universe would be the Hubble time $t_H=1/H_0\approx14$ Gyr, astonishingly near the true 13.8 Gyr.
Try this in the sim above
Drag the Hubble constant $H_0$ slider higher and watch the v–d line tilt up steeply — a bigger slope means faster recession, and the $1/H_0$ readout (the implied age) shrinks: a faster-expanding universe is a younger one. Switch to 🍞 Raisin Bread mode and notice every raisin drifts away from the highlighted "YOU" — the picture would look identical centred on any raisin, which is the whole point: no centre. Finally push the redshift $z$ slider up and watch the $\lambda_{\text{obs}}/\lambda_{\text{emit}}$ readout grow — that is light from the most distant, fastest galaxies arriving stretched and reddened.
3Equation Derivation
Hubble's Law
$$v = H_0\,d$$
Cosmological Redshift
$$1+z = \frac{\lambda_{\text{obs}}}{\lambda_{\text{emit}}} = \frac{a(t_{\text{obs}})}{a(t_{\text{emit}})}$$
Hubble Time (rough age estimate)
$$t_H = \frac{1}{H_0}$$
Symbol Definitions
| Symbol | Meaning | SI Unit |
|---|
| $v$ | Recession velocity of a galaxy | km s⁻¹ |
| $d$ | Proper distance to the galaxy | Mpc |
| $H_0$ | Hubble constant (present-day expansion rate) | km s⁻¹ Mpc⁻¹ |
| $z$ | Redshift | dimensionless |
| $\lambda_{\text{obs}}$ | Observed wavelength | m |
| $\lambda_{\text{emit}}$ | Emitted (rest-frame) wavelength | m |
| $a(t)$ | Cosmic scale factor (today, $a=1$) | dimensionless |
| $t_H$ | Hubble time, ≈ age of universe in matter-dominated case | s (or Gyr) |
1
Empirical observation. In 1929 Hubble plotted recession velocity (from spectral redshift) vs. distance (from Cepheid variables) for nearby galaxies and found a linear relation $v=H_0 d$. The slope $H_0$ has the same value in every direction.
2
Why linearity is natural. If space itself stretches uniformly, the proper distance between any two comoving points is $d(t)=a(t)d_0$. Taking the time derivative: $\dot d=\dot a d_0=(\dot a/a)d=H(t)\,d$. The velocity is automatically proportional to the distance — no central point needed.
3
Redshift connection. Light emitted at scale factor $a_e$ and observed at $a_o$ has wavelength stretched by the same ratio: $\lambda_o/\lambda_e=a_o/a_e=1+z$. For small $z$ this reduces to a Doppler-like form $z\approx v/c$, hence $v\approx cz=H_0 d$.
4
From $H_0$ to age. If expansion had been constant, the age would be exactly $1/H_0$. With $H_0=70$ km s$^{-1}$ Mpc$^{-1}$, $t_H\approx14.0$ Gyr. The actual age is corrected by the deceleration/acceleration history; for the standard $\Lambda$CDM model it is 13.8 Gyr — remarkably close to $1/H_0$.
5
Mapping to the simulation. Slider $H_0$ sets the slope of the v–d line. Galaxies are placed in a 2D comoving grid; their proper distances are $a(t)$ times their initial separation. Speed colour-coding makes it visually clear that more distant galaxies recede faster. The 'Raisin Bread' mode shows the same expansion from a different observer's frame — every observer sees the same Hubble flow.
Primary references: Halliday, Resnick & Walker 10th Ed., §44-3; Ryden — Introduction to Cosmology 2nd Ed., Ch. 2 & 6.
4Frequently Asked Questions
💡 Concept Are galaxies moving through space, or is space itself expanding?▼
It is space itself that is expanding. Galaxies are largely stationary in the comoving frame; the space between them stretches. This distinction matters: in the expansion picture there is no central point, no edge, and recession velocities can exceed the speed of light at large distances without violating special relativity (which forbids only local motion through space faster than $c$).
Key: Space stretches between galaxies — there is no centre, and 'superluminal recession' is allowed.
🎯 Simulation What exactly is the simulation showing?▼
The grid mode displays galaxies on an expanding 2D comoving grid: every galaxy moves outward from every other galaxy at a rate proportional to distance. Velocity arrows show $v=H_0 d$. The redshift mode plots the linear v–d relation. The raisin-bread mode demonstrates that any observer sees the same Hubble flow. Live readouts give $H_0$, the Hubble time $1/H_0$, and the implied age in Gyr.
Key: Animated cosmic expansion with live $v$, $d$, $z$, and Hubble-time readouts.
🌍 Real Life How do astronomers measure $H_0$ in practice?▼
By measuring the distance to a galaxy with a 'standard candle' (Cepheid variable, Type Ia supernova) and its redshift from the spectrum. The slope of $v=cz$ versus $d$ gives $H_0$. The Hubble Space Telescope Key Project and SH0ES use Cepheids+SNe Ia to obtain $H_0\approx73$, while CMB analysis (Planck) gives $\approx67$ — the famous 'Hubble tension'.
Key: Distance ladder (Cepheids+SNe) gives $H_0\approx73$; CMB gives $\approx67$ — Hubble tension.
🔬 Non-Obvious If everything is moving away, where is the centre?▼
There is no centre. In the expanding-space picture, every observer sees every other galaxy receding with a velocity proportional to distance. This is the cosmological principle: the universe is homogeneous and isotropic on large scales. The raisin-bread analogy makes it visual — pick any raisin, every other raisin moves away, and yours is no more 'central' than any other.
Key: No centre — the cosmological principle says every observer sees the same expansion.
📐 Mathematical Why doesn't Hubble's Law violate relativity at large distances?▼
At distance $d_H=c/H_0$ (the Hubble radius, ≈14 billion light-years), $v=c$. Beyond this, the recession 'velocity' exceeds $c$. This is not motion through space — it is the rate at which space itself expands. Special relativity forbids only local velocity > $c$, which is preserved everywhere. Light from a galaxy currently 16 Gly away can still reach us because its proper distance to us decreases as photons traverse and emit time goes by.
Key: Recession is expansion of space, not motion through space — SR is not violated.
🧠 Deep Will the universe expand forever?▼
Yes, according to current observations. The 1998 SNe Ia results showed expansion is accelerating, attributed to dark energy with equation-of-state $w\approx-1$ (cosmological constant). With $\Omega_\Lambda\approx0.69$, $\Omega_m\approx0.31$, the universe will expand forever and approach exponential (de Sitter) expansion. Distant galaxies will eventually redshift beyond observability — the 'Big Rip' is unlikely under the current $\Lambda$CDM model.
Key: Yes — accelerating expansion driven by dark energy means perpetual expansion; visible universe will shrink in the far future.
Explanatory resources: Halliday, Resnick & Walker 10th Ed., §44-3; Ryden — Introduction to Cosmology 2nd Ed., Ch. 2 & 6.
5Common Misconceptions
❌ Misconception: Hubble's Law means we are at the centre of the universe.
✅ Correction: Every observer sees the same Hubble flow. From any galaxy, every other galaxy recedes with $v=H_0d$. This is a direct consequence of homogeneous expansion: no point is special. The observation is symmetric, like raisins in a rising loaf — every raisin sees others receding.
📖 Reference: Weinberg — Cosmology, §1.2; HRW 10th Ed., §44-3.
❌ Misconception: Galaxies receding faster than the speed of light violate Einstein's theory.
✅ Correction: Recession velocities are not motion through space; they describe the rate at which space itself expands. Special relativity prohibits local material/signal velocity > $c$, which holds everywhere. Galaxies beyond the Hubble radius ($d>c/H_0$) recede at $v>c$ — perfectly compatible with general relativity.
📖 Reference: Davis & Lineweaver (2004), 'Expanding Confusion', PASA 21, 97; Misner-Thorne-Wheeler — Gravitation, §27.
❌ Misconception: $1/H_0$ is exactly the age of the universe.
✅ Correction: $1/H_0\approx14.0$ Gyr (for $H_0=70$) is only the age if the expansion rate has been constant. In reality, expansion was decelerating (matter-dominated era) then accelerating (dark-energy era); the actual age is 13.8 Gyr, very close but not equal to $1/H_0$. The match is a coincidence of the current epoch.
📖 Reference: Ryden — Introduction to Cosmology, 2nd Ed., §6.2; Planck 2018 results, A&A 641, A6.
❌ Misconception: The redshift is just an ordinary Doppler shift due to galaxies moving away.
✅ Correction: Cosmological redshift comes from the stretching of wavelengths along with expanding space, not from local Doppler motion. The relation $1+z=a_o/a_e$ involves the scale factor, not a velocity. For small $z$, the formulas coincide ($z\approx v/c$), but for $z\gtrsim1$ the Doppler formula gives wrong distances.
📖 Reference: Harrison — Cosmology: The Science of the Universe, 2nd Ed., §11; HRW 10th Ed., §44-3.
❌ Misconception: If the universe is expanding, then atoms, the Earth, and the Solar System are also expanding.
✅ Correction: Local gravitational and electromagnetic forces overwhelm cosmic expansion at galactic and smaller scales. Bound systems (atoms, planets, galaxies) do not expand. Only unbound, large-scale structures separate with the Hubble flow. Cooperstock et al. (1998) showed the local impact of expansion is utterly negligible.
📖 Reference: Cooperstock, Faraoni & Vollick (1998), ApJ 503, 61; Carrera & Giulini (2010), Rev. Mod. Phys. 82, 169.
Misconception research: Trumper (2001), Sci. Educ. 10; Bailey et al. (2012), Astron. Educ. Rev. 11; Hansson & Redfors (2006), Sci. Educ. 90.