Help · WarpLab

What WarpLab is

Each module runs a piece of real physics as a model you can change: two black holes merging, photons through a double slit, an electron tunnelling through a wall, a qubit and an entangled pair, the Higgs boson in real CMS collisions, dark matter halos fitted to real galaxy rotation curves, neutrinos changing flavour with a fit to Daya Bay's final data, a planet found by microlensing in real OGLE light curves, a ray-traced black hole set beside the EHT's images of M87* and Sgr A*, a flight through SDSS's 3D map of 623 379 galaxies and Laniakea's flow lines, and Maldacena's picture of the Higgs field as money changing hands.

Where a module uses data, it is public data from the instrument that took it, saved as a dated snapshot. No page calls an observatory while you read it.

Everything is in Spanish, English and Italian, and works on a phone. No account is needed.

Partner institutions get their own page with their logo, the modules they curate and their credits, and those modules carry their badge. A partner can keep modules for its members: they open with its email domain or an invite it sent.

How to read a module

A module page has the view at the top, then its sources and its benchmarks. Press ? or H, or the What am I looking at? button, for the module's help: what you are looking at, how its controls work on your device, and a glossary of every term and number on screen.

Numbers in amber are live: the model computes them as you change it. Everything else is a label or fixed data.

A source badge next to a number opens where it came from: source, licence, version and the date of the snapshot. The Sources list under the view names every dataset the module uses; a purely analytic module says it uses none.

Benchmarks are the checks the model must pass against known physics. They run as automatic tests on every change. A module a scientist checked names them, with what they checked and when.

Without 3D graphics, or when your device asks for reduced motion, a module shows a still or step-by-step version with the same physics and numbers. A note above it says which and why. Add ?fallback=1 to a module's address to see that version on purpose.

How the live feed works

The feed lists gravitational-wave alerts, notices from NASA's General Coordinates Network (GCN) on gamma-ray bursts from Fermi and Swift, high-energy neutrinos from IceCube and X-ray flashes from Einstein Probe, microlensing events from OGLE's Early Warning System, new arXiv papers and lab news, newest first. Tap a source's name on an item to see only that source. Each item has one plain-language line and, when one fits, a link to the module that explains it: a gravitational-wave alert opens the merger, an IceCube neutrino the neutrino oscillation module, and an OGLE microlensing event (Fit this event) opens the microlensing module with that event's light curve loaded and fitted. Open an alert to see each of its fields explained. Tap a topic on an item to follow it; following needs a sign-in.

Preliminary: a first automatic alert, sent within minutes and before anyone has checked it. Later analysis confirms it or withdraws it.

Retracted: an alert the observatory withdrew, usually because the signal was a glitch in a detector. It stays on the list, marked, so the record stays complete.

Stale: a source that has not updated within its usual interval. Its last items stay on the list with the time of its last update until it catches up.

Significance: a significant alert passed the bar for a public alert, so astronomers were asked to look for light from it. A low-significance one is shared openly but may well be noise. For GCN notices, significant means WarpLab's instant-alert list: an IceCube Gold track, or a gamma-ray burst from Fermi or Swift placed in a circle small enough for telescopes to follow it up.

Accounts and privacy

An account is optional: every public page works signed out. Signed out, WarpLab sets one cookie, your language. It counts page views with Vercel Analytics, which sets no cookie and keeps nothing that identifies you, and runs no ads. The privacy policy, linked at the foot of every page, has the details and your rights.

You sign in with a link sent to your email. Signed in, WarpLab keeps your language and time zone, your place in each module, the topics you follow and your digest choice, and opens a partner's members-only modules when your email address or an invite gives you access.

WarpLab emails nothing yet but the sign-in link. The topic alerts and the digest you choose are saved for when they start.

You can delete the account from its page. That deletes your settings, progress, follows and partner memberships.

VR and AR

The double slit and the gravitational-wave merger also open in VR and AR; the black hole opens in VR, where you stand ten Schwarzschild radii from it; the cosmic web opens in VR, where you float inside the galaxy map, 1 m is 10 Mpc and the left stick flies you the way you look.

VR needs a headset whose browser runs WebXR, such as a Meta Quest 3. The Enter VR button appears only where VR works. Point with a controller or your hand, press with the trigger or a pinch, and aim at the floor to teleport.

AR needs an Android phone with Chrome and Google Play Services for AR, or a Quest in passthrough. Aim at a table until a ring appears, place the model, then drag and pinch it on a phone or move and resize it from the panel in a headset.

The first time you enter VR or AR, a short intro teaches the controls by doing them: point at the panel, press a button, then teleport in VR or place the model in AR. Each step moves on when you do it, with controllers, hands or a phone's screen, and Skip closes it. Later sessions show a reminder for a few seconds, and Controls on the panel opens the intro again.

iPhone and iPad browsers do not run WebXR, so they show a note instead of a button. The 3D view on the page works there.

Questions

Is WarpLab free?

Yes. Every public module, the live feed and this help are free, and none of them needs an account.

Is the data real?

Yes, where a module uses data. Tap the source badge next to a number for its source and date. Some modules run the model with no data at all, and their Sources section says so.

Why is the gravitational wave drawn so big?

A real wave changes lengths by about one part in 10²¹: a 4 km arm of LIGO moves about a thousandth of a proton's width. The drawings exaggerate it, and each view says by how much.

Why can I not see Enter VR?

The button appears only on a headset whose browser can start VR. On a desktop or a phone the page itself is the whole module.

Why does a module show a still version?

Your browser cannot draw 3D, or your device asks for reduced motion. The still or step-by-step version teaches the same physics with the same controls.

Can an alert disappear from the feed?

No. A withdrawn alert stays, marked retracted, and its details stay readable.

Do I need an account?

No. Sign in only to keep your place in the modules, follow topics or open a partner's members-only modules.

Who checks the physics?

Every model passes its benchmarks against known results on every change, and they are listed on its page. A module checked by a scientist also names them.

Can I use WarpLab in class?

Yes. Every module works on a projector and on a phone, and ?fallback=1 gives a version where nothing moves on its own. The gravitational-wave module's guided tours tell a story in four or five stops; the address of a stop opens the class at that stop.

Glossary

Every term the modules' help panels explain, module by module. Each module's help panel links here.

A black hole's shadow

Black hole: A region where gravity is so strong that nothing, not even light, gets back out. Its boundary is the event horizon.; Why it matters Every picture here is of the light round it: the hole itself gives none.
GM/c²: The natural length of a black hole of mass M: 1.5 km for the Sun's mass, 64 AU for M87*. Every distance in the model is measured in it, so one picture fits any mass.
Event horizon: The surface of no return, at r₊ = (1 + √(1 − a²)) GM/c²: 2 GM/c² (the Schwarzschild radius) with no spin, down to 1 GM/c² at the fastest spin.
Spin a: How fast the hole turns, from 0 to just under 1. A spinning hole drags space round with it, so orbits and light on one side differ from the other.; Why it matters The EHT and simulations favour a fast spin for M87*, near 0.9.
Shadow: The dark patch where light would come from the hole's direction but none can: rays aimed there fall in. With no spin it is 2√27 ≈ 10.4 GM/c² across, 2.6 times the horizon.; Why it matters Its size depends almost only on mass over distance, so it weighs the hole.
Photon ring: Light that went round the hole once or more before reaching you, piled into a thin ring at the shadow's edge. The Photon ring layer lights the disk's light that crossed its plane before landing.
Photon sphere: Where light can orbit, unstably: 3 GM/c² with no spin. Rays that skim it can wind round many times; seen from far away its edge is the shadow's, √27 GM/c² from the centre.
Innermost stable circular orbit (ISCO): The closest circular orbit that lasts: 6 GM/c² with no spin, 1 GM/c² at the fastest. Gas inside it plunges in, so the disk ends there.; Why it matters How deep the disk reaches sets how much energy it can release: 6% of its mass with no spin, up to 32% spinning fast.
Accretion disk: Gas orbiting the hole in a thin, flat disk, heated by its own friction as it spirals in. The model uses the Novikov–Thorne disk, with the Page–Thorne formula for the light from each ring.
Doppler beaming: Gas coming towards you looks brighter and bluer, gas going away dimmer and redder. At a third of light speed the difference is more than tenfold.
Gravitational redshift: Light climbing out of a deep well loses energy. Together with beaming it sets g, the ratio of the light's energy here to its energy at the gas; brightness goes as g⁴.
Gravitational lensing: Gravity bending light. Near the hole it bends so much that the far side of the disk shows over the top of the shadow, and the sky behind is stretched into rings: the Sky grid layer shows it.
Viewing angle: The angle between your line of sight and the spin axis: 0° looks down on the disk, 90° along its edge. M87*'s jet points about 17° from us.
Microarcsecond (µas): A millionth of a second of arc: the width of a doughnut on the Moon, seen from Earth. M87*'s ring is 42 µas across.; Why it matters Seeing it took telescopes spread over the Earth, working as one dish the planet's size.
θg: GM/c² seen from Earth: GM/(c²D) as an angle. The ring is about 11 θg across; the papers give 3.8 µas for M87* and 4.8 µas for Sgr A*.
Event Horizon Telescope (EHT): Radio telescopes on several continents observing together at 1.3 mm, combined by very long baseline interferometry. Its resolution is about 20 µas.; Why it matters Its 2017 campaign gave the first images of a black hole's shadow: M87* (released 2019) and Sgr A* (2022).
Resolution (beam): The smallest detail the array can see, about 20 µas: images are blurred by a spot that size. The circle in the corner of each picture shows it.
Ring diameter: The emission ring's size, measured from the brightest radius round a centre chosen to make it the same in every direction (the EHT's REX method). Blurring pulls it inwards a little.
Hawking temperature: Quantum theory gives every black hole a faint glow at T = ħc³/(8πGMk_B): 6 × 10⁻⁸ K for one solar mass, colder for heavier ones, zero at the fastest spin.; Why it matters M87*'s is about 10⁻¹⁷ K, far below the 2.7 K of the cosmic background: it absorbs more than it emits.
Kerr space-time: The exact solution of Einstein's equations for a spinning black hole (Roy Kerr, 1963). The picture traces light through it; with no spin it is Schwarzschild's.
Ray tracing: Following each pixel's light backwards, step by step along its curved path, until it falls in, hits the disk or reaches the sky. The web view does it on the graphics card; the headset reads the bends from a precomputed table.

The cosmic web

Redshift z: How much a galaxy's light has been stretched on its way: every wavelength arrives 1 + z times longer. SDSS measures it from the lines in each galaxy's spectrum.; Why it matters It is the one number SDSS measures for every galaxy; the map turns it into a distance.
Hubble's law: Nearby, a galaxy's redshift times the speed of light (cz) is proportional to its distance: d = cz / H0. Here H0 = 67.66 km/s per Mpc (Planck 2018).
Comoving distance: How far a galaxy is now, along the expanding grid of space. It adds up the expansion rate over the light's journey, so far away it falls below Hubble's straight line.; Why it matters Each point in the map sits at this distance.
Light-travel time: How long ago the light we see left the galaxy: 0.33 billion years for Coma, 2.5 billion at z = 0.2. The galaxy has moved on since, so it is not its distance divided by the speed of light.
Mpc: A megaparsec: 3.26 million light-years, about 30 times the Milky Way's width.
Cosmic web: The pattern galaxies make on the largest scales: filaments and walls with clusters where they cross, round voids that are nearly empty.; Why it matters Gravity grew it from tiny ripples in the early Universe, so its shape tests how gravity and dark matter work.
Void: A region with far fewer galaxies than average, tens of Mpc across, such as the Boötes void. In the density layer, cells with under a fifth of the expected galaxies.
Survey footprint: The part of the sky SDSS measured: about 8300 square degrees, a fifth of the sky. Outside it the map is empty because nobody looked, not because there is nothing.; Why it matters It tells an empty wedge from a void.
Fingers of God: Clusters look stretched along the line of sight because galaxies orbiting inside them add their own speed, hundreds of km/s, to the expansion's redshift.
Laniakea: Our home supercluster as Tully et al. (2014) drew it: the basin whose galaxies all drift towards the Great Attractor on top of the expansion. Hawaiian for "immense heaven".
Peculiar velocity: A galaxy's own motion on top of the expansion, pulled by the matter round it. Cosmicflows-4 finds it by measuring distances without redshift (Tully–Fisher, supernovae) and subtracting Hubble's law.; Why it matters The flow lines follow these motions, mapped into a 3D field.
Supergalactic plane: The flat sheet most nearby clusters lie in, the equator of the coordinates the map uses. The 100 Mpc rings sit in it, with its pole up.
g − r colour: A galaxy's brightness in green light minus red, in magnitudes. Above about 0.7 it is red, mostly old stars; below, blue, still forming stars.; Why it matters Red galaxies crowd into clusters, blue ones line the filaments.

Double slit

Photon: The smallest amount of light there is. Each photon lands at one point on the screen, never spread out, and where it lands is random.; Why it matters The pattern only shows once many photons have landed, so it is a pattern of chances.
Photons on the screen: How many photons have landed so far, up to 10 000. The same photons land in the same places on every visit.
Screen: photon hits: The dark band at the top. Each dot is one photon where it hit the screen. While there are under 1000, a white ring marks the newest one.
Position on screen (mm): Distance across the screen from the point straight behind the middle of the two slits, in millimetres. The view shows 40 mm either side.
Hits: The bars under the screen: how many photons landed in each narrow strip of it. A bar full to its share meets the prediction curve.
Prediction: The solid curve: the model's chance of a photon landing at each point on the screen. It does not need any photons; the hits grow into it.; Why it matters Comparing bars to the curve is how you test the model.
Interference: What happens when two waves meet. Where they are in step they add up to bright; where one is up and the other down they cancel to dark.
Fringes: The bright and dark bands on the screen. Bright fringes are where the waves from the two slits arrive in step, dark ones where they cancel.
Fringe spacing: Distance between the centres of two neighbouring bright fringes: Δy = λL/d. At 100 µm it is 5.32 mm. With the detector on it reads No fringes.; Why it matters Move the slits closer and the fringes spread out; move them apart and they crowd together.
Slit separation: Distance d from the centre of one slit to the centre of the other, from 50 to 500 µm. A micrometre (µm) is a thousandth of a millimetre.
Slit width: How wide each slit is: 20 µm here, fixed. Each slit alone spreads light into a broad band about 27 mm either side of the middle.; Why it matters That band is the outline the fringes sit inside, so they fade towards the edges.
Wavelength: Distance λ from one crest of the light wave to the next: 532 nm, the green of a laser pointer. A nanometre is a millionth of a millimetre.
Photon energy: The energy one photon carries, E = hc/λ: 2.33 eV for 532 nm light. The constants h, c and e come from CODATA 2022; the badge shows the source.
Screen distance: Distance L from the slits to the screen: 1.0 m. A farther screen would spread the fringes further apart.
ψ₁, ψ₂: The wave amplitudes for a photon reaching a point through slit 1 or through slit 2. They can be positive or negative, so two of them can cancel.
|ψ₁ + ψ₂|²: The chance of a hit when nothing records the slit: add the two amplitudes first, then square. The cancelling gives the dark fringes.
|ψ₁|² + |ψ₂|²: The chance of a hit when the detector records the slit: square each amplitude, then add. Nothing cancels, so the light falls in one broad band.
Which-path detector: A detector that records which slit each photon took. When it is on, the fringes go, the same amount of light lands in one broad band, and the slit separation no longer matters.; Why it matters Knowing the path and seeing interference cannot both happen.
No detector: The faint dashed curve shown while the detector is on: the prediction with the detector off, kept for comparison.
Scale: In AR, the size of the model compared with the real bench. It starts at 1:10, about 28 by 40 cm, and goes from 1:40 up to 1:1.

Find the Higgs

Collision (event): Two protons hitting head on inside the LHC, and everything the CMS detector saw come out. The module calls each recorded collision an event.; Why it matters Only a tiny share of collisions makes a Higgs boson, so you need many of them before a bump shows.
CMS Open Data, records 5200 and 5500: Record 5200 is the 278 four-lepton collisions from 2011 and 2012 that CMS released for teaching, free to use (CC0). Record 5500 is the analysis example whose cuts selected them.; Why it matters Every point on the histogram is a real collision you can download yourself.
Lepton: A light particle that the detector measures very cleanly. Here it means an electron (e) or a muon (μ), a heavier cousin of the electron.; Why it matters Four clean leptons are one of the clearest traces a Higgs boson can leave.
Channels: 4μ, 4e, 2e2μ: Which leptons a collision left: four muons, four electrons, or two of each. With the analysis cuts there are 113 in 4μ, 48 in 4e and 117 in 2e2μ.; Why it matters Electrons and muons are measured differently, so each channel is a separate check.
Four-lepton mass m₄ℓ: Add up the four leptons' energies and momenta and you get one number, their invariant mass. It is the mass of whatever made them, whichever way they flew.; Why it matters Collisions that made the same particle pile up at the same mass, so a particle shows as a bump.
GeV: Giga-electronvolt, the unit for energy and mass in particle physics. A proton weighs about 0.94 GeV; the Higgs about 125.
Z boson: A heavy particle that carries the weak force, with a mass of 91.2 GeV. It lives too briefly to see, and often decays to two leptons of the same kind and opposite charge.; Why it matters A Higgs boson decays to two Z bosons here, and each Z to a lepton pair.
Z₁: Of the two lepton pairs in a collision, the one whose mass is nearest the Z's, 91.2 GeV. The Z₁ mass above slider throws out collisions whose Z₁ is lighter than you set.; Why it matters Raise it above 75 GeV and the bump at 91 GeV goes, while most Higgs collisions stay.
Z₂: The other lepton pair. In a Higgs decay it is usually far lighter than a real Z, because a 125 GeV Higgs leaves it at most about 34 GeV.; Why it matters That is why asking Z₂ for 40 GeV removes the Higgs bump.
pT (transverse momentum): How much momentum a lepton carries across the beam, in GeV. The analysis keeps muons above 5 GeV and electrons above 7; the slider sets one floor for every lepton.; Why it matters Leptons with little pT are more often fakes or come from other processes.
Cuts: Rules a collision must pass to count, such as Z₁ heavier than 40 GeV. The analysis cuts of record 5500 already kept these 278; your sliders can only make them tighter, never looser.; Why it matters Good cuts remove collisions that only look like a Higgs and keep most real ones.
Try buttons: Four ready-made cut settings. Remove the Z peak sets Z₁ above 75 GeV, Ask Z₂ for 40 GeV sets Z₂ above 40, Hard leptons only sets pT above 20 GeV, and The analysis cuts goes back to the start.
Bin (Events / 3 GeV): The mass axis from 70 to 181 GeV is cut into slices 3 GeV wide, as in the CMS 2012 figure. Each point is the number of kept collisions whose mass falls in that slice. The faint outline is all 278 with the analysis cuts.
Error bars (68%): The range of true average counts that could have given the count seen, 68 times in 100. They are exact Poisson intervals: 10 events give 6.89 to 14.27.; Why it matters With so few events per bin, a single bin can jump by several just by chance.
Collisions shown and events kept: Collisions shown counts the collisions on the plot so far, up to 278. Events kept is how many of them pass your cuts.
121–130 GeV, below 110, above 181: Counts of kept events in three mass ranges. 121–130 GeV is the window round the Higgs, where the analysis cuts keep 12; below 110 GeV holds the single-Z bump; above 181 GeV, off the plot, sit pairs of real Z bosons.
Peak, 110–150 GeV: Where the bump sits: the highest bin between 110 and 150 GeV and its two neighbours, averaged by their counts. With the analysis cuts it lands at about 125 GeV; none means no event is left in that range.
Higgs boson, 125.3 GeV: The particle that gives the W and Z bosons their mass, found by CMS and ATLAS in 2012. The dashed H line marks the mass CMS published then, 125.3 GeV.; Why it matters If your peak lands on this line, the data you are cutting are finding the same particle.
Background fit: The dashed curve: a smooth falling shape, e^(a + bm), fitted to the kept masses from 100 to 170 GeV in 1 GeV bins. It stands for everything that is not a Higgs. With fewer than 5 events in that range there is no fit.; Why it matters A bump only means something against what you would see without it.
Bump (background + bump fit): The solid curve adds to the background a bump 2 GeV wide, about the detector's resolution, whose size and mass the fit chooses. Bump mass is where it lands; Bump, fitted is how many events it holds.
Background, 121–130 GeV: How many events the background curve alone expects in the Higgs window. With the analysis cuts it is about 5.6, where 12 were seen.; Why it matters The gap between expected and seen is what the significance measures.
Local significance Z: How much better the bump explains the data than the background alone, in standard deviations (σ), from the ratio of the two fits' likelihoods (Wilks' formula). With the analysis cuts it is 2.8σ at 125 GeV, between record 5500's about 2σ and CMS 2012's 3.2σ in this channel.; Why it matters It says how often the background alone would fake a bump this big at this one mass.
σ (standard deviations): A scale for how unlikely a result is by chance. 1σ happens about 1 time in 6, 3σ about 1 in 740, and 5σ about 1 in 3.5 million.; Why it matters Physicists claim a discovery only at 5σ.
Global significance and the look-elsewhere effect: The fit tried every mass from 110 to 150 GeV, and a fluke could have turned up at any of them. The global figure pays for that search by counting how often the scan climbs past 1σ (Gross and Vitells): 2.8σ local becomes 1.5σ global with the analysis cuts.; Why it matters The more places you look, the easier it is to find a bump by chance.
Cut settings tried: Each new mix of the three sliders and the channels you try counts once. The explainer gives the best local significance among them on all 278 collisions, against the analysis cuts'.; Why it matters Tuning cuts until the bump grows is looking elsewhere too, which is why an analysis fixes its cuts before it looks.

Dark matter in galaxies

Rotation curve: How fast stars and gas circle a galaxy's centre at each radius. In a disk galaxy it rises quickly and then stays about flat, far past where the light fades.; Why it matters If the visible mass were all, the speed would fall off far out, as the planets' speeds do round the Sun.
SPARC: Spitzer Photometry and Accurate Rotation Curves: 175 disk galaxies with infrared photometry from the Spitzer space telescope and rotation curves from radio and optical data (Lelli, McGaugh and Schombert 2016). The module uses the 163 with quality 1 or 2.; Why it matters One homogeneous sample, so every galaxy is modelled the same way.
Measured speed, with error bars: The rotation speed at each radius, from the Doppler shift of hydrogen's 21 cm radio line and of Hα light, corrected for the disk's tilt. The bar is its measured uncertainty.
Radius (kpc): Distance from the galaxy's centre in kiloparsecs. One kiloparsec is about 3260 light-years; the Sun sits about 8 kpc from the centre of the Milky Way.
Stars and gas (baryons): The ordinary matter we can see. The stars line comes from the 3.6 µm infrared light of the disk and bulge; the gas line from the atomic hydrogen measured, with helium added. Stars and gas together is V_bar² = V_gas|V_gas| + Υ_d V_disk² + Υ_b V_bul².; Why it matters Its gap below the points is what a halo has to fill.
Mass-to-light ratio Υ: How many solar masses of stars each solar luminosity of 3.6 µm light stands for. Stellar population models give about 0.5 for disks and 0.7 for bulges; the fit lets it move with a prior of 0.1 dex round those values.; Why it matters Heavier stars could fill some of the gap, but not as far out as the points go.
Dark matter halo: A roughly round cloud of matter that gives off no light, larger than the visible galaxy. The model adds its speed to the baryons': V² = V_bar² + V_halo².
NFW halo (V₂₀₀, c): The density profile cold dark matter simulations give (Navarro, Frenk and White 1996), steep to the centre. V₂₀₀ is the speed at r₂₀₀, where the halo's mean density is 200 times the universe's critical density; the concentration c says how much of its mass sits near the middle.; Why it matters It is what the standard cosmology predicts, so how well it fits is a test.
Pseudo-isothermal halo (ρ₀, r_c): A halo with a flat-density core of radius r_c and central density ρ₀, falling as 1/r² outside. It is an empirical shape, not a prediction.; Why it matters Dwarfs such as DDO 154 often fit a core better than the NFW cusp: the cusp-core problem.
Fit: Least squares (Levenberg–Marquardt): the halo's two numbers, Υ, distance and inclination move together until the model passes as close to the points as their error bars allow. Distance and inclination may move within the catalogue's errors, as Li et al. 2020 let them.
χ²/ν (reduced chi-squared): The sum of each point's miss squared, in units of its error bar, divided by the number of points minus the numbers the fit sets. Near 1 is a good fit; far above 1 the model misses.; Why it matters It turns 'does the line go through the points' into one number.
Halo mass inside the last point: How much dark matter the model puts inside the last measured radius: V_halo² r / G, in solar masses (M☉).
Dark share: Of all the mass inside the last measured radius, the share the halo provides. In most galaxies it is well over half.; Why it matters It says how much of the galaxy is matter no telescope has seen.
Distance and inclination used: The galaxy's distance sets its size and the stars' mass; its tilt converts the speed seen along the line of sight into the true one. The catalogue gives each with an error; the fit may move them within it.
Radial acceleration relation: At every measured radius of every galaxy, the acceleration the rotation shows, g_obs = V²/r, against what stars and gas give by Newton, g_bar = V_bar²/r. All 153 galaxies fall on one curve (McGaugh, Lelli and Schombert 2016).; Why it matters One curve for galaxies of every size is a strong clue to how dark matter, or gravity, behaves.
g† (acceleration scale): The acceleration where the relation bends away from g_obs = g_bar, about 1.2 × 10⁻¹⁰ m/s²: a hundred-billionth of the Earth's gravity. The module fits it to the sample as you change Υ.
Scatter (dex): How far the points sit from the fitted curve, on average, in powers of ten: 0.13 dex is about 35%.
MOND: Modified Newtonian Dynamics (Milgrom 1983): instead of unseen mass, gravity itself grows stronger than Newton's below an acceleration a₀ about equal to g†.; Why it matters It predicts the relation from the visible mass alone, but struggles in galaxy clusters and with the cosmic microwave background.

Gravitational-wave chirp

Solar mass (M☉): The mass of our Sun, about 2 × 10³⁰ kg. Black holes and neutron stars are weighed in it.; Why it matters GW150914's black holes weighed 36 and 29 Suns, each packed inside a horizon about 200 km across.
Mass 1 and mass 2: The masses of the two bodies before they merge. In the chirp tab they are as the detector sees them, and the model ignores redshift. In the catalogue and the fit they are source-frame masses, as the bodies would weigh where they are; in the catalogue, mass 1 is the heavier.; Why it matters Heavier binaries merge sooner and at a lower frequency, so their chirp is shorter and deeper.
Distance: Luminosity distance to the source, in megaparsecs. 1 Mpc is about 3.26 million light-years, so GW150914's 410 Mpc (the 2016 paper) is about 1.3 billion light-years. The strain falls as 1/distance: twice as far, half the strain.; Why it matters In the chirp tab distance changes only how loud the signal is, never its shape.
Chirp mass: The mass combination 𝓜 = (m₁m₂)^(3/5) / (m₁ + m₂)^(1/5). It sets how fast the frequency sweeps up during the inspiral. For 36 + 29 M☉ it is 28.1 M☉.; Why it matters The signal pins it down far better than either mass: in the fit, pairs with nearly the same chirp mass fit almost equally well.
Total mass: Mass 1 plus mass 2. It sets where the inspiral ends and how high the merger frequency gets: the heavier the pair, the lower the pitch at the end. In the catalogue, the median of the sum is not exactly the sum of the two medians.
Final mass: The mass of the single black hole left after the merger. It is always less than mass 1 plus mass 2: GW150914 went from 36 + 29 to 62 M☉. The chirp tab works it out from the two masses with fits to simulations of non-spinning black holes.; Why it matters The gap between the two is the mass that left as gravitational waves.
Energy radiated: Mass turned into gravitational waves, by E = mc². GW150914 radiated about 3 M☉, most of it in its last few orbits.; Why it matters At its peak that was more power than all the stars in the visible universe give off as light.
Remnant spin: How fast the final black hole spins, as a share of the most a black hole of its mass can spin (0 to 1). GW150914's remnant spins at 0.67. Most of it comes from the orbit: the two holes' circling becomes the remnant's rotation.; Why it matters Mass and spin together set the ringdown's pitch.
Effective spin: The mass-weighted spin of the two bodies along the orbital axis, from −1 to +1. Positive means spinning with the orbit, negative against it; near 0 the spins are small or tilted into the orbital plane. GW150914 reads −0.04.; Why it matters Spins aligned with the orbit let the pair circle longer before merging; the chirp model and the fit's templates leave spin out.
Redshift: The expanding universe stretched the wave by a factor 1 + z on its way here. That lowers every frequency, so at the detector the binary looks (1 + z) times heavier than it is. GW150914 has z ≈ 0.1, so the fit stretches its template masses by 1.1.; Why it matters It is why the catalogue's source-frame masses and the chirp tab's detector masses do not compare one to one.
Time to merger: In the chirp tab, how long the signal takes from 20 Hz, where the detectors start to hear it, to its peak. In 3D, real seconds to the end of the plunge: negative before it, positive after. It scales as 𝓜^(−5/3): lighter binaries stay in band much longer.; Why it matters GW150914 lasts under a second from 20 Hz; two neutron stars take minutes.
Cycles from 20 Hz: How many full wave cycles the signal goes through from 20 Hz to the merger. Each cycle is half an orbit, since every orbit sends out two crests.; Why it matters Searches add the signal up cycle by cycle, so a faint signal with many cycles can still rise out of the noise.
Frequency: How many wave crests pass per second, in Hz. It is twice the orbital frequency and rises as the orbit shrinks. In the chirp plot it is the white line, read on the right-hand axis.
Heard frequency: Play chirp plays the signal at real speed, with every frequency ×4: two octaves up. "Heard at" is the pitch that comes out of the speaker. Loudness is scaled so the merger plays at full volume.; Why it matters The signal starts at 20 Hz, too low for a phone speaker to play.
Strain h: How much the wave stretches and squeezes space, as a fraction of length. The plot shows h₊, the plus polarisation at Earth, for a binary seen face-on. GW150914 peaks near 10⁻²¹.; Why it matters 10⁻²¹ of the distance from the Earth to the Sun is about one atom's width.
Ringdown: After the merger, the single misshapen black hole rings like a struck bell at one frequency, and the ringing dies away in a few milliseconds. The readout is that frequency, set by the remnant's mass and spin.; Why it matters GW150914's 62 M☉ remnant at spin 0.67 rings at about 272 Hz: that pitch alone gives away its mass and spin.
Phases: The 3D readout's stage. Inspiral: the orbit shrinks from 30 Hz. Merger: a plunge of about 4 ms. Ringdown: one black hole rings. Settled: one quiet black hole until the loop restarts. Merger and ringdown are approximate.
Separation: The distance between the two black holes' centres in km, from Kepler's law at the current frequency. Once they merge it reads as one black hole.; Why it matters Sizes are to scale with it: at 30 Hz GW150914's holes are about 1000 km apart, each about 100 km in radius.
Orbital speed: How fast the two black holes move relative to each other, as a share of light speed: v = (πGMf)^(1/3) / c. About 0.31 at 30 Hz, past half near the merger.
Playback: How many times slower than real the 3D view runs. The default speed stretches the inspiral from 30 Hz, a fraction of a second in reality, to about 16 s. Speed switches between Real and two slow motions, the second four times slower.
Display scaling: What is drawn bigger than life. In 3D, sizes, separation, wavelength and wave speed are to scale with each other, and only the wave height is drawn about 10²⁰ times larger. In the detector the arms move with the strain ×10²⁰, the light is drawn as a share of its peak, and time runs 40 times slower.; Why it matters Drawn to true scale, the waves and the arms would not move a single pixel.
Dark fringe: LIGO sets its output where the two returning laser beams meet out of step and cancel, so with no wave no light reaches the photodetector. A passing wave makes one arm longer and the other shorter, the beams stop cancelling, and light leaks out.; Why it matters On a dark output any light at all is the signal, instead of a tiny change on a bright background.
Arm change: How far each 4 km arm (3995 m) moves: hL/2, one arm longer and the other shorter. At GW150914's peak that is about 4.5 × 10⁻¹⁸ m, 1/370 of a proton's width. In VR and AR it is the arm change of the detector standing in the scene.; Why it matters No instrument had measured a length change this small before LIGO.
Light out: The share of the laser power that reaches the photodetector: P/P₀ = sin²(φ/2), where φ = 4πhL/λ is the beams' phase difference and λ = 1064 nm. Zero with no strain, growing as h². At GW150914's true peak it is of order 10⁻²¹ of the laser power.; Why it matters Growing as h² makes a bare Michelson nearly blind, so real LIGO holds its output a hair off dark.
Whitened data: The real strain each LIGO site recorded, with each frequency divided by its noise level, then filtered to 35–350 Hz. A quarter of a second, 1024 samples per second, around GW150914.; Why it matters Raw strain is swamped by low-frequency noise far larger than the signal; whitened, the chirp shows by eye.
σ of noise: The unit of the fit's vertical axis: one standard deviation of the whitened noise. 1 is a typical noise wiggle; GW150914 climbs to several σ near the merger.
Template: The predicted signal for one pair of masses: IMRPhenomD from PyCBC, non-spinning, one for each pair of whole source-frame masses from 24 to 44 M☉. Each is redshifted by 1 + z = 1.1 and whitened the same way as the data. The fit picks arrival time, amplitude and phase per detector; you pick the masses.
Match: How much of the data the template explains, over both detectors: √(explained power / data power), from 0 to 1.; Why it matters Noise is in the data, never in the template, so even the right masses stay below 1.
Residual: Data minus template, drawn under each detector. The readout is the share of the data's power left over, which equals 1 − match². Smaller is a better fit, but the noise keeps it above zero.
Livingston ahead: How many milliseconds earlier the fitted template sits in Livingston than in Hanford. The fit lets each detector's arrival time move within ±10 ms, about the light travel time between the two sites. GW150914 reached Livingston about 7 ms first.; Why it matters The gap and its sign narrow down where in the sky the wave came from.
GPS time: Seconds since midnight UTC on 6 January 1980, counted without leap seconds, so it now runs 18 s ahead of UTC. Detectors stamp their data with it. GW150914 arrived at 1126259462.4.
Type: Two black holes, a neutron star and a black hole, or two neutron stars. Read here from the masses: a body under 3 M☉ counts as a neutron star, a heavier one as a black hole. A rule of thumb, not the release's own label.
Signal-to-noise ratio: How loud the signal stood above the noise, across the detector network. Searches start to trust a signal around 8 to 10. GW150914 reads 26 in GWTC-2.1.; Why it matters The louder the signal, the tighter the ranges on its masses and distance.
False alarm rate: How often detector noise alone would make a signal at least this loud, per year. Smaller is more convincing.; Why it matters For GW150914 the 2016 paper put it at less than once in 200,000 years.
Probability it is real (p_astro): The probability, from 0 to 1, that the signal came from space rather than from detector noise. Recent releases list an event as a detection above 0.5.
90% credible interval: Each value is a median with a small + and − after it. The analysis gives a 90% chance that the true value lies between median − minus and median + plus: 34.6 +4.4 −2.6 M☉ means 32.0 to 39.0 M☉.; Why it matters A single number hides how unsure the measurement is; many events' ranges overlap.
Release: The catalogue release the values come from, such as GWTC-2.1 or GWTC-3. Later releases re-analyse earlier events, so values shift: the chirp tab uses the 2016 paper (36 + 29 M☉ at 410 Mpc), the catalogue GWTC-2.1 (34.6 + 30.0 M☉ at 470 Mpc).
Strain axis (×10ⁿ): The strain plots label their axis in a power of ten picked from the peak. Under h (×10⁻²¹), a tick at 1 means h = 10⁻²¹.
Post-Newtonian (2PN): Newton's gravity plus Einstein's corrections in powers of v/c, here up to (v/c)⁴. The model uses it for the inspiral; merger and ringdown are a smooth bridge to the final black hole's ringing, not a numerical-relativity fit.
Stretched and squeezed: The two colours on the 3D sheet mark where the passing wave stretches space and where it squeezes it. Stretching one way comes with squeezing across it, and the two swap every half cycle.
Test ring: A ring of free particles standing across the wave's path, a layer in the 3D view. As the wave passes it turns into an ellipse one way, then the other.; Why it matters That stretch-and-squeeze is exactly what moves LIGO's mirrors.
Lensing: The black holes bend the light of the stars behind them, so the sky warps around them. Turn it on under Layers and look with the Against sky view.
Detector network (H1, L1, V1): The detectors that saw an event: LIGO Hanford (H1) and LIGO Livingston (L1) in the United States, Virgo (V1) near Pisa, and KAGRA (K1) in Japan. A tour's Virgo stops show them as chips, Virgo highlighted.; Why it matters Arrival times at three or more sites place a source on the sky: Virgo shrank GW170814's patch from 1160 to 60 square degrees.

Currencies and the Higgs

Map (lattice): The countries sit on a square grid, 2, 3 or 4 per side. Neighbours are joined by routes, and every square of four countries is a block: a 3 × 3 map has 12 routes and 4 blocks.; Why it matters Physicists cut space into a grid like this to calculate fields on a computer.
Country: A dot on the map with a three-letter code, such as PES for the peso. In the dictionary below the map, a country is a point in space.
Currency: Each country's own money and its own choice of what one unit is worth. Physicists call such a local choice of reference a gauge.; Why it matters Because each country can rename its unit, no single rate is a fact about the world.
Route: The line between two neighbouring countries, where you can change one currency for the other. Its number is the exchange rate.
Exchange rate: How many units of the right (or lower) country's currency one unit of the left (or upper) country's buys. Very small or large rates are written short: 1.3E-3 means 0.0013.; Why it matters In the physics, the rates are the gauge field: the electromagnetic potential.
Nudge this rate: A slider that moves the picked route's rate up to 25% up or down. The two blocks beside the route change: one starts to pay clockwise, the other anticlockwise.
Block: A square of four countries and the four routes round it. Each block shows what one trip round it gains, with ↻ or ↺ for the way that pays.
Trip round a block: Leave with 100 units, change money at each of the four corners and come back to where you started. The ledger lists each step and what you hold after it.; Why it matters Coming back to the same currency is what makes the result comparable, whatever the units.
Loop gain: How much richer a trip round a block leaves you, in percent, going the way that pays. A gain like this, made only by changing money, is arbitrage; below 0.05% the module counts it as none.; Why it matters The loop gain is the field strength, the magnetic field: the part no convention can change.
F, the log gain round a block: Take the natural log of each rate; F is the top and right ones minus the bottom and left ones. A trip clockwise multiplies your money by e^F, so the gain is e^|F| − 1: F = 0.1 is about 10.5%. F above 0 pays clockwise, below 0 anticlockwise.
Best loop pays, Loops that pay: Best loop pays is the largest loop gain on the map. Loops that pay counts the blocks with any gain, out of all blocks; Find the best loop opens the best one.
Trader: The dot that circles each block that pays, the way that pays, faster the more it pays. The still version leaves it out.; Why it matters In the physics it is a charged particle, like an electron, pushed round by a magnetic field.
Change of currency (gauge transformation): Every country picks a new unit, worth up to ten times more or less than the old one. The rates move; the counts below show how many changed, while the loop gains that changed stay at 0.; Why it matters Rates are convention; what a loop gains is what you can measure.
Remove a zero: One country makes a new unit worth 10 old ones. Every rate out of it grows tenfold and every rate into it shrinks tenfold; ×10 or ÷10 on the country, and Zeros removed, keep count. Add a zero undoes it.; Why it matters It is the simplest change of currency: one country, and still no loop gain moves.
Calm world (pure gauge): A world whose rates are not 1 but where no block and no closed trip of any length gains anything. It is what you get from rates of 1 everywhere after a change of currency.; Why it matters It shows that rates far from 1 can still hide no field at all.
Random world: A calm world with every rate pushed up or down by up to about 13%. Most blocks then pay a little, some more than others.
Gold price (Au), the Higgs field: Each country also sells gold at its own price in its own money, shown as Au on the country. The module sets the prices from the rates, so that gold trips gain as little as the rates allow, with the top-left country at 40.; Why it matters In Maldacena's picture this price is the Higgs field.
Gold trip: Buy an ounce of gold on one side of a route, sell it on the other and change the money back: one route, no block. The percentage under each route's rate is what it gains; Best gold route pays is the largest.; Why it matters Once gold is sold, a single route can pay, so the rates are no longer free.
Gold at 1 everywhere (unitary gauge): Each country takes one ounce of gold as its unit, as in the essay: a gold peso of 40 pesos, a gold dollar of 5 dollars. The rates move, no gold trip or loop does, and any rate other than 1 pays on its own route.; Why it matters In these units the gold-trip gain is plain to read off each rate.
Energy of a wave of rates: Step 3 takes a wave of rates, rising and falling across the countries, and adds up what it pays. Energy from loops falls towards 0 as the wave gets longer; energy from gold is the same at every length. Total with gold is the two together.; Why it matters How energy depends on wave length tells physicists whether a particle has a mass.
How much gold weighs in the economy: A slider from 0 to 0.8 for how much gold trips count against loops. The energy from gold is this number squared, the dashed floor on the chart.
Mass: Without gold a long wave costs almost nothing: a particle with no mass, like the photon. With gold no wave costs less than the floor: a particle with mass, like the W and Z bosons of the weak force.; Why it matters This is the job the Higgs field does in nature.
Gold / silver (the Higgs boson): How many times dearer gold is than silver in one country. Gold at 2000 pesos and silver at 1000 make it 2; in australes of 1000 pesos they cost 2 and 1, and it is still 2.; Why it matters A quantity the same in every currency is a physical field, and its ripples are the Higgs boson found at the LHC in 2012.

Planets by microlensing

Microlensing: A star's gravity bending the light of a star behind it into images too close together to see apart. Their total is brighter than the source alone, so the background star brightens and fades as the lens passes.; Why it matters It needs no light from the lens or its planet, so it finds planets round faint stars thousands of light-years away.
Einstein radius: The ring a source exactly behind the lens would form. Every length here is in Einstein radii; for a star half the Sun's mass halfway to the Galactic centre it is about 2 AU on the lens's plane.
Einstein time t_E: The days the source takes to cross one Einstein radius. It mixes the lens's mass, the distances and the relative speed, so alone it does not give the mass.; Why it matters It sets how long the event lasts; most bulge events take 10 to 100 days.
Closest approach u₀: The source's smallest distance from the lens, in Einstein radii, at the time t₀. A point lens magnifies by A(u) = (u² + 2)/(u √(u² + 4)): 1.34 at u = 1, about 1/u when u is small.
Magnification A: How many times brighter the source looks: the sum of its images' brightnesses. Its curve against time is the light curve.
Images and parity: A single lens makes two images, a binary lens three or five. Each is the source seen magnified and distorted; half of them are mirror-flipped (negative parity, hollow on the sky view).
Mass ratio q: The planet's mass over its star's. Jupiter round the Sun is 10⁻³, the Earth 3 × 10⁻⁶. OGLE-2012-BLG-0406Lb's is 5.8 × 10⁻³: about 3.6 Jupiter masses round a 0.59 M☉ star.; Why it matters The blip's length and size give q directly; the planet's mass in kilograms needs the star's.
Separation s: The planet's distance from its star projected on the sky, in Einstein radii. Planets near s = 1 make the largest caustics and are found most often.; Why it matters s and 1/s give much the same central caustic, so some events have two answers; the planetary caustic of OGLE-2012-BLG-0406 settles it at 1.35.
Caustic: The closed curves on the sky where a point source's magnification is infinite and two extra images appear or vanish. A planet beyond s = 1 has a small caustic about s − 1/s from its star.; Why it matters Crossing or passing one makes the planet's blip.
Source radius ρ: The background star's radius in Einstein radii. Near a caustic the star's disc is not a point: its edge crosses first, and the spike is rounded off.
Source and blend flux: The light is F = F_s A + F_b: the source's flux, magnified, plus light from stars in the same seeing disc that is not. The fit solves both exactly.
χ²/ν: The sum of (data − model)² / error² over the points, divided by the degrees of freedom (points less the parameters fitted). Near 1, the model fits within the errors.; Why it matters A single lens on the planet's event leaves χ²/ν far above 1; adding the planet brings it near 1.
OGLE Early Warning System: OGLE's real-time pipeline (Udalski 2003): it finds microlensing events in the bulge each night with the 1.3 m Warsaw Telescope at Las Campanas and publishes their photometry and a single-lens fit.
Roman's bulge survey: NASA's Nancy Grace Roman Space Telescope will watch 1.7 deg² towards the Galactic centre once every 12.1 minutes for six seasons of 70.5 days (STScI), expecting more than a thousand planets (NASA).

Neutrino oscillation

Flavour: Which charged lepton a neutrino makes when it interacts: an electron, a muon or a tau. A neutrino is born and detected as a flavour, but it travels as a mix of three mass states.; Why it matters Because the mass states drift out of step, the flavour mix changes on the way: that is oscillation.
L/E: Distance travelled over energy, here in km/GeV (the same number as m/MeV). Each term of the oscillation goes as sin²(1.267 Δm² L/E).; Why it matters Experiments with very different distances and energies see the same physics when their L/E matches.
Mixing angle θ₁₃: One of three angles that say how the flavours are made of the mass states. Reactors measure sin²2θ₁₃, the depth of the electron antineutrino's dip at about 1.6 km for MeV energies.; Why it matters Daya Bay found it non-zero in 2012; it is the smallest of the three angles and the best measured.
Mass splitting Δm²: The difference of two mass states' squares, in eV². Δm²₃₂ ≈ 2.5 × 10⁻³ eV² sets the atmospheric oscillation; Δm²₂₁ = 7.53 × 10⁻⁵ eV² the slow solar one.; Why it matters It sets where along L/E the dip falls; oscillation needs neutrinos to have mass.
Survival probability: The chance a neutrino arrives as the flavour it started as. For reactor antineutrinos: 1 − cos⁴θ₁₃ sin²2θ₁₂ sin²Δ₂₁ − sin²2θ₁₃ (cos²θ₁₂ sin²Δ₃₁ + sin²θ₁₂ sin²Δ₃₂), with Δᵢⱼ = 1.267 Δm²ᵢⱼ L/E.
Two-flavour formula: 1 − sin²2θ₂₃ sin²Δ₃₂: the survival of muon neutrinos with only two flavours mixing, as atmospheric and IceCube analyses often fit it. The dashed line on the plot; the three-flavour lines differ by a few per cent.
Inverse beta decay: An electron antineutrino turns a proton into a neutron and a positron. The positron's flash (the prompt energy) and the neutron's capture on gadolinium a few tens of microseconds later make the pair Daya Bay counts.; Why it matters The neutrino's energy is the prompt energy plus about 0.78 MeV.
Far/near ratio: Events in the far hall over events in the near halls, each scaled by what six reactors would send it without oscillation. Without oscillation it is 1 at every energy.; Why it matters The reactors' flux, known only to a few per cent, cancels.
Effective baseline L_eff: The far hall's distance to the six reactors, averaged by the flux each sends it: about 1.63 km. The plot's horizontal axis is L_eff over the neutrino energy.
χ²: The sum over the points of (data − model)² / error². A good fit has χ² near the number of points less the parameters fitted: 24 here.; Why it matters Fit moves the sliders to where χ² is lowest.
± σ: How far a fitted parameter moves before χ² rises by 1: its one-standard-deviation uncertainty. This fit counts statistics only, so its σ is smaller than the published one, which adds every systematic.
Daya Bay: A reactor experiment in southern China, 2011 to 2020: eight identical 20-tonne detectors in three halls, between 360 m and 1.9 km from six reactor cores. Its full data release (Zenodo, 2025) holds 5.55 million IBD candidates over 3158 days.

Qubit and Bell test

Qubit: The quantum version of a bit. Its state is α|0⟩ + β|1⟩, a mix of the two values that is only settled when you measure it.; Why it matters Every quantum computer is built from qubits like this one.
|0⟩ and |1⟩: The two basic states of a qubit, like 0 and 1 of an ordinary bit. On the sphere |0⟩ is the top and |1⟩ the bottom.
Amplitudes α and β: The two numbers, possibly complex, that say how much of |0⟩ and of |1⟩ the state holds. |α|² + |β|² = 1, and |α|² is the chance of +1 when you measure along z.; Why it matters A complex amplitude carries a phase, which no single measurement along z can see but a gate can turn into a difference.
Bloch sphere: A ball on which every state of one qubit is an arrow from the centre to the surface. Opposite points are opposite outcomes of one measurement.; Why it matters It turns two complex amplitudes into one picture you can rotate.
Polar angle θ: How far the arrow tips down from the top, from 0° (|0⟩) to 180° (|1⟩). At 90° it lies on the equator, an equal mix of |0⟩ and |1⟩.
Azimuth φ: How far the arrow is turned round the vertical axis, from −180° to 180°. At 0° it points along x, at 90° along y. It is the phase between α and β.
Bloch vector (x, y, z): The arrow's three coordinates, a vector of length 1. Each one is the average outcome you would get measuring many copies along that axis.; Why it matters z = 1 means +1 every time along z; z = 0 means half and half.
Gate: An operation that turns the arrow about an axis by a fixed angle. It changes the state without measuring it, so nothing is lost.
H (Hadamard): A half turn about the axis halfway between x and z. It swaps |0⟩ and |+⟩, so H on |0⟩ lands on the equator.; Why it matters It is the usual way to put a qubit into an equal mix of |0⟩ and |1⟩.
X, Y and Z: Half turns about the x, y and z axes. X swaps |0⟩ and |1⟩, the quantum NOT. Z flips the sign of |1⟩, which a measurement along z cannot see.
S and T: Smaller turns about z: S a quarter turn, T an eighth. S takes |+⟩ to |+i⟩. Both leave the chances along z as they were.
Measurement axis: The direction you measure along: x, y or z. The answer is +1 or −1; along z they mean |0⟩ and |1⟩, along x |+⟩ and |−⟩, along y |+i⟩ and |−i⟩.
Fresh copies: Qubits prepared the same way and each measured once. The histogram counts how many gave +1 and how many −1; the same copies give the same outcomes on every visit.
Born rule: The chance of +1 along an axis n is |⟨+n|ψ⟩|², set by the angle between the arrow and the axis. The tick on each bar marks it, and the measured share settles on it as copies add up.; Why it matters Quantum mechanics predicts chances, not single outcomes, and this is how.
Collapse: Measuring this qubit gives +1 or −1 with the Born rule's chances and moves the arrow onto that end of the axis. Measure again along the same axis and you get the same answer every time.
Entangled pair Φ⁺: Two qubits in the state (|00⟩ + |11⟩)/√2: neither has an answer of its own, but measured along the same direction they always agree. Alice's outcomes alone stay 50/50 whatever Bob does.; Why it matters The agreement shows only when the two lists are put side by side, so it cannot carry a message.
Settings and rounds: Each round a referee hands Alice and Bob a random setting, 0 or 1. They win when their outcomes agree, except when both settings are 1: then they must differ. Rounds counts the pairs sent so far.
Detector angles a, a′, b, b′: The direction each detector turns to in the x–z plane: a for Alice's setting 0, a′ for her setting 1, b and b′ for Bob's. 0° points along z, 90° along x.; Why it matters The angles, not the pair, decide how far the score can go.
Correlation E: For one pair of settings, how often the outcomes agree minus how often they differ, as a share of those rounds. It runs from 1 (always agree) to −1 (always differ); for Φ⁺ it is cos(a − b).
CHSH value S: S = E(a, b) + E(a, b′) + E(a′, b) − E(a′, b′), one number from the four correlations. S measured comes from the rounds sent; S expected is what the theory gives for these angles.
S ± σ: The measured S with one standard deviation σ, the spread that chance alone gives with this many rounds. Four times the rounds halves σ.
Win rate: The share of rounds won. It follows S: win rate = 1/2 + S/8, so S = 2 is 75% and S = 2√2 about 85%.
Classical limit 2: No local hidden-variable rule can make S larger than 2, so none wins more than 75% of rounds. Every fixed strategy scores exactly 2 or −2.; Why it matters Going past 2 rules out every explanation in which each particle carries its answers with it.
Quantum bound 2√2: The most S any quantum state can reach, 2√2 ≈ 2.83. The entangled pair reaches it exactly at Tsirelson's angles.
Tsirelson's angles: a = 0°, a′ = 90°, b = 45°, b′ = −45°: each pair of settings 45° apart, except a′ and b′, which are 135° apart. They give the entangled pair S = 2√2. All aligned puts every detector at 0°, where both sources give S = 2.
Local hidden-variable rule: Each pair leaves with a hidden direction, and each detector answers +1 if it points within 90° of it, on its own. Its correlation is 1 − 2d/π for detectors d apart, and it reaches exactly 2 at Tsirelson's angles.; Why it matters It is the best a local rule can do, so it is the fair rival.
Beats 2 by nσ: Shown when |S| sits more than 3σ above 2; a smaller excess counts as within the classical limit. Until σ falls under 0.5 the panel asks you to send more pairs.; Why it matters A local run can land a little above 2 by chance, so only a clear margin counts.

Quantum tunnelling

Tunnelling: A quantum particle getting through a barrier that it does not have the energy to climb. The wave leaks into the barrier, and if the barrier is thin some of it comes out the other side.; Why it matters It powers the scanning tunnelling microscope and the flash memory in phones.
Wavepacket: A short burst of wave that stands for one electron. Its shape tells you where the electron is likely to be found; here it is a Gaussian, a bell curve.
V(x): The potential: the energy the electron would need to be at each position x. The grey filled shape is the barrier; everywhere else it is zero.
Barrier shape: Rectangle is one flat-topped wall. Double wall is two walls of width a with a 1.5 nm gap between them. Smooth bump is a bell-shaped hill whose width at half its height is a.
Resonant energy: With the double wall, the gap between the walls acts as a trap. At certain energies a wave with exactly that energy crosses both walls completely. Move E₀ to find them.; Why it matters Two walls can let through more than one wall alone.
Packet energy E₀: The electron's average energy of motion, from 0.3 to 2.5 eV. It is the dashed line on the plot, and the packet is drawn riding on it.
Packet width σ: How spread out the packet is in position, from 0.8 to 2 nm. A narrower packet holds a wider range of energies.; Why it matters That range is why Theory, this packet differs from Plane wave at E₀.
Barrier height V₀: The energy at the top of the barrier, from 0 to 3 eV. When E₀ is below V₀ a classical electron always bounces back.
Barrier width a: How thick the barrier is, from 0.1 to 1.5 nm: the rectangle's width, each wall's width, or the bump's width at half height.; Why it matters Below the top, the share that gets through drops very fast as the barrier gets thicker.
|ψ|²: The filled shape: how likely it is to find the electron at each position. The whole area always adds up to 1, one electron.
Re ψ: The thin wavy line: one part of the wave itself, which goes up and down. Higher energy means shorter ripples.
Transmitted: The share of the chance of finding the electron that is now right of the barrier, including what has left the box on the right. It grows during the run and then settles.
Reflected: The share that is left of the barrier, including what has left the box on the left. It reads Not yet until the packet has finished leaving the barrier.
Theory, this packet: The share the textbook formula predicts for this packet: the transmission of every energy in it, weighted by how much of the packet has that energy.; Why it matters When the run ends, Transmitted should land on this number.
Plane wave at E₀: The share that would get through for an endless wave with exactly the energy E₀ and no spread. A wide packet comes close to it.
Decay length 1/κ: Inside the barrier the wave shrinks as e^(−κx): every 1/κ it drops to about a third. It reads Over the top when E₀ is at or above V₀, because then the wave does not shrink.; Why it matters A barrier a few decay lengths thick lets almost nothing through.
Time: The electron's own time since release, in femtoseconds (fs), millionths of a billionth of a second. The moving view slows it so the packet takes about 6 seconds to cross.
Left, Right, Theory: The bar under the plot. The grey part on the left is what is left of the barrier, the coloured part on the right what is right of it, the faint middle what is still inside. The white mark is the theory.
eV: The electronvolt: the energy an electron gains crossing 1 volt. It is the energy scale on the left of the plot.
Position (nm): Where along the line the electron is, in nanometres, millionths of a millimetre. The barrier starts at 0; the view shows 20 nm either side.
Schrödinger equation: The rule for how a quantum wave changes over time. Each run solves it for the electron on 4160 points, using a method called Crank–Nicolson.
Absorbing layers: Strips past both edges of the view that soak up the wave instead of bouncing it back. The solver counts what each one takes, so the three parts of the bar always add up to 1.