CHRONOSCOPE · time, at every scale Home
◌ Workbench
ConceptsThe big ideas, with the evidence — and every lab that shows them The mathsEvery model's equations, constants and worked examples — each checked by the test suite For teachersClass links, completion codes and printable worksheets Your passportTour stamps, missions, your prediction record — and save it to a file Quizzes and reviewTour quizzes, and questions that come back to check what stuck About ChronoscopeWho made it, how it was built, and why you can trust it
Mode

∑ The maths › Cosmic horizons · open the lab →

Cosmic horizons: the maths

The same smooth ΛCDM universe as the Expanding-universe lab, drawn in conformal time so light travels at 45°. Tables of conformal time built by integration. Computed live.

Equations

η(a) = ∫₀a da′ / (a′² H(a′))
Conformal time: how far light can have travelled (in today's distances) by the time the universe had size a.
particle horizon = c · η(1)
The edge of the observable universe: the farthest matter whose light has had time to reach us.
event horizon = c · [η(∞) − η(1)]
The farthest galaxy a signal sent today can ever reach, because accelerating expansion carries the rest away.

Constants

Ωm, ΩΛ, Ωr0.315, 0.685, 9.1 × 10⁻⁵Planck Collaboration (2020)
H₀67.4 km/s/MpcPlanck (2020)

Worked examples ✓ checked on every change

Observable universe (particle horizon), today46.1 billion lythe integral above, done independently — about 46 billion light-years
Event horizon, today16.7 billion lythe integral above — about 16.7 billion light-years

The app's test suite puts the lab in each setup, reads the lab's own result, and fails if it strays from these values.

What's simplified

  • A perfectly smooth universe.
  • Distances are 'proper distance today' (comoving).

Where it breaks

Only if the ΛCDM model does — for example if dark energy isn't constant.

Sources: T. M. Davis & C. H. Lineweaver, PASA 21, 97 (2004); Planck Collaboration (2020).

New here?