Dichotomy
To run 1 m, you first have to run half of it, then half of the rest, then half of what's left… Zeno of Elea (5th century BC) asked: how can you ever finish infinitely many steps?
- Halvings done
- 0
- Distance left
- 1 m
- That's roughly
- —
- This step takes
- —
- Total time
- 0 s
- ½ + ¼ + ⅛ + … (n terms)
- 0
From 1 m to the Planck length: log2(1 m / ℓP) ≈ 115.6, i.e. 116 halvings.
The mathematical solution
Infinitely many steps can take a finite time, because their durations form a convergent geometric series:
12 + 14 + 18 + … = ½1 − ½ = 1A runner at speed v finishes after t = 1 m / v. Limits and series are explained in “Derivative and integral”.
What about the Planck length?
If space were grainy at a scale of about ℓP, halving would physically stop after about 116 steps. But the paradox is already solved by mathematics, without that assumption. Whether space is grainy remains an open question — we don't know.
Horizons for the observer
Computed on this page from the Friedmann equation: flat ΛCDM, H0 = 67.7 km/s/Mpc, Ωm = 0.31, ΩΛ ≈ 0.69, Ωr = 9·10−5.
Why 46 billion light-years if it's 13.8 billion years?
Light from the most distant regions traveled to us for about 13.8 billion years, but space kept stretching all the while. The place it left is today about 46 billion light-years away — that's the particle horizon, the radius of the observable Universe.
The Hubble sphere (c/H0) is the distance at which galaxies recede at the speed of light. We see galaxies beyond it — it is not a horizon.
Event horizon
Light emitted now by galaxies farther than about 16–17 billion light-years (today) will never reach us, because dark energy is accelerating the expansion. We still see their old light, though.
The CMB — the cosmic microwave background — was released about 380 thousand years after the Big Bang (z ≈ 1100) from places that are now about 45–46 billion light-years from us.
A smaller Universe in the future
The particle horizon keeps growing, but ever more slowly, toward a finite limit (in this model about 62.8 billion light-years). Yet what the sky contains shrinks: galaxies beyond the Local Group cross the event horizon, and their light is redshifted exponentially and fades away.
After about 100–200 billion years, an observer in Milkomeda (the merged Milky Way and Andromeda) will in practice see only their own galaxy, and the CMB will be stretched to undetectable wavelengths — “the return of the static Universe” (L. Krauss and R. Scherrer, 2007).
The collision with Andromeda is usually estimated at about 4–5 billion years from now, but a 2025 study (Sawala et al.) gives only about a 50% chance of a merger within 10 billion years. This page's model assumes it does happen.
The Universe's budget (Planck 2018)
The Hubble tension: measurements from the CMB give about 67 km/s/Mpc, while Cepheids and supernovae give about 73 km/s/Mpc — the cause of the difference is unknown. DESI (2024–2025) gives early hints that dark energy may change over time.
What does inflation solve?
The horizon problem: the CMB has the same temperature to within 1 part in 100,000, even though without inflation opposite regions of the sky could never have been “in touch”. Flatness: inflation stretches space flat. Monopoles: it dilutes hypothetical particles from early epochs.
Quantum fluctuations stretched to cosmic size became the seeds of galaxies. Next to this: a random field — schematic, without the acoustic peaks of the real CMB.
How certain is this?
Inflation is the leading model: among other things, it predicts a nearly scale-invariant spectrum of fluctuations, matching the measured ns ≈ 0.965. But it is not proven: primordial gravitational waves (B-mode polarization) have not been detected yet — the BICEP2 signal from 2014 turned out to be dust in our Galaxy. Alternative models exist too.
The times and the growth factor (≥ e60 ≈ 1026) depend on the version of the model.
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Challenges
0 / 7Did you know…?
If a proton were 1 cm across, a hydrogen atom would be about 600 m wide. Almost the entire atom is empty space.
The geometric mean of the Planck length and the diameter of the observable Universe is about 0.1 mm — roughly the thickness of a hair. On a logarithmic scale, humans sit not far from the middle of this ladder.
The most distant galaxies seen (JWST, e.g. MoM-z14) shone about 300 million years after the Big Bang, and their light has been stretched about 15 times.
How does it work?
Logarithmic scale
Each step of 1 on the slider means multiplying by 10. An object's position is the exponent of a power of ten:
n = log10d1d2If d1/d2 = k, then k smaller objects fit in a row, and by volume about k3 (for similar shapes, with no gaps between them).
Planck units
Built only from the constants ħ, G and c:
ℓP = √ ħG/c3 ≈ 1.616·10−35 mtP = √(ħG/c5) ≈ 5.39·10−44 s, mP = √(ħc/G) ≈ 2.18·10−8 kg, TP = mPc2/kB ≈ 1.42·1032 K.
It's the scale at which quantum-gravity effects become important and our current theories stop working. A heuristic argument: to probe such a tiny distance, you'd have to concentrate so much energy that a black hole would form. It has not been proven to be a “pixel of space” or the smallest possible length; only some hypotheses (e.g. loop quantum gravity) suggest a minimum length.
The Friedmann equation
H(a) = H0√ Ωra−4 + Ωma−3 + ΩΛThe scale factor a tells how many times space has stretched (today a = 1). The page numerically integrates t = ∫ da/(aH) and the conformal time η = ∫ c da/(a2H). The particle horizon is η(today), the event horizon is η(∞) − η(today). Acceleration begins when Ωma−3/2 = ΩΛ.
SI prefixes
| a | atto | 10−18 | k | kilo | 103 |
| f | femto | 10−15 | M | mega | 106 |
| p | pico | 10−12 | G | giga | 109 |
| n | nano | 10−9 | T | tera | 1012 |
| µ | micro | 10−6 | P | peta | 1015 |
| m | milli | 10−3 | E | eksa | 1018 |
1 light-year = c · 1 year ≈ 9.461·1015 m; 1 AU = 149,597,870,700 m.
Simplifications
Sizes are typical and rounded, and shapes are stylized. Objects stand side by side as in a display case, not in their real locations. The quark and electron have only an upper limit on their size. The heliosphere is not a sphere. Galaxies in the Local Group and Laniakea are placed schematically. The ΛCDM model leaves out massive neutrinos and evolving dark energy; the number of visible galaxies in the future is an illustrative model (the brightness threshold of a JWST-class telescope), not a precise forecast.