What an “ekpyrotic cycle” means in this toy model
The ekpyrotic (and related cyclic) cosmology is a speculative alternative to standard inflationary stories.
In many versions, our observable universe is associated with a three-dimensional “brane” embedded in a higher-dimensional space.
A cycle is driven by the branes separating, then approaching, then colliding (a “bounce”), after which the process repeats.
The key qualitative claim is that a phase with a very stiff equation of state (often written as w ≫ 1) can strongly suppress anisotropy and curvature during contraction.
This page provides a calculator-style sandbox for exploring that idea with simple, transparent arithmetic.
You enter an interbrane separation D (in micrometers), a relative speed v/c (as a fraction of the speed of light), and an equation-of-state parameter w.
The calculator then estimates the time to collision, the full cycle period, a heuristic effective smoothing e-fold count, a corresponding smoothing factor, and a toy residual curvature parameter.
The emphasis is on units and scaling: how does the output change if you increase the separation by a factor of 10, or if you reduce the speed by a factor of 10?
In this toy setup, the time scales depend linearly on D and inversely on v, while the smoothing proxy depends on (1 + w) times a logarithm of a length ratio.
Those relationships are easy to compute and easy to sanity-check.
Important: This calculator is not a research-grade cosmology solver.
It does not simulate perturbations, entropy production, or the microphysics of a bounce.
Treat the outputs as illustrative numbers that help you reason about orders of magnitude.
How to use the calculator
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Enter Interbrane separation D in micrometers (μm). For reference: 1 μm = 10−6 m.
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Enter Relative speed v/c as a number between 0 and 1 (exclusive). Example: 0.1 means 10% of the speed of light.
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Enter w (equation-of-state parameter). This toy model expects w ≥ 1; ekpyrotic discussions often consider much larger values.
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Select Compute Cycle to update the results panel. Use Copy Summary to copy a plain-text summary of the computed metrics.
The results are shown in scientific notation when values are extremely small or large.
If you enter invalid values (for example, v/c ≥ 1), the calculator will display a validation message and disable copying.
For keyboard users, after you compute, focus moves to the results panel so you can read the updated table immediately.
The calculator uses a deliberately simplified “bulk frame” picture:
two parallel branes start separated by distance D and approach each other at relative speed v.
The speed of light is treated as constant c = 299,792,458 m/s.
The Planck length is treated as constant ℓP = 1.616 × 10−35 m.
1) Collision time and cycle period
Convert micrometers to meters: Dm = D × 10−6.
Then the time to collision is:
Formula: t_coll = D_m / (v c)
A full cycle is modeled as “approach + rebound,” so:
Formula: T_cycle = 2 t_coll
In other words, if you double the separation you double the time, and if you reduce the speed by a factor of 10 you increase the time by a factor of 10.
This is the simplest possible kinematic estimate and ignores gravitational backreaction, warping, and any time dependence of the speed.
2) Effective smoothing e-folds (heuristic)
To represent “how much smoothing” occurs, the calculator uses a toy effective e-fold count:
Formula: N_eff = 1.5 × (1 + w) × ln (D_m / ℓ_P)
This is not a derived prediction; it is a compact way to encode the intuition that larger separations and larger w can correspond to stronger suppression of unwanted components.
The logarithm appears because many “how many e-folds?” estimates compare a macroscopic length to a microscopic reference scale.
The factor 1.5 × (1 + w) is a convenient knob that makes the dependence on w explicit.
Practical note: because ln(Dm/ℓP) is positive for any everyday length scale, increasing D increases Neff slowly (logarithmically), while increasing w increases Neff quickly (linearly).
That difference in scaling is why the smoothing proxy can change dramatically when you adjust w.
3) Smoothing factor and residual curvature proxy
The calculator reports a smoothing factor S as:
S = exp(Neff).
It also reports a toy residual curvature parameter:
Ωk = exp(−2 Neff).
In this simplified proxy, smaller Ωk indicates a flatter, more homogeneous post-bounce universe.
Because exponentials can overflow floating-point numbers, the results panel switches to a compact display when Neff is very large.
If you see output like e^712.3, that is a formatting choice to avoid printing Infinity.
Worked example (with realistic expectations about scale)
Suppose you choose D = 1 μm, v/c = 0.1, and w = 100.
The calculator first converts D to meters (10−6 m) and computes the collision time:
tcoll = Dm / (v c).
It then doubles that to get the full period Tcycle.
Finally, it computes Neff from the logarithm of Dm/ℓP, and uses exponentials to produce S and Ωk.
If you keep D fixed and reduce the speed from 0.1 to 0.01, the collision time increases by a factor of 10.
If you keep v fixed and increase D from 1 μm to 10 μm, the collision time also increases by a factor of 10.
In contrast, Neff changes only modestly when you change D by a factor of 10 (because of the logarithm), but it changes strongly when you change w.
Because exponentials grow (and shrink) extremely fast, you should expect S to become enormous and Ωk to become tiny for large w.
That behavior is a feature of the toy proxy, not a guarantee about any real cosmological model.
Sample scenarios table
The table below is auto-filled using the same JavaScript functions as the main calculator.
It provides three scenarios to compare how changing v/c, D, and w affects the time scales and smoothing proxies.
Scientific notation is used because the values can span many orders of magnitude.
| D (μm) |
v/c |
w |
tcoll (s) |
Tcycle (s) |
Neff |
Ωk |
| 1 |
0.1 |
100 |
|
|
|
|
| 1 |
0.01 |
100 |
|
|
|
|
| 10 |
0.05 |
50 |
|
|
|
|
Limitations and interpretation (important)
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Not a full ekpyrotic model: Real ekpyrotic/cyclic scenarios involve scalar-field dynamics, matching conditions through a bounce, entropy production, and constraints from perturbations.
This calculator does not attempt to model those.
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Heuristic smoothing: Neff, S, and Ωk here are proxies designed for intuition.
They should not be treated as observational predictions.
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Parameter ranges: v/c is restricted to (0, 1) and w ≥ 1 for numerical sanity.
Extremely large w will produce exponentials that overflow typical floating-point ranges.
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Units matter: D is entered in micrometers; internally it is converted to meters.
If you interpret D as another unit, the results will be wrong by many orders of magnitude.
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Interpretation tip: If you want to compare scenarios, change one input at a time and watch which outputs respond linearly (time scales) versus exponentially (smoothing proxies).
Even with these limitations, the calculator is useful for building intuition about how quickly exponential measures can change with w and with the logarithm of a length scale.
If you are comparing scenarios, focus on relative changes when you adjust one input at a time.
FAQ
Does this calculator prove or disprove ekpyrotic cosmology?
No. It is a numerical illustration of a few simplified relationships.
Whether any ekpyrotic or cyclic scenario matches observations depends on detailed model building and comparison to data (for example, the spectrum of primordial perturbations).
Introduction: Why does the smoothing factor become so large?
The smoothing factor is defined as an exponential of Neff.
Exponentials grow rapidly, so even moderate increases in Neff can produce huge values.
This is intentional: it mirrors the common cosmology intuition that a sufficient number of e-folds can strongly suppress unwanted components.
What should I do if I see “Infinity” or a very large exponent?
Very large w can push exp(Neff) beyond what JavaScript can represent as a finite number.
The results panel tries to avoid that by displaying e^N when N is extremely large.
If you want finite numeric output, reduce w or reduce D.
Is v/c a realistic parameter for brane motion?
In this toy model, v/c is simply a convenient way to set a time scale.
Realistic brane dynamics (if they exist) would be governed by a higher-dimensional theory and could involve time-dependent velocities.
Here, v/c is treated as constant to keep the calculator understandable.
Glossary (quick reference)
- Brane
- A membrane-like object in higher-dimensional theories; in some scenarios our universe is associated with a brane.
- Equation-of-state parameter (w)
- The ratio of pressure to energy density, w = p/ρ. “Stiff” matter corresponds to large w in ekpyrotic discussions.
- E-fold
- A logarithmic measure of change in a scale factor or related quantity. One e-fold corresponds to a factor of e.
- Planck length (ℓP)
- A fundamental length scale (~1.616 × 10−35 m) used here only as a reference in a logarithm.
- Ωk
- A curvature density parameter in cosmology. Here it is used as a toy proxy defined by exp(−2Neff).
If you want to explore further, try these quick experiments: (1) keep D fixed and vary v/c to see purely kinematic scaling;
(2) keep v/c fixed and vary w to see how strongly the smoothing proxy responds;
(3) increase D by factors of 10 and observe that Neff changes by an additive amount (logarithmic growth), not a multiplicative one.
These patterns are the main educational value of the page.
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