Ekpyrotic Cycle Duration Calculator

What an “ekpyrotic cycle” means in this toy model

This ekpyrotic-cycle calculator uses a speculative alternative to standard inflationary accounts as its starting point. In many ekpyrotic and cyclic proposals, the observable universe is associated with a three-dimensional “brane” embedded in a higher-dimensional space. A cycle is described as branes separating, then approaching, then colliding in a “bounce,” after which the sequence repeats. A central qualitative idea is that contraction with a very stiff equation of state, often written as w ≫ 1, can suppress anisotropy and curvature.

The inputs on this page turn that brane-cycle picture into transparent toy arithmetic rather than a physical simulation. You supply an interbrane separation D in micrometers, a relative speed v/c as a fraction of light speed, and an equation-of-state parameter w. The reported values are the estimated time to collision, full cycle period, heuristic effective smoothing e-fold count, associated smoothing factor, and a toy residual curvature parameter.

For this ekpyrotic brane-cycle model, the useful lesson is how the chosen scales propagate through the outputs. Time varies directly with D and inversely with v. The smoothing proxy instead uses (1 + w) times a logarithm of a length ratio. Those intentionally simple relationships make it straightforward to compare parameter choices without treating the result as a prediction of a complete cosmological theory.

How to use the ekpyrotic cycle calculator

1. For the ekpyrotic brane-cycle estimate, enter Interbrane separation D in micrometers (μm). One micrometer is 10⁻⁶ m.
2. Enter Relative speed v/c as a number strictly between 0 and 1. For example, 0.1 represents 10% of the speed of light.
3. Enter w, the equation-of-state parameter. This toy calculation accepts w ≥ 1; discussions of ekpyrotic contraction often use much larger values.
4. Select Compute Cycle to calculate the brane-collision and smoothing outputs. Copy Summary copies the displayed metrics as plain text.

Ekpyrotic-cycle results use scientific notation where the collision times or curvature proxy are extremely small or large. Invalid inputs, such as v/c ≥ 1, trigger a validation message and leave copying unavailable. After calculation, keyboard focus moves to the results panel so the updated cycle table can be read immediately.

Ekpyrotic-cycle formulas, units, and assumptions

The formulas behind this ekpyrotic brane-cycle calculator assume a deliberately simple bulk-frame picture: two parallel branes begin a distance D apart and move together at relative speed v. The page uses c = 299,792,458 m/s for light speed and ℓ_P = 1.616 × 10⁻³⁵ m for the Planck-length reference scale.

1) Ekpyrotic brane collision time and cycle period

For the brane-separation input, first convert micrometers to meters: D_m = D × 10⁻⁶. The model then calculates collision time as:

Formula: t_coll = D_m / (v c)

The toy ekpyrotic cycle treats a full period as approach plus rebound:

Formula: T_cycle = 2 t_coll

Thus, doubling the starting brane separation doubles these time scales, while reducing v/c by a factor of 10 makes them ten times longer. This is only a constant-speed kinematic estimate; it omits gravitational effects, warping, and changing brane velocity.

2) Ekpyrotic effective smoothing e-folds (heuristic)

To express the toy model’s smoothing behavior, the calculator defines an effective e-fold count as:

Formula: N_eff = 1.5 × (1 + w) × ln (D_m / ℓ_P)

This ekpyrotic smoothing expression is not a derived observational prediction. It is a compact proxy that makes larger separations and larger w yield stronger suppression. The logarithm compares the input length with a microscopic reference length, while the factor 1.5 × (1 + w) makes the chosen equation of state explicit.

In this calculator, ln(D_m/ℓ_P) is positive for ordinary input scales. Raising D therefore raises N_eff gradually because the dependence is logarithmic, whereas raising w raises it linearly. This distinction explains why changes to w can dominate the displayed smoothing proxy.

3) Ekpyrotic smoothing factor and residual-curvature proxy

The ekpyrotic-cycle output defines smoothing factor S as S = exp(N_eff) and its residual-curvature proxy as Ω_k = exp(−2 N_eff). Within this deliberately simplified convention, a smaller Ω_k represents a flatter and more homogeneous post-bounce result.

Because the smoothing exponential can exceed ordinary floating-point range, the ekpyrotic results panel uses a compact exponent display when N_eff is very large. An entry such as e^712.3 is a display convention that avoids showing Infinity for the smoothing factor.

Ekpyrotic brane-cycle worked example

Consider the page’s first ekpyrotic brane-cycle scenario: D = 1 μm, v/c = 0.1, and w = 100. The calculation converts D to 10⁻⁶ m, obtains t_coll = D_m / (v c), and doubles that value for T_cycle. It then uses the logarithm of D_m/ℓ_P to form N_eff, from which S and Ω_k follow by exponentiation.

Holding the separation fixed while changing v/c from 0.1 to 0.01 makes the collision time ten times longer. Holding speed fixed while changing D from 1 μm to 10 μm also makes collision time ten times longer. By contrast, a tenfold increase in D changes N_eff only through the logarithm, while changing w has a much stronger linear effect on that proxy.

Large w values can make the displayed ekpyrotic smoothing factor enormous and the residual-curvature proxy extremely small. That is the intended behavior of these exponential toy definitions, not evidence that a physical cyclic cosmology must produce those values.

Ekpyrotic brane-cycle sample scenarios table

This ekpyrotic-cycle table is filled with the same JavaScript calculation used by the main form. Its three brane-separation, speed, and w combinations make the contrast between kinematic timing and heuristic smoothing visible. Scientific notation is used because several outputs span many orders of magnitude.

Ekpyrotic-cycle limitations and interpretation

• Not a full ekpyrotic model: Real ekpyrotic and cyclic scenarios involve scalar-field dynamics, bounce matching conditions, entropy production, and perturbation constraints. This brane-cycle calculator models none of those ingredients.
• Heuristic smoothing: N_eff, S, and Ω_k are intuition-building proxies in this page’s toy setup, not observational predictions.
• Parameter ranges: The form restricts v/c to (0, 1) and requires w ≥ 1. Very large w can drive the smoothing exponential outside typical floating-point range.
• Units matter: D is entered in micrometers and converted internally to meters. Reading that input as any other distance unit changes the calculated brane-collision time by many orders of magnitude.
• Interpretation tip: To compare toy ekpyrotic scenarios, vary one input at a time and distinguish the linear timing response from the exponential smoothing outputs.

Despite those restrictions, this ekpyrotic-cycle calculator can clarify how its chosen exponential measures react to w and to the logarithm of a brane-separation scale. When comparing cases, concentrate on relative changes produced by one controlled input adjustment rather than on the absolute values as physical forecasts.

Ekpyrotic cycle calculator FAQ

Does this ekpyrotic-cycle calculator prove or disprove ekpyrotic cosmology?

No. This page numerically illustrates a small set of explicitly simplified brane-cycle relationships. Assessing whether an ekpyrotic or cyclic scenario fits observations requires detailed model building and comparison with data, including primordial-perturbation predictions.

Why does the ekpyrotic smoothing factor become so large?

The calculator defines its smoothing factor as the exponential of N_eff. Exponentials grow rapidly, so even moderate increases in this heuristic e-fold count can produce very large values. That behavior is intentional within the page’s proxy and reflects the general idea that e-fold-like suppression can be powerful.

What should I do if the ekpyrotic output shows “Infinity” or a very large exponent?

A sufficiently large w can push exp(Neff) beyond JavaScript’s finite numeric range. The results panel normally avoids that for the smoothing factor by displaying e^N at large N. To keep that output finite, reduce w or reduce D.

Is v/c a realistic parameter for ekpyrotic brane motion?

Here, v/c is only a convenient constant-speed control for the brane-collision time. If brane dynamics exist in a realistic model, they would arise from a higher-dimensional theory and could involve changing velocities. The fixed-v/c assumption is used solely to keep this toy calculation legible.

Ekpyrotic-cycle glossary (quick reference)

Brane

A membrane-like object in higher-dimensional theories; some scenarios associate our universe with a brane.

Equation-of-state parameter (w)

The ratio of pressure to energy density, w = p/ρ. Ekpyrotic discussions use large w to describe a very stiff component.

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 (ℓ_P)

A fundamental length scale (~1.616 × 10⁻³⁵ m) used here only as a reference in a logarithm.

Ω_k

A curvature density parameter in cosmology. This calculator uses it as a toy proxy defined by exp(−2N_eff).

To investigate the ekpyrotic-cycle relationships, keep D fixed while varying v/c to isolate the kinematic scaling; keep v/c fixed while varying w to see the smoothing proxy respond; then raise D by factors of 10 to observe additive logarithmic growth in N_eff. Those contrasts are the page’s main educational purpose.