What this calculator is really estimating
In many discussions of eternal inflation, the false vacuum does not decay everywhere at once. Instead, small regions tunnel into a lower-energy state. Each successful tunneling event creates a bubble that expands into the surrounding inflating space. Inside that bubble, inflation may end or change character, opening the door to a hot Big Bang-like history. Because the background can continue inflating elsewhere, bubble formation can keep happening again and again.
Once that picture is on the table, a natural question follows: if many bubbles can form, how often should they hit one another? A complete answer depends on spacetime geometry, causal structure, the choice of measure, and the detailed field theory of the vacuum transition. This calculator does not try to solve that full problem. Instead, it offers a stripped-down counting model that lets you see how the answer scales when you vary the rate of nucleation, the speed of the bubble wall, and the length of time available for growth.
That makes the tool most useful for intuition. If you increase Γ, collisions should become more common. If your bubble expands faster, it sweeps out a larger region in which encounters can happen. If you wait longer, the relevant volume rises sharply because the radius itself grows with time and the volume grows with the cube of that radius. The calculator is built to make those relationships visible with as little mystery as possible.
How to use the calculator
Start by entering a bubble nucleation rate Γ in the calculator's simplified units of per cubic light-year per billion years. Next, enter the wall speed v as a fraction of the speed of light, so a value of 0.9 means the wall expands at 90% of c in this toy setup. Then enter the elapsed time T in billions of years. When you press the compute button, the page estimates the number of collisions expected in the swept region and the chance that at least one collision occurs.
If you want to explore rare events, try a very small nucleation rate such as 1e-10 or 1e-12. If you want to see the probability climb toward certainty, increase Γ, increase T, or do both together. Because the dependence on the bubble radius enters through a cubic volume, modest changes in v or T can matter more than they first appear to matter.
Assumptions, units, and what is being simplified
The model assumes that nucleation is homogeneous and random, so bubbles appear uniformly through the background with a constant effective rate Γ. It also assumes that our bubble expands approximately as a sphere and that its effective wall speed can be treated as constant over the interval you chose. That is a strong simplification, but it keeps the calculation easy to inspect.
The units here are deliberately pedagogical rather than fully relativistic. In more formal treatments, Γ is often expressed per unit four-volume in an inflating spacetime, often in Hubble-scale units, and careful attention is paid to the background metric. This page instead treats Γ as an effective rate per spatial volume per time. That means the outputs are best read as toy-model estimates for scaling intuition, not as direct observational forecasts.
Another way to say this is that the calculator answers a local counting question: if a bubble grows for time T with speed v, what volume has it effectively exposed to new nucleation events, and how many such events would you expect on average if the background rate were Γ? That is narrower than the full cosmological question, but it is also a very teachable starting point.
Formula
The bubble radius after time T is treated as r = vT. In this page's simplified unit system, that means the radius tracks linearly with both speed and time. Once the radius is known, the calculator estimates the swept volume using the ordinary volume of a sphere.
The expected number of other bubbles nucleating inside that volume is then written as the rate times the volume.
Finally, if collisions are treated as Poisson-distributed events with mean N, the probability of at least one collision is one minus the probability of zero collisions.
The results panel also shows a rough waiting-time proxy, T divided by N when N is positive. That number is included only as a quick piece of intuition. It is not a rigorous first-passage time in an expanding spacetime, and it should not be read as a precise collision clock.
How to interpret the outputs
The expected number of collisions is an average over many hypothetical realizations of the same setup. An expected value below 1 does not mean collisions are impossible; it means most realizations would have zero collisions while a smaller fraction would have one or more. Likewise, an expected value above 1 does not guarantee a collision in every realization, but it does push the probability upward quickly.
The probability output translates that mean into a more intuitive question: what is the chance that at least one collision occurs? For small N, the probability is approximately N itself, which is why rare-event probabilities closely track the expected count. As N grows larger than 1, the probability rises rapidly toward 100%, even though the expected number can keep increasing well beyond that point.
This distinction matters because two different parameter sets can produce the same qualitative story for different reasons. A moderate Γ with a very long time horizon may give a similar expected count to a larger Γ over a shorter interval. The calculator lets you explore those tradeoffs directly.
Worked example
Suppose you choose Γ = 1×10−6 per ly3 per Gyr, v = 0.9, and T = 5 Gyr. The radius is r = vT = 0.9 × 5 = 4.5 in the page's simplified distance units. The swept volume is then V = (4/3)π(4.53) ≈ 381.7. Multiplying by the nucleation rate gives N ≈ 3.82×10−4, and the probability of at least one collision is about 0.038%.
That result is tiny, which is exactly what you should expect from a low event rate applied to a modest volume. The worked example is useful because it shows the logic of the calculator without hiding the scaling. If you kept the same Γ and v but doubled T, the radius would double, the volume would increase by a factor of eight, and the expected collision count would do the same.
Illustrative scenarios
The table below holds v = 0.9 and T = 5 Gyr fixed while varying only Γ. Because N = ΓV in this toy model, the expected count increases linearly with the nucleation rate. The probability does not rise linearly forever, though; once N becomes a few or more, the probability is already close to saturation.
| Γ (per ly3 per Gyr) | Expected collisions N | Collision probability P |
|---|---|---|
| 1×10−8 | 3.82×10−6 | 0.00038% |
| 1×10−6 | 3.82×10−4 | 0.038% |
| 1×10−4 | 3.82×10−2 | 3.75% |
| 1×10−2 | 3.82 | 97.8% |
Choosing sensible toy inputs
If you are using the calculator for learning, it helps to think in terms of regimes rather than trying to guess the one true value of Γ. Extremely small rates represent highly suppressed tunneling, where collisions remain rare even over long intervals. Larger rates represent more aggressive vacuum decay, where encounters become common quickly. The value of v usually matters through the volume term, so raising the speed can have a strong effect even though it may feel like a smaller change at first glance.
Time T is also worth special attention. Because the model uses a spherical volume with radius proportional to T, the collision expectation grows like T3. That means a tenfold increase in time can produce a thousandfold increase in the swept volume. In a serious cosmological treatment you would want more careful geometry, but for intuition this cubic growth is exactly the sort of scaling you want to notice.
Limitations and interpretation
This page is intentionally simplified, and the simplifications matter. It ignores the full spacetime geometry of eternal inflation, where causal structure and available four-volume are central to the actual collision problem. It does not track bubbles that formed before our reference bubble. It treats the wall speed as a constant effective parameter, even though realistic bubble walls can accelerate, interact, and backreact on the surrounding spacetime.
The model also sidesteps the famous measure problem. In an eternally inflating spacetime, global probability statements can depend on the cutoff or measure you choose. None of that complexity is encoded here. Instead, the calculator uses a local, finite counting picture designed to reveal how outcomes scale with the basic knobs you control.
So the safest reading is this: use the outputs to build intuition about dependence on Γ, v, and T. Do not treat the numbers as precision forecasts for our observable universe. They are toy-model answers to a toy-model question, and that is exactly what makes them useful for fast exploration.
Additional context
Eternal inflation is often motivated by inflationary potentials with metastable regions. In those models, the decay rate can be exponentially suppressed by an instanton action, making collisions fantastically rare on timescales comparable to the age of our universe. On the other hand, if the effective decay rate is high enough, the inflating phase can become short-lived in any given region because bubbles form and merge readily.
The wall speed is usually close to relativistic in many discussions, but the practical meaning of speed depends on the coordinates and geometric quantities being used. This calculator deliberately compresses all of that into one transparent control. That is not a bug; it is the teaching strategy. You can turn one knob and watch the swept volume grow.
Finally, even if collisions are common somewhere in the larger multiverse picture, detectability is another question entirely. We only observe within our past light cone, and any signature would have to survive later cosmic evolution. The step from 'collisions can happen' to 'collisions could be measured' is much larger than the simple formulas above might suggest.
Mini-game: Collision Observatory
The calculator above does the actual math. This optional mini-game turns the same ideas into a quick observational challenge. Your current Γ input raises or lowers nucleation pressure, your v input controls how quickly your bubble grows, and your T input sets the observing window length. None of that changes the calculator result; it simply gives you a more physical feel for why collision opportunities pile up as the accessible region expands.
Why this mechanic fits the calculator: a larger effective volume means more opportunities for impacts. In the game, you feel that pressure as the round progresses. Higher Γ throws more nucleation events at you, higher v makes your own bubble expand faster, and larger T extends the interval over which those collisions can occur. That is not a substitute for the calculation, but it is a memorable way to build intuition for the same variables.