Eternal Inflation Bubble Collision Calculator

Understanding Bubble Collisions in Eternal Inflation

Eternal inflation proposes that our observable universe is merely one pocket region among an unfathomable multiverse of "bubble" universes. Each bubble forms when a metastable inflating state decays, releasing its vacuum energy and allowing conventional Big Bang cosmology to unfold inside. Because inflation never completely ends, these bubbles continue to nucleate randomly throughout a rapidly expanding false vacuum background. The probability that any given bubble will encounter another depends on how quickly bubbles appear and how large a domain each bubble sweeps out over time. This calculator models that situation with a handful of idealized parameters.

The starting point is the bubble nucleation rate Γ, interpreted as the expected number of new bubbles per unit four-volume of the inflating background. Specialists often quote this rate in terms of events per cubic Hubble length per Hubble time; here we rescale it into more approachable units of cubic light-years per billion years. Because the physics of tunneling through a potential barrier determines Γ, different theories of high-energy physics predict wildly different values. Some models anticipate a wildly suppressed rate on the order of $e^\{-10^\{5\}\}$, while others allow rates many orders of magnitude larger. The rate you select strongly influences how soon collisions become likely.

Once a bubble forms, its wall races outward at nearly the speed of light as the new vacuum interior expands. We approximate that expansion with an effective wall velocity parameter v, expressed as a fraction of light speed. In realistic treatments one might consider the bubble wall acceleration, interactions with ambient radiation, or anisotropic growth. However, a simple constant v captures the intuitive picture: a faster wall sweeps through more space and thus samples a larger portion of the inflating background. Setting v close to one reflects the expectation that bubble walls rapidly become ultra-relativistic.

The elapsed time T represents how long the bubble has had to expand since its nucleation. A newly formed bubble occupies a modest volume and is unlikely to encounter others right away. But as T grows, the volume explored grows as the cube of radius. By measuring time in billions of years we keep the inputs accessible, although nothing prevents you from using fractions for short intervals or large numbers for extremely old bubbles.

Under the assumptions that bubbles nucleate homogeneously in space and that our bubble expands as a perfect sphere, the volume swept out over time is given by the standard formula for a sphere:

Here the radius r is simply vT in units where the speed of light is one. Because volume scales as r³, doubling the expansion time increases potential collision volume eightfold. Multiplying this volume by the nucleation rate provides the expected number of foreign bubbles that will appear in the region our bubble inhabits. We label this expectation value N:

Although N may be less than one for modest times or low nucleation rates, the distribution of collisions follows the Poisson law. The probability of witnessing at least one collision is therefore $P=1-e^{-<mi>N</mi>}$. In the limit of rare events the chance is approximately equal to N, but at large N it saturates toward certainty. The calculator presents both values so you can explore how different regimes behave.

A subtle issue arises when considering collisions from bubbles nucleated in the external false vacuum before our bubble formed. Those preexisting neighbors may already be large enough to intersect our region almost immediately, complicating naive estimates. The simplified model here assumes that Γ is low enough that such earlier bubbles are sparse or that the creation of our bubble effectively resets the region. Realistic treatments that account for global spacetime geometry use more elaborate light-cone calculations. Nevertheless, the spherical volume approach conveys the essential scaling and serves as a rough guide.

Why study bubble collisions at all? One motivation is the possibility of detectable imprints. If our universe experienced a collision in its early history, the interaction could leave subtle anisotropies in the cosmic microwave background or generate characteristic gravitational waves. These signatures provide a rare observational window into physics beyond our horizon. By playing with the nucleation rate and expansion speed here, one can get intuition for how plausible such collisions might be in a given theoretical framework.

The table below shows example scenarios. Each row varies the nucleation rate while holding the expansion speed at v = 0.9 and the time at five billion years. The expected number of collisions N rises linearly with the rate, while the probability approaches one when N exceeds several.

Even for extraordinarily high nucleation rates the probability of encountering another bubble within a few billion years can remain surprisingly small. This insight underscores why many cosmologists still consider our sky remarkably smooth: unless the vacuum decay rate is near catastrophic levels, collisions are a rare spectacle. Yet over immense timescales, or with accelerated expansion speeds, collisions become essentially inevitable.

Another point of contemplation is the aftermath of a collision. Depending on the energy difference between vacua and relative wall velocities, collisions may produce domain walls, trigger reheating events, or simply pass through each other with little consequence. Some analyses suggest that collisions could truncate inflation within the impacted region, potentially spawning anisotropic universes. Others posit that only gently colliding bubbles preserve a hospitable interior like our own. The model in this calculator stops short of those complexities but provides a baseline for how often such events might transpire.

One might also wonder about the measure problem in eternal inflation. If the inflating background grows faster than bubbles can eat into it, the total physical volume of false vacuum may diverge, making probabilities ill-defined. The local approach used here sidesteps global ambiguities by focusing on a finite region around a single bubble. Nevertheless, the broader question of assigning probabilities in an eternally inflating spacetime remains unsolved and continues to inspire heated debate. By adjusting the inputs you can simulate how different cutoff prescriptions might influence encounter rates.

Exploring these scenarios highlights the profound difference between expectation and observation. While the calculator might show a high expected number of collisions over trillions of years, our observational window is minuscule. Detecting evidence for even a single collision would revolutionize cosmology, yet the odds of such evidence falling within our past light cone may be slim. Therefore the calculator also functions as a humbling reminder of our limited vantage in the vast multiverse.

Finally, the model offers an intriguing pedagogical exercise. Students can manipulate the parameters to grasp how cubic volume scaling and exponential probability combine. Researchers can use it as a sanity check for back-of-the-envelope estimates when exploring novel models of vacuum decay. Science fiction authors might even employ it to gauge how often neighboring universes crash together in their multiversal epics. However you use it, keep in mind that the numbers generated are not definitive predictions but rather guides through a speculative landscape where our physical intuitions strain to keep pace with mathematical possibilities.

As with all calculators in this speculative series, the assumptions are deliberately simple to keep computations transparent. Real multiverse dynamics involve general relativity, quantum field theory, and the thorny issue of initial conditions. Nevertheless, even simplified tools can foster intuition. By experimenting with extreme values—such as bubble walls creeping along at a mere fraction of light speed or nucleation rates set so high that collisions become quotidian—you can probe the boundaries of conceptual plausibility. Enjoy the exploration and ponder how many unseen bubbles might be racing toward ours even now.