MAC Big Bang Theory Matter-Antimatter Collision Big Bang Theory A three-dimensional mathematical model of how a collision between matter and antimatter could have ignited the universe ignition Concept: the author's original idea Mathematical formalization, simulations and write-up: developed with Claude (Anthropic) Version 1.0 · October 17, 2026 Status of this document This is a toy model . Every equation is derived or checked, and every simulation result is computed, but the model is a proposal under test, not established physics. Chapter 11 lists where it is known to be in trouble, and that list is part of the result. MAC Big Bang Theory 2 Abstract The MAC Big Bang Theory proposes that the universe began when matter and antimatter, present in the quantum vacuum in roughly equal amounts, collided at a single point. The released energy ignited neighbouring fuel, and the resulting chain reaction spread outward like a flame through a cloud of gas. Almost all the antimatter was consumed, a small remainder of matter survived, and as the fireball cooled that remainder formed the visible universe. This document turns that picture into a quantitative three-dimensional model with five parts: (1) a random initial draw of matter and antimatter in a patch of vacuum, (2) annihilation kinetics with an exact analytic solution, (3) an ignition front that spreads through the patch in 3D, (4) energy release and cooling, and (5) the leftover matter that is compared with the observed baryon-to-photon ratio η ≈ 6.1 × 10 -10 Main results • Collisions alone cannot leave matter. Annihilation is one-for-one, so the surviving matter fraction equals the initial imbalance δ = (N m - N a )/(N m + N a ), up to the photon-yield factor κ. • A natural source for δ. If each particle is independently matter or antimatter with probability ½, a patch of N particles has a typical imbalance 1/√N. Reproducing η requires N ≈ 1/(κη) 2 , about 2.7 × 10 18 for κ = 1. • Exact kinetics. The annihilation equations are solved in closed form. Antimatter falls as a power law (N a ∝ 1/τ) and then exponentially once Δτ exceeds 1. • A 3D front. The burn spreads as a Fisher–KPP front with speed 2√(Dr). In a diffusion estimate the speed is about 0.8 c. In a 3D lattice simulation the front forms and moves at 86% of that speed over the length of the run. • The main difficulty. One random 50/50 draw over the entire observable universe would give an imbalance of only about 2.6 × 10 -45 , some 35 orders of magnitude below η. The theory therefore needs a patch structure, a coherent sign, or an extra mechanism. The model is internally consistent and makes definite, checkable statements. It does not yet explain why the universe is matter-dominated, because the size and coherence of the ignition patch must be supplied from outside. Chapter 12 sets out the steps that would settle the question. MAC Big Bang Theory 3 Contents . 2 Abstract . 2 Main results . 6 Notation and symbols . 7 1. Introduction . 7 1.1 The puzzle . 7 1.2 The MAC idea in plain words . 7 1.3 What this document does and does not do . 7 1.4 Structure of the model . 9 2. Observational background . 9 2.1 The numbers to be explained . 9 2.2 Expansion and cooling . 10 2.3 How annihilation works . 10 2.4 Sakharov's conditions and where MAC sits . 11 3. Postulates of the MAC theory . 11 3.1 What each postulate rules in and out . 12 3.2 The central logical chain . 13 4. The three-dimensional model . 13 4.1 Geometry . 13 4.2 Fields . 14 4.3 The initial condition: a random draw . 14 4.4 The rate equations . 15 4.5 Dimensionless form . 16 5. Statistical origin of the imbalance . 16 5.1 The distribution of the imbalance . 17 5.2 From imbalance to the observed ratio . 18 5.3 How likely is a big enough imbalance? . 19 5.4 The whole observable universe is too big . 19 5.5 Independent patches and domains . 21 6. Annihilation kinetics . 21 6.1 Reduction to one equation MAC Big Bang Theory 4 . 21 6.2 The exact solution . 22 6.3 Two regimes . 22 6.4 How much annihilation is enough? . 24 7. Ignition and the spreading front . 24 7.1 Why the burn spreads . 24 7.2 The Fisher–KPP equation . 25 7.3 The three-dimensional lattice simulation . 28 7.4 An ignition threshold . 30 8. Energy release and cooling . 30 8.1 The burn temperature . 30 8.2 The photon yield κ . 30 8.3 Energy release in a uniform patch . 31 8.4 After the burn . 32 9. The combined model and its results . 32 9.1 The model in five steps . 32 9.2 Quantitative results . 32 9.3 What the model predicts . 33 9.4 What the model does not determine . 34 10. Predictions and tests . 34 10.1 Predictions that can be checked . 34 10.2 An honest limitation . 36 11. Challenges and objections . 36 11.1 The coherence problem . 36 11.2 The energy budget . 36 11.3 Net baryon number and the vacuum . 36 11.4 The fuel: real or virtual? . 37 11.5 Other limitations . 37 11.6 What would count as success . 38 12. Conclusions and roadmap . 38 12.1 What has been established . 38 12.2 What has not . 38 12.3 Roadmap MAC Big Bang Theory 5 . 39 Appendix A. Derivations . 39 A.1 Mean absolute imbalance . 39 A.2 Solving the annihilation equation . 39 A.3 The comoving form of the rate equations . 39 A.4 Front speed on a lattice . 39 A.5 Kinetic diffusion constant . 41 Appendix B. Numerical methods and source code . 41 B.1 Methods . 41 B.2 Monte Carlo test of the 1/√N law . 41 B.3 Exact annihilation solution and ODE check . 42 B.4 Three-dimensional front simulation . 43 Appendix C. Glossary . 44 References MAC Big Bang Theory 6 Notation and symbols Table 1. Symbols used throughout. Natural units (ħ = c = k B = 1) are used in formulas unless stated. Symbol Meaning Unit / value n m , n a number density of matter, antimatter cm -3 N m , N a comoving numbers (n a 3 ), per comoving volume dimensionless N = N m0 + N a0 total number of particles in the initial patch dimensionless Δ = N m - N a net matter number, conserved by annihilation dimensionless δ = Δ/N fractional initial imbalance dimensionless η observed baryon-to-photon ratio 6.1 × 10 -10 κ photons produced per initial particle model parameter ⟨σv⟩ thermally averaged annihilation rate ≈ 6 × 10 -16 cm 3 /s τ annihilation clock, ∫⟨σv⟩ a -3 dt cm 3 s -1 (scaled) u(x,t) burn fraction: 0 = fuel, 1 = burnt dimensionless D effective diffusion constant of the burn cm 2 /s r local burn growth rate s -1 v front speed, 2√(Dr) cm/s a(t), H scale factor and Hubble rate H = ȧ/a T, g * temperature, effective relativistic degrees of freedom MeV, 10.75 (used) m nucleon mass 0.938 GeV ε, E energy density, comoving energy GeV cm -3 R c critical ignition radius cm Throughout, “matter” means baryons and “antimatter” means antibaryons. Leptons are ignored, which is adequate for a model whose only output is the net baryon number. Numerical values come from standard cosmology references (Planck 2018 results and Kolb & Turner). MAC Big Bang Theory 7 1. Introduction 1.1 The puzzle Every particle of ordinary matter has an antimatter twin with the same mass and opposite charges. When a particle meets its twin, both vanish into energy. The laws of physics treat the two almost symmetrically, so the early universe should have made them in equal numbers. If it had, they would have annihilated completely and left a universe of light and nothing else. We live in a universe that is almost entirely matter. Measured against the photons of the cosmic microwave background, there is one baryon for every 1.6 billion photons (η ≈ 6.1 × 10 -10 ). This small leftover is everything: stars, planets and people. The question this document addresses is how a collision between matter and antimatter could explain both the Big Bang and that leftover. 1.2 The MAC idea in plain words • Before the Big Bang, the quantum vacuum was never empty. It held matter and antimatter in about equal amounts, spread thinly, like a gas. • By a one-in-an-enormous-number chance, a particle and antiparticle collided and annihilated. • The energy released was enough to make neighbouring particles collide, and those collisions released more energy. Like a spark in a cloud of flammable gas, the reaction spread. • The antimatter was used up. A small excess of matter was left because the two species were never perfectly equal. • The fireball cooled, and the leftover matter became the universe. 1.3 What this document does and does not do The aim is to find out what the idea implies when it is written as mathematics. Three questions are asked in order. Does the collision picture work at all, given that annihilation is one-for-one? How would a burn spread through three-dimensional space, and how fast? And what must be true of the initial patch for the leftover to equal η? Honest scope The model computes consequences of the MAC assumptions. It does not prove them. It finds one strong result, that the leftover is fixed by the initial imbalance, and one strong difficulty, that a random initial draw gives an imbalance far too small if it covers the whole observable universe. Both are developed in detail. 1.4 Structure of the model MAC Big Bang Theory 8 The model is a pipeline of five stages. Chapters 4 to 8 develop one stage each, and Chapter 9 combines them. Figure 1 gives the overview. Figure 1. The five stages of the MAC model. The output of the pipeline is the leftover matter fraction, which is compared with η. Chapter 2 collects the standard cosmology that the model relies on. Chapter 3 states the postulates. Chapters 10 and 11 list predictions, tests and objections. The appendices give derivations, the numerical methods, the source code and a glossary. MAC Big Bang Theory 9 2. Observational background This chapter lists the measured quantities and standard results the MAC model uses. None of it is new. 2.1 The numbers to be explained Table 2. Observed quantities used by the model (rounded). Quantity Value Comment Baryon-to-photon ratio η 6.1 × 10 -10 Planck 2018 (CMB and BBN agree to within errors) CMB photon density n γ ≈ 411 cm -3 T = 2.725 K Observable universe radius ≈ 4.4 × 10 28 cm about 46.5 billion light-years Observable universe volume ≈ 3.6 × 10 86 cm 3 4πR 3 /3 Photons in observable universe ≈ 1.5 × 10 89 n γ times volume Baryons in observable universe ≈ 10 80 η times photon number The baryon-to-photon ratio is defined by (1) where n B is the baryon density, n B̄ the antibaryon density (essentially zero today) and n γ the photon density. Because the ratio is measured late, after annihilation is finished, it is the number the MAC model must reproduce. 2.2 Expansion and cooling In a radiation-dominated universe the expansion rate is set by the Friedmann equation, (2) with solution a(t) ∝ t 1/2 and T ∝ 1/a. The time–temperature relation, for g * = 10.75, is (3) Figure 2 plots this relation. In MAC the same expansion law applies to the burnt region once it has thermalized. The value g * = 10.75 is correct below about 1 MeV for photons, electrons, positrons and three neutrino species. At higher temperatures g * is larger, which slightly changes the numbers but not the scaling. MAC Big Bang Theory 10 Figure 2. Radiation-era cooling, T versus t, for g * = 10.75. 2.3 How annihilation works A nucleon and an antinucleon annihilate mostly into pions, which decay into photons, electrons and neutrinos. The cross-section at low energy is set by the pion size, so the thermally averaged rate is of order (4) (Kolb & Turner). With that rate, the density of antimatter at which a typical antiparticle finds a partner in one Hubble time is easily reached in the early universe, so annihilation is efficient whenever matter and antimatter coexist at high density. 2.4 Sakharov's conditions and where MAC sits In 1967 Sakharov listed three conditions for a universe that starts symmetric to end up with more matter than antimatter: baryon-number violation, C and CP violation, and departure from thermal equilibrium. Standard baryogenesis models try to supply all three. The MAC model works differently. It does not generate the asymmetry by dynamics. It inherits an initial imbalance from the random draw and tests whether collisions can carry that imbalance through to the present. The departure from equilibrium is supplied by the explosive burn itself. Where the model stands relative to standard theory MAC is a carrier model, not a generator. Annihilation takes whatever imbalance exists at the start and delivers it, almost exactly, as the final baryon excess. Everything depends on the starting imbalance. MAC Big Bang Theory 11 3. Postulates of the MAC theory The model rests on seven postulates. Each is stated first in words and then in the form used in the equations. P1. A fuel of matter and antimatter. Before ignition, a region of the quantum vacuum contains real matter and antimatter particles at low density. The region is called the patch and contains N particles in total. P2. Equal odds for each particle. Each particle is independently matter or antimatter with probability ½. No species is favoured on average. P3. A rare ignition event. A collision between one matter and one antimatter particle occurs at a point x 0 . Its probability is small but not zero. The point is the ignition seed. P4. Collisions release energy. Each annihilation converts the rest mass of the pair, 2mc 2 , into radiation that thermalizes locally. P5. The burn spreads in three dimensions. Heated fuel ignites neighbouring fuel. The burn fraction u(x,t) obeys a reaction–diffusion law with growth rate r and diffusion constant D. P6. Expansion and cooling. The burnt region expands as a radiation-dominated universe, a ∝ t 1/2 , and cools as T ∝ 1/a. P7. The leftover is the observed matter. After annihilation is complete, the surplus of matter over antimatter is the baryon content of the universe. 3.1 What each postulate rules in and out Table 3. Origin and status of each postulate. Postulate Taken from Status P1 quantum vacuum picture assumed; origin of real particles is open (Ch. 11) P2 symmetry between matter and antimatter assumed; testable through its 1/√N consequence (Ch. 5) P3 rare-event statistics assumed; no probability is computed P4 standard annihilation physics established P5 reaction–diffusion theory (Fisher, KPP) standard mathematics, applied by analogy MAC Big Bang Theory 12 Postulate Taken from Status P6 standard radiation-era cosmology established P7 baryon number conservation after annihilation established 3.2 The central logical chain Postulates P2, P4 and P7 combine into a short argument that drives the rest of the document. • Annihilation removes one matter particle for every antimatter particle (P4). • Therefore the net matter number Δ = N m - N a does not change during the burn. • If the burn is complete, all antimatter is gone and N m = Δ is what remains (P7). • Under P2, Δ is a random number with typical size √N, so the leftover fraction is about 1/√N. This chain is exact within the model. It turns the question “how did matter survive?” into the question “how large, and how coherent, was the initial patch?” Chapters 4 to 6 develop the chain in three dimensions. MAC Big Bang Theory 13 4. The three-dimensional model 4.1 Geometry The model lives in comoving Cartesian coordinates x = (x, y, z). Comoving coordinates are fixed to the expanding fluid, so the expansion factor a(t) appears in the equations instead of in the geometry. The initial patch is a ball of comoving radius R p that contains N particles. Outside the patch we assume no fuel, so the burn cannot escape. Figure 3 shows the geometry at three stages: unburnt fuel in the outer region, a thin burning front, and a burnt, cooling core. The ignition seed sits at the centre. Because the equations are rotationally symmetric about the seed, the real front is a sphere; the lattice simulations in Chapter 7 test how well this holds on a grid. Figure 3. The MAC universe model in three dimensions. A black seed at the centre ignites an expanding orange front. Inside the front the fuel is spent and the burnt core cools (blue). Outside the front the fuel is still intact. 4.2 Fields Four fields describe the state at every point and time: • n m (x, t) and n a (x, t), the number densities of matter and antimatter, MAC Big Bang Theory 14 • u(x, t) ∈ [0, 1], the burn fraction, which is 0 in intact fuel and 1 where the antimatter has been consumed, • T(x, t), the temperature of the burnt plasma, obtained from the energy density ε. 4.3 The initial condition: a random draw Postulate P2 says that each of the N particles is independently matter or antimatter with probability ½. The number of matter particles is then a binomial random variable: (5) Figure 4 shows one such draw with N = 1800 particles placed uniformly at random. This particular draw produced 851 matter and 949 antimatter particles, an imbalance of 5.4%, and antimatter happened to win. Because the draw is symmetric, antimatter wins as often as matter. By convention we call whichever species wins “matter”, which is why the theory is conditioned on our universe being matter-dominated. Figure 4. One random initial draw in a 3D patch. Each point is a particle, blue for matter and red for antimatter. The star marks the ignition seed. 4.4 The rate equations The densities obey transport equations with annihilation. In an expanding universe, with Hubble rate H, diffusion constant D and annihilation rate ⟨σv⟩: MAC Big Bang Theory 15 (6) (7) Subtract the second from the first. The annihilation terms are identical and cancel, leaving an equation for the net matter density n Δ = n m - n a that has no source or sink: (8) Net matter can only move, by diffusion, and be diluted by expansion. It is never created or destroyed by collisions. This is the equation behind the central logical chain of Chapter 3. The comoving net matter number Δ = ∫ n Δ a 3 d 3 x is exactly conserved. The annihilation energy is tracked by an energy-density equation. Each annihilation converts 2m of rest energy into radiation: (9) The factor 4H (rather than 3H) appears because radiation redshifts as well as dilutes. The thermal diffusivity χ describes how fast the radiation spreads. 4.5 Dimensionless form A natural unit of length is the mean free path ℓ = 1/(nσ), and a natural unit of time is the crossing time ℓ/c. In these units the kinetic-theory diffusion constant D = cℓ/3 becomes D̃ = 1/3, and the growth rate becomes r̃ = ν - 1, where ν is the average number of new annihilations triggered by one annihilation (Chapter 7). Only these dimensionless combinations enter the front dynamics, so the 3D simulations in Chapter 7 are run in dimensionless units. MAC Big Bang Theory 16 5. Statistical origin of the imbalance Chapter 4 showed that annihilation preserves the net matter number. This chapter asks how large that number is if the initial draw is random, and how large the patch must be for the leftover to match the observed η. 5.1 The distribution of the imbalance For N independent particles, each matter with probability ½, the probability of exactly k matter particles is (10) The net number Δ = 2N m0 - N has mean 0 and variance N. For large N the distribution is Gaussian, so the fractional imbalance δ = Δ/N is normally distributed with zero mean and variance 1/N: (11) The mean absolute imbalance follows from the properties of the half-normal distribution (Appendix A.1): (12) A Monte Carlo test confirms this scaling. For each N from 10 2 to 10 12 , 4000 random draws were made, and the mean |δ| was recorded. The fitted log–log slope is -0.5006, in agreement with the predicted -0.5 (Figure 5). MAC Big Bang Theory 17 Figure 5. Monte Carlo check of the 1/√N law. Blue points are simulations, the red line is the theory. 5.2 From imbalance to the observed ratio The observed η is a ratio of net baryons to photons. After complete annihilation the leftover is N m = Δ, while the photon number depends on how many photons the annihilation energy produced. Let κ denote the number of photons produced per initial particle, so that N γ = κN. Then (13) The factor κ is a property of the energy release and thermalization, and Chapter 8 estimates it. In the simplest case κ = 1 and the required imbalance equals η itself. Setting the typical imbalance 1/√N equal to the required value gives the size of patch that makes a matter-dominated universe typical rather than rare: (14) For κ = 1 and η = 6.1 × 10 -10 this is N req = 2.69 × 10 18 particles. Table 4. Patch size needed for the leftover to equal η, for different photon yields κ. The first two rows are the values that Chapter 8 obtains for burn temperatures of 1 GeV and 100 MeV. κ (photons per initial particle) required δ = κη N req = 1/δ 2 0.065 3.97 × 10 -11 6.36 × 10 20 0.65 3.96 × 10 -10 6.36 × 10 18 1 6.10 × 10 -10 2.69 × 10 18 10 6.10 × 10 -9 2.69 × 10 16 MAC Big Bang Theory 18 κ (photons per initial particle) required δ = κη N req = 1/δ 2 100 6.10 × 10 -8 2.69 × 10 14 1000 6.10 × 10 -7 2.69 × 10 12 5.3 How likely is a big enough imbalance? The probability that a patch of N particles has an imbalance at least as large as the required value is (15) Figure 6 plots this for x = 6.1 × 10 -10 . The probability is close to 1 for N well below 1/x 2 and falls off rapidly above it. So the theory works if the ignition patch contains about 10 18 to 10 20 particles, and it fails dramatically if the patch is much larger. Figure 6. Probability that a random patch has an imbalance of at least 6.1 × 10 -10 , versus the number of particles N. Table 5. Probability of a large enough imbalance, for several patch sizes (κ = 1). N P(|δ| ≥ 6.1 × 10 -10 ) 1 × 10 16 9.51 × 10 -1 1 × 10 17 8.47 × 10 -1 1 × 10 18 5.42 × 10 -1 1 × 10 19 5.37 × 10 -2 1 × 10 20 1.06 × 10 -9 MAC Big Bang Theory 19 N P(|δ| ≥ 6.1 × 10 -10 ) 1 × 10 21 6.53 × 10 -83 5.4 The whole observable universe is too big The observable universe holds about 1.5 × 10 89 photons. If matter and antimatter were drawn independently for that many particles, the typical imbalance would be only 2.6 × 10 -45 . The required value, 6.1 × 10 -10 , is about 35 orders of magnitude larger. Figure 7 shows the two points on the 1/√N curve. Figure 7. The typical imbalance 1/√N as a function of patch size. The green point is the MAC requirement. The red point is the whole observable universe drawn as one random patch. The conclusion is sharp. A single random 50/50 draw cannot produce η if it applies to the whole observable universe. Either the ignition patch was small (about 10 18 to 10 20 particles) and the rest of the universe arose from it in some way that kept its asymmetry, or there is a further mechanism. 5.5 Independent patches and domains If many independent patches each have a random imbalance, the average over many patches cancels. Figure 8 shows the difference. Forty by forty independent patches with random signs average to 0.043, close to zero, while one coherent patch keeps its full imbalance. A universe built from many independent ignition patches would show regions dominated by matter and regions dominated by antimatter. MAC Big Bang Theory 20 Figure 8. Left: independent patches, each with a random winning species (blue = matter, red = antimatter). Right: a single coherent patch. Gamma-ray background measurements constrain such boundaries strongly: a matter–antimatter interface would emit characteristic annihilation radiation (Steigman 1976; Cohen, De Rújula and Glashow 1998). These constraints are the reason MAC needs a single coherent patch, and Chapter 11 returns to this point.