MAC Big Bang Theory · version 1.1 1 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, with a coherence mechanism and real lattice simulations in 2D and 3D Concept: the author’s original idea. Mathematical formalization, simulations and write-up: developed with Claude (Anthropic). Version 1.1 (corrected and extended) · October 17, 2026 Status of this document This is a toy model. Every equation was derived or re-checked and every simulation result was computed again for this version, but the model is a proposal under test, not established physics. Chapter 12 lists where it is known to be in trouble, and that list is part of the result. Version 1.1 fixes the errors found in version 1.0 (listed in Appendix D), adds a mechanism for the coherent patch (Chapter 9), and compares real 2D and 3D lattice runs with the theory (Sections 6.5, 7.4 and 9.2). Appendices E to H add parameter sweeps of the real runs, a worked numerical example, a list of objections with answers, and extended derivations. 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 six parts: (1) a random initial draw of matter and antimatter in a patch of vacuum, (2) a sign-locking and enlargement mechanism that makes one coherent patch, (3) annihilation kinetics with an exact analytic solution, (4) an ignition front that spreads through the patch, (5) energy release and cooling, and (6) the leftover matter, compared with the observed baryon-to-photon ratio η ≈ 6.1 × 10 -10 Main results • Collisions alone cannot leave matter, but they can carry it. Annihilation is one-for-one, so the net matter number Δ is conserved and the leftover matter fraction equals the initial imbalance δ (up to the photon yield κ). New in this version: real 2D and 3D lattice runs show this holds only if the surplus matter and antimatter can mix. With no mixing the burn leaves 2.3 times too much matter and a large antimatter residue (14% of the initial antimatter). • 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 needs N ≈ 1/(κη) 2 , about 2.69 × 10 18 for κ = 1. MAC Big Bang Theory · version 1.1 2 • Exact kinetics, tested. The annihilation equations are solved in closed form. In fast-mixing 2D and 3D lattice runs the time to reduce the antimatter to 10 -3 of its start is within 7% (2D) and 3% (3D) of the theory. • A 3D front. The burn spreads as a Fisher–KPP front with asymptotic speed 2√(Dr). Lattice runs in 2D and 3D agree with the continuum theory for the same equation to 1.4% and 0.9% in radius. The earlier “86% of the asymptotic speed” is a finite-time effect, not a failure: the continuum theory itself is only at about 0.89 of its asymptotic speed at the same time. • The main difficulty, and a mechanism. One random 50/50 draw over the whole observable universe gives an imbalance of about 2.6 × 10 −45 , some 35 orders of magnitude below η. Chapter 9 adds two assumptions: a sign-locking step (domain coarsening, tested in 2D and 3D) that makes one patch coherent, and an inflation-like enlargement by about 54 e-folds that carries the patch to cosmological size. Both are added assumptions, not derivations. The model is internally consistent and makes definite, checkable statements. It still does not explain why the universe is matter-dominated, because the origin of the fuel, the probability of ignition and the enlargement mechanism are supplied from outside. Chapter 13 sets out what would settle the question. MAC Big Bang Theory · version 1.1 3 Contents 4 Notation and symbols 5 1. Introduction 7 2. Observational background 9 3. Postulates of the MAC theory 10 4. The three-dimensional model 12 5. Statistical origin of the imbalance 16 6. Annihilation kinetics 19 7. Ignition and the spreading front 24 8. Energy release and cooling 26 9. A mechanism for the coherent patch 29 10. The combined model and its results 30 11. Predictions and tests 31 12. Challenges and objections 33 13. Conclusions and roadmap 34 Appendix A. Derivations 35 Appendix B. Numerical methods and source code 40 Appendix C. Glossary 41 Appendix D. Corrections made in version 1.1 41 Appendix E. Parameter sweeps: real runs against theory 44 Appendix F. A worked example from start to finish 45 Appendix G. Objections that are often raised, and the answers 46 Appendix H. Extended derivations 47 Appendix I. Simulation settings at a glance 48 Appendix J. Reproducing the results 49 Appendix K. Index of equations 50 References MAC Big Bang Theory · version 1.1 4 List of figures 6 Figure 1. 8 Figure 2. 10 Figure 3. 11 Figure 4. 13 Figure 5. 14 Figure 6. 15 Figure 7. 15 Figure 8. 17 Figure 9. 18 Figure 10. 18 Figure 11. 21 Figure 12. 21 Figure 13. 22 Figure 14. 23 Figure 15. 23 Figure 16. 25 Figure 17. 27 Figure 18. 27 Figure 19. 42 Figure E1. 43 Figure E2. Notation and symbols MAC Big Bang Theory · version 1.1 5 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 scaled u(x,t) burn fraction: 0 = fuel, 1 = burnt dimensionless D, r, v* diffusion constant, local growth rate, front speed 2√(Dr) cm 2 /s, 1/s, cm/s λ* front decay constant √(r/D) 1/length φ(x,t) coarse-grained sign field of the imbalance (Chapter 9) dimensionless, −1 to 1 N e number of e-folds of linear enlargement (Chapter 9) dimensionless 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 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). 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. In the plain picture this fuel is thinly spread, like a gas. The quantitative model needs something different: Chapter 8 shows that a burn hot enough to make the observed photons requires fuel at roughly nuclear density or above. The gas picture is therefore kept only as an analogy for how the burn spreads, not as a description of the fuel. • 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. MAC Big Bang Theory · version 1.1 6 • 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. Four 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? What must be true of the initial patch for the leftover to equal η? And can the patch be made coherent and large enough, and does a real 2D or 3D run behave like the theory? 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. Chapter 9 adds a mechanism for that difficulty, but the mechanism is itself an assumption that has been tested only as a toy simulation. 1.4 Structure of the model The model is a pipeline of six stages. Chapters 4 to 9 develop one stage each, and Chapter 10 combines them. Figure 1 gives the overview. Figure 1. The six 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 11 and 12 list predictions, tests and objections. The appendices give derivations, the numerical methods with the complete source code, a glossary and the list of corrections made in this version. MAC Big Bang Theory · version 1.1 7 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) Correction. Version 1.0 printed 1.32 s here. That coefficient belongs to g * = 3.36 (photons and neutrinos only, after electron–positron annihilation). The standard relation t = 2.42 g * -1/2 s (1 MeV/T) 2 gives 0.74 s for g * = 10.75. Figure 2 uses the corrected value. 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 only if the neutrinos are still coupled, with photons, electrons and positrons. At higher temperatures g * is larger, which changes the numbers but not the scaling. MAC Big Bang Theory · version 1.1 8 Figure 2. Radiation-era cooling, T versus t, for g * = 10.75 (t = 0.74 s at 1 MeV). 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 and are mixed. 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 · version 1.1 9 3. Postulates of the MAC theory The model rests on nine postulates. P1 to P7 are those of version 1.0. P8 and P9 are new and are the price of fixing the coherence problem. Each is stated first in words. P1. A fuel of matter and antimatter. Before ignition, a region of the quantum vacuum contains real matter and antimatter particles. 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. P8 (new). Sign locking. Before ignition the coarse-grained sign of the local imbalance relaxes by curvature-driven domain coarsening until one sign fills the patch. The magnitude of the imbalance is then a property of the stable state, δ 0 P9 (new). Enlargement that preserves η. After the burn the coherent patch is enlarged by an inflation-like expansion of N e e-folds of linear size, with particle number and photon number growing together, so that η does not change. Table 3. Origin and status of each postulate. Postulate Taken from Status P1 quantum vacuum picture assumed; origin of real particles is open (Ch. 12) 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 P6 standard radiation-era cosmology established P7 baryon number conservation after annihilation established P8 Allen–Cahn domain coarsening assumed; simulated in 2D and 3D (Ch. 9); it fixes the sign, not the value of η P9 inflation-like expansion assumed; only the required size, about 54 e-folds, is computed (Ch. 9) 3.1 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). MAC Big Bang Theory · version 1.1 10 • Therefore the net matter number Δ = N m − N a does not change during the burn. • If the burn is complete and the surplus can mix, all antimatter is gone and N m = Δ is what remains (P7). The words “and the surplus can mix” are new; Section 6.5 shows why they are needed. • 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. 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 model in three dimensions. A seed at the centre ignites an expanding front. Inside the front the fuel is spent and the burnt core cools. Outside the front the fuel is still intact. 4.2 Fields Four fields describe the state at every point and time: the number densities n m (x, t) and n a (x, t); the burn fraction u(x, t) ∈ [0, 1], which is 0 in intact fuel and 1 where the antimatter has been consumed; and the temperature T(x, t) of the burnt plasma, obtained from the energy density ε. Chapter 9 adds a fifth, the sign field φ(x, t). 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) MAC Big Bang Theory · version 1.1 11 Figure 4 shows one such draw with N = 1800 particles placed uniformly at random. This particular draw produced 909 matter and 891 antimatter particles, an imbalance of 1.0%. 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⟩: (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 Section 3.1. The comoving net matter number Δ = ∫ n Δ a 3 d 3 x is exactly conserved. Equation (8) also carries the warning that Section 6.5 quantifies: net matter is moved only by diffusion, so surplus matter and surplus antimatter in different places cancel only if diffusion brings them together. The annihilation energy is tracked by an energy-density equation. Each annihilation converts 2m of rest energy into radiation: MAC Big Bang Theory · version 1.1 12 (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. 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 -0.5002, in agreement with the predicted −0.5 (Figure 5). MAC Big Bang Theory · version 1.1 13 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 lists other photon yields. Its first two rows use the exact values of κ from Chapter 8 (0.0646 and 0.646 for burn temperatures of 1 GeV and 100 MeV), so they now agree with Table 8. Version 1.0 had rounded κ to 0.065 and 0.65 here, which gave slightly different numbers. Table 4. Patch size needed for the leftover to equal η, for different photon yields κ. κ (photons per initial particle) required δ = κη N req = 1/δ 2 0.0646 3.94 × 10 −11 6.44 × 10 20 0.646 3.94 × 10 −10 6.44 × 10 18 1 6.10 × 10 −10 2.69 × 10 18 10 6.10 × 10 −9 2.69 × 10 16 100 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 x is MAC Big Bang Theory · version 1.1 14 (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. Table 5 gives the numbers. The theory works if the ignition patch contains up to about 10 18 to 10 19 particles, where the probability is still 54% to 5%. At 10 20 it is about 10 -9 , so the range 10 18 to 10 20 quoted in version 1.0 was too generous: the upper end is effectively excluded for a single random draw. 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 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. MAC Big Bang Theory · version 1.1 15 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 16 to 10 19 particles) and the rest of the universe arose from it in some way that kept its asymmetry, or there is a further mechanism. Chapter 9 takes the first route. 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 have an average imbalance of 0.0012 (typical size 1/√1600 = 0.025), 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. 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. Chapter 9 gives a mechanism that produces one, and Chapter 12 returns to the remaining objections. MAC Big Bang Theory · version 1.1 16 6. Annihilation kinetics This chapter solves the annihilation equations exactly in a uniform, well-mixed patch and then tests the result against real stochastic runs on 2D and 3D lattices. The solution tells us how fast the antimatter disappears and what condition guarantees that it is gone. 6.1 Reduction to one equation In a uniform patch, diffusion drops out. In comoving variables N m = n m a 3 and N a = n a a 3 , the rate equations become (16) Define the annihilation clock (17) which measures how much annihilation opportunity has accumulated. Using N m = N a + Δ with Δ constant, the equation for antimatter alone is (18) 6.2 The exact solution Equation (18) is a Bernoulli equation. Substituting y = 1/N a turns it into a linear equation dy/dτ = Δy + 1, whose solution gives (Appendix A.2) (19) The ratio of antimatter to matter has an even simpler form: (20) Equation (19) was checked against a stiff numerical ODE solver for four values of δ from 10 -1 to 10 -4 The largest relative difference was 1.0 × 10 −11 , which is at the level of the solver tolerance (Figure 9, black dots). MAC Big Bang Theory · version 1.1 17 Figure 9. Antimatter remaining, N a /N a0 , versus the annihilation clock τ for four values of the imbalance δ. Black dots are the numerical solution of the ODE. 6.3 Two regimes • Power-law stage (Δτ ≪ 1). The imbalance is irrelevant, matter and antimatter annihilate as equals, and N a ≈ N a0 /(1 + N a0 τ). The density falls as 1/τ, which is the usual second-order annihilation law. • Exponential stage (Δτ ≳ 1). Once the surplus matter dominates, each remaining antiparticle finds a partner easily and the antimatter disappears exponentially fast, N a /N m ∝ e -Δτ The crossover occurs at τ ≈ 1/Δ. Smaller imbalances need longer to finish the burn, which is visible in the figure as the curves leaving the power law at later and later times. 6.4 How much annihilation is enough? To suppress the antimatter-to-matter ratio below a threshold ρ th , Equation (20) requires (21) Table 6. Required annihilation clock for different antimatter residues. Threshold ρ th Required Δτ Interpretation 1 × 10 -3 6.9 antimatter is 0.1% of matter 1 × 10 -6 13.8 one part per million 1 × 10 -10 23.0 comparable to η 1 × 10 -12 27.6 far below observational limits In a well-mixed plasma the requirement is mild. A value of Δτ near 23 already brings the ratio down to 10 -10 . In physical terms Δτ is roughly ⟨σv⟩ n Δ t, the number of annihilation times that the surplus matter experiences. That product is large in any dense early plasma, so for a well-mixed fuel the kinetics pose no obstacle. They only fix the order of events: annihilation must finish before nucleosynthesis starts at about 1 MeV. The next section tests how much the words “well-mixed” matter. MAC Big Bang Theory · version 1.1 18 Figure 10. Antimatter-to-matter ratio, normalized to its initial value, versus Δτ. Dots mark the thresholds of Table 6. 6.5 Real lattice runs versus the theory (2D and 3D) The exact solution assumes a well-mixed patch. To test it, a real stochastic A + B → 0 reaction was run on a 2D lattice (200 × 200 cells) and a 3D lattice (40 × 40 × 40 cells). Each cell starts with an independent Poisson number of matter and antimatter particles (the real random draw), with mean 20 per cell and an imposed imbalance δ = 0.05, so that Δ = 1 per cell and Δτ = τ. At each step every particle hops to a neighbouring cell with probability h, and then pairs in each cell annihilate at a rate proportional to N m N a . Three mixing strengths were run: h = 0.5 (fast), h = 0.05 (slow) and h = 0 (none). The mean-field theory, Equation (19), is the prediction for perfect mixing. The code is in Appendix B. Figure 11. Antimatter remaining in real 2D and 3D lattice runs (coloured) against the mean-field theory (black). Fast mixing follows the theory. Slow mixing is slower. With no mixing, antimatter survives in the cells where it started in the majority. MAC Big Bang Theory · version 1.1 19 Table 7. Real lattice runs against theory. t -3 is the clock τ at which N a /N a0 first falls below 10 -3 ; the last two columns are measured at the end of the run (τ = 40) and compared with the conserved net matter Δ/N a0 Case t -3 antimatter left matter left / N a0 Δ / N a0 mean-field theory 4.6 0 — — 2D, fast mixing 4.9 0.000 N a0 0.106 0.107 2D, slow mixing 10.4 0.000 N a0 0.102 0.102 2D, no mixing never 0.138 N a0 0.244 0.105 3D, fast mixing 4.7 0.000 N a0 0.107 0.107 3D, slow mixing 6.6 0.000 N a0 0.104 0.104 3D, no mixing never 0.137 N a0 0.244 0.107 Three results follow. First, with fast mixing the real runs follow the theory: t -3 is 4.9 (2D) and 4.7 (3D) against 4.6 predicted, and the matter left equals Δ to within a few percent. Second, with slow mixing the antimatter falls more slowly than the mean-field law (t -3 = 10.4 in 2D and 6.6 in 3D), but it still reaches zero and the final matter is again Δ. Third, and most important, with no mixing the burn leaves 0.137 N a0 of antimatter and 0.244 N a0 of matter. That is 2.3 times the conserved surplus. Net matter is conserved in this case too (the difference between the two leftovers is Δ), but the surplus of matter in some cells and of antimatter in others is never brought together. Consequence for the theory The statement “the leftover equals the initial imbalance” needs a mixing condition: the surplus must be able to diffuse across the correlation length of the initial random draw during the burn. In the model this holds if the burn products (pions, photons) move on mean free paths long compared with the cell size of the initial fluctuations. This condition is added to the central logical chain in Section 3.1. 7. Ignition and the spreading front 7.1 Why the burn spreads In a uniform mixture of matter and antimatter, annihilation would proceed everywhere at once, with no front. MAC needs something that keeps the fuel cold and slow until a seed ignites it. Suppose each annihilation releases energetic pions and other products. These travel about one mean free path ℓ and trigger, on average, ν further annihilations in the fuel they pass through. If ν is greater than 1, the process is a branching chain reaction, like a nuclear chain reaction, and the disturbance grows. Two numbers describe the coarse-grained dynamics. With one generation per crossing time ℓ/c, the net growth rate is r = (ν − 1) c/ℓ. Kinetic theory for products moving at speed c with mean free path ℓ gives the diffusion constant D = cℓ/3. 7.2 The Fisher–KPP equation When many branching particles are averaged, the burn fraction u(x, t) obeys the Fisher–Kolmogorov–Petrovsky–Piskunov (KPP) equation, first written down in 1937 for the spread of an advantageous gene: (22) The term r u(1 − u) captures growth into unburnt fuel and saturation when the fuel is used up. A front solution travels with a speed fixed by the leading edge, where u is small. There the equation linearizes to ∂u/∂t = D∂²u/∂x² + ru. Try u ∝ exp[−λ(x − vt)]: MAC Big Bang Theory · version 1.1 20 (23) A front started from a compact seed selects the slowest allowed speed, the minimum of v(λ), reached at λ* = √(r/D): (24) Substituting D = cℓ/3 and r = (ν − 1)c/ℓ gives a speed that does not depend on the density or the mean free path: (25) Table 8. Front speed from Equation (25) for several multiplication factors ν. Comments corrected from version 1.0, which called every row valid. Multiplication ν Front speed v*/c Comment 1.05 0.26 diffusion estimate reasonable 1.10 0.37 diffusion estimate reasonable 1.25 0.58 marginal: v is more than half of c 1.50 0.82 marginal: v is above 0.8 c, treat as an upper estimate 1.75 1.00 at the limit: the diffusion estimate reaches c 2.00 1.00 (capped at 1) diffusion estimate exceeds c; true front is ballistic, v = c A moderately supercritical burn (ν = 1.5) moves at about 0.82 c. The diffusion estimate loses meaning as the speed approaches c, because the diffusion equation allows arbitrarily fast signals; for ν above 1.75 it would give a speed above that of light. In that regime the products move ballistically and the front speed is limited by c. Either way, the burn crosses a patch of radius R p in a time of order R p /v, which for the patch sizes of Chapter 5 is a tiny fraction of a second. 7.3 The lattice simulations To check that the equation really produces an expanding front, Equation (22) was integrated on cubic (3D) and square (2D) lattices with grid spacing 1, D = 1, r = 0.25, time step 0.1, forward Euler stepping and the standard nearest-neighbour Laplacian. A ball of radius 3 with u = 1 was placed at the centre as the ignition seed. The continuum prediction for the asymptotic front speed is v* = 2√(Dr) = 1.00 in both cases. The 3D grid was 121 3 and the 2D grid 401 2 ; runs stopped before the front reached the periodic boundary. The front radius is R = (3V/4π) 1/3 in 3D and R = (V/π) 1/2 in 2D, where V is the volume (area) where u > 0.5.