The windows: which strong and weak forces let a four-dimensional universe burn hydrogen

A universe with four dimensions of space has to choose its nuclear forces; nothing carries ours over. The book fixes the charge law (paper 1) and keeps two numbers open for the nucleus: the strength $s$ of the attraction between nucleons, a factor on a borrowed force of fixed shape, and the weak coupling $G_4$, Fermi’s constant’s four-dimensional counterpart, which has units of mass$^{-3}$ and so no value inherited from ours. Figure 1 lays out the plane of those two choices and draws on it every constraint the book has computed, each from its own calculation. Each one rules out a part of the plane, and what is left is narrower than any one constraint suggests.

2.1 Results

The strong force’s window. The deuteron binds once $s > 1.4002$, and the spin-singlet pair of nucleons binds once $s > 1.8909$. Above that, two protons (or two neutrons) hold together by the strong force alone, and the slow weak step that paces hydrogen burning in our Sun disappears. Between them the deuteron’s binding rises from zero to 16.5 MeV. At $s = 1.5152$ it binds by 2.224 MeV, as ours does. These two edges bound the plane’s horizontal axis.

Proton + proton. In our Sun, hydrogen burning starts with $p + p \to d + e^+ + \nu$, which needs the deuteron to bind by more than the neutron–proton mass difference plus an electron mass, 1.804 MeV. That happens only above $s = 1.4984$, so the lowest 20% of the strong window (shaded) has no positron-emitting route to deuterium. The electron-capturing route $p + e^- + p \to d + \nu$ needs only 0.782 MeV and opens at $s = 1.4516$; in our Sun it runs at a few parts in a thousand of $p + p$ (Vinyoles et al. 2017), so across the 10% of the window between the two thresholds deuterium is made, but slowly. The figure shades the whole strip, where $p + p$ itself is closed. Of the plane drawn, 78% has $p + p$ open and no bound diproton.

The first minutes. The universe’s first minutes turn every neutron that survives them into helium-4. A weak force too feeble freezes the neutrons out of balance with the protons while there are still many of them, and the universe starts as mostly helium. For each clock of the first minutes (the universe’s age when it is 1 MeV hot, which gravity sets and the book has not yet fixed) the coloured line marks where exactly half the hydrogen survives; above it more does. The lines are flat in $s$, and they step down by exactly half a decade in $G_4$ for every decade of the clock: $\log_{10} G_4 / (G_F/M_W) = 5.43, 4.93, 4.43, 3.93, 3.43$ at clocks of 0.01, 0.1, 1, 10, 100 s (a slope of 0.500 per decade; each line flat in $s$ to 0.004 wherever $p + p$ is open). Both facts are the physics of freeze-out (section 2.2.4). The rate that converts neutrons and protons grows as $G_4^2 T^{2d-1}$ and the expansion rate as $T^{(d+1)/2}/t_1$, so the freeze-out temperature depends only on $G_4^2 t_1$. At the edge the frozen neutron fraction is a quarter, $T_f = Q/\ln 3 = 1.18$ MeV (computed: 1.18 MeV). The free neutron then lives 6 days, far longer than the wait for deuterium, so the deuteron’s binding hardly enters. The lines bend down only near the deuteron’s threshold, where the wait for deuterium becomes long enough for neutrons to decay. Depending on the clock, the first minutes leave half their hydrogen on 32–69% of the plane.

The weak window’s lower edge says something about $G_4$ in units of the electron: half the hydrogen survives only if $G_4 m_e^3$ is within a few orders of ours’ $G_F m_e^2$. The figure’s reference $G_F/M_W$, the weak force “as strong as ours at its own scale”, lies 3.4–5.4 decades below the edge and leaves a universe of helium. Matching ours at the electron’s energies, $G_4 = G_F/m_e$ (5.20 on the figure’s scale), gives a neutron that lives 4.2 hours and a freeze-out at 0.52 MeV. It keeps more than half the hydrogen for any clock longer than 0.029 s.

What hydrogen burning makes. At 56 points, the 8 strengths where the book’s reaction network has been built ($s$ = 1.5152, 1.55, 1.6, 1.65, 1.7, 1.75, 1.8, 1.85) across $\log_{10} G_4/(G_F/M_W)$ = 4, 4.5, 5, 5.5, 6, 6.5, 7, the network burns a core of hydrogen and primordial helium at the temperature where half its hydrogen goes in 4.6 billion years. That is the pace of our Sun’s centre, the book’s convention until stellar lifetimes are computed. Each point is coloured by what that core ends as; the number above it is the core’s temperature in millions of kelvin. The two mass models the book carries agree at every point (0 differ; section 2.2.5):

• 27 points end as helium-3 (1.0–3.6 MK). A fast $p + p$ makes the core cool, and then $^3$He + $^3$He, which has a high barrier and no weak step to wait on, is too slow to finish the chain in 4.6 billion years.
• 15 points end as helium-4 (4.1–10.4 MK).
• 14 points climb past helium, to elements 6–13, at 7.8–18.8 MK: the hottest cores, at the low-$s$ end and at $s = 1.65$ with a weak force 6–16× weaker than the electron-matched one.

$p + p$’s weak factor against ours runs from 0.023 at $s = 1.5152$ to 9.1 at 1.85 at the electron-matched $G_4$: four-dimensional phase space grows faster with the positron’s energy, which grows with the deuteron’s binding. So the strong and weak forces trade through $p + p$. A weaker $G_4$ or a weaker $s$ means a slower $p + p$, a hotter core at the same pace, and helium in place of helium-3.

No CNO-like cycle. The network’s one decisive uncertain mass was 5-11, a triton-like cluster on ceryllium-8. If 5-11 plus a proton could throw out a helium-4 and give ceryllium-8 back, the primordial helium would make its own catalyst (two helium-4s are bound as ceryllium-8, and protons climb from there to 5-11) and get it back: a cycle like our CNO cycle, needing no earlier stars’ carbon. Treated with every cluster free, in one calculation for all three nuclei, that step costs at least 19 MeV at $s = 1.5152$, rising to 69 at 1.85 (section 2.2.5). There is no cycle, and the low-$s$ cores climb past helium.

Our world’s point. The star marks both forces as ours, carried over: the deuteron bound by 2.224 MeV, as ours is ($s$ = 1.5152), and the weak force matched to ours at the electron’s energies, $G_4 = G_F/m_e$ (5.20 on the figure’s scale). Its first minutes leave helium $Y$ = 0.59, 0.38, 0.14, 0.01, 0.00 at clocks of 0.01, 0.1, 1, 10, 100 s (ours: 0.245). A core burning at the Sun’s pace there ends past helium (gluorine-22 81%, gluorine-23 17%) at 18.5 MK, in either mass model. So our own forces, carried into four dimensions, sit inside every window drawn for any clock longer than 0.029 s, and their stars climb past helium.

What is left. A strong force between the diproton’s edge and $p + p$’s threshold; a weak force above the first minutes’ line for the universe’s clock; and within that, hydrogen burning that ends as helium rather than stranded as helium-3, which at the Sun’s pace needs $s$ below about 1.7 at the electron-matched $G_4$, and lower for a stronger weak force. The window is real and not small, but each force is held to within about an order of magnitude of a value set by another: the strong force by the deuteron, the weak force by the electron, and both together by $p + p$.

2.2 Methods

2.2.1 The plane

$s$ multiplies the attractive terms of the Minnesota nucleon force (Thompson, Lemere, and Tang 1977), whose two Gaussians and repulsive core are kept as functions of distance in both worlds. The force’s shape is a borrowing, and its strength is the choice. The plane is drawn on 50 strengths from 1.40 to 1.891 and 17 couplings from $10^{ 3 }$ to $10^{ 7 }$ times $G_F/M_W$ in steps of 0.25 decades, at clocks of 0.01, 0.1, 1, 10, 100 s. $G_F/M_W$ (1.45 × 10⁻¹⁶ MeV$^{-3}$) is a reference, not a claim. It is what a 4 + 1 gauge theory gives if its coupling is ours at its own scale ($g_5^2 M_W = g^2$), and quoting every $G_4$ against it keeps the axis dimensionless. Produced by fusion/tools/windows4.mjs, drawn by papers/p2/figures/windows.mjs.

2.2.2 The strong window’s edges

Two nucleons in an s-wave with the triplet (deuteron) or singlet (diproton) Minnesota force, solved as a radial problem in four dimensions. The s-wave carries the centrifugal-like term $\hbar^2 (d-1)(d-3)/8\mu r^2$, a barrier in 4D that our deuteron does not have. Bound states are found by Sturm bisection on a finite-difference grid (atom/js/nuclear.js, levels; the deuteron with its wavefunction by deuteron), and each edge by bisection in $s$ (window). The singlet edge is computed without the Coulomb force, so it is strictly the dineutron’s. The diproton binds a little above it, and the dashed line is the conservative edge.

2.2.3 Proton + proton

The positron route needs $B_d > (m_n - m_p) + m_e$ and the electron-capture (pep) route $B_d > (m_n - m_p) - m_e$, each with the deuteron’s binding from 2.2.2. Where it is open, the $p + p$ rate is computed, not assumed. The nuclear part is $|\int u_d\, u_E\, dr|^2$, the overlap of the deuteron with the two protons’ Coulomb scattering state, both from the same force. The lepton part is the positron’s phase space at that world’s endpoint, and the constant is set on our $S(0) = 4.01 \times 10^{-22}$ keV b (Adelberger et al. 2011) (nuclear.js ppOverlap; on ours it gives $\Lambda^2 = 7.04$ against the evaluated 7.035 of the same review).

2.2.4 The first minutes

Weak rates are Fermi’s contact theory (Fermi 1934) in $d$ dimensions (atom/js/weak.js). The coupling $G_d$ has units of mass$^{-(d-1)}$, and a decay’s rate is

$$\Gamma = \frac{G_d^2\, m_e^{2d-1}\, |M|^2\, \Omega_d^2\, f_d(W_0)}{(2\pi)^{2d-1}}, \qquad f_d(W_0) = \int_1^{W_0} p^{\,d-2}\, W\, (W_0 - W)^{d-1}\, dW,$$

with $W$ the electron’s energy and $p$ its momentum in units of $m_e$, $\Omega_d$ the unit sphere’s area, and no Fermi function (it costs 7% on our neutron). The free neutron’s $|M|^2 = |V_{ud}|^2(1 + 3g_A^2)$ is kept as ours: in 4D the nucleon is still the smallest spinor, and $g_A$ belongs to the strong force. The hot universe converts neutrons to protons through $n + \nu \to p + e^-$ and $n + e^+ \to p + \bar\nu$, with thermal, Pauli-blocked leptons and the electron’s mass kept. The reverse rate follows by detailed balance. The neutron fraction $X$ is integrated as the universe cools from equilibrium to a thousandth of the expansion rate (neutronFraction), and it fixes an equivalent freeze-out temperature $T_f = Q/\ln(1/X - 1)$.

The clock is $t = t_1 (1\,\mathrm{MeV}/T)^{(d+1)/2}$, the radiation-dominated law in $d$ dimensions with $t_1$ the free parameter that gravity will set, and one clock serves freeze-out and the bottleneck alike (declared; ours’ two clocks, before and after electron–positron annihilation, differ by 1.8×). The deuterium bottleneck breaks where Saha’s equilibrium puts as many deuterons as neutrons, with baryons per photon η = 6.1 × 10⁻¹⁰ (ours, declared), and the helium made is $Y = 2X_f\, e^{-t_D/\tau_n}$ (atom/js/bbn.js firstMinutes). The edge drawn is $Y = 1/2$, found on the grid and interpolated linearly in $\log G_4$.

The scaling of section 2.1 follows from the rate’s dimensions. With massless leptons the conversion rate is $\propto G_d^2 T^{2d-1}$ and the expansion rate $H = T^{(d+1)/2}/(q\, t_1)$, so $\Gamma = H$ at $T_f \propto (G_d^2 t_1)^{-2/(3d-3)}$, a function of $G_d^2 t_1$ alone. Contours of constant helium are lines of slope $-1/2$ in $\log G$ against $\log t_1$. The computed slope, with the electron’s mass and the integration kept, is 0.500.

2.2.5 Hydrogen burning at the network’s points

Nuclei. The light 4D nuclei, by protons and neutrons, are $k$ helium-4s plus at most one small cluster (a nucleon, two nucleons, a triton-like or helium-3-like cluster). Their bindings come from three calculations:

• helium-4 and mass 3 by diffusion Monte Carlo, exact for $A \le 4$;
• the helium-4 “ladder” ($k$ helium-4s by Monte Carlo on their pair potential);
• the clusters on fixed draws of a core of $k$ helium-4s in Brink’s antisymmetrized-Gaussian model (Brink 1966; Freer et al. 2018) (Hill–Wheeler (Hill and Wheeler 1953) over the cluster’s placements).

The ladder and Brink’s grid are two methods, and on our own nuclei they err in opposite directions. Against AME2020 (Wang et al. 2021), for $k + 1$ helium-4s relative to free ones:

(MeV.) The ladder underbinds, and the grid overbinds more as the core grows. Mixing grid clusters onto ladder cores, which the network first did, puts nuclei from the two methods side by side and decides reactions by their difference.

The adopted model keeps the ladder’s nuclei and scales every grid separation on core $k$ by $r_k = S_4^{\mathrm{ladder}}(k)/S_4^{\mathrm{grid}}(k)$. That is the ratio of the two methods’ energies to remove one more helium-4 from the same core, so each cluster’s binding is measured against the grid’s own helium-4 binding and carried to the ladder. On ours it gives the 22 measured separations of a nucleon, two nucleons, a triton or a helium-3 from cores of ⁸Be to ²⁰Ne to a median 30%, against 91% for the grid alone. The grid alone is kept as the other mass model (fusion/tools/nuc4.mjs, MASS=ladder|grid; fusion/tools/masscheck.mjs).

The one nucleus where they differed in kind was 5-11 on its ceryllium-8 core: the adopted model’s scaling opened its (p,α) step at $s \le 1.6$, Brink’s grid closed it, and ours (¹¹B + p → 3α, $Q = 8.59$ MeV measured; 7.2 in the grid, 15.5 in the adopted model) could not say which to trust. It is decided by a third calculation that treats all three nuclei alike (brink.js freeClusterGCM, nuclei.mjs qbound): helium-4 as one determinant, exact in the model; the two-helium-4 core by angle-projected Hill–Wheeler, converged (two grids agree to 0.03 MeV; ours’ ⁸Be comes out 0.085 MeV unbound, measured 0.092, which is how the force’s exchange mixture was set); and 5-11 by Hill–Wheeler over $K$ placements of all three clusters, every centre free, screened by each placement’s own energy. 5-11’s energy is variational in $K$, so $Q(K) = E(2\alpha + t) - E(2\alpha) - E(\alpha)$ is an upper bound on the model’s own $Q$, falling as $K$ grows. The bound is negative at every strength already at $K = 120$; at $K = 480$ it is −15.7, −20.6, −27.9, −35.5, −43.5, −51.8, −60.5, −69.5 MeV at $s$ = 1.5152 … 1.85, and doubling to $K = 960$ moves $s = 1.5152$ from −15.7 to −18.7. Ours’ bound, 12.25 MeV at $K = 960$, sits above the measured 8.59 as a bound must. The adopted model takes 5-11 from the bound (fusion/qbound4.json), the least-closed value it allows.

Rates. Two-body rate constants at the density of our Sun’s centre, on a temperature grid (fusion/tools/rates4.mjs → fusion/data4.js):

• computed where the book has computed the reaction: $p + p$ (2.2.3, with 4D’s weak factor), helium-4 + helium-4 by E2 capture, and protons and helium-4s onto two to four helium-4s by E1 and E2 capture on folded potentials;
• carried otherwise: our analogue’s nuclear factor (its REACLIB (Cyburt et al. 2010) rate, from pynucastro’s snapshot (Smith et al. 2023), against our $p + p$’s, through our Coulomb barrier) put through 4D’s barrier;
• β decays from their own endpoints with ¹³N’s matrix element and 4D’s phase space;
• (p,α) branches wherever the compound nucleus sits above its helium-4 threshold, with our ¹⁵N(p,α)’s factor.

Every weak reaction is baked at the electron-matched $G_4$. A point at another $G_4$ multiplies all of them by $(G_4/G_{4,\mathrm{baked}})^2$, exactly how the weak factor scales.

The core. Hydrogen 0.75 and helium-4 0.25 by mass, burned at constant temperature and density by backward Euler with Newton’s method on the analytic Jacobian (fusion/js/net4.js). The temperature is the one at which half the hydrogen goes in 4.6 billion years, found by bisection. It ends as helium-3 if that holds more than half the mass, as helium if helium-4 does, and past helium if nuclei of mass 8 and above do (no point fell short of all three).

Not in it. Electron screening; neutron captures; reverse rates’ partition functions; the star around the core (no structure, no mixing, no lifetime: the pace is the convention). Pinned gravity’s clock is a parameter, not a result.

2.2.6 What is computed, carried and chosen

2.2.7 Checks on our own world

2.2.8 Reproducing it

node four/fusion/tools/rates4.mjs <s,…> bakes the network (about 2.5 minutes a strength; MASS=grid for the other model, into a copy of four/), node four/fusion/tools/windows4.mjs (with ALT=<that copy>) computes the plane, and node four/papers/p2/figures/windows.mjs draws it. node four/fusion/tools/masscheck.mjs writes ours’ check, and node four/papers/p2/build.mjs fills this text from all of them. The light nuclei come from the runner’s masses-1 and plusmore-1 runs.

References