Appendix N: Observable Phase Diagram, Numerical Simulation Code, and Computational Framework

Revision note (v13.0). This appendix is rebuilt around the environment-dependent range $\lambda(\rho)$ and the screening and range factor $f$ of the screened scalar sector. The v12 phase diagram in $S_r(r)\,S_\rho(\rho)$ with a fixed $\lambda_c=1\ \mu$m is retired. Retired: the Yukawa kernel $-4\pi G/(k^2+k_c^2)$ with $k_c=1/\lambda_c$ (it removes Newtonian gravity on all astronomical scales), the GADGET-4 pseudocode, the Bullet Cluster toy calculation, and every simulation output that was never run (the "softer core $\rho\propto r^{-0.7}$", the "10–20% fewer subhaloes", the quoted suppression logarithm). The scripts in scripts/ implement the legacy v12 model and are kept for the audit trail only. The Earth and Sun thin-shell entries below are the first-pass estimate; the full calculation is open. See the architecture section of the main document.

1. The range $\lambda(\rho)$ and the factor $f$

1.1 Definitions

For the benchmark potential $V(\phi)=\Lambda^{4+n}/\phi^{n}$ and the coupling $A(\phi)=e^{\beta\phi/M_{\rm Pl}}$, the field sits at the minimum of $V_{\rm eff}=V+\rho\,e^{\beta\phi/M_{\rm Pl}}$:

The first term of $m_{\rm eff}^2$ dominates for the parameters used below, and it scales as $\rho^{(n+2)/(n+1)}$, so $m_{\rm eff}\propto\rho^{(n+2)/[2(n+1)]}$, which is $\rho^{3/4}$ for $n=1$. For $n=1$ it also scales as $\Lambda^{-5/4}$, so a larger $\Lambda$ gives a longer range.

The observable is $\Delta a/a\simeq2\beta_0^2\,\Delta\delta\,f$. For small perturbations of the field by unscreened bodies, the linearised equation is $(\nabla^2-m_{\rm eff}^2)\,\delta\phi=(\beta/M_{\rm Pl})\,\delta\rho$. For a point source of mass $M_s$ its solution gives a potential energy $-2G\beta_i\beta_s\,m_iM_s\,e^{-r/\lambda}/r$ for a test mass $m_i$, using $1/(4\pi M_{\rm Pl}^2)=2G$. This reproduces the force ratio $2\beta_i\beta_s$ and the simplest estimate $f\approx e^{-r/\lambda(\rho)}$, with $\lambda$ evaluated at the density of the medium in the gap. It is valid only in the linear regime. Thin-shell screening of a dense body requires the nonlinear solution (Section 2).

1.2 Illustration

The table uses $n=1$, $\Lambda=2.4$ meV (the dark-energy scale, a reference value and not a fit) and $\beta=0.05$. The code in Section 6 reproduces it. The conversion $1\ {\rm kg/m^3}=4.31\times10^{15}\ {\rm eV^4}$ and the reduced Planck mass $2.435\times10^{27}$ eV are used. The last column is $1-f$ at $r=1\ \mu$m with $f=e^{-r/\lambda}$ and $\lambda$ taken at the density in the first column.

The densities inside a body set the range inside it, which controls the screening of the body. The density in the gap between an atom and a surface sets the range for the force across the gap. In an ultra-high-vacuum chamber the gap range is at least the $7.1\times10^{7}$ m of the first row for these parameters, because the range grows as the density falls. The range factor alone then gives $f\approx1$ at 1 μm. The column $1-f$ is that linear range factor at $\beta=0.05$. The first-pass Earth and Sun factors in Section 1.4 are values of $3\,\Delta R/R$ at $\beta_0=0.053$.

1.3 Dependence on $\Lambda$ and the micrometre scale

Two results follow for $n=1$ and $\beta=0.05$.

• A range of 1 μm inside gold needs $\Lambda=2.65\times10^{-5}$ eV. The range in aerogel is then 0.29 mm and the range in a vacuum chamber is $2.5\times10^{5}$ m (about 158 miles).
• At $\Lambda=2.4$ meV the density at which the range falls to 1 μm is $3.5\times10^{7}$ kg/m³, about 1,800 times the density of gold.

The micrometre scale of v12 is therefore not a consequence of the mechanism. A micrometre-scale signal depends on the experimental geometry and the field solution.

1.4 Regimes of the observable

The v12 phase diagram divided the $(r,\rho)$ plane using $S_r$ and $S_\rho$. The regimes are now set by two quantities: the ratio $r/\lambda(\rho_{\rm gap})$, and the screening of the source, $s=3\,\Delta R/R$.

Where each experiment sits depends on parameters that are not fixed, so the table gives only the qualitative regime and what is needed.

2. Field-equation solver specification

A screened scalar needs the solution of a nonlinear field equation. A linear kernel in Fourier space cannot represent the density-dependent mass. This section specifies the solver. No solution of this solver has been computed, so no laboratory value of $f$ is quoted. The analytic first-pass thin-shell estimate for the Earth and the Sun is a separate calculation, recorded in Section 1.4 and in the screened scalar document.

Equation. In the quasi-static limit,

Variables. The potential is singular at $\phi\to0$, so the solver works with a positive variable, for example $\psi=\ln(\phi/\phi_{\min}(\rho_{\rm ref}))$, which keeps $\phi>0$.

Geometries, in order of increasing cost: (i) one-dimensional, an atom perpendicular to a planar slab; (ii) axisymmetric, a finite disc or sphere target; (iii) three-dimensional, with the vacuum-chamber walls and the support structure of the targets.

Boundary conditions. Far from the apparatus the field tends to $\phi_{\min}$ at the density of the residual gas, or of the chamber walls if the range exceeds the chamber size. Inside dense walls the field sits close to $\phi_{\min}(\rho_{\rm wall})$. The treatment of the walls as a boundary condition has to be justified for each geometry.

Method. Newton relaxation with multigrid acceleration, with the density map taken from a measured or modelled target.

Validation checks, all with known answers:

1. A uniform medium returns $\phi_{\min}(\rho)$ and $m_{\rm eff}(\rho)$ of Section 1.1.
2. A small unscreened source returns the linearised potential of Section 1.1, with the force ratio $2\beta_i\beta_s$ and the factor $e^{-r/\lambda}$.
3. The solution converges under mesh refinement and is independent of the position of the outer boundary where the range allows.

Outputs. The map $f(r,\rho,\text{geometry})$ for each target and chamber, and the screening factor $s$ of each body.

3. N-body and structure-formation work

Why the Yukawa kernel is retired. The kernel $-4\pi G/(k^2+k_c^2)$ with $k_c=1/\lambda_c$ is the Fourier transform of a Yukawa potential, $\Phi=-GM\,e^{-r/\lambda_c}/r$. For $\lambda_c=1\ \mu$m the exponent at a distance of 1 AU is $1.5\times10^{17}$. The potential is negligible beyond a few micrometres, so the kernel removes Newtonian gravity on every astronomical scale. In v13 gravity comes from the metric sector (Level 0), and the scalar is an additional force. A fixed-range Yukawa kernel is not the force law of a screened scalar either.

What is needed. A screened scalar requires a nonlinear field-equation solver coupled to the particle dynamics, solved on the density field at each step. Codes for screened modified gravity exist. Candidates, all to be verified before use, are ECOSMOG, MG-GADGET, ISIS and MG-AREPO. None has been run for MCE parameters, and no result is claimed.

Scale of the effect. In the linear regime the scalar changes the force by a factor at most $G_{\rm eff}/G=1+2\beta^2$, which is $1.0057$ for $2\beta_0^2=5.7\times10^{-3}$. The v12 outputs of tens of per cent in subhalo counts and a changed central density slope were not produced by any run. Linear growth is treated in Appendix P.

4. Retired material

Bullet Cluster. The v12 calculation multiplied an impact velocity of $3\times10^{6}$ m/s by a collision time of 0.2 Gyr to get 614 kpc. The arithmetic is correct. It contains no MCE input and does not locate a lensing mass. A conformally coupled scalar does not bend light beyond general relativity, so Level 1 cannot place a lensing mass offset from the baryons. The Bullet Cluster is unexplained by Level 1.

5. Scripts in scripts/

Figures produced by these scripts are not predictions of the v13 model and should not be cited as such.

6. Code for the $\lambda(\rho)$ table

The script below uses only the Python standard library and reproduces the table of Section 1.2. It was run for this version.

import math

HBAR_C = 1.973269804e-7          # eV m
C_LIGHT = 2.99792458e8           # m/s
EV = 1.602176634e-19             # J per eV
M_PL = 2.435e27                  # reduced Planck mass in eV
KG_M3_TO_EV4 = C_LIGHT**2 / EV * HBAR_C**3    # 1 kg/m^3 in eV^4


def m_eff(rho_kg_m3, lam_eV, n, beta):
    """Effective mass in eV at the minimum of V_eff, V = Lam^(4+n)/phi^n."""
    rho = rho_kg_m3 * KG_M3_TO_EV4
    phi = (n * lam_eV ** (4 + n) * M_PL / (beta * rho)) ** (1.0 / (n + 1))
    m2 = n * (n + 1) * lam_eV ** (4 + n) * phi ** (-(n + 2)) + beta ** 2 * rho / M_PL ** 2
    return math.sqrt(m2)


environments = [
    ("Air at 1e-6 Pa", 1.2e-11),
    ("Air at atmospheric pressure", 1.2),
    ("Aerogel", 10.0),
    ("Water", 998.0),
    ("Aluminium", 2700.0),
    ("Gold", 19320.0),
    ("Cosmic mean matter", 2.9e-27),
]

print("n=1, Lambda=2.4 meV, beta=0.05")
for name, rho in environments:
    m = m_eff(rho, 2.4e-3, 1, 0.05)
    lam = HBAR_C / m                        # range in metres
    one_minus_f = -math.expm1(-1e-6 / lam)  # 1 - exp(-r/lambda) at r = 1 micrometre
    print(f"{name:28s} rho={rho:9.3g}  m_eff={m:9.3e} eV  lambda={lam:9.3e} m  1-f={one_minus_f:8.1e}")

7. Status