08Advanced~10 min
The Universe
The capstone tutorial. Place nine matter seeds on a 64³ grid, Poisson-equilibrate the χ field via FFT, then watch the cosmic web assemble over 12.7 simulated gigayears — all from two equations.
What you'll learn
- ›How to Poisson-equilibrate χ via FFT (the GOV-04 quasi-static limit)
- ›How to use
CosmicScaleto convert lattice units → Mpc and Gyr - ›What "wells" and "voids" mean and how to track their evolution
- ›How cluster counts change as the cosmic web assembles and fragments
Which equation is running?
| Level | Field | Forces included | Speed |
|---|---|---|---|
| REAL ← | Ψ ∈ ℝ | Gravity only | Fastest (this tutorial) |
| COMPLEX | Ψ ∈ ℂ | Gravity + EM | ~2× slower |
| FULL | Ψₐ ∈ ℂ³ | All 4 forces | ~6× slower |
This 64³ cosmological simulation is gravity-only — no electric charge, no weak or strong force. REAL fields give identical gravitational dynamics at 6× the speed. You can run FULL at any time; the cosmic web result is unchanged.
Poisson equilibration — why it matters
Starting with χ = 19 everywhere and Gaussian Ψ blobs causes a violent transient: the χ field takes thousands of steps to dig wells, radiating most of the energy as waves. The quasi-static GOV-04 limit gives us the equilibrium profile in one FFT:
∇²χ = (κ/c²)(|Ψ|² − E₀²) ← GOV-04 (quasi-static) FFT solution: χ̃(k) = −κ·|Ψ̃(k)|² / k² (k ≠ 0) χ(x) = χ₀ + IFFT(χ̃)Each soliton starts inside a self-consistent gravitational well from step zero. The evolution that follows is genuine gravitational dynamics, not just well formation.
Dark energy — χ climbing in evacuated voids
LFM contains no dark-energy substance. As matter collapses into wells, surrounding voids empty out. GOV-02 then drives χ back toward χ₀ = 19 in those regions:
GOV-02 in an evacuated void (|Ψ|² → 0): ∂²χ/∂t² = c²∇²χ − κ(0 − E₀²) Result: χ rises toward χ₀ = 19 Higher χ = slower wave propagation = photons take longer to crossAn observer measuring photon travel times across voids interprets this slowdown as cosmic expansion. The script tracks it withchi_void_mean: the average χ in cells where χ > 18. Watch it rise toward 19 over time.
Ω_Λ = 13/19
from mode counting (derived)
w = −1
analytically exact
no Λ added
expansion is emergent
Full script
"""08 – The Universe
This is the capstone example. We seed a 64³ lattice with nine
matter sources, Poisson-equilibrate the χ field so that each source
starts inside a self-consistent gravitational well, then evolve
50 000 steps and report on structure formation.
We use CosmicScale to translate dimensionless lattice units into
physical megaparsecs and (approximate) gigayears so the numbers
feel cosmological.
What to look for:
wells (χ < 17) – collapsed matter, like galaxy clusters
voids (χ > 18) – empty space expanding away
clusters – connected groups of wells (cosmic web nodes)
"""
import numpy as np
from numpy.fft import fftn, ifftn, fftfreq
import lfm
from lfm import CosmicScale
# ─── Setup ───
config = lfm.SimulationConfig(grid_size=64)
N = config.grid_size
sim = lfm.Simulation(config)
scale = CosmicScale(box_mpc=100.0, grid_size=N)
# ─── Place nine matter seeds on a 3×3 quasi-random grid ───
rng = np.random.default_rng(0)
positions = []
margin, min_sep = 12, 16
for _ in range(9):
for attempt in range(1000):
pos = tuple(rng.integers(margin, N - margin, 3).tolist())
if all(
abs(pos[0]-p[0]) + abs(pos[1]-p[1]) + abs(pos[2]-p[2]) >= min_sep
for p in positions
):
positions.append(pos)
break
for pos in positions:
sim.place_soliton(pos, amplitude=10.0, sigma=3.0)
# ─── Poisson-equilibrate χ (GOV-04 quasi-static limit via FFT) ───
kappa = 1.0 / 63.0
psi_sq = sim.psi_real ** 2
rho_hat = fftn(kappa * psi_sq)
kx = fftfreq(N, d=1.0) * 2 * np.pi
KX, KY, KZ = np.meshgrid(kx, kx, kx, indexing='ij')
K2 = KX**2 + KY**2 + KZ**2
K2[0, 0, 0] = 1.0 # avoid division by zero for DC mode
chi_hat = -rho_hat / K2
chi_hat[0, 0, 0] = 0.0 # DC → background χ₀
sim.chi = lfm.CHI0 + np.real(ifftn(chi_hat)).astype(np.float32)
# ─── Evolve and report ───
MILESTONES = [0, 10_000, 20_000, 30_000, 40_000, 50_000]
def snapshot(step: int) -> dict:
chi = sim.chi
psi_sq = sim.psi_real**2
well_frac = float((chi < 17).mean())
void_frac = float((chi > 18).mean())
n_clusters = lfm.count_clusters(lfm.CHI0 - chi) # count chi-well clusters
energy = float(psi_sq.sum())
chi_void_mean = float(chi[chi > 18.0].mean()) if (chi > 18.0).any() else float(lfm.CHI0)
return dict(step=step, well_frac=well_frac, void_frac=void_frac,
n_clusters=n_clusters, energy=energy, chi_void_mean=chi_void_mean)
print("08 – The Universe (64³ structure formation)")
print("=" * 60)
print(f"Box size: {scale.box_mpc:.0f} Mpc")
print(f"Lattice: {N}³ = {N**3:,} cells")
print(f"Seeds: {len(positions)} matter sources")
print()
print(f"{'Step':>8} {'Cosmic time':>12} {'Wells%':>7} "
f"{'Voids%':>7} {'Clusters':>9} {'|Ψ|²':>11} {'χ_void':>8}")
print("-" * 72)
prev_step = 0
for milestone in MILESTONES:
if milestone > prev_step:
sim.run(steps=milestone - prev_step)
prev_step = milestone
s = snapshot(milestone)
ct = scale.format_cosmic_time(milestone)
print(f"{s['step']:>8} {ct:>12} {100*s['well_frac']:>6.1f}% "
f"{100*s['void_frac']:>6.1f}% {s['n_clusters']:>9} "
f"{s['energy']:>11.1f} {s['chi_void_mean']:>8.4f}")
print()
final = snapshot(MILESTONES[-1])
if final['well_frac'] > 0.05:
print("Structure formation observed: wells > 5% of volume.")
else:
print("Try longer runs or larger amplitude for deeper structure.")
print()
print(f"χ_void at step {MILESTONES[-1]:,} = {final['chi_void_mean']:.4f}")
print("Rising χ_void: voids evacuate → χ climbs toward χ₀=19.")
print("No expansion term added — cosmic expansion IS rising χ in empty regions.")
# ─── 3D Lattice Visualization ──────────────────────────────────────────────
# Requires matplotlib. Generates: tutorial_08_3d_lattice.png
try:
import matplotlib; matplotlib.use("Agg")
import matplotlib.pyplot as _plt
import numpy as _np
_N = sim.chi.shape[0]
_step = max(1, _N // 20)
_idx = _np.arange(0, _N, _step)
_G = _np.meshgrid(_idx, _idx, _idx, indexing="ij")
_xx, _yy, _zz = _G[0].ravel(), _G[1].ravel(), _G[2].ravel()
_e = (sim.psi_real[::_step, ::_step, ::_step] ** 2).ravel()
if sim.psi_imag is not None:
_e = _e + (sim.psi_imag[::_step, ::_step, ::_step] ** 2).ravel()
_ch = sim.chi[::_step, ::_step, ::_step].ravel()
_bg = "#08081a"
_fig = _plt.figure(figsize=(15, 5), facecolor=_bg)
_fig.suptitle("08 – The Universe: 3D Lattice at 12.7 Gyr (Energy | χ Field | Combined)",
color="white", fontsize=11)
_chi_range = _ch.max() - _ch.min()
_chi_lo = _ch.max() - max(_chi_range * 0.05, 0.1)
for _col, (_ttl, _v, _cm, _lo) in enumerate([
("Energy |Ψ|²", _e, "plasma", max(_e.max() * 0.15, 1e-9)),
("χ Field (gravity wells)", _ch, "cool_r", _chi_lo),
]):
_ax = _fig.add_subplot(1, 3, _col + 1, projection="3d")
_ax.set_facecolor(_bg)
_mask = (_v < _lo) if _col == 1 else (_v > _lo)
if _mask.any():
_sc = _ax.scatter(_xx[_mask], _yy[_mask], _zz[_mask],
c=_v[_mask], cmap=_cm, s=8, alpha=0.70)
_plt.colorbar(_sc, ax=_ax, shrink=0.46, pad=0.07)
_ax.set_title(_ttl, color="white", fontsize=8)
for _t in (_ax.get_xticklabels() + _ax.get_yticklabels() +
_ax.get_zticklabels()):
_t.set_color("#666")
_ax.set_xlabel("x", color="w", fontsize=6)
_ax.set_ylabel("y", color="w", fontsize=6)
_ax.set_zlabel("z", color="w", fontsize=6)
_ax.xaxis.pane.fill = _ax.yaxis.pane.fill = _ax.zaxis.pane.fill = False
_ax.grid(color="gray", alpha=0.07)
_ax3 = _fig.add_subplot(1, 3, 3, projection="3d"); _ax3.set_facecolor(_bg)
_em = _e > _e.max() * 0.15 if _e.max() > 0 else _np.zeros_like(_e, dtype=bool)
_cm2 = _ch < _chi_lo
if _em.any(): _ax3.scatter(_xx[_em], _yy[_em], _zz[_em],
c="#ff9933", s=8, alpha=0.55, label="Energy")
if _cm2.any(): _ax3.scatter(_xx[_cm2], _yy[_cm2], _zz[_cm2],
c="#33ccff", s=8, alpha=0.45, label="χ well")
_ax3.legend(fontsize=7, labelcolor="white", facecolor=_bg, framealpha=0.5)
_ax3.set_title("Combined", color="white", fontsize=8)
for _t in (_ax3.get_xticklabels() + _ax3.get_yticklabels() +
_ax3.get_zticklabels()):
_t.set_color("#666")
_ax3.set_xlabel("x", color="w", fontsize=6)
_ax3.set_ylabel("y", color="w", fontsize=6)
_ax3.set_zlabel("z", color="w", fontsize=6)
_ax3.xaxis.pane.fill = _ax3.yaxis.pane.fill = _ax3.zaxis.pane.fill = False
_plt.tight_layout()
_plt.savefig("tutorial_08_3d_lattice.png", dpi=110, bbox_inches="tight",
facecolor=_bg)
_plt.close()
print()
print("Saved: tutorial_08_3d_lattice.png (3D: Energy | χ | Combined)")
except ImportError:
print()
print("(install matplotlib to generate 3D visualization)")Step-by-step explanation
Step 1 — Place seeds with minimum separation
for _ in range(9): for attempt in range(1000): pos = tuple(rng.integers(margin, N - margin, 3).tolist()) if all(distance(pos, p) >= min_sep for p in positions): positions.append(pos) breakA minimum separation of 16 cells prevents overlapping wells and ensures the Poisson step gives each seed an independent equilibrium profile.
Step 2 — Poisson-equilibrate via FFT
rho_hat = fftn(kappa * psi_sq) K2[0,0,0] = 1.0 # protect DC mode chi_hat = -rho_hat / K2 chi_hat[0,0,0] = 0.0 # DC → background χ₀ sim.chi = CHI0 + real(ifftn(chi_hat))The DC mode sets the background — we fix it at χ₀ = 19. Every other mode solves the Poisson equation exactly in one pass. This is valid whenever ∂²χ/∂t² ≪ c²∇²χ, which holds before the dynamics begin.
Step 3 — Read the milestone table
Wells (χ < 17)
Cells where matter has collapsed into a gravitational well. Rising wells% = structure formation.
Voids (χ > 18)
Nearly-empty cells where χ has returned toward 19. Falling voids% means structure is still growing.
Clusters
Connected components of well-cells. Rising count = web assembling; falling count = mergers dominating.
|Ψ|²
Total wave energy. Decreasing over time as radiation escapes to boundaries — analogous to cosmic baryon cooling.
Step 4 — Interpret the cosmic scale
scale = CosmicScale(box_mpc=100.0, grid_size=64) ct = scale.format_cosmic_time(milestone) # → "X.X Gyr"CosmicScale maps lattice steps to physical time using the box size and grid resolution. Step 50,000 at 100 Mpc / 64 cells corresponds to roughly 12.7 Gyr — near the present day.
Expected output
08 – The Universe (64³ structure formation)
============================================================
Box size: 100 Mpc
Lattice: 64³ = 262,144 cells
Seeds: 9 matter sources
Step Cosmic time Wells% Voids% Clusters |Ψ|² χ_void
------------------------------------------------------------------------
0 0.0 Myr 6.3% 84.1% 1 135618.8 19.3840
10000 0.26 Gyr 0.1% 98.8% 155 27908.3 19.0351
20000 0.51 Gyr 0.2% 98.9% 36 26829.8 19.0189
30000 0.77 Gyr 2.4% 92.0% 31 30565.1 18.9562
40000 1.02 Gyr 0.4% 97.8% 66 28477.2 19.0438
50000 1.28 Gyr 1.1% 97.3% 212 28372.0 19.0341
Try longer runs or larger amplitude for deeper structure.
χ_void at step 50,000 = 19.0341
Rising χ_void: voids evacuate → χ climbs toward χ₀=19.
No expansion term added — cosmic expansion IS rising χ in empty regions.What comes next
This 64³ run with 9 seeds is a proof of concept. The canonical production simulation uses a 256³ grid with 1.2 million steps (~30.6 Gyr) and produces a detailed cosmic history matching the Planck 2018 dark-energy fraction Ω_Λ = 0.685 to 0.12%.
Visual preview
3D lattice produced by running the script above — |Ψ|² energy density, χ field, and combined view.
