One constant-Lagrangian galaxy, grown from the earliest moment it can exist to today in a flat ΛCDM background. The CL mass M grows by a standard cosmological inflow (mass-accretion) history, H(t) and a(t) follow ΛCDM, and the critical radius rc = (2GM/H²)1/3, the radius where the radial river stalls, opens up with the expansion. Press play or drag the time slider. This is the derived baseline; the same galaxy with core resets and fossil stages is in CL Galaxy Growth with Core Resets.
rc and a(t). For fixed M, rc ∝ a exactly while matter dominates, and freezes at (2GM/H₀²ΩΛ)1/3 once Λ takes over. With mass growth, rc ∝ a·M1/3 in the matter era.
rc as a density radius. The mean density inside rc is 3M/4πrc³ = 3H²/8πG, exactly the critical density. The ΛCDM virial radius R200 is the radius of 200 times that density, so R200 = rc/2001/3 ≈ rc/5.85 for the same mass.
The CL curve at each epoch. X = √(2GM/R) − HR, vL² = ³⁄₂X²; v² = ½X²r²/R² inside R and vL² − (√(2GM/r) − Hr)² outside, with the epoch's own M, R and H. The spiral streamlines follow tan α = vr,eff/vφ and wind up into the circle rc; the view draws them from 0.7 rc inward, because closer to rc they wind so tightly that the turns merge.
Optical disc and dark-matter edges (switch, off by default). The halo mass today follows from the plateau speed, V200 = vL/(Vmax/V200) and M200 = V200³/(10GH₀), and grows with the same accretion history. The dark-matter edge is RΔ = (2GMh/ΔH²)1/3, with Δvir(z) from Bryan & Norman (1998) or Δ = 200. The optical edge is Ropt = 3.2 Rd, with Rd from r1/2 = 0.015 R200c, so Ropt ≈ 0.029 R200c. Note that rc has the same form with Δ = 1 and the CL mass M in place of the halo mass.
Earliest possible onset. The galaxy starts at the latest of three conditions: M(t) ≥ the minimum galaxy mass, R(t) < rc(t) (otherwise X ≤ 0 and no CL structure exists), and z ≤ zmax for the first structures.
What is assumed
imposed Flat ΛCDM with Ωm = 0.31, H₀ = 67.9 km/s/Mpc (= 2.2×10⁻¹⁸ s⁻¹, the value used in the SPARC CL fits); radiation neglected, so the earliest ~1 Myr are not exact.
calibrated The growth laws are halo mass-accretion histories from ΛCDM simulations (McBride et al. 2009; Wechsler et al. 2002) or the power law of de Haas (2026a). Applying them to the CL mass M, which tracks the inner (bulge-like) mass rather than the halo, is an assumption of this model.
calibrated Average size relations: Ropt = 3.2 Rd (Persic & Salucci 1991), r1/2 ≈ 0.015 R200c (Kravtsov 2013), Δvir(z) (Bryan & Norman 1998), Vmax/V200 ≈ 1.2–1.4 for Milky-Way-mass haloes. These are population averages with about 0.2 dex scatter, applied here to one galaxy. Both edges are the standard dark-matter picture, drawn for comparison with rc; the CL model itself has no halo.
assumed The core radius R follows one of three simple histories; none is derived from the CL theory yet.
derived rc(t), vL(t), the rotation curve and the spiral at each epoch follow from M(t), R(t) and H(t) with no further freedom.
not included The dynamical foreground (bars, rings, mergers, virial zones) and nested Lagrangians; this is the single-L kinematic background only.