During inflation the scale factor follows the real de Sitter / slow-roll solution for exponential (inflationary) expansion, with the Hubble parameter H held constant:
a(t) = a0 · e^(H·t) (a0 = 1, inflationary epoch)
N = ln(a(t)/a0) = H·t (e-folds — the standard metric)
d(t) = d0 · a(t) (physical separation of two
comoving points, comoving
separation d0 held fixed)
Most inflationary models require N ≥ 60 e-folds — set H and Δt so H·Δt ≥ 60 to reproduce the textbook minimum used to solve the horizon and flatness problems. After Δt, the field decays and expansion switches to the slower, decelerating radiation-era law a(t) ∝ √t (matter/radiation-dominated growth), continuous with a(Δt):
a(t>Δt) = a(Δt) · sqrt(1 + k·(t-Δt))
- Grid — comoving coordinates are fixed; every vertex's on-screen position is multiplied by the live scale factor a(t), so the grid itself visibly expands.
- Marked pair — two points start at comoving separation d0; their physical separation d(t)=d0·a(t) is read out live and matches the exponential curve, not an animation trick.
- Scale-factor chart — log10(a) vs. time, so the exponential inflation segment renders as a straight line (a real signature of exponential growth on a log plot) before bending into the shallower post-inflation curve.