H₀
km/s/Mpc
ΩM
ΩΛ
ΩK
flat
w
z₂
amber dot on diagram
The comoving distance light has had time to travel since the Big Bang. At each cosmic time on the y-axis, anything outside this curve is causally disconnected from us — light from it hasn’t reached us yet.
The comoving distance from which light emitted now could ever reach us in the future. In an accelerating universe this is finite (about 16–17 Glyr for Planck 2018) — galaxies beyond it will redshift out of contact forever.
The surface where the recession velocity vrec = H(t) D equals c. In comoving coordinates the Hubble sphere shrinks in the de Sitter era — one of the more counter-intuitive features Davis & Lineweaver use this diagram to clarify.
The past light cone (solid) traces the worldline of every photon arriving at us right now. The future light cone (dotted) traces every photon we emit today — it asymptotes to the event horizon, never crossing it.
Galaxies sit at constant comoving distance, so their worldlines are vertical lines in this view. We mark the worldlines for objects we observe today at z = 1, 3, 10, and the cosmic microwave background last-scattering surface at z ≈ 1100.
The horizontal dashed line at t = t₀ is the present epoch. The white star at (0, t₀) is our worldline crossing “now”. Following the past light cone down from that point traces back every direction in the sky.
Everything is computed from the wCDM expansion function
E(z) = √[ΩM(1+z)3 + ΩK(1+z)2 + ΩΛ(1+z)3(1+w)].
The diagram requires four integrals, each evaluated with Simpson’s rule:
For the high-z particle-horizon integral the calculator substitutes u = ln(1+z) so Simpson’s rule samples both the low-z bulk and the high-z tail evenly — without that change, the linear-z rule undersamples z < 5 and the particle horizon comes out roughly 3× too large. Cross-check (Planck 2018): Dp(0) ≈ 46.8 Glyr, De(0) ≈ 16.7 Glyr, DH(0) = c/H₀ ≈ 14.5 Glyr.