Why does χ differ from c/H?

c/H is a local expansion timescale distance. The particle horizon involves the full integral of c/a back to the early universe and typically grows to many times c/H today in ΛCDM.

Omitted for simplicity; at z ≫ 10³ radiation matters for precision, but the schematic curves here are dominated by matter and Λ for late-time display.

Cosmological Expansion (FLRW)

a(t), z, χ and c/H vs time; flat Ω_m + Λ (toy ΛCDM).

About this model

The Friedmann equation for a spatially flat universe with matter density Ω_m and cosmological constant Ω_Λ = 1 − Ω_m gives H(a) = H₀ √(Ω_m a⁻³ + Ω_Λ). Integrating dt = da/(aH) yields cosmic time vs scale factor; integrating χ = ∫ c dt/a = ∫ c da/(a²H) gives the comoving radius of the particle horizon. The Hubble length c/H is shown for comparison: it is not the same as the particle horizon. This toy integrator is for qualitative teaching, not Planck-level parameter estimation.

Who it's for: After redshift Doppler and cosmic distance ladder; before rigorous CMB physics.

Key terms

scale factor
redshift
FLRW
particle horizon
Hubble length

How it works

Homogeneous isotropic expansion is modeled with a scale factor a(t) (set to 1 today). Redshift is z = 1/a − 1 when photons left a source. In flat FRW with only matter and Λ, Ω_m + Ω_Λ = 1. The comoving particle horizon χ is how far light could have traveled since the hot early universe (integral of c dt/a); it is not the same as c/H, though both grow with time in Λ-dominated eras. This page integrates Friedmann kinematics for teaching; it is not a substitute for CMB/ladder precision cosmology.

Key equations

H² = H₀² (Ω_m a⁻³ + Ω_Λ), flat: Ω_Λ = 1 − Ω_m

dt = da / (a H(a)), χ = ∫ c dt / a = ∫ c da / (a² H)

z = 1/a − 1 (today a = 1)

Frequently asked questions

Why does χ differ from c/H?: c/H is a local expansion timescale distance. The particle horizon involves the full integral of c/a back to the early universe and typically grows to many times c/H today in ΛCDM.
Where is radiation?: Omitted for simplicity; at z ≫ 10³ radiation matters for precision, but the schematic curves here are dominated by matter and Λ for late-time display.

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How it works

Key equations

Frequently asked questions

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How it works

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Frequently asked questions