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)

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.

Other simulators in this category — or see all 51.

View category →

NewUniversity / research

Hubble Diagram (SN Ia & ΛCDM fit)

Apparent magnitude vs redshift; flat ΛCDM distance modulus; grid fit for H₀ and Ω_m.

Launch Simulator

NewUniversity / research

Hydrogen Saha Equilibrium (recombination)

n_p n_e / n_H vs T and ionized fraction X; steep Boltzmann knee ~3000 K scale (density dependent).

Launch Simulator

NewUniversity / research

Chandrasekhar limit (Lane–Emden toy)

White-dwarf M–R track for n = 3/2 cold degenerate electrons; UR n = 3 marks the ~1.44 M☉ scale (μ_e dependent).

Launch Simulator

NewUniversity / research

Neutron star TOV: toy M–R (polytrope)

Schwarzschild TOV for polytropes n = 1 or 3/2; K normalized so the ρ_c scan peaks near ~2 M☉ (schematic only).

Launch Simulator

NewSchool

Galaxy Rotation Curve

Keplerian decline vs flat v(r); toy halo slider (dark matter motivation).

Launch Simulator

NewSchool

Stellar Life Cycle

Cloud → MS → giant/SN → WD / NS / BH vs initial mass (schematic).

Launch Simulator

Cosmological Expansion (FLRW)

How it works

Key equations

Frequently asked questions

More from Astronomy & The Sky

Hubble Diagram (SN Ia & ΛCDM fit)

Hydrogen Saha Equilibrium (recombination)

Chandrasekhar limit (Lane–Emden toy)

Neutron star TOV: toy M–R (polytrope)

Galaxy Rotation Curve

Stellar Life Cycle

Cosmological Expansion (FLRW)

How it works

Key equations

Frequently asked questions