Apparent magnitude vs redshift; flat ΛCDM distance modulus; grid fit for H₀ and Ω_m.
About this model
Type Ia supernovae (after standardization) map to a luminosity distance D_L(z) in a spatially flat ΛCDM cosmology with Ω_m + Ω_Λ = 1. The comoving line-of-sight distance is χ = (c/H₀)∫₀^z dz′/E(z′) with E(z) = √(Ω_m(1+z)³ + Ω_Λ), and D_L = (1+z)χ. The distance modulus μ = 25 + 5 log₁₀(D_L / 1 Mpc) is plotted against redshift z — the classic Hubble diagram. This page uses a trapezoid rule integrator and a coarse grid plus greedy local steps to fit (H₀, Ω_m) to scattered toy points; it omits Malmquist bias, host-galaxy extinction, and the full SALT2/STRETCH calibration chain used in real cosmology papers.
Who it's for: Introductory cosmology after distance modulus and FLRW kinematics; pairs with the FLRW expansion and cosmic distance ladder pages.
Key terms
Hubble diagram
Type Ia supernova
distance modulus
luminosity distance
flat ΛCDM
dark energy
How it works
Type Ia supernovae trace the luminosity distanceD_L(z) through their standardized apparent magnitudes. In a spatially flatΛCDM model with Ω_m + Ω_Λ = 1, the line-of-sight comoving distance is χ = (c/H₀)∫₀^z dz′/E(z′) with E = √(Ω_m(1+z)³ + Ω_Λ), and D_L = (1+z)χ. The distance modulusμ = m − M = 25 + 5 log₁₀(D_L / 1 Mpc) is plotted vertically vs z. Adjust H₀ and Ω_m to slide the gold curve, or press Fit flat ΛCDM for a coarse grid + local search that minimizes the mean squared residual in m (toy errors only — no Malmquist bias, no stretch correction).
Frequently asked questions
Why is the fit only in (H₀, Ω_m)?
Absolute magnitude M is degenerate with a vertical shift in m at fixed z; this toy holds M fixed inside the generated “truth” and fits geometry only. Real analyses marginalize over nuisance parameters and use larger compilations.
Does this include radiation or curvature?
No — flatness is enforced with Ω_Λ = 1 − Ω_m, and radiation is ignored so the integrand matches the same late-universe teaching model as the FLRW expansion page.