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
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.
More from Astronomy & The Sky
Other simulators in this category — or see all 28.
Galaxy Rotation Curve
Keplerian decline vs flat v(r); toy halo slider (dark matter motivation).
Stellar Life Cycle
Cloud → MS → giant/SN → WD / NS / BH vs initial mass (schematic).
Exoplanet Radial Velocity
K from masses & P; sinusoidal V_r(t); M sin i.
Exoplanet Transit (light curve)
Uniform disk overlap; R_p/R_*; impact b; F(t) vs period.
Sphere of Influence (Hill)
r_H ≈ a (m/3M)^(1/3): schematic secondary orbit and Hill radius vs masses and a.
Measuring c (ToF toy)
c ≈ 2D/Δt round-trip; schematic path + Fizeau/Foucault context.