iCosmos › Reference

Equations, methods and validation

What iCosmos calculates, how, and how accurately.

The model

A homogeneous and isotropic (FLRW) universe with matter, radiation, curvature and dark energy with the equation of state w(a) = w₀ + w_a(1 − a). The expansion rate is H(z) = H₀ E(z), with

E²(z) = Ω_r(1+z)⁴ + Ω_m(1+z)³ + Ω_k(1+z)² + Ω_Λ(1+z)^{3(1+w₀+w_a)} exp[−3 w_a z/(1+z)].

Curvature is not an input: Ω_k = 1 − Ω_m − Ω_Λ − Ω_r.

Distances and volume

• Line-of-sight comoving distance: D_C = (c/H₀) ∫₀^z dz′/E(z′).
• Transverse comoving distance D_M: equal to D_C in a flat universe, and (c/H₀) sinh(√Ω_k D_CH₀/c)/√Ω_k in an open one (sin and |Ω_k| in a closed one).
• Angular diameter distance D_A = D_M/(1+z); luminosity distance D_L = (1+z) D_M.
• Whole-sky comoving volume: (4π/3) D_M³ in a flat universe, with the curved-space forms otherwise.

Time

Lookback time t_L = (1/H₀) ∫₀^z dz′/[(1+z′) E(z′)]; the age of the universe is the same integral taken to infinite redshift.

Growth and survey observables

The linear growth factor is found by solving the growth equation numerically. The sound horizon r_d, the BAO ratios D_M/r_d, D_H/r_d and D_V/r_d, and σ₈(z), fσ₈(z) and S₈ are described on the BAO and σ₈ pages.

Validation

Every number is checked automatically against independent calculations. On the published benchmark of 21 universes, distances and ages agree with SciPy and mpmath reference values to about 1 part in 10¹⁴ and the growth factor to about 2 × 10⁻⁹; in stress tests over 400 random matter + Λ + curvature universes (H₀ = 70, Ωm 0.005–3, ΩΛ −1 to 3.5, including near-loitering and very low-density open models), the growth factor agrees with the exact Heath integral to better than 10⁻⁹. A further 200 random evolving-dark-energy universes are checked for finite or explained results, not for accuracy. Where a quantity cannot be computed reliably, the calculator says so instead of showing a number. The validation tables are on the Reference page.

Also on this page

• Constants and units
• Numerical methods
• Data sources (DESI DR2 BAO measurements)
• Differences from classic iCosmos (2008–2015)
• Preset models and their sources
• Further reading

How to cite

Vardanyan, Mihran. iCosmos V4: Cosmology Calculator. https://icosmos.co.uk/. Include your access date, model parameters and exported data when reporting calculations.

Open the iCosmos cosmology calculator

iCosmos 4.5.0 · updated 10 November 2026