Matter-Radiation Equality Calculator

The Matter-Radiation Equality Calculator is a precise scientific tool that determines the exact moment in cosmic history when the energy density of matter (baryons + cold dark matter) equaled the energy density of radiation (photons + neutrinos). This pivotal epoch, known as matter-radiation equality, occurred approximately 50,000–70,000 years after the Big Bang and marks the transition from a radiation-dominated to a matter-dominated universe.

About the Matter-Radiation Equality Calculator

The Matter-Radiation Equality Calculator uses the well-established Friedmann-Lemaître-Robertson-Walker (FLRW) cosmology and the standard ΛCDM model backed by observations from Planck, WMAP, and other major surveys. The core formula for the redshift of matter-radiation equality is derived from equating the matter and radiation energy densities:

ρₘ(a) = ρₘ,₀ × a⁻³
ρᵣ(a) = ρᵣ,₀ × a⁻⁴

→ ρₘ(aₘᵣ) = ρᵣ(aₘᵣ) → Ωₘ,₀ a⁻³ = Ωᵣ,₀ a⁻⁴
→ 1 + zₘᵣ = Ωₘ,₀ / Ωᵣ,₀ ≈ 3400 – 3600 (Planck-era value)

Where Ωᵣ,₀ ≈ 9.2 × 10⁻⁵ (including photons and neutrinos with effective neutrino temperature) is calculated using current CMB temperature T₀ = 2.7255 K.

Why Is Matter-Radiation Equality Important?

Matter-radiation equality is one of the most crucial milestones in cosmic evolution because:

• It marks the transition from radiation-dominated to matter-dominated expansion
• Before equality, potential wells could not grow → suppressed structure formation
• After equality, gravitational instability rapidly amplifies density perturbations
• It sets the turnover scale in the matter power spectrum (kₑq ≈ 0.01 h/Mpc)
• It affects the height of the first acoustic peak in the CMB power spectrum
• It influences early Integrated Sachs-Wolfe (ISW) effect and baryon acoustic oscillations

When and Why Should You Use This Calculator?

Use this Matter-Radiation Equality Calculator when you need precise values for:

• Cosmological simulations and N-body modeling
• Teaching university-level cosmology or astrophysics courses
• Research papers requiring exact zₘᵣ for different cosmologies
• Understanding the impact of Ωₘh² on structure formation
• Comparing different cosmological datasets (Planck vs WMAP)

Scientific Background and Formulas Used

The radiation density parameter today is:

Ωᵣ,₀ h² = 4.153 × 10⁻⁵ (photons + neutrinos, Nₑff = 3.046)

Thus the redshift of equality is:

1 + zₘᵣ = Ωₘ,₀ / Ωᵣ,₀ = Ωₘ,₀ h² / 4.153×10⁻⁵

For Planck 2018 best-fit parameters (Ωₘ,₀ = 0.315, h = 0.674), this yields zₘᵣ ≈ 3402, occurring ~57,000 years after the Big Bang at a temperature of ~9300 K.

User Guidelines

1. Select “Planck 2018” for the most accurate current cosmology
2. Use “Custom Parameters” only if you have specific Ωₘ,₀ and h values from your analysis
3. The calculator includes neutrino contribution with Nₑff = 3.046 (standard in modern cosmology)
4. All results are based on peer-reviewed papers and official collaboration releases

This Matter-Radiation Equality Calculator is built with the latest cosmological parameters (2024–2025) and strictly follows peer-reviewed methodologies used by the Planck Collaboration.