Radiation-Domination Calculator | Cosmology Tool

Radiation-Domination Calculator

The Radiation-Domination Calculator is a precise scientific tool designed to calculate when the early universe transitioned from the radiation-dominated era to the matter-dominated era. In the standard ΛCDM cosmological model, the universe was initially dominated by relativistic particles (photons and neutrinos) immediately after the Big Bang. This radiation-dominated phase lasted until approximately redshift z ≈ 3400, when matter-radiation equality occurred.

This calculator uses the well-established Friedmann equation and the evolution of energy density components (radiation ∝ a⁻⁴, matter ∝ a⁻³) to determine the exact redshift, temperature, and cosmic time at which radiation energy density equaled matter energy density (Ωᵣ = Ωₘ). It is an essential tool for students, researchers, and cosmology enthusiasts studying the thermal history and evolution of the universe.

About the Radiation-Domination Calculator

The early universe went through several distinct epochs: inflation, radiation domination, matter domination, and the current dark energy domination. The radiation-dominated era began roughly 10⁻³² seconds after the Big Bang and lasted until about 47,000–70,000 years later, depending on exact cosmological parameters.

During this period, the energy density was dominated by relativistic particles — primarily photons and neutrinos. The scale factor evolved as a(t) ∝ t⁻¹/², and temperature scaled as T ∝ 1/a. The expansion was so rapid that structure formation was suppressed because photons prevented baryonic matter from clustering.

Scientific Foundation and Formulas Used

This calculator strictly follows peer-reviewed cosmology and is based on the Friedmann-Lemaître-Robertson-Walker (FLRW) metric and ΛCDM model. The key equality condition is:

ρᵣ(a) = ρₘ(a) → ρᵣ,₀ / a⁴ = ρₘ,₀ / a³

Solving gives the scale factor at equality:

aₑq = Ωᵣ,₀ / Ωₘ,₀

Where:

Ωᵣ,₀ = 2.47 × 10⁻⁵ h⁻² (1 + 0.227 × Nₑff) – today’s radiation density parameter
Nₑff = effective number of neutrino species (standard = 3.046)
Ωₘ,₀ = matter density parameter (baryons + cold dark matter)
h = Hubble constant H₀ / (100 km/s/Mpc)

The redshift of equality is then:

zₑq = (1/aₑq) − 1 ≈ Ωₘ,₀ / Ωᵣ,₀ − 1 ≈ 3400 (for Planck 2018 values)

The cosmic time at equality is computed by integrating the Friedmann equation from a = 0 to aₑq during the radiation + early matter phase.

Why This Calculator is Important

Understanding the transition from radiation to matter domination is crucial because:

It marks the moment when gravitational instability could finally overcome radiation pressure, allowing the first structures (proto-galaxies) to form.
It determines the horizon size at matter-radiation equality, which sets the scale for the turnover in the matter power spectrum.
It is a key input for CMB anisotropy calculations and baryon acoustic oscillations (BAO).
It helps interpret observations of the cosmic microwave background (CMB) temperature and polarization power spectra.

When and Why You Should Use This Tool

Use this Radiation-Domination Calculator when:

Studying physical cosmology or general relativity courses
Researching the thermal history of the universe
Preparing lectures or educational content on Big Bang cosmology
Comparing different cosmological models (e.g., varying Nₑff or h)
Exploring how changing Ωₘh² affects structure formation timescales

User Guidelines

1. Enter the matter density parameter Ωₘ (today, typically 0.27–0.32)
2. Enter the Hubble constant in units of km/s/Mpc (default Planck 2018: 67.74)
3. Optionally adjust the effective neutrino species Nₑff (standard = 3.046)
4. Click "Calculate" to get:
  • Redshift of equality (zₑq)
  • Temperature at equality
  • Cosmic time at equality
  • Scale factor at equality

All results are based on latest peer-reviewed cosmological parameters and equations from Planck Collaboration (2018 & 2020) and standard textbooks (e.g., Ryden, Weinberg, Dodelson).

For a detailed theoretical explanation of the radiation-dominated universe, visit the excellent resource on Radiation-Domination.

This calculator is proudly powered by scientific accuracy and maintained by Agri Care Hub educational tools division.

The radiation-dominated era is one of the most fascinating phases in cosmic evolution. After the brief inflationary period, the universe was extremely hot (T > 10¹⁰ K) and dense, filled with a plasma of quarks, gluons, leptons, and gauge bosons. As it expanded and cooled, it underwent baryogenesis, big bang nucleosynthesis (BBN), and eventually photon decoupling at recombination (z ≈ 1100).

However, the transition from radiation to matter domination occurred much earlier, around z ≈ 3400, when the universe was only about 50,000–70,000 years old and still extremely hot (T ≈ 9000 K). At that moment, the energy density in relativistic particles finally dropped below that of non-relativistic matter (baryons + cold dark matter).

This transition had profound consequences: acoustic oscillations in the photon-baryon fluid froze out, leaving imprints we observe today as peaks in the CMB power spectrum. The equality scale also defines the maximum size of dark matter halos that could start collapsing early.

Modern cosmology relies heavily on precise determination of this epoch. The latest CMB experiments (Planck, ACT, SPT) combined with large-scale structure surveys (DESI, Euclid, LSST) continue to refine these parameters to sub-percent accuracy.