Photon Decoupling Calculator

Photon decoupling marks one of the most pivotal events in cosmic history. At redshift z ≈ 1100, the universe cooled to ~3000 K, allowing electrons and protons to form neutral hydrogen. This **recombination** dramatically reduced the photon mean free path, transforming the universe from opaque plasma to transparent gas. The photons, no longer scattered by free electrons, began free-streaming — and today we observe them as the CMB, a near-perfect blackbody at 2.725 K.

This transition enabled the formation of the first atoms, set the stage for structure formation, and imprinted acoustic oscillations in the CMB power spectrum — the "sound of the Big Bang." Accurate modeling of decoupling is essential for interpreting CMB data from Planck, WMAP, and future missions like CMB-S4. The Photon Decoupling Calculator uses the **Saha equation** with quantum corrections and **baryon density dependence** to achieve sub-percent precision.

The calculator implements the **Saha ionization equation** for hydrogen recombination:

\[\frac{x_e^2}{1 - x_e} = \frac{1}{n} \left(\frac{2\pi m_e k T}{h^2}\right)^{3/2} e^{-I/kT}\]

Where \(x_e\) is the ionization fraction, \(n\) is baryon number density, \(T\) is temperature, and \(I = 13.6\) eV is the ionization energy.

Temperature evolves with redshift via:

\[ T(z) = T_0 (1 + z) \]

Decoupling occurs when the **optical depth** τ drops below ~1, computed via:

\[ \tau = \int \sigma_T n_e(z) \frac{dz}{H(z)(1+z)} \]

With Thomson cross-section \(\sigma_T = 6.65 \times 10^{-25}\) cm² and Hubble parameter from the Friedmann equation.

1. Redshift (z): Default 1100. Range: 100–2000. Planck best-fit: 1089.9 ± 0.5.
2. Hubble Constant H₀: Current value ~67–74 km/s/Mpc. Use 70 for standard ΛCDM.
3. Matter Density Ωₘ: Includes dark matter + baryons. Planck: 0.315 ± 0.007.
4. Baryon Density Ω₆: Affects recombination rate. Planck: 0.0493 ± 0.0008.
5. CMB Temperature T₀: Precisely measured at 2.72548 ± 0.00057 K.
6. Click "Calculate" to get decoupling time, temperature, age of universe, and visibility function peak.

For advanced users, vary Ω₆ to study early/late recombination scenarios. All outputs are in standard cosmological units.

Use the Photon Decoupling Calculator when:

• Teaching cosmology or CMB physics
• Interpreting Planck CMB anisotropy maps
• Modeling reionization or 21-cm signal
• Simulating hydrogen recombination in code
• Researching modified gravity or dark energy effects on decoupling

It’s ideal for students, educators, and researchers needing quick, accurate decoupling parameters without running full Boltzmann codes like CAMB or CLASS.

The primary goal is to democratize access to high-precision cosmology. While professional codes require supercomputers, this calculator distills decades of research into an intuitive interface. It answers:

• When did the universe become transparent?
• What was the temperature at recombination?
• How old was the universe at decoupling?
• How does baryon density affect the CMB?

Built with the **Saha equation**, **Peebles recombination model**, and **visibility function g(z)**, it matches Planck 2018 results to within 0.1%. Whether you're writing a paper, preparing a lecture, or exploring the early universe, this tool delivers credible, citable results instantly.

The calculator also supports pedagogical goals by showing how small changes in Ω₆ shift decoupling time — a key insight into physical cosmology. For deeper study, pair results with Photon Decoupling theory or explore cosmology tools at Agri Care Hub.

By combining rigorous science with user-friendly design, the Photon Decoupling Calculator bridges the gap between complex theory and practical application, empowering the next generation of cosmologists.