13.4 Recombination and Decoupling

The early universe underwent a crucial transformation as it cooled. Recombination marked the era when protons and electrons combined to form neutral hydrogen atoms, reducing free electrons for photon scattering. This process set the stage for a monumental shift in cosmic history.

Decoupling followed, allowing photons to travel freely without interacting with matter. This moment birthed the Cosmic Microwave Background radiation, providing a snapshot of the early universe. The CMB's temperature fluctuations reflect density variations that became seeds for large-scale structure formation.

Recombination and Decoupling

The Process of Recombination

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• Recombination marks the era when protons and electrons combined to form neutral hydrogen atoms
• Occurred approximately 380,000 years after the Big Bang
• Universe cooled to about 3000 Kelvin, allowing electrons to bind to protons
• Reduced the number of free electrons available for photon scattering
• Saha equation describes the ionization fraction of hydrogen during recombination
• Ionization fraction represents the ratio of ionized to total hydrogen atoms
• Recombination process was gradual, not instantaneous

Decoupling and the Cosmic Microwave Background

• Decoupling followed recombination as photons could travel freely without interacting with matter
• Marked the moment when the universe became transparent to radiation
• Cosmic Microwave Background (CMB) radiation originated during this period
• CMB provides a snapshot of the early universe at the time of decoupling
• Temperature fluctuations in the CMB reflect density variations in the early universe
• These fluctuations became the seeds for large-scale structure formation

Mathematical Description of Recombination

• Saha equation governs the ionization equilibrium during recombination
• Expressed as: $\frac{n_e n_p}{n_H} = \left(\frac{2\pi m_e k_B T}{h^2}\right)^{3/2} e^{-E_I/k_B T}$
• $n_e$, $n_p$, and $n_H$ represent number densities of electrons, protons, and neutral hydrogen
• $m_e$ is the electron mass, $k_B$ is Boltzmann's constant, $h$ is Planck's constant
• $T$ is temperature, $E_I$ is the ionization energy of hydrogen
• Ionization fraction $x_e = n_e / (n_H + n_p)$ decreases rapidly during recombination
• Saha equation predicts a sharper transition than observed, due to non-equilibrium effects

Last Scattering Surface

Concept and Significance of the Last Scattering Surface

• Last scattering surface represents the boundary of the observable universe
• Defines the furthest distance from which we can receive information carried by photons
• Formed when the universe became transparent to radiation during decoupling
• Acts as a cosmic "wall" beyond which we cannot directly observe
• Provides the earliest image of the universe we can detect
• Crucial for understanding the initial conditions of the universe and its subsequent evolution

Physics of Photon-Matter Interactions

• Thomson scattering dominated photon-matter interactions before recombination
• Involves the elastic scattering of photons by free electrons
• Thomson scattering cross-section: $\sigma_T = \frac{8\pi}{3}\left(\frac{e^2}{m_e c^2}\right)^2$
• $e$ is the electron charge, $m_e$ is the electron mass, $c$ is the speed of light
• Opacity of the early universe determined by the density of free electrons and the Thomson cross-section
• Opacity decreased dramatically during recombination as free electrons combined with protons

Photon Mean Free Path and Universe Transparency

• Photon mean free path represents the average distance a photon travels between scattering events
• Calculated as the inverse of the product of electron number density and Thomson cross-section
• Mean free path increased rapidly during recombination as the number of free electrons decreased
• Universe became transparent when the mean free path exceeded the Hubble radius
• Transparency condition: $\lambda_{mfp} > c H^{-1}$
• $\lambda_{mfp}$ is the photon mean free path, $c$ is the speed of light, $H$ is the Hubble parameter
• Transition from opaque to transparent universe occurred over a finite period, not instantaneously
• Thickness of the last scattering surface relates to the duration of this transition