The Epoch of Reionization (EoR) marks a fundamental transition in the history of the Universe during which the neutral intergalactic medium (IGM) was transformed into an ionized state by the radiation from the first generations of stars and active galactic nuclei (AGNs).

After the Big Bang, the Universe was a hot plasma of free electrons and nuclei. Approximately 380,000 years after the Big Bang ($z \sim 1100$), the temperature dropped to roughly 3000 K, allowing electrons to combine with protons to form neutral hydrogen atoms. This event, known as recombination, made the Universe transparent to photons, which we observe today as the cosmic microwave background (CMB). Following recombination, the Universe entered the “Dark Ages,” a period where no luminous sources yet existed and the IGM was almost entirely neutral.

Reionization began as the first structures collapsed under gravity to form the earliest stars (Population III) and AGNs.

The First Stars (Population III): These stars formed in low-mass dark matter halos ($M \sim 10^4 M_{\odot}$). Because the gas was metal-free, cooling—a prerequisite for star formation—could only occur via molecular hydrogen ($H_2$).
Mathematical Condition (Jeans Mass): For gas to fall into dark halos and collapse, the mass must exceed the Jeans mass ($M_J$). For $z \lesssim 140$, this is given by:

$$M_J = 5.7 \times 10^3 \left( \frac{\Omega_m h^2}{0.15} \right)^{-1/2} \left( \frac{\Omega_b h^2}{0.022} \right)^{-3/5} \left( \frac{1+z}{10} \right)^{3/2} M_{\odot}$$.

Molecular Dissociation: The first Population III stars produced photons that dissociated $H_2$ at energies between 11.26 eV and 13.6 eV. This “self-regulation” temporarily suppressed further star formation until more massive halos ($T_{vir} > 10^4$ K) formed, allowing for cooling via atomic hydrogen.

As more stars and quasars formed, they created expanding bubbles of ionized hydrogen (H II regions) around them.

Equilibrium of Ionization: The density of neutral hydrogen ($n_{HI}$) in the IGM is determined by the balance between the photoionization rate ($\Gamma_{HI}$) and the recombination rate ($\alpha$):

$$n_{HI} = \frac{\alpha}{\Gamma_{HI}} n_p^2$$.

Recombination Rate: The number of recombinations per unit volume per second is given by:

$$-\frac{dn_e}{dt} = n_e^2 \alpha(T_e)$$.

Completion: The epoch ended when these H II regions overlapped, rendering the entire IGM effectively ionized.

The physics of reionization is fundamentally a race between two processes: ionization (radiation tearing atoms apart) and recombination (electrons and protons finding each other to become neutral again).

We quantify the progress of reionization using the volume filling factor of ionized hydrogen, denoted as $Q_{HII}$. This represents the fraction of the universe’s volume that has been ionized.

The evolution of reionization is governed by a differential equation balancing the production of ionizing photons against the rate at which atoms recombine:

$$\frac{dQ_{HII}}{dt} = \frac{\dot{n}_{ion}}{\langle n_H \rangle} - \frac{Q_{HII}}{\bar{t}_{rec}}$$

Where: * $\dot{n}_{ion}$ is the comoving emission rate of ionizing photons (how fast galaxies are pumping out UV light). * $\langle n_H \rangle$ is the mean comoving density of hydrogen atoms. * $\bar{t}_{rec}$ is the average recombination time (how long it takes an electron to recombine with a proton).

The Recombination Time and Clumping: Recombination happens faster in dense regions. Because the universe is not perfectly smooth, we must introduce the clumping factor ($C$), defined as $C = \langle n_H^2 \rangle / \langle n_H \rangle^2$. The recombination time is expressed as:

$$\bar{t}_{rec} = \frac{1}{C \alpha_B \langle n_e \rangle (1+z)^3}$$

Where $\alpha_B$ is the “Case B” recombination coefficient (which accounts for electrons cascading down energy levels) and $z$ is the redshift. If the gas is highly clumped ($C > 1$), recombination happens much faster, meaning galaxies have to work much harder to keep the universe ionized.

The Photon Budget: To figure out $\dot{n}_{ion}$, astronomers must model the sources of early light. The production of ionizing photons is typically modeled as:

$$\dot{n}_{ion} = f_{esc} \xi_{ion} \dot{\rho}_{SFR}$$

Where: * $\dot{\rho}_{SFR}$ is the cosmic Star Formation Rate density (how much stellar mass is being born per unit volume per year). * $\xi_{ion}$ is the efficiency of those stars at producing ionizing photons (dependent on the types of stars; early, hot stars have a higher $\xi_{ion}$). * $f_{esc}$ is the escape fraction: the critical percentage of ionizing photons that actually escape the dense gas of their host galaxy and make it into the IGM.

1. The Gunn-Peterson Test: Spectra of high-redshift quasars ($z \gtrsim 6$) show a “Gunn-Peterson trough,” where all radiation blueward of the $Ly\alpha$ emission line is absorbed by diffuse neutral hydrogen. The optical depth ($\tau$) for this absorption is expressed as:

$$\tau \approx 4.14 \times 10^{10} h^{-1} \frac{n_{HI}(z)}{\text{cm}^{-3}} (1+z)^{-1} (1+\Omega_m z)^{-1/2}$$.

The disappearance of this trough at $z \lesssim 5.8$ indicates the Universe was fully reionized by that time.

2. CMB Polarization: Results from the WMAP satellite detected unexpectedly high polarization in the CMB on large angular scales. This polarization results from the Thomson scattering of CMB photons by free electrons in the reionized IGM. These measurements suggest that reionization occurred much earlier than previously thought, at a redshift of $z \sim 15$.

3. Metal Enrichment: Even the most distant quasars ($z \sim 6$) show metallicities near 1/10th of the solar value, proving that significant star formation and supernova explosions had already enriched the IGM during the reionization process.