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====== Epoch of Reionization ======

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).

===== Context: From Recombination to the Dark Ages =====

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.


[{{ :courses:ast403:reionization.png? | Fig 1: Reionization era.}}]
===== The Onset of Reionization =====

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.

===== The Reionization Process =====

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. 


[{{ :courses:ast403:eor_process.webp | Fig 2: Artist impression of the evolution of the Universe.}}]

===== Observational Evidence =====

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 ($\tau_{GP}$) for this absorption is expressed as:

 
$$\tau_{GP} \approx 10^5 x_{HI} \left( \frac{1+z}{7} \right)^{3/2}$$

Here, $x_{HI}$ is the fraction of neutral hydrogen. Notice the massive $10^5$ coefficient. This means that if even $0.1\%$ ($10^{-3}$) of the IGM is neutral, $\tau_{GP}$ becomes very large, and *all* the light is absorbed, creating a massive blackout in the spectrum known as the Gunn-Peterson trough    
    
The disappearance of this trough at $z \lesssim 5.8$ indicates the Universe was fully reionized by that time.

2.  **CMB Polarization:** When the Universe became ionized, it filled with free electrons. As the photons from the Cosmic Microwave Background (CMB) traveled toward us, some scattered off these free electrons via Thomson scattering. 
We measure the integrated optical depth ($\tau_e$) to the CMB:

$$\tau_e = \int_0^{z_{reio}} n_e(z) \sigma_T \frac{c}{(1+z) H(z)} dz$$

Where $n_e(z)$ is the electron density at redshift $z$, $\sigma_T$ is the Thomson scattering cross-section, and $H(z)$ is the Hubble parameter. By measuring the polarization of the CMB, satellites like Planck have measured $\tau_e \approx 0.054$, which tells us that the midpoint of reionization occurred around $z \approx 7.7$.

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.




===== Mapping EoR with 21-cm Hydrogen Line =====

To map the Epoch of Reionization (EoR), astronomers have to completely flip their perspective. Instead of using optical or infrared telescopes to look for the brilliant light of the first galaxies, they use radio telescopes to look for the vast, cold oceans of neutral hydrogen gas that surround them. 

Rather than looking at the light, they are looking at the shadows. The key to this is the 21-cm hydrogen line.

Here is a detailed look at how this signal is generated, how it maps the ionized bubbles, and how massive radio arrays are attempting to detect it.

**The Physics: The Spin-Flip Transition
**
The hydrogen atom consists of a single proton and a single electron. According to quantum mechanics, both of these particles have a property called "spin." When the spins of the proton and electron are **parallel** (spinning in the same direction), the atom is in a slightly higher energy state. When the electron flips so its spin is **anti-parallel** to the proton, the atom drops to a lower energy state. When this "spin-flip" occurs, the atom releases a tiny amount of energy in the form of a radio wave. The wavelength of this photon is exactly 21.1 cm, which corresponds to a rest-frame frequency of roughly 1420 MHz. 

Because the early Universe is filled with unimaginably massive clouds of neutral hydrogen, these rare, individual spin-flips add up to a faint but ubiquitous background glow across the cosmos.


[{{ :courses:ast403:hydrogen-spinflip.svg | Fig 3: Spin-flip transition.}}] 

**Redshift and the Radio Window:
**
Just like the Lyman break, the 21-cm signal is redshifted as the Universe expands. To figure out where we should tune our radio dials to hear the EoR (which occurred between redshifts $z = 6$ and $z = 10$), we use the frequency redshift equation:

$$\nu_{\text{obs}} = \frac{1420 \text{ MHz}}{1+z}$$

If we want to map neutral hydrogen at $z = 8$ (deep in the EoR), the frequency we observe on Earth is shifted down to roughly 157 MHz. 

This presents an interesting quirk for observational astronomy: this frequency falls squarely within the VHF band used for FM radio and television broadcasts on Earth. To map the early Universe, astronomers must build telescopes far away from human interference, in places like the remote deserts of Australia or the Karoo in South Africa.

**The Mathematics of the Signal: Brightness Temperature
**\\

Radio astronomers do not just measure the raw brightness of the 21-cm line; they measure its contrast against the omnipresent backlight of the Cosmic Microwave Background (CMB). This contrast is expressed as the **differential brightness temperature** ($\delta T_b$). 

The full equation for the 21-cm brightness temperature during the EoR is complex, but its core dependencies are elegantly described as:

$$\delta T_b \approx 27 x_{HI} (1 + \delta_b) \left( \frac{\Omega_b h^2}{0.023} \right) \sqrt{\frac{0.15}{\Omega_m h^2} \frac{1+z}{10}} \left( 1 - \frac{T_{\gamma}}{T_S} \right) \text{ mK}$$

Let's break down the two most critical variables for mapping the bubbles:\\

* $x_{HI}$: This is the neutral fraction of the hydrogen gas. If the gas is completely ionized by a nearby galaxy, $x_{HI} = 0$, and the entire equation collapses to zero.\\

* $T_S$ vs $T_{\gamma}$: $T_S$ is the "spin temperature" of the gas, and $T_{\gamma}$ is the temperature of the CMB. The signal is only visible if the gas temperature decouples from the CMB ($T_S \neq T_{\gamma}$).\\

**The Tomographic Map:** Because $\delta T_b$ drops to zero wherever $x_{HI}$ drops to zero, ionized bubbles surrounding early galaxies appear as **"dark holes"** in the 21-cm signal. By tuning their receivers to different frequencies (which correspond to different redshifts, and therefore different "slices" of time), astronomers can build a 3D tomographic map of these dark, growing bubbles eating away at the glowing neutral Universe.


[{{ :courses:ast403:global21cm.png | Fig 4: The global 21cm neutral hydrogen signal over redshift (and frequency) where the top image is the spatially fluctuating 21cm neutral hydrogen and the bottom is the globally averaged 21cm signal. The 21cm signal is strongly coupled to the CMB, which means it is undetectable until the temperature of the neutral hydrogen deviates from the CMB. Borrowed from http://pritchardjr.github.io/research.html.}}]

**The Telescope Arrays: HERA and the SKA:**

A single radio dish cannot resolve these ancient bubbles. To get the necessary resolution, astronomers use interferometry—networking dozens or hundreds of antennas together so they act like a single, massive telescope.

//HERA (Hydrogen Epoch of Reionization Array):// Located in South Africa, HERA consists of hundreds of large, densely packed parabolic dishes pointing straight up. HERA is designed to measure the power spectrum of the 21-cm signal. Rather than taking a high-definition picture of individual bubbles, it measures the statistical variance — telling us the average size of the bubbles and how fast they are growing at specific times.

//SKA (Square Kilometre Array):// Currently under construction across Australia and South Africa, the SKA will be the largest radio telescope ever built. SKA-Low (in Australia) will consist of over 130,000 wire antennas. It will have the raw sensitivity and resolution to go beyond statistics and actually capture true tomographic imaging — giving us direct, 3D video-like maps of the ionized bubbles expanding and merging.

**The Ultimate Challenge: The Foreground Problem:
**
The reason we don't have perfect maps of the EoR yet is due to a monumental data analysis challenge known as the foreground problem.

The 21-cm signal from the EoR is incredibly weak, typically measuring around **$10$ to $30 \text{ mK}$** (millikelvins). However, our own Milky Way galaxy emits massive amounts of synchrotron radiation (electrons spiraling in magnetic fields) at these exact same low frequencies. The Milky Way foreground is upwards of $10^4$K (tens of thousands of degrees).

This means the foreground noise is about **100,000 times louder** than the signal astronomers are trying to find. Finding the 21-cm EoR signal is often compared to trying to hear a mosquito buzzing while standing next to a roaring jet engine. It requires incredibly precise calibration and complex algorithms to mathematically peel away the smooth foreground of the Milky Way to reveal the crinkly, bubbly 21-cm signal hiding underneath.