Question

At about \(10^{-6}\) s after the Big Bang, the universe had cooled to a temperature of approximately \(10^{13} \mathrm{~K}\). a) Calculate the thermal energy \(k_{\mathrm{B}} T\) of the universe at that temperature. b) Explain what happened to most of the hadrons-protons and neutrons-at that time. c) Explain what happened to electrons and positrons in terms of temperature and time.

Short answer

The thermal energy of the universe at a temperature of \(10^{13} K\) was \(1.381 \times 10^{-10} J\). b) What happened to most of the hadrons (protons and neutrons) in the early universe? Most of the hadrons in the early universe underwent a rapid process called "hadron annihilation." The majority of these hadrons and antihadrons collided, annihilating each other and releasing energy in the form of photons. As a result, the number of hadrons reduced significantly, leaving only a small excess of protons and neutrons that would later combine to form atomic nuclei during nucleosynthesis. c) What occurred with electrons and positrons in terms of temperature and time? As the universe expanded and cooled, electrons and positrons underwent pair production and annihilation processes. Eventually, the temperature dropped below the threshold for efficient pair production, and the remaining electrons and positrons annihilated each other, releasing energy in the form of photons. By about \(10^{11}\) K and at around \(1\) s after the Big Bang, almost all electron and positron pairs had annihilated, leaving a small excess of electrons which later combined with protons and neutrons to form atoms during the epoch of recombination, about 380,000 years after the Big Bang.

Step-by-step solution

Calculate thermal energy

First, we need to calculate the thermal energy of the universe at that time. To do this, we will use the formula for thermal energy: \(k_B T\), where \(k_B\) is the Boltzmann constant (\(1.38064852 \times 10^{-23} J\cdot K^{-1}\)), and \(T\) is the given temperature (\(10^{13} K\)). \(k_B T = (1.38064852 \times 10^{-23} \mathrm{J\cdot K^{-1}}) \cdot (10^{13} \mathrm{K})\) Calculating this, we get: \(k_B T = 1.381 \times 10^{-10} \mathrm{J}\)

Describe what happened to hadrons

At about \(10^{-6}\) s after the Big Bang, most hadrons (protons and neutrons) in the universe underwent a rapid process called "hadron annihilation." During this time, the universe was filled with a hot, dense soup of quarks, gluons, and other particles. As the universe cooled, quarks and antiquarks combined to form protons, neutrons, and their antiparticles (antiprotons and antineutrons). The majority of these hadrons and antihadrons collided, annihilating each other and releasing energy in the form of photons. As a result, the number of hadrons reduced significantly, leaving only a small excess of protons and neutrons that would later combine to form atomic nuclei during nucleosynthesis.

Explain the fate of electrons and positrons

The electrons and positrons in the early universe also underwent a similar process as the hadrons. At high temperatures, these particles were produced and annihilated through pair production and pair annihilation processes. As the universe expanded and cooled, the rate of production of electron-positron pairs dropped, and the rate of annihilation increased. Eventually, the temperature dropped below the threshold for efficient pair production, and the remaining electrons and positrons annihilated each other, releasing energy in the form of photons. This process took longer for electrons and positrons than it did for hadrons due to their smaller mass, which means that they required a lower temperature to undergo pair production and annihilation. By about \(10^{11}\) K and at around \(1\) s after the Big Bang, almost all electron and positron pairs had annihilated, leaving a small excess of electrons which later combined with protons and neutrons to form atoms during the epoch of recombination, about 380,000 years after the Big Bang.

Key concepts

Hadron Annihilation

The early universe was a hotbed of activity, teeming with elementary particles shortly after the Big Bang. Among these particles, hadrons—a category that includes protons and neutrons—played a pivotal role. During a period known as hadron annihilation, the universe's temperature was so high that quarks and antiquarks frequently combined into hadrons and their corresponding antiparticles.

As the primordial soup cooled, these matter-antimatter pairs collided, leading to annihilation. This process released vast amounts of energy, which contributed to the universe's thermal energy. Despite the extensive annihilation, a slight asymmetry favored matter over antimatter, leaving behind the protons and neutrons that would eventually contribute to the formation of the universe's first atomic nuclei. These events paved the way for the complex structures we observe today, from stars to galaxies.

Pair Production and Annihilation

In the scorching heat of the early universe, conditions were ripe for a dazzling phenomenon called pair production. This refers to the creation of a particle and its antiparticle from energy—think of it as energy transforming into mass, according to Einstein's famous equation, E=mc^2. Electrons and their antiparticles, positrons, popped into existence by drawing on the intense thermal energy available at the time.

Pair annihilation was the inevitable flip side. As particles and antiparticles met, they obliterated each other, converting their mass back into energy in the form of high-energy photons. This interplay continued in the cosmic dance until the universe expanded and cooled to the point where pair production dwindled, tilting the balance toward annihilation and reducing the population of these particles substantially.

Cosmic Background Radiation

The remnants of these early universe annihilations are not lost to time—they linger on as cosmic background radiation. This is the afterglow of the Big Bang and encompasses the entire cosmos as a faint microwave radiation. Detected serendipitously by Arno Penzias and Robert Wilson in 1965, this cosmic microwave background (CMB) provides us with a snapshot of the universe just 380,000 years after its birth when atoms first formed and photons could travel freely without being scattered by electrons and protons.

Today, studying the CMB gives scientists a wealth of information about the universe's early moments, its composition, and its evolution over billions of years.

Primordial Nucleosynthesis

After the universe went through its hadron annihilation period and the pair production and annihilations dwindled, it set the stage for primordial nucleosynthesis. This epoch marks a significant chapter in cosmic history as the first step toward creating a chemically varied universe from the fundamental building blocks that had survived earlier events.

Over time, as the temperature continued to drop, protons and neutrons began to collide and fuse into deuterium and other light nuclei, like helium and lithium. This process heavily depended on the delicate interplay between the expansion rate of the universe and the nuclear forces at play within these early particles. Primordial nucleosynthesis lasted for only a few minutes, but it had an enduring legacy, seeding the universe with the initial ingredients needed for the development of stars, galaxies, and eventually life.