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. b) Explain what would happen to most of the hadronsprotons and neutrons. c) Explain also about the electrons and positrons in terms of temperature and time.

Short answer

Additionally, describe the behavior of hadrons and the behavior of electrons and positrons during this time. Answer: The thermal energy of the universe at this time was approximately 2.0715 × 10^-9 J. Most hadrons, such as protons and neutrons, began to combine via strong nuclear force to form more complex atomic nuclei in a process called nucleosynthesis, resulting in the formation of light elements like hydrogen and helium. The universe was also filled with a mix of particles and antiparticles, including electrons and positrons. As the universe expanded and cooled, electron-positron pairs began to annihilate, producing photons in a process called electron-positron annihilation. Eventually, the remaining electrons combined with the nuclei formed during nucleosynthesis, leading to the formation of neutral atoms.

Step-by-step solution

Calculate the thermal energy

To calculate the thermal energy, we will use the ideal gas equation: \(E = \dfrac{3}{2}k_B T\), where \(E\) is the thermal energy, \(k_B\) is Boltzmann's constant (\(1.381 \times 10^{-23} \mathrm{J~K^{-1}}\)), and \(T\) is the temperature of the gas. Given the temperature \(T = 10^{13} \mathrm{~K}\), we can calculate the thermal energy. \(E = \dfrac{3}{2}\times (1.381 \times 10^{-23} \mathrm{J ~K^{-1}}) \times (10^{13} \mathrm{~K})\) \(E = 2.0715 \times 10^{-9} \mathrm{J}\) So, the thermal energy of the universe at this time is approximately \(2.0715 \times 10^{-9} \mathrm{J}\).

Explain the behavior of hadrons

At \(10^{-6}\) s after the Big Bang, the universe was incredibly dense and hot. Most of the hadrons such as protons and neutrons began to combine via strong nuclear force to form more complex atomic nuclei in a process called nucleosynthesis. The formation of light elements such as hydrogen and helium occurred at this time. As the universe continued to expand and cool, the density decreased, and protons and neutrons could no longer interact via the strong nuclear force. Consequently, nucleosynthesis could not proceed further, leaving the universe with a specific composition of light elements (mostly hydrogen and helium), which remains up to the present day.

Discuss the behavior of electrons and positrons

At this temperature and time, the universe was filled with a mix of particles and antiparticles, including electrons and positrons (electron-positron pairs). As the universe expanded and cooled, the energy density decreased. When the energy density of the universe fell below the threshold for electron-positron pair production, these pairs began to annihilate, producing photons. This process is called electron-positron annihilation. As the universe continued to cool, the rate of electron-positron pair production decreased significantly. Most of the remaining electrons and positrons annihilated each other, producing photons and leaving a small number of electrons behind. These electrons would later combine with the nuclei formed during nucleosynthesis, leading to the formation of neutral atoms.

Key concepts

Understanding Thermal Energy Calculation Post-Big Bang

Immediately after the Big Bang, the universe was a scorching hotspot with temperatures around 10 trillion Kelvin (\(10^{13}\text{ K}\)). To fathom the intense thermal energy at that moment, we wield the ideal gas equation, which correlates temperature with thermal energy.

Thermal energy, denoted by E, leaps from the rule: \(E = \dfrac{3}{2}k_B T\) where \(k_B\) stands for the Boltzmann constant (\(1.381 \times 10^{-23} \text{J K}^{-1}\) and T embodies the temperature. Inserting our colossal temperature value, one gets a thermal energy of approximately \(2.0715 \times 10^{-9} \text{J}\) per particle.

Why is this important? Thermal energy underpins the various particle interactions that emerged. This thermal landscape set the stage for crucial processes like nucleosynthesis and, as will be noted soon, particle annihilation. But also, knowing how to calculate thermal energy helps us comprehend how an incredibly dense and hot universe began to evolve into the one we inhabit today.

Deciphering Hadron Behavior Post-Big Bang

Right after the Big Bang, when temperatures were some of the highest ever experienced, hadrons—which include protons and neutrons—were exceptionally frantic. They interlaced to compose nucleons, essentially picking dance partners in the cosmic ballroom of nucleosynthesis.

Binding the Cosmic Fabric Together

Hadrons fused due to the strong nuclear force, crafting more complex atomic nuclei. It's within this primeval cauldron that the universe's light element recipes—mostly hydrogen and helium–were penned. As the universe inflated and the thermal hubbub subsided, so did the nucleosynthesis rave; protons and neutrons found themselves too far apart to sustain the energetic frolic of fusion.

Hence, the universe's elemental tapestry stopped expanding, bequeathing us the signature spread of elements we measure today. Knowledge of hadron behavior offers a cornerstone to unravel the universe's evolutionary path, especially in understanding why the cosmos is not an endless sea of ever-more-complex atoms.

Exploring Electron-Positron Annihilation

The early universe wasn't just about hadrons and nuclei; it was also rife with electrons and their antimatter twins, positrons. When matter and antimatter meet, they cancel each other out—a process known as annihilation. After the Big Bang, this was happening on an immense scale.

Particles to Photons: The Transformation

Electron-positron pairs annihilated, fueling the universe with photons—the fundamental particles of light. As the universe's temperature took a dive, fewer pairs were produced, leading to an annihilation spree. This didn't simply result in a fireworks show, but played a pivotal role in thinning the population of electrons and positrons, setting up a scenario for remaining electrons to eventually mingle with nuclei to form stable atoms.

This annihilation is a critical concept for students to grasp as it clarifies why the universe is not filled with equal parts matter and antimatter. It also illuminates the origins of the Cosmic Microwave Background Radiation, which we observe as the afterglow of the Big Bang. Delving into electron-positron annihilation enriches our understanding of the early universe and its gradual journey from a chaotic hot soup to a structured, life-sustaining cosmos.