In the very early stages of the universe, it is usually a...

In the very early stages of the universe, it is usually a good approximation to neglect particle masses and chemical potential compared with $k T$. (a) Write down the average number and energy densities of a gas of non-interacting fermions in thermal equilibrium under these conditions. (You need not evaluate dimensionless integrals of order 1.) (b) If the gas expands adiabatically while remaining in equilibrium, how do the average number and energy densities depend on the dimensions of the system? Assume that the fermions are predominantly electrons and positrons when $T \simeq 10^{11} \mathrm{~K}$ in parts (c) and (d) below. (c) Is the assumption made in (a) that the particles are non-interacting reasonable? Why? [Hint: What is the average coulomb interaction energy? Positron charge $=1.6 \times 10^{-10}$ coulomb; Boltzmann's constant $k=1.38 \times$ $\left.10^{-16} \mathrm{erg} / \mathrm{K}\right]$. (d) If the interaction cross sections in the electron-positron gas are typically of order of magnitude of the Thompson cross section $\sigma_T=8 \pi r_0^2 / 3$ (classical electron radius $r_0=2.8 \times 10^{-13} \mathrm{~cm}$ ), estimate the mean free time between collisions of the particles. If the expansion rate in part (b) $\approx$ $10^4 \mathrm{sec}^{-1}$, is the assumption that the gas remains in equilibrium reasonable? Why?

In the very early stages of the universe, it is usually a good approximation to neglect particle masses and chemical potential compared with $k T$.
(a) Write down the average number and energy densities of a gas of non-interacting fermions in thermal equilibrium under these conditions. (You need not evaluate dimensionless integrals of order 1.)
(b) If the gas expands adiabatically while remaining in equilibrium, how do the average number and energy densities depend on the dimensions of the system?

Assume that the fermions are predominantly electrons and positrons when $T \simeq 10^{11} \mathrm{~K}$ in parts (c) and (d) below.
(c) Is the assumption made in (a) that the particles are non-interacting reasonable? Why? [Hint: What is the average coulomb interaction energy?
Positron charge $=1.6 \times 10^{-10}$ coulomb; Boltzmann's constant $k=1.38 \times$ $\left.10^{-16} \mathrm{erg} / \mathrm{K}\right]$.
(d) If the interaction cross sections in the electron-positron gas are typically of order of magnitude of the Thompson cross section $\sigma_T=8 \pi r_0^2 / 3$ (classical electron radius $r_0=2.8 \times 10^{-13} \mathrm{~cm}$ ), estimate the mean free time between collisions of the particles. If the expansion rate in part (b) $\approx$ $10^4 \mathrm{sec}^{-1}$, is the assumption that the gas remains in equilibrium reasonable? Why?

Key Concepts

Mean Free Path and Mean Free Time

The mean free path is the average distance a particle travels before undergoing a collision, while the mean free time is the corresponding average time interval between collisions. These concepts are determined using the interaction cross section and particle densities. They are critical in assessing whether a system remains in equilibrium; if the collision rate (inverse of mean free time) is much faster than any dynamical expansion rate, the system can maintain equilibrium.

Coulomb Interaction Energy in a Plasma

Coulomb interaction energy refers to the potential energy between charged particles due to their electric charge. In a high-temperature plasma, even if the particles are charged (like electrons and positrons), the average Coulomb interaction energy can be very small compared to the thermal energy (kT), justifying the assumption of non-interaction under certain early-universe conditions.

Scaling of Densities with Expansion

When a gas expands adiabatically, its number density scales inversely with the volume, typically as L?³ in a three-dimensional space. Similarly, the energy density depends on both the number density and the temperature evolution, often scaling as L?? for a radiation-dominated gas. Such scaling relations are key to understanding the evolution of the early universe where energy and particle distributions change with the expanding metric.

Thomson Cross Section

The Thomson cross section gives the effective cross-sectional area for low-energy scattering of electromagnetic radiation by free charged particles, such as electrons or positrons. In an early-universe context, even though the interactions are electromagnetic, the magnitude of this cross section helps estimate collision frequencies, and if the resulting mean free time is short compared to the expansion time scale, the equilibrium assumption holds.

Thermal Equilibrium

Thermal equilibrium implies that the system's properties, such as temperature, are spatially uniform and time-independent under slow or adiabatic evolution. In equilibrium, the statistical distributions (like the Fermi-Dirac distribution) reliably describe the population of energy states, making it possible to derive macroscopic quantities such as number density and energy density from microscopic interactions.

Ultra-Relativistic Approximation

In the ultra-relativistic limit, particle masses can be neglected compared to the kinetic energies (kT), simplifying the energy-momentum relation to E = pc. This approximation is valid at very high temperatures and allows integrals for number and energy densities to be expressed in a dimensionless form, significantly simplifying the statistical mechanics formulas for particles such as electrons and positrons in the early universe.

Fermi-Dirac Statistics

Fermi-Dirac statistics describe the occupancy of energy states by fermions at finite temperature, taking into account the Pauli exclusion principle which prohibits more than one fermion from occupying the same quantum state. This distribution is crucial for calculating average number and energy densities in systems of non-interacting fermions in thermal equilibrium, especially in conditions where quantum effects dominate.

Adiabatic Expansion

Adiabatic expansion refers to the process where a system expands without exchanging heat with its surroundings. In cosmological contexts, this means that as the universe expands, the entropy remains constant. As a result, thermodynamic quantities like temperature and densities evolve as functions of the scale factor, allowing us to relate the decrease of number and energy densities to the increasing volume.

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