(II) At approximately what time had the universe cooled...

(II) At approximately what time had the universe cooled below the threshold temperature for producing (a) kaons $\left(M \approx 500 \mathrm{MeV} / c^{2}\right)$ (b) $\Upsilon\left(M \approx 9500 \mathrm{MeV} / c^{2}\right)$ and $(c)$ muons $\left(M \approx 100 \mathrm{MeV} / c^{2}\right) ?$

Key Concepts

Temperature-Time Relationship in a Radiation-Dominated Universe

In the early universe, when radiation dominated the energy density, a specific relation existed between time and temperature, often characterized by T ? t^(-1/2). This relationship enables physicists to estimate the time at which the universe cooled to a given temperature threshold necessary for the production or freeze-out of particles, linking particle physics scales with cosmological evolution.

Freeze-Out and Decoupling

As the universe cools below the particle production threshold, the production of certain particle types becomes exponentially unlikely, effectively 'freezing out' their abundances. This decoupling from the thermal bath signifies that the particle's interactions no longer keep pace with the expansion, leading to a relic abundance that remains largely unchanged in later cosmic evolution.

Thermal History of the Universe

This concept describes the evolution of the universe from an extremely hot and dense initial state to its current cooler and more diffuse condition. As the universe expands, the cooling process undergoes several key phases, during which different particles can be created or become unable to be produced. Understanding the thermal history is crucial to linking cosmic time with the energy conditions that allowed the formation or freeze-out of various particles.

Particle Production Threshold

Particles in the early universe are produced when the ambient temperature is comparable to or exceeds their rest mass energy. This threshold condition means that only when the thermal energy of the universe is high enough can a certain type of particle be generated, setting a limit on the energy (or temperature) required for the production of particles like muons, kaons, and heavy mesons.

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