What is the difference between hot and cold dark matter? How do astronomers decide which was more important in the formation of large-scale structures such as clusters of galaxies?

Dark matter remains one of the most profound mysteries in modern astrophysics. Among its proposed constituents is hot dark matter (HDM), a term that refers to particles that moved at relativistic speeds during the formation of the universe’s structures. These lightweight, 'hot' particles, like neutrinos, are so named because their high velocities correspond to high temperatures according to the kinetic theory of gases.

To understand HDM, imagine the early universe as a soup of particles moving rapidly in all directions. Due to their swift speeds, hot dark matter would not clump easily to form smaller structures, like stars and galaxies. Instead, it could only form the largest of structures later on, leading to what is known as a 'top-down' model of structure formation. This means that HDM would first create vast, diffuse structures which would subsequently break down to form smaller objects.

Contrasting hot dark matter, cold dark matter (CDM) consists of particles moving at sub-relativistic speeds during the epoch of structure formation. These 'cold' particles, possibly including the theoretical Weakly Interacting Massive Particles (WIMPs), are believed to be much heavier than their hot counterparts.

The slower movement of CDM particles allows them to coalesce and form structures 'from the ground up.' Small clumps of dark matter would have formed first and then merged to create galaxies and larger cosmic structures. This 'bottom-up' formation model aligns with the observed distribution of galaxies and the cosmic web, making CDM a favored candidate in explaining the universe's architecture. Additionally, the hypothesized properties of CDM make it easier to detect indirectly through gravitational effects on visible matter, gravitational lensing, and perturbations in the cosmic microwave background radiation.

The term large-scale structure formation refers to the processes that shaped the universe’s vast network of galaxies, galaxy clusters, and even larger cosmic features such as superclusters and filaments. A major question in cosmology is how these intricate structures came to be from the initially almost uniform distribution of matter after the Big Bang.

According to the cold dark matter paradigm, small structures formed early and eventually built up into the grand cosmic structures we observe. This is supported by the large quantity of structure on small scales, such as the existence of dwarf galaxies and the detailed weaving of the cosmic web. Through gravitational attraction, dark matter halos acted as 'seeds,' pulling together normal matter to form the galaxies and larger structures seen today. This understanding of structure formation is corroborated by a plethora of observations, including the distribution of galaxies and clusters in the universe.

The cosmic microwave background radiation (CMB) is a pivotal source of evidence for cosmology. It encompasses the thermal radiation left over from the period of recombination in Big Bang cosmology and offers a snapshot of the universe when it was only about 380,000 years old.

Small fluctuations in the CMB’s temperature reveal patterns that are connected to the distribution of matter in the early universe. By analyzing these patterns, cosmologists can deduce how matter clumped together over time. The precision measurements of the CMB by space observatories like the Wilkinson Microwave Anisotropy Probe (WMAP) and the Planck satellite have shown that the fluctuations are consistent with a universe dominated by cold dark matter, and have provided crucial support for the CDM model of large-scale structure formation. The imprint of these fluctuations on the CMB serves as a supporting pillar for our understanding of cosmic history, alongside observations of distant supernovae, galaxy clusters, and large-scale galaxy surveys.