(a) If the Hubble constant is \(73 \mathrm{~km} / \mathrm{s} / \mathrm{Mpc}\), the critical density \(\rho_{c}\) is \(1.0 \times 10^{-26} \mathrm{~kg} / \mathrm{m}^{3}\). The average density of dark matter is known to be about \(0.20\) times the critical density. Suppose that massive neutrinos constitute this dark matter, and the average density of neutrinos throughout space is 100 neutrinos per cubic centimeter. (In fact, the density of neutrinos is far less than this.) Under these assumptions, what must be the mass of the neutrino? Give your answers in kilograms and as a fraction of the mass of the electron. (b) Why do astronomers think that massive neutrinos are not the dominant type of dark matter in the universe?

The Hubble constant, denoted as 'H0', is a measure of the current expansion rate of the universe. Named after the astronomer Edwin Hubble, it quantifies how fast galaxies are receding from each other as a function of distance. This constant is not only foundational for cosmology but also aids in estimating the size and age of the universe.

To put it simply, the farther away a galaxy is, the faster it seems to be moving away from us, with the Hubble constant providing the rate of this expansion per megaparsec of distance. Observations have produced various estimates of the Hubble constant, typically in the units of kilometers per second per megaparsec (km/s/Mpc). In the context of our exercise, the given value is 73 km/s/Mpc and plays a key role in calculating the critical density of the universe, which is crucial to understanding the balance between the expansion and potential contraction of the universe.

Critical density represents the threshold value for which the universe is perfectly balanced between re-collapsing due to its own gravity and expanding indefinitely. If the actual density of the universe is greater than the critical density, gravity will eventually win, causing the universe to collapse in a 'Big Crunch'. Conversely, if the density is less than the critical density, the universe will keep expanding.

Understanding critical density is imperative as it allows us to evaluate different scenarios for the ultimate fate of the universe. In the exercise given, we use the critical density \( \rho_c \) to determine the density of dark matter, by assuming it to be a fraction of the critical density. Mathematically, the critical density is derived using the Hubble constant and the gravitational constant, reflecting the close relationship between cosmic expansion and gravitational forces.

Dark matter is one of the most mysterious components of the cosmos. It does not emit, absorb, or reflect light, making it invisible and detectable only through its gravitational effects on visible matter, radiation, and the large-scale structure of the universe.

Dark matter is believed to make up about 27% of the universe, critical for holding galaxies and clusters together due to its gravitational pull. In the exercise, we consider that the average density of dark matter is about 0.20 times the critical density, and this value helps us to infer properties of dark matter particles, such as neutrinos, when we consider them as potential dark matter candidates.

The electron is a subatomic particle with a negative electric charge and is a fundamental constituent of matter. Its mass, denoted as \( m_e \), is a physical constant that is influential in a wide range of physics calculations, including atomic and particle physics.

The electron's mass is approximately \(9.109 \times 10^{-31} \) kilograms, and it serves as a reference point when comparing masses of other subatomic particles, like neutrinos. In our exercise, the mass of the neutrino is calculated and then expressed as a fraction of the mass of the electron to provide a sense of scale and to highlight the incredibly small mass of neutrinos in comparison to electrons.