"What evidence suggests that the Milky Way contains dark matter?"
Atomic hydrogen clouds far away from the center of the galaxy (beyond the Sun) orbit much faster than expected, indicating the majority of the galaxy's mass to be beyond the Sun's orbit. However, most of the light in the galaxy comes from the stars near the center, therefore there must be unseen material, or dark matter in the halo.
"Briefly describe how we construct rotation curves for spiral galaxies and how these curves lead us to conclude that spiral galaxies contain dark matter."
A rotation curve is created by talking into account the tilt of a galaxy then plotting the shifts of its 21-cm atomic hydrogen emission line from where the galaxy rotates away and towards us. The rotation curves of spiral galaxies are surprisingly flat after an initial rise. Because the speeds of gas clouds stay fairly constant despite their growing distance from the galaxy's center, we can conclude that most of the galaxy's matter is far from the center. However, as with the Milky Way, most of the light is near the center, indicating the presence of dark matter beyond.
"What is gravitational lensing? Why does it occur? How can we use it to estimate the masses of lensing objects?"
Gravitational lensing is when a massive object causes nearby light to bend as that light passes through its gravitational field. An object behind a massive object might therefore appear to be distorted and appear in multiple places because its light couldn't travel straight past the foreground object. Gravitational lensing is useful for estimating mass because how much the lensing object distorts the light around it is relative to its gravity.
"What do we mean by MACHOs? Describe several possibilities for the nature of MACHOs. Based on evidence from gravitational lensing events, can MACHOs explain all the dark matter in the Milky Way? Why or why not?"
MACHOs are Massive Compact Halo Objects. These are a form of ordinary dark matter that may include brown dwarfs, faint red stars, and Jupiter-size objects left over from when the Milky Way formed. MACHOs can't explain all the dark matter because if they could then they there would be more frequent occurrences of lensing when they came across our line of sight to a distant star.
"What do we mean by WIMPs? Why do many astronomers believe that dark matter consists largely of WIMPs?"
WIMPs are Weakly Interacting Massive Particles, a form of extraordinary dark matter. WIMPs are like neutrinos because they have no electrical charge and aren't bound to charged particles and therefore are dark, but they're more massive than neutrinos. WIMPs would be too weak to collapse with the protogalactic cloud, which would explain why dark matter in a galaxy doesn't behave like visible matter.
"Explain how the total amount of dark matter in the universe holds the key to its final fate. According to current evidence about the amount of dark matter, what is the fate of the universe?"
The more mass the universe has, the greater the pull of gravity, and the greater the likelihood that this pull will slow the universe's expansion and lead to collapse. The more we know about the amount of dark matter, the better we can predict gravity's influence. Current evidence relies mostly on the mass-to-light ratio seen in galaxy clusters, which has the universe at 25% of critical density, the point at which there is sufficient mass to stop expansion and begin the process of collapse. Therefore the universe will probably continue to expand.
"Briefly describe the four possible patterns for the expansion of the universe. What is the fate of the universe in each case? How does the current age of the universe differ among the four cases?"
The four patterns of universe expansion are recollapsing universe, critical universe, coasting universe, and accelerating universe.
In the recollapsing universe, the density of the universe is greater than the critical density and the universe eventually collapses. This is a closed universe. It requires the least amount of time and would mean the universe is currently 5 billion years old.
In the critical universe, the density of the universe is equal to the critical density and the expansion of the universe first slows then stops. This is a flat universe. The current age of the universe is 10 billion years in this scenario.
In the coasting universe, the density of the universe is less than the critical density and the universe expands forever with no change in the expansion rate. This is an open universe and the age of the universe would be about 15 billion years.
In the accelerating universe, the expansion rate increases due to a repulsion force and becomes cold and dark more quickly than a coasting universe. An accelerating universe can be open, flat, or closed. If our universe is accelerating, then its current age is about 16 billion years.
"Briefly describe the recent evidence suggesting that the expansion of the universe is accelerating."
Recent evidence for an accelerating universe is found in white dwarf supernovae with large redshifts. The shifts are dimmer than expected for a coasting universe and therefore also don't fit the brighter appearances that would be found in a recollapsing or critical universe. What causes the acceleration is unknown.

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