Explain why it is important to measure how the expansion rate of the universe has changed over time. How is this rate measured?

To understand the universe's expansion, scientists observe the phenomenon known as redshift. This is akin to the Doppler effect, where sounds shift in pitch as a source moves towards or away from us. But for stars and galaxies, instead of sound, we look at light. As the universe expands, light from distant galaxies stretches out, making it appear redder than it would if the galaxy were stationary relative to us.

This change in color, or redshift, correlates to how fast these celestial bodies are moving away. By measuring redshift, astronomers can determine the velocity at which a galaxy is receding, which provides clues about the rate at which the universe is expanding.

The cosmic microwave background radiation (CMBR) is a critical piece of evidence in the study of the universe's evolution. It's the afterglow of the Big Bang, a faint cosmic whisper from nearly 13.8 billion years ago. The CMBR fills the entire universe and provides a snapshot of the infant cosmos, showing tiny fluctuations in temperature that correspond to regions of slightly different densities.

These fluctuations hold information about the universe's early conditions and its subsequent rate of expansion. By analyzing the patterns in this radiation, scientists can map the early universe's structure and measure its expansion over vast cosmic timescales. It's like examining the universe's birth certificate to understand its past and predict its future.

Dark energy and matter are the universe's most elusive constituents. Dark matter, which does not emit or absorb light, accounts for about 27% of the universe, influences the motion of galaxies but is detectable only through its gravitational effects. Dark energy is even more mysterious and makes up approximately 68% of the universe. Thought to be responsible for the universe's accelerating expansion, its true nature remains one of the greatest enigmas in physics.

Understanding these components is crucial because they shape the cosmos on the largest scales. They affect the universe's overall density, its rate of expansion, and ultimately, its fate. Current models of the universe's expansion include the influence of dark matter's gravitational pull and dark energy's push, making the accurate measurement of these dark entities essential for cosmology.

The fate of the universe is a profound question that has fascinated humanity for centuries. Current observations suggest that the universe is not only expanding but doing so at an accelerating rate, which hints at an endless expansion leading to a cold, dark, and sparse cosmos, often referred to as the 'Big Freeze.'

Alternatively, if dark energy were to change in nature, it could lead to a 'Big Crunch,' where the universe would ultimately collapse upon itself. Another fate could be the 'Big Rip,' where the expansion accelerates to the point where all matter is torn apart. By understanding how the expansion rate has changed and will change, scientists aim to predict which of these outcomes, if any, will come to pass. The study of redshift, cosmic microwave background radiation, and the nature of dark energy and matter help piece together this cosmic puzzle.