For all problems, unless otherwise stated, use \(\mathrm{H}_{0}=\) \(70 \mathrm{km} / \mathrm{s} / \mathrm{Mpc}\). Show that, if the universe were infinite in age and extent the cosmological redshift is not sufficient to get us out of Olbers's paradox.

The cosmological redshift is a key concept in understanding our universe. It happens due to the expansion of space. As light travels through space, the fabric of space itself stretches. This stretching increases the wavelength of light. Longer wavelengths mean the light shifts towards the red end of the spectrum. Hence, it's termed 'redshift'.
For example, imagine light from a distant galaxy. When this light leaves the galaxy, it starts its journey towards us. Over time, as space expands, this light's wavelength gets stretched. By the time it reaches us, it appears redder than when it started.
This phenomenon helps astronomers figure out how far away objects are. The more redshifted the light, the farther away the object is.
Cosmological redshift not only helps us measure distances but also shows evidence that the universe is expanding.

When we talk about an infinite universe, we mean one that has no bounds and is eternal in age; it has existed forever. In such a universe, every line of sight should eventually end on a star. This situation brings us to a famous problem known as Olbers's Paradox.
According to Olbers's Paradox, if the universe were infinite and filled uniformly with stars, the night sky should be bright. This is because there are an infinite number of stars, and their light should fill up every part of the sky.
But in reality, our night sky is dark, not bright. The paradox challenges the idea of an infinite and static universe. Astronomers have debated this for centuries, and current understanding points towards solutions involving the universe’s expansion and its finite age, among other factors.

The expansion of the universe is another crucial idea in modern cosmology. Our universe isn't static; it's continuously expanding. This discovery was one of the significant ideas that shifted our understanding of the cosmos.
Edwin Hubble, an astronomer, observed that galaxies are moving away from us, and the farther away they are, the faster they're receding. This observation led to the conclusion that the universe itself is expanding. This expansion affects everything, including the light traveling through space.
As space expands, it stretches the wavelengths of light. This stretching causes the cosmological redshift we talked about earlier. The expansion means that galaxies and stars are constantly moving away from each other, which impacts how we perceive light over long distances.
In terms of Olbers's Paradox, the expansion helps explain why the night sky is dark. As the light from distant stars stretches and redshifts over time, it reaches us with diminished energy, ensuring the sky isn’t as bright as we'd expect in an infinite, static universe.

The intensity of starlight is essential in resolving questions about the night sky's darkness. Intensity refers to the brightness or energy from a star that reaches us. In an infinite universe, you would think the combined light from all stars should make the night sky very bright.
However, several factors affect this intensity:

• Cosmological Redshift: Due to the universe's expansion, the light from distant stars gets redshifted, which reduces its energy when it arrives.
• Finite Age of the Universe: The universe is not infinitely old. Light from the farthest stars hasn't had enough time to reach us yet.
• Absorption and Scattering: Interstellar dust and gas can absorb or scatter starlight, diminishing its intensity.

Even if we integrate the light from an infinite number of stars, the combined effect of all these factors significantly mitigates the brightness we observe, leading to a dark night sky. Therefore, while individual stars contribute light, the overall observed intensity drops, helping resolve Olbers's Paradox.