2014PUPC普林斯顿大学物理竞赛Online真题与答案免费下载-Cosmology

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2014PUPC普林斯顿大学物理竞赛

Cosmology Online Part 完整版真题免费下载

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If you choose to submit your solutions for the cosmology problem, you are NOT required to submit the condensed matter problem!

This part will contain background material in several important topics in cosmology, including some that are at the forefront of groundbreaking research in present day.

Some sections offer background material and are followed by several conceptual and/or applied questions that you are expected to answer in any order you choose (but the material is structured in a way that would lead you through the assignment sequence logically).

Good luck!

Cosmology is the scientific study of the large scale properties of the universe as a whole. It aims to understand the origin, evolution and ultimate fate of the universe. The prevailing theory about the origin and evolution of our universe is the so-called Big Bang theory. In this problem, you will explore the main concepts in cosmology.

  1. The Scale Of the Universe

Cosmological observations indicate that the universe is homogeneous and isotropic on large scales. Ho­mogeneity is the statement that the universe looks the same at each point, while isotropy states that the universe looks the same in all directions. This is referred to as the Cosmological Principle and we will assume it to be true for the following problem.

A key piece of observational evidence in cosmology is that almost everything in the Universe appears to be moving away from us, and the further away something is, the more rapid its recession appears to be. Yet the universe looks just the same from any point in space. A common analogy is to imagine baking a cake with raisins in it. As the cake rises, the raisins move apart. But from each raisin’s perspective, it seems that all the others are receding, and the further away they are the faster that recession is. Because everything is flying away from everything else, we conclude that in the distant past everything in the Universe was much closer together. For all the following questions, we will consider the widely accepted theory that the universe started with an explosion, called the Big Bang.

Hence, the expansion of the universe as seen from any point in the space, can be described by a scalar factor, a(t), which is independent of position. Given a point in space with vector position r0, its position at any time t will be characterized by the equation:

2014PUPC普林斯顿大学物理竞赛真题答案(1)

At the time of Big Bang, a(0) = 0, while at the present time, a(t0) = 1.

1.1. Finding Hubble’s Law

Hubble’s law expresses the relationship between the vector position r and velocity v of any point in space, for which the expansion of the universe is homogeneous and isotropic. The dependence of r on v is expressed through a multiplicative factor called the Hubble constant. Derive the Hubble law, v = Hr. Is the Hubble constant the same at all times or does it change with time? Is the Hubble law true for an observer at any point in space?

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