Cosmology Lecture 4

The lecturer begins by addressing the availability of last week's notes and a mistake with this week's notes, which were written on long pads. A discussion on inherited pads from the math department leads to a broader conversation about the universe's structure, touching on the concept of a torus and its implications for observable phenomena. The lecture also covers the role of dark energy in the universe's expansion, clarifying misconceptions about its direct impact on cosmic growth.

This paragraph delves into the idea that despite the universe's expansion, objects like atoms and solar systems remain intact due to forces like electrostatic and gravity, which are stronger than the expansion force. The lecturer explains that energy conservation principles differ in an expanding universe and introduces the concept that changes in energy translate into kinetic energy of expansion.

The lecturer explores the geometric properties of space, discussing the idea of a homogeneous universe where every point is the same as every other point. The paragraph examines the concept of a metric to describe space and how coordinate transformations can reveal the true nature of space's geometry, whether it's flat, spherical, or hyperbolic.

This section discusses the scale-dependent nature of homogeneity in the universe. While the universe appears homogeneous on large scales, it is not on smaller scales where fluctuations in density occur. The lecturer uses the analogy of the Earth's surface to illustrate this concept and introduces the idea of classifying geometries based on their intrinsic properties.

The lecturer clarifies misconceptions about the torus's isotropic properties and explains why a torus, while homogeneous, is not isotropic due to preferred axes. The paragraph emphasizes the importance of understanding space's intrinsic geometry, independent of how it might be visualized in higher dimensions.

This part of the lecture addresses the difficulty humans face in visualizing dimensions higher than three. The lecturer suggests that our brains are wired for three-dimensional navigation, making it hard to conceptualize additional dimensions. The focus is on intrinsic geometry and the importance of not relying on extrinsic embeddings for understanding.

The lecturer uses the thought experiment of a one-dimensional world with light propagation to illustrate the intrinsic versus extrinsic properties of space. The discussion highlights that two-dimensional surfaces cannot be flattened without stretching, but one-dimensional spaces (like lines) can be, emphasizing the unique nature of different dimensions.

This paragraph establishes the relationship between gravitational fields and space-time curvature, as posited by general relativity. The lecturer explains that tidal forces, which are indicative of curvature, arise from masses, and thus, gravitational fields are a direct result of space-time curvature.

The lecturer outlines the assumptions of homogeneity and isotropy in cosmology, leading to three candidate space-time geometries named after their curvature (K=1, K=0, K=-1). The paragraph discusses the implications of these geometries for the expansion of the universe and how they relate to observable phenomena.

This section introduces the space-time metric and its significance in general relativity. The lecturer explains the process of calculating the Einstein tensor and energy momentum tensor to derive equations that describe how the scale factor 'a' varies with time, which is central to understanding the universe's expansion.

The lecturer discusses the components of the energy-momentum tensor and their physical interpretations, highlighting the tensor's sensitivity to the material nature of the universe. The paragraph explores how the left-hand side of Einstein's field equations, involving geometry, relates to the right-hand side, which involves energy and momentum.

This part of the lecture focuses on the importance of the equation of state in determining how energy density changes with the scale factor. The lecturer provides examples of different equations of state for matter-dominated and radiation-dominated scenarios and emphasizes the need to understand the nature of the material in the universe to solve the equations.

The lecturer defines the equation of state as a relationship between energy density and pressure, which is crucial for understanding the behavior of different types of materials in the universe. The paragraph explains the concept of adiabatic change and how it applies to the expansion of the universe, leading to a general formula for energy density variation.

The lecture concludes with an introduction to dark energy, characterized by a constant energy density that does not change with the expansion of the universe. The lecturer discusses the unusual property of dark energy, where pressure and energy density have opposite signs, leading to a universe that continues to expand without dilution of its energy density.

The final paragraph touches on the concept of the cosmological constant and its implications for the expansion of the universe. The lecturer suggests that the cosmological constant can be modeled as a force proportional to distance between particles, which can be repulsive or attractive, depending on its sign, and significantly influence the fate of the universe.