## MATH3531 Cosmology

### 10 creditsClass Size: 100

Taught: Semester 1 (Sep to Jan) View Timetable

Year running 2005/06

### Pre-requisites

(MATH1382 or MATH1410 or PHYS1030), and (MATH1350 or MATH1355 or MATH1400 or MATH1460 or PHYS1030) and (MATH2360 or MATH2420 or MATH1470) or equivalent.

This module is approved as an Elective

### Objectives

To introduce the student to the mathematics and physics of the large scale structure and motion of the universe. On completion of this module, students should be able to: a) State the main connections between the theories of classical mechanics, special and general relativity, and quantum mechanics; b) Explain the main observational features of galactic motion using the ideas of measurement of distance, parallax, Doppler shift and luminosity; c) Calculate particle motion in the framework of Newtonian gravitation; d) Demonstrate the relation between Hubble's law and the Cosmological Principle; e) Derive the differential equation for the cosmological scale factor and produce solutions in special cases.

### Syllabus

Observations over the last decades indicate that the Universe is expanding, that is, galaxies are generally moving apart from one another. If we accept that the Universe started in a Big Bang, we would like to know how old the Universe is, whether it will expand for ever, or stop expanding and collapse in a Big Crunch. We can only begin to address these problems if we know what the density of matter in the Universe is, and how fast the Universe is expanding at the present time. The course will start with a description of how we make measurements of the distances and velocities of galaxies . The large-scale force dominating the expansion of the Universe is gravity, and a proper treatment of gravity is given through the General Theory of Relativity. However, there is a mathematical model of cosmology based on the classical gravity of Newton, and the course will be based on this approach because it turns out that the equations for the expansion of the Universe derived in this way are essentially correct even if the full relativistic theory are taken into account. No knowledge of the theory of relativity is assumed, but a description of this theory will be given and results derived from the classical theory will be discussed in the light of relativistic theory when the need arises. Newtonian gravitational theory will be introduced and equations for the rate of expansion of the Universe will be derived. Conditions on mass density, current expansion rate etc will be derived for various expansion models. Among topics to be discussed will be the effect of Einstein's cosmological constant, 'missing matter', and black holes. The topics covered include: 1. Observations on the Universe: measurement of distance, parallax, Doppler shifts, luminosity. 2. Brief descriptions of (a) special relativity, the Lorentz transformation, length contraction and time dilation, relativistic mechanics, energy-mass relation; (b) general relativity- curvature of space-time, description of the geometry of space-time in terms of the metric elements, Einstein's field equations; (c) quantum mechanics, photons. 3. Newtonian gravitation. Inverse square law, conservative forces, energy conservation, gravitational potential, potential and force inside and outside a solid sphere, escape velocity, classical black holes. 4. Hubble's law of the expansion of the Universe and the Cosmological Principle consistency, making allowances for the time of light propagation, the cosmological scale factor R(t). 5. The cosmological differential equation based on the particle model. Kinetic and potential energy of a system of particles, the cosmological constant ? and cosmic repulsion, the differential equation for R( t), the deceleration parameter. 6. The cosmological differential equation based on the continuum model. The exact form of the cosmic repulsion term. 7. Simple Friedmann models. Models with ?=0: expanding and oscillating models, critical mass density, Einstein-de Sitter model, dark matter, general theory of ?=0, and Milne model. Models with zero total energy. The shape of R(t) for different models.

### Teaching methods

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

Lectures: 20 hours. 6 example classes.

### Methods of assessment

Due to COVID-19, teaching and assessment activities are being kept under review - see module enrolment pages for information

One 2 hour examination at end of semester (100%).