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## MATH3532 Relativity and Cosmology

### 15 creditsClass Size: 110

Module manager: Prof Carmen Molina-Paris
Email: C.MolinaParis@leeds.ac.uk

Taught: Semester 2 (Jan to Jun) View Timetable

Year running 2019/20

### Pre-requisites

 MATH2365 Vector Calculus PHYS1300 Maths 2- Multivariable Calculus

Module replaces

MATH3531 Cosmology

This module is not approved as a discovery module

### Module summary

This module introduces students to the basic mathematics and physics of relativity and modern cosmology. The module will introduce the student to the geometrical theories underpinning our current knowledge of both black holes and the large-scale structure of our universe.

### Objectives

To introduce the student to the mathematics and physics of relativity and the large scale structure and motion of the universe.

Learning outcomes
a) Understand the physical basis of light-speed invariance and the replacement of Galilean transformation by the special Lorentz transformation;
b) understand the connection between geometric concepts of curvature and distance and the large-scale matter content of the universe;
c) explain the main observational features of the expansion of the universe using the ideas of measurement of distance, parallax, Doppler shift and luminosity;
d) derive the Hubble Law from homogeneity and isotropy;
e) understand the meaning of Friedmann's equations (fluid equation and acceleration equation), solve these differential equations for the cosmological factor, and derive solutions for different matter content and curvatures.

### Syllabus

The following core topics will be covered:
1. Outline of the current astronomical results on structure and composition of the Universe, the cosmological redshift, and the Hubble law.
2. The differential geometry of special and general relativity: Riemannian and Lorentzian geometries, four-dimensional manifolds.
3. Space, time, and gravity in Special Relativity, and General Relativity - an outline of key ideas and basic results. The concept of spacetime and its curvature. 3+1 representation of spacetime metric. The light cone, causality, space-time intervals.
4. Einstein's equations: curvature of spacetime and its matter content.
5. Cosmological solutions to Einstein's equations: the Walker-Robertson metric of uniform Universe and Friedmann's equations.
6. Expansion of the Universe. The Hubble constant and the deceleration parameter. The critical density.
7. Friedmann's models of open, flat and closed universes. The Big Bang theory. The cosmic microwave background radiation.

In addition to the above, further topics will be drawn from the following, or similar, as time allows:
8. Distances to sources with given redshift. The angular size and luminosity "distances". The cosmological horizon.
9. The Cosmological constant solution to Einstein's equations. The accelerating expansion of the Universe. Dark matter and dark energy.
10. Black hole solutions to Einstein's equations.
11. Gravitational waves as solutions to Einstein's equations. LIGO detection of gravitational waves.

### Teaching methods

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

 Delivery type Number Length hours Student hours Lecture 33 1.00 33.00 Private study hours 117.00 Total Contact hours 33.00 Total hours (100hr per 10 credits) 150.00

### Private study

Studying and revising of course material.
Completing of assignments and assessments.

### Opportunities for Formative Feedback

Regular problem solving assignments

### Methods of assessment

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

Exams
 Exam type Exam duration % of formal assessment Standard exam (closed essays, MCQs etc) 2 hr 30 mins 100.00 Total percentage (Assessment Exams) 100.00

Normally resits will be assessed by the same methodology as the first attempt, unless otherwise stated