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Schedule Spring 2021

    Milestone I - General Relativity and Background cosmology: Milestone II - Thermodynamics/statistical mechanics and the thermal history of our Universe: Milestone III - Cosmological perturbation theory and inflation: Milestone IV - From perturbations to statistical observables:
    • Week 10 (17-18 Mar)
    • Week 11 (24-25 Mar)
    • Week 12 (31-1 Apr)
    • Week 13 (7-8 Apr)
    • Week 14 (14-15 Apr)
    • Week 15 (21-22 Apr)
    • Week 16 (28-29 Apr)
    • Week 17 (5-6 May)
    • Week 18 (12-13 May)
    • Week 19 (19-20 May)
    • Week 20 Exam


    Schedule Spring 2021


    This page will show what we are meant to go through (and what we actually did go through) every week. It will be updated as we go along. Lectures are given on Zoom Wednesday 1415-1600 and Thursday 1215-1400.


    Week 1 (13-14 Jan)


    Lectures:

    Problems:

    If you want to start practice doing calculation with GR take a look at Calculations with General Relativity. We will go through how to do these calculations in the lectures next week when deriving the Friedmann equations.

    Summary:

    We give an overview of what you are supposed to learn in this course and give some practical information about the course and the project. We then start on a crash course in General Relativity. We will first go through Newtonian gravity (the differenial formulation) and then go on to introduce the concepts of tensors, a metric, connections, parallel transport, curvature, the geodesic equation and finally the Einstein Equations. Finally we show how Newtonian gravity arises as a limit of General Relativity. The main aim here (since we don't have much time for this) is simply to give you the operational knowledge of doing calculations with GR.

    Learning goals:

    Know what a tensor is and be able to evaluate simple tensorial expressions. Know the Einstein summations convention. You are supposed to know the algorithm for evaluating the Einstein equations and (after next week when we have gone through this in detail) be able to perform this kind of calculation, i.e. start from a given metric and compute the inverse metric, the connections coefficients, the Ricci tensor, the Ricci scalar and use this to evaluate the left hand side of the Einstein equation. You should also know about the geodesic equation: the key equation in GR that tells us how particles move in a given spacetime (but we will get back to this in more detail later in the course).

    Video:


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 200 MB.


    Week 2 (20-21 Jan)


    Lectures:

    Problems:

    Calculations with General Relativity, Solve the continuity equation for a perfect fluid, Simplified form for the Friedman equation, Curvature in the Friedman equations. There are also problems in Baumann and at the end of Chapter 2 in Dodelson.

    We are now done with the theory behind Milestone I. If you want to take a look at exam problems from this part of the course you can see Home-exam 2020 Problem 1b, Exam 2018 (postponed) Problem 1c, 1e, Exam 2018 Problem 2, Exam 2016 Problem 1a, 1b, Exam 2015 Problem 1e. Exam 2012 Problem 1b, 1c. Solutions to these problems can be found here.

    Summary:

    We gave an introduction to theoretical cosmology at the background level. The cosmological principle and its implication for the geometry of the Universe. We went through how to model matter/radiation at the background (perfect fluids) and how different components evolve in time and derived the Friedmann equations. From this we presented the standard model of cosmology today: the $\Lambda$CDM model.

    Learning goals:

    The aim is to know the basics of background cosmology: the cosmological principle $\to$ Friedmann metric and going from the Friedmann metric to the Einstein equations and obtaining the Friedmann equations. You should be able to do the different parts of this calculation. You should know how each energy component evolves in an expanding Universe, the expressions for the Hubble function, what density parameters are, the usual cosmological parameters and the $\Lambda$CDM model. You should also know a bit about different measures of time and distance in our Universe.

    Video:


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 200 MB.


    Week 3 (27-28 Jan)


    Lectures:

    Problems:

    Temperature of neutrinos, Evaluating Boltzmann Integrals (only if you like trying to evaluate integrals), Photon and neutrino density parameters, Baryon to photon ratio, The ideal gas law and the equation of state for matter. There are also problems in Baumann and at the end of Chapter 2 in Dodelson.

    Summary:

    We gave an introduction to the first milestone in the numerical project. We introduced the Boltzmann formalism for dealing with thermodynamical systems in and out of equilibrium. We first introduced the key quantity - the distribution function - and the key equation for how this evolves - the Boltzmann equation - and then talked about how we can compute standard macroscopic quantities like energy density and pressure from the distribution function. We talked about how to include interactions and how to integrate the Boltzmann equation (taking moments) to get more familiar fluid equations. We briefly introduced the master equation (integrated Boltzmann equation in a smooth Universe) for a $1+2\leftrightharpoons 3+4$ process which we will do in much more detail next week.

    Learning goals:

    You should know what the distribution function is and how we can compute macroscopic quantities like number-density, energy density and pressure from it. You should know what the Boltzmann equation is and what it represents and how to expand it in terms of partial derivatives of the phase-space coordinates. You should know how we can get conservation equations for macroscopic quantities by taking moments of the Boltzmann equation and know what these equations represents.

    Video:


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 200 MB.


    Week 4 (3-4 Feb)


    Lectures:

    • [Lecture 7] The Boltzmann equation for a $1+2\leftrightharpoons 3+4$ process. The Saha approximation. Summary of the thermal history of our Universe. Recombination in our Universe (Saha equation).
    • [Lecture 8] Recombination beyond the Saha approximation (Peebles equation). The optical depth. Dark matter production if time.

    Problems:

    The problems from last week and The Saha Equation (Problem 3 Exam 2016) (this is basically the derivation covered in the lectures). If you want a more challenging problem see: Exam problem - Recombination including Helium (for a solution see this). If you want a (much more extensive) problem (theoretical and numerical) on dark matter freeze-out you can take a look at the project given in AST3220 last year.

    Summary:

    In the first lecture we talked about how to deal with a general $1+2\leftrightharpoons 3+4$ interaction in the Boltzmann formalism. We applied this to recombination of the Universe. First by apply the Saha approximation and then in the second lecture we finished the discussion of recombination by talking about the full Peebles equation and introduced the quantities that we get from this that tell us about how much photons scatter in our Universe: the optical depth and the visibility function. These things are what you are now going to implement numerically in Milestone II. We then did a short summary of the thermal history of the Universe and finished by talking about one possible scenario (freeze-out) for how the observed dark matter abundance in the Universe could have been created. We are now done with the theory related to Milestone II.

    Learning goals:

    You should know the Saha equation and how to apply it for hydrogen recombination to get a quadratic equation for the free electron fraction. You should know the (integrated) Boltzmann equation for a general $1+2\leftrightharpoons 3+4$ interaction and what each term represents. You should know what the Peebles equation is and be able to physically describe the process of recombination. You should know what the optical depth and the visibility function represent physically. You should have an overview of the thermal history of the Universe knowing about important events like neutrino decoupling, electron positron annhilation, Big Bang Nucleosynthesis (BBN) and recombination.


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 300 MB.


    Week 5 (10-11 Feb)


    Lectures:

    Problems:

    The inverse of the perturbed metric, Christoffel symbols for perturbed metric, Trajectories of photons in a perturbed Universe and if you haven't worked with Fourier transforms: Fourier transform basics.

    Summary:

    We gave an overview of what we are going to go through over the next month which is to study the evolution of structures in the Universe. We talked about perturbation theory of the fluids and the metric (including the Scalar, Vector, Tensor decomposition). We also gave a summary of Fourier transforms and why working in Fourier space is so useful for us. In the next lecture we wil start doing the Boltzmann equation for photons.

    Learning goals:

    You should know what scalar-vector-tensor perturbations are and know that can treat them seperately (SVT decomposition theorem). You should know how the perturbed metric for scalar perturbations in the Newtonian gauge and what the metric potentials represent (Newtonian gravitational potential). You should know how to perturb the Boltzmann equation for photons, expand it in partial derivatives and use the geodesic equation to compute how photons propagate.


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 300 MB.


    Week 6 (17-18 Feb)


    Lectures:

    Problems:

    Boltzmann equation for dark matter (this is basically the derivation covered in the lectures, but also ask you to derive the Euler equation), The dark matter equations in Fourier space. For another approach of deriving the dark matter evolution equations see Exam problem - Cold Dark Matter Perturbations ( excluding e) ).

    Summary:

    We went through the Boltzmann equation for photons and found the evolution equation for the temperature perturbation $\Theta$. We then did the same analysis for dark matter using a slightly different strategy by taking moments and deriving the continuity and Euler equation. We briefly discussed the collision term for Thompson scattering. In the second lecture we gave an introduction to Milestone II.

    Learning goals:

    You should know (the key steps) how to work with the Boltzmann equation in linear perturbation theory and get the evolution equations for photons and cold dark matter. You should know the physical significance of these equations.


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 300 MB.


    Week 7 (24-25 Feb)


    Lectures:

    Problems:

    Boltzmann equation for baryons, Exam problem - Scattering processes involving baryons (basically what we did in the lectures). Legendre multipoles math, The perturbed Einstein equations: The left hand side, The perturbed Einstein equations: The right hand side.

    Summary:

    We went through the Boltzmann equation for baryons and neutrinos and talked about how to transform the equation set we have derived to Fourier space. We then introduced Legendre multipoles. We then talked about how to expand the photon temperature perturbation into multipoles. We have now derived all the Boltzmann equations we need and next week we will move on to the final piece of the perturbation system - the perturbed Einstein equations.

    Learning goals:

    You should know the physical arguments for what scattering processes including baryons are relevant and which are not and if we expect them to contribute to the continuity and/or Euler equation. You should be able to derive the neutrino equation from the photon equation. You should be able to expand the photon distribution into multipoles.


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 300 MB.


    Week 8 (3-4 Mar)


    Lectures:

    Problems:

    The perturbed Einstein equations: The left hand side (this is just what we did in the lectures), The perturbed Einstein equations: The right hand side (this is just what we did in the lectures), Momentum Flux - Velocity in terms of the multipoles for photons, Estimating how much inflation we need, The flatness problem.

    Summary:

    We derived the evolution equations for the metric potentials $\Phi,\Psi$ completing the Einstein-Boltzmann system and gave a summary of the full system. We talked about the theory of inflation, discussing some problems in the standard Big Bang model and how inflation solves them. We then went through how we can set the initial conditions for the density perturbations and the metric potentials using the key assumption of adiabatic initial conditions.

    Learning goals:

    You should know how to derive the perturbed Einstein equations. You should about some problems with the standard Big Bang model and why we need something like inflation, how inflation works, the preditions of it and roughly how it solves these problems. You should know how to set the initial conditions for $\Phi,\Theta_0,\delta_b,\delta_{\rm CDM}$ from the initial condition for $\Psi$.


    Download: Lecture 1, Lecture 2. NB: Videos have a size of $\sim$ 300 MB.


    Week 9


    Lectures:

    • Lecture 1:
    • Lecture 2:

    Problems:

    Summary:

    This week we will probably focus on the project and solving some problems.

    Learning goals: