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 and evolution of perturbations: Review of course:

# 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)

### 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)

### 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)

### 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)

### 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)

### 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)

### 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)

### 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 (10-11 Mar)

### Lectures:

• [Lecture 1] Help with the project at ITA / over Zoom.
• [Lecture 2] Lecture cancelled.

### Problems:

No new problems for this week.

### Summary:

This week we will focus on working with the project. Bring laptops to class and work on the project there and get help if there is any issues.

## Week 10 (17-18 Mar)

### Problems:

No new problems this week. I reccomend instead trying to redo the derivation of our expression for the power-spectrum $C_\ell$ from the temperature perturbation $\Theta$ (but this derivation is not relevant for the exam).

### Summary:

We gave an overview over what CMB experiments measure, how to expand the temperature field in spherical harmonics and gaussian random fields. Then we derived an expression of the CMB angular power-spectrum in terms of the photon multipole perturbations. We also gave an introduction to Milestone III.

### Learning goals:

You should know what spherical harmonics are and what a gaussian random field is and what the key observables for a random field are: correlation functions and that for a gaussian random field all the information is contained in the two point correlation function. You should know the expression for the CMB power-spectrum in terms of the photon multipoles.

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

## Week 11 (24-25 Mar)

### Problems:

I reccomend trying to redo the derivation of the line of sight integral (can also try to include the quadrupole term $P_2(\mu)\Pi$ on the right hand side). If you want a more challenging problem then try to do the same for neutrinos: Exam problem - Line of sight integration for neutrinos.

### Summary:

We introduced the method of line of sight integration to compute the photon multipoles $\Theta_\ell$ and talked about what this tells us about the CMB anisotropies. We learned that to know the CMB power-spectrum we mainly need to understand what the effective photon temperature perturbation $\Theta_0+\Psi$ is at the last scattering surface. We then went through how to understand how the effective photon temperature perturbation evolves and how this maps into the CMB power-spectrum and discussed the effect of baryons, radiation driving and photon diffusion.

### Learning goals:

You should know how to derive the line of sight integral expression for the photon multipoles $\Theta_\ell$. You should know what the different terms in the line of sight intergral represent physically. You should know how $\Theta_0+\Psi$ evolves prior to recombination and what this tells us about the structure of the CMB power-spectrum. You should know about baryon loading, radiation driving and diffusion damping and how this manifest itself in the power-spectrum. In particular the relative height of the peaks as a measure of the baryon density is important to know.

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

## Week 12 (31-1 Apr)

### Lectures:

No lectures this week due to Easter.

### Problems:

No problem sets this week. Work on Milestone III instead.

## Week 13 (7-8 Apr)

### Problems:

Problem 2 Exam 2015, Problem 2 Exam 2013.

### Summary:

We reviewed how to go from the perturbations to the CMB power-spectrum and went through the different parts of the CMB power-spectrum (1) Sachs-Wolfe plateu, 2) First three peaks 3) damping tail) and discussed how it changes when we vary the cosmological parameters $A_s,n_s,\Omega_b h^2,\Omega_M h^2,\tau$. We briefly talked about reionization and gravitational lensing and the degeneracy between $A_s$ and $\tau$ and how it could be broken by the lensing effect (and polarization data we will talk about later). Finally we went through some exam problems related to this.

### Learning goals:

You should be able to explain how the peturbations are mapped into the CMB power-spectrum and give an approximate expression for $\Theta_\ell$ for the Sachs-Wolfe term. You should be able to explain how a given mode gets mapped onto the CMB power-spectrum. You should know what the Sachs-Wolfe platau, the first three peaks and the damping tail can tell us about cosmological parameters. Key things to know 1) the relative peak heights as a measure of the amount of baryons and dark matter 2) the position of the first peak as a measure of curvature (and be able to give a simple explaination for this based on how photon move in a curved Universe) and dark energy 3) how the power-spectrum changes when we tune the inflationary parameters. You should know about some key parameter degeneracies.

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

## Week 14 (14-15 Apr)

### Problems:

Problem 2 Exam 2018 postponed, Problem 2 Exam 2010 , Exam problem - The CMB and matter power-spectrum and if you want some more math Growth of matter perturbations on subhorizon scales. See also: Problem 1 and Problem 2 at the Astro Forum (look at plots of CMB and matter power-spectrum and figure out what parameters I have varied).

### Summary:

We talked about the growth of matter perturbations and the key observable quantity for this: the matter power-spectrum. We explained why we have a peak in the power-spectrum. Finally we talked about the evolution of a density peak in real space to explain why we get an excess of matter at the BAO scales $r_s \sim 150$ Mpc and how this translates into a peak in the galaxy two point function and equivalently osciallations in the power-spectrum.

### Learning goals:

You should know how matter perturbations grow in the different regimes: superhorizon scales and subhorizon scales in both the radiation and matter era and on the basis of this be able to explain the peak of the matter power-spectrum. You should be able to explain how a density peak evolves in real-space.

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

## Week 15 (21-22 Apr)

### Lectures:

• [Lecture 1] Help with Milestone III.
• [Lecture 2] Help with Milestone III.

### Problems:

No new problems this week, see previous week.

### Summary:

This week we just used the lectures for help with Milestone III. We postponed the introduction of Milestone IV till next week to not overload you with new stuff when trying to finish the current milestone (though the slides are ready and can be found on Milestone IV if you want to start earlier).

## Week 16 (28-29 Apr)

### Problems:

No new problems this week, see previous week.

### Summary:

This week we introduced Milestone IV and went trough polarization and gravitational waves.

### Learning goals:

You should know the basics of CMB polarization, the two polarization patterns E and B modes and what generates them (E: Thompson scattering, B: gravitational lensing of E modes + gravity waves), the effects of reionization on polarization spectra (reionization bump at low $\ell$) and what the signatures of gravitational waves on the B mode power spectra (signature around $\ell \sim 100$ whose amplitude is given by the tensor-to-scalar ratio $r$).

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

## Week 17 (5-6 May)

### Lectures:

• [Lecture 1] Review of General Relativity and background cosmology.
• [Lecture 2] Review of thermodynamics and recombination.

### Problems:

Exam problems related to General Relativity and background cosmology: Problem 1c,1e (Geodesic equation and tensors), Problem 1a (Equivalence principle), Problem 1e (Einstein Equations), Problem 1b (How energy density evolves with time), Problem 1b (Conformal time), Problem 1b (Equation of state).

### Summary:

This week we review important things we have gone through previously (General Relativity and the theory behind milestone I and II).

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

## Week 18 (12-13 May)

### Lectures:

• [Lecture 1] Review of cosmologial perturbation theory.
• [Lecture 2] Holiday so no lecture.

### Problems:

Exam problems related to perturbation theory: Problem 1a, 1e, Problem 1e, Problem 1a, Problem 1e, Problem 1d, 1e, Problem 1b, 1c, Problem 1c, 1d, Problem 1b, Problem 1c, 1g.

### Summary:

This week we review important things we have gone through previously (the theory behind milestone III - cosmological perturbation theory).

Download: Lecture 1. B: Videos have a size of $\sim$ 300 MB.

## Week 19 (19-20 May)

### Lectures:

• [Lecture 1] Help with project.
• [Lecture 2] Review of evolution of perturbation, the CMB and matter power-spectra.

### Summary:

This week we review important things we have gone through previously (the theory behind milestone IV).

Download: Lecture 1. B: Videos have a size of $\sim$ 300 MB.

## Week 20 Oral Exam (4 June)

The University decided early on there would be no live written exams this semester so we will have an oral exam over Zoom this year (though now it will be possible to do it live at ITA if you prefer that). We will therefore focus more on physical understanding instead of performing long calculation, but you should of course also know the theory behind this and be able to explain stuff related to that. The structure of the exam will be as follows:

• ~20 min - About the project. We go through some result you have made in the project (e.g. density parameters, visibility functions, free electron fraction, evolution of different perturbations etc.) and you get to explain a bit the physics behind those plots. If you have written well about the results in the report (and understand what you have written) then you are well prepared already.
• ~20 min - General questions about the curriculum. No calculations needed. E.g. what is cosmic variance? Why do we need inflation? Why do we solve in Fourier space instead of real space? What is the effect of primordial gravitational waves on the CMB? We will likely also do a "What parameters have changed?" quiz here showing you some CMB and matter power-spectra and you get to explain what parameter have changed.
• ~20 min - More advanced topic. You draw a topic and then we go through it in more detail. You might need to do some basic calculations on the side, but it won't be major due to the time limit and due to us being over Zoom - but knowing the theory and be able to explain certain things related to key derivations is relevant. The topics you can get are:
• Recombination - You should be able to explain the process of recombination in detail. What is the relevant interaction? What is the key equation (the master equation) that tells us how the free electron fraction evolve and where does this come from? What is the Saha equation and the key steps in deriving this for recombination? When do we naively expect recombination to happen? What is Peebles model for recombination?
• The Boltzmann formalism - General questions about the formalism we use. What is the distribution function? How do we compute fluid quantities from this? What is the Boltzmann equation and how do we work with this? How do we perturb the Boltzmann equation (for photons and CDM) and explain how to compute certain terms / explain why certain terms can be neglected. What is taking moments of the Boltzmann equation and what does this give us?
• Evolution of matter perturbations and the matter power-spectrum - You should be able to explain how matter perturbations (dark matter and baryons) grow in different regimes (inside/outside horizon, radiation-matter-DE era). Based on this be able to explain the shape and features of the matter power-spectrum and what this can tell us about cosmological parameters.
• Evolution of photon perturbations and the CMB power-spectrum - You should be able to explain how photon perturbations evolve in different regimes (inside/outside the horizon). You should be able to explain what the naive solution is and the corrections to this (Baryon loading, Radiation driving, Diffusion damping). You should know what line of sight integration is and how this tells us how the perturbations are mapped into the CMB power-spectrum. You should be able to explain how different parts of the CMB power-spectrum can be used to tell us about cosmological parameters.