# Table of contents

Back to the main page 1. Lecture notes 2. Literature 3. Useful resources

# Lecture notes

A brief summary of some useful math needed in this course (will be introduced as we go along):

- 1 Math
- 1.1 3D Integrals
- 1.2 Boltzmann Integrals
- 1.3 Fourier transforms
- 1.4 Legendre Multipoles
- 1.5 Power-Spectrum
- 1.6 Spherical Harmonics
- 1.7 Spherical Bessel Functions

A brief introduction to General Relativity:

- 1 General Relativity
- 1.1 Newtonian gravity
- 1.2 Tensors
- 1.2.1 3D Rotations
- 1.2.2 Einstein Summation Convention
- 1.3 Vectors and Tensors in spacetime
- 1.4 Derivatives in spacetime
- 1.5 The metric tensor
- 1.6 Parallel transport and the Riemann curvature tensor
- 1.7 The Ricci tensor, the Ricci scalar and the Einstein equation
- 1.8 Minimal coupling and the Geodesic equation
- 1.9 Newtonian limit
- 1.10 Summary

A brief introduction of the background cosmology and thermodynamics/statistical physics in General Relativity:

- 1 Background cosmology
- 1.1 Derivation of the Friedman equations
- 2 Thermodynamics and statistical physics
- 2.1 Distribution function
- 2.2 Boltzmann equation in general relativity
- 2.3 Moments of the Boltzmann equation
- 2.4 Boltzmann equation in a smooth Universe
- 2.5 Boltzmann equation for the number density in a smooth Universe
- 3.1 The optical depth and the visibility function
- 3.2 Hydrogen Recombination: Saha approximation
- 3.3 Hydrogen Recombination: Peebles equation

# Literature

## Textbooks

For a textbook that covers the theoretical background needed for this course see Dodelson "Modern Cosmology".

For cosmology generally, Weinberg is a good reference. For most people it will not be a good book to learn cosmology from, since it goes into tedious detail in all the arguments, but if you want a rigorous treatment or want to understand some detail that is skipped or handwaved away in a usual textbook, then this is the book to look at: S. Weinberg, "Cosmology", Oxford University Press (2008).

For learning more about General Relativity I highly recommend Sean Carrol's book, thorough and clear with a good section on cosmology: S. M. Carroll, Spacetime and Geometry: An Introduction to General Relativity", Addison Wesley (2004).

## Online material

There are some very good lecture notes online for example Daniel Baumann's notes from Cambridge (which is basically a book) is very good. The presentation here is more theoretical level than Dodelson so it depends on what you like best. I strongly reccomend both this and Dodelson's book.

You can find a PDF with the lecture notes Sean Carroll's book is built on here. For a more compressed introduction to General Relativity I reccommend A No-Nonsense Introduction to General Relativity by Sean Carroll. For a summary of what you need to know about General Relativity in this course see this note by Haavard Ihle or this note by Oystein Elgaroy. There is also some material given in the leture notes below.

Most of the teaching in this course will be on the blackboard, but for the things that we do as slides we will upload those here.

- PDF Slides: Introduction to the numerical project and Milestone 1 by Hans A. Winther (2020)
- PDF Slides: Introduction to Milestone 2 by Hans A. Winther (2020)
- PDF Slides: Introduction to Milestone 3 by Hans A. Winther (2020)
- PDF Slides: Introduction to Milestone 4 by Hans A. Winther (2020)
- PDF Slides: Introduction to theoretical cosmology by Hans A. Winther (2018)
- PDF Slides: Theoretical cosmology: Thermal history and the early Universe by Hans A. Winther (2018)
- PDF Slides: Summary of recombination (what to do in Milestone II) by Hans Kristian Eriksen (2015)
- PDF Slides: An introduction to the CMB power spectrum by Hans Kristian Eriksen (2015)
- PDF Slides: Acoustic oscillations and $C_\ell$ peaks Haavard Ihle (2018)

# Useful resources

## Videos

The Perimeter Institute in Canada has put some lectures that overlaps fairly nicely with a lot that we will go through. You can find these lectures on this website.

## Papers

Petter Callin has written a great low-level review "How to compute the CMB spectrum" that has all the equations, some programming techniques and some plots that you can use to compare your results to. This is suitable for everybody and we will use the same notation as they do in this paper.

Another great paper is Chung-Pei Ma and Edmund Bertschinger "Cosmological Perturbation Theory in the Synchronous and Conformal Newtonian Gauges". It's written at a much higher level than the paper above so if you are new to cosmology stick with the one above. The abstract: *"This paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. It differs from others in the literature in that we give, in both gauges, a complete discussion of all particle species that are relevant to any flat cold dark matter (CDM), hot dark matter (HDM), or CDM+HDM models (including a possible cosmological constant). The particles considered include CDM, baryons, photons, massless neutrinos, and massive neutrinos (an HDM candidate), where the CDM and baryons are treated as fluids while a detailed phase-space description is given to the photons and neutrinos (...)"*

In Milestone IV we will do so-called line-of-sight integration. The original paper describing this key technique is "A Line of Sight Approach to Cosmic Microwave Background Anisotropies" by Uros Seljak and Matias Zaldarriaga.

For a more analytical approach to the evolution of perturbations and the CMB power-spectra see the great papers by Hu & Sugiyama, Hu, Sugiyama & Silk and Eisenstein & Hu ([1], [2], [3], [4], [5]) These papers are great for getting a deeper understanding of the features in the power-spectra and the results in these papers can also be used to check that the results we get from our code are reasonable.

## Physics of the CMB

As for getting physical intuition about the CMB and how different physical processes manifests themselves in the power-spectra there are no better online resource than Wayne Hu's CMB tutorials.

## Codes

Once you get to the end of this project and have your prediction for the CMB power spectra. Then its useful to compare your results to public codes and see how well it agrees. For this NASA has made an online tool that you can use to compute this, see CAMB Web Interface.

You can also download and run CAMB or CLASS yourself.

There are several codes that does the recombination history. We have Recfast++, this is the same code that is (most commonly) used within CAMB and CLASS, and if you want a more accurate recombination analysis (that takes many excited states into account which again has a $\sim 1\%$ effect on $X_e,T_b$) then there is CosmoRec and HyRec.