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Table of contents

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Back to the main page 1. Lecture notes 2. Literature

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Lecture notes

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The theory needed in this course is covered in Dodelson and Baumann, but we also provide a comprehensive set of lecture notes given below including videos where we go through the lectures. The topics covered are grouped by what you need to know to complete each milestone in the numerical project.

Overview of the course

• 1.1 Overview of Part I: Background Cosmology
• 1.2 Overview of Part II: Recombination History
• 1.3 Overview of Part III: Evolution of structure in the Universe
• 1.4 Overview of Part IV: The CMB and matter power-spectra

[Milestone I] A Crash course in General Relativity - The Rise of the Einstein tensor:

• 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

[Milestone I] Background cosmology - Revenge of the Cosmological Constant:

• 1 Introduction to theoretical cosmology
• 2 Background cosmology
• 2.1 The Cosmological Principle
• 2.2 Mass-Energy content of the Universe
• 2.3 Derivation of the Friedman equations
• 2.4 Cosmological parameters and the standard model of cosmology
• 2.5 Measures of time in cosmology
• 2.6 Measures of distances in cosmology
• 2.7 Summary

[Milestone II] Thermodynamics/statistical mechanics and the thermal history of our Universe - The Photons Strikes Back:

• 1 Thermodynamics and statistical mechanics
• 1.1 Distribution function
• 1.2 Boltzmann equation in general relativity
• 1.3 Moments of the Boltzmann equation
• 1.4 Boltzmann equation in a smooth Universe
• 1.5 Boltzmann equation for the number density in a smooth Universe
• 1.6 Decoupling, freeze-out and the Saha approximation
• 2 Thermal history of the Universe
• 2.1 Recombination
• 2.2 The optical depth and the visibility function
• 2.3 Hydrogen Recombination: Saha approximation
• 2.4 Hydrogen Recombination: Peebles equation
• 2.5 Summary

[Milestone III] Cosmological perturbation theory - Return of the Einstein Tensor:

• 1.1 Metric perturbations
• 1.2 The Newtonian gauge
• 1.3 Perturbations in real and fourier space
• 1.4 Our plan
• 1.5 Perturbations of the Einstein-Boltzmann equations
• 1.5.1 Geodesic equation in a perturbed Universe
• 1.5.2 Boltzmann equation for Photons
• 1.5.3 Boltzmann equation for Baryons
• 1.5.4 Boltzmann equation for Cold Dark Matter
• 1.5.5 Boltzmann equation for Massless neutrinos
• 1.5.6 Multipole expansions
• 1.5.7 Einstein equations
• 1.6 Summary

[Milestone III] Initial conditions and the theory of inflation - A New Hope for Isotropy:

• 1 Initial conditions
• 1.1 Inflation
• 1.2 Why do we need inflation and how does it work?
• 1.3 Predictions of inflation: primordial power-spectrum
• 1.4 Connecting inflation to initial conditions: Conserved curvature perturbation
• 2 Adiabatic initial conditions
• 3 Truncating the Boltzmann hierarchy
• 4 Summary

[Milestone IV] From perturbations to statistical observables and evolution of the perturbations - The Power-Spectrum Awakens:

• 1 From perturbations to statistical Observables
• 1.1 What do CMB experiments measure?
• 1.2 The CMB power-spectrum
• 1.3 Line of sight integration
• 1.4 The matter power-spectrum
• 1.5 Polarisation and gravitational waves
• 2 Physical understanding of the observables
• 2.1 The CMB power-spectrum
• 2.1.1 What the line of sight integrals tell us
• 2.1.2 Acoustic oscillations
• 2.1.2.1 The free oscillator
• 2.1.2.2 Baryon loading
• 2.1.2.3 Radiation driving
• 2.1.2.4 Diffusion damping
• 2.1.2.5 Summary
• 2.1.5 Cosmological parameter dependence of the CMB power-spectra
• 2.2 Matter power-spectrum
• 2.2.1 Growth of matter perturbations
• 2.2.2 Perturbations in real space
• 2.3 Summary

[Milestone V] Non-linear structure-formation (not in the curriculum):

• 1 Non-linear structure-formation
• 1.1 The Vlasov-Poisson equations
• 1.2 Lagrangian perturbation theory
• 1.3 Algorithms for density assignment
• 1.4 Algorithms for solving the Poisson equation
• 1.5 Algorithms for time-integration
• 1.6 Algorithms for power-spectrum estimation
• 1.7 Algorithms for halo-finding

[Appendix] Units in cosmology:

• 1 Units
• 1.1 Particle Physics Units
• 1.2 Planck Units
• 2 Dimensional analysis

[Appendix] Numerical methods:

• 1 Numerical methods
• 1.1 Solving coupled ODEs
• 1.2 Spline interpolation
• 1.3 Spherical Bessel Functions
• 1.4 External codes: Recombination solvers
• 1.5 External codes: Einstein-Boltzmann solvers
• 1.6 Generating a gaussian random field (not in the curriculum)
• 1.7 Generating a CMB map (not in the curriculum)
• 1.8 List of various cosmology codes

[Appendix] Some 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 Gaussian random fields
• 1.6 Power-Spectrum
• 1.7 Spherical Harmonics
• 1.8 Spherical Bessel Functions
• 1.9 Lagrangian formulation of General Relativity (not in the curriculum)

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Literature

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Textbooks

For a textbook that covers the theoretical background needed for this course see Dodelson "Modern Cosmology" (2003). A brand new second edition of this book, written by Scott Dodelson and Fabian Schmidt, was released in 2020 and contains many modern developments.

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 a bit more theoretical than Dodelson so it depends on what you like best. I strongly reccomend both this and Dodelson's book.

For cosmology generally, Weinberg is a great 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).

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 Øystein Elgarøy. There is also some material given in the leture notes below.

Here are some lecture notes from PhD winther/summer schools that are very nice. They are very dense, but contains a good summary of all the material we are going to go through in this course. More relevant for PhD students and to use as reference materials. Lecture Notes on CMB Theory: From Nucleosynthesis to Recombination by Wayne Hu. Covariant Linear Perturbation Formalism by Wayne Hu. Physics of the Cosmic Microwave Background Radiation by David Wands, Oliver F. Piattella and Luciano Casarini. Wandering in the Background: A CMB Explorer by Wayne Hu (PhD Thesis).

Here are some very good lecture notes that covers most of the theory we are going through in this course written in Norwegian:

• En kort introduksjon til generell relativitetsteori by Øystein Elgarøy
• Kosmologisk perturbasjonsteori: forpostfektninger by Øystein Elgarøy
• Kosmologisk perturbasjonsteori: I ditt ansikts sved skal du ete ditt brød by Øystein Elgarøy
• Kosmologisk perturbasjonsteori: Einsteintensoren vender tilbake by Øystein Elgarøy
• Initialbetingelser: I begynnelsen var $\varphi$ by Øystein Elgarøy
• Skalarfelt og kvantefluktuasjoner by Øystein Elgarøy

Research 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. You should read this paper for the numerical project.

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 paper presents a systematic treatment of the linear theory of scalar gravitational perturbations in both the synchronous gauge and the conformal Newtonian gauge. It gives a complete discussion of all particle species that are relevant to any flat cold dark matter model: cold dark matter, baryons, photons, massless neutrinos and massive neutrinos.

A more modern version of what the paper above does and also a much more complete description of how to solve for the CMB including all possible geometries for both scalar, vector and tensor perturbations is A Complete Treatment of CMB Anisotropies in a FRW Universe by Wayne Hu, Uros Seljak, Martin White and Matias Zaldarriaga.

A great classical paper on cosmological perturbation theory is Gauge-independent cosmological perturbations by James Bardeen.

Some other good papers for including additional physical effects (that we don't cover in this course):

• Efficient Computation of CMB anisotropies in closed FRW models by Antony Lewis, Anthony Challinor and Anthony Lasenby. How to do the CMB in a closed Universe.
• Lensed CMB power spectra from all-sky correlation functions by Anthony Challinor and Antony Lewis. How to include the effect of gravitational lensing to the CMB spectrum.
• A Line of Sight Approach to Cosmic Microwave Background Anisotropies by Uros Seljak and Matias Zaldarriaga. The paper proposing the key line of sight technique.

Some papers from experiments that have measured the CMB and the cosmological parameters constraints they obtained (see also List of cosmic microwave background experiments for a complete list of 50+ experiments performed to date):

• The Cosmic Microwave Background Spectrum from the Full COBE/FIRAS Data Set by Fixsen et al. (1996). The best we had back in 1996. The CMB spectrum up to $\ell \sim 30$.
• Cosmology from Maxima-1, Boomerang and COBE/DMR CMB Observations by Jaffe et al. (2001). State of the art back in 2001. Combination of data from COBE and the baloon experiments Maxima and Boomerang.
• Cosmological parameters from SDSS and WMAP by Tegmark et al. (2004). Combination of CMB and galaxy clustering data. State of the art in 2004 getting $\ell \sim 800$ for temperature and $\ell \sim 400$ for polarisation.
• Nine-Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Cosmological Parameter Results by Hinshaw et al. (2013). The final WMAP results. State of the art in 2013 with $\ell \sim 2000$.
• Planck 2018 results. VI. Cosmological parameters by the Planck collaboration (2018). State of the art from Planck today with $\ell \sim 2500$ for temperature and $\ell \sim 2000$ for polarisation.
• BICEP2 / Keck Array VIII: Measurement of gravitational lensing from large-scale B-mode polarization by Keck array and BICEP collaboration (2016).
• Measurements of Degree-Scale B-mode Polarization with the BICEP/Keck Experiments at South Pole by Buder et al. (2018). Ground based polarisation measurements from BICEP/Keck.
• The Atacama cosmology telescope: CMB polarisation at $200 \lt \ell \lt 9000$ by Sigurd Naess et al. (2018). Ground based polarisation measurements from ActPol.

Various slides used in lectures

• 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)

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