This is my first report for the course “Data Science Laboratory” in the second semester in 2020-2021 on ELTE. In this article, I will briefly summarize all the relevant information about my semester project and then detail my work so far on it.
As an MSc Physics student on ELTE enrolled in the Data Science specialization, we need to pass the mandatory course called “Data Science Laboratory”. The exact goals of the course are meticulously articulated in the subject’s GitHub repo as “…to instill the practical skills needed for exploratory data analysis. With the acquired knowledge the student shall be able to perform independent research requiring the handling of big data.”
At the beginning of the semester each student must select a task from a list of existing projects and make reports and oral presentations about the topic every other week. I’ve chosen the task involving work with the Cosmic Microwave Background radiation (or CMB for short), which I need to study both from the computer simulation and the data analysis side.
Even those with basic interest and knowledge in astronomy have heard of the CMB probably, at least on an elementary level. The notoriety of this phenomenon implies it correctly, that it is an actively studied field in astronomy. The reason is that the CMB has an essential role in the understanding of fundamental questions and topics in physics. It is used eg. to determine cosmological parameters or to better understand the events following the Big Bang.
In its core nature CMB is an immensely redshifted black-body radiation with average colour temperature of $2.725$ K. By saying “CMB” we actually refer to the collection of those photons, which was created in the Big Bang and thus already existed during the recombination period of the universe. For a long period, the universe was opaque. Photons created in the Big Bang constantly scattered on electrons in this very dense and hot so-called “gas-universe”. After approximately $370\,000$ years after the Big Bang the universe cooled to a point (to $\approx 3000$ K), where the formation of hydrogen atoms was energetically favorable. We say, that at this point the existing photons “decoupled from the matter” (they stopped being constantly scattered on electrons), their mean free path increased by an intangible extent, they stopped being in constant thermodynamical equilibrium with free electrons and the universe become electromagnetically transparent. Due to the expansion of the universe since then, the originally $3000$ K hot CMB photons redshifted by a factor of $z = 1100$ to the already mentioned $2.725$ K, which we observe today. These photons travel through space since the era of the recombination. Looking at the sky, the light of the CMB reaches us from all direction at the same time. Effectively, we see this light coming to us from a very distant spherical shell surrounding us, while we’re sitting in the middle. Since this shell represents the place of origin of the CMB at the time of the recombination event, we call this as the “Last Scattering Surface” (or LSS for short)[1].
This radiation carries cardinal information about the cosmology of the universe. First the primordial fluctuations of the universe imprinted into its structure (this is called as the primary anisotropy). Later as the CMB traveled through space, it came into contact with objects and structures in the universe, which locally changes the CMB and imprints the data of cosmological objects into it (which component is called as the secondary anisotropy).
During the observations we measure the temperature fluctuations of this light in every direction from which we’re able to create a temperature map. This map was first created using the data of the COBE space observatory and was announced in 1992[2]. This was followed by the WMAP spacecraft in 2003[3], and later by the Planck mission in 2013[4].
This temperature fluctuation could be explained by that the universe had - an already mentioned - primordial anisotropic structure that built up these fluctuations starting somewhere until the recombination event, which we observe in the CMB radiation today. The information or the quantity what we measure is a scalar function over a spherical surface (this is the LSS mentioned the previous subsection). First and foremost it’s treated as a temperature, so
\begin{equation} f \left( \vartheta, \varphi \right) \equiv T \left( \vartheta, \varphi \right) \end{equation}
where $f$ could is an arbitrary scalar quantity over a spherical surface. I’ll consider the quantity $f$ only as a temperature in this project.
My project goals could be divided into two big sections.
The first one involves the simulation of artificial CMB maps and the determination of their power spectrum. This consist of the analysis of the CMB’s power spectrum both with, and without the additional noises and effects originated from the measuring instruments and other sources. To successfully perform this task, I’ll need to dive into the generation process of the CMB temperature map. To make my life easier, I can use the supplementary materials recommended on the course for this project.
The second section of the project consist of the analysis of actual observational data. As it was already mentioned in part II., the Planck space observatory made the latest and most precise observations on the CMB until now. Also, since its data can be accessed freely from the official site of the mission[5] in numerous different formats, I’ll use the observations of the Planck space observatory in my project. My goal in this second section is to reverse engineer the power spectrum from the real dataset and compare it against the simulated one. This requires first to filter out all the superimposed effects (noise, secondary anisotropy, etc.) in the raw data as much as possible. Optionally I also intend to determine some of the cosmological parameters using these observations, if that is possible within the confines of this course.
The Internet holds a wealth of information about both the computational and analytic aspects of the CMB research. I’ve started with the first part of my project by simulating CMB maps from an arbitrary power spectrum. My main lead to learn the process to simulate CMB maps were the Jupyter notebooks and other materials made for the McMahon Cosmology Lab’s ACT CMB Summer School in 2015[6]. These notebooks covers the required part of the CMB simulations from a theoretical point of view almost entirely. Unfortunately the actual coding is more or less obsolete and unfinished in many ways.
My workflow so far consists of the complete rewrite of the ACT CMB School notebooks to make them considerably more versatile even in their core functionality. Also I need to completely revamp the rudimentary visualization capabilities to be able to create at least acceptable figures for the project.
After finishing the work on the most basic functions I was able to generate my first full-sky CMB map from the input simulated power spectrum. My first acceptable attempt could be seen on Figure 4., below. This was made by simply generating a rectangular map of the CMB and then casting it on a figure with Mollweide projection.
All the codes are available in my GitHub repo made for this course.
Generating a map which could be seamlessly projected to a sphere, or to a projection of a sphere is not trivial. Simply generating a 2D, rectangular matrix and casting it on a spherical surface simply do not work, or at least isn’t looking good. The map distorts closer to the poles, which makes it considerably uncomfortable to look at. Because of this, I’ll make my primary goal to generate small, rectangular, but geometrically precise snippets of the CMB map. If I’ll have enough time, I’m going to try to recreate these maps on a spherical surface too.