This last term I taught Introduction to Cosmology, a course for graduate students at the University of Chicago (although some undergrads typically do take it). While I have been known to pass myself off as a professional cosmologist, I’ve never found it a particularly easy course to teach. The straightforward reason is that it’s a big universe out there, and not much time to cover it. The UofC is on a quarter system, so a term is only ten weeks of classes; this makes it hard to squeeze in as much material as you would be able to in an ordinary semester.
More importantly, though, cosmology is a mess. Unlike most other subjects that would have a course devoted to them, there is no sense in which cosmology is a single logical structure that is built up from a small number of axioms. To discuss various crucial topics, you need to bring in general relativity, thermodynamics, particle physics, astrophysics, and occasionally the kitchen sink. In particular, neither GR nor particle physics are prerequisites for taking the course, so the basics of those subjects need to be covered when necessary.
A substantial fraction of contemporary cosmology is devoted to investigating structure formation and the cosmic microwave background. To do those subjects any justice requires not only the basics of general relativity, but a pretty well-grounded understanding of relativistic perturbation theory, which is an intricate and subtle discipline all its own. So the prospective cosmology instructor has a choice: go whole-hog in doing structure formation and perturbation theory, skipping past many of the fun topics in early-universe cosmology, or do the converse, putting some effort into inflation and relic abundances and nucleosynthesis while waving hands briefly about large-scale structure and the CMB. Since there is a separate cosmology course taught in the Astronomy department, which inevitably concentrates on structure formation and the CMB, I chose the latter route. We covered the basics of general relativity (enough to derive the Friedmann equation for the expansion of the universe) and particle physics (enough to understand the basics of cross-sections and calculate relic abundances). Hopefully, the students who don’t decide to become full-time working cosmologists will nonetheless, ten or twenty years down the line, recall the basic ideas of how to calculate the density of a dark matter candidate or why primordial nucleosynthesis provides such strong constraints on the physics of the early universe.
There were problem sets every week, but no final exam. Instead, the students each wrote a final paper, and the good news is that you get to read them. The final papers have been put on a web page (as pdf files), and you could do much worse by way of reviewing the hot topics in current cosmology research than to read through these papers. (The indended audience for the papers was “people who have just taken this course,” so they do tend to get a little technical.) In the past I’ve asked each student to pick a somewhat narrow topic and do a little review on it. This year, as an experiment, I instead asked them to find one specific research paper that had appeared in the past year or so, and write an overview of it that explained the main results as well as some of the background. Topics include:
- Nucleosynthesis contstraints on the variation of constants
- Origin of cosmological magnetic fields
- Alternatives to dark energy
- Anomalies in the cosmic microwave background
- Methods for probing cosmic acceleration
- Properties of perturbations generated by inflation
- Thermal field theory in the early universe
- Quantum-computational cosmology
- Characterizing CMB polarization
- The quantum vacuum and the cosmological constant
- Origin of supermassive black holes
- The birth of the universe in string theory
- Searches for dark matter
- The topology of the universe
- Limits on primordial gravitational waves
Overall, they did a fantastic job, and I’m proud to share the results with the wider world.
Cosmology must be relatively fun these days. It’s probably true that it is hard to teach it because it relies on many things that the students don’t necessarily know in advance. But so do other fields. Condensed matter physics also more or less requires everyone to know thermodynamics, quantum mechanics, electromagnetism, mechanics of solids, etc. String theory requires more or less all of previous physics plus some branches of maths.
I feel that all these interesting papers should have been written in a more publicly usable and organizable form. What about asking students in similar courses all over the world to write the paper as a set of technical math-ready Wikipedia articles? It would be more useful for the rest of the world, and the students might be more motivated.
Hi, these term papers are really very good! Bravo for teaching such a good course and for bringing the students up to such a high level. They surely were happy with the course, since you gave them a chance to learn about very interesting and timely topics. How much better that must be than simply passing a take-home exam! Perhaps we need to find a way to lure you up to Evanston to teach for a quarter…
Lubos:
That’s an interesting sugestion! One problem with this is that currently wikipedia doesn’t handle mathematical text too well. You have very limited LaTex commands available. You can’t use automatically numbered equations etc…
Lubos,
A rather tasty idea… Then I could digest Wikipedia all day! 🙂
Maybe someone can ask Chris Hillman and the other mathematicians how they manage the equations in their Wikipedia pages (an example).
I think they use MathML.
thanks for the link, the reviews of topics too recent to be covered by books are really useful and well done.
dear Lubos, I regret that I spent two years studying strings, when in two months one can learn cosmology and have some relative fun
Dear AO, there might be even more straightforward ways for you to have fun. Others, thanks for constructive replies to the proposal. I am not 100% quite sure whether equations in online encyclopedias should be numbered. Is this really a killing problem? 😉
Lubos, well it is annoying if you have a lot of equations in an article and want to make changes. Don’t forget that the intended public are people who need more explanations so you need to show intermediate steps that to us can look trivial. Being able to link to specific equations from other articles could be very useful too.
PK: Thanks for the MathML info regarding Wikipedia. I knew that there was a reason that I should upgrade my MathType. Sean: Your students’ papers (all of the ones I looked at so far) are really great. As a person who works in a science field 10^{34] scales away from cosmology, the papers give me something with meat to chew on to learn some of the key methods/questions/answers.
The most interesting paper is http://pancake.uchicago.edu/~carroll/371/papers06/Caldwell-BBN.pdf
Gravitational compression and electrostatic repulsion of charges control the fusion rate. So the analysis is showing that the ratio of gravity to electromagnetic force was similar in the first seconds of the BB.
So the evidence that gravity strength constant G was the similar to the current value in the big bang is not there.
Example: suppose gravity was x times weaker at 1 second after BB. Nucleosynthesis would be the same, because electromagnetism and gravity strength is a constant.
Changing G doesn’t vary the fusion rate of charges, because the effect of extra gravitational compression is simply offset by the extra Coulomb repulsion.
Fusion can only occur where charges approach closely enough for the strong attractive nuclear force to cause they to fuse together. This means attractive gravity must overcome the repulsive Coulomb barrier.
The paper is very useful when you take it as showing that the ratio of gravity to electromagnetism was constant. (This is quite a different claim than saying gravity is constant.)
It’s actually not a change in G but a change in a dimensionless coupling that is constrained, see here.
Wikipedia does not currently output MathML. There is a project to add MathML support to MediaWiki (the wiki software that powers Wikipedia). Currently, MediaWiki just outputs PNGs. BlahTeX will add the option of outputting MathML instead.
Count Iblis,
Thanks for that reference, as it confirms what I just said:
“The paper is very useful when you take it as showing that the ratio of gravity to electromagnetism was constant. (This is quite a different claim than saying gravity is constant.)”
A ratio of force strengths is dimensionless, and this ratio isn’t changing. The absolute strengths of the forces can vary.
Dear Prof. Carroll,
I think a combination of the two books: Mukhanov’s new cosmology book and Dodelson’s “Modern cosmology” is good for students. However, they won’t fit in a quater of UC. 🙂
Best, Y
Thanks, Yidun. I haven’t seen Mukhanov’s book yet, I’m looking forward to it. Dodelson’s book is fantastic, although it focuses on the microwave background — it would be perfect for a course that took the opposite tack from mine, neglecting the early universe for the later universe. There is a relatively new book by Lars Bergstrom and Ariel Goobar that is quite good, although at crucial places it doesn’t go into as much detail as you would like from a text.
Prof. Carroll: Thanks! I think your TASI lecture notes is very good for an intro course on cosmology, although students still need to read many reference papers. But they need to read references anyway. I strongly agree that cosmology is very hard to teach, unlike those regular grad courses such as QM, GR, and so on. However, I think a student should have better learned advanced GR and QFT before he sit in a real cosmology class. Otherwise, it would be hard for both of the student himself and the teacher.
Science,
I don’t think gravitational compression has anything to do with primordial nucleosynthesis. It matters to nucleosynthesis in stars and I don’t know where else.
Primordial nucleosynthesis is supposed to depend on G in two ways: first, via the expansion rate H which determines the temperature at which the weak interactions, that interconvert protons and neutrons, freeze out and the relative neutron abundance is determined; second, by determining how long the neutron population spends decaying into protons while it waits for the universe to get cold enough that the formation of deuterium outpaces its photodissociation by photons in the high-energy tail of the Planck spectrum.
Now those arguments involve dimensional quantities. If their measurement is “operationally meaningless” then it is news to me. I doubt if I quite understand the paper that Count links to, but I am not convinced by a quick persual of it.
Shane
Shane, what Michael Duff means is that you can always eliminate the dimensional constants and formulate your results entirely in terms of dimensionless constants.
Newton’s constant enters the equations via dimensionless combinations of the Planck mass, Fermi’s constant and masses of the proton, neutron etc. The ratio of the abundances of the light elements are dimensionless numbers, so these are functions of these dimensionless combinations.