The Sciences

Anatomy of a Paper: Part II, Calculation

Cosmic VarianceBy Sean CarrollJul 31, 2007 3:38 AM


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In the exciting cliffhanger that was Part One, we saw how the idea behind a paper came to be -- nurtured from a meandering speculation into a somewhat well-defined calculational question. In particular, Lotty Ackerman and Mark Wise and I were asking what would happen if there were a preferred direction during inflation -- an axis in the sky along which primordial perturbations were just a little bit different than in the perpendicular plane. We guessed, even in the absence of a specific model, that such a statistical anisotropy would show up as a nearly scale-invariant modulation of the power spectrum. Now we need to turn such ideas into something more concrete. In fact, our phenomenological guess was enough to go and start calculating how this new effect will show up on the CMB, and we all set about doing exactly that. None of us -- Mark, Lotty, and I -- are really experts at this sort of thing, but that's why they make books and review articles. (Without Scott Dodelson's book, I would have been in trouble.) As it turns out, many years ago Mark had written one of the very first papers on deriving CMB anisotropies from inflationary perturbations, so he had a head start on calculating things. But the analysis that he and Larry Abbott had done way back when had concentrated on the gravitational redshift/blueshift of the CMB (the Sachs-Wolfe effect), which is only the most important contribution on large angular scales. Lotty and I realized that we should be able to calculate the effect at every scale all at once, which turned out to be right. It's true that messy astrophysical effects (acoustic oscillations) become important at medium and small scales, and it would take a real cosmologist to understand them. But all we were doing was changing the initial amplitude of the perturbations, in a direction-dependent way. The eventual effect is simply a product of the initial amplitude and a "transfer function" that encodes the messy fluid dynamics once and for all; since our new primordial power spectrum left the transfer function unaffected, we didn't have to worry about it. (More generally, Lotty and I were full contributors when it came to ideas, but Mark is very fast when it comes to calculations. We would have to occasionally distract him with something shiny while we sat down to catch up with the equations.)

So we read up on calculating CMB anisotropies, and applied it to our model. Since everyone usually assumes that all directions are created equal, we couldn't simply plug and chug; we had to re-do the usual calculations from the start, keeping the extra degree of complexity introduced by our preferred direction. That provided a good excuse to educate ourselves about some of the nitty-gritty involved in turning primordial density perturbations into a signal on the CMB sky. In particular, we had to play with spherical harmonics, which are the conventional way to encode information spread over a sphere -- for example, the temperature of the microwave background as a function of position on the sky. Every good physicist knows the basic properties of spherical harmonics, but we had to do some particular integrals that were not that common. I don't know about you, but when I'm faced with a nontrivial integral, I try Mathematica first, ask questions later. But Mathematica didn't know these integrals, so actual work was required. At some point it dawned on me that we could use a recursion equation -- relating one spherical harmonic to a set of others -- to turn the integral into something doable. No special points for me; my collaborators figured it out independently. Still, it's always fun to crack a knotty calculational problem. A few amusing footnotes to the recursion-equation episode. First footnote: I figured it out while sampling a martini at the Hilton Checkers lounge in downtown L.A. This was last fall, while I was still relatively new to the area, and was spending time checking out the various local establishments. Verdict: a pretty good martini, I must say. The bartender was intrigued by all the equations I was happily scribbling, and asked me what was going on. I explained just a bit about the CMB etc., and she was genuinely interested. But then, alas, she mentioned something about astrology. So I had to explain that this was actually very different etc. I got the impression that she ultimately did appreciate the difference between astronomy and astrology, once it was laid right out there. Now if only we could replace the horoscopes in daily newspapers with charts of the night sky. Second footnote: it is one thing to think "maybe a recursion equation would be useful here," it's another thing to actually remember the damn equation. I'm generally not a good equation-rememberer, and I wasn't lugging any reference books with me. (I was in a bar, remember?) But I was lugging my laptop with me, and there was wireless internet. So naturally I looked up the equation on Wikipedia, and there it was! I checked it against some more conventionally reliable resource once I got home, but the Wikipedia page was perfectly accurate. (Nobody finds it worthwhile to vandalize pages on special mathematical functions.) Final footnote: something interesting is revealed about the nature of a technical education. The point is, I didn't know much about the particular recursion equation that I ended up using; I wasn't sure that an appropriate equation even existed. There was just a vague feeling in my mind that Legendre polynomials (which I did know how to relate to spherical harmonics) were the kind of thing that probably obeyed some recurrence relations. But the last time that I actually had dealt directly with Legendre polynomials, if ever, was most likely when I took an undergraduate class in quantum mechanics or mathematical physics, a good twenty years (half my life) ago. In other words, my physics education worked exactly as it is supposed to -- it stuck a vague idea into my head that persisted undisturbed for a couple of decades, that was there when I needed it, and provided me with just enough information that I knew where to turn when the occasion arose. This is one of the reasons I feel such antipathy toward GRE's and grad-school qualifying exams and the like: they set up a testing environment that bears absolutely no relationship to the way that real research is done, and end up valorizing a certain kind of cleverness and calculational speed over real insight and creativity. On the other hand, they do provide a way to quantify something, even if it's not something very important, and we can then proceed to deploy these scientific-looking numbers to separate the men from the sheep, secure in the knowledge that our quantifications are highly precise, if completely inaccurate. Okay, rant over. So we all managed to turn an interesting collection of cranks, showing how to convert a set of direction-dependent primordial density perturbations into a set of quantities one could observe in the cosmic microwave background. All in all, an impressive-looking bunch of equations resulted, and that was basically half of the paper we ended up writing. The other half dealt with the question we had started with in the first place -- is there some compelling, or at least plausible, physical model that would actually lead to such perturbations? Rotational invariance is a subset of Lorentz invariance, a cornerstone of relativity and thus of all of modern physics. Fortunately, however, violating Lorentz invariance is one of the things I am especially good at. I've already blogged about a paper I wrote with Eugene Lim on the cosmological consequences of Lorentz-violating vector fields. The big difference is that Eugene and I, following in the footsteps of Ted Jacobson and David Mattingly and others, had taken advantage of the fact that the real universe already has a preferred cosmological reference frame -- the one in which the CMB is statistically isotropic. We imagined that there were some hypothetical vector field that had a nonzero value in empty space, but which (basically) pointed along a timelike direction, orthogonal to hypersurfaces of constant cosmological time. What Lotty, Mark and I needed was a vector field that pointed in some preferred direction in space, i.e. a spacelike vector. But that's not so hard; just take the theory with a timelike vector and change some minus signs to plus signs. We didn't put too much tender loving care into constructing the world's most compelling theory, because it wasn't the theory that was our primary concern -- it was the robust predictions that theories of that sort might end up making. But we were able to write down a model that seemed to have all the properties we wanted. Within the assumptions of that model, we could make a very specific calculation of the predicted density perturbations, and compare them with the model-independent guess we had started with. There was pretty good agreement; our guess was that the perturbations would be basically scale-invariant, and the particular model we considered produced perturbations that only varied by about 10% over phenomenologically interesting scales. I should mention that, while working on the vector-field idea, I found myself in another bar -- this one across the puddle, a neighborhood pub in London. Guinness this time, not a martini. And wouldn't you know it, the bartender sees my equations spread out there and asks what it is I'm doing. (By the time I retire, every bartender in the Western hemisphere is going to have at least a passing acquaintance with the basics of contemporary cosmology.) This guy was really into it, and wanted to write down not just the title but also the ISBN number of the book I was reading. Since it was Dodelson's cosmology text, which is a gripping read but full of equations, I scribbled a short list of more accessible books he could check out, about which he seemed truly excited. Now if only the London pubs would stay open past ten p.m., we'd have an excellent situation all around. Stay tuned for our exciting conclusion in Part Three!

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