Anatomy of a Paper: Part II, Calculation

By Sean Carroll | July 30, 2007 11:38 pm

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.)

CMB map 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!

  • Eric

    The danger with getting things like recursion relations for special functions from Wikipedia seems to be not so much vandalism as the fact that it’s entirely possible that the person who wrote the article has made mistakes that no one has caught yet. If you ever want a source for such things to check Wikipedia against on the fly, Abramowitz and Stegun is available online.

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  • Aaron Bergman

    There’s also Wolfram’s site.

  • Nonnormalizable


    I don’t know about you, but when I’m faced with a nontrivial integral, I try Mathematica first, ask questions later.

    I’m with you all the way!

  • Anon

    Gradshteyn and Ryzhik (used to) rule.

  • Carl Brannen

    The spherical harmonics show up in the time independent solutions to Schroedinger’s equation for rotationally symmetric potential functions. But Laplace’s equation is simpler. So I used Laplace’s equation when I wanted to write up an introduction to how quantum numbers arise from differential equations.

  • Joe Zuntz

    Fun paper.

    The resulting expressions (10,12,13) can be directly compared with observations to probe the existence of small Lorentz-violating effects in the very early universe.

    Will you be doing that?

    Eq 10 would be pretty easy to tack onto a Boltzmann code like CAMB or CMBEASY. THe longest part would be typing in equation 13!

  • Anonymous

    “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”

    But Sean, don’t you recognize that it was very likely the whole process of studying for a grad-school qualifying exam that stuck that vague idea into your head to begin with? The whole point of a hard exam is to make you study much harder than you normally would. Most of the physics I still remember I learned from my qual.

  • Sean

    Joe, we’re not going to be doing that ourselves, but we’re hoping that someone is. (In fact, someone is.)

    Anonymous, in my case that’s not true, since in the astronomy dept at Harvard we didn’t have any such test; they were much more interested in getting us into research as soon as possible, which I think is a good idea. It’s certainly true that studying from qualifying exams can be very educational; what I object to is using the results of such exams to rate the ability of the examinee, even to the extent of kicking them out of the department. (I also don’t think the mental anguish that often accompanies such tests is worth it.) Empirically, these tests are worthless in judging how good somebody is at doing physics, and they shouldn’t be used for that purpose.

  • Count Iblis

    In many cases when dealing with special functions the methods described in the book
    The book A=B can be used. Most special functions are special cases of hypergeometric function. If you just do some brute force calculation by plugging in the definition of the function in terms of an infinite series, you’ll typically end up with a binomial summation. The book gives general algorithms for simplifying such binomial summations that always work (you’ll either get an answer or you’ll have the proof that it cannot be simplified).

  • Neil B.

    Hmmm… interesting to hear about preferred frames of any kind. Yes, of course there’s something special about being roughly at rest compared to the overall matter content of the universe, which means you see isotropic CMB radiation. I heard there is some motion (what, a few dozen km/sec? – which is reasonable for a random stirring around of different little bits of gas etc.) of our system relative to that frame. (I also heard from poster/blogger “Island” of some interesting coincidences about the direction of that, relative to the solar system – true?) Well, how does that compare with inertial/tidal issues of motion relative to that material? I recently started a discussion “GR support for a preferred frame 2” about inertial markers of “real motion” in the universe to the NG sci.physics.relativity, which attracted many decent quality responses. (The good stuff is not all on sci.physics.research, but thank _______ for that NG!) Below are some edited and rearranged excerpts of my own postings. Note well my point about how this relates to geometric shapes and motion on them per se, over and above any true physical characteristics or theory:


    For example, if space is hyperspherical (which maybe it isn’t due
    to current expansion, but in principle it could have been) then consider
    what happens inside a box “actually moving” through space: since internal test bodies follow great circle routes (geodesics) on the hypersphere, they will oscillate back and forth inside the container as it travels around the hypersphere at a rate proportional in part to “velocity” relative to the geometric construct itself. (Consider that their individual geodesics can’t stay parallel; they must intersect like longitude circles on the

    This is simply unavoidable, and means that once you have rules about
    “following” this or that path on a curved surface, then it becomes a
    standard for referencing motion on that surface. Well, relative to what? I
    would suppose, the average motion of matter, but this issue really isn’t
    adequately clarified, and not adequately presented in discussion and
    textbooks. (Sure, Taylor and Wheeler show ships coming together while
    traveling the surface of the earth in _Spacetime Physics_, but don’t really
    apply that to the implications for standard of motion in the universe.)

    … the environment inside the moving box really is a special frame, because objects experience a tidal field with no internal sources, or specific external ones (it’s not like passing by a planet….) because this is an effect from the overall density of the universe … small test particles spontaneously fall towards and away from the center of the box even though there is no mass density *within the box* (well, I assume we can keep other matter out of the box as it moves, and still maintain sufficiently geodesic motion for the test particles.) But yes, OTOH, particles falling to the center of the earth do that also, so the inside of the box is a sort of tidal field. The important thing to me is not whether the box environment should be called “special”, but the very fact we can tell how fast we are “moving through the universe” – IOW, the preferred frame for velocity, not “special” in any sense beyond that. See the interesting comments by Ben Rudiak-Gould [of UC Berkeley] on this
    issue [in this sci.physics.relativity thread.] (In particular, it isn’t
    what can be simulated by acceleration or rotation in an inertial space.)

    Note, this goes beyond physics, because it means we can define “motion” on a featureless geometric body once we have rules like the following of a geodesic. That is counterintuitive given there not being “marks” on the hypersphere to reference relative motion ….we can indeed tell that we are moving on the featureless surface in this case, that’s the whole point of my illustration. (This post has two points, the first is about motion in a real physical universe – even if ours isn’t curved that particular way. The second point is about motion “on a mathematical sphere”, where oddly we can also tell “how fast we are moving”!
    We can tell how fast we are moving by the rate of acceleration of the test bodies inside the box. We aren’t moving relative to the frame except to imagine that something of the surface is absolute as a holder of rest, since we aren’t moving farther or closer to anything (uniform curvature
    everywhere, and we stay inside the manifold.) This *is* odd and
    counterintuitive, since a “mathematical entity” like a sphere isn’t suppose
    to be thought of the way we’d consider a steel ball, that we know how fast we are sliding along because there’s a “real surface” to be moving past constituent particles of. I think it shows something about “time” being special as a concept.

  • Sam Gralla


    It’s like a searchable, more extensive, A&S.

  • citrine


    The fact that the Laplacian is seperable in many coordinate systems, including ellipsoidal ones, has always fascinated me. I presume that perturbations along some direction turns the initial spherical space into an ellipsoid. Did you get to use this result in any form?

  • Sean

    citrine — actually, no. The perturbations are stretched in the process of their initial formation, but the sphere at recombination that we observe today is still unstretched. (What mathematicians charmingly call the “round sphere.”) We were interested in projections of statistically anisotropic perturbations onto that round sphere, so we used ordinary spherical harmonics, but they had unconventional correlations between different modes.

  • Eric

    My main reason for mentioning Abramowitz and Stegun was Sean’s comment about checking the info from Wikipedia against a more “conventionally reliable” source. Although, I suppose that if one is inclined to trust Mathematica, one may as well also trust Wolfram’s website.

  • Joe Fitzsimons

    Sean, you seem to get a lot of work done in pubs. I’m completely the opposite. If I drink, I have to stop myself from working, because if I do, I’ll make a mistake so bizarre it’ll take me days to route out exactly what’s gone wrong. The flawed assumption always looks so reasonable, even in the sober light of day.

    Oh, and I second Anon’s recommendation of Gradshteyn and Ryzhik. Every theorist should have a copy.

  • cynic


    If you haven’t done so, check out ‘Symmetry and Separation of Variables’ by W Miller, Jr., Addison Wesley, 1977 .

    It’s good to see the blogosphere lit up by an enthusiastic discussion of a topic as old school as spherical harmonics. There is something rather satisfying about that stuff; even our host seems to get a thrill out of it.

    As for the blast from the past phenomenon – the spotting and exploitation of analogies is a big part of physics research; this episode could be a case study in the way it is done. I remember solving a problem in wave propagation in a stratified medium by retrieving and brushing up a half-forgotten exercise in Landau and Lifshitz Quantum Mechanics. Is that an examinable skill? I fear that, back in the day when both examiners and candidates were real men, it would have been regarded as bad form, if not outright cheating. A referee shared this view, demanding that any mention of matters quantum mechanical be excised from a hydro paper.

    BTW Whitaker and Watson’s ‘Modern Analysis’ is the real deal when it comes to special functions Chapter 18 dispatches ‘the equations of mathematical physics’ in less than 20 pages, and invents twistor theory in the process.

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About Sean Carroll

Sean Carroll is a Senior Research Associate in the Department of Physics at the California Institute of Technology. His research interests include theoretical aspects of cosmology, field theory, and gravitation. His most recent book is The Particle at the End of the Universe, about the Large Hadron Collider and the search for the Higgs boson. Here are some of his favorite blog posts, home page, and email: carroll [at] .


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