An Integral Appendix

After the conference dinner at the Ripples in the Cosmos meeting in Durham I attended recently, a group of us adjourned to the Castle bar for a drink or several. I ended up chatting to one of the locals, Richard Bower, mainly on the subject of beards. I suppose you could call it a chinwag. Only later on did  we get onto the subject of a paper we had both worked on a while ago. It was with some alarm that I later realized that the paper concerned was actually published twenty years ago. Sigh. Where did all that time go?

Anyway, Richard and I both remembered having a great time working on that paper which turned out to be a nice one, although it didn’t exactly set the world on fire in terms of citations. This paper was written before the standard “concordance” (LCDM) cosmology was firmly established and theorists were groping around for ways of reconciling observations of the CMB from the COBE satellite with large-scale structure in the galaxy distribution as well as the properties of individual galaxies. The (then) standard model (CDM with no Lambda) struggled to satisfy the observational constraints, so in typical theorists fashion we tried to think of a way to rescue it. The idea we came up with was “cooperative galaxy formation”, as explained in the abstract:

We consider a model in which galaxy formation occurs at high peaks of the mass density field, as in the standard picture for biased galaxy formation, but is further enhanced by the presence of nearby galaxies. This modification is accomplished by assuming the threshold for galaxy formation to be modulated by large-scale density fluctuations rather than to be spatially invariant. We show that even a weak modulation can produce significant large-scale clustering. In a universe dominated by cold dark matter, a 2 percent – 3 percent modulation on a scale exceeding 10/h Mpc produces enough additional clustering to fit the angular correlation function of the APM galaxy survey. We discuss several astrophysical mechanisms for which there are observational indications that cooperative effects could occur on the scale required.

I have to say that Richard did most of the actual work on this paper, though all four authors did spend a lot of time discussing whether the idea was viable in principle and, if so, how we should implement it mathematically. In the end, my contribution was pretty much limited to the Appendix, which you can click to make it larger if you’re interested.

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As is often the case in work of this kind, everything boiled down to evaluating numerically a rather nasty integral. Coincidentally, I’d come across a similar problem in a totally different context a few years previously when I was working on my thesis and therefore just happened to know the neat trick described in the paper.

Two things struck me looking back on this after being reminded of it over that beer. One is that a typical modern laptop is powerful enough to evaluate the original integral without undue difficulty, so if this paper had been written nowadays we wouldn’t have bothered trying anything clever; my Appendix would probably not have been written. The other thing is that I sometimes hear colleagues bemoaning physics students’ lack of mathematical “problem-solving” ability, claiming that if students haven’t seen the problem before they don’t know what to do. The problem with that complaint is that it ignores the fact that many problems are the same as things you’ve solved before, if only you look at them in the right way. Problem solving is never going to be entirely about “pattern-matching” – some imagination and/or initiative is going to required sometimes- but you’d be surprised how many apparently intractable problems can be teased into a form to which standard methods can be applied. Don’t take this advice too far, though. There’s an old saying that goes “To a man who’s only got a hammer, everything looks like a nail”. But the first rule for solving “unseen” problems has to be to check whether you might in fact already have seen them…

6 Responses to “An Integral Appendix”

  1. Anton Garrett Says:

    You can rephrase it as an integral over all parameter space, with the cutoffs expressed as Heaviside step functions in the integrand. Then integrate by parts so as to turn those step functions into their first differential, the delta-function. Then the integration is trivial. Does this help?

    • telescoper Says:

      Oops. Missed this reply earlier. I think that doesn’t help much because you end up with error functions that you can’t do anything much with. In the case we have here, the problem can be reduced to a one-dimensional integral with simple limits so it does speed up the numerical evaluation a lot.

  2. “It was with some alarm that I later realized that the paper concerned was actually published twenty years ago. Sigh. Where did all that time go?”

    Where, indeed. I was back in Blighty myself last week. Interestingly, six separate people (only two of whom know each other), from working-class bloke to intellectuals (one of whom was a Nobel-Prize winner on the radio; the others I heard personally), at no prompting from me, spontaneously discoursed on the nature of time, our memories, the relation between the two etc.

  3. drewancameron Says:

    This is a neat little trick. Do you know if this is the one used by modern codes for computing truncated multivariate normal densities? From my experience the time spent on this task by tmvtnorm for R is non-trivial (in modern terms; i.e., few seconds) for, say, a 100 dimension problem from which the truncated density is to be evaluated like 10000 times.

  4. […] announcement of this awarded reminded me that I was one of the co-authors of a paper with Simon White, but looking it up I realized that was way back in 1993! Where does the time […]

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