Waiter, There's a Derivative in my Cereal

By Sean Carroll | July 1, 2008 8:10 pm

You can learn a lot by reading scientific papers. For example, I’ve known for a while now that the Hubble parameter tells you how fast the universe is expanding — it’s the (conveniently normalized) first derivative of the scale factor, for you calculus-philes out there. The deceleration parameter tells you how fast the universe is decelerating — it’s the second derivative, telling us how the expansion rate is changing with time. (Of course now we know that it’s really accelerating, but we didn’t know that back when the phrases were introduced.)

Less well known, but more amusing, is that the third derivative — how is the acceleration changing as a function of time? — is characterized by the “jerk.” Makes sense, when you think about it — when you jerk at something, you are not just pulling it (causing acceleration), but pulling at it faster and faster. It nevertheless leads to fun if predictable jokes, with this person or that being labeled a “cosmic jerk.”

Do we really need a name for the fourth derivative, telling us how the jerk is changing with time? Apparently we do, as it has been denoted the “snap.” I just learned this from this new paper:

Cosmic Jerk, Snap and Beyond
Authors: Maciej Dunajski, Gary Gibbons

Abstract: We clarify the procedure for expressing the Friedmann equation in terms of directly measurable cosmological scalars constructed out of higher derivatives of the scale factor. We carry out this procedure for pure dust, Chaplygin gas and generalised Chaplygin gas energy-momentum tensors. In each case it leads to a constraint on the scalars thus giving rise to a test of General Relativity. We also discuss a formulation of the Friedmann equation as unparametrised geodesic motion and its connection with the Lagrangian treatment of perfect fluids coupled to gravity.

The best part is this footnote:

The analogous expressions involving 5th and 6th derivatives are known as crackle and pop. This terminology goes back to a 1932 advertisement of Kellogg’s Rice Crispies which `merrily snap, crackle and pop in a bowl of milk.’

I suppose there is also some interesting science in there. But now I really want to write a paper that makes use of the 5th time derivative of the scale factor. The words are too delicious to resist.


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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] cosmicvariance.com .


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