# December 25, 2006 {#week243}

Today I'd like to talk a bit about the first stars in the Universe, and
some hotly contested possible observations of these stars. Then I want
to describe a new paper by my student Derek Wise. But first --- if anyone
gave you a gift certificate for a bookstore this holiday season, here
are two suggestions.

The first one is really easy and fun:

1) William Poundstone, _Fortune's Formula: The Untold Story of the Scientific Betting System that Beat the Casinos and Wall Street_, Farrar, Strauss and Giroux, New York, 2005.

Packed with rollicking tales of gangsters, horse-racing, blackjack, and
insider trading, this is secretly the story of how Claude Shannon
developed information theory --- and how he and his sidekick John Kelly
Jr. used it to make money in casinos and Wall Street. I'd known about
Shannon's work on information... but not that he beat 99.9% of mutual
fund managers, making an average compound return of 28% for many years ---
as compared to 27% for Warren Buffett!

This book has just a few equations in it. I was delighted by one
discovered by Kelly, which I'd never seen before. Translating into my
own favorite notation, it goes like this:
$$S = \log M$$
It's the fundamental equation relating gambling to information! Let me
explain it --- in language far more complicated than you'll see in
Poundstone's book.

What's $M$? It's the best possible average growth of a gambler's money.
For example, if his best possible strategy lets him triple his money on
average, then $M = 3$.

What's $S$? This is the amount of "inside information" the gambler has:
information he has, that the people he's betting against don't.

Some technical stuff: First, the above "average" is a geometric mean,
not an arithmetic mean. Second, if we measure information in bits, we
need to use base 2 in the logarithm. Physicists would probably prefer to
use base $e$, which means measuring information in "nits". It doesn't
really matter, but let's use base 2 for now.

To get a feeling for why Kelly's theorem is true, it's best to start
with the simplest example. If $S = 1$, then $M = 2$. So, if a gambler
receives one bit of inside information, he can double his money!

This sounds amazing, but it's also obvious.

Suppose you have one bit of inside information: for example, whether a
flipped coin will land heads up or tails up. Then you can make a bet
with somebody where they give you \$1,000,000 if you guess the coin flip
correctly, and you give them \$1,000,000 if you guess wrong. This is a
fair bet, so they will accept. That is, they'll *think* it's fair if
they don't suspect you have inside information! But since you do have
this information, you'll win the bet, and double your money on this
coin flip.

Kelly's equation is usually phrased in terms of the *rate* at which the
gambler gets inside information, and the *rate* at which his money
grows. So, for example, to earn 12% interest annually, you only need to
receive
$$\log(1.12) = 0.163$$
bits of inside information --- and find some dupe willing to make bets
with you about this.

The last part is the hard part: the "inside information" really needs
to be information people don't believe you have. I must learn hundreds
of bits of information about math each year --- stuff only I know --- but I
haven't found anyone simultaneously smart enough to understand it and
dumb enough to make bets with me about it!

Still, I like this relation between information theory and gambling,
because one stream of Bayesian probability theory says probabilities are
subjectively defined in terms of the bets you would accept.

The argument for this is called the "Dutch book argument". It
basically shows how you can make money off someone who makes bets in
ways that correspond to stupid probabilities that don't add to 1, or
fail to be coherent in other ways:

2) Carlton M. Caves, _Probabilities as betting odds and the Dutch book_, available at `http://info.phys.unm.edu/~caves/reports/dutchbook.pdf`

So, there's a deep relation between gambling and probability --- no news
here, really.

But, there's also a deep relation between probability and information
theory, discovered by Shannon. Briefly, it goes like this: the
information you obtain by learning the value of a random variable is
$$S = -\sum_i p_i \log(p_i)$$
where the sum is taken over all the possible values of this random
variable, and $p_i$ is the probability that it takes its $i$th value. So,
for example, if you flip a fair coin, where $p_1 = p_2 = 1/2$, the
information you get by looking at the coin is
$$-\left[\frac12 \log\left(\frac12\right) + \frac12 \log\left(\frac12\right)\right] = 1$$
One bit!

So: gambling is related to probability, and probability is related to
information. Kelly's result closes the circle by providing a direct
relation between gambling and information!

But, apparently some of Kelly's ideas are still controversial in the
world of economics and stock trading. If you read Poundstone's book,
you'll learn why.

The next book takes more persistence to read:

3) Avner Ash and Robert Gross, _Fearless Symmetry: Exposing the Hidden Patterns of Numbers_, Princeton U. Press, Princeton, 2006.

The authors do a creditable job of what might at first seem utterly
impossible: explaining heavy-duty modern number theory to ordinary
mortals. The formal prerequisites are little more than high school
algebra, and the style is expository, but anyone except an expert will
need to stop and think at times.

They start by explaining modular arithmetic --- you know, stuff like
adding and multiplying "$\mod 7$". Then they tackle groups, and
permutations, since the main theme of the book is symmetry. Then they
move on to algebraic varieties, in a simple no-nonsense style cleverly
adapted from Grothendieck's later work (without terrifying the reader
by mentioning this fact).

Next they tackle some serious number theory: quadratic reciprocity,
Galois groups, and elliptic curves. Then they describe more general
forms of reciprocity, leading up to a taste of the Langlands program.
They conclude with a sketch of how Fermat's last theorem was proved.

These days mathematical physicists are all excited about a variant of
the Langlands program: the so-called "geometric" Langlands program,
which is related to string theory. Drinfeld has been running a seminar
on this at Chicago for years, but that's not what got the physicists
interested --- it's these papers by Witten that did it:

4) Anton Kapustin and Edward Witten, "Electric-magnetic duality and the geometric Langlands program", 225 pages, available as [`hep-th/0604151`](https://arxiv.org/abs/hep-th/0604151).

5) Sergei Gukov and Edward Witten, "Gauge theory, ramification, and the geometric Langlands program", 160 pages, available as [`hep-th/0612073`](https://arxiv.org/abs/hep-th/0612073).

So, if you're trying to learn this geometric Langlands stuff, and you
want to fit it into the grand landscape of mathematics, the book
Fearless Symmetry could be a fun way to learn some the math underlying
the ordinary Langlands stuff.

I started girding myself for a discussion of the Langlands program in
["Week 217"](#week217), ["Week 218"](#week218) and
["Week 221"](#week221), but then I got distracted. I'll get back
to it someday, but right now I'm in the mood for lighter stuff... so
let me tell you a bit about the first stars.

The story starts around 380,000 years after the Big Bang, when the hot
hydrogen and helium forming our Universe cooled down to 3000 kelvin -
just cool enough for the electrons to stick to the atomic nuclei instead
of zipping around on their own.

When the electrons in a gas are hot enough for some to zip around on
their own, we say the gas is "ionized". When a *lot* of them are
zipping around, we call it a "plasma". Because charged particles
interact with the electromagnetic field, light doesn't pass through
plasma cleanly: it keeps getting absorbed and re-emitted.

So, before our story started, you couldn't see very far: it would be
like trying to look through a wall of fire. But, around 380,000 years
after the Big Bang, the gas became transparent!

What would it have looked like? Nobody ever seems to talk about this.
So, I'll just guess, and hope some expert corrects me.

Back when the gas filling the Universe was 5000 kelvin in temperature,
just a bit cooler than the surface of the Sun, everything was yellow.
You couldn't see far at all: you would have been blinded by a yellow
glare.

But when it cooled to 4000 kelvin in temperature, the Universe became
orange.

And when it cooled to 3000 kelvin, the Universe became red.

And when it cooled a tiny bit further, it became infrared. As far as
visible light goes, the Universe became transparent!

This would happen everywhere more or less at once. But since light takes
time to travel, you'd see a transparent sphere around you, expanding
outwards at the speed of light, with reddish walls.

It's been sort of like this ever since.

So, when we look far away with our best telescopes, we look back in time
to the time when the Universe became transparent --- but no further.
We're surrounded by a distant, ancient wall of fire. It's now about
13.3 billion light-years away --- or 13.3 billion year back in time, if
you prefer. And, it's receding at a rate of one light-year per year.

But by now, the light from this wall of fire has been severely
redshifted. In other words, it's been stretched along with the
expansion of the Universe --- stretched by a factor of 1100, in fact!

So, what had been the hot infrared glow of 3000-kelvin plasma is now a
feeble microwave glow corresponding to an icy temperature of 2.7 kelvin.
This is the famous "cosmic microwave background radiation".

But let's go back in time....

From the moment the hot gas became transparent to the time when the
first stars formed, the Universe was dark except for the dimming
infrared glow of that distant wall of fire. This era is called the
"Dark Ages".

During the Dark Ages, gas cooled down and clumped under its own gravity
--- apparently with a lot of help from cold dark matter of some unknown
sort. Without postulating this matter, nobody can figure out how
galaxies formed as soon as they did.

As befits their name, the Dark Ages are still shrouded in mystery. There
are a lot of unanswered questions besides the nature of dark matter.
Which formed first --- individual stars, or galaxies? And, when did the
Dark Ages end?

It's currently believed that the first stars formed sometime between
150 million and 1 billion years after the Big Bang.

At the later end of that range, the Universe could have gotten quite
cold before starlight warmed up the interstellar gas and reionized it.
There's even a spooky theory that the Universe was full of hydrogen
snowflakes near the end of the Dark Ages --- see
["Week 196"](#week196) for more on this, and a timeline of the
earlier history of the Universe.

But, the current best guess, based on data from the Wilkinson Microwave
Anisotopy Probe, says that reionization happened 400 million years after
the Big Bang:

6) Marcelo A. Alvarez, Paul R. Shapiro, Kyungjin Ahn and Ilian T. Iliev, "Implications of WMAP 3 year data for the sources of reionization", _Astrophys. J._ **644** (2006), L101--L104. Also available as [`astro-ph/0604447`](https://arxiv.org/abs/astro-ph/0604447).

This would be too early for hydrogen snow, since my rough calculation
says the microwave background radiation was 30 kelvin then, while
hydrogen freezes at 14 kelvin.

What were the first stars like? Without heavier elements to catalyze
nuclear fusion, they could have been larger than current-day stars:
perhaps hundreds of times the size of our Sun! These so-called
Population III stars have not actually been seen. But, it's possible
that we've finally caught a glimpse of them, not individually but in a
sort of statistical sense:

7) A. Kashlinsky, R. G. Arendt, J. Mather and S. H. Moseley, "New measurements of cosmic infrared background fluctuations from early epochs", to appear in _Ap. J. Letters_. Available as [`astro-ph/0612445`](https://arxiv.org/abs/astro-ph/0612445).

8) A. Kashlinsky, R. G. Arendt, J. Mather and S. H. Moseley, "On the nature of the sources of the cosmic infrared background", to appear in _Ap. J. Letters_. Available as [`astro-ph/0612447`](https://arxiv.org/abs/astro-ph/0612447).

Using delicate techniques to carefully sift through the *infrared* (not
microwave) background radiation, the authors claim to find radiation not
accounted for by previously known sources. Assuming the standard
cosmological scenario, the sources of this radiation date back to less
than 1 billion years after the Big Bang, and were individually much
brighter than current-day stars.

Here's a picture of their data:
$$\href{http://www.spitzer.caltech.edu/Media/releases/ssc2005-22/ssc2005-22a.shtml}{\includegraphics[max width=0.65\linewidth]{../images/infrared_background_first_stars.jpg}}$$

9) NASA / JPL-Caltech / A. Kashlinsky, "Infrared background light from first stars", `http://www.spitzer.caltech.edu/Media/releases/ssc2005-22/`

On top is a photograph taken by the Spitzer Space Telescope: a 10-hour
infrared exposure of a tiny patch of sky, $6 \times 12$ arcminutes across,
chosen for having a bare minimum of foreground stars, galaxies and dust.
(For comparison, the Moon is 30 arcminutes across.) On the bottom is the
same picture with known sources of infrared subtracted. What's left may
be the severely redshifted light from early stars!

Or, it may not. In the following news story, Ned Wright of UCLA said,
"I'm very skeptical of this result. I think it's wrong. I think what
they're seeing is incompletely subtracted residuals from nearby
sources."

10) Dinesh Ramde, Associated Press, "Hints of early stars may have been found", `http://www.usatoday.com/tech/science/space/2005-11-02-early-stars_x.htm`

So, we'll have to see how it goes....

But in the meantime, we can think about mathematical physics. My student
Derek Wise is graduating this year, and he's doing his thesis on Cartan
geometry, MacDowell-Mansouri gravity and $BF$ theory. Let me say a little
about this paper of his:

11) Derek Wise, "MacDowell-Mansouri gravity and Cartan geometry", available as [`gr-qc/0611154`](https://arxiv.org/abs/gr-qc/0611154).

Elie Cartan is one of the most influential of 20th-century geometers. At
one point he had an intense correspondence with Einstein on general
relativity. His "Cartan geometry" idea is an approach to the concept
of parallel transport that predates the widely used Ehresmann approach
(connections on principal bundles). It simultaneously generalizes
Riemannian geometry and Klein's Erlangen program (see
["Week 213"](#week213)), in which geometries are described by their
symmetry groups:
$$
  \begin{tikzcd}[row sep=huge]
    \text{Euclidean geometry} \rar \dar
    & \text{Klein geometry} \dar
  \\\text{Riemannian geometry} \rar
    & \text{Cartan geometry}
  \end{tikzcd}
$$
Given all this, it's somewhat surprising how few physicists know about
Cartan geometry!

Recognizing this, Derek explains Cartan geometry from scratch before
showing how it underlies the so-called MacDowell-Mansouri approach to
general relativity. This plays an important role both in supergravity
and Freidel and Starodubtsev's work on quantum gravity (see
["Week 235"](#week235)) --- but until now, it's always seemed like a
"trick".

What's the basic idea? Derek explains it all very clearly, so I'll
just provide a quick sketch. Cartan describes the geometry of a lumpy
bumpy space by saying what it would be like to roll a nice homogeneous
"model space" on it. Homogeneous spaces are what Klein studied; now
Cartan takes this idea and runs with it... or maybe we should say he
*rolls* with it!

For example, we could study the geometry of a lumpy bumpy surface by
rolling a *plane* on it. If our surface is itself a plane, this rolling
motion is trivial, and we say the surface is "flat" in the sense of
Cartan geometry. But in general, the rolling motion is interesting and
serves to probe the geometry of the surface.

Alternatively, we could study the geometry of the same surface by
rolling a *sphere* on it. Derek illustrates this with a picture of a
hamster crawling around in a plastic "hamster ball", which is
something you can actually buy for your pet hamster to let it explore
your house without escaping or getting in trouble:
$$\includegraphics[max width=0.65\linewidth]{../images/cartan_geometry_1.jpg}$$
(I've read about falling cats in papers on gauge theory, but this is
the first mathematical physics paper I've read containing the word
"hamster".)

If our surface is itself a sphere of the same radius, this rolling
motion is trivial, and we say the surface is flat in the sense of Cartan
geometry --- but now it's a different sense than when we used a plane as
our "model geometry"!

Which model geometry should we use in a given problem? It depends on
which one best approximates the lumpy bumpy space we're studying!

The ordinary formulation of general relativity fits into this framework,
with a little work. Two well-known mathematical gadgets called the
"Lorentz connection" and "coframe field" fit together to describe
what would happen if we rolled a copy of Minkowski spacetime over the
lumpy bumpy spacetime we live in.

That's great if Minkowski spacetime is the best homogeneous
approximation to the spacetime we live in. But nowadays we think the
cosmological constant is nonzero, so the Universe is expanding in a
roughly exponential way. This makes another model geometry, "deSitter
spacetime", the best one to use!
$$\includegraphics[max width=0.65\linewidth]{../images/cartan_geometry_2.jpg}$$
So, if we know Cartan geometry, we can use that... and we get something
called the MacDowell-Mansouri formulation of gravity. Or, if we don't
want our spacetime to have lumps and bumps --- if we want it to look
locally just like the Klein model geometry --- we can use a different
theory, a topological field theory called $BF$ theory (see
["Week 232"](#week232)).

In short, the passage from a topological field theory describing a
"locally homogeneous" spacetime to full-fledged gravity with all its
lumps and bumps is nicely understood in terms of how Cartan's approach
to geometry generalizes Klein's!

For more details, you'll just have to read Derek's paper. You might
also try these:

12) Michel Biesunski, "Inside the coconut: the Einstein-Cartan discussion on distant parallelism", in _Einstein and the History of General Relativity_, eds. D. Howard and J. Stachel, Birkhauser, Boston, 1989.

This describes the correspondence between Cartan and Einstein. I believe
this centered, not on Cartan geometry per se, but on the
"teleparallel" formulation of gravity (see
["Week 176"](#week176)). But, they're somewhat related.

13) Richard W. Sharpe, _Differential Geometry: Cartan's Generalization of Klein's Erlangen Program_, Springer-Verlag, New York, 1997.

This is the main textbook on Cartan geometry. But, it's probably best
to read a few chapters of Derek's paper first, since the key ideas are
presented more intuitively.

My friend the geometer and analyst Rafe Mazzeo, whom I recently saw at
Stanford, told me that Cartan geometry was all the rage these days. I'm
embarrassed to say I hadn't known this! I think the kinds of Cartan
geometry being intensively studied are related to conformal geometry, CR
structures and stuff like that...

Merry Christmas!

------------------------------------------------------------------------

**Addenda:** I thank Chris Weed for catching typos. For more discussion,
go to the [$n$-Category Café](http://golem.ph.utexas.edu/category/2006/12/this_weeks_finds_in_mathematic_3.html).

------------------------------------------------------------------------

> *The Universe has as many different centers as there are living beings
in it.*
> 
> --- Alexander Solzhenitsyn