# June 1, 2003 {#week196}

Today I'd like to talk about the Big Bang and Pythagorean spinors. But
first, a book! If you want to start learning general relativity without
first mastering the intricacies of tensors, here's a way to get going:

1) James B. Hartle, _Gravity: an Introduction to Einstein's General Relativity_, Addison-Wesley, San Francisco, 2003.

Hartle is an expert on general relativity, but here he avoids showing
off. He gets to the physics of general relativity as quickly and simply
as possible, avoiding the usual route of doing huge amounts of math
first. In particular, he works out the physics of specific solutions of
Einstein's equations --- like those describing black holes and the Big
Bang --- before he introduces the equations. This puts off the hard
abstract stuff until later, when the student has more feeling for it.
The purists may grumble, but I have a feeling it's pedagogically sound.

Now... what happened in the first second after the Big Bang?

This may sound like an insanely ambitious question, but in fact we seem
to have a fairly good idea of what happened all the way back to the
first microsecond --- unless, of course, there's some important physics
we're missing. This paper tells the story quite nicely:

2) Dominik J. Schwarz, "The first second of the universe", available as [`astro-ph/0303574`](https://arxiv.org/abs/astro-ph/0303574).

But, since the physics gets weirder as we approach the Big Bang, I'll
tell this story in reverse order --- and I'll start from *now*, just to
set the stage.

So, here's a quick reverse history of the universe:

- 13.7 billion years after the Big Bang: now.\
    Temperature: 2.726 K

According to data from the Wilkinskon Microwave Anisotropy Probe (WMAP),
our best estimate of the age of the universe is 13.7 billion years, plus
or minus 150 million years or so. Previous estimates were similar, but
with an uncertainty of about half a billion years.

The temperature listed here is that of the cosmic microwave background
radiation. Slight deviations from this average figure were first
detected in 1992 by the Cosmic Background Explorer (COBE). This
satellite-based experiment found hot and cold patches in the microwave
background that differ from the mean temperature by an amount on the
order of 30 microkelvin. WMAP is a more refined experiment along the
same lines, which came out with a lot of exciting new results in
February, 2003.

- 200 million years after the Big Bang: reionization.\
    Temperature: roughly 50 K.

"Reionization" is the name for when the hydrogen in the universe,
which had cooled after the Big Bang, became hot and ionized again. The
most likely cause is radiation from the very first stars. So, we now
think the first stars ignited about 200 million years after the Big
Bang.

Again this is a result from WMAP. The error bars are quite large, so the
numbers above could be off by a factor of two or so. Nonetheless, a lot
of people were surprised by this result, since they thought that the
clumping of matter due to gravity would have taken *much* longer to form
stars!

There's a lot we don't know about star and galaxy formation, because
the current conventional wisdom requires that the gravitational clumping
was seeded by "cold dark matter" --- but nobody knows what this stuff
is.

- 380 thousand years after the Big Bang: recombination.\
    Temperature: 3000 K, or .25 eV of energy per particle.

"Recombination" is the usual stupid name for when the universe cooled
down enough for electrons and protons to stick together and form
hydrogen atoms. It should really be called "combination", because the
electrons and protons were never combined *before* this time. Before
this time the universe was always full of plasma --- that is, electrically
charged particles running around loose. Afterwards it was full of
electrically neutral hydrogen... at least until the stars lit up and
reionized a lot of this hydrogen.

Plasma absorbs light of all frequencies, while electrically neutral
gases tend to be transparent except for certain frequency bands. Thus,
recombination was the first time when light could start travelling for
long distances without getting absorbed! For this reason, the cosmic
background radiation we see now consists of the photons emitted right at
the time of recombination. When emitted it had a temperature of about
3000 kelvin, but it has cooled with the expansion of the universe.

The era between recombination and the ignition of the first stars goes
by a romantic name: the Dark Ages. Adding to the romance, Pfenniger and
Puy hypothesized that hydrogen could have frozen into crystalline flakes
before the stars lit up and began warming the universe again. A cold,
dark, eerie universe full of hydrogen snowflakes... the thought sends
shivers right up my spine! Unfortunately, if stars formed as early as
WMAP says they did, the universe would probably not get cold enough for
these crystals to form.

By the way, the figure of 380 thousand years for the time of
recombination is another result from WMAP, consistent with previous
estimates, but presumably more accurate.

- 10 thousand years after the Big Bang: end of the radiation-dominated
    era.\
    Temperature: 12,000 K, or 1 eV per particle.

Before this time, the energy density due to light exceeded that due to
matter, so we say the universe was "radiation-dominated". Afterwards
the universe became "matter-dominated" --- at least until considerably
later, when matter spread out so thin that the dominant form of energy
became "dark energy", as it seems to be now.

(The best estimate due to WMAP says that currently the energy in the
universe is 4% ordinary matter, 23% cold dark matter and 73% dark
energy. For more on the latter two concepts, see
["Week 167"](#week167).)

The end of the radiation-dominated era is important because this is when
gravity began to amplify small fluctuations in the density of matter. In
other words, this is when stuff began to form clumps of various sizes,
eventually leading to stars, galaxies, galaxy clusters, and so on.
During the radiation-dominated era, density fluctuations were mainly
made of *light*, and these could not grow because the light was moving
too fast to form clumps. People believe that as soon as this era ended,
cold dark matter began clumping up under its own gravity. Ordinary
matter started clumping up later, after recombination --- since before
that it was in the form of plasma, which stayed smoothed out by its
interaction with light.

- 1000 seconds after the Big Bang: decay of lone neutrons.\
    Temperature: roughly 500 million K, or about 50 keV per particle.

A lone neutron is not a stable particle: with a mean lifetime of 886
seconds, it will decay into a proton, electron and antineutrino. So, any
neutrons created early in the history of the universe must fuse with
protons to form nuclei by roughly this time, or they are doomed to
decay.

- 180 seconds after the Big Bang: nucleosynthesis begins.\
    Temperature: roughly 1 billion K, or about 100 keV per particle.

At about this time, the temperature dropped to the point where a proton
and neutron could stick together forming a deuterium nucleus, and the
process of "nucleosynthesis" began, in which deuterium nuclei stick
together to form nuclei of helium. This is responsible for the fact that
even before the stars started processing hydrogen into heavier elements,
the universe was about 25% helium, the rest being almost all hydrogen.

- 10 seconds after the Big Bang: annihilation of electron-positron
    pairs.\
    Temperature: roughly 5 billion K, or about 500 keV per particle.

Apart from the neutrinos and the photon, the lightest particle in nature
is the electron. The rest mass of an electron corresponds to an energy
of 511 keV, so it only takes twice that much energy to create an
electron-positron pair. If we multiply 511 MeV by Boltzmann's constant,
we get a temperature of roughly 5 billion kelvin. That means that at
this temperature, two particles colliding head-on will often have enough
kinetic energy to create a electron-positron pair. So, when it's this
hot or hotter, collisions between particles generate a thick stew of
electrons and positrons!

But as temperatures cool below this point, the density of this stew
drops off exponentially: electron-positron pairs annihilate each other,
leaving radiation. This happened roughly 10 seconds after the Big Bang.

- 1 second after the Big Bang: decoupling of neutrinos.\
    Temperature: roughly 10 billion K, or about 1 MeV per particle.

Neutrinos can easily zip through light-years of lead, but the very early
universe was so compressed that they interacted vigorously with other
forms of matter. But around a second after the Big Bang, the density of
the universe decreased to about 400,000 times that of water, and
neutrinos "decoupled" from other matter.

Since these neutrinos were not reheated by nucleosynthesis, they should
now be cooler than the cosmic microwave background radiation --- about 2
kelvin instead of 2.726 kelvin. We are currently unable to detect such
unenergetic neutrinos, but detecting them would be a major confirmation
that our theories of the early universe are correct.

- 100 microseconds after the Big Bang: annihilation of pions.\
    Temperature: roughly 1 trillion K, or about 100 MeV per particle.

Particles made of quarks and antiquarks are called "hadrons", and they
interact via the strong nuclear force. The only hadrons we encounter in
daily life are protons and neutrons, made of 3 quarks each. But the
lightest hadrons are the pions, which come in positive, negative and
neutral forms. The positive and negative ones are antiparticles of each
other, while the neutral one is its own antiparticle. They all have mass
on the order of 100 MeV.

Just as I described for electron-positron pairs, at a high enough
temperature everything is always awash in a sea of pions, while below
this temperatures the pions quickly disappear by annihilation. To
estimate the relevant temperature, we can just convert its mass to a
temperature following the rough rule 1 MeV \~ 10 billion kelvin. So,
when the temperature of the early universe dropped below 1 trillion
kelvin, pions went away. This happened around 100 microseconds after the
Big Bang. Before this, hadrons ruled!

- 50 microseconds after the Big Bang: QCD phase transition.\
    Temperature: 1.7-2.1 trillion K, corresponding to 150-180 MeV per
    particle.

At normal temperatures, quarks and antiquarks are confined within
hadrons by the strong force. The strong force is carried by gluons, so
you can vaguely visualize a hadron as a bag-like thing in which quarks
and antiquarks wiggle about, constantly exchanging virtual gluons, which
also exchange virtual gluons, quarks and antiquarks of their own. The
details are described by "quantum chromodynamics", or QCD. Since QCD
says the strong force gets stronger with increasing distance, if you try
to pull a quark out of this bag, it takes enough energy to create a
whole new bag!

But if you have a bunch of hadrons at temperatures above 2 trillion
kelvin or so, they'll be smashing into each other so furiously that the
distinction between the "bags" and the "space between the bags",
never completely sharp, dissolves entirely. At this point, all you've
got is a bunch of quarks, antiquarks and gluons zipping around. This is
a new state of matter: a "quark-gluon plasma". In
["Week 76"](#week76) and ["Week 117"](#week117) I described
how how people at the Relativistic Heavy Ion Collider in Brookhaven are
making quark-gluon plasmas by smashing nuclei at each other at high
speeds.

A lot of Dominik Schwarz's paper is about the "QCD phase transition"
which happened about 50 microseconds after the Big Bang, when the
universe cooled down enough for the quark-gluon plasma to condense into
the confined phase. Though the subject is controversial, most people
think this phase transition is a "first-order" transition, meaning
that heat is emitted as the transition happens, just as when water vapor
condenses to form liquid droplets. If so, the quark-gluon plasma would
probably supercool until small bubbles of hadron phase formed. As these
bubbles grew, latent heat would be emitted. This would tend to reheat
the quark-gluon plasma, limiting the speed at which the bubbles expand.
Heat would mainly be dispersed by means of neutrinos and acoustic waves
--- i.e., sound.

- 10 picoseconds after the Big Bang: electroweak phase transition.\
    Temperature: 1-2 quadrillion K, corresponding to 100-200 GeV per
    particle.

At high enough temperatures, there should be no difference between the
electromagnetic force and weak force. This difference should only arise
when things cool down enough for the Higgs field to settle into a fixed
position, breaking the symmetry between these forces --- a bit like how
ice crystallizes, breaking the rotational symmetry of liquid water. At
higher temperatures the Higgs field wiggles around too much to settle
down. Or in the language of particles rather than fields: collisions
between particles create a stew of Higgs bosons!

The mass of Higgs seems to be somewhere around 130 GeV. If so, the
electroweak phase transition would have occured roughly $10^{-11}$ seconds
after the Big Bang. But, since we haven't actually gotten direct
evidence for the Higgs boson yet, this is still a bit speculative. Right
now people say the Large Hadron Collider at CERN will come online and
start looking for the Higgs in 2007. But the LHC project has gotten some
nasty budget cuts recently, so I wouldn't be surprised if there were
delays. If and when the Higgs is found, maybe I'll return to this topic
and say what people think happened *before* the electroweak phase
transition. People have thought about this a lot. But for now, I'll
quit here!

If you want to learn more about the early universe, start with this
classic:

3) Steven Weinberg, _The First Three Minutes_, Basic Books, New York, 1977.

Then catch up with recent developments by reading these websites:

3) "Ned Wright's Cosmology Tutorial", `http://www.astro.ucla.edu/~wright/cosmolog.htm`

4) Martin White, "The Cosmic Rosetta Stone", `http://astron.berkeley.edu/~mwhite/rosetta/rosetta.html`

To dig deeper, try these books:

5) P. Coles and F. Lucchin, _Cosmology: The Origin and Evolution of Cosmic Structure_, Wiley, New York, 1995.

6) Edward W. Kolb and Michael Turner, _The Early Universe_, Addison-Wesley, Reading, Massachusetts, 1990.

For a detailed description of some of WMAP's results, try these:

7) C. L. Bennett et al, "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Preliminary Maps and Basic Results", available as [`astro-ph/0302207`](https://arxiv.org/abs/astro-ph/0302207).

8) D. N. Spergel et al, "First Year Wilkinson Microwave Anisotropy Probe (WMAP) Observations: Determination of Cosmological Parameters", available as [`astro-ph/0302209`](https://arxiv.org/abs/astro-ph/0302209).

These are two of thirteen related papers produced by the WMAP team! Both
of them have lots of coauthors, one of which is Ned Wright, author of
the nice website mentioned above.

Here is Pfenniger and Puy's paper on hydrogen "snowflakes":

9) D. Pfenniger and D. Puy, "Possible flakes of molecular hydrogen in the early Universe", available as [`astro-ph/0211393`](https://arxiv.org/abs/astro-ph/0211393).

I should also thank Ted Bunn for telling me some stuff about converting
between times, temperatures and redshifts. Cosmologists of the early
universe use "redshift z" to stand for the time when the universe was
$1/(z+1)$ times as big as it is now --- by which I mean, distances between
faraway objects were multiplied by this factor. Equivalently, this is
the time when the temperature of the background radiation was $z+1$ times
as big as it is now. So, converting between temperatures and redshifts
is easier. Converting these to times is less trivial, and indeed the
times listed above are more likely to suffer from inaccuracies than the
temperatures.

By the way, some people say it's confusing to use numbers like
"billion", "trillion" and "quadrillion" to mean $10^9$, $10^{12}$ and
$10^{15}$, respectively --- because these are American usages, and Europeans
(they claim) use "milliard", "billion" and "billiard" for these
numbers. These people say that, for example "gigakelvin",
"terakelvin" and "exakelvin" are less ambiguous than "billion
kelvin", "trillion kelvin" and "quadrillion kelvin".

This is probably true, but it's also true that fewer people know what
an "exakelvin" is than a "quadrillion kelvin". Since I was trying to
explain cosmology to the unwashed masses, I opted to use number words
above, and explain them here. I was using the American system... and
I'm sort of betting this system will take over, because I've *never*
heard anyone use the word "billiard" to mean $10^{15}$. More importantly,
I really hope that *some* system takes over, because it's a bit sad not
to be able to use words for numbers.

Speaking of numbers... now for some math!

The volume in honor of Penrose's 65th birthday is full of fun stuff
about spin networks, twistors, and so on --- but I particularly liked this
paper by Trautman on "Pythagorean spinors":

10) Andrzej Trautman, "Pythagorean spinors and Penrose twistors", in _The Geometric Universe: Science Geometry and the Work of Roger Penrose_, eds. Huggett, Mason, Tod, Tsou and Woodhouse, Oxford U. Press, Oxford, 1998. Also available at `http://www.fuw.edu.pl/~amt/amt.html`

If you're a physicist you'll have heard about Dirac spinors, Weyl
spinors, Majorana spinors, and maybe even Majorana-Weyl spinors. I've
you haven't, you can read my explanations in ["Week 93"](#week93).
But what in the world are "Pythagorean" spinors? The basic idea is
that from two spinors you can make a vector, and Trautman points out
that a special case of this idea gives a famous old formula for getting
Pythagorean triples --- that is, integers $a,b,c$ with
$$a^2 + b^2 = c^2.$$
I think I'll explain this in detail....

Spinors are used to describe spin-$1/2$ particles, so-called because they
don't come back to where they were when you turn them around 360
degrees --- you have to rotate them *twice* to get back where you started!
Thus, mathematically, spinors are representations of the double cover of
the rotation group, or the double cover of the Lorentz group if you take
special relativity into account.

In 4d spacetime, the double cover of the Lorentz group is $\mathrm{SL}(2,\mathbb{C})$, the
group of $2\times2$ complex matrices with determinant $1$. We can take a spinor
to be just a pair of complex numbers, but there are actually two ways
such a thing can transform under $\mathrm{SL}(2,\mathbb{C})$. One way is obvious, but for
the other we take the *complex conjugate* of the matrix before letting
it act on the spinor. We get two sorts of spinors, called left- and
right-handed "Weyl spinors". In physics, we use these to describe
massless particles that spin either clockwise or counterclockwise along
their line of motion as they zip along at the speed of light.

In 3d spacetime, the double cover of the Lorentz group is $\mathrm{SL}(2,\mathbb{R})$, the
group of $2\times2$ *real* matrices with determinant $1$. In this dimension, we
can take a spinor to be a pair of *real* numbers. But since we don't
have complex conjugation at our disposal, we don't get left- and
right-handed versions of these spinors, and we don't call them Weyl
spinors. Since they are real, we call them "Majorana spinors".

Since Pythagoras had a strong fondness for number theory, if he were
alive today he might want to simplify things even further and consider
$\mathrm{SL}(2,\mathbb{Z})$, the group of $2\times2$ *integer* matrices with determinant $1$. This
acts on "Pythagorean spinors", namely pairs of integers.

We could also go up to higher dimensions using the quaternions and
octonions: $\mathrm{SL}(2,\mathbb{H})$ is the double cover of the Lorentz group in 6d
spacetime, and $\mathrm{SL}(2,\mathbb{O})$ is the double cover of the Lorentz group in 10d
spacetime. But I explained this in my octonion webpage:

11) John Baez, "$\mathbb{OP}^1$ and Lorentzian geometry", `http://math.ucr.edu/home/baez/octonions/node11.html`

so I won't talk about it now.

In each case, there's a trick for turning a spinor into a lightlike
vector. In 4 dimensions we do it like this: we take a left-handed spinor
$\psi$, take its conjugate transpose to get a right-handed spinor $\psi^*$, and
form
$$\psi \otimes \psi^*$$
which we can think of as a $2\times2$ hermitian matrix. If you're a fancy
mathematical physicist you know that the space of $2\times2$ hermitian matrices
is the same as 4d Minkowski spacetime, with the matrices of determinant
zero corresponding to the lightlike vectors, so you're done! Otherwise,
you can work out the above matrix explicitly:
$$
  \begin{gathered}
    \underbrace{\psi = \left(\begin{array}{c}a\\b\end{array}\right)}_{\mbox{a column vector}}
    \qquad\quad
    \underbrace{\psi^* = (a^*,b^*)}_{\mbox{a row vector}}
  \\\underbrace{\psi\otimes\psi^* = \left(\begin{array}{cc}aa^*&ab^*\\ba^*&bb^*\end{array}\right)}_{\mbox{a $2\times2$ matrix}}
  \end{gathered}
$$
This matrix is hermitian, so you can write it as a real linear
combination of Pauli matrices:
$$\psi \otimes \psi^* = t \sigma_t + x \sigma_x + y \sigma_y + z \sigma_z$$
where
$$
  \begin{gathered}
    \sigma_t = \left(
      \begin{array}{cc}
        1&0\\0&1
      \end{array}
    \right)
  \end{gathered}
  \qquad\quad
  \begin{gathered}
    \sigma_x = \left(
      \begin{array}{cc}
        0&1\\1&0
      \end{array}
    \right)
  \\\sigma_y = \left(
      \begin{array}{rr}
        0&-i\\i&0
      \end{array}
    \right)
  \end{gathered}
  \qquad\quad
  \begin{gathered}
    \sigma_z = \left(
      \begin{array}{cc}
        1&0\\0&-1
      \end{array}
    \right)
  \end{gathered}
$$
You get a vector in Minkowski spacetime, $(t,x,y,z)$. If you check that
this vector is lightlike:
$$t^2 = x^2 + y^2 + z^2$$
you'll be done.

The trick in 3 dimensions is just the same except that now the
components of $\psi$ are real numbers, so things simplify: we don't need
complex conjugation, and $\psi \otimes \psi^*$ will be a *real* hermitian matrix. Real
hermitian matrices are the same as vectors in 3d Minkowski spacetime,
since we can write them as linear combinations of the three Pauli
matrices without i's in them --- namely, all of them except $\sigma_y$. So, we
get a lightlike vector in 3d Minkowski spacetime: say, $(t,x,z)$ with
$$t^2 = x^2 + z^2$$
Now for the really fun part: the trick works the same with Pythagorean
spinors except now everything in sight is an integer...

... so $(t,x,z)$ is a Pythagorean triple!!! And in fact, we get every
Pythagorean triple this way, at least up to an integer multiple. And in
fact, this trick was already known by Euclid.

Explicitly, if
$$\psi = \left(\begin{array}{c}a\\b\end{array}\right)$$
then
$$
  \begin{aligned}
    2\psi\otimes\psi^*
    &= \left(
      \begin{array}{cc}
        2a^2&2ab
      \\ab&2b^2
      \end{array}
    \right)
  \\&= (a^2+b^2)\sigma_t + 2ab\sigma_x + (a^2-b^2)\sigma_z
  \end{aligned}
$$
so we get the Pythagorean triple
$$(t,x,z) = (a^2 + b^2, 2ab, a^2 - b^2)$$
For example, if we take our spinor to be
$$\psi = \left(\begin{array}{c}2\\1\end{array}\right)$$
we get the famous triple
$$(t,x,z) = (5,4,3)$$

By the way, you'll notice I had to insert a fudge factor of "$2$" in
that formula up there to get things to work. I'm not sure why.

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

**Addendum:** Thanks to Andy Everett for catching a typo. Noam Elkies
sent me the following:

> Dear J. Baez,
>
> In article  you write:
>
> > \[......\] so $(t,x,z)$ is a Pythagorean triple!!!   And in fact,
> > we get every Pythagorean triple this way, at least up to
> > an integer multiple.  And in fact, this trick was already
> > known by Euclid.
>
> Are you sure of this?  The formula was surely known by Euclid's time --
> I've even seen claims that it must have been known by the Babylonians
> (perhaps not a coincidence, since Pythagoras spent some time in Babylonia)
> -- but did Euclid have anything like this interpretation, or the proof
> that every "Pythagorean triple" is proportional to one of this form?
>
> \[For that matter there's apparently some controversy about just who
> Pythagoras might have been and what he might have known, believed,
> and/or proved, since secrecy was one of the Pythagoreans' tenets.\]
>
> > By the way, you'll notice I had to insert a fudge factor of "$2$"
> > in that formula up there to get things to work.  I'm not sure why.
>
> This is presumably an artifact of the ambiguity in the
> "symmetric square of $\mathbb{Z}$".  In general, the symmetric square
> of a module $M$ can be formed as either a submodule or a quotient
> module of the tensor square of $M$.  These two symmetric squares
> are (canonically) isomorphic if $2$ is invertible, but not in general.
>
> The parametrization of Pythagorean triples is also closely related
> with the "half-angle substitution" of elementary calculus.
> For yet another interpretation, see
>
> - `http://www.math.harvard.edu/~elkies/Misc/hilbert.dvi`
>
> \[also .pdf instead of .dvi\].
>
> The dissections that illustrate the Pythagorean theorem
> can be generalized to the law of cosines; see
>
> - `http://www.math.harvard.edu/~elkies/Misc/cos1.ps`
> - `http://www.math.harvard.edu/~elkies/Misc/cos2.ps`
>
> for the acute and obtuse cases respectively.
>
> Enjoy,
> --- Noam D. Elkies

I replied saying that by Euclid knowing "this trick", I only meant he
knew this formula for Pythagorean triples:
$$(a^2 + b^2, 2ab, a^2 - b^2)$$

I don't know who proved it gives *all* of them (up to multiples).

Rob Johnson then noted:

>
> There is not any high powered math involved in showing that these are
> all the pythagorean triples up to scalar multiples.  See
>
> - `http://www.whim.org/nebula/math/pythag.html`
>
> Nothing more than the fundamental theorem of arithmetic is used (to
> justify the statement "Since $b^2 = 4MN$ and $\operatorname{gcd}(M,N) = 1$, both $M$ and $N$
> must be perfect squares."), and Euclid knew that.
>
> So my guess is that Euclid probably knew that this formula gives all
> pythagorean triples up to scalar multiples, but it is just a guess.
>
> Rob Johnson\
> `rob@whim.org`