Baryogenesis

Baryogenesis is the generic designation for the physical processes that generate matter (more specifically, a class of fundamental particle called baryon) from an otherwise matter-empty state (such as it is generally believed to be the state of the Universe at its onset, the so-called Big Bang).

The baryogenesis theories deal with different theories of physics to describe the possible mechanisms for generating baryons. They essentially incorporate the following areas:

The fundamental difference between baryogenesis theories is the description of the interaction between fundamental particles. Among the baryogenesis theories are:

The next step after baryogenesis, is the much better understood nucleosynthesis, the forming of atomic nuclei.

Contents

1 Background

2 The Sakharov conditions

3 Matter content in the Universe

3.1 The baryon asymmetry parameter
3.2 A naïve estimation of the baryon asymmetry of the Universe

Background

The Dirac equation was first announced by Paul Dirac around 1928, and it describes the dynamics of a single fermion. This equation predicts the existence of antiparticles as possible solutions, along with the expectable solutions for the corresponding particles. Nevertheless, this equation deals only with the dynamics of point-like, ½-spin charged particles such as the electron, saying nothing on the rate of production of either particles or antiparticles. In other words, the rate of production of electron might as well be the same as of the anti-electron (or positron). In principle, one concludes that in a situation of chemical balance between the two, there would be (at best) very localized regions in space with more of either matter or antimatter. This, however, is not enough to explain the amount of matter that survived the matter-formation era at the beginning of the Universe.

Prior to 1967, there were two main 'philosophical' interpretations to the state of the Universe right at its beginning: either there was already a small preference for matter, with the total baryonic number of the Universe different from zero (<math>B(time=0) \neq 0<math>); or, the Universe at its beginning was perfectly symmetric (<math>B(time=0) = 0<math>), but somehow a set of phenomena contributed to a small unbalance. The second point of view is preferred, although there is no clear experimental evidence indicating either of them to be the correct one. The aforementioned preference is merely based on the following philosophical point-of-view: if the Universe encompasses everything (time, space, and matter), nothing exists outside of it and therefore nothing existed before it, leading to the baryonic number <math>B=0<math>. One challenge then is to explain how the Universe evolves to produce <math>B \neq 0<math>.

The Sakharov conditions

In 1967, Andrei Sakharov proposed a set of three conditions that a baryon-generating particle interaction to produce matter and antimatter at different rates. These conditions were inspired in recent discoveries: the cosmic background radiation (Penzias and Wilson, 1965), and the CP-symmetry violation in the neutral kaon system (Cronin, Fitch and collaborators, 1964). The three conditions on a baryon-generating interaction are the following:

Baryon number <math>B<math> violation.
C-symmetry and CP-symmetry violation.
Withdrawal from thermal equilibrium.

The first condition may seem trivial, but to this day there is no experimental evidence on particle interactions where the baryon number is violated: so far, all observed particle interactions are so that the baryon number before and after such reactions is the same. Technically, this translates as the commutator of the baryon number quantum operator with the Standard Model hamiltonian operator is zero: <math>[B,H] = BH - HB = 0<math>. This is a strong indication that the Standard Model of Particle Physics is not a finalized theory, and other extensions to it are under active investigation. Among the possible extensions are supersymmetry and Grand Unification Theories.

The second condition(s), however, has been known since the late 1950s and early 1960s. Violation of CP-symmetry is currently one important area of investigation in particle physics. This symmetry violation is related with the time inversion symmetry T, assuming that the CPT-symmetry is valid. In layman terms this translates into the rate of a given reaction is not the same if it evolved backwards in time.

The last condition involves cosmology, and the usual frame for describing the Universe at its early stages in the form of the inflation theory. In essence, this condition states that the rate of a baryon-asymmetry generating reaction has to be lower than the rate of expansion of the Universe. In this situation, the particles and their corresponding antiparticles do not get the opportunity to achieve thermal equilibrium due to the fast expansion rate, and therefore the chances for catastrophic annihilation are reduced.

These three conditions have to occur at the same time in order to produce different contents of matter and antimatter.

Matter content in the Universe

The baryon asymmetry parameter

The challenges to the physics theories are then to explain how to produce this preference of matter over antimatter, and also the size of this asymmetry. An important quantifier is the asymmetry parameter,

<math>\eta = \frac{n_B - n_{\bar B}}{n_\gamma}<math>.

This quantity relates the overall number density difference between baryons and anti-baryons (<math>n_B<math> and <math>n_{\bar B}<math>, respectively) and the number density of cosmic background radiation photon <math>n_\gamma<math>. Because baryon number violating particle interactions have not yet been observed in the energy ranges obtained in laboratory, it is assumed that, after the Big Bang, no baryogenesis occurs explicitly, wherefore the asymmetry should not change.

According to the Big Bang model, matter decoupled from the cosmic background radiation (CBR) at a temperature of roughly 3000 kelvins, corresponding to an average kinetic energy of <math>3000\ \mathrm{K} / (10.08 \times 10^4 \ \mathrm{K/eV}) = 0.3\ \mathrm{ eV}<math>. After the decoupling, the total number of CBR photons remains constant. Therefore due to space-time expansion, the photon density decreases. The photon density at equilibrium temperature <math>T<math>, per cubic kelvin and per cubic centimeter, is given by

<math>n_\gamma = \frac{1}{\pi^2} {\left(\frac{k_B T}{\hbar c}\right)}^3 \int_0^\infty \frac{x^2}{\exp^x - 1} dx \simeq 20.3 T^3 ,<math>

with <math>k_B<math> as the Boltzmann constant, <math>\hbar<math> as the Planck constant divided by <math>2\pi<math> and <math>c<math> as the speed of light in vacuum. In the numeric approximation at the left hand side of the equation, the convention <math>c = \hbar = k_B = 1<math> was used (natural units), and for T in kelvins the result is given in K^-3 cm^-3. At the current CBR photon temperature of T = 2.73 K, this corresponds to a photon density <math>n_\gamma<math> of around <math>411<math> CBR photons per cubic centimeter.

Therefore, the asymmetry parameter η, as defined above, is not the "good" parameter. Instead, the preferred asymmetry parameter uses instead the entropy density s,

because the entropy density of the Universe remained reasonably constant throughout most of its evolution. The entropy density is

<math>s \equiv \frac{\mathrm{entropy}}{\mathrm{volume}} = \frac{p + \rho}{T} = \frac{2\pi^2}{45}g_{*}(T) T^3<math>

with <math>p<math> and <math>\rho<math> as the pressure and density from the energy density tensor <math>T_{\mu\nu}<math>, and <math>g_*<math> as the effective number of degrees of freedom for "massless" (<math>mc^2 <\!\!< k_B T<math>) particles, at temperature <math>T<math>,

<math>g_*(T) = \sum_\mathrm{i=bosons} g_i{\left(\frac{T_i}{T}\right)}^3 + \frac{7}{8}\sum_\mathrm{j=fermions} g_j{\left(\frac{T_j}{T}\right)}^3<math>,

for bosons and fermions with <math>g_i<math> and <math>g_j<math> degrees of freedom at temperatures <math>T_i<math> and <math>T_j<math> respectively. At the present era, <math>s = 7.04 n_\gamma<math>.

A naïve estimation of the baryon asymmetry of the Universe

Observational results yield that η is approximately equal to 10⁻¹⁰ — more precisely, 2.6 < η × 10¹⁰ < 6.2. This means that for every 10 billion pairs of particle and antiparticle, there was one extra particle that was left without an antiparticle with which to annihilate into background radiation. This is a very small number, and explaining how to obtain it is very difficult: one is trying to make predictions to the very large (cosmology) based on the laws of the very small (particle physics)!

A reasonable idea of how this number is found experimentally follows. The Hubble Space Telescope surveys report that the observable Universe contains approximately 125 billion (1.25×10¹¹) galaxies. Assuming that they are, in average, similar to our own Galaxy, then each would contain around 100 billion suns (10¹¹). The weight of our Sun, which is a typical star, is around 2×10³⁰ kilograms. Making the gross approximation that our Sun is composed of hydrogen, which weighs approximately 1.67×10⁻²⁷ kilogram, then the last figure is equal to 1.2×10⁵⁷ atoms (write the number 12 and then 56 zeros). The total number of atoms in the visible Universe is then approximately 1.5×10⁷⁹. For a Universe with 14 billion (1.4×10¹⁰) years of age, its spatial radius would be around 1.3×10²⁶ meters, or a sphere of 9.2×10⁷⁸ cubic meters. The atom density would then be around 1.6/m³. On the other hand, statistical physics tells us that a gas of photons in thermal equilibrium at the temperature of the cosmic background radiation, 2.73 kelvins, has a number density of 4.1×10⁸ per cubic meter. The resulting estimation for the parameter η would then be approximately 4×10⁻⁹. This is not a bad approximation; it is only an order of magnitude above the value quoted in the literature. The exact experimental value involves measuring the concentration of chemical elements in the Universe not originating from stellar synthesis.

Baryogenesis

Background

The Sakharov conditions

Matter content in the Universe

The baryon asymmetry parameter

A naïve estimation of the baryon asymmetry of the Universe

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