Baryogenesis

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Baryogenesis is the generic designation for the hypothetical physical processes that generated an asymmetry between baryons and anti-baryons in the very early universe.

Baryogenesis theories deal with different sub-fields of physics to describe the possible mechanisms for generating baryons. Most important are:

The fundamental difference between baryogenesis theories is the description of the interactions 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

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Background

The Dirac equation, formulated by Paul Dirac around 1928 as part of the development of relativistic quantum mechanics, predicts the existence of antiparticles along with the expected solutions for the corresponding particles. Since that time, it has been verified experimentally that every particle has a corresponding antiparticle. The CTP Theorem guarantees that a particle and its anti-particle have exactly the same mass and lifetime, and exactly opposite charge. Given this symmetry, it is unsettling that the universe does not have equal amounts of matter and antimatter. Indeed, there is no evidence that there are any significant concentrations of antimatter in the observable universe.

There are two main interpretations for this disparity: either when the universe began 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 was originally 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>.

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The Sakharov conditions

In 1967, Andrei Sakharov proposed a set of three conditions that a baryon-generating particle interaction must satisfy to produce matter and antimatter at different rates. These conditions were inspired by 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). These three conditions are the following:

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.

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Matter content in the Universe

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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,

<math>\eta_s = \frac{n_B - n_{\bar B}}{s}</math>

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>.

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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, each contains around 100 billion (1011) stars. The mass of our Sun, which is a typical star, is around 2×1030 kilograms. Making the approximation that our Sun is composed only of hydrogen atoms, each of which weighs approximately 1.67×10−27 kilograms, the Sun contains 1.2×1057 atoms. The total number of atoms in the visible Universe is then approximately 1.5×1079. The Universe is 14 billion (1.4×1010) years old, so the farthest away we can see is 14 billion light years, or 1.3×1026 meters. This means that the visible Universe is a sphere of 9.7×1078 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 estimate of η is 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.

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See also

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Textbooks

  • {{cite book 
| author = Kolb, Edward W. and Turner, Michael S.
| title = The Early Universe
| publisher = Perseus Publishing
| year = 1994
| id = ISBN 0-201-62674-8
}}
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Articles

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External links

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