Definitions

# Baryogenesis

In physical cosmology, baryogenesis is the generic term for hypothetical physical processes that produced an asymmetry between baryons and anti-baryons in the very early universe, resulting in the substantial amounts of residual matter that make up the universe today.

Baryogenesis theories — the most important being electroweak baryogenesis and GUT baryogenesis — employ sub-disciplines of physics such as quantum field theory, and statistical physics, to describe such possible mechanisms. The fundamental difference between baryogenesis theories is the description of the interactions between fundamental particles.

The next step after baryogenesis is the much better understood Big Bang nucleosynthesis, during which light atomic nuclei began to form.

## 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 known kind of particle has a corresponding antiparticle. The CPT 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 puzzling that the universe does not have equal amounts of matter and antimatter. Indeed, there is no experimental 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 ($B\left(time=0\right) neq 0$); or, the universe was originally perfectly symmetric ($B\left(time=0\right) = 0,$), but somehow a set of phenomena contributed to a small imbalance. 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 $B=0$. From a more scientific point-of-view, there are reasons to expect that any initial asymmetry would be wiped out to zero during the early history of the universe. One challenge then is to explain how the universe evolves to produce $B neq 0$.

## The Sakharov conditions

In 1967, Andrei Sakharov proposed a set of three necessary conditions that a baryon-generating interaction must satisfy to produce matter and antimatter at different rates. These conditions were inspired by the recent discoveries of the cosmic background radiation and CP-violation in the neutral kaon system. The three necessary "Sakharov conditions" are:

Currently, there is no experimental evidence of particle interactions where the conservation of baryon number is broken perturbatively: this would appear to suggest that all observed particle reactions have equal baryon number before and after. Mathematically, the commutator of the baryon number quantum operator with the (perturbative) Standard Model hamiltonian is zero: $\left[B,H\right] = BH - HB = 0$. However, the Standard Model is known to violate the conservation of baryon number non-perturbatively: a global U(1) anomaly. Baryon number violation can also result from physics beyond the Standard Model (see supersymmetry and Grand Unification Theories).

The second condition — violation of CP-symmetry — was discovered in 1964 (direct CP-violation, that is violation of CP-symmetry in a decay process, was discovered later, in 1999). If CPT-symmetry is assumed, violation of CP-symmetry demands violation of time inversion symmetry, or T-symmetry.

The last condition states that the rate of a reaction which generates baryon-asymmetry must be less than the rate of expansion of the universe. In this situation the particles and their corresponding antiparticles do not achieve thermal equilibrium due to rapid expansion decreasing the occurrence of pair-annihilation.

## 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 magnitude of this asymmetry. An important quantifier is the asymmetry parameter,
$eta = frac\left\{n_B - n_\left\{bar B\right\}\right\}\left\{n_gamma\right\}$.
This quantity relates the overall number density difference between baryons and anti-baryons ($n_B$ and $n_\left\{bar B\right\}$, respectively) and the number density of cosmic background radiation photon $n_gamma$.

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 $3000 mathrm\left\{K\right\} / \left(10.08 times 10^4 mathrm\left\{K/eV\right\}\right) = 0.3 mathrm\left\{ eV\right\}$. 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 $T$, per cubic kelvin and per cubic centimeter, is given by

$n_gamma = frac\left\{1\right\}\left\{pi^2\right\} \left\{left\left(frac\left\{k_B T\right\}\left\{hbar c\right\}right\right)\right\}^3 int_0^infty frac\left\{x^2\right\}\left\{exp\left(x\right) - 1\right\} dx simeq 20.3 T^3$,
with $k_B$ as the Boltzmann constant, $hbar$ as the Planck constant divided by $2pi$ and $c$ as the speed of light in vacuum. In the numeric approximation at the left hand side of the equation, the convention $c = hbar = k_B = 1$ was used (natural units), and for $T$ in kelvins the result is given in $text\left\{K\right\}^\left\{-3\right\} ; text\left\{cm\right\}^\left\{-3\right\}$. At the current CBR photon temperature of $T = 2.73 text\left\{K\right\}$, this corresponds to a photon density $n_gamma$ of around $411$ CBR photons per cubic centimeter.

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

$eta_s = frac\left\{n_B - n_\left\{bar B\right\}\right\}\left\{s\right\}$
because the entropy density of the universe remained reasonably constant throughout most of its evolution. The entropy density is
$s stackrel\left\{mathrm\left\{def\right\}\right\}\left\{=\right\} frac\left\{mathrm\left\{entropy\right\}\right\}\left\{mathrm\left\{volume\right\}\right\} = frac\left\{p + rho\right\}\left\{T\right\} = frac\left\{2pi^2\right\}\left\{45\right\}g_\left\{*\right\}\left(T\right) T^3$
with $p$ and $rho$ as the pressure and density from the energy density tensor $T_\left\{munu\right\}$, and $g_*$ as the effective number of degrees of freedom for "massless" particle (inasmuch as $mc^2 ll k_B T$ holds) at temperature $T$,
$g_*\left(T\right) = sum_mathrm\left\{i=bosons\right\} g_i\left\{left\left(frac\left\{T_i\right\}\left\{T\right\}right\right)\right\}^3 + frac\left\{7\right\}\left\{8\right\}sum_mathrm\left\{j=fermions\right\} g_j\left\{left\left(frac\left\{T_j\right\}\left\{T\right\}right\right)\right\}^3$,
for bosons and fermions with $g_i$ and $g_j$ degrees of freedom at temperatures $T_i$ and $T_j$ respectively. At the present era, $s = 7.04 n_gamma$.