Baryogenesis

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 microwave background and CP-violation in the neutral kaon system. The three necessary "Sakharov conditions" are:

• Baryon number ${\displaystyle B}$ violation.
• C-symmetry and CP-symmetry violation.
• Interactions out of thermal equilibrium.

Baryon number violation is a necessary condition to produce an excess of baryons over anti-baryons. But C-symmetry violation is also needed so that the interactions which produce more baryons than anti-baryons will not be counterbalanced by interactions which produce more anti-baryons than baryons. CP-symmetry violation is similarly required because otherwise equal numbers of left-handed baryons and right-handed anti-baryons would be produced, as well as equal numbers of left-handed anti-baryons and right-handed baryons. Finally, the interactions that produce asymmetry must occur before thermal equilibrium because the thermal average of the standard model baryon asymmetry is zero.

In the Standard Model

The Standard Model can incorporate baryogenesis, though the amount of net baryons (and leptons) thus created may not be sufficient to account for the present baryon asymmetry. There is a required one excess quark per billion quark-antiquark pairs in the early universe in order to provide all the observed matter in the universe. This insufficiency has not yet been explained, theoretically or otherwise.

Baryogenesis within the Standard Model requires the electroweak symmetry breaking to be a first-order cosmological phase transition, since otherwise sphalerons wipe out any baryon asymmetry that happened up to the phase transition. Beyond this, the remaining amount of baryon non-conserving interactions is negligible.

The phase transition domain wall breaks the P-symmetry spontaneously, allowing for CP-symmetry violating interactions to break C-symmetry on both its sides. Quarks tend to accumulate on the broken phase side of the domain wall, while anti-quarks tend to accumulate on its unbroken phase side. Due to CP-symmetry violating electroweak interactions, some amplitudes involving quarks are not equal to the corresponding amplitudes involving anti-quarks, but rather have opposite phase (see CKM matrix and Kaon); since time reversal takes an amplitude to its complex conjugate, CPT-symmetry is conserved in this entire process.

Though some of their amplitudes have opposite phases, both quarks and anti-quarks have positive energy, and hence acquire the same phase as they move in space-time. This phase also depends on their mass, which is identical but depends both on flavor and on the Higgs VEV which changes along the domain wall. Thus certain sums of amplitudes for quarks have different absolute values compared to those of anti-quarks. In all, quarks and anti-quarks may have different reflection and transmission probabilities through the domain wall, and it turns out that more quarks coming from the unbroken phase are transmitted compared to anti-quarks.

Thus there is a net baryonic flux through the domain wall. Due to sphaleron transitions, which are abundant in the unbroken phase, the net anti-baryonic content of the unbroken phase is wiped out as anti-baryons are transformed into leptons. However, sphalerons are rare enough in the broken phase as not to wipe out the excess of baryons there. In total, there is net creation of baryons (as well as leptons).

In this scenario, non-perturbative electroweak interactions (i.e. the sphaleron) are responsible for the B-violation, the perturbative electroweak Lagrangian is responsible for the CP-violation, and the domain wall is responsible for the lack of thermal equilibrium and the P-violation; together with the CP-violation it also creates a C-violation in each of its sides.

Relation to Big Bang nucleosynthesis

The central question to baryogenesis is what causes the preference for matter over antimatter in the universe, as well as the magnitude of this asymmetry. An important quantifier is the asymmetry parameter, given by

$${\displaystyle \eta =\frac{n_{\text{B}}-n_{\overline{\text{B}}}}{n_{\gamma }},}$$ where n_B and n_B refer to the number density of baryons and antibaryons respectively and n_γ is the number density of cosmic background radiation photons.

According to the Big Bang model, matter decoupled from the cosmic background radiation (CBR) at a temperature of roughly 3000 kelvin, corresponding to an average kinetic energy of 3000 K / (10.08×10³ K/eV) = 0.3 eV. 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 is given by $${\displaystyle \begin{array}{cc}{n}_{\gamma }& =\frac{1}{{\pi }^2}{\left(\frac{k_{\text{B}}T}{\hslash c}\right)}^3{\int }_0^{\mathrm{\infty }}\frac{x^2}{e^x-1}dx\\ & =\frac{2\zeta (3)}{{\pi }^2}{\left(\frac{k_{\text{B}}T}{\hslash c}\right)}^3\\ & \approx 20.3{\left(\frac{T}{1\mathrm{K}}\right)}^3{\text{cm}}^{-3},\end{array}}$$ with k_B as the Boltzmann constant, ħ as the Planck constant divided by 2π and c as the speed of light in vacuum, and ζ(3) as Apéry's constant. At the current CBR photon temperature of 2.725 K, this corresponds to a photon density n_γ of around 411 CBR photons per cubic centimeter.

Therefore, the asymmetry parameter η, as defined above, is not the "best" parameter. Instead, the preferred asymmetry parameter uses the entropy density s, $${\displaystyle {\eta }_s=\frac{n_{\text{B}}-n_{\overline{\text{B}}}}{s}}$$ because the entropy density of the universe remained reasonably constant throughout most of its evolution. The entropy density is $${\displaystyle s\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\frac{\mathrm{e}\mathrm{n}\mathrm{t}\mathrm{r}\mathrm{o}\mathrm{p}\mathrm{y}}{\mathrm{v}\mathrm{o}\mathrm{l}\mathrm{u}\mathrm{m}\mathrm{e}}=\frac{p+\rho }{T}=\frac{2{\pi }^2}{45}{g}_{\text{⁎}}(T)T^3,}$$ with p and ρ as the pressure and density from the energy density tensor T_μν, and g_⁎ as the effective number of degrees of freedom for "massless" particles at temperature T (in so far as mc² ≪ k_BT holds), $${\displaystyle g_{\text{⁎}}(T)=\sum _{i=\mathrm{b}\mathrm{o}\mathrm{s}\mathrm{o}\mathrm{n}\mathrm{s}}{g}_i{\left(\frac{T_i}{T}\right)}^3+\frac{7}{8}\sum _{j=\mathrm{f}\mathrm{e}\mathrm{r}\mathrm{m}\mathrm{i}\mathrm{o}\mathrm{n}\mathrm{s}}{g}_j{\left(\frac{T_j}{T}\right)}^3,}$$ for bosons and fermions with g_i and g_j degrees of freedom at temperatures T_i and T_j respectively. At the present epoch, s = 7.04 n_γ.

Other models

B-meson decay

Another possible explanation for the cause of baryogenesis is the decay reaction of B-mesogenesis. This phenomenon suggests that in the early universe, particles such as the B-meson decay into a visible Standard Model baryon as well as a dark antibaryon that is invisible to current observation techniques.

Asymmetric dark matter

The asymmetric dark matter proposal investigates mechanisms that would explain the abundance of dark matter but lack of dark antimatter as the consequence of the same effect as would explain baryogenesis.