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To set a context for the confrontation of theoretical predictions with observational data it is useful to review the physics and cosmology of the early evolution of the Universe, touching on the specifics relevant for the synthesis of the light nuclides during the first ~ 20 minutes. In this section is presented an overview of this evolution along with the predicted primordial abundances, first in the standard model and then for two examples of nonstandard models which involve variations on the early-Universe expansion rate (Steigman, Schramm, & Gunn 1977) or asymmetries between the number of neutrinos and antineutrinos (e.g., Kang & Steigman 1992, and references therein).

2.1. Early Evolution

Discussion of BBN can begin when the Universe is a few tenths of a second old and the temperature is a few MeV. At such an early epoch the energy density is dominated by the relativistic (R) particles present, and the Universe is said to be "radiation-dominated". For sufficiently early times, when the temperature is a few times higher than the electron rest-mass energy, these are photons, e± pairs, and, for the standard model of particle physics, three flavors of left-handed (i.e. one helicity state) neutrinos (and their right-handed antineutrinos).

Equation 1 (1)

where rhogamma is the energy density in CBR photons (which, today, have redshifted to become the CBR photons at a temperature of 2.7 K).

In standard BBN (SBBN) it is assumed that the neutrinos are fully decoupled prior to e± annihilation and do not share in the energy transferred from the annihilating e± pairs to the CBR photons. In this approximation, in the post-e± annihilation Universe, the photons are hotter than the neutrinos by a factor Tgamma / Tnu = (11/4)1/3, and the relativistic energy density is

Equation 2 (2)

During these radiation-dominated epochs the age (t) and the energy density are related by (32 pi G / 3) rhoR t2 = 1, so that once the particle content (rhoR) is specified, the age of the Universe is known (as a function of the CBR temperature Tgamma). In the standard model,

Equation 3 (3)

Equation 4 (4)

Also present at these early times are neutrons and protons, albeit in trace amounts compared to the relativistic particles. The relative abundance of neutrons and protons is determined by the charged-current weak interactions.

Equation 5 (5)

As time goes by and the Universe expands and cools, the lighter protons are favored over the heavier neutrons and the neutron-to-proton ratio decreases, initially following the equilibrium form (n / p)eq propto exp(- Deltam / T), where Deltam = 1.29 MeV is the neutron-proton mass difference. As the temperature drops the two-body collisions in Equation 5 become too slow to maintain equilibrium and the neutron-to-proton ratio, while continuing to decrease, begins to deviate from (exceeds) this equilibrium value. For later reference, we note that if there is an asymmetry between the numbers of nue and bar{nu}e ("neutrino degeneracy"), described by a chemical potential µe (such that for µe > 0 there are more nue than bar{nu}e), then the equilibrium neutron-to-proton ratio is modified to (n/p) propto exp(- Deltam / T - µe / T). In place of the neutrino chemical potential, it is convenient to introduce the dimensionless degeneracy parameter xie ident µe / T, which is invariant as the Universe expands.

Prior to e±  annihilation, at T approx 0.8 MeV when the Universe is ~ 1 second old, the two-body reactions regulating n/p become slow compared to the universal expansion rate and this ratio "freezes in", although, in reality, it continues to decrease, albeit more slowly than would be the case for equilibrium. Later, when the Universe is several hundred seconds old, a time comparable to the neutron lifetime (taun = 885.7 ± 0.8 s), the n / p ratio resumes falling exponentially: n / p propto exp(- t/taun). Since there are several billion CBR photons for every nucleon (baryon), the abundances of any complex nuclei are entirely negligible at these early times.

Notice that since the n / p ratio depends on the competition between the weak interaction rates and the early-Universe expansion rate (as well as on a possible neutrino asymmetry), any deviations from the standard model (e.g., rhoR -> rhoR + rhoX or xie neq 0) will change the relative numbers of neutrons and protons available for building more complex nuclides.

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