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As the Universe evolved from its early, hot, dense beginnings (the "Big Bang") to its present, cold, dilute state, it passed through a brief epoch when the temperature (average thermal energy) and density of its nucleon component were such that nuclear reactions building complex nuclei could occur. Because the nucleon content of the Universe is small (in a sense to be described below) and because the Universe evolved through this epoch very rapidly, only the lightest nuclides (D, 3He, 4He, and 7Li) could be synthesized in astrophysically interesting abundances. The relic abundances of these nuclides provide probes of conditions and contents of the Universe at a very early epoch in its evolution (the first few minutes) otherwise hidden from our view. The standard model of Cosmology subsumes the standard model of particle physics (e.g., three families of very light, left-handed neutrinos along with their right-handed antineutrinos) and uses General Relativity (e.g., the Friedman equation) to track the time-evolution of the universal expansion rate and its matter and radiation contents. While nuclear reactions among the nucleons are always occurring in the early Universe, Big Bang Nucleosynthesis (BBN) begins in earnest when the Universe is a few minutes old and it ends less than a half hour later when nuclear reactions are quenched by low temperatures and densities. The BBN abundances depend on the conditions (temperature, nucleon density, expansion rate, neutrino content and neutrino-antineutrino asymmetry, etc.) at those times and are largely independent of the detailed processes which established them. As a consequence, BBN can test and constrain the parameters of the standard model (SBBN), as well as probe any non-standard physics/cosmology which changes those conditions.

The relic abundances of the light nuclides synthesized in BBN depend on the competition between the nucleon density-dependent nuclear reaction rates and the universal expansion rate. In addition, while all primordial abundances depend to some degree on the initial (when BBN begins) ratio of neutrons to protons, the 4He abundance is largely fixed by this ratio, which is determined by the competition between the weak interaction rates and the universal expansion rate, along with the magnitude of any nue - bar{nu}e asymmetry. 1 To summarize, in its simplest version BBN depends on three unknown parameters: the baryon asymmetry; the lepton asymmetry; the universal expansion rate. These parameters are quantified next.

1.1. Baryon Asymmetry - Nucleon Abundance

In the very early universe baryon-antibaryon pairs (quark-antiquark pairs) were as abundant as radiation (e.g., photons). As the Universe expanded and cooled, the pairs annihilated, leaving behind any baryon excess established during the earlier evolution of the Universe [1]. Subsequently, the number of baryons in a comoving volume of the Universe is preserved. After e ± pairs annihilate, when the temperature (in energy units) drops below the electron mass, the number of Cosmic Background Radiation (CBR) photons in a comoving volume is also preserved. As a result, it is useful (and conventional) to measure the universal baryon asymmetry by comparing the number of (excess) baryons to the number of photons in a comoving volume (post-e ± annihilation). This ratio defines the baryon abundance parameter etaB,

Equation 1 (1)

As will be seen from BBN, and as is confirmed by a variety of independent (non-BBN), astrophysical and cosmological data, etaB is very small. As a result, it is convenient to introduce eta10 ident 1010 etaB and to use it as one of the adjustable parameters for BBN. An equivalent measure of the baryon density is provided by the baryon density parameter, ΩB, the ratio (at present) of the baryon mass density to the critical density. In terms of the present value of the Hubble parameter (see Section 1.2 below), H0 ident 100 h km s-1 Mpc-1, these two measures are related by

Equation 2 (2)

Note that the subscript 0 refers to the present epoch (redshift z = 0).

From a variety of non-BBN cosmological observations whose accuracy is dominated by the very precise CBR temperature fluctuation data from WMAP [2], the baryon abundance parameter is limited to a narrow range centered near eta10 approx 6. As a result, while the behavior of the BBN-predicted relic abundances will be described qualitatively as functions of etaB, for quantitative comparisons the results presented here will focus on the limited interval 4 leq eta10 leq 8. As will be seen below (Section 2.2), over this range there are very simple, yet accurate, analytic fits to the BBN-predicted primordial abundances.

1.2. The Expansion Rate At BBN

For the standard model of cosmology, the Friedman equation relates the expansion rate, quantified by the Hubble parameter (H), to the matter-radiation content of the Universe.

Equation 3 (3)

where GN is Newton's gravitational constant. During the early evolution of the Universe the total density, rhoTOT, is dominated by "radiation" (i.e., by the contributions from massless and/or extremely relativistic particles). During radiation dominated epochs (RD), the age of the Universe (t) and the Hubble parameter are simply related by (Ht)RD = 1/2.

Prior to BBN, at a temperature of a few MeV, the standard model of particle physics determines that the relativistic particle content consists of photons, e ± pairs and three flavors of left-handed (i.e., one helicity state) neutrinos (along with their right-handed, antineutrinos; Nnu = 3). With all chemical potentials set to zero (very small lepton asymmetry) the energy density of these constituents in thermal equilibrium is

Equation 4 (4)

where rhogamma is the energy density in the CBR photons (which have redshifted to become the CBR photons observed today at a temperature of 2.7K). In this case (SBBN: Nnu = 3), the time-temperature relation derived from the Friedman equation is,

Equation 5 (5)

In SBBN it is usually assumed that the neutrinos are fully decoupled prior to e ± annihilation; if so, they don't share in the energy transferred from the annihilating e ± pairs to the CBR photons. In this very good approximation, the photons are hotter than the neutrinos in the post-e ± annihilation universe by a factor Tgamma / Tnu = (11/4)1/3, and the total energy density is

Equation 6 (6)

corresponding to a modified time-temperature relation,

Equation 7 (7)

Quite generally, new physics beyond the standard models of cosmology or particle physics could lead to a non-standard, early Universe expansion rate (H'), whose ratio to the standard rate (H) may be parameterized by an expansion rate factor S,

Equation 8 (8)

A non-standard expansion rate might originate from modifications to the 3+1 dimensional Friedman equation as in some higher dimensional models [3], or from a change in the strength of gravity [4]. Different gravitational couplings for fermions and bosons [5] would have similar effects. Alternatively, changing the particle population in early Universe will modify the energy density - temperature relation, also leading, through eq. 3, to S neq 1. While these different mechanisms for implementing a non-standard expansion rate are not necessarily equivalent, specific models generally lead to specific predictions for S.

Consider, for example, the case of a non-standard energy density.

Equation 9 (9)

where rho'R = rhoR + rhoX and X identifies the non-standard component. With the restriction that the X are relativistic, this extra component, non-interacting at e ± annihilation, behaves as would an additional neutrino flavor. It must be emphasized that X is not restricted to additional flavors of active or sterile neutrinos. In this class of models S is constant prior to e ± annihilation and it is convenient (and conventional) to account for the extra contribution to the standard-model energy density by normalizing it to that of an "equivalent" neutrino flavor [6], so that

Equation 10 (10)

For this case,

Equation 11 (11)

In another class of non-standard models the early Universe is heated by the decay of a massive particle, produced earlier in the evolution [7]. If the Universe is heated to a temperature which is too low to (re)populate a thermal spectrum of the standard neutrinos (TRH ltapprox 7 MeV), the effective number of neutrino flavors contributing to the total energy density is < 3, resulting in Delta Nnu < 0 and S < 1.

Since the expansion rate is more fundamental than is Delta Nnu, BBN for models with non-standard expansion rates will be parameterized using S (but, for comparison, the corresponding value of Delta Nnu from eq. 11 will often be given for comparison). The simple, analytic fits to BBN presented below (Section 2.2) are quite accurate for 0.85 leq S leq 1.15, corresponding to -1.7 ltapprox DeltaNnu ltapprox 2.0

1.3. Neutrino Asymmetry

The baryon asymmetry of the Universe, quantified by etaB, is very small. If, as expected in the currently most popular particle physics models, the universal lepton and baryon numbers are comparable, then any asymmetry between neutrinos and antineutrinos ("neutrino degeneracy") will be far too small to have a noticeable effect on BBN. However, it is possible that the baryon and lepton asymmetries are disconnected and that the lepton (neutrino) asymmetry could be large enough to perturb the SBBN predictions. In analogy with etaB which quantifies the baryon asymmetry, the lepton (neutrino) asymmetry, L = Lnu ident Sigmaalpha Lnualpha, may be quantified by the neutrino chemical potentials µnualpha (alpha ident e, µ, tau) or, by the degeneracy parameters, the ratios of the neutral lepton chemical potentials to the temperature (in energy units) xinualpha ident µnualpha / kT, where

Equation 12 (12)

Prior to e ± annihilation, Tnu = Tgamma, while post-e ± annihilation (Tnu / Tgamma)3 = 4/11. Although in principle the asymmetry among the different neutrino flavors may be different, mixing among the three active neutrinos (nue, nuµ, nutau) ensures that at BBN, Le approx Lµ approx Ltau (xie approx xiµ approx xitau) [8]. If Lnu is measured post-e ± annihilation, as is etaB, then for xinu << 1, Lnu approx 3Lnue and, for xi ident xie << 1, Lnu approx 0.75xi.

Although any neutrino degeneracy (xinualpha < 0 as well as > 0) increases the energy density in the relativistic neutrinos, resulting in an effective Delta Nnu neq 0 (see eq. 10), the range of |xi| of interest to BBN is limited to sufficiently small values that the increase in S due to a non-zero xi is negligible. However, a small asymmetry between electron type neutrinos and antineutrinos (xie gtapprox 10-2; L gtapprox 0.007), while large compared to the baryon asymmetry, can have a significant impact on BBN since the nue affect the interconversion of neutrons to protons. A non-zero xie results in different (compared to SBBN) numbers of nue and bar{nu}e, altering the n/p ratio at BBN, thereby changing the yields (compared to SBBN) of the light nuclides.

Of the light, relic nuclei, the neutron limited 4He abundance is most sensitive to a non-zero xie; 4He is a good "leptometer". In concert with the abundances of D, 3He, and 7Li, which are good baryometers, the 4He abundance provides a test of the consistency of the standard model along with constraints on non-standard models. The analytic fits presented below (Section 2.2) are reasonably accurate for xie in the range, -0.1 ltapprox xie ltapprox 0.1, corresponding to a total lepton number limited to |L| ltapprox 0.07. While this may seem small, recall that a similar measure of the baryon asymmetry is orders of magnitude smaller: etaB approx 6 × 10-10.

1 A lepton asymmetry much larger than the baryon asymmetry (which is very small; see Section 1.1 below) would have to reside in the neutrinos since charge neutrality ensures that the electron-positron asymmetry is comparable to the baryon asymmetry. Back.

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