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The baryon asymmetry is an excess of net baryon number over zero, conveniently measured by ${\eta }_B=(n_B-n_{\bar{B}})/n_{\gamma }$. Ordinary matter contains abundant baryons, whereas no comparable primordial antibaryon population is observed. Large nearby domains of matter and antimatter would produce radiation through particle-antiparticle annihilation at their boundaries. Antiparticles produced in energetic collisions are not evidence for a cosmological antimatter domain. The small nonzero baryon-to-photon ratio inferred from Big Bang nucleosynthesis is independently tested by baryon-density consistency between nucleosynthesis and the microwave background.

Starting with zero net baryon number, the Sakharov conditions require baryon-number violation, C and CP violation, and departure from thermal equilibrium. Conservation of baryon number would keep the initial value zero. C symmetry or CP symmetry would pair baryon production with equal antibaryon production; CP rate cancellation in baryogenesis explains why C violation alone does not suffice. In thermal equilibrium with zero imposed charge chemical potentials, CPT symmetry forbids the required net baryon number. An asymmetric rate in one reaction must not be mistaken for net production after all reverse reactions are included.

A simple example is heavy-boson decay baryogenesis. Take equally many heavy $X$ bosons and their antiparticles, with decay channelsThe symbols in parentheses are branching fractions. The two $X$ channels carry baryon numbers $2/3$ and $-1/3$, so both cannot conserve one assigned baryon number for $X$. Their mean baryon number is $r-1/3$, while the mean for $\bar{X}$ is $-\bar{r}+1/3$. Consequently,where $ϵ$ is the asymmetry per decaying particle. CP violation permits $r\ne \bar{r}$ without violating the equality of total decay widths required by CPT symmetry. Microscopically a direct CP asymmetry requires interference between decay amplitudes with both relative weak phases and absorptive phases; specifying complex couplings alone would not establish unequal partial decay widths.

For radiation domination, a sufficient decay condition is ${\Gamma }_X\lesssim H(T=M_X)=1.66\sqrt{g_{\ast }}{M}_X^2/m_P$. The heavy population can then survive as the equilibrium abundance drops, before decaying while inverse production is suppressed. Its actual abundance differs from its thermal equilibrium value. If subsequent baryon-number washout is negligible and cosmological entropy conservation holds, an initial total abundance $Y_X=(n_X+n_{\bar{X}})/s$ producesHere $s$ is the entropy density. Residual baryon-number washout multiplies this expression by an efficiency factor below one. The example realizes all three Sakharov conditions; it illustrates a mechanism, rather than identifying the actual cosmological source of the baryon asymmetry.

Deuterium and helium-4 broadly favor the same baryon-density range in standard Big Bang nucleosynthesis, after systematic and reaction-rate uncertainties are allowed. Near the illustrative value ${\eta }_{10}\sim 6$, the baryon-density consistency between nucleosynthesis and the microwave background givesDeuterium constrains this particularly sharply; helium-4 gives a weaker baryon-density constraint and a useful test of the expansion rate. Helium-3 observations do not supply an equally precise independent consistency test because of stellar processing. Lithium inferred directly from the metal-poor stellar plateau is lower than the standard prediction at this density by a factor of a few. Thus broad concordance is real, but complete four-element concordance without qualifications is false.

An independent baryon density comes from the cosmic microwave background acoustic pattern, especially the relative odd and even peak heights set by baryon loading of the photon–baryon fluid. For a numerical comparison appropriate to this paper's era, the 2003 microwave-background and large-scale-structure analysis obtained ${\Omega }_b{h}^2=0.0224\pm 0.0009$, consistent with the density inferred from deuterium. This comparison spans the first minutes and the much later recombination epoch, and tests the thermal history and conservation of baryon number relative to photon entropy between them; it is more substantial than comparing several abundance fits sharing the same nuclear model.

Ionization-corrected Lyman-alpha forest gas inventories and baryon fractions in galaxy clusters provide further checks, with their own radiation, gas-physics and total-mass uncertainties. A total matter fraction of order ${\Omega }_m\sim 0.3$ is much larger than ${\Omega }_b\sim 0.045$, so the abundance result also supports a predominantly nonbaryonic matter component. The lithium discrepancy and abundance systematics motivate continued tests; they do not erase the agreement between two physically distinct baryon-density determinations.

In standard Big Bang nucleosynthesis, with the radiation content and reaction rates fixed, the main adjustable abundance parameter is the baryon-to-photon ratio $\eta$. The baryon-density consistency between nucleosynthesis and the microwave background usesIncreasing cosmological baryon density makes deuterium burn more efficiently, so D/H falls steeply. The primordial helium mass fraction grows only slowly, because its dominant control is the surviving neutron fraction. Near the observationally favored density, lithium-7 production, much of it initially as beryllium-7, rises with density. At lower densities the full lithium curve has a valley; the sketch focuses on the branch relevant to the baryon-density comparison.

For the historical meaning of “current” in this 2008 paper, representative 2007–2008 fits near ${\eta }_{10}\simeq 6$ areThese local fits, with nuclear-rate uncertainties, and representative D/H and lithium ranges are documented in Steigman's 2007 nucleosynthesis analysis. They are not a replacement for a nuclear network outside their calibrated range.

The cosmic microwave background acoustic peak pattern measures baryon loading at recombination, independently of the abundance measurements. For example, the five-year WMAP cosmological analysis gives ${\Omega }_{b,0}{h}^2=0.02273\pm 0.00062$ from WMAP alone, corresponding to ${\eta }_{10}\simeq 6.23\pm 0.17$. For $h\simeq 0.7$, this is ${\Omega }_{b,0}\simeq 0.046$. Galaxy-cluster baryon fractions combined with independent total-matter density provide another, more astrophysically model-dependent cross-check.

At ${\eta }_{10}\simeq 6.2$, the fitted predictions are D/H $\simeq 2.5\times 10^{-5}$, $Y_p\simeq 0.249$, and ${}^{7}Li/H\simeq 4.6\times 10^{-10}$. A representative observed D/H range around $(2.7\pm 0.3)\times 10^{-5}$ overlaps the prediction; its inferred density ${\eta }_{10}\simeq 6.0\pm 0.4$ is compatible with the cosmic microwave background. The helium result $Y_p=0.2477\pm 0.0029$ from the 2007 helium analysis also agrees. Other helium analyses had lower central values, illustrating why ionization and temperature systematics matter.

In contrast, a representative plateau estimate $12+{\log }_{10}({}^{7}Li/H)\simeq 2.1\pm 0.1$ corresponds to roughly $(1.0$–$1.6)\times 10^{-10}$, a factor of several below the prediction at the same density. This is the cosmological lithium problem. The three abundance ranges therefore should not all be described as exact simultaneous agreement: deuterium and helium broadly support the independently measured baryon density, while lithium has a substantial discrepancy. The plotted ranges explicitly refer to the 2007–2008 comparison, not present-day parameter estimates.

The agreement of the main light-element abundances with a hot early nuclear-burning phase, and the independent baryon-density consistency between nucleosynthesis and the microwave background, gives strong support to the hot Big Bang framework. These measurements connect the first minutes to recombination, despite their very different temperatures, densities and observables. In particular, the large primordial helium mass fraction and residual deuterium are natural outputs of Big Bang nucleosynthesis, rather than arbitrary products of subsequent star formation.

The cosmological lithium problem prevents claiming that the simplest abundance calculation explains every observation without qualification. One must distinguish nuclear-rate uncertainties, stellar depletion, atmosphere and abundance systematics, and possible new particle physics. Thus the broad hot Big Bang framework is strongly supported, while lithium tests the completeness of its simplest nucleosynthesis and stellar-abundance interpretation. An unresolved lithium discrepancy alone does not erase the independent successes of deuterium, helium and the cosmic microwave background.