2. Light element abundances

The best evidence that the expansion and cooling of the universe traces back to high redshift is the success of the standard model for the origin of deuterium and isotopes of helium and lithium, by reactions among radiation, leptons, and atomic nuclei as the universe expands and cools through temperature T ~ 1 MeV at redshift z ~ 10¹⁰. The free parameter in the standard model is the present baryon number density. The model assumes the baryons are uniformly distributed at high redshift, so this parameter with the known present radiation temperature fixes the baryon number density as a function of temperature and the temperature as a function of time. The latter follows from the expansion rate Eq. (11), which at the epoch of light element formation may be written as

(61)

where the mass density _r counts radiation, which is now at T = 2.73 K, the associated neutrinos, and e^± pairs. The curvature and terms are unimportant, unless the dark energy mass density varies quite rapidly.

Independent analyses of the fit to the measured element abundances, corrected for synthesis and destruction in stars, by Burles, Nollett, and Turner (2001), and Cyburt, Fields, and Olive (2001), indicate

(62)

both at 95% confidence limits. Other analyses by Coc et al. (2002) and Thuan and Izotov (2002) result in ranges that lie between the two of Eq. (62). The difference in values may be a useful indication of remaining uncertainties; it is mostly a consequence of the choice of isotopes used to constrain _B0 h². Burles et al. (2001) use the deuterium abundance, Cyburt et al. (2001) favor the helium and lithium measurements, and the other two groups use other combinations of abundances.

The baryons observed at low redshift, in stars and gas, amount to (Fukugita, Hogan, and Peebles, 1998)

(63)

It is plausible that the difference between Eqs. (62) and (63) is in cool plasma, with temperature T ~ 100 eV, in groups of galaxies. It is difficult to observationally constrain the idea that there is a good deal more cool plasma in the large voids between the concentrations of galaxies. A more indirect but eventually more precise constraint on _B0, from the anisotropy of the 3 K thermal cosmic microwave background radiation, is discussed in test (11).

It is easy to imagine complications, such as inhomogeneous entropy per baryon, or in the physics of neutrinos; examples may be traced back through Abazajian, Fuller, and Patel (2001) and Giovannini, Keihänen, and Kurki-Suonio (2002). It seems difficult to imagine that a more complicated theory would reproduce the successful predictions of the simple model, but Nature fools us on occasion. Thus before concluding that the theory of the pre-stellar light element abundances is known, apart from the addition of decimal places to the cross sections, it is best to wait and see what advances in the physics of baryogenesis and of neutrinos teach us.

How is general relativity probed? The only part of the computation that depends specifically on this theory is the pressure term in the active gravitational mass density, in the expansion rate equation (8). If we did not have general relativity, a simple Newtonian picture might have led us to write down /a = -4 G _r / 3 instead of Eq. (8). With _r ~ 1/a⁴, as appropriate since most of the mass is fully relativistic at the redshifts of light element production, this would predict the expansion time a / is 2^1/2 times the standard expression (that from Eq. [61]). The larger expansion time would hold the neutron to proton number density ratio close to that at thermal equilibrium, n/p = e^-Q/kT, where Q is the difference between the neutron and proton masses, to lower temperature. It would also allow more time for free decay of the neutrons after thermal equilibrium is broken. Both effects decrease the final ⁴He abundance. The factor 2^1/2 increase in expansion time would reduce the helium abundance by mass to Y ~ 0.20. This is significantly less than what is observed in objects with the lowest heavy element abundances, and so seems to be ruled out (Steigman, 2002). ⁽⁷⁰⁾ That is, we have positive evidence for the relativistic expression for the active gravitational mass density at redshift z ~ 10¹⁰, a striking result.

⁷⁰ There is a long history of discussions of this probe of the expansion rate at the redshifts of light element production. The reduction of the helium abundance to Y ~ 0.2 if the expansion time is increased by the factor 2^1/2 is seen in Figs. 1 and 2 in Peebles (1966). Dicke (1968) introduced the constraint on evolution of the strength of the gravitational interaction; see Uzan (2002) for a recent review. The effect of the number of neutrino families on the expansion rate and hence the helium abundance is noted by Hoyle and Tayler (1964) and Shvartsman (1969). Steigman, Schramm, and Gunn (1977) discuss the importance of this effect as a test of cosmology and of the particle physics measures of the number of neutrino families. Back.