## Nucleosynthesis in a hot big bang

Hydrogen is the most abundant of the chemical elements (apart from places like Earth where the heavier elements have collected and condensed), helium amounts to about 25% by mass, and only about 2% of the baryon mass is in heavier elements. What produced this mix? In the 1930s people were

exploring two main lines of thought, that the chemical elements might have formed in stars or in the early universe. The former was suggested by the growing evidence that the Sun and other stars radiate energy released by the fusion of atomic nuclei into heavier nuclei. One could imagine that the heavy elements produced in stars by this nuclear burning were ejected by stellar winds or explosions and that the debris formed new stars and planets. The other picture assumes that temperatures and densities in the early stages of expansion of the universe were large enough to have forced nuclear reactions among atomic nuclei that produced a mix of elements (that might have been adjusted by what happened later in stars). The later well-tested theory combines these ideas: the heavier elements originated in stars while most of the helium is a fossil remnant of the hot big bang, along with the thermal CMBR.

We review here the main steps in the development of the hot big bang part of this theory. Alpher and Herman (2001) describe the history and present recollections of the introduction of main features of the concepts by them and their colleagues in the 1940s and 1950s. Kragh (1996, Chapter 3: Gamow's Big Bang) presents helpful details about the events and the people involved in the research. The essays in Chapter 4 add to the story of what happened later, in the 1960s.

The earliest discussions of element formation in the big bang picture considered the idea that the relative abundances of the chemical elements and their isotopes might have been determined by relaxation to thermal equilibrium at some hot early stage of expansion of the universe. The concept can be compared to that of the blackbody radiation discussed in Chapter 2. We remarked that at thermal equilibrium the intensity of the radiation at each wavelength is determined by just one quantity, the temperature. At equilibrium the relative abundances of the elements and their isotopes would be fixed by two quantities, the temperature and the density of matter. The analysis by von Weizacker (1938) showed that the situation has to be at least a little more complicated than that. He found that a rough fit to the observed pattern of abundances of the elements would follow if particle reactions generally ceased to be important - the pattern of element abundances were close to "frozen in" - when the expanding universe had cooled to a temperature of about T ~ 2 x 1011 K, but that residual reactions after that would have to have shaped the variation of the abundance from one atomic weight to the next in the middle part of the periodic table to that characteristic of a lower temperature, T ~ 5 x 109 K. Chandrasekhar and Henrich (1942) repeated the analysis. Their more complete computations based on better data for the nuclear physics and element abundances indicated that the abundances of the heavier elements would had to have been frozen at about von Weizacker's higher temperature, while the abundances of the lighter elements were determined by an approach to thermal equilibrium later, at about von Weizacker's lower temperature. For our purpose the important thoughts are that the early universe might have been hot, expanded and cooled at a rate characteristic of thermonuclear reactions, and left an interesting variety of chemical elements.

The physicist George Gamow took the leading role in improving these thoughts. Gamow (1942) and Gamow and Fleming (1942) argued that the picture of near thermal equilibrium seems less plausible than a distinctly nonequilibrium process in an expanding universe. That might involve a "rapid breaking-up of the original superdense nuclear matter... Even in ordinary uranium-fission a number of free neutrons are being emitted in each breaking-up process, and this number most probably increases in the case of the more violent fission of superheavy nuclei. Neutrons produced this way will turn spontaneously into protons, and will contribute to a larger abundance of hydrogen" (Gamow 1942).

All these papers assumed we live in an expanding, evolving universe (in von Weizacker's case one that is finite but large enough to include the most distant observed galaxies) but did not explicitly take account of the relativis-tic theory for the rate of expansion. Gamow (1946) took this important step. He remarked that in the early stages of an expanding universe the mass density would be large, and that would make the rate of expansion rapid:1 "we see that the conditions necessary for rapid nuclear reactions were existing only for a very short time, so that it may be quite dangerous to speak about an equilibrium state" of the kind people had considered earlier. Gamow (1946) also noted that the positive electric charges of atomic nuclei tend to slow their fusion by pushing the nuclei apart, while the free neutrons he had mentioned earlier (Gamow 1942), which have no electric charge, react rapidly with protons and heavier atomic nuclei. That is wanted for a rapid build-up of the elements. One notices a roughly parallel development of ideas here and in the nuclear weapons program: both were thinking about neutron production and capture (a point Smirnov elaborates beginning on page 92).

Gamow's (1946) proposal was that the heavy elements were built by the "coagulation" of neutrons followed by nuclear beta decays that convert neutrons to protons (the decays being accompanied by the emission of electrons, or what is known as "beta" or "ft" radiation). Ralph Alpher, who was Gamow's graduate student, made the coagulation idea more specific. In his doctoral dissertation (at The George Washington University, Alpher 1948a) the proposal is that the elements were built up by sequences of radiative captures of neutrons (that is, neutron capture accompanied by the emission of a photon, a quantum of electromagnetic radiation) and nuclear beta decays. In a preliminary report of this building-up idea, by Alpher, Bethe and Gamow (1948), Hans Bethe's name was added to produce an approximation to the first three letters of the Greek alphabet.

1 The expansion rate has to be large to escape the strong gravitational attraction of the large mass density. One sees from equation (G.1) on page 518 that when the mass density is really large its value fixes the expansion rate, because the mass density term is by far the largest in the right-hand side of this equation. If the mass density is dominated by matter with relatively low pressure then when the density has dropped to the value p the model universe has been expanding for the time t = 890p-1/2 s, (3.1)

where the value of p is measured in gcm-3. If the mass density is dominated by radiation the time is three quarters of the value in this equation. The shorter time follows because the mass density in radiation falls more rapidly with the expansion of the universe than does the mass density in matter (for the reason in footnote 9 on page 29), and a larger earlier density requires a greater expansion rate to escape the stronger gravitational pull.

Alpher (1948a) and Alpher, Bethe and Gamow (1948) could cite a piece of evidence for their picture. They pointed out that in the building-up process the more readily an atomic nucleus of given mass and charge can absorb a neutron, to be promoted to a larger mass, the lower the expected cosmic abundance of that species of nuclei. And from measured nuclear reaction rates they concluded that if the elements were built up by exposure to a gas of neutrons that is hot - moving with velocities characteristic of a temperature of about 109 K - it would produce relative abundances of the elements that suggest a promising match to what is observed.2 This encouraging result is still seen to apply in part of the periodic table, and the interpretation is the neutron capture building-up process, but transferred from the hot big bang to exploding stars, in what has become known as the "r-process." And basic parts of this building-up picture figure in the now well-tested theory for the origin of the lightest elements - the isotopes of hydrogen and helium - in the hot big bang. This was a memorable advance. But we must consider the introduction of several other important ideas.

Alpher (1948a,b) pointed out a problem with the theory. We remarked in footnote 1 that the relativistic big bang cosmology sets a relation between the mass density and the expansion rate. The condition that the rate of capture of neutrons produces a significant but not excessive amount of heavier elements sets another relation between the mass density and the expansion rate when the building-up reactions were at their peak. And there is a third condition, that the process must be completed in a few hundred seconds, before the neutrons have decayed.3 Alpher showed that it is not possible to satisfy these three relations by the choice of two quantities, the characteristic matter density and characteristic expansion time when the building-up process occurred.4

2 Alpher pointed out that if the velocities were much smaller than this then large rates of resonance capture by some nuclei would spoil the anticorrelation of element abundances and neutron absorption cross sections; if much larger the radiation would tend to break up the nuclei.

3 In these exploratory discussions Gamow and Alpher generally left open the origin of the neutrons, and more broadly what was happening prior to the build-up of the elements. Alpher, Bethe and Gamow (1948) did note that the density of the universe might never have exceeded the density at element build-up, "which can possibly be understood if we use the new type of cosmological solution involving the angular momentum of the expanding universe (spinning universe)." On the origin of the neutrons, Gamow (1942) remarked that the fission of uranium produces free neutrons, and that the "rapid breaking up of the original superdense nuclear matter" might produce a large proportion of them. The simpler solution soon recognized is that if the universe had expanded from a very high temperature then heavy nuclei would have thermally evaporated and free neutrons would have been produced by the thermonuclear reactions to be discussed.

4 In a little more detail the problem Alpher identified goes as follows. The condition that the build-up process produces an interesting but not excessive heavy element abundance is that the product avnt is of order unity. Here a is the radiative neutron capture cross section, v is the relative velocity of neutron and nucleus, n is the number density of nuclei, and t is

Alpher (1948a) also pointed to an effect that was missing in the calculation, which turned out to be the solution to his problem. The neutron gas was supposed to be hot (for the reason indicated in footnote 2), the hot matter ought to be accompanied by thermal blackbody radiation at the same temperature, and the mass density in this radiation would be much larger than the mass density in matter. But it was Gamow (1948a, with more detail in Gamow 1948b) who put these points together and solved the problem:5 the large mass density in radiation speeds the expansion of the early universe so the build-up process can happen before the neutrons decay. This is the first analysis of the modern picture of the role of thermal radiation in element formation in the early universe.6

Gamow's argument begins with the thermal blackbody radiation present in a hot big bang. There would be a time, early enough in the expansion, when the temperature was high enough that the radiation would evaporate the atomic nuclei of any heavy elements, producing a gas of free protons and neutrons. As the universe expanded and cooled heavier elements could start to form in appreciable amounts, starting with captures of neutrons by protons to make deuterons (the nuclei of the stable heavy isotope of hydrogen). Each capture would be accompanied by the release of a photon (a quantum of electromagnetic radiation; at this energy usually written as Y) in the reaction7

the expansion time, all evaluated during the build-up process. The measurements in nuclear reactors by Hughes (1946), as summarized by Alpher (1948a), indicate that the product av is not very sensitive to v, and for the lightest nuclei amounts to av ~ 10~19 cm3 s_1. If the mass density is dominated by baryons then equation (3.1) gives nt2 ~ 1030 cm 3 s2. It would follow from these two relations that build-up had to have occurred when the universe had been expanding for about t ~ 103 years. But that is absurdly large compared to the neutron lifetime, about 15 minutes.

5 The Alpher, Bethe, and Gamow paper, which was submitted for publication on February 18, 1948, did not take note of the problem. They instead wrote that it "is necessary to assume" a much larger value of nt, that allowed the build-up process to happen at t ~ 20 s, well within the neutron lifetime. Alpher's thesis, which was accepted in April 1948, has a reasonable value of nt from the point of view of nuclear physics, and the consequent problem with the neutron lifetime. Gamow's paper, submitted on June 21, 1948, does not mention the problem but it presents the solution: take account of the mass density in radiation, which considerably increases the expansion rate. The published version of Alpher's dissertation, submitted July 2, 1948, states the problem but not Gamow's solution. It can take time to straighten out ideas.

6 In the physical situation assumed in the calculations by von Weizacker (1938) and Chan-drasekhar and Henrich (1942) it is implicitly assumed that the baryons, being at near thermal equilibrium, are in a sea of blackbody radiation at the same temperature. We have found no one who took notice of the consequences of the presence of the mass density in this radiation prior to Gamow (1948a).

7 Gamow's nonequilibrium calculation for this reaction assumes thermal equilibrium for the radiation but not for the relative abundances of the atomic nuclei. This is consistent: the interactions between radiation and electrons are fast enough to guarantee that the radiation remains very close to thermal, while the reaction in equation (3.2) is slow enough to break equilibrium.

The two-headed arrow means the reaction can go either way: a sufficiently energetic photon can break up a deuteron. Gamow noted that the critical temperature for the survival of deuterons, and hence the accumulation of appreciable numbers of them, is

At higher temperatures radiation breaks up deuterium as fast as it forms. When the temperature has fallen below Tcrit the dissociation reaction going from right to left in equation (3.2) markedly slows because the cooler radiation does not have many photons energetic enough to break apart deuterons. This means deuterium starts to accumulate. As it does the deuterium can rapidly burn to helium by particle exchange reactions.8

It is essential for the consistency of this picture that at the critical temperature Tcrit in equation (3.3) the total mass density is dominated by the energy of the thermal radiation that would accompany the hot plasma.9 We know the radiation temperature, Tcrit, when deuterons can start accumulating. The temperature tells us the mass density in radiation, which has to be very close to the total mass density. As we have noted, the mass density sets the expansion rate. The expansion time, shortened by the mass density in radiation, turns out to be comfortably less than the neutron lifetime, so neutrons could be available for the deuterium-producing reaction

8 The most important reactions are d + d « 3He + n, d + d « t + p, 3He + n « t + p, t + d « 4He + n, (3.4)

where tritium (t) is the unstable isotope of hydrogen that contains two neutrons. The reaction in equation (3.2) is slower, in conditions of interest here, because the electromagnetic interaction is weaker. That means the rate of equation (3.2) controls the rate of formation of helium by these reactions.

9 At the present epoch the mass density in radiation is smaller than in matter, as is indicated in Table 2.1, but at the time of light element formation the mass in radiation was the largest component. This is because the energy of each photon, and its equivalent in mass, decreases as the universe expands, an effect of the cosmological redshift. The wavelength of a CMBR photon is increasing, as A « a(t), where a(t) is the expansion parameter in equation (2.3) (and in equation 2.4 for the density and equation 2.7 for the temperature). Thus the photon energy is decreasing as e = hv = hc/A « a(t)-1, where c is the velocity of light and h is Planck's constant. The number densities of baryons and photons decrease as the volume of the universe increases, in proportion to a(t)-3. Putting this together, we see that the mass densities in baryons and radiation vary as

as the universe expands. The middle part of the second expression is inserted as a reminder of a general relation: the energy density in thermal radiation at temperature T is u = aT4, where a is Stefan's constant (not to be confused with the expansion parameter a(t)). When the temperature had fallen to Tcr;t (equation 3.3) the mass density would have been dominated by the radiation. This mass density sets the time elapsed, tcrit ~ 230 s (from equation 3.1), from a really hot beginning to T = Tcrit.

in equation (3.2). The matter density when the temperature has fallen to Tcrit dictates how much deuterium could accumulate.

The density of neutrons and protons would have to have been large enough to allow production of an appreciable amount of deuterium, but not so much that it converts more hydrogen into heavier elements than is observed.10 This consideration led Gamow to conclude that, if the neutron capture picture for element formation were right, then when the temperature in the early expanding universe had fallen to Tcrit the number density of baryons -neutrons and protons - would have to have been ncrit — 1018 cm-3.

Alpher and Herman (1948) took the next bold step: from the conditions required for element production in the early universe, predict the present temperature of the fossil radiation left from the hot early universe. When the temperature was Tcrit, and elements heavier than hydrogen could start accumulating, Gamow had found an estimate of the baryon mass density ncrit that would allow production of a reasonable abundance of the heavier elements. The mass density and temperature drop as the universe expands, in the proportion n <x T3 (as one sees in equation 3.5). This means that when the temperature drops by a factor of 10 the mass density in matter drops by a factor of 1000. And one can similarly compute the temperature when n has dropped to its present value. Alpher and Herman found that, at present matter density,11

the radiation temperature would be

In view of all the uncertainties this is strikingly close to what was measured many years later, T0 = 2.725 K.

One should not take the consistency of numerical values for the theory and measurement of T0 too seriously, because there were problems with estimates of the mass density in equation (3.6). It was later learned that an error in the early estimates of the scale of distances to the galaxies introduced a

10 As in the discussion in footnote 4, the Gamow condition is expressed as Ccritn-crifVcriticrit ~ 1, where a"crit is the radiative capture cross section for equation (3.2), vcrit is the relative neutron—proton velocity, and racrit and icrit are the baryon number density and the expansion time, all evaluated when the temperature is T = Tcrit.

11 Alpher and Herman (1948) did not state this quantity. It is the mass density the group used in other papers, including Gamow (1946), Alpher (1948a,b), and Alpher and Herman (1949). Within rounding error it agrees with the indication in Alpher and Herman (1948) that the mass densities in matter and radiation are equal at T = 600 K. There is the complication that their reported application of the Gamow condition yields a matter density at Tcrit that extrapolates to To = 20 K at mass density 10-30 gcm-3, not To = 5 K. Either there is an error in the paper or they adopted a present mass density well below the value they and Gamow generally used.

considerable overestimate of all the mass density measurements.12 Also, it was known at the time that different methods yield quite different values for the mass density.13 The point of lasting value is that in the hot big bang cosmology there is a relation between measurable conditions in the universe as it is now and conditions in the early universe when light elements could have been produced. The details of this relation have since been refined, as will be described, but this consideration by Alpher and Herman remains part of our standard cosmology.

There are several names to describe physical conditions in the early expanding universe, at the epoch of light element formation. In the published version of his thesis, Alpher (1948b) offered this opinion:

According to Webster's New International Dictionary, 2nd Ed., the word "ylem" is an obsolete noun meaning "The primordial substance from which the elements were formed." It seems highly desirable that a word of so appropriate a meaning be resurrected.

Alpher's ylem left us fossils, the light elements and the thermal cosmic background radiation (CMBR).

Gamow recognized that the thermal radiation in the ylem would be present after the early episode of element formation, and it would remain an important dynamical actor.14 Alpher and Herman (1948) went further: they clearly stated that the thermal radiation would be present now. And

12 Hubble's distance estimates were low by a factor of about 7.6. A mass density estimate from that time, corrected for the distance scale while leaving all other data unchanged, should be divided by the factor 7.62, which would divide the predicted present temperature of the CMBR by the factor 7.62/3 ~ 4.

13 The range of estimates of the mass density is an early indication of the dark matter problem. Hubble (1936) reported that the mean mass density is no less than about pmin = 1 X 10_30 gcm~3 and may be as large as pmax = 1 x 10_28 gcm~3. Gamow and colleagues used the lower value; Alpher (1948a) attributes it to Hubble (1936, 1937). It corresponds to the density parameter (defined in equation G.1) Qm¡n = 0.002. This number, which is independent of the distance scale, is comparable to the mass density in stars entered in Table 2.1. That makes sense, because Hubble's lower mass density used observations of the masses in the luminous parts of individual galaxies, which are dominated by the mass in stars and include most of the stars. Hubble's larger estimate, pmax, corresponds to density parameter Qmax = 0.2. It is comparable to the total mass density in matter entered in the table. This also makes sense, because Hubble based it on the mass per galaxy in clusters of galaxies, which he attributed to Smith (1936) (though Zwicky (1933) had made the point earlier). We know that clusters contain a close to fair sample of baryonic and dark matter, so the cluster mass per galaxy multiplied by the number density of galaxies gives a pretty good measure of the cosmic mean mass density. If Alpher and Herman had used pmax, it would have increased their estimate of To by the factor ~ which happens to about cancel the effect of the distance error. Let us notice, however, that the Gamow condition relates the CMBR temperature to the baryon mass density. The baryon mass is smaller than the total represented by Hubble's pmax, which is dominated by nonbaryonic dark matter, and larger than Hubble's pmin, which does not include the plasma in and around groups and clusters of galaxies.

14 The paper Gamow (1948a) presents an estimate of the time — well after element formation — when the mass densities in matter and radiation were equal and, as Gamow recognized, the expanding universe became unstable to the gravitational growth of nonrelativistic concentrations of matter that eventually became galaxies.

in subsequent papers (Alpher and Herman 1949, 1950), they converted the present temperature to the present mass density in radiation, producing the first estimates of the third quantity in Table 2.1. Less easy to judge is whether they saw indications of the experimental methods to be described later in this chapter that might have been capable of detecting the radiation. Burke, on page 182, gives an assessment of the experimental situation.

The next important refinement in the theory is the process that fixes the relative number n/p of the neutrons and protons that enter the first step of element-building in equation (3.2). In the paper Gamow (1948a) the value of n/p is left open: Gamow was content to establish orders of magnitude. Hayashi (1950), on the other hand, recognized that in this hot big bang cosmology n/p may be computed from well-determined physics, as follows.

When the temperature in the early stages of expansion of the universe was above about 1010 K (and, by the argument in footnote 1 on page 26, the large mass density of the radiation caused the universe to have been expanding for about a second), the radiation was hot enough to produce a thermal sea of electrons and their antiparticles, positrons, and a sea of neutrinos and antineutrinos (v and v), mainly by the reactions

These particles convert protons to neutrons and back again by the reactions p + e- ^ n + v, n + e+ ^ p + v, n ^ p + e- + v. (3.9)

At temperatures above 1010 K the reactions drive the ratio n/p of numbers of neutrons and protons to its thermal equilibrium value, n = e-Q/kT, (3.10)

p at temperature T.15 Here Q = (mn — mp)c2, where mn — mp is the difference of mass of a neutron and of a proton, and k is the Boltzmann constant.

Hayashi found that as the universe expanded and cooled below 1010 K the reactions in equation (3.9) slowed to the point that the value of n/p froze,

15 To be more accurate, we should note that the equilibrium value of n/p also depends on the lepton number, which is the sum of the numbers of e— and v particles minus the sum of the numbers of e+ and v. The reactions in equations (3.8) and (3.9) do not change the lepton number: its value had to have been set by initial conditions very early in the expansion of the universe. Equation (3.10) assumes the absolute value of the lepton number density is small compared to the number density of CMBR photons. A positive and large lepton number suppresses n/p, and a strongly negative lepton number increases n/p. This point figures in the cold big bang model we discuss beginning on page 35. The present observational constraints are consistent with the small lepton number assumed in equation (3.10). To be even more accurate we should take notice of the three families of neutrinos, but that does not figure in the history in this chapter.

and then n/p more slowly decreased as neutrons freely decayed to protons (by the last reaction in equation 3.9 going to the right). By the time the temperature had dropped to Tcrit the ratio of neutrons to protons (n/p) would have fallen to ~0.2. In the standard cosmology most of the neutrons present at this time combined with protons to form deuterons, and most of the deuterium burned to the heavy isotope of helium, 4He, with a trace amount of the lighter isotope 3He.

The paper Alpher, Follin and Herman (1953) presents a detailed application of Hayashi's idea. Their analysis of how the ratio n/p varies as the universe expands and cools is essentially the modern computation. Enrico Fermi and Anthony Turkevich (in work that is not published but is reported in Gamow (1949), ter Haar (1950), and in more detail in Alpher and Herman 1950, 1953) worked out the chains of particle exchange reactions that burn deuterium along with neutrons and protons to helium and trace amounts of heavier elements. These analyses essentially completed the formulation of all the pieces of what was much later established as the standard model for the origin of most of the isotopes of hydrogen and helium, along with the CMBR.

We can reconsider now the question in footnote 3 on page 27: what was the nature of the universe before the build-up of the light elements? The theory just described assumes the expansion traces back to temperatures above 1010 K, when the distributions of the radiation and the baryons are supposed to have been very close to spatially uniform - homogeneous even on small scales. The baryon density at that epoch is chosen to fit the observed light abundances. That is one way to determine the value at the present epoch listed in Table 2.1. (Another way is examined in Chapter 5.) The baryons have to have been created, but that is assumed to have happened still earlier, at much higher temperatures than we are considering. General relativity theory gives the rate of expansion of the universe. The early expansion is rapid, but at temperature 1010 K the exchanges of energy among particles and radiation are even faster. This means conditions then would have forced relaxation to thermal equilibrium, including the thermal ratio of neutrons to protons. Thus for the purpose of the theory of light element formation that commences at T ~ 109 K we need only these assumptions: we need not enquire about conditions at still earlier times. The question is fascinating, of course, and there are ideas: a favorite is the inflation picture (described in Guth 1997 and outlined on page 520). But for the story of the CMBR we need not consider the weight of evidence of whether inflation is a useful approximation to what actually happened in the exceedingly early universe.

We are very interested in the development of the empirical checks of these ideas on how the CMBR got its thermal spectrum (shown on page 16), and how the light elements formed. In the 1960s it was not at all obvious whether we would be able to find convincing tests, or, if we did, whether these ideas would pass the tests. People felt free - perhaps even compelled -to cast about for other ideas that might be philosophically attractive and perhaps better approximate reality. The debate over these alternatives is an important part of the story of how we arrived at the standard cosmology. We turn now to some of the ideas.

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