REVIEW ON EFFECTS OF LONG-LIVED NEGATIVELY CHARGED MASSIVE PARTICLES ON BIG BANG NUCLEOSYNTHESIS

Review on Effects of Long-Lived Negatively Charged Massive Particles on Big Bang Nucleosynthesis

Motohiko KusakabeaaaJSPS Postdoctoral Fellow for Research Abroad, Grant. J. Mathews    Toshitaka Kajino    Myung-Ki Cheoun
Abstract

We review important reactions in the big bang nucleosynthesis (BBN) model involving a long-lived negatively charged massive particle, , which is much heavier than nucleons. This model can explain the observed Li abundances of metal-poor stars, and predicts a primordial Be abundance that is larger than the standard BBN prediction. In the BBN epoch, nuclei recombine with the particle. Because of the heavy mass, the atomic size of bound states is as small as the nuclear size. The nonresonant recombination rates are then dominated by the -wave 2P transition for Li and Be. The Be destruction occurs via a recombination with the followed by a proton capture, and the primordial Li abundance is reduced. Also, the Be production occurs via the recombination of Li and followed by deuteron capture. The initial abundance and the lifetime of the particles are constrained from a BBN reaction network calculation. We estimate that the derived parameter region for the Li reduction is allowed in supersymmetric or Kaluza-Klein (KK) models. We find that either the selectron, smuon, KK electron or KK muon could be candidates for the with TeV, while the stau and KK tau cannot.

Keywords: Negatively charged massive particle; Big Bang Nucleosynthesis; Cosmology.

Received 4 July 2015

Revised 12 April 2016

PACS Nos.: 26.35.+c, 95.35.+d, 98.80.Cq, 98.80.Es

1 Introduction

The primordial light element abundances calculated in standard big bang nucleosynthesis (SBBN) are more or less consistent with those inferred from astronomical observations. A large discrepancy, however, exists in the primordial Li abundance. Spectroscopic measurements of metal-poor stars (MPSs) indicates a roughly constant abundance ratio, Li/H, as a function of metallicity. However, the theoretical primordial abundance in the SBBN model is larger than the observational value by about a factor of (e.g., Refs. ?, ?) when the baryon-to-photon ratio in the CDM model is taken from observations of the cosmic microwave background radiation by the Wilkinson Microwave Anisotropy Probe or the Planck observatory). This discrepancy suggests that unknown physics is present to reduce the Li abundance during or after BBN. The early Galaxy might have had a Li abundance smaller than the cosmic average value because of a chemical separation induced by the primordial magnetic field. Rotationally induced mixing, and a combination of atomic and turbulent diffusion might have reduced the Li abundance in stellar atmospheres. As another possibility, an exotic particle might have existed during big bang nucleosynthesis (BBN), and affected the primordial abundances.

A late-decaying negatively charged massive particle (CHAMPs or Cahn-Glashow particles) has been considered as a solution to the Li problem.. Constraints on supersymmetric models have also been derived through BBN calculations. Long-lived CHAMPs have been searched for in collider experiments. No signature of an has been observed, and limits on the mass have been placed by measurements at the Large Hadron Collider. Lower limits on the mass of the scalar leptons (staus) are typically several hundred GeV, and depend on parameters of particle models.

The particles and nuclei can form bound atomic systems ( or -nuclei) with binding energies  MeV in the limit that the mass of , , is much larger than the nucleon mass. The -nuclei are exotic chemical species with very heavy masses and chemical properties similar to normal atoms and ions. The present existence of the superheavy stable (long-lived) particles have been searched for in experiments.

Table 1 shows constraints on the abundances of -nuclei derived from experiment. The first column shows upper limits on the number ratio of to nucleons. The second column shows the element composing the sample used in the experiment, and the third column shows the mass region where the derived constraint is applicable. The fourth column shows the reference for the experiment.

Table 1: Experimental constraints on CHAMP.

  sample element mass of Ref. H GeV H GeV H GeV Na GeV C GeV Pb GeV   H Li Be B GeV C O F  

If the particle exists during the BBN epoch, it binds with nuclei and triggers nuclear reactions. The formation of most -nuclei proceeds through radiative recombination of nuclides and . Importantly, the Be formation proceeds also through the non-radiative Be charge exchange reaction between a Be ion and an .

The Li abundance can significantly increase through the -catalyzed transfer reaction He(, )Li, where 1(23)4 signifies a reaction .

A relatively weak destruction of Be can also proceed, and the primordial Li abundance bbbBe produced during the BBN is transformed into Li by electron capture in the epoch of the recombination of Be and electron much later than the BBN epoch. The primordial Li abundance is, therefore, the sum of abundances of Li and Be produced in BBN. In SBBN with the baryon-to-photon ratio inferred from the Planck, Li is produced predominantly as Be during the BBN. is reduced. Bound states of B and include atomic states composed of nuclear ground and excited states of B. In the reaction Be(,)B, the first atomic excited state of B, and the atomic ground state of B(,0.770 MeV) consisting of the nuclear excited state of B and an , work as resonances that reduce Be abundance.

Effects on other nuclear abundances have been investigated in a large reaction network calculation for typical values of the abundance. Especially, productions of nuclei with mass number via several reactions including Be+ BB+ through the B atomic excited state of B were studied. No effect was, however, found in the abundances of nuclei with .

The resonant reaction Be(, )Be through the atomic ground state of Be(, 1.684 MeV) has been found to be nonexistent since the state Be(, 1.684 MeV) is not a resonance but a bound state located below the Be+ separation channel. This has been confirmed by a four-body calculation for an system and another three-body calculation.

The most important reaction of Be production has been found to be Li(, )Be. This reaction is the key reaction since a signature of the particle is left on primordial Be abundance through this reaction. This model can, therefore, be tested by observations of Be abundances in MPSs in the future. Therefore, realistic calculations of the reaction rate with quantum many-body models are needed.

In this paper we review the BBN model with a CHAMP, based upon our recent extensive study. We then estimate the possibility that the particle which affects the primordial Li abundance corresponds to particles in supersymmetric or extra-dimensional models. In Sec. 2, the adopted model for the nuclear charge density and the calculation of binding energies of the -nuclei are explained. In Sec. 3, the radiative recombination of light nuclides with the particles are reviewed. In Sec. 4, a calculation of the radiative proton capture reactions Be(, )B is shown. In Sec. 5, a reaction for Be production is described. In Sec. 6, our reaction network calculation is described. In Sec. 7, the latest observational constraints on the primordial nuclear abundances are described. In Sec. 8, we show the evolution of elemental abundances as a function of cosmic temperature, and derive updated constraints on the initial abundance and the lifetime of the . In Sec. 9, constraints on the particle for the reduction of the Li abundance are discussed. In Sec. 10 we summarize this review.

Throughout the paper, we use natural units, , for the reduced Planck constant , the speed of light , and the Boltzmann constant . We use the usual notation 1(23)4 for a reaction .

2 Model

2.1 Nuclear Charge Density

We assume that a CHAMP with a single negative charge and spin zero, , exists during the BBN epoch. The mass of the , i.e., is set to be GeV.

In this review, we take the standard case of Ref. ?. It is assumed that the nuclear charge density is given by a Woods-Saxon shape,

(1)

where is the distance from the center of mass of the nucleus, is the charge of the nucleus, is the parameter characterizing the nuclear size, is nuclear surface diffuseness, and is a normalization constant. The value is fixed by the equation of charge conservation, , and it is given by

(2)

For a given value of diffuseness , the size parameter can be constrained so that the parameter set of (, ) satisfies the root-mean-square (RMS) charge radius measured in nuclear experiments. The value of fm is chosen.

The potential between an and a nucleus ( potential) is calculated by folding the Coulomb potential with the charge density:

(3)

where is the position vector from an to the center of mass of , is the position vector from the center of mass of , is the displacement vector between the and the position, and is the charge density of the nucleus. Under the assumption of a Woods-Saxon charge distribution , the potential reduces to the form

See Ref. ? for a discussion of the dependences of binding energies, reaction rates, and BBN on , the nuclear charge density, and experimental uncertainties in the RMS charge radii.

2.2 Binding Energy

Binding energies and wave functions for bound states of nuclei , i.e., -nuclei or , have been calculated using both numerical integrations of the Schrdinger equation with RADCAP and variational calculations with the Gaussian expansion method.

The reduced mass of the system is given by

(5)

where is the mass of nuclide . In the limit of a heavy particle, i.e,. , . The binding energies, therefore, become independent of .

3 Radiative Recombination With

The recombination reactions of Be, Li, Be, and He with are important particularly, as seen in the BBN network calculation of Sec. 8. The recombination rates for these four nuclides are therefore, precisely calculated. Rates for other nuclides are, on the other hand, approximately given (Sec. 3.5).

3.1 Be

3.1.1 Energy Levels

Binding energies of Be atomic states with main quantum numbers ranging from one to seven have been calculated. Binding energies in the case of two point charges are given analytically by , where is the fine structure constant. Since the Be nuclear charge distribution has a finite size, the amplitude of the Coulomb potential at small is less than that for two point charges. Wave functions at small radii and binding energies of tightly bound states with small values, therefore, deviate from those of the Bohr model. On the other hand, the binding energies of loosely bound states with large values are similar to those of the Bohr model.

Figure 1 shows the energy level diagram of Be in the case of GeV. The latest calculation indicates that the nonresonant rate of the recombination of Be and is larger than the resonant rate. Red and blue arrows show the three important transitions in the nonresonant recombination reaction (Sec. 3.1.2), while purple and green arrows show decays of the two important resonances into bound states of Be in the resonant reaction (Sec. 3.1.1).

Fig. 1: Energy level diagram of Be in the case of GeV. Red and blue arrows show the three important transitions in the nonresonant recombination reaction of Be(, )Be, while purple and green arrows show decays of the two important resonances into bound states of Be in the resonant reaction.

3.1.2 Be(, )Be Resonant Rate

The resonant rate of the reaction Be(, )Be has been calculated taking into account the change of the E1 effective charge as a function of . The E1 effective charge is given by

(6)

where and are the mass and the charge number of species and 2.

The recombination of Be and Li with proceed via resonant reactions through atomic states , composed of a nuclear excited state and an . There are an infinite number of atomic states of Be, composed of the first nuclear excited state Be[Be(0.429 MeV, )] and an . However, the resonant reaction rate is suppressed by a factor of with the resonant energy [cf. Eq. (20)]. Therefore, only resonances whose energy levels are close to that of the entrance channel Be+ are important. In the case of GeV, important resonances are then the 3P and the 3D states.

The thermal reaction rate is derived as a function of temperature by numerically integrating the cross section over energy,

(7)

where is the center of mass kinetic energy, and is the reaction cross section as a function of .

The resonant rate is derived to be

(8)

The first term corresponds to the atomic transition from the resonance Be(3D) to Be(2P), while the second term corresponds to sums of the atomic transitions from the resonance Be(2P) to Be(2S) and Be(1S).

3.1.3 Be(, )Be Nonresonant Rate

The nonresonant rate for the reaction Be(, )Be in the temperature region of has been calculated and fitted to be

(9)

The nucleosynthesis for He and heavier nuclei as well as nuclear recombinations with proceed after the temperature of the universe decreases to . The reaction rates for higher temperatures are, therefore, not necessary in BBN calculations.

Nonresonant cross sections have been calculated with RADCAP taking into account the multiple components of partial waves for scattering states. We show continuum wave functions at the CM energy MeV, which is the average energy corresponding to the temperature of the recombination of Be+ for the case of GeV, i.e., with K.

The total cross section for the absorption of an unpolarized photon with frequency via an E1 transition from a bound state (, ) to a continuum state () is given by

where is the wave number, and

(11)

is the dipole radial matrix element for the radius , and wave functions are normalized as

(12)

and asymptotically at large

(13)

where is defined by

(14)

with the Bohr radius, is the Coulomb phase shift, and is the phase shift due to the difference in Coulomb potential between cases of the point charge and finite size nuclei. The important point is that we must use exact values of the reduced mass [Eq. (5)] and the effective charge of E1 transitions [Eq. (6)] compared to the case of hydrogen-like electronic ions.

The calculated cross sections are compared with those for the recombination of two point charges. For a system of two point charges, wave functions of scattering and bound states, and the bound-free absorption cross section have been derived analytically. Again, appropriate values of and are to be used in the analytic formulae for the present system.

Figure 2 shows bound-state wave functions (upper panel) and continuum wave functions (middle panel) at MeV for the Be+ system as a function of radius for the case of GeV. Thick solid lines correspond to calculated wave functions. Thin solid lines show calculated wave functions for the case of point-charge nuclei, and dotted lines correspond to the analytic formula for point-charge nuclei. In fact, dotted lines overlap thin solid lines almost completely and no difference is seen. In the upper panel, wave functions for the bound GS (1S state), 2S, 2P, 3P, 3D, and 4F states are plotted. The wave functions for the GS and 2S state in the finite charge distribution case (thick lines) significantly deviate from those of the point charge case (thin lines). While a certain degrees of deviations are seen in wave functions of bound 2P and 3P states also, the wave functions of 3D and 4F states agree with those for the point charge case. The scattering wave functions for the -, -, -, and -waves are plotted in the middle panel. The wave functions of the and states for the finite charge distribution case deviates from those of the point charge case.

Fig. 2: Bound-state wave functions (upper panel) and continuum wave functions at MeV (middle panel) for the Be+ system as a function of radius for the case of GeV. The bottom panel shows the recombination cross section as a function of CM energy . The thick solid lines correspond to calculated results for the case of a finite size Be charge distribution. The thin solid lines in the upper and middle panels show calculated results for the point charge Be. The dotted lines in the bottom panel correspond to analytic formulae for hydrogen-like atomic states composed of two point charges. This figure is reprinted from Ref. ?.

The bottom panel shows the recombination cross section as a function of the energy . Solid lines correspond to the calculated results, while the dotted lines correspond to the analytic solution for the two point charges. Partial cross sections for the following transitions are drawn: scattering -wave bound 1S state (black lines); -wave 2S (red); -wave 2P (green); -wave 2P (blue); -wave 3P (gray); -wave 3P (sky blue); -wave 3D (orange); -wave 3D (cyan); -wave 4F (violet); and -wave 4F (magenta).

Because of the large difference in the scattering -wave function, the cross sections for transitions from an initial -wave, i.e., -wave 2P and -wave 3P are much smaller than those in the point charge case. Partial cross sections for transitions from an initial -wave to bound 1S, 2S and 3D states are also altered by the finite-size charge distribution. The cross sections for transitions to 1S and 2S states are affected additionally by differences in binding energies of the states between the finite- and point-charge cases.

An important characteristic of the Be+ recombination has been found.: In the limit of a heavy particle, i.e., GeV, the most important transition in the nonresonant recombination is the -wave 2P. This is different from the case of the point charge model, where the transition -wave 1S is predominant (see dotted lines). In the case of a finite size charge distribution, in addition to the main pathway of -wave 2P, cross sections for the transitions -wave 3D and -wave 3P are also larger than that for the GS formation.

The latest rate is more than 6 times larger than the previous rate. This large difference is caused since the most important transition from the scattering -wave to the bound 2P state, and many other transitions are taken into account in the latest calculation.

3.2 Li

Figure 3 shows the energy level diagram of Li in the case of GeV. The nonresonant rate of the recombination of Li and is larger than the resonant rate. Red, black, and blue arrows show the three important transitions in the nonresonant recombination reaction, while a purple arrow shows the decay of the important resonance into a bound state of Li in the resonant reaction.

Fig. 3: Energy level diagram of Li in the case of GeV. Red, black, and blue arrows show the three important transitions in the nonresonant recombination reaction of Li(, )Li, while a purple arrow shows the decay of the important resonance into a bound state of Li in the resonant reaction.

Similarly to the recombination of Be+, the recombination can efficiently proceed via resonant reactions involving atomic states of composed of the first nuclear excited state Li[Li(0.478 MeV, )].

The resonant rate for the reaction Li(, )Li has been calculated to be

(15)

The rate correspond to the atomic transition from the resonance Li(2P) to Li(1S).

The thermal nonresonant rate is given by

(16)

This Li+ system has the important characteristic that the transition -wave 2P is the most important for GeV, as in the Be+ system.

3.3 Be

In the recombination of Be and , there are no important resonances of atomic states composed of the nuclear excited states for Be. Then, only the nonresonant rate is needed. The nonresonant rate is then taken from Ref. ?. The transition, -wave 2P, is most important for GeV in this Be+ system, also.

3.4 He

Atomic states composed of nuclear excited states He are never important resonances in the recombination process. The nonresonant rate is then taken from Ref. ?.

Because of the small amplitude of the Coulomb potential, the effect of the finite-size nuclear charge does not significantly affect the wave functions and cross sections. As a result, even in the case of heavy particles ( GeV), the rate for the recombination of He and is contributed by the transition -wave 1S most, similarly to the case of the Coulomb potential for point charges.

3.5 Other nuclei

For other nuclides, we approximately adopt the Bohr atom formula in the estimation of recombination rates. We adopt cross sections in the limit that the CM kinetic energy, , is much smaller than the binding energy, . This is justified since the condition, with the relative velocity of a nucleus and , always holds when the bound state formation is more efficient than its destruction. The cross sections are given by

(17)

where is the base of the natural logarithm. The thermal reaction rate is then given by

(18)

where is the -value in units of MeV, and we defined a rate coefficient . The -value for the recombination is equal to the binding energy of the -nucleus .

Rates of radiative recombination of and and photoionization of are taken from Ref. ?.

4 Resonant Proton Capture Reactions

Two important resonant reactions are

(19)

where (2P) indicates the atomic 2P state, and and are masses of nucleus and -nucleus , respectively. The states B(2P) and B(2P) are the first atomic excited states. The resonant reactions through the atomic excited states are important since they result in Be destruction and B production. The superscript indicates an atomic excited state, that is different from a nuclear excited state indicated by a superscript . Resonant rates for these radiative capture reactions can be calculated as follows.

The thermal reaction rate for isolated and narrow resonances is given by

(20)

where is Avogadro’s number, is the reduced mass in atomic mass units (amu) given by with and the masses of two interacting particles, 1 and 2, in amu, is the temperature in units of K. The parameter is a statistical factor defined by

(21)

where is the spin of the particle , is the spin of the resonance, and is the Kronecker delta necessary to avoid a double counting of identical particles. The quantity is defined by

(22)

where and are the partial widths for the entrance and exit channels, respectively. is the total width for a resonance with resonance energy , is the factor in units of MeV, and is the resonance energy in units of MeV.

When as in the reactions considered here, and the radiative decay widths of B and B, , are much smaller than those for proton emission (as assumed here), the thermal reaction rate is given by

(23)

where is the radiative decay width in units of MeV, and is a rate coefficient determined from and .

The rate for a spontaneous emission via an electric dipole (E1) transition is given by

(24)

where is the E1 effective charge [Eq. (6)], is the energy of the emitted photon, is the angular momentum of the initial state, and are magnetic quantum numbers of initial and final states with the subscript . and are wave functions of the initial and final states, respectively, and is the dipole spherical surface harmonic.

In the present system of B+, the effective charge is in the limit of , where and are the charge numbers of B and the , respectively.

Resonant rates for the proton capture reactions are adopted from Ref. ?, and the nonresonant rates are taken from Ref. ?.

5 Be PRODUCTION FROM Li

The dominant reaction of Be production in the BBN model with the particle is Li(, )Be. Both resonant and nonresonant components can contribute to the reaction rate. Realistic calculations for this reaction rate are not available yet. The current rate is based on the assumption that the astrophysical factor for the reaction is taken from the existing data for Li(, )He, i.e, MeV b. This reaction is the key reaction since a signature of the particle is left on the primordial Be abundance through this reaction. This model can, therefore, be tested in the future by observations of Be abundances in MPSs. Therefore, realistic calculations of this reaction rate with quantum many-body models are needed.

Figure 4 shows the energy level diagram of Be in the case of GeV. The red arrow shows the reaction for the Be production, Li(, )Be. Because of the relatively large abundances of Li and (see Fig. 5 below) and the large reaction rate, Be is produced significantly through this reaction. It has been shown in a large network reaction calculation that other reactions cannot be responsible for significant Be production.

Although the resonant reaction Be(, )Be through the atomic ground state of Be(, 1.684 MeV), has been suggested for Be production, it was found that Be(, 1.684 MeV) is located below the Be+ threshold by a revised estimation using a more realistic nuclear charge radius. Energy levels of Be, Be, and Be in Fig. 4 correspond to the best estimate. The state Be is below the state of the Be+, and does not operate as a resonance. This reaction is, therefore, unimportant. We note that even if the state Be is barely higher than the separation channel, this reaction rate is suppressed by an extremely small Coulomb penetration factor as shown [Sec. 2.3.2 in Ref. ?] for the resonance B(,0.770 MeV) in the reaction Be(,)B.

Fig. 4: Energy level diagram of the system of Be+ in the case of GeV. Red arrow shows the most important reaction for the Be production, Li(, )Be.

6 Bbn Reaction Network

We adopted the Kawano reaction network code, and utilized a modified version. The free particle and bound -nuclei are encoded as new species, and reactions involving the particle are encoded as new reactions. The -decay rates of -nuclei () are taken from Ref. ?. Thermonuclear reaction rates of -nuclei are adopted from Refs. ?, ?, ?. Two parameters are updated in this review. First, the neutron lifetime is the central value of the Particle Data Group,  s based upon improved measurements. The baryon-to-photon ratio is , corresponding to the baryon density determined from the Planck observation of the cosmic microwave background, for the base CDM model (Planck+WP+highL+BAO).

Be production through Be, i.e., He(, )Be(, )Be, depends significantly on the energy levels of Be and Be, and precise calculations with a quantum four body model by another group is under way. In this paper, we neglect that reaction series, and leave that discussion to a future work. The reaction He(, )Be is thus not included, and the abundance of Be is not shown in the figures below.

7 Abundance Constraints

Observational constraints on the deuterium abundance are taken from the weighted mean value of D/H. When the central values of adopted reaction rates, the neutron lifetime, and the baryon-to-photon ratio are used, the calculated abundances in the SBBN model is out of the observational limit. The range is then adopted.

Constraints on the primordial He abundance are taken from the mean value of Galactic HII regions measured through the 8.665 GHz hyperfine transition of He, i.e., He/H=.

Constraints on the He abundance are taken from observational values of metal-poor extragalactic HII regions, i.e, , and adopt its range since the range is inconsistent with the theoretical abundances in the SBBN model.

We take the observational constraint on the Li abundance from the central value of log(Li/H) derived in the 3D NLTE model of Ref. ?.

Detections of Li abundances of MPSs have been reported. The measured abundance of Li/H, is 1000 times higher than the SBBN prediction, and is also significantly higher than the prediction by a standard Galactic cosmic-ray nucleosynthesis model. In a subsequent detailed analyses, however, it was found that most of the previous Li absorption feature could be attributed to a combination of 3D turbulence and nonlocal thermal equilibrium (NLTE) effects in the model atmosphere. We adopt the least stringent (95% C.L.) upper limit among all limits reported in Ref. ?, i.e., Li/H= for the G64-12 (NLTE model with 5 free parameters).

Spectroscopic observations of Be in MPSs show that the Be abundance scales linearly with Fe abundances generally. The linear trend is explained with Galactic cosmic-ray nucleosynthesis models. Primordial abundances of Be before the start of the cosmic-ray nucleosynthesis, however, may be found by future observations. We adopt the strongest upper limit on the primordial Be abundance, log(Be/H) derived from observations of carbon-enhanced MPS BD+44493 of an iron abundance [Fe/H] ccc[A/B], where is the number density of and the subscript indicates the solar value, for elements A and B. with Subaru/HDS.

8 Results

We show calculated results of BBN for GeV. See Ref. ? for results for various mass values. First, we analyze the time evolution for abundances of normal and -nuclei. Then we update constraints on the parameters characterizing the particle.

The two free parameters in this BBN calculation are the ratio of number abundance of the particles to the total baryon density, , and the decay lifetime of the particle, . The lifetime is assumed to be much smaller than the age of the present universe, i.e., Gyr. The primordial -particles from the early universe are thus by now, long extinct.

As for the fate of -nuclei, it is assumed that the total kinetic energy of products generated from the decay of is large enough so that all -nuclei can decay into normal nuclei plus the decay products of . The particle is detached from -nuclei with its rate given by the decay rate. The lifetime of -nuclei is, therefore, given by the lifetime of the particle itself.

8.1 Nucleosynthesis

Figure 5 shows the calculated abundances of normal nuclei (a) and nuclei (b) as a function of . Curves for H and He correspond to the mass fractions, (H) and (He) in total baryonic matter, while the other curves correspond to number abundances with respect to that of hydrogen. The dotted lines show the result of the SBBN model. The abundance and the lifetime of the particle are assumed to be and , respectively, for this example.

Fig. 5: Calculated abundances of normal nuclei (a) and -nuclei (b) as a function of . and are the mass fractions of H and He, in total baryonic matter, while the other curves correspond to number abundances with respect to that of hydrogen. The abundance and lifetime of the particle are taken to be and , respectively. The dotted lines show the results of the SBBN model.

Four lines are shown corresponding to different reaction rates for Be(, )B and Be(, )B. The two reactions are dominantly via resonant reactions whose rates are very sensitive to binding energies of -nuclei. The binding energies are affected by adopted nuclear charge distributions. Therefore, we show results for the four different cases of charge distribution. Three cases correspond to Gaussian (thick dashed lines), WS40 (solid lines), and well (dot-dashed lines) models, while one case corresponds to the previous calculation in which the reaction rate for Be(, )B derived with a quantum many-body model was adopted. The amount of Be destruction varies significantly when the nuclear charge distributions are changed. The result of our Gaussian charge distribution model (thick dashed line) is close to that of the quantum many-body model. The differences in the curves for Be thus indicate the effect of uncertainties in the charge density, which are estimated from measurements of RMS charge radii.

Early in the BBN epoch (), is the only -nuclide with an abundance larger than H. Its abundance is the equilibrium value determined by the balance between the recombination of and and the photoionization of . When the temperature decreases to , He is produced as in SBBN (panel a). Simultaneously, the abundance of He increases through the recombination of He and particles (panel b). As the temperature decreases further, the recombination of nuclei with gradually proceeds in order of decreasing binding energies of , similar to the recombination of nuclei with electrons.

Be first recombines with via the Be(, )Be reaction at . The produced Be nuclei are then partially destroyed via the Be(, )B reaction. In a later epoch at , the Be abundance increases mainly through the reaction He(