# General limit on the relation between abundances of D and Li in big bang nucleosynthesis with nucleon injections

###### Abstract

The injections of energetic hadrons could have occurred in the early universe by decays of hypothetical long-lived exotic particles. The injections induce the showers of nonthermal hadrons via nuclear scattering. Neutrons generated at these events can react with Be nuclei and reduce Be abundance solving a problem of the primordial Li abundance. We suggest that thermal neutron injection is a way to derive a model independent conservative limit on the relation between abundances of D and Li in a hadronic energy injection model. We emphasize that an uncertainty in cross sections of inelastic scattering affects the total number of induced neutrons, which determines final abundances of D and Li. In addition, the annihilations of antinucleons with He result in higher D abundance and trigger nonthermal Li production. It is concluded that a reduction of Li abundance from a value in the standard big bang nucleosynthesis (BBN) model down to an observational two upper limit is necessarily accompanied by an undesirable increase of D abundance up to at least an observational 12 upper limit from observations of quasi-stellar object absorption line systems. The effects of antinucleons and secondary particles produced in the hadronic showers always lead to a severer constraint. The BBN models involving any injections of extra neutrons are thus unlikely to reproduce a small Li abundance consistent with observations.

###### pacs:

13.75.-n, 26.35.+c, 98.80.Cq, 98.80.Es## I Introduction

Many environments have been considered regarding the origin of deuterium Epstein et al. (1976). They include pregalactic cosmic rays (CRs) from quasars and collapsing objects, shock waves, and neutron stars. In general, the CRs induce nuclear reactions producing D, He, Li, Be, and B nuclides Reeves (1970); Meneguzzi et al. (1971); Reeves (1974). Pregalactic CRs or cosmological CRs generated before the Galaxy formation also produce Li (via the fusion Montmerle (1977a)) and He (via He+ nuclear spallation Montmerle (1977b)). Li productions have been calculated for the CRs in specific environments: the CRs accelerated in structure formation shocks at the Galaxy formation epoch Suzuki and Inoue (2002) and the CRs from supernova remnants at the pregalactic epoch Rollinde et al. (2005, 2006). Since a metal pollution proceeds along with a stellar activity in the universe, the CRs would come to contain metals such as C, N, and O. Therefore, the pregalactic CR nucleosynthesis would also produce Be and B through reactions of (C, N, or O)+( or ) Kusakabe (2008); Rollinde et al. (2008) and (He or )+He or Li)+ followed by (He or Li)+Be+ with byproducts and Kusakabe and Kawasaki (2013).

Another possible source of the CR is an energy injection at decay and annihilation of exotic long-lived particles Lindley (1979); Chechetkin et al. (1982); Khlopov and Linde (1984); Balestra et al. (1984); Ellis et al. (1985); Lindley (1986); Sedelnikov et al. (1995); Reno and Seckel (1988); Levitan et al. (1988); Dimopoulos et al. (1988a, b); Dimopoulos et al. (1989); Terasawa et al. (1988); Kawasaki et al. (1994); Khlopov et al. (1994); Kawasaki and Moroi (1995); Holtmann et al. (1996); Jedamzik (2000); Kawasaki et al. (2001); Cyburt et al. (2003); Kawasaki et al. (2005a, b); Jedamzik (2004a, b); Jedamzik et al. (2006); Ellis et al. (2005); Kusakabe et al. (2006); Kanzaki et al. (2007); Jedamzik (2006); Cumberbatch et al. (2007); Kusakabe et al. (2009); Kawasaki et al. (2008); Kawasaki and Sato (2009); Cyburt et al. (2009); Pospelov and Pradler (2010a); Pospelov:2010kq (); Cyburt et al. (2010); Ellis et al. (2011); Kang:2011vz (); Olive et al. (2012); Kusakabe:2013sna (); Ishida:2014wqa (). A constraint on the mass of a hypothetical stable heavy neutrino has been derived through calculation of its present cosmological energy density Hut (1977); Lee and Weinberg (1977). An unstable heavy neutrino was then considered, and constraints on its mass and lifetime were derived Sato and Kobayashi (1977); Dicus et al. (1977); Vysotsky et al. (1977). The electromagnetic decay of the unstable particle is constrained through distortions in the energy spectrum of cosmic microwave background radiation Sato and Kobayashi (1977). The constraints on hypothetical heavy neutrino Miyama and Sato (1978a) and primordial black holes Miyama and Sato (1978b) were then derived from the effect on light element abundances through energy densities in detailed calculations of big bang nucleosynthesis (BBN). The decay of unstable heavy neutrinos also affects nuclear abundances through nonthermal photodissociation of nuclei Lindley (1979). The radiative decay induces electromagnetic cascades of energetic photons, electrons, and positrons during the propagation of the nonthermal photon emitted at the decay Ellis et al. (1985).

Effects of hadronic injections at the decay were studied Levitan et al. (1988); Dimopoulos et al. (1988a); Dimopoulos et al. (1989); Dimopoulos et al. (1988b); Reno and Seckel (1988). Levitan et al. investigated hadronic cascades of proton and antiproton and dissociations of He Levitan et al. (1988). Dimopoulos et al. Dimopoulos et al. (1988a); Dimopoulos et al. (1989); Dimopoulos et al. (1988b) extensively studied the effects on abundances of nuclei up to Li and Be. They considered the reaction, i.e., H()H, for D production, and the reaction, i.e., Be()Li, for Be destruction, where 1(23)4 stands for a reaction . Antiprotons injected at decays of exotic long-lived particles could dissociate He and produce D and He Chechetkin et al. (1982); Khlopov and Linde (1984); Balestra et al. (1984). The cross sections of He annihilation have been measured Balestra et al. (1988), and the yields of D, H, and He at the annihilation were calculated as a function of energy of antiproton Sedelnikov (1999). Effects of exotic particles on nuclear abundances through hadronic showers have been extensively studied with realistic initial spectra of injected hadrons Kawasaki et al. (2005b); Jedamzik (2006).

The standard BBN (SBBN) model explains primordial light element abundances inferred from astronomical observations well Fields (2011). Modifications of the BBN model are then constrained from the consistency between theoretical predictions and observations of abundances. Among light elements produced during the BBN, however, the lithium has an unexplained discrepancy between SBBN prediction and observational determinations of its primordial abundances Melendez and Ramirez (2004); Asplund et al. (2006). Spectroscopic observations of metal-poor stars (MPSs) indicate an abundance measured by number relative to hydrogen, i.e., Li/H Spite and Spite (1982); Ryan et al. (2000); Melendez and Ramirez (2004); Asplund et al. (2006); Bonifacio et al. (2007); Shi et al. (2007); Aoki et al. (2009); Hernandez et al. (2009); Sbordone et al. (2010); Monaco et al. (2010, 2011); Mucciarelli et al. (2011) ^{1}^{1}1Surface Li abundances of metal-poor red giant branch stars do not depend on parameters of standard stellar models as much as dwarf stars do. Mucciarelli et al. Mucciarelli et al. (2011) determined Li abundances of metal-poor halo red giant branch stars, and estimated initial abundances, which were also 2–3 lower than SBBN prediction. ^{2}^{2}2Monaco et al. Monaco et al. (2011) reported that one star, 37934, among 91 stars of the globular cluster M4 has a high lithium abundance (Li/H=) consistent with the abundance of the SBBN model.. This abundance is a factor of 2–4 higher than the SBBN prediction when we adopt the baryon-to-photon ratio determined from the observation of the cosmic microwave background radiation with Wilkinson Microwave Anisotropy Probe (WMAP) Hinshaw:2012aka ().

After the lithium problem was recognized, the neutron injection during the BBN was suggested to be a solution since it can reduce Be abundance via Be()Li()He, although it increases D abundance via H()H simultaneously Jedamzik (2004a); Albornoz Vasquez
et al. (2012). Such a neutron injection is realized in the hadronic decay of exotic long-lived massive particles Jedamzik (2004a); Kawasaki
et al. (2005b); Jedamzik (2006). Important reactions caused by injected nonthermal hadrons have been identified in a statistical study, which are shown to be closely associated with resulting elemental abundances Cyburt et al. (2010). A wide parameter region of the lifetime and the abundance of a long-lived particle was studied, and a parameter region for Li reduction has been found Kawasaki
et al. (2005b); Jedamzik (2006); Cyburt et al. (2009) ^{3}^{3}3If long-lived exotic particles of sub GeV-scale mass exist, and their decay products do not include nucleons, another route of additional neutrons operates Pospelov and
Pradler (2010a). When mesons such as and are generated by the particle decays, they can convert protons to neutrons, and a reduction of Li abundance realizes along with an enhancement of D abundance. When the decays do not generate any mesons, and muons and neutrinos are generated, on the other hand, induced electron antineutrinos convert protons to neutrons. In this case, the dissociation of once enhanced D by nonthermal photons can reduce D abundance to the level consistent with observations.. In this paper, we focus solely on the parameter region for Li reduction, and derive a model independent constraint on a relation between abundances of D and Li, by using recent D abundance data.

In Sec. II, we describe input physics and assumptions adopted in this paper. We prove that the assumption of thermal neutron injection (TNI) leads to a conservative lower limit on the ratio of the increase of D abundance to the decrease of Li abundance. In Sec. III, we describe the TNI model and the BBN model, as well as adopted observational constraint on primordial nuclear abundances. The TNI is assumed to occur instantaneously, and the injection time and the abundance of injected neutron are used as parameters in this model. In Sec. IV, results of the BBN calculations are shown, and a relation between abundances of D and Li is derived. In Sec. V, we estimate an effect of antinucleon annihilation with He on the abundance relation. In Sec. VI, we estimate amounts of Li production induced by the antinucleon+He annihilation. In Sec. VII, conclusions are done finally. In Appendix A, we list important nuclear reactions which work in a parameter region for the reduction of primordial Li abundance. In Appendix B, approximate analytic estimates of D and Li abundances are shown.

In this paper, we adopt notation of with a real number and an integer , and with a parameter and a real number . The Boltzmann’s constant (), the reduced Planck’s constant (), and the light speed () are normalized to be unity.

## Ii input physics

In this paper, we concentrate on a production of D and a reduction of Be and Li induced by hadronic energy injection at temperature (or cosmic time s). This injection epoch corresponds to that of the solution to the Li problem by the hadronic energy injection model Jedamzik (2004a, b); Kawasaki
et al. (2005b). The injection produces energetic nucleons, antinucleons, and mesons. Such hadrons can scatter background nuclei so that many energetic hadrons are generated and hadronic showers composed of energetic hadrons are developed Dimopoulos
et al. (1988a); Dimopoulos et al. (1989); Kawasaki
et al. (2005b); Jedamzik (2006). Main reactions changing abundances of D and (Li+Be) ^{4}^{4}4A sum of Li and Be abundances gives primordial Li abundance before the start of the early stellar activity. This is because Be produced at BBN epoch is transformed to Li by electron capture after the recombination of Be ion. in this parameter range Jedamzik (2004a); Kawasaki
et al. (2005b); Jedamzik (2006) are

(1) |

(2) |

respectively, where H, , or is a byproduct. If the neutron injection time is s as considered here, effects of long-lived mesons are negligible Reno and Seckel (1988).

In Sec. II.1, we comment that energetic proton, antiproton, and nuclei quickly thermalize while an energetic neutron can induce inelastic scatterings off background proton. In Sec. II.2, we present that the BBN calculation for the case of the TNI provides a lower limit on the ratio, i.e., D/Li, where is a difference between final number densities of nuclide in this model () and the SBBN model (). A more precise estimation of the ratio D/Li should include the annihilation of antineutron with He. Nuclear data on the annihilation, however, contains a large uncertainty. It is shown that effects of hadronic showers composed of energetic neutron and antineutron always enhance the ratio D/Li above that of the TNI model. In Sec. II.3, we see that the assumption of instantaneous thermalization of nonthermal neutron leads to a lower limit on the ratio.

### ii.1 Hadronic shower

#### ii.1.1 Stopping of energetic proton

We assume instantaneous thermalizations of nonthermal , , and nuclei for the following reason.

An inelastic scattering of two nucleons can be triggered by incident nucleons with energies of GeV (Fig. 1 in Ref. Meyer (1972)). Such incident nucleons are thus relativistic to a certain degree. Because of the Coulomb interaction via electric charge, a relativistic proton undergoes Coulomb energy loss. The loss rate for is given (Eq. [A.18] in Ref. Reno and Seckel (1988) ^{5}^{5}5We checked this equation. The Eq. (B6) in Ref. Kawasaki
et al. (2005b) may show an erroneously larger rate by a factor of two.) by

(3) |

where and is the kinetic energy and the charge number of proton, respectively, is the fine-structure constant, and is the electron mass. is a parameter associated with Coulomb divergence (Eqs. [13.13] and [13.43] in Ref. Jackson (1975)). Here, is the velocity of the proton, and is the Lorentz factor. is the plasma frequency of background plasma composed of electron and positron Jackson (1975). and are the total number density and the energy density, respectively, of electron and positron plasma. The total number density is given by for and for with the mass fraction of He to total baryon, the baryon-to-photon number ratio , and the number density of background photon Reno and Seckel (1988).

Cross sections for inelastic scattering of two nucleons are mb Meyer (1972). The reaction rate is then given by

The rate of energy degradation via Coulomb scattering is, on the other hand, given by

where the numerical factor in the second line corresponds to the case of . Nonthermal protons generated at hadron injections hardly trigger an inelastic collision before they lose energies because of quick thermalization, i.e., for . The same holds true for antiprotons, and nuclei with larger charge numbers.

#### ii.1.2 Inelastic scattering of energetic neutron

Nonthermal protons effectively stop without inducing hadronic scatterings. Hadronic showers then contain only neutrons and antineutrons as mediator particles which can interact with background nuclei nonthermally by the energy injected at the particle decay. Main reactions between a nonthermal neutron and a background proton, which is much more abundant than background neutron at , are

(6) | |||||

(7) | |||||

(8) |

where and are nonnegative integers. The elastic scattering corresponds to in Eq. (6). For a same set of and values, reaction thresholds of the second reaction are higher than those of the third by , where and are the masses of neutron and proton, respectively.

The first reaction does not change the combination of nucleon isospins so that the number of energetic particle, i.e., neutron, is not changed. The second reaction could increase the number of energetic neutron, while the third decreases it both by the unit of one. Two protons from the third reaction stop instantaneously. If the sum of rates for the second reaction over and is larger than that for the third, nonthermal neutron abundance goes up from the abundance of originally injected neutron. If the total rate for the second is smaller than that for the third, however, the nonthermal neutron abundance goes down. If the both rates balance approximately, the nonthermal neutron abundance does not change during developments of hadronic showers.

Cross sections of the second and third reactions have been measured, and they equate within the statistical errors Dunaitsev and D. (1960); Bystricky et al. (1987). Although an isospin symmetry in the two reactions seems to exist, it is not yet verified experimentally. Uncertainties in reaction rates affect a net number of neutrons which are generated in the universe. The net abundance of nonthermal neutron is the most important quantity determining abundances of D and Li. Then, one should be cautious about the uncertainties in reaction rates when a parameter space for Li reduction is searched. Recent previous BBN calculations including hadronic particle injection were based on biased network codes in which either reaction of the second and third types is included for some sets of and Kawasaki et al. (2005b); Jedamzik (2006). The present study escapes from these uncertainties, and obtains a conservative lower limit on D/Li.

### ii.2 Production of neutron and D

In this subsection, we focus on the processes occurring at the time of neutron injection, , and omit the index for the time on physical quantities for simplicity. Firstly we describe changes in D and Li abundances caused by injections of neutrons and antineutrons by the following two equations. The amount of Li reduction is approximately proportional to the total abundances of injected nonthermal neutron, i.e., since Be is destroyed by neutron [Eq. (2)]. The equation for is

(9) | |||||

where and are the abundances of primary neutron and antineutron, respectively, injected at the considered event, and are the probabilities that the -th generation neutrons and antineutrons, respectively, generate the -th generation species for or . We note that is a sum of components for multiple reactions (). If no neutron is emitted at a reaction induced by a -th neutron, the net number of neutron changes by . The value is then for this reaction . The first and second terms of the right hand side (RHS) correspond to neutrons originating from primary neutrons and antineutrons, respectively. We neglect effects of the scattering off background and He. Since annihilation cross sections of and He reactions are significant in comparison with total cross sections Bendiscioli and Kharzeev (1994), generated are typically lost after at most a few reactions unaccompanied with annihilations.

The change in D abundance is described as

The first term of the RHS is for deuterons produced via H()H. Note that the injected neutrons are mostly captured by proton, and converted to D for s. The second term is for the sum of the -th deuterons produced mainly via He spallation by the -th neutrons which originate from primary neutrons. The third term includes deuterons produced at annihilations with He, and the sum of the -th deuterons produced mainly via He spallation by the -th generation neutrons originating from primary antineutrons.

The present model is constrained by an overproduction of D as described below. We then conserve the model by keeping D abundances low while reducing Li abundances. When instantaneous thermalizations of energetic and are assumed, no secondary or higher order energetic particles would be generated. Then, an equation, i.e., , holds. Accordingly, one obtains , and , where subscript 1 in and indicates that the amounts count only particles originating from primary neutrons and not higher order neutrons.

The ratio is estimated as follows: First, we assume the symmetry in injected amounts of neutron and antineutron (). The following relation then holds:

(11) |

The value is given by

(12) |

where and are number densities of H and He, respectively. In the epoch after the He production, the ratio is . and are cross sections for annihilation by hydrogen and particle, respectively. The ratio in the parenthesis with subscript indicates the value for annihilation of . is the fraction of the He annihilation into exit channels including species .

#### ii.2.1 Effect of annihilation

Although an estimation of [Eq. (12)] is associated with uncertainties, an example estimation is shown as follows:

Nuclear data on He annihilation at low energies indicate fractions for the production of and , i.e., – and Balestra et al. (1988). We then assume the similarity of the fractions for and , and take values of and . In addition, we assume the simple scaling of with the mass number , and Levitan et al. (1988). In this case, the equation, , holds, and Eq. (11) becomes

(13) |

#### ii.2.2 Effect of secondary neutron

Here the assumption of instantaneous thermalization is removed, i.e., . A relation between yields of the -th generation neutron and deuteron derives from Eqs. (9) and (LABEL:eq10) as

(14) |

The quantity is described by an integration of a distribution function in energy of the -th generation neutron multiplied by a rate for production of species . A lower limit on is estimated utilizing experimental data on cross sections Meyer (1972) as

(15) | |||||

where for and , and and represents an effective cross section for production of at the reaction with , as explained below. We defined

(16) | |||||

which is a sum of cross sections for final states weighted according to the net increase in deuteron number. Similarly we defined

(17) |

and

(18) | |||||

as the sums of cross sections weighted according to the net increase in neutron number.

The value in the second line of Eq. (15) was estimated as follows: We adopt values of mb from the mirror reaction, i.e., (Fig. 7 of Ref. Meyer (1972)), and mb (Fig. 6 of Ref. Meyer (1972)). In addition, an asymmetry in cross sections of [Eq.(7)] and [Eq.(8)] was allowed conservatively by 20 % of the total inelastic cross section at maximum, i.e., mb Meyer (1972).

### ii.3 Neutron thermalization

In this model, in addition to the SBBN, we consider an extra production of D and a destruction followed by some degrees of reproduction of Be (Sec. IV).

We write the ratio between changes of (Li+Be) and D as a function of the kinetic energy of neutron, i.e., , and . It is given by

(19) |

where is the change of abundance caused by neutrons with energy in the universe of temperature . is the cross section for the reaction as a function of .

is the destruction fraction of Li, which is produced via the reaction Be()Li, during its propagation in the cooling universe. is the survival fraction of D, which is produced via the reaction H()H, during its propagation. If energetic Li and nuclei are produced by the respective reactions, they instantaneously lose their energies through the Coulomb scattering, and are thermalized soon after the productions Cyburt et al. (2003); Kawasaki et al. (2005b). The quantities and should then be taken as values for thermal Maxwell-Boltzmann distribution of Li and D (see Appendix B). Note that although the quantities, and , depend on , we omit to express the argument.

The ratio of is roughly speaking smaller at higher energies as seen hereinbelow while the ratio of is larger at higher energies (see Fig. 6 in Appendix B).

Figure 1 shows the ratio of thermonuclear reaction rates estimated with recommended rates given by Descouvemont et al. Descouvemont et al. (2004) [for Be()Li] and Ando et al. Ando et al. (2006) [for H()H]. Because of a decrease in the Be()Li rate at high energies, the ratio decreases at high temperatures. At low temperatures (), Li is not destroyed, i.e., , although Be is transformed to Li via Be()Li. The amount of Li reduction is, therefore, small [Eq. (19)]. An efficient destruction of Li then prefers an operation of Be()Li at higher temperature. At high temperatures (), on the other hand, the Be production in the SBBN is not yet completed. Although Be nuclei are converted to Li, the same nuclei are produced via the reaction He()Be later in lower temperatures until the reaction stops (Appendix B). In a white region at , therefore, the reduction of Li is most efficient.

When energetic neutrons are injected, they experience an energy loss, especially the Coulomb scattering off the background electrons and positrons through interaction via their magnetic moments Reno and Seckel (1988); Kawasaki et al. (2005b); Jedamzik (2006). Nonthermal neutrons are then quickly thermalized. Nevertheless, a small abundance of energetic neutrons can react with background H and Be before they could be thermalized. At high neutron energies, the ratio of cross sections for Be()Li and H()H is small. Although the ratio of rates averaged over Maxwell-Boltzmann distribution is shown in Fig. 1, the trend in reaction rate as a function of temperature roughly traces that in cross section as a function of energy. Neutrons with higher energies thus relatively prefer the production of D over the destruction of Be.

In order to obtain a conservative lower limit on Li, we assume that nonthermal neutrons instantaneously thermalize, and cause a preferential reduction of Li. Even if energetic hadrons induced by a hadronic energy injection were instantaneously thermalized, thermalized antinucleon can destroy background He nuclei through annihilation processes (Sec. V).

## Iii Model (thermal neutron injection)

We assume that the TNI occurs at time instantaneously with a number density of injected neutron . ^{6}^{6}6Although the instantaneous injection is assumed in this study, injections of finite durations can be supposed. The finite durations realize in spontaneous decays and annihilations of long-lived exotic particles or evaporations of exotic objects.. The abundance is measured as the number density relative to that of total baryons, i.e., .

### iii.1 Method

We use the BBN code by Kawano Kawano (1992); Smith et al. (1993) with the Sarkar’s correction Sarkar (1996) to He abundance. Reaction rates relating to light nuclei of mass number are updated with the JINA REACLIB Database V1.0 Cyburt et al. (2010). We adopt the neutron lifetime of s Serebrov and Fomin (2010).

### iii.2 Observational limits

We adopt an upper limit on the abundance ratio Li/H from a recent observation of MPSs, i.e., log(Li/H) derived with the 3D nonlocal thermal equilibrium model Sbordone et al. (2010). Taking the two (standard deviation) uncertainty, we assume the primordial abundance of . A consistency between a theoretical prediction and observations of MPSs requires a reduction of (Li+Be) during the BBN in amounts of at least .

The SBBN prediction of deuterium abundance is (D/H). The final value of D/H after the D production caused by the neutron injection should not deviate from primordial abundance inferred from observations of Lyman- absorption system in the foreground of quasi-stellar objects (QSO). Recent measurement of a damped Lyman system QSO Sloan Digital Sky Survey (SDSS) J1419+0829 was performed most precisely of all QSO absorption systems ever found Pettini and Cooke (2012). We adopt the best measured abundance, log(D/H)= (best), and a mean value of ten QSO absorption line systems including J1419+0829, log(D/H)= (mean) Pettini and Cooke (2012). ^{7}^{7}7Recently Olive et al. Olive et al. (2012) have considered possible effects of cosmic chemical evolution of D and Li, and suggested a solution to the Li problem by the Be destruction at the hadronic decay of a long-lived exotic particle followed by a depletion of D in the cosmic chemical evolution..

## Iv Result

Figure 2 shows calculated abundances of H and He, i.e., and , respectively, (mass fractions), and other nuclides (number ratios relative to H) as a function of the temperature . Solid lines correspond to cases of different injection times of , , , and s, for the same injected abundance . Dashed lines show fiducial abundances of the SBBN model. In all cases, final values of baryon-to-photon ratios are the WMAP9 value (model CDM; WMAP data only) Hinshaw:2012aka (). In Appendix A, we describe important reactions through which nuclear abundances are affected. It is seen that abundances of T, Li, and Be are changed much by the TNI, and that increases in abundances of T and Li depend significantly on . At a large value of , the destruction reaction of T, i.e., H(, )He, is ineffective because of a low temperature. The final T abundance is then large. T nuclei produced during the BBN epoch decay to He with the half life of yr Purcell et al. (2010). The final He abundance is, therefore, given by a sum of abundances of T and He at BBN. Since the He abundance is much larger than the T abundance in the BBN epoch, increases of T abundance change the final He abundance by only negligible amounts.

Figure 3 shows contours for final abundances of D (solid lines) and Li (dashed lines) in the (, ) plane. In a narrow region indicated at , s, ) by points, the primordial abundance inferred from observations of Li Sbordone et al. (2010) is reproduced within the two uncertainty keeping the D abundances close to the observed value Pettini and Cooke (2012). This region is, therefore, the most preferred region. Dark (black) points correspond to calculated D abundances in the 12 range of the best observed value, while light (green) points correspond to those in the 5 range of the mean value. It is found that D abundances in the 11 range of the best value and the 4 range of the mean value are never accompanied with Li abundances in the observational 2 range in this model. The recent precise determination of D abundance in the QSO absorption line systems thus completely excludes the solution to the Li problem in this model. This calculation itself should be similar to a recent calculation for neutron injection which concluded that this model can provide a solution to the Li problem Albornoz Vasquez et al. (2012). Our different conclusion results from the use of the new observational constraints on primordial D abundance. Effects on abundances of , D, H, He, Li, Li, and Be are different in different parameter cases. Reasons for that are described in Appendixes A and B.

Figure 4 shows a region on the parameter plane of (Li/H, D/H) which can be occupied in this model. The lines with arrows indicate the regions which satisfy observational constraints on abundances of D (12 for the best value, and 5 for the mean value) Pettini and Cooke (2012) and Li (2 ) Sbordone et al. (2010). A lower limit on as a function of Li/H can be read from this figure. The points at (Li/H, D/H) , ) satisfy the constraints. Abundances in this parameter region give close agreement with those found in a recent detailed study on effects of hadronic decay Cyburt et al. (2010) as their most favorable results of abundances.

In our preferred parameter region, the abundance of D is related to that of Li for the adopted nuclear reaction rates (Sec. III.1) and baryon-to-photon ratio Hinshaw:2012aka () as described by

(20) |

This constraint is free of many uncertainties related to nuclear and electromagnetic reactions for nonthermal particles produced by the neutron injection. Although primary antinucleons, and secondary and higher order neutrons always increase the ratio D/Li (see Secs. II.2 and V), their effects depend [Eqs. (15) and (21))] on information of relative injected amounts of , , and , and their injected energy spectra. The information itself depends on the decay property of the long-lived exotic particle such as its mass and decay modes. Equation (20) then corresponds to the most conservative model independent lower limit on D/H as a function of Li/H.

### iv.1 dependence

In the case of earliest neutron injection at s (Fig. 2), effects of additional neutrons are removed by efficient nuclear reactions. Especially, although the Be abundance reduces right after the neutron injection, the reaction He()Be enhances Be again.

In the best case of injection, i.e., s, the Be abundance decreases and the D abundance increases a little less efficiently.

In the case of later injection, i.e., s, the D abundance increases via H()H, and is not affected by already inefficient D destruction reactions. The resulting D abundance is thus larger than in the best case. The Be conversion to Li by neutron capture efficiently proceeds. However, the reaction Li()He is no longer operative. The resulting decrease in the mass-number-seven (Li+Be) abundance is, therefore, very small.

In the case of the latest injection, i.e., s, some portion of injected neutrons decay with the lifetime s Serebrov and Fomin (2010) before they could trigger the D production via H()H. This leads to a suppressed D production. The reduction of (Li+Be) is not operative as in the previous case.

If the neutron injection has a duration, deviations in final abundances would be approximately given by weighted average over time of deviations obtained in this instantaneous injection model.

### iv.2 dependence

When amounts of neutron injection are small, i.e., , both of the Be destruction and the D production are efficient. If the injection is strong, i.e, , however, the efficiency of Be destruction plateaus since it gets difficult for neutrons to find Be nuclei with an already small abundance [cf. Eq. (LABEL:eqb7)]. The efficiency in the D production, on the other hand, is not suppressed since the target of neutrons at the reaction H()H is proton whose abundance is very large, and dose not change significantly in this model for parameter values of and considered here.

## V Antinucleon+He annihilation

The antinucleon ()+He annihilation (as considered in Ref. Chechetkin et al. (1982)) is an important process which always operates when ’s are produced. The annihilation of (thermalized) and He affects the elemental abundances even when productions of secondary particles via He spallations by energetic hadrons can be neglected. The annihilations produce light mesons, , , , , and He.

Generated neutrons of abundance are almost completely captured by protons, and produce deuterons if the time of neutron injection is s. The final abundance of D produced through the +He annihilation is then given by

(21) |

where H is the number densities of primary injected simultaneously at the injection of neutrons [cf. Eq. (9)] relative to that of background hydrogen. The ratio in the parenthesis with subscript is the value for annihilation of species [cf. Eq. (12)]. is the fraction of annihilation into final states including a deuteron to that for total annihilation.

Equation (21) is transformed to an equation:

where is the number abundance of generated neutron relative to that of H. is the effective number of primary antinucleons per primary neutron, and is the number ratio between the neutron produced secondarily by the annihilation of plus He, and the primary neutron. The square bracket in the second line is the quantity averaged over and with weights of . The equation, i.e., , is satisfied.

We try an example estimation. We assume that abundances of nonthermal primary antinucleons are twice as large as those of primary neutron. This leads to . We assume , Balestra et al. (1988) for both and , Levitan et al. (1988), and , as done in deriving Eq. (13). The following equation is then derived:

Using Eqs. (LABEL:eq19) and (LABEL:eq20), we obtain

(24) |

This component should add to the production of D in the present model in which only effects of neutron were taken into account. The total change of D abundance is, therefore, given by . In this case, the abundance of D in the preferred parameter region (Sec. IV) is

(25) |

Figure 5 shows contours for final abundances of D (solid lines) and Li (dashed lines) on the (, ) plane in the case that the additional D production from the annihilation is taken into account by the lower limit, i.e., Eq. (24). We find that this small fraction of additional D production narrows the best parameter region in Figs. 3 and 4 without moving contours in Fig. 3 significantly.

## Vi Li production from primary antinucleons

The He annihilation produces nonthermal H and He. The nuclides with mass number three can react with background He, and produce Li. The decay of H can be neglected since its half life, i.e., y Purcell et al. (2010), is much longer than time scales of related processes [e.g., inverse of Eq. (LABEL:eq4)] in the relevant temperature range. The abundance from this -induced Li production is then estimated as

(26) | |||||

where is the ratio of the cross section for annihilation into final states including a nuclide to that for total annihilation. and are the cross section and the threshold energy for the reaction ()Li, and are the kinetic energy and the velocity of , and is the distribution function of secondary produced at the annihilation of +He as a function of . is the total reaction rate of nuclide as a function of , and is the survival fraction of Li, which is produced via the secondary reaction ()Li, during its propagation.

We try an example estimation for this component of nonthermal Li production. We assume , , and Balestra et al. (1988) for both and , Levitan et al. (1988), and , as done in Sec. V. The energy spectra of mass-three-nuclides, i.e., were assumed to be given by an result of experiment measuring the spectrum for He at He annihilation Balestra et al. (1988). In the experiment, no dependence of the spectrum on the initial energy has been observed, and the nuclide He in the final state can be identified without being confused with other hadronic species.

The reaction cross sections are taken from Ref. Cyburt et al. (2003). The total rate is assumed to be the Coulomb loss rate since the Coulomb loss dominates as long as the energy is not too high. At temperature , the Coulomb loss rate of relativistic charged particles is given by Eq. (3). The rate of non-relativistic charged particles is given Reno and Seckel (1988); Kawasaki et al. (2005b) by

(27) | |||||

The Li survival fraction is calculated in our BBN code.

In Fig. 5, dotted lines correspond to abundance ratios, i.e., Li/HLi/HLi/H, , , and (from bottom to top). In high temperature environments, Li nuclei produced in the reaction ()Li are effectively destroyed via proton burning, i.e., . A significant production of Li then occurs at relatively low temperature when the Li destruction is ineffective and the energy loss rate of secondary nuclides H and He through Coulomb scattering off background is diminished because of the reduced abundances of through their pair annihilation.

## Vii Conclusions

The injections of energetic hadrons could have occurred in the early universe by hypothetical events of decays or annihilations of long-lived exotic particles, or evaporations of exotic objects. The injections cause scattering of thermal nuclei by energetic hadrons, and showers of nonthermal nucleons, antinucleons, and nuclei can develop. Neutrons generated at the exotic events can react with Be and reduce final abundances of Li (which are mainly produced via the electron capture of Be). It has been suggested that the Be reduction can be a solution to a discrepancy between theoretical Li abundances of the SBBN model and that inferred from observations of Galactic metal-poor stars. The theoretical abundance is about a factor of three larger than the observational one.

Based on an analysis of related physical processes, we prove that the assumption of instantaneous thermalization of injected neutron provides the way to derive a conservative limit on the relation between abundances of D and Li in the hadronic energy injection model, which is independent of uncertainties in generations and reactions of nonthermal hadrons originating from the injections (Sec. II). Furthermore, two important points are stressed: 1) An uncertainty in cross sections of inelastic scattering [Eqs. (6), (7), and (8)] affects the total number of neutrons generated from the primary neutron injection, which is critical for resulting abundances of D and Li. 2) One must include effects of annihilations of antinucleons with He on a primordial D abundance even if antinucleons generated with neutron were instantaneously thermalized.

We then consider a simple model in which extra thermal neutrons are injected in a late epoch of the BBN. We estimate the probability that primordial abundances of Li in this model can be consistent with observed abundances. Relations between primordial abundances of D and Li are obtained in a manner to conserve the probability securely.

We perform a BBN calculation, and find a very small parameter region of the neutron injection time () and the number density () of injected neutron in which Li abundances are within the 2 uncertainty range determined from observation and changes in D abundance are minimum. In the preferred parameter region, the injection time is s, and its number density is times as large as that of total baryonic matter. A typical pattern of nucleosynthesis in the parameter region is analyzed (Appendix A). Situations of D production and Li reduction are observed especially (Appendix B).

We derive a model-independent result [Eq. (20)] that a reduction of Li abundance from the SBBN value down to the observational two upper limit is necessarily accompanied by an undesirable increase of D abundance up to at least the 12 upper limit (best observed value) and the 5 upper limit (mean observed value). When effects of antinucleons+He annihilations are considered utilizing a possible example case, the preferred parameter regions become narrower in the present model. BBN models involving any injections of extra neutron are, therefore, not likely to accommodate alone a reduction of primordial Li abundance to the observed level.

## Appendix A Important reactions

We analyzed nucleosynthesis with a BBN code, and found important reactions operating in the case of extra neutron injection of at