Observational Constraint on Heavy Element Production in Inhomogeneous Big Bang Nucleosynthesis

Observational Constraint on Heavy Element Production in Inhomogeneous Big Bang Nucleosynthesis

Riou Nakamura 111E-mail: riou@phys.kyush-u.ac.jp Department of Physics, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan    Masa-aki Hashimoto Department of Physics, Faculty of Sciences, Kyushu University, Fukuoka 812-8581, Japan    Shin-ichiro Fujimoto Department of Control and Information Systems Engineering, Kumamoto National College of Technology, Kumamoto 861-1102, Japan    Nobuya Nishimura National Astronomical Observatory, 2-21-1, Osawa, Mitaka, Tokyo 181-8588, Japan    Katsuhiko Sato Institute for the Physics and Mathematics of the Universe, The University of Tokyo, Chiba, 277-8568, Japan School of Science and Engineering, Meisei University, Tokyo 191-8506, Japan
July 1, 2019

Based on a scenario of the inhomogeneous big-bang nucleosynthesis (IBBN), we investigate the detailed nucleosynthesis that includes the production of heavy elements beyond Li. From the observational constraints on light elements of He and D for the baryon-to-photon ratio given by WMAP, possible regions found on the plane of the volume fraction of the high density region against the ratio between high- and low-density regions.

In these allowed regions, we have confirmed that the heavy elements beyond Fe can be produced appreciably, where - and/or -process elements are produced well simultaneously compared to the solar system abundances. We suggest that recent observational signals such as He overabundance in globular clusters and high metallicity abundances in quasars could be partly due to the results of IBBN. Possible implications are given for the formation of the first generation stars.

26.35.+c, 98.80.Ft, 13.60.Rj

I Introduction

Big bang nucleosynthesis has been investigated mainly on the context of the standard cosmological model (SBBN), where origin of light elements of He, D, and Li have been discussed in detail Iocco:2008va (). While observations of He are still in debate with the uncertainty of 20-30 % in the abundance Luridiana2003 (); OliveSkillman04 (); Izotov:2007ed (), those of D constrain severely the possible range of the abundance production in the early universe Kirkman2003 (); OMeara2006 (); Pettini2008 (). Contrary to the above standard BBN, the heavy element nucleosynthesis beyond the mass number has been proposed from twenty years ago IBBN0 (); IBBN1 (); TerasawaSato89 (); Alcock1987 (); 2zone (); Jedamzik1994 (); Matsuura:2004ss (), where the model is called the inhomogeneous BBN (IBBN). This model relays on the inhomogeneity of baryon concentrations that could be induced by baryogenesis (e.g. Ref. Matsuura:2004ss ()) or some phase transitions such as QCD or electro-weak phase transition Alcock1987 (); Fuller1988 (); IBBN_QCD () during the expansion of the universe. Although a large scale inhomogeneity is inhibited by many observations WMAP3 (); WMAP5 (), small scale one has been advocated within the present accuracy of the observations. Therefore, it remains a possibility for IBBN to occur in some degree during the early era.

On the other hand, Wilkinson Microwave Anisotropy Probe (WMAP) has derived critical parameters concerning the cosmology of which the present baryon-to-photon ratio is determined to be  WMAP5 (). This value is almost consistent with that obtained from the observation of D. Therefore, considering the uncertainty of the He abundance, we can fix the ratio in the discussion of the nucleosynthesis in the early universe. If the present ratio of is determined, BBN can be performed along that line in the thermodynamical history with use of the nuclear reaction network. On the other hand, peculiar observations of abundances for heavy elements and/or He could be understood in the way of IBBN. For example, the quasar metallicity of C, N, and Si could have been explained from IBBN Juarez2009 (). Furthermore, from recent observations of globular clusters, possibility of inhomogeneous helium distribution is pointed out Moriya2010 (), where some separate groups of different main sequences in blue band of low mass stars are assumed due to high primordial helium abundances compared to the standard value Bedin2004 (); Piotto2007 ().

Despite a negative opinion against IBBN due to insufficient consideration of the scale of the inhomogeneity Rauther2006 (), Matsuura et al. have found that the heavy element synthesis for both - and -processes is possible if Matsuura2005 (), where they have also shown that the high regions are compatible with the observations of the light elements, He and D Matsuura2007 (). However, their analysis is only limited to a parameter of a specific baryon number concentration. Therefore, it should be needed to constrain the possible regions from available observations in the wide parameter space that describes the IBBN.

In §II, we review and give the adopted model of IBBN Matsuura2007 (). Constraints on the critical parameters of IBBN due to light element observations are shown in §III, and the productions of possible heavy element nucleosynthesis is presented in §IV. Finally, §V is devoted to the summary and discussion.

Ii Cosmological Model

We adopt the two-zone model for the inhomogeneous BBN, where the early universe is assumed to have the high- and low- baryon density regions IBBN1 () under the background temperature . For simplicity we ignore the diffusion effects before and during the primordial nucleosynthesis , where the plausibility will be discussed in §V. After the epoch of BBN, all the elements are assumed to be mixed homogeneously.

Let us define the notations, , and as averaged-, high-, and low- baryon number densities. is the volume fraction of the high baryon density region. and are mass fractions of each element in averaged-, high- and low-density regions, respectively, Then, basic relations are written as follows:


Here we assume the baryon fluctuation to be isothermal as was done in previous studies (e.g., Refs. TerasawaSato89 (); Alcock1987 (); Fuller1988 ()). Under that assumption, since the baryon-to-photon ratio is defined by the number density of photon in standard BBN, , Eqs.(1) and (2) are rewritten as follows:


where s with subscripts are the baryon-to-photon ratios in each region. In the present paper, we fix from the cosmic microwave background observation WMAP3 (); WMAP5 (). and are obtained from both and the density ratio between high- and low-density region: .

To calculate the evolution of the universe, we solve the following Friedmann equation,


where is the cosmic scale factor and is the gravitational constant. The total energy density is the sum of decomposed parts:

Here the subscripts , and indicate photons, electrons/positrons, neutrinos, and baryons, respectively. We note that is the average value of baryon density obtained from Eq. (1).

The energy conservation law is used to get the time evolution of the temperature and the baryon density,


where is the pressure of the fluid.

Iii Constraints from light-element observations

In this section, we calculate the nucleosynthesis in high- and low-density regions with use of the BBN code Hashimoto1985 () which includes 24 nuclei from neutron to O. We adopt the reaction rates of NACRE NACRE (), the neutron life time sec Hagiwara:2002fs (), and take account of the number of species of the massless neutrinos .

Figure 1 illustrates the light element synthesis in the high- and low-density regions with and that correspond to and . In the low-density region the evolution of the elements is almost the same as that of standard BBN. In the high-density region, while He is more abundant than that in the low-density region, Li (or Be) is much less produced. It implies that heavier nuclei such as O, hardly synthesized in SBBN, are synthesized at high-density region.

Fig. 1: Illustration of the nucleosynthesis in the two-zone IBBN model with and . The baryon-to-photon ratios in the high- (left panel) and low- (right panel) density regions are and , respectively.

For , the heavier elements can be synthesized in the high-density regions as discussed in Ref. Jedamzik1994 (). For , contribution of the low-density region to can be neglected and therefore to be consistent with observations of light elements, we need to impose the condition of . Now, we put constraints on and by comparing the average values of He and D obtained from Eq. (4) with the following observational values. First we adopt the primordial He abundance reported in Ref. OliveSkillman04 ():


Next, we take the primordial abundance from the D/H observation reported in Ref. PDG2008 ()


where the systematic error given in Ref. OMeara2006 () is adopted.

Figure 2 illustrates the constraints on the plane from the above light-element observations with contours of constant . The solid and dashed lines indicate the upper limits from Eqs. (7) and (8), respectively. As the results, we can obtain approximately the following relations between and :


As shown in Figure 2, we can find the allowed regions which include the very high-density region such as .

Matsuura et al. Matsuura2007 () defined a parameter of the baryon number concentration in the high density region instead of two parameters of and that are needed to solve Eqs.(3) and (4):

However, they have only examined the case of and , where for . Our constraints in Eq. (9) correspond to . Since we have fixed the value of , we can obtain physically more reasonable regions on the plane of .

Naturally, as takes larger value, nuclei which are heavier than Li are synthesized more and more. Then we can estimate the amount of total CNO elements in the allowed region. Figure 3 illustrates the contours of the summation of the average values of the heavier nuclei (), which correspond to Fig. 2 and are drawn using the constraint from He and D/H observations . As a consequence, we get the upper limit of total mass fractions for heavier nuclei as follows: .

We should note that abundance flows proceed beyond the CNO elements thanks to the larger network for high -values as shown in Table 2 of the following section.

Fig. 2: Constraints on the plane from the observations of light element abundances. The region below the solid line is allowed one obtained from He observation OliveSkillman04 (). Constraints from the D/H observation PDG2008 () are shown by the region below the dot-dashed line. The shaded region is the allowed parameters determined from the two observations of He and D/H. The dotted lines show the contours of the baryon-to-photon ratio in the high-density region. Filled squares indicate the parameters for heavy element nucleosynthesis adopted in §IV.
Fig. 3: Contours of the averaged total mass fractions which are the sum of nuclei heavier than Li, where we find consistent regions with He and D observations.

Iv Heavy element Production

In the previous section, we have obtained the amount of CNO elements produced in the two-zone IBBN model. However, it is not enough to examine the nuclear production beyond because the baryon density in the high-density region becomes so high that elements beyond CNO isotopes can be produced Wagoner1967 (); 2zone (); Matsuura:2004ss (); Matsuura2005 ().

In this section, we investigate the heavy element nucleosynthesis in the high-density region considering the constraints shown in Fig. 2. The temperature and density evolutions are the same as used in the previous section. Abundance change is calculated with a large nuclear reaction network, which includes 4463 nuclei from neutron , proton to Americium (Z = 95 and A = 292). Nuclear data, such as reaction rates, nuclear masses, and partition functions, are the same as used in fujimoto () except for the neutron-proton interaction; We take the weak interaction rates between n and p from Kawano code Kawano (), which is adequate for the high temperature epoch of  K. We note that mass fraction of He and D obtained with the large network are consistent with those in in §III within the accuracy of few percents.

As seen in Fig. 3, heavy elements of are produced nearly along the upper limit of . Therefore, to examine the efficiency of the heavy element production, we select five models with the following parameters: , and corresponded to , , , , and . Adopted parameters are indicated by filled squares in Fig. 2.

Figure 4 shows the results of nucleosynthesis in the high-density regions with and . For , the nucleosynthesis paths are classified with the mass number Matsuura2005 (). For nuclei of mass number , proton captures are very active compared to the neutron capture of  K and the path moves to the proton rich side, which began by breaking out of the hot CNO cycle. For nuclei of , the path goes across the stable nuclei from proton to neutron rich side, since the temperature decreases and the number of seed nuclei of the neutron capture process increase significantly. Concerning heavier nuclei of , neutron captures become much more efficient. In Figure 4(a), we see the time evolution of the abundances of Gd and Eu for the mass number 159. First Tb (stable -element) is synthesized and later Gd and Eu are synthesized through the neutron captures. After sec, Eu decays to nuclei by way of Eu Gd Tb, where the lifetimes of Eu and Gd are min and h, respectively. These neutron capture process is not similar to the canonical process, since the nuclear processes proceed under the condition of the high-abundance of protons.

For , the reactions first proceed along the stable line, because triple- reactions and other particle induced reactions are very effective. Subsequently, the reactions directly proceeds to the proton rich region, through rapid proton captures. As shown in Fig. 4(b), Sn which is proton-rich nuclei is synthesized. After that, stable nuclei Cd is synthesized by way of Sn In Cd, where the lifetimes of Sn and In are min and min, respectively. In addition, we notice the production of radioactive nuclei of Ni and Co, where Ni is produced at early times, just after the formation of He. Usually, nuclei such as Ni and Co are produced in supernova explosions, which are assumed to be the events after the first star formation (e.g. Ref. Hashimoto1995 ()). In IBBN model, however, this production can be found to occur at extremely high density region of as the primary elements without supernova events in the early universe.

To explain differences of the nuclear reactions which depend on the baryon density, we focus on the neutron abundances. Figure 5 shows the evolutions of the neutron abundances in the SBBN and IBBN models. For , neutron abundance decreases rapidly at 10 sec to the formation of He and Ni. Thus, neutron abundance is not enough to induce the neutron capture producing heavy nuclei of . On the other hand, neutron abundance tends to remain even at the high temperature for the lower value of . We can see the case of , where there remain much neutrons to occur the neutron capture reaction. Thus the neutron capture process to produce heavy elements of can become active.

Time scales in the decrease for the neutron abundances change drastically the flow of the abundance production. Figures 6 and 7 show the flows for . Before the significant decrease in the neutron abundances before sec, the nucleosynthesis proceeds already along the stable line by way of the neutron included reactions (Fig. 6). At that time, the nuclear reactions are stuck around with , since it takes time to synthesize heavier nuclei because Nd () and Sm () have some stable isotopes. As time goes, neutron captures of these nuclei start, where the neutron captures proceed significantly and -elements can be synthesized. After the depletion of neutrons ( sec), nuclei around the neutron numbers are produced through proton induced reactions such as Sm (Fig. 7).

Final results ( K) of nucleosynthesis calculations are shown in Tables 1 and 2. Table 1 shows the abundances of light elements, He, D, and Li, in high- and low-density regions with their average values. Abundances of the low-density side (the third and sixth columns) are obtained from the calculation by BBN code used in §III, because abundance flows beyond are negligible. We should note that the average abundances of He and D are consistent with their observational values of (7) and (8). Table 2 shows the amounts of heavy elements. When we have calculate the average values, we set the abundances of as zero for low-density side. For , a lot of nuclei of are synthesized whose amounts are comparable to that of Li. Produced elements in this case include both -element (i.e. Ba) and -elements (for instance, Ce and Nd), since moderate amounts of neutrons remain as shown in Fig. 5

For , there are few -elements while both -elements (i.e. Kr and Y) and -elements (i.e Se and Kr) are synthesized such as the case of supernova explosions. Although heavy nuclei of are not synthesized appreciably, those of are produced well owing to the explosive nucleosynthesis under the high density circumstances (). The most abundant element is found to be Ni whose production value is much larger than the estimated upper limit of the total mass fraction (shown in Fig.3) derived from the BBN code calculations. This is because our BBN code used in §III includes the elements up to and the actual abundance flow proceeds to much heavier elements.

Figure 8 shows the abundances averaged between high- and low-density region using Eq. (4) compared with the solar system abundances Anders1989 (). For , abundance productions of are comparable to the solar values. For , those of have been synthesized well. In the case of , there are outstanding two peaks; one is around and the other can be found around . Abundance patterns are very different from that of the solar system, because IBBN occurs under the condition of significant abundances of both neutrons and protons.

Fig. 4: Time evolution of the mass fractions in high-density regions of (a) and (b) .
Fig. 5: Time evolution of the neutron abundance in SBBN () and IBBN in the high-density region.
Fig. 6: Abundance distribution at K ( sec) in . The gray regions are those of stable nuclei.
Fig. 7: . Abundances distribution at K ( sec) in .
(, ) (, )
sec,  K sec,  K
elements high low average high low average

(a) For cases of and .

(, ) (, )
sec,  K sec,  K
elements high low average high low average

(b) For cases of and .

Table. 1: Mass fractions of light elements for follow cases : , , , and . and is the time and temperature at the final stage of the calculations.
element high average element high average element high average
Ni Nd Nd
Co Ni Ca
Sr Sm Mn
Sr Pm Eu
Se Sm Ce
Se Pm Cr
Sr Sm Ce
Kr Pm Ni
Kr Sm Nd
Ge Pm Eu
Kr Pm Nd
Kr Nd Fe
Kr Sm Tb
Ge Nd La
Se Sm N
Br Pm Cr
Se Sm Ba
Y Ni C
Zr Cd Dy
Rb Eu C
Rb Eu O
Y Cd Gd
Zr Eu Cs
As Eu Nd
Ga Ce Ho
Se Eu Pr
Nb Cd Ce
Table. 2: Mass fractions of heavy elements for three cases of , and .
Fig. 8: Comparison of the averaged mass fractions in the two-zone model with the solar system abundances Anders1989 () (indicated by dots).
Fig. 9: Same as Fig. 8, but is fixed as .

V Summary and Discussion

We have investigated the consistency between inhomogeneous BBN and the observation of He and D/H abundances under the standard cosmological model having determined by WMAP. We have adopted the two-zone model, where the universe has the high- and low- baryon density regions at the BBN epoch.

First, we have calculated the light element nucleosynthesis using the BBN code having 24 nuclei for the high- and low-density regions. We have assumed that the diffusion effect is negligible. There are significant differences for the time evolution of the light element between the high- and low-density regions; In the high-density region, the nucleosynthesis begins faster and He is more abundant than that in the low density region as shown in Figure 4. From He and D/H observations, we can put severe constraint on two parameters of the two-zone model: the volume fraction of the high-density region and the density ratio between the two regions, where we have assumed that abundances in the two regions are mixed homogeneously.

Second, using the allowed parameters constrained from the light element observations, we calculate the nucleosynthesis that includes 4463 nuclei in the high-density regions. Qualitatively, results of nucleosynthesis are the same as those in Ref. Matsuura2005 (). In the present results, we showed that - and -elements are synthesized simultaneously at high-density region with .d Such a curious site of the nucleosynthesis have never been known in previous studies of nucleosynthesis.

As the results, we have obtained the average values of mass fractions from the nucleosynthesis in high-density and that in low density regions. The total averaged mass fractions beyond the light elements are constrained to be (for ) and (for ). We find that the average mass fractions in IBBN amount to as much as the solar system abundances. As see from Fig. 8, there are over-produced elements around (for ) and (for ). It seems to be conflict with the chemical evolution of the universe. However, we show only the results of the upper-bounds on diagram. Since and are free-parameters, over-production can be avoided by the adjustment of and/or . Figure 9 illustrates the mass fraction in with various sets. It is shown that the abundance pattern can be lower than the solar system abundance. Although we showed here only the result of case, it is possible to avoid producing over-abundance in other parameters, and . If we put constraint on the plane from the heavy element observations, the limit of those parameters should be tightly.

In our calculation, the radioactive nuclei are produced much in the high-density region. Especially, we should note that Ni decays into Fe (Ni Co Fe), where the existence of Fe surely affects the process of the formation of the first generation stars. Therefore, it may be also necessary for IBBN to be constrained from the star formation scenarios, because opacity change due to IBBN will affect them.

Recent observational signal of over-abundances of He mass fractions in globular clusters could motivate the IBBN scenario toward the detailed modeling. The over-abundances of He are suggested to be in the range of where estimated from the H-R diagram of the blue Main-Sequence of NGC2808 in Ref.Piotto2007 (). If the origin of He in globular clusters is due to IBBN, must be greater than in some regions during the epoch of BBN. Then, the averaging procedure could be constrained from the more detailed observations of abundances. Since the history of changes in abundances has been investigated in detail through the chemical evolution of galaxies Anderson2009 (), further plausible constrains on the averaging process should be studied in the next step.

In our study, we ignore the diffusion effects. However, it is shown that the diffusion affects the primordial nucleosynthesis significantly IBBN1 (). Matsuura et al. Matsuura2007 () has estimated the size of the high-baryon density island to be  m –  m at the BBN epoch. The upper bound is obtained from the maximum angular resolution of CMB and the lower is from the analysis of comoving diffusion length of neutron and proton given in Ref. IBBN0 (). In our case, we can estimate the scale of the high-density and the effects of the diffusion from .

The neutron diffusion effects can be discussed with use of the results obtained in the previous section by comparing the scale of the high-density region with the diffusion length. The present value of the Hubble length is  m. We may estimate the scale of the high-density region from the Hubble length multiplied by . From ranges of the volume fraction adopted in §IV, , we obtain the scale of the high-density regions at present epoch as  m  m. We can estimate the scale at redshift from the relation . As the result, we expect at BBN era as  m  m. We can say that the nucleon diffusion effects would be neglected because the diffusion length is much smaller than .

On the other hand, the high-density region is expected to be smaller than  m. It seems to be very bad that the upper bound of is larger than the value as far as our two zone model is concerned. However, the high-density island cannot be observed directly, since we assume that the high- and low-density regions become homogeneous after the nucleosynthesis.

Finally, distances between high density regions are difficult to derive without specific models beyond the two-zone model. We will plan to calculate the nucleosynthesis with the diffusion of abundances and/or more plausible averaging process included.

This work has been supported in part by a Grant-in-Aid for Scientific Research (18540279, 19104006, 21540272) of the Ministry of Education, Culture, Sports, Science and Technology of Japan, and in part by a grant for Basic Science Research Projects from the Sumitomo Foundation (No. 080933).


  • (1) G. Steigman, Ann. Rev. Nucl. Part. Sci. 57, 463 (2007);
    F. Iocco, G. Mangano, G. Miele, O. Pisanti and P. D. Serpico, Phys. Rept. 472, 1 (2009)
  • (2) V. Luridiana,A. Peimbert, M. Peimbert, & M. Cervino, Astrophys. J. 592, 846 (2003)
  • (3) Y. I. Izotov, T. X. Thuan and G. Stasinska, Astrophys. J. 662, 15 (2007)
  • (4) Olive & Skillman, Astrophys. J., 617, 29–40 (2004)
  • (5) D. Kirkman, D. Tytler, N. Suzuki, J. M. O’Meara and D. Lubin, Astrophys. J. Suppl. 149, 1 (2003) [arXiv:astro-ph/0302006].
  • (6) M. Pettini, B. J. Zych, M. T. Murphy, A. Lewis, & C. C. Steidel, Mon. Not. R. Astron. Soc. 391, 1499, (2008)
  • (7) J. M. O’Meara, S. Burles, J. X. Prochaska, G. E. Prochter, R. A. Bernstein and K. M. Burgess, Astrophys. J. 649, L61 (2006)
  • (8) C. Alcock, G.M. Fuller, and G.J. Mathews, Astrophys. J. 320, 439 (1987)
  • (9) N. Terasawa and K. Sato, Phys. Rev. D 39, 2893 (1989)
  • (10) K. Jedamzik, and J.B. Rehm, Phys. Rev. D64, 023510 (2001)[astro-ph/0101292];
    T. Rauscher, H. Applegate, J. Cowan, F. Thielmann, and M. Wiescher, Astrophys. J. 429, 499 (1994).
  • (11) J. H. Applegate, C. J. Hogan, and R. J. Scherrer, Phys. Rev. D35, 1151 (1987)
  • (12) R. M. Malaney and W. A. Fowler, Astrophys. J 333, 14 (1988);
    J. H. Applegate, C. J. Hogan, R. J. Scherrer, Astrophys. J. 329, 572 (1988);
    N. Terasawa and K. Sato, Astrophys. J. 362, L.47 (1990);
    D. Thomas, D. N. Schramm, K.A. Olive, G. J. Mathews, B. S. Meyer, and B. D.  Fields, Astrophys. J. 430, 291 (1994);
  • (13) K. Jedamzik, G. M. Fuller, G. J. Mathews, and T. Kajino, Astrophys. J. 422, 423 (1994);
  • (14) S. Matsuura, A. D. Dolgov, S. Nagataki and K. Sato, Prog. Theor. Phys. 112, 971 (2004)
  • (15) G. M. Fuller, G. J. Mathews and C. R. Alcock, Phys. Rev. D 37, 1380 (1988);
  • (16) H. Kurki-Suonio and R. A. Matzner, Phys.Rev. D39, 1046 (1989);
    H. Kurki-Suonio and R. A. Matzner, Phys.Rev. D42, 1047 (1990);
  • (17) C.L. Bennett, et al., Astrophys. J. Suppl. 148, 1 (2003)
    D. N. Spergel et al., Astrophys. J. Suppl. 170, 377 (2007)
    J. Dunkley et al. Astrophys. J. Suppl. 180, 306 (2009)
  • (18) E. Komatsu et al., arXiv:1001.4538 [astro-ph.CO].
  • (19) Y. Juarez, R. Maiolino, R. Mujica, M. Pedani, S. Marinoni, T. Nagao, A. Marconi, & E. Oliva, Astron. & Astrophys., 494, L25, (2009)
  • (20) L. R. Bedin et al., Astrophys. J., 605, L125 (2004);
  • (21) G. Piotto et al., Astrophys. J., 661 L53, (2007)
  • (22) T. Rauscher, Phys. Rev. D 75, 068301 (2007)
  • (23) S. Matsuura, S. I. Fujimoto, S. Nishimura, M. A. Hashimoto and K. Sato, Phys. Rev. D 72, 123505 (2005)
  • (24) S. Matsuura, S. I. Fujimoto, M. A. Hashimoto and K. Sato, Phys. Rev. D 75, 068302 (2007).
  • (25) K. Jedamzik [astro-ph/9911242].
  • (26) S. Fujimoto,M. Hashimoto, O. Koike,K. Arai, & R. Matsuba, Astrophys. J. 585, 418 (2003),
    O. Koike, M. Hashimoto, R. Kuromizu, & S. Fujimoto, Astrophys. J. 603, 592 (2004),
    S. Fujimoto, M. Hashimoto, K. Arai, & R. Matsuba, Astrophys. J. , 614, 847 (2004),
    S. Nishimura, K. Kotake, M. Hashimoto, S. Yamada, N. Nishimura, S. Fujimoto and K. Sato, Astrophys. J. 642, 410 (2006).
  • (27) M. Hashimoto & K. Arai, Physics Reports of Kumamoto University, 7, 47, (1985).
  • (28) B. Fields and S. Sarkar, arXiv:astro-ph/0601514.
  • (29) L. Kawano, FERMILAB-Pub-92/04-A
  • (30) E. Anders and N. Grevesse, Geochim. Cosmochim. Acta 53, 197 (1989).
  • (31) C. Angulo, M. Arnould, M. Rayet, P. Descouvemont, D. Baye, C.  Leclercq-Willain, A. Coc, S. Barhoumi, P. Aguer, C. Rolfs, et al., Nuclear Physics A 656, 3 (1999).
  • (32) M. Hashimoto, Progress of Theoretical Physics, 94, 663, (1995).
  • (33) K. Hagiwara et al. [Particle Data Group], Phys. Rev. D 66, 010001 (2002).
  • (34) R. V. Wagoner, W. A. Fowler, & F. Hoyle, Astrophys. J. , 148, 3 (1967)
  • (35) M. E. Anderson, J. N. Bregman, S. C. Butler and C. R. Mullis, Astrophys. J. 698, 317 (2009)
  • (36) T. Moriya and T. Shigeyama, Phys. Rev. D 81, 043004 (2010)
Comments 0
Request Comment
You are adding the first comment!
How to quickly get a good reply:
  • Give credit where it’s due by listing out the positive aspects of a paper before getting into which changes should be made.
  • Be specific in your critique, and provide supporting evidence with appropriate references to substantiate general statements.
  • Your comment should inspire ideas to flow and help the author improves the paper.

The better we are at sharing our knowledge with each other, the faster we move forward.
The feedback must be of minimum 40 characters and the title a minimum of 5 characters
Add comment
Loading ...
This is a comment super asjknd jkasnjk adsnkj
The feedback must be of minumum 40 characters
The feedback must be of minumum 40 characters

You are asking your first question!
How to quickly get a good answer:
  • Keep your question short and to the point
  • Check for grammar or spelling errors.
  • Phrase it like a question
Test description