Journal of Astrophysics
Volume 2013, Issue 1 587294
Research Article
Open Access

Constraint on Heavy Element Production in Inhomogeneous Big-Bang Nucleosynthesis from the Light Element Observations

Riou Nakamura, 

Corresponding Author

Riou Nakamura

Department of Physics, Graduate School of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan kyushu-u.ac.jp

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Masa-aki Hashimoto, 

Masa-aki Hashimoto

Department of Physics, Graduate School of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan kyushu-u.ac.jp

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Shin-ichiro Fujimoto, 

Shin-ichiro Fujimoto

Department of Control and Information Systems Engineering, Kumamoto National College of Technology, 2659-2 Suya, Koshi, Kumamoto 861-1102, Japan ac.jp

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Katsuhiko Sato, 

Katsuhiko Sato

Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan u-tokyo.ac.jp

National Institutes of Natural Sciences, Kamiyacho Central Place 2F, 4-3-13 Toranomon, Minato-ku, Tokyo 104-0001, Japan nins.jp

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First published: 01 September 2013
https://doi.org/10.1155/2013/587294
Academic Editor: G. Chincarini
Academic Editor: X. Dai
Academic Editor: A. De Rosa
Academic Editor: Milan S. Dimitrijevic
Academic Editor: G. Wegner

Abstract

We investigate the observational constraints on the inhomogeneous big-bang nucleosynthesis that Matsuura et al. (2005) suggested that states the possibility of the heavy element production beyond 7Li in the early universe. From the observational constraints on light elements of 4He and D, possible regions are 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 Ni can be produced appreciably, where p- and/or r-process elements are produced well simultaneously.

1. Introduction

Big-bang nucleosynthesis (BBN) has been investigated to explain the origin of the light elements, such as  4He, D,  3He, and  7Li, during the first few minutes [1–4]. Standard model of BBN (SBBN) can succeed in explaining the observation of those elements,  4He [5–9], D [10–13], and  3He [14, 15], except for  7Li. The study of SBBN has been done under the assumption of the homogeneous universe, where the model has only one parameter, the baryon-to-photon ratio η. If the present value of η is determined, SBBN can be calculated from the thermodynamical history with the use of the nuclear reaction network. We can obtain the reasonable value of η by comparing the calculated abundances with observations. In the meanwhile, the value of η is obtained as η = (5.1 − 6.5) × 10−10 [1] from the observations of  4He and D. These values agree well with the observation of the cosmic microwave background: η = (6.19 ± 0.14) × 10−10 [16].

On the other hand, BBN with the inhomogeneous baryon distribution also has been investigated. The model is called as inhomogeneous BBN (IBBN). IBBN relies on the inhomogeneity of baryon concentrations that could be induced by baryogenesis (e.g., [17]) or phase transitions such as QCD or electro-weak phase transition [18–21] during the expansion of the universe. Although a large-scale inhomogeneity is inhibited by many observations [16, 22–24], a 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. In IBBN, the heavy element nucleosynthesis beyond the mass number A = 8 has been proposed [17, 18, 25–35]. In addition, peculiar observations of abundances for heavy elements and/or  4He could be understood in the way of IBBN. For example, the quasar metallicity of C, N, and Si could have been explained from IBBN [36]. Furthermore, from recent observations of globular clusters, a possibility of inhomogeneous helium distribution is pointed out [37], 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 [38, 39]. Although baryogenesis could be the origin of the inhomogeneity, the mechanism of it has not been clarified due to unknown properties of the supersymmetric Grand Unified Theory [40].

Despite a negative opinion against IBBN due to insufficient consideration of the scale of inhomogeneity [41], Matsuura et al. have found that the heavy element synthesis for both p- and r-processes is possible if η > 10−4 [42], where they have also shown that the high η regions are compatible with the observations of the light elements,  4He and D [43]. However, their analysis is only limited to a parameter of a specific baryon number concentration. In this paper, we extend the investigations of Matsuura et al. [42, 43] to check the validity of their conclusion from a wide parameter space of the IBBN model.

In Section 2, we review and give the adopted model of IBBN which is the same one as that of Matsuura et al. [43]. Constraints on the critical parameters of IBBN due to light element observations are shown in Section 3, and the possible heavy elements of nucleosynthesis are presented in Section 4. Finally, Section 5 is devoted to the summary and discussion.

2. Model

In this section, we introduce the model of IBBN. We adopt the two-zone model for the inhomogeneous BBN. In the IBBN model, we assume the existence of spherical high-density region inside the horizon. For simplicity, we ignore in the present study the diffusion effects before (1010 K < T < 1011 K) and during the primordial nucleosynthesis (107 K < T < 1010 K), because the timescale of the neutron diffusion is longer than that of the cosmic expansion [25, 37].

To find the parameters compatible with the observations, we consider the average abundances between the high- and low-density regions. We get at least the parameters for the extreme case by averaging the abundances in two regions. Let us define the notations, nave, nhigh, and nlow as average,  high, and low baryon number densities. fv is the volume fraction of the high baryon density region. , , and are mass fractions of each element i in average, and high- and low-density regions, respectively. Then, basic relations are written as follows [43]:
()
Here, we assume the baryon fluctuation to be isothermal as was done in previous studies (e.g., [18, 19, 30]). Under that assumption, since the baryon-to-photon ratio is defined by the number density of photon in standard BBN, (1) is rewritten as follows:
()
()
where ηs with subscripts are the baryon-to-photon ratios in each region. In the present paper, we fix ηave = 6.19 × 10−10 from the cosmic microwave background observation [16]. The values of ηhigh and ηlow are obtained from both fv and the density ratio between high- and low-density regions: R ≡ nhigh/nlow = ηhigh/ηlow.
To calculate the evolution of the universe, we solve the following Friedmann equation:
()
where x is the cosmic scale factor and G is the gravitational constant. The total energy density ρ in (4) is the sum of decomposed parts:
()
Here, the subscripts γ, ν, and e± indicate photons, neutrino, and electrons/positrons, respectively. The final term is the baryon density obtained as ρb≃mpnave.
We should note the energy density of baryon. To get the time evolution of the baryon density in both regions, the energy conservation law is used as follows:
()
where p is the pressure of the fluid. When we solve (6), initial values in both regions are obtained from (2) with fv and R fixed. For ηhigh ≥ 2 × 10−4, the baryon density in the high-density region, ρhigh, is larger than the radiation component at T > 109 K. However, we note that the contribution to (5) is not ρhigh, but fvρhigh. In our research, the ratio of fvρhigh to ργ is about 10−7 at BBN epoch. Therefore, we can neglect the final term of (5) in the same way as it has been done in SBBN during the calculation of (4).

3. Constraints from Light Element Observations

In this section, we calculate the nucleosynthesis in high- and low-density regions with the use of the BBN code [44] which includes 24 nuclei from neutron to  16O. We adopt the reaction rates of Descouvemont et al. [45], the neutron lifetime τN = 885.7 sec [1], and consider three massless neutrinos.

Let us consider the range of fv. For fv ≪ 0.1, the heavier elements can be synthesized in the high-density regions as discussed in [33]. For fv > 0.1, contribution of the low-density region to ηave can be neglected, and therefore to be consistent with the observations of light elements, we need to impose the condition of fv < 0.1.

Figure 1 illustrates the light element synthesis in the high- and low-density regions with fv = 10−6 and R = 106 that corresponds to ηhigh = 3.05 × 10−4 and ηlow = 3.05 × 10−10. Light elements synthesized in these calculations are shown in Table 1. In the low-density region, the evolution of the elements is almost the same as the case of SBBN. In the high-density region, while  4He is more abundant than that in the low-density region,  7Li (or  7Be) is much less produced. In this case, we can see that average values such as  4He and D are overproduced as shown in Table 1. However, this overproduction can be saved by choosing the parameters carefully. We need to find the reasonable parameter ranges for both fv and R by comparing with the observation of the light elements.

Table 1. The numerical abundances of light elements synthesized as shown in Figure 1.
Elements Xi
p 0.608 0.759 0.684
D 3.07 × 10−18 1.19 × 10−4 5.95 × 10−5
T + 3He 1.15 × 10−13 3.41 × 10−5 1.71 × 10−5
4He 0.392 0.241 0.316
7Li + 7Be 8.2 × 10−13 6.29 × 10−10 3.14 × 10−10
Details are in the caption following the image
Figure 1 (a)
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Illustration of the nucleosynthesis in the two-zone IBBN model with fv = 10−6 and R = 106. The baryon-to-photon ratios in the high (a) and low (b) density regions are ηhigh = 3.05 × 10−4 and ηlow = 3.05 × 10−10, respectively.
Details are in the caption following the image
Figure 1 (b)
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Illustration of the nucleosynthesis in the two-zone IBBN model with fv = 10−6 and R = 106. The baryon-to-photon ratios in the high (a) and low (b) density regions are ηhigh = 3.05 × 10−4 and ηlow = 3.05 × 10−10, respectively.
Now, we put constraints on fv and R by comparing the average values of  4He and D obtained from (3) with the following observational values. First, we consider the primordial  4He abundance reported in [8]:
()
and [9]:
()
We adopt  4He abundances as follows:
()
Next, we take the primordial abundance from the D/H observation reported in [12]:
()
and [13]:
()
Considering those observations with errors, we adopt the primordial D/H abundance as follows:
()
Figure 2 illustrates the constraints on the fv − R plane from the above light element observations with contours of constant ηhigh. The solid and dashed lines indicate the upper limits from (9) and (12), respectively. From the results, we can obtain approximately the following relations between fv and R:
()
The  4He observation (9) gives the upper bound for fv<7.4×10−6, and the limit for fv>7.4 × 10−6 is obtained from D observation (12). As shown in Figure 2, we can find the allowed regions which include the very high-density region such as ηhigh = 10−3.
Details are in the caption following the image
Figure 2
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Constraints on the fv − R plane from the observations of light element abundances. The region below the red line is an allowed one obtained from  4He observation (9). Constraints from the D/H observation (12) are shown by the region below the blue line. The gray region corresponds to the allowed parameters determined from the two observations of  4He and D/H. There is another region which is still consistent with only D/H in the upper right direction. This is the contribution of the low-density region with ηlow ~ 10−12. The D abundance tends to decrease against the baryon density for η > 10−12. 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 Section 4.
We should note that ηhigh takes a larger value, nuclei which are heavier than  7Li 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 (A > 7), which correspond to Figure 2 and are drawn using the constraint from  4He and D/H observations. As a consequence, we get the upper limit of total mass fractions for heavier nuclei as follows:
()
Details are in the caption following the image
Figure 3
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Contours of the average total mass fractions which are the sum of the nuclei heavier than  7Li, where we find a consistent region of the produced elements with  4He and D observations.

4. 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 A > 8 because the baryon density in the high-density region becomes so high that elements beyond CNO isotopes can be produced [17, 31, 32, 34, 42]. In this section, we investigate the heavy element nucleosynthesis in the high-density region considering the constraints shown in Figure 2. Abundance change is calculated with a large nuclear reaction network, which includes 4463 nuclei from neutron (n) and proton (p) to Americium (Z = 95 and A = 292). Nuclear data, such as reaction rates, nuclear masses, and partition functions, are the same as the ones used in [46–49] except for the neutron-proton interaction. We use the weak interaction of Kawano code [50], which is adequate for the high-temperature epoch of T > 1010 K.

As seen in Figure 3, heavy elements of X(A > 7) > 10−9 are produced nearly along the upper limit of R. Therefore, to examine the efficiency of the heavy element production, we select five models with the following parameters: ηhigh = 10−3, 5.1 × 10−4, 10−4, 5.0×10−5, and 10−5 corresponded to (fv, R) = (3.24 × 10−8,  1.74 × 106), (1.03  ×  10−8,  9.00  ×  105), (5.41 × 10−7,  1.84 × 105), (1.50 × 10−6,  9.20 × 104), and (5.87 × 10−6,  1.82 × 104). Adopted parameters are indicated by filled squares in Figure 2.

First, we evaluate the validity of the nucleosynthesis code with 4463 nuclei. Table 2 shows the results of the light elements, p, D,  4He,  3He, and  7Li. The results of the high-density region are calculated by the extended nucleosynthesis code, and the abundances in the low-density region are obtained by BBN code. The average abundances are obtained by (3). Since the average values of  4He and D are consistent with the observations, there is no difference between BBN code and the extended nucleosynthesis code in regard to the average abundances of light elements.

Table 2. Mass fractions of light elements for the four cases: ηhigh≃10−3, ηhigh = 5 × 10−4, ηhigh≃10−4, and ηhigh = 10−5. tfin and Tfin are the time and temperature at the final stage of the calculations.
fv, R 3.23 × 10−8, 1.74 × 106 1.03 × 10−7, 9.00 × 105
(ηhigh, ηlow) (1.02 × 10−3, 5.86 × 10−10) (5.10 × 10−4, 5.67 × 10−10)
(tfin, Tfin) 1.0 × 105 sec, 4.2 × 107 K 1.1 × 105 sec, 4.9 × 107 K
Elements High Low Average High Low Average
p 0.586 0.753 0.744 0.600 0.753 0.740
D 1.76 × 10−21 4.50 × 10−5 4.26 × 10−5 3.43 × 10−21 4.75 × 10−5 4.34 × 10−5
3He + T 2.91 × 10−14 2.18 × 10−5 2.07 × 10−5 2.77 × 10−14 2.23 × 10−5 2.04 × 10−5
4He 0.413 0.247 0.256 0.400 0.247 0.260
7Li + 7Be 1.63 × 10−13 1.78 × 10−9 1.68 × 10−9 6.80 × 10−14 1.65 × 10−9 1.52 × 10−9
fv, R 5.41 × 10−7, 1.84 × 105 5.87 × 10−6, 1.82 × 104
(ηhigh, ηlow) (1.04 × 10−4, 5.62 × 10−10) (1.02 × 10−5, 5.59 × 10−10)
(tfin, Tfin) 1.2 × 105 sec, 4.3 × 107 K 1.2 × 105 sec, 4.5 × 107 K
Elements High Low Average High Low Average
p 0.638 0.753 0.742 0.670 0.753 0.745
D 6.84 × 10−22 4.79 × 10−5 4.36 × 10−5 1.12 × 10−22 4.48 × 10−5 4.37 × 10−5
3He + T 1.63 × 10−13 2.23 × 10−5 2.04 × 10−5 1.49 × 10−9 2.25 × 10−5 2.03 × 10−5
4He 0.362 0.247 0.258 0.330 0.247 0.254
7Li + 7Be 7.42 × 10−13 1.64 × 10−9 1.49 × 10−9 6.73 × 10−8 1.62 × 10−9 7.96 × 10−9

Figure 4 shows the results of nucleosynthesis in the high-density regions with ηhigh≃10−4 and 10−3. In Figure 4(a), we see the time evolution of the abundances of Gd and Eu for the mass number 159. First,  159Tb (stable r-element) is synthesized and later  159Gd  and  159Eu are synthesized through the neutron captures. After t = 103 sec,  159Eu decays to nuclei by way of  159Eu →  159Gd→ 159Tb, where the half-lifes of  159Eu and  159Gd are 26.1 min and 18.479 h, respectively.

Details are in the caption following the image
Figure 4 (a) ηhigh = 1.02 × 10−4
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Time evolution of the mass fractions in high-density regions of (a) ηhigh = 1.02 × 10−4 and (b) ηhigh = 1.06 × 10−3.
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Figure 4 (b) ηhigh = 1.06 × 10−3
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Time evolution of the mass fractions in high-density regions of (a) ηhigh = 1.02 × 10−4 and (b) ηhigh = 1.06 × 10−3.

For ηhigh≃10−3, the result is seen in Figure 4(b).  108Sn, which is a proton-rich nuclei is synthesized. After that, stable nuclei  108Cd is synthesized by way of  108Sn→ 108In → 108Cd, where the half-lifes of  108Sn and  108In  are 10.3 min and 58.0 min, respectively. These results are qualitatively the same as Matsuura et al. [42].

In addition, we notice the production of radioactive nuclei of  56Ni and  57Co, where  56Ni is produced at early times, just after the formation of  4He. Usually, nuclei such as  56Ni and  57Co are produced in supernova explosions, which are assumed to be the events after the first star formation (e.g., [51]). In IBBN model, however, this production can be found to occur at an extremely high-density region of ηhigh ≥ 10−3 as the primary elements without supernova events in the early universe.

Final results (T = 4×107 K) of nucleosynthesis calculations are shown in Table 3. When we calculate the average values, we set the abundances of A > 16 to be zero for low-density side. For ηhigh≃10−4, a lot of nuclei of A > 7 are synthesized whose amounts are comparable to that of  7Li. Produced elements in this case include both s-element (i.e.,  138Ba) and r-elements (for instance,  142Ce and  148Nd). For ηhigh≃10−3, there are few r-elements while both s-elements (i.e.,  82Kr and  89Y) and p-elements (i.e.,  74Se and  78Kr) are synthesized such as the case of supernova explosions. For ηhigh = 10−3, the heavy elements are produced slightly more than the total mass fraction (shown in Figure 3) derived from the BBN code calculations. This is because our BBN code used in Section 3 includes the elements up to A = 16 and the actual abundance flow proceeds to much heavier elements.

Table 3. Mass fractions of heavy elements (A > 7) for three cases of ηhigh≃10−3, ηhigh = 5.33 × 10−4, and ηhigh≃10−4.
fv = 3.23 × 10−8, R = 1.74 × 106 fv = 1.03 × 10−7, R = 9.00 × 105 fv = 5.41 × 10−7, R = 1.84 × 105
(ηhigh = 1.06 × 10−3) (ηhigh = 5.33 × 10−4) (ηhigh = 1.02 × 10−4)
Element High Average Element High Average Element High Average
  
Ni56 1.247 × 10−4 6.658 × 10−6 Nd142 2.051 × 10−5 1.738 × 10−6 Nd145 3.692 × 10−7 3.342 × 10−8
Co57 1.590 × 10−5 8.487 × 10−7 Ni56 1.270 × 10−5 1.077 × 10−6 Ca40 2.706 × 10−7 2.450 × 10−8
Sr86 1.061 × 10−5 5.662 × 10−7 Sm148 1.059 × 10−5 8.976 × 10−7 Mn52 2.417 × 10−7 2.188 × 10−8
Sr87 9.772 × 10−6 5.214 × 10−7 Pm147 6.996 × 10−6 5.930 × 10−7 Eu155 2.374 × 10−7 2.149 × 10−8
Se74 9.745 × 10−6 5.200 × 10−7 Pm145 6.559 × 10−6 5.559 × 10−7 Ce140 1.931 × 10−7 1.748 × 10−8
Sr84 9.172 × 10−6 4.894 × 10−7 Sm146 6.539 × 10−6 5.542 × 10−7 Cr51 1.546 × 10−7 1.400 × 10−8
Kr82 8.910 × 10−6 4.754 × 10−7 Nd143 4.146 × 10−6 3.514 × 10−7 Ce142 1.114 × 10−7 1.008 × 10−8
Kr81 7.797 × 10−6 4.160 × 10−7 Pr141 3.957 × 10−6 3.354 × 10−7 Ni56 1.100 × 10−7 9.964 × 10−9
Ge72 7.674 × 10−6 4.095 × 10−7 Nd144 3.952 × 10−6 3.350 × 10−7 Nd146 1.049 × 10−7 9.501 × 10−9
Kr78 7.602 × 10−6 4.057 × 10−7 Sm147 3.752 × 10−6 3.180 × 10−7 Eu156 9.436 × 10−8 8.542 × 10−9
Kr80 7.063 × 10−6 3.769 × 10−7 Sm149 3.322 × 10−6 2.815 × 10−7 Nd148 9.361 × 10−8 8.474 × 10−9
Kr83 6.252 × 10−6 3.336 × 10−7 Pm146 2.629 × 10−6 2.228 × 10−7 Fe52 8.974 × 10−8 8.124 × 10−9
Ge73 6.144 × 10−6 3.278 × 10−7 Sm144 2.207 × 10−6 1.870 × 10−7 Tb161 8.956 × 10−8 8.108 × 10−9
Se76 5.929 × 10−6 3.164 × 10−7 Sm150 1.683 × 10−6 1.426 × 10−7 La139 8.804 × 10−8 7.971 × 10−9
Br79 5.904 × 10−6 3.150 × 10−7 Pm144 1.581 × 10−6 1.340 × 10−7 N14 8.736 × 10−8 7.909 × 10−9
Se77 5.345 × 10−6 2.852 × 10−7 Pm143 1.575 × 10−6 1.335 × 10−7 Cr48 8.561 × 10−8 7.750 × 10−9
Y89 4.759 × 10−6 2.539 × 10−7 Sm145 1.010 × 10−6 8.568 × 10−8 Ba138 7.955 × 10−8 7.202 × 10−9
Zr90 4.412 × 10−6 2.354 × 10−7 Co57 8.643 × 10−7 7.326 × 10−8 C12 7.672 × 10−8 6.945 × 10−9
Rb85 4.324 × 10−6 2.307 × 10−7 Eu153 5.563 × 10−7 4.715 × 10−8 Dy162 6.835 × 10−8 6.188 × 10−9
Rb83 4.082 × 10−6 2.178 × 10−7 Ce140 4.944 × 10−7 4.191 × 10−8 C13 6.428 × 10−8 5.819 × 10−9
Y88 3.845 × 10−6 2.052 × 10−7 Nd145 4.376 × 10−7 3.709 × 10−8 O16 6.301 × 10−8 5.704 × 10−9
Zr88 3.546 × 10−6 1.892 × 10−7 Eu155 4.224 × 10−7 3.581 × 10−8 Gd158 5.845 × 10−8 5.292 × 10−9
As73 3.519 × 10−6 1.878 × 10−7 Eu151 4.106 × 10−7 3.480 × 10−8 Cs137 5.559 × 10−8 5.033 × 10−9
Ga71 3.388 × 10−6 1.808 × 10−7 Cr52 4.071 × 10−7 3.450 × 10−8 Nd147 3.962 × 10−8 3.587 × 10−9
Se75 2.933 × 10−6 1.565 × 10−7 Cd108 3.596 × 10−7 3.048 × 10−8 Ho165 3.770 × 10−8 3.413 × 10−9
Nb91 2.896 × 10−6 1.545 × 10−7 Gd156 3.368 × 10−7 2.854 × 10−8 Pr143 3.111 × 10−8 2.817 × 10−9
As75 2.856 × 10−6 1.524 × 10−7 Cd110 3.103 × 10−7 2.630 × 10−8 Ce141 2.998 × 10−8 2.714 × 10−9
Mo92 2.442 × 10−6 1.303 × 10−7 Eu152 2.809 × 10−7 2.381 × 10−8 Gd160 2.950 × 10−8 2.670 × 10−9
Ge70 2.318 × 10−6 1.237 × 10−7 Sm151 2.795 × 10−7 2.369 × 10−8 Xe136 2.771 × 10−8 2.509 × 10−9
Sr88 2.021 × 10−6 1.078 × 10−7 Eu154 2.759 × 10−7 2.339 × 10−8 Xe134 2.238 × 10−8 2.026 × 10−9
  
3.010 × 10−4 1.652 × 10−5 1.062 × 10−4 9.006 × 10−6 3.850 × 10−6 3.485 × 10−7

Figure 5 shows the average abundances between high- and low-density regions using (3) in comparing with the solar system abundances [52]. For ηhigh≃10−4, abundance productions of 120 < A < 180 are comparable to the solar values. For ηhigh≃10−3, those of 50 < A < 100 have been synthesized well. In the case of ηhigh = 5   ×   10−4, there are two outstanding peaks: one is around A = 56  (N = 28) and the other can be found around A = 140. Abundance patterns are very different from that of the solar system ones, because IBBN occurs under the condition of a significant amount of abundances of both neutrons and protons.

Details are in the caption following the image
Figure 5
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Comparison of the average mass fractions in the two-zone model with the solar system abundances [52] (indicated by dots).

5. Summary and Discussion

We extend the previous studies of Matsuura et al. [42, 43] and investigate the consistency between the light element abundances in the IBBN model and the observation of  4He and D/H.

First, we have done the nucleosynthesis calculation using the BBN code with 24 nuclei for both regions. The time evolution of the light elements at the high-density region differs significantly from that at the low-density region. The nucleosynthesis begins faster and  4He is more abundant than that in the low-density region. By comparing the average abundances with the  4He and D/H observations, we can get the allowed parameters of the two-zone model: the volume fraction fv of the high-density region and the density ratio R between the two regions.

Second, we calculate the nucleosynthesis that includes 4463 nuclei in the high-density regions. Qualitatively, results of nucleosynthesis are the same as those in [42]. In the present results, we showed that p- and r-elements are synthesized simultaneously at high-density region with ηhigh≃10−4.

We find that the average mass fractions in IBBN amount to as much as the solar system abundances. As seen from Figure 5, there are overproduced elements around A = 150 (for ηhigh = 10−4) and A = 80 (for ηhigh = 10−3). Although it seems to conflict with the chemical evolution in the universe, this problem could be solved by the careful choice of fv and/or R. Figure 6 illustrates the mass fractions with ηhigh = 1.0 × 10−4 for three sets of fv − R. It is shown that the abundances can become lower than the solar system abundances. If we put a constraint on the fv − R plane from the heavy element observations [53–56], the parameters in IBBN model should be tightly determined.

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Figure 6
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Same as Figure 5, but ηhigh is fixed as 10−4.

In the meanwhile, we would like to touch on the consistency against the primordial  7Li. We have obtained interesting results about  7Li abundances in our model. For the recent study of  7Li, the lithium problem arises from the discrepancy among the  7Li abundance predicted by SBBN theory, the baryon density of WMAP, and abundance inferred from the observations of metal-poor stars (see [57, 58]). As seen in Table 2,  7Li is clearly overproduced such as ×10−9 for ηhigh = 10−5, although we adopt the highest observational value  7Li/H = (2.75 − 4.17)×10−10 [59]. However, for cases of ηhigh = 10−3, 5×10−4 and 10−4, the values of agree with the observation. Usually, the consistency with BBN has been checked using observations of  4He, D/H, and  7Li/H. Then, the parameters such as ηhigh = 10−5 ought to be excluded. However, the abundance of is sensitive to the values of both ηhigh and ηlow. As for the future work of IBBN, we will study in detail the  7Li production. In addition, recent  4He observation could suggest the need for a nonstandard BBN model [8]. IBBN may also give a clue to the problems.

Acknowledgments

This work has been supported in part by a Grant-in-Aid for Scientific Research (no. 24540278) 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).

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