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Journal of Astrophysics

Volume 2013 (2013), Article ID 587294, 9 pages

http://dx.doi.org/10.1155/2013/587294

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

^{1}Department of Physics, Graduate School of Sciences, Kyushu University, 6-10-1 Hakozaki, Higashi-ku, Fukuoka 812-8581, Japan^{2}Department of Control and Information Systems Engineering, Kumamoto National College of Technology, 2659-2 Suya, Koshi, Kumamoto 861-1102, Japan^{3}Institute for the Physics and Mathematics of the Universe, University of Tokyo, Kashiwa, Chiba 277-8568, Japan^{4}National Institutes of Natural Sciences, Kamiyacho Central Place 2F, 4-3-13 Toranomon, Minato-ku, Tokyo 104-0001, Japan

Received 30 March 2013; Accepted 29 July 2013

Academic Editors: G. Chincarini, X. Dai, A. De Rosa, M. S. Dimitrijevic, and G. Wegner

Copyright © 2013 Riou Nakamura et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### 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 ^{7}Li in the early universe. From the observational constraints on light elements of ^{4}He 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 , D, , and , during the first few minutes [1–4]. Standard model of BBN (SBBN) can succeed in explaining the observation of those elements, [5–9], D [10–13], and [14, 15], except for . 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 . If the present value of 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 by comparing the calculated abundances with observations. In the meanwhile, the value of is obtained as [1] from the observations of and D. These values agree well with the observation of the cosmic microwave background: [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 has been proposed [17, 18, 25–35]. In addition, peculiar observations of abundances for heavy elements and/or 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 - and -processes is possible if [42], where they have also shown that the high regions are compatible with the observations of the light elements, 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 and during the primordial nucleosynthesis , 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, , , and as , , and low baryon number densities. is the volume fraction of the high baryon density region. , , and are mass fractions of each element 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 s with subscripts are the baryon-to-photon ratios in each region. In the present paper, we fix from the cosmic microwave background observation [16]. The values of and are obtained from both and the density ratio between high- and low-density regions: .

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 in (4) is the sum of decomposed parts: Here, the subscripts , , and indicate photons, neutrino, and electrons/positrons, respectively. The final term is the baryon density obtained as .

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 is the pressure of the fluid. When we solve (6), initial values in both regions are obtained from (2) with and fixed. For , the baryon density in the high-density region, , is larger than the radiation component at K. However, we note that the contribution to (5) is not , but . In our research, the ratio of to is about 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 . We adopt the reaction rates of Descouvemont et al. [45], the neutron lifetime sec [1], and consider three massless neutrinos.

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

Figure 1 illustrates the light element synthesis in the high- and low-density regions with and = that corresponds to and . 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 is more abundant than that in the low-density region, (or ) is much less produced. In this case, we can see that average values such as 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 and by comparing with the observation of the light elements.

Now, we put constraints on and by comparing the average values of and D obtained from (3) with the following observational values. First, we consider the primordial abundance reported in [8]: and [9]: We adopt 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 plane from the above light element observations with contours of constant . 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 and : The observation (9) gives the upper bound for , and the limit for 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 .

We should note that takes a larger value, nuclei which are heavier than 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 Figure 2 and are drawn using the constraint from and D/H observations. As a consequence, we get the upper limit of total mass fractions for heavier nuclei as follows:

#### 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 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 and proton 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 K.

As seen in Figure 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 Figure 2.

First, we evaluate the validity of the nucleosynthesis code with nuclei. Table 2 shows the results of the light elements, , D, , , and . 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 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.

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

For , the result is seen in Figure 4(b). , which is a proton-rich nuclei is synthesized. After that, stable nuclei is synthesized by way of , where the half-lifes of and are min and min, respectively. These results are qualitatively the same as Matsuura et al. [42].

In addition, we notice the production of radioactive nuclei of and , where is produced at early times, just after the formation of . Usually, nuclei such as and 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 as the primary elements without supernova events in the early universe.

Final results ( = K) of nucleosynthesis calculations are shown in Table 3. When we calculate the average values, we set the abundances of to be zero for low-density side. For , a lot of nuclei of are synthesized whose amounts are comparable to that of . Produced elements in this case include both -element (i.e., ) and -elements (for instance, and ). For , there are few -elements while both -elements (i.e., and ) and -elements (i.e., and ) are synthesized such as the case of supernova explosions. For , 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 and the actual abundance flow proceeds to much heavier elements.

Figure 5 shows the average abundances between high- and low-density regions using (3) in comparing with the solar system abundances [52]. For , abundance productions of are comparable to the solar values. For , those of have been synthesized well. In the case of , there are two outstanding peaks: one is around and the other can be found around . 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.

#### 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 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 is more abundant than that in the low-density region. By comparing the average abundances with the and D/H observations, we can get the allowed parameters of the two-zone model: the volume fraction of the high-density region and the density ratio 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 - and -elements are synthesized simultaneously at high-density region with .

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 (for = ) and = (for = ). Although it seems to conflict with the chemical evolution in the universe, this problem could be solved by the careful choice of and/or . Figure 6 illustrates the mass fractions with for three sets of . It is shown that the abundances can become lower than the solar system abundances. If we put a constraint on the plane from the heavy element observations [53–56], the parameters in IBBN model should be tightly determined.

In the meanwhile, we would like to touch on the consistency against the primordial . We have obtained interesting results about abundances in our model. For the recent study of , the lithium problem arises from the discrepancy among the 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, is clearly overproduced such as for , although we adopt the highest observational value [59]. However, for cases of , and , the values of agree with the observation. Usually, the consistency with BBN has been checked using observations of , D/H, and . Then, the parameters such as ought to be excluded. However, the abundance of is sensitive to the values of both and . As for the future work of IBBN, we will study in detail the production. In addition, recent 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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