Reply to 'Comment on 'Heavy element production in inhomogeneous big bang
  nucleosynthesis''
Reply to ’Comment on ’Heavy element production in inhomogeneous big bang
nucleosynthesis”
Shunji Matsuura
Department of Physics, School of Science, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
Shin-ichirou Fujimoto
Department of Electronic Control, Kumamoto National College of Technology, Kumamoto 861-1102, Japan
Masa-aki Hashimoto
Department of Physics, School of Sciences, Kyushu University, Fukuoka 810-8560, Japan
Katsuhiko Sato
Department of Physics, School of Science, University of Tokyo,
7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan and
Research Center for the Early Universe, University of Tokyo, 7-3-1 Hongo, Bunkyo, Tokyo 113-0033, Japan
This is a reply report to [1]. We studied heavy element production in the high baryon density
region in the early universe[3]. However it is claimed by [1] that a small scale but high baryon density
region contradicts observations for the light element abundance or, in order not to contradict to the
observations the high density region must be so small that it cannot affect the present heavy element
abundance.
In this paper we study big bang nucleosynthesis in the high baryon density region and show that
in certain parameter spaces it is possible to produce enough amount of the heavy element without
contradiction to cosmic microwave background and light element observations.
PACS numbers: 26.35.+c, 98.80.Ft, 13.60.Rj
I. INTRODUCTION
In a standard scenario, big bang nucleosynthesis (BBN) can produce only light elements, up to 7Li, and all heavy
elements have been synthesized in stars. However, many phase transitions in the early universe could have printed
their trace in a non-standard way. For example, some baryogenesis models[2] predict very high baryon density islands
in ordinary low density backgrounds.
In the previous paper[3], we studied heavy element production in inhomogeneous BBN from this point of view. 1
However we limited ourselves to the heavy element abundance and did not discuss about the light element abundance
and consistency with observations. This is because we assumed that high baryon density region is very local and
do not affect the global light element abundance. In [1], Rauscher pointed out that in order not to contradict
to observations, the high baryon density region must be very small and cannot affect the present heavy element
abundance. In this paper, we show that there is a parameter region in which the heavy element can be produced
enough to affect observation while keeping the light element abundance consistent with observations. We consider
that the disagreement between Rauscher’s opinion and our opinion comes from two points. One is that we are looking
at some parameter regions in which neutrons in high baryon density do not diffuse so much as to cause disaster in
standard BBN. We would like to emphasize this point. The other is that the relevant quantity is not the spatial size
of the high baryon density region but the amount of baryon in high density regions.
We will discuss the following issues: In section II, we discuss the light element abundance in the homogeneous high
baryon density region and after mixing the high and the low baryon density region. In section III, we study the heavy
element(Ru,Mo) abundance in high and averaged baryon density and show that heavy elements can be produced
enough without contradicting the light element observation. In section IV, we briefly comment on the diffusion scale
of the high baryon density region.
1 For previous works on the inhomogeneous big bang nucleosynthesis, see [4, 5]. Heavy elements production is also mentioned in [6].
http://arxiv.org/abs/0704.0635v3
II. LIGHT ELEMENT ABUNDANCE
A. Homogeneous BBN
We calculate homogeneous BBN with various values of η(baryon photon ratio). In Table.I and II, we show the
numerical result of the mass fraction and the number fraction of each light element for η = 10−3 and 3.162× 10−10.
η = 10−3
name mass fraction number fraction
H 5.814 × 10−1 8.475 × 10−1
4He 4.185 × 10−1 1.525 × 10−1
3He 4.842 × 10−13 1.614 × 10−13
7Li+7Be 1.559 × 10−12 2.227 × 10−13
D 1.577 × 10−22 7.883 × 10−23
Table. I: The mass and the number fractions of light elements for the homogeneous BBN with η = 10−3
η = 3.162 × 10−10
name mass fraction number fraction
H 7.58 × 10−1 9.26× 10−1
4He 2.419 × 10−1 7.39× 10−2
3He 4.299 × 10−5 1.433 × 10−5
7Li + 7Be 8.239 × 10−10 1.177 × 10−10
D 1.345 × 10−4 6.723 × 10−5
Table. II: The mass and the number fractions of light elements for the homogeneous BBN with η = 3.162 × 10−10
As baryon density becomes higher, more protons and neutrons are bounded to form 4He. At η = 10−3, most of the
final product of 7Li comes from 7Be which decays to 7Li after BBN. Details on light element production for various η
can also be found in [7]. In this paper we almost concentrate on a case in which the high baryon density region has
η = 10−3. We expect that compared to η ≥ 10−3, the profile of the abundance for η = 10−3 is more different from
standard BBN because most of the light element abundances change monotonically with respect to η and if this case
does not contradict to observations, other cases would also be consistent. Briefly, the amount of H decreases and 4He
increases monotonically as η become larger. The number fraction of D is less than 10−20 for η greater than 10−7. For
3He, the number fraction drastically decreases around η = 10−4 down to O(10−13), and for 7Li, the number fraction
increases until η = 10−6 and drastically decreases for a larger value of η. In the following sections, we will see that
this non-standard setup does not strongly contradict to the observations. For simplicity we ignore the diffusion effect
before and during BBN, and after BBN both high and low baryon density regions are completely mixed. Detailed
analysis such as the case in which the high baryon density region doesn’t completely mixed, or taking into account
diffusion effects are left for future work.
B. Parameters and Basic equations
In this section, we summarize the relations among parameters.
Notations : n, nH , nL are averaged, high, and low baryon number density. fH , fL are the volume fractions of the
high and the low baryon density region. yi, y
i , y
i are the mass fractions of each element (i) in averaged-, high- and
low-density regions. The basic relations are
fH + fL = 1 (1)
fHnH + fLnL = n (2)
yHi f
HnH + yLi f
LnL = yin. (3)
Under the assumption that the temperature of the universe is homogeneous, the above equation can be written as
fHηH + fLηL = η (4)
yHi f
HηH + yLi f
LηL = yiη (5)
where η = n
,ηH,L = n
Conventional parameters for inhomogeneous BBN are η, f and density ratio R = n
. Here
we use a different combination of parameters. Relevant values for the abundance analysis are products fH,L × ηH,L
and ηH,L. fH,Lv × η
H,L determines the amount of baryon from high- and low- density regions. ηH,L determines the
mass fraction of each species of nuclei. For convenience, we write the ratio of baryon number contribution from high
density region as a, i.e., fHηH : fLηL = a : (1 − a). There are 5 parameters(nH,L, n and fH,L) and 2 constraints
(Eq.(1) and Eq.(2)). We calculate the light element abundance for various values of ηH,L. η can also take any value,
but in order not to contradict observational constraints, we choose η from 3.162 × 10−10 to 10−9. a is determined
by Eq(4). The aim of the analysis in this section is not to find parameter regions which precisely agree with the
observational light element abundance and η from CMB. Our model is too simple to determine the constraints to
parameters. For example, we completely ignore the diffusion effect before and during BBN. Instead we see that at
least our analysis in previous paper is physically reasonable.
C. Theoretical predictions and observations of light elements
We consider the cases of ηH = 10−3 and ηL = 3.162 × 10−10. The mass fractions of and H and 3He in the
high density region are 0.5814 and 4.842 × 10−13, respectively, while those in the low density region are 0.758 and
4.299× 10−5. From Eq.(5), we have
fHηHyH3He + f
LηLyL3He = ηy3He (6)
4.842× 10−13 × a+ 4.299× 10−5 × (1− a) = y3He (7)
fHηHyH
+ fLηLyL
= ηyH (8)
0.5814× a+ 0.758× (1− a) = yH. (9)
We can calculate an averaged value of the abundance ratio of 3He to H as
4.842× 10−13 × a+ 4.299× 10−5 × (1− a)
0.5814× a+ 0.758× (1 − a)
. (10)
where a is related to η as
η − ηL
ηH − ηL
η − 3.162× 10−10
10−3 − 3.162× 10−10
η − 3.162× 10−10
. (13)
Here a varies from 0 to 0.9 for reasonable values of η, or 3.162× 10−10 − 10−9. Similarly, for ηH = 10−3 the number
fractions are
1.577× 10−22 × a+ 1.345× 10−4 × (1− a)
0.5814× a+ 0.758× (1 − a)
0.0e+00
2.0e-05
4.0e-05
6.0e-05
8.0e-05
1.0e-04
1e-97e-105e-10
Fig. 1: Averaged ratio of D to H,(D/H) vs η
0.0e+00
5.0e-06
1.0e-05
1.5e-05
2.0e-05
1e-97e-105e-10
Fig. 2: Same as Fig.1 but for (3He/H)
0.0e+00
5.0e-11
1.0e-10
1.5e-10
2.0e-10
1e-97e-105e-10
Fig. 3: Same as Fig.1 but for (7Li/H)
1.559× 10−12 × a+ 8.239× 10−10 × (1− a)
0.5814× a+ 0.758× (1− a)
. (15)
Fig.1,2 and 3 represent the averaged abundance ratio, (D/H), (3He/H) and (7Li/H) respectively.
We can see that the light element abundance is the same order around η ∼ 5 × 10−10 − 10−9 as observations
[8, 9, 10, 11, 12, 13, 14, 15].
)obs = (1.5− 6.7)× 10
−5 (16)
)obs = (0.59− 4.1)× 10
−10. (17)
We do not discuss detail about diffusion here. But at least above result suggest that our analysis is not beside the
point.
III. THEORETICAL PREDICTIONS AND OBSERVATIONS OF HEAVY ELEMENTS (92,94MO, 96,98RU)
The same analysis can be applied for heavy elements such as 92Mo, 94Mo, 96Ru and 98Ru. We are interested in
these elements because in many models of supernovae nucleosynthesis, these p-nuclei are less produced. We will see
that some amount of these heavy elements can be synthesized in BBN.
η = 10−3
name mass fraction
H 5.814 × 10−1
4He 4.185 × 10−1
92Mo 1.835 × 10−5
94Mo 4.1145 × 10−6
96Ru 1.0789 × 10−5
98Ru 1.0362 × 10−5
Table. III: The mass fractions of nuclei for homogeneous BBN with η = 10−3
From Table.III, we can derive the expected value of these elements.
1.835× 10−5 × a
0.5814× a+ 0.758× (1 − a)
4.1145× 10−6 × a
0.5814× a+ 0.758× (1 − a)
1.0789× 10−5 × a
0.5814× a+ 0.758× (1− a)
1.0362× 10−5 × a
0.5814× a+ 0.758× (1− a)
. (21)
We plot expected value of these quantities in Fig.4.
These values should be compared with the solar abundance(Table.IV)[17].
0.0e+00
5.0e-08
1.0e-07
1.5e-07
2.0e-07
2.5e-07
1e-97e-105e-10
Mo/H)
Mo/H)
Ru/H)
Ru/H)
Fig. 4: (92Mo/H),(94Mo/H),(96Ru/H) and (98Ru/H) vs η. Red, green, blue and pink lines represent the ratio
(92Mo/H),(94Mo/H),(96Ru/H),(98Ru/H) respectively.
name number fraction ratio to H
H 7.057280 × 10−1 1
92Mo 8.796560 × 10−10 1.2465 × 10−9
94Mo 5.611420 × 10−10 7.9512 × 10−10
96Ru 2.501160 × 10−10 3.5441 × 10−10
98Ru 8.676150 × 10−11 1.2294 × 10−10
Table. IV: The abundances of 92,94Mo and 96,98Ru in the solar system[17]
Compared those observational values with Fig.4, it is clear that the heavy element produced in BBN can affect the
solar abundance heavy element. Some of them are produced too much. But this is not a problem of the previous work
[3], because we assumed that high density regions are very small and do not disturb standard BBN. The analysis here
suggest that even if we assume the density fluctuations are completely mixed, heavy element can have enough affect
to the solar abundance.
IV. DIFFUSION DURING BBN
In the previous analysis, we assumed that the diffusion effect can be ignored during BBN and both high density
regions and low density regions are completely mixed after BBN. In this section, we determine the scale of high baryon
density island in which the diffusion effect during BBN is very small enough and our assumption is valid. We do not
discuss the diffusion after BBN here.
A detail analysis of the comoving diffusion distance of the baryon, the neutron and the proton is in [16]. From Fig.1
in [16], in order to safely ignore the diffusion effect, it is necessary for the high baryon density island to be much larger
than 105cm at T=0.1MeV(1.1× 109K). Notice that T ∝ 1
, where A is a scale factor. For scale d now corresponds to
d/(4.0× 108) at BBN epoch. Present galaxy scale is O(1020)cm, which corresponds to O(1012)cm >> 105cm at BBN
epoch.
temperature and scale
temperature scale
1.1× 109K (BBN) d
3000K (decouple) 3.7× 106 × d
2.725K (now) 4.0× 108 × d
Table. V: Relation between temperature and scale
The maximum angular resolution of CMB is lmax ∼2000. The size of universe is ∼ 5000Mpc. In order not to
contradict to CMB observation, the fluctuation of baryon density must be less than ∼ 16Mpc now. This corresponds
to 1017cm at BBN.
Since the density fluctuation size in Dolgov and Silk’s model[2] is a free parameter, the above brief estimation sug-
gests that we can take the island size large enough to ignore the diffusion effect without contradicting to observations,
i.e., the reasonable size of 105cm −1017cm at the BBN epoch. We can choose distances between high density islands
so that we obtain a suitable value of f .
V. SUMMARY
In this paper, we studied the relation between the heavy element production in high baryon density regions during
BBN and the light element observation. By averaging the light element abundances in the high and the low density
regions we showed that it is possible to produce a relevant amount of heavy element without contradicting to observa-
tions. However we should stress that in this paper we restricted ourselves to some parameter regions where neutrons
in high baryon density regions do not destroy the standard BBN. So our setup is different from the conventional
inhomogeneous BBN studies. We also studied the size of the density fluctuation to show that there is a parameter
region in which the neutron diffusion is negligible and which is much smaller than CMB observation scale. It is
worthwhile to investigate further how the produced heavy elements can be related to the detailed observations.
VI. ACKNOWLEDGEMENTS
We thank R.H. Cyburt, R. Allahverdi and R. Nakamura for useful discussions. This research was supported in
part by Grants-in-Aid for Scientific Research provided by the Ministry of Education, Science and Culture of Japan
through Research Grant No.S 14102004, No.14079202. S.M.’s work was supported in part by JSPS(Japan Society for
the Promotion of Science).
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	Introduction
	Light element abundance
	Homogeneous BBN
	Parameters and Basic equations
	Theoretical predictions and observations of light elements
	Theoretical predictions and observations of heavy elements (92,94Mo, 96,98Ru)
	Diffusion during BBN
	Summary
	Acknowledgements
	References
