KEK Report 87-26
January 1988
H
PROCEEDINGS OF THE FIRST WORKSHOP ON THE NEUTRON LIFETIME
KEK, T s u k u b a , November 1 9 , 1 9 8 7
Edited by
S. YASUMI
NATIONAL LABORATORY FOR
HIGH ENERGY PHYSICS
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TIONAL LABORA
TORY FOR
HIGH ENERGY PHYSICS
© National Laboratory for High Energy Physics, 1988
KEK Reports are available from:
Technical Information Office
National Laboratory fjr High Energy Physics
1-1 Oho, Tsukuba-Shi
Ibaraki-ken, 305
JAPAN
Phone: 0298-64-1171
Telex: 3652-534 (Domestic)
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Cable: KEK0H0
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Nationa1 Laboratory for High Energy Physics. 1988
KEK Reports are availab1e from:
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Na
七iona1 Laboratory
f~rHigh Energy Physics
1-1 Oho
，
Tsukuba-Shi
Ibaraki-ken
，
305
JAPAN
Phone:
Te1ex:
Cable:
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(0)3652-534 (lnterna
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シーデイングズである。
弘の経験からすると.中性子の寿命の測定というと今更そんな必廷があるのですか? というのが
大方の反応であった。しかしながら実際
K
は中性子寿命の実験データの測定精度は他の粒子のそれに
比べると遥
ζ
i
感く，現在における基本的原子核常散のうちで最も精度の惑いものの一つ(J.
Byrne)
である。その意味では乙のテーマば“古いがしかし今尚新しい研究テーマ"と云う乙とができょう。
これに関する詳細と，本研究を
J
思いたつに到った直接の動機については本プロシーディングズの
1
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9
月京大原子炉実験所(以下
KURI
と略す)において開催された中
性子寿命に関するセミナーを契機として
KEK
と
KURI
とを中心にして中性子寿命研究ク勺レープが徐
々に形成ぎれてきた。乙の間互に連携を保ちながら討論を重ねて出来上ったのが
1n
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ζ
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WorkingTime
(1)
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て頂く。
最後に本研究の出発
ζ
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際して高エネルギー物理学研究所西川哲治所長，同物理研究部
菅原克芋:
総主幹及び京都大学原子炉実験所
岡本朴所長をはじめとする所内所外の数多くの方々が示された御
理解と御支持とし対して深甚の謝意を表する次第である。
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PRIMORDIAL NUCLEOSYNTHESIS, SOLAR NEUTRINO AND
NEUTRON LIFETIME
Nobuo TERASAWA
Department of Physics, Faculty of Science, I The University of Tokyo, Bunkyo-ku, Tokyo 113, Japan
Abstract
Primordial nucleosynthesis and the solar neutrino problem are reviewed with rela-tion to the neutron lifetime. Ambiguities of the theoretical predicrela-tions arising due to the uncertainty of the neutron lifetime is summarized quantitatively. In primordial nucle-osynthesis, the uncertainty in the neutron lifetime seriously affects the predictions and preecesions within 1 % are required for definite predictions. If error analyses are suffi-cient, precisions up to 2 % are acceptable.
Introduction
Astronomical observations rarely provide quantitative and crucial tests of theoretical pre-dictions except for a few subjects. Primordial nucleosynthesis and the solar neutrino problem are examples of such a few exceptions. Unfortunately the uncertainty in the neutron lifetime affects the theoretical predictions in both the cases. In this article, I review the present status of the comparisons between theoretical predictions and astro-nomical observations in primordial nucleosynthesis and in the solar neutrino problems, and present the quantitative dependences of several values predicted theoretically on the uncertainty in the neutron lifetime.
In primordial nucleosynthesis, the 4Heabundance, which is not only the major prod-uct of primordial nucleosynthesis but the key value in the close comparison between the theoretical predictions and observations, sensitively depends on the neutron lifetime. Hence the neutron lifetime has been often treated as if it were a free parameter of the theory and sometimes it was even checked what a degree of the value is probable as the true neutron lifetime. The only intrinsic parameter of the standard Big Bang nucleosyn-thesis is the baryon-to-photon ratio T|(= (rifl —r»fl)/n,), but the number of the neutrino species N„, also affect the * Heabundance. We will be able to derive definite constraints on TJ and JV„if the uncertainty in the neutron lifetime is negligibly small.
We do not know few subjects in astrophysics except for the solar neutrino problem in which the discrepancy between the theoretical prediction and the observational data is so serious. The predicted neutrino capture rate has been improved not a little in the sense that the discrepancy with the observed rate by Davis's experiments is decreased. This is completely owing to the revision of the thermonuclear reaction rates and the neutron トー.
PRIMORDIAL NUCLEOSYNTHESIS
，
SOLAR NEUTRINOM
1
>
NEUTRON LIFETIME Nobuo TERASAWA Dep釘tmcntof Physics. FacuIty of Scicnce. 百leUnivcrsity ofTokyo， Bunkyo・加，Tokyo113.Japan Abstract Primordia1 nucIeosynthcsis and thc solar ncutrino prob1em釘cr'ev:ewed with rela-tiontoぬcncu回目Iifctimc.Ambiguitics of脱 出 閃 毘tica1predictions nrising due 10 Ihe unccnainty of the neu回nIife白neis summarized qu釦titatively.In primordia1 nuc1c -osyn出 回5，出cunccrtainty in白巴目cu回 日lifelImcseri口uslyaffωtsthc predictions and preecesions wi由加1% are required for definitc predictions. If eπロranalyscs are suffi -cienl
，
prccisions up 10 29
o
釘
eacccptabIc. Introduction As位。.nomicalobservations ta冗lyprovidequantitative釘叫crucial包S句。fthe日 間IIcalpre -diClIons except for a few su吋eC隠.Prirnor司ialnucleosynL1:!csis and thc solar ncunino prob1em are examp1es of such a few exceptions. Unf.町tunate1y山 uncertaintyin the neutron 1ifctime affcc白 血thc::oreticalpn:dictions in both thc cascs. In this articIc. 1 revicw thc pπscnt status of Ihe compnrisons bclWecn出c:orctica
I
predictions and as甘 か nomica1 observations in primordia1 nuc1c
:
o
sy柚csisand in lhe solar neu凶noproblems，
andpresent由equantitalivedcp
.
:
ndencesロ
fscvc叫va1ucspr官diclcdthcoretica11y on曲E mαnainty in thc neu釘onIifetime. In primordial nuc1cosynthesis，白e4 Hcabundanα. which is not on1y出emajorprod・ ucl of primordial nucleosynthesis but the lcey va1uc in thc cIosc compnrison bctwecn 白ethcoretical pn:dictions and obscrvations，
scnsitivc1y depcnds on thc ncutronIifc白nc. Hence恥 neu回nIifctimc bas bccn often町atedぉifil明 rca frecpnrame町 ofthc 曲 目ryandsome出lCSitwas cvcn ch巴dccdwhat a degrccロf白cvalue is probablcllSthc trucncutron lifclIn1e.The on1y iIl官in.icparan1ctcr of thc standard Big Bang nucIeosyn・ thesis is thc b町on-to
-
photonmtio可{三(n8-n9>/吋.
but the numbcr of the ncutrino S戸ciesN，削a1soaffecI thc4 Hcabundancc・
Wcwill bc ab1c 10 derive definitc cons町ainlS on可 制dN"笹 山cuncenainty in伽 neu回 目Iifctime is ncgligibly small. Wc do not know fcw subjec凶inastrophysics CXCCpl ti町 thc鈎lar目eu凶noproblcm in which thc discrepancy bctwecn出ctheoretical pr吋ictionand thc observational data is soscrious.百lCprcdic惚dncu凶nocaplぽerate has bccn improvcd not a littlc in thc sensc 白at白ediscrepancy with the observed rate by Davis's expcriments is decreased百lis is complctely owing to出erevision of thc Ihcrmonuclc:ar reaclion ralcs and thc neu回 目lifetime. Although we cannot be so optimistic to expect that the discrepancy may be
re-moved only by the improvement of the nuclear data, it will be clear where the essence
of the problem is if the ambiguities in the thermonuclear reaction rates and the neutron
lifetime become negligibly small. The predicted neutrino Mux, which is expected to be
detected in the Davis's underground experiments, depends most sensitively on the
reac-tion rates 8Be(e+i08Be and die uncertainty in the neutron lifetime does not affect the prediction so seriously. More accurate measurements of tire neutron lifetime, however,
will be required, when other types of solar neutrino detectors which have a lower
thresh-old and sensitive to the low energy neutrinos produced in die p-p reactions such as a 71 Ga detector are in practical use in the near future.
The plan of this article is as follows. Primordial nucleosynthesis is reviewed in the
co section A. In die section A.l, fundamental assumptions of the standard hot Big Bang
nucleosynthesis are presented and die result of the numerical calculations is revealed.
Observations of the light element abundances and estimation of the primordial
abun-dances are reviewed briefly in the section A.2. The predicted abunabun-dances are compared
with the primordial abundances in the section A.3. In the section B, the solar neutrino
problem is reviewed. The standard solar model is described and the expected energy
spectrum of the solar neutrinos is shown in the section B.l. Uncertainties in the expected
neutrino capture rates are presented in the section B.2. Finally required accuracy in the
. neutron lifetime is revealed as a conclusion.
A Primordial Nucleosynthesis
Primordial nucleosynthesis provides a crucial test of the standard hot Big Bang model as
a probe of die physical conditions in die early universe. In this section, the present status
of the comparison between theoretical implications and observations is reviewed. On the
details both of die standard model and the present observational status, see the review
by Boesgaard and Steigman( 19S7)[1], The most recent result of calculations
incorporat-ing the revised thermonuclear reaction rates and comparisons with observational data is
provided by Terasawa and Sato (1987)[2J.
A.1 T h e Predicted Primordial Abundances A.l.l The Fundamental Assumptions
The standard model of hot Big Bang nucleosynthesis rests on some fundamental
assump-tions:
1. Gravitation is described by a metric theory and general relativity is valid.
2. The Universe was homogeneous and isotropic.
3. The Universe was at a sufficiently high temperature that all particles present were
Li statistical equilibrium.
4. Only known particles were present.
5. All particles were nondegenerate.
6. The baryon-to-photon ratio is positive.
According to these assumptions the space-time is described by the Robertson-Walker lifetime. Although we cannot be so optimistic to expect that the discrepancy may be re・ moved only by the improvement of the nuclear data. it w辺1be clear where the essence of the problem is笹 山ambiguitiesin恥 thennonuclearreaction rates and蜘 neutron lifetime恥comenegligibly small. The predicted neu甘inoflux. which is expected to be detectedin白eDavis's undergroun
ヨ
experiments.deper
:
dsmost sensitively0白血巴陀ac -tion rates 8Be(e+/I)8Be and the uncertainty in the neu回目!ifetimcdoes not affect the prediction50 seriously. More accuratc measur芭mentsof the neutron lifetime. however. willbe required，when 0出ertypes of solar neutrino detectors which have a lower thresh -old and sensitive to the low energy neutrinos produced in thep
-
p reactions such as a 71Ga detector are in pr冨cticaluse in the near future. 百leplanof由isarticle is as fol1ows.P
r
i
mordial nuclec町nthesisis reviewed in the N section A. In the section A.l. fundamentalぉsumptionsof the standard hot Big Bang nucleosynthesis釘epresen俗dand the result of the numerical calculations is revealed. Observations of the light element abundances and estimation of the primordial abun -dances are reviewed briesy in thc section A.2.百epredicted abundanccs are compared wi曲theprimordial abundances in the section A.3. In the scction B. the solar neu釘ino problem is reviewed.百leStandard solar model is described and the exp喧ctedenergy spectrum of the501訂neutrinosisshown in白es目tionB.1.Uncertainties in theexpected neu凶nocapture rates釘epresented in the section B.2. Finally required acc町acyin出E neutron lifetime is revcaled as a conclusion. A Primordial Nucleosynthesis Pnmordial nucle沿synthesisprovides a crucialほstof the standard hot Big Bang model凶 a probe ofthe physical conditions in the early universe. In this section. the present status of the comparison between theoretical implications and observations is reviewed. On the details both of the sロ
ndardmodel and the present observational status. see the review by Boesgaard and Steigman( 19&7)[1].百lCmost reccnt resull of calculations incorporat -ing白crevised thennonuclear reaction ratcs and comparisons wi由。bservationaldJtais provided by Terasawa and Sato (19幻)[2].A.l The Predicted Primordial Abundances A.l.l The Fundamental Assumptions The Standard m叫elof hot Big Dang nucleosynthesis rests on some fundamental assump -tions: 1.Gravitatio!lis described by a metric白eoryand gcneral relativity is valid. 2.百leUniverse was homogeneous alld isotropic. 3.τne Universe w鑓 3ta st:fficiently high temperat町ethat all p回iclespresent were :'1 statistical equllibrium. 4. Only known particles were p陀sent. 5. A
I
1
particles were nondegenerate. 6.τne baryon・to
-
photonrallOispositive. According to these assum ptions the space-白neis dcscribed by the Robenson-Walkermetric and the evolution of me scale factor in die early Universe is described by die
equation for die Robcrtson-Walkcr-Friedmann model,
[;(§)]*=>
While die total energy density of die universe is dominated by relarivisric matter,
pta pn<x a~*, Equation 1 is integrated to
^ j W = 1. (2)
Since the energy density of radiation varies as; p
1cc T*, it follows from Equation2
that X « t~
xfl. To a good approximation, when die cosmic time t ~ 1 s, die radiation
temperature is T ~ 10
10K or ~ 1 MeV.
A.L2 The Thermal History of The Universe and Nucleosynthesis
We summarize primordial nucleosyndiesis in die followings by tracing down die
evolu-tion of die Universe. The history of Big Bang nucleosyndiesis begins at t ~ 10~
2,T'~
10"K.
i ) t ~ 1 0 -
2s , r ~ 1 0
nK
The Universe consists of photons O7), neutrinos (u„v,; i = e,v,r...),
electron-positron pairs (e*), and nucleons (N), and all of diem are in equilibrium through die
electroweak interactions. The neutron-to-proton ratio is also maintained at its
equilib-rium by die charge current weak interactions,
n/p = exp(-&m/T), (Aw = t n » - m , ) . (3)
i i ) t ~ 0 . 1 s , r ~ 3 x l 0
1 0K
When the temperature drops at a few MeV, die neutral current weak interaction
become too slow to keep up with the cosmic expansion and die neutrinos decouple from
matter.
i i i ) i ~ l s , r ~ 1 0
1 0K
The charged current interactions become too slow to maintain die equilibrium
be-tween neutron and proton and the neutron-to-proton ratio effectively freezes out,
n/p ~ exp( - A m/T.). (4)
The freeze-out temperature T. is determined by die competition between die weak
interaction rate(r = nu<r
B») and die expansion rate(/f ~ i"'). The expansion rate is
determined by die total energy density of relativisdc particles
PR = Pi + p, + p
v= [l + j + ^ ] p
r(5)
On the odier hand, die neutron lifetime is die measure of die weak interaction between
neutron and proton, r a r„
_'. Thus die longer die neutron lifetime is or the greater die
number of neutrino species is, the greater die freeze-out value of die neutron-to-proton
ratio and hence die greater die
4He abundance turn out to be.
i v ) t ~ 1 0 s , 2 " ~ 3 x 10'K
The electron-positron pairs annihilate to heat up radiation. Since die neutrinos have
been already decoupled from matter, die radiation^) temperature turns out to be higher
than die neutrino temperature'!',,) after die annihilation,
i y i W l l / 4 ) "
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Since the threshold energy of the deuteron photodissociation is small, nucleosynthe-sis is delayed to such a later time. The number of photons capable of dissociating the deuteron per nucleon varies as 7j-Iezp(—2.2/T). Hence the greater nucleon/photon ratio t), the higher the temperature at which the deuteron formation effectively begins. The 4He abundance, therefore, increases with increasing JJ.
The synthesis of heavier elements follows the deuteron formation as shown schemati-cally in Fig. 1. Due to the gaps of stable nuclei between mass-5 and mass-8, the synthesis of the elements heavier than helium is prevented and some trace amounts of7 Li and 7 Be are produced.
v i ) t > 1 03s , r < 3 x l 08K
The coulomb barriers become so large that nucleosynthesis is effectively terminated. In Fig. 2, we show an example of the time evolution of the abundances of elements forn,o = 3(j),0 = v/W~X0), N„ = 3 , T , = 3 .
A.L3 The Predicted Abundances of Light Elements
The predicted abundances of D , T ,3He , 4He ,and7LiareshowninFig.3forM- = 3 and in Fig.4 for JV„ = 4 . The neutron lifetime is taken to be r„ = 898s and r„ = 898 ± 16s which are deviated la from the mean given in the Particle Data(1986) [3]. The 4He abundance increases with increasing TJ as described above. On the other hand, the D and 3He abundances decrease with increasing t) because they are consumed to produce 4 He more effectively for larger rj. The dip of the 7Li abundance is formed by the following reason;7 Li is produced directly via4 He ( T . f )7 Li in the range TJW<3 and in the range rjio>3 the indirect process 4He (3He,7)7Be,7Be(e-,i/,)7Li dominate*
the productions.
In Table 1, the abundances of major elements are listed for various values of r, for n = 10~1 0. In the range 2 < mo < 10, the prrdicted primordial abundance of 4He is well fit by
Yp = 0 224+ 0.011 In 7j,o + 0.013(iV„ - 3 ) + 0 . 0 0 3 2 ^ ^ - = — ^ - )
= 0.224+0.011 In 7|io+ 0 013(iVv - 3 ) + 0 017(71/2 - 10.4min), (7) where TI/2 isthehalflifeofnemron. Though the abundances of D ,3He ,7Li sensitively depend on 17 and dependences on T„ and Nv are almost negligible within the ranges of the expected uncertainties of these two values, the ambiguity of the4 He abundance due to the uncertainties in these values is serious. As we can see from eq. (7), 1 a uncertainty of the neutron lifetime 16 s corresponds to the difference of a factor 2 in n. Such a large ambiguity cannot be neglected in deriving constraints on n by comparing the predicted abundances with the primordial abundances implicated by observations. If the ambiguity of i) arising from the uncertainty in T„ is suppressed within 10% of Ig n i.e. factor 1.3 of t), the uncertainty in T„ will be almost negligible. It corresponds to a precision in T» within ~ 0.6 %. Concerning this point it will be discussed in detail in the conclusions.
A.2 T h e Primordial Abundances
As the primordial abundances of the light elements to be compared with the predicted abundances, we adept the following values.
D/H > 1 x lO"3 (8) .t:>. Since白ethreshold energy ofthe deuteron phot凶iss
∞
iationis sma11， nuc1eosynthe -sis is delayed to such a later time.The number of photons capable of diss町iatingthe dcutcron per nuclcon varics凶η-1exp( -2.2/
T
)
.
Hcnce白cgreater nuclcon/photon ratioη.出chigher出etemperatw唱atwhich the deuteron formation effectively begins. 百le4 He abundance，白ercforc，inc問aseswith incr官asingη. 百esynthesis ofheavier elements follows the deuteron formation as shown schema!I -cally in Fig. 1.Due 10出egaps of slable nuclei between mass-5 and mass-8，
the synthesis of the elements heavier than helium is prcvented and some trace amounts of7Li and 7 Be are pr凶uced. vi) t~ 1O ]s.T;53 x 108K The coulomb baniers hecomc so large that nucleosynthesis is cffectively terminated. In Fig. 2，
we show an examplc of the timc cvolution of thc abund:mccs of clcmcnts for可10= 3(可10=
=
，
，
/
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4
forN
v = 4_百leneutron lifetime is taken to beT. 898 s andT .. 898士165which are deviatedla合'Omthe mean given in the Particle Data(1986) [3J. 羽te4He abundance incr官ase5with incrcasingηas described abuve. On the other hand， the 0a
n
d
3 He abundances decr官asewith incrcasingηbecause they arc consumed 10 produce 4 He morc effectively for larger年 百ledip of the 7 Li abundance is formed by thcfo日owingreason; 7 Li is pr叫uceddircctlyvia 4He (T ，"1)7Li in therange'110;
5
3 and in themnge可10と
3白eindircct process 4He eHeマ
，
)7Be，
7Be(e九v.)7 Lidominate~the pr凶uctions. In Table 1， the abundanc""s of m司jorelements are listcd for various valucs of九for η= 10-10 • In the range 2 <η10 < ¥0，thc
p
n
-
dicted prim口氏lialabundance of 4 He is well fit by ITn - 898S¥Y
p = 0 幻224糾ω +叫0.01山1lIh
恥川
引nn川η川T引'1附1l 0.224+ O.β0111n 1司1'悶0+00ω13沢(N，ν一
3勾)+00似17穴(η /ρz一
1ωO.
4
mi毘吋)， (7) whercη11is the halflife ofn:
c
u!ron. Though the abundances ofD .
3He • 7Li sensitively dependonηand dependences on九and比 四 凶mostnegligible within thc mngcs日f the expected uncertainties of these two values， the ambiguity of the 4 He abundance duc 10 the uncertainties in Ihese values is serious. As we can see from eq.σ)，
10 unce此ainty of the neutron lifetime ) 6 s coπ官sponds10山ediffercnce of a factor 2 in η. 5uch a largc ambiguity cannot be neglected in deriving constraints onηby comparing山eprcdictcd abundances wi白theprimordial abundances implicated by obscrvations. Ifめcambiguity ofηarising合omthc uncertainty in九isSlIppressed wi山in10% oflgη i.e. factor 1.3 ofη. the uncertainty in・
r"wlll泌a1moslnegligible. It corrcs卯nds10 a pn
:
cision泊T. withinrv 0.6 % • Concemin;
g
1トispoint it wi
U
be discusscd in delail in the conc1usions.A.2 The Primordial Abundances
As the primordial ab叩dancesof Ihe
i
I
ghl elements 10 be compared with thc prcdictedabundances
，
we &;!~pt Ihe fo1
l
owing vaJues.D(n,7)T
D(d,p)T
I
T —I
T(p,
7)
4He
T(d,n)
4He
p,n
p(n,7)D
I
D
T fan)
3He
T(c" i7,)
3He
4He
_ L
D(p,7)
3Hc
D(d,n)
3He
I
— • 3H c 3Hc(n,7)
<He
3He(d,p)
4He
I
4
He(
3He,7)'Be
4HefT.7)
7U
I I
7
Be
7Be(e-, v.fli
7Li
Figure 1: Main paths of nucleosynthesis.
C
q
u
C/)ro
10"
10
1 w
- 2 :.
10
-3 ^
10
10
10
10
- 4 _
-5 ;.
- 6 ,.
-7 „
1 0
_ aio-
9
10
- 1 0
10
1 t • ' — ' ' ' ' " 1 pF \
V
r ff 1.
r s/
r Jy /
s i / . i >r, .i
= a , 1 1 1 1 1 1 | 1 J 1 1 1 i • • | ^ j r—iT ^ i
/—A 1
ii nl m/1 / \ \ "^
I J
/ - ^ X
J
H
e
^
/ /X \ i
/ V_ \ ^ •
./ . . . ^rrv
_
_ ZB&J
10'
Time
10'
10'
sec.
Figure 2: Time evolution of elements.
— 5 —
c o
z o
ω
﹂ 比ω
ω
町 三
pin p(n，
，
)D D D(nマ
，
)T D(，p)T dj
wEFHe T(c-ii.)3恥 T(p，1')4Hc T(，nd)‘
Hc D(p・'1)3Hc D(d，n)JHcJ
3He lHc(n，'1)4Hc 3Hc(，dp)4HcI
4 勺Hc 4附
守
T
や
.
々
守
陶
4H恥ぷ
eσ
ぷ
!
守
ザ
)7 7B巴-一一
7百Bc(令c-司勺.ν均
'
.
)
'
L
i
一
一
-7L
i
Figurc 1: Main pa血sof nuclcosyn出巴sis.10-
1
10-
2
10-
3
10-
4
10-
5
k
-10-
6
‘ 、‘ 、、
‘ 、、
、
‘ 、、
、
、
、
、 、、
、
‘ 、、
、
‘ 、‘ 、、
目 、 ヘ ・ ・ 戸1
O~
"
1
10-
7
10-
8
10-
9
1
0
-
1
0
L
1
0
'
Tlme
1
0
3
1
1
1
1
0
2
Figurc 2: nme evolution of clcmcnts.5
a.
0.30
0.25
0.20
rnr
N
v=
3
1
__ 898+16 S
^ — . 898
S
- 8 9 8 - 1 6 S
D+
3
rie
Figure 3: Predicted abundances ofD,
3He,
4He,and
7Li versus tjibriy,^ 3 . The
4He
abundace is given by its mass fraction and the abundances of the other elements are the
number ratios to the hygrogen.
0.30
- i 1 11—i—i—i—r-N
v
=4
0.30
0.25
> -
: ^ ^ "
898+16 s
898 s
898-16 s
0 . 2 5
0.20
:
0.20
10
-10
i , * t I i i i i % . i10
-10
10
- 9
Figure 4: Same figure with Fig. 3 for JV„ = 4 .
o'l
0
.
3
0
0.30
ト
ly=3
0.25
-"--三Z
-
-
-
竺
Zニー'一三τ孟 --・・-~ーーτニー--ーー ---てエニーーてごトーー-」ー・てニ:---:---一 ---~.
.
.
-
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=
:
:
.
ニ
ゴ
エ
戸
-
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-
-
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-
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-"二".，.，.-:... ，.こ-'二J -r _.
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0
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一ーー~ι
____
898+16 s
898
s
ー
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:
898-16
s
ヱ
¥
︿
10-
1
0
0.20
1
0
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3
0
.
2
0
1
0
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1
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-
4
1
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1
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工
¥
<(
1
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円
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i
g
u
r
e
3
:
P
r
e
d
i
c
t
e
d
a
b
u
n
d
a
n
c
e
s
ofD
，
3He
，
4
H
e
.
a
n
d
7L
i
v
e
r
s
u
S
η
f
o
r
N
II=
3
.
百l
e4He
a
b
u
n
d
a
c
e
i
s
g
i
v
e
n
b
y
I
t
s
m
a
s
s
f
r
a
c
t
I
o
n
a
n
d
t
h
e
a
b
u
n
d
a
n
c
e
s
o
f
t
h
e
o
t
h
e
r
e
l
e
m
c
n
t
s
a
r
e
t
h
e
n
u
m
b
e
r
r
a
t
i
o
s
10t
h
e
h
y
g
r
o
g
e
n
.
l
¥
I
v
=
4
.
o
.
a
o
一ーーョニ 一ーー士三二ニー.-，，::;戸田 -一ー・---二τ
-
二
-
-
- -・・・ヲ三二-ー'一二--- ---与~二回--一...~_...-~ - - 三Zと三=戸ー「ー・...~~・回
-_...二，戸-...~ .;~，.-，， -. ...-0.25
ー--・696+16
5
_ _ 898
s
-・
698-16
s
0
.2
.
0
1
0
-
3
D ¥
史
ア
J
-
1
0
-4
¥
¥
¥
:
_
:
h
o
-
S
11
0-
9
1
0
-
1
0
1
0
-
1
0
1
0
-
9
円
Fig
u
r
c
4:Same f
i
g
u
r
c
w
i
t
h
Fig
.
3f
o
r
Nν=4.，
( D +3H e ) / H < l x l 0- 4 (9) 0.22 < y „ < 0.26 (10) 1 x 10-1 0<7Li/H < 1 x 10"' (11) where Yp is the mass fraction of 4He and the abundances of the other elements are the
number ratio to that of hydrogen.
Though these values are taken rather loose to derive firm constraints on the Iepton and baryon asymmetries, recent develops in the observational studies enable us to find more crucial limits on the primordial abundances of 4Hc and 7Li.
Steigman, Gallagher, and Schramm [4] have found by extrapolating the correlation between the 4He abundance and the carbon abundance of HII regions
yp = 0.235 ± 0 . 0 1 2 (3tr). (10') This value is consistent with other recent estimates [5,6] as declared in [4]. Since Spite and Spite[7] claimed that the extreme Pop II stars give the best guess on the primordial abundance of 7 Li, it has been recognized generally that the 7 Li abundance also provides the crucial test of the consistency of the primordial nucleosynthesis. Recent two independent studies [8,9] have obtained the same result each other,
7Li/H = 1 . 2 ± 0 . 9 x l O -1 0 (3a), (11') which is consistent with7Li/H RS 1.1 x 1 0- 1 0 obtained by Spite and Spite 1982[7]. Adding to (10') and (11'), the "best bet values" for the primordial abundances of D and 3He given by Yang et al. (1984) [10] are employed as a stringent set of constraints,
D/H > 1 x lO"5, (8') (D + 3He)/H < 1 x l O "4. (9')
The set of ihc bounds on the primordial abundances (8')~ (11') will provide more crucial restrictions on JJ.
A.3 Comparison of The Predicted Abundances with Observations
By comparing the predicted abundances of D , 3He, 4He, and 7Li with the primordial abundances inferred from observations, constraints on T; are found. In Table 2, constraints on 7) are summarized for N„ = 3, 4. The baryorvphoton ratio i) is related to the baryon density parameter by
(lfl = 0.00353 ( i o T i o ) ^ ^ , d 2 ) where 6a is the present temperature of the cosmic microwave background radiation
( 9 = To/2.7°K ) and ho is the present value of Hubble constant normalized as fto = /fn/lOOMpc-'kms"1. Hence the bounds 3 < JJIO < 10 forJV„ = 3 correspond to 0.01 < Da < 0.03. It is consistent with the so-called "luminous mass" of galaxies n,„i = 0.01 - 0.02, which is estimated from the dynamics of galaxies.
‘司 (D +3He)/H < 1 X 10-4 0辺<}ち予 <0.2 1 以x1ω0-10
，
5
忘
:
7Li/畑
Hく1x 1ω0-9 9 (9) (10) (11)where
y
p
is山 massfrac由n0{4He and白 abundancesof脚 叫crelemcnts are the numbcrratio toUlat of hydrogcn. η10ugh these values are taken mther 1∞
'sc to derive finn cons回intson白clepton and baryo目 白;ymme凶es，
recent develops in the observational studies enable us to find moreロuciallimits00出epr加10抵ialllbundancesof‘
He田dヲU. S包igm回 .Gallagher， 回dSchramm [4} have foundby extrapolating the correlation bc!Ween恥 4He abundance and山 由 加nabun必n倍。fHll開gionsむ =
0.235士0.012 (3σ) . ( 10') This va1ue is∞
nsis飽ntw抽 o出erreαntes白 田 健s[5，句asdeclared in μJ. SinceSpi飽andSpi旬[7]claimed that the extreme Pop IIs国 富glVe由ebcst guess on thepr加口>r:Iialabun白nceof7 U，
it has been recognizedgcnemlly that the 7Liabundance dωprovides the crucia1test of the consistency of the primordial nucleosyn出esIS.Re氾ent two indcpendent studies [8，
9] have obtained出CS創nercsult cach othcr，
7U/H=
1.2土0.9X 10-10(3a)，
(11') which is consistent with 7Li /H回1.1X 10-10 obtainedbySpiteandSpite 1982[7]. Adding to (10・)and (11')，也s“
bestbet va1ues" for thc primordia1abundances of0and 3He givcn by Yang eta1.(1984)[1OJ are employed as a stringcnt sel of constraints，
D/H>lx
l
O
-
'
，
(8') (D + 3He)/H < 1 x 10-4. (9')The sct OfUl巴加，undson the戸泊。凶i必abundances(8・)~(U') wi且戸口，videmore crucial restrictions 00η.
A.3 Comparison of The Pred icted Abundances with Observations
By comparing曲epredicted abuodanccs of D
，
3Hc，
4Hc，
and 7Liwith thc primordial abundanccs infcrred from obse刊a!Ions，constr垣n也白n可arefound.InTable 2， coostt討n隠onηare summanzed for
N
II=
3，
4.T
h
c baryon
/
p
hoton mtioηis related to白cbaryondensity parameter by On "，
0
∞
3
5
3
(品かl~hõ2
(12) where 00i
s
the pre罰 則自emp訂atureof the cosmic microwave background m必ation (9=
T
o
/
2
アK) andh
o
is the presem value of Hubble constant nonnalizcd ash
o
=
H
o
/
l
∞
M
p
c
-
I
kms-I
.
Hence the加unds3く可10< 10{orN
II=
3 correspond 10 0.01 < Os < 0.03. Jt is consistent wi血the符-c剖凶“luminousm師;"of galaxicsn
，
.l=
0.01 -0.02，
which is estimated from the dynamics of gal副es.Table 1: Predicted abundances of D , 3He, "He ,and 7Li for Af„ = 3 andtj = 1 0- 9 T„(SCC) 898-48 -40 -32 -24 -16 - 8 + 0 + 8 +16 +24 +32 +40 +48 D/H x 103 1.07 1.08 1.08 1.09 1.09 1.10 1.10 1.11 1.11 1.12 1.12 1.13 1.13 3Hc/H x 106 9.09 9.10 9.11 9.11 9.12 9.13 9.14 9.14 9.15 9.16 9.17 9.18 9.18 7Li/H x 10l 0 9.81 9.85 9.89 9.94 9.98 10.0 10.1 10.1 10.1 10.2 10.2 10.3 10.3 Y 2.39 2.41 2.43 2.44 2.46 2.48 2.49 2.51 2.52 2.54 2.55 2.57 2.58 Table 2: Constraints on JI 1) 1') 2 ) 2') 3) 3') 4) 4') Bounds on abundance 0.22 <YP< 0.26 Yp= 0.235 ± 0 . 0 1 2 D/H > 1 x lO"5 D/H > 2 x lO"5 (D + 3He)/H < 10 x 10-5 ( D +3H e ) / H < 6 x l O "5 1 0 -1 0<7U / H < 1 0 - ' 7L i / H = ( 1 . 2 ± 0 . 9 ) x l O -1 0 Nu = 3 1 <1io< 30 1 < 7 J , 0 < 7 7J,o<10 1io<8 3<»jio 5<7JI0 »lio< 10 2 < T J I O < 6 Af„ = 4 1io<7 JJIO<3 TJ10< 10 JJio<8 3 < 7)10 5<7Jl0 ijio< 10 2 < j j i o < 5 1 + 2 + 3 + 4 3<7)io<10 3<T(io<7 l ' + 2,+ 3,+ 4' 5 < i j ,0< 6 Table 1:Predicted abundances of D
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