BIG BANG NUCLEOSYNTHESIS
I.
EQUILIBRIUM ABUNDANCES
The binding energies of the first four light nuclei, 2 H, 3 H, 3 He and 4 He are 2.22 MeV, 6.92
MeV, 7.72 MeV and 28.3 MeV, respectively (see Table I). As the universe cools below these
temperatures, one expects these bound structures to form. The abundance of light elements,
which are synthesized in the early universe, can be used to obtain important constraints on
the cosmological parameters.
Though the energy considerations suggest that these nuclei could be formed when the
temperature of the universe is in the range (1 − 30) MeV, the actual synthesis takes place
only at a much lower temperature, TBBN ≃ 0.1 MeV. The main reason for this delay is the
‘high entropy’ of our universe, i.e., the high value for the photon-to-baryon ratio, η −1 .
Let us provisionally assume that the nuclear (and other) reactions are fast enough to
maintain thermal equilibrium between various species of particles and nuclei. In thermal
equilibrium, the number density of a nuclear species A NZ with atomic mass A and charge
Z will be
mA T 3/2
µA − mA
.
exp
2π
T
In particular, the equilibrium number densities of protons and neutrons are
nA = gA
np ≃ 2
mB T
2π
3/2
(1)
µp − mp
,
T
(2)
exp
µn − mn
mB T 3/2
.
(3)
exp
2π
T
The mass difference between the proton and the neutron Q ≡ mn − mp = 1.293 MeV has
nn ≃ 2
3/2
to be retained in the exponent but can be ignored in the prefactor mA . We have set in the
prefactor mn ≃ mp ≃ mB , an average value.
Since the chemical potential is conserved in the reactions producing A NZ out of Z protons
and (A − Z) neutrons, µA for any species can be expressed in terms of µp and µn :
µA = Zµp + (A − Z)µn .
(4)
Writing
µA − mA
exp
T
"
#
mA
Zµp + (A − Z)µn
= exp
exp −
T
T
1
= [exp(µp /T )]Z [exp(µn /T )](A−Z) exp(−mA /T )
(5)
and substituting for exp(µp /T ) and exp(µn /T ) from (2) and (3) we get
µA − mA
exp
T
=
2−A nZp nn(A−Z)
2π
mB T
3A/2
Zmp + (A − Z)mn − mA
exp
T
=
2−A nZp nn(A−Z)
2π
mB T
3A/2
exp(BA /T ),
"
#
(6)
where
BA ≡ Zmp + (A − Z)mn − mA
(7)
is the binding energy of the nucleus. Therefore, the number density (1) becomes
nA = gA 2
−A
3/2
A
2π
mB T
3(A−1)/2
nZp nn(A−Z) exp(BA /T ).
(8)
Since particle number densities in the expanding universe decrease as a−3 (for constant
number per comoving volume), it is useful to use the total baryon density, nb = nn + np +
P
i (AnA )i ,
as a fiducial quantity and to consider the ‘mass fraction’ contributed by nuclear
species A(Z),
XA ≡
Note that
P
i
AnA
.
nb
(9)
Xi = 1. Substituting for nA , np and nn in (8) by nA = nb (A−1 XA ) =
ηnγ (A−1 XA ), np = ηnγ Xp and nn = ηnγ Xn , where
η≡
nb
= 2.68 × 10−8 (Ωb h2 )
nγ
(10)
is the baryon-to-photon ratio and nγ = [2ζ(3)/π 2]T03 is the number density of photons, we
get
XA = F (A)(T /mB )3(A−1)/2 η A−1XpZ XnA−Z exp(BA /T )
(11)
F (A) = gA A5/2 [ζ(3)A−1π (1−A)/2 2(3A−5)/2 ].
(12)
where
Eq. (11) shows why the high entropy of the universe, i.e. small value of η, hinders the
formation of nuclei. (For purposes of comparison, in a star like our sun, nγ /nb ∼ 10−2 ;
even in the post-collapse core of a supernova, nγ /nb is only a few. Indeed, the entropy of
the univesre is enormous.) To get XA ≃ 1, it is not enough that the univesre cools to the
temperature T <
∼ BA ; it is necessary that is cools still further so as to offset the small value
2
of the η A−1 factor. The temperature TA at which the mass fraction of a particular species
A will be of order unity (XA ≃ 1) is given by
TA ≃
BA /(A − 1)
.
ln(1/η) + 1.5 ln(mB /T )
(13)
This temperature is much smaller than BA ; for 2 H, 3 He and 4 He the value of TA is 0.07
MeV, 0.11 MeV and 0.28 MeV, respectively. Comparison with the binding energy of these
nuclei shows that these values are lower than BA by a factor of about 10, at least.
II.
FALLING OUT OF EQUILIBRIUM
Thus, even when the thermal equilibrium is maintained, significant synthesis of nuclei can
occur only at T <
∼ 0.3 MeV and not at higher temperatures. If such is the case, then we would
expect significant production (XA ≃ 1) of nuclear species A at temperatures T <
∼ TA . It
turns out, however, that the rate of nuclear reactions is not high enough to maintain thermal
equilibrium between various species. We have to determine the temperatures up to which
thermal equilibrium can be maintained and redo the calculations to find non-equilibrium
mass fractions.
In particular, we used the equilibrium densities for np and nn in the above analysis. In
thermal equilibrium, the interconversion between n and p is possible through weak interaction processes:
n ↔ p + e− + ν̄e ,
νe + n ↔ p + e− ,
e+ + n ↔ p + ν̄e .
(14)
When the rates for these interactions are rapid enough compared to the expansion rate H,
chemical equilibrium obtains,
µn − µp = µe − µν ,
(15)
from which it follows that in chemical equilibrium
nn
Xn
=
= exp[−Q/T + (µe − µν )/T ].
np
Xp
(16)
Based upon the charge neutrality of the universe, we can infer that µe /T ∼ ne /nγ = np /nγ ∼
η, from which it follows that µe /T ∼ 10−10 . The electron-neutrino number of the universe is
3
similarly related to µν /T ; however, since this relic background has not been detected, none
of the neutrino numbers is known. We will assume that the lepton numbers, like the baryon
numbers, are small, so that µν /T ≪ 1. With this assumption,
nn
np
!
= exp(−Q/T ).
(17)
EQ
The ratio will be maintained as long as the n − p reactions are rapid enough. But when the
reaction rate Γ falls below the expansion rate, H = 1.66g 1/2T 2 /mPl ≃ 5.5(T 2/mPl ) at some
temperature TD , the ratio (nn /np ) will get frozen at the value exp(−Q/TD ). The only process
which can continue to change this ratio thereafter will be the beta decay, n → p + e + ν̄, of
the free neutron. The neutron decay will continue to decrease the ratio until all neutrons
are used up in forming bound nuclei.
To determine TD we have to estimate Γ and use the condition Γ = H. The rate Γ can
be calculated from the theory of the weak interactions. The rates are found by integrating
the square of the matrix element for a given process, weighted by the available phasespace densities of particles (other than the initial nucleon), while enforcing four-momentum
conservation. One obtains
Γ(nνe → pe− ) = A
Γ(ne+ → pν̄e ) = A
Γ(n → pe− ν̄e ) = A
Z
0
Z
0
Z
0
∞
∞
q0
dqν qν2 qe Ee (1 − fe ) fν ,
Ee = Eν + Q,
dqe qe2 qν Eν (1 − fν ) fe ,
Eν = Ee + Q,
dqe qe2 qν Eν (1 − fν ) (1 − fe ) ,
q0 =
q
Q2 − m2e .
(18)
Here A is an effective coupling while fν and fe are the distribution functions for electrons and
neutrinos. Although the weak interaction coupling GF is known quite accurately from muon
decay, the value of A or, equivalently, the neutron lifetime, cannot be directly determined
from this alone because neutrons and protons also interact strongly, hence the ratio of
nucleonic axial vector (GA ) and vector (GV ) couplings is altered from unity. Moreover,
relating these couplings to the corresponding experimentally measured couplings of the u
and d quarks is complicated by weak isospin violating effects. If we assume conservation of
the weak vector current (CVC), then GV = GF cos θc where sin θc = 0.22 = |Vus |. However,
the weak axial current is not conserved and GA for nucleons differs from that for the first
generation quarks. These non-perturbative effects cannot be calculated reliably, hence GA
(in practice, GA /GV ) must be measured experimentally. The neutron lifetime is then given
4
by
τn−1
m5
= e3 G2V
2π
G2
1 + 2A f,
GV
!
(19)
where f = 1.715 is the integral over the final state phase space (including Coulomb corrections) and GV is usually determined from superallowed 0+ → 0+ pure Fermi decays of
suitable light nuclei. It is thus more reliable to measure the neutron lifetime directly, and
then relate it to the coupling A in (18) in order to obtain the other reaction rates. The
experimental value is τn = 885.7 ± 0.8 sec. (In the literature one often finds the neutron half
life τ1/2 (n) = τn ln 2.) Thus we obtain, for example,
Γ(pe → νn) ≃

 τn−1 (T /me )3 exp(−Q/T )

2)
7π(1+3gA
60
G2F T 5
T ≪ Q, me ,
T ≫ Q, me .
(20)
Clearly τn Γ is a function of temperature alone. Similar calculations can be performed for
all other reactions allowing us to compute τn Γ(n → p) and τn Γ(p → n) as functions of T .
At high temperatures, T ≫ Q ≃ 1.3 MeV, both rates vary as T 5 ; in the range T ≃ (0.1 − 1)
MeV, the rate Γ(n → p) ∝ T 4.42 while Γ(p → n) decreases faster; for T < 0.1 MeV,
Γ(n → p) ≃ τn−1 is essentially dominated by the neutron decay while Γ(p → n) drops
exponentially.
These rates have to be compared to the Hubble factor, H(T ) ≃ 5.5(T 2 /mPl ) ≃
0.179(T /me )2 s−1 :
Γ/H ∼ (T /0.8 MeV )3 for T >
∼ me .
(21)
Thus thermal equilibrium between neutrons and protons exists only for T >
∼ TD ≃ 0.8 MeV.
At TD , when the assumptions of thermal equilibrium becomes invalid, the n/p ratio will be
Q
Xn
= exp −
Xp
TD
1
≃ ,
6
(22)
giving Xn ≃ (1/7), Xp ≃ (6/7). Since TA calculated in (13) is lower than TD for all the
light nuclei, none of the light nuclei will exist in significant quantities at this temperature
TD . For example, 2 H contributes only a mass fraction X2 ≃ 10−12 at TD .
To summarize: the time t ≃ 1 s, when T ≃ 1 MeV is a very interesing time in the
history of the universe. Shortly before this epoch, the three neutrino species decouple
from the plasma. A little later, the e± pairs annihilate, transferring their entropy to the
photons alone and thereby rising Tγ relative to Tν by a factor of (11/4)1/3 . At about
this time the weak interactions that interconvert neutrons and protons freeze out, with
5
(nn /np )D = exp(−Q/TD ) ≃ (1/6). After freeze-out, the neutron-to-proton ratio does not
remain truly constant, but slowly decreases due to occasional weak interactions (eventually
dominated by free neutron decays). The light elements are still in statistical equilibrium
with very small abundances (see Table I).
III.
SYNTHESIS OF THE LIGHT ELEMENTS
As the temperature falls further to T = THe ≃ 0.28 MeV, a significant amount of 4 He could
have been produced if nuclear rates were high enough. These reactions [D(D,n) 3 He(D,p)
4
He; D(D,p) 3 H(D,n) 4 He; D(D,γ) 4 He] are all based on D, 3 He and 3 H and do not occur
rapidly enough because the mass fractions are still too small at T ≃ 0.3 MeV: X2 H ∼ 10−12 ,
X3 He ∼ 10−19 , and X3 H ∼ 5 × 10−19 .
These abundances become nearly unity at T <
∼ 0.1 MeV; therefore, only at T <
∼ 0.1
MeV can these reactions be fast enough to produce an equilibrium abundance of 4 He. (The
delay from 0.3 to 0.1 MeV is known as ‘the deuterium bottleneck’.) When the temperature
2
3
becomes T <
∼ 0.1 MeV, the abundance of H and H builds up and these elements further
react to form 4 He. A good fraction of 2 H and 3 H is converted to 4 He. The resultant
abundance of of 4 He can be easily calculated by assuming that almost all neutrons end up
in 4 He. Since each 4 He nucleus has two neutrons, (nn /2) helium nuclei can be formed (per
unit volume) if the number density of neutrons is nn . Thus the mass fraction of 4 He will be
Y ≡ X4 He =
4(nn /2)
2(nn /np )NS
4n4
=
=
.
nb
np + nn
1 + (nn /np )NS
(23)
The ratio (nn /np ) at the time of ‘freeze-out’ (tD ) was (1/6); from tD till the time of nucleosynthesis (tNS ) a certain fraction of neutrons would have decayed, lowering this ratio. Since
the freeze-out occured at TD ≃ 0.8 MeV, tD ≃ 1 s and 4 He synthesis occured at TNS ≃ 0.1
MeV, tNS ≃ 3 min, the decay factor will be exp(−tNS /τn ) ≃ 0.8. Therefore, we obtain
(nn /np )NS ≃ 0.8 × (1/6) ≃ (1/7) =⇒ Y ≃ 0.25.
(24)
As the reactions converting 2 H and 3 H to 4 He proceed, the number density of 2 H and 3 H
is depleted and the reaction rates, which are proportional to Γ ∝ XA (ηnγ )hσvi, becomes
small. These reactions soon freeze-out leaving a residual fraction of 2 H and 3 H (a fraction
of about 10−5 − 10−4 ). Since Γ ∝ η it is clear that the fraction of (2 H, 3 H) left unreacted
6
will decrease with η. In contrast, 4 He synthesis, which is not limited by any reaction rate,
is fairly independent of η and depends only on the (n/p) ratio at T ≃ 0.1 MeV.
The production of still heavier elements, even those like
16
C and
16
O which have higher
binding energies than 4 He, is highly suppressed in the early universe. Two factors are
responsible for this suppression:
(i) Direct reactions between two He nuclei or between H and He will lead to nuclei with
atomic masses 8 or 5. Since there are no tightly bound isotopes with masses 8 or 5, these
reactions do not lead to any further synthesis. (The three body interaction, 4 He + 4 He +
4
He → 12 C is suppressed by the low number density of 4 He nuclei; it is this ‘triple-α’ reaction
which helps further synthesis in stellar interiors.)
(ii) For nuclear reactions to proceed, the participating nuclei must overcome their
Coulomb repulsion. The probability to tunnel through the Coulomb barrier is governed
by the factor
h
F = exp −2Ā1/3 (Z1 Z2 )2/3 (T /1 MeV )−1/3
i
(25)
where Ā = A1 A2 /(A1 + A2 ). For heavier nuclei (with larger Z), this factor suppresses the
reaction rate.
Small amounts (about 10−10 − 10−9 by mass) of 7 Li are produced by 4 He(3 H,n)7 Li or
by 4 He(3 He,γ)7 Be followed by the decay of 7 Be to 7 Li. The first process dominates if η <
∼
−10
−10
3 × 10
and the second process for η >
∼ 3 × 10 . In the second case, a small amount
(10−11 ) of 7 Be is left as a residue.
A good approximation to the BBN predictions is the following:
η
,
5 × 10−10
−1.6
D
η
−5±0.06
= 3.6 × 10
,
H p
5 × 10−10
!
−0.63
3
η
He
= 1.2 × 10−5±0.06
,
H p
5 × 10−10
Yp (4 He) = 0.245 + 0.01 ln
7
Li
H
!
p
= 1.2 × 10
−11±0.2
"
η
5 × 10−10
7
−2.38
(26)
η
+ 21.7
5 × 10−10
2.38 #
.
IV.
A.
FROM OBSERVATIONS TO PRIMORDIAL ABUNDANCES
Introduction
To test the SBBN it is necessary to confront predictions of BBN with the primordial
abundances of light nuclides which are not ‘observed’, but are inferred from observations.
The path from observational data to primordial abundances is long and twisted and often
fraught with peril. In addition to the usual statistical and insidious systematic uncertainties,
it is necessary to forge the conncetion between the derived abundances and their primordial
values. It is fortunate that each of the key elements is observed in different astrophysical sites
using very different astronomical techniques and that the corrections for chemical evolution
differ and, even more important, can be minimized. For example, deuterium is mainly
observed in cool, neutral gas (HI regions) via resonant UV absorption from the ground state
(Lyman series), while radio telescopes allow helium-3 to be studied via the analog of the
21 cm line for 3 He+ in regions of hot, ionized gas (HII regions). Helium-4 is probed via
emission from its optical recombination lines in HII regions. In cotrast, lithium is observed
in the absorption spectra of hot, low-mass halo stars. With such different sites, with the
mix of absorption/emission, and with the variety of telescopes involved, the possibility of
correlated errors biasing the comparison with the prediction of BBN is unlikely.
This favorable situation extends to the obligatory evolutionary corrections. For example,
although until recently observations of D were limited to the solar system and the Galaxy,
mandating uncertain corrections to infer the pregalactic abundance, the Keck and Hubble
Space telescopes have opened the window to deuterium in high redshift, low metallicity,
nearly primordial regions (Lyman-α clouds). Observations of 4 He in low-metallicity (∼ 1/50
of solar) extragalactic HII regions permit the evolutionary correction to be reduced to the
level of the statistical uncertainties. The abundances of lithium inferred from observations
of the very metal-poor halo stars (∼ 1/1000 of solar and even lower) require almost no
correction for chemical evolution.
On the other hand, the status of 3 He is in contrast to that of the other light elements.
Although all prestellar D is converted to 3 He during pre-main sequence evolution, 3 He is
burned to 4 He and beyond in the hotter interiors of most stars, while it survives the cooler
exteriors. For lower mass stars, a greater fraction of the prestellar 3 He is expected to survive
8
and, indeed, incomplete burning leads to buildup of 3 He in the interior which may, or may
not, survive to be returned to the interstellar medium. In fact, some planetary nebulae
have been observed to be highly enriched in 3 He, with abundances 3 He/H∼ 10−3 . Although
such high abundances are expected in the remnants of low mass stars, if all stars in the
low mass range produced comparable abundances, we would expect solar and present ISM
abundances of 3 He to greatly exceed their observed values. It is therefore necessary that
at least some low mass stars are net destroyers of 3 He. For example, there could be extra
mixing below the convection zone in these stars when they are on the red giant branch.
Given such complicated histories of survival, destruction and production, the use of current
Galactic and solar system data to infer the primordial abundance of 3 He should be taken
with caution.
B.
1.
Observations
D
Observations of deuterium in the solar system and in the interstellar medium (ISM) of
the Galaxy provide interesting lower bounds to its primordial abundance. Measuring the
primeval deuterium - rather than finding a lower bound - became possible only in the last
few years. The idea is that deuterium abundance in a high redshift hydrogen cloud can be
measured. Distant hydrogen clouds are observed in absorption against even more distant
quasars. Many absorption features are seen - the Lyman series of hydrogen and the lines of
various ionization states of carbon, oxygen, silicon, magnesium and other elements. Because
of the large hydrogen abundabce, Ly-α is very prominent. In the rest frame, Ly-α occurs at
1216 Å, so that for a cloud at redshift z, Ly-α is seen at 1216(1 + zcloud)Å. The isotopic shift
for deuterium is −0.33(1 + z)Å, or expressed as a Doppler velocity, −82 km s−1 . The idea is
to detect the deuterium Ly-α feature in the wing of the hydrogen feature. For z >
∼ 3, Ly-α
is shifted into the visible part of the spectrum and thus can be observed from Earth; “Ly-α
clouds” are ubiquitous with hundreds being seen along the line of sight to a quasar of this
redshift, and judged by their metal abundance, anywhere from 10−2 of that seen in the solar
system to undetectably small levels, these clouds represent nearly pristine samples of cosmic
material. There are technical challenges: Because the expected deuterium abundance is
9
17
−2
small, D/H∼ 10−5 − 10−4, clouds of very high column density, nH >
∼ 10 cm , are needed;
because hydrogen clouds are ubiquitous, the probability of another, low coloumn-density
cloud sitting in just the right place to mimic deuterium - an interloper - is not negligible;
many clouds have broad absorption features due to large internal velocities or complex
velocity structure; and to ensure sufficient signal-to-noise bright QSOs and large-aperture
telescopes are a must.
At present (2007) we have six high redshift, low metallicity, QSO absorption line systems
with deuterium detections leading to reasonably reliable abundance determinations. There
is excessive dispersion among these determinations (central values vary between 1.6 and 8
in units of 10−5), suggesting that systematic errors, whose magnitudes are hard to estimate,
may have contaminated the determinations of at least some of the D and/or H column
densities. This dispersion serves to mask the anticipated primordial deuterium plateau.
The weighted average of these 6 best detections in high redshift Ly-α gives
−5
(D/H)p = (2.68+0.27
−0.25 ) × 10 .
2.
(27)
D + 3 He
Chemical evolution issues have been woven into the study of BBN from the start. In
order to extrapolate contemporary abundances to primordial abundances the use of stellar
and Galactic chemical evolution models is unavoidable. The difficulties are well illustrated
by 3 He: generally the idea that the sum of D+3 He is constant or slowly increasing seems
to be true, but the details, e.g., predicted increase during the last few Gyr, are inconsistent
with a measurement of 3 He in the local ISM.
Beginning with deuterium, the assumed primeval abundance of Eq. (27) is a factor of two
larger than the present ISM abundance, D/H = (1.5±0.2)×10−5 , determined by the Hubble
Space Telescope observations. This implies (i) little nuclear processing over the history of
the Galaxy; and/or (ii) significant infall of primordial material into the disk of the Galaxy.
The metal composition of the Galaxy, which indicates significant processing through stars,
together with the suggestion that even more metals may have been made and ejected into
the IGM, means that option (i) is less likely than option (ii). Even more intriguing is the fact
that the inferred abundance of deuterium in the pre-solar nebula, D/H = (2.6 ± 0.4) × 10−5 ,
indicates less processing in the first 10 Gyr of Galactic history than in the past 5 Gyr.
10
Moving on to 3 He, the primeval value corresponding to our assumed abundance is 3 He/H≃
10−5 . The pre-solar value, measured in meteorites and more recently in the outer layer of
Jupiter, is 3 He/H = (1.2 ± 0.2) × 10−5 , comparable to the primeval value. The value in the
present ISM, 3 He/H = (2.2 ± 0.8) × 10−5 , is about twice as large as the primeval value. On
the other hand, the primeval sum of deuterium and 3 He,
(D + 3 He)/H = (3.7 ± 1.0) × 10−5 ,
(28)
is essentially equal to that determined in the pre-solar nebula and for the present ISM. This
indicates little net 3 He production beyond the burning of deuterium to 3 He, in conflict with
conventional models for the evolution of 3 He which predict a significant increase in D +3 He
due to 3 He production by low-mass stars.
3.
7 Li
For the primordial 7 Li abundance, one has to correct the data from measurements in the
atmospheres of old halo stars by including empirical corrections for cosmic ray production,
stellar depletion:
−10
(7 Li/H) = (1.2+0.35
.
−0.20 ) × 10
4.
(29)
4 He
Helium-4 plays a different role and presents different challenges. First, the primeval yield
of 4 He is relatively insensitive to the baryon density. Second, the chemical evolution of 4 He
is straightforward; the abundance of 4 He increases due to stellar production: As gas cycles
through generations of stars, hydrogen is burned to helium-4 (and beyond), increasing the
4
He abundance above its primordial value. As a result, the present 4 He mass fraction, Y0 ,
has received a significant contribution from post-BBN, stellar nucleosynthesis, and Y0 > YP .
However, since the “metals” such as oxygen are produced by short-lived, massive stars,
and 4 He is synthesized (to a greater or lesser extent) by stars of all masses, at very low
metallicity the increase in Y shoould lag that in, e.g., O/H, so that as O/H→ 0, Y → YP .
Therefore, although 4 He is observed in the Sun and in Galactic HII regions, the crucial data
for inferring the primordial abundance of 4 He comes from measurements of He/H in regions
11
of hot, ionized gas (HII regions) found in metal-poor, dwarf emission-line galaxies. One
takes a sample and extrapolates to zero metallicity. The present (2007) inventory of such
regions studied for their helium content exceeds 80. Thus, statistics is no longer a problem,
but systemetic corrections and/or errors become the limiting factor.
Indeed, many effects have to be considered to achieve the desired accuracy: corrections
for doubly ionized 4 He and neutral 4 He have to be made; absorption by dust and by stars
have to be accounted for; collisional excitation must be accounted for; potential systematic
errors in the input atomic physics; and extrapolation to zero metallicity must be made in
the absence of a well motivated model.
A recent analysis gives
Yp = 0.238 ± 0.005.
V.
(30)
THE BARYON-TO-PHOTON RATIO
BBN and light element abundances can be used to fix the baryon-to-photon ratio at the
end of BBN (t ∼ 200 sec). Using the above values one obtains
η = (5.6 ± 0.4) × 10−10 .
(31)
This value is driven almost entirely by deuterium.
The data may hint that there has been more depletion of 7 Li in old halo stars than
the conventional models suggest, by a factor of about three. On the other hand, the data
support the simple hypothesis of only pre-MS 3 He production.
To relate η to the present baryon density, Ωb h2 , one needs to know the present photon
temperature,
T0 = 2.725 ± 0.001 K,
(32)
and the average mass per baryon number,
m̄ = 1.6700 × 10−24 g.
(33)
With the assumption that the expansion has been adiabatic since BBN, we obtain
Ωb h2 = (3.650 ± 0.004) × 107 η.
12
(34)
Then the BBN determination of η, Eq. (31), translates to
Ωb h2 = 0.0204 ± 0.0015 (95% CL).
(35)
For h = 0.70 ± 0.07, this implies a baryon fraction
Ωb = 0.041 ± 0.009,
(36)
with the error dominated by that in H0 .
Measurements of the cosmic microwave background (CMB) anisotropy have recently also
determined the baryon density. The physics underlying the method is very different - gravitydriven acoustic oscillations in the Universe at 500,000 yrs - but the result is similar, Ωb h2 =
0.0223 ± 0.0008.
The BBN measurement of η, now confirmed by CMB, requires nonbaryonic dark matter
with Ωdm h2 ∼ 0.12.
VI.
A.
BEYOND SBBN
Neutrino Degeneracy
We now consider the possible role of neutrino degeneracy. A chemical potential in electron
neutrinos can alter neutron-proton equilibrium, as well as increase the expansion rate, the
latter effect being less important.
The possible excess of electron neutrinos over antineutrinos is parametrized by a dimensionless chemical potential,
ξνe ≡ µνe /T,
(37)
which remains constant for freely expanding neutrinos. Anticipating that ξνe will be constrained to be sufficiently small, we neglect the slight increase in expansion rate due to the
increased energy density of the neutrinos and consider only the effect on neutron-proton
interconversions. (We do not consider a chemical potential for other neutrino types, which
would only add to the energy density without affecting the weak reactions.) Consequently,
only the abundance of 4 He is significantly affected and the allowed range of η is still determined by the adopted primordial abundances of the other elements.
13
The resultant increase in the rate of nνe → pe− alters the detailed balance equation to
Γp→n = Γn→p
"
#
Q
exp −
− ξνe ,
T (t)
(38)
and, hence, lowers the equilibrium neutron abundance to
h
Xneq (t, ξνe ) = 1 + e(y+ξνe )
i−1
,
y ≡ Q/T (t).
(39)
This alters the asymptotic neutron abundance by the same factor,
Xn (ξνe , y → ∞) = e−ξνe Xn (y → ∞).
(40)
It is now easy to see that the synthesized helium mass fraction depends on ξνe as follows:
Yp (4 He) ≃ 2Xn (tNS ) ≃ 2Xn (y → ∞)e−tNS /τn e−ξνe
= Ypξνe =0 (4 He)e−ξνe ≃ Ypξνe =0 (4 He)(1 − ξνe )
≈ 0.245 − 0.25ξνe .
(41)
Imposing 0.228 ≤ Yp ≤ 0.248 requires
−0.012 <
∼ ξνe <
∼ 0.068.
(For orientation, a value of ξν ≡ µν /T ≈
(42)
√
2 is equivalent to adding an additional neutrino
flavor (with ξν = 0) which we consider in the next subsection.)
B.
Constraints on New, Light Particles
The number of relativistic degrees of freedom at the epoch of nucleosynthesis can be
constrained. The relevant effects are related to the dependence of the expansion rate, H,
and time, t on the number of relativistic degrees of freedom:
H=
s
2
8πρr
1/2 T
=
1.66
g
,
3m2Pl
mPl
(43)
and, using H 2 (t) = 1/(4t2 ) in the RD epoch,
t ≃ 0.3g −1/2
mPl
T
∼ 1 sec
2
T
1 MeV
14
−2
g −1/2 .
(44)
The counting of the effective number of degrees of freedom goes through the following
summation over relativistic species:
g=
X
gi
bosons
Ti
T
4
+
7 X
Ti
gi
8 fermions
T
4
.
(45)
In the Standard Model, the relativistic degrees of freedom at T ∼ 1 MeV include the photons
(gγ = 2), electrons and positrons (ge = 4) and three neutrino generations (gν = 2), all with
the same temperature. Thus
gSBBN (T ∼ 1 MeV ) = 2 +
7
(4 + 3 × 2) = 10.75.
8
(46)
At T <
∼ me , the relativistic degrees of freedom include the photons and the three neutrino
generations with lower temperature (Tν /T = (4/11)1/3 ):
7
4
gSBBN (T ∼ 0.1 MeV ) = 2 + × 3 × 2 ×
8
11
4/3
≃ 3.36.
(47)
The freeze-out of the weak interactions that keep the (n/p) ratio is determined by the
ratio between the weak interaction rate and the expansion rate. If g at that time is different
from the standard one, eq. (21) is modified:
Γ/H ∼ (T /0.8 MeV )
3
10.75
gh
!1/2
for T >
∼ me .
(48)
Thus the freeze-out temperature TD becomes larger with g:
TD
gh
=
0.8 MeV
10.75
1/6
.
(49)
Consequently, the (n/p) ratio [see eq. (22)] becomes larger with g:
n
p
!
D
Q
= exp −
TD
Q
= exp −
0.8 MeV
1/6

!SBBN [(10.75/gh ) ]
n

.
= 
p D
0.8
M eV /TD
(50)
Nucleosynthesis starts when the deuterium bottleneck is overcome, that is TNS ≈ 0.086
MeV. If g is different at that time from the standard one, this temperature will be reached
at earlier time:
tN S
=
180 sec
15
3.36
gl
!1/2
.
(51)
Then the depletion in the (n/p) ratio due to neutron decay will be weaker:
−tNS /τn
e
180 sec
= exp −
τn
=
−tNS /τn SBBN
e
tNS /180
sec
[(3.36/gl )1/2 ]
.
(52)
Combining (50) and (52), we learn that eq. (24) is modified as follows:
1/2
(nn /np )NS ≃ 0.8(3.36/gl )
C.
1/6
× 0.15(10.75/gh )
.
(53)
Additional Neutrino Generations
Let us consider the effect of additional neutrino generations, ∆Nν = Nν − 3:
gh = 10.75 + 1.75∆Nν ,
gl = 3.36 + 0.46∆Nν .
(54)
A single additional neutrino generation leads to an increase of about 6% in the ratio (n/p)NS ,
leading to a similar increase in the helium abundance. More accurate calculations give
Yp (4 He) = 0.245 + 0.013∆Nν .
(55)
Imposing, for example, Yp (4 He) < 0.25, we obtain
∆Nν ≡ Nν − 3 < 0.4.
(56)
In particular, nucleosynthesis excludes the possibility of additional massless (or, more precisely, significantly lighter than 1 MeV) active neutrino generations.
At present, the measurement of the decay width of the Z 0 boson into neutrinos makes
the existence of three, and only three, light (that is, mν <
∼ mZ /2) active neutrinos an
experimental fact. When expressed in units of the SM prediction for a single neutrino
generation, one gets
Nν = 2.994 ± 0.012 (Standard Model fits to LEP data),
Nν = 3.00 ± 0.06 (Direct measurement of invisible Z width).
16
(57)
D.
Superweakly Interacting Particles
While the BBN bound on the number of active neutrino generations is, at present, weaker
than the bound from laboratory measurements, the bound on the number of relativistic
degrees of freedom at the time of BBN is more generic than that and applies in an interesting
way to many other extensions of the Standard Model.
The interaction rate keeping a superweakly interacting particle in thermal equilibrium will
typically fall behind the Hubble expansion rate at much larger decoupling temperature than
the value of a few MeV for active neutrinos. If the comoving entropy increases afterwards
due to annihilation of massive particles, the abundance of the new particle will be diluted
relative to that of neutrinos or, equivalently, their temperature will be lowered relatively
since neutrinos have the same temperature as that of photons down to T ∼ me . Hence
the energy density during nucleosynthesis of new massless (or, more generally, m ≪ MeV )
particles i is equivalent to an effective number ∆Nν of additional doublet neutrinos:
∆Nν = fB,F
X
i
gi
2
Ti
Tν
4
,
(58)
where fB = 8/7 (bosons) and fF = 1 (fermions), and
"
Ti
gI (T )
=
Tγ
gI (TD )
#1/3
,
(59)
and gI is found by summing over particles that are in thermal equilibrium with the photons.
Thus the number of such particles allowed by a given bound on ∆Nν depends on how
small Ti /Tν is, i.e. on the ratio of number of interacting relativistic degrees of freedom at
TD (when i decouples) to its value at a few MeV (when neutrinos decouple). Using table
III we see that TD > mµ implies Ti /Tν < 0.910 while TD > T qh bounds Ti /Tν < 0.594. The
smallest possible value of Ti /Tν in the SM is 0.465, for TD > T EW .
For example, consider a new singlet fermion F (‘sterile neutrino’). Its decouping temperature can be approximately calculated by equating the interaction rate to the Hubble rate.
For definiteness, consider annihilation to leptons, and parametrize the cross section as
hσviℓℓ̄↔F F̄ ≡ G2X T 2 .
(60)
Them equating the annihilation rate Γeq
ann = nF hσvi to the expansion rate H gives
(TD )F
GF
=
(TD )ν
GX
2/3
17
g[(TD )F ]
g[(TD )ν ]
!1/6
.
(61)
Taking into account that GF = 1.166 × 10−5 GeV −2 , (TD )ν ∼ 2 MeV and g[(TD )ν ] = 10.75,
we find that
−7
−2
(TD )F >
∼ mµ ⇔ GX <
∼ 3 × 10 GeV ,
−8
−2
(TD )F >
∼ 0.3 GeV ⇔ GX <
∼ 10 GeV .
(62)
The energy density of the new particle during nucleosynthesis is then equivalent to, respectively, ∆Nν = 0.69 and ∆Nν = 0.12. Thus, if we impose, for example, ∆Nν < 0.4, we must
require that the interaction strength of the new particle, GX , is weaker than a few times
10−8 GeV −2 . Taking GX ∼ g 2 /m2X , we find that the mass scale of new intermediate vector
bosons should be mX >
∼ 10 T eV .
E.
Sterile Neutrinos
We now consider extra fermions that have no new gauge interactions. Their only interactions with Standard Model particles are through their Yukawa couplings to the active
neutrinos. The consequence of that is that the neutrino mass eigenstates are mixtures of
active and sterile interaction eignestates. Consider for simplicity a single active neutrino νa
and a single sterile neutrino νs . Then, the two mass eigenstates, ν1 and ν2 , are related to
these states through a unitary mixing matrix:
ν1
ν2
!
=
cos θ
sin θ
− sin θ cos θ
!
νa
νs
!
.
(63)
We also define the mass-squared difference,
∆m2 = m22 − m21 .
(64)
Because interaction eigenstates are not eigenstates of free propagation, an initial νa state
could oscillate into a νs state. The transition probability is a function of the mixing angle
sin θ, of the mass-squared difference ∆m2 , of the energy E and distance traveled L:
2
P (νa → νs ) = sin 2θ sin
2
∆m2 L
.
4E
!
(65)
In dense plasma of the early universe, the interactions of the active neutrino with the plasma
change this transition probability. The production of sterile neutrinos by oscillations is given
by
Γs ∝ sin2 θm Γ0 feq ,
18
(66)
TABLE I: The binding energies of light nuclei.
AZ
BA (MeV) TA (MeV) gA XA (T ≫ Q) XA (T = TD )
2H
2.22
3H
6.92
3 He
7.72
0.11
4 He
28.3
0.28
12 C
92.2
0.07
6 × 10−12
10−12
2
2 × 10−23
10−23
1
2 × 10−34
10−28
1
2 × 10−126
10−108
3
2
TABLE II: Light element abundances - SBBN predictions and ‘observed’ values (η̂ ≡ η/5 × 10−10 ).
Predictions
Observations
(D/H)
3.6 × 10−5±0.06 η̂ −1.6
(2.7 ± 0.3) × 10−5
(3 He/H)
1.2 × 10−5±0.06 η̂ −0.63
(3.7 ± 1.0) × 10−5 −(D/H)
(7 Li/H) 1.2 × 10−11±0.2 η̂ −2.38 + 21.7 η̂ 2.38
Y(4 He)
0.245 + 0.01 ln η̂
−10
(1.2+0.35
−0.20 ) × 10
0.238 ± 0.010
where
tan 2θm =
sin 2θ
,
cos 2θ + (2EVeff /∆m2 )
(67)
where Veff is the effective potential due to forward scattering in matter, and
Γ0 =
180ζ(3)ga 2 4
GF T p.
7π 4
(68)
Sterile neutrinos would never be abundant in the universe if Γs /H < 1. But what we
have learnt above is that can impose a stronger condition: the energy density should be
smaller than the energy density of, say, one light neutrino species at BBN (T ∼ 1 MeV).
This leads to a bound of the form
5
2
∆m2 sin2 2θ <
∼ 2.3 × 10 eV .
19
(69)
TABLE III: Thermodynamic history of the RD era.
T
Threshold (GeV)
Particle Content
< me
0.0005
γ(+3 decoupled ν’s)
2
1
me − (TD )ν
0.0023
add e±
11/2
2.75
(TD )ν − mµ
0.106
ν’s become interacting
43/4
2.75
mµ − mπ
0.135
add µ±
57/4
3.65
mπ − T qh
∼ 0.15
add π ± , π 0
69/4
4.41
T qh − ms
0.194
¯ 8g 205/4
γ, 3ν, e± , µ± , u, ū, d, d,
13.1
ms − mc
1.3
add s, s̄
247/4
15.8
mc − mτ
1.78
add c, c̄
289/4
18.5
mτ − mb
4.3
add τ ±
303/4
19.4
mb − mW
80.4
add b, b̄
345/4
22.1
mW − mt
175
add W ± , Z 0
381/4
24.4
mt − mH
?
add t, t̄
423/4
27.1
mH − T EW
300
add H 0
427/4
27.3
20
gI (T ) Nγ (T0 )/Nγ (T )