Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying $\Lambda$ term related to WMAP

Abstract

We investigate the big-bang nucleosynthesis in a Brans-Dicke model with a varying $\Lambda$ term using the Monte-Carlo method and likelihood analysis. It is found that the cosmic expansion rate differs appreciably from that of the standard model. The produced abundances of $^4$He, D, and barely Li are consistent with the observed ones within the uncertainties in nuclear reaction rates when the baryon to photon ratio $\eta = (5.47 - 6.64)\times 10^{-10}$, which is in agreement with the value deduced from WMAP. Comment: Submitted to A&A, 5 pages, 7 figures

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arXiv:astro-ph/0509076v1 5 Sep 2005
Astronomy & Astrophysics manuscript no. Bdl˙6
(DOI: will be inserted by hand later)
February 5, 2008
Big-bang nucleosynthesis in Brans-Dicke cosmology
with a varying Λ term related to WMAP
R. Nakamura1, M. Hashimoto1, S. Gamow1, and K. Arai2
1Department of Physics, Kyushu University, Fukuoka, 810-8560, Japan
2Department of Physics, Kumamoto University, Kumamoto, 860-8555 Japan
Received / Accepted
Abstract. We investigate the big-bang nucleosynthesis in a Brans-Dicke model with a varying Λ term using the
Monte-Carlo method and likelihood analysis. It is found that the cosmic expansion rate differs appreciably from
that of the standard model. The produced abundances of4He, D, and barely Li are consistent with the observed
ones within the uncertainties in nuclear reaction rates when the baryon to photon ratio η = (5.47 − 6.64)×10−10,
which is in agreement with the value deduced from WMAP.
Key words. general:nuclear reactions – nucleosynthesis – abundances – cosmology:early universe – cosmic mi-
crowave background
1. Introduction
The standard model of the big-bang nucleosynthesis
(SBBN) has succeeded in explaining the origin of light ele-
ments4He, D, and7Li. Although the value of the baryon-
to-photon ratio η has been derived from the observations
of the Wilkinson Microwave Anisotropy Probe (WMAP)
(Bennett et al. 2003) to be
seems to be inconsistent with the results of SBBN (Coc
et al. 2004). Contrary to the excellent concordance with
η of WMAP for D, the abundance of4He by SBBN is
rather low compared to that from WMAP. Therefore, non-
standard models of BBN have been proposed with the
Friedmann model modified (Steigman 2003).
η10= 6.1+0.3
−0.2, the value
For non-standard models, scalar-tensor theories have
been investigated (e.g., Bergmann 1968; Wagoner 1970;
Endo & Fukui 1977; Fukui et al. 2001). For a simple
model with a scalar φ, it is shown that a Brans-Dicke
(BD) generalization of gravity with torsion includes the
low-energy limit string effective field theory (Hammond
1996). Related to the cosmological constant problem, a
Brans-Dicke model with a varying Λ(φ) term (BDΛ)
has been presented, and also investigated from the point
of an inflation theory (Berman 1989). Moreover, it is
found that the linearized gravity can be recovered in the
Randall-Sundrum brane world (Garriga & Tanaka 2000).
Furthermore, scalar-tensor cosmology is constrained by χ2
test for WMAP spectrum (Nagata et al. 2004) where the
present value of the coupling parameter ω0 = ω(φ0) is
bounded to be ω0> 50 (4σ) and ω0> 1000 (2σ) in the
limit to BD cosmology.
In the mean time, BBN has been studied in BDΛ (Arai
et al. 1987; Etoh et al. 1997). A relation between BBN
and scalar-tensor gravity is investigated with the inclusion
of e+e−annihilation to the equation of state, where the
present value of the scalar coupling has been constrained
(Damour & Pichon 1999). On the other hand, it is sug-
gested that a decaying Λ modifies the evolution of the
scale factor and affects the temperature Trof the cosmic
microwave background at redshift z ≤ 104, when the re-
combination begins due to the decrease in Tr(Kimura et
al. 2001), while a decaying Λ is found to be consistent with
temperature observations of the cosmic microwave back-
ground for z < 4 (Puy 2004). Therefore, it is worthwhile
to check the validity of BDΛ related to the recent obser-
vations. In the present paper, we investigate how extent
BBN in the BDΛ model can be reconciled with η from
WMAP.
In §2, the formulation for BDΛ is given and the
evolution of the universe in BDΛ is shown. Our re-
sults of BBN are presented in §3 using the Monte-Carlo
method (Cyburt et al. 2001), and constraints are given to
the parameters inherent in BDΛ. We examine in §4 the
evolution of the scale factor and the resulting abundances
with taking into account the deviation from the equation
of state p = ρ/3 during the stage of e+e−annihilation. In
§5, the likelihood analysis (Fields et AB. 1996) is adopted
to get most probable values and the accompanying errors.

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2Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term
-1
-0.5
0
0.5
1
1.5
2
2.5
3
log t [sec]
-10
-9.5
-9
-8.5
-8
log x
BDΛ model (B*= −10)
BDΛ model (B*= −2.5)
Friedmann model
Fig.1. Evolution of the scale factor for BDΛ (µ = 0.7, t0= 13.7 Gyr, and η10= 6.1) and Friedmann model.
2. Brans-Dicke cosmology with a varying Λ term
The field equations for BDΛ are written as follows (Arai
et al. 1987):
Rµν−1
2gµνR + gµνΛ =8π
φTµν+ω
+1
φ(φ,µ;ν− gµν2φ),
φ2(φ,µ;ν−1
2gµνφ,αφ,α)
(1)
R − 2Λ − 2φ∂Λ
∂φ=
ω
φ2φ,µφ,µ−2ω
φ2φ,(2)
where ω is the coupling constant.
The equation of motion is obtained with use of the
Friedmann-Robertson-Walker metric:
ds2= −dt2+ a(t)2
?
dr2
1 − kr2+ r2dθ2+ r2sin2θdφ2
?
,(3)
where a(t) is the scale factor and k is the curvature con-
stant. Let x be a scale factor normalized to its present
value, i.e., x = a/a0, then we get from the (0,0) compo-
nent in Eq. (1)
?˙ x
x
?2
+k
x2−Λ
3−ω
6
?˙φ
φ
?2
−˙ x
x
˙φ
φ=8π
3
ρ
φ,
(4)
where ρ is the energy density.
We assume the simplest case of the coupling between
the scalar and matter fields:
2φ =
8π
2ω + 3µTν
ν, (5)
where µ is a constant. Assuming the perfect fluid for Tµν,
Eq. (5) reduces to
d
dt(˙φx3) =
where p is the pressure.
A particular solution of Eq. (2) is obtained from
Eqs. (1) and (5):
8πµ
2ω + 3(ρ − 3p)x3, (6)
Λ =2π (µ − 1)
φ
ρm0x−3, (7)
where ρm0is the matter density at the present epoch.
The gravitational “constant” G is expressed as follows
G =1
2
?
3 −2ω + 1
2ω + 3µ
?1
φ.
(8)
The radiation density ρr contains the contributions
from photons, neutrinos, electrons and positrons at t ≤
1 s. The total energy density is given as
ρ = ρm+ ρr, ρr= ρrad+ ρν+ ρe±. (9)
Here the energy density of matter varies as ρm= ρm0x−3.
The radiation density ρr= ρr0x−4except e+e−epoch
where e+e−annihilation changes the relation Tr ∼ x−1.
We assume that the pressure satisfies p = ρ/3, which is
legitimated only for relativistic particles. Then, Eq. (6) is
integrated to give
˙φ =
?
8πµ
2ω + 3ρm0t + B
?
1
x3,(10)
where B is a constant (Arai et al. 1987). Although the re-
lation p = ρ/3 does not hold during the epoch of e+e−an-
nihilation, as pointed out by Damour and Pichon (1999),

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Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term3
the inclusion of φ measures small deviation from SBBN
in the case of our interest, so that our solution (10) can
reasonably describe the evolution of φ except the annihi-
lation epoch. We consider that B affects the evolution of
x significantly from the early epoch to the present com-
pared to the contribution from e+e−annihilation. As the
consequence, neutron to proton ratio is affected seriously
by the initial value of B. Considering the important con-
tribution of ρe± to BBN, we examine the effects of e+e−
annihilation in §4. Hereafter we use the normalized values:
B∗= B/(10−24g s cm−3), and η10= 1010η. The coupled
equations (4), (7), and (10) can be solved numerically
with the specified quantities: as for macroscopic quanti-
ties, G0 = 6.672 × 10−8dyn cm2g−2, H0 = 71 km s−1
Mpc−1(Bennett et al. 2003), and Tr0= 2.725 K (Mather
et al. 1999); as for microscopic quantities, the number of
massless neutrino species is 3 and the half life of neu-
trons is 885.7 s (Hagiwara et al. 2002). Though we adopt
ω = 500, the epoch of the appreciable growth of |˙G/G|
is t < 103s regardless of the value ω (Arai et al. 1987).
Therefore, even if we adopt a value ω > 500 (Will 2001),
we can get qualitatively the same conclusion by chang-
ing the parameters µ and B∗. We impose the condition
|˙φ/φ|0= |˙G/G|0< 10−13yr−1which is the most severe
observational limit (M¨ uller et al. 1991).
We remark that BDΛ is an extension of the original
form of BD and reduces to the Friedmann model when φ
= constant, µ = 1, and ω ≫ 1. We have Λ < 0 if µ < 1,
and φG > 0 if both µ > 3 and ω ≫ 1. Figure 1 shows the
evolution of the scale factor for BDΛ with the relevant
parameters in the present study and for the Friedmann
model. Note that the difference in the expansion rate at
t < 10 s in BDΛ. In particular, around t = 5 s, the curve
x in BDΛ crosses that of the Friedmann model, which will
have sensitive effects on BBN. Since Λ is proportional to
ρm0, µ affects the evolution of the scale factor around the
present epoch. In our BDΛ model, if |B∗| increases, the
expansion rate increases at t < 10 − 100 s. It is remarked
that the change in G between the recombination and the
present epoch is less than 0.05 (2σ) from WMAP (Nagata
et al. 2004), which is consistent with BDΛ since |(G −
G0)/G0| < 0.005 at t > 1 yr.
3. Big-bang nucleosynthesis
Changes in the expansion rate compared to the standard
model affect the synthesis of light elements at the early
era, because the neutron to proton ratio is sensitive to the
expansion rate.
For the BBN calculation, we use the reaction
rates (Cyburt et al. 2001) based on NACRE (Angulo et
al. 1999). We adopt the observed abundances of4He, D/H
and7Li/H as follows; Yp= 0.2391± 0.0020 (Luridiana et
al. 2003), D/H= 2.78+0.44
7Li/H=(2.19 ± 0.28) × 10−10(Bonifacio et al. 2002).
Since the results of WMAP constrain cosmological pa-
rameters, we calculate the abundance of4He, D and7Li
paying attention to the value η10= 6.1. First, we carry
−0.38× 10−5(Kirkman et al. 2003),
0.2
10-3
0.22
0.24
0.26
0.28
0.3
Yp
10-5
10-9
10-4
D/H
B*=10
B*=-10
B=-2.5
B*=0
10-10
10-9
η
10-10
7Li/H
Fig.2. Light element abundances against η in BDΛ for
µ = 0.7 and possible values of B∗.
0.15
10-3
0.2
0.25
0.3
Yp
10-5
10-9
10-4
D/H
µ=2
µ=1
µ=0
µ=−1
10-10
10-9
η
10-10
7Li/H
Fig.3. Same as Fig. 2 but for B∗= 0 and various values
of µ.
out the BBN calculations with use of the adopted experi-
mental values of nuclear reaction rates given in NACRE.
Figure 2 illustrates4He, D/H, and7Li/H for µ = 0.7. The
abundance of4He is very sensitive to both B∗and µ; it
increases if |B∗| or µ increases. On the other hand, D and
7Li are more sensitive to µ than B∗as seen from Fig. 3. As
a result,4He and D/H are consistent with η obtained from
WMAP in the range −0.5 ≤ µ ≤ 0.8 and −10 ≤ B∗≤ 10.
Next, we perform the Monte-Carlo calculations to
obtain the upper and lower limits to individual abun-
dance using the uncertainties in the nuclear reaction rates
(Cyburt et al. 2001). Figure 4 illustrates4He, D/H and
7Li/H with 2σ uncertainties for B∗= −2.5 and µ = 0.7.
The light-shaded areas denote the regions of observed
abundances, and the dark-shaded area indicates the limit
obtained from WMAP. While the obtained values of4He

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4Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term
Fig.4. Light-element abundances vs. η in BDΛ for µ =
0.7 and B∗= −2.5. Dashed lines show ±2σ uncertainties
in nuclear reaction rates. The dark-shaded area indicates
the constraint by WMAP and light-shaded areas denote
regions of observational abundances.
and D are consistent with η by WMAP, the lower limit in
7Li is barely consistent.
4. Effects of e+e−annihilation on BBN
In the previous sections, we have assumed the equation of
state p = ρ/3 in Eq. (6) to obtain Eq. (10) at the epoch
of e+e−annihilation. Let us discuss the effects of e+e−
annihilation on the evolution of the scalar field and the
scale factor due to the deviation from the relation p =
ρ/3. The electron-positron pressure and energy density
are written with the variable ζ = me/kBTras follows
pe=2m4
e
π2?3
∞
?
n=1
(−1)n+1
?1
nζ
?2
K2(nζ), (11)
ρe= 3 pe+2m4
e
π2?3
∞
?
n=1
(−1)n+1
?1
nζ
?
K1(nζ) , (12)
where ? is Planck’s constant in units of 2π, kBis
Boltzmann’s constant, and meis the electron rest mass.
Ki(i = 1 and 2) are modified Bessel functions of order
i (e.g. Damour & Pichon 1999). In the numerical calcu-
lations, the summations in Eqs. (11) and (12) are taken
over n = 1−10. We can get the scale factor by integrating
Eq. (4) with the aid of Eq. (6). To see the effects of e+e−,
we take the form:
˙φx3=
8πµ
2ω + 3
?t
(ρe− 3pe)x3dt + B.(13)
A direct comparison is made for the evolution of the
scalar field. The results are shown in Fig. 5, where the solid
line indicates the case of Eq. (13) with B∗= −2.43 and
µ = 0.7, and the broken line is the case of Eq. (10) with
-10123
log ( t [sec] )
7.23
7.24
7.25
7.26
7.27
7.28
log( φ [g sec2/ cm3])
Eq. (13)
Eq. (10)
Fig.5. Evolution of the scalar field. The solid line refers
the integration of Eq.(13) with B∗= −2.43, and the bro-
ken line is for Eq. (10) with B∗= −2.50.
B∗= −2.50 and µ = 0.7. These sets of parameters yield
the same macroscopic quantities given in §2. Although we
can appreciate the slight difference at t < 103s, it remains
small during and after the stage of BBN. The effects on
the evolution of the scale factor are minor and the change
in Ypis found to be at most 0.1% compared to that ob-
tained in §3. We can conclude that since the effects of B
in the range −10 ≤ B∗≤ 10 is much larger compared to
those of e+e−annihilation, the deviation from the relation
p = ρ/3 due to e+e−does not change our results qualita-
tively. However, we note that even small differences in Yp
may affect the detailed statistical analysis combined with
theoretical and observational uncertainties performed in
the previous sections.
5. Discussion and conclusions
We have carried out the BBN calculations in the µ − B∗
plane and obtain the ranges −0.5 ≤ µ ≤ 0.8 and −10 ≤
B∗≤ 10 that are consistent with both the abundance
observations and η obtained from WMAP.
To evaluate uncertainties of theory and observations,
we calculate normalized likelihood distributions in BBN
(Fields et al.1996; Hashimoto et al. 2003). In Fig. 6, we
show the likelihood functions for4He, D, and7Li. The
combined distributions, L47= L4· L7and L247= L2·
L4·L7are shown in Fig. 7. As the result, we get the 95%
confidence limit of η: 5.47 ≤ η10≤ 6.64.
The consistency holds within 1σ error for4He and D,
and 2σ for4He, D, and7Li. Though new reaction rates
recently published (Descouvemont et al. 2004) will change
the errors to some extent in the likelihood analysis, our
conclusion holds qualitatively.
Our previous studies (Etoh et al. 1997) showed 1 <
µ < 3 if Λ > 0 for large value of ω. In the present case,
the Λ term becomes negative from Eq. (7) for µ < 1: this

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Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term5
02468 1012
η10
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Likelihood
L4 ( Yp )
L2( D/H )
L7( 7Li/H )
WMAP
Fig.6. Likelihood function as a function of η10 for
4He (L4), D (L2) and7Li (L7). The vertical lines indi-
cate upper and lower limit to η by WMAP.
02468 10
η10
0
0.2
0.4
0.6
0.8
1
Likelihood
L247
L47
WMAP
Fig.7. Combined Likelihood function for two (L47) and
three-elements (L247).
would not conflict with available observations and/or ba-
sic theory (Vilenkin 2004). Alternatively, if we consider
Λ = Λ0+ Λ(φ) with |Λ(φ0)/Λ0| < 0.01, then the cosmo-
logical term becomes consistent with the present observa-
tions. Though the evolutionary path in the early universe
can deviate from the Friedmann model (Arai et al. 1987),
parameters in BDΛ must be searched in detail for val-
ues of ω > 500 to get quantitative results of BBN. We
note that it is shown that negative energies are present in
scalar-tensor theories, though it is not clear how to iden-
tify them definitely (Faraoni 2004).
To save the apparent inconsistency for SBBN, effects of
neutrino degeneracy, changes in neutrino species, or other
new physical processes have been included (Steigman
2003). In our model, we need only a scalar field that could
be related to a string theory (Hammond 1996). It is noted
that the original BD cosmology (µ = 1) would be limited
severely by the more accurate observation of light elements
and/or the future constraints for η as shown in the present
investigation.
Acknowledgements. Data
outona
the Astronomical Data Analysis Center of the National
Astronomical Observatory of Japan.
analysis were
computer
in part carried
generalcommon user systemat
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