Big-Bang Nucleosynthesis in a Brans-Dicke Cosmology with a Varying Λ Term Related to WMAP

A&A 448, 23–27 (2006) Astronomy DOI: 10.1051/0004-6361:20042618 & c ESO 2006 Astrophysics

Big-bang nucleosynthesis in a Brans-Dicke cosmology with a varying Λ term related to WMAP

R. Nakamura1, M. Hashimoto1,S.Gamow1, and K. Arai2

1 Department of Physics, Kyushu University, Fukuoka, 810-8560, Japan 2 Department of Physics, Kumamoto University, Kumamoto, 860-8555, Japan

Received 28 December 2004 / Accepted 11 October 2005

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 of 4He, D, and 7Li 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. nuclear reactions, nucleosynthesis, abundances – cosmology: early Universe – cosmology: cosmic microwave background

1. Introduction equation of state, where the present value of the scalar coupling has been constrained (Damour & Pichon 1999). On the The standard model of big-bang nucleosynthesis (SBBN) has other hand, it is suggested that a decaying Λ modifies the evo- 4 succeeded in explaining the origin of the light elements He, lution of the scale factor and affects the temperature T of the 7 r D, and Li. Although the value of the baryon-to-photon ra- cosmic microwave background at redshift z ≤ 104, when the tio η has been derived from the observations of the Wilkinson recombination begins due to the decrease in Tr (Kimura et al. Microwave Anisotropy Probe (WMAP) (Bennett et al. 2003) 2001), while a decaying Λ is found to be consistent with tem- η = . +0.3 to be 10 6 1−0.2, the value seems to be inconsistent with perature observations of the cosmic microwave background for the results of SBBN (Coc et al. 2004). Contrary to the ex- z < 4 (Puy 2004). Therefore, it is worthwhile to check the va- cellent concordance with η of WMAP for D, the abundance lidity of BDΛ related to the recent observations. In the present 4 of He by SBBN is rather low compared to that from WMAP. paper, we investigate to what extent BBN in the BDΛ model Therefore, non-standard models of BBN have been proposed can be reconciled with η from WMAP. with the Friedmann model modified (Steigman 2003). In Sect. 2, the formulation for BDΛ is given and the evo- For non-standard models, scalar-tensor theories have been lution of the universe in BDΛ is shown. Our results of BBN investigated (e.g., Bergmann 1968; Wagoner 1970; Endo & are presented in Sect. 3 using the Monte-Carlo method (Cyburt Fukui 1977; Fukui et al. 2001). For a simple model with et al. 2001), and constraints are given to the parameters inher- a scalar φ, it is shown that a Brans-Dicke (BD) general- ent in BDΛ. We examine in Sect. 4 the evolution of the scale ization of gravity with torsion includes the low-energy limit factor and the resulting abundances taking into account the de- ff string e ective field theory (Hammond 1996). Related to the viation from the equation of state p = ρ/3 during the stage of cosmological constant problem, a Brans-Dicke model with e+e− annihilation. In Sect. 5, a likelihood analysis (Fields et al. Λ Λ a varying (φ)term(BD ) has been presented, and also in- 1996) is adopted to obtain the most probable values and the vestigated from the point of inflation theory (Berman 1989). accompanying errors. Moreover, it is found that the linearized gravity can be recov- ered in the Randall-Sundrum brane world (Garriga & Tanaka 2000). Furthermore, scalar-tensor cosmology is constrained by 2. Brans-Dicke cosmology with a varying Λ term 2 a χ test for the WMAP spectrum (Nagata et al. 2004) where The field equations for BDΛ are written as follows (Arai et al. = the present value of the coupling parameter ω0 ω(φ0)is 1987): bounded to be ω0 > 50 (4σ)andω0 > 1000 (2σ) in the limit to BD cosmology. 1 8π ω 1 R − g R + g Λ= T + φ − g φ φ,α BBN has been studied in BDΛ (Arai et al. 1987; Etoh et al. µν 2 µν µν φ µν φ2 ,µ;ν 2 µν ,α 1997). The relation between BBN and scalar-tensor gravity 1 + − + φ − g φ , (1) is investigated with the inclusion of e e annihilation in the φ ,µ;ν µν

Article published by EDP Sciences and available at http://www.edpsciences.org/aa or http://dx.doi.org/10.1051/0004-6361:20042618 24 R. Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term Λ ∂ ω ,µ 2ω -8 R − 2Λ − 2φ = φ,µφ − φ, (2) * ∂φ φ2 φ BDΛ model (B = −10) Λ * − where ω is the coupling constant. BD model (B = 2.5) -8.5 Friedmann model The equation of motion is obtained with use of the Friedmann-Robertson-Walker metric: 2 dr -9 ds2 = −dt2 + a(t)2 + r2dθ2 + r2 sin2 θdφ2 , (3) x 1 − kr2 log where a(t) is the scale factor and k is the curvature constant. Let x be a scale factor normalized to its present value, i.e., x = -9.5 a/a0, then we get from the (0, 0) component in Eq. (1) x˙ 2 k Λ ω φ˙ 2 x˙ φ˙ 8π ρ -10 + − − − = , (4) x x2 3 6 φ x φ 3 φ -1 -0.5 0 0.5 1 1.5 2 2.5 3 where ρ is the energy density. log t [sec] We assume the simplest case of the coupling between the Λ = = scalar and matter fields: Fig. 1. Evolution of the scale factor for BD (µ 0.7, t0 13.7Gyr, and η = 6.1) and Friedmann model. 8π 10 φ = µT ν, (5) 2ω + 3 ν where µ is a constant. Assuming a perfect fluid for Tµν,Eq.(5) reduces to the present compared to the contribution from e+e− annihi- πµ d ˙ 3 = 8 − 3 lation. As a consequence, the neutron to proton ratio is seri- φx + (ρ 3p)x , (6) dt 2ω 3 ously affected by the initial value of B. Considering the im- where p is the pressure. portant contribution of ρe± to BBN, we examine the effects of + − A particular solution of Eq. (2) is obtained from Eqs. (1) e e annihilation in Sect. 4. Hereafter we use the normalized ∗ −24 −3 10 and (5): values: B = B/(10 gscm )andη10 = 10 η. The cou- pled Eqs. (4), (7) and (10) can be solved numerically with the − 2π (µ 1) −3 = × Λ= ρm0 x , (7) specified quantities: for macroscopic quantities, G0 6.672 φ −8 2 −2 −1 −1 10 dyn cm g , H0 = 71 km s Mpc (Bennett et al. 2003), and T = 2.725 K (Mather et al. 1999); for micro- where ρm0 is the matter density at the present epoch. r0 The gravitational “constant” G is expressed as follows scopic quantities, the number of massless neutrino species is 3 and the half life of neutrons is 885.7 s (Hagiwara et al. 2002). 1 2ω + 1 1 Although we adopt ω = 500, the epoch of the appreciable G = 3 − µ · (8) 2 2ω + 3 φ growth of |G˙/G| is t < 103 s regardless of the value ω (Arai et al. 1987). Therefore, even if we adopt a value ω>500 The radiation density ρ contains the contributions from pho- r (Will 2001), we can arrive qualitatively the same conclusion tons, neutrinos, electrons and positrons at t ≤ 1 s. The total ∗ by changing the parameters µ and B . We impose the condi- energy density is given as | ˙ | = | ˙ | −13 −1 tion φ/φ 0 G/G 0 < 10 yr which is the most severe = + = + + ± observational limit (Müller et al. 1991). ρ ρm ρr,ρr ρrad ρν ρe . (9) BDΛ is an extension of the original form of BD and re- = −3 Here the energy density of matter varies as ρm ρm0 x .The duces to the Friedmann model when φ = constant, µ = 1 = −4 + − + − radiation density ρr ρr0 x except e e epoch where e e an- and ω 1. We have Λ < 0ifµ<1, and φG > 0if ∼ −1 nihilation changes the relation Tr x . We assume that the both µ>3andω 1. Figure 1 shows the evolution of = pressure satisfies p ρ/3, which is legitimate only for rela- the scale factor for BDΛ with the relevant parameters in the tivistic particles. Then, Eq. (6) is integrated to give present study and for the Friedmann model. Note that the dif- Λ 8πµ 1 ference in the expansion rate at t < 10 s in BD . In partic- φ˙ = ρ t + B , (10) ular, around t = 5s,thecurvex in BDΛ crosses that of the 2ω + 3 m0 x3 Friedmann model, which will have noticeable effects on BBN. where B is a constant (Arai et al. 1987). Although the relation Since Λ is proportional to ρm0, µ affects the evolution of the p = ρ/3 does not hold during the epoch of e+e− annihilation, scale factor around the present epoch. In our BDΛ model, if as pointed out by Damour & Pichon (1999), the inclusion of φ |B∗| increases, the expansion rate increases at t < 10−100 s. measures small deviation from SBBN in our interest, so that The change in G between the recombination and the present our solution (10) can reasonably describe the evolution of φ epoch is less than 0.05 (2σ) from WMAP (Nagata et al. 2004), except during the annihilation epoch. We consider that B af- which is consistent with BDΛ since |(G − G0)/G0| < 0.005 at fects the evolution of x significantly from the early epoch to t > 1yr. R. Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term 25

0.3 0.28 0.26 p Y 0.24 0.22 * 0.2 B =10 -3 * 10 B =-10 B=-2.5 * B =0 -4 D/H 10

-5 10 -9 10 Li/H 7

-10 10 -10 -9 10 10 η

Fig. 2. Light element abundances against η in BDΛ for µ = 0.7and Fig. 4. Light-element abundances vs. η in BDΛ for µ = 0.7andB∗ = possible values of 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. 0.3

p 0.25

Y with η obtained from WMAP in the range −0.5 ≤ µ ≤ 0.8and 0.2 −10 ≤ B∗ ≤ 10. µ=2 0.15 Next, we perform the Monte-Carlo calculations to obtain -3 µ=1 10 µ=0 the upper and lower limits to individual abundance using the uncertainties in the nuclear reaction rates (Cyburt et al. 2001). -4 µ=−1 10 4 7 D/H Figure 4 illustrates He, D/Hand Li/H with 2σ uncertainties ∗ = − = -5 for B 2.5andµ 0.7. The light-shaded areas denote 10 -9 the regions of observed abundances, and the dark-shaded area 10 indicates the limit obtained from WMAP. While the obtained values of 4He and D are consistent with η by WMAP, the lower Li/H 7 limit in 7Li is barely consistent. -10 10 -10 -9 10 10 + − η 4. Effects of e e annihilation on BBN Fig. 3. Same as Fig. 2 but for B∗ = 0 and various values of µ. 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 3. Big-bang nucleosynthesis evolution of the scalar field and the scale factor due to the deviation from the relation p = ρ/3. The electron-positron pressure Changes in the expansion rate compared to the standard model = and energy density are written with the variable ζ me/kBTr affect the synthesis of light elements in the early era, because as follows the neutron to proton ratio is sensitive to the expansion rate. ∞ 2 2m4 1 For the BBN calculation, we use the reaction rates (Cyburt p = e (−1)n+1 K (nζ) , (11) e π23 nζ 2 et al. 2001) based on NACRE (Angulo et al. 1999). We n=1 adopt the observed abundances of 4He, D/Hand7Li/Hasfol- 4 ∞ lows: Y = 0.2391 ± 0.0020 (Luridiana et al. 2003), D/H = 2m + 1 p ρ = 3 p + e (−1)n 1 K (nζ) , (12) +0.44 × −5 7 / = ± × e e π23 nζ 1 2.78−0.38 10 (Kirkman et al. 2003), Li H (2.19 0.28) n=1 10−10 (Bonifacio et al. 2002). Since the results of WMAP constrain cosmological pa- where is Planck’s constant in units of 2π, kB is Boltzmann’s = rameters, we calculate the abundance of 4He, D and 7Li pay- constant, and me is the electron rest mass. Ki (i 1 and 2) = are modified Bessel functions of order i (e.g. Damour & ing attention to the value η10 6.1. First, we carry out the BBN calculations with use of the adopted experimental values Pichon 1999). In the numerical calculations, the summations = − of nuclear reaction rates given in NACRE. Figure 2 illustrates in Eqs. (11) and (12) are taken over n 1 10. We can obtain 4 / 7 / = 4 the scale factor by integrating Eq. (4) with the aid of Eq. (6). He, D Hand Li Hforµ 0.7. The abundance of He is very + − ∗ | ∗| To see the effects of e e , we take the form: 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∗ t 3 8πµ 3 4 / φ˙x = (ρe − 3pe)x dt + B. (13) as seen from Fig. 3. As a result, He and D H are consistent 2ω + 3 26 R. Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term

7.28 1 L ( Y ) WMAP Eq. (13) 0.9 4 p Eq. (10) L ( D/H ) 7.27 0.8 2 7 L ( Li/H ) ]) 7 3 0.7

/ cm 7.26 0.6 2 0.5 [g sec

7.25 Likelihood 0.4 φ 0.3 log( 7.24 0.2 0.1

7.23 0 -1 0 1 2 3 024681012η log ( t [sec] ) 10 4 Fig. 5. Evolution of the scalar ﬁeld. The solid line refers the integration Fig. 6. Likelihood function as a function of η10 for He (L4), D (L2) ∗ 7 of Eq. (13) with B = −2.43, and the broken line is for Eq. (10) with and Li (L7). The vertical lines indicate the upper and lower limit to η B∗ = −2.50. by WMAP.

1 A direct comparison is made for the evolution of the scalar WMAP field. The results are shown in Fig. 5, where the solid line indi- L247 cates the case of Eq. (13) with B∗ = −2.43 and µ = 0.7, and the 0.8 L47 broken line is the case of Eq. (10) with B∗ = −2.50 and µ = 0.7. These sets of parameters yield the same macroscopic quantities given in Sect. 2. Although we can appreciate the slight differ- 0.6 ence at t < 103 s, it remains small during and after the stage of BBN. The effects on the evolution of the scale factor are

Likelihood 0.4 minor and the change in Yp is found to be at most 0.1% compared to that obtained in Sect. 3. We can conclude that since the effects of B in the range −10 ≤ B∗ ≤ 10 is much larger 0.2 then those of e+e− annihilation, the deviation from the relation p = ρ/3 due to e+e− does not change our results qualitatively. ff ff However, we note that even small di erences in Yp may a ect 0 the detailed statistical analysis combined with theoretical and 0246810η 10 observational uncertainties performed in the previous sections.

Fig. 7. Combined likelihood function for two (L47) and three- elements (L ). 5. Discussion and conclusions 247 We have carried out BBN calculations in the µ − B∗ plane and obtain the ranges −0.5 ≤ µ ≤ 0.8and−10 ≤ B∗ ≤ 10 that are Λ=Λ +Λ |Λ Λ | consistent with both the abundance observations and η obtained Alternatively, if we consider 0 (φ) with (φ0)/ 0 < from WMAP. 0.01, then the cosmological term becomes consistent with the To evaluate the uncertainties in theory and observation, we present observations. Although the evolutionary path in the calculate normalized likelihood distributions in BBN (Fields early universe can deviate from the Friedmann model (Arai Λ et al. 1996; Hashimoto et al. 2003). In Fig. 6, we show the et al. 1987), parameters in BD must be searched in detail for likelihood functions for 4He, D and 7Li. The combined distri- values of ω>500 to obtain quantitative results of BBN. It is = · = · · shown that negative energies are present in scalar-tensor the- butions, L47 L4 L7 and L247 L2 L4 L7 are shown in Fig. 7. ≤ ≤ ories, although it is not clear how to identify them definitely We obtain the 95% confidence limit of η:5.47 η10 6.64. The consistency holds within 1σ errors for 4He and D, (Faraoni 2004). and 2σ for 4He, D and 7Li. Although new reaction rates re- To avoid the apparent inconsistency for SBBN, effects of cently published (Descouvemont et al. 2004) will change the neutrino degeneracy, changes in neutrino species or other new errors to some extent in the likelihood analysis, our conclusion physical processes have been included in models (Steigman holds qualitatively. 2003). In our model, we need only a scalar field that could be Our previous studies (Etoh et al. 1997) showed 1 <µ<3if related to string theory (Hammond 1996). The original BD cos- Λ > 0 for large values of ω. In the present case, the Λ term be- mology (µ = 1) would be limited severely by the more accurate comes negative in Eq. (7) for µ<1: this would not conflict with observation of light elements and/or future constraints for η as available observations and/or basic theory (Vilenkin 2004). shown in the present investigation. R. Nakamura et al.: Big-bang nucleosynthesis in Brans-Dicke cosmology with a varying Λ term 27

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