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SCIENCE ^DIRECT'
Physics Letters B 597 (2004) 1-10
www.elsevier.com/locate/physletb
Solving the discrepancy among the light elements abundances and WMAP
Kazuhide Ichikawa, Masahiro Kawasaki, Fuminobu Takahashi
Institute for Cosmic Ray Research, University of Tokyo, 5-1-5 Kashiwa-no-Ha, Kashiwa City, Chiba 277-8582, Japan
Received 10 June 2004; accepted 7 July 2004 Available online 20 July 2004 Editor: T. Yanagida
Abstract
Within the standard Big Bang nucleosynthesis (BBN) and cosmic microwave background (CMB) framework, the baryon density measured by the Wilkinson Microwave Anisotropy Probe (WMAP) or the primordial D abundance is much higher than the one measured by the 4He or 7Li abundances. To solve the discrepancy, we propose a scenario in which additional baryons appear after BBN. We show that simply adding the baryons cannot be a solution but the existence of a large lepton asymmetry before BBN makes the scenario successful. These extra baryons and leptons, in addition to the initial baryons which exist before the BBN, can be all produced from g-balls. © 2004 Elsevier B.V. All rights reserved.
PACS: 26.35.+c; 98.80.-k; 98.80.Cq; 98.80.Ft
Keywords: Baryon density; Big Bang nucleosynthesis; CMBR; Lepton asymmetry; 2-balls
1. introduction
The baryon density is one of the most important cosmological parameters. Especially, it is the only one input parameter for the standard Big Bang nucleosynthesis (BBN) theory, which predicts the abundances of light elements, D, 4He and 7Li. Meanwhile, for the cosmic microwave background (CMB), it also plays an important role in determining the shape of
E-mail address: kazuhide@icrr.u-tokyo.ac.jp (K. Ichikawa).
the acoustic peaks. Observations of the three light elements and the CMB on the whole indicate equal amount of baryons and make us confirm the validity of the standard cosmology.
Recently, following the precise measurement of the CMB by the Wilkinson Microwave Anisotropy Probe (WMAP), its concordance with the BBN and the light elements observations has been investigated in Refs. [1-4]. The baryon density measured from the WMAP data is = ttbh2 = 0.024 ± 0.001 (with the power-law ACDM model) [5], where Qb is the baryon energy density divided by the critical energy
0370-2693/$ - see front matter © 2004 Elsevier B.V. All rights reserved. doi:10.1016/j.physletb.2004.07.011
density today and h is the Hubble constant in units of 100 kms-1 Mpc-1. The uncertainty is very small because the WMAP has detected the first and second peaks accurately in the temperature angular spectrum [6]. This corresponds to n = nb/nY = (6.6 ± 0.3) x 10-10, where nb and nY are baryon and photon number densities, via the relation n = oib/(3.65 x 107). Refs. [1-4] take this well-determined WMAP a>b as the BBN input1 and calculate the light elements abundances and their theoretical errors using improved evaluations of nuclear reaction rates and uncertainties. The results are compared with the received measurements of the primordial abundances of three light elements, D (Ref. [7]), 4He (Refs. [8] or [9]) and 7Li (Refs. [10] or [11]). Although there are small differences concerning their adopted reaction rates or observation data, their conclusions agree: from the WMAP baryon density, the predicted abundances are highly consistent with the observed D but not with 4He or 7Li. They are produced more than observed. Especially, the 7Li-WMAP discrepancy is severer and it may require an explanation.
The most conservative and likely interpretation of such discrepancy is that systematic errors in the primordial 4He and 7Li measurements are underestimated in spite of the thorough analysis hitherto. Or, as Ref. [2] has pointed out, it is possible that since the cross section of the reaction 7Be(d, p)24He has not been measured for the BBN energy range, it could reduce the 7Li yield to match with the observation if the rate turns out to be hundreds times more than the often neglected value obtained by the extrapolation.
If the discrepancy cannot be attributed to systematic errors, we would have to invoke new physics to reduce their abundances. Actually, some approaches to reduce 7Li by astrophysical means, such as nonstandard depletion mechanism inside stars, are discussed in Refs. [1,3,12] and references therein.
What we seek in this Letter is a cosmological solution. For a long time, not a few non-standard models are known to affect 4He abundance but none of them
1 To be more precise, they adopt n = (6.14 ± 0.25) x 10-10 which is obtained from the running spectral index model value Mb = 0.0224 ± 0.0009 via the Mb—n relation. Since this value is inferred from the combination of CMB, galaxy and Lyman a forest data, we adopt the value quoted in the text which is inferred using only the WMAP data.
seem to be able to solve the discrepancy. For example, non-standard expansion rate (extra relativistic degrees of freedom) and/or large lepton asymmetry change 4He too much while adjusting 7Li to its observed value [13-15]. A varying fine structure constant (electromagnetic coupling constant) can relieve the tension between either D and 4He or D and 7Li but not the three elements together [16]. At this time, only bolder attempt such as to combine non-standard expansion rate and a varying fine structure constant investigated by two of the authors in Ref. [17] seems to be able to accommodate D, 4He and 7Li simultaneously, but whether the introduction of these non-standard ingredients can fit the WMAP data is not fully checked yet.2
In this Letter, we investigate a solution which allows different amount of baryons for the BBN and the CMB. Unfortunately this only reconciles the WMAP, D and either 4He or 7Li. We choose to make 7Li consistent with the observation in this way and consider a large lepton asymmetry in addition in order to accommodate 4He too. One of the advantages of this scenario is that both the additional baryons and lepton asymmetry can be produced from Q--balls, which also generate the original baryons before the BBN.
In next section, we explain the discrepancy briefly and investigate how the prediction of the primordial light element abundances is affected by the additional baryons after the BBN. After we show that is not enough to solve the discrepancy, we introduce the lep-ton asymmetry and calculate how much extra baryons and lepton asymmetry are needed to be a solution. In Section 3, we describe how to generate required baryons and leptons. We conclude in Section 4.
2. A solution to the discrepancy
In this section, we show that three elements abundances and the WMAP data are not consistent within the standard framework of cosmology but they are reconciled with the appearance of additional baryons after BBN and the existence of the large lepton asym-
2 A varying deutron binding energy may have the capacity to
render internal agreement between the light element abundances and
with WMAP [18].
metry before the BBN. We also compute how much of those are necessary to be a solution.
2.1. The discrepancy among the light elements abundances and WMAP
First of all, we summarize the measured baryon density from the BBN and CMB in the left three panels in Fig. 1. We compute the theoretical abundances of 4He, D and 7Li and their uncertainties by Monte Carlo simulations using the values obtained in Ref. [19] based on the reaction rates compiled in Ref. [20]. These are compared with the observations of three light elements. We adopted the following data:
Y4He = 0.238 ± 0.002 ± 0.005,
(D/H) = 2.78
+°.44 10 0.38 x 10
(7Li/H) = 1.23-0.32 x 10-10 (95%)
(1) (2) (3)
where Eq. (1) for 4He mass fraction is taken from Ref. [8] (FO), Eq. (2) for D to H ratio in numbers from Ref. [7], and Eq. (3) for 7Li to H ratio in numbers from Ref. [10] (R). In Eq. (1), the first uncertainty is statistical and the second one is systematic, which are added in quadrature to be the total observational uncertainty. The systematic errors are already included in the error bars of the other two elements. In Eq. (3), quoted uncertainty is 95% and we take its half value to be 1a uncertainty. As indicated by the three boxes in the figure, the 4He and 7Li measurements indicate n ^ (2-4) x 10-10, but much higher value n ^ (6-7) x 10-10 is necessary to explain the D measurement.
On superposing the WMAP measured value,3
n = (6.6 ± 0.3) x 10
the discrepancy seems to be severer. The range of Eq. (4) is expressed by the narrow vertical bar in Fig. 1, overlapping only with the n range deduced from D. The baryon density measured by the WMAP and the one by either 4He or 7Li abundances are not very consistent.
3 The WMAP group quotes n = (6.5+0 3) x 10-10 as mean and 68% confidence range [5]. Although our adopted values are slightly different since we derive them from Mb as explained in Section 1, the difference scarcely affect our results.
Fig. 1. In the left column, familiar (namely, n at present, n0, is same as the one at the BBN, nBBN ) standard BBN calculations of 4He, D and 7Li abundances as functions of n are expressed by three dark-shaded curves whose widths show theoretical 1a uncertainties. The bar penetrating the figures displays n measured by the WMAP with 1a uncertainty. In the right column, the light-shaded curves indicate the abundances modified by the baryon appearance after the BBN. In this case, we assume that n is increased from nBBN to be the WMAP measured value, n0 = nWMAP = 6.6 x 10-10. On top of the figures, we mark the scale to tell the amount of added n after the BBN, where n0 = nBBN + nadd. In both columns, the observational 1a uncertainties are expressed by the vertical extension of the boxes, which are drawn to overlap the theory curves so that their horizontal extension shows allowed ranges of n.
We quantify the discrepancy by calculating x 2 as a function of n,
(ath(n) - O2 , (n - nWMAP)
,,obs\2
(ath)2 + (aobS)
where a and ai are, respectively, abundances and their 1 a uncertainty of the element i . Their theoretical values are calculated from Monte Carlo simulations and observational values are those of Eqs. (1)-(3). For asymmetric errors, we adopt conservatively the larger one as 1a error (for 7Li, we divide the error in Eq. (3) by two as explained above). We use the value of Eq. (4) for the second term. By taking the sum over three elements D, 4He and 7Li, we can investigate overall
consistency and by choosing a particular element as i, we can study whether that element is consistent with the WMAP result.
The results are Figs. 2 and 3. Fig. 2 shows the two degree of freedom x22) calculated from each element measurement and WMAP. Fig. 3 shows the four degree of freedom x24) from three elements measurements and WMAP. As expected from Fig. 1, Fig. 2 shows that D is highly consistent with WMAP but not for 4He and 7Li. When we analyze with the combined data of three elements and WMAP, we see in Fig. 3 that there is no baryon density range to explain the light elements abundances and the CMB. For details, 4He-WMAP discrepancy is not very severe but 7Li is definitely inconsistent with WMAP and this 7Li discrepancy mainly raises the overall x2. Therefore, a task worth challenging would be to look for a cos-mological model which can adjust 7Li abundance to the observed value while not amplifying the mild 4He-WMAP discrepancy. Of course, it is all the better to alleviate the tension of 4He in addition.
We here comment on the other 4He and 7Li measurements. Ref. [9] (IT) reports Y4He = 0.2421 ± 0.0021 and Ref. [11] (B) reports log[(7Li/H)] + 12 = 2.34 ±0.056 ±0.06. The systematic effects are already included in the uncertainty in Ref. [9] 's 4He data. For Ref. [11]'s 7Li data, the statistical error and systematic error are added in quadrature so that [(7Li/H)] = 2.19-0.46 x 10-10.The results using those data instead of Ref. [8]'s 4He and/or Ref. [10]'s 7Li are also shown in Figs. 2 and 3. Ref. [9]'s 4He is less consistent with WMAP than Ref. [8] 's because of much less uncertainty in spite of the higher central value. It is tempting to address a strong 4He-WMAP discrepancy issue on the basis of Ref. [9] 's 4He but we adopt conservative Ref. [8]'s value to focus on 7Li discrepancy problem. For 7Li, we see that the use of Ref. [11] 's value relaxes the discrepancy with WMAP. We adopt Ref. [10]'s measurement because they analyze the observations with lower and broader range of metallicity, but we note that considerable discrepancy still exists for Ref. [11]'s 7Li.
2.2. Effects of additional baryons after BBN
Roughly speaking, the discrepancy exists because the WMAP data needs more baryons than those required to account for the primordial light elements
Fig. 2. X(2) calculated from a single element measurement and WMAP. The different types of lines use the data of D [7] (solid line), 4He of Fields and Olive (FO) [8] (thick dashed line), 4He of Izotov and Thuan (IT) [9] (thin solid line), 7Li of Ryan et al. (R) [10] (thick dotted line), and 7Li of Bonifacio et al. (B) [11] (thin dotted line). The horizontal lines correspond to, from bottom to top, 1a, 95%, 99% and 99.9% confidence levels derived from two degree of freedom x 2
Fig. 3. x(4) calculated from three elements measurements and WMAP Expressing each two measurements of 4He and 7Li by the abbreviations introduced in the caption to Fig. 2, the solid line uses FO for 4He and R for 7Li, the dashed line uses FO and B, the dotted line uses IT and R, and the dotted-dashed line uses IT and B. The horizontal lines correspond to, from bottom to top, la, 95%, 99% and 99.9% confidence levels derived from four degree of freedom x 2.
abundances, especially 4He and 7Li. Then a naive solution would be allowing to increase the baryons after the BBN to the amount required to explain the WMAP data before the physics which form the acoustic peaks takes place.
However, such increase in the baryons (in the form of protons, i.e., H nucleus) considerably affects the observation of the light elements abundances because they are always measured in terms of the ratio to H.
Since the numerator does not change and the denominator increases, the abundances decrease in general. To quantify this effect, let nBBN denote the baryon-to-photon ratio during the BBN and let it increase by nadd (without entropy production) to be nWMAP = 6.6 x 10-10 after the BBN but well before the matterradiation equality. They of course satisfy nWMAP = nBBN + nadd. For 4He, the effect is very simple because its abundance is conventionally expressed by a mass fraction: Y4He = 4n4He/nb, where n4He is the number density of 4He. Noting that since entropy is not produced, the photon number density scales as a-3, where a is the scale factor, the 4He mass fraction is modified from the value when the BBN ends, Y
(BBN) 4 He
to the one we observe, Y4Hs), as
Y (obs) _ „ (BBN) 'ffiBN
4He ~~ 4He " ^ '
He He nWMAP
For D and 7Li, since the abundance is expressed by the ratio to H in numbers,
(D/H)0bs = (D/H)bbn-
RBBN + nadd
where RBBN denotes the ratio of H to photon in num bers during the BBN,
RBBN =
= ^(1 - Y^).
In Eq. (8), we regard abundances of the light elements other than H and 4He are negligibly small.
Using the Eqs. (6)-(8), we can predict the observed abundances modified by the baryon appearance after the BBN as shown in the right three panels in Fig 1. Since we only consider the increase in n, they are truncated at nWMAP and nadd increases to the left. Compared with the standard case in the panels on the left-hand side, the abundances become smaller for greater nadd as expected and it is conspicuous for 4He.
Similar to the standard BBN case, the comparison with the observations is made by three boxes in the figure. We see D and either 4He or 7Li can be reconciled with appropriate additional baryons but not for three elements together.
To obtain a solution to reconcile three elements abundance measurements simultaneously, we first choose to make D and 7Li consistent by the adding baryons nadd ^ 1.5 x 10-10. Then we introduce a (negative) lepton asymmetry before the onset of the
BBN in order to enhance 4He abundance (about 1.5 times) while retaining D and 7Li abundances. We next quantify how much lepton asymmetry is needed.
2.3. Plus a lepton asymmetry before BBN
The lepton asymmetry is parametrized by the degeneracy parameter = t^e/T where ¡xe is the electron-type neutrino chemical potential. Taking into account the tendency toward flavor equilibration [21], we assume every flavor has the same amount of asymmetry for concreteness.
Since we only consider | < O(1), its contribution as the extra relativistic degree of freedom is very small and the effect on the nuclear beta equilibrium, p + e- ^ n + ve dominates. This property is important because it ensures that the baryon density measurement by CMB as Eq. (4) is not disrupted. When there is a negative lepton asymmetry (fe < 0), neutrinos are less than anti-neutrino and the equilibrium shifts to increase n relative to p. This is formulated as the following equation. Since their number densities obey Boltzmann statistics, an equilibrium number ratio at temperature T is [22]
where Am is the neutrino-proton mass difference and Am « 1.29 MeV. From this expression it is easy to see 4He is sensitive to non-zero He because its abundance is approximately written as
^He = '
+ np 2
f 1 + (np/nn)f
l + eA m/Tf+^e' (10)
where the subscript f means the values at the weak interaction freeze-out and Tf « 0.7 MeV, showing 4He mass fraction depends exponentially on He. This is illustrated in the top panel of Fig. 4. The dependences of D and 7Li on He are also shown but they are very small. This is important because we do not want to spoil the consistency between D and 7Li by the introduction of the lepton asymmetry.
From Eq. (10), we estimate He ~ -0.3 is necessary in order to make the 4He abundance larger and achieve the scenario mentioned at the end of the previous subsection. To confirm such an estimation, we
search for the allowed parameter region on nadd-£e plane by calculating x2 as a function of those two parameters. Since we fix the baryon density after BBN at the WMAP central value, the x2 here is calculated using three light elements data as Eq. (5) with the last term omitted. The inclusion of the small uncertainty in WMAP baryon density would make the allowed region just a little larger. The results are displayed in Fig. 5. The allowed regions determined from each element are understood from Figs. 1 and 4, and their product set makes the region shown in the bottom-right panel, which accommodates three elements. Especially, there is no solution for £e = 0 or nadd = 0, and the best solution is located at about (nadd,£e) = (2 x 10—10, -0.4), as expected during the argument thus far.
3. g-ball baryo- and leptogenesis
Fig. 4. £e-dependence of light element abundances. The cases for £e = 0, +0.1, —0.1 are drawn with solid, dashed, and dotted lines.
Fig. 5. 1a (dotted lines) and 95% (solid lines) confidence levels in the n-£e plane determined by each light elements observations alone and by three elements combined.
In this section, we show a model that explains the large lepton asymmetry, the initial baryon asymmetry before BBN, and additional baryon appearance after the BBN all together. To accomplish this let us consider the Affleck-Dine (AD) mechanism [23] and the subsequent g-ball formation [24-26] in the gauge-mediated supersymmetry (SUSY) breaking model. Before turning to a close examination of the model, it will be useful to summarize our basic strategy. The mechanism generating large lepton asymmetry we adopt here is based on Ref. [27]. The AD mechanism can produce large (and negative) lepton asymmetry by choosing an appropriate leptonic flat direction such as e2L1L3, where the subindices represent the generation. After the flat direction starts oscillating, the AD field experiences the spatial instability and deforms into non-topological solitons, g-balls (L-balls). Then, almost all the produced lepton numbers are absorbed into the L-balls [26,28]. If the lifetime of such L-balls is longer than the onset of electroweak phase transition but shorter than the epoch of BBN, the large lepton asymmetry is protected from sphaleron effects [29] and later released into the universe by the decay of the L-balls. On the other hand, small (negative) lepton numbers are evaporated from the L-balls due to thermal effects before the electroweak phase transition, which are transformed into small baryon asymmetry through the sphaleron effect. Thus generated baryon
number is the source for light element synthesis. Additional baryon number after the BBN can also be explained by the use of g-balls. If the dimension of a flat direction is not so large, other flat directions can obtain non-zero vevs simultaneously. For instance, udd direction is compatible with eLL direction. The baryon asymmetry generated by the udd direction is also absorbed into g-balls (B -balls). If the B -balls decay after the relevant BBN epoch, they can explain the late-time baryon appearance. In order to avoid dissociating the synthesized light elements with the decay products, the mass per unit baryon charge of the B -ball must be tacitly chosen to be slightly above the neutron or proton mass. To this end, we take m3/2 = 1 GeV in the following discussion (see Eq. (18)). It should be noted that the AD mechanism and associated g-ball formation can account for both the large lepton asymmetry and the baryon asymmetry before and after the BBN.
3.1. Affleck-Dine mechanism and g-balls
First off, let us review the AD mechanism and several properties of g-balls. In the minimal supersym-metric standard model (MSSM) there exist flat directions, along which there are no classical potentials in the supersymmetric limit. Since flat directions consist of squarks and/or sleptons, they carry baryon and/or lepton numbers, and can be identified as the Affleck-Dine (AD) field. For a definite discussion, we adopt eLL and udd directions as the AD fields. It is of use to parametrize the flat direction with a single complex scalar field W, so we express eLL and udd directions as WL and WB, respectively.
The flat directions are lifted by supersymmetry breaking effects. In the gauge-mediated SUSY breaking model, the potential of a flat direction is parabolic at the origin, and almost flat beyond the messenger scale [24,28,30],
fects [31],
V ~ Vgauge
MU log^
|W| «MS, |W| »Ms,
where m$ is a soft breaking mass ~ 0(1 TeV), MF is the SUSY breaking scale, and MS is the messenger mass scale.
Since gravity always exists, flat directions are also lifted by gravity-mediated SUSY breaking ef-
Vgrav — m
1 + K log
Iff Ml
where K is the numerical coefficient of the one-loop corrections and M* is the gravitational scale (— 2.4 x 1018 GeV). This term can be dominant only at high energy scales because of small gravitino mass - 0(1 GeV).
The non-renormalizable terms, if allowed by the symmetries of Lagrangian, can exist and lift the flat directions;
M2n-6 '
where M is a cut off scale of the non-renormalizable term. In our scenario described in the next subsection, is lifted by the non-renormalizable potential, while feels only the SUSY breaking potentials.
The baryon and lepton number is usually created just after the AD field starts coherent rotation in the potential, and its number density nB L is estimated as
nB,L(tosc) — eœ$o
where e (< 1) is the ellipticity parameter, which represents the strongness of the A term, and œ and $osc are the angular velocity and amplitude of the AD field at the beginning of the oscillation (rotation) in its effective potential.
Actually, however, the AD field experiences spatial instabilities during its coherent oscillation, and deforms into non-topological solitons called g-balls [24-26]. When the zero-temperature potential Vgauge dominates at the onset of coherent oscillation of the AD field, the gauge-mediation type g-balls are formed. Their mass Mg and size Rg are given by [32]
Mg — MF Q
M-1g1/4.
From the numerical simulations [26,28], the produced g-balls absorb almost all the charges carried by the AD field and the typical charge is estimated as [28]
with p « 6 x 10-4.
There are also other cases where Vgrav dominates the potential at the onset of coherent oscillation of the
AD field. If K is positive, Q-balls do not form until the AD field leaves the Vgrav dominant region. Later it enters the Vgauge dominant region and experiences instabilities so that the gauge-mediation type Q -balls are produced (delayed-type Q-balls) [28]. The charge of the delayed-type Q-ball is
( mf y
\'"3/2/
with 0eq ~ MF/m3/2. Here the subscript "eq" denotes a value when the gauge- and the gravity-mediation potentials become equal. Thus the delayed-type Q-balls are formed at Heq ~ M2F/M*.
On the other hand, if the coefficient of the one-loop correction K is negative, the gravity-mediation type Q-balls ("new" type) are produced [33]. The typical value of their mass, size, and charge are estimated as
Mq ~ m3/2Q, Rq ~
\m3/2/
with ¡3 = 6.0 x 10-3.
Finally let us mention the decay of Q-balls. In the case of L--balls, they decay into leptons such as neutrinos via wino exchanges. Also B -balls can decay into nucleons such as protons and neutrons via gluino exchanges. The decay rate of Q-balls is bounded as [34]
dQ_ dt
of A 192tt2'
where A is a surface area of the Q-ball. For L-balls, the decay rate is estimated as a value of the order of the upper limit, while it must be somewhat suppressed for B-balls, if the mass per unit baryon number, Mq/Q, is close to the nucleon mass ~ 1 GeV.
3.2. B -and L-balls
Now we detail the scenario. In order to generate 0(1) lepton asymmetry, the AD field responsible for large lepton asymmetry should start to oscillate from the gravitational scale, i.e., 0L,osc = M*, which leads to the formation of delayed-type L-balls. Meanwhile the value of at the onset of the oscillation must be suppressed due to the existence of the non-renormalizable term. Also SB is assumed to form new
type B -balls, which decay after the BBN. Let us see if this setup can be naturally realized step by step.
First of all, of the two flat directions, only should be lifted by the non-renormalizable term, which can be explained by the R-symmetry. For in-
stance, assigning R-charges i to e, L, //„. and II, to u and d, and 1 to Q, non-renormalizable terms for eLL are forbidden, while the following superpotential can lift the udd direction;
(udd) NR =
9(udd) _ 2M3 ~ 6M3'
l^l10 M6
During inflation, thus sits at the minimum ~ (HM3)1/4 determined by balancing the non-renor-malizable term and the negative Hubble-induced mass term.
Second, the abundance of the late-time baryon appearance must be of the order of nadd ~ O( 10-10). According to the result of Ref. [27], both the large lepton asymmetry, | ~ 0(0.1), and the baryon asymmetry necessary for the BBN, nBBN ~ 0( 10-10), can be successfully generated.4 Therefore it is enough to estimate the relative abundance of the additional baryon number to the lepton asymmetry. Using Eq. (14), the baryon number of the udd direction is suppressed by ($B/$L)2, which is calculated as
(19) (<PB_
where we assumed M ~ M* and m3j2 ~ 1 GeV. Thus we can naturally obtain the desired abundance, nadd ~ 0(10-10).
Thirdly, the sign of K should be positive for while negative for &B. Unless the flat direction includes the third generation superfields, the sign of K tends to be negative due to the contribution from the gauge interactions. This is because the contribution to
4 To obtain positive baryon asymmetry, the total lepton asymme-
try must be negative. The aim of Ref. [27] was to produce "positive"
lepton asymmetry of electron type, with the total lepton asymmetry
being negative. Here what we want is %e ~ 0(-0.1) < 0, there-
fore the flavor equilibration due to neutrino oscillation [21] does not
spoil our scenario. However, for recent work of suppressing such
flavor equilibration, see Ref. [35].
K from Yukawa (gauge) interactions is positive (negative). Note that the Yukawa coupling not only for top but also for bottom and tau particles can be O(1) for large tan j. Therefore, if we choose the flat directions as, e.g., e2L1L3 and u1d2d1, the sign of K for the two directions is the desired one.
Lastly, we need to show that the new type B -balls decay after the relevant BBN epoch. The decay rate of the B -ball is
i5/2 3/2
48^j3|K |M3/2'
where we used Eq. (19). Simply equating this quantity with the Hubble parameter, we have the relatively high decay temperature,
Td ~ 0.08 MeV
l£l 0.1
around which the light elements are being synthesized. However, as noted above, the decay rate is suppressed if the mass per unit baryon charge is very close to the nucleon mass, leading to smaller decay temperature. Therefore we expect the new type B -balls decay after the light element synthesis ceases to proceed.
To sum up the major characteristics of our scenario, the g-balls protect not only lepton asymmetry from the sphaleron effects, but also baryon asymmetry from the BBN processes, which leads to the late-time baryon appearance.
4. Conclusion
Concerning recent cosmological observations, there seems to be conflicting measurements of the baryons
in the universe, namely, the baryon density deduced from the CMB observation by the WMAP or the primordial D abundance is much higher than the one
derived from 4He or 7Li abundances. In this Letter, we have proposed a scenario to reconcile such inconsistency by adding baryons after the BBN and assuming a lepton asymmetry before the BBN. The introduction of the additional baryons leads to success in explaining the observed low abundance of 7Li without recourse to special models of stellar depletion. Also it was shown that the scenario can be naturally implemented in the AD mechanism, where g-balls play an essential role
in preserving extra lepton and baryon asymmetries. We should note that this is the first cosmological scenario that can completely remove the long-standing tension among the three light elements and CMB. According to our result, amazingly, about one third of the baryon we observe in the present universe might have been "sterile" during the BBN epoch. Such illuminating history of the universe might be confirmed through further observational results with better accuracy in the future.
Acknowledgements
This work was partially supported by the JSPS Grant-in-Aid for Scientific Research No. 10975 (F.T.).
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