A solution to the baryon and dark-matter coincidence puzzle in a $\tilde{N}$ dominated early universe

If a bosonic partner of a right-handed neutrino dominates the early universe sufficiently before its decay, important ingredients in the present universe are related to physics of the right-handed neutrino sector. In particular, we find that the ratio of the baryon to the dark-matter densities is given only by low-energy parameters such as a neutrino mass and a gravitino mass if the reheating temperature of inflation is much higher than $10^{12}$ GeV. Here, the gravitino is assumed to be the lightest supersymmetric particle and the dominant component of the dark matter. The observed ratio, $\Omega_{B}/\Omega_{DM} \simeq 0.21\pm 0.04$, suggests the mass of the gravitino to be in the range of $\cal O$(10) MeV provided the CP violating phase is of the order 1.


Introduction
The seesaw mechanism [1] is very attractive, since it explains naturally not only the observed small neutrino masses but also the baryon asymmetry in the present universe [2].
The important ingredient in the seesaw mechanism is the presence of right-handed neutrinos N i (i = 1 − 3) whose Majorana masses M i are very large such as M i ≃ 10 9−15 GeV. In a supersymmetric (SUSY) extension of the seesaw mechanism the right-handed neutrinos N i are necessarily accompanied with SUSY-partner bosonsÑ i (right-handed sneutrinos), and it is quite plausible [3] that theÑ i have very large classical values during inflation if the masses of the right-handed neutrinos are smaller than the Hubble constant of the inflation. If it is the case and coherent oscillations of the bosonsÑ i dominate the early universe sufficiently before their decays, some of important parameters in the present universe are determined by the physics of the right-handed neutrino sector. In this letter, we point out that if a boson partner of a right-handed neutrino N 1 dominate once the early universe it may solve the coincidence puzzle of the baryon and dark-matter densities provided that the mass of gravitino is O(10) MeV.
Before discussing the physics ofÑ 1 we should note a generic problem in supergravity, that is the gravitino problem [4]. If the gravitino is unstable, it has a long lifetime and decays during or after the big-bang nucleosynthesis (BBN). The decay products destroy the light elements created by the BBN and hence the abundance of the relic gravitino is constrained from above. This leads to an upper bound of the reheating temperature T R of inflation. The recent detailed analysis [5] shows a stringent upper bound such as T R < 10 4 GeV for the gravitino having hadronic decay modes. In the present scenario this reheating temperature means the temperature just after the decay of the coherent N oscillation (i.e. the decay temperature T d ). Such a low decay temperature is nothing unnatural in the scenario, but the produced lepton (baryon) asymmetry is too small [3].
A solution to this gravitino problem is to assume that the gravitino is the lightest SUSY particle (LSP) and hence stable [6]. This solution is very interesting in the present scenario, since the ratio of Ω B to Ω 3/2 is independent of the unknown temperature T d , but it is given by only low-energy parameters if the reheating temperature of inflation is sufficiently high. Here, Ω B and Ω 3/2 are mass density parameters of the baryon and the gravitino, respectively. We find that the ratio is determined by masses of a neutrino, the gluino and the gravitino and an effective CP-violating phase (as shown in Eq. (20)). The observation Ω B /Ω DM ≃ 0.21 ± 0.04 [7] suggests m 3/2 = O(10) MeV. (m 3/2 is the mass of the gravitino.) Here, we have assumed that the gravitino is the dominant component of the cold dark matter, that is Ω DM ≃ Ω 3/2 . The gravitino of mass in the range of O(10) MeV will be testable in future experiments as discussed in Ref. [8].
2 Matter from a coherent right-handed sneutrino 2.1 Baryon asymmetry from a coherent right-handed sneutrino We consider a frame work of the minimal supersymmetric standard model (MSSM) with three generations of heavy right-handed neutrinos N i (i = 1 − 3). The N i couple to the MSSM particles through a superpotential, where M i denote masses of the right-handed neutrinos and L α (α = e, µ, τ ) and H u are the supermultiplets of lepton doublets and a Higgs doublet which couples to up-type quarks.
The small left-handed neutrino masses are obtained via the seesaw mechanism [1].
The right-handed sneutrinos may have large classical values during inflation if their effective masses are smaller than the Hubble parameter H inf [3]. Hereafter, we restrict our discussion to the lightest right-handed sneutrinoÑ 1 , for simplicity, and treat the amplitudeÑ init 1 during the inflation as a free parameter.
After the end of the inflation, the Hubble parameter H decreases and theÑ 1 starts to oscillate when H becomes smaller than its mass M 1 . 1 The coherent oscillation of theÑ 1 decays into LH u orLH u and their CP-conjugates when H ≃ Γ N 1 , where Γ N 1 ≃ (1/4π) α |h 1α | 2 M 1 is the decay rate of theÑ 1 . The decay produces the lepton number density as, n L = ǫ × nÑ 1 , where nÑ 1 is the number density of theÑ 1 at the decay time, and ǫ is the lepton asymmetry produced in theÑ 1 decay. Assuming M 1 ≪ M 2 , M 3 , the explicit form of ǫ is given by [9,10] ǫ ≃ (1 − 2) × 10 −10 M 1 10 6 GeV where δ eff is an effective CP violating phase and m ν3 corresponds to the heaviest neutrino mass, we have used H u = 174 GeV × sin β, assuming sin β ≃ 1/ √ 2 − 1.
When theÑ 1 dominates the universe, we can write the energy density (ρ) and the entropy density (s) of the universe at the decay time as Here, T d is the temperature of radiation right after theÑ 1 decay, g * the number of effective degrees of freedom which is 230 for the temperature T ≫ 1 TeV in the MSSM and M pl ≃ 2.4 × 10 18 GeV the reduced Planck scale. In the above equation, we have assumed instantaneous decay of theÑ 1 and used the energy conservation. Hereafter, we only focus on the scenario in which theÑ 1 domination is the case.
The resultant lepton number is converted to the baryon-number asymmetry [2], which is given by [3,11] n B n γ = − 8 23 where we have used s/n γ ≃ 7.04 at the present and Eqs. (3) in the last equation. We take m ν3 ≃ 0.05 eV as suggested from the atmospheric neutrino oscillation and assume sin δ eff ≃ 1. Then, the observed baryon asymmetry n B /n γ = 6.5 +0.
Before closing this subsection, we should mention washout effects of the lepton asymmetry. When the decay temperature of theÑ 1 is close to its mass, T d ≃ M 1 , the produced lepton-number asymmetry is washed out by lepton-number violating interactions mediated by N 1 . Thus, in order to avoid the washout effect, we require T d < M 1 , and this condition is rewritten by using Yukawa coupling constants in Eq. (1) as [11], Here, we have used Eq. (3) to relate M 1 and T d . We require the Yukawa couplings h 1α to be as small as the Higgs coupling to the electron. We may explain naturally such small Yukawa coupling constants by a spontaneously broken discrete Z 6 flavor symmetry [11,12].

Conditions forÑ domination
In the previous subsection, we consider theÑ 1 to dominate the energy density of the early universe. We discuss, here, conditions for theÑ domination.
We classify the history of the energy density of the early universe by the reheating temperature T R of inflation, the initial amplitude |Ñ init 1 | and the decay temperature T d of the right-handed sneutrino. 2 If the Hubble parameter at the end of the reheating process of inflation is smaller than M 1 , theÑ 1 starts to oscillate around its minimum before the end of the reheating. The domination of theÑ 1 starts at a temperature T dom which is estimated as Thus, a condition for the domination of theÑ 1 is On the other hand, if the Hubble parameter at the end of the reheating of inflation is larger than M 1 , theÑ 1 starts to oscillate after the end of the reheating process. The temperature of the background radiation when theÑ 1 oscillation starts is given by In this case, theÑ 1 dominates the universe soon after it starts the oscillation. As in the previous case the temperature T dom at which the domination of theÑ 1 begins is estimated 2 T R is defined as a temperature of the radiation right after the end of the reheating process of inflation. as Thus, the condition for theÑ 1 to dominate the universe is, As we have seen in the previous subsection, we consider T d ≃ 10 6 GeV − 10 7 GeV, and hence the above conditions Eqs. (8) or (11) can be satisfied for a wide range of the initial amplitude of theÑ 1 , T R and M 1 .

Dark-matter genesis
As discussed in the introduction, we assume the gravitino to be the LSP and the dominant component of the cold dark-matter (CDM). As we see below, the relic gravitino density is proportional to the decay temperature of theÑ 1 if the gravitinos are produced mainly by theÑ 1 decay. Thus, the ratio between Ω B h 2 and Ω 3/2 h 2 becomes independent of the decay temperature T d (see Eq. (4)) and is determined only by low-energy parameters.
However, the gravitino is forced into thermal equilibrium by the scattering process if the decay temperature T d is sufficiently high. If it is the case, the density of the gravitino is not proportional to T d , making the above argument invalid. The freeze-out temperature of the gravitino from the thermal bath is given by [13] T f ≃ 10 9 GeV g * (T f ) 230 where m gluino denotes the mass of the gluino. We should note here that our conclusion does not change as long as T d < T f . We check in the next subsection that this condition is satisfied.
On the other hand, when the reheating temperature T R of inflation is higher than T f , the gravitino is kept in the thermal equilibrium and its resultant density is estimated as If T R is lower than T f , the gravitino cannot be in the thermal equilibrium and its resultant density is given by [14] Ω 3/2 h 2 ≃ 2.1 × 10 3 T R 10 10 GeV 10 MeV However, the gravitino density from the reheating process of the inflation is diluted by entropy production from theÑ 1 decay. By assuming the instantaneous decays of thẽ N 1 , which is accurate enough for the present purpose, we obtain the dilution factor (from the energy conservation) as where we have used Eq. (8) and (11). As a result, the present gravitino density is written as where Ω 3/2 (T )h 2 denote the gravitino density in Eqs. (13) or (14) at each temperatures T = T d or T R . The first term in Eq. (16) represents the density of the gravitino produced in theÑ 1 decay, while the second term is the resultant density of the gravitino produced in the reheating process of inflation.

A solution to the coincidence puzzle
As we have seen, the baryon asymmetry in the present universe comes from theÑ 1 decay, and the resultant baryon density Ω B h 2 is given by where k N is defined as for each values of T R , T f and T osci .
For the third and the fourth cases in Eq. (19), the gravitino densities depend on the initial amplitudes of theÑ 1 . On the other hand, the second terms are negligible for the first and the second cases in Eq. (19) if the reheating temperature T R or the oscillation temperature T osi are much higher than 10 12 GeV. (The model discussed in the next section gives most likely |Ñ init 1 | ≃ M pl .) In Fig. 1, we plot the ratio Ω B /Ω DM as a function of T R,osci for the first and the second cases in Eq. (19). We find that the ratio becomes independent of the |Ñ init 1 | and T R,osci for sufficiently high temperatures T R,osci and it is determined only by the low-energy parameters. In those regions, the ratios Ω B /Ω DM are given by Comparing Eq. (20) with the WMAP result Ω B /Ω DM ≃ 0.21 ± 0.04 [7], we obtain the mass of the gravitino as which suggests a gauge mediation SUSY breaking (GMSB) [15]. Therefore, the coincidence puzzle between the baryon and the dark-matter densities can be naturally solved in the GMSB model when the both of the densities come dominantly from theÑ 1 decay.
Notice that we obtain T f ≃ 10 10 GeV from Eq. (12) in the parameter region Eq. (21) and hence the condition T d < T f discussed in the previous subsection is satisfied since Thus, the NLSP can escape from the BBN constraints [16].
3 Some discussion

A model for the right-handed neutrino sector
In the previous section, we have used a potential for theÑ 1 , V ≃ M 2 1 |Ñ 1 | 2 . However, if we assume the broken U(1) B−L gauge symmetry to generate the Majorana masses of N i , the other MSSM fields are destabilized through a D-term potential of U(1) B−L during thẽ N 1 oscillation. 3 In this case, the evolution of the scalar fields becomes rather complex to trace and hence it becomes difficult to predict the cosmic baryon asymmetry. 4 To avoid the above problem, we consider a model with U(1) R × Z B−L 4 symmetry whose charge assignments are given in Table 1. 5 Here, U(1) R is the R symmetry. A simple superpotential allowed by the symmetry is where we have added two MSSM singlets X and S, y and f i denote the Yukawa coupling constants, the parameter v the breaking scale of the Z B−L 4 symmetry, and W MSSM the superpotential consists of the MSSM fields. As we see below, the evolution of theÑ 1 can be analyzed by using the potential M 2 1 |Ñ 1 | 2 as long as the Hubble parameter during the inflation is much smaller than v ≃ 10 15 GeV.
From the superpotential Eq. (23), the scalar potential which is relevant to the dynamics of theÑ 1 is given by where φ denotes the flat direction in the MSSM defined by H u = 1/ √ 2(0, φ) T ,L = 1/ √ 2(φ, 0) T , and we have omitted the flavor index from the Yukawa coupling constants for abbreviation. 6 In the following discussion, we focus on the evolution of theÑ 1 ,X and S, assuming M 1 ≪ H inf ≪ M 2 , M 3 and φ = 0. The dynamics of φ is discussed in the next subsection, where we see that the thermal mass term sets φ to the origin.
If H inf ≪ v,S andX are fixed to their minima during inflation (we have required y be not too small). We also require f |Ñ init 1 | ≪ v not to destabilize the minimum ofS. 7 Thus, the scalar fields are fixed in the end of inflation at 4 If B or L violating non-renormalizable terms exist in the MSSM superpotential, the Affleck-Dine baryogenesis [17] may work, which changes our result in the previous section. Even if there is no such B or L violating terms, decay processes of the multi-field oscillations are not so simple and the fate of thẽ N 1 oscillation is difficult to be predicted. 5 The three right-handed neutrinos are required to cancel Z B−L 4 gauge anomalies. 6 We can easily extend our discussion to the case where the Hubble mass terms are induced by the supergravity effects. 7 Even for |Ñ init 1 | ≃ M pl this condition can be easily realized by a spontaneously broken discrete Z 6 symmetry [11,12], where f may be as small as 10 −5 .
After the end of inflation, the Hubble parameter H becomes smaller than M 1 and thẽ N 1 starts to oscillate around its origin. Since the time scale of the motion ofX andS (∼ 1/(yv)) is much smaller than the one of theÑ 1 oscillation (∼ 1/M 1 ),X andS trace their minima along with theÑ 1 oscillation; Therefore, we find that our assumption in the previous sections to take the scalar potential of theÑ 1 as V ≃ M 2 1 |Ñ 1 | 2 is valid. Thus, we expect the initial amplitude of theÑ init 1 to be of the order of M pl . 8

Stability of the LH u flat direction
We give a comment on stability of the LH u flat direction φ during theÑ 1 oscillation.
Instability of the LH u direction comes from a cross term in the scalar potential between the LH u direction and theÑ 1 in Eq. (24). 9 However, we find that thermal effects stabilize the LH u direction φ The LH u flat direction φ is at the origin when theÑ 1 has a large amplitude, since it has a large positive mass term |hÑ 1 | 2 |φ| 2 . After theÑ 1 starts to oscillate, the positive mass term |hÑ 1 | 2 |φ| 2 decreases and the cross term between φ and theÑ 1 becomes more significant than the positive mass term. When theÑ 1 becomes smaller than M 1 /h, (see the last two terms in Eq. (24)), φ seems to depart from the origin for theÑ 1 < ∼ M 1 /h. However, we should note here that there is a thermal mass term for φ from the thermal background, and hence the effective potential for φ is given by where T denotes the temperature of the thermal background. Here, the coefficient α is estimated as α 2 ≃ 3g 2 2 /8 + g 2 1 /8 ≃ 1/4 for φ ≪ T , and we have omitted the thermal effects for theÑ 1 . 10 As we see below, the flat direction φ is still stabilized at the origin by the thermal mass term in the course of theÑ 1 oscillation.
If T R > T osci (see Eq. (9)), theÑ 1 starts to oscillate during the radiation dominated era, and hence |Ñ 1 | and T decrease with a(t) −3/2 and a(t) −1 , respectively. Here, a(t) denotes the scale factor of the universe. To discuss the stability of φ, it is convenient to define the temperature T back ∝ a(t) −1 during theÑ 1 domination, which corresponds to the temperature without theÑ 1 decay. 11 Since (T back ) 2 decreases faster than M 1 |Ñ 1 |, φ = 0 is a stable point until the decay time of theÑ 1 , if the condition, is satisfied at theÑ 1 decay time. Here, T back d is a background temperature at theÑ 1 decay time, which is given by where t decay denotes the decay time of theÑ 1 , and H d, dom ∝ T 2 d, dom /M pl the Hubble parameters at the decay time of theÑ 1 and at the beginning of theÑ 1 domination, respectively. From the energy conservation at the decay time of theÑ 1 , we find that the amplitude |Ñ decay 1 | satisfies Thus, the condition Eq. (28) can be written as 10 Possible thermal effects to the motion of theÑ 1 are discussed in Ref. [11] which shows that those effects are irrelevant as long as M 1 > ∼ T d . 11 The actual background temperature is much higher than the temperature T back , since the decay of theÑ 1 reheats up the radiation. Thus, the condition in Eq. (28) is a sufficient one to stabilize the LH u flat direction by the thermal effects.
• T d < M 1 , to avoid washout effect of the lepton asymmetry.
• v ≫ H inf , to fixX andS as in Eq. (25) during the inflation.