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Affleck-Dine Baryogenesis and Dark Matter Production after High-scale Inflation

The discovery of the primordial B-mode polarisation by the BICEP2 experiment indicates inflation with a relatively high energy scale. Taking this indication into account, we propose consistent scenarios to account for the observed baryon and dark matter densities in gravity and gauge mediated supersymmetry breaking models. The baryon asymmetry is explained by the Afflck-Dine mechanism, which requires relatively low reheating temperature to avoid a sizable baryonic isocurvature perturbation. The low reheating temperature then requires non-thermal production of dark matter to account for the correct relic density of dark matter. Our scenarios can account for the observations of baryon and dark matter density in gravity and gauge mediation and predict some parameters, including the mass of dark matter.


I. INTRODUCTION
Inflationary cosmology is now almost confirmed by the discovery of the primordial B-mode polarization by the BICEP2 experiment [1]. Then we confront a mystery of the origins of baryon asymmetry and dark matter (DM), which are erased by exponential expansion of the Universe during inflation. For a consistency of the inflationary cosmology, we have to account for the origin of baryon asymmetry and DM after inflation, 1 taking experimental facts into account, including the result of the BICEP2.
The recent result of the BICEP2 experiment implies inflation with a relatively high energy scale: where H inf is the Hubble parameter during inflation, and r is the tensor-to-scalar ratio. To explain the origins of baryon asymmetry and DM after such a high-scale inflation, we focus on models of supersymmetry (SUSY). In these models, the lightest SUSY particle (LSP) is a good candidate for DM, and the baryon asymmetry can be explained by the Affleck-Dine mechanism [4,5], in which a baryonic scalar field with a flat potential, called the AD field, plays an important role. However, the Affleck-Dine baryogenesis after high-scale inflation results in a sizable baryonic isocurvature fluctuation [6][7][8][9], 2 unless the vacuum expectation value (VEV) of the AD field is very large during inflation. The Affleck-Dine baryogenesis with such a large VEV of the AD field often requires too low reheating temperature to produce DM thermally. It is necessary to investigate a scenario for non-thermal production of DM.
In this paper, we construct consistent scenarios to account for the observed baryon and DM densities in the cases with and without Q-ball formation in models of gravity and gauge mediation. In gravity mediation, non-thermal production of DM in a low reheating temperature has been investigated in detail in Ref. [33] (see also Ref. [34]), where DM is produced mainly by two mechanisms: direct decay of inflaton into SUSY particles [34][35][36][37][38][39] and inelastic scatterings during reheating process [33,34]. In addition, if Q-balls are formed after the Affleck-Dine baryogenesis, DM is also produced from the decay of Qballs [16,[22][23][24][25][26]32]. In this case, since the branching fractions of the Q-ball into quarks and gauginos are related with each other by a simple relation, we may overcome the baryon-DM coincidence problem [32]. In gauge mediation, the gravitino is the LSP, which is produced by scatterings between gluinos and gluons just after the end of reheating and is thus related with the reheating temperature [40]. Besides, any flat direction other than the one contains Higgs fields results in the formation of Q-balls after the Affleck-Dine baryogenesis. Then Q-balls provide another source of gravitinos [27][28][29][30][31]. Although gravitinos can be directly produced from the decay of Q-balls, they are mainly produced from the next-to-lightest SUSY particle (NLSP) into which Q-balls decay. We find that these scenarios in gravity and gauge mediation can be consistent with the observations of baryon and DM densities as well as the result of the BICEP2 experiment, and have predicted some parameters, including the mass of DM.
This paper is organized as follows. In the next two sections, we calculate the baryon density and baryonic isocurvature perturbation resulting from the Affleck-Dine baryogenesis, and derive an upper bound on reheating temperature by consistency with observations. Then we briefly explain the Q-ball and its properties. In Sec. V, we construct scenarios to account for the observed baryon and DM density in the context of low reheating temperature. We consider the cases with and without Q-ball formation, in gravity and gauge mediation.
Section VI is devoted to the summary and conclusion.

II. AFFLECK-DINE BARYOGENESIS
Typical SUSY models, including the Minimal SUSY Standard Model (MSSM), contain many flat directions, whose F and D term potentials are absent in renormalizable levels [41].
Such flat directions are sometimes referred to as AD fields, named after Affleck and Dine.
The following combination of an up-type right-handed squark and two down-type righthanded squarks, denoted as φ, is an example of a flat direction, calledūdd flat direction: where subscripts and superscripts represent family indices and color indices, respectively. In this paper, we do not restrict ourselves toūdd flat direction, but use it as an illustration in some cases.
Let us consider classical dynamics of an AD field with B (or B − L) charge, such asūdd flat direction. During inflation, we assume that the AD field obtains a large VEV. Nonrenormalizable terms, which break CP and B (or B−L) symmetry in general, are relevant for the dynamics of the AD field due to the large VEV. After inflation ends, the AD field starts to oscillate and rotate in the complex plane around the low energy vacuum, whose dynamics is far from thermal equilibrium. In this way, Sakharov's conditions for baryogenesis [42] are satisfied and B (or B − L) asymmetry is generated. Since the amplitude of the oscillation decreases due to the Hubble expansion, the non-renormalizable terms becomes irrelevant and B (or B − L) symmetry is approximately restored. If the AD field releases its charge into the standard model particles after the sphaleron process [43] freezes out, the AD field should have B charge to account for the baryon asymmetric Universe. On the other hand, if the AD field releases its charge before the sphaleron process freezes out, the AD field should have B − L charge so that the sphaleron process does not wash out the asymmetry. In this section, we investigate the Affleck-Dine baryogenesis, taking into account the energy scale of inflation given in Eq. (1).
Since we assume that a flat direction obtains a large VEV in the early Universe, higherdimensional terms should be taken into consideration. We assume an R-parity symmetry to avoid catastrophic proton decay, and superpotentials such as W =ūdd ⊃ φ 3 disappears.
Promoting this symmetry into a discrete R-symmetry which controls higher-dimensional terms for the flat direction, we assume the superpotential of the AD field as where M pl ( 2.4 × 10 18 GeV) is the reduced Planck scale, λ is a coupling constant, and n ≥ 4 is a certain integer which is determined by the R-charge of the AD field. For example, n = 6, 9, 12, ... forūdd flat direction, depending on the discrete R-symmetry.
The AD field has usual soft terms as well as the F -term potential from the superpotential of Eq. (6) as where m φ (|φ|) is the mass of the AD field at the energy scale of |φ|. The term proportional to an O(1) parameter a g is an A-term mediated by Planck-suppressed interactions, and m 3/2 is the mass of the gravitino. Hereafter, we set the phase of the AD field and the SUSY-breaking F-term such that Im[λ] = 0 and Im[a g ] = 0.
The AD field acquires Hubble induced terms during inflation [5], because inflation is associated with non-zero vacuum energy V = 3H 2 M 2 pl , which largely breaks SUSY. We write these Hubble-induced terms as where c H and a H are O(1) constants. Hubble-induced A-terms are absent during inflation if the field which has a non-zero F -term during inflation is charged under some symmetry and its VEV is less than the Planck scale during inflation [9]. These conditions are usually satisfied for models of high-scale inflation in supergravity [44,45] and thus we set a H = 0 in this paper. 3 As we show below, we have to take into account higher-dimensional terms V K coming from a Kähler potential. Let us consider the following Kähler potential as an illustration for the origin of V K : where I is a field which has a non-zero F -term during inflation (i.e. |F I | 2 = 3H 2 M 2 pl ). This operator gives the AD field a potential as where "· · · " denotes higher-dimensional Planck-suppressed terms. The parameter a H 2 is an To sum up, the potential of the AD field is given by the sum of the soft SUSY breaking terms V s , the F -term potential V F , the Hubble-induced terms V H , and the potential coming from a Kähler potential V K : In some cases, thermal potentials affect on the dynamics of the flat direction. As we see below, however, we are interested in the case that the VEV of the AD field is so large to evade the baryonic isocurvature constraint, which results in a low reheating temperature.
Hence thermal potentials are irrelevant in the following discussion. In this paper, we set all unknown O(1) parameters as one: |a g | , |c H | , |a H 2 | = 1. We also assume c H < 0, which makes the AD field obtain a large VEV during and after inflation as 4 |φ| |c H | (n − 1) with the Hubble parameter being H inf and H(t), respectively. Note that if λ = O(10 −4 ), the VEV of the AD field during inflation is as large as the Planck scale. Curvature of the phase direction of AD field, θ, is dominantly given by V K ; 4 In the case of D-term inflation (i.e. c H = 0), the AD field can stay anywhere V H is satisfied due to the Hubble friction. Eq. (14) is correct even in this case as long as the initial field value of the AD field is sufficiently large.
After inflation ends, the energy density of the Universe is dominated by coherently oscillating inflaton and the Hubble parameter decreases as the Universe expands. When the Hubble-induced mass becomes less than the soft mass (i.e. H(t) = H osc m φ (φ)), the AD field begins to oscillate around the low-energy vacuum, φ = 0. Since the phase of the AD field just before the oscillation begins is generally different from the one determined by the A-term of Eq. (7), the AD field also begins to rotate in the complex plane. Baryon number is generated by the rotation because baryon density is given by where b is the baryon charge of the AD field and the dot above the AD field denotes the derivative with respect to the time. For example, b = −1/3 forūdd flat direction. The time evolution of the baryon density is written aṡ We estimate the solution of this equation as where θ is the initial phase of the AD field, which stays at a certain non-zero phase θ due to the Hubble friction before oscillation begins. We use |φ osc | n ∼ H osc |φ osc | 2 M n−3 pl /λ in the last line. We define an ellipticity parameter as where n φ √ V |φ| 2 is the number density of the AD field. In this paper, we assume = m 3/2 /H osc , because the initial phase of the AD field θ is of the order of unity unless it is fine-tuned to coincide with the low energy vacuum θ = 0. Thus we obtain the present baryon-to-entropy ratio as where we use ρ inf | osc 3M 2 pl H 2 osc in the last line. Here we have assumed that there is no entropy production other than the decay of inflaton, which also implies that the AD field does not dominate the Universe. Note that if the AD field releases its charge into the standard model particles before the sphaleron process freezes out, there is an O(1) correction to Eq. (25) [46]. This is the case that we consider in Secs. V A and V C.

III. BARYONIC ISOCURVATURE PERTURBATION
Since there is no sizable Hubble-induced A-term, the phase direction of the AD field develops quantum fluctuations during inflation, and as a result a baryonic isocurvature fluctuation, which is tightly constrained by recent observations, is produced. In this section, we consider the dynamics of the phase direction of the AD field in detail.
First we consider the case of λ 10 −4 , in which the VEV of the AD field is so small that the potential V K is negligible and the curvature of the phase direction is much less than the Hubble parameter during inflation (see Eq. (16)). The phase direction of the AD field therefore acquires quantum fluctuations during inflation as [6][7][8][9] |δθ| √ 2H inf 2π |φ inf | .
Since the baryon number is related to the initial phase (see Eq. (21)), this fluctuation induces a sizable baryonic isocurvature fluctuation as The baryonic isocurvature perturbation is constrained by observations of the cosmic microwave background, which have shown that the density perturbations are predominantly adiabatic [47,48]. The P lanck Collaboration puts an upper bound on the totally uncorrelated isocurvature fraction as [49] P SS (k * ) where P RR and P SS are power spectra of the adiabatic fluctuation and isocurvature fluctuation, respectively, and k * (= 0.05 Mpc −1 ) is a pivot scale. Since we are interested in the baryonic isocurvature fluctuation, we use the following relation: where Ω b and Ω DM are the density parameter of the baryon and DM, respectively. Thus we obtain an upper bound on the baryonic isocurvature fluctuation as where we have used P RR 1/2 2.2 × 10 −9 [48]. The upper bound and the value of H inf indicated by BICEP2 put a lower bound on φ inf as This means that the VEV of the AD field should be as large as the Planck scale, which implies that the higher-dimensional operator in the superpotential should be suppressed, (14)). When the AD field obtains such a large VEV, the potential V K is effective and the phase direction of the AD field obtains a mass of the order of the Hubble parameter during inflation (see Eq. (16)). 5 In this case, the baryonic isocurvature fluctuation is absent from the beginning. To summarize, in order to avoid a sizable baryonic isocurvature fluctuation, the VEV of the AD has to be as large as the Planck scale, in which case baryonic isocurvature fluctuation is absent due to the potential originated from a Kähler potential. Thus we assume λ 10 −4 hereafter. 6 Let us discuss the implication of the baryonic isocurvature fluctuation on the reheating temperature using Eq. (25). Since the VEV of the AD field at the beginning of its oscillation is related with that during inflation via |φ osc | = (H osc /H inf ) 1/(n−2) |φ inf |, a large VEV during inflation results in a relatively large VEV at the onset of its oscillation, and then the AD field tends to dominate the Universe. Therefore, in order to account for today's baryon-toentropy ratio without additional entropy production except for the decay of inflaton, the reheating temperature tends to be small to dilute the AD field successfully. 5 In contrast, if λ = O(1), the lower bound in Eq. (31) requires about 1% and 10% tuning on the initial phase θ for n = 4 and n = 6 flat directions, respectively. This tuning would not be explained by the anthropic principle because human life would be able to exist whether or not baryonic isocurvature fluctuation exists. Taking this tuning seriously, we consider the case of λ 10 −4 in this paper. 6 The smallness of λ might be understood by some flavor symmetry.
From Eq. (14), the VEV of the AD field at the beginning of its oscillation is given by where H osc is the Hubble parameter at the oscillation time. Thus the observed baryon density requires the reheating temperature of the Universe as where we have used Y B 8.7 × 10 −11 for the observed baryon-to-entropy ratio [50], and Here we comment on the gauge symmetry breaking and its effect on the reheating process [51]. Since the non-zero VEV of the AD field breaks gauge symmetries, one may consider that the reheating process might be hindered. The hindrance of the reheating process occurs if the AD field continues to oscillate coherently and if the Standard Model gauge symmetry is completely broken by the VEV of the AD field. The former condition is not satisfied in the case that Q-balls are formed, which we explain in the next section. The latter condition is not satisfied forūdd and LH u flat directions, for example. As explained in Sec. V, Q-balls are usually formed in gauge mediation except for the case of LH u flat direction.
Thus, the suppression of the reheating process may be realized only in gravity mediation without Q-ball formation. This is the case we consider in Sec. V A, in which we check that the reheating process is not affected by the AD field at least in the case we are interested in (see Eq. (52)).
Finally, we comment on the case that the superpotential of the AD field is absent due to a discrete R-symmetry. In this case, the VEV of the AD field during inflation naturally becomes the Planck scale and thus a baryonic isocurvature fluctuation is absent. However, the VEV at the beginning of its oscillation is also the Planck scale, at which the AD field still feels V K . Since the potential of the AD field V K is so complicated that the ellipticity parameter becomes O(1), Eq. (25) implies that the reheating temperature has to be much lower than 1 MeV in this case, which spoils the success of the BBN scenario. Thus we consider the case of W = 0.

IV. Q-BALLS
In this section, we explain the dynamics of the Q-ball, which is a non-topological soliton formed after the Affleck-Dine mechanism in many SUSY models [14][15][16][17][18][19][20]. We first explain the condition of the Q-ball to be formed and then review decay processes of the Q-ball.

A. Formation of Q-ball
After the AD field starts to oscillate and rotate around the low energy vacuum, the amplitude of the oscillation decreases due to the Hubble expansion. Since baryon numberviolating terms are higher-dimensional ones, their effects become irrelevant and the generated baryon number is conserved soon after the beginning of the oscillation. Thus, in this section, we assume baryon number to be conserved and investigate the stable configuration of the AD field in a system with non-zero baryon charge.
The energy of the AD field is given as We are interested in the case with sufficiently small value of the AD field, for which the potential is approximated by V = m 2 φ (φ) |φ| 2 . Since the baryon density is already produced by the Affleck-Dine mechanism, we consider a system with non-zero baryon charge. The baryon charge is given by where we have omitted the factor b for notational simplicity. The scalar field configuration which minimizes the energy given in Eq. (34) with a fixed baryon charge Q is obtained by minimizing the following combination; where ω 0 is a Lagrange multiplier. Terms with time derivatives are rewritten aṡ The minimization condition determines the time dependence of the AD field as Assuming a spherically symmetric ansatz, ϕ(r) = ϕ(r), we obtain the following equation which determines ϕ(r): The boundary condition is ϕ (0) = 0 and ϕ(∞) = 0 since we are interested in smooth and localized configurations. Regarding φ and r as a position and a time variable, Eq. (39) is interpreted as the equation of motion of a particle in one dimension with a friction term (2/r)∂ϕ/∂r. Using this analogy, one can find the following condition for existence of a spatially localized configuration, referred to as Q-ball [20]: In most cases we are interested in, the energy of the Q-ball per unit charge is well approximated by ω 0 .
Linear analyses indicate that there are instability bands during the oscillation of the AD field which corresponds to a typical size of the Q-ball R if the condition of Q-ball formation Eq. (40) is satisfied [15,16]. This means that the coherently oscillating AD field is unstable and fragments into Q-balls soon after the onset of its oscillation. A typical charge of Q-balls is roughly estimated by the charge which is contained in the volume of a typical Q-ball size R at the formation time:  (79) and (117)) [17][18][19]52].

B. Decay of Q-ball
To explain the decay of Q-balls, let us focus on a Q-ball which consists only of squarks.
Numerical simulations have shown that almost all of the baryon charge of the AD field are transferred into Q-balls [17][18][19]52]. Then Q-balls decay and release their baryon charge into quarks if they are unstable. The AD field interacts with quarks via gauge interactions and thus Q-balls lose their baryon charge by emitting quarks from their surfaces [21]. 7 The condition for Q-ball decay is that the energy of the Q-ball per unit baryon charge, ω 0 / |b|, is larger than masses of baryons in the hadron phase, m b 1 GeV.
Usually, the baryon charge density inside a Q-ball is so large that a naive rate estimated by squark decay exceeds an upper limit by the Pauli blocking effect. The rate of particle emission from a Q-ball surface is therefore determined by the Pauli blocking effect on its surface and is given as where n is the outward-pointing normal vector, j is particle flux, andR is the effective radius of the Q-ball given by φ(R) ∼ ω 0 [53]. The interaction energy E i is given by the energy of the Q-ball per unit charge, ω 0 , when the relevant elementary process is squark decay, such asq → q + (gaugino). In addition, the baryon charge density inside the Q-ball is so large that the scattering process via gaugino (or Higgsino) exchangeq +q → q + q occurs efficiently. It has been shown that the rate of this process is also saturated by the Pauli blocking effect, and the interaction energy E i is given by 2ω 0 in this case [53]. The rate of Q-ball decay is dominated by the latter process and thus its lifetime Γ −1 Q is given as 7 Far inside Q-balls, field values of squarks are large and hence gauginos and quarks are heavy. Therefore, Q-balls cannot decay into them. 8 Here we assume that quarks and gauginos are massless. A correction for the flux of massive particles is derived in the Appendix.
where n q is the number of species for quarks interacting with the AD field and is typically O(10).
Q-balls completely lose their charge and energy when the condition Γ Q ∼ H is satisfied.
The decay temperature of Q-ball is thus determined as where the first line is the case where Q-balls decay before reheating completes while the second one is the case where Q-balls decay after reheating completes. In the latter case, the energy density of Q-balls may dominate the Universe. Since Q-balls are localized lumps much smaller than the horizon scale, their energy density decreases as a −3 , where a is a scale factor. Thus the energy density of Q-balls dominates that of the Universe when the following condition is satisfied: where ρ I is the inflaton energy. We checked that the energy density of Q-balls never dominate that of the Universe in the case we consider.
As explained above, Q-balls dominantly decay into quarks. However, they decay into SUSY particles if the decay process is kinematically allowed. From kinematics and the conservation of baryon charge, Q-ball can decay only into particles lighter than the energy of the Q-ball per unit charge, ω 0 . Since ω 0 is less than the mass of the AD field (see Eq. (40)), Q-balls cannot decay into the AD field itself. This is another explanation of the stability of Q-ball. On the other hand, Q-balls decay into gauginos and/or Higgsinos if they interact with the Q-balls and their masses are less than ω 0 . However, in contrast to the case of quarks, gauginos and Higgsinos cannot be produced through a scattering process likeq +q → (gaugino) + (gaugino) due to the conservation of baryon charge. Thus if we could neglect their masses, their production rate from Q-ball decay is given by Eq. (43) with In addition, there is a correction coming from non-zero masses of gauginos and Higgsinos as explained in the Appendix.

V. SCENARIOS FOR DARK MATTER PRODUCTION IN LOW REHEATING TEMPERATURE
To construct a consistent cosmological scenario, it is necessary to account for the following observed DM density as well as the baryon density: where we have used Ω DM h 2 0.12 for the observed DM abundance in the last line [50]. As explained in Sec. III, the reheating temperature of the Universe must satisfy Eq. (33) to account for the observed baryon density by the Affleck-Dine baryogenesis. In this section, we propose scenarios to account for the DM density in that reheating temperature.
Assuming models of gravity mediation and gauge mediation, we investigate scenarios in each model with and without Q-ball formation. In Sec. V A, we consider a model of gravity mediation without Q-ball formation, in which DM is mainly produced from two processes; decay and inelastic scatterings during reheating era [33,34]. In Sec. V B, we consider a model of gravity mediation with Q-ball formation. In this case, DM can be produced from Q-balls [16,[22][23][24][25][26]32] as well as by the above two processes. In Sec. V C, we consider a model of gauge mediation without Q-ball formation, which is only the case for LH u flat direction. In this case, gravitino DM is produced from the thermal plasma at the time of reheating [40] as well as from decay and inelastic scatterings during reheating era. A model of gauge mediation with Q-ball formation is investigated in Sec. V D. Also in this case, Q-balls provide another source of DM [27][28][29][30][31].

A. Gravity mediation without Q-ball
In models of gravity mediation, the mass of the AD field m φ (|φ|) logarithmically depends on |φ| due to renormalization group running of squark masses, and the condition for Qball formation, Eq. (40), is satisfied when the mass decreases with increasing the VEV of the AD field (i.e. dm φ /dφ < 0) [16]. While the strong interaction makes squarks light with increasing energy scale, Yukawa interactions make them heavy. In a typical model of gravity mediation, stops become heavy with increasing energy scale while the first and second family squarks become light. Thus there may be no Q-ball solution if the AD field consists mainly of third family squarks. In this subsection, we consider the case without Q-ball formation in gravity mediation.
Since gravitino mass is of the same order as squark masses in gravity mediation, the ellipticity parameter, (= m 3/2 /H osc m 3/2 /m φ (φ osc )), is of the order of one (see Eq. (23)).
Thus we obtain the VEV of the AD field at the beginning of its oscillation and the reheating temperature of the Universe from Eqs. (32) and (33). Assuming that the AD field continues to oscillate until reheating completes, we calculate the amplitude of the oscillation at the time of reheating as Thus we find that the amplitude of the oscillation at the time of reheating is much less than the reheating temperature. This indicates that the AD field cannot affect the reheating process even if it could continue to oscillate coherently. Of course, the coherent oscillation dissipates through interactions with the thermal plasma much before reheating completes, but in this paper we do not investigate this issue in detail (see Ref. [51,55] and references therein).
The Affleck-Dine baryogenesis with n = 4 flat direction predicts the reheating temperature as T RH 4 × 10 5 GeV for λ = 10 −4 , in which case DM is produced thermally. Since the LSP is mostly bino-like in a typical model of gravity mediation, thermally produced binos over-closes the Universe unless the mass of bino is fine-tuned to co-annihilate [56,57] with the stau [58]. On the other hand, when we consider n = 4 flat direction with λ 10 −10 or n = 6, 8 flat direction with λ 10 −4 , the required reheating temperature is less than 1 GeV (see Eq. (33)). Such a low reheating temperature results in a non-thermal production of DM, which we have investigated in Ref. [33]. Here we review this calculation and apply it in our situation. In a low reheating temperature, DM is produced mainly through two contributions: direct decay of inflaton into SUSY particles [34][35][36][37][38][39] and inelastic scatterings between high energy particles and thermal plasma during reheating process [33,34]. 9 9 DM is also produced thermally if the maximal temperature of the Universe is larger than the freeze-out temperature of the LSP [54,59]. However, this contribution is always subdominant as shown in Ref. [33], and we neglect it in this paper.
DM can be produced directly by the decay of inflaton [35][36][37][38][39] and its abundance is 0.4 eV m DM 1 TeV where m DM is the mass of the LSP, and m I is the mass of the inflaton. We write Br I as the average number of DM produced by each inflaton decay. One may expect Br I = 1 due to SUSY. In Ref. [39], however, it was pointed out that SUSY particles are produced through a cascade shower from inflaton decay, like pions in a QCD shower, and the effective branching ratio Br I deviates from 1. They calculated Br I and found that Br I is as large as O(10 0−1 ) for m I = 10 10 GeV. They also extrapolated the results from m I ≤ 10 10 GeV to larger inflaton mass and estimated Br I = O(10 0−2 ) for m I = 10 13 GeV.
Next, let us explain the other process of DM production; inelastic scattering process during the reheating era [33,34]. At the first stage of reheating, the inflaton decays into high energy particles with the energy of the order of the inflaton mass m I . The daughter particles interact with each other and produce many low energy particles almost without losing their energy. Then low energy particles thermalize by their own interactions, but the energy of the thermal plasma is still much smaller than that of the inflaton. Once the thermal plasma is created in the Universe, the high energy particles produced from the inflaton decay inelastically interact with the thermal plasma, through which the number of high energy particles drastically increases [60,61] (see also Ref. [62][63][64]). Throughout this process, DM is produced in a certain amount as we explain below. 10 Let us estimate an inelastic scattering cross section between a high energy particle and the thermal bath. It is enhanced by t-channel contribution and is given as where an infrared divergence is naturally regulated by the thermal mass of an internal field, α 1/2 T . However, in estimating the scattering rate, it is necessary to take into account an interference effect between a daughter particle and its parent particle, known as the Landau-Pomeranchuk-Migdal (LPM) effect [65][66][67][68][69][70][71]. The interference effect forbids subsequent scattering processes while their phase spaces overlap with each other. When we write the position vector of a parent particle as x µ = (∆t, ∆tẑ), the interference effect remains until the phase factor varies significantly as where k and k ⊥ are the four momentum and the perpendicular momentum of the daughter particle, respectively. 11 We write the emission angle of the daughter particle as θ (= k ⊥ /k 0 ).
From Eq. (57), we find that the LPM effect suppresses subsequent inelastic scattering processes during the interval ∆t(k 0 ) ∼ k 0 /k 2 ⊥ . We therefore determine the inelastic scattering rate as , .
Let us estimate the perpendicular momentum of the daughter particle ∆k ⊥ . The perpendicular momentum evolves as random walk due to elastic scatterings with the thermal plasma as whereq el is a diffusion constant given by the elastic scattering rate Γ el ( αT ) aŝ Using Eqs. (59) and (60), we obtain The rate of inelastic scatterings is determined by Eqs. (58) and (61), from which we find that the energy loss rate increases with increasing the energy of the daughter particle.
Therefore high energy particles from the inflaton decay continue to split into high energy particles whose energy is less than but the same in order of magnitude as parent particle's energy. Throughout this splitting process, the number density of high energy particles 11 Here and hereafter we assume that the daughter particle is charged under a non-Abelian gauge group.
increases exponentially [60,61]. Given a certain time when their energy is of the order of E, we can estimate their number density n h from the energy conservation as where n I is the number density of the inflaton [33].
During the above splitting process, inelastic scatterings between a high energy particle and the thermal plasma can produce DM, when the center-of-mass energy √ 4T E is larger than the mass of DM m DM . The production rate is given as [34] for a reaction whose center-of-mass energy is just above the threshold (i.e. E E th ≡ m 2 DM /4T ). Although the fine structure constant α for the DM production process is in general different from the one for the inelastic scattering process, we neglect the difference in this paper for simplicity. The energy density of DM which is produced during the splitting process of high energy particles is therefore obtained as Note that the DM abundance is independent of the mass of the inflaton. In general, the gauge coupling constants in the second line cannot be canceled with each other, but our conclusion is still correct with an uncertainty within one order of magnitude.
Here we comment on some assumptions which we have implicitly used in the above calculation. We assume that the inflaton is so heavy that its daughter particles can produce DM at the time of T = T RH (i.e. m I ≥ m 2 DM /2T RH ∼ 10 7 GeV). Since the energy scale of inflation is large, it is expected that the inflaton mass is also large. The other case has been calculated in the original paper [33]. In addition, the result in Eq. (67) The LSP is mostly bino-like in a typical model of gravity mediation Its annihilation effect is negligible as far as its abundance is consistent with the observed DM abundance.
In summary, DM can be produced from two sources, and its abundance is given by the sum of Eqs. (55) and (67). Taking the annihilation of DM into account, we conclude that the DM abundance is given by In Fig. 1, we show the constraint on the DM mass m DM and the reheating temperature  12 In large field models discussed in Refs. [72,73], the inflaton mass can be as large as 10 15 GeV.
where we assume = 1. Eq. (70) is the case where the DM abundance is determined by the contribution from the decay of inflaton, while Eq. (71) is the case where the DM abundance is determined by the contribution from inelastic scattering process during reheating. 13 Note that H osc is roughly given by the squark mass at the energy scale φ osc ∼ 10 15 GeV, and the parameter λ has an upper bound as λ 10 −4 to avoid a sizable baryonic isocurvature perturbation as explained in Sec. II.
Interestingly, by combining Eqs. (55) and (67), we obtain a lower bound on the inflaton mass as a function of reheating temperature as m I 1.3 × 10 12 GeV Br I 5 to obtain the correct DM density, Eq. (50). Although we can evade this constraint for T RH m DM , only n = 4 flat direction with λ 10 −10 is consistent with such a high reheating temperature. Otherwise, the DM abundance exceeds the observed one whatever the DM mass is.
Finally, we comment on the testability of this scenario. For SUSY particles with masses of O(1) TeV, the elastic scattering cross section between the bino-like LSP and nucleon is as large as 10 −46 − 10 −45 cm 2 , which is detectable in future direct detection experiments of DM such as XENON1T [74]. Since the thermal relic of the bino-like LSP is usually much larger than the observed DM abundance, the above scenario would be favored if DM is detected in that region.

B. Gravity mediation with Q-balls
As we have mentioned in the previous subsection, the mass of the AD field m φ (|φ|) logarithmically depends on |φ| due to the renormalization group running of squark masses. 13 As we have mentioned, the prediction given in Eq. (71) involves an uncertainty within about one order of magnitude. We basically write the potential of the AD field in gravity mediation as where the second term in the parenthesis represents renormalization group running. In many cases in gravity mediation, the strong interaction dominates quantum corrections for a typical flat direction, and we obtain K < 0 and |K| ∼ 0.01 − 0.1 [16], which satisfies the condition for Q-ball formation in Eq. (40). The configuration of the AD field is obtained by solving Eq. (39) with the above potential. The solution is well approximated by [16] φ(r, t) where R, ω 0 , and φ 0 are given as where m φ (φ 0 ) is the mass of the AD field at the energy scale of φ 0 . Since the energy of the Q-ball, M Q , is calculated from Eq. (34) as we find that the energy of the Q-ball per unit charge is approximately equal to ω 0 m φ 1 GeV and thus Q-balls decay into quarks.
Using ω 0 m φ and R ∼ m −1 φ in Eq. (41), we estimate a typical charge of Q-ball formed after the Affleck-Dine baryogenesis as The numerical simulations have shown that the coefficient β is approximately given by 2 × 10 −2 [17,18,52], which we have used in the second line.
The decay rate of the Q-ball is calculated from Eq. (45) withR R(2 log(φ 0 / √ 2ω 0 )) 1/2 7R. For n = 4, the Q-ball decays after the reheating as T decay 10 3 GeV λ 10 −4 where we assume that the effective number of relativistic degrees of freedom at the decay where we assume that the effective number of relativistic degrees of freedom at the decay time g * is 10.75. We find that Q-balls decay after DM freezes out for n = 6 and n = 8.
Let us consider DM production mechanisms. In order to suppress the baryonic isocurvature perturbation (see Eq. (33)), the reheating temperature is required as T RH 1 GeV for n = 4 flat direction with λ 10 −10 or n = 6, 8 flat directions. In such a low reheating temperature, the two contributions, from the direct decay of the inflaton and from inelastic scatterings during the thermalization, explained in the previous subsection can generate DM, and the results is again given by Eqs. (55) and (67). In addition, the decay of Q-balls also generate a sizable DM abundance [16,[22][23][24][25][26]32] because they decay after DM freezes out (see Eq. (82)).
In the rest of this subsection, we focus on a scenario of baryon and DM co-genesis from Q-ball decay. This scenario can be realized in the non-shaded region in Fig. 1 where the two contributions of DM abundance, from the direct decay of inflaton and from inelastic scatterings, are negligible. This is the case with, for example, a flat direction with n = 8 and λ 10 −4 .
As we have explained in Sec. IV, the decay rate of Q-ball is saturated and determined by the Pauli blocking effect. While SUSY particles are produced from the Q-ball surface only through elementary decay process likeq → q + (gaugino), quarks are dominantly produced through scattering process via gaugino or Higgsino exchange likeq +q → q + q [53]. Thus, the ratio of the Q-ball decay into sparticles and quarks is calculated as where a factor of 8 is due to the difference of the elementary processes as we have explained in Sec. IV. The factor n s is the effective number of sparticles into which Q-balls can decay.
Since the flux of massive particles from a Q-ball surface is smaller than that of massless particles, there is a correction due to non-zero sparticle masses, which is derived in the Appendix. Thus we obtain where m s is the mass of the sparticle s, g s is the number of species for the sparticle and f is a function given in the Appendix. For example, g s = 1, 3, and 8 for the bino, wino, and gluino, respectively. 14 However, Q-balls can decay only into particles lighter than the energy 14 Note that g s also depends on the flat direction. In the case ofūdd flat direction, Q-balls do not decay into winos, g wino = 0, since they consists of only right-handed squarks [32].
of the Q-ball per unit charge, ω 0 , which is approximately equal to the mass of squarks at the energy scale of |φ| (∼ 10 15 GeV). In a typical model of gravity mediation, a mass of squarks at the energy scale of 10 15 GeV is mostly smaller than the mass of the gluino and larger than that of the bino (LSP) [76]. Thus in the typical models Q-balls can decay into binos, and not into gluino. 15 Depending on a model, Q-balls can also decay into winos and Higgsinos. Hereafter, we assume Q-balls can not decay into Higgsinos, for simplicity (see Ref. [76] for complete discussion).
Since all sparticles eventually decay into binos, 16 we obtain the following formula for the baryon-to-DM ratio: mb bm p We should emphasize that this ratio is independent of the reheating temperature and the charge of Q-balls. As an illustration, let us calculate two asymptotic solutions. When where we assume n q = 10 and b = 1/3 in the first line. The bino mass of 10 GeV is unrealistic for ordinary models in gravity mediation. We conclude that we can explain the observed baryon-to-DM ratio if the bino mass is close to below ω 0 .
In Fig. 2, we show the constraint on ω 0 and mb. On the boundary of and inside the blue (dark gray) shaded region, the DM density produced by the decay of Q-balls is equal and larger than the observed value respectively. Here, we have assumed the grand unified theory (GUT) relation, where the masses of the wino and gluino are two and six times larger than that of the bino. Under the GUT relation, the red (light gray) shaded region is already 15 Even in this case Eq. (84) is valid since f (x > 1) = 0. 16 For the case of the axino LSP, see Refs. [25,26]. Exclusion plot in a model of gravity mediation with Q-ball formation. We assume that the number of quarks interacting with Q-ball, n q , is 10 and baryon charge of the AD field, b, is −1/3. We also assume that the masses of the wino and gluino are two and six times larger than that of the bino, which is a typical case in gravity mediation with the grand unified theory relation.
The abundance of DM produced from Q-ball decay (ρ Q DM ) is larger than that observed in the blue (dark gray) shaded region. In the red (light gray) shaded region, the gluino mass is smaller than excluded by the gluino search at the ATLAS Collaboration [75]. Detailed calculations in the constrained MSSM will be performed in the near future [76].

C. Gauge mediation without Q-ball
In models of gauge mediation, SUSY breaking in a hidden sector is transmitted to the standard model sector by gauge interactions mediated by a messenger sector. Since the AD field has some gauge charge, gauge fields acquire effective masses of the order of g |φ|, where g generically stands for the Standard Model gauge coupling. The transmission of SUSY breaking effect is therefore suppressed for g |φ| M s , where M s is a messenger scale [77], and thereby the soft mass of the AD field is suppressed. The condition for Q-ball formation, Eq. (40), is therefore satisfied if there is no term in the potential of the AD field other than the soft mass term. However, if the AD field is LH u flat direction, the µ term of Higgs multiplets prevents the flat direction forming Q-balls [78]. 17 In this subsection, we consider the case of LH u flat direction in gauge mediation and construct a consistent scenario to account for the observed DM density as well as baryon density.
Since the mass of the gravitino m 3/2 is much smaller than the electroweak scale, and since the AD field starts to oscillate at H osc ∼ µ ∼ 1 TeV, the ellipticity parameter defined in Eq. (23) is much smaller than one, Taking into account ∆B after sphaleron 0.3 × ∆(B − L) before sphaleron due to the sphaleron effect [46] and b = −1/2 for B − L charge of LH u flat direction, we find that Eq. (33) is rewritten as Note that we assume a discrete R-symmerty which controls higher-dimensional terms in the superpotential as well as the term with n = 2. Therefore, in contrast to the case with the R-patiry, LH u flat direction with n = 6 or 8 is also possible.
Next, let us estimate the abundance of the gravitino, which is the LSP in gauge mediation. There are two contributions to the gravitino production, depending on the reheating temperature. If the reheating temperature is larger than the mass scale of MSSM particles, MSSM particles are in the thermal equilibrium and gravitinos are produced from thermal bath through scatterings between gluons and gluinos, as is discussed in Refs. [40,[79][80][81][82].
Here we quote the result from Ref. [79]: where mg is the gluino mass. The parameter dependences of this result can be understood by ρ th where n r = ζ(3)T 3 /π 2 is the number density of radiation in the thermal bath, and g 3 is a strong coupling constant.
If the reheating temperature is smaller than the mass scale of MSSM particles, MSSM particles are produced by the mechanism explained in Sec. V A. Then, MSSM particles eventually decay into gravitinos. The abundance of the gravitino is given by where m SUSY is the mass scale of MSSM particles whose production by inelastic process is efficient. Typically, m SUSY can be identified with the mass of the NLSP. Note that if the NLSP is produced too much, its annihilation is effective and the gravitino abundance decreases accordingly. When the NLSP is right-handed stau, which is the case with typical gauge mediation models, its annihilation cross section is given as [83] σ ann v l R 3 × 10 −11 GeV −2 ml where we have omitted bino mass dependence for simplicity. In analogy with the derivation of Eq. (68), the annihilation effect of the NLSP puts an upper bound on the gravitino abundance produced from the decay of the NLSP: In summary, the gravitino abundance is given by Note that gravitinos and MSSM particles can be produced in inflaton decay as is discussed in Sec. V A. If the reheating temperature is lower than the mass scale of MSSM particles, the gravitino abundance is given by Eq. (55). We can neglect this contribution at least if m φ 10 12 GeV. On the other hand, if the reheating temperature is higher than the mass scale of MSSM particles, the MSSM particles are soon thermalized and Eq. (90) holds. The gravitino abundance produced in inflaton decay depends on the branching ratio of inflaton decay into gravitino. In this subsection, we assume that the branching into gravitino is so small that its abundance from inflaton decay is much less than the observed DM abundance.
In Fig. 3, we show the constraint on the gravitino mass and the reheating temperature.
We assume that the mass of gluino is 2 TeV, µ = 1 TeV, and m SUSY = m NLSP = 1 TeV.
We also assume that the NLSP is right-handed stau and its annihilation cross section is given by Eq. (95). On the boundary of and inside the shaded region, the DM abundance is equal to and larger than that observed, respectively. In the red (light gray) shaded region, DM is produced from scatterings betweens gluons and gluinos. The upper bound on the blue (dark gray) shaded region is determined by the NLSP abundance produced through inelastic scatterings (that is, Eq. (94) with T RH given by Eq. (89) If the DM abundance is produced by scatterings from thermal bath (the lower boundary of the red (light gray) shaded region in Fig. 3), from Eqs. (89) and (90), the mass of the gravitino is predicted to be The case with n = 4 and λ ∼ 10 −4 requires large gravitino mass. The large gravitino mass contributes to electric dipole moments (EDMs) through supergravity effect in general. For gravitino mass larger than O(100) MeV, suppression of CP phases due to the supergravity effect is necessary [84].
Let us consider the case where the DM abundance is explained by the contribution from inelastic scatterings during the thermalization process (the boundary of the blue (dark gray) shaded region in Fig. 3), focusing on the free-streaming velocity of the gravitino. The NLSP decays into gravitinos with a decay rate given by The decay temperature of the NLSP is thus given by (100) The present-day free-streaming velocity of the gravitino DM is calculated as [27,85,86] v 0 m NLSP /2 m 3/2 9 × 10 −9 m NLSP 5 TeV where T 0 ( 2.3 × 10 −4 eV) is the temperature at the present time, and g * s is the effective number of relativistic degrees of freedom for entropy. The observation of the Lyman-α forest puts an upper bound on the free-streaming velocity as v 0 2.5 × 10 −8 [87] (see Ref. [86] for review). This results in m NLSP 3 TeV from Eq. (102). It is expected that future observations of redshifted 21 cm line would improve the upper bound to v 0 2 × 10 −9 [88].
Hence this scenario would be tested by future observations of redshifted 21 cm line if the mass of the NLSP is less than about 14 TeV.
Finally, we comment on the BBN bound on the gravitino mass [89,90]. Since the NLSP is long-lived in gauge mediation, they may produce high energy hadrons or photons after the BBN epoch. These high energy particles induce hadro-and photo-dissociation processes of light elements, which spoil the success of the BBN scenario. This puts a stringent lower bound on the NLSP mass especially for the case of m 3/2 1 GeV. When the NLSP is the stau, which is a typical case in gauge mediation, the NLSP mass has to be larger than about 700 GeV for m 3/2 10 GeV [90].

D. Gauge mediation with Q-balls
In this subsection, we consider a flat direction other than LH u flat direction. As explained in the previous subsection, the soft mass term for the AD field is absent for a VEV larger than the messenger scale because the transmission of SUSY breaking effect is suppressed for such a large VEV. Thus we obtain the following potential for the AD field [77]: where M s is a messenger mass, and g generically stands for the Standard-Model gauge coupling. A parameter M 2 F is proportional to the VEV of the F component of a gaugesinglet chiral multiplet in the messenger sector as where y is a coupling constant for the interaction between the gauge singlet chiral multiplet and the messenger field. The mass of the gravitino is related to the SUSY breaking F -term as where a factor k is less than one when the messenger sector indirectly couple to the SUSY breaking sector. Hereafter we redefine the combination of yk as k ( 1).
Since the curvature of the potential at the energy scale of |φ osc | is roughly given by respectively. For n = 8 flat direction, the reheating temperature is so small that the success of the BBN is spoiled. Hereafter, we consider only the cases with n = 4 and 6.
The reheating temperature for n = 4 flat direction may be so large that gravitino is produced from the thermal plasma. From Eq. (90), we obtain the gravitino abundance produced from the thermal plasma as Although gravitinos and MSSM particles can be produced from inflaton decay directly in the same way in Sec. V A, we neglect its contribution assuming a large inflaton mass. For n = 6 flat direction, the reheating temperature is smaller than the freeze-out temperature of the NLSP and thus the NSLPs are produced at the reheating epoch as explained in the previous subsection (see Eq. (94)). This is also the case with n = 4 and λ 10 −9 . Since the NLSPs may annihilate and eventually decay into gravitinos, the abundance of the gravitino is again given by the second line of Eq. (97) with the reheating temperature of Eq. (109).
Thus we obtain the abundance of the gravitino as In addition, Q-balls may provide another source of gravitinos as shown below.
For the potential in Eq. (103), there exists a Q-ball solution, approximated to be [91] where ω 0 and φ 0 are given as We omit O(1) logarithmic corrections on Q-ball parameters for simplicity [94]. Since the energy of the Q-ball is calculated as the energy of the Q-ball per unit charge dM Q /dQ is approximated to be ω 0 . 18 If φ osc or φ 0 is less than about the messenger mass M s ( gM 2 F /m φ ), the suppression on the transmission of SUSY breaking effect is absent and the situation is similar to models of gravity mediation [28,92]. We have checked that φ osc and φ 0 is larger than M s in the case we are interested in, if the mass of the AD field m φ is larger than 10 TeV or that of gravitino m 3/2 is less than 10 GeV. 19 If φ 0 M 2 F /m 3/2 , the potential of the AD field is dominated by the soft mass of the form m 2 3/2 |φ| 2 , which is induced by gravity mediated SUSY breaking effect. In this case, a Q-ball solution is known as a "new type Q-ball" [93], which is stable and is a DM candidate for the case of m 3/2 / |b| 1 GeV. In this paper, we focus on the case of the LSP DM and leave that case for a future work. Note that if k ∼ 1, a "new type Q-ball" is never formed.
Now, let us calculate the NLSP abundance from Q-ball decay, which is another source of gravitinos. A detailed analysis has been done in Ref. [31] (see also Ref. [29,30]), where gravitino production from the direct decay of Q-balls via Planck-suppressed operators [27][28][29]31] is taken into account. There, the ellipticity parameter is regarded as a free parameter.
In this paper, we use = m 3/2 /H osc without fine-tuning. In this case, gravitinos directly produced in Q-ball decay is too inefficient to account for the observed DM density.
Q-balls decay into quarks at T = T decay and lose their charges. Since the energy of Q-ball per unit charge, ω 0 , is proportional to Q −1/4 (see Eq. (114)), a Q-ball can decay into the next-to-lightest SUSY particle (NLSP) once its charge decreases down to Q cr ≡ where m NLSP is the mass of the NLSP. 21 Thus we estimate the total number of the NLSP from the decay of each Q-ball as 20 Q-balls with baryon charge less than about 10 18 is completely disappeared by the dissipation effect in the thermal plasma [95,96] (see also Ref. [19]). In this case Q-balls do not affect the DM abundance. 21 In the case ofūdd flat direction, Q-balls does not interact with sleptons and thus cannot decay into them.
In this case, the mass of the NLSP for the condition ω 0 ≶ m NLSP in Eq. (120) has to be replaced with that of the lightest MSSM particle which Q-balls can decay into.
where the branching into the NLSP, Br NLSP , is of the order of 0.01. The first line is for the case that Q-balls can decay into the NLSP from the beginning, while the second one is for the case that Q-balls can decay into the NLSP only after their charges decrease down to Q cr . The combination of Q cr /Q is rewritten as We find that Q cr /Q > 1 for relatively large gravitino mass. The number density of the NLSP from Q-ball decay is then given as If these NLSPs do not annihilate, the gravitino abundance from the Q-ball through the NLSP decay is given as The NLSPs produced from Q-balls are soon thermalized and may annihilate. We assume that the NLSP is right-handed stau and hence its annihilation cross section is given by Eq. (95). The annihilation effect of the NLSP results in the upper bound on the NLSP abundance given by n decay (ann) NLSP s 45 The gravitino abundance from the Q-ball through the NLSP decay is given by ρ To sum up, the gravitino abundance is given by where ρ th 3/2 is absent for the case of n = 6. We show the results in Figs. 4 and 5. Inside blue (dark gray) shaded region, the gravitino abundance produced through inelastic scatterings during the reheating epoch is larger than the observed DM abundance. In the green (middle gray) shaded region, the abundance of DM produced from Q-ball decay mediated by the NLSP (ρ decay 3/2 ) is larger than that observed. Q-balls can decay into the NLSPs from the first time (that is, ω 0 > m NLSP ) below the green line. The lower boundary of the blue (dark gray) and green (middle gray) shaded regions are determined by the annihilation of the NLSP (Eqs. (112) and (127)). In the red (light gray) shaded region, the abundance of DM produced from the thermal plasma (ρ th 3/2 ) is larger than that observed. The DM abundance is consistent with that observed on the boundary of the non-shaded regions.
We find that the gravitino mass has to be larger than O(1) GeV to account for the observed baryon and DM abundance for the case of n = 6. For the case of n = 4, while the DM abundance can be explained by the gravitino production from the thermal plasma in any value of gravitino mass, the gravitino production from Q-ball decay mediated by the NLSP can account for the DM abundance only if m 3/2 O(1) GeV. As we have mentioned, this large value of the gravitino mass in general induces EDMs [84], which will be detected in near future unless CP phases due to supergravity effect is suppressed by some reasons or tunings. In addition, since gravitinos are produced from the NLSP decay, the gravitino DM obtains a sizable free-streaming velocity derived by Eq. (102). This again constraints the mass of the NLSP as m NLSP 3 TeV [87], and also this scenario would be tested by future observations of redshifted 21 cm line if m NLSP 14 TeV [88].

VI. SUMMARY AND CONCLUSIONS
We have proposed scenarios which account for the observed baryon asymmetry and DM density in models of gravity and gauge mediation, taking into account an implication on the energy scale of inflation H inf 10 14 GeV by the recent result of the BICEP2 experiment [1].
We have considered the Affleck-Dine baryogenesis without Hubble induced A-term potentials, which is indeed the case for most models of high-scale inflation in supergravity [9,44,45] and for D-term inflation [6][7][8]. In this case, the Affleck-Dine baryogenesis is excluded by an experimental upper bound on baryonic isocurvature perturbation, unless the initial phase of the AD field is tuned or the VEV of the AD field is as large as the Planck scale during We assume that n = 4 and k = 1. We also assume that λ = 10 −7 (left panel) and 10 −9 (right panel). In the green (middle gray) shaded region, the abundance of DM produced from Q-ball decay mediated by the NLSP (ρ decay 3/2 ) is larger than that observed. In the blue (dark gray) shaded region, the NLSP is produced too much through inelastic scatterings during the reheating epoch, that is, ρ inela 3/2 > ρ obs DM .
In the red (light gray) shaded region, the abundance of DM produced from the thermal plasma (ρ th 3/2 ) is larger than that observed. Here we have assumed that the mass of the gluino is five times larger than that of the NLSP. The lower boundaries of blue (dark gray) and green (middle gray) shaded regions are determined by the annihilation effect of the NLSP. The DM abundance is consistent with that observed at the boundary of the non-shaded regions. Q-balls can decay into the NLSPs from the first time (that is, Q < Q cr ) below the green line.
inflation. We have investigated the latter possibility in detail.
Since the AD field with large VEV results in larger energy density ratio for AD field/inflaton, it indicates lower reheating temperature of the Universe to account for the present baryon density successfully without additional entropy production. Hence, this scenario predicts a relatively low reheating temperature. When the reheating temperature is low, DM is dominantly produced in non-thermal processes. Therefore, it is necessary to investigate the non-thermal processes in detail. In addition, the Affleck-Dine baryogenesis often results in formation of Q-balls, which decay into light SUSY particles as well as quarks at later time. Based on these issues, we have constructed consistent scenarios to account for the observed baryon and DM densities in the cases with and without Q-ball formation in models of gravity and gauge mediation.
In gravity mediation, DM is produced mainly from two sources: direct decay of inflaton into MSSM particles [34][35][36][37][38][39] and inelastic scatterings during reheating process [33,34]. While the former process depends on unknown inflaton mass, the latter process depends only on the reheating temperature and DM mass. From the observation of the DM abundance, we have predicted the DM mass around the TeV scale. In addition, if Q-balls are formed after the Affleck-Dine baryogenesis, they emit gauginos lighter than squarks and thus they can be another source of DM. Interestingly, since the Pauli blocking effect at the Q-ball surface gives a simple relation between the branching into quarks and gauginos from Q-ball decay, we can overcome the baryon-DM coincidence problem in this scenario [32]. We have predicted the mass of the bino, which is the LSP in typical gravity mediation models, as O(1) TeV. These scenarios would be tested by direct detection experiments in the near future [34].
In gauge mediation, gravitino DM is produced by scatterings between gluon and gluino in the thermal bath at the reheating epoch [40]. We have confirmed that the baryon and DM density can be simultaneously explained in the Affleck-Dine baryogenesis when the mass of the gravitino is about O(1) MeV or O(1) GeV, depending on the power of superpotential for the AD field. However, any flat directions other than LH u fragment into Q-balls after the Affleck-Dine baryogenesis. Then Q-balls decay into NLSPs as well as quarks and the NLSPs eventually decay into gravitinos [30,31]. We have found that this scenario can also explain the observed DM density, and have predicted that the gravitino and the NLSP masses are proportional to v χ × dp χ = p χ /E × E/p χ dE = dE, where we use p 2 χ = (ω 0 − E) 2 − m 2 χ . We obtain their flux at the Q-ball surface as n · j χ 1 8π 2 This integral can be performed analytically and we obtain the following correction to the flux given in Eq. (43): Note that while the flux of sparticles is given by the above formula, the flux of quarks takes a different form 8 × ω 3 0 /(96π 2 ), because they are produced through scattering process inside the Q-ball (see the discussion below Eq. (43).