Inflaton as the Affleck-Dine Baryogenesis Field in Hilltop Supernatural Inflation

In this paper, we investigate the parameter space in the framework of hilltop supernatural inflation in which the inflaton field can play the role of Affleck-Dine (AD) field to produce successful baryogenesis. The suitable value of reheating temperature could coincide with the reheating temperature required to produce LSP dark matter. The baryon isocurvature perturbation is shown to be negligible. We consider $p=3$, $p=4$ and $p=6$ type III hilltop inflation and discuss how to connect the models to supersymmetric theories.

In this paper, we investigate the parameter space in the framework of hilltop supernatural inflation in which the inflaton field can play the role of Affleck-Dine (AD) field to produce successful baryogenesis. The suitable value of reheating temperature could coincide with the reheating temperature required to produce LSP dark matter. The baryon isocurvature perturbation is shown to be negligible. We consider p = 3, p = 4 and p = 6 type III hilltop inflation and discuss how to connect the models to supersymmetric theories.

I. INTRODUCTION
Inflation Refs. [1][2][3] (see e.g. Ref. [4] for a textbook review) is a vacuum-dominated epoch in the early Universe when the scale factor grew exponentially. This scenario is used to set the initial condition for the hot big bang model and to provide the primordial density perturbation as the seed of structure formation. In the framework of slow-roll inflation, the slow-roll parameters are defined by where M P = 2.4 × 10 18 GeV is the reduced Planck mass. The spectral index can be expressed in terms of the slow-roll parameters as The latest Planck 2018 data analysis Ref. [5] provide n s = 0.9649 ± 0.0042 (68% CL) and we will assume n s = 0.96 (because in our model r ∼ 0, this would more or less lower the preferable n s .) in this paper. The spectrum is given by By using slow-roll approximation, the number of e-folds is given by From observation P 1/2 R ≃ 5 × 10 −5 at N ≃ 60 (we call this CMB normalization). Hilltop supernatural inflation Refs. [6][7][8][9] is very interesting because it can produce a red spectrum for primoridal curvature perturbation as observed from the latest Cosmic Microwave Background (CMB) probe. It can also evade both thermal and non-thermal gravitino problem Ref. [9], produce primordial black holes Ref. [17] or induced gravity waves Ref. [18].
In this paper, we show that the inflaton field in hilltop supernatural inflation can play the role of the Affleck-Dine field such that successful baryogenesis can be produced after inflation.

II. HILLTOP SUPERNATURAL INFLATION
Type III hilltop inflation can be realized by a modification of hybrid inflation. The potential for a conventional hybrid inflation is given by 1 where ψ r is the inflaton field and φ r is the waterfall field. The effective mass of the waterfall field is During inflation, the field value of ψ r gives a large positive mass to φ r therefore it is trapped to φ r = 0 and the potential during inflation is of the form where V 0 = κ 2 Λ 4 . The end of inflation is determined by m 2 φ = 0 when the watefall field starts to become tachyonic which implies The idea of supernatural inflation Ref. [19] is to consider TeV scale SUSY breaking which can be realized by V 0 = M 4 S where M S ∼ 10 11 GeV is the gravity mediated SUSY breaking scale and m ψr ∼ TeV is the soft mass. Unfortunately, for original supernatural inflation, because the potential in Eq. (7) is concave upward, a blue spectral index n s > 1 is obtained.
We now consider a term in the superpotential, for p different superfields ψ p , and D-flat direction for ψ i (|ψ 1 | = |ψ 2 | = · · · = |ψ p |). The real inflaton field is a linear combination of the scalar component of the p fields: The scalar potential during the inflation in terms of the kinetic normalized field ψ is given as where m 2 ψ = m 2 ψ i /p for the average of the SUSY breaking squared masses of ψ i , and A is a SUSY breaking parameter. The SUSY breaking mass m ψ and A are typically of the order of the gravitino mass m 3/2 Refs. [20,21]. The detailed values of these parameters depend on specific mechanisms of SUSY breaking.
We will neglect the last term on the right hand side and approximate the potential as Ref. [22] where We assume that aA is real and positive in the convention that all the ψ i configuration is aligned to be real The integral of Eq. (4) can be done analytically and we obtain x Substitute these into Eqs. (1), (2), and (3), we have .
From the above equations we can obtain We are going to need the field values during inflation and at the end of inflation for the following discussion. From Eqs. (14), (15) and (17), we can obtain and ψ ψ end After inflation, the inflaton field starts to oscillate and eventually decays when the magnitude of its decay width reaches the Hubble parameter. An interesting question is whether the inflaton can play the role of Afflect-Dine (AD) field Refs. [21,23]. The possibility was investigated for supernatural inflation in Ref. [19] but it was shown that it does not work for the inflaton field in that model. A quadratic potential of the inflaton field for chaotic inflation which plays the role of an AD field after inflation is investigated in Refs. [24][25][26][27]. In [28], a similar idea is realized in a large field inflation model where the inflaton is a linear combination of right-handed sneutrino fields is considered.
Recently, [29] considered the inflaton field as the Affleck-Dine field by adding a nonminimal coupling to gravity in order to evade the severe experimental constraints from tensor to scalar ratio. We investigate the case for hilltop supernatural inflation in the framework of small field inflation in this paper.
Our inflaton field is assumed to be a flat direction, therefore it may play the role of Afflect-Dine field and produce baryon asymmetry if it carries non-zero B − L 2 . We consider the cases p = 3, p = 4 and p = 6. 3 If the inflaton carries non-zero baryon (or lepton) number, baryon number density is produced when the inflaton start to oscillate. The oscillation in general is accompanied with a spiral motion due to the potential for the angular direction. The baryon number density n B is given by the angular motion of ψ as where q is the baryon number carried by the AD field, and we have defined ψ = |ψ|e iθ . The evolution of the baryon number density is given bẏ where In order to solve the above equation, we multiply both sides of the equality by the cube of the scale factor R(t) 3 and integrating with respect to the time t to obtain where q i is a charge of ψ i . The integration is done in a short period of time from the end of inflation to the onset of spiral motion of the AD field, because we have assumed that baryon number has been diluted during inflation and the contribution to the integral is small after the onset of spiral motion since the sign of Im[ψ p ] is changing rapidly and the amplitute of |ψ| would shrink due to its decay into other particles (with the baryon number conserved). Just after the end of inflation, the universe is matter dominated and the scale factor goes like R(t) ∝ t 2/3 . Unlike the common case where the amplitude of the AD field |ψ| p decreases with time as H p/(p−2) ∝ r −p/(p−2) due to its trapping by a large negative Hubble induced mass, our AD field is the inflaton field, therefore its energy density is dominated by the oscillation of the quadratic potential which behaves like matter Hence the integrand is proportional to t (2−p) . By using ψ = |ψ|e iθ the integration gives 4 Note that if sin[pθ sp + arg(A)] = 0, there will be no baryon number generated. Here we will assume where the Hubble parameter is evaluated at the end of inflation when the inflaton (as the AD field) starts to oscillate. By using the definition of λ from Eq. (13), the above equation can be written as In general, it can be expressed as when the inflaton (AD field) starts to oscillate, where q A is the charge carried by V A . For simplicity, in the following we will set q A ∼ 1. In our model, the mechanism of inflation determines the initial conditions of the AD field and there is no need to assume it in some ad hoc way.
The baryon number density will then be diluted due to the expansion of the universe. When reheating happens, Γ ∼ H and hence the reheating temperature is given by T R ∼ √ ΓM P . At reheating, the baryon number density is given by 4 Here the result is obtained for p = 3. When p = 3, a factor ∝ ln M P φsp is obtained which is not very sensitive to the value of φsp.
The baryon asymmetry Y = n B /s = n B /(2π 2 g * T 3 R /45) = 0.9 × 10 −10 as required by BBN and CMB gives The rehating temperature is given by We assume the SUSY breaking scale such that m ψ and A are between TeV and 100 TeV. We will see that CMB constraints requires η 0 ≡ m 2 ψ /V 0 ∼ 0.01 or so, therefore 10 −26 As we will see in the following section, in general higher SUSY breaking scale makes baryogenesis easier, but we do not wish to deviate from TeV scale too much. This is the reason behind choosing this range of energy scales.
In the analysis so far, we were assuming that the AD condensate evolves homogeneously after it formed. In general, there is a possibility that the AD condensate becomes unstable with respect to spacial perturbations and turns into non-topological solitons called Q-balls Refs. [31][32][33][34][35]. If Q-balls are formed, our scenario for the evolution of the universe may need to be modified. Q-balls are not formed if m ψ ≫ m 1/2 , where m 1/2 is the mass scale for the gauginos Ref. [36]. In order for the Q-balls to be formed, it is necessary that the potential for the AD scalar is flatter than |ψ| 2 at large field values. After taking account the one-loop correction, the potential for the AD scalar looks like where the coefficient K is determined from the renormalization group equations, see e.g. Refs. [34,37,38]. Loops containing gauginos make a negative contribution proportional to m 2 1/2 , while loops containing sfermions make a positive contribution proportional to m 2 AD . Thus when the spectrum is such that the gauginos are much lighter than the sfermions i.e. m AD ≫ m 1/2 , K is likely to be positive and thus Q-balls will not be formed. More complete analysis of the Q-balls is beyond the scope of the current paper and is left to the future investigations.

A. p = 3 case
For p = 3, we have From Eq. For the case V 1/2 0 = 10 −12 which corresponds to 100 TeV SUSY breaking scale, the parameter λ as a function of η 0 is plotted in Fig. 1. We can see from the plot that a small value of λ and a value of η 0 which is not far from unity is required. Therefore even if there is a Hubble induced mass, as long as it is not very large, our results is not affected. As we will see, this also applys to p = 4 and p = 6 cases.
5 It seems we can make λ arbitrarily small by fine-tuning η 0 to approach η 0 = 0.02 because when η 0 = 0.02, λ = 0 as can be seen from Eq. (35). However, this is not correct since when λ = 0 the inflaton potential is concave upward and the condition ns = 0.96 cannot be achieved. The reason behind this discrepancy is that in this case the small field approximation breaks down and Eq. (35) is no longer valid.
we can obtain the required reheating temperature for successful AD baryogenesis as a function of the inflaton mass m ψ . The result for p = 3 case is shown in Fig. 3. We also include upper and lower bounds for the reheating temperature correspond respectively to thermal and non-thermal gravitino production [9][10][11][12][13][14][15][16]. Here it is assumed that m ψ = m 3/2 . In SUSY breaking scenarios where the gravitino mass is much larger than squark / slepton masses, the plots of the constraints would shift to the left and tends to become weaker. Therefore the constraint we use here is conservative. As we can see from the figure, generally speaking, for higher soft mass and SUSY breaking scale, the required reheating temperature is lower. Note that if we choose a larger η 0 which corresponds to a smaller ψ end /M P , we will need a larger reheating temperature. On the other hand, if we choose η 0 approaches 0.02, ψ end /M P can be bigger and a lower reheating temperature is resulted. However, as η 0 approaches 0.02, ψ end /M P becomes sensitive to η 0 . As can be seen from Eqs. (35) and (47), η 0 = 0.02 cannot be achieved because the formula becomes singular and our small field approximation breaks down.    Note that for p = 4, λ does not depend on V 0 . The parameter λ as a function of η 0 is plotted in Fig. 4. We can see from the plot that a small value of λ is required. However, since λ is the ratio of SUSY breaking A-term and the Planck mass, its value is natually small.
From Eq. (32), for p = 4, we have From Eqs. (19) and (20), we have and For the case V The result for p = 4 case is shown in Fig. 6. As we can see from the figure, the reheating temperature predicted for p = 4 appears to be higher than that for p = 3. In addition, for higher soft mass and SUSY breaking scale, the required reheating temperature is lower. Note that if we choose a larger η 0 which corresponds to a smaller ψ end /M P , we will need a larger reheating temperature. On the other hand, if we choose η 0 approaches 0.01, ψ end /M P can be bigger and a lower reheating temperature is resulted. However, as η 0 approaches 0.01, ψ end /M P becomes sensitive to η 0 . As can be seen from Eqs. (40) and (47) η 0 = 0.01 cannot be achieved because the formula becomes singular and our small field approximation breaks down.

C. p = 6 case
For p = 6, we have For the case V 1/2 0 = 10 −12 which corresponds to 100 TeV SUSY breaking scale, the parameter λ as a function of η 0 is plotted in Fig. 7. From the plot we can see that λ is not necessarily a small number and can be quite large depends on η 0 . It may be interesting to note that λ ∼ O(1) can be achieved. On the other hand, from Eq. (44), we can see that it seems for TeV < A < 100 TeV, λ has to be very small unless coupling a is very large 6 . From Eq. (32), for p = 6, we have From Eqs. (19) and (20) , we can obtain the required reheating temperature for successful AD baryogenesis as a function of the inflaton mass m ψ . The result for p = 6 case is shown in Fig. 9. For p = 6, it is required that η 0 > 0.005.

IV. BARYON ISOCURVATURE PERTURBATION
Since the AD field (as the inflaton in our model) is a complex field, it is possible that during inflation the quantum fluctuations of the phase would induce isocurvature perturbation. The fluctuations of 6 A larger effective λ may be obtained if we introduce a singlet χ with the superpotential W ψ,χ = a M P ψ 3 χ + 1 2 Mχχ 2 . We will leave detailed model building in our future work. Here it is assumed that m ψ = m 3/2 . If m ψ < m 3/2 , the gravitino bound shifts to the left and becomes weaker or even negligible. We also include a bound of LSP production from Eq. (55).
the phase of the AD field is given by where H and ψ are the values obtained during inflation at N = 60. The baryon isocurvature perturbation is defined as where ρ B and ρ γ are the energy densities of the baryons and photons. From Eq. (25), we have In the second equality of the above equation, we assume cot[pθ sp +arg(A)] ∼ O(1) . From the lastest PLANCK 2018 data for dark matter isocurvature perturbation S cγ [5], where β iso < 0.038 is used. We can obtain S bγ 5.33 × 10 −5 , where we have used Ω c /Ω b = 5.33. By using Eq. (51) and V If the reheating temperature required for successful AD baryogenesis is high, and the parameter space is such that thermal gravitino problem is evaded, e.g. large gravitino mass, there is another upper bound for the reheating temperature from LSP production given by T R < 2 × 10 10 GeV 100 GeV m LSP .
It suggests an interesting possibility that when the reheating temperature required for successful baryogenesis is high, dark matter can also be generated. There is another possible way to generate dark matter in our model. For higher scale SUSY breaking, the required reheating temperature is lower, this helps to evade gravitino problem and if we choose a 100 TeV inflaton mass, the decay could lead to a non-thermal origin for dark matter Refs. [39][40][41][42][43][44].

VI. CONCLUSION
In this paper, we have shown that it is possible for the inflaton field to play the role of the Affleck-Dine field to produce successful baryogenesis in the model of hilltop supernatural inflation. Depends on the reheating temperature, both the thermal and non-thermal gravitino bound can be satisfied. We calculated the baryon isocurvature perturbation and found that it can be neglected. We also explore the interesting possibility that the reheating temperature for successful baryogenesis can also be resposible for LSP dark matter production. If the inflaton mass is around 100 TeV, dark matter could have been produced non-thermally via inflaton decay.