Minimal Supergravity, Inflation, and All That

We consider an inflationary model in the hidden-sector broken supergravity with an effectively large cutoff. The inflaton decay into right-handed neutrinos naturally causes the observed baryon asymmetry of the universe with a reheating temperature low enough to avoid the gravitino overproduction. We emphasize that all the phenomenological requirements from cosmology and particle physics are satisfied in the large-cutoff theory.


Introduction
The landscape of many vacua 1 is a plausible structure in the fundamental theory of physical laws in nature. In particular, this structure is expected as one of the theoretical ingredients to understand the observed small cosmological constant [1]. However, the anthropically allowed region of vacua in the landscape seems too large to be predictive enough in the presence of a variety of couplings. Thus, it is a challenging problem to derive further physical consequences from the landscape of vacua.
The (non-)presence of inflationary dynamics is a promising candidate as the first criterion to select realistic vacua [2]. We can naturally expect that macroscopic universe is realized through inflation from fundamental-scale physics. Moreover, under the dynamics of inflation, mediocrity principle [3] may prefer long-lasting inflations which result in larger-volume universes where more habitable galaxies are produced. In this respect, multiple inflations [4] give a remarkable possibility to be considered [5].
In a recent article [5], we have pointed out that the inflationary dynamics possess a potential to select minimal supergravity as a large-cutoff theory, where the gravitational scale M G is smaller than the cutoff scale M * stemming from the fundamental theory. Such a large-cutoff supergravity naturally causes multiple slow-roll inflations, which possibly meet mediocrity principle.
The large-cutoff theory is also attractive from the viewpoint of particle-physics phenomenology: First of all, the suppression of the flavor-changing neutral currents is automatic in the large-cutoff theory, since all of the higher-dimensional operators are suppressed by the large cutoff M * except for the genuine gravitational interactions. In detail, the large-cutoff supergravity predicts a hierarchical spectrum [5] of supersymmetric particles as m 0 ≫ |M i |, where m 0 is the universal soft mass for sfermions and M i the gaugino masses (i = 1, 2, 3). Thus, the current chargino mass bound suggests heavy sfermions at several TeV. Such a soft mass parameter belongs to the parabolic [5] or hyperbolic [6] regime allowed for a given µ parameter. Indeed the recent detailed analysis [5] has confirmed that the region with large sfermion masses along the small-µ-parameter curve continued from the focus point [7] is consistent with the electroweak symmetry breaking [8]. In the region of heavy sfermions and light gauginos (an order of magnitude lighter than the sfermions), the constraint from CP violation is rather weak and even order one CP-violating phases are allowed for m 0 > ∼ 10TeV [9]. Furthermore, the lightest supersymmetric particle can explain the dark matter density of the present universe in the above mass region for supersymmetric particles [10].
In this paper, we discuss a minimal new inflation model 2 as an example in the framework of the large-cutoff supergravity with emphasis on baryon asymmetry generated by leptogenesis [11] to complete a model of the large-cutoff hypothesis. 3 We claim that all the phenomenological requirements from cosmology and particle physics are satisfied in a certain parameter region of the large-cutoff theory.

Supergravity new inflation
We adopt a new inflation model considered in Ref. [13,14]. As an effective field theory for an inflaton chiral superfieldφ, the superpotential is given by for n ≥ 3 and the Kähler potential is given by where we have taken the unit with the reduced Planck scale M G ≃ 2.4 × 10 18 GeV equal to one. The positive parametersK,g, andk are of orders 1, 10 −(n−2) , and 10 −2 , respectively, for our large-cutoff hypothesis M * ≃ 10M G . The tiny scaleṽ 2 > 0 can be generated dynamically [15] and the ellipsis denotes higher-dimensional operators which may be neglected in the following analysis.
For the canonically normalized field φ = √Kφ , the superpotential is given by [16] and the Kähler potential is given by where we have definedṽ The effective potential for the lowest component of φ is given by where Thus, the potential of the inflaton field ϕ = √ 2Re φ is approximately given by for the inflationary period near the origin ϕ > 0.
The inflationary regime is determined by the slow-roll condition [17] ǫ(ϕ) = 1 2 where For the potential Eq.(8), we obtain The slow-roll condition Eq.(9) is satisfied for ϕ ≤ ϕ f where which yields the value of the inflaton field at the end of inflation.
The value ϕ Ne of the inflaton corresponding to the e-fold number N e is given by This leads to Hence the spectral index of the density fluctuations is given by [14] Note that the spectral index does not depend on v 2 and g explicitly. We show the k dependence of the spectral index n s for n = 4 and N e = 45, 50, 55 [13,14] in Fig.1.
Now we proceed to determine the inflation scale v from the density fluctuations. The amplitude of primordial density fluctuations is given by where ϕ N 0 is the value of inflaton field at the epoch of the present-horizon exit. Thus we Owing to the COBE normalization for n = 4, N 0 ≃ 50, and k < ∼ 0.01, where C(k, N 0 ) is a function of order unity. On the other hand, the e-fold number of the present horizon is also given by where H denotes the Hubble scale at the horizon exit and T R the reheating temperature.
By means of Eq. (17), (19), and (22), we can determine v and N 0 from g, k, and T R .
For n = 4, g ∼ 0.1, k ∼ 0.01, and T R ∼ 10 5−9 GeV, the inflation scale v is given by O(10 12 )GeV, and the e-fold number N 0 of the present horizon is given by 47.6 − 50.6. In Fig.2, we show the k dependence of the spectral index n s for the reheating temperature T R = 10 5 , 10 7 , and 10 9 GeV. We conclude that the implicationg ∼k ∼ 0.01 of the largecutoff hypothesis 4 is consistent with an experimental value n s = 0.95 ± 0.02 [18] of the spectral index for a wide range of the reheating temperature. 5 4 For instance, Eq.(5) yields g = 0.1 and k = 0.01 forK = 0.5,g = 0.018 andk = 0.0025. 5 The inflaton as a massless scalar field in the de Sitter background has quantum fluctuations whose amplitude is given by ∆ϕ ∼ H/(2π). Thus the amplitude ∆ϕ at ϕ = ϕ N0 is given by For n = 4, g ∼ 0.1, k ∼ 0.01, and T R ∼ 10 5−9 GeV, the fluctuation amplitude ∆ϕ| ϕN 0 takes a value of order 10 −6 ϕ N0 , which is much less than the mean-field value ϕ N0 to justify the above slow-roll analysis.

The gravitino mass
In the previous section, we have confirmed that the new inflation model in the large-cutoff hypothesis is consistent with the cosmological observations. In this section, we discuss the gravitino problem under such an inflationary scenario.
As considered in Ref. [13], we assume that the positive energy Λ 4 SUSY of the SUSY breaking is dominantly canceled out by the negative energy at the inflaton potential minimum. Namely we impose where φ 0 is the minimum point of φ in Eq.(6).
Then we obtain the gravitino mass as The value of φ 0 is approximately given by Consequently the gravitino mass is given by More precisely, by means of Eq. (19) and (22), the gravitino mass can be expressed as a function of g, k, and T R , although the dependence on T R is very weak, as can be seen from Eq. (19) and (22). The result is shown in Fig.3. For g < ∼ 0.2 and k < ∼ 0.035, the gravitino mass is larger than 4TeV, which may avoid the gravitino overproduction for a reheating temperature T R ∼ 10 6−7 GeV [19].
In contrast, the sfermion soft mass is given as m 0 ≃ m 3/2 if no D-term contributes to the SUSY breaking. Thus, m 0 < 10TeV implies g > ∼ 0.07 for k < ∼ 0.035.

Reheating for baryogenesis
Now we are ready to consider the baryon asymmetry in the present new inflation model with the large cutoff.
We assume the baryon asymmetry is generated by leptogenesis [11] through nonthermal production of right-handed neutrinos, as investigated in Ref. [10,11], which pro-vides a numerical estimate Here m N , m φ , and m ν 3 are the masses of the right-handed neutrino N, the inflaton φ and the heaviest (active) neutrino, respectively. The phase δ ef f is the effective CP phase defined in Ref. [11] and tan β is the ratio of the vacuum expectation value of up-and down-type Higgs bosons in the MSSM. The reheating temperature is given by where Γ φ is the decay width of the inflaton and g * is the effective number of massless degrees of freedom to be taken as g * = 228.75 numerically. Note that the inflaton mass in our new inflation model also weakly depends on the k and the reheating temperature T R , as is the case for the gravitino mass in Eq.(26).
Let us introduce the following superpotential interaction as the dominant source of the N production: 6 where h is a positive parameter of the order of the inflaton self-coupling g. 7 The coupling Eq.(30) gives a decay width From this decay width the reheating temperature after inflation for n = 4 is given by 8 6 The inflaton field is also expected to decay through couplings with light fields ψ i in the Kähler potential such as i c i |φ| 2 |ψ i | 2 . However, the decay width Γ ∼ i |c i | 2 φ 2 0 m 3 φ through these couplings is so small that we neglect such contributions. 7 Here, we assign the same charge for φ and N under Z 2n R-symmetry, while we assign the matter parity + for φ and − for N . Hence we expect the presence of such operators as φ n−3 N 4 in addition to Eq.(30). We do not include such operators since the operator Eq.(30) with the smallest number of N dominates the reheating and leptogenesis. 8 The cross term between φ n−1 N 2 and v 2 φ in the superpotential gives a comparable decay width. We neglect this contribution since it does not essentially affect our conclusions.
where we have omitted the weak dependence on k in Eq. (21). Therefore, the reheating temperature T R ∼ 10 6−7 GeV is typical in this model. As mentioned above, this reheating temperature is low enough to avoid the gravitino overproduction.
Note that the operator Eq.(30) also gives the Majorana mass to the neutrino: Thus the mass inequality 2m N < m φ , namely, is satisfied with a typical parameter set g ∼ h. This is appropriate for the non-thermal production of neutrinos which leads to the non-thermal leptogenesis.
Based on the above setup, we now estimate the baryon asymmetry due to the decay of inflaton 9 in our model as a function of the couplings g, k, and the reheating temperature The baryon asymmetry is determined by four independent parameters g, k, v, and h. In terms of the observed density fluctuations, we can represent v with the other parameters. We further use the reheating temperature as an input parameter instead of h by means of Eq.(28), (29) and (31): Then the baryon asymmetry n B /s is given in terms of g, k, and T R by where h(g, k, T R ) is given by Eq. (35) with v determined by Eq. (19) and (22).
In Fig.4, we plot the contours of m 3/2 and ( n B s )/( n B s ) 0 for T R = 4 × 10 6 GeV, m ν 3 = 0.05eV, δ ef f = 1, sin β = 1, where (n B /s) 0 is the baryon asymmetry of the universe suggested by WMAP [21]: We note that the baryon asymmetry and the gravitino mass for different reheating temperatures can also be seen from Fig.4: The baryon asymmetry is proportional to the square of the reheating temperature T 2 R , since the coupling h is approximately proportional to the reheating temperature. As for the gravitino mass, its value is almost independent of T R , since v is almost independent of T R . This figure shows that the sufficient baryon asymmetry is produced in a typical parameter region of the large-cutoff hypothesis: k ∼ 0.01, g ∼ 0.01 − 0.1, and T R ∼ 10 6 GeV, which turns out to be low enough to avoid the gravitino overproduction. Thus it is revealed that the large-cutoff hypothesis is also consistent with the observed baryon asymmetry.

Discussion
We have studied the large-cutoff hypothesis from the viewpoint of cosmology. We first confirmed that the spectral index in the new inflation model has an upper bound n s < ∼ 0.95 (see Ref. [14]) and the large-cutoff hypothesis implies its boundary value, which remarkably agrees with the present experimental suggestion n s = 0.95 ± 0.02 [18]. Secondly, we found a concrete setup where the sufficient baryon asymmetry can be produced via nonthermal leptogenesis with the reheating temperature low enough to avoid the gravitino overproduction in a typical parameter region of large-cutoff hypothesis.
We again emphasize that the large cut-off hypothesis has several advantages from the viewpoint of particle-physics phenomenology. It solves the FCNC problem and produces the mass spectrum m 0 ∼ 10m 1/2 ∼ 10µ, which yields the correct electroweak symmetry breaking [5]. Furthermore, the spectrum realized in the large-cutoff hypothesis accommodates the appropriate amount of the dark matter density [10].
We also mention CP violations in the visible-sector supersymmetric standard model as a sensitive low-energy probe of the supersymmetry breaking. Phases of the theory would be limited severely if the scalar masses were to be less than the TeV scale. In contrast, for m 0 ∼ 10TeV, such a constraint is far milder, with the very heavy scalar masses expected to be realized in the large-cutoff hypothesis from the viewpoint of electroweak symmetry breaking and dark matter, as mentioned above.
The heavy scalar masses are remarkably consistent with the cosmological constraint, as we saw in this paper. Thus we conclude that the large-cutoff theory with the supergravity new inflation and non-thermal leptogenesis is consistent with all the phenomenological requirements from cosmology and particle physics.
Finally we comment on other types of inflations. The presence of the large cutoff seems advantageous for other inflationary models such as hybrid inflation and chaotic inflation.
In particular, large-field inflations imply the presence of a larger scale (see Ref. [22]) than the reduced Planck scale. In fact, we suspect that multiple inflations may be so generic as to include various types of inflations as their components, whose slow-roll conditions are realized by the large-cutoff mechanism [5].

Appendix: Another source of baryon and gravitino
In section 4, we put aside the baryon asymmetry and the gravitino produced through the coherent oscillation of right-handed sneutrino. Here we argue that this contribution can be small enough to be neglected.
Firstly we explain the motion of right-handed sneutrino field which is the source of the baryon asymmetry and gravitino. During inflation, the right-handed sneutrino is fixed at the origin due to the Hubble mass. After the inflaton starts to roll down to the vacuum, the mass of the right-handed sneutrino changes along the motion trajectory of the inflaton.
As the oscillation energy of the inflaton decreases, the origin of right-handed sneutrino becomes unstable, and right-handed sneutrino also starts oscillation. Then the decay of right-handed sneutrino becomes significant. 10 The baryon asymmetry and gravitino are provided by the decay of this right-handed sneutrino [20].
Let us estimate the yields of the baryon asymmetry and gravitino provided through the coherent oscillation of right-handed sneutrino. As mentioned above, the decay of right-handed sneutrino becomes significant when the motion of right-handed sneutrino is induced by that of inflaton. Then the yields of the baryon asymmetry n N B /s and the grvitino number n N 3/2 /s produced at the decay time of right-handed sneutrino are given by Here, ε denotes the CP-asymmetry in right-handed sneutrino decay defined in Ref. [11], T N is the temperature of radiation produced by right-handed sneutrino decay, Y φ 3/2 is the yield of gravitino produced by inflaton decay, and ρ N is the energy of the right-handed sneutrino at the right-handed sneutrino decay.
After the inflaton decay, these yields are diluted by the dilution factor ∆ estimated as where ρ φ is the energy of the inflaton at the right-handed sneutrino decay. Thus n N B /s and n N 3/2 /s after the inflaton decay are given by These values are smaller than the yields produced at inflaton decay for ρ N ≪ ρ φ (see Eq.(36)), which we assume in the main text.
In fact, we checked that ρ N ≪ ρ φ is realized in a typical parameter region by solving the equations of motion numerically for n = 4. We note a possibility that parametric resonance occurs in specific points, and the energy of right-handed sneutrino ρ N becomes comparable to that of inflaton ρ φ in such a case.