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Spontaneous baryogenesis in spiral inflation

We examined the possibility of spontaneous baryogenesis driven by the inflaton in the scenario of \textit{spiral inflation}, and found the parametric dependence of the late-time baryon number asymmetry. As a result, it is shown that, depending on the effective coupling of baryon/lepton number violating operators, it is possible to obtain the right amount of asymmetry even in the presence of a matter-domination era as long as such era is relatively short. In a part of the parameter space, the required expansion rate during inflation is close to the current upper-bound, and hence can be probed in the near future experiments.


Introduction
It is known that our visible world is made of either matter or anti-matter only, depending on how we define them. Observations indicate that the asymmetry between matter and anti-matter in terms of the ratio of baryon-to-entropy density is about 10 −10 [1,2]. Such an asymmetry could be an initial condition of the universe evolving to our present universe. However, in the presence of inflation [3][4][5] which is now believed to be a crucial ingredient of the thermal history of the universe at a very early epoch, typically well before the conventional electroweak phase transition, an initial asymmetry which might have existed is expected to be diluted to a totally negligible level, and hence there should be a process, called baryogenesis, able to generate an asymmetry after inflation.
Typically, when it works through particle-interactions, a baryo/leptogenesis mechanism is required to satisfy the so called Sakharov conditions [6], i.e., (i) baryon(B)/lepton(L) number violating process, (ii) C-and CP -violation, and (iii) out-of-equilibrium decay of particles producing baryon/lepton number. However, when the dynamics of a background field is involved, the above conditions can be relaxed. Spontaneous baryogenesis [7,8] (see also Ref. [9] for cosmological aspects), is a specially interesting case as the asymetry can be generated in equilibrium. The key feature of spontaneous baryogenesis is that, when a baryon/lepton current is coupled to a background evolution of a field, the time-dependence of the field can provide an effective chemical potential associated with baryon/lepton number. As a result, in the presence of B-or L-violating processes in thermal equilibrium, an asymmetry of B-or L-number can be generated even in the thermal bath. The main question in this novel scenario is the identity of the background field and its precise nature which should allow B-or L-violating processes in thermal equilibrium. In principle, the background field can be any scalar field which has a sizable time-evolution at the epoch of baryo/leptogenesis as long as the symmetries of the theory allow a time-dependent coupling of the field to the baryonic/leptonic current.
An additional possibility related to spontaneous baryogenesis is the production of an asymmetry from the decay of the oscillating scalar field associated with spontaneous baryogenesis. Typically, the nature of the deriving field for spontaneous baryogenesis is an angular degree of freedom of a complex field. Hence, when the complex field carries a charge, a motion of the phase field implies an asymmetry of the charge. Even though an oscillation of the angular degree with respect to a true vacuum can not provide an asymmetry with a definite sign, its decay can results in a net baryon/lepton asymmetry with a specific sign, thanks to the expansion of the universe [7] 1 . A model-dependent question is if the net asymmetry can be large enough to match observations. On the other hand, Spiral inflation [13,14] was proposed as a phenomenological scenario of inflation circumventing the flatness and trans-Planckian issues of the inflaton potential. One of the key features of such a scenario is that the inflaton trajectory is spiraling-out and inflation ends by a waterfall-like drop. Such a spiral motion is something similar to an angular motion of a complex field. Hence, a natural question is whether the inflaton in the Spiral inflation scenario can be responsible for generating the right amount of baryon number asymmetry through either spontaneous baryogenesis or the second possibility mentioned in the previous paragraph, i.e. through its decays in an expanding Universe.
In this work, within the framework of spiral inflation, we show that a right amount of baryon number asymmetry can be achieved in a certain parameter space not by spontaneous baryogenesis but by the remnant of the decays of the inflaton. This paper is organized as follows. In section 2, the general form for the potential for spiral inflation is introduced. In section 3, spiral inflation is described, including the post-inflation behavior of the field configuration. In section 4, the genesis of charge asymmetry by means of the inflaton after inflation through either spontaneous baryogenesis or decays of inflaton is considered, searching for the parameter space for a right amount of baryon number asymmetry at the present universe. In section 5, conclusions are drawn.

The model
The general form of the potential responsible for spiral infaltion can be written as and p, q > 0. We may take p and q to be non-negative integers, and consider the case of p ≥ 4 and 0 < q ≤ 2 which might be theoretically plausible. Clearly φ and θ can be considered as the modulus and the phase of a complex field given by Φ = φe iθ / √ 2. This potential is a hilltop potential having a trench spiraling-out from the hilltop.
Starting from the hilltop, the field configuration is expected to follow closely the minimum of the trench as long as the curvature along the orthogonal direction is large enough, satisfying V = 0 with ' ' denoting a derivative with respect to φ. Hence, from one finds that along the trajectory leading to V dφ = − ∂V m /∂θ dθ (2.10) Therefore, for |V φ | |V m | which is expected to be satisfied in the vicinity of the minimum of the trench, We denote the trajectory following the minimum of trench and the direction orthogonal to the trajectory as I and ψ, respectively. Then, an infinitesimal displacement along I can be written as with the unit vectors along I and ψ given by, so that the directional derivatives are found to be respectivcely Hence, one finds and where the elements of the mass-square matrix (M 2 ) are found to be Spiral motion ends as the field configuration leaves the trench, falling along the φ direction at a point satisfying There are two solutions of Eq. (2.25), denoted as φ e and φ r , The smaller one is φ e , the end point of the slow-roll inflation. For p ≥ 4 which is the case we are interested in, f e 1 unless φ e is quite close to φ 0 . In this case, which is determined by Λ for a given choice of the other parameters. Note that, as long as κ 1, the approximation in Eq. (2.26) is good enough for our purpose.
The other solution, the largest one, φ r , represents the location of the re-trapping of the field in the trench. It satisfies When φ < φ e , inflation consistent with observations (in slow-roll regime) takes place due to the gentle spiral dynamics of the field configuration. For where qh 1 was used. Hence, using Eq. (2.9), we find Note that in order for I to follow closely the minimum of the trench during inflation, the mass scale along ψ should be large enough or at least comparable to the expansion rate, that is, m 2 ψ /3H 2 * 1 which constrains κ to satisfy where ' * ' denotes a quantity associated with a pivot scale of observations. It turns out that this constraint is easily satisfied in the parameter space we are interested in. The slow-roll parameters are given by where we used f (φ) 1 and defined a function g(M, φ 0 ) which will prove to be convenient later. At φ = φ * associated with, for example, Planck pivot scale, f (φ * ) 1 is expected (see Eq. (2.26)). Hence, * |η * | as long as p/ [p − (q + 1)] ∼ O(1), and η * is nearly fixed by the observed spectral index of the density power-spectrum as n obs s 1 + 2η * in order to match observations. The power spectrum is given by where P obs R = 2.1 × 10 −9 [2] is the observed amplitude of the density power spectrum. If Eq. (3.1) is satisfied for most of the region of (φ * , φ e ), the number of e-foldings generated is found to be For a given comoving scale k * and the present horizon k 0 , observations require such number of e-foldings to be N obs e (k * ) = 62 + ln where we took V e = V * and ρ R is the radiation energy density when reheating is efficient enough to recover a radiation-dominant universe. Specifically, we take ρ R to be the energy density of the universe when H = (2/3)Γ ψ with Γ ψ being the decay rate of ψ-particles. For a given set of (p, q), if the model-dependent couplings of φ and θ to other matter fields are fixed, the model parameters which still remain free are : Also, there are three observable constraints: which can be re-used to find φ * , using Eq. (3.10). Note that κ is treated as a free parameter in this case although it should satisfy Eq. (3.6). Also, H * depends dominantly on κ due to the factor 1/|η * | in front of ln κ in Eq. (3.15). In Fig. 1, we show the allowed parameter spaces. As can be clearly seen in the figure, the φ 0 dependence of each parameter except M is quite weak, so, for simplicity we took φ 0 = 10 17 GeV as a representative value.   1), we ignore such a dependence for simplicity. Such an assumption is equivalent to setting κ = 0 in Eq. (3.21) when N obs e is estimated. Eq. (3.16) can be satisfied by adjusting the ratio Λ 4 /V 0 as a free parameter replacing Λ. In Fig. 2, the parameter space matching inflationary observables is depicted for (p, q) = (8, 1) with κ = 10 −2 and 10 −3 as an example.

Post inflation
As the inflaton leaves the trench at φ e , the dominant dynamics turns to the oscillation along φ with respect to φ 0 as long as the oscillation amplitude δφ satisfies δφ |φ 0 − φ r | = (κ/p)φ 0 . At this period, V φ governs the dynamics, i.e., the time scale of the oscillation is determined by the mass scale along φ at φ ≈ φ 0 , As the motion along φ becomes sufficiently small, the field configuration can be re-trapped in the trench and the dynamics would be again along the minimum of the canal in the vicinity of the true vacuum. In this case, V φ = −V m 0, and Eqs. (3.1) and (3.2) are applicable again. At φ ≈ φ 0 , the mass-squared of each orthogonal direction, defined as m 2 i ≡ d 2 V /di 2 (i = I, ψ), is found to be In Eq. (3.18) we have not applied Eq. (2.9) for the term with sin(h − θ) because the field configuration does not follow the trench unless it is trapped again on it. During the oscillation phase which takes place mostly along the φ direction, the value of this term would vary. In Eq. (3.20), g(M, φ 0 ) is a free parameter determined mainly by the set of (M, φ 0 ). It can be either larger or smaller than unity, but lower-bounded as g(M, φ 0 ) > |η * | for p ≥ 2(q + 1) (see Eq. (3.7)) which is the region we are interested in. Hence, depending on M and φ 0 , in the vicinity of φ ≈ φ 0 we can have g(M, φ 0 ) > 1 leading to m 2 I,0 > 3H 2 * . However, m 2 I depends on φ and the slope along I changes its sign across φ 0 . As a result, the angular motion after inflation is commenced only when the oscillation amplitude along φ is significantly reduced so as to have m 2 I 3H 2 . In the vicinity of φ 0 , for the oscillation amplitude of φ denoted as δφ( φ 0 ), Therefore, the onset of the oscillation along I is expected to happen as δφ is reduced to δφ osc , and we find which is valid only for g(M, φ 0 ) 1. For g(M, φ 0 ) O(1), we notice that m 2 I changes its sign as the oscillation amplitude becomes smaller than and rapidly approaches to m 2 I,0 . We take this crossing point as the onset of the oscillation phase of I in this case, and the expansion rate around the epoch is found to be A comment on the possibility of a second inflationary period caused by re-trapping is in order. From Eqs. (2.28), (3.23) and (3.25), Min g(M, φ 0 ), 1/2 > κ and m 2 I > 3H 2 is expected around the epoch of re-trapping, i.e., the angular motion after inflation would take place before the field configuration is trapped in the trench. Hence, a second stage of inflation would not take place.
Eventually, particles I and ψ would decay. We express the decay rate of i-particle as 2 where γ i (i = I, ψ) is a numerical constant taking allowed decay channels into account. From Eqs. (3.18) and (3.19), m ψ,0 m I,0 and generically we may expect ψ decays earlier than I as long as γ ψ ∼ γ I . Also, if Γ ψ ≥ m I,0 , the oscillation of I field after inflation would take place in a universe dominated by radiation, otherwise it will happen in a universe dominated by ψ-particles. In terms of our model parameters, the ratio of interest is given by If Γ ψ Γ I , there is a possibility for I-particles to eventually dominate the universe around the epoch of its decay. In order to check this possibility, we compare the energy density of I-particles and that of the background radiation as follows. When I starts its oscillation, the oscillation amplitude is expected to be with α = O(1) < π. As ψ decays, the universe is dominated by radiation. During this epoch, the energy density of I before its decay is given by Therefore, I particles would be subdominant at the epoch of their decays only if Otherewise, an era of I-particle domination appears, and entropy release due to the lat-time decay of I takes place. This causes a dilution of pre-existing particle densities relative to entropy density. In the sudden decay approximation, the dilution factor ∆ is approximately given by where T × and T I are respectively temperatures when H = H × and H I,d .

Charge asymmetry after inflation
In this section, we consider the possibility of generating a B-or L-asymmetry from the dynamics or decay of the I-field. From now on, we assume that the fields φ and θ in the potential Eq. (2.1) are components of a complex field such as Φ ≡ φe iθ / √ 2 which can carry a charge denoted as Q.

Spontaneous baryogenesis
A slow motion of I after inflation might be considered for spontaneous baryogenesis by introducing a derivative interaction of θ to baryonic/leptonic current in Lagrangian such as where λ is a dimensionless coupling constant, and j µ is a baryonic/leptonic current. In the current scenario we are considering, a slow evolution of θ after inflation takes place around the epoch of H ∼ H osc which is given by either Eq. where That is, T osc and T e differ only by a factor of a few since g(M, φ 0 ) |η * | ∼ O(10 −2 ). Also, as Γ ψ becomes closer to or even larger than H osc , the epoch between T e and T osc becomes narrower. Hence, spontaneous baryogenesis is unlikely to occur in our scenario.

Chage asymmetry from the decay of I particles
As another possibility of generating a charge asymmetry, which was already discussed in the original paper of spontaneous baryogenesis (Ref. [7]), we consider the decay of I at its oscillation phase, assuming B or L-violating processes were decoupled already before the onset of I's oscillations. Here we do not specify B or L-violating operators, but simply assume the branching fraction of relevant channels to be close to unity. During the era of ψ-particle domination with the sudden decay approximation of ψ, approximately the solution of I's EOM can be taken to be 3 and the charge density associated with I is given by where we used φ φ 0 and θ I/φ 0 in the vicinity of the true vacuum. Then, the charge asymmetry from the decay of I is obtained as 4 where x ≡ m I,0 t and  where s ψ is the entropy density around the epoch of ψ-particle decays, β ≡ 2π 2 g * S (T ψ ) 45 π 2 g * (T ψ ) 90 with T ψ being the background temperature at ψ-particle decay, and we set g * S (T ) = g * (T ψ ), ignoring their temperature-dependence. (4.14) Paying attention to the transfer mechanism, we found that in spiral inflation the decays of the inflaton can produce the right amount of baryon number asymmetry while obtaining inflationary observables consistent with observations. In contrast to the naive expectation that it would be difficult to obtain the right amount of baryon number asymmetry, it is found that even in the presence of matter-domination era a sufficient amount of baryon number asymmetry can be obtained as long as the matter-domination era right after inflation is terminated rather soon. Definitely this is a model-dependent statement, since (as expected) the late-time baryon number asymmetry in our scenario depends on model-dependent decay rate(s) (or branching fraction(s)) of the inflaton to baryonic/leptonic particles through operators (maybe) violating baryon/lepton number.
In a part of the parameter space, the expansion rate during inflation is required to be close to the current upper-bound, and hence it would be easily probed in the near-future experiments, for example CMB-S4 [15], PIXIE [16], and LiteBIRD [17].