Natural and Multi-Natural Inflation in Axion Landscape

We propose a landscape of many axions, where the axion potential receives various contributions from shift symmetry breaking effects. We show that the existence of the axion with a super-Planckian decay constant is very common in the axion landscape for a wide range of numbers of axions and shift symmetry breaking terms, because of the accidental alignment of axions. The effective inflation model is either natural or multi-natural inflation in the axion landscape, depending on the number of axions and the shift symmetry breaking terms. The tension between BICEP2 and Planck could be due to small modulations to the inflaton potential or steepening of the potential along the heavy axions after the tunneling. The total duration of the slow-roll inflation our universe experienced is not significantly larger than $60$ if the typical height of the axion potentials is of order $(10^{16-17}{\rm \,GeV})^4$.


I. INTRODUCTION
The discovery of the primordial B-mode polarization of cosmic microwave background (CMB) by BICEP2 [1] provides us with valuable information on the early universe. The measured tensor-to-scalar ratio reads r = 0.20 +0.07 −0.05 , which, taken at face value, implies largefield inflation models such as quadratic chaotic inflation [2], natural inflation [3], or their extensions to polynomial [4,5] or multi-natural inflation [6]. In particular, the inflaton field excursion exceeds the Planck scale, which places a tight constraint on the inflation model building.
The observed large tensor-to-scalar ratio, however, has a tension with the Planck results, r < 0.11 (95% CL) [7]. The tension can be relaxed if the scalar perturbations are suppressed at large scales [8], which might be a result of the steep potential after the false vacuum decay via bubble nucleation [9,10]. Alternatively, the tension can be relaxed by a negative running of the spectral index [1,11]. The running can be generated if there are small modulations to the inflaton potential [12,13]. We shall see that the BICEP2 result and its apparent tension with Planck can be naturally explained in a landscape of many axions.
In the landscape paradigm [14,15], there are numerous false vacua where eternal inflation occurs, continuously producing universes by bubble nucleation. Our universe is considered to be within a single bubble inside which slow-roll inflation took place after the tunneling event. This paradigm has various implications for cosmology and particle physics and seems to have gained further momentum after BICEP2. It however remains unanswered why and how the slow-roll inflation took place after the false vacuum decay. It might be due to some fine-tuning of the potential. Such fine-tuning may be justified because, most probably, there is an anthropic bound on the duration of slow-roll inflation after the bubble nucleation [16].
Still, it is uncertain how a very flat inflaton potential extending beyond the Planck scale is realized in the landscape. Also, there is no clear connection between the slow-roll inflation and the landscape paradigm.
In this Letter we propose a landscape of axions where the eternal inflation occurs in the false vacua and the slow-roll inflation regime naturally appears after the tunneling events.
See Fig.1 for illustration of this concept. Most important, we find that there is very likely to be a direction along which the effective decay constant is super-Planckian, because of accidental alignment of axions known as the Kim-Nilles-Peloso (KNP) mechanism [17]. 1 The important parameters that characterize the size and shape of the axion landscape are the number of axions, N axion , and that of shift symmetry breaking terms, N source . In the case of N source > N axion , the vacuum structure of the axions generates the so-called landscape, in which there are valleys and hills in the axion potential; most important, there exist many local vacua where the eternal inflation occurs, leading to a multiverse. On the other hand, in the case of N source ≤ N axion , all the vacua are degenerate in energy, and we need to embed the axion landscape into the string landscape to induce eternal inflation. In both cases the KNP mechanism works and the super-Planckian decay constant can be generated from sub-Planckian ones. We will estimate the probability for obtaining a super-Planckian decay constant for a wide range of values of N axion and N source based on a simplified model. When N axion = N source , we will show that the enhancement of the decay constant by a factor of 10 3 happens with a probability of about 0.8%, 8%, and 24% for N axion = 10, 100 and 300, respectively. Thus, the existence of such a flat inflationary potential over super-Planckian field values is a built-in feature of the axion landscape. We will also study the other cases; when N source > N axion , it becomes less likely to obtain a large enhancement of the decay constant, whereas numerous local minima are generated. When N source < N axion , there appear massless (or extremely light) axions, which may play an important cosmological role.
Here let us mention the work relevant to the present study. Recently, it was shown in [19] that the KNP mechanism works for the case of N axion = N source > 2 with smaller values of anomaly coefficients. They estimated the probability for a given direction to have an effective super-Planckian decay constant, and they found that it was roughly given by the inverse of the enhancement factor. Our result looks significantly larger than their estimate, but this is because we have estimated a probability that the decay constant corresponding to the lightest direction happens to be super-Planckian for each configuration of the axion landscape. We obtained a consistent result when we used the same values of N axion = N source adopted in [19].
The inflation dynamics along the plateau is given by either natural inflation [3] or multinatural inflation [6], depending on N axion and N source as well as the height of each terms. In a sufficiently complicated landscape with N source N axion , we expect that the latter will be more generic. As we shall discuss shortly, the decay constant, inflaton mass, and duration of inflation etc. are determined by N axion and N source , i.e., the size and shape of the axion landscape. The axion landscape thus provides a unified picture of the slow-roll inflation in the landscape paradigm. In string theory, axions tend to be lighter than geometric moduli owing to gauge symmetries, and so, the axion landscape can be thought of as a low-energy branch of the string landscape paradigm. In this case, the role of the axion landscape is to generate an axion with a super-Planckian decay constant.

II. NATURAL AND MULTI-NATURAL INFLATION
We must have a good control of the inflaton potential over more than about 10 times the reduced Planck scale for successful large-field inflation suggested by the BICEP2 result [1].
One way for realizing a sufficiently flat potential over large field ranges is to impose a shift symmetry on the inflaton: where φ is an inflaton field and C is a real transformation parameter. In the string theory there appear many moduli fields through compactifications, and their axionic partners respect such shift symmetry and therefore axions tend to be lighter than the moduli fields.
Thus, string axions are a good candidate for the inflaton. 2 In order to have a graceful exit of the inflation, we need to break the shift symmetry. One plausible way is to break the continuous shift symmetry down to a discrete one by adding a sinusoidal function: where f is the decay constant and Λ denotes a dynamically generated scale violating the continuous shift symmetry. Then, the natural inflation [3] takes place for a sufficiently large f . In particular, f 5M P l is required by the Planck result [7], and a similar bound is obtained by a combined analysis of BICEP2 and Planck [24], where M P l ≃ 2.4 × 10 18 GeV is the reduced Planck mass.
One can extend the natural inflation to include multiple sinusoidal functions of different height and decay constants; where θ i denotes a relative phase between different sinusoidal functions and a constant term V 0 is to make the cosmological constant vanish in the present vacuum. This is the multinatural inflation [6,25,26]. 3 This potential is naturally generated if the axion is coupled to multiple gauge theories and/or stringy instantons [30,31]. Note that the potential contains many local vacua [6]. The multi-natural inflation can accommodate a wide range of values of the spectral index n s and the tensor-to-scalar ratio r [6,25,26]. If there is a mild hierarchy of order 10 − 100 among the decay constants, it is possible to generate a sizable running of the spectral index, dn s /d ln k ≃ −0.03 ∼ −0.02, which remains approximately constant over the CMB scales without any conflict with large-scale structure data [12,13]. In the multi-natural inflation, at least one of the decay constants must be greater than the Planck scale to explain the BICEP2 result, although the lower bound on f i is relaxed compared to the natural inflation.

III. KIM-NILLES-PELOSO MECHANISM
The central issue for successful natural or multi-natural inflation is how to realize the effective decay constant greater than the Planck scale. There is an interesting possibility to generate such a large decay constant from sub-Planckian decay constants, proposed by Kim, Nilles and Peloso [17]. Their idea is very simple. Let us consider two axions with a shift symmetry, φ i → φ i + 2πf i . We assume that the shift symmetry is broken by the following potential: where i, j run over 1, 2 and a ij is an integer-valued anomaly coefficient which depends on the sources for the shift symmetry breaking. The typical value of the decay constants, f 1 and The required enhancement is of order 10 2−3 for the decay constant around the string scale.
The KNP mechanism is very useful to embed the natural and multi-natural inflation in UV theory such as string theory where the typical values of the decay constant is below the Planck scale. For instance, multi-natural inflation in string-inspired setting was discussed in [25,26] based on the KNP mechanism. Recently, the extension of the KNP mechanism to multiple axions was proposed in [19], where they showed that the KNP mechanism works in the presence of multiple axions, and they estimated the probability to realize an effective large decay constant, f eff in the case of N axion = N source , assuming that the other directions are sufficiently heavy. They showed that the probability for the alignment of axions is approximately given by the inverse of the enhancement f eff /f i . Later in this letter we will directly calculate the distributions of the axion mass for a wide range of values of N axion and N source , based on a simplified model of the axion landscape. There we will show that the required enhancement can be generically realized in the axion landscape.

IV. FALSE VACUUM DECAY
In the landscape paradigm, eternal inflation occurs in numerous false vacua, creating various universes through the tunneling, leading to a multiverse. As pointed out in Ref. [16], if we live in a bubble created from the false vacuum decay followed by the slow-roll inflation with the e-folding number N e = 50 − 60, we may be able to observe a negative curvature as a remnant of the bubble nucleation. The detection of the negative spatial curvature will be more likely if there is a pressure toward shorter inflation. Also, if the inflaton potential after the bubble nucleation is sufficiently steep, scalar density perturbations at large scales can be suppressed [32,33]. This could explain the low ℓ-anomaly of CMB. After BICEP2, the problem of the low-ℓ suppression was sharpened [10]. Recently, the landscape and its implications were also discussed in the sneutrino chaotic inflation model [34]. As we shall see shortly, these expected properties are built-in features of the axion landscape.
The Coleman-De-Luccia (CDL) instanton [35] requires a rather large curvature of the potential, V ′′ ≫ H 2 in the false vacuum; otherwise the tunneling occurs as in the Hawking-Moss case [36]. In order to realize both the eternal inflation in the false vacuum, CDL instanton, and the subsequent slow-roll inflation in terms of a single scalar field, one needs a rather contrived functional form of the inflaton potential. If one considers multiple fields, on the other hand, we can naturally realize mass hierarchy between the false vacuum and the slow-roll inflation regime [37]. Still, the existence of the inflationary plateau had no direct relation with the landscape so far. This questions is also answered naturally in the axion landscape.

V. AXION LANDSCAPE
Let us now consider an axion landscape. For simplicity we will focus on a landscape of axions with sub-Planckian decay constants, but the application to other cases is straightforward. The shift symmetries of axions are explicitly broken by various sources such as gaugino condensation and/or instanton: with φ i is an axion, a ij integer-valued anomaly coefficients, and f i the decay constant. 4 The axion is assumed to have a periodicity, The potential height Λ i is naively expected to be of order M string = O(10 17 ) GeV in the string theory, whereas some of them can be smaller if they are dynamically generated. In general, N source does not necessarily coincide with N axion . For the moment we assume N source is comparable to N axion , and we shall discuss the other cases and its cosmological implications later.
Here and in what follows, N source counts only the shift symmetry breaking terms which can affect the vacuum structure and/or the inflaton dynamics, and hierarchically small contributions are to be considered separately in the low energy. For a sufficiently large N source , they form a complicated landscape with numerous local minima, where eternal inflation occurs, continuously creating new universes by bubble nucleation (cf. Fig. 1). On the other hand, when N source is equal to or smaller than N axion , all the vacua are degenerate in energy. This is not an obstacle for generating a direction with a super-Planckian decay constant. In order to induce eternal inflation, however, we would need to either introduce another kind of shift symmetry breaking to lift the degeneracy or embed the axion landscape into the string landscape which contains many local vacua. In the latter case, the role of the axion landscape is to provide an axion with the super-Planckian decay constant.
In the axion landscape, a super-Planckian decay constant for the inflaton is likely realized by accidental alignment of axions in the axion landscape. As will be shown below, the 4 It is possible to add another functional form of the axion potential [38][39][40], but it does not significantly change our arguments, as long as there are sufficient number of axions which form the landscape like (5).
In fact, such an additional potential will help to generate many false vacua.
probability to obtain an enhancement of the decay constant by a factor of 10 3 is about 1%(10%) for N axion = N source = 10(100). On the other hand, the probability decreases as N source becomes larger than N axion . The case of N axion = N source was recently studied in detail in [19], and they showed that the KNP mechanism works for multiple axions with smaller values of the anomaly coefficients. Contrary to [19], however, we believe that no hierarchy in the potential height is needed to ensure the mass hierarchy between the inflaton and the other axions. This has an important implication for the duration of the inflation as we discuss later. Thus the KNP mechanism generically works for a wide range of values of N axion , N source , and the potential height Λ 4 i . The slow-roll inflation is a natural outcome of the axion landscape.
If the axion landscape has many local vacua, the CDL instantons are formed at the tunneling. This is because the typical decay constants are sub-Planckian and the axions other than the inflaton tend to be so heavy that the barrier between the local minima is narrow. After the tunneling, the universe will be dominated by the curvature term and the heavy axions for a while. The slow-roll inflation by the lightest axion starts when its energy density dominates the universe.
In order to get the feeling of the probability for a sufficiently flat direction to arise by the accidental alignment, let us consider a simplified model. We set Λ i = Λ, f i = f and θ = 0 in Eq. (5). 5 We take a ij as an integer valued random matrix satisfying −n ≤ a ij ≤ n to scan various realization of the axion landscape. In order to find the flattest direction, we Taylor expand the cosine function up to the second order of axions. Then the mass matrix for axions {φ i } is proportional to Of course, as the field values change, the contribution of each shift symmetry breaking term to the mass term changes. Since we are interested in the case that all the axions other than the lightest one are stabilized in one of the false vacua, such an expansion can be justified.
We numerically estimate the eigenvalue distribution of the integer-valued random matrix 5 In general, some of the phases are physical and cannot be absorbed by the redefinition of the parameters or the shift of axions. In fact, they will play an important role in the multi-natural inflation [6,25,26] to obtain a wide range of values of n s and r.
squared, A ij . Let us denote the eigenvalues of A ij as 0 < a 2 1 ≤ a 2 2 ≤ · · · ≤ a 2 N axion . 6 Here we exclude the case of detA = 0, in which case there is always massless axion(s). This should be included in the case of N source < N axion , as some of the source terms are not independent (as far as the axion mass is concerned). The effective decay constant for each axion is enhanced by a /a i with respect to its typical value f . For instance, the effective decay constant for the lightest axion is given by and similarly for the second or third lightest axions. Here a ≡ a 2 i is the squared root of the averaged eigenvalues of A ij . Note that a scales as √ N axion , which however does not change the relation (8) as it only affects the typical mass scale, not the decay constants.
Before going to the detail, we give the result on the integrated probability that the effective decay constant for the lightest axion is enhanced by more than f eff /f i , for the case of N source = N axion : as long as it is much smaller than unity. Here P(x) denotes the probability that the enhancement factor f eff /f i for the lightest axion becomes larger than x. This implies that the the effective decay constant for the lightest axion is enhanced as N axion increases.
In Fig. 2 we show the integrated probability distribution functions for the enhancement factor of the effective decay constant for the lightest, the second lightest, and the third lightest axions. We have generated 10 5 random matrices A ij with detA = 0 for N axion = 10 and n = 2. For instance we can see that the probability that the lightest axion mass happens to be 10 3 (10 2 ) times lighter than the averaged mass is about 0.8%(8%). In other words, the decay constant for the lightest axion is enhanced by a factor of 10 3 than the typical value with a probability of about 1% for N axion = 10. Surprisingly, even for a relatively small axion landscape that consists of only 10 axions, we can realize the enhancement of 10 3 with a probability about 1%. 7 As one can see from the figure, it is much more unlikely to have 6 We focus on a situation that a 1 is smaller than a 2 , and identify the lightest axion with the inflaton.
On the other hand, assisted inflation [41] or N-flation [42] will be possible if some of the eigenvalues are degenerate. 7 We are aware that it is non-trivial to interpret the probability because of the measure problem. two (or more) relatively flat directions simultaneously. This suggests that our inflaton is the lightest axion in the axion landscape.
Note that our result on the probability is about one order of magnitude larger than the result of Ref. [19]. This is because they estimated a probability for a given combination of axions to have a super-Planckian decay constant assuming the rest of axions are hierarchically heavier. Instead, we have evaluated all the mass eigenvalues for each realization of the axion landscape, and then estimated the probability for the lightest one to have super-Planckian decay constants. This corresponds to the probability that, for a given axion landscape, we find at least one flat direction with the super-Planckian decay constant. In addition, we can easily extend our analysis to the case of N source > N axion as we directly estimate the mass eigenvalue distribution.
We show in Fig. 3 the integrated probability distribution function for f eff /f in the case of N source = N axion , N axion + 1, N axion + 2. As the number of source terms increases, it becomes more difficult to have a light axion with a super-Planckian decay constant. Roughly, the probability distribution scales as P N axion +1 ∼ (P N axion ) 2 and P N axion +2 ∼ (P N axion ) 3 . This can be intuitively understood as follows. Note that, if one replaces one of the rows of the random matrix A ij with another random-valued row, the resultant matrix will be almost statistically independent from the initial one. Therefore, adding another source term is approximately equivalent to considering another sets of axions with N axion = N source . In order to have a suppression of the axion mass (or equivalently, the enhancement of the decay constant), the accidental cancellation must happen in the two sets of axions simultaneously. Therefore, we expect P N axion +1 ∼ (P N axion ) 2 , and similarly for a larger value of the source terms. Now let us go onto a larger axion landscape. To simplify our analysis we focus on the case of N axion = N source . We show in Fig. 4 the probability to have a flat direction with the decay constant more than 10 2 (upper), 10 3 (middle), 10 4 (bottom) times larger than the typical value in one realization of the axion landscape, as a function of N axion . We can see that the probabilities for the enhancement of 10 2 , 10 3 , and 10 4 are about 70%, 8%, and 0.8%, respectively for N axion ≈ 100, and they further increase in proportion to N axion . Note that the enhancement by a factor of 10 4 is less likely unless the number of axions becomes much larger than O(100). If we compare our result at N axion = 10 with that of [19], ours is about one order of magnitude larger, probably because we focus on the lightest direction; if we randomly choose one direction from the 10 directions, the probability to realize the The effective potential for the inflaton generically contains a combination of various sinusoidal functions. Note that the height of the inflaton potential is generically comparable to the typical height in the axion land scape, although it can be suppressed if there is a hierarchy in the size of the shift symmetry breaking. This is the case if some of them are dynamically generated. If the inflaton potential is dominated by a single sinusoidal function, the inflaton dynamics is well approximated by the natural inflation (2). On the other hand, if there are multiple sinusoidal functions with slightly different decay constants and heights, it will be the multi-natural inflation (3). This is the case if N source > N axion . In particular, if N source is slightly larger than N axion and some of the source terms have hierarchically small height, the inflaton potential receives small modulations, leading to a sizable negative run-  ning of the spectral index [12,13]. Thus, the running of the spectral index, if detected, will constrain the relation between N source and N axion . Now let us consider the eternal inflation which occurs in the false vacua before the slowroll inflation takes place. The tunneling rate is not sensitive to the value of the inflaton, since the couplings between the inflaton and the heavy axions are weak [37]. Therefore, the inflaton field value at the tunneling is generically deviated from the inflaton potential minimum by O(f eff ), which determines the typical duration of the slow-roll inflation. 8 The mass hierarchy between the false vacuum and the inflation plateau is also determined by It sensitively depends on the duration of inflation whether we will be able to observe the negative spatial curvature. The detection is more likely if there is a pressure toward shorter inflation. In the axion landscape, this is generically the case because the flat direction arises from the accidental alignments. In particular, the pressure toward shorter inflation will be significant if the number of axions is relatively small, or if the number of source terms tends to be (much) larger than the number of axions. On the other hand, the pressure might be weakened if there are many axions with N axion ∼ N source , since the large effective decay constant can be generated more easily. Therefore, detection or non-detection of the negative spatial curvature by future observations will provide us with information on the size and shape of the axion landscape.
So far we have assumed N source ∼ N axion . If N source N axion , it will be more difficult 8 If the energy difference is small, the inflaton might be more or less stabilized near the potential minimum.
This situation can be avoided if the typical energy density in the false vacua is larger than the inflaton potential height.
to realize a super-Planckian decay constant, as we have seen in Fig. 2. For N source < N axion , there appear massless (or extremely light) N axion − N source axions in the low energy, while the above discussion on the inflation remains unchanged. They may behave as dark radiation or hot dark matter, ameliorating the tension between BICEP2 and Planck [43,44]. They are produced through various mechanisms, e.g., coherent oscillations, thermal production, and non-thermal production from decays of long-lived states [45][46][47]. If they get tiny masses, stable axion dark matter can generate isocurvature fluctuations, which is strongly constrained by the BICEP2 result [48][49][50]. If there exist axions with masses at (more than) PeV scales, they may play a role in generating fluctuations as curvatons [51].
Another interesting implication of the axion landscape is that the decompactification problem during the inflation will be naturally evaded because the axions are the lightest moduli fields in the string theory owing to the shift symmetry. Although it is a non-trivial task to infer the supersymmetry breaking scale in the axion landscape, there is a lower bound on the gravitino mass, m 3/2 H inf ∼ 10 14 GeV [52], in a known set-up of moduli stabilization. Here m 3/2 is the gravitino mass, which is typically comparable to other heavy moduli masses. Note that there exist moduli fields heavier than the gravitino, which do not play any cosmological role in our context. In fact, such a high supersymmetry breaking is consistent with the Standard Model Higgs mass [53][54][55].

VI. REHEATING
For successful inflation, the inflaton energy is transfered into the Standard Model particles, and the baryon asymmetry is generated after the inflation. To this end, we shall discuss the reheating temperature and the leptogenesis after the inflaton decay. The mass of the inflaton m φ in the natural or multi-natural inflation is given by The decay width of the inflaton reads where N g = 8 + 3 + 1 counts the number of Standard Model gauge bosons. Note that the decay constant f a can be different from the those in the scalar potential. Then, the reheating temperature T R is estimated as Here, we used T R = (90/π 2 g * (T R )) Γ φ M Pl and g * (T R ) = 106.75 counts the number of degree of the relativistic particles in plasma. Therefore successful thermal leptogenesis [56] is possible when the right-handed neutrinos is produced from the thermal bath: where M N 1 is the mass of the lightest right-handed neutrino.
For T R M N i m φ /2, non-thermal leptogenesis will be also possible, if there exists a direct coupling to the right-handed neutrinos, say, L ∼ (φ/f a )M N i N i N i , where N i is the right-handed neutrinos. Then, the partial decay width reads and the reheating temperature does not change significantly. The right amount of the baryon asymmetry is then obtained through the decay of N i produced from the inflaton and the sphaleron process: Here, Br is the branching fraction of the decay mode of φ → 2N i .

VII. CONCLUSIONS
We have proposed a landscape of many axions where the axion potential receives various Interestingly, there is very likely to be a direction along which the effective decay constant exceeds the Planck scale owing to the accidental alignment of axions, i.e., the KNP mechanism. Therefore, in the axion landscape, the existence of the slow-roll inflation regime is a natural outcome of the complicated vacuum structure. We have also argued that the effective inflation model in the axion landscape will be either natural inflation or multinatural inflation, depending the values of N axion and N source . In the latter case, a wide range of (n s , r) as well as the running of the spectral index can be realized. The size and shape of the axion landscape are parametrized by N axion and N source , which determines the effective super-Planckian decay constant, and therefore the typical duration of the slow-roll inflation.
In a certain case, there might be a strong pressure toward shorter inflation, and it will be more likely that we can measure the negative spatial curvature as a remnant of the CDL tunneling. Conversely, non-detection of the negative curvature will constrain the size and shape of the axion landscape. Also, the size of density perturbations and the inflaton potential height will be useful to extract information on the axion landscape. A more quantitative study on this issue will be given elsewhere.
Note added: After completing this work, there appeared the papers [57][58][59][60], in which the alignment of two axions was discussed based on the KNP mechanism as well. We also note that the charge assignment studied in Refs. [57,59] leads to the suppression of the lighter axion mass, similarly to the seesaw mechanism for the light neutrino mass.