Role of QCD in moduli stabilization during inflation and axion dark matter

Ignorance of the initial condition for the axion dynamics in the early Universe has led us to consider an O(1) valued initial amplitude, and that prefers the decay constant, Fa, of the QCD axion to be an intermediate scale such as 1012 GeV in order to explain the dark matter abundance. We explore a cosmological scenario of Fa being much larger than 1012 GeV by considering the axion and modulus dynamics during inflation to set the initial amplitude. We show that if the volume modulus (radion) of the extra-dimension is stabilized mainly by the QCD contribution to the modulus potential during inflation, the QCD axion with the string-scale decay constant obtains a mass around the inflationary Hubble parameter. This means that the axion rolls down to the θ = 0 minimum during the inflation realizing almost vanishing initial amplitude, and the inflationary quantum fluctuation can be the dominant source of the current number density of axions. We find natural parameter regions where the axion explains the cold dark matter of the Universe, while the constraint on the isocurvature perturbation is avoided. The presence of the axion miniclusters or axion stars are predicted in a wide range of parameters, including the one explains the Subaru-HCS microlensing event.

Recently, it was shown in Ref. [38] (see also the last part of the introduction for the earlier works) that the abundance and isocurvature problems can be avoided by an enhanced axion mass generated by small instantons.In particular, in an extra-dimensional set-up, one can naturally have a large QCD coupling during inflation due to the modified potential for the volume moduli (radion). 1This scenario is self-consistent since the realization of the axion from compactification requires the stabilization of the size of the extra-dimension both in the current Universe and in the inflationary era.The moduli is generically light compared to the size of the extra-dimension and it is mandatory to discuss its stabilization.In the context of supersymmetry (SUSY), a light QCD axion implies that there should be a scalar superpartner, or string modulus with a mass generated by the SUSY breaking, couples to the gluon.The moduli stabilization in the perturbative regime, i.e., within the effective theory, means that we need delicate balance among terms which are in the different orders in the weak coupling expansion.For example, in the case of the volume moduli, the different order terms in the inverse volume expansion should balance to find a minimum.Moreover, the cancellation of the cosmological constant in the current Universe requires additional delicate balance among those term.It is then natural to assume that this detail balance is badly broken during the inflation where we have non-vanishing vacuum energy.
If the moduli have different values during inflation, there is a chance that the QCD gets stronger during the inflation, and contribute significantly to the moduli potential as well as the axion potential.In this paper, we find that if the radius of the extra-dimension is stabilized between the Hubble-induced mass during inflation and the QCD induced potential, the QCD axion mass is naturally around the Hubble parameter during inflation, thanks to the fact that the up-type Yukawa coupling y u in the Standard Model happens to satisfy |y u | ∼ (F a /M P ) 2 .We also find that consistent with the high-scale inflation, the inflationary fluctuations of the axion field can generate the correct the DM abundance, without introducing the isocurvature problem due to the time-varying axion mass.We study the stabilization of the moduli in general set-up, and show that the dangerous CP violating effects to regenerate a large initial amplitude of the axion is naturally avoided due to the symmetry of the extra-dimension spacetime.
A heavy axion due to stronger QCD during the inflation was first pointed out by Dvali [39] in the context of alleviating the axion abundance by using the moduli field dependent gluon coupling.It was, however, soon pointed out that there are CP problems in general for this solution to the over-production problem [40].
For example, in the setup of the minimal SUSY standard model (MSSM), a large field value of the Higgs field during inflation makes the quarks heavier than those today and thus QCD gets stronger via the loop effect.In that case, it was pointed out that generic CP phases of soft breaking term spoil the solution to the over-production problem [41].On the other hand, not as a solution to suppress the initial amplitude, there have been a discussion of suppressing the isocurvature perturbation by stronger QCD [42].More recently, this idea was studied in [43] by assuming approximate CP symmetry in MSSM soft breaking parameter, while in [44,45] (in the context of the stochastic axion scenario mentioned below), a standard model setup has the CP violation accidentally absent.Compared to previous works we will show: • in a relatively generic setup with an extra dimension, the volume modulus can be stabilized during the inflation by a potential whose main contribution is induced by the QCD dynamics.The heavier QCD axion during inflation is then naturally realized with an accidentally CP-safe structure thanks to the five-dimensional spacetime.Namely, the CP violation is suppressed by the volume of the extra dimension.
In the paper [39], it was commented that the quantum fluctuation of the axion field during inflation may play an important role in explaining its abundance.However, now it is widely accepted that such a scenario is excluded by the problem of too large isocurvature perturbation.One possible way out is the scenario of the stochastic axion [46,47], in which the small quantum fluctuation with H inf ∼ Λ QCD , accumulates to form an equilibrium state, which favors a parametrically small initial axion misalignment explaining the axion dark matter.This scenario requires a moderately low inflation scale and a very long inflation period for the fluctuation to accumulate.Compared to the previous work, we show that • axion dark matter from the large quantum fluctuation can be realized if the heavy axion becomes lighter than the Hubble parameter during the last few e-folds of the inflation.
In contrast to the stochastic axion scenario, the axion fluctuation is produced in a very short period with a large inflation scale.The mode evolution is very different, and the mechanism may be probed from the observation of tensor-to-scalar mode and further measurement of the dark matter isocurvature.Miniclusters and axion stars can be formed due to the dominant fluctuation in the intermediate scale.
The organization of the paper is as follows.In Sec. 2, we revisit the scenario of stronger QCD and relate it to the moduli stabilization during inflation.In Sec. 3, we discuss the axion dark matter by especially focusing the new scenario of the production by isocurvature-safe axion fluctuation and discuss the minicluster formations.In Sec. 4 and Sec.5, we construct models for the moduli stabilization with QCD induced potential during inflation.The final section is devoted to discussion and conclusions.

Moduli and axion stabilizations by stronger QCD during inflation
In this section, we describe the main idea of the stabilization of the moduli and axion fields by the non-perturbative QCD effects during inflation.It turns out that both fields are simultaneously stabilized in the early stage of inflation and the axion mass is comparable to the Hubble scale.
We consider the cosmological scenario where a modulus field, T ,2 has different field values during and after inflation due to the modified potential shape such as by the Hubble induced mass term.In supergravity and superstring theories, a kinetic term of a gauge field in the Standard Model is obtained via where f (X) denotes a function of X, and F µν denote a gauge field strength tensor.f ( T ) = 1/(4g 2 ) with g is the measured gauge coupling in the present vacuum.X denotes the expectation value of X.However, g may be different in the early Universe depending on T inf , where we use sub/super script of inf to denote the value during inflation, here and hereafter.
In addition, an axion, a, couples to the gauge field, where Fµν denotes the dual of F µν , which may be either abelian or non-abelian, in which case we omit the gauge index.In SUSY extensions, the axion is regarded as the superpartner of a modulus, while we do not restrict ourselves to SUSY models.In the following, we will always assume that the (perhaps time-varying) masses of SUSY partners except for the modulus are well above the energy scale, e.g. the Hubble parameter, under consideration for simplicity of discussion.Alternatively, the axion may also originate from the extra-dimensional Wilson loop of the dark gauge field.In these cases, a natural scale of the axion decay constant is the compactification scale, that is naturally, We are interested in the QCD effects to the moduli potential during inflation.Indeed, if the SU (3) c gauge coupling during inflation is larger than the current one, the QCD confinement scale, Λ QCD , become much larger than the ordinary QCD scale ∼ 100 MeV.The axion mass is obtained due to the non-perturbative effect, both during inflation and after inflation, where m a denotes the axion mass, and m u is the up-quark mass.This form should be valid as long as m u is smaller than Λ QCD and it is the lightest quark.Namely, it does not depend on how many quarks are lighter than the QCD scale.It should be noted that such a quantity may depend on cosmic time during inflation.In (2.4), the up-quark mass is where h inf is the expectation value of the SM Higgs field during inflation, and y u ∼ 10 −5 is the up-type Yukawa coupling.When Λ inf QCD is high enough and the chiral symmetry is broken in the 6 flavor QCD with all the SM quarks "light", the QCD itself can induce an electroweak symmetry breaking by the top quark condensation that gives a tad pole term of the Higgs field, Here, the self-coupling of the Higgs field and the top quark Yukawa coupling are assumed to be of order unity.The axion mass during inflation is Here, for simplicity, we assume that F a does not depend on time, which will be revisited when we discuss concrete models.We note that the Higgs field should not acquire a much larger (positive or negative) mass squared, (M inf h ) 2 , during inflation than the inflationary QCD scale.As we will see, the QCD scale during inflation is comparable to the inflation potential scale in order to stabilize the modulus.If the |(M inf h ) 2 | were too large, it would contribute to the inflation potential via Coleman-Weinberg corrections, which usually spoil the inflation dynamics, or, at least, it requires a fine-tuning among contributions to the total potential.We also note that the Higgs field may be driven to be very large value around the Planck scale during inflation in SUSY models [41], while it usually does not work well without a fine-tuning of the CP phase [41] (however, the model in Sec.5 may work 3 ).Although we have in mind that SUSY is in some high energy scale, we do not consider the possibility that the some MSSM field excursion is so large that it introduces a misalignment of the QCD axion potential minima during and after the inflation.Now let us consider the moduli stabilization during inflation by taking into account of the back reaction from the QCD dynamics, which is assumed to be a dominant source for the stabilization.If the QCD contribution is dominant, we expect that its contribution to the moduli mass is comparable to the Hubble induced mass, namely the Hubble scale, with m moduli,inf , H inf being the mass scale of the moduli and the Hubble scale during inflation, respectively.For the axion mass during inflation, we find With F a ∼ 10 15 GeV during inflation, it is natural that the first factor is order 1 and therefore we find As a consequence, we notice that if the QCD potential plays an important role in stabilizing the moduli, the axion mass during inflation is comparable to the Hubble parameter. 3The Higgs VEV during inflation can be much larger than Λ inf QCD due to the coupling with e.g. the inflaton field [45], and the lightest quark mass (up-quark mass) becomes larger than Λ inf QCD .Then, the axion mass is given by m 2 a,inf ∼ In this case, the coincidence found in this paper does not apply.The condition m a > H inf is satisfied in a broader parameter space because the axion mass is not suppressed by the factor of y u .We need an additional CPV source for explaining dark matter abundance since it is then natural to have the axion mass to be always heavier than the Hubble parameter during inflation.
Motivated by this observation, in the following we study axion productions when Eq. (2.10) is satisfied.The initial amplitude is almost zero in this case as the axion is settled into its minimum during inflation, while it is also sensitive to the inflationary fluctuations.We also discuss the moduli stabilization during inflation in a realistic set-up.
3 Dark matter axion from isocurvature fluctuations

Axion fluctuation during inflation
The axion is stabilized around its potential minimum when m a,inf is large enough compared to the Hubble constant H inf during inflation, We consider the case where this is satisfied at the moment of the horizon exit of the scale of the cosmic-microwave background (CMB).If there is no new source of CP violation in the axion potential, which is the assumption to be justified, the initial misalignment angle of the axion is approximately zero, i.e. the axion potential minimum is the same as that at low temperatures 4 .The absence of new CP violating source is a crucial assumption in order for the minimum of the potential during inflation to be zero.Potentially dangerous possibilities are CP violating configurations of the moduli/inflaton during inflation and the presence of CP violating interactions which become important when the Higgs VEV is large and/or QCD coupling is strong.The moduli/inflaton configurations can easily be CP conserving by assuming that the moduli and inflaton are both CP even fields.For example, if the moduli we consider is the moduli of an extra-dimension, it is CP even as it is a component of the metric.We discuss this possibility in more detail in Sec.5.The enhancement of the small instanton effects together with the higher dimensional operators generically give a large CP violating contribution to the axion potential [38].This contribution, however, depends heavily on the particle content of the theory as well as the evolution of the coupling constant at high energies.The contribution can be negligible, especially, with a large (but not too large) extra-dimension (and SUSY) we consider later in this paper.
Under those assumptions, the axion abundance from the usual misalignment or realignment mechanism is almost zero which means the upper bound on F a disappears if there were no other contributions.In the following, we discuss the contribution to the axion abundance from the quantum fluctuation of the axion field during inflation.We will see that this contribution can be sizable enough to explain dark matter of the Universe without contradicting the isocurvature constraints from the CMB observations.
The QCD confinement scale during inflation changes from moment to moment because the moduli field may also be rolling through the change of its potential, and because the QCD scale is sensitive to the moduli field in an exponential way.Less than O(10%) change of the modulus field value would alter the QCD scale extremely because the large coefficient of the beta function of the gauge coupling.Thus, even if m a,inf H inf is satisfied at an early stage of the inflation, m a,inf may become smaller than H inf before the end of inflation.The constraint from the isocurvature fluctuation can be avoided if the time of m a,inf being smaller than H inf happens sometime after when the inflaton fluctuation of the CMB scale exits the horizon.
In general the axion field can be decomposed by with a 0 being the homogeneous mode in the observable Universe, which is set to zero due to Eq. (3.1) around the horizon exit of the CMB scale, Once the axion becomes lighter than the Hubble parameter, the axion field fluctuation, δa, gets of the order δa ∼ H inf /2π.We define k cutoff as the comoving momentum of the axion fluctuation when the axion mass becomes comparable to the Hubble parameter during inflation, i.e. m a = H inf , and k cutoff denotes the smallest comoving momentum with the axion fluctuation. 5 We use the power spectrum with the cutoff of k cutoff : where δa k is the Fourier transform of the canonically normalized axion field, Θ is the heaviside step function, and k denotes the comoving momentum.Note that the spectrum around the 5 The relation between the e-fold and the comoving momentum k is given as GeV where the scale factor is taken as R = 1 at present.
cutoff may depend on the detail of the moduli potential, and how the axion mass becomes smaller than the Hubble parameter, while the behavior with k k cutoff after the inflation, where the axion mass is neglected, is almost model-independent.It represents the wellknown scale-invariant power spectrum in the de-Sitter space in the wave number range.The fluctuation is frozen until it reenters the horizon or the axion mass becomes important.The fluctuation characterized with the power-spectrum Eq. (3.4) is considered as the initial condition set at the end of inflation.

Axion fluctuation after inflation
For simplicity, we consider the case where the Universe after inflation is soon dominated by radiation (the extension with a matter-dominated-reheating era is straightforward).The fluctuation evolves via the equation of motion after the end of inflation, where R is the scale factor, and m a (t) denotes the time dependent axion mass.In the regime of (k/R) 2 + m 2 a H 2 , we can neglect the gradient contribution and δa k is frozen at the value soon after inflation.
The mode k starts to oscillate when (k/R) 2 + m 2 a H 2 gets satisfied.By noting that k/R and m a change slowly, the differential "number density" starts to decrease due to the red shift, If the mode k cutoff reenters the horizon much before the quark/hadron transition, i.e. if k cutoff /R = H is satisfied at H m a , the number density to entropy ratio for each k mode is estimated as, The axion abundance is (3.8) where s 0 (ρ c ) is the present entropy density (critical density), g (g ,s ) is the relativistic degrees of freedom for energy (entropy) density, and m 0 a ≡ m a [t → ∞] is the axion mass in the vacuum.We have defined T cutoff as the temperature H = k cutoff /R.A remarkable difference from the standard misalignment mechanism is that the above form of the abundance depends on the decay constant inversely.This is because neither the "initial amplitude" nor the cosmic time for the onset of oscillation depends on the axion mass.Only when we convert the number density into the energy density the axion mass enters in the estimation, giving the inverse dependence of F a .
On the other hand, when the mode of k cutoff reenters the horizon much after the quark/hadron transition, i.e. k cutoff /R = H happens at H m a , the dominant component is from a production similar to the misalignment mechanism.The difference is that we have a cutoff in the IR modes, which will be important to alleviate the isocurvature bound.We call the production in this region, quantum misalignment mechanism.By defining the mode k a = m a R a at H = m a , when the scale factor is R a , all the modes in the range k cutoff < k < k a can be regarded as "zero" mode and they start to oscillate at the moment according to Eq. (3.5).The number density to entropy density ratio from this component is evaluated as ). (3.9) If this component dominates, by redefining the misalignment angle where ∆N is defined as In summary, the axion abundance is given in two regimes, It should be noted that a simple power spectrum in (3.4) is assumed in our estimation as mentioned before, but the detail distribution around the mode k cutoff is model-dependent and it may affect the final abundance estimation.Nonetheless, we shall use the estimation in (3.12) in the following discussion.Isocurvature problem In the regime of log( ka k cutoff ) 0, we have to care about the isocurvature bound.Indeed, if k cutoff is too small, we encounter the isocurvature problem because our scenario approaches to the usual pre-inflationary PQ breaking scenario of the QCD axion.In addition to the CMB measurement, the isocurvature perturbation is also constrained by the Lyman α data because the matter power spectrum is distorted [56,57].To avoid these constraints, we require that a large amount of the axion isocurvature perturbation is only generated for the scale much below O(0.1) Mpc.This reduces to the condition ∆N 20.This implies a lower bound k cutoff 3 Mpc −1 , where we have defined that R = 1 in the present Universe.
The isocurvature perturbations at smaller scale can be also probed by the CMB spectral distortions [58][59][60] which are the deviations from the black body spectrum of the CMB frequency spectrum. 7The distortions have the information before the recombination, and it can probe small-scale isocurvature perturbations [62]. 8The current bound on the amplitude of the power spectrum of the CDM isocurvature mode has been studied in [62] by using the limits from COBE/FIRAS, and O(1) amplitude is still allowed for k 1 Mpc −1 if e.g. the scale invariant power spectrum is assumed.Further probes of the distortions are prospected by several proposals, e.g.PIXIE [63], PRISM [64], Super-PIXIE [65], Voyage 2050 [66], BISOU [67].For example, [62] shows that PIXIE-type experiments can detect O(1) amplitude for the scale-invariant power spectrum.
In Fig. 2, we have shown the current abundance of the axion dark matter on H inf − F a plane.The black solid lines denote Ω a h 2 /0.12 = 1 for ∆N = −6, −4, −2, 1, 2.5, 5, 10, 20 from upper left to lower right, respectively.The darker gray shaded region denotes the bound from the BICEP/Keck Observations [68].The lighter gray region denotes ∆N > 20, which is constrained from the Lyman-alpha and the CMB measurements as discussed before.We also note that the perturbativity condition of m a F a is satisfied in the figure.Usually satisfying the bound of the isocurvature and explaining the axion abundance by the fluctuation requires a fine-tuning of the (time-dependent) potential so that the axion becomes light after the horizon exit of the CMB scale and before the end of inflation.We emphasize again in our scenario that this is as natural as the parameter region that axion remains always light or heavy during inflation thanks to Eq. (2.10).
Axion clumps Given that we have large amplitudes for small scales, there can be overdense regions of axions at later times; the ratio of the fluctuation to the total dark matter energy density, parameterized by δ, can be of order 1.More concretely, after when the k cutoff mode starts to oscillate, the fluctuation for each mode, δ[k], is obtained from Eq. (3.5) as where δ log k ρ a ≡ k 3 ×m a n a,k and we neglect the component without the logarithmic enhancement in ρ a for the case k a k cutoff . 9We plotted δ[k] in Fig. 3.
8 While the analysis assumes the simple power law spectrum, our scenario gives more complicated spectrum, in particular, when k ∼ k a . 9For k k a , we have used (k in the radiation dominated Universe, where we defined R k as the scale factor when the oscillation starts for the k mode.Since R k ∝ k −1 , we obtain (k 3 n a,k )/s ∝ k −1 .Such a density perturbation can be a seed for the axion minicluster/star or axiton formations as in the case of the axions from topological defects [69] [70-73] (see Refs. [74][75][76][77][78] for more recent simulations).The difference is the distribution of δ[k] which has the overdensity in a wider range of k, especially for the case k cutoff k a , where δ ∼ 1 is satisfied in a scale independent way in the range k cutoff < k < k a .In addition, as we will see, our scenario does not have the stringent bound from the axion decay constant c.f. Ref. [79].
While a detailed study is beyond our scope in this paper, those axion clumps are possible to be explored by such as lensing [80][81][82][83], and even may explain the one indicated in the Subaru HSC microlensing [84,85].In addition, the axion search experiments can be also have distinguished behavior, because axionic dense clumps going through the detector can enhance the signal rate by many orders of magnitude, although such events may be rare.Besides, axinovae is also an interesting phenomenon [86].

A 5D model of axion and moduli
Let us now discuss a concrete setup of moduli/axion which allows us to calculate the behavior of the axion during inflation.We consider a 5-dimensional (5D) scenario where the moduli is the radius of the extra-dimension.
Background metric To explain the setup, we first consider that the metric is a (nondynamical) background, i.e. the moduli is infinitely heavy.The fifth direction is compactified on an orbifold S 1 /Z 2 , and the background metric is given by where y is the extra space coordinate in the interval of [0, L].We assume that QCD is propagating in the bulk, so that the gauge coupling constant depends on L. The gluon kinetic term in the bulk is given by with g 2 5 being a dimension −1 coupling, and G M N is the field strength of the gluon field and we use M, N, R, • • • to denote the 5D space-time index.By decomposing the 5D gluons as (A a µ , A a 5 ), we take their boundary conditions as (N, N ) for A a µ and (D, D) for A a 5 , where N (D) denotes the Neumann (Dirichlet) boundary condition, and the first and the second label correspond to the boundary conditions for the two orbifold fixed points.At low energy, the zero mode only appears for the 4D part, A a µ , while the pseudo-scalar particle A a 5 obtains a mass due to the boundary condition.We now have the Standard Model as the low energy effective theory with the QCD coupling as a function of the radius.
The axion can be introduced in various ways.One is to consider SUSY realization as in the next section (Sec.5),so that the axion is identified as the superpartner of the moduli whose value determines L. In this case we need to assume that there is no other dynamics than QCD to give a mass to the axion.We can also consider a scenario where the axion arises from the gauge field in the extra-dimension via compactification.By introducing a 5D U (1) gauge field (A µ , A 5 ), and taking (D, D) for A µ and (N, N ) for A 5 , we obtain a zero mode of A 5 , which is identified as the axion field.The axion-gluon coupling is obtained by the 5D Chern-Simons term, where b CS denotes an integer.Note that the CS term is invariant under the gauge transformation of e.g.A 5 → A 5 + ∂ 5 α(x, y) for α(x, 0) = α(x, L) = 0. 10 After the integration of the fifth direction, we obtain the effective Lagrangian consistent with the one in Sec. 2, where g s [L] is the QCD gauge coupling constant given as 1/g 2 s = L/g 2 5 from the 5D gauge coupling g 5 .Importantly both the g s and F a are functions of the size of the extra dimension, L, and thus change when the moduli has a different value during inflation.For instance, in the case where axion is originated from a 5D gauge field, F a [L] is obtained as 10 These conditions of α are required to erase the boundary term.
Dynamical metric and moduli Let us discuss the dynamics of the moduli, which appears as a degree of freedom from 5D gravity. 11The moduli field φ is introduced as a component in the 5D metric, (4.5) Here, the vector fluctuation is omitted because it does not have zero mode due to the orbifold projection.The physical size of the extra dimension is given as Lφ 1/3 .The five dimensional Einstein action reduces to the four dimensional one such as where LM 3 5 = M 2 P .There is a freedom to choose L to an arbitrary value by rescaling φ.For usefulness, we took L to obtain φ = 1 at the present vacuum.Then, L denotes the physical size of the extra dimension in the current Universe.During inflation, it may be useful to go to the basis where the kinetic terms are canonically normalized, In the basis with φ (φ), the present vacuum is given by φ = 0 ( φ = 1).g s with arbitrary φ can be obtained as This can be different at different epoch of the cosmological history due to the moduli dynamics.The decay constant also depends on the moduli.In the case of the axion as 5D gauge boson, we have , where F a scales by φ −1/6 for a fixed physical extra dimension length L φ 1/3 .As we will discuss below, we consider 0 < φ < 1 ( φ < 0 ) so that the QCD gauge coupling constant during inflation becomes much larger than the present value, and the QCD-induced potential plays important role in the moduli stabilization.

Moduli stabilization in the current Universe
Now we are ready to discuss the moduli stabilization.The moduli stabilization can be performed in more or less generic ways.This is because by integrating out various heavy fields, the moduli 1PI effective potential can be generically expanded as where α, β, η are constants with mass dimension five, four, three respectively.The overall factor of φ −2/3 is originated from the transformation from the Jordan to the Einstein frame, and we have written the leading three terms in the 1/(Lφ 1/3 ) expansion.The expansion is justified as long as we have four dimensional effective theory in the weakly coupled regime.The α term represents the 5D cosmological constant, and the β term scales as the brane tension which has no volume dependence.Terms including higher powers of 1/(Lφ 1/3 ) are obtained in general, and we only include up to the η term in the following discussion.Indeed, the same form of the potential has been discussed in the LARGE volume scenario [87] in which the third term in (4.15) appears from the α corrections to the Kähler potential. 12To find out the vacuum in the current Universe, the following two conditions are required, These two conditions give us two relations among coefficients, One can see that we need |α| |β| |η| in the Planck unit to have L 1/M P .The moduli mass around the present vacuum is obtained to be which means we need α > 0, β < 0 and η > 0. Since η is at most O(M 3 P ), the moduli is generically light compared to the Planck scale for L 1/M P .In SUSY models, η is further suppressed by the SUSY breaking scale.For example, if we want to have a solution with β and γ needs to be The moduli mass squared is calculated to be around 5 × 10 −9 M P .Note that we have taken φ = 1 at the present vacuum so that L represents the physical volume today.
It is important to note that we need delicate balance among terms in the potential in order to obtain a vacuum of a weakly coupled theory with vanishing cosmological constant.We need at least three terms to satisfy the conditions in Eq. (4.10) at a finite value of Lφ 1/3 1/M P .The leading two terms should be suppressed somehow to be balanced with the third term.The moduli is generically much lighter than M P , and has a fragile potential under the modification of the V = 0 condition as well as the different background field values.

Moduli during inflation
Given that the moduli potential in the current Universe is under the delicate balance, it is quite natural to assume that the moduli potential has a different shape during inflation.We parametrize the potential during inflation as where the subscript inf , again, denotes the coefficients during inflation, and the coefficients depend on time, i.e. we assume α inf , β inf , η inf , respectively, settle into α, β, η after the end of inflation.In addition to the potential in (4.15), the moduli potential receives a contribution from non-perturbative QCD effects.We estimate the QCD confinement scale Λ, which can be understood as Λ inf QCD in Sec. 2, by where M * denotes the UV cutoff scale where it satisfies g 2 s (M * ) = g 2 5 /L at the present vacuum ( φ = 1) 13 .Λ may be rewritten as We include non-perturbative QCD effects by The moduli find its minimum during inflation with the conditions of where V inf denotes the total potential during inflation.These two conditions give us the relations where φ inf denotes the value of φ at the local potential minimum during inflation. 14n Fig. 4, we show the parameter region where the moduli and the axion can be heavy enough to be stabilized (non-colored region) on the H inf − Λ plane.For small Λ, the axion and/or the moduli is lighter than H inf so that the they cannot find their minimum during inflation (blue and thicker blue).We introduce a parameter C mass which represents the uncertainty associated with the axion mass during inflation, for example, from the uncertainty of the expectation value of the Higgs field during inflation.We parameterize m a during inflation (cf.Eq. (2.7)) as with C mass = [0.1 − 10].In Fig. 4, the solid green lines denote m a,inf = H inf for C mass = 0.1, 1, 10 from upper left to lower right, respectively.In the blue shaded region, the condition m moduli,inf > H inf is not satisfied.Here, we took , and we also took α inf = −H 2 inf for demonstration.β inf and η inf are determined by the conditions in (4.21) and (4.22) for given H inf .In the lower region for each green line, the axion mass is less than the Hubble parameter, m a,inf < H inf .In the gray shaded region with horizontal boundary, we have 1/ Λ < Lφ 1/3 .The black solid, dashed, dotted lines, respectively, denote Λ4 /V inf = 1, 10, 10 2 , which implies a cancellation between the potential terms of 1, 10%, 1%, respectively.Above the lines we need more fine-tuning.We can see that the m a,inf ∼ H inf region indeed coincides with the less fine-tuned region.Here F a ∼ 5 × 10 16 GeV in the whole range of the figure.

Scenario for axion dark matter
We discuss the evolution of the moduli field value during inflation and scenarios that the axion dark matter becomes the dominant one at present.We assume that the local minimum of the moduli potential continuously changes during inflation.In particular, we consider that the moduli field value at the minimum gradually becomes large and approaches the value at present ( φ = 1).To make the moduli follow the local minimum, we require m moduli,inf > H inf and i.e. m moduli,inf dφ min /dt φ min , where φ min denotes the moduli field value at the potential local minimum [48,89].The moduli mass is much larger than the inverse of the typical time scale of the change of the potential minimum, and thus the moduli can quickly settle down to the potential minimum.This can prevent the moduli from overshooting into the runaway regime.
As the moduli field value become larger, the gauge couplings becomes smaller, and then the QCD confinement scale or the axion mass becomes smaller.The axion mass has an exponential sensitivity to the change of the moduli field value, and the axion mass can drastically change compared to the moduli mass.If the axion mass becomes much smaller than the Hubble scale during inflation, the axion starts to fluctuate, and its fluctuation contributes to the current axion dark matter abundance.This is the scenario discussed in Sec. 3. We expect the scenario to be probed in the future CMB experiments and axion DM search, which may have a significant enhanced signal event rates with a time dependence due to the minicluster/axionstar formation.
We also note another possibility so that m a,inf > H inf is satisfied until around the end of inflation.This is realized in a large region in the Fig. 4 but it is with fine-tuning among the potential terms for the moduli stabilization.In this case, the current axion abundance is too small to explain the dominant DM.Even in this case, once we have a new source of CP violation in the axion potential during inflation.For instance, one can explain the large enough abundance by enhancing the small instanton effect [38].

A SUSY model and suppressed CP violation
The scenario in the previous sections assumes that there is no CP violation that becomes important in the moduli or axion potentials during inflation.The extra-dimension model is in this sense suitable as the space-time symmetry restricts the CP violation to appear.Here we provide a possible UV model with many additional particles in the context of supergravity and show that the behaviors discussed so far as well as the CP safe nature hold.To this end, we consider T is a moduli multiplet, in which the previously discussed moduli field Re T is embedded.As a minimal possibility, we consider the axion is Im T. We will consider that Re T is stabilized purely by Kähler potential contribution with SUSY breaking for satisfying the quality of the "Peccei-Quinn symmetry", while during inflation, the QCD contribution becomes stronger, providing an alternative source of the stabilization of Re T.
Let us consider the low energy SUSY theory in four dimension.The Kähler potential is given as The first term represents the Planck mass in terms of the extradimensional radius.In addition, we introduce a SUSY breaking field Z SB on a brane, and F , the Kählar potential, appears in the logarithm i.e. the sequestering formalism.∆K is the correction of the Kählar potential, and it may be from Casimir energy and/or leading α correction motivated from the string theory.For later convenience, we consider the following super-potential, Here c is the constant term that may arise from gaugino condensation on a brane.∆W can appear from bulk QCD condensation, which is negligible in the present vacuum because it is from the ordinary QCD.However, it is non-negligible during inflation and plays an important role in stabilizing the moduli and axion at the era.This is the feature of our scenario and the new point of this model.We also put a generic superpotential for Z SB , otherwise, the F-term would be vanishing.At the leading order of ∆K, F, ∆W [T ] we get the supergravity potential 3) Here we neglect the higher order terms, e.g., in the present vacuum (see the following).We further imposed for simplicity of discussion.These conditions can be easily realized when the SUSY breaking field carries a charge under some symmetry.The last condition is the kinetic normalization for Z SB .
As we have mentioned, which can arise from Casimir energy e.g.[90].Here f (T + T † ) represents a finite mass effect from the bulk matter.
In addition, we may consider the D-term contribution by assuming the presence of a gauge field in the bulk.We get In particular, if this D-term is from an abelian gauge group (or asymptotically non-free gauge coupling) we do not need to care about the non-perturbative effect to the moduli potential.The kinetic term of the four dimensional gravity in this formalism is already normalized, but the kinetic term of T in this formalism is By noting the kinetic normalization condition, 3  4 log Re(T ) = 1 12 log φ → Re T ∝ φ 1/3 , we get the potential in terms of φ as (5.7) α originating from V D,coeff is vanishing if the VEV of φ i is absent.Here, we take the redundant parameter L=1 for simplicity of discussion.At the loop level, the bulk gauge coupling g 2 5 / Re[T ] enters into the running of the Kähler potential.Thus in general we have contribution scaling with 1/(Re T ) 3 from loop corrections from either α and β terms, Since it is a loop suppressed effect we do not consider it in the present Universe but it will be more important than the γ term when Re T is small during inflation.It is known that in the absence of Λ, η, and V D,coeff , which is assumed in the present Universe, that the moduli can be stabilized [91].Namely, the stabilization can be successful only with the presence of β and γ(φ) term with certain bulk fields, thanks to the non-trivial φ dependence in the Casimir energy.In this case, for the vanishing cosmological constant, |∆Y Z SB | 2 is suppressed compared with |c| 2 since M −2 5 ∆K is both loop-and M 5 /M P suppressed,

P
(5.9) We put for F Z SB to emphasize that it is the condition for cancelling the cosmological constant in the vacuum.This is different from ordinary supergravity because the gravitino mass is heavier than the SUSY breaking, which is a well known feature of the (almost) no-scale supergravity [92,93,91].
During inflation, on the other hand, many fields, including T , can be away from the potential minimum.Since we consider a small Re T , then the loop suppressed O(1/φ), Λ, and α-term from a D-term contribution can be important depending on the inflation model building.Thus by using the small Re T expansion, we have the dominant terms (5.10)This is the generic form that we have used.As we have mentioned, we can also take account of the α correction from string theory, e.g.Refs.[94,95], to get the last term.One can replace T + T † by T + T † + ξ and expand the potential in terms of ξ.
Another interesting feature is the CP-safety in the colored sector.Indeed, it is also known that the soft mass spectrum can be CP-conserving if T dominantly mediates the SUSY breaking (with R-symmetry) [96][97][98].Thus a good quality of the CP symmetry is guaranteed in our scenario, if during and after inflation the moduli F-term has the dominant SUSY breaking contribution to the particles that are relevant to the QCD. 15In this scenario, the shift of the axion minimum during and after inflation is naturally suppressed.Thus during inflation the axion is stabilized very close to the CP-conserving minimum.
We consider this parity safety to be more generic.This is due to the fact that the parity in 4D effective theory is a part of the rotational invariance in the 5D theory and thus conserves.At this stage, one can assign parities for moduli and axion as even and odd, respectively.The orbifold projection would not disturb this assignment as both fields are not projected out.The parity violating contribution to the moduli/axion potential can be in principle generated by the boundary effects, which, however, are suppressed by the volume of the extra dimension.Since the gluon, axion, and the moduli live in the bulk, the parityviolating couplings of moduli and gluon are naturally suppressed by the 5th dimensional volume and thus our setup can naturally preserve the feature needed for suppressing the axion misalignment angle during inflation. 16

Conclusions and discussion
The axion abundance is quadratically sensitive to the initial amplitude.In the case where the axion exists as an effective degree of freedom before the inflation era, it is commonly assumed that the initial value is set by a random choice, and also the model is subject to the constraint from the isocurvature perturbation due to the fluctuations of the axion field during inflation.
coupling and gauge coupling may give relevant contributions.During inflation, the colored particles can decouple either due to some large expectation values of a combination of charged fields respecting SUSY or due to the decoupling of heavy SUSY partners.Thanks to the CP-safety, the additional CP phases do not shift the minimum of the strong CP phase in a CP violating way.In some CP-conserving case, the phase may be shifted by π which is the exception of the argument [99][100][101].We do not consider this possibility in this paper. 16A caveat may arise from the fact that the scale of 1/g 2 5 is close to the extra-dimensional scale to enhance the inflationary QCD.We implicitly assumed that we do not have the CP violating higher dimensional terms with a scale around or smaller than this, i.e. the QCD has one of the strongest couplings.Since a less than O(0.1)CP violating contribution to the misalignment angle during inflation is still consistent with our scenario, we expect this is not a very strong assumption.
This discussion is justified under the assumption that the inflation dynamics would not affect the axion potential.We show, in this paper, the axion can actually be heavier than the Hubble parameter during the inflation so that the axion field value and the fluctuations are stabilized.For example, we discussed a string-theory motivated scenario of our spacetime, where there is an extra dimension as small as the size of the string/GUT scale.As it is usually the case, the volume of the extra dimension is delicately tuned in order to realize our almost flat spacetime by balancing various potential terms such as the Casimir energy and the scale/volume dependent coupling constants.The inflation dynamics, i.e., the extra energy source to bring the flat spacetime to de Sitter space, generically interferes with this delicate tuning during inflation, and modifies the size of the extra dimension.It is possible to make the size smaller by a factor of O (10), which in turn modifies the coupling constants in the Standard Model.If the gluon fields propagate into the extra dimension, the QCD coupling is enhanced, and a very large axion mass can be realized.Since the source of the axion potential is dominated by the QCD dynamics, the minimum of the potential during the inflation is the same as the current one as long as there is no new CP violating contribution to the axion potential during inflation.We discussed the possible sources of the new CP phases, and found that the scenario of the volume modulus naturally avoids such phases.
This scenario gives a unified picture of the axion and the spacetime.The size of the extra dimension and the axion decay constant can both be of the order of the GUT/string scale.One can have a natural picture of identifying the axion as a component of a gauge field which have the Chern-Simons coupling to QCD in the bulk for the super partner of the moduli.It is interesting that the abundance of dark matter may be telling us what happened at the very beginning of our Universe as it is sensitive to the initial condition.
So far, we have considered the moduli and inflaton are different degrees of freedom.A simple unified possibility is, however, that the moduli is inflaton.In more detail, we have assumed a particular motion of the moduli, i.e. the field value of the moduli is continuously changing during inflation and it is stabilized soon after the end of inflation.Such a motion indeed evokes inflaton.

Figure 1 :
Figure 1: The axion dark matter abundance as a function of ∆N (black solid line).In the upper figure, we took F a = 10 13 GeV and H inf = 5 × 10 13 GeV.In the lower figure, we took F a = 5 × 10 15 GeV and H inf = 5 × 10 13 GeV.Uncertainty of the estimation of the dark matter abundance around ∆N (k cutoff ∼ k a ) is denoted by black dashed lines.

Fig. 1
Fig. 1 shows the abundance of the axion dark matter by varying k cutoff or ∆N (k max ) (black solid lines).In the upper figure, we took F a = 10 13 GeV and H inf = 5 × 10 13 GeV for which the effective theory description is barely satisfied by noting the Gibbons Hawking temperature is H inf /2π and the cutoff scale is 2πF a .The current total dark matter abundance is explained only by the high momentum modes of k ≥ k cutoff k a .In the lower figure, we took F a = 5 × 10 15 GeV and H inf = 5 × 10 13 GeV.The black dashed line denote uncertainties of our estimation around ∆N ∼ 0 (k cutoff ∼ k a ).

Figure 2 :
Figure 2: The parameter region for QCD axion dark matter from inflationary fluctuations in H inf − F a plane.The black solid lines denote Ω a h 2 /0.12 = 1 for the case that axion becomes light during inflation at ∆N = −6, −4, −2, 0, 1, 2.5, 5, 10, 20 from upper left to lower right, respectively.The darker gray shaded region is the constraint from the BICEP/Keck.The light gray shaded region or more precisely ∆N 20 is constrained from the Lyman-alpha and CMB results.

Figure 3 :
Figure 3: δ[k] for k cutoff k a (left) and k cutoff k a (right).We plotted δ[k] around k = k a by the dashed gray line due to the uncertainty of n fluc a [k] as discussed around Sec. 3.2.

Figure 4 :
Figure 4: The green solid lines denote m a,inf = H inf for C mass = 0.1, 1, 10, and we have m a,inf < H inf in the lower region for each line.The blue shaded region denotes m moduli,inf < H inf .The gray shaded region with horizontal boundary is given by 1/ Λ < Lφ 1/3 .The black solid, dashed, dotted lines denote Λ4 /V inf = 1, 10, 10 2 , respectively.See the main text for details.