Axion Misalignment Driven to the Hilltop

The QCD axion serves as a well-motivated dark matter candidate and the misalignment mechanism is known to reproduce the observed abundance with a decay constant $f_a \simeq \mathcal{O}(10^{12})$ GeV for a misalignment angle $\theta_{\rm mis} \simeq \mathcal{O}(1)$. While $f_a \ll 10^{12}$ GeV is of great experimental interest, the misalignment mechanism requires the axion to be very close to the hilltop, i.e. $\theta_{\rm mis} \simeq \pi$. This particular choice of $\theta_{\rm mis}$ has been understood as fine-tuning the initial condition. We offer a dynamical explanation for $\theta_{\rm mis} \simeq \pi$ in a class of models. The axion dynamically relaxes to the minimum of the potential by virtue of an enhanced mass in the early universe. This minimum is subsequently converted to a hilltop because the CP phase of the theory shifts by $\pi$ when one contribution becomes subdominant to another with an opposite sign. We demonstrate explicit and viable examples in supersymmetric models where the higher dimensional Higgs coupling with the inflaton naturally achieves both criteria. Associated phenomenology includes a strikingly sharp prediction of $3 \times 10^9~{\rm GeV} \lesssim f_a \lesssim 10^{10}$ GeV owing to anharmonic effects, the absence of isocurvature perturbations, and possible formation of axion miniclusters due to attractive self-interactions near the hilltop.


INTRODUCTION
Measurements of the neutron electric dipole moment indicate an unnaturally small value of the CP-violating QCD θ parameter [1,2], which is known as the strong CP problem [3].
Shortly after recognizing this discrepancy, the Peccei-Quinn (PQ) mechanism [4,5] was developed as a resolution; an anomalous U (1) PQ symmetry is spontaneously broken at a scale f a and the resulting pseudo Nambu-Goldstone mode, called the axion a, dynamically relaxesθ = θ− a/f a to a vanishing value consistent with experiments. The value of f a plays a critical role in observables related to the axion so its precise determination, theoretically and experimentally, is crucial.
A model with a weak scale decay constant was initially proposed [6,7] but immediately ruled out by laboratory searches. Today, supernovae cooling is the most competitive lower bound giving f a 10 8 GeV [8][9][10][11][12]. Since the axion is very light and stable on cosmological time scales, one can imagine a scenario where its relic abundance accounts for the observed dark matter (DM) abundance Ω DM h 2 = 0.12. A thermal axion relic abundance is too hot and scarce to be consistent with cold DM. Two non-thermal production mechanisms are commonly considered. The relic abundance from the misalignment mechanism [13][14][15], namely coherent oscillations due to an initial axion field value θ mis f a , is Ω mis h 2 0.12 θ 2 mis F(θ mis ) f a 5 × 10 11 GeV where F(θ mis ) is the anharmonicity factor. With the natural assumption of O(1) initial misalignment, f a = 10 11 − 10 12 GeV is compatible with the observed DM abundance. If the PQ symmetry is broken after inflation and the domain wall number is unity, the abundance of axions emitted from the string-domain wall network is [16][17][18] Ω string,DW h 2 0.04 − 0.3 f a 10 11 GeV 7/6 . (1. 2) The decay constant f a ∼ 10 11 GeV reproduces the DM abundance.
As ongoing axion experiments are about to reach sensitivity required to probe small decay constants of 10 8 GeV < f a < 10 12 GeV [19][20][21][22][23][24][25][26][27][28], exploring the theoretical landscape pertaining to small f a is important. Some studies have been successful in allowing small f a in a natural setting, such as parametric resonance from a PQ symmetry breaking field [29] and decays of quasi-stable domain walls [17,[30][31][32]. The misalignment mechanism can reproduce the observed DM abundance for f a to π [33][34][35][36][37], where the anharmonicity factor F(θ mis ) becomes important. In this study, we propose a scenario which dynamically predicts θ mis π and thus small f a in the context of axion DM from the misalignment mechanism.
It is commonly assumed that no misalignment angles are special in the early universe, and θ mis π requires a fine-tuned initial condition. This is not the case given two conditions are met: 1) the axion field dynamically relaxes to the minimum of the potential in the early universe and 2) the model possesses a non-trivial prediction between the minima of the axion potential in the early and today's epochs. We refer to the axion relaxation with the fulfillment of these requirements as Dynamical Axion Misalignment Production (DAMP).
We study DAMP by the dynamics of the Higgs fields during inflation. The mechanism follows from suspending the assumption that axion's late-time dynamics is agnostic to inflationary dynamics. To be concrete, we study the Minimal Supersymmetric Standard Model (MSSM).
The Higgs fields H u and H d in general couple to the inflaton potential energy via higher dimensional operators, which lead to so-called Hubble induced masses. The Higgs fields can acquire a large field value in the early universe by virtue of the Hubble induced mass. This large field value gives large quark masses, which enhance the confinement scale to Λ QCD during inflation. Since m a is proportional to Λ QCD , this raises the axion mass to allow for earlier relaxation to the minimum. Note that we need to assume the Higgs fields are not charged under PQ symmetry; otherwise, the decay constant will be as high as the Higgs VEV and suppress the axion mass. For context, early studies [38][39][40] have made use of moduli fields to raise the QCD confinement scale Λ QCD → Λ QCD during inflation. This avoids fine-tuning problems that arise under the assumption of an O(1) initial misalignment with large values f a > 10 12 GeV. Later studies [41,42] used Higgs fields as the moduli fields and refined the scope of the mechanism to reduce isocurvature perturbations for models with large inflation scales, which comes at the cost of an inability to suppress the axion abundance.
This loss of abundance predictability is because no assumptions are made about the evolution of the axion minimum through inflation. In the MSSM for example, we have [40] θ eff = θ QCD + arg(det λ u λ d ) + 3 arg(mg) + 3 arg(Bµ), (1.3) where λ u , λ d are the Yukawa coupling matrices, mg denotes the gluino mass, and Bµ is the soft breaking mass for Higgs scalars. Although a large Λ QCD can help fulfill the first DAMP criterion, we should also explain how a Hubble induced mass fits in with Eq (1.3) to fulfill the second criterion.
The Kähler potential can give rise to a Hubble induced Bµ term. If the argument of the term is different from the vacuum Bµ term by π and dominates, a shift of π relative to the vacuum value is induced in the axion potential. 1 The difference of π in the arguments can be understood by the (approximate) CP symmetry of the theory, such that the Bµ terms are real and the difference of π is simply the opposite signs of the terms. An approximate CP symmetry is also invoked in Ref. [39], where a relaxation to θ mis 0 is considered. Note that the shift of π in the axion potential occurs only if the number of generations is odd.
A large m a allows the axion field to relax to the bottom of the potential during inflation, and a π shifted axion potential means this minimum coincides with today's hilltop. Without additional particles beyond the MSSM, Λ QCD and consequently m a cannot be arbitrarily large; we find m a 10 TeV. Thus, in the minimal scenario, we consider TeV scales for Hubble during inflation H I to allow for the relaxation of the axion misalignment during inflation.
We also explore non-minimal models where m a and thus H I can be larger. Relaxing the axion arbitrarily close to today's hilltop may cause overproduction of axion DM, but we find that the running of Yukawa terms in the Standard Model (SM) gives a sufficient CP phase change O(10 −16 ) to avoid the scenario. An exciting implication of this mechanism is that f a is fixed to roughly 3 × 10 9 GeV by the observed DM abundance and CP-violating phase renormalization in the theory. We impose CP symmetry in the Higgs and inflaton sectors. Additional CP violation (CPV) of up to O(10 −4 ) only induces O(1) changes in the prediction of f a . In summary, by the inflationary dynamics of the Higgs fields as well as the (approximate) CP symmetry, we can fulfill both criteria of a DAMP scenario; in particular in this paper we explore the case where the inflationary minimum is shifted by π from today's minimum, which is referred to as DAMP π .
We now elaborate on the approximate CP symmetry. Although the O(1) amount of CPV measured in the SM must be generated in the theory, a small CPV in the extended sectors can be a consequence of the suppressed couplings with the source of CP violation.
Such hierarchical couplings can result from the protection of additional symmetries or the geometric separation in the extra dimensions. Additionally, any quantum corrections that 1 If the arguments are the same, we may dynamically relax the axion to today's minimum during inflation as discussed in a separate paper [43]. is exactly what can give rise to a shift of π in the axion potential.
In Sec. 2 we briefly review the axion misalignment mechanism and the role of the anharmonicity factor. We also discuss how the amount of CPV in a theory can be connected to the axion abundance in a DAMP π model. In Sec. 3 we show how a Hubble induced mass for the Higgs in the early universe can induce an axion mass enhancement and a phase shift of π in the axion potential, fulfilling the DAMP π criteria. In Sec. 4 we discuss both a set of minimal models with the cosmology fully evaluated, and extended models with a larger viable parameter space and a simplified discussion of the post-inflationary cosmology.
Finally, in Sec. 5 we summarize and discuss the possible phenomenological implications of this model as well as future directions.

AXION MISALIGNMENT & EARLY RELAXATION
We first review the axion misalignment mechanism. The equation of motion and energy density of axions are given byθ with θ 0 (θ i ) the angle at the end (onset) of inflation, unless the number of e-folding is exceedingly large N e ∼ (H I /m a ) 2 as pointed out by Refs. [48][49][50]. As a result of inflation, the misalignment angle takes a random but uniform value θ mis in the observable universe.
Around the QCD phase transition, the axion acquires a mass from the QCD non-perturbative effects and starts to oscillate, when 3H m a , from ϕ mis = θ mis f a towards the minimum today. Without fine-tuning, θ mis is expected to be order unity. The coherent oscillations of axions contribute to the cold dark matter abundance where F(θ mis ) 1 for θ mis π and, for θ mis 0.9 π, is analytically approximated by [34] F(θ mis ) 16 √ 2 π 3 ln (2.5) Several numerical studies have been devoted to the determination of F(θ mis ) [33,35,36,51] but DAMP π calls for a dedicated study for the extreme limit of π − θ mis 1. The exponents in Eqs. (2.4) and (2.7) assume the topological susceptibility of QCD given by the dilute instanton gas approximation (see the lattice results in Refs. [52][53][54][55][56]) but our results are insensitive to this uncertainty.
We now discuss how our framework, by relaxing the above assumptions, makes a prediction for f a using the DM abundance and the CP-violating phase δθ CP of the theory. The axion mass arises from QCD dynamics. There is no a priori reason that the axion mass during inflation is given by the exact same QCD effect observed today. In fact, there are numerous scenarios where the axion is enhanced in the early universe, e.g. a large QCD confinement scale [38][39][40], explicit PQ breaking [57,58], and magnetic monopoles [59,60]. If the axion mass is larger than the Hubble scale during inflation, the axion starts oscillations and is rapidly relaxed towards the minimum, Additionally, if the CP phase of the model has a phase shift of π after inflation, as explained and elaborated in Secs. 1 and 3, the location of this minimum is then converted into the maximum of the potential, making the effective misalignment angle θ mis π. There is however a limit on how close θ mis can be to π because the quantum correction to the θ parameter from the CP violation in the SM Yukawa couplings is δθ CP ∼ 10 −16 [61] and the running of the Yukawa couplings necessarily induces a phase shift of similar order between the inflationary and low energy scales. This small deviation from the hilltop δθ = π − θ mis allows for the prediction of f a due to the anharmonic effects. In the limit θ mis → π, F(θ mis ) and thus Ω a h 2 are only logarithmic dependent on δθ so one can predict f a in terms of the deviation δθ by requiring DM abundance using Eqs. (2.4) and (2.7) f a 2.4 × 10 9 GeV Ωh 2 0.11 where we assume ln δθ

DYNAMICAL AXION MISALIGNMENT PRODUCTION AT THE HILLTOP
We would like to show that allowing H u and H d to acquire large VEVs during inflation can lead to a DAMP π scenario. To guide the reader, we first restate the conditions under which the DAMP model is applicable: 1) the axion field dynamically relaxes to the minimum of the potential in the early universe and 2) the model possesses a non-trivial prediction between the minima of the axion potential in the early and today's epochs. Throughout our discussion of DAMP π models, we have in mind a minimal model as a proof of principle and extended models to further explore viable parameter space. Generically, we can include inflaton-Higgs dynamics with the effective operators suppressed by the cutoff scale M in the Kähler potential where X is the chiral field whose F -term provides the inflaton potential energy. We omit O(1) coupling constants here and hereafter. For illustration purposes, we only show lower dimensional operators relevant for the following discussion. Higher dimensional operators do not change the discussion.
During inflation, the inflaton F -term gives the Higgs fields Hubble induced terms, where c = (M Pl /M ) 2 and H I is the Hubble scale during inflation. We assume that the Hubble induced mass terms are negative. They push the Higgs fields in the D-flat direction |H u | = |H d | up to the cutoff scale M , and as we will see in the following sections the large Higgs VEVs realize DAMP π .

Axion Mass During Inflation
Together with the effective terms from the Kähler potential in Eq. (3.1), the MSSM Higgs potential reads We assume that the Higgs sector is nearly CP symmetric, which is anyway required from the measurements of the electric dipole moment for TeV scale supersymmetry. See Refs. [62] and [63] for the latest measurement and its implication to supersymmetric theories, respectively. We also assume a CP symmetry in the inflaton-Higgs coupling. The Higgs fields break SU (2) L × U (1) Y → U (1) EM by both the early universe VEV and today's VEV. Parameterizing the Higgs field space in terms of a radial mode φ ≡ |H u | = |H d | along the D-flat direction and an angular mode ξ = arg (H u H d ), which is the relative phase of the Higgs fields, allows us to write The large VEV φ i gives quarks very large masses during inflation. In the MSSM, the

1-loop renormalization group equation (RGE) is
where µ r is the renormalization scale, N = 3 is the gauge group index, and F is the number of active fermions in the theory. Solving the RGE from the TeV scale up to the scale φ i , and from the scale down while pretending that all quarks are above the scale where the gauge coupling diverges, we obtain the fiducial dynamical scale This is the physical dynamic scale Λ QCD if all quarks (including the KSVZ quarks) are above the scale. If some quarks are below the scale, the physical dynamical scale Λ QCD is given by The axion mass vanishes when the gluino is massless since strong dynamics gives the mass dominantly to the R-axion. The axion mass is hence given by m a 1 4π where we assume that the gluino mass is below the physical dynamical scale and that the large Higgs VEV does not break the PQ symmetry. We include the factor of 4π expected from the naive dimensional analysis [64][65][66][67]. Here mg is the RGE invariant one, mg ,phys /g 2 .
The holomorphy of the gauge coupling guarantees that we may use the fiducial dynamical We may raise the dynamical scale further by introducing additional particles. One possibility is to introduce a moduli field whose field value controls the gauge coupling [38][39][40][41][42], and assume that the moduli field value during inflation raises the gauge coupling constant.
Another possibility is to introduce additional SU (3) c charged fields and assume that their masses are large during inflation as considered in Ref. [41]. A field whose field value con- Combining Eqs. (3.8) and (3.10) we find that for appropriate values of H I , the early universe axion mass is large enough for relaxation of the axion field to its minimum. Since the largeness of the dynamical scale Λ QCD depends on the VEV of φ, the decay of the inflaton and proceeding relaxation of φ to today's VEV means that the post-inflationary cosmology is non-trivial. Prior to exploring this complex cosmology, however, we turn our attention to the relative π phase shift of the axion potential.

Shifted Axion Potential
Another consequence of a large Higgs VEV during inflation from the Kähler potential in Eq. (3.1) is that the relative phase between the Higgs fields is shifted by π as can be seen explicitly in Eq.

COSMOLOGICAL EVOLUTION
In Sec. 3 we demonstrated that, in the early universe, both a large axion mass and a phase shift of π are possible due to a large Higgs VEV with an opposite phase from today. The remaining question is whether there exists a viable cosmology with a consistent evolution between the two periods without spoiling predictions. We first explore the inflationary and post-inflationary constraints in the minimal model, and then later comment on the broader parameter space allowed by extended models.

Minimal Models
The first consistency check we should perform is to ensure the axion mass is larger than One needs to carefully consider the evolution of the axion potential after inflation ends.
There must be a transition of the value of ξ from the inflationary minimum toward today's minimum. This transition necessarily induces the transition of the minimum from π to 0 in the axion potential. If this transition occurs at a time when the enhanced axion mass is still comparable to or larger than Hubble, the misalignment angle could relax to a value very different from π. To understand this constraint, we turn to the post-inflationary evolution of the Higgs fields.
The Hubble induced mass terms which stabilize φ at a large VEV are tied to the inflaton energy density. If the sign of the Hubble induced terms remains the same after inflation, MSSM soft terms of the corresponding mode as the inflaton energy density redshifts and/or decays.
A second possibility for this evolution is that the sign of the Hubble induced mass flips after inflation (except for the H u H d term.) This may occur in two-field inflation models. For example, with K = (−c Z |Z| 2 +cZ|Z| 2 )|φ| 2 with c Z > cZ and W = mZZ, we assume that the scalar component of Z acquires a large field value and drives inflation. It isZ whose F -term, During inflation, the kinetic energy of φ Z is much smaller than its potential energy, i.e. |∂ µ Z| 2 F 2 Z , and thus the Hubble induced mass for φ is negative. As inflation ends, Z s potential and kinetic energies become comparable but, since c Z > cZ, the sign of the Hubble induced mass for the Higgs radial mode flips to positive. Consequently, φ is no longer trapped at a large VEV but oscillates towards the origin immediately after inflation. The early onset of radial oscillations helps because a longer period of redshifting in φ suppresses Λ QCD , which leads to the desired post-inflationary suppression of the axion mass.
It is only necessary to track the ratio m a /H between the onset of angular oscillations at cH 2 = Bµ and thermalization of the Higgs fields. During this period, the Higgs phase ξ can evolve and with it comes the shift in the axion potential. If the axion mass is subdominant to Hubble friction, however, the axion field is overdamped and remains agnostic to this evolution. When the Higgs is finally thermalized, its energy density is depleted and the field is quickly set to the minimum today, removing the axion mass enhancement. As a result, to preserve the prediction of the axion misalignment angle, m a /H needs to stay under unity during this period.
Thermalization of the Higgs is mediated by scattering with gluons via a loop-suppressed operator [69,70] with B 10 −2 and φ identified as the oscillation amplitude. Interestingly, due to the scaling properties during a matter-dominated era, the Higgs scattering generates a radiation energy density that is constant in time, whose contribution to thermal bath's temperature is with φ i as the field value of the Higgs at the end of inflation. This radiation persists throughout the evolution until the Higgs fields are thermalized at H Γ h . This radiation is important because in some cases it can dominate over the radiation produced from the inflaton decay and cause a period of a constant temperature in the cosmological evolution.
This has the effect of maintaining the finite temperature suppression to the axion mass where the temperature dependence is determined by the contribution from the gauge multiplets, while the contribution from chiral multiplets vanishes because of the cancellation between the RGE contribution and the fermion mass suppression. In the extended models, Λ QCD may be different from the estimate in Eq. (3.7) because of the backreaction from strong QCD dynamics. We note that the value of m a evolves after inflation not only due to this temperature suppression, but also its explicit dependence on Λ fid ∝ φ Another requirement of DAMP π comes from avoiding PQ symmetry restoration. In a thermal environment, the PQ breaking field saxion P acquires a thermal mass y 2 T 2 P 2 because of the Yukawa coupling yP QQ with the PQ quarks Q,Q. This thermal mass can be relevant at large temperatures and stabilize P at a vanishing value to restore PQ symmetry. This can be easily prevented if the symmetry breaking is enforced by the following superpotential, The F -term of S stabilizes P andP in the moduli space PP = f 2 a to break PQ. The coupling constant λ may be as large as 4π in strongly coupled models [71,72]. To ensure that PQ is not thermally restored after inflation where the maximum temperature achieved during reheating is T max (H I M Pl T 2 R ) 1/4 , 4 the thermal mass must be less than λf a < 4πf a at this time, giving an upper bound on the Yukawa coupling Finally, the purple regions in Fig. 2 are excluded by energy conservation which restricts the reheat temperature T R to a maximum value dictated by the energy in the inflaton, In the red regions, the axion starts to oscillate from the hilltop towards today's minimum during a matter-dominated era by the inflaton, in which case reheating produces entropy, dilutes the axion abundance, and spoils the prediction of a small f a . Even though the minimal model has proven to provide a viable cosmology for DAMP π , we explore extended models in the following subsection to further broaden the parameter space.

Extended Models
The most stringent constraint in the minimal model is a relatively low upper bound on H I due to difficulties in enhancing the axion mass during inflation. As shown in Eq. (3.10), we can enhance Λ QCD , and consequently m a , by introducing additional matter content. As we raise the value of Λ QCD , the values of fields in the PQ sector may be shifted from the one in the vacuum and impact the evaluation of the axion mass. The field value of the Higgs may be also affected.
We consider a simple model where a PQ symmetry breaking field P couples to KSVZ quarks QQ by a Yukawa coupling y. We need to reliably evaluate the VEVs of both P and QQ, as both are PQ charged and the decay constant during inflation f I is given by the larger of P and (QQ) 1/2 . The superpotential of P and QQ is where the first term is the non-perturbative Affleck-Dine-Seiberg superpotential [74]. We omit the factors of 4π except for the evaluation of the axion mass. The scaleΛ is related The relation can be obtained by comparing the effective superpotential after integrating out QQ from Eq. (4.6) and the effective potential where the effect of a P field value different from f a is included. Note that the potential of P from strong dynamics exhibits a runway behavior, ∂W eff /∂P ∝ P −2/3 . For this reason, we introduce a higher dimensional term |P | 6 /M 2 to further stabilize P at a large field value which can come from a superpotential term (χ/M )P 3 , where χ is a chiral field. We also consider the Hubble induced mass of P and QQ. Explicitly we take We stress that these terms are used in our analysis but they are not the only possible extensions to DAMP π . The sign of the Hubble induced mass of Q andQ are taken to be positive to ensure that Q is not destabilized by the Hubble induce mass, whereas that of P is positive so that the PQ symmetry is not restored.
As the Hubble induced mass of Q breaks the supersymmetry of the QCD sector, the axion mass may non-trivially depends on the parameters. When yP is larger than Q, QQ can be integrated out at the mass threshold yP and the effective potential is given by Eq. (4.8).
The axion mass is given by If Q is larger than yP , the theory below the mass threshold Q is a supersymmetric, pure SU (2) gauge theory with a dynamical scaleΛ(Λ/Q) 1/3 . The axion mass is then given by m a 1 4π Note that the two formulae agree with each other when the field value of Q is determined by the F -term condition of QQ from the superpotential in Eq. (4.6). We find that in the allowed parameter space, either yP or Q is larger than the dynamical scale and the procedure of integrating out QQ can be safely done.
Strong dynamics also affects the Higgs. The effective superpotential of φ is given by  This gives the Higgs a mass Λ 3 eff /(16π 2 M 2 ), which should be smaller than √ cH I .
By computing and comparing the axion mass to H I , we put an upper bound on the allowed values of H I such that DAMP π 's first criterion is fulfilled during inflation, which is shown in Fig. 3. In deriving the blue-shaded region, we integrate out QQ, obtain the scalar potential of P from the effective potential Eq. (4.8), and add the potential of P in Eq. (4.9), determine the field value of P during inflation, and compute the axion mass. This corresponds to the case where yP Λ , √ cH I . The gray contours show the constraint using the full potential described above. As y becomes larger the constraints approach to the blue-shaded region. An additional constraint shown in the red regions arises because strong dynamics drives the Higgs to the origin. We now discuss the post-inflationary evolution and constraints similar to Sec. 4.1. Although this study is comprehensive in evaluating the inflationary constraints, the thermal masses for P and Q dramatically complicate PQ dynamics during and after reheating.
Nonetheless, we are able to identify a large allowed parameter space in the H I , T R plane in the following way. There exists a wide region where the Higgs fields thermalize before the Higgs angular mode ξ begins to oscillate so that Λ QCD is quickly set to today's value Λ QCD before ξ has a chance to evolve and shift the axion potential. Physically, this means that the axion potential turns off before the location of its minimum shifts. Therefore, the prediction of the misalignment angle is automatically preserved without the need to track the post-inflationary evolution of P and QQ. This is the case when the Higgs scattering rate in Eq. (4.1) equals the Hubble rate before cH 2 drops below Bµ. This region is described by an allowed window of T R for a given H I where the two distinct formulae come from thermalization during the matter-and radiation-dominated epochs respectively. This window becomes wider as H I increases so as long as a consistent range of T R exists. The upper bound on H I ultimately enters from the inflationary constraint shown in the blue regions of Fig. 3.
We now comment on one plausible extension to further open up the parameter space with higher H I . The gluino mass affects the axion mass as in Eq. (3.8) and is assumed to stay invariant between inflation and today. If mg is also larger during inflation, the axion mass and the upper bound on H I can be raised by as much as (Λ fid /mg) 1/2 , making high scale inflation easily compatible with DAMP π .
With respect to PQ restoration in the extended models, we can assume the same PQ breaking mechanism as in Sec. 4.1 so the constraint in Eq. (4.5) applies equally here.
While the post-inflationary constraints are not fully evaluated, these extended models have been shown capable of fulfilling the criteria of DAMP π while extending the allowed parameter space to much higher H I than in Sec. 4.1.

CONCLUSION
It has been widely known that the misalignment mechanism can source axion dark matter in the early universe with a decay constant f a O(10 12 ) GeV. For f a 10 12 GeV as is of interest to many experimental searches, the observed DM abundance can be obtained if the misalignment angle θ mis is taken sufficiently close to π, where the anharmonic effect becomes important. In particular, when the axion is very close to the hilltop of the potential, the onset of oscillations is delayed so the axion abundance is less redshifted and thus more enhanced. As demonstrated in Fig. 2 and Eq. (2.7), f a 10 10 GeV corresponds to δθ ≡ π − θ mis O(10 −3 ), while f a 4 × 10 9 GeV already requires δθ O(10 −9 ). Such a small δθ has generically been understood as fine-tuning of the initial condition. In this paper, we offer an explanation to this small δθ using axion dynamics in the early universe.
We point out that a class of models violates the canonical assumption that the axion field is overdamped by Hubble friction and takes a random value during inflation. Instead, there exist numerous possibilities wherein the axion is large compared to Hubble during inflation and thus relaxes to the minimum of the potential. We refer to this mechanism as Dynamical Axion Misalignment Production (DAMP). Additionally, if the model possesses an approximate CP symmetry, then the axion potential may receive a phase shift of π because the nearly real parameters for setting the axion minimum can flip the sign between inflation and the QCD phase transition. This shift converts the potential minimum into a maximum and explains why the axion is very close to the hilltop-a mechanism we dub DAMP π .
We explicitly construct models for DAMP π , where the higher dimensional coupling between the Higgs in the MSSM and the inflaton gives rise to both a large axion mass and the phase shift of the axion potential. Specifically, a negative Hubble induced mass drives the Higgs to a large field value that enhances the quark masses, which in turn raise the QCD scale. The axion is larger than usual due to stronger QCD dynamics. Lastly, a Hubble induced Bµ term that carries an opposite sign from that of the MSSM necessarily induces a shift in the axion potential by π. Together, renormalization from the SM Yukawa couplings and any additional CP violating phases in the model can provide the desired finite phase shift between O(10 −16 -10 −3 ). The mechanism works only if the number of generations is odd.
Strikingly, due to the anharmonic effects of the axion potential, the prediction of f a from the DM abundance has an extraordinarily mild logarithmic dependence on δθ 1.
Other phenomenological features of DAMP π are as follows. Thanks to early relaxation, axion dark matter has exponentially damped isocurvature perturbations and instead has nearly pure adiabatic perturbations. Due to attractive self-interactions near the hilltop, the axion adiabatic perturbations can be greatly amplified by the anharmonic effects and grow into axion miniclusters [33,35,75,76]. The axions from the misalignment mechanism are necessarily cold-a feature to distinguish from other non-thermal production mechanisms.
It is also potentially interesting to study the imprints of maximal CP violation on the QCD phase transition as well as Big Bang nucleosynthesis.