Diluting the inflationary axion fluctuation by a stronger QCD in the early Universe

We propose a new mechanism to suppress the axion isocurvature perturbation, while producing the right amount of axion dark matter, within the framework of supersymmetric axion models with the axion scale induced by supersymmetry breaking. The mechanism involves an intermediate phase transition to generate the Higgs \mu-parameter, before which the weak scale is comparable to the axion scale and the resulting stronger QCD yields an axion mass heavier than the Hubble scale over a certain period. Combined with that the Hubble-induced axion scale during the primordial inflation is well above the intermediate axion scale at present, the stronger QCD in the early Universe suppresses the axion fluctuation to be small enough even when the inflationary Hubble scale saturates the current upper bound, while generating an axion misalignment angle of order unity.

well as a restored PQ phase until some moment after the primordial inflation.
Another scenario which we will focus on in this paper is that U(1) PQ is spontaneously broken during the primordial inflation and never restored afterwards. Then the model is not subject to the condition N DW = 1, but is constrained by the axion isocurvature perturbation [4][5][6]. For instance, from the observed CMB power spectrum, one finds [7], where θ mis and δθ denote the average misalignment angle and the angle fluctuation, respectively, for the axion field right before the conventional QCD phase transition when m a (t QCD ) ≈ H(t QCD ) with a temperature T (t QCD ) ∼ 1 GeV. The relic axion density is given by with DM ≈ 0.24 being the total dark matter fraction. Here we have assumed that |δθ| |θ mis | and there is no significant evolution of f a from t QCD to the present time t 0 so that f a (t QCD ) ≈ f a (t 0 ). In inflationary cosmology, the primordial quantum fluctuation of the axion field results in δθ ≡ δθ(t QCD ) = γ δθ(t I ) = γ where f a (t I ) and H(t I ) denote the axion scale and the Hubble parameter, respectively, during the primordial inflation epoch t I , and the factor γ is introduced to take into account the evolution of δθ from t I to t QCD . Note that the inflationary Hubble scale H(t I ) is bounded by the tensor-to-scalar ratio of the CMB perturbation as r 0.16 and the weak gravity conjecture [8] suggests that generic axion scales are bounded as where M Pl 2.4 × 10 18 GeV is the reduced Planck mass.
To discuss the implication of the isocurvature constraint (1), one needs to specify the cosmological evolution of the axion scale after the primordial inflation is over. If f a (t I ) ∼ f a (t 0 ) as has been assumed in most of the previous studies, it requires that either H(t I ) is smaller than its upper bound ∼ 10 14 GeV by at least five orders of magnitude, so that the CMB tensor mode is too small to be observable, or δθ should experience a large suppression after the primordial inflation, which appears to be difficult to be implemented.
The above observation suggests a more attractive scenario real- [9] in a natural manner. Indeed supersymmetric axion models offer a natural scheme to realize such a scenario, generating the axion scale through the competition between the tachyonic SUSY breaking mass term and a supersymmetric, but Planck-scale-suppressed higher dimensional term in the scalar potential [10][11][12][13]. One then finds which explains elegantly the origin of an intermediate axion scale at present, while giving a Hubble-induced inflationary axion scale well above the present axion scale, if the supersymmetry (SUSY) breaking mass m SUSY at present is around TeV scale. Furthermore, this type of axion models can be successfully embedded into string theory. Specifically, they can be identified as a low energy limit of string models involving an anomalous U(1) A gauge symmetry with vanishing Fayet-Illiopoulos term [12,14]. In such string models, the U(1) A gauge boson is decoupled from the low energy world by receiving a heavy mass M A ∼ g 2 M Pl /8π 2 through the Stückelberg mechanism, while leaving the global part of U(1) A as an unbroken PQ symmetry in the supersymmetric limit. Once SUSY breaking is introduced properly, in both the present Universe and the inflationary early Universe, the residual PQ symmetry can be spontaneously broken to generate the axion scales as (6).
In this paper, we discuss a novel mechanism to suppress the axion isocurvature perturbation, while producing the right amount of axion dark mater, within the framework of supersymmetric axion models with the axion scales given by (6). The isocurvature constraint (1) and the relic axion density (2) suggest that for H(t I ) near the current upper bound ∼ 10 14 GeV, the allowed amount of axion dark matter is maximal when f a (t 0 ) ∼ 10 11 -10 13 GeV, while f a (t I ) nearly saturates the weak gravity bound (5), e.g. f a (t I ) ∼ 10 16 -10 17 GeV. Interestingly, the axion scales generated by SUSY breaking as (6) automatically realize such pattern if m SUSY is around TeV scale. More specifically, for the case the isocurvature bound (1) reads off H(t I ) 10  To suppress δθ through its cosmological evolution, one needs a period with m a (t) > H(t) well before t QCD . On the other hand, usually this is not easy to be realized because the axion mass should be generated mostly by the QCD anomaly in order for the strong CP problem solved by the PQ mechanism. (See Refs. [15][16][17][18] for an alternative possibility.) In the following, we propose a simple scheme to achieve such a cosmological period by having a phase of stronger QCD in the early Universe. Although the simplest model that realizes our scenario suffers from a new domain-wall problem, we can address it by adding PQ-charged superfields such that the discrete symmetry of the model is broken before inflation.
Our scheme is based on a phase transition at t = t μ t I , which will be called the μ-transition in the following as it generates the Higgs μ-parameter through the superpotential term [19], where X is a PQ-charged gauge-singlet superfield. Specifically, so that With this transition, the weak scale experiences an unusual evolution in a way that the weak scale before the μ-transition is comparable to the axion scale (6), as will be discussed below.
To proceed, let us discuss first the key features of the scheme, and later present an explicit model to realize the whole ingredients. Including the Hubble-induced contribution, the mass of the D-flat Higgs direction H u H d is generically given by where c φ and ξ φ are model-dependent parameters of order unity. In our scheme, both c φ and ξ φ are assumed to be negative, so the competition between the tachyonic m 2 φ |φ| 2 and a supersymmetric term of O(|φ| 6 On the other hand, after the μ-transition, m 2 φ > 0 due to μ ∼ m SUSY . The resulting weak scale and axion scale at present are given by A simple consequence of the above evolution of H u H d is that the weak scale is comparable to the axion scale before the μ-transition: This results in a higher QCD scale, i.e. a stronger QCD, and therefore a heavier axion mass which might be even bigger than the Hubble scale for a certain period. Let us estimate the QCD scale ˜ QCD before the μ-transition, which is defined as the scale where the 1-loop QCD coupling blows up, as well as the resulting axion mass m a . For the case with ˜ QCD <m˜g(m˜g) < 10 −5φ , where m˜g denotes the gluino mass before the μ-transition, we find QCD ≈ 23 TeV where tan β = H u / H d at present, and m˜g/g 2 for the gluino mass mg at present. Here we assume that On the other hand, if m˜g (˜ np ) <˜ QCD < 10 −5φ , the resulting QCD scale is estimated as with the axion mass m a ≈m Here ˜ np denotes the scale where the stronger QCD becomes nonperturbative, i.e. around g 2 3 = 8π 2 /N c with N c = 3. Note that the axion potential for the axion mass (18) can be obtained by a single insertion of the SUSY breaking spurion m˜gθ 2 to the nonperturbative superpotential W np ∼˜ 3 QCD induced by the gluino condensation.
If the stronger QCD scale ˜ QCD is high enough, there could be a period with m a (t) > H(t) well before the conventional QCD phase transition. As is well known, in such a period the axion field experiences a damped oscillation, with an amplitude ā (averaged over each oscillation period) evolving as a ∝ R −3/2 (t), (19) where R(t) is the scale factor of the expanding Universe. Then the spatially averaged vacuum value of the axion field is settled down at the minimum of the axion potential induced by the stronger QCD, while the axion angle fluctuation is diluted according to where t = t i denotes the moment when the damped axion oscillation begins, and t = t μ is the moment when it is over. Note that, after the μ-transition, the weak scale and the QCD scale quickly roll down to the present values, so the axion mass becomes negligible compared to H(t) until t ∼ t QCD when the Universe undergoes the conventional QCD phase transition. Also, the minimum of the axion potential induced by the stronger QCD is generically different from the minimum of the axion potential at present. As a result, our scheme generates an axion misalignment angle of order unity: over the period t i t t μ with a temperature ratio: Then the resulting δθ given by (20) can be small enough to satisfy the isocurvature bound (1) even when H(t I ) saturates its upper bound ∼ 10 14 GeV. Note that during t i t t μ , This means that in this period the Universe is dominated by the vacuum energy density with the Hubble scale given by (23), which is often called the thermal inflation [20]. It should be stressed that in our scheme the axion isocurvature perturbation is suppressed by two steps. The first suppression is Let us now present an explicit model implementing the mechanisms discussed above. As a simple example, we consider a model with the following superpotential, where X and Y are PQ-charged gauge singlets responsible for the μ-transition, L is the MSSM lepton doublet, and + c are U (1) Y -charged exotic matter fields introduced to give a thermal mass to Y . Then the scalar potential for the μ-transition is given where For simplicity, we will assume that all the dimensionless parameters appearing in the superpotential and the SUSY breaking scalar masses are of order unity. However it should be noted that these parameters can have a variation of O(0. [1][2][3][4][5][6][7][8][9][10] easily. In particular, the superpotential parameters κ n can have a much wider variation without invoking fine-tuning. This gives us a rather large room to get an enough suppression of the axion angle fluctuation δθ through a stronger QCD before the μ-transition. At any rate, assuming that c X,Y > 0, ξ X > 0 and ξ Y < 0, the scalar potential (26) indeed yields the desired μ-transition as with T (t μ ) ∼ m SUSY . 1 Now the Higgs and slepton fields can have a nontrivial evolution along the following flat direction: 1 In our example (25), the cosmological evolution of X and Y can cause a domain-wall problem associated with the discrete symmetry generated by X → −X and Y → e iπ /3 Y . This problem can be avoided by extending the model to make not restored. For instance, one can introduce additional PQ-charged fields Z and ϕ with Z ϕ 2 + Y ϕ 3 /M Pl in the superpotential. If ϕ has a tachyonic SUSY breaking mass (ξ ϕ < 0), as well as a tachyonic Hubble-induced mass (c ϕ < 0), the discrete symmetry of the model, which is now involving ϕ 3 → e −iπ /3 ϕ 3 , is never restored in the early Universe, and the model is free from the domain-wall problem, while implementing our mechanism. It is also possible to construct a model in which the μ-transition is triggered by a PQ singlet in a way not having any discrete symmetry. which satisfies the F and D flat conditions. The relevant terms of the scalar potential of φ d, are given by Again, assuming c d, < 0 and ξ d, < 0, but m 2 φ d ,φ (t 0 ) > 0 due to μ(t 0 ) ∼ m SUSY , the above scalar potential yields and To summarize, under a reasonably plausible assumption on the SUSY breaking during the primordial inflation and in the present Universe, the model with the superpotential (25) can successfully realize the desired cosmological evolution of the three relevant scales: the axion scale, the weak scale, and the QCD scale as given by (28), (32) and (33). Being generated by SUSY breaking, an inflationary axion scale f a (t I ) ∼ √ H(t I )/m SUSY f a (t 0 ) is determined to be well above the present axion scale f a (t 0 ) ∼ √ m SUSY M Pl , and a stronger QCD in the early Universe is realized to yield an enough suppression of the axion angle fluctuation even when H(t I ) saturates its upper bound. We note that the minimum of the axion potential induced by the stronger QCD depends on arg(κ 3 A 3 ), but not on arg(Bμ), while the minimum of the axion potential at present depends on arg(Bμ), but not on arg(κ 3 A 3 ). As a result, the stronger QCD generates an axion misalignment angle θ mis = O(1), so that the axion dark matter with a = DM arises naturally in our scheme.
There is a remaining issue which should be addressed to complete our scheme. As we have noticed, the μ-transition is foregone by a late-time thermal inflation. This suggests that the scheme should be accompanied by a late-time baryogenesis operating after the μ-transition. In fact, the model of (25) offers an elegant mechanism to generate the baryon asymmetry through the rolling flat direction L H u [21]. More detailed cosmology of our scheme, including the leptogenesis by rolling L H u , will be discussed elsewhere [22].