A Minimal Sub-Planckian Axion Inflation Model with Large Tensor-to-Scalar Ratio

We present a minimal axion inflation model which can generate a large tensor-to-scalar ratio while remaining sub-Planckian. The modulus of a complex scalar field $\Phi$ with a $\lambda |\Phi|^4$ potential couples directly to the gauge field of a strongly-coupled sector via a term of the form $(|\Phi|/M_{Pl})^{m} F \tilde{F}$. This generates a minimum of the potential which is aperiodic in the phase. The resulting inflation model is equivalent to a $\phi^{4/(m+1)}$ chaotic inflation model. For the natural case of a leading-order portal-like interaction $\Phi^{\dagger}\Phi F \tilde{F}$, the model is equivalent to a $\phi^{4/3}$ chaotic inflation model and predicts a tensor-to-scalar ratio $r = 16/3N = 0.097$ and a scalar spectral index $n_{s} = 1-5/3N = 0.970$. The value of $|\Phi|$ remains sub-Planckian throughout the observable era of inflation, with $|\Phi| \lesssim 0.01 M_{Pl}$ for $N \lesssim 60$ when $\lambda \sim 1$.


II. A MINIMAL SUB-PLANCKIAN AXION INFLATION MODEL WITH LARGE r
The model is similar to the heavy quark axion (KSVZ) model [14]. We introduce a complex field Φ, which is a gauge singlet, and a chiral U(1) A global symmetry. We also introduce a heavy fermion Q in the fundamental representation of the gauge group, where Φ interacts with Q via The Φ scalar potential is * Electronic address: j.mcdonald@lancaster.ac.uk 1 M Pl = (8πG) −1/2 . 2 In [5] it is shown that a hybrid inflation model based on unification energy can fit all observations with φ ∼ √ 8πM Pl . 3 For an alternative shift-symmetry mechanism motivated by extra-dimensions, see [8].
where λ ∼ 1 is expected dimensionally. Q gains its mass from the VEV of Φ. During inflation we can assume that |Φ| ≫ µ. Therefore we will set µ = 0 in the following and consider V (Φ) = λ|Φ| 4 . Φ has charge +1 under U(1) A and Q has charge +1/2. The phase θ of Φ (= φe iθ / √ 2) in Eq. (1) can be rotated away via a U(1) A transformation. This results in a U(1) A -breaking interaction of θ with the gauge fields due to the chiral anomaly, Here FF = F µνF µν ,F µν = 1 2 ε µνρσ F ρσ and g is the gauge coupling of the strongly-coupled gauge sector. (Gauge indices are suppressed.) In general, we can also include a U(1) A -symmetric non-renormalizable interaction of the form where ξ is a dimensionless parameter. The combination of Eq. (3) and Eq. (4) can then be written as where Λ = M Pl /(32π 2 ξ) 1/m . We define the strong coupling scale to be Λ sc . The potential term generated by the strongly-coupled gauge sector is then Therefore we can define the full potential for Φ during inflation to be where we have added a constant term Λ 4 sc so that the potential equals zero at the global minimum 4 . This potential has a minimum which is aperiodic in θ. Along the φ direction for a given θ, the strong-coupling term modulates the |Φ| 4 potential. For a range of parameters which we will determine below, there are local minimum of the potential as a function of |Φ|, which correspond to the cosine term being close to 1. The value of |Φ| at these minima satisfies where n is an integer. This is a good approximation if (φ/Λ) m ≫ 1. This results in a spiralling groove inscribed on the |Φ| 4 potential in the complex Φ plane. Inflation can occur along this groove, allowing the field to traverse a long distance in field space while |Φ| remains sub-Planckian throughout.
The inflation dynamics of this model are the same as that of the model presented in [15], where a multiplicative modulation of the |Φ| 4 potential was considered. The distance a along the minimum in field space is related to θ by From Eq. (8), If ( √ 2Λ/φ) 2m ≪ 1 then to a good approximation da = φ(θ)dθ. In this case we can consider a to be a canonically normalized field along the minimum of the potential. The model will behave as a single field inflation model if the field φ orthogonal to a has a mass much larger than H. (We will derive the condition for this to be true later.) Using da = φ(θ)dθ, we find Therefore where we have defined a = 0 at φ = φ 0 . We can then define a new slow-roll field,â, given bŷ Along the minimum ("groove"), the potential is The model will therefore have the same inflaton dynamics as a φ 4/(m+1) chaotic inflation model. The spectral index n s and the tensor-to-scalar ratio r as a function of the number of e-foldings N are therefore and a and φ are related to N byâ and A case of particular interest is where the field Φ is the fundamental object out of which the effective theory is constructed, by which we mean that only integer powers of Φ and Φ † occur in the effective theory at low energies. This excludes, for example, terms proportional to |Φ|. In this case the natural U(1) A -invariant combination is the bilinear Φ † Φ. If Planck-scale physics generates a U(1) A -symmetric interaction of the form is assumed to be expandable with a leadingorder term proportional to x when x ≪ 1, there will be a portal-like leading-order interaction of the form Φ † ΦFF, corresponding to Eq. (4) with m = 2. In the following we will focus attention on the m = 2 model, which we consider to be the most likely form of coupling to the gauge sector.
The m = 2 model is equivalent to a φ 4/3 chaotic inflation model. In this case the predictions for N = 55 are n s = 1 − 5/3N = 0.970 and r = 16/3N = 0.097, with a negligible running of n s . The spectral index is in reasonable agreement with the value determined by Planck, n s = 0.9624 ± 0.0075 (Planck + WP, assuming negligible running and including a tensor component [16]). The values φ and Λ are determined by the curvature perturbation power spectrum, where P . (21) For m = 2 and N = 55 these become and Thus we find that |Φ| < ∼ 0.01M Pl throughout the observable era of inflation when λ ∼ 1. Therefore the model can produce a large value for the tensor-to-scalar ratio while remaining sub-Planckian throughout. The model also allows conventional particle physics strength |Φ| 4 potentials with λ ∼ 1 to serve as basis for the inflaton potential.
The value of Λ is small compared with M Pl . This means that the dimensionless coupling ξ is necessarily large when λ ∼ 1, Such large dimensionless couplings are not without precedent in inflation models. In non-minimally coupled models of inflation based on a |Φ| 4 potential [17], the coupling ξ between Φ † Φ and the Ricci scalar R is given by ξ ≈ 10 5 √ λ [17]. In both models, the large value of ξ effectively replaces the small scalar coupling of conventional φ 4 chaotic inflation models.
Therefore ξ = 1 implies that |Φ|/M Pl = 0.35. However, if we expect the leading-order Planck correction to the potential to be of the order of |Φ| 6 /M 2 Pl , then this will dominate the λ|Φ| 4 term due to the small value of λ. On the other hand, if we simply wish |Φ| to be sub-Planckian so that the potential is calculable with respect to Planck corrections, then ξ ∼ 1 is possible.
We next determine the conditions for the underlying assumptions of the model to be consistent. We have assumed that a local minimum in the radial direction exists. This requires that the derivative of the potential in the φ direction is dominated by the strong-coupling term, This imposes a lower bound on Λ sc , For m = 2 this becomes Using Eq. (22) and Eq. (23) we obtain We have also assumed that the effective mass squared at the minimum in the radial direction, which is dominated by the strong coupling term, is large enough to reduce the dynamics of the model to a single-field inflation model in theâ direction. This requires that V ′′ sc (φ) ≫ H 2 at the minimum. This also imposes a lower bound on Λ sc , For m = 2 this becomes, Using Eq. (22) and Eq. (23) we obtain This is a weaker lower bound than that from the existence of the minimum, Eq. (29). Therefore if Λ sc > 3.4 × 10 15 λ −1/16 GeV then the m = 2 model can consistently account for sub-Planckian inflation while generating a large value for r. The assumptions underlying the model with λ ∼ 1 will therefore be well-satisfied if 5 Λ sc > ∼ 10 16 GeV.

III. CONCLUSIONS
We have presented a minimal axion inflation model which is consistent with the observed value of n s and which can generate a large tensor-to-scalar ratio r ∼ 0.1 while remaining sub-Planckian throughout. The model also allows λ|Φ| 4 potentials with λ ∼ 1 to serve as basis for the inflaton potential. The model requires only a single complex field and a single strongly-coupled gauge group. For the case where the effective theory at low energies is constructed from integer powers of Φ and Φ † and all non-renormalizable terms are part of an expansion in inverse powers of the Planck scale, the most likely coupling of Φ to the gauge sector has the portal-like form Φ † ΦFF. In this case the model is dynamically equivalent to a φ 4/3 chaotic inflation model, with n s = 0.970 and r = 0.097. The model is explicitly sub-Planckian throughout the observable era of inflation, with |Φ| < ∼ 0.01M Pl for N < ∼ 60. The strong coupling scale must be greater than 10 16 GeV for the model to be consistent. If the λ|Φ| 4 coupling takes the dimensionally natural value expected in conventional particle physics theories, λ ∼ 1, then the dimensionless coupling ξ of Φ † Φ to the gauge fields must be large, ξ ∼ 10 5 , in order to reproduce the observed CMB temperature anisotropies. Values of ξ ∼ 1 also result in sub-Planckian |Φ|, with |Φ|/M Pl = 0.35 ξ −1/3 . This case requires additional suppression of Planck corrections to the potential but allows the potential to be calculable with respect to such corrections.