UV-Protected Inflation

In Natural Inflation, the Inflaton is a pseudo-Nambu-Goldstone boson which acquires a mass by explicit breaking of a global shift symmetry at scale f. In this case, for small field values, the potential is flat and stable under radiative corrections. Nevertheless, slow roll conditions enforce f>>M_p making the validity of the whole scenario questionable. In this letter, we show that a coupling of the Inflaton kinetic term to the Einstein tensor allows f<<M_p by enhancing the gravitational friction acting on the Inflaton during inflation. This new unique interaction, a) keeps the theory perturbative in the whole inflationary trajectory, b) preserves the tree-level shift invariance of the pseudo-Nambu-Goldstone Boson and c) avoids the introduction of any new degrees of freedom with respect the standard Natural Inflation.


INTRODUCTION
Inflation, a rapid expansion of the early Universe, is a beautiful solution of the homogeneity, isotropy and flatness puzzles [1]. Although an isotropic expansion of the Universe might be obtained by considering general nonminimally coupled p-forms [2], the minimally coupled zero-form (a scalar field) is the most simple and natural source for inflation [3][4][5]. In this case, the scalar field during inflation generates an almost DeSitter (exponential) expansion of the early Universe.
In a Friedmann-Robertson-Walker spacetime (FRW) with metric ds 2 = −dt 2 +a(t) 2 d x·d x, a minimally coupled scalar field (φ) to gravity, with a potential V (φ) > 0, produces an accelerationä ∼ −(φ 2 − V ), where (˙) = d/dt. It is then clear that to get an accelerated expansion (ä > 0) for a long time, the field φ has to "slow roll" in its own potential, i.e.φ 2 ≪ V . Unfortunately though, while solving the cosmological puzzles, this seemly innocuous condition threaten the whole inflationary paradigm, as we shall see.
During slow roll, the Hubble equation and field equations for the Inflaton are where H =ȧ/a is the Hubble "constant", ( ′ ) = d/dφ and M p is the Planck scale. Eqs. (1) are found by considering the slow roll conditions By using (1) and pluging into (2) we find the two independent conditions A common problem of standard inflationary scenarios is the "eta" problem. Gravity is not a renormalizable theory as its perturbative expansion breaks down at the Planck scale M p . Therefore, operators suppressed by the scale M p , although not known, are generically expected to complete the theory at UV. In particular, one would expect the Inflaton potential to be generically "corrected" by higher dimensional operators. Consider for example the following dimension six correctioñ where c is an unknown coefficient expected to be of O (1). With this correction, we have that the η parameter is modified asη ≃ η + c 2 3 + . . ., therefore, if η ≪ 1, in order to keepη ≪ 1 we need c ≪ 1. However, this coefficient cannot be calculated unless the full UV completed theory of gravity is known. Note that for small field scenarios this is the leading correction to η. In large field scenario the problem is clearly far more severe as an infinite amount of non-renormalizable operators has to be set exactly to zero.
The "eta" problem might be nevertheless naturally solved in small fields scenarios by introducing new symmetries in the Inflaton lagrangian. These symmetries might in fact force the potential to be flat even under radiative corrections. There are two possible symmetries to achieve this goal: global and local.
Local symmetries (such as local supersymmerty) might stabilize the Inflaton potential in supergravity. Still, in this framework, supersymmetry is explicitly broken by the gravitational background making the Inflaton potential generically too steep to produce inflation [6]. Additional local symmetries related to the matter content of the full theory in which the Inflaton is embedded might nevertheless change this no-go result, as proposed in [7]. However, to date, this direction is still under development.
The other possibility for a radiative stability of the Inflaton potential is the existence of global symmetries. A use of global symmetries in inflation is encoded in the so called Natural Inflation [8]. In Natural Inflation, the Inflaton is a pseudo-Nambu-Goldstone boson acquiring a mass by explicit breaking of a global shift symmetry at scale f . This happens by instanton effects, as in the Peccei-Quinn mechanism [9]. In this case, for small field values, the potential is flat and stable under radiative corrections. Nevertheless, the slow roll condition η ≪ 1 implies f ≫ M p making the whole scenario unreliable.
In the following, we will show that the scale f might be safely taken to be much smaller than the Planck scale by introducing a non-minimal coupling of the Inflaton kinetic term to the Einstein tensor. In particular, we will show that this new theory is in the weak coupling regime during the whole inflationary evolution and does not propagate any more degrees of freedom than the ones already existing in Natural Inflation.

CONSTRUCTING THE ACTION
We now consider the most generic theory such that • It is diffeomorphisms invariant; • It only propagates a (non-ghost) massless spin 2 and a spin 0 particle on any background; Such action has been found in [10] and it is In (3) we used the notation where G αβ ≡ R αβ − 1 2 Rg αβ and M are respectively the Einstein tensor and a new mass scale, not necessarily related to the Planck mass. Note that the minus sign in the definition (4) is crucial to avoid ghosts, whenever weak energy conditions are satisfied [11]. One may wonder whether an infinite series of curvatures non-minimally coupled to the kinetic term of the scalar, can be added to (3). We claim that it is unlikely that a fine tuning exist in which both metric and scalar equation of motions are second order in derivatives.
In Natural Inflation the massless field θ is a pseudo-Nambu-Goldstone field with decay constant f and periodicity 2π. Let us consider the following tree-level lagrangian for θ where ψ is a fermion charged under the (non-abelian) gauge field with field strength F αβ , D = γ α D α is the gauge invariant derivative and m is a mass scale.
The chiral symmetry of the system is however broken at one loop level [12] giving the effective interaction θF ·F , whereF µν = √ −gǫ αβµν F αβ and ǫ αβµν is Levi-Civita antisymmetric symbol. Instanton effects related to the gauge theory F introduce a potential K(F ·F ) [13]. In the zero momentum limit we can integrate out the combination F ·F and obtain a periodic potential for the field θ (note that this is independent upon the canonical normalization of θ) which has a stable minimum at θ = 0 [14]. If we expand the potential around its own maximum at θ = θ max , we get where Λ is the strong coupling scale of the gauge theory F [15] and φ = f (θ − θ max ). The approximation (6) is valid as long as φ ≪ f and it is precisely in this regime that the Universe can naturally inflate, as it is shown later on. What is very important to note is that since the nonlinearly realized symmetry is restored as Λ → 0, the potential (6) is natural and no UV corrections may spoil its flatness.
Finally, the action we will use for the inflationary background is therefore

STRONG COUPLING SCALE
In order to find the validity range of the effective action (7) we should find the energy scale in which nonrenormalizable operators become strong. Obviously, in a non-minimal coupled model, this is a background dependent question as already noted in [10,16]. In a nontrivial background for the scalar field configuration one might employ the gauge δφ = 0 (at all orders), which is quite useful in cosmological perturbations theory [17]. In this case the metric is perturbed as g αβ = γ αβ + h αβ Mp , where γ αβ and h αβ are respectively the background and the perturbed metrics. We now use the ADM formalism where the metric is generically written as ds 2 = −N 2 dt 2 + h ij dx i − N i dt dx j − N j dt . In this form the degrees of freedom of the graviton are encoded into h ij and N, N i are lagrange multiplier in the action (7) [10]. In this formalism one might define the extrinsic curvature K ij = 1 N ḣ ij − D (i N j) and the three curvature R where the covariant derivative D i and R are both calculated with the three-metric h ij [11]. The perturbed action (7) is then [18] As we shall see, during slow roll inflation,φ 2 M 2 M 2 p ≪ 1 and considered roughly constant. Therefore (8) is well approximated as so that the strong coupling scale of (9) is manifestly M p . Note that in the case of multi-fields coupled to (4), as in the New Higgs Inflation [10], the strong coupling scale is lower [10]. Another source for a strong coupling scale is the operator S int = d 4 x √ −g φ f F ·F , which was integrated out to get the action (7) [24].
In ADM formalism, and during slow roll, the scalar perturbations of the metric are codified into the canonically normalized scalar perturbationζ in h ij = a(t) 3 (1 + 2 Ṁ φζ )δ ij [18]. After integrating by parts S int in the δφ = 0 gauge, we get the perturbed action where δC αβγ is the perturbed Chern-Simons three-form relater to the gauge field F [13]. This interaction has a renormalizable and a non-renormalizable term. The renormalizable term is always in the weak regime as long as f ≫ M . The non-renormalizable interaction however, introduce a new strong coupling scale M new = H f M . The strong coupling scale of the theory will then be M * = min (M p , M new ). Note that in the Minkowski limit the canonical normalization ofζ is different and it boils down into the replacement of H instead of M in M new , so that M new → f , as expected. Moreover, because slow roll conditions are violated in the Minkowski limit (M p H ∼ φ → 0), the unitary violating scale related to the operator ∆ αβ smoothly converge to M * ∼ (M p M 2 ) 1/3 . What is important to note is that during the whole evolution, from the Inflationary to the Minkowski background, the curvature is always below the scale M * , so that our theory can be perturbatively trusted.

UV-PROTECTED INFLATION
The variation of the action (7) with respect to the lapse N and the field φ give rise to the two Hubble and field equations [10] Another issue is related to the global symmetry of the tree-level action (5). It is widely believed that Quantum Gravity does not allow global symmetries. This is due to the fact that Black Hole evaporation democratically emit any particle coupled to gravity. The only constraint on this emission is to conserve the total energy and/or fluxes at infinity. Global charges do not carry any flux and therefore cannot be conserved. In our scenario, however, the global (shift) symmetry is already broken by gauge instanton effects. Therefore, the only way that gravity may participate to the quantum breaking of the global chiral symmetry of (5) is through gravitational anomalies. The latter, can only couple to the axion via the interaction θRR = θ √ −gǫ αβµν R γ δαβ R δ γµν , related to the gravitational Chern-Simons three-form [13]. A potential K G (RR) might then be generated by instanton effects if and only if the zero momentum limit of the instanton correlator RR, RR does not vanish [13]. In any case, even if the potential K G (RR) is generated, it is suppressed by the factor e −S ≪ 1, where S is the instanton action. Quantum gravity effects can therefore only produce a small correction to the mass of the Inflaton field by redefining a new effective Λ.
Although the potential we introduced is stable under radiative corrections, one might wonder about derivative terms generated by loops corrections. During slow roll inflation, all these corrections are negligible as proportional to the slow roll parameters and suppressed by the scale M * .
Concluding, in our case in which f, Λ, M, H ≪ M * , there are no substantial quantum (gravity) corrections to the inflationary evolution.

CONCLUSIONS
A pseudo-Nambu-Goldstone scalar (the axion), whose potential is obtained by a global symmetry breaking at scale f via gauge field instanton effects, has a naturally flat potential, as long as f ≪ M p . Unfortunately though, slow roll conditions for the axion require f ≫ M p . This requirement might be softened by introducing a plethora of N ≫ 1 axions as in [20], if large cross interaction among the fields can be avoided. In this case, the effective friction acting on the radial direction in the field space, is boosted by a factor N so that slow roll conditions require only f ≫ M p / √ N . However, in this framework, the strong coupling scale of the theory is lowered to M * ∼ M p / √ N [21]. This nullifies the attempt of [20] to produce a natural inflationary scenario [22]. This problem may however be solved if only two axions are considered. In this case, by a fine adjustment of their coupling to the anomalous currents, one can find f ≪ M * [23].
In this paper we showed an alternative way to increase the friction in a Natural Inflationary scenario without introducing any new degrees of freedom. In our model the friction of the axion is gravitationally enhanced. In this case, in order for the axion to slow roll down its own potential, a natural value for the global symmetry breaking scale f ≪ M p (or more precisely f ≪ M * ) can be easily obtained. This feature is uniquely obtained by the interaction of the Einstein tensor to the kinetic term of the axion which keeps nevertheless the theory perturbative during the whole inflationary evolution. This interaction is unique in the sense that it does not propagate more degrees of freedom than a massless spin 2 and a scalar, while keeping the tree-level shift invariance of the axion untouched.