Scalar-tensor extension of Natural Inflation
SScalar-tensor extension of Natural Inflation
Guillem Simeon ∗ Facultat de F´ısica, Universitat de Barcelona, Mart´ı i Franqu`es 1, 08028 Barcelona, Spain
We show that embedding Natural Inflation in a more general scalar-tensor theory, with non-minimal couplings to the Ricci scalar and the kinetic term, alleviates the current tension of NaturalInflation with observational data. The coupling functions respect the periodicity of the potentialand the characteristic shift symmetry φ → φ + 2 πf of the original Natural Inflation model, andvanish at the minimum of the potential. Furthermore, showing that the theory exhibits a rescalingsymmetry in the regime where the coupling to the Ricci scalar is small and the coupling to thekinetic term is large, we obtain that the agreement with cosmological data can take place at anarbitrarily low periodicity scale f , solving at tree-level the problem of super-Planckian periodicityscales needed in Natural Inflation. I. INTRODUCTION
Natural Inflation [1] is a model for inflation protected against radiative corrections, based on a pseudo-Nambu-Goldstone inflaton, with a potential of the form V ( φ ) = Λ (1 + cos ( φ/f )). Successful inflation is achieved bydemanding the periodicity scale f to take super-Planckian values and usually it is considered that Λ ∼ GeV,that is, Λ is taken to be of the order of the Grand Unification Theories (GUT) scale.A super-Planckian periodicity scale f might be problematic due to corrections coming from gravitational instantons[2-5]. Besides this, in the light of the Planck collaboration results, standard Natural Inflation is in tension withcosmological data [6]. Some proposals have been made in order to tackle one or both issues (see [7-11], for example).Furthermore, in the context of inflationary cosmology, modifications of the gravitational sector have been consideredin order to extend the single-field scenario and make it compatible with observations. These modifications give riseto a plethora of models. One special proposal, relevant for our present work, is to introduce a coupling of the inflatonto the Ricci scalar, thought to arise from quantum gravitational corrections (see [12] and references therein). Oneremarkable example of this class of theories is the Higgs inflation proposed by Bezrukov and Shaposhnikov [13]. Thesemodels are a subclass of the so-called scalar-tensor theories, where different couplings of curvature and a scalar field,the inflaton, are considered. We are also interested in the case where a derivative coupling to the Einstein tensor isintroduced. These subclasses of theories have been studied in a cosmological setting ([14], [15], [16], [17], [18], [19]),and are known to give rise to accelerated expansion and equations of motion for the metric tensor and the scalar fieldwith derivatives of order not higher than two.In the framework of Natural Inflation, as we have mentioned before, some proposals have been put forward in orderto solve the problems of the original model. One particular proposal, in the line of considering a coupling to the Ricciscalar, was to consider this coupling to be dynamic and to evolve with the same periodicity of the potential, thuspreserving at tree-level the shift symmetry of Natural Inflation [20]. In the Einstein frame and in some specific cases f ∼ M P and a better fit to Planck data could be obtained. Recently, the covariant one-loop quantum gravitationalcorrections to the effective potential of the model have been studied in [21], being a first step to roughly understandthe implications of the presence of this periodic coupling to the Ricci scalar coming from the gravitational UV regime.On the other hand, the authors in [22] considered a non-canonical kinetic term with a coupling to the Einstein tensor,where f (cid:28) M P could be obtained. In this case, the model is UV-protected, in the sense that all involved scales duringinflation are much lower than the Planck scale.In this letter we consider the extension of Natural Inflation in the context of a scalar-tensor theory, where dynamiccouplings to the Ricci scalar and to the Einstein-tensor-coupled kinetic term are introduced. This proposal could beregarded as the minimal combination of [20] and [22], exploring the synergies between the two models, and motivatedby the fact that from the EFT point of view shift-symmetric couplings should be considered in the action. Thecouplings are defined in such a way that respect the periodicity and the shift symmetry of standard Natural Inflation,and furthermore vanish at the minimum of the potential. We will show that this extension allows the scale f to havearbitrarily low values and to obtain results in agreement with observational data from Planck, therefore solving attree-level both of the aforementioned issues at the same time. ∗ Electronic address: [email protected] a r X i v : . [ a s t r o - ph . C O ] J un II. SET-UP
In [23] the authors studied a general scalar-tensor theory and performed an analysis on the scalar and tensorperturbations of the model. In this letter, the theory will be restricted to the tree-level action: S = (cid:90) d x √− g (cid:20) F ( φ ) R − g µν ∂ µ φ∂ ν φ + F ( φ ) G µν ∂ µ φ∂ ν φ − V ( φ ) (cid:21) , (1)where G µν = R µν − g µν R is the Einstein tensor and V ( φ ) = Λ (1 + cos φf ). We have not considered the couplingto the Gauss-Bonnet invariant also included in [23], since we are interested in the minimal combination of the models[20] and [22]. We set M P = 1 and we take the couplings to the Ricci scalar R and the kinetic term to be, respectively, F ( φ ) = 1 + α (cid:18) φf (cid:19) = 1 + α Λ V (2) F ( φ ) = β Λ (cid:18) φf (cid:19) = β Λ V, (3)with two a priori arbitrary parameters α and β measuring the strength of the couplings being introduced. Anotherreason for the exclusion of the Gauss-Bonnet coupling is to have more parametric control over the model, since thereare already three free parameters ( α , β and f ), and we therefore leave the study of the Gauss-Bonnet coupling forfuture work. The choice of these coupling functions allows the theory described by (1) to conserve the characteristicshift symmetry φ → φ +2 πf of Natural Inflation. Furthermore, these couplings vanish at the minimum of the potential φ = πf , reducing the theory to Einstein’s gravity with a minimally coupled scalar field.Variation of the action (1) with respect to the metric g µν and φ and considering a flat FRLW metric given by ds = − dt + a ( t ) ( dx + dy + dz ) gives the equations of motion of the model, describing an expanding homogeneousand isotropic universe. Defining the following slow-roll parameters (cid:15) = − ˙ HH , (cid:15) = ˙ (cid:15) H(cid:15) (4) l = ˙ FHF , l = ˙ l Hl (5)where dots denote differentiation with respect to time and H ≡ ˙ a/a is the Hubble parameter, and demanding | (cid:15) i | , | l i | (cid:28)
1, the equations of motion in the slow-roll regime are obtained [23]:3 H F ≈ V (6)2 ˙ HF − H ˙ F ≈ − ˙ φ − H F ˙ φ (7)3 H ˙ φ + V (cid:48) − H F (cid:48) + 18 H F ˙ φ ≈ φ . Equations (6) and (8) can be used to express the slow-rollparameters (4) and (5) as follows: (cid:15) ≈ V (cid:48) V (1 − ˜ αV )(1 + ˜ αV + 2 ˜ βV ) , (cid:15) = (cid:15) (cid:48) (cid:15) ˙ φH (9) l = F (cid:48) F ˙ φH , l = l (cid:48) l ˙ φH (10)with ˜ α ≡ α/ Λ , ˜ β ≡ β/ Λ and ˙ φH ≈ − V (cid:48) V (1 + ˜ αV )(1 − ˜ αV )(1 + ˜ αV + 2 ˜ βV ) (11)It is important to notice that all the quantities defined in (9), (10) and (11) are independent of the scale Λ.According to the analysis of the scalar and tensor perturbations done in [23] in the general framework of a scalar-tensor theory, one can write the spectral tilt n s and the tensor-to-scalar ratio r in our model as n s ≈ − (cid:15) − l − (cid:15) (cid:15) + l l (cid:15) + l (12) r ≈ (cid:15) + l ) . (13)Slow-roll inflation ends whenever one of the aforementioned slow-roll parameters becomes (cid:39)
1, and we define thevalue of φ that fulfills this condition as φ e . Finding a value φ such that N = (cid:90) φ e φ H ˙ φ dφ ≈ − , (14)that is to say, finding the value φ that allows to drive 50 −
60 e-folds of slow-roll inflation, and evaluating n s and r at this field value, allows us to obtain the predictions of the model that can be contrasted to the Planck collaborationresults [6]. III. ANALYSIS AND RESULTS
Before examining the parameter space spanned by α , β and f , we consider the regime α (cid:28) β (cid:29)
1. Underthis assumption, we find the slow-roll parameter (cid:15) (9) to be approximately: (cid:15) ≈ V (cid:48) βV (15)This approximation will be valid while 1 + cos φf (cid:29) √ β . (16)Performing a change of variables u ≡ φ/f , (cid:15) reads: (cid:15) ≈ f ˜ β ( ddu V ) V (17)where we have factored out the dependence of f given the form of V considered in Natural Inflation. Under thefollowing rescalings: f → Cf, β → C − β (18)where C is an arbitrary constant, it is explicit that (cid:15) remains invariant.Consider now the number of e-folds N given by (14) and the expression (11). Under the approximation α (cid:28) β (cid:29)
1, taking into account the range of validity (16), we get: N ≈ (cid:90) φ e φ − βV V (cid:48) dφ (19)Suppose that, given φ e , the value of the field φ which gives a desired number of e-folds N is known. Using againthe definition (11) and under f → Cf and β → C − β :˜ N ≈ (cid:90) φ (cid:48) e φ (cid:48) − βC − V (cid:0) φCf (cid:1) ddφ V (cid:0) φCf (cid:1) dφ = (cid:90) φ (cid:48) e /Cφ (cid:48) /C − β V (cid:0) ˜ φf (cid:1) dd ˜ φ V (cid:0) ˜ φf (cid:1) d ˜ φ (20)where we defined ˜ φ ≡ φ/C . Demanding ˜ N = N and taking into account that both integrands are equal, we findthat φ (cid:48) e = Cφ e and φ (cid:48) = Cφ . That is, the values of φ giving the desired number of e-folds N transform as φ e → Cφ e , φ → Cφ under f → Cf and β → C − β .All the quantities defined in (9) and (10), approximated using (11) and under the conditions α (cid:28) β (cid:29) (cid:15) = (cid:15) (cid:48) (cid:15) ˙ φH ≈ − f ˜ β ddu (cid:15) ddu V(cid:15) V (21) l = F (cid:48) F ˙ φH ≈ − f ˜ β ddu F ddu VF V (22) l = l (cid:48) l ˙ φH ≈ − f ˜ β ddu l ddu Vl V . (23)This means that n s (12) and r (13) are not changed under f → Cf , β → C − β which, in turn, will be evaluatedat the invariant value u = φ /f = Cφ /Cf . Therefore, we can conclude that after finding a pair of parameters f and β that give n s and r in agreement with the results of the Planck collaboration [6], the model with parameters Cf and C − β will also agree with them.By inspecting numerically the parameter space using the full expressions (9), (10) and (11) with α (cid:28) f (cid:28)
1, we find that for f = 0 .
01 and β in the range 3 . × − . × the predicted n s and r are in agreement with the Planck collaboration results, as shown in Fig. 1, meaning that they are found within the95% and 68% confidence level (CL) contours. We find that (cid:15) is the first slow-roll parameter to become approximatelyequal to 1, being the others less than unity, and therefore defining the end of slow-roll inflation (i.e. (cid:15) ( φ e ) (cid:39) β we find φ e /f to have values from 1.72 to 2.13. This implies that condition (16) will be fulfilled allthe way down from φ /f to φ e /f . Using then the approximate symmetry under rescalings (18) we can arbitrarilydecrease f by increasing β and, for instance, we would obtain β in the range 3 . × − . × for f = 10 − , and3 . × − . × for f = 10 − . In Fig. 1 the results for α = 0 . P ξ ∼ H π (cid:15) = V π F (cid:15) (cid:39) . × − (24)giving Λ ∼ × − − × − for f and the range of β considered before. IV. CONCLUSIONS
We have shown that the embedding of Natural Inflation in a more general scalar-tensor theory, with couplingsrespecting the periodicity and the shift symmetry of the original model, can drive slow-roll inflation for f (cid:28) f = 0 .
01 and β in the range 3 × − . × weobtain spectral indices n s and tensor-to-scalar ratios r within the 95% and 68% CL contours of Planck data, givingΛ ∼ − . FIG. 1: Results obtained by running β from 3 . × to 2 . × (from left to right) for several values of α and f = 0 . N = 50 (upper lines) and N = 60 (lower lines). The cases of α = 0 .
01 and α = 0 .
001 are almost indistinguishable. Lightand dark shading correspond, respectively, to 95% C.L. and 68% C.L. of Planck collaboration constraints.
Furthermore, in the limit of small coupling to the Ricci scalar, i.e. α (cid:28)
1, given that β (cid:29) f → Cf and β → C − β , leaving the values of n s and r invariant. This remarkable property allows to arbitrarily decrease the value of the periodicity scale f byincreasing the coupling to the kinetic term, solving the issue of Natural Inflation super-Planckian periodicity scales f and alleviating the current tension of Natural Inflation with Planck data.Finally, we are aware of the fact that the UV completion of the model has not been studied in the present paper,but the analysis of this issue is beyond the scope of this letter. Therefore, this issue is expected to be addressed infuture work. Acknowledgments
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