Natural Inflation with a periodic non-minimal coupling
Natural inflation is an attractive model for primordial inflation, since the potential for the inflaton is of the pseudo Nambu-Goldstone form, , and so is protected against radiative corrections. Successful inflation can be achieved if and where can be seen as the strong coupling scale of a given non-abelian gauge group. However, the latest observational constraints put natural inflation in some tension with data. We show here that a non-minimal coupling to gravity , that respects the symmetry and has a simple form, proportional to the potential, can improve the agreement with cosmological data. Moreover, in certain cases, successful inflation can be achieved even for a periodicity scale smaller than the Planck scale.
Inflationary models where the inflaton is a pseudo-Goldstone boson can naturally provide a flat potential, which is protected against radiative corrections. Natural Inflation (NI) Freese et al. (1990) realizes this idea using a continuous shift symmetry of an axion-like field broken down to a discrete shift symmetry by non-perturbative effects associated with a non-abelian gauge field that becomes strongly coupled at a scale , thus, generating a potential of the form .
In this letter we consider a minimal extension of the original NI model by considering the simplest non-minimal coupling of the inflaton to gravity , which respects the discrete shift symmetry . We require Einstein gravity to be recovered at the minimum of the potential and we also assume the non-minimal coupling function to be proportional to . This last requirement encodes the fact that gravity feels the field value through its potential energy, and so the process generating such non-minimal coupling should have that information. In other words, any coupling to which is non-derivative in should vanish if goes to zero. A similar logic of adding shift symmetric corrections to NI was also proposed in Germani and Kehagias (2011), using only derivative interactions, and in Alonso-Ãlvarez and Jaeckel (2017), using non-standard kinetic terms.
The motivation to consider this extension is three-folded. First, from and EFT point of view all the couplings allowed by the symmetries should be considered. Second, and on a more practical level, the predictions of natural inflation are in tension with the latest Planck results Ade et al. (2016a, 2015) and so it is interesting to understand what kind of extensions could alleviate the tension. Third, in standard NI the axion decay constant needs to be super-Planckian in order to fit cosmological data. This feature might be problematic due to the presence of gravitational instanton corrections Banks et al. (2003); Rudelius (2015a, b); Montero et al. (2015); Hebecker et al. (2017). Several extensions of the original NI model have been proposed to overcome one or both of the last two issues (for an incomplete list see Arkani-Hamed et al. (2003); Kim et al. (2005); Dimopoulos et al. (2008); Anber and Sorbo (2010); Bachlechner et al. (2015); Ferreira and Notari (2017, 2018)), although some of the proposals still seem to inherit the gravitational instanton problem Rudelius (2015b, a); Montero et al. (2015); Hebecker et al. (2017).
As we will show, the simple extension we consider here comes with a set of interesting predictions which can address both issues.
Ii Non-minimal coupling to gravity
We consider natural inflation with a non-minimal coupling to gravity described by the action:
where . In standard NI Freese et al. (1990) turns out to be of the order of the GUT scale and the periodicity scale has to take values larger than the (reduced) Planck mass, GeV, in order to achieve successful inflation. Here we add the simplest non-minimal coupling compatible with the periodicity of the original potential, that gives standard Einstein gravity at the minimum of the potential and such that the deviation is proportional to the potential of ,
where the dimensionless number is the only new parameter. Note also that for the Planck scale to be well-defined.
The usual way to get rid of the non-minimal coupling from the action is to change variables, going from the so-called Jordan frame to the Einstein frame, by means of a conformal transformation of the metric:
Such a transformation leads to a non-canonical contribution to the kinetic term and a rescaling of the potential. Explicitly, the action in the Einstein frame reads:
where quantities computed with the transformed metric are denoted with a tilde and
where a prime denotes . By redefining the scalar field using the transformation
we obtain the action in the Einstein frame in terms of a new canonical field with an effective potential :
making the standard slow-roll analysis possible.
Iii Slow-roll analysis
The potential in the Einstein frame , shown in fig. 1, flattens as approaches 0.5, while new extrema appear for . The potential is still periodic, but with a new periodicity scale, , which turns out to be different from the scale , present in the original potential, although still of the same order of magnitude. In fig. 2 we show the full dependence of on . When comparing the periodicity scale to the observed Planck scale we will always refer to their ratio in the Einstein frame, where the axion is canonical and the Planck mass is a constant. In this frame the Friedmann equation and the axion equation of motion yield simply
Inflation starts when the slow-roll conditions, i.e. and , are satisfied, where
and it ends when , which determines the field value at the end of inflation, .
The number of e-foldings of inflation from a given value to is given by:
The observable scales correspond to e-folds before the end of inflation. We will use , the uncertainty being, as usual, due to model-dependence in the reheating history. The observational constraint imposes a relation between the values of , and . We use this constraint to eliminate the scale , and we vary the two independent parameters and . We will, however, present the results as a function of and . Finally, we compute the scalar spectral index and the tensor-to-scalar ratio .
In figs. 3 and 4 we show the results for and for different values of and after solving numerically eqs. (8-10). In the same figures we also show the 68% and 95% C.L. contours coming from the Planck TT, TE, EE and low P data Ade et al. (2016a) and from the combination of Planck TT and low P data with BAO Ade et al. (2016a, b) and Bicep-Keck data Ade et al. (2015). Unless otherwise stated, we will take the former constraints as a benchmark for the rest of the analysis.
The results show that suppresses the amount of tensor modes with respect to standard NI. This is expected since increasing lowers and flattens the potential, as can be seen from fig. 1. This suppression alleviates some of the current tension of NI with the observational constraints: the values for and with a non-minimal coupling to gravity are found to be well within the 95% C.L. region for a wide range of parameters and, for , the values reach the 68% C.L. region. On the other hand, negative values of worsen the compatibility with observations, compared to the ones predicted by NI, with predictions excluded from the 95% C.L. region. In all further analysis, this case will be omitted.
For we find, for instance, that: gives results in the 68% C.L. region, when ; for this happens for ; and in the case of the region is reached when . Values of larger than 20 show a saturation and give roughly the same curve in the plane, although with different values of (or ). The reason is that if we increase for fixed , the potential does not support inflation. Therefore, also needs to increase. On the other hand for large the relevant region for observations inflation is the one close to the bottom of the potential. Thus, if we expand around the minimum, in the Jordan frame, both the potential and the non-minimal coupling become quadratic in , the latter with a strength given by . From this we can see that in this limit increasing can be compensated with a larger to get the same prediction. Generically, and except for close to 0.5, in order to obtain predictions in agreement with Planck data, the scale needs to be increased when is also increased.
When gets close to the threshold value 0.5 both the scales and can be slightly sub-Planckian. In fig. 4 we show such cases, which are characterized by very small tensors, , for . This range of tensor modes is unobservable by near future experiments future.
For the values of and that give spectral indices and tensor-to-scalar ratios lying inside the 68% CL region of the Planck data (see Fig. 1), GeV is obtained in accordance to the expected value for the GUT scale. For example, for and we obtain GeV, while for and we obtain GeV. However, for values of larger than those plotted in the figures, in order to be inside the 68% CL contour also needs to increase such that , the scale of the potential in the Einstein frame, remains roughly constant. For example, for we get .
One can also estimate the reheating temperature, and so the exact number of efolds , assuming a given decay process for the inflaton. For example, using the typical perturbative decay of an axion into gauge bosons with a rate we get, assuming that the inflation oscillates until reheating is complete, GeV and , where we used and GeV, a number of relativistic species and a gauge coupling . Note, however, that larger decay rates are expected if the axion decays at tree level into fermions with a large Yukawa coupling Ferreira and Notari (2017, 2018), or if it decays through a parametric resonance Adshead et al. (2015, 2017); Notari and Tywoniuk (2016), leading to a larger . For instantaneous reheating, in fact, one gets for the same value of and , GeV and .
We have considered in this Letter a non-minimal coupling between the inflaton and gravity, in the context of Natural inflation, that respects the symmetry . Assuming a simple cosine form, proportional to the potential itself, with only one extra dimensionless parameter, , we have obtained a model that gives rise to predictions for the spectral index and the tensor-to-scalar ratio that lay within the 68% C.L. region of Planck data. This is an improvement over the predictions made by minimally coupled Natural Inflation, which are excluded from this region. The parameters that give rise to these results yield a scale GeV, consistent with the expected value of the GUT scale.
Another interesting consequence of the non-minimal coupling is that inflation can be observationally viable also for sub-Planckian values of both and the periodicity scale in the Einstein frame . We have shown that for we can have , within the 68% C.L. (this case corresponds to smaller values of the tensor-to-scalar ratio ). This would alleviate possible issues that might arise due to gravitational instantons corrections Banks et al. (2003); Rudelius (2015b); Montero et al. (2015), which have been estimated to be exponentially small only if the periodicity scale is sub-Planckian.
Acknowledgements.We would like to acknowledge Jaume Garriga, Cristiano Germani and Javier Rubio for discussions. This work is supported by the grants EC FPA2010-20807-C02-02, AGAUR 2009-SGR-168, ERC Starting Grant HoloLHC-306605 and by the Spanish MINECO under MDM-2014-0369 of ICCUB (Unidad de Excelencia “Maria de Maeztu”).
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