Natural metric-affine inflation

We consider here natural inflation in the low energy (two-derivative) metric-affine theory containing only the minimal degrees of freedom in the inflationary sector, i.e. the massless graviton and the pseudo-Nambu-Goldstone boson (PNGB). This theory contains the Ricci-like and parity-odd Holst invariants together with non-minimal couplings between the PNGB and the above-mentioned invariants. The Palatini and Einstein-Cartan realizations of natural inflation are particular cases of our construction. Explicit models of this type featuring non-minimal couplings are shown to emerge from the microscopic dynamics of a QCD-like theory with an either sub-Planckian or trans-Planckian confining scale and that is renormalizable on Minkowski spacetime. Moreover, for these models, we find regions of the parameter space where the inflationary predictions agree with the most recent observations at the 2σ level. We find that in order to enter the 1σ region it is necessary (and sufficient) to have a finite value of the Barbero-Immirzi parameter and a sizable non-minimal coupling between the inflaton and the Holst invariant (with sign opposite to the Barbero-Immirzi parameter). Indeed, in this case the potential of the canonically normalized inflaton develops a plateau as shown analytically.


Introduction
Inflation is currently accepted as the best theory of the earliest observable stages of our universe.It explains the horizon and flatness problems and provides us with a microscopic origin for the small inhomogeneities and anisotropies in the cosmic microwave background (CMB).Such fluctuations include tensor perturbations that may show up in the future as primordial gravitational waves in detectors like the DECi-hertz Interferometer Gravitational wave Observatory (DE-CIGO) [1], the Big Bang Observer (BBO) [2,3] and the Advanced Laser Interferometer Antenna (ALIA) [2,4].In practice, inflation is often realized by introducing a scalar field, the inflaton, with a potential that is nearly flat for a large enough range of field values.Indeed, this acts approximately as a vacuum energy and the universe inflates.
From the particle-theory point of view, however, it is difficult to realize such potential flatness without introducing a high level of fine tuning.One way to efficiently circumvent this problem is to impose a symmetry that protects the inflaton potential from large radiative corrections.Natural inflation [5] is a concrete way to realize this idea: the inflaton is identified with the PNGB of an approximate and spontaneously broken symmetry.In the limit where the explicit symmetrybreaking terms vanish the PNGB becomes an exact Nambu-Goldstone boson and, therefore, acquires an exact shift symmetry, which is the manifestation of Goldstone's theorem.As a result the flatness of the inflaton potential turns out to be natural in 't Hooft's sense [6].
Another field-theoretic attractive feature of natural inflation is the fact that it admits an explicit UV completion within an asymptotically free theory.One considers a QCD-like theory with a large (Planckian) confining scale and identifies the inflaton with the composite particle analogous to the lightest meson 1 [5,7].The effective inflaton action should then be periodic in the inflaton field 2 .So far a simple cosine potential and non-minimal coupling between the Ricci scalar and the inflaton has been shown to emerge from such a UV complete theory [7].This has been achieved by assuming gravity to be formulated ab initio in the "metric" way: namely, the (gravitational) connection was assumed to be equal to the Levi-Civita (LC) one, Γ ρ µ σ .The corresponding inflationary predictions were obtained in [11].
It is, however, well known that, in a purely geometric formulation of gravity, the metric, g µν , and the connection, A ρ µ σ , can be independent and so generically A ρ µ σ ̸ = Γ ρ µ σ .Such formulation, known as metric-affine gravity, can be used to find physically different actions for the metric and the matter fields.For example, metric-affine gravity admits two rather than just one twoderivative curvature invariants: the usual Ricci-like scalar and the Holst invariant [12][13][14], which can be used to construct new models 3 [19][20][21][22][23].In particular, this scenario may then be used to find more UV complete realizations of natural inflation.
Thus, the purpose of this paper is to study natural inflation in metric-affine gravity focusing on models that admit a UV completion.This alternative realizations of natural inflation is also motivated by the fact that a simple minimally-coupled (PNGB) inflaton with a cosine potential is in tension with the most recent CMB observations performed by the Planck, Background Imaging of Cosmic Extragalactic Polarization (BICEP) and Keck collaborations [24][25][26], within Einstein's general relativity.Other possible ways of restoring the agreement between natural inflation and CMB observations have been discussed in [7,8,11,[27][28][29][30][31].
The paper is organized as follows.In Sec. 2 we construct the low-energy metric-affine theory for the metric and the PNGB inflaton.For the sake of minimality, we restrict the analysis to metric-affine models where the field content taking part in inflation includes only the graviton and a single PNGB.In Appendix A we also investigate possible UV completions of this effective action.Sec. 3 is devoted to an analytical study of the corresponding effective potential of the canonically normalized inflaton.The inflationary predictions are then analyzed in Sec. 4. Finally, in Sec. 5 we offer a detailed summary of the results and our conclusions.

The action and the dynamical degrees of freedom
Here we build a natural inflationary scenario in a metric-affine theory containing only the minimal degrees of freedom in the inflationary sector (the massless graviton and the PNGB).So we demand the distorsion, i.e. the difference between the full connection and the Levi-Civita one to be non dynamical.The torsion T µνρ is defined in terms of the distorsion by T µνρ ≡ C µνρ − C ρνµ ,.
In this case, as shown in Ref. [21], the general effective action including the metric, the distorsion and all matter fields (scalars ϕ, fermions ψ and gauge fields A I µ with field strength in the theory is Here F ρ µν σ is the curvature associated with 4 A ρ µ σ , Φ represents the set of fields that are independent of C ρ µ σ , namely the dots are curvatures and covariant derivatives of the previous fields constructed with the LC connection, T µνρσ (Φ) is a rank-four contravariant tensor that depends on Φ only (not on its derivatives) and Σ(Φ, DΦ, C) is a scalar that depends on Φ and C ρ µ σ only.In the dependence of Σ we have stressed that the covariant derivative DΦ built with A ρ µ σ can appear, but this can be expressed in terms of C and Φ.Note that T µνρσ (Φ) and Σ(Φ, DΦ, C) should also be invariant under all gauge and global symmetries present in the theory.
Let us now consider the part of S EFT that contains only the massless graviton and the PNGB.Restricting ourselves to terms that feature only two derivatives, which are expected to be the leading ones in the low energy limit, we obtain the following action of natural metric-affine inflation: where here ϕ is the PNGB field, is the natural-inflaton potential [5], Λ and f are two energy scales and Λ cc accounts for the (tiny and positive) cosmological constant responsible for the observed dark energy and is negligible during inflation, which occurs at a much larger energy scale.In (2.5) α and β are (not yet specified) functions of ϕ and R and R are, respectively, a scalar and pseudoscalar contraction of the curvature, where ϵ µνρσ is the totally antisymmetric Levi-Civita symbol with ϵ 0123 = 1.We refer to R as the Holst invariant (see [12][13][14]).The pseudoscalar R vanishes for C ρ µ σ = 0 (that is when the connection is the LC one) thanks to the cyclicity property R µνρσ + R νσρµ + R σµρν = 0, where R µνρσ represents the Riemann tensor.This is the reason why in standard Riemannian geometry R is absent.Therefore, R can be considered as a manifestation of a connection that is independent of the metric.
The distorsion is non dynamical (does not introduce additional degrees of freedom) if and only if the action has the form in (2.2) [21].So, in our case the distorsion can be integrated out to obtain an equivalent metric theory, i.e. a gravity theory where the connection is the LC one and the components of the distorsion are expressed in terms of the other fields, Φ. Performing this integration for the inflationary action in (2.5) leads to [20,21,32] where R is the Ricci scalar (remember that R = R when A ρ µ σ = Γ ρ µ σ ).In the expression above we have a non-minimal coupling α(ϕ)R.This can be removed by performing the Weyl transformation where M P is the reduced Planck mass.Such transformation requires α > 0 in order for the transformation to be regular and to preserve the signature of the metric.One then obtains This action can be rewritten as follows where F (ϕ) ≡ 2α(ϕ)/M 2 P and where a prime represents a derivative with respect to ϕ (from now on a prime on a function represents a derivative with respect to its argument).Since k(ϕ) is always positive, ϕ is never a ghost.
Note that in the particular case β = 0, the expression above simply becomes In the case of natural inflation formulated directly in the metric formalism (as opposed to the metric-affine formalism), namely starting from the action and then going to the Einstein frame, one finds (2.11) but with [11] k This k(ϕ) differs from the one in (2.14) because of the extra positive term 3M 2 P F ′2 (ϕ)/(2F 2 (ϕ)).Therefore, we can see that the two theories differ even if one sets β = 0 in the metric-affine case.
The scalar kinetic term in (2.11) can be brought into a canonical form through a field redefinition ϕ → χ satisfying (2.17) Since k(ϕ) is always positive, χ is a monotonically growing function of ϕ and can be inverted to obtain ϕ as a function of χ.Then we can express the action in terms of the metric and χ: where we have defined The inflationary predictions of the theory can be now extracted from the action in the form (2.18) through standard methods once the functions α(ϕ) and β(ϕ) are specified.
In Appendix A we provide a microscopic origin for the following α and β, This choice is, therefore, well motivated from the particle-physics perspective and we will thus assume it in the rest of the paper.The quantity M 2 P /(4β 0 ) is called the Barbero-Immirzi parameter [33,34].

An analytical study of the inflationary potential
As we have seen in the previous section, the inflationary potential U (χ) (for the canonically normalized inflaton χ), defined in (2.19) is the result of an integration and an inversion of a function.In our context we are unable to perform these operations analytically, expect in some very specific cases.Nevertheless, we provide an analytical discussion here in order to obtain an understanding of the numerical calculations that will be reported in Sec. 4.
First of all, we start spending a few words on U as a function of ϕ and then we move to discuss the impact of the canonical normalization of the scalar field.For generic ξ, but neglecting the tiny Λ cc , (2.19) becomes (3.1) (dotted) and ξ > 1 2 (dashed).
Some reference plots are given in Fig. 1.Since the potential is an even function of ϕ with period 2πf , let us focus on the ϕ ∈ [0, πf ] domain.
The quantity in (3.1) has been studied as function of ϕ in [11].The stationary points are and, when ξ > 1/2, The value of the second derivative on those points is So, ϕ 2 is always a minimum.The stationary point ϕ 3 is always a maximum.For ξ > 1 2 the value ϕ 1 = 0 is a minimum, otherwise it is a maximum.In the latter case inflation happens as usual from the maximum to the minimum.We stress that for ξ = 1 2 , ϕ 1 = ϕ 3 = 0 is a maximum.On the other hand, when ξ > 1 2 , there seems to be two possible regions available for inflation: 0 < ϕ < ϕ 3 and ϕ 3 < ϕ < ϕ 2 .In the latter inflation happens as before just with the replacing of 0 with ϕ 3 .In the former, one could think to inflate from the maximum in ϕ 3 to the minimum in ϕ 1 = 0 and then use U (ϕ 1 ) as the cosmological constant.Unfortunately, the potential value in those points is so that there is only a factor of ξ (in the big ξ limit) of difference between the two scales i.e. not enough to account for the inflationary scale and the cosmological constant at the same time.Therefore, from now on we will consider only inflation happening towards the minimum in ϕ 2 .More details about U will be given in the next section.
Let us now proceed with the discussion of the effect of the change of field from ϕ to χ.As discussed around (2.17) the kinetic function, k(ϕ), defines the canonically normalized inflaton, χ(ϕ), and the inverse function ϕ(χ).The latter, as well as the potential V (ϕ)/F 2 (ϕ), determines the potential of the canonically normalized field, U (χ), through (2.19).Therefore, k(ϕ) is a key ingredient to determine the inflationary observables and in this section we study it for the theoretically well-motivated values of α(ϕ) and β(ϕ) in (2.20).
In Fig. 2 the function k(ϕ) defined in Eq. (2.13) is plotted.We find that a peak appears in k increasing ξ (the higher ξ the higher the peak).For ϕ around the location of the peak, ϕ peak , the function χ(ϕ) increases more rapidly.This corresponds to a quasi-flat region in the function ϕ(χ): those values of χ that are mapped to a ϕ around ϕ peak .Eq. (2. 19) tells us that a quasi-flat region in the function ϕ(χ) is also a quasi-flat region in the potential of the canonically normalized inflaton, U (χ).The higher the peak in k(ϕ) is the flatter and larger this quasi-flat region turns out to be.This analytical understanding will be checked through numerical calculations in Sec. 4, and the plateau in U (χ) will help in reaching the agreement with CMB observations.It is, therefore, interesting to analytically study how the location and height of the peak depends on the parameters of the model.
For the sake of simplicity we explicitly present this analysis in the case ξ = 0, for which α = M 2 P /2 and thus F = 1.For any β(ϕ), Eq. (2.13) then tells us that the kinetic function is Setting β(ϕ) as in (2.20), we can now look for the location of ϕ peak by requiring that k ′ (ϕ peak ) = 0.
Barring values of ϕ such that β ′ (ϕ) = 0, which are points of minimum according to (3.10), the equation k ′ (ϕ) = 0 leads to a second-order algebraic equation for cos(ϕ/f ).There is only one admissible solution (respecting | cos(ϕ/f )| ≤ 1), which gives therefore, the square root in (3.11) is always real.From (3.12) one can also easily show that the modulus of the right-hand side of (3.11) is always less than or equal to one and thus can be equal to a cosine.We can now determine the height of the peak by inserting this result in (3.10): Note that there is a symmetry β(ϕ) → −β(ϕ) (see Eqs. (2.13) and (3.10)).Therefore, only the relative sign between β 0 and ξ matters.From now on we will use the convention where β 0 can change sign, while ξ remains positive.In Fig. 3 we show cos(ϕ peak /f ) and k(ϕ peak ) as functions of ξ and β 0 .The quantity cos(ϕ peak /f ) is independent of f , while k(ϕ peak ) does depend on it and we set f = 5M P in Fig. 3; however, as clear from Eq. (3.13), for a high peak, k(ϕ peak ) simply scales as 1/f 2 .
Before ending this section let us observe that in Fig. 2 we have chosen trans-Planckian values of f .As we will see in Sec. 4, among others, it is also for such values that we reach the agreement with CMB observations.Nevertheless, it is important to note that this setup do not necessarily invalidate the effective-field-theory treatment of gravity, which we are relying on.Indeed, the condition that allows us to do so is that the energy density should be much less than the Planck energy density [35] |U (χ)| ≪ M 4 P , for all relevant values of χ.As we will show in Sec. 4, this condition is amply satisfied.It should be kept in mind, however, than in some UV completions of all interactions such as string theory [36] having a trans-Planckian f can spoil the validity of natural inflation, at least in its simplest incarnation [37,38] (see [39] for a review).In some others, such as asymptotically free/safe theories [40] featuring higher-derivative terms (see e.g.[41] and references therein) a trans-Planckian f can be consistent [8].We find this aspect interesting as observing the predictions of natural inflation may give us information on the theory that UV completes Einstein gravity.For this reason, in Sec. 4 we will consider both trans-Planckian and sub-Planckian values of f and, as we will show, agreement with data can be achieved in both cases.

Inflationary predictions
In this section we present the inflationary predictions of the model.Using the slow-roll approximation, all the inflationary observables can be derived from the inflaton Einstein-frame potential (2.19) and its derivatives.The potential slow-roll parameters are defined as: 3) The expansion of the Universe is evaluated in number of e-folds, which is where the field value at the end of inflation is given by ϵ(χ end ) = 1, while the field value χ N at the time a given scale left the horizon is given by the corresponding N e .The tensor-to-scalar ratio r, the spectral index n s and its running α s ≡ dn s /d ln k are: ) Finally, the amplitude of the scalar power spectrum is which needs to satisfy the experimental constraint [25] A s ≃ 2.1 × 10 −9 .(4.9) In order to get a better understanding of the impact of α(ϕ) and β(ϕ) we consider three different scenarios: 1. ξ ̸ = 0 and β(ϕ) = 0.

ξ ̸ = 0 and β(ϕ) = 0
This is the scenario that reproduces the non-minimal natural inflation without the Holst invariant.Before starting our analysis we introduce the following two parameters that allow to measure the natural inflation mass scales Λ and f in terms of M P .[26].For reference the predictions of quadratic inflation for N e ∈ [50, 60] (brown) and standard natural inflation for N e = 60 (black).

ξ < 0 and β(ϕ) = 0
As long as α(ϕ) in Eq. (2.20) stays strictly positive, ξ is allowed to take negative values.This sets the lower bound ξ > − 1 2 .In Fig. 4 we plot the corresponding results for r and n s versus the parameters ξ and δ f when N e = 60 and δ Λ is fixed so that the constraint (4.9) is satisfied.
We can see that with ξ increasing in absolute value r (n s ) increases (decreases) leading the prediction even more away from the allowed region.This was somehow expected because a negative ξ increases the height of maximum of U in Eq. (3.8) and this usually comes with an increase in r.Therefore, we conclude that the configuration with ξ < 0 and β(ϕ) = 0 is excluded by data and the computation of the predictions for α s is not needed.5.The pink areas represent the 1,2σ allowed regions coming from the Planck legacy data [43], while the gray areas are the same as in Fig. 4.  In Fig. 5 we plot the corresponding results for r and n s as functions of the parameters ξ and δ f when N e = 60 and δ Λ is fixed so that the constraint (4.9) is satisfied.
Regardless of δ f , the predictions share some common properties.First, as expected, r decreases by increasing ξ, while n s first reaches a maximum (whose numerical value is dependent on the actual value of δ f ) and then decreases.A similar behaviour is also shown by δ Λ as a function of ξ.Moreover, δ Λ also increases by increasing δ f .At selected δ f values, by increasing ξ the predictions manage to reach the 2σ allowed region.In particular for δ f = 6, they touch the border of the 1σ region.This happens for ξ values around 1 or smaller.For δ f ≳ 12, no predictions enter the allowed region.
The results can be better understood by giving a look at the Einstein frame potential U (χ), which is plotted in Fig. 7 for selected benchmark points.For clarity the lines have been truncated at half period i.e. χ = χ(ϕ 2 ).First of all we can see that the potential is lowered and flattened as an effect of the non-minimal coupling.As anticipated in Sec. 3, the condition in (3.14) to rely on an effective-field-theory treatment of gravity is amply satisfied (see Fig. 7).The main reason why this happens is the smallness of Λ in Planck units, δ Λ ≪ 1, (shown in Fig. 5d); in the construction of Appendix A such smallness is natural because it is generated by the smallness of quark-like masses (the only source of a chiral symmetry breaking) compared to a QCD-like confining scale.The flattening of the potential usually comes with a lowering of r, which is confirmed by the predictions in Fig. 5.We also see that by increasing δ f (at fixed ξ), χ N moves further and further away from the maximum of the potential towards steeper regions, implying an increase on r, which is confirmed by Fig. 5b.
Since a relevant region of the parameters space falls within the 2σ boundary of r vs. n s , we study whether the constraints on the running of the spectral index α s are satisfied.In Fig. 6 we plot the corresponding results as functions of the parameters ξ and δ f .We considered only the parameters space that predicts r ≲ 0.0383, which is the highest r-value in the 2σ boundary line.The results for δ f = 12, 14, 16 are absent because they do not exhibit any region compatible with the r vs. n s constraints.This allows us to appreciate the different lines corresponding to the other values of δ f .We can see that in this case the stronger constraint is the one coming from r vs. n s and that all the points in agreement with the r vs. n s constraint are also in agreement with the α s vs. n s one.We also notice that in the allowed region, α s is always predicted to be negative.
Overall we can conclude that the presence of the non-minimal coupling ξ is beneficial to restore compatibility with data of natural inflation.This is also the case in the metric theory of natural inflation in the presence of the non-minimal coupling [11], where k(ϕ) is given by (2.16) rather than (2.14).
4.2 ξ = 0 and ξ > 0 In Fig. 8 we plot the corresponding results for r and n s versus the parameters ξ and δ f when β 0 ≥ 0, N e = 60 and δ Λ is fixed so that the constraint (4.9) is satisfied.
We can see that the β 0 > 0 and β 0 = 0 results overlap with each other.Therefore, the effect of having β 0 > 0 is just to shift the results at higher ξ values.Unfortunately, with ξ increasing in both cases r (n s ) increases (decreases) leading the prediction even more away from the allowed region.Because of this, we truncated the study at ξ = 10000 and concluded that the configuration with ξ = 0 and β 0 ≥ 0 is excluded by data and the computation of the predictions for α s is not needed.

β 0 < 0
In Fig. 9 we plot the corresponding results for r and n s versus the parameters ξ and δ f when β 0 = −6M 2 P , N e = 60 and δ Λ is fixed so that the constraint (4.9) is satisfied.We can see that by increasing ξ, r decreases in most of the chosen configuration but δ f = 4, 5 where, instead, r vs. ξ exhibits first an increasing behaviour and then a decreasing one like in all the other δ f 's.On the other hand, the behaviour of n s vs. ξ is more oscillating (showing first a short increasing phase, then a decreasing one and finally a new increasing phase) for all the chosen configurations but for δ f = 4 where n s appears to just increase with ξ.At big enough ξ, the results in the region r ≲ 0.015 and n s ≳ 0.958 overlap, but they are still dependent on f .This is visible in Fig. 9(c).This means that by changing f , we can find a ξ so that the results instead do not change.
Once again, it is worth studying the predictions for α s .In Fig. 10 we plot the corresponding results as functions of the parameters ξ and δ f .The same cut in r as in Fig. 6 has been implemented.We can see that the constraint on α s reduces the allowed parameter space.Again all the predicted values for α s are negative but now larger in absolute value than those of the β = 0 case in Fig. 6.
To get a better understanding of the large ξ behaviour, we plot for selected benchmark points U (χ) centered at the origin and centered around the corresponding inflection point respectively in Fig. 11(a) and in Fig. 11(b).For clarity the lines have been truncated at half period i.e. χ = χ(ϕ 2 ).For the selected points, χ N is close to the inflection point of the potential.Moreover, it is hard to distinguish the different potentials around (and after) the inflection points.The equation for such a points is U ′′ (χ flex ) = 0 (4.11) In terms of ϕ (4.11) can be rewritten as Using Eq. (3.1) and imposing ξ = 0, (4.12) becomes The right-hand side of this equation turns out to be strongly suppressed, such that we can very well approximate (4.12) as k ′ (ϕ) ≃ 0. As a result, the inflection point of U and the maximum point of k are numerically almost indistinguishable.As we can see in Fig. 11(b) a change in f can be compensated by a change in ξ so that the shape of U around ϕ peak stays unchanged (the constraint between f and ξ can be formalized as U (ϕ peak ) = constant).This is reflected in the inflationary results as well, where all the lines overlap when ϕ N is around ϕ peak .P for the same δ f values and color code as in Fig. 9.The pink and gray areas are the same as in Fig. 6.As anticipated in Sec. 3, once again we note that the condition in (3.14) to rely on an effectivefield-theory treatment of gravity is amply satisfied, as shown by Fig. 11.Also here the main reason why this happens is the smallness of δ Λ , (shown here in Fig. 9d), which is natural in the construction of Appendix A.
Overall we can conclude that the presence of the non-minimal coupling ξ combined with a negative β 0 can restore the compatibility with data of natural inflation.running α s to be out of the allowed region.The issue can be solved by using ξ > 0, which restores compatibility with data by moving the large α s region towards smaller n s values.

Conclusions
In this paper we have constructed and analysed a metric-affine realization of natural inflation.
We have focused on the low-energy (two-derivative) metric-affine inflationary theory containing only the graviton and the PNGB inflaton in the particle content.In particular, the connection does not carry additional degrees of freedom.This theory, whose action is given in 5 (2.5), features, besides the PNGB potential, non-minimal couplings between the PNGB and the two linear-in-curvature invariants that one can construct in metric-affine gravity: R and R, defined in (2.7).We have integrated out the distorsion to find an explicit metric theory, with action given in (2.11), where the connection is given by the LC formula.Moreover, in Appendix A we have found an explicit UV completion for the cosine potential and both non-minimal couplings in (2.6) and (2.20), respectively: by UV completion we mean that these functions emerge from the microscopic dynamics of a quantum field theory that is renormalizable on Minkowski spacetime.The UV completion of Einstein gravity is beyond the scope of the present paper; perhaps it can be achieved in string theory or in asymptotically free/safe theories featuring higher-derivative terms.
We have performed an analytical study of the potential of the canonically normalized inflaton, χ, which is defined in terms of the original PNGB ϕ through the kinetic function k, see Eqs. (2.13) and (2.17).We have analytically shown that k develops a high peak for large and opposite values of β 0 and ξ, defining the non-minimal coupling between the PNGB and the Holst invariant, the second formula in (2.20).This corresponds to a plateau in the potential of χ, which has been found explicitly.
Finally, we have performed an analysis of the inflationary predictions of this theory for the P for the same values and color codes of δ f and ξ as in Fig. 14.The pink and gray areas are the same as in Fig. 6. potential and non-minimal couplings in (2.6) and (2.20), respectively, because, as shown, they admit an explicit UV completion.For these models, we have found regions of the parameter space where the inflationary predictions agree with the most recent observations performed by the Planck, BICEP and Keck collaborations at the 2σ level.We have found that in order to enter the 1σ region it is necessary (and sufficient) to have a finite value of the Barbero-Immirzi parameter (a sizable |β 0 |) and a sizable ξ (with sign opposite to the Barbero-Immirzi parameter).This is precisely the region of parameter space for which the potential of χ develops a plateau.It is indeed the presence of this plateau that renders the theory observationally viable.As far as δ f is concerned, the results of this work are obtained considering the window δ f ∈ [10 −2 , 16].We have shown that this scenario can be compatible with observations for both trans-Planckian and sub-Planckian values of f .

Figure 3 :
Figure 3: Left plot: the location of the peak of the kinetic function, cos(ϕ peak /f ), for ξ = 0. Right plot: the height of the peak, k(ϕ peak ), for ξ = 0 and f = 5M P .

Figure 5 :Figure 6 :
Figure 5: r vs. n s (a), r vs. ξ (b), ξ vs. n s (c), δ Λ vs. ξ (d) for N e = 60 and δ f ranging from 4 (purple) to 16 (red) with steps of 2, displayed in rainbow colors.The gray, brown and black color codes are the same as in Fig. 4.

Figure 15 :
Figure 15: α s vs. n s (a), α s vs. ξ (b) for N e = 60 with β 0 = −10M 2P for the same values and color codes of δ f and ξ as in Fig.14.The pink and gray areas are the same as in Fig.6.