Studying G-axion Inflation model in light of PLANCK

# Studying G-axion Inflation model in light of PLANCK

## Abstract

With the Planck 2015 result, most of the well known canonical large field inflation models turned out to be strongly disfavored. Axion inflation is one of such models which is becoming marginalized with the increasing precession of CMB data. In this paper, we have shown that with a simple Galileon type modification to the marginally favored axion model calling G-axion, we can turn them into one of the most favored models with its detectable prediction of and within its PLANCK range for a wide range of parameters. Interestingly it is this modification which plays the important role in turning the inflationary predictions to be independent of the explicit value of axion decay constant . However, dynamics after the inflation turned out to have a non-trivial dependence on . For each G-axion model there exists a critical value of such that for we have the oscillating phase after inflation and for we have non-oscillatory phase. Therefore, we obtained a range of sub-Planckian value of model parameters which give rise to consistent inflation. However for sub-Planckian axion decay constant the inflaton field configuration appeared to be singular after the end of inflation. To reheat the universe we, therefore, employ the instant preheating mechanism at the instant of first zero crossing of the inflaton. To our surprise, the instant preheating mechanism turned out to be inefficient as opposed to usual non-oscillatory quintessence model. For another class of G-axion model with super-Planckian axion decay constant, we performed in detail the reheating constraints analysis considering the latest PLANCK result.

\affiliation

Indian Institute of Technology Guwahati, Guwahati 781039, India

\emailAdd

debu@iitg.ac.in

\emailAdd

pankaj.saha@iitg.ac.in

\keywords

Galileon Cosmology, Natural Inflation, Instant Preheating

## 1 Introduction

The Inflationary paradigm has been an essential part of standard model cosmology since its inception [1, 2, 3]. With increasingly precise measurements of various cosmological observables, the inflationary mechanism as the initial condition of standard CDM cosmology is gradually becoming unique in nature. However, it is the source of inflation, which remains obscured till today because of neatly universal tree level prediction of the inflationary observables for a large number of models, [4]. According to the latest cosmological measurements made by Planck[5], Keck Array, and BICEP2 Collaborations[6], the scalar spectral index of curvature perturbation is and the scale dependence of scalar spectral index is tightly constrained to . The upper bound on the tensor-to-scalar ratio, for 95 GHz Data From Keck Array, is . All these cosmological observables can be obtained by introducing the inflation mechanism driven by either single or multiple scalar fields known as inflaton with a large variety of potential. It is based on the idea of slow-roll dynamics where the inflation rolls down an almost flat potential for long time to solve the so-called horizon and homogeneity problem(see[7] for a review). In terms of classical model building, it is very simple to construct such a flat potential. However, such an effective flat potential is very difficult to achieve in the framework of quantum field theory due to naturalness. Therefore, it is instructive to invoke shift symmetry in the inflaton field space. Importantly the constant shift symmetry naturally provides nearly flat potential and makes it stable against radiative correction. This was the idea behind the so-called natural inflation model[8] introduced in the early 90s. The inflaton, in this case, is known as axion whose nearly flat potential is of the form

 V(ϕ)=Λ4[1−cos(ϕf)].

Where and are two mass scales characterizing the height() and width() of the potential. The scale is known as axion decay constant. Though it is one of the best theoretically motivated model of inflation, it turned out to be observationally tightly constrained. Furthermore, it’s predictions closest to the observations become quantum field theoretically implausible for the super-planckian axion decay constant . Nonetheless, because of its naturalness, many different modifications have been proposed to make it observationally favored and simultaneously bring down the value of to a sub-Planckian value. String inspired -flation [9], axion monodromy inflation[10] are the various variant of this natural inflation model where there exist multiple fields with sub-Planckian values of . Multiple axionc fields are aligned in such a way that the effective axion decay constant could be sub-Planckian and make the model cosmologically viable. Recently proposed Weak-Gravity Conjecture(WGC)[11],as well as some string theory construction [12] put severe constraints on such models of alignment mechanism[13, 14]. However, considerations of covariant entropy bound seem to relax the bound[15].

In the present paper based on our previous works [16, 17], we point out that by introducing a specific form of higher derivative Galileon term for the axion, not only we can make inflation compatible with PLANCK, but also we can have sub-Planckian axion decay constant. We will call it as G-axion inflation model to make it compatible with the name already exists in the literature. However, for sub-Planckian axion decay constant, the price we have to pay is that the axion does not have any oscillatory phase after the inflation. Therefore, the usual reheating mechanism will be inapplicable. Hence we employ instant preheating mechanism proposed by Felder, Kofman and Linde [18] as our rescue. We have also found a bound on such that our model can be strictly sub-Planckian. For comparison, we will also study models with super-Planckian [19]. We must emphasize at this point that our G-axion inflation scenario with super-planckian axion decay constant turns out be more favorable than the conventional model with regard to the PLANCK observation.

We have organized the paper as follows: following our previous work, we first introduce Galilon type modified axion inflation model. We also discuss in detail their predictions for all the cosmological observables considering some simple choices of kinetic functions. In section-3, we discuss the background dynamics of inflaton during and after the inflation. It can be seen that generically for sub-Planckian axion decay constant inflaton field encounters singularity after some e-folding number. Those models which complete the inflation and become divergent after first zero crossing, we call them non-oscillating model. However for super-Planckian axion decay constant all the models have oscillatory behavior. In the subsequent sections, we will discuss the dynamics after inflation. In section-4, we consider the oscillating model for super-Planckian , and perform the reheating constraint analysis. For concreteness, we also compare our result with the usual axion inflation predictions. In section-5, we consider non-oscillating models for sub-Planckian . In order to reheat our universe, we employ the instant preheating mechanism for two different phenomenological coupling of the reheating field with the inflaton and compare the outcomes. In the end, we discuss our results and possible future directions.

## 2 Model and its cosmological dynamics

In the usual inflationary scenario, the action consists of a canonical kinetic term and a potential which is sufficiently flat to ensure inflation. However, a non-canonical inflationary model with a higher derivative term in the Lagrangian has special significance from the theoretical point view. One such popular model is Dirac-Born-Infeld (DBI) inflation [20, 21, 22], where non-canonical Kinetic terms also appear. It has been found out that apart from conventional inflationary prediction, these kinds of non-canonical models have other interesting observable predictions such as large non-gaussianity, variable sound speed. They are constructed in such a way that it does not lead to any ghosts. A very intriguing such class of models exhibiting ‘Galilean’ symmetry () has been dubbed as Galileon inflation models [23]. These Galileon Models has been further extended by Deffayet et. al.[24], where the Galilean symmetry was broken in order to preserve the second order nature of the equation of motions. A moments pause will ensure us that such a breaking of symmetry is indeed necessary for a viable model of inflation in order to end the inflation and to reheat the universe. Here we will not be talking about the symmetry breaking mechanism, rather, we will discuss a specific class of inflationary models with a Galileon symmetry broken down to discrete shift symmetry.

### 2.1 G-axion

In this section, we will be essentially reviewing the main construction based on our previous work [16, 17]. The Lagrangian for G-axion field is

 S = ∫d4x√−g[M2p2R−X−M(ϕ)X□ϕ−Λ4[1−cos(ϕf)]] (1)

where and , and are the axion decay constant and axionic shift symmetry breaking scale respectively. is also known to be associated with the scale of inflation. is the reduced Planck mass. This specific form of the higher derivative term sometimes known as kinetic gravity braiding (KGB). With the usual FLRW-background ansatz for the spacetime

 ds2=−dt2+a(t)2(dx2+dy2+dz2), (2)

one gets the following Einstein’s equations by varying the action with respect to the metric

 3M2pH2=−3H˙ϕ3M(ϕ)−X+2X2M′(ϕ)+Λ4[1−cos(ϕf)] (3)
 M2p˙H=−X(1−3M(ϕ)H˙ϕ+M(ϕ)¨ϕ+M′(ϕ)˙ϕ2) (4)

and by varying scalar field:

 1a3ddt[a3(1−3HM˙ϕ−2M′X)˙ϕ]+˙ϕddt(M′X)+Λ4fsin(ϕf)=0. (5)

Where, is the Hubble constant. Following[25] the slow-roll equation for the axion turns out to be,

 3H˙ϕ(1−3M(ϕ)H˙ϕ)+Λ4fsin(ϕf)=0. (6)

From the form of the above equation, we can consider two different ways to inflate our universe. The condition will give the standard axion inflation scenario, while the condition , provides alternative scenario where the higher derivative term comes into play. As emphasized in the introduction, the usual canonical axion inflation scenario is tightly constrained from the PLANCK observation. We will see for a wide range of parameter space higher derivative term will play main roll in G-axion-inflation. Solving the slow-roll equation of motion for we get

 |˙ϕ|≃Mp(V′3M(ϕ)V)1/2. (7)

Using Eq.(7), the condition for the KGB term to dominate the usual slow-roll term can be monitored by defining a parameter

 τ=M(ϕ)V′(ϕ)=M(ϕ)Λ4fsin(ϕf)≫1. (8)

The slow roll parameters in terms of potential function, denoting as , are

 ϵ = 12Asin32(~ϕ)√~M(~ϕ)(1−cos(~ϕ))2 ; η=12A√cos(~ϕ)cot(~ϕ)√~M(~ϕ)(1−cos(~ϕ)) α = 2√A~M′(~ϕ)√2ϵ4√~M(~ϕ)5sin(~ϕ)   ;   β=136A~M(~ϕ)2ϵ√~M(~ϕ)3sin(~ϕ). (9)

We will call as the KGB function and is an associated mass scale which will control the strength of the higher derivative term. We define a parameter which will greatly simplify further calculations. In case of potential driven inflation [26], the KGB function has a significant influence on the inflation dynamics. Here our aim is to identify the form of the KGB function based on the principle of constant shift symmetry and try to satisfy the experimental observation. At this point let us also define an additional parameter corresponding to the higher order slow roll parameter which is related to another measurable quantity called running of scalar spectral index,

 ξ=M4p12MV′(V′′′V′V2)=−1A2sin(~ϕ)~M(~ϕ)(1−cos(~ϕ))2. (10)

For our later convenience, we note the following relation between the KGB function and the slow roll parameter,

 M(ϕ)˙ϕ3M2PH≃−23ϵ. (11)

Which tells us that, though the higher derivative term dominates the dynamics of slow-roll inflation, the standard part of the Lagrangian remains much larger than the Galileon part during inflation and they both became comparable only after the end of inflation .

### 2.2 Cosmological quantities: (ns,r,dnks)

It is worth mentioning that inflation not only solves some of the important problems of Big-Bang Cosmology but it is known to be an important mechanism for the generation of seed for the large-scale structure of our universe(see [7] for a review). The formation of the this seed of structure formation and their subsequent evolution is known as ‘The Cosmological Perturbation Theory’(See [27] and [28] for some recent review. The perturbation treatment of Galileon models has been done extensively after the emergence of the model [29, 30]. In this section, we will note down some of the important cosmological quantities which are being measured in cosmological experiments. Following [25], the amplitude of the power spectrum of the curvature perturbation is written as

 PR=3√664π2H2M2pϵ. (12)

The spectral tilt and its running can be easily found out using the relation, 1

 Missing dimension or its units for \hskip (13) dnsdlnk≡dnks=−3ξ+24ϵη−24ϵ2−3α2−8αϵ+4αη+3η2+18β.

The power spectrum and the spectral index of the primordial gravitational wave are:

 PT=8M2p(H2π)2  ;  nT = −2ϵ. (14)

Another important quantity of cosmological importance is the tensor to scalar ratio

 r=−32√69nT (15)

The idea of inflation was introduced in the first place to solve the horizon problem and the flatness problem. It has been commonly said that we need sufficient amount of inflation to solve the aforementioned two problems. This idea of ‘sufficient’ inflation is quantified by what is known as the ‘e-folding’ number which will be described next.

### 2.3 Number of e-folds and modified Lyth bound

The ‘e-folding’ number is given by by the following expression,

 N=∫t2t1Hdt=∫t2t1H˙ϕdϕ=1Mp∫ϕinϕendτ14√2ϵdϕ. (16)

From the cosmological observation, the e-folding number is found to be . It is apparent that the amount of inflation must be proportional to the amount of field excursion during the slow roll inflation. This quantity is known as Lyth bound [31](also see [32, 33]). Lyth bound is assumed to play an important role in constraining the parameter space of a model under consideration from the low energy effective field theory point of view. In the slow roll approximation, one can compute the amount of field excursion by series expansion in terms of e-folding number calculated from the onset of inflation as follows

 ϕ(δN)=ϕin+∂ϕ∂NδN+12∂2ϕ∂N2δN2+…. (17)

Therefore, up to second order in slow roll, one can write down the general expression for the amount of field excursion for e-folding number as

 Δϕ=|ϕ(δN)−ϕin|=δN(∂ϕ∂N)[1+δN2(2ϵ−η+α2)]. (18)

Therefore, by using eq.(16), we get

 Δϕ=δN∣∣ ∣∣√2ϵτ(ϕ)14∣∣ ∣∣[1+δN(2ϵ−η+α2)]. (19)

From the above expression for the e-folding number assuming, that the slow-roll parameters and behave monotonously, we can write

 N≲ΔϕMp∣∣ ∣∣τ(ϕ)14√2ϵ∣∣ ∣∣max=ΔϕMp∣∣ ∣ ∣∣τ14max√2ϵmin∣∣ ∣ ∣∣. (20)

In deriving the above expression, we have kept only the leading order term in slow roll. However, above expression can get significant correction near the end of inflation where slow roll parameters are not small. We will do the detailed study on this issue and its consequence on the model for our future publication. Now, Eqs(12-15) can be used to relate the field excursion during inflation with the scalar-to-tensor ratio as,

 Δϕ≳ (NMp)∣∣ ∣ ∣∣√2ϵminτ14max∣∣ ∣ ∣∣=f√ANTmax√9r36√6, (21)

where,

 Tmax=(s3M(~ϕin)sin~ϕin)14.

As we can see there is an important difference in the above expression for Lyth Bound with usual slow-roll inflation case via the presence of the term which contains the KGB term . This feature will help us to construct a model with detectable tensor-to-scalar ratio with sub-planckian axion decay constant. Therefore, we are in a position to compare the Lyth bound between G-axion inflation and the usual canonical slow-roll inflation, which can be written in a model independent way as . For, and , . Now since we have freedom in choosing , the modified Lyth bound can be made very small. As an example, for , if we choose , one gets . In the subsequent section, we will consider some simple phenomenological form of the KGB functions keeping the constant shift symmetry of the Lagrangian intact, and construct both super and sub-Planckain inflation models in compatible with PLANCK.

### 2.4 Simple choices of M(ϕ) and determination of cosmological parameters

As we have emphasized in the beginning, we will be considering some simple functional form of exhibiting shift symmetry of the axion field. As we will see from our analysis, the CMB observables are not sufficient to constrain the value of the axion decay constant for the model under consideration. Therefore, the dynamics after the inflation will be important and we will indeed see the existence of a critical value of axion decay constant which separates super and sub-Planckian scenarios in terms of oscillation dynamics. However, in this section, we will mainly compute the inflationary observables and compare our result with the current observational bounds coming from PLANCK for some simple choices of . We consider the following forms of :

1. : ,

2. : ,

3. : ,

4. : ,

5. : ,

The CMB observables with the above choices of function have been presented in figs.(1) and (2). For comparison, we also plotted the standard axion inflation model. From our analysis and also clear from the plot, we have two different categories of model based on their predictions. The best fit models are of type . One can clearly see that within region of , natural inflation model is almost ruled out. On the other hand the our aforementioned best fit model predicts as low as within the same region. Other three types are marginally fitting with the data. All these models predict very high value of tensor to scalar ratio which can be measured in the near future CMB experiments. Particularly, for , which is the best out of this group predicts, for which is within region of CMB data. Interestingly, as we will discuss in detail, those two types of models show distinct behavior after the inflation depending on the value of axion decay constant.

All the inflationary prediction will only depend on the emergent parameter . Therefore, we need further condition to constrain the value of . We determine the height of the axion potential from the WMAP normalization for the scalar power spectrum [34]

 PR=A√632π2(ΛMp)4(1−cos~ϕ1)3√s3M(~ϕ1)sin32~ϕ1≃2.4×10−9,.

The value of turned out to be for all the above models under consideration. Therefore, by using the values of , the following expression of ,

 s3f3=1A2Λ4M4p, (22)

will provide us all possible values from super-Planckian to sub-Planckian for . From particle physics, we can set the lower bound on as a symmetry breaking scale which should be higher than the inflationary scale implying . However, the bound on the values of from the any fundamental theory is not evident to us and obviously needs further study. Hence, we chose as a free parameter which measures the strength of the higher derivative operator. Up to this point, we have all the necessary ingredients for a successful inflation producing the correct values of the scalar spectral index within the observational limit. Our next task would be to go beyond and study the axion dynamics after inflation when the particle production from inflaton field will be initiated. At this point let us re-emphasize the fact that during the inflation the change in axion field turned out to be

 Δϕ=(~ϕ1−~ϕ2)×f

Therefore, for a wide range of axion decay constant , we will have sub-planckian field excursion with the required value of e-folding number. Consequently our model will not have any serious trans-planckian problem.

## 3 Evolution of the scalar field: sub-Planckian (f,s)

It must be clear from previous discussions that the value of will be constrained from appropriate slow roll condition. For our G-axion or more generally Galileon inflation models, the nature of initial slow roll condition is dependent upon the choice of as it has to satisfy a non-trivial relation (8). In this section we will try to understand, qualitatively, the evolution of the scalar field depending on the initial conditions as well as our model parameters. The equation of motion for the scalar field (5) combined with eq.(4, in unit of ) can be written as

 Δ(t)¨ϕ(t)+[3H−9M(ϕ)H2˙ϕ+12M′′(ϕ)˙ϕ3−92M2(ϕ)H2˙ϕ4+32M(ϕ)M′(ϕ)˙ϕ5]˙ϕ(t)+V′(ϕ)=0 (24)

Where, the coefficient of is denoted as ,
From the above equation, it is evident that the solution may encounter singular behavior depending upon the initial condition when the coefficient becomes zero. The initial conditions in turn depend upon the parameters . In our numerical calculation, it has been found that the scalar field does show singularity or oscillatory behavior depending on the values of the above mentioned parameters. The variation of with time for three different values of has been shown in the fig.3.

It can be seen from the figure that the minimum value of the coefficient tends to zero with decreasing . The smallest value of axion decay constant which yields an oscillatory solution is when the coefficient is about to touch the zero axes. These numerical results have been confirmed in four different numerical environments. Fig.4 shows an illustration of the scalar field behavior for . We consider three different values of in unit of Planck when the scalar field solution makes a transition from being singular to oscillatory behavior. For all different functional form of , we found similar behavior for different critical values of . However, it is interesting to note that models of type , which are marginally fitting with the PLANCK observation, always complete the inflation before the field dynamics becomes singular for sub-Planckian decay constant. On the other hand best fit models of type do not show such behavior as singularity appears well before the completion of inflation for sub-Planckian . This behavior can be inferred also from the fact that for aforementioned models the slow roll condition is violated well before the inflation ends for sub-Planckian . To this end we emphasize the following observation: all the models under consideration will have coherent oscillation for super-Planckian axion decay constant. Two of them can explain CMB observation for sub-Planckian . From our numerical analysis, we found can even take super sub-Planckian value without changing the properties of the solution. However, from the effective field theory point view, the value of axion decay constant should be limited to . None the less, the price we pay for those sub-Planckian model is that after inflation the inflaton field shows singular behavior, we call them non-oscillating axion models. Hence, as will be subsequently discussed, for those models we will employ the instant preheating mechanism to reheat the universe.

An interesting point about the solution is that there exists a minimum value of the field excursion, above which we have the oscillatory solution. This value is independent of the parameters for a particular form of . In the table 1, we have shown the critical value of and the minimum field excursion when the oscillatory behavior sets in.

The PLANCK-2015 observation suggests the value of should be within , considering different models under consideration. Using the condition for dominating higher derivative term during inflation eq.8, aforementioned range of constrains the value of axion decay constant to be .

It is now clear that for a wide range of axion decay constant our G-axion model fits extremely well with the CMB observation. For further understanding, it is of prime importance to go beyond the inflationary dynamics. However, for the singular behavior of the inflaton field which we have been discussing in the previous sections, we need a detailed analysis considering the additional reheating field coupled with the inflaton. Thanks to the instant preheating mechanism [18] which has been proposed based on the mechanism of parametric resonance[35]. Instead of considering the full dynamics, for non-oscillatory model we employ the aforementioned instant preheating mechanism which acts at the instant of the first zero crossing of the inflaton field [36, 39, 37, 38]. However, for the oscillatory solution, the usual treatment of parametric resonance as well as perturbative reheating by solving the appropriate Boltzmann equation[40, 41] must be applicable. For those oscillatory models, we compute the behavior of density and pressure of the inflaton field in terms of background expansion,

 ρϕ=12˙ϕ2−3M(ϕ)H˙ϕ3+12M′(ϕ)4+Λ4[1−cos(ϕf)] (25) Pϕ=12˙ϕ2+12M′(ϕ)4+M(ϕ)˙ϕ2¨ϕ−Λ4[1−cos(ϕf)]. (26)

As expected, by using the numerical fitting, we found , with the equation state . Thus inflaton behaves as a matter(pressure-less dust) field during the coherent oscillating phase.

As emphasized earlier, in the subsequent sections our main goal is to study the dynamics of G-axion model during reheating phase for both oscillatory and non-oscillatory cases. For the oscillating axion model, we do the general reheating constraint analysis and analyze the reheating temperature and duration of perturbative reheating considering the constraints from CMB. Finally for the non-oscillatory model we employ instant reheating scenario to reheat the universe.

## 4 Oscillating axion: Constraints from rehating predictions

As we have seen, depending upon the value of axion decay constant we have two different possibilities for the axion field dynamics after the inflation. In this section, we consider studying the reheating constraints for the oscillating models based on the analysis of [42]. The evolution of cosmological scales throughout the evolution history of our universe, and the entropy conservation [43, 42, 44] have been proved to be an important way to constraining the inflationary model. One of the important steps towards this is to parametrize the reheating phase by its characteristic reheating temperature , the equation of state parameter , and reheating e-folding number . Following our previous work [45], aforementioned parameters are inter-related through the following equations,

 Nre=4(1+γ)(1−3ωre1)+γ(1−3ωre2)⎡⎢ ⎢⎣61.6−ln⎛⎜ ⎜⎝V14endHk⎞⎟ ⎟⎠−Nk⎤⎥ ⎥⎦ (27) Tre=⎡⎣(4311gre)13a0T0kHke−Nk⎤⎦3[(1+ωre1)+γ(1+ωre2)](3ωre1−1)+γ(3ωre2−1)[32.5Vendπ2gre]1+γ(1−3ωre1)+γ(1−3ωre2). (28)

In the above expressions we have considered two stage reheating process as will be subsequently explained. In this two stage reheating process, we parametrize the reheating parameters as . Where, are efolding number during the first and second stage of the reheating phases with the equation of state parameters respectively. In the derivation of above two formulas, we also assume the change of reheating stage from the first to the second one to be instantaneous.

With all the aforementioned ingredients we study the possible constraints on our G-axion inflation model. As we have seen from the previous analysis, all the cosmological quantities during inflation can be expressed in terms of the emergent parameter . Therefore, we will have a range of value of for which cosmological predictions will be within the range of observed values of .

A particular scale exiting the horizon during inflation and re-entering the horizon during usual cosmological evolution provides us an important relation . Where, are the cosmological scale factor at the end of the reheating phase and at the present time respectively. are the efolding number and the Hubble parameter respectively for the aforementioned scale which exits the horizon during inflation. is the number of relativistic degrees of freedom after the end of reheating phase. is the current value of the CMB temperature. For further calculation, we define a quantity, . We identify the scale of cosmological importance as the pivot scale of PLANCK, , and the corresponding estimated scalar spectral index to be .

The background inflationary dynamics of all the models under consideration do not explicitly depend upon axion decay constant but on an emergent parameter . Hence our subsequent discussions on constraints from reheating will remain same for any value of above critical value while keeping constant. Before we quantify the constraints on our model, general description of our figure are as follows: Throughout the analysis we will consider two physically realizable values of . It can be shown that if we consider single phase reheating process, for , both the quantities, , become indeterministic. This fact corresponds to all the vertical solid red lines in and plots. All the dotted or solid blue curves correspond to . On the other hand, all the dotted and solid magenta curves correspond to the two stage reheating process with , and . It is worth mentioning that by tuning either , or within the magenta curve will always remain within the blue curve and the vertical red line. On the same plot of , we also plotted corresponding to the brown curves. The solid brown curve corresponds to , for the sub-Planckian models. For the super-Planckian models, solid brown curve corresponds to the bunch of curves which are in the middle of the three bunches. For super-Planckian model, -prediction is almost independent of value. If we consider PLANCK’s bound of , one can easily discard some region of the parameter . Each bunch of (red, magenta, blue) curves apparently emanating from a point corresponds to a particular value of . Therefore, at least for the super-Planckina models, as we go towards higher value of it will given the overall description of all the plots. We now set to constrain the model parameters and corresponding prediction for for two different kind of models.

### 4.1 Sub-Planckian models: M(ϕ)={constant,sin(ϕ/f),sin2(ϕ/f)}

As we have described already, sub-Planckian models are those which can explain CMB observation even for sub-Planckian axion decay constant. For those models we can have axion decay constant as low as Pecci-Quinn symmetry breaking scale . However, from the effective field theory point of view, we must limit the value of above the inflationary energy scale in the unit of Planck. As explained before, for these models, the axion field will not have an oscillating phase after inflation. Therefore, we have to employ the instant preheating scenario which we have extensively discussed in the next section. Important to mention that for all these ’s, we will certainly have coherent oscillation, but with the axion decay constant . For illustration, we have plotted only for first two models. As we have found, for both the models the critical value at which the axion field starts to oscillate after the inflation are in Planck unit for respectively. For illustration we have considered for and for in the fig.6. For both the models, corresponds to the left bunch of curves emanating from a point on line. As we further increase the value of , the left bunch or in other words the line does not move further left. This essentially means, for a particular value of efolding number N, we have a minimum value of as a function of . This can also be seen from the Fig.(1).

### 4.2 Super-Planckian models: M(ϕ)={1−sin(ϕ/f),cos2(ϕ/f)}

In this section, we discuss those models which fit well with the PLANCK observation for super-Planckian axion decay constant, Fig.(1). Let us re-emphasize again that for these models also we have critical values of . However, below this value the inflaton encounters singularity before the inflation ends. Therefore, we only have oscillatory solution for these models which are observationally viable. In order to compare, we have also plotted with respect to for usual natural inflation as shown in Fig.(8). Therefore, from the (brown curve) superimposed with the curve, clearly disfavored the conventional natural(axion) inflation. Going beyond the natural(axion) inflation is necessary. In any case for illustration we have considered for and for as shown in Fig.(7). Where increasing the value of is equivalent to going from the right bunch of curve to left bunch. Therefore, within , one can clearly discard for models respectively. One can easily see that our generalized natural inflationary models fall within the region in space provided by PLANCK as opposed to the natural inflation with usual kinetic term.

## 5 Non-oscillating axion: Instant preheating

Reheating phase after the inflation is a stage when the energy stored in the inflaton field is transferred into the matte field we see today. As we have noted, we get two types of scalar field solution after inflation. For oscillatory solution, the standard mechanism of reheating works well in our model[17, 48]. But for the cases when the scalar field does not have coherent oscillation after the inflation, we need to invoke alternative mechanism to reheat our universe. We see that instant preheating, originally proposed as an alternative to preheating mechanism for no-oscillatory inflation models, could be important. This mechanism has been successfully implemented to reheat the universe in quintessential inflation [36, 37, 38, 39]. In the following sections, we study instant preheating for sub-Planckian model for . As mentioned before, we consider two types of coupling between the reheating field and the inflaton and compare their results. We will see how the higher derivative coupling term plays the important roll in our analysis.

### 5.1 Conventional Shift Symmetry Breaking Coupling

Let us start by considering the interaction Lagrangian as of the original work of Felder-Kofman-Linde. Where the inflation interacts with another scalar field which decays to a fermion field . We write the interaction Lagrangian as

 Lint=−12g2ϕ2χ2−h¯ψψχ, (29)

where the couplings are supposed to be positive with in order for the perturbation treatment to be valid. It is evident that the above Lagrangian does not respect the shift symmetry of our original Lagrangian. We will consider a special shift symmetric case in the next subsection. For simplicity, we will consider the particles do not have a bare mass, while its effective mass is provided by the inflation field as

 mχ(ϕ)=g|ϕ|. (30)

The production of the initiate as starts changing non-adiabatically after the end of inflation.

 |˙mχ|≳m2χ  or,|˙ϕ|≳gϕ2 (31)

In the Fig.9-a, we see the region where the adiabatic condition is violated for different values of dimensionless coupling parameter . Above condition implies that

 |ϕ|≲|ϕprod|=(˙ϕendg)12 (32)

Now, to estimate analytically, we assume the slow-roll condition to hold till the end of the inflation. This is where the higher derivative term in our model plays the role. Using eq.[11], we find that

 |˙ϕend|≃(2ϵendMp3√3M(ϕ))13(Vend)16=(2Mp3√3M(ϕ))13(Vend)16=P(ϕ)(Vend)16, (33)

where, we have denoted

 P(ϕ)=(2Mp3√3M(ϕ))13

It would more intuitive to express the above expression of in terms of the derived parameter, that we have defined earlier. This reads as

 P(ϕ)=0.721A23fMp[Λ4~M(~ϕ)]13 (34)

The production time for particles can be estimated as

 Δtprod∼|ϕprod||˙ϕ0|∼1√gP(ϕ)(Vend)16 (35)

where is the velocity of the field near the minimum of the effective potential. Let us now make an estimate of the numerical values of the quantities involved. We can use the relation (12) and the observed value of the scalar power spectrum to determine the value of the inflaton potential at the end of the inflation. Taking a sample value of the KGB scale , one finds . In order for the particle production to occurs within a very short period of time, we must satisfy which provides lower bound on as . Where is the axion decay constant in unit of . Using these values, we find the value of . Hence to get , we must choose . Using uncertainty relation one can estimate the momentum of particles which will be be created non-adiabatically, and [18, 35], the occupation number of particles jumps from zero to

 nk≃exp(−πk2/k2prod) (36)

during the time interval . The number density can be estimated to be

 nχ=12π2∫∞0k2nkdk≃k3prod8π3≃(gPend)32V14end8π3 (37)

and the energy density of the particles can be found to be

 ρχ=mχnχ(aenda)3=(gPendV16)328π3g|ϕ|(aenda)3 (38)

where the term corresponds to the dilution of the energy density due to cosmic expansion.

In numerical calculation, we found that the value of as small as satisfy the adiabaticity condition mentioned above. And with this value of coupling constant, we can readily check that production of the particle is indeed instantaneous. Now, if the quanta of the -field were to thermalized into radiation instantly, the radiation energy density would become

 ρr≃ρχ∼(gPend)V16)328π3gϕprod∼g2P2(ϕ)V13end8π3. (39)

Now the efficiency of instant preheating can be parametrized by the following ratio,

 ρrρϕ ≃ 2×10−3(gA23)2(fMp)2(1~Mend)23≃5×10−6g2.

Where, in the final numerical value we considered order of magnitude values of all the parameters , and . To this end let us quote the above ratio of energy densities for the Quintecence inflation model where also one does not have oscillatory phase after the inflation [38]

 ρrρϕ≃10−2g2. (40)

This result can be proved to be generically true for conventional canonical inflation model with the aforementioned interaction eq.29 term. Therefore, comparing with the usual model, we conclude that for G-axion inflation model instant preheating will not be efficient enough as it is suppressed by the same emergent parameter which played the important role in obtaining the sub-Planckian axion decay constant . In addition, we also have dimensionless coupling constant which is constrained to be small to avoid strong coupling problem.

So far we have not mentioned anything about the fermionic coupling parameter . Even though the particle production at the instant of first zero crossing of inflaton, does not seem efficient, for completeness let us constraint the possible value of for the instant preheating mechanism to work. The decay rate reheating field into radiation is given by , with . Now for instant preheating to work it must be larger than the expansion rate of the universe at instant of preheating

 Γψ¯ψ≳Hprod⟹h2≳8πHprodgϕprod (41)

The order of magnitude of can be estimated for some sample value of we can have

 h≳0.3forg∼10−2 h≳0.9forg∼10−3

As we have concluded from our analysis, instant preheating mechanism turns out to be inefficient for the conventional shift symmetry breaking coupling considered above. We want to see if any other type such as shift symmetric coupling can give some enhancement in the efficiency.

### 5.2 Shift Symmetric Coupling

As we have started with shift symmetric theory, for completeness, we express the main result for the following shift symmetric coupling

 Lint=−12M2 sin2(ϕf)χ2−h¯ψψ.χ (42)

Where dimensionful coupling parameter. In the fig.9-b, we plotted the region where the adiabaticity is lost. Follow the same methodology discussed before and under some reasonable assumption and approximation, we arrived at the following ratio of produced energy density over the inflaton energy density,

 ρrρϕ≃2×10−3(MMpA2)(ΛMp)2(1~Mend)≃2×10−7M, (43)

Where again for final numerical value, we have chosen , and . Therefore, we again see the suppression by the factor .

Our observation in this section is that the instant preheating mechanism seemed to be inefficient in transferring the energy from inflaton to reheating field for higher derivative driven G-axion inflation. This negative result can be attributed to the fact that the velocity of the inflaton field near the end of inflation is suppressed by the parameter as opposed to the usual axion inflation scenario. Therefore, the adiabaticity violation turned out to be very weak near the zero crossing of the inflaton field. Hence, the particle production became inefficient compared to the usual inflation model.

## 6 Conclusions

In this paper we have studied the cosmology of a modified axion inflation model with a specific form of higher derivative kinetic term. We call it G-axion. As we have already discussed, the usual axion inflation is disfavored for its prediction of larger tensor to scalar ratio within range of of PLANCK. From our two component reheating constraint analysis shown in fig.8, we also noticed that the canonical axion inflation model is outside the PLANCK limit. Therefore, modification of axion inflation is essential. In this paper we extended our previous analysis [16, 17] taking into account some simple choices of Galileon interaction term in understanding more on the sub-Planckian dynamics. One of the important motivations of studying such a modification, of course, is to explain the current observational data by PLANCK. However, another reason is to make the model consistent in the framework of effective field theory. With such a higher derivative modification, our model turned out to predict inflationary parameter within the range of Planck data for sub-Planckian values of all its parameters. However, we have found for some specific choices of functions, , the sub-planckian axion decay constant yields singular behavior in the scalar field dynamics specifically after the inflation. We dubbed them as non-oscillatory G-axion model. The existence of such kind of singular behavior in cosmological context has been discussed in recent studies[49, 50]. The singularity behavior emerges from the kinetic function when approaching towards zero. This may be related to the strong coupling problem. Understanding this properties will be interesting, and we left it for our future studies. Nonetheless, it has been found that these sub-Planckian models provide successful inflation and right after crossing the zero in the field space it hits the singularity. Since these models give successful inflation, we employed the instant preheating mechanism to reheat our universe. However, it turned out that the mechanism under study is not efficient enough. The amount of energy transferred from inflaton to the reheating field is suppressed by the same factor which helped us to make the model consistent with the PLANCK observation on inflationary observables. We have studied two different types of coupling namely, shift symmetric and conventional broken shift symmetric, to study preheating. For both types of coupling energy transfer turned out the be inefficient. Therefore, we need further study to understand the sub-Planckian G-axion model. Interestingly, with regards to the latest observation made by PLANCK, we found out super-Planckain G-axion models for which are very well fitted as opposed to the conventional axion inflation. For those two models we have prediction of within the range of which could be detectable in the near future CMB experiments. For super-Planckian G-axion models, the inflaton oscillates after the end of inflation. We, therefore, have studied model independent reheating constraint analysis.

A particularly interesting point we would like to mention is the role played by the axion decay constant in the dynamics of G-axion inflation. Depending on the value of , dynamics could be either usual kinetic term dominated or the galileon term dominated. Therefore, an interesting regime of exists when both terms play the role. In terms of the emergent parameter , the condition for inflation dominated by KGB term translated into the following approximate relation: . Although this bound on has a certain degree of dependence on the choice of . If the value of the axion decay constant violates the above bound, depending on the initial condition, our numerical computation shows that the inflaton dynamics is dominated by the higher derivative term at the initial stage and subsequently it becomes canonical kinetic term dominated around the end of inflation. This fact of two phases of inflation will certainly have interesting consequences on the observable quantities and CMB spectrum. For instance, in some models of inflation[51, 52, 53] a phase of super inflationary is introduced at the initial stage of inflation to explain the power suppression in the large angular scales of the CMB. This interesting effect can be naturally explained in our model. We left it for our future study.

## 7 Acknowledgment

We are very thankful for numerous vibrant discussions with our HEP and GRAVITY group members.

### Footnotes

1. In case of KGB inflation,the slow-roll scalar field equation can be used to find

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