1 Introduction

Motivated by recent cosmological observations of a possibly unsuppressed primordial tensor component of inflationary perturbations, we reanalyse in detail the 5D conformal SUGRA originated natural inflation model of Ref. [1]. The model is a supersymmetric variant of 5D extra natural inflation, also based on a shift symmetry, and leads to the potential of natural inflation. Coupling the bulk fields generating the inflaton potential via a gauge coupling to the inflaton with brane SM states we necessarily obtain a very slow gauge inflaton decay rate and a very low reheating temperature  GeV. Analysis of the required number of e-foldings (from the CMB observations) leads to values of in the lower range of present Planck 2015 results. Some related theoretical issues of the construction, along with phenomenological and cosmological implications, are also discussed.

January 15, 2015

Natural Inflation from 5D SUGRA

and Low Reheat Temperature

Filipe Paccetti Correia111E-mail: fcorreia@deloitte.pt,  Michael G. Schmidt222E-mail: m.g.schmidt@thphys.uni-heidelberg.de, and  Zurab Tavartkiladze333E-mail: zurab.tavartkiladze@gmail.com

Deloitte Consultores, S.A., Praça Duque de Saldanha, 1 - 6, 1050-094 Lisboa, Portugal444Disclaimer: This address is used by F.P.C. only for the purpose of indicating his professional affiliation. The contents of the paper are limited to Physics and in no ways represent views of Deloitte Consultores, S.A.

Institut für Theoretische Physik, Universität Heidelberg, Philosophenweg 16,

69120 Heidelberg, Germany

Center for Elementary Particle Physics, ITP, Ilia State University, 0162 Tbilisi, Georgia

1 Introduction

Inflation solves the problems of early cosmology in a natural way [2] and besides that produces a primordial fluctuation spectrum [3] which allows to discuss structure formation successfully. In detailed models (i) a sufficient number of -folds for the inflationary phase has to be produced, (ii) guided by bounds presented recently by the Planck Collaboration [4], the cosmic background radiation and a spectral index should be generated.555In the original version of this paper (see v1 of arXiv:1501.03520) we had the 2013 value [5] which being about a standard deviation below this value makes quite a difference for our analysis. And (iii), the normalization of fluctuations has to be reproduced. Rather flat potentials for the inflaton field lead to the “slow roll” needed for (i). Such potentials appear naturally in (tree level) global supersymmetric models; higher loop corrections can be controlled, but the inclusion of supergravity easily produces an inflaton mass of the order of the Hubble scale.

In models with an extra dimension the fifth component of a gauge field entering in a Wilson loop operator can act as an inflaton field of pseudo Nambu-Goldstone type which is protected against gravity corrections and avoids a transplanckian scale [6], [7], present in the original model of “natural inflation” [8]. We have presented such a model [1] based on 5D conformal SUGRA on an orbifold with a predecessor based on global supersymmetry with a chiral “radion” multiplet on a circle in the fifth dimension  [9]. We also made the interesting observation that a spectral index as observed recently [different from a value very close to one usually obtained in straightforward SUSY hybrid inflation [11]], is obtained rather generically in gauge inflation. Actually, in the supersymmetric formulation we have a complex scalar field which besides the gauge inflaton contains a further “modulus” field which also might allow for successful inflation [1]. The main difference between the two inflation types is that gauge inflation leads to a large tensor to scalar ratio in [1]) whereas modulus inflation leads to very small in [1]).666Genuine two field inflation was discussed in ref. [10]. The two basic inflation types depending on initial conditions turn out to be still like in [1]. Since inclusion of the modulus into the inflation process is fully legitimate, one can reserve this scenario as an alternative with a tiny tensor perturbations, if it should be. Recently the BICEP2 data [12] gave strong indication of a large ratio though recent joint analysis of BICEP2/Keck and Planck [13] gave a reduced upper bound ,777Earlier, Planck’s intermediate results [14] noted about a possible ordinary dust contribution instead of the light polarization effect really due to gravitational waves. with the likelihood curve for having a maximum for . Because of this, we here consider the gauge inflation of ref. [1] again with particular emphasis on the required length of inflation. The well known -folds solving the horizon problem will turn out to require a substantial expansion during the reheating period within the natural inflation scenario emerged from 5D SUGRA.

Let us present the organization of the paper and summarize some of the results. In Sec. 2 we perform a detailed analysis of natural inflation with -type potential. For the calculation of the spectral index and the tensor to scalar ratio , we use a second order approximation with respect to slow roll (SR) parameters. Since these quantities () are determined at the point where the SR parameters are tiny, this approximation is sufficient for all practical purposes. However, near the end of the inflation, when SR breaks down, we perform an accurate numerical determination of the point via the condition on the Hubble slow roll parameter (see [15]-[17] for definitions). This is needed to compute, with desired precision, the number of -foldings () before the end of the inflation. We carry out our analysis by using recent fresh data [4] for , , the amplitude of curvature perturbations and bounds for [13]. Our results are in agreement with a recent analysis of natural inflation by Freese and Kinney (see citation in [8]). For various cases we have also calculated the reheat temperature, which we use later on for contrasting with our 5D SUGRA emerged inflation scenario requiring a very low reheat temperature.

In Sec. 3 and Appendix A we shortly review our model of ref. [1] in a more self-contained way and discuss how natural inflation emerges from D SUGRA. Using a superfield formulation, we do not need to go into the details of the component expressions in conformal 5D SUGRA of Fujita, Kugo and Ohashi (FKO) [18]. Indeed this emerged from our discussion [19, 20] (see also Ref. [23]) bringing the 5D conformal SUGRA formulation closer to the 4D global SUSY language [20]. We concentrate here on gauge inflation, i.e. on the case (stabilized moduli in the origin or a choice of initial conditions888For a discussion of moduli stabilization in the superfield formalism within 5D SUGRA see [24]. For a choice of initial conditions leading approximately to see Ref. [10].). In section 3, discussing the realization of natural inflation within 5D SUGRA, we present a new mechanism for inflaton decay, which eventually leads to the reheat of the Universe. Note that, besides a specific string theory realization [25], the inflaton decay and reheating has never been discussed before in the context of natural inflation. We show that the inflaton’s slow decay is a natural consequence of the 5D construction (with consistent UV completion), being realized by couplings of the heavy bulk supermultiplets generating the inflaton potential through their gauge coupling with brane SM states. Since the inflaton decay proceeds by 4-body decay and the decay width is strongly suppressed by the 2-nd power of the tiny gauge coupling constant999From a very recent paper [26] we learned that the ‘weak gravity conjecture’ (going back to Ref. [27]) based on magnetically charged black hole considerations and the dangerous neighborhood to a global symmetry, applies in disfavor of gauge (extranatural) inflation and might explain difficulties to embed the model in string theory. (of the gauge inflaton-charged fields) and a relatively small inflaton mass coupled to the intermediate bulk fields, a strong suppression of the reheat temperature comes out naturally. Our 5D SUGRA construction allows us to make an estimate  GeV (where is a brane Yukawa coupling). At the end of Sec. 3 we show that, by the parameters we are dealing with, preheating is excluded within the considered scenario.

Appendix A discusses the Kaluza-Klein spectrum of the fields involved, as well as the SUSY breaking effects for brane fields. We also perform a derivation of higher dimensional operators involving the inflaton and light (MSSM) states relevant for the inflaton decay. As it turns out, the dominant decay channel is (with and denoting SM lepton and Higgs doublets respectively). Sec. 4 includes a discussion and concluding remarks about some related issues.

2 Natural inflation

In this section we analyse inflation with the potential of natural inflation [8] given by:


where is a canonically normalized real scalar field of inflation. In the concrete scenario of Ref. [1], we focus later on, the inflaton originates from a D gauge superfield, while the parameters/variables of (1) are derived through the underlying 5D SUGRA. See Eqs. (24), (25), (A.17) and also the comment underneath Eq. (A.17).

The slow roll parameters (”VSR” - derived through the inflaton potential) are given by


where  GeV is the reduced Planck mass. In order to make notations compact, for the VSR parameters we do not use the subscript ‘V’ (denoting them by ). However, for HSR parameters (derived through the Hubble parameter) we use subscript ‘H’ (e.g. ), as adopted in literature [15], [17], [16]. The number of e-foldings during inflation, i.e. during exponential expansion, denoted further by , is calculated as


In this exact expression the HSR parameter (defined below), participates. The point , at which inflation ends, is determined by the condition . The point corresponds to the begin of the inflation. Also further, symbols with superscript or subscript ’i’ will correspond to values at the beginning of the inflation, while superscript/subscript ’e’ will indicate end of the inflation.

The observables and depend on the value of (the point at which scales cross the horizon). This allows to determine as follows. Via HSR parameters, the expressions for and are given by [15], [17], [16]:


where we have limited ourself with second order corrections. The HSR parameters are given by:


with the Hubble parameter and it’s derivative with respect to the inflaton field. The subscript in (4) indicates that the parameter is defined at the point at which scales cross the horizon. As it turns out, at this scale the slow roll parameters are small and second order corrections in and are small and the approximations made in (4) are pretty accurate. Exact relations between VSR () and HSR parameters () are given by [15], [17], [16]:


When the slow roll parameters are small, from (6), the HSR parameters to a good approximation can be expressed in terms of VSR parameters as


Using these approximations in (4), we can write and in terms of VSR parameters:


where we have still restricted the approximations up to the second order. Applying these expressions, for the model (determining and as given in Eq. (2)), we arrive at:




From Eq. (10) we can express in terms of and . As will turn out, the latter’s value is small, so to a good approximation we find:


Plugging this into Eq. (9) for the spectral index we get:


Using the recent value from Planck [4]101010The central value of , is larger, though the range (within ) is consistent with Planck’s old result [5]. This required modification of our first version appeared before the new results.111111Recent joint analysis of BICEP2/Keck and Planck [13] gave an upper bound , while the likelihood curve for has a maximum for . Note that the value of reported before by BICEP2 collaboraton [12] was although, later on, Planck’s intermediate results [14] warned about possible ordinary dust contribution instead of the light polarization effect really due to the gravitational waves. relation (12) provides an upper bound for the value of :


This will be used as orientation for further analysis and various predictions.

So far, we have performed calculations in a regime of small slow roll parameters, determining the value of via Eq. (11). As was mentioned, the value of is determined from the condition . Near this point both and parameters turn out to be large and instead of an expansion we need to perform numerical calculations. This will be relevant upon the calculation of the number of e-foldings .

Figure 1: Dependence of and on the value of , for .

Since, within our model, via Eq. (2) VSR parameters are related to each other as


the three equation in (6) can be rewritten as


where has been dropped because of it’s smallness. From the system of (15), for a fixed value of , the parameters and can be found in terms of the single parameter . The dependance of these parameters on the value of , for are shown in Fig. 1 (for different values of shapes of the curves are similar). We see that is achieved when and thus, the expansion with respect to within this stage of inflation is invalid. On the other hand, the values of and remain relatively small. From the relation one derives:


Using this, the integral in (3) can be rewritten as


Having the numerical dependence (depicted in Fig. 1), we can evaluate the integral in (17) and find for various values of . The results are given in Fig. 2. While BICEP2/Keck and Planck [13] reported the bound , upon generating the curves of Fig. 2 we also allowed larger values of . Curves in Fig. 2 and also Table 1 demonstrate that, within natural inflation, with values (the previous Planck 2013 value) and or and there is an upper bound on :


Turned around this also implies that violating the bound (18), say , indicates larger and/or smaller . Present Planck 2015 data seems to favor this. Bound (18) (if realized) would lead to another striking prediction and constraint.

Figure 2: Number of e-foldings. Solid lines correspond to the values of which fit with the current experimental data within error bars (with no restriction on ). Shaded areas correspond to the marginalized joint CL regions, given recently in [4] for pairs, mapped by us to the pairs for natural inflation. Gray background corresponds to the Planck TTlowP, while red and blue colors represent Planck TTLowPBKP and Planck TTLowPBKPBAO respectively (see Ref. [4] for an explanation of these combinations).

As discussed in Refs. [28], [5], the , guaranteeing causality of fluctuations, should satisfy:


where for the scale we take , while the present horizon scale is . The factor accounts for the dynamics of the inflaton’s oscillations [29], [30] after inflation, and can be for our model approximated as (will turn out to be a pretty good approximation).

To reconcile the first two entries () of Eq. (19) with the bound of Eq. (18) (see also Fig. 2), the remaining entries of Eq. (19) should be significant enough to bring down (at least) to . The and entries on the r.h.s. of Eq. (19) can be calculated with help of another observable - the amplitude of curvature perturbation , which according to the Planck measurements [4], [5], should satisfy (this value corresponds to the CDM model). Generated by inflation, this parameter is given by:

Table 1: Numerical Results for different values of and . For all cases

In order to obtain the observed value of , for typical and we need to have . This, on the other hand, gives and . Using these values in (19) we see that the sum of the and terms is. Thus, the last term should be responsible for a proper reduction of . Namely, during the reheating process, the universe should expand by nearly (or even more) e-foldings. This means that, for this case, the model should have a significant reheat history with  GeV.121212The reheating process can continue even till the epoch of nucleosynthesis. In this case one should have  GeV. Within the scenario of natural inflation, this has not been appreciated before.131313See however some recent analysis in Ref. [21]. For lower and appropriate values of (and ) the reheating temperature can be big. The concise numerical results (compared to the rough evaluation below Eq. (20)) are given in Table 1, where we considered cases with not smaller than  GeV, and . The values of the spectral index running are also presented. The first three row-blocks correspond to the values of within ranges of the current experimental data. The first three cases of the bottom block correspond to the within range, while the last three lines of this block have lower values of (beyond the deviation). Since the issue for the value of is not fully settled yet, we have included moderately large values of (). At the bottom block of the table we gave results for , which corresponds to the peak of the ’s likelihood curve presented by the joint analysis of BICEP2/Keck and Planck [13]. Note that the results presented here are consistent with the analysis for natural inflation carried out before [8] (see citation of this Ref.).

Below we will show that within our scenario of natural inflation, a low reheat temperature is realized naturally.

3 Natural inflation from 5D SUGRA

In order to address the details of inflaton decay, related to the reheat temperature, we need to specify the underlying theory natural inflation emerged from. A very good candidate is a higher dimensional construction [6]. Here we present a 5D conformal SUGRA realization [1] of this idea, using the off-shell superfield formulation developed in Refs. [19], [20].141414For the component formalism of 5D conformal SUGRA see the pioneering work by Fujita, Kugo and Ohashi [18]. Note also, that the component off shell 5D SUGRA formulation, discussed by Zucker [22], was used in many phenomenologically oriented papers.

Lagrangian couplings, for the bulk hypermultiplets’, components are:


where the odd fields are set to zero.151515The bulk hypermultiplet action of Eq. (21), derived from 5D off shell SUGRA construction [19], including coupling with a radion superfield , in a rigid SUSY limit coincides with the one given in Ref. [32]. is the even component of the 5D vector supermultiplet. With the parity assignments


the KK decomposition for and superfields is given by


With these decompositions, and steps given in Appendix A, we can calculate the mass spectrum of KK states, their couplings to the inflaton and with these, the one loop order inflation potential (dropping higher winding modes) having the form of (1) with


The 4D inflaton field is related to the 5D gauge field as:


Since the model is well defined, we also can write down the inflaton coupling with the components of . The latter, having a coupling with the SM fields, would insure the inflaton decay and the reheating of the Universe. In our setup, we assume that all MSSM matter and scalar superfields are introduced at the brane. Since is even under orbifold parity and a singlet under all SM gauge symmetries, it can couple to the MSSM states through the following brane superpotential couplings


where and are 4D SUSY superfields corresponding to lepton doublets and up type higgs doublet superfields respectively. In Eq. (26), without loss of generality, only one lepton doublet (out of three lepton families) is taken to couple with the ,


where now denotes the fermionic lepton doublet and an up-type higgs doublet. States and stand for their superpartners respectively. and in Eq. (27) indicate scalar and fermionic components of the superfield .161616In Eq. (27) we have omitted and type terms, which because of the smallness of the term(fewTeV) and suppressed lepton Yukawa couplings() can be safely ignored in the inflaton decay proccess.

Upon eliminating all -terms and heavy fermionic and scalar states (in the and superfields), we can derive effective operators containing the inflaton linearly. As it will turn out within the model considered (see discussion in Appendix A.1), the states are heavier than the inflaton and operators containing are irrelevant for the inflaton decay. Thus, the effective operators, needed to be considered, are


These terms should be responsible for the inflaton decay. Derivation and form of the -coefficients are given in Appendix A.

3.1 Inflaton Decay and Reheating

Figure 3: Diagram responsible for the inflaton’s dominant decay.

As was mentioned above and shown in Appendix A.1, the slepton states have masses and thus are heavier than the inflaton. Indeed, the latter’s mass, obtained from the potential, is:


( for successful inflation). Thus the inflaton decay in channels containing is kinematically forbidden. Anticipating, we note that the preheating process by inflaton decay in heavy states is excluded within our scenario with parameters we consider (this is shown at the end of this subsection). Thus, the reheating proceeds by perturbative 4-body decay of the inflaton.

Among operators generated via exchange of heavy fermionic and scalar states, only those given in Eq. (28) are relevant. For calculating the decay widths (in a pretty good approximation) it is enough to have the form of the coefficients.

As shown in Appendix A, within our model and the corresponding operator does not play any role. Moreover, according to Eqs. (A.26) and (A.30) we have (due to a ) and (with , dictated from the inflation). Thus, we get an estimate for the following branching ratio


This means that the inflaton decay is mainly due to the operator [see Eqs. (28 and (A.26), with gauge coupling and Yukawa coupling ], i.e. in the channel (the diagram in Fig. 3. Remember: denotes the SM lepton doublet and the scalar up type higgs doublet). For simplicity we assume that the state includes the light SM higgs doublet with weight nearly equal to one, i.e. .

For the decay width we get:171717For 4-body phase space we have used an expression of [31] derived for the decay, setting and replacing


The factor in the numerator accounts for the multiplicity of final states. (The final channel includes three combinations , , and for each pair of identical final states a factor should be included.) The denominator factors in (31) come from the phase space integration. Using the form of , given by Eq. (A.26), in expression (31), we get:


Expressing through the reheat temperature [33]


( is the number of relativistic degrees at temperature ) and using expressions (32) and (29), we get


From this, with , and we obtain  GeV.

Table 2: Values of , and for different cases of successfull inflation.

Our 5D SUGRA construction allows more accurate estimates, because some of the parameters are related to each other. For instance, from (24) we have


From (35) we see that in order to have we need  GeV.181818For adequate suppression of undesirable non local operators the large volume is needed [6]. The latter value suites well with most of the values of given in Table 1 (calculated from the inflation potential). At the same time, we see from (35) that can not be suppressed and should be . Using Eqs. (35) and (36) in (34), we obtain


This expression is useful to find the maximal value of . Using the pairs of given in Table 1, from Eq. (37) it turns out that  GeV. This is an upper bound on the reheating energy density obtained within our 5D SUGRA scenario. In Table 2 we give the values of , and for various cases. Input values of and were taken from Table 1, which correspond to successful inflation. Also, we have selected the values of in such a way as to get . We see that within deviations of we have  GeV, corresponding to reheat temperatures  GeV. These values can be easily reconciled with those low values of , given in Table 1, by natural selection of the brane Yukawa coupling in a range .

Excluding Preheating

Since in the presented 5D SUGRA scenario the inflaton has direct couplings with heavy KK states of and superfields, we need to make sure that after inflation, during the inflaton oscillation there is no production of these heavy states and no reheat is anticipated by the preheating process. Below we show that indeed, within our model preheating does not take place.

Starting from the fermionic KK states (which turned out to dominate in reheating), their masses are given by Eq. (A.16). with and shift of the inflaton around the vacuum


for fermion masses we get


The is the quantum part oscillating around the potential’s minimum (after the end of inflation) and finally relaxing to . Our aim is to see if either of the masses in (39) become zero during inflaton damped oscillation. As was shown in Ref. [34], this is the criterion for the fermionic preheating.

The amplitude of has a well defined value at the end of the inflation when slow roll breaks down, i.e. at the point . With , for times we have


On the other hand, from our Table 2 we have . Using this in (40), we get


The kinetic energy of the oscillation is still at most comparable at the end of inflation and there is also damping. Thus, Eq. (41) is a good estimate for the maximal amplitude of . With this bound, we can see that the term in Eq. (39) will not be able to nullify fermion masses during inflaton oscillations. This fact, as was shown in Ref. [34], prevents KK fermion production and no fermionic preheating takes place.

Now we turn to the scalar KK states. With Eqs. (A.10), (38) and for the scalar masses we get


Therefore, with Eq. (41) and the values of given in Table 2, we see that masses in (42) never cross zero and for the positively defined mass of the states we have


where the inflaton mass  GeV. Therefore, all modes from the scalar KK tower are much heavier (by a factor ) than the inflaton mass for any time during the inflaton’s oscillation. As was shown in Refs. [33], [35] for these conditions the amplification and/or production of the scalar modes never happens. This excludes preheating also via the scalar production. Within our scenario this result is insured by the gauge symmetry, because the inflaton in the heavy KK states’ masses contributes in the combination [see e.g. Eqs. (39) and (42)].

Thus, we finally conclude that within our scenario reheating occurs by the perturbative inflaton four-body decays discussed at the beginning of Sec. 3.1

4 Discussion and Concluding Remarks

In the effective action of our 5D conformal SUGRA model the -th component () of a vector supermultiplet couples to a charged hypermultiplet . This, due to a fixed compactification radius leads to the potential of natural inflation for the CP odd part of , neglecting the suppressed higher winding modes. We analysed this potential like in [8] putting emphasis on the potential of inflation and the number of e-folds of perturbations leaving the horizon. This we compared with the number of e-folds required by a causal connection between the observed universe background fluctuations and by the size of observed curvature perturbations. For a large tensor component and or and