Saddle point inflation in string-inspired theory

The observed value of the Higgs mass indicates the possibility that there is no supersymmetry below the Planck scale and that the Higgs can play the role of the inflaton. We examine the general structure of the saddle point inflation in string-inspired theory without supersymmetry. We point out that the string scale is fixed to be around the GUT scale $\sim10^{16}$GeV in order to realize successful inflation. We find that the inflaton can be naturally identified with the Higgs field.


Introduction
The recently observed particle at ATLAS [1] and CMS [2] experiments at the Large Hadron Collider (LHC) is consistent with the Standard Model (SM) Higgs with the mass around 125 GeV.Up to now, there has been observed no significant deviation from the SM nor a hint of new physics.Once the Higgs mass is determined, we have fixed all the parameters in the SM and can extrapolate it up to its ultraviolet (UV) cutoff scale.In particular, the quadratically divergent bare Higgs mass is found to be suppressed when the UV cutoff is at around the Planck scale [3], see also Ref. [4].Furthermore, the quartic Higgs coupling becomes tiny at the same time, see e.g Refs.[3,5].This opens up the possibilities of identifying the Higgs field as the inflaton [6,7], and of the absence of supersymmetry below the Planck scale.Although non-supersymmetric vacua are ubiquitous in string theory [8,9], their phenomenology has not been well studied.It becomes important to explore the phenomenology starting from non-supersymmetric theory.
In this letter, we consider the saddle point inflation scenario starting in string-inspired theory without supersymmetry.The potential is generated perturtabatively in contrast to the supersymmetric case where the potential comes only non-perturbatively.Then, we calculate the cosmological parameters by assuming that the potential is tuned in such a way that the first n derivatives vanish at some point.The predicted cosmological parameters are consistent with the recent Planck 2015 result [11].Furthermore, we can estimate the order of the string scale from the height of the potential that is roughly given by the string scale to the fourth multiplied by the rather small ten-dimensional one-loop factor.
To realize the saddle point, some amount of the fine-tuning is needed.This fine-tuning would be achieved by some principles which are beyond the ordinary local field theory, e.g. the multiple point criticality principle [13] and the maximum entropy principle [14].
This letter is organized as follows.In the next section, we consider the potential that has a saddle point where the first n derivatives vanish.Then we calculate the cosmological parameters of the model.In Sec. 3, we estimate the stirng scale in the case of the non-supersymmetric heterotic-like string model.In Sec. 4, we summarize our result.

Saddle point inflation and observables
We start with a general potential V as a function of an inflaton field ϕ.We will discuss the possibility of identifying it as the SM Higgs in the next section.
where we have assumed |δϕ end | ≫ |δϕ| in the last step.From Eqs. (3)(4)(5) (7), we obtain The cosmological observables, namely the scalar perturbation A s , spectral index n s , tensor-to-scalar ratio r, and running index dn s /d ln k3 are constrained by the Plnack 2015 data [11] A s ≃ 2.2 × 10 −9 , 0.954 < n s < 0.980, r < 0.168, −0.03 < dn s d ln k < 0.007, (13) at the 95% CL. 4 The e-folding number corresponds to the stage of inflation observed by the Planck experiment.We note that this model gives a concave potential , η < 0, which is favored by the recent Planck data. 5nflaton is absent.This is realized if the inflaton comes from the extra component of the gauge field/metric, for example.Then the dominant contribution to the potential is the one loop correction, which is suppressed compared to the string scale by the loop factor: For d = 10, we obtain the following numerical value (16) In fact, the 10 dimensional cosmological constant of SO( 16) × SO( 16) heterotic string theory [12] is calculated as Λ SO( 16)×SO( 16) ≃ 3.9 × 10 −6 M 10 s . ( Because we assume that the tree potential of the inflaton vanishes, the effective action below the string scale becomes Here χ is the dimensionless inflaton field, g s is the string coupling, V (χ) is the one loop potential, and V 6 is the compactification volume.Because M s is the only mass scale of the theory, A(χ), B(χ) and V (χ) should be functions of order one with a i 's, b i 's and v i 's being order one constants.Next let us move to the Einstein frame.Namely, we redefine the metric in such a way that A(χ) becomes 1.In the Einstein frame, we have Here where c i 's and u i 's are order one constants.In terms of the dimensionless canonical field φ, the action becomes where W (φ) is a function of order one.
The argument so far is quite general.In the following, we assume that the potential has a saddle point where the first n derivatives vanish as in Sec. 2. This may happen by some mechanism beyond the ordinary local field theory such as the multiple point criticality principle [13] and the maximum entropy principle [14].Here, we take as a simple possibility.We expect that φ c is the order one quantity.In terms of the canonical field ϕ = M P φ, the potential V (ϕ) becomes Then, from Eq. ( 8), we get (25) Furthermore, Eq. ( 25) and the COBE normalization Eq. ( 9) fix the value of V c = C loop g 2 s M 2 P M 2 s W 0 , from which we can obtain the string scale.In Table 1, we present the predictions of the cosmological parameters taking C loop = 10 −7 , N = 60.From this table, we can see that n ≥ 4 is favored by the current observation Eq. ( 13).The tensor to scalar ratio is very small compared to the current limit provided that φ c is of order one.As we vary n from 2 to 6, g s M s takes from 4×10 14 GeV to 3×10 15 GeV for W 0 = 1.If g s takes O(0.1), the result indicates that the M s is around the GUT scale ∼ 10 16 GeV.
Finally let us discuss the possibility of identifying the inflaton as the SM Higgs.The recent analysis shows that the Higgs potential for the large values of the Higgs field h is roughly given by n  24) and Eq. ( 26).
in the SM [3,5] and its simple extensions [15] when the top mass is around 171-172 GeV.We examine whether V SM can be connected to the potential V in Eq. ( 24) under the assumption that ϕ is identified as h.In Fig. 1, Eqs. (24)(26) are plotted.Here we take n = 4, φ c = 1, W 0 = 1 as an example.One can see that two lines are crossed at around ϕ ≃ 10 16 GeV, which we call ϕ 0 .We interpret this as the indication that the potential is given by the SM at lower energies, and becomes stringy, Eq. ( 24), above the string scale ∼ 10 16 GeV.
We also show ϕ 0 as a function of ϕ c = φ c M P in Fig. 2. ϕ 0 takes the order of 10 16 GeV for ϕ c = O(M P ).

Summary
We have examined the possibility of the saddle point inflation in the context of non-supersymmetric string theory, which is ubiquitous and becomes more realistic in light of the recent LHC result.Contrary to supersymmetric theory, the potential is generated perturbatively.We have assumed that the potential of the inflaton is identically zero at the tree level, and it is radiatively generated by the loop effect.We have estimated the string scale that realizes a successful inflation assuming that the potential is tuned so that it has a saddle point where first n derivatives vanish.Interestingly, the string scale becomes around the GUT scale, ∼ 10 16 GeV, if the string coupling is O(0.1).Furthermore, we have found that it is reasonable to identify the inflaton as the Higgs field.It is interesting that, in addition to the LHC results, the scale of the inflation supports non-supersymmetric string theory.

Figure 2 :
Figure 2: ϕ 0 as a function of ϕ c .ϕ 0 is the value of ϕ for which V SM equals V .