A new class of inflation models within the context of G-inflation is proposed, in which the standard model Higgs boson can act as an inflaton thanks to Galileon-like non-linear derivative interaction. The generated primordial density perturbation is shown to be consistent with the present observational data. We also make a general discussion on potential-driven G-inflation models, and find a new consistency relation between the tensor-to-scalar ratio and the tensor spectral index , , which is crucial in discriminating the present models from standard inflation with a canonical kinetic term.
Primordial inflation (1); (2) is now regarded as a part of the “standard” cosmology because it not only solves the flatness and the horizon problems but also accounts for the origin of primordial fluctuations (3). To construct a model of inflation, one usually assumes a scalar field that drives inflation (called an inflaton) outside the standard model (SM) of particle physics. This is because there are no scalar fields in the SM except for the Higgs boson and it has been found that the SM Higgs boson cannot be responsible for inflation as long as its kinetic term is canonical and it is minimally coupled to gravity (4). The difficulty here lies in the fact that the self interaction of the SM Higgs boson is so strong that the resultant primordial density fluctuation would be too large to be consistent with the present observational data (5).
To construct inflation models within the SM,
several variants of Higgs-driven inflation
have been proposed so far. They include models with a non-minimal coupling term to gravity
(6) and with a non-minimal coupling of the Higgs kinetic term with
the Einstein tensor (7).
The simplest way to enhance the kinetic energy would be to add a non-canonical higher order kinetic term. A number of novel inflation models with non-standard kinetic terms have been proposed, such as k-inflation (9), ghost condensate (10), and Dirac-Born-Infeld inflation (11). When incorporating higher order kinetic terms special care must be taken in order to avoid unwanted ghost instabilities. Since newly introduced degrees of freedom will lead easily to ghosts, it would be desirable if the scalar field does not give rise to a new degree of freedom in spite of its higher derivative nature. It has recently been shown that special combinations of higher order kinetic terms in the Lagrangian produce derivatives no higher than two both in the gravitational and scalar field equations (12); (13). A scalar field having this property is often called the Galileon because it possesses a Galilean shift symmetry in the Minkowski background. Such a scalar field has been studied in the context of modified gravity and dark energy in (14). Recently, an inflation model dubbed as “G-inflation” was proposed (15), in which inflation is driven by a scalar field with a Galileon-like kinetic term. In Ref. (15), the background and perturbation dynamics of G-inflation were investigated, revealing interesting features brought by the Galileon term. For example, scale-invariant scalar perturbations can be generated even in the exactly de Sitter background, and the tensor-to-scalar ratio can take a significantly larger value than in the standard inflation models, violating the standard consistency relation. Other aspects of G-inflation have been explored in Refs. (16); (17) (see also (18)).
In this paper, we propose a new Higgs inflation model by adding a Galileon-like kinetic term to the standard Higgs Lagrangian. We show that a self coupling constant of the order of the unity is compatible with the present observational data thanks to the kinetic term enhanced by the Galileon effect. We however start with a general discussion on G-inflation driven by the potential term because our potential-driven G-inflation is not restricted only to the Higgs field. We first give a criterion to determine which term becomes dominant in the kinetic term. Then, the slow-roll parameters and the slow-roll conditions are concretely given in terms of the potential and the function characterizing the Galileon term. We also derive the expressions for primordial fluctuations in terms of the slow-roll parameters, and find a new model-independent consistency relation for a potential-driven G-inflation model, which is quite useful for discriminating it from the standard inflation model with a canonical kinetic term. It turns out, however, that primordial non-Gaussianity of the curvature fluctuation is not large in potential-driven G-inflation. Finally, as a concrete example of a potential-driven G-inflation model, we propose a Higgs G-inflation model. This model predicts that the scalar spectral index and the tensor-to-scalar ratio for the number of -folds , which, together with the new consistency relation , makes our Higgs G-inflation model testable in near future.
This paper is organized as follows. In the next section, we make a general discussion on the potential-driven G-inflation model. In Sec. III, we apply it to more concrete examples, which have chaotic-type, new-type, and hybrid type potential forms. In Sec. IV, we present a new class of inflation model that regards the standard model Higgs boson as an inflaton in the context of G-inflation. Final section is devoted to conclusions and discussion.
Ii potential-driven G-inflation
The general Lagrangian describing lowest-order G-inflation is of the form (15)
where is the reduced Planck mass, is the Ricci scalar and . The main focus of the present paper is G-inflation driven by the potential term with the kinetic term modified by the term. We therefore take the “standard” form of the function ,
while for simplicity we assume the following form of the term,
ii.1 The background dynamics
Taking the homogeneous and isotropic metric , we have the following basic equations governing the background cosmological dynamics:
where the dot represents derivative with respect to and the prime with respect to . In the above we have defined
We assume that all of these quantities are small:
Thus, the energy density is dominated by the potential under the slow-roll conditions:
The slow-roll equation of motion for the scalar field is given by
One can consider two different limiting cases here. The case corresponds to standard slow-roll inflation, while in the opposite limit, , the Galileon effect alters the scalar field dynamics. We are interested in the latter case. Since , it is required that in order for this regime to be realized. The slow-roll equation of motion can be solved for to give
We have fixed the sign of so that , i.e., the scalar field rolls down the potential. This seems to be a natural situation for the scalar field dynamics. As we will see below, ghost instabilities are avoided provided that , and hence only in this branch the Universe can be stable. From Eq. (15) we see
Therefore, the condition that the kinetic term coming from is much bigger than the usual linear kinetic term is equivalent to
Using the slow-roll equations, one can rewrite the slow-roll parameters in terms of the potential as
where and are the slow-roll parameters conventionally used for standard slow-roll inflation,
Equations (18) and (19) clearly show that the Galileon term effectively flatten the potential thanks to the factor . This implies that in the presence of the Galileon-like derivative interaction slow-roll inflation can take place even if the potential is rather steep.
For later convenience we define . It will be also useful to note that
This means that even if the Galileon dominates the dynamics of slow-roll inflation, the standard part of the Lagrangian remains much larger than the Galileon interaction term,
Let us make a brief comment on the initial condition for the scalar field. The field may initially be off along the slow-roll trajectory (15). As long as , the field safely approaches the trajectory (15). If initially, the situation is more subtle, because the solution would approach another branch of the slow-roll attractor and the field would go on to climb up the potential. This is what indeed happens if at the initial moment, signaling ghost instabilities [see Eqs. (28)–(30) below]. Note, however, that in Eq. (30) and are evaluated along the slow-roll trajectory; it is therefore possible in principle that but still one has at the initial moment. In this case the solution approaches the healthy branch of the slow-roll attractor.
ii.2 Primordial fluctuations
Let us investigate the properties of scalar cosmological perturbations in potential-dominated G-inflation. The quadratic action for the curvature perturbation in the unitary gauge, , is given by (15)
where is the conformal time and
The above expressions are for general and , but in the present case we simply have
Evaluating the power spectrum from the quadratic action (25) is a standard exercise; we arrive at
The spectral tilt, , can be evaluated as
where the relation was used.
The tensor perturbations are generated in the same way as in the usual canonical inflation models, and hence the power spectrum and the spectral index of the primordial gravitational waves are given by
Thus, we obtain a new, model-independent consistency relation between the tensor-to-scalar ratio and the tensor spectral index:
Iii Galilean symmetric models
In this section, we shall clarify slow-roll dynamics of G-inflation for three representative forms of the potential. We consider the simplest case where the Galileon-type kinetic term respects not only the Galilean shift symmetry in the Minkowski background, but also the shift symmetry const (19) during inflation, i.e.,
where is a mass scale. Here the sign of should be chosen to coincide with that of . For (36) the term is odd, and hence the slow-roll solution (15) can be realized only in one side of a -symmetric potential. Note, however, that the following results can be generalized qualitatively to the cases with more general having weak dependence on , because may be practically constant for slowly-rolling . Note also that we assume (36) only in the inflationary stage; may change globally in -space and the detailed shape of would play an important role during the reheating stage after inflation. From the conservative point of view, reheating will proceed in the same way as in the usual inflation models by taking such that around the minimum of the potential . In this section we focus on the dynamics of in the inflationary stage, and we will come back to the issue of reheating in Sec. IV.
iii.1 Chaotic inflation
with being an integer. We assume that the field is moving in the side and hence . In this case, the condition is equivalent to
Since the slow-roll parameters for are given by
potential-driven G-inflation proceeds as long as
If , one can consider the scenario in which standard chaotic inflation follows slow-roll G-inflation. This scenario is possible if
If, on the other hand, , i.e., , slow-roll G-inflation ends at and standard chaotic inflation does not follow. In this case, G-inflation is possible even in the region where the potential is too steep to support standard chaotic inflation. In this case, the number of -folds reads
From this we obtain the field value evaluated -folds before the end of inflation,
The situation is summarized in Fig. 1.
Now we investigate the primordial perturbation. In the present case we find
The scalar-to-tensor ratio is given by
For and , we obtain and . The COBE/WMAP normalization, at (5), is attained by taking
For and we find and , which are also compatible with WMAP (5). In this case we obtain
under the COBE/WMAP normalization (5), showing that can easily be . This motivates us to study Higgs G-inflation, which will be discussed in the later section.
iii.2 New inflation
Next, let us consider the new inflation model (21) where the potential is given by
with . Since we consider the range and there, we take . In this case, the condition is equivalently written as
The slow-roll parameters can be expressed as
Both of the above quantities are smaller than unity in the range
with . From this we see that potential-driven G-inflation can occur if
Slow-roll inflation ends anyway at
If then the Galileon effect never operates during inflation. To have a G-inflationary phase we therefore require , i.e.,
If (i.e., ), standard new inflation is followed by G-inflation, as shown in Fig. 2. In this case, however, the Galileon term does not help to support an inflationary phase in the region where slow-roll inflation would otherwise be impossible. This case is summarized in Fig. 2.
If , and hence standard inflation would be impossible. Nevertheless, slow-roll G-inflation can take place with the help of the Galileon term, as shown in Fig. 3.
The number of -folds during new G-inflation is given by
Then, we find the field value, , evaluated -folds before the end of inflation as,
Except for the special case with we may have . If this is satisfied then we find
Now we investigate the primordial perturbation. In this case, we have
From Eq. (32) the power spectrum of the primordial density perturbation generated during the potential-dominated new G-inflation is given by
More concretely, one can evaluate as
iii.3 Hybrid inflation
Finally, let us study the hybrid inflation model (22) where the potential is effectively given by
We consider the range , and hence take . Hybrid inflation ends when the waterfall field becomes tachyonic. Let be the value of where this occurs. We will therefore focus on the range . For , the constant piece in the potential can be ignored, and hence the situation reduces to chaotic inflation studied in Sec. III.1.
The situation here is analogous to the case of new inflation. The Galileon effect operates for , where is given in Eq. (51). The slow-roll parameters for are given by
so that the slow-roll conditions are satisfied in the range , where and are the quantities defined in Eq. (54). Thus, the inflaton dynamics can be summarized in Figs. 2 and 3, depending on the values of and . Since , G-inflation is followed by standard hybrid inflation if . If then G-inflation can occur even though standard inflation would be impossible as . We remark that hybrid inflation ends at , which is not indicated explicitly in the figures, and for hybrid inflation reduces to chaotic inflation, which is not shown in the figures either.
The primordial perturbation is described in the same way as in new inflation,
The number of -folds during hybrid G-inflation reads
We then obtain the field value evaluated -folds before the end of inflation as,
In the case where
In the opposite case where
Iv Higgs G-inflation
Let us now construct a Higgs inflation model in the context of G-inflation. The tree-level SM Higgs Lagrangian is
where is the covariant derivative with respect to the SM gauge symmetry, is the SM Higgs boson, is the vacuum expectation value (VEV) of the SM Higgs and is the self coupling constant. Since we would like to have a chaotic inflation-like dynamics of the Higgs boson, we consider the case where its neutral component is very large compared with to : . In this situation, we have only to consider a simpler action,
In addition to the above action, we consider a Galileon-type interaction, which breaks Galilean shift symmetry weakly,
where is a mass parameter. Here we assume . Note that gauge fields that couples to receive heavy mass from the field value of the Higgs boson and hence we can neglect the effect of gauge fields when we consider the inflationary trajectory. This setup corresponds to the case
The Galileon effect operates provided that , i.e.,
In this regime the slow-roll parameters are given by
and , so that the slow-roll conditions are satisfied if
If , Higgs G-inflation proceeds even if standard Higgs inflation would otherwise be impossible. One can draw essentially the same diagram as Fig. 1 for Higgs-G inflation.
The number of -folds is given by
from which we obtain the field value evaluated -folds before the end of inflation as
As will be mentioned at the end of this section, reheating after Higgs G-inflation proceeds in the same way as in the standard inflationary models, and hence the history of the Universe after inflation will not be altered. Therefore, we use the value .
Now let us turn to the primordial perturbation in this model. Using the slow-roll approximation, we obtain
We thus arrive at
According to WMAP observations, (5), and hence
The scalar-to-tensor ratio is given by
For this yields , which is large enough to be detected by the forthcoming observation by PLANCK (23).
Note that in the above discussion we have neglected the quantum corrections. In order to have the precise relation between the potential of the Higgs field and the observational signatures such as the tensor-to-scalar ratio , we must know/assume the complete theory valid up to the inflationary scale, which is left for future study. However, the qualitative argument will not be changed even if we take into account quantum effects, because it can be absorbed by the variation of .
Before closing this section, let us take a brief look at reheating after Higgs G-inflation. The dynamics of the inflaton field during the reheating stage is non-trivial in general when one considers non-standard kinetic terms. In the present case, however, the effect of Galileon-like interaction can safely be ignored during reheating because is suppressed around the minimum of the potential, . In Fig. 4 we show a numerical example of the evolution of and in the final stage of Higgs G-inflation and in the begining of the reheating stage, when the Higgs field oscillates rapidly. We have confirmed that the Galileon terms become ineffective very soon after inflation ends, leading to reheating in the same way as in the usual case, , for the quartic potential.
The Galileon-like nonlinear derivative interaction, , opens up a new arena of inflation model building while keeping the models healthy, and the novel class of inflation thus developed —G-inflation— possesses various interesting aspects to be explored. In (15) the extreme case was emphasized where G-inflation is driven purely by the kinetic energy, though the background and the perturbation equations have been derived without assuming any specific form of and . In this paper, we have studied the effects of the Galileon term on potential-driven inflation. Although the energy density is dominated by the potential anyway, the dynamics of the inflaton is nontrivial when the term participates more dominantly than the usual linear kinetic term . We have demonstrated that the Galileon term makes the potential effectively flatter so that slow-roll inflation proceeds even if the potential is in fact too steep to support conventional slow-roll inflation. In light of this fact, we have constructed a viable model of Higgs inflation, i.e., Higgs G-inflation, showing that the power spectrum of the primordial density perturbation is compatible with current observational data. The tensor-to-scalar ratio is large enough to be detected by the Planck satellite.
In this paper we have focused on the power spectrum of the curvature perturbation and the consistency relation relative to the amplitude of tensor perturbations. Primordial non-Gaussianity would be another powerful probe to discriminate G-inflation among others. Unfortunately, however, non-Gaussianity arising from the present potential-driven models is estimated to be not large. This is because is composed of and slow-roll suppressed terms with in the present model, leading to . (Detailed computation of primordial non-Gaussianity from G-inflation will be given elsewhere (24); see also (16); (17); (18).) We would thus conclude that the smoking gun of potential-driven G-inflation is the consistency relation which is unique enough to distinguish G-inflation from standard canonical inflation and k-inflation.
We have focused on the (generalized form of the) leading order Galileon term. As demonstrated in (14), higher order terms play an important role in the cosmological dynamics of Galileon dark energy models. Therefore, it would be interesting to consider the effects of the higher order Galileon terms in the context of primordial inflation, which is left for further study.
This work was partially supported by JSPS through research fellowships (K.K.) and the Grant-in-Aid for the Global COE Program “Global Center of Excellence for Physical Sciences Frontier”. This work was also supported in part by JSPS Grant-in-Aid for Research Activity Start-up No. 22840011 (T.K.), the Grant-in-Aid for Scientific Research Nos. 19340054 (J.Y.), 21740187 (M.Y.), and the Grant-in-Aid for Scientific Research on Innovative Areas No. 21111006 (J.Y.).
- preprint: RESCEU-28/10
- Inflation models where Higgs multiplets act as an inflaton in the context of supersymmetric extensions of the SM, e.g., the next-to-minimal supersymmetric standard model, have also been proposed in (8). In these models, a non-canonical Khler potential for the Higgs multiplet is assumed.
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