Hillclimbing Higgs inflation

TL;DR

This paper introduces a novel Higgs inflation model utilizing hillclimbing inflation, which predicts different inflationary outcomes compared to traditional Higgs inflation, especially in scenarios with degenerate vacua.

Contribution

It presents a new inflationary framework based on hillclimbing inflation with the Higgs field and degenerate vacua, expanding the landscape of Higgs inflation models.

Findings

Inflationary predictions differ significantly from ordinary Higgs inflation.

The model accommodates degenerate vacua in the Higgs potential.

Potential for observable deviations in cosmic microwave background signatures.

Abstract

We propose a realization of cosmic inflation with the Higgs field when the Higgs potential has degenerate vacua by employing the recently proposed idea of hillclimbing inflation. The resultant inflationary predictions exhibit a sizable deviation from those of the ordinary Higgs inflation.

Figures

Tables

Table 1. Table 1: Allowed region and inflationary predictions for β 2 = 2 × 10 − 5 subscript 𝛽 2 2 superscript 10 5 \beta_{2}=2\times 10^{-5} . The left and right values correspond to N = 50 𝑁 50 N=50 and 60 60 60 , respectively. Model 1 and Model 2 are given in Eq. ( 13 ). Note that the allowed region of M 𝑀 M scales as β 2 − 1 / 4 superscript subscript 𝛽 2 1 4 \beta_{2}^{-1/4} (see the main text). Also, the values of n s subscript 𝑛 𝑠 n_{s} and r 𝑟 r do not depend on β 2 subscript 𝛽 2 \beta_{2} significantly.

$\Omega$	Model 1	Model 2
$M/M_{\text{P}}$	$[0.1005,0.0923]$	$[0.0907,0.0837]$
${\varphi }_{\text{J},\mathrm{end}}/M_{\text{P}}$	$[0.0635,0.0583]$	$[0.0562,0.0519]$
${\varphi }_{\text{J},\mathrm{CMB}}/M_{\text{P}}$	$[0.0991,0.0912]$	$[0.0854,0.0791]$
$n_s$	$[0.9628,0.9688]$	$[0.9647,0.9703]$
$r$	$[0.00381,0.00272]$	$[0.000646,0.000468]$

Equations

$$S$$

$$R_{\text{J}}$$

$$S$$

$$K$$

$$-\frac{K}{2}(\partial {\varphi }_{\text{J}}{)}^2\simeq -\frac{3}{4}(\partial \ln \Omega {)}^2.$$

$$V_{\text{J}}$$

$$V$$

$${ϵ}_V$$

$$N$$

$$n_s$$

$$V_{\text{J}}({\varphi }_{\text{J}})$$

$${\lambda }_{\text{eff}}({\varphi }_{\text{J}})$$

$$\Omega$$

$$V$$

Full text

††preprint: CTPU-17-18††preprint: OU-HET/935

I Introduction

Inflation plays an essential role in modern cosmology Starobinsky (1980); Sato (1981); Guth (1981), not only by addressing the horizon and flatness problems Guth (1981), but also by giving primordial seeds for late-time structures Mukhanov and Chibisov (1981). The properties of primordial perturbations have been strongly constrained by precision cosmology, especially by the cosmic microwave background (CMB) observations Ade et al. (2016), and such observations are expected to explore the inflationary physics much further in the forthcoming decade. Nevertheless, the identity of the inflaton, the scalar field causing inflation, is still veiled in mystery.

The Higgs particle—the quantum fluctuation of the Higgs field around its potential minimum—had long been the last missing element of the Standard Model (SM), and was finally discovered in 2012 Aad et al. (2012); Chatrchyan et al. (2012). Ever since, the Higgs field has been the only (possibly) elementary scalar field observed by human beings. The possibility of realizing inflation with this Higgs field has been studied extensively, and it has turned out that the Higgs field can indeed be identified as the inflaton with the help of a large non-minimal coupling $\xi\sim 10^{5}$ to the Ricci scalar Bezrukov and Shaposhnikov (2008).aaa In earlier Ref. Salopek et al. (1989), the Higgs inflation with essentially the same parameters $\xi\sim 10^{4}$ and $\lambda\sim\left(\xi/10^{5}\right)^{2}\sim 10^{-2}$ has also been sketched; see also Refs. Lucchin et al. (1986); Futamase and Maeda (1989); van der Bij (1994, 1995); Cervantes-Cota and Dehnen (1995). It is noted that we may also cope with a smaller $\xi\sim 10$–$10^{2}$ under the SM criticality Hamada et al. (2014a); Bezrukov and Shaposhnikov (2014); Hamada et al. (2015a). See also Ref. Ema et al. (2017) for the explosive production of longitudinal gauge bosons and possible strong coupling issues under the presence of the large non-minimal coupling.

This scenario, now called the Higgs inflation, has been found to fit in the most favored region by CMB observations Ade et al. (2016).

Regarding the mass of the Higgs particle $m_{H}=125.09\pm 0.24\,\text{GeV}$ Aad et al. (2015), there was an interesting prediction based on the Multiple Point Principle (MPP) Froggatt and Nielsen (1996).bbbSee Appendix D in Ref. Hamada et al. (2015b) for a review, and Ref. Nielsen (2012) for possible generalizations. The MPP requires that there exist another vacuum in the Higgs potential around the Planck scale, in addition to the electroweak one. This means that the Higgs quartic coupling and its beta function both vanish there, $\lambda\sim\beta_{\lambda}\sim 0$.ccc See e.g. Refs. Hamada et al. (2013); Buttazzo et al. (2013) for more recent analyses. Especially, it is intriguing that the bare Higgs mass can also vanish around the Planck scale, and hence there can be a triple coincidence Hamada et al. (2013).

The observed Higgs mass has turned out to be almost within 1$\sigma$ from the value $m_{H}=135\pm 9\,\text{GeV}$ predicted in this way Froggatt and Nielsen (1996).

Even though these two scenarios, the Higgs inflation and the MPP, seem attractive, difficulties arise when it comes to combining them. The MPP requires a degenerate vacuum around the Planck scale, which spoils the monotonicity of the Higgs potential which is necessary for a successful inflation Hamada et al. (2014b). On this regard, an interesting proposal has recently been made by two of the present authors: the hillclimbing inflation Jinno and Kaneta (2017). This is a general framework which enables a successful inflation with an inflaton potential with multiple vacua. This idea opens up a new possibility of identifying the Higgs field as the inflaton while having degenerate vacua in the Higgs potential. The aim of this paper is to pursue this possibility.ddd The gauge-Higgs unification models fit in the periodic potential case in the general consideration of the hillclimbing inflation Jinno and Kaneta (2017). Such a possibility will be pursued in a separate publication.

This paper is organized as follows. In Sec. II we briefly summarize inflationary behavior and predictions in the hillclimbing inflation. Then in Sec. III we propose an inflation model using the Higgs field as the inflaton. We conclude in Sec. IV.

II Hillclimbing inflation and its predictions

In this section, we briefly summarize the inflaton behavior and inflationary predictions in the general hillclimbing inflation. We start from the Jordan-frame action that has a non-minimal coupling between the inflaton and gravity:

[TABLE]

where (and throughout the paper) we work in the Planck units $M_{\text{P}}=1/\sqrt{8\pi G}=1$ unless otherwise stated; the subscript J indicates that the quantity is given in the Jordan frame; $\phi_{\text{J}}$, $R_{\text{J}}$ and $V_{\text{J}}(\phi_{\text{J}})$ are the inflaton, the Ricci scalar and the inflaton potential, respectively; and we assume that the conformal factor $\Omega(\phi_{\text{J}})$ is positive for the inflaton field values we consider. Under the Weyl rescaling $g_{\mu\nu}=\Omega g_{\text{J}\mu\nu}$, the Ricci scalar transforms as

[TABLE]

and we obtain the Einstein-frame action that has a canonically normalized Ricci scalar:

[TABLE]

where

[TABLE]

$\left(\partial\phi_{\text{J}}\right)^{2}=g^{\mu\nu}\partial_{\mu}\phi_{\text{J}}\partial_{\nu}\phi_{\text{J}}$, and the Einstein-frame potential reads $V=V_{\text{J}}/\Omega^{2}$. If the second term dominates in Eq. (4), the kinetic term in the action (3) reduces to Kallosh et al. (2014a)

[TABLE]

This means that $\phi\simeq\sqrt{3/2}\ln\Omega$ works as a canonically normalized inflaton. If the potential can be expanded as a series of $\Omega$, it will become exponentially flat in terms of $\ln\Omega$ as we will see below.

In Ref. Jinno and Kaneta (2017) it has been proposed to use $\Omega\ll 1$ for the inflationary dynamics, instead of $\Omega\gg 1$. In this class of models, called the hillclimbing inflation, it is assumed that $V_{\text{J}}$ and $\Omega$ vanish at the same point $\phi_{\text{J}*}$ in the field space, and the Taylor expansion of the potential starts from $\Omega^{n}$ $(n\geq 2)$ around $\phi_{\text{J}}=\phi_{\text{J}*}$:

[TABLE]

with $\mathcal{V}_{\text{J},k}$ being constants. One of the examples is the case where the potential has a quadratic (or higher) minimum with vanishing cosmological constant at $\phi_{\text{J}}=\phi_{\text{J}*}$, and $\Omega$ vanishes linearly in $\phi_{\text{J}}-\phi_{\text{J}*}$ there.eee In fact, we have $\text{d}V_{\text{J}}/\text{d}\Omega=(\text{d}V_{\text{J}}/\text{d}\phi_{J})/(\text{d}\Omega/\text{d}\phi_{J})=0$ at the minimum in this case. However, it should be noted that this is just one of the possible realizations and that the expansion (6) is possible even if $\text{d}\Omega/\text{d}\phi_{\text{J}}=0$ at the vanishing point $\phi_{\text{J}}=\phi_{\text{J}*}$. Also, if we allow $\Omega$ to be proportional to a fractional power of $\phi_{\text{J}}-\phi_{\text{J}*}$ around the vanishing point, it is even possible to remove the assumption that $V_{\text{J}}$ at $\phi_{\text{J}}=\phi_{\text{J}*}$ is a local minimum. In Sec. III we seek for a realization of this inflationary setup with the Higgs field.fff In the context of MPP, the existence of another local minimum of $V_{\text{J}}$ is an essential requirement, as explained in Sec. I. The vanishing $\Omega$ at the coincident point can be regarded as a realization of the generalized MPP discussed in Ref. Nielsen (2012).

With the Jordan-frame potential (6), the corresponding Einstein-frame potential $V=V_{\text{J}}/\Omega^{2}$ reads

[TABLE]

at $\Omega<1$, where we have written $V_{0}:=\mathcal{V}_{\text{J},2}$ and $\eta_{k}:=-\mathcal{V}_{\text{J},k+2}/\mathcal{V}_{\text{J},2}$ and the leading exponent $n\geq 1$ dominantly determines the inflationary predictions.ggg Having $n\geq 2$ means that we assume $\mathcal{V}_{\text{J},k}=0$ for $k=3,\dots,n+1$.

The last expression in Eq. (7) tells that the potential is exponentially flat for the canonical inflaton field. In Sec. III we will see that the leading power depends on the explicit form of the conformal factor we take.

It is remarkable that the Einstein-frame potential $V=V_{\text{J}}/\Omega^{2}$ has been lifted up by the small $\Omega$ and made monotonic, even around a local minimum of $V_{\text{J}}$. As is pointed out in Ref. Jinno and Kaneta (2017), inflation at $\Omega\ll 1$ means that the Jordan-frame potential $V_{\text{J}}=\Omega^{2}V$ increases in time, that is, the inflaton climbs up the Jordan-frame potential hill. This observation is crucial in making successful inflation with inflaton potentials having multiple vacua, as stressed in that paper. In Sec. III we propose taking the SM Higgs field as the inflaton.

For the inflationary predictions, this class of models show attractor behavior called $\eta$-attractor.hhh It can be shown that these models share the inflationary predictions with some branch of $\alpha$-attractor Kallosh et al. (2013, 2014b) at the leading order in the $e$-folding Jinno and Kaneta (2017). However, there are several reasons to distinguish $\xi$- and $\eta$-attractors from $\alpha$-attractor: First of all, their actions generically differ even after the Weyl transformation. Second, the reheating and preheating processes depend on the preferred frame in which the canonically normalized matter fields are introduced; see Ref. Ema et al. (2017) for example. Finally, such a distinction is important in constructing inflation models with particle-physics motivated potentials, as stressed in Ref. Jinno and Kaneta (2017).

Following the standard procedure, the slow-roll parameters with the potential (7) are obtained as

[TABLE]

where we used the following expression for the $e$-folding number $N$:

[TABLE]

The inflationary predictions at the leading order in $N$ become

[TABLE]

where $n_{s}$ and $r$ are the spectral index and the tensor-to-scalar ratio, respectively.

III Hillclimbing Higgs inflation

Now let us take the Higgs field as the inflaton. We write its effective potential as

[TABLE]

Around $\phi_{\text{J}}=M\sim 10^{17\text{--}18}\,\text{GeV}$, the effective coupling can be approximated by Hamada et al. (2015a)

[TABLE]

where $\beta_{2}\simeq 2\times 10^{-5}=:\beta_{2}^{\text{SM}}$ in the SM Hamada et al. (2014b). The cubic and higher order terms are loop-suppressed, $\beta_{3},\dots\ll\beta_{2}$, and will be neglected hereafter.

In the following we set $\lambda_{\text{min}}=0$ so that the potential becomes zero at $\phi_{\text{J}}=M$ by assuming the MPP. In the SM, this is realized with the top quark mass $m_{t}\simeq 171.4\,\text{GeV}$ for the strong coupling $\alpha_{s}\simeq 0.1185$, leading to $M\simeq 4\times 10^{18}\,\text{GeV}$ Hamada et al. (2015a).iii The precisely-measured top mass $m_{t}^{\text{MC}}=173.1\pm 0.6$ GeV Patrignani et al. (2016) is the Monte-Carlo mass, which is just a parameter in the Monte-Carlo code. The mass relevant to our discussion is the pole mass. A theorists’ naive average gives a combined result $m_{t}^{\text{pole}}=173.5\pm 1.1$ GeV Patrignani et al. (2016), and the MPP value of the top pole mass $171.4\,\text{GeV}$ is within $1.9\,\sigma$ of this bound.

However, the precise values of the $\beta_{2}$ and $M$ that realize $\lambda_{\text{min}}=0$ are altered by extra particles such as the heavy right-handed neutrinos and the Higgs-portal dark matter; see e.g. Refs. Davoudiasl et al. (2005); Iso et al. (2009); Haba et al. (2014a); Hamada et al. (2014c); Haba et al. (2014b); Hamada et al. (2017). Therefore we take them as free parameters hereafter.

Also, we consider the following forms for the conformal factor in this paper:jjj We take $\Omega=0$ at $\phi_{\text{J}}=M$ as discussed in footnote f. Then Eq. (13) is the two simplest possibilities that lead to $\Omega=1$ at $\phi_{\text{J}}=0$ at the same time.

[TABLE]

We summarize the setup in Fig. 1. Given this setup, the Einstein-frame potential is expanded as

[TABLE]

Therefore, the leading exponent is given by $n=1$ and $2$ for Model 1 and 2, respectively, and the potential height in the Einstein frame is given by $V_{0}\sim\beta_{2}M^{4}$. Taking Eq. (8) and the curvature perturbation $A_{s}\sim V_{0}/\epsilon_{V}$ into account, one sees that the observed value $A_{s}\simeq 2.2\times 10^{-9}$ constrains the model parameters along $M\propto\beta_{2}^{-1/4}$. Figure 2 shows such a constraint for each of Model 1 and 2. The two bands correspond to Model 1 and 2, and the upper and lower lines for each band correspond to $N=50$ and $60$, respectively. In making this figure we numerically solved for the $e$-folding $N$ under the slow-roll assumption, defining the end of inflation by ${\rm max}(\epsilon_{V},\eta_{V})=1$. It should be mentioned that while we have investigated only two simple models, there are various possible choices of $\Omega$ which gives different viable parameter spaces. In addition, as mentioned above, the values of $\beta_{2}$ and $M$ may easily change in models beyond the SM by the existence of additional particles and associated intermediate scales; see e.g. Refs. Davoudiasl et al. (2005); Iso et al. (2009); Haba et al. (2014a); Hamada et al. (2014c); Haba et al. (2014b).

Figure 3 shows the inflationary predictions in the hillclimbing Higgs inflation. It is seen that the prediction of the tensor-to-scalar ratio differs between Model 1 and 2 because of the difference in the leading exponent. See also Table 1. These predictions fall within the planned sensitivities by near-future experiments such as the POLARBEAR-2 Inoue et al. (2016), LiteBIRD Matsumura et al. (2013) and CORE Delabrouille et al. (2017), and in Model 1 there is a good possibility of distinguishing from the ordinary Higgs inflation by these experiments.

Note that the prediction for $r$ differs from the rough estimate (10) by ${\mathcal{O}}(10)\%$. This is because Eq. (10) is derived by taking only the leading term in Eq. (14) into account, while higher order terms can contribute to the inflaton dynamics as the conformal factor grows towards the end of inflation. Such a contribution is larger if the coefficient of the leading term is smaller, and this is why Model 2 shows a larger deviation from Eq. (10) compared to Model 1.

In Table 1 we summarize the allowed value for $M$ and corresponding inflationary predictions for $\beta_{2}=2\times 10^{-5}$. One sees that $M\sim 0.1\,M_{\text{P}}$ is favored for this value of $\beta_{2}$ and also that $\phi_{\text{J}}$ at the CMB scale corresponds to $\sim 0.01\,M$ away from the potential minimum at $\phi_{\text{J}}=M$.

IV Conclusion

In this paper we have proposed a realization of cosmic inflation using the Higgs field with degenerate vacua. This realization utilizes the recently proposed idea of hillclimbing inflation Jinno and Kaneta (2017), which is a general framework to enable a successful inflation using an inflaton potential with multiple vacua. It has been shown that a successful inflation occurs while the inflaton is climbing up the potential hill from the high-scale vacuum around the Planck scale to the electroweak vacuum, and that the resulting inflationary predictions come well within the region favored by the CMB observations, while showing a sizable deviation from those of the ordinary Higgs inflation.

Though in this paper we have considered only the case where the Higgs field has degenerate vacua, the original proposal in Ref. Jinno and Kaneta (2017) can work also when the Higgs potential becomes negative at some scale. Such a study will be presented in a separate publication.

Acknowledgments

The authors are grateful to Y. Hamada, H. Kawai, Y. Nakanishi, T. Onogi, and S. Rusak for useful discussions. The work of R.J. and K.K. is supported by IBS under the project code, IBS-R018-D1. The work of K.O. is supported in part by JSPS KAKENHI Grant Nos. 16J06151 and 23104009, 15K05053.