eISSN: 2576-4543 PAIJ

Physics & Astronomy International Journal
Reseach Article
Volume 2 Issue 5

Electroweak symmetry braking and quantization of the higgs field in early universe
Yoshimasa Kurihara
The High Energy Accelerator Organization (KEK), Japan
Received: August 28, 2018 | Published: October 12, 2018

Correspondence: Yoshimasa Kurihara, The High Energy Accelerator Organization (KEK), Oho 1–1, Tsukuba, Ibaraki 305–0801, Japan, Tel +8129 8796 088, Email

Citation: Kurihara Y. Electroweak symmetry braking and quantization of the higgs field in early universe. Phys Astron Int J. 2018;2(5):468‒474. DOI: 10.15406/paij.2018.02.00126

Abstrat

In this paper, we propose a novel method to treat the electroweak symmetry braking. In this method, a conformal metric is employed and the Higgs field is scaled owing to the conformal function and the mass parameters of the quadratic term of the Higgs potential has a time dependence through the conformal function, and it induces the phase transition. Quantization of the Higgs field is induced associated with the canonical quantization of general relativity. The cosmic inflation and the electroweak phase–transition are discussed in a framework of the scaled field. The Friedmann equations are numerically solved and an example of a possible solution to match with the cosmic inflation scenario is given in this research.

Keywords: electroweak symmetry, conformal metric, higgs potential, cosmic inflation, phase transition, friedmann equations

Introduction

After the discovery of the Higgs boson1,2 in 2012, the standard theory of particle physics (SM) is established as the canon of a fundamental physics. According to the SM, the electroweak symmetry is spontaneously broken owing to the Higgs mechanism,3–5 and the current universe is considered to be filled with the Higgs filed which has a finite vacuum expectation–value. On the other hand, the electroweak symmetry is expected to be in the unbroken phase in the early universe before the cosmic inflation. A standard scenario of the big–bang cosmology is that the electroweak phase–transition from the unbroken to the broken phases might occur during (or at the beginning of) the cosmic inflation, and the universe was re–heated after the cosmic inflation, and then the big–bang started.

The cosmic inflation was proposed to solve the flatness and horizon problems by several authors independently6–9 in 1980. In these “old” models, the cosmic inflation is induced by the Higgs filed (or some scalar filed which is referred to as the inflaton field), and it is terminated when the electroweak phase–transition of the first–kind occurred. These models are suffered by the vacuum–bubble problem which destroyed isotropy of the universe. In 1982, the “new” inflation models10–12 are proposed which utilize the phase–transition of second–kind to avoid the vacuum–bubble problem. The inflation terminated moderately in the “new” models. Although these “new’ models can solve the vacuum–bubble problem, they require a fine tunning of initial parameters to realize the cosmic inflation for a enough time duration. Yet other models of the cosmic inflation was proposed such as the chaotic inflation,13 the Higgs inflation with non–minimal coupling to gravity14 and so on.

In any inflation scenario, the electroweak phase–transition is critical to induce and terminate the cosmic inflation. However, a widely accepted mechanism to induce the symmetry braking is not established yet. The early scenario15 of the symmetry braking using higher–order radiative corrections is now excluded in precise measurements of the SM parameters. The radiative braking scenario is re–examined and concluded that it is still viable if an additional scalar field is introduced.16 This scenario is intensively investigated in literatures.17,18 Recently, this scenario is extended19 using the classically conformal MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFSeIqqaaaaaaa aaWdbiabgkHiT8aacqWFsectaaa@4338@ extension of the SM.20

We propose a new and novel mechanism of the electroweak symmetry braking in this study. In the proposed method, which is referred to as the “scaled scalar–field method”, a conformal metric is employed, and the Higgs field is scaled owing to the conformal function. Quantization of the Higgs field is induced due to the canonical quantization of general relativity. In consequence, a mass parameter of the Higgs field (a quadratic term of the Higgs potential) has the time dependence through the conformal function, and it causes the phase transition. The scaled scalar–field method is introduced in section 2 after a brief explanation of our geometrical setups. The Friedmann equation which governs the cosmic inflation is formulated using the scaled scalar–field method in section 3.1. After some appropriate approximation, the Friedmann equations are numerically solved and an existence of a possible solution to match with the cosmic inflation scenario is shown in section 3.2. A summary of the method and consequences on the inflation scenario is provided in section 4.

Scalar field in conformal metric

Geometrical setups

A scalar field defined on a four dimensional space time manifold MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ with a GL(1,3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4rai aadYeacaaIOaGaaGymaiaaiYcacaaIZaGaaGykaaaa@3BAA@ symmetry is considered in this study. First, classical general relativity and the scalar field defined on MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ are summarized using a vierbein formalism. The formalism and terminology in this study follow our previous works.21–23 At each point on MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ , a local Lorentz manifold M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ with a Poincaé symmetry ISO(1,3)=SO(1,3) T 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamysai aadofacaWGpbGaaGikaiaaigdacaaISaGaaG4maiaaiMcacaaI9aGa am4uaiaad+eacaaIOaGaaGymaiaaiYcacaaIZaGaaGykamrr1ngBPr wtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae8xYIiUaamivaSWa aWbaaKqaGeqabaqcLbmacaaI0aaaaaaa@5185@ is associated, where T 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaWbaaKqaGeqabaqcLbmacaaI0aaaaaaa@3996@ is a four–dimensional translation group. On a open neighborhood around any point xU MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEai abgIGiolaadwfacqGHckcZtuuDJXwAK1uy0HwmaeHbfv3ySLgzG0uy 0Hgip5wzaGqbaiab=ntinbaa@4681@ , a trivial frame vector is expressed as x μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiEaS WaaWbaaKqaGeqabaqcLbmacqaH8oqBaaaaaa@3AB2@ , and a trivial vector bundle (frame bundle) can be introduced. An orthonormal basis of μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOaIy 7cdaWgaaqcbasaaKqzadGaeqiVd0gajeaibeaaaaa@3B44@ in T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivam rr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaGae83mH0ea aa@4203@ and d x μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai aadIhalmaaCaaajeaibeqaaKqzadGaeqiVd0gaaaaa@3B9B@ in T * MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaWbaaKqaGeqabaqcLbmacaaIQaaaamrr1ngBPrwtHrhAXaqeguuD JXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@44CB@ are also introduced. A short–hand notations of μ =/ x μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOaIy Bcfa4aaSbaaKqaGeaajugWaiabeY7aTbWcbeaajugibiaai2dacqGH ciITcaaIVaGaeyOaIyRaamiEaSWaaWbaaKqaGeqabaqcLbmacqaH8o qBaaaaaa@44BB@ are used through out this study. The Einstein’s equivalent theorem insists an existence of an isomorphism M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestcqGHsgIR caWGnbaaaa@43E9@ at any point in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ . A metric tensor g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaS WaaSbaaKazba2=baqcLbmacqGHIaYTcqGHIaYTaKazba2=beaaaaa@3F64@ in MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ is mapped to η MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4TdG 2cdaWgaaqcbasaaKqzadGaeyOiGCRaeyOiGClajeaibeaaaaa@3CDE@ in M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ using a vierbein μ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFWesrlmaaDaaa jeaibaqcLbmacqaH8oqBaKqaGeaajugWaiaadggaaaaaaa@4698@ as g μν = η ab μ a ν b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaS WaaSbaaKqaGeaajugWaiabeY7aTjabe27aUbqcbasabaqcLbsacaaI 9aGaeq4TdG2cdaWgaaqcbasaaKqzadGaamyyaiaadkgaaKqaGeqaam rr1ngBPrwtHrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWF WesrlmaaDaaajeaibaqcLbmacqaH8oqBaKqaGeaajugWaiaadggaaa qcLbsacqWFWesrlmaaDaaajeaibaqcLbmacqaH9oGBaKqaGeaajugW aiaadkgaaaaaaa@5AD2@ . An orthogonal basis in T * M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaWbaaKqaGeqabaqcLbmacaaIQaaaaKqzGeGaamytaaaa@3AED@ and TM MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivai aad2eaaaa@3825@ are respectively expressed as d x a = μ a d x μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai aadIhalmaaCaaajeaibeqaaKqzadGaamyyaaaajugibiaai2datuuD JXwAK1uy0HwmaeHbfv3ySLgzG0uy0Hgip5wzaGqbaiab=btifTWaa0 baaKqaGeaajugWaiabeY7aTbqcbasaaKqzadGaamyyaaaajugibiaa dsgacaWG4bWcdaahaaqcbasabeaajugWaiabeY7aTbaaaaa@51EF@ and a = a μ μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOaIy 7cdaWgaaqcbasaaKqzadGaamyyaaqcbasabaqcLbsacaaI9aWefv3y SLgznfgDOfdaryqr1ngBPrginfgDObYtUvgaiuaacqWFWesrlmaaDa aajeaibaqcLbmacaWGHbaajeaibaqcLbmacqaH8oqBaaqcLbsacqGH ciITlmaaBaaajeaibaqcLbmacqaH8oqBaKqaGeqaaaaa@5141@ using a vierbein and its inverse. The Einstein convention for repeated indices is used though out this study. In addition, Greek and Roman indices are used for a coordinate on MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ and M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ , respectively. A local Lorentz metric and the Levi Civita tensor are respectively defined as η =diag(1,1,1,1) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4TdG 2cdaWgaaqcbasaaKqzadGaeyOiGCRaeyOiGClajeaibeaajugibiaa i2dacaWGKbGaamyAaiaadggacaWGNbGaaGikaiaaigdacaaISaGaey OeI0IaaGymaiaaiYcacqGHsislcaaIXaGaaGilaiabgkHiTiaaigda caaIPaaaaa@4B17@ and ε MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyTdu 2cdaWgaaqcbasaaKqzadGaeyOiGCRaeyOiGCRaeyOiGCRaeyOiGCla jeaibeaaaaa@3FE3@ with ε 0123 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyTdu 2cdaWgaaqcbasaaKqzadGaaGimaiaaigdacaaIYaGaaG4maaqcbasa baqcLbsacaaI9aGaaGymaaaa@3ECE@ . Dummy Roman–indices are often abbreviate to dots (or asterisks), when the index pairing of the Einstein convention is obvious, such as η ab μ a ν b = η μ ν MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4TdG 2cdaWgaaqcbasaaKqzadGaamyyaiaadkgaaKqaGeqaamrr1ngBPrwt HrhAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFWesrlmaaDa aajeaibaqcLbmacqaH8oqBaKqaGeaajugWaiaadggaaaqcLbsacqWF WesrlmaaDaaajeaibaqcLbmacqaH9oGBaKqaGeaajugWaiaadkgaaa qcLbsacaaI9aGaeq4TdG2cdaWgaaqcbasaaKqzadGaeyyXICTaeyyX ICnajeaibeaajugibiab=btifTWaa0baaKqaGeaajugWaiabeY7aTb qcbasaaKqzadGaeyyXICnaaKqzGeGae8hmHu0cdaqhaaqcbasaaKqz adGaeqyVd4gajeaibaqcLbmacqGHflY1aaaaaa@6DC8@ . When multiple dots appear in an expression, pairing must be a left–to–right order at both upper and lower indices, e.g. a b = a ab b ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Pb8HTWaaSba aKqaGeaajugWaiabgwSixlabgwSixdqcbasabaqcLbsacqGHNis2cq WFBaVylmaaCaaajeaibeqaaKqzadGaeyyXICTaeyyXICnaaKqzGeGa aGypaiab=Pb8HTWaaSbaaKqaGeaajugWaiaadggacaWGIbaajeaibe aajugibiabgEIizlab=Tb8ITWaaWbaaKqaGeqabaqcLbmacaWGHbGa amOyaaaaaaa@61F7@ . A principal connection of the fiber bundle so(1,3)M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Xc8Zjab=Hc8 VjaaiIcacaaIXaGaaGilaiaaiodacaaIPaGaeyOKH4Qaamytaaaa@4BAD@ is represented as ω μ b a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyYdC 3cdaqhaaqcbasaaKqzadGaeqiVd0MaaGiiaiaadkgaaKqaGeaajugW aiaaiccacaWGHbaaaaaa@3FFB@ , which is referred to as the spin connection. The spin connection satisfies a metric compatibility condition as ω μ a η b = ω μ ab = ω μ ba MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyYdC xcfa4aa0baaSqaaKqzadGaeqiVd0wcLbsacaaIGaGaeyyXICnaleaa jugibiaaiccajugWaiaadggaaaqcLbsacqaH3oaAjuaGdaahaaqcba sabeaajugWaiabgwSixlaadkgaaaqcLbsacaaI9aGaeqyYdCxcfa4a a0baaKqaGeaajugWaiabeY7aTbqcbasaaKqzadGaaGiiaiaadggaca WGIbaaaKqzGeGaaGypaiabgkHiTiabeM8a3LqbaoaaDaaajeaibaqc LbmacqaH8oqBaKqaGeaajugWaiaaiccacaWGIbGaamyyaaaaaaa@6082@ . A vierbein for e a = μ a d x μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=5b8LTWaaWba aKqaGeqabaqcLbmacaWGHbaaaKqzGeGaaGypamrr1ngBPrwtHrhAXa qehuuDJXwAKbstHrhAG8KBLbacgaGae4hmHu0cdaqhaaqcbasaaKqz adGaeqiVd0gajeaibaqcLbmacaWGHbaaaKqzGeGaamizaiaadIhalm aaCaaajeaibeqaaKqzadGaeqiVd0gaaaaa@5CC3@ and a GL(1,3) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4rai aadYeacaaIOaGaaGymaiaaiYcacaaIZaGaaGykaaaa@3BAA@ invariant volume form v= ε e e e e /4! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=9c82jaai2da cqaH1oqzlmaaBaaajeaibaqcLbmacqGHflY1cqGHflY1cqGHflY1cq GHflY1aKqaGeqaaKqzGeGae8NhWxwcfa4aaWbaaSqabeaajugibiab gwSixdaacqGHNis2cqWFEaFzjuaGdaahaaWcbeqaaKqzGeGaeyyXIC naaiabgEIizlab=5b8LLqbaoaaCaaaleqabaqcLbsacqGHflY1aaGa ey4jIKTae8NhWxwcfa4aaWbaaSqabeaajugibiabgwSixdaacaaIVa GaaGinaiaaigcaaaa@6E84@ are introduced. Similarly, the three–dimensional volume form and two–dimensional surface form are also introduced as V a = ε a e e e /3! MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=vb8wTWaaSba aKqaGeaajugWaiaadggaaKqaGeqaaKqzGeGaaGypaiabew7aLTWaaS baaKqaGeaajugWaiaadggacqWIVlctaKqaGeqaaKqzGeGae8NhWxwc fa4aaWbaaSqabeaajugibiabgwSixdaacqGHNis2cqWFEaFzjuaGda ahaaWcbeqaaKqzGeGaeyyXICnaaiabgEIizlab=5b8LLqbaoaaCaaa leqabaqcLbsacqGHflY1aaGaaG4laiaaiodacaaIHaaaaa@63DE@ and S ab = ε ab e e /2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=jb8tTWaaSba aKqaGeaajugWaiaadggacaWGIbaajeaibeaajugibiaai2dacqaH1o qzlmaaBaaajeaibaqcLbmacaWGHbGaamOyaiabgwSixlabgwSixdqc basabaqcLbsacqWFEaFzjuaGdaahaaWcbeqaaKqzGeGaeyyXICnaai abgEIizlab=5b8LLqbaoaaCaaaleqabaqcLbsacqGHflY1aaGaaG4l aiaaikdaaaa@6066@ , respectively. The volume form V a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=vb8wTWaaSba aKqaGeaajugWaiaadggaaKqaGeqaaaaa@45AF@ is a three–dimensional volume perpendicular to e a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=5b8LTWaaWba aKqaGeqajqwaG9FaaKqzadGaamyyaaaaaaa@476B@ , and the surface form S ab MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=jb8tTWaaSba aKqaGeaajugWaiaadggacaWGIbaajeaibeaaaaa@4690@ is a two–dimensional plane perpendicular to both e a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=5b8LTWaaWba aKqaGeqabaqcLbmacaWGHbaaaaaa@459E@ and e b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=5b8LLqbaoaa CaaaleqajeaibaqcLbmacaWGIbaaaaaa@462D@ . Fraktur letters are used for differential forms. A unit of c=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4yai aai2dacaaIXaaaaa@38E4@ is used while the reduced Planck constant MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeS4dHG gaaa@37A3@ and Newtonian gravitational constant G MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4raa aa@3746@ (or the Einstein constant κ=4πG MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOUdS MaaGypaiaaisdacqaHapaCcaWGhbaaaa@3C3A@ in our convention) written explicitly. In this units, there are two physical dimensions, the length and mass dimensions, which are denoted as L MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaa aa@374B@ and M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ , respectively.

The Lagrangian for pure gravity without the cosmological term and matter fields is expressed as

L G = 1 2 S , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Ta8mLqbaoaa BaaajeaibaqcLbmacaWGhbaaleqaaKqzGeGaaGypaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGOmaaaacqWFsa=ulmaaBaaajeai baqcLbmacqGHflY1cqGHflY1aKqaGeqaaKqzGeGaey4jIKTae8hhHi vcfa4aaWbaaKqaGeqabaqcLbmacqGHflY1cqGHflY1aaqcLbsacaaI Saaaaa@5DD0@ (1)

ab =d w ab + w a w b , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=XrisTWaaWba aKqaGeqabaqcLbmacaWGHbGaamOyaaaajugibiaai2dacaWGKbGae8 hmWF3cdaahaaqcbasabeaajugWaiaadggacaWGIbaaaKqzGeGaey4k aSIae8hmWF3cdaqhaaqcbasaaKqzadGaaGiiaiabgwSixdqcbasaaK qzadGaamyyaaaajugibiabgEIizlab=bd83TWaaWbaaKqaGeqabaqc LbmacqGHflY1caWGIbaaaKqzGeGaaGilaaaa@61E6@ (2)

where w MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=bd83TWaaWba aKqaGeqabaqcLbmacqGHIaYTcqGHIaYTaaaaaa@47E6@ is the spin one–form, which is defined as w ab = ω μ a η b d x μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=bd83TWaaWba aKqaGeqabaqcLbmacaWGHbGaamOyaaaajugibiaai2dacqaHjpWDju aGdaqhaaWcbaqcLbmacqaH8oqBjugibiaaiccacqGHflY1aSqaaKqz GeGaaGiiaKqzadGaamyyaaaajugibiabeE7aOLqbaoaaCaaaleqaje aibaqcLbmacqGHflY1caWGIbaaaKqzGeGaamizaiaadIhalmaaCaaa jeaibeqaaKqzadGaeqiVd0gaaaaa@6175@ . A two form MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=XrisTWaaWba aKqaGeqabaqcLbmacqGHIaYTcqGHIaYTaaaaaa@46DE@ is referred to as the curvature form, that is a rank– 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGOmaa aa@3736@ Lorentz tensor on M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ .

A Lagrangian of a scalar field φ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOXdO MaaGikaiaadIhacaaIPaaaaa@3A99@ can be expressed in the vierbein formalism as

L S = 1 2 1 3! S ( η ι s ι s V(φ) e e ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Ta8mLqbaoaa BaaajeaibaqcLbmacaWGtbaaleqaaKqzGeGaaGypaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGOmaaaajuaGdaWcaaGcbaqcLbsa caaIXaaakeaajugibiaaiodacaaIHaaaaiab=jb8tTWaaSbaaKqaGe aajugWaiabgwSixlabgwSixdqcbasabaqcLbsacqGHNis2juaGdaqa daGcbaqcLbsacqaH3oaAlmaaCaaajeaibeqaaKqzadGaey4fIOIaey 4fIOcaaKqzGeGaeqyUdK2cdaWgaaqcbasaaKqzadGaey4fIOcajeai beaajugibiab=Xc8ZTWaaWbaaKqaGeqabaqcLbmacqGHflY1aaqcLb sacqGHNis2cqaH5oqAjuaGdaWgaaqcbasaaKqzadGaey4fIOcaleqa aKqzGeGae8hlWp3cdaahaaqcbasabeaajugWaiabgwSixdaajugibi abgkHiTiaadAfacaaIOaGaeqOXdOMaaGykaiab=5b8LTWaaWbaaKqa GeqabaqcLbmacqGHflY1aaqcLbsacqGHNis2cqWFEaFzlmaaCaaaje aibeqaaKqzadGaeyyXICnaaaGccaGLOaGaayzkaaqcLbsacaaISaaa aa@8DD8@ (3)

where V(φ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOvai aaiIcacqaHgpGAcaaIPaaaaa@3A77@ is a potential energy. A scalar–field two–form s MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Xc8ZTWaaWba aKqaGeqabaqcLbmacqGHIaYTaaaaaa@4659@ is defined as

s a =dφ e a =( φ ) e e a . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Xc8ZTWaaWba aKqaGeqabaqcLbmacaWGHbaaaKqzGeGaaGypaiaadsgacqaHgpGAcq GHNis2cqWFEaFzlmaaCaaajeaibeqaaKqzadGaamyyaaaajugibiaa i2dajuaGdaqadaGcbaqcLbsacqGHciITjuaGdaWgaaWcbaqcLbsacq GHflY1aSqabaqcLbsacqaHgpGAaOGaayjkaiaawMcaaKqzGeGae8Nh Wx2cdaahaaqcbasabeaajugWaiabgwSixdaajugibiabgEIizlab=5 b8LTWaaWbaaKqaGeqabaqcLbmacaWGHbaaaKqzGeGaaGOlaaaa@6964@ (4)

Here ι a = ι ξ a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyUdK wcfa4aaSbaaKqaGeaajugWaiaadggaaSqabaqcLbsacaaI9aGaeqyU dK2cdaWgaaqcbasaaKqzadGaeqOVdG3cdaahaaqcbasabeaajugWai aadggaaaaajeaibeaaaaa@4402@ is a contraction operator with respect to the trivial coordinate–vector field ξ a = η a μ μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOVdG 3cdaahaaqcbasabeaajugWaiaadggaaaqcLbsacaaI9aGaeq4TdG2c daahaaqcbasabeaajugWaiaadggacqGHflY1aaWefv3ySLgznfgDOf daryqr1ngBPrginfgDObYtUvgaiuaajugibiab=btifTWaa0baaKqa GeaajugWaiabgwSixdqcbasaaKqzadGaeqiVd0gaaKqzGeGaeyOaIy 7cdaWgaaqcbasaaKqzadGaeqiVd0gajeaibeaaaaa@59C9@ . A physical dimension of the scalar field is set as L 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamitaS WaaWbaaKqaGeqabaqcLbmacqGHsislcaaIXaaaaaaa@3A78@ in this study. In consequence, a Lagrangian form has null physical dimension as

[ φ ]= L 1 { [ ι s ]= L 1 [ V ]= L 4 [ L S ]=1, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajugibiabeA8aQbGccaGLBbGaayzxaaqcLbsacaaI9aGaamitaSWa aWbaaKqaGeqabaqcLbmacqGHsislcaaIXaaaaKqzGeGaeyOKH4Acfa 4aaiqaaOqaaKqzGeqbaeaabiqaaaGcbaqcfa4aamWaaOqaaKqzGeGa eqyUdKwcfa4aaSbaaKqaGeaajugWaiabgkci3cWcbeaatuuDJXwAKz KCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsacqWFSa=Clmaa CaaajeaibeqaaKqzadGaeyOiGClaaaGccaGLBbGaayzxaaqcLbsaca aI9aGaamitaSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaIXaaaaaGc baqcfa4aamWaaOqaaKqzGeGaamOvaaGccaGLBbGaayzxaaqcLbsaca aI9aGaamitaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislcaaI0aaa aaaaaOGaay5EaaqcLbsacqGHsgIRjuaGdaWadaGcbaqcLbsacqWFla ptjuaGdaWgaaqcbasaaKqzadGaam4uaaWcbeaaaOGaay5waiaaw2fa aKqzGeGaaGypaiaaigdacaaISaaaaa@7A30@ (5)

where [] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaG4wai abgkci3kaai2faaaa@39CB@ shows the physical dimension of a quantity MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyOiGC laaa@37FF@ . On the other hand, the gravitational Lagrangian has a length square dimension [ L G ]= L 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaG4wam rr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaacqWFlapt lmaaBaaabaqcLbmacaWGhbaaleqaaKqzGeGaaGyxaiaai2dacaWGmb WcdaahaaqabeaajugWaiaaikdaaaaaaa@4B42@ . An action integral is defined as

= ( 1 κ L G +2 L S ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFqesscaaI9aqc fa4aa8qaaOqabSqabeqajugibiabgUIiYdGaaGikaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaeqOUdSMaeS4dHGgaamrr1ngBPrMr Yf2A0vNCaeXbfv3ySLgzGyKCHTgD1jhaiyaacqGFlaptjuaGdaWgaa qcbasaaKqzadGaam4raaWcbeaajugibiabgUcaRiaaikdacqGFlapt lmaaBaaajeaibaqcLbmacaWGtbaajeaibeaajugibiaaiMcacaaIUa aaaa@62DC@ (6)

Physical constants in front of the gravitational and scalar–field Lagrangian are chosen to adjust the physical dimension of the action integral to be null. Owing to require a stationary condition on a variation of the action integral with respect to the vierbein form δ e =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq 2cdaWgaaqcbasaamrr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKCHTgD 1jhaiuaajugWaiab=5b8LbqcbasabaWefv3ySLgznfgDOfdarCqr1n gBPrginfgDObYtUvgaiyaajugibiab+brijjaai2dacaaIWaaaaa@5320@ , one can obtain the Euler–Lagrange equation as

1 2 ε a e =κ t a , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGOmaaaacqaH1oqzlmaaBaaajeai baqcLbmacaWGHbGaeyyXICTaeyyXICTaeyyXICnajeaibeaatuuDJX wAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsacqWFCeIu lmaaCaaajeaibeqaaKqzadGaeyyXICTaeyyXICnaaKqzGeGaey4jIK Tae8NhWx2cdaahaaqcbasabeaajugWaiabgwSixdaajugibiaai2da cqGHsislcqaH6oWAcqWIpecAcaaIGaGae8xlWtxcfa4aaSbaaKqaGe aajugWaiaadggaaSqabaqcLbsacaaISaaaaa@6AE7@ (7)

where t a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=1c80TWaaSba aKqaGeaajugWaiaadggaaKqaGeqaaaaa@45E5@ is the energy–momentum three–form of the scalar field, which can be represented as

t a = 1 3! ε a ( ι s )( ι s ) e +V(φ) V a . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=1c80TWaaSba aKqaGeaajugWaiaadggaaKqaGeqaaKqzGeGaaGypaiabgkHiTKqbao aalaaakeaajugibiaaigdaaOqaaKqzGeGaaG4maiaaigcaaaGaeqyT du2cdaWgaaqcbasaaKqzadGaamyyaiabl+UimbqcbasabaqcLbsaca aIOaGaeqyUdK2cdaWgaaqcbasaaKqzadGaey4fIOcajeaibeaajugi biab=Xc8ZTWaaWbaaKqaGeqabaqcLbmacqGHflY1aaqcLbsacaaIPa Gaey4jIKTaaGikaiabeM7aPTWaaWbaaKqaGeqabaqcLbmacqGHxiIk aaqcLbsacqWFSa=ClmaaCaaajeaibeqaaKqzadGaeyyXICnaaKqzGe GaaGykaiabgEIizlab=5b8LTWaaWbaaKqaGeqabaqcLbmacqGHflY1 aaqcLbsacqGHRaWkcaWGwbGaaGikaiabeA8aQjaaiMcacqWFvaVvlm aaBaaajeaibaqcLbmacaWGHbaajeaibeaajugibiaai6caaaa@805A@ (8)

Here, rising and lowering Roman indices are performed using a Lorentz metric tensor, e.g. ι a s b = η a ι e b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyUdK 2cdaahaaqcbasabeaajugWaiaadggaaaWefv3ySLgzgjxyRrxDYbqe guuDJXwAKbIrYf2A0vNCaGqbaKqzGeGae8hlWp3cdaahaaqcbasabe aajugWaiaadkgaaaqcLbsacaaI9aGaeq4TdG2cdaahaaqcbasabeaa jugWaiaadggacqGHflY1aaqcLbsacqaH5oqAjuaGdaWgaaWcbaqcLb sacqGHflY1aSqabaqcLbsacqWFEaFzlmaaCaaajeaibeqaaKqzadGa amOyaaaaaaa@5CEA@ . A torsionless condition and Klein–Gordon equation can be obtained from δ (w,dw) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq 2cdaWgaaqcbasaaKqzadGaaGikamrr1ngBPrMrYf2A0vNCaeHbfv3y SLgzGyKCHTgD1jhaiuaacqWFWa=DcaaISaGaamizaiab=bd83jaaiM caaKqaGeqaamrr1ngBPrwtHrhAXaqehuuDJXwAKbstHrhAG8KBLbac gaqcLbsacqGFqesscaaI9aGaaGimaaaa@5864@ and δ (φ,dφ) =0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq 2cdaWgaaqcbasaaKqzadGaaGikaiabeA8aQjaaiYcacaWGKbGaeqOX dOMaaGykaaqcbasabaWefv3ySLgznfgDOfdaryqr1ngBPrginfgDOb YtUvgaiuaajugibiab=brijjaai2dacaaIWaaaaa@4CE4@ , respectively.

The scalar–filed Lagrangian L S MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=Ta8mTWaaSba aKqaGeaajugWaiaadofaaKqaGeqaaaaa@458D@ given in (3) can be expressed using a trivial frame vector in M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ or MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ as

(3)=v[ 1 2 η φ φV(φ) ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aaiodacaaIPaGaaGypamrr1ngBPrMrYf2A0vNCaeHbfv3ySLgzGyKC HTgD1jhaiuaacqWFVaVDjuaGdaWadaGcbaqcfa4aaSaaaOqaaKqzGe GaaGymaaGcbaqcLbsacaaIYaaaaiabeE7aOTWaaWbaaKqaGeqabaqc LbmacqGHflY1cqGHflY1aaqcLbsacqGHciITjuaGdaWgaaWcbaqcLb sacqGHflY1aSqabaqcLbsacqaHgpGAcaaIGaGaeyOaIyBcfa4aaSba aSqaaKqzGeGaeyyXICnaleqaaKqzGeGaeqOXdOMaeyOeI0IaamOvai aaiIcacqaHgpGAcaaIPaaakiaawUfacaGLDbaajugibiaaiYcaaaa@69E6@

= g d x 0 d x 1 d x 2 d x 3 [ 1 2 g μ 1 μ 2 μ 1 φ μ 2 φV(φ) ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGypaK qbaoaakaaakeaajugibiabgkHiTiaadEgaaSqabaqcLbsacaWGKbGa amiEaSWaaWbaaKqaGeqabaqcLbmacaaIWaaaaKqzGeGaey4jIKTaam izaiaadIhalmaaCaaajeaibeqaaKqzadGaaGymaaaajugibiabgEIi zlaadsgacaWG4bWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacq GHNis2caWGKbGaamiEaSWaaWbaaKqaGeqabaqcLbmacaaIZaaaaKqb aoaadmaakeaajuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaik daaaGaam4zaSWaaWbaaKqaGeqabaqcLbmacqaH8oqBlmaaBaaajeai baqcLbmacaaIXaaajeaibeaajugWaiabeY7aTTWaaSbaaKqaGeaaju gWaiaaikdaaKqaGeqaaaaajugibiabgkGi2UWaaSbaaKqaGeaajugW aiabeY7aTTWaaSbaaKqaGeaajugWaiaaigdaaKqaGeqaaaqabaqcLb sacqaHgpGAcaaIGaGaeyOaIyBcfa4aaSbaaKqaGeaajugWaiabeY7a TTWaaSbaaKqaGeaajugWaiaaikdaaKqaGeqaaaWcbeaajugibiabeA 8aQjabgkHiTiaadAfacaaIOaGaeqOXdOMaaGykaaGccaGLBbGaayzx aaqcLbsacaaISaaaaa@7E96@ (9)

where g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaa aa@3766@ is a determinant of a metric tensor g MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4zaS WaaWbaaKqaGeqabaqcLbmacqGHIaYTcqGHIaYTaaaaaa@3BF5@ . Here a relation ι a e b = δ a b MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyUdK wcfa4aaSbaaKqaGeaajugWaiaadggaaSqabaWefv3ySLgzgjxyRrxD YbqeguuDJXwAKbIrYf2A0vNCaGqbaKqzGeGae8NhWx2cdaahaaqcba sabeaajugWaiaadkgaaaqcLbsacaaI9aGaeqiTdq2cdaqhaaqcbasa aKqzadGaamyyaaqcbasaaKqzadGaamOyaaaaaaa@527A@ is used. The Einstein equation (7) can be expressed using components of a trivial basis on M MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamytaa aa@374C@ as

R ab 1 2 η ab R=2κ T ab , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaS WaaSbaaKqaGeaajugWaiaadggacaWGIbaajeaibeaajugibiabgkHi TKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGaaGOmaaaacqaH3o aAlmaaBaaajeaibaqcLbmacaWGHbGaamOyaaqcbasabaqcLbsacaWG sbGaaGypaiabgkHiTiaaikdacqaH6oWAcqWIpecAcaaIGaGaamivaS WaaSbaaKqaGeaajugWaiaadggacaWGIbaajeaibeaajugibiaaiYca aaa@51AA@ (10)

T ab = a φ b φ 1 2 η ab φ φ+ η ab V, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaKqaGeaajugWaiaadggacaWGIbaajeaibeaajugibiaai2da cqGHciITjuaGdaWgaaqcbasaaKqzadGaamyyaaWcbeaajugibiabeA 8aQjaaiccacqGHciITjuaGdaWgaaqcbasaaKqzadGaamOyaaWcbeaa jugibiabeA8aQjabgkHiTKqbaoaalaaakeaajugibiaaigdaaOqaaK qzGeGaaGOmaaaacqaH3oaAlmaaBaaajeaibaqcLbmacaWGHbGaamOy aaqcbasabaqcLbsacqGHciITjuaGdaWgaaWcbaqcLbsacqGHflY1aS qabaqcLbsacqaHgpGAcaaIGaGaeyOaIyBcfa4aaWbaaSqabeaajugi biabgwSixdaacqaHgpGAcqGHRaWkcqaH3oaAlmaaBaaajeaibaqcLb macaWGHbGaamOyaaqcbasabaqcLbsacaWGwbGaaGilaaaa@6AE7@ (11)

where R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaS WaaSbaaKqaGeaajugWaiabgkci3kabgkci3cqcbasabaaaaa@3C09@ and R MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOuaa aa@3751@ are the Ricci and scalar curvature, respectively. An energy–momentum tensor T MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaKqaGeaajugWaiabgkci3kabgkci3cqcbasabaaaaa@3C0B@ is defined through the relation t a = V T a MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=1c80TWaaSba aKqaGeaajugWaiaadggaaKqaGeqaaKqzGeGaaGypaiab=vb8wLqbao aaBaaaleaajugibiabgwSixdWcbeaajugibiaadsfalmaaDaaajeai baqcLbmacaaIGaGaamyyaaqcbasaaKqzadGaeyyXICnaaaaa@54D8@ .

Conformal metric and scaled field method

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric24–27 is considered in the homogeneous and isotropic universe using a coordinate (t,r,θ,ϕ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aadshacaaISaGaamOCaiaaiYcacqaH4oqCcaaISaGaeqy1dyMaaGyk aaaa@3F6F@ on MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFZestaaa@412A@ such as;

d s 2 =d t 2 Ω 2 (t)( f 2 (r)d r 2 + r 2 d θ 2 + r 2 sin 2 θd ϕ 2 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai aadohalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaai2dacaWG KbGaamiDaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0 IaeuyQdC1cdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacaaIOaGa amiDaiaaiMcajuaGdaqadaGcbaqcLbsacaWGMbWcdaahaaqcbasabe aajugWaiabgkHiTiaaikdaaaqcLbsacaaIOaGaamOCaiaaiMcacaWG KbGaamOCaKqbaoaaCaaaleqajeaibaqcLbmacaaIYaaaaKqzGeGaey 4kaSIaamOCaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaamiz aiabeI7aXTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaS IaamOCaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqbaoaavacakeqa leqajeaibaqcLbmacaaIYaaakeaajugibiGacohacaGGPbGaaiOBaa aacqaH4oqCcaWGKbGaeqy1dy2cdaahaaqcbasabeaajugWaiaaikda aaaakiaawIcacaGLPaaajugibiaaiYcaaaa@73CE@ (12)

where f 2 (r)=1K r 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzaS WaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaaGikaiaadkhacaaI PaGaaGypaiaaigdacqGHsislcaWGlbGaamOCaKqbaoaaCaaaleqaje aibaqcLbmacaaIYaaaaaaa@4396@ , and K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4saa aa@374A@ is a constant related the space–time curvature. This metric can be expressed using a conformal time τ(t)= d t ˜ /Ω( t ˜ ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiXdq NaaGikaiaadshacaaIPaGaaGypaKqbaoaapeaakeqaleqabeqcLbsa cqGHRiI8aiaadsgaceWG0bGbaGaacaaIVaGaeuyQdCLaaGikaiqads hagaacaiaaiMcaaaa@451F@ as

(12)= Ω 2 (τ)( d τ 2 f 2 (r)d r 2 r 2 d θ 2 r 2 sin 2 θd ϕ 2 ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aaigdacaaIYaGaaGykaiaai2dacqqHPoWvlmaaCaaajeaibeqaaKqz adGaaGOmaaaajugibiaaiIcacqaHepaDcaaIPaqcfa4aaeWaaOqaaK qzGeGaamizaiabes8a0TWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqz GeGaeyOeI0IaamOzaSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaIYa aaaKqzGeGaaGikaiaadkhacaaIPaGaamizaiaadkhalmaaCaaajeai beqaaKqzadGaaGOmaaaajugibiabgkHiTiaadkhajuaGdaahaaWcbe qcbasaaKqzadGaaGOmaaaajugibiaadsgacqaH4oqClmaaCaaajeai beqaaKqzadGaaGOmaaaajugibiabgkHiTiaadkhalmaaCaaajeaibe qaaKqzadGaaGOmaaaajuaGdaqfGaGcbeWcbeqcbasaaKqzadGaaGOm aaGcbaqcLbsaciGGZbGaaiyAaiaac6gaaaGaeqiUdeNaamizaiabew 9aMTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaaGccaGLOaGaayzkaaqc LbsacaaIUaaaaa@73A9@ (13)

From the metric (13) and the torsion–less condition, the spin form can be obtained as

w =( 0 H f 1 dr H rdθ H rsinθdϕ 0 f dθ f sinθdϕ 0 cosθdϕ 0 ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=bd83TWaaWba aKqaGeqajeaqbaqcLbmacqGHIaYTcqGHIaYTaaqcLbsacaaI9aqcfa 4aaeWaaOqaaKqzGeqbaeqabuabaaaaaOqaaKqzGeGaaGimaaGcbaqc LbsacaWGibGaamOzaKqbaoaaCaaaleqajeaibaqcLbmacqGHsislca aIXaaaaKqzGeGaaGiiaiaadsgacaWGYbGaaGiiaaGcbaqcLbsacaWG ibGaaGiiaiaadkhacaWGKbGaeqiUdehakeaajugibiaadIeacaaIGa GaamOCaiGacohacaGGPbGaaiOBaiabeI7aXjaadsgacqaHvpGzaOqa aKqzGeGaaGiiaaqcaasaaKqzadGaaGimaaGcbaqcLbsacqGHsislca WGMbGaaGiiaiaadsgacqaH4oqCaOqaaKqzGeGaeyOeI0IaamOzaiaa iccaciGGZbGaaiyAaiaac6gacqaH4oqCcaWGKbGaeqy1dygakeaaju gibiaaiccaaOqaaKqzGeGaaGiiaaGcbaqcLbsacaaIWaaakeaajugi biaaiccacaaIGaGaeyOeI0Iaci4yaiaac+gacaGGZbGaeqiUdeNaam izaiabew9aMbGcbaqcLbsacaaIGaaakeaajugibiaaiccaaOqaaKqz GeGaaGiiaaGcbaqcLbsacaaIWaaakeaaaeaaaeaaaeaaaaaacaGLOa GaayzkaaqcLbsacaaISaaaaa@8F29@ (14)

where H=((dΩ/dτ)/Ω)(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisai aai2dacaaIOaGaaGikaiaadsgacqqHPoWvcaaIVaGaamizaiabes8a 0jaaiMcacaaIVaGaeuyQdCLaaGykaiaaiIcacqaHepaDcaaIPaaaaa@4627@ . The lower half is omitted because it is obvious due to antisymmetry of the spin form. From (14) and the surface form, we can obtain the classical Hamiltonian as22

FLRW = 1 2 w w S = Ω 2 [ 3 H 2 ( f r ) 2 ] dτ dr f (rdθ)(rsinθdϕ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrMrYf 2A0vNCaeHbfv3ySLgzGyKCHTgD1jhaiuaajugibiab=XqiiTWaaSba aKqaGeaajugWaiaabAeacaqGmbGaaeOuaiaabEfaaKqaGeqaaKqzGe GaaGypaiabgkHiTKqbaoaalaaakeaajugibiaaigdaaOqaaKqzGeGa aGOmaaaacqWFWa=DlmaaDaaajeaibaqcLbmacaaIGaGaey4fIOcaje aibaqcLbmacqGHflY1aaqcLbsacqGHNis2cqWFWa=DlmaaCaaajeai beqaaKqzadGaey4fIOIaeyyXICnaaKqzGeGaey4jIKTae8NeWp1cda WgaaqcbasaaKqzadGaeyyXICTaeyyXICnajeaibeaajugibiaai2da cqqHPoWvjuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaajuaGdaWada GcbaqcLbsacaaIZaGaamisaSWaaWbaaKqaGeqabaqcLbmacaaIYaaa aKqzGeGaeyOeI0scfa4aaeWaaOqaaKqbaoaalaaakeaajugibiaadA gaaOqaaKqzGeGaamOCaaaaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqa baqcLbmacaaIYaaaaaGccaGLBbGaayzxaaqcLbsacaaIGaGaamizai abes8a0jabgEIizNqbaoaalaaakeaajugibiaadsgacaWGYbaakeaa jugibiaadAgaaaGaey4jIKTaaGikaiaadkhacaWGKbGaeqiUdeNaaG ykaiabgEIizlaaiIcacaWGYbGaci4CaiaacMgacaGGUbGaeqiUdeNa amizaiabew9aMjaaiMcacaaISaaaaa@9C13@ (15)

and the Liouville form as

1 2 w S = Ω 2 [ cotθ r dτ dr f (rsinθdϕ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaaigdaaOqaaKqzGeGaaGOmaaaatuuDJXwAKzKCHTgD1jha ryqr1ngBPrgigjxyRrxDYbacfaGae8hmWF3cdaahaaqcbasabeaaju gWaiabgwSixlabgwSixdaajugibiabgEIizlab=jb8tTWaaSbaaKqa GeaajugWaiabgwSixlabgwSixdqcbasabaqcLbsacaaI9aGaeuyQdC 1cdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacaaIBbGaeyOeI0sc fa4aaSaaaOqaaKqzGeGaci4yaiaac+gacaGG0bGaeqiUdehakeaaju gibiaadkhaaaGaaGiiaiaadsgacqaHepaDcqGHNis2juaGdaWcaaGc baqcLbsacaWGKbGaamOCaaGcbaqcLbsacaWGMbaaaiabgEIizlaaiI cacaWGYbGaci4CaiaacMgacaGGUbGaeqiUdeNaamizaiabew9aMjaa iMcaaaa@7A42@

+ 2f r dτ(rdθ)(rsinθdϕ)+3H dr f (rdθ)(rsinθdϕ)]. MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGiiai abgUcaRKqbaoaalaaakeaajugibiaaikdacaWGMbaakeaajugibiaa dkhaaaGaaGiiaiaadsgacqaHepaDcqGHNis2caaIOaGaamOCaiaads gacqaH4oqCcaaIPaGaey4jIKTaaGikaiaadkhaciGGZbGaaiyAaiaa c6gacqaH4oqCcaWGKbGaeqy1dyMaaGykaiabgUcaRiaaiodacaWGib GaaGiiaKqbaoaalaaakeaajugibiaadsgacaWGYbaakeaajugibiaa dAgaaaGaey4jIKTaaGikaiaadkhacaWGKbGaeqiUdeNaaGykaiabgE IizlaaiIcacaWGYbGaci4CaiaacMgacaGGUbGaeqiUdeNaamizaiab ew9aMjaaiMcacaaIDbGaaGOlaaaa@6D00@ (16)

A canonical quantization of general relativity requires the commutation relation between the spin form and surface from as22

[ w ^ a 1 a 2 (x), S ^ b 1 b 2 (y) ]=iG δ (4) (xy) δ b 1 [ a 1 δ b 2 a 2 ] , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajuaGdaqiaaGcbaWefv3ySLgzgjxyRrxDYbqeguuDJXwAKbIrYf2A 0vNCaGqbaKqzGeGae8hmWFhakiaawkWaaSWaaWbaaKqaGeqabaqcLb macaWGHbWcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaqcLbmacaWG HbWcdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaaaaKqzGeGaaGikai aadIhacaaIPaGaaGilaKqbaoaaHaaakeaajugibiab=jb8tbGccaGL cmaajuaGdaWgaaqcbasaaKqzadGaamOyaSWaaSbaaKqaGeaajugWai aaigdaaKqaGeqaaKqzadGaamOyaSWaaSbaaKqaGeaajugWaiaaikda aKqaGeqaaaWcbeaajugibiaaiIcacaWG5bGaaGykaaGccaGLBbGaay zxaaqcLbsacaaI9aGaeyOeI0IaamyAaiabl+qiOjaadEeacqaH0oaz lmaaCaaajeaibeqaaKqzadGaaGikaiaaisdacaaIPaaaaKqzGeGaaG ikaiaadIhacqGHsislcaWG5bGaaGykaiabes7aKTWaa0baaKqaGeaa jugWaiaadkgalmaaBaaajeaibaqcLbmacaaIXaaajeaibeaaaeaaju gWaiaaiUfacaWGHbWcdaWgaaqcbasaaKqzadGaaGymaaqcbasabaaa aKqzGeGaeqiTdqwcfa4aa0baaKqaGeaajugWaiaadkgalmaaBaaaje aibaqcLbmacaaIYaaajeaibeaaaSqaaKqzadGaamyyaSWaaSbaaKqa GeaajugWaiaaikdaaKqaGeqaaKqzGeGaaGyxaaaacaaISaaaaa@8E55@ (17)

and otherwise zero, where δ b 1 [ a 1 δ b 2 a 2 ] = δ b 1 a 1 δ b 2 a 2 δ b 1 a 2 δ b 2 a 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiTdq 2cdaqhaaqcbasaaKqzadGaamOyaSWaaSbaaKqaGeaajugWaiaaigda aKqaGeqaaaqaaKqzadGaaG4waiaadggalmaaBaaajeaibaqcLbmaca aIXaaajeaibeaaaaqcLbsacqaH0oazlmaaDaaajeaibaqcLbmacaWG IbWcdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaaabaqcLbmacaWGHb WcdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaqcLbmacaaIDbaaaKqz GeGaaGypaiabes7aKTWaa0baaKqaGeaajugWaiaadkgalmaaBaaaje aibaqcLbmacaaIXaaajeaibeaaaeaajugWaiaadggalmaaBaaajeai baqcLbmacaaIXaaajeaibeaaaaqcLbsacqaH0oazlmaaDaaajeaiba qcLbmacaWGIbWcdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaaabaqc LbmacaWGHbWcdaWgaaqcbasaaKqzadGaaGOmaaqcbasabaaaaKqzGe GaeyOeI0IaeqiTdq2cdaqhaaqcbasaaKqzadGaamOyaSWaaSbaaKqa GeaajugWaiaaigdaaKqaGeqaaaqaaKqzadGaamyyaSWaaSbaaKqaGe aajugWaiaaikdaaKqaGeqaaaaajugibiabes7aKTWaa0baaKqaGeaa jugWaiaadkgalmaaBaaajeaibaqcLbmacaaIYaaajeaibeaaaeaaju gWaiaadggalmaaBaaajeaibaqcLbmacaaIXaaajeaibeaaaaaaaa@7FC3@ . Here, w ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaHaaake aatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsa cqWFWa=DaOGaayPadaWcdaahaaqcbasabeaajugWaiabgkci3kabgk ci3caaaaa@494A@ and S ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaHaaake aatuuDJXwAKzKCHTgD1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsa cqWFsa=uaOGaayPadaqcfa4aaSbaaKqaGeaajugWaiabgkci3kabgk ci3cWcbeaaaaa@4995@ are operators respectively corresponding to the spin and surface forms, which are formally described as

{ w ^ = w , S ^ =iG δ δ w . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaceaake aajugibuaabeqaciaaaOqaaKqbaoaaHaaakeaatuuDJXwAKzKCHTgD 1jharyqr1ngBPrgigjxyRrxDYbacfaqcLbsacqWFWa=DaOGaayPada WcdaahaaqcbasabeaajugWaiabgkci3kabgkci3caaaOqaaKqzGeGa aGypaiab=bd83TWaaWbaaKqaGeqabaqcLbmacqGHIaYTcqGHIaYTaa qcLbsacaaISaaakeaajuaGdaqiaaGcbaqcLbsacqWFsa=uaOGaayPa daWcdaWgaaqcbasaaKqzadGaeyOiGCRaeyOiGClajeaibeaaaOqaaK qzGeGaaGypaiaadMgacqWIpecAcaWGhbqcfa4aaSaaaOqaaKqzGeGa eqiTdqMaaGiiaiaaiccaaOqaaKqzGeGaeqiTdqMae8hmWFxcfa4aaW baaSqabKqaGeaajugWaiabgkci3kabgkci3caaaaqcLbsacaaIUaaa aaGccaGL7baaaaa@70BF@ (18)

The commutation relation (17) can be represented using terms of the FLRM metric. The metric tensor is a functional of two functions Ω(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiabes8a0jaaiMcaaaa@3B32@ and f(r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOzai aaiIcacaWGYbGaaGykaaaa@39C1@ other than the integration measure. On the other hand, the Liouville–form (16) includes a derivative Ω ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafuyQdC Lbaiaaaaa@3811@ , and it does not have a term df(r)/dr MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai aadAgacaaIOaGaamOCaiaaiMcacaaIVaGaamizaiaadkhaaaa@3D43@ . Therefore, quantization of the system can be performed by replacing a scale faction Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC faaa@3808@ by the corresponding operator as Ω ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaHaaake aajugibiabfM6axbGccaGLcmaaaaa@396C@ . The conformal–time derivative1 Ω ˙ =dΩ/dτ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafuyQdC LbaiaacaaI9aGaamizaiabfM6axjaai+cacaWGKbGaeqiXdqhaaa@3EB6@ is included only in the last term of (16). Thus, a non–zero component of the commutation relation can be obtained from (16) and (17) as

[ w ^ (r), S ^ ( r ) ] | τ= τ =23[ Ω ^ (τ), Ω ˙ ^ (τ) ] d 3 r' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajuaGdaqiaaGcbaWefv3ySLgzgjxyRrxDYbqeguuDJXwAKbIrYf2A 0vNCaGqbaKqzGeGae8hmWFhakiaawkWaaSWaaWbaaKqaGeqabaqcLb macqGHflY1cqGHflY1aaqcLbsacaaIOaGaamOCaiaaiMcacaaISaqc fa4aaecaaOqaaKqzGeGae8NeWpfakiaawkWaaKqbaoaaBaaaleaaju gibiabgwSixlabgwSixdWcbeaajugibiaaiIcaceWGYbGbauaacaaI PaaakiaawUfacaGLDbaajugibiaaiYhajuaGdaWgaaqcbasaaKqzad GaeqiXdqNaaGypaiqbes8a0zaafaaaleqaaKqzGeGaaGypaiaaikda cqGHflY1caaIZaqcfa4aamWaaOqaaKqbaoaaHaaakeaajugibiabfM 6axbGccaGLcmaajugibiaaiIcacqaHepaDcaaIPaGaaGilaKqbaoaa HaaakeaajugibiqbfM6axzaacaaakiaawkWaaKqzGeGaaGikaiabes 8a0jaaiMcaaOGaay5waiaaw2faaKqzGeGaamizaKqbaoaaCaaaleqa jeaibaqcLbmacaaIZaaaaKqzGeGaaCOCaiaaiEcaaaa@831C@ (19)

where r=(τ,r) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCai aai2dacaaIOaGaeqiXdqNaaGilaiaahkhacaaIPaaaaa@3D13@ , r =( τ ,r') MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmOCay aafaGaaGypaiaaiIcacuaHepaDgaqbaiaaiYcacaWHYbGaaG4jaiaa iMcaaaa@3DDC@ and r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaCOCaa aa@3775@ , r' MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaCOCai aaiEcaaaa@3826@ are three–dimensional spacial vectors. The three–dimensional integration can be expressed as

d 3 r'= d r f( r ) ( r dθ)( r sinθdϕ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizaS WaaWbaaKqaGeqabaqcLbmacaaIZaaaaKqzGeGaaCOCaiaaiEcacaaI 9aqcfa4aaSaaaOqaaKqzGeGaamizaiqadkhagaqbaaGcbaqcLbsaca WGMbGaaGikaiqadkhagaqbaiaaiMcaaaGaey4jIKTaaGikaiqadkha gaqbaiaadsgacqaH4oqCcaaIPaGaey4jIKTaaGikaiqadkhagaqbai GacohacaGGPbGaaiOBaiabeI7aXjaadsgacqaHvpGzcaaIPaGaaGOl aaaa@5678@ (20)

On the other hand, the tight hand side of (17) is represented as

(17)=i(2G) δ 3 (rr'). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aaigdacaaI3aGaaGykaiaai2dacqGHsislcaWGPbGaaGikaiaaikda cqWIpecAcaWGhbGaaGykaiabes7aKTWaaWbaaKqaGeqabaqcLbmaca aIZaaaaKqzGeGaaGikaiaahkhacqGHsislcaWHYbGaaG4jaiaaiMca caaIUaaaaa@4A3A@ (21)

Therefore, the commutation relation is obtained as

[ Ω ^ (τ), Ω ˙ ^ (τ) ] d 3 r'=i G 3 δ 3 (rr'). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajuaGdaqiaaGcbaqcLbsacqqHPoWvaOGaayPadaqcLbsacaaIOaGa eqiXdqNaaGykaiaaiYcajuaGdaqiaaGcbaqcLbsacuqHPoWvgaGaaa GccaGLcmaajugibiaaiIcacqaHepaDcaaIPaaakiaawUfacaGLDbaa jugibiaadsgalmaaCaaajeaibeqaaKqzadGaaG4maaaajugibiaahk hacaaINaGaaGypaiabgkHiTiaadMgajuaGdaWcaaGcbaqcLbsacqWI pecAcaWGhbaakeaajugibiaaiodaaaGaeqiTdq2cdaahaaqcbasabe aajugWaiaaiodaaaqcLbsacaaIOaGaaCOCaiabgkHiTiaahkhacaaI NaGaaGykaiaai6caaaa@5EF2@ (22)

This can be understood as the equal–time commutation relation of the conformal metric. While this commutation relation is obtained from the commutation relation (17), one can obtain the same result based on quantization by Nakanishi28 as shown in Appendix A.

The scalar field can be defined in the conformal metric with K=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4sai aai2dacaaIWaaaaa@38CB@ in the Cartesian coordinate for three–dimensional space

d s 2 = Ω 2 ( τ )( d τ 2 ( d x 1 ) 2 ( d x 2 ) 2 ( d x 3 ) 2 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai aadohalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaai2dacqqH PoWvlmaaCaaajeaibeqaaKqzadGaaGOmaaaajuaGdaqadaGcbaqcLb sacqaHepaDaOGaayjkaiaawMcaaKqbaoaabmaakeaajugibiaadsga cqaHepaDlmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgkHiTK qbaoaabmaakeaajugibiaadsgacaWG4bWcdaahaaqcbasabeaajugW aiaaigdaaaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaG OmaaaajugibiabgkHiTKqbaoaabmaakeaajugibiaadsgacaWG4bWc daahaaqcbasabeaajugWaiaaikdaaaaakiaawIcacaGLPaaalmaaCa aajeaibeqaaKqzadGaaGOmaaaajugibiabgkHiTKqbaoaabmaakeaa jugibiaadsgacaWG4bqcfa4aaWbaaSqabKqaGeaajugWaiaaiodaaa aakiaawIcacaGLPaaajuaGdaahaaWcbeqcbasaaKqzadGaaGOmaaaa aOGaayjkaiaawMcaaaaa@6CDF@ (23)

Instead of a polar–coordinate because virbeins are now independent of r MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOCaa aa@3771@ . The action integral of a scalar Lagrangian (9) in the conformal FLRW metric can be obtained as29

s = L S = dτd x 1 d x 2 d x 3 [ 1 2 η ( χ )( χ )+ 1 2 Ω ¨ Ω χ 2 V(χ) ] MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaamrr1ngBPrwtHr hAXaqeguuDJXwAKbstHrhAG8KBLbacfaqcLbsacqWFqesslmaaBaaa jeaibaqcLbmacaWGZbaajeaibeaajugibiaai2dajuaGdaWdbaGcbe WcbeqabKqzGeGaey4kIipatuuDJXwAKzKCHTgD1jharCqr1ngBPrgi gjxyRrxDYbacgaGae43cWZucfa4aaSbaaKqaGeaajugWaiaadofaaS qabaqcLbsacaaI9aqcfa4aa8qaaOqabSqabeqajugibiabgUIiYdGa amizaiabes8a0jabgEIizlaadsgacaWG4bWcdaahaaqcbasabeaaju gWaiaaigdaaaqcLbsacqGHNis2caWGKbGaamiEaSWaaWbaaKqaGeqa baqcLbmacaaIYaaaaKqzGeGaey4jIKTaamizaiaadIhalmaaCaaaje aibeqaaKqzadGaaG4maaaajuaGdaWadaGcbaqcfa4aaSaaaOqaaKqz GeGaaGymaaGcbaqcLbsacaaIYaaaaiabeE7aOTWaaWbaaKqaGeqaba qcLbmacqGHflY1cqGHflY1aaqcfa4aaeWaaOqaaKqzGeGaeyOaIyBc fa4aaSbaaSqaaKqzGeGaeyyXICnaleqaaKqzGeGaeq4XdmgakiaawI cacaGLPaaajuaGdaqadaGcbaqcLbsacqGHciITjuaGdaWgaaWcbaqc LbsacqGHflY1aSqabaqcLbsacqaHhpWyaOGaayjkaiaawMcaaKqzGe Gaey4kaSscfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqcLbsacaaIYaaa aKqbaoaalaaakeaajugibiqbfM6axzaadaaakeaajugibiabfM6axb aacqaHhpWylmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgkHi TiaadAfacaaIOaGaeq4XdmMaaGykaaGccaGLBbGaayzxaaaaaa@A605@ (24)

using a local conformal coordinate, where χ(x)=Ω(τ)φ(x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm MaaGikaiaadIhacaaIPaGaaGypaiabfM6axjaaiIcacqaHepaDcaaI PaGaeqOXdOMaaGikaiaadIhacaaIPaaaaa@4431@ . We note that det{ g } = Ω 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaakaaake aajugibiabgkHiTiaadsgacaWGLbGaamiDaiaaiUhacaWGNbWcdaWg aaqcbasaaKqzadGaeyOiGCRaeyOiGClajeaibeaajugibiaai2haaS qabaqcLbsacaaI9aGaeuyQdC1cdaahaaqcbasabeaajugWaiaaisda aaaaaa@484C@ for the conformal metric (23). For further discussions, we specify the potential energy as the Higgs–type field as;

V( χ )= Ω 4 V(φ)= 1 2 ( Ω μ ) 2 χ 2 + λ 4 χ 4 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOvaK qbaoaabmaakeaajugibiabeE8aJbGccaGLOaGaayzkaaqcLbsacaaI 9aGaeuyQdC1cdaahaaqcbasabeaajugWaiaaisdaaaqcLbsacaWGwb GaaGikaiabeA8aQjaaiMcacaaI9aGaeyOeI0scfa4aaSaaaOqaaKqz GeGaaGymaaGcbaqcLbsacaaIYaaaaKqbaoaabmaakeaajugibiabfM 6axLqbaoaalaaakeaajugibiabeY7aTbGcbaqcLbsacqWIpecAaaaa kiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibi abeE8aJTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaey4kaSsc fa4aaSaaaOqaaKqzGeGaeq4UdWgakeaajugibiaaisdaaaGaeq4Xdm 2cdaahaaqcbasabeaajugWaiaaisdaaaqcLbsacaaISaaaaa@6525@ (25)

where μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 gaaa@3830@ and λ>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaaGOpaiaaicdaaaa@39B0@ is real constants. Here the physical dimension of each term is given as;

[ μ ]=M, [ ]=LM[ μ ]= L 1 , [ Ωμ χ ]= L 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajugibiabeY7aTbGccaGLBbGaayzxaaqcLbsacaaI9aGaamytaiaa iYcacaaIGaqcfa4aamWaaOqaaKqzGeGaeS4dHGgakiaawUfacaGLDb aajugibiaai2dacaWGmbGaamytaiabgkziUMqbaoaadmaakeaajuaG daWcaaGcbaqcLbsacqaH8oqBaOqaaKqzGeGaeS4dHGgaaaGccaGLBb GaayzxaaqcLbsacaaI9aGaamitaSWaaWbaaKqaGeqabaqcLbmacqGH sislcaaIXaaaaKqzGeGaaGilaiaaiccajuaGdaWadaGcbaqcfa4aaS aaaOqaaKqzGeGaeuyQdCLaeqiVd0gakeaajugibiabl+qiObaacqaH hpWyaOGaay5waiaaw2faaKqzGeGaaGypaiaadYealmaaCaaajeaibe qaaKqzadGaeyOeI0IaaGOmaaaajugibiaai6caaaa@6728@ (26)

This field χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm gaaa@3831@ is provided owing to scale the scalar field φ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOXdO gaaa@3837@ by the conformal function Ω(τ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiabes8a0jaaiMcaaaa@3B32@ , which is referred to as the scaled scalar field (SSF). While a quartic term has no corrections, a quadratic term receives corrections from the conformal function and its second derivative. According to a standard procedure for field quantization (canonical quantization), the field and its conjugate momentum are replaced by corresponding operators. Here, the non–zero component of equal–time commutation relations of the SSF are naturally introduced from (21) as;

[ χ ^ (τ,x), χ ˙ ^ (τ,x') ]= φ ^ (τ,x) φ ^ (τ,x')[ Ω ^ (τ), Ω ˙ ^ (τ) ], MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaadmaake aajuaGdaqiaaGcbaqcLbsacqaHhpWyaOGaayPadaqcLbsacaaIOaGa eqiXdqNaaGilaiaahIhacaaIPaGaaGilaKqbaoaaHaaakeaajugibi qbeE8aJzaacaaakiaawkWaaKqzGeGaaGikaiabes8a0jaaiYcacaWH 4bGaaG4jaiaaiMcaaOGaay5waiaaw2faaKqzGeGaaGypaKqbaoaaHa aakeaajugibiabeA8aQbGccaGLcmaajugibiaaiIcacqaHepaDcaaI SaGaaCiEaiaaiMcajuaGdaqiaaGcbaqcLbsacqaHgpGAaOGaayPada qcLbsacaaIOaGaeqiXdqNaaGilaiaahIhacaaINaGaaGykaKqbaoaa dmaakeaajuaGdaqiaaGcbaqcLbsacqqHPoWvaOGaayPadaqcLbsaca aIOaGaeqiXdqNaaGykaiaaiYcajuaGdaqiaaGcbaqcLbsacuqHPoWv gaGaaaGccaGLcmaajugibiaaiIcacqaHepaDcaaIPaaakiaawUfaca GLDbaajugibiaaiYcaaaa@7376@

=i G 3 | φ ^ (x) | 2 δ (3) (xx')i δ (3) (xx'), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGypai aadMgajuaGdaWcaaGcbaqcLbsacqWIpecAcaWGhbaakeaajugibiaa iodaaaGaaiiFaKqbaoaaHaaakeaajugibiabeA8aQbGccaGLcmaaju gibiaaiIcacaWH4bGaaGykaiaacYhalmaaCaaajeaibeqaaKqzadGa aGOmaaaajugibiabes7aKTWaaWbaaKqaGeqabaqcLbmacaaIOaGaaG 4maiaaiMcaaaqcLbsacaaIOaGaaCiEaiabgkHiTiaahIhacaaINaGa aGykaiabgkziUkaadMgacqaH0oazlmaaCaaajeaibeqaaKqzadGaaG ikaiaaiodacaaIPaaaaKqzGeGaaGikaiaahIhacqGHsislcaWH4bGa aG4jaiaaiMcacaaISaaaaa@6177@ (27)

where φ ^ (x) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaHaaake aajugibiabeA8aQbGccaGLcmaajugibiaaiIcacaWH4bGaaGykaaaa @3C90@ is a scaler field operator. The overall factor G/3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeS4dHG Maam4raiaai+cacaaIZaaaaa@39E5@ can absorbed by re–definition of the scalar field. Although the φ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaHaaake aajugibiabeA8aQbGccaGLcmaaaaa@399B@ is an operator, it is assumed to commutes each other as [ φ ^ , φ ˙ ^ ]=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaG4waK qbaoaaHaaakeaajugibiabeA8aQbGccaGLcmaajugibiaaiYcajuaG daqiaaGcbaqcLbsacuaHgpGAgaGaaaGccaGLcmaajugibiaai2faca aI9aGaaGimaaaa@4304@ . Non–commutativity of χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm gaaa@3831@ and χ ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafq4Xdm Mbaiaaaaa@383A@ is induced by that of Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC faaa@3808@ and Ω ˙ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafuyQdC Lbaiaaaaa@3811@ .

Since the commutation relation (27) for the scaled–field operator χ ^ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaHaaake aajugibiabeE8aJbGccaGLcmaaaaa@3995@ is the same as the standard Klein–Gordon field–operator, the standard procedure of the field quantization in a momentum space using the Fock Hilbert–space can be performed as usual. We note that, in the SSF formalism, the commutation relation is required only on the space–time metric (vierbein). Any observers in the local space time cannot observe the scalar field independently form the space time metric. This is one of a realization of the concept given in Kurihara30 such that “Classical mechanics in the stochastic space is equivalent to quantum mechanics on the standard space time manifold”. According to this concept, only the vierbein is quantized using the quantum commutation relations with keeping the scalar field classical. Even if the local observer is in the flat space–time, one may observe the SSF which is quantized due to the quantum space time. From equations (24) and (25), the effective potential can be obtained as;

V ˜ (χ)= 1 2 ( μ ˜ ) 2 χ 2 + λ 4 χ 4 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaaGaaake aajugibiaadAfaaOGaay5adaqcLbsacaaIOaGaeq4XdmMaaGykaiaa i2dacqGHsisljuaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaik daaaqcfa4aaeWaaOqaaKqbaoaalaaakeaajugibiqbeY7aTzaaiaaa keaajugibiabl+qiObaaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqaba qcLbmacaaIYaaaaKqzGeGaeq4Xdm2cdaahaaqcbasabeaajugWaiaa ikdaaaqcLbsacqGHRaWkjuaGdaWcaaGcbaqcLbsacqaH7oaBaOqaaK qzGeGaaGinaaaacqaHhpWylmaaCaaajeaibeqaaKqzadGaaGinaaaa jugibiaaiYcaaaa@5A02@ (28)

where

( μ ˜ ) 2 = Ω ¨ Ω + ( Ωμ ) 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaabmaake aajuaGdaWcaaGcbaqcLbsacuaH8oqBgaacaaGcbaqcLbsacqWIpecA aaaakiaawIcacaGLPaaalmaaCaaajeaibeqaaKqzadGaaGOmaaaaju gibiaai2dajuaGdaWcaaGcbaqcLbsacuqHPoWvgaWaaaGcbaqcLbsa cqqHPoWvaaGaey4kaSscfa4aaeWaaOqaaKqbaoaalaaakeaajugibi abfM6axjabeY7aTbGcbaqcLbsacqWIpecAaaaakiaawIcacaGLPaaa lmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiaai6caaaa@5233@ (29)

An effective mass of the SSF has time dependence through the relation (29), and the electroweak phase–transition from unbroken to broken phases can occur through it. Although the SSF mass was very small, which corresponds to the unbroken phase, owing to a small value of Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC faaa@3808@ and Ω ¨ /Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafuyQdC LbamaacaaIVaGaeuyQdCfaaa@3A59@ at the early universe, it can be large as the same as that in the current universe after the inflation due to the second term with a large value of Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC faaa@3808@ . In the current universe, the first term of (29) is negligibly small compared with the second term, and the time dependence of the mass term through Ω(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiaadshacaaIPaaaaa@3A66@ is much smaller than the current accuracy of Higgs mass measurements.

A vacuum expectation value v 0 = μ ˜ / λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamODaK qbaoaaBaaajeaibaqcLbmacaaIWaaaleqaaKqzGeGaaGypaiqbeY7a TzaaiaGaaG4laKqbaoaakaaakeaajugibiabeU7aSbWcbeaaaaa@410B@ and higgs mass m h = 2 μ ˜ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaS WaaSbaaKqaGeaajugWaiaadIgaaKqaGeqaaKqzGeGaaGypaKqbaoaa kaaakeaajugibiaaikdaaSqabaqcLbsacaaIGaGafqiVd0MbaGaaaa a@4059@ can be extracted from the effective potential. The Klein–Gordon equation obtained from the action (24) with respect to the scaled field χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm gaaa@3831@ can be expressed as

χ ¨ Δχ+ V ˜ χ =0, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafq4Xdm MbamaacqGHsislcqqHuoarcqaHhpWycqGHRaWkjuaGdaWcaaGcbaqc LbsacqGHciITjuaGdaaiaaGcbaqcLbsacaWGwbaakiaawoWaaaqaaK qzGeGaeyOaIyRaeq4Xdmgaaiaai2dacaaIWaGaaGilaaaa@4875@ (30)

where Δ= i=1,2,3 i i MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiLdq KaaGypaKqbaoaaqabakeqajeaibaqcLbmacaWGPbGaaGypaiaaigda caaISaGaaGOmaiaaiYcacaaIZaaaleqajugibiabggHiLdGaeyOaIy 7cdaWgaaqcbasaaKqzadGaamyAaaqcbasabaqcLbsacqGHciITlmaa BaaajeaibaqcLbmacaWGPbaajeaibeaaaaa@4AE5@ . For the uniform and isotoropic universe, one can set Δχ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiLdq Kaeq4XdmMaaGypaiaaicdaaaa@3B18@ . In this case, the energy–momentum tensor (11) is represented as

T 00 = 1 2 χ ˙ 2 + V ˜ , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaK qbaoaaBaaajeaibaqcLbmacaaIWaGaaGimaaWcbeaajugibiaai2da juaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikdaaaGafq4Xdm MbaiaalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgUcaRKqb aoaaGaaakeaajugibiaadAfaaOGaay5adaqcLbsacaaISaaaaa@48FB@ (31)

T ii = 1 2 χ ˙ 2 V ˜ , i=1,2,3 (not summed), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaKqaGeaajugWaiaadMgacaWGPbaajeaibeaajugibiaai2da juaGdaWcaaGcbaqcLbsacaaIXaaakeaajugibiaaikdaaaGafq4Xdm MbaiaalmaaCaaajeaibeqaaKqzadGaaGOmaaaajugibiabgkHiTKqb aoaaGaaakeaajugibiaadAfaaOGaay5adaqcLbsacaaISaGaaGiiai aaiccacaaIGaGaamyAaiaai2dacaaIXaGaaGilaiaaikdacaaISaGa aG4maiaaiccacaaIOaGaaeOBaiaab+gacaqG0bGaaeiiaiaabohaca qG1bGaaeyBaiaab2gacaqGLbGaaeizaiaaiMcacaaISaaaaa@5C3C@ (32)

and other components are zero. This energy–momentum tensor can be interpreted as a density ( ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqyWdi haaa@383A@ ) and pressure ( p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiCaa aa@376F@ ) of a perfect fluid as T 00 =ρ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaKqaGeaajugWaiaaicdacaaIWaaajeaibeaajugibiaai2da cqaHbpGCaaa@3D8B@ and T ii =p MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamivaS WaaSbaaKqaGeaajugWaiaadMgacaWGPbaajeaibeaajugibiaai2da caWGWbaaaa@3D28@ , respectively.

Cosmic inflation due to the scaled scalar field

Friedmann equation

The Einstein equation (10) for perfect fluid with the conformal metric (13) can be expressed as follows:

3( H 2 +K)=+κ ρ, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaG4mai aaiIcacaWGibWcdaahaaqcbasabeaajugWaiaaikdaaaqcLbsacqGH RaWkcaWGlbGaaGykaiaai2dacqGHRaWkcqaH6oWAcqWIpecAcaaIGa GaeqyWdiNaaGilaaaa@458F@ (33)

2 H ˙ + H 2 +K=κ p, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGOmai qadIeagaGaaiabgUcaRiaadIealmaaCaaajeaibeqaaKqzadGaaGOm aaaajugibiabgUcaRiaadUeacaaI9aGaeyOeI0IaeqOUdSMaeS4dHG MaaGiiaiaadchacaaISaaaaa@4521@ (34)

where the curvature K MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaam4saa aa@374A@ is put buck in this subsection. These equations, refereed to as the Friedmann equations, can be rearranged as

H 2 = 2κ 3 ρK, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisaS WaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaaGypaKqbaoaalaaa keaajugibiaaikdacqaH6oWAcqWIpecAaOqaaKqzGeGaaG4maaaacq aHbpGCcqGHsislcaWGlbGaaGilaaaa@4535@ (35)

H ˙ = κ 3 ( ρ+3p ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmisay aacaGaaGypaiabgkHiTKqbaoaalaaakeaajugibiabeQ7aRjabl+qi ObGcbaqcLbsacaaIZaaaaKqbaoaabmaakeaajugibiabeg8aYjabgU caRiaaiodacaWGWbaakiaawIcacaGLPaaajugibiaai6caaaa@46C1@ (36)

When the density and pressure are provided from the SSF, the Friedmann equations have a form

H 2 =+ 2κ 3 ( 1 2 χ ˙ 2 + V ˜ )K, MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisaS WaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaaGypaiabgUcaRKqb aoaalaaakeaajugibiaaikdacqaH6oWAcqWIpecAaOqaaKqzGeGaaG 4maaaajuaGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGaaGymaaGcbaqc LbsacaaIYaaaaiqbeE8aJzaacaWcdaahaaqcbasabeaajugWaiaaik daaaqcLbsacqGHRaWkjuaGdaaiaaGcbaqcLbsacaWGwbaakiaawoWa aaGaayjkaiaawMcaaKqzGeGaeyOeI0Iaam4saiaaiYcaaaa@528E@ (37)

H ˙ = 2κ 3 ( χ ˙ 2 V ˜ ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGabmisay aacaGaaGypaiabgkHiTKqbaoaalaaakeaajugibiaaikdacqaH6oWA cqWIpecAaOqaaKqzGeGaaG4maaaajuaGdaqadaGcbaqcLbsacuaHhp WygaGaaSWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqzGeGaeyOeI0sc fa4aaacaaOqaaKqzGeGaamOvaaGccaGLdmaaaiaawIcacaGLPaaaju gibiaai6caaaa@4B6A@ (38)

If appropriate initial conditions are given, the history of the universe can be obtained by solving equations (30), (37) and (38), simultaneously. Further simplification is possible in this case as follows: A conformal–time derivative of (37) provides a equation

d H 2 dτ = 2κ 3 ( χ ¨ + V ˜ χ ) χ ˙ κ 3 d μ ˜ 2 dτ χ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgacaWGibqcfa4aaWbaaSqabKqaGeaajugWaiaaikda aaaakeaajugibiaadsgacqaHepaDaaGaaGypaKqbaoaalaaakeaaju gibiaaikdacqaH6oWAcqWIpecAaOqaaKqzGeGaaG4maaaajuaGdaqa daGcbaqcLbsacuaHhpWygaWaaiabgUcaRKqbaoaalaaakeaajugibi abgkGi2MqbaoaaGaaakeaajugibiaadAfaaOGaay5adaaabaqcLbsa cqGHciITcqaHhpWyaaaakiaawIcacaGLPaaajugibiqbeE8aJzaaca GaeyOeI0scfa4aaSaaaOqaaKqzGeGaeqOUdSgakeaajugibiaaioda cqWIpecAaaqcfa4aaSaaaOqaaKqzGeGaamizaKqbaoaaGaaakeaaju gibiabeY7aTbGccaGLdmaalmaaCaaajeaibeqaaKqzadGaaGOmaaaa aOqaaKqzGeGaamizaiabes8a0baacqaHhpWylmaaCaaajeaibeqaaK qzadGaaGOmaaaajugibiaaiYcaaaa@6D7F@

= 2κ 3 Δχ χ ˙ κ 3 d μ ˜ 2 dτ χ 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGypaK qbaoaalaaakeaajugibiaaikdacqaH6oWAcqWIpecAaOqaaKqzGeGa aG4maaaacqqHuoarcqaHhpWycuaHhpWygaGaaiabgkHiTKqbaoaala aakeaajugibiabeQ7aRbGcbaqcLbsacaaIZaGaeS4dHGgaaKqbaoaa laaakeaajugibiaadsgajuaGdaaiaaGcbaqcLbsacqaH8oqBaOGaay 5adaqcfa4aaWbaaSqabeaajugibiaaikdaaaaakeaajugibiaadsga cqaHepaDaaGaeq4Xdm2cdaahaaqcbasabeaajugWaiaaikdaaaqcLb sacaaISaaaaa@58EA@ (39)

where the Klein–Gordon equation (30) is used. Under the assumption that the SSF is the uniform and isotoropic, a spacial derivative are set to zero again as Δχ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiLdq Kaeq4XdmMaaGypaiaaicdaaaa@3B18@ . Therefore, the equation (39) can be expressed as

d H 2 dτ = κ 3 d μ ˜ 2 dτ χ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgacaWGibWcdaahaaqcbasabeaajugWaiaaikdaaaaa keaajugibiaadsgacqaHepaDaaGaaGypaiabgkHiTKqbaoaalaaake aajugibiabeQ7aRbGcbaqcLbsacaaIZaGaeS4dHGgaaKqbaoaalaaa keaajugibiaadsgajuaGdaaiaaGcbaqcLbsacqaH8oqBaOGaay5ada qcfa4aaWbaaSqabKqaGeaajugWaiaaikdaaaaakeaajugibiaadsga cqaHepaDaaGaeq4Xdm2cdaahaaqcbasabeaajugWaiaaikdaaaqcLb sacaaIUaaaaa@569A@ (40)

Due to definitions of μ ˜ 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafqiVd0 MbaGaalmaaCaaajeaibeqaaKqzadGaaGOmaaaaaaa@3A80@ in (29), its conformal time derivative is expressed as

d μ ˜ 2 dτ = 2 d dτ ( Ω ¨ Ω )+2Ω Ω ˙ μ 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgajuaGdaaiaaGcbaqcLbsacqaH8oqBaOGaay5adaWc daahaaqcbasabeaajugWaiaaikdaaaaakeaajugibiaadsgacqaHep aDaaGaaGypaiabl+qiOTWaaWbaaKqaGeqabaqcLbmacaaIYaaaaKqb aoaalaaakeaajugibiaadsgaaOqaaKqzGeGaamizaiabes8a0baaju aGdaqadaGcbaqcfa4aaSaaaOqaaKqzGeGafuyQdCLbamaaaOqaaKqz GeGaeuyQdCfaaaGccaGLOaGaayzkaaqcLbsacqGHRaWkcaaIYaGaeu yQdCLafuyQdCLbaiaacqaH8oqBlmaaCaaajeaibeqaaKqzadGaaGOm aaaajugibiaai6caaaa@5C85@ (41)

Under the assume that higher derivatives of the scale function Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC faaa@3808@ are much smaller than both the scale function itself and first derivative of that, the first term on the righthand side can be ignored compared with the second term. The validity of this assumption will be confirmed later in this section. In this case, the equation (40) can be approximated as

Ω ˙ Ω (τ)= κ 3 μ Ω(τ)χ(τ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiqbfM6axzaacaaakeaajugibiabfM6axbaacaaIOaGaeqiX dqNaaGykaiaai2dajuaGdaGcaaGcbaqcfa4aaSaaaOqaaKqzGeGaeq OUdSgakeaajugibiaaiodacqWIpecAaaaaleqaaKqzGeGaeqiVd0Ma aGiiaiabfM6axjaaiIcacqaHepaDcaaIPaGaeq4XdmMaaGikaiabes 8a0jaaiMcacaaISaaaaa@522A@ (42)

where H(τ)>0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisai aaiIcacqaHepaDcaaIPaGaaGOpaiaaicdaaaa@3BF3@ is assumed through any τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiXdq haaa@383F@ . This equation can be express using the proper time t MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiDaa aa@3773@ using a relation Ω(τ)dτ=dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiabes8a0jaaiMcacaWGKbGaeqiXdqNaaGypaiaadsgacaWG 0baaaa@4089@ as

dΩ(t) dt = κ 3 μ Ω(t)χ(t). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgacqqHPoWvcaaIOaGaamiDaiaaiMcaaOqaaKqzGeGa amizaiaadshaaaGaaGypaKqbaoaakaaakeaajuaGdaWcaaGcbaqcLb sacqaH6oWAaOqaaKqzGeGaaG4maiabl+qiObaaaSqabaqcLbsacqaH 8oqBcaaIGaGaeuyQdCLaaGikaiaadshacaaIPaGaeq4XdmMaaGikai aadshacaaIPaGaaGOlaaaa@50FC@ (43)

The second Friedmann equation (38) can be approximated under the same assumptions as

dΩ(t) dt = κ 3 [ Ω (t) 2 ( 2 ( dχ(t) dt ) 2 + ( μ χ(t) ) 2 ) λ 2 χ (t) 4 ] 1 2 . MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgacqqHPoWvcaaIOaGaamiDaiaaiMcaaOqaaKqzGeGa amizaiaadshaaaGaaGypaKqbaoaakaaakeaajuaGdaWcaaGcbaqcLb sacqaH6oWAaOqaaKqzGeGaaG4maaaaaSqabaqcfa4aamWaaOqaaKqz GeGaeuyQdCLaaGikaiaadshacaaIPaWcdaahaaqcbasabeaajugWai aaikdaaaqcfa4aaeWaaOqaaKqzGeGaaGOmaKqbaoaabmaakeaajuaG daWcaaGcbaqcLbsacaWGKbGaeq4XdmMaaGikaiaadshacaaIPaaake aajugibiaadsgacaWG0baaaaGccaGLOaGaayzkaaWcdaahaaqcbasa beaajugWaiaaikdaaaqcLbsacqGHRaWkjuaGdaqadaGcbaqcfa4aaS aaaOqaaKqzGeGaeqiVd0gakeaajugibiabl+qiObaacqaHhpWycaaI OaGaamiDaiaaiMcaaOGaayjkaiaawMcaaSWaaWbaaKqaGeqabaqcLb macaaIYaaaaaGccaGLOaGaayzkaaqcLbsacqGHsisljuaGdaWcaaGc baqcLbsacqaH7oaBaOqaaKqzGeGaaGOmaaaacqaHhpWycaaIOaGaam iDaiaaiMcalmaaCaaajeaibeqaaKqzadGaaGinaaaaaOGaay5waiaa w2faaSWaaWbaaKqaGeqabaWcdaWcaaqcbasaaKqzadGaaGymaaqcba saaKqzadGaaGOmaaaaaaqcLbsacaaIUaaaaa@7EA2@ (44)

Above two Friemann equations (43) and (44) must be consistent each other within the approximation. It leads a following differential equation

χ ˙ (τ)=± λ 2 χ (τ) 2 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGafq4Xdm MbaiaacaaIOaGaeqiXdqNaaGykaiaai2dacqGHXcqSjuaGdaWcaaGc baqcfa4aaOaaaOqaaKqzGeGaeq4UdWgaleqaaaGcbaqcLbsacaaIYa aaaiabeE8aJjaaiIcacqaHepaDcaaIPaWcdaahaaqcbasabeaajugW aiaaikdaaaqcLbsacaaISaaaaa@4B73@ (45)

with respect to the conformal time τ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiXdq haaa@383F@ . This equation can be solved as

χ(τ)= ( λ 2 τ+ 1 χ 0 ) 1 , MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm MaaGikaiabes8a0jaaiMcacaaI9aqcfa4aaeWaaOqaaKqzGeGaeS4e I0wcfa4aaSaaaOqaaKqbaoaakaaakeaajugibiabeU7aSbWcbeaaaO qaaKqzGeGaaGOmaaaacqaHepaDcqGHRaWkjuaGdaWcaaGcbaqcLbsa caaIXaaakeaajugibiabeE8aJLqbaoaaBaaajqwaG9FaaKqzadGaaG imaaWcbeaaaaaakiaawIcacaGLPaaalmaaCaaajqwaG9FabeaajugW aiabgkHiTiaaigdaaaqcLbsacaaISaaaaa@566F@ (46)

where χ 0 =χ(τ=0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm wcfa4aaSbaaKqaGeaajugWaiaaicdaaSqabaqcLbsacaaI9aGaeq4X dmMaaGikaiabes8a0jaai2dacaaIWaGaaGykaaaa@42B5@ . This solution does not give any oscillating fields because of the requirement Δχ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuiLdq Kaeq4XdmMaaGypaiaaicdaaaa@3B18@ .

Numerical calculations

Next the scaling function for the cosmic inflation era is treated in this section. During the cosmic inflation, a condition dΩ/dtdχ/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai abfM6axjaai+cacaWGKbGaamiDaebbfv3ySLgzGueE0jxyaGqbaiab =TMi=iaadsgacqaHhpWycaaIVaGaamizaiaadshaaaa@46B4@ is expected naturally. Due to the definition of the SSF, this condition can be satisfied if φ1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqOXdO weeuuDJXwAKbsr4rNCHbacfaGae8NAI0JaaGymaaaa@3EDC@ and dφ/dt(dΩ/dt)/Ω MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai abeA8aQjaai+cacaWGKbGaamiDaebbfv3ySLgzGueE0jxyaGqbaiab =PMi9iaaiIcacaWGKbGaeuyQdCLaaG4laiaadsgacaWG0bGaaGykai aai+cacqqHPoWvaaa@4A63@ during the cosmic inflation. A validity of this assumption will be discussed later in this section. The Friedmann equation (43) can be solved under the assumption as;

Ω(t)= Ω 0 exp( H 0 t ), MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiaadshacaaIPaGaaGypaiabfM6axTWaaSbaaKqaGeaajugW aiaaicdaaKqaGeqaaKqzGeGaciyzaiaacIhacaGGWbqcfa4aaeWaaO qaaKqzGeGaamisaSWaaSbaaKqaGeaajugWaiaaicdaaKqaGeqaaKqz GeGaamiDaaGccaGLOaGaayzkaaqcLbsacaaISaaaaa@4B49@ (47)

where H 0 = κ/3 μ χ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisaS WaaSbaaKqaGeaajugWaiaaicdaaKqaGeqaaKqzGeGaaGypaKqbaoaa kaaakeaajugibiabeQ7aRjaai+cacaaIZaGaeS4dHGgaleqaaKqzGe GaaGiiaiabeY7aTjabeE8aJTWaaSbaaKqaGeaajugWaiaaicdaaKqa Geqaaaaa@47A6@ with an initial condition of Ω(t=0)= Ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiaadshacaaI9aGaaGimaiaaiMcacaaI9aGaeuyQdCvcfa4a aSbaaKqaGeaajugWaiaaicdaaSqabaaaaa@4108@ . Due to the assumption above, the SSF stays constant at χ(0<t< t e )= χ 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm MaaGikaiaaicdacaaI8aGaamiDaiaaiYdacaWG0bqcfa4aaSbaaKqa GeaajugWaiaadwgaaSqabaqcLbsacaaIPaGaaGypaiabeE8aJTWaaS baaKqaGeaajugWaiaaicdaaKqaGeqaaaaa@463F@ during the inflation. An inflation starting time is set to t=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiDai aai2dacaaIWaaaaa@38F4@ . An initial value of the SSF may be given by a quantum fluctuation of the field, which is naturally expected to be an order unity. Here the initial value χ 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm 2cdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacaaI9aGaaGym aaaa@3CAA@ in the Planck units is assumed. On the other hand, the Higgs potential parameters, μ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeqiVd0 gaaa@3830@ and λ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW gaaa@382E@ , at the beginning of the universe are set to be the same as the current universe at the tree level for simplicity. From the recent measurement of the Higgs mass,31 the quadratic term can be obtained as Ωμ/=90 GeV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaeqiVd0MaaG4laiabl+qiOjaai2dacaaI5aGaaGimaabaaaaaaaaa peGaaiiOaiaadEeacaWGLbGaamOvaaaa@41B9@ and λ=0.134 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4UdW MaaGypaiaaicdacaaIUaGaaGymaiaaiodacaaI0aaaaa@3C9D@ . Therefore, the initial value H o MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisaS WaaSbaaKqaGeaajugWaiaad+gaaKqaGeqaaaaa@39E9@ is obtained in the Planck units as H 0 =90/( 3 m p ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisaS WaaSbaaKqaGeaajugWaiaaicdaaKqaGeqaaKqzGeGaaGypaiaaiMda caaIWaGaaG4laiaaiIcajuaGdaGcaaGcbaqcLbsacaaIZaaaleqaaK qzGeGaamyBaKqbaoaaBaaajeaibaqcLbmacaWGWbaaleqaaKqzGeGa aGykaaaa@45B6@ , where m p =1.22× 10 19  GeV MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamyBaK qbaoaaBaaajeaibaqcLbsacaWGWbaaleqaaKqzGeGaaGypaiaaigda caaIUaGaaGOmaiaaikdacqGHxdaTcaaIXaGaaGimaSWaaWbaaKqaGe qajqwaG9FaaKqzadGaaGymaiaaiMdaaaqcLbsaqaaaaaaaaaWdbiaa cckacaWGhbGaamyzaiaadAfaaaa@4AD5@ is the Planck mass in the natural units. In order to make the inflation scenario working well, the e–folding number N e MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOtaK qbaoaaBaaajeaibaqcLbmacaWGLbaaleqaaaaa@3A49@ defined as exp( N e )=Ω( t e )/ Ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaciyzai aacIhacaGGWbGaaGikaiaad6ealmaaBaaajeaibaqcLbmacaWGLbaa jeaibeaajugibiaaiMcacaaI9aGaeuyQdCLaaGikaiaadshajuaGda WgaaqcbasaaKqzadGaamyzaaWcbeaajugibiaaiMcacaaIVaGaeuyQ dC1cdaWgaaqcbasaaKqzadGaaGimaaqcbasabaaaaa@4BA1@ must be greater than or around 70. When the normalization Ω( t e )=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiaadshalmaaBaaajeaibaqcLbmacaWGLbaajeaibeaajugi biaaiMcacaaI9aGaaGymaaaa@3F0F@ and the e–folding number N e =70 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOtaS WaaSbaaKqaGeaajugWaiaadwgaaKqaGeqaaKqzGeGaaGypaiaaiEda caaIWaaaaa@3CB6@ are required, the initial value of the scaling function can be obtained as Ω 0 = e 70 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC 1cdaWgaaqcbasaaKqzadGaaGimaaqcbasabaqcLbsacaaI9aGaamyz aSWaaWbaaKqaGeqabaqcLbmacqGHsislcaaI3aGaaGimaaaaaaa@409D@ . The inflation termination–time can be obtained by solving the Ω( t e )=1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC LaaGikaiaadshalmaaBaaajeaibaqcLbmacaWGLbaajeaibeaajugi biaaiMcacaaI9aGaaGymaaaa@3F0F@ such as t e =0.89× 10 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamiDaS WaaSbaaKqaGeaajugWaiaadwgaaKqaGeqaaKqzGeGaaGypaiaaicda caaIUaGaaGioaiaaiMdacqGHxdaTcaaIXaGaaGimaSWaaWbaaKqaGe qabaqcLbmacqGHsislcaaIYaGaaGinaaaaaaa@45D0@ second. In our scenario, a speed of changing a vacuum expectation value was very high, and thus, the SSF vacuum is staying at the origin V(χ(t=0))=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamOvai aaiIcacqaHhpWycaaIOaGaamiDaiaai2dacaaIWaGaaGykaiaaiMca caaI9aGaaGimaaaa@3FD1@ for a short period. Then, delayed explosion of the SSF field χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm gaaa@3831@ caught the vacuum expectation value up and the inflation was terminated. Therefore, the electroweak phase–transition must be the second kind.

The evolution of the scaling function can be fixed completely using above parameters under the approximations. Before a starting time of the inflation, the first term of (29) was much smaller than the second term. At that period, the potential energy of the SSF had a minimum at χ=0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeq4Xdm MaaGypaiaaicdaaaa@39B2@ as shown in Figure 1 (t=0) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aadshacaaI9aGaaGimaiaaiMcaaaa@3A59@ . During the inflation, the SSF had a finite vacuum expectation value (Figure 1 (0<t< t e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aaicdacaaI8aGaamiDaiaaiYdacaWG0bqcfa4aaSbaaKqaGeaajugW aiaadwgaaSqabaqcLbsacaaIPaaaaa@3FA2@ ). At the end of the inflation, the vacuum expectation value arrived at the same value as that in the current universe (Figure 1 (t= t e ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aadshacaaI9aGaamiDaSWaaSbaaKqaGeaajugWaiaadwgaaKqaGeqa aKqzGeGaaGykaaaa@3DBF@ ). Although the vacuum expectation value stayed almost constant at the beginning inflation, it grew very rapidly after the second term of (29) became dominant in the potential energy as shown in Figure 2. The inflation was terminated when the SSF arrived at the vacuum expectation value, and it fixed the e–folding number. A precise value of the inflation ending time must be evaluated by solving equations (43) and (45) (or equivalently (46)), simultaneously. We note that the relation between the conformal and proper times can be fixed only after the solution of the scaling function is obtained. One cannot solve equations analytically without the approximation that the SFF is constant during the inflation. When the inflation duration changed ±1% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyySae RaaGymaiaaiwcaaaa@39D2@ around its nominal value, the e–folding number varied about ±2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeyySae RaaGOmaaaa@3924@ as shown in Figure 3. On the other hand, the same variation for the duration of the inflation affects the vacuum expectation value about 50% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGynai aaicdacaaILaaaaa@38A2@ to 500% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGynai aaicdacaaIWaGaaGyjaaaa@395C@ , as shown in Figure 2. A tolerance of the inflation duration is rather narrow to realize a current observed value of the vacuum expectation value.

Figure 1 The potential energy of the SSF before the inflation (solid line), during the inflation (dotted line), and at the end of the inflation (dashed line).

Figure 2 The potential–energy term of the SSF normalized using the SM value in the current universe. If we require that the cosmic inflation terminated when the potential–energy term of the SSF arrived at the current value ( V 0 = V 0 ( SM )=1 ) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMipgYlb91rFfpec8Eeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aabmaabaqcLbsaqaaaaaaaaaWdbiaadAfakmaaBaaajeaibaqcLbma caaIWaaaleqaaKqzGeGaeyypa0JaamOvaSWaaSbaaKqaGeaajugWai aaicdaaKqaGeqaaOWdamaabmaabaqcLbsapeGaam4uaiaad2eaaOWd aiaawIcacaGLPaaajugib8qacqGH9aqpcaaIXaaak8aacaGLOaGaay zkaaaaaa@4B3C@ , a duration of the in ation is 0.89× 10 24 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMipgYlb91rFfpec8Eeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaK qzGeGaaGimaiaac6cacaaI4aGaaGyoaiabgEna0kaaigdacaaIWaWc daahaaqcbasabeaajugWaiabgkHiTiaaikdacaaI0aaaaaaa@4584@ second.

Figure 3 The e–folding number different from nominal value N e 70 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMipgYlb91rFfpec8Eeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaK qzGeaeaaaaaaaaa8qacaWGobGcpaWaaSbaaKqaGeaajugWa8qacaWG Lbaal8aabeaajugib8qacqGHsislcaaI3aGaaGimaaaa@41BB@ is shown with respect to the duration time of the ination. When the ination duration changed ±1% MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMipgYlb91rFfpec8Eeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaK qzGeaeaaaaaaaaa8qacqGHXcqScaaIXaGaaiyjaaaa@3E8D@ around its nominal value, the e–folding number varied about ±2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMipgYlb91rFfpec8Eeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaaK qzGeaeaaaaaaaaa8qacqGHXcqScaaIYaaaaa@3DE5@ .

The solution (47) is obtained under the assumption of dΩ/dtdχ/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamizai abfM6axjaai+cacaWGKbGaamiDaebbfv3ySLgzGueE0jxyaGqbaiab =TMi=iaadsgacqaHhpWycaaIVaGaamizaiaadshaaaa@46B4@ . The validity of this assumption during the cosmic inflation must be confirmed. If the time dependence of the SSF is put back in the solution as H 0 H(t)= κ/3 μχ(t) MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaamisaS WaaSbaaKqaGeaajugWaiaaicdaaKqaGeqaaKqzGeGaeyOKH4Qaamis aiaaiIcacaWG0bGaaGykaiaai2dajuaGdaGcaaGcbaqcLbsacqaH6o WAcaaIVaGaaG4maiabl+qiObWcbeaajugibiaaiccacqaH8oqBcqaH hpWycaaIOaGaamiDaiaaiMcaaaa@4CB4@ , the time derivative of the scaling function becomes

dΩ(t) dt = Ω 0 H(t)( 1+ 1 χ(t) dχ(t) dt t )exp( H(t)t ). MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqbaoaalaaake aajugibiaadsgacqqHPoWvcaaIOaGaamiDaiaaiMcaaOqaaKqzGeGa amizaiaadshaaaGaaGypaiabfM6axTWaaSbaaKqaGeaajugWaiaaic daaKqaGeqaaKqzGeGaamisaiaaiIcacaWG0bGaaGykaKqbaoaabmaa keaajugibiaaigdacqGHRaWkjuaGdaWcaaGcbaqcLbsacaaIXaaake aajugibiabeE8aJjaaiIcacaWG0bGaaGykaaaajuaGdaWcaaGcbaqc LbsacaWGKbGaeq4XdmMaaGikaiaadshacaaIPaaakeaajugibiaads gacaWG0baaaiaadshaaOGaayjkaiaawMcaaKqzGeGaciyzaiaacIha caGGWbqcfa4aaeWaaOqaaKqzGeGaamisaiaaiIcacaWG0bGaaGykai aadshaaOGaayjkaiaawMcaaKqzGeGaaGOlaaaa@6718@ (48)

In calculations for the cosmic inflation, the time derivative term in the right hand side is neglected. The validity of this approximation can be examined using the equation of motion. The time evolution of the SSF is governed by the equation (46) with respect to the conformal time. For a conversion from the conformal time to the proper time, the approximated solution of (47) is used for a numerical calculation. A numerical result of a time evolution of the tern (dχ/dt)t/χ MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaaGikai aadsgacqaHhpWycaaIVaGaamizaiaadshacaaIPaGaamiDaiaai+ca cqaHhpWyaaa@4083@ is shown in Figure 4. It is shown that that term is less than unity during the cosmic inflation, and thus the effect on the result is a factor of two on Ω 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVCI8FfYJH8YrFfeuY=Hhbbf9v8qqaqFr0xc9pk0xbb a9q8WqFfeaY=biLkVcLq=JHqpepeea0=as0Fb9pgeaYRXxe9vr0=vr 0=vqpWqaaeaabiGaciaacaqabeaadaqaaqaaaOqaaKqzGeGaeuyQdC 1cdaWgaaqcbasaaKqzadGaaGimaaqcbasabaaaaa@3A70@ at most.

Figure 4 A term ( t/χ )dχ/dt MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbdfwBIj xAHbqedmvETj2BSbqefm0B1jxALjhiov2DaerbuLwBLnhiov2DGi1B TfMBaebbnrfifHhDYfgasaacPi=BMipgYlb91rFfpec8Eeeu0xXdbb a9frFj0=OqFfea0dXdd9vqaq=JfrVkFHe9pgea0dXdar=Jb9hs0dXd bPYxe9vr0=vr0=vqpWqaaeaabiGaciaacaqabeaadaabauaaaOqaam aabmaabaqcLbsaqaaaaaaaaaWdbiaadshacaGGVaGaeq4Xdmgak8aa caGLOaGaayzkaaqcLbsapeGaamizaiabeE8aJjaac+cacaWGKbGaam iDaaaa@4614@ is shown with respect to the inflation duration. The approximation used in this report requires a value must be small compare with unity.

Summary

We propose a novel method to treat the electroweak symmetry braking, which is named the scaled scalar–field method. In this method, a conformal metric is employed and the Higgs field is scaled owing to the conformal function. In consequence, a mass parameter of the Higgs field (a quadratic term of the Higgs potential) has the time dependence through the conformal function, and it causes the phase transition. Quantization of the Higgs field is induced associated with the canonical quantization of general relativity. In the context of the scaled field, only the vierbein is quantized owing to the quantum commutation relations with keeping the scalar field classical.
The cosmic inflation and the electroweak phase–transition are investigated in a framework of the scaled field. The Friedmann equations and their appropriate approximations are provided using the scaled field method. The Friedmann equations are numerically solved and an example of a possible solution to match with the cosmic inflation scenario is shown. The electroweak phase–transition induced by the scaled field is the second kind, and thus, the fine–tuning problem is still exists.

Acknowledgements

I appreciate the kind hospitality of all members of the theory group of Nikhef, particularly Prof. J. Vermaseren and Prof. E. Laenen. A major part of this study has been conducted during my stay at Nikhef in 2017. In addition, I would like to thank Dr. Y. Sugiyama for his continuous encouragement and fruitful discussion.

Conflict of interest

Author declares there is no conflict of interest.

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