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New Phys.: Sae Mulli 2025; 75: 103-107

Published online January 31, 2025 https://doi.org/10.3938/NPSM.75.103

Copyright © New Physics: Sae Mulli.

Primordial Gravitational Wave in de Sitter Space and the Ermakov-Lewis Invariant

Seoktae Koh*

Department of Science Education, Jeju National University, Jeju 63243, Korea

Correspondence to:*kundol.koh@jejunu.ac.kr

Received: October 9, 2024; Accepted: November 11, 2024

This is an Open Access article distributed under the terms of the Creative Commons Attribution Non-Commercial License(http://creativecommons.org/licenses/by-nc/4.0) which permits unrestricted non-commercial use, distribution, and reproduction in any medium, provided the original work is properly cited.

The governing equation of primordial gravitational waves in de Sitter space takes the form of a harmonic oscillator with a time-dependent frequency. The Ermakov-Lewis invariant of this time-dependent harmonic oscillator is obtained using the mode solutions of the primordial gravitational wave and an auxiliary equation, which is dual to the mode equations, in de Sitter space. Additionally, the dual symmetry of the mode functions of the primordial gravitational wave is briefly mentioned by employing the invariance of the Schwarzian derivative.

Keywords: Time-dependent harmonic oscillator, Ermakov-Lewis invariant, Schwarzian derivative, Primordial gravitational wave

It is widely accepted that an accelerating phase in the early Universe is crucial for understanding the history of the Universe and explaining cosmic background radiation (CMB) anisotropy, large-scale structure (LSS) and primordial gravitational waves.

The CMB anisotropy, LSS formation and the primordial gravitational wave are known to have been seeded by the quantum fluctuations in an inflationary period.

The governing equations, commonly referred to as the Sasaki-Mukhanov equations[1, 2], which are responsible for the CMB, LSS, or primordial gravitational waves are presented in the form of a time-dependent harmonic oscillator (TDHO). The TDHO appears quite often in the branch of physics for example in optics, black hole physics and cosmology. The dynamics of a scalar field in an expanding background can also be described by a damped harmonic oscillator or a Caldirola-Kanai harmonic oscillator[3, 4].

We use the Ermakov-Lewis invariant approach to study a TDHO problems in this work. The time-dependent equations of motion can be transformed into the time-independent equations by introducing an auxiliary field satisfying the Penny equation[5]. The Ermakov-Lewis invariant method has been used to define an invariant vacuum for the quantum fluctuations in inflation[6].

In Ermakov systems for the TDHO, there exists an invariant quantity whose physical meaning is proposed to be a conserved energy in a time-independent harmonic oscillator systems[7] or a conservation of an angular momentum[8]. This invariant quantity can provide useful tools for constructing the theoretical models of inflation. We can trace the underlying symmetries of the time-dependent harmonic oscillator problems from the Schwarz equation, which can be obtained from the Ermakov systems using the nonlocal transformations. The Schwarzian derivative is known to be invariant under the SL(2,R) transformation.

In this study, we present a solution to the auxiliary equations with the mode solutions of the primordial gravitational wave in de Sitter space. We then show that these solutions yield the constancy of the Ermakov-Lewis invariant. Finally, we provide a brief overview of the Schwarz equation through the nonlocal transformation from the Ermakov systems. The invariance of the Schwarz derivative under the SL(2,R) transformation provides the dual symmetry of the mode functions of the primordial gravitational wave[9].

With a linearised Friedmann-Lemaitre-Robertson-Walker metric for the tensor modes

ds2=a2(η)[-dη2+(δij+hij)dxidxj,

where hij is a traceless and transverse tensor perturbation,

the quadratic action for hij for the Einstein-Hilbert action in de Sitter space is given by

δ2S=Mp28dηd3xa2[hijhij-(khij)(khij)],

where the prime denotes the derivative with respect to the conformal time η and we have used the mode decomposition of hij as

hij(x,η)=λ=+,×d3k(2π)3/2εijλ(k)hkλ(η)eik·x,

where εijλ is a polarization tensor and Mp2=1/8πG is the reduced Planck mass.

Varying (2) with respect to hij leads to the equation of motion for each polarization mode of the gravitational wave

hk+2aahk+k2hk=0,

where we have ignored the polarization superscript λ. This equation reminds us the equation of motion of a massless scalar field. If we introduce uk=hka, the equation of uk becomes

uk+k2-aauk=0.

This takes the form of the time-dependent harmonic oscillator with a time-dependent frequency ω(η)=k2-aa. This equation consists of Ermakov systems with the auxiliary equation

ρk+ω2(η)ρk=Ω2ρk3,

where Ω2 is an arbitrary constant. Multiplying ρk to (5) and uk to (6) and then subtracting (6) from (5) yields

ρkuk-ukρk=ddη(ρkuk-ρkuk)=-Ω2ukρk3.

Multiplying the integration factor (ρkuk-ρkuk) on the both sides, we have

(ρkuk-ukρk)ddη(ρkuk-ukρk)=-Ω2ukρk3(ρkuk-ukρk),

or

12ddη(ρkuk-ukρk)2=-12Ω2ddη(uk/ρk)2

then we obtain the invariant quantities (Ermakov-Lewis invariant) by integrating

I=12(ρkuk-ukρk)2+Ω2ukρk2.

The physical meaning of this invariant is suggested in several literature; conservation of an angular momentum[8] , conservation of energy[7] and so on.

In the next section we will briefly introduce how to solve the time-dependent harmonic oscillator using the Ermakov-Lewis invariant method.

Equation (5) describes the time-dependent harmonic oscillator with a time-dependent frequency ω2(η)=k2-a/a and with a unit mass. Using the Ermakov-Lewis invariant method that will be described here shortly, we can obtain the solution to the time dependent harmonic oscillator.

Transforming to a new variable by introducing an auxiliary function ρ(η)

vk=uk/ρk(η),

the Eq. (5) is transformed as

d2vkdτ2+Ω2vk=0,

with ρ satisfying Eq. (6), where we have used

τ(η)=η1ρk2(η)dη.

Because the solution to (12) is given by

vk=e±iΩτ,

the solution to the time-dependent harmonic oscillator becomes

uk(η)=ρk(η)e±iΩ1ρk2dη.

Once we get the solution to (6), then we can obtain the solution to (5) from (15).

The Hamiltonian for vk is given by

Hk(v,πτ)=12πk2+12Ω2vk2const.,

where πk is a conjugate momentum to v

πk=vkτ=(ρkuk-ρkuk).

In the original variable, the Hamiltonian (16) becomes

Ik=Hk(v,πτ)=12(ρkuk-ρkuk)2+Ω2ukρk2,

and this is the Ermakov-Lewis invariant (10) which is conserved

dIkdt=Ikt+{Ik,Hk}=0

where {,} is a Poisson bracket for the classical systems and can be thought of as a Dirac bracket for the quantum systems.

This implies the conservation of energy in the transformed coordinate space.

The solutions to (6) is given by[10, 5]

ρ=aψ2+bφ2+2cψφ,

where ab-c2=Ω2W2 and ψ and φ are two linearly independent solutions of (5) and W is a Wronskian of ψ and φ. If we choose c=0 and a=1, then we can set to b=Ω2/W2, and then (20) can be written as

ρ=ψ2+Ω2W2φ21/2.

As an example, we consider the de Sitter space with a=eHt/H with a constant H, where t is a physical time related to the conformal time as dt=a(η)dη, and we have

aa=2η2,η=-1He-Ht=-1aH,

then (5) can be solved exactly[11]

uk(η)=|η|CkJ3/2(k|η|)+DkY3/2(k|η|),

where J3/2(x) and Y3/2(x) are the Bessel and Neumann functoin with the order 3/2, respectively. The Wronskian of the two independent solutions of (23) are

W=u1u2-u1u2=2π.

With the solution (23), the solution of the auxiliary equation (6) from (20) leads to

ρk=[2kπsin(x)x-cos(x)2+πΩ22ksin(x)+cos(x)x2]1/2,

where x=k|η| and we have used

J3/2(x)=2πx-cos(x)+sin(x)x,Y3/2(x)=-2πxcos(x)x+sin(x).

If we choose Ω as (Ω is an arbitrary constant)

Ω2=W2=4π2,

then we get

ρk=2πk1+1k2|η|21/2.

With (23) and (25), we can check

the Ermakov-Lewis invariant Ik (10) becomes constant,

Ik=12α2Ω2+2β2π2.

In this section we will briefly mention the underlying symmetries of the time-dependent harmonic oscillator. The Ermakov systems are transformed by the nonlocal transformation into the Schwarz equation. When we perform the nonlocal transformation

z=u-2,

where we have omitted the subscript k of uk in this section, then (5) is transformed into the third-order differential equations and it turns out that the Schwarz equation

S[z]=2ω2(η),

where S[z] is the Schwarzian derivative

S[z]=zz-32zz2=zz-12zz2.

With the similar transformation ξ=ρ-2, (6) becomes

S[ξ]=-2Ω2ξ2+2ω2,

which is a variant of the Schwarz equation.

The Schwarzian derivative S[z] in (32) is known to be invariant under the SL(2,R) transformations

z˜=az+bcz+d,ad-bc0,

i.e. S[z˜]=S[z].

The Schwarz equation (31) has a solution of the quotient form

z=φ(t)ψ(t),

where φ(t) and ψ(t) are the two linearly independent solution of (5).

Since S[z] is invariant under the transformation (34), it can be proved that u”/u is also invariant under the transformation (30)[12]

u˜u˜=-12S[z˜]=-12S[z]=uu,

which leads to[12]

u˜(η)=C1u(η)+C2u(η)ηdηu2(η).

These properties of dual symmetry have been employed in the investigation of the scale invariance[9] and the enhancement of the power spectrum[13].

We have considered the primordial gravitational wave in de Sitter space and found that the governing equation, the Sasaki-Mukhanov equation, of the primordial gravitational waves takes the form of the harmonic oscillator with a time-dependent frequency. The Ermakov-Lewis invariant method is used to study the TDHO problems in de Sitter space. With the solutions to the gravitational wave modes, uk and the auxiliary field, ρk, we have shown that the Ermakov-Lewis invariant becomes constant and then is conserved.

The nonlocal transformation of the variables transforms the gravitational wave mode equations and the auxiliary field equation into third-order differential equations, known as the Schwarz equation. The Schwarz derivative is known to have a symmetry under the SL(2,R) transformation. This dual symmetry allows to study the scale invariance as well as the enhancement of the power spectrum in an early phase of the universe.

Although we have focused on the primordial gravitational wave for this work, we can extend it to the scalar mode equations responsible for the large-scale structures and the cosmic background radiation anisotropy. It would be interesting to study the role of the Ermakov-Lewis invariant and the symmetric properties of the Schwarzian derivative to constrain the inflationary model or to provide additional observables to help understand the evolution of our universe.

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