Ex) Article Title, Author, Keywords
Ex) Article Title, Author, Keywords
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.
Seoktae Koh*
Department of Science Education, Jeju National University, Jeju 63243, Korea
Correspondence to:*kundol.koh@jejunu.ac.kr
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
where
the quadratic action for
where the prime denotes the derivative with respect to the conformal time η and we have used the mode decomposition of
where
Varying (2) with respect to
where we have ignored the polarization superscript λ. This equation reminds us the equation of motion of a massless scalar field. If we introduce
This takes the form of the time-dependent harmonic oscillator with a time-dependent frequency
where
Multiplying the integration factor
or
then we obtain the invariant quantities (Ermakov-Lewis invariant) by integrating
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
Transforming to a new variable by introducing an auxiliary function
the Eq. (5) is transformed as
with ρ satisfying Eq. (6), where we have used
Because the solution to (12) is given by
the solution to the time-dependent harmonic oscillator becomes
Once we get the solution to (6), then we can obtain the solution to (5) from (15).
The Hamiltonian for
where
In the original variable, the Hamiltonian (16) becomes
and this is the Ermakov-Lewis invariant (10) which is conserved
where
This implies the conservation of energy in the transformed coordinate space.
The solutions to (6) is given by[10, 5]
where
As an example, we consider the de Sitter space with
then (5) can be solved exactly[11]
where
With the solution (23), the solution of the auxiliary equation (6) from (20) leads to
where
If we choose Ω as (Ω is an arbitrary constant)
then we get
With (23) and (25), we can check
the Ermakov-Lewis invariant
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
where we have omitted the subscript k of
where S[z] is the Schwarzian derivative
With the similar transformation
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
i.e.
The Schwarz equation (31) has a solution of the quotient form
where
Since S[z] is invariant under the transformation (34), it can be proved that u”/u is also invariant under the transformation (30)[12]
which leads to[12]
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,
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.
This research was supported by the 2024 scientific promotion program funded by Jeju National University.