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A Harmonic Oscillator Universe
Jinn-Ouk Gong
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DOI: 10.22661/AAPPSBL.2018.28.6.49

A Harmonic Oscillator Universe

JINN-OUK GONG
KOREA ASTRONOMY AND SPACE SCIENCE INSTITUTE, DAEJEON 34055, KOREA

ABSTRACT

We briefly review the quantum mechanical aspects of cosmological perturbations. As the leading parts of the pertubations can be described as quantum harmonic oscillators, the standard wisdom of quantum mechanics can be directly applied. In addition, time-dependent background brings interesting consequences relevant for the generation of cosmological structure accessible by cosmological observations.

INTRODUCTION

One of the most splendid triumphs of scientific knowledge is that we are able to understand some aspects of the Universe that we reside in quantitatively; this has been one of the oldest wishes of humanity, present ever since we looked up into the sky and wonderd about the night. We know the age of the Universe with sub-percent level accuracy; we can estimate which kind of energy is occupying how much of a fraction in the universe; and we understand how the Universe has been evolving throughout its history [1]. These advances in our understanding of the Universe are greatly indebted to developments in both experimental and theoretical physics: with sensitive telescopes on the ground and in space we can amass a huge amount of precise data on the luminosity, spectrum and position of various cosmological observables such as stars, galaxies, clusters of galaxies and the cosmic microwave background. Modern physics, including quantum mechanics and general relativity, allows us to interpret such data precisely, which can eventually lead to deeper understanding of the Universe. We now know more about the Universe than before.

However, this does not mean our ultimate qoal for understanding the Universe has been fulfilled. Rather, because we understand the Universe better, the remaining problems are even more prominent. How did our Universe begin? How is it possible to describe the Big Bang? What are dark matter and dark energy? What will be the fate of our universe? Such questions are related to the physics beyond our current understanding, like the quantum theory of gravity, and will keep driving us to search for fundamental laws of physics. The quest is still ongoing, and we have gained fruitful lessons during the last decades.

One lesson is that before the onset of the standard hot Big Bang evolution, the Universe is likely to have experienced a period of accelerated expansion, called inflation, during which the Universe has expanded more than e60 1026 times [2]. Due to such tremendous expansion, the subsequent universe at the beginning of the hot Big Bang evolution stage was extremely homogeneous and isotropic, as is confirmed by the cosmic microwave background. More importantly, during inflation, transient quantum fluctuations in space-time and matter contents are stretched to cosmological distance scales, and become the seed of subsequent observable cosmic structures in the Universe [3]. The oldest signature that we can observe at the moment - the temperature fluctuations in the cosmic microwave background - have been measured over decades with ever-increasing accuracy, and the inflationary primordial perturbations are consistent with all observations until now [4].

Thus, cosmological perturbations during inflation provide an important path toward understanding the initial conditions and subsequent evolution of the Universe. Further, by creating access to more accurate future cosmological observations, we hope to test the yet unknown physics beyond the standard model of particle physics in which inflation would be embedded. This, at present, seems to be one of few opportunities for particle physics given that a groundbreaking, novel discovery at the Large Hadron Collider is unlikely. A striking feature of cosmological perturbations is that while their description is surprisingly simple, the consequences of such simple description are surprisingly surprising. After briefly summarizing a description of observationally relevant cosmological perturbations, we will examine, concisely, their two most notable consequences.

Primordial perturbations constrained by observations
To describe the evolution of the Universe, we resort to general relativity. From the observations on the cosmic microwave background, we assume that the Universe is at the background level homogeneous and isotropic. The Friedmann-Robertson-Walker metric that corresponds to such a universe is as follows:

(1)

The last expression is written without any perturbation. We can include perturbations in the above metric tensor that can be classified according to their spatial transformation: as there are 2 spatial indices, we can include rank-0 (scalar), rank-1 (vector) and rank-2 tensor perturbations. With perturbations, explicitly splitting different types, we have

(2)

The metric tensor describes the geometric part of the Einstein equation, which can be derived from the Einstein-Hilbert action:

(3)

where mPl=(8πG)-1/2 2.435횞1018 GeV is the reduced Planck mass and R is the Ricci scalar. So we need the counterpart of the equation; i.e., the matter contents that resides in the Universe. There exist various types of matter, such as photons and relativistic species, non-relativistic ordinary matter that in cosmology is called (though incorrectly) baryons and other pressureless matter, the cosmological constant that was originally introduced by Einstein himself to achieve a static universe but is now supposed to be the driving force of the current acceleration as observed from the supernovae, and so on. For inflation, a simple choice is to introduce a hypothetical scalar field called inflaton 𝜙. If single-component and canonical, the action is written as

(4)

where V(𝜙) is the potential of the scalar field. We can include the field fluctuation by splitting the background part 𝜙0 as 𝜙= 𝜙0+δ𝜙. Note that then there are 4 scalar perturbations (A, B, 𝜑 and ) in the metric and 1 in the matter sector (δ𝜙), 2 vector perturbations in the metric (Si and Fi) and 1 tensor perturbation (hij).

Given the action, it is straightforward to the relevant action of the perturbations of our interest. But before we proceed, we should note that there are more variables than physically meaningful, viz. some perturbations are physically irrelevant. This is because - as is more obvious in the 1+3 decomposition known as the Arnowitt-Deser-Misner formulation [5] - there are two constraint equations that should always be valid, and further because the general coordinate transformation xx+ξ with ξ some infinitesimal 4-vector is allowed without changing physics. Thus, we can eliminate 4 out of 5 scalar perturbations, leaving only 1 physically meaningful one. While solving the constraints is straightforward, we need to determine the manner of fixing the coordinates to use in the study of cosmological perturbations. This is due to the fact that there are many different ways to fix the coordinate transformation scalar ξ0 and ξ (relevant for spatial transformation via iξ) by setting certain perturbation variables to zero [6]. One important choice is the so-called comoving gauge, set by

(5)

This means we do not give physical degrees of freedom to the matter sector1. Two important reasons why the comoving gauge is special is 1) the spatial curvature of constant-time hypersurfaces, 𝜑 |com ≡ 𝓡, retains the same value irrespective of which gauge it is evaluated, and 2) the value of 𝓡, on the length scales greater than the horizon, is conserved. Moreover, such a conserved value of 𝓡 remains intact until the moment when the cosmic microwave background was generated long after inflation, satisfying

(6)

which is known as the Sachs-Wolfe effect [7]. This is how the primordial curvature perturbation can be constrained by observations on the temperature fluctuations in the cosmic microwave background.


1 Further, vanishing δ𝜙 enables us to treat 𝜙=𝜙0(t) as the time itself upon appropriate redefinition, implying that there is a unique choice of constant-time hypersurfaces. This observation leads us to the effective field theory of inflation, which is an interesting subject but will not be pursued here any more.

To find the action for cosmological perturbations, the basic steps are straightforward. We just expand the full action, (3) and (4), as a perturbative series, and collect the terms of the order we are interested in. This is in practice a bit tedious but, after all, we can find the quadratic action for the comoving curvature perturbation 𝓡 and tensor perturbation hij as [8]

(7)


(8)

where ϵ ≡-/H2. Note that since the scale factor a(t) is factored out for hij as in (2), the indices of hij are raised and lowered by δij, so =ijij and so on. Since the structure of the action is the same, now we will only focus on the scalar perturbation: the same discussions are applied to tensor perturbation. While (7) denotes a massless scalar field 𝓡 (i.e. there is no potential) in a non-trivial background as the overall coefficient shows, we can transform (7) into a more familiar form as

(9)

where =dt/a is the so-called conformal time, a prime denotes a derivative with respect to τ, u = z𝓡 and z = 0/H. This is a surprising result; if we derive the equation of motion for u, we find in terms of the Fourier mode uk,

(10)

Thus, identifying k2-z"/z=ω2k(τ), we find a system of quantum harmonic oscillator with a time-dependent frequency ωk(τ) in the Minkowski space-time. That is, the Universe contains only quantum harmonic oscillators.

Particle production from gravity
Since the form of the action (9) is identical to that of a 1) free and 2) canonical scalar field in the Minkowski space, the quantization procedure is standard. That is, we promote u and the conjugate momentum Π=δ𝓛/δu'=u' to operators û and and impose the canonical commutation relation between them.

1. Since u is a free field, we can expand the operator û in terms of the creation and annihilation operators in the Fourier space. That is, the Fourier mode, given by

(11)

is promoted to the operator û(τ, k), which can be expanded in terms of the creation and annihilation operators

(12)

where the creation and annihilation operators satisfy the standard commutation relation

(13)

otherwise zero.

2. Now we require that the canonical conjugate variables and satisfy the equal time canonical commutation relation,

(14)

Thus, the mode function uk satisfies the normalization condition

(15)

Still, we need to determine the mode function u(τ, k), which would fix the vacuum state |0> defined by

(16)

In the Minkowski space, the vacuum state is such that the Hamiltonian operator of the system is minimized. In fact, in our case we have only 1 single sensible situation where this could occur: when k z"/z, so that the frequency is practically time-independent. Thus, we can straightly apply the standard procedure to find the mode function solution, i.e. the vacuum state. The Lagrangian in this limit, say τ = τ0, is approximated by

(17)

which gives the Hamiltonian operator

(18)

Evaluating the expectation value of with respect to the vacuum state |0>0, we find

(19)

Thus, our task is to find the mode function uk that minimizes this expression. Such mode function solution is

(20)

which corresponds to the vacuum state with the frequency ωk = k. In fact, we can see that this is exactly the solution of a massless scalar field, Ek = ωk = k. Moreover, using (20) the Hamiltonian (18) is written as

(21)

which is precisely that of a harmonic oscillator (barring the factors coming from the Fourier mode convention and infinite spatial volume).

This mode function solution, or the vacuum state, however, does not remain as the solution (or vacuum state) all the time. Remember that the mode function solution (20) is found when the frequency is simply k. In general, as we can read from ωk =ωk(τ), the frequency is time-dependent. Thus, the Hamiltonian operator is time-dependent and in turn the mode function solution (or the vacuum state) that minimizes the Hamiltonian is no longer the same. Let us write the Fourier mode expansion at some later time τ1>τ0 in terms of a new set of creation and annihilation operators as well as new mode function:

(22)

and we can define a new vacuum state as

(23)

In general, the new mode function υk is related to another mode function uk via a linear transformation, the so-called Bogoliubov transformation:

(24)

Note that (22) is also the solution of the equation (10), provided that the complex coefficients k and k satisfy

(25)

to satisfy (15), given that uk is a solution. That is, in general the vacuum state is time-dependent:

(26)

What this tells us is: the notion of vacuum state is dependent on time and there is no unique vacuum state throughout all the time.

This has a profound consequence. Let us consider that at τ1>τ0 we can expand the Fourier mode of the rescaled curvature perturbation u(τ, k) as (22), with the mode function υk being related to the one at τ0, uk, by (24). If we evaluate the expectation value of the number operator with respect to |0>0, the vacuum at τ0, we find

(27)

That is, even if we start with a vacuum state |0>0, which contains no particle at an initial time τ0, at a later time τ1 we find that |0>0 contains a non-vanishing number of b-particles. That is, we have something out of nothing. This is how quantum fluctuations are generated in the gravitational background.

Evolution of the vacuum state
What is important in the Hamiltonian formulation is that a canonical variable and its conjugate momentum are independent degrees of freedom. In that sense, it is instructive to rewrite (9) up to total derivative as [9]

(28)

Then, we define the conjugate momentum Π as usual:

(29)

The Hamiltonian becomes

(30)

An important feature of the form of the quadratic action (28) is that, unlike (18), the conjugate momentum Π given by (29) is coupled to the conjugate variable set u. Thus, while the name is the same, the conjugate momentum Π is different depending on which form of the quadratic action we use, so the evolution in the phase space (u, Π) becomes totally different. But again, this never means one is correct while the other is incorrect. Just the conjugate momenta are different and so are the corresponding phase portraits.

To proceed, we first promote the Fourier modes k and uk to the (time-dependent) operators k(τ) and ûk(τ). In terms of the creation and annihilation operators, k(τ) and ûk(τ) are given by

(31)

Note that with the operators being time-dependent, we are in the Heisenberg picture, which is as usual in inflation. One important difference from the conventional approach in the Heisenberg picture is that, the time dependence is given to the creation and annihilation operators, ak= ak(τ) and q = a(τ). Instead, the mode functions are set up at an initial time, τ = τ0 as in the standard approach the creation and annihilation operators are defined initially.

To set up the initial conditions at τ = τ0, we assume that the modes are deep inside the horizon, or the interaction between k and ûk is not effective. That is, we may only consider the decoupled part of to set up the initial conditions. Then, we can determine the initial mode function solutions by demanding that the expectation value of the free Hamiltonian operator is minimized at τ=τ0, and we find

(32)

Barring the time dependence of the creation and annihilation operators, these solutions are of the same form as those for the time-independent textbook quantum mechanical harmonic oscillator.

Then, we can find the Hamiltonian equations for the operators as [10]

(33)

The general solution of these equations are given by the linear combination of the initial ones:

(34)

which is again the Bogoliubov transformation, with the constraint |k(τ)|2-|k(τ)|2 = 1. Thus, they could be parametrized in terms of the hyperbolic functions as

(35)

where without loss of generality we take the time-dependent functions rk(τ), θk(τ) and 𝜑k(τ) real. Assuming a perfect de Sitter background so that a'/a = -1/τ, we finally the reach the exact solutions

(36)


(37)


(38)

With these solutions, the evolution operator Ûk(τ) is factorized as

(39)

where2

(40)


(41)

k and Ŝk are called rotation and squeezing operators respectively.

What do these separate operators do on the vacuum state? With the vacuum state defined by

(42)

the action of k on |0>0 is

(43)

Thus, the rotation operator k does on the initial vacuum state |0>0 nothing but produce an irrelevant phase θk. This is natural, since k is coming from the decoupled, free part of the Hamiltonian for which the expansion of the universe is neglected, so that we should find the same results as in the Minkowski space. That is, the vacuum remains a vacuum. Meanwhile, applying the squeezing operator Ŝk on |0>0 gives

(44)

where

(45)

is the 2-mode (with momenta k and -k for momentum conservation) state with the same occupation number n, i.e., n-particle state. Thus the squeezing operator Ŝk, which originated from the interacting part of the Hamiltonian, is responsible for the creation of two quanta with momenta k and -k. In other words, the cosmological vacuum fluctuations are amplified. This is the same description as before, viz. the generation of cosmological perturbations out of the vacuum, but in a different disguise.


2 Here, we implicitly "discretize" of the momentum, ∫ d3k/(2π)3L-3k where L3 is the volume in which the modes of our interest reside. With a given volume L3, we may now isolate the volume from the canonical creation and annihilation operators (13) in such a way that, since ak has a mass dimension of M-3/2, akL3/2 âk and likewise for a. That mean that the new (dimensionless) operators âk and â now satisfies
[âk, â] = 1.

OUTLOOK

In this article, we have examined the consequences arising from primordial cosmological perturbations. The leading quadratic parts of the perturbations are described in a manner almost identical to textbook quantum mechanical harmonic oscillators, therefore the standard wisdom and practices of quantum mechanics can be directly applied. Nevertheless, there are critical differences due to the time-dependent background of primordial cosmological perturbances, like the generation of cosmological perturbations out of a vacuum and squeezing of the vacuum states filled with 2-mode quanta of cosmological perturbations. Thus, behind these phenomena is gravity - the driving force of the time-dependent background. More subtle and interesting issues are also coming from gravity at a non-linear level. We anticipate that these areas will be observed in the coming cosmological observations in the next decade.

Acknowledgements: I thank Min-Seok Seo for discussions on the topics discussed here. I am supported in part by the Basic Science Research Program through the National Research Foundation of Korea Research Grant 2016R1D1A1B03930408, and by a TJ Park Science Fellowship of POSCO TJ Park Foundation.

References

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Jinn-Ouk Gong is a senior research scientist at Korea Astronomy and Space Science Institute (KASI). He received his Ph.D. in physics from KAIST in 2005. Before he joined KASI, he worked at Asia Pacific Center for Theoretical Physics (APCTP) from November 2012 till November 2017 as a Junior Research Group Leader. His main research field is theoretical cosmology.

 
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