Structure and Expansion of a Universe whose Vacuum is modelled as a Set of identical, bi-modal, frequency-quantised, simple quantum harmonic oscillators.

Abstract

The ‘fabric’ of the model presented here consists of identical, contiguous cubes of space, each of which is occupied by a simple quantum harmonic oscillator. A possible structure is proposed for the ensemble of cubes, and it is shown also that other configurations, on all scales can exist in the ensemble, even although the members of that collection are absolutely identical.

The expansion of space is considered initially from the standpoint of a universe which is not accelerating, and equations are developed which relate to this ‘coasting’ condition, and, in which the Hubble parameter, H, is shown to be a function of time. These equations are extended by the inclusion of the deceleration parameter of cosmology, q, treating q as a parameter.

The observations of Schmidt, Riess et al* and Perlmutter et al** have shown that approximately five billion years ago the universe experienced a jerk and went from a decelerating state to one of acceleration. From these findings it is argued that the deceleration parameter is not constant, but is a function of time. A distribution for q of the form: q = e ^k(t₀^-t) + B is proposed, where k and B are constants and the subscript 0 on t refers to time ‘now’. This guess proves to be reassuringly productive for, with the appropriate boundary conditions, it captures the deceleration phase before the jerk and the coasting condition through which the universe must pass in order to attain the accelerating state. In addition, the equation shows that the acceleration is increasing after the jerk. This is consonant with the observations of the above-mentioned experimenters.

The magnitude for q favoured by some theorists is ½. From the first extended equations (see the text for explanation) it is then shown that this q is located in time 2/3H_o before the present, where H₀ is the current magnitude of Hubble’s parameter (2.38 x 10^-18s^-1). However, it is seen that this time is determined from the set of extended equations mentioned above, where q is considered to be a parameter.

By incorporating the expression proposed for q to form the second extended equations it is shown that q = ½ is located at a time in the past close to 0.925/H₀. Two methods are outlined which will permit the speed of expansion of space at the jerk to be determined, together with a rough estimate of that speed.

It is shown that the Hubble parameter, which should, strictly, be termed ‘Hubble’s function’ in view of its time dependence can be interpreted in two different ways, one of which is that Hubble’s function is equal to one third of the volumetric strain rate of space.

It is concluded, for this model that the universe is infinite and observation of such is observer-centred. Further, it is argued that each observer can, in principle see a different part of the universe, but, due to the current size calculated for the part of the universe examined here, spatially, but closely- separated observers do not possess sufficient precision to distinguish any differences in their respective far-field observations.

List of Contents

Abstract.. 2

Introduction.. 4

A stacking structure for the universe.. 5

The expansion of space.. 8

The cosmic jerk.. 17

The Hubble parameter.. 24

The expansion of space relative to an observer.. 25

Discussion.. 28

References.. 29

Introduction

It is accepted generally that the shape and size of the universe is unknown, or, indeed if it is even finite. Calculation by others of the size of the universe which, in principle, is capable of being observed yields a spherical shape with a diameter in the region of 93 x 10 ⁹ light years. It is of interest to note that, for a universe containing 10⁸⁰ baryons and modelled as a collection of contiguous cubes, it is calculated in ((r2)) that the side of this array is 76 x 10⁹ light years. The volume of this cube may be contained within a sphere of diameter, 94.3 x 10 ⁹ light years. In addition, the boundaries of current models of large-scale galaxy formation, simulated numerically in supercomputers, are the sides of very large cubes.

The speed of the expansion of space has only been confirmed at great distances from the point of observation and was determined experimentally by Edwin Hubble. Hubble established a linear relationship between the speed of the expansion at a point in space and the distance of an observer from that point. The parameter which reduces the proportionality between the speed of expansion and the distance is called, probably incorrectly, Hubble’s constant, for it is thought to be a function of time; further, the magnitude of the parameter has been the subject of much debate over many decades. The latest estimate seen by the author for Hubble’s parameter is 73.8 x 10³ /3.28 x 10⁶, neglecting the error band; the units of this quantity are km/s/light year. When used in this form the distance to the point of observation should be expressed in light years. The magnitude of the parameter, if the distance is expressed in metres, is 2.38 x 10^-18s^-1.

This hybrid model, which employs results from relativity and quantum mechanics, is based entirely upon average conditions, takes no account of the gravitational effect (if any) of the mass/energy which may occupy space. Indeed, at the current time, from the data garnered by the Planck mission, it has been determined that space contains, on average, slightly less than the equivalent of 6 protons per cubic metre, composed of baryonic matter, Dark matter and what is sometimes called the Dark energy, but in (r1) is shown to be related to the ground state of a quantum harmonic oscillator. It is acknowledged that the first two of these have effects on space opposite to that of the third.

It may be argued that, since the current mass/energy density is so low, it is not necessary to take account of any countervailing gravitational mechanism at the present time, and hence the expansion of space at all of the levels in this model may be considered.

Indeed, since this model and also that developed in (r1,r2,r3) considers only average conditions, then the highly diffuse conditions which are concomitant with the averaging process are extant on all scales of these models.

A stacking structure for the universe

The model of the universe presented here consists of a set of identical, simple, frequency-quantised, quantum harmonic oscillators, each of which occupies a cubic space of side equal to the prevailing wavelength; we describe the cubic space as an enclosure and designate such an enclosure as a fundamental cube. Aspects of the model used here are discussed at length in (r1), (r2) and (r3).

Now, a simple harmonic oscillator oscillates only in one direction, and a simple quantum harmonic oscillator is no exception. If the cubes are stacked together at random to form a larger cubical structure then it is possible that sub-structures may be formed where, amongst other things, there are predominant directions of oscillation due to cubes of the same oscillatory direction meeting along a face, the normal to which is in the direction of the oscillation. This would imply a preferred direction in such instances, and would constitute a wave train of length proportional to the number of cubes.

If it is posited that space, on large scales, is very closely isotropic, then configurations such as this violate isotropy. It follows that if space is to be closely isotropic, then we must consider how structures such as the above may be minimised, or even avoided, and also possible arrangements of cubes which will render space isotropic. An overall structure in which space is both homogeneous and isotropic is characteristic of a Friedman-Lemaitre-Robertson-Walker (FLRW) universe.

Within the constraints of the foregoing we now proceed to investigate cubical structures. To facilitate this we assign colours, red (R), yellow (Y) and blue (B) to the three possible directions of oscillation with respect to a set of Cartesian axes located at the vertex of a cube. It should be noted that these distinctions only find meaning in the context of their assembly into larger cubical structures. It is helpful to envisage the now distinct cubes as a set of child’s coloured building blocks.

On the accompanying diagram sheet the fundamental cubes and all combinations thereof are, for clarity, shown in plan view.

The basic elements are as shown in Fig1.

Figs1(a),(b),(c) and (d) show a set of arrangements which all will form an ‘8’ cube. It is immediately evident that all of the arrangements shown conform to the requirement that no two cubes of the same colour meet along a face.

However, we see, for example, in (a) that, whilst cubes of all of the colours are represented, there are twice as many cubes of one colour as there is of cubes of the other colours, whilst for the arrangements shown in Figs (b),(c) and (d), in each example, one of the colours is not present.

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