The Cause of Expansion

While we have been generally running along with the idea that the universe expanded, we have not really asked the most obvious question:

How and why did the universe expand?

In contradiction to the implication associated with its name, the ‘Big Bang model does not actually describe the process of expansion in terms of an explosion, where matter is thrown outwards with kinetic energy with the debris travelling through existing space, but rather as the uniform expansion of every cubic metre of space as a function of time.

1

If we initially assume that the universe is a closed system that represents the totality of everything, nothing could have existed outside the singularity and therefore expansion must have occurred from within the ‘inner space’ of the singularity. As such, we might question whether the simplicity of the Friedmann equation, as shown in its reduced form in [1], is valid throughout all of time; especially as the previously described ‘material universe  is rolled back to its quantum beginnings

[1]      1   

Note: As a speculative consideration, the density [ρ=ωPc2] in [1] can also be translated into a pressure [P=ρ/ωc2], see Fluid equation and equations of state discussions. While the introduction of dark energy having negative pressure [ω=-1] is often assumed to drive the accelerated expansion of the universe after 7 billion years, we might reverse this idea by assuming that some positive pressure within the universe in comparison to a lower pressure within some wider quantum universe might be the cause of expansion - see diagram in the discussion entitled the 'evolution of inflation' for more details. These references are intended as complementary ideas to those outlined below, which are also discussed further under the heading Energy & Pressure Model.

At face value, [1] appears to have more to do with the gravitation of some given density [ρ], which we might assume would only slow expansion, if the universe has some overall centre of gravity. Alternatively, we might consider the expansion of space in terms of a metric, e.g. the FRW metric, to see if it tells us anything about the cause of expansion.

[2]      2

Based on the assumption of a homogeneous energy-density at the largest scale of the material universe, the spatial curvature [k] is often set to zero. At smaller scales, matter can be bounded into high concentrations due to gravitational, electromagnetic and strong nuclear forces that might be assumed to have helped form galaxies, solar systems, atoms and atomic nuclei. However, it would appear that neither [1] or [2] really provide any great insight to the question:

What caused space to expand?

To be honest, the general description of this aspect of the cosmological model can often appear frustratingly ambiguous, but it is clearly of central importance to any real understanding. While we shall return to this issue below, we are also trying to extend the basic cosmological model to include the idea of inflation, which is thought to describe a very short period of exponential expansion within the very first second of existence. The cause of this expansion is sometimes described in terms of a ‘scalar field’, although the physics that supports this concept is not necessarily well understood. It is also unclear how the expansion of space continued after the inflation period, as the scalar field is thought to have decayed during this process. Within the wider discussion of what is referred to as the Concordance Model, we will introduce the idea of dark energy, which has the strange attribute of negative pressure, which in the present-era is thought to account for the accelerated expansion of the universe. However, immediately following the inflationary era, the density of dark energy is thought to have been so low that it would have had no effect on the expansion of the universe until after some 7 billion years of elapsed time. At this point, some vague idea of the inertia of space is sometimes introduced to suggest that all the matter within the universe, put in motion by the initial inflation process, continues presumably based on some concept of inertial momentum.

But can space expand due inertia?

Well, the first problem associated with this question is whether we really understand what causes inertia, because in many ways, even today, science only really describes the effect and not the cause of inertia. The second problem is that we usually associate inertia with the mass of an object moving through space and not with the expansion of ‘empty’ space. The word ‘empty’ is highlighted because in the context of the vacuum of space, quantum theory forwards the idea that the vacuum might consist of a form of energy that science is still debating, which might have some form of inertial energy-mass. However, if the singularity is assumed to contain all of existence, then nothing can exist outside the universe and the expansion of space has to be driven from within. As such, it seems difficult to reconcile the idea of inertia with the expansion of space itself, at least, when described in these terms. Alternatively, we might consider the possibility that our universe is not the sum total of existence, but simply part of a larger system that is not yet perceived - see note above for pressure idea in this context.

So is there anything more we can say about expansion at this point?

Given that we have not really answered the previous questions, we might wish to pursue the issue a little further by first considering the implications within the accepted scope of the Friedmann and Acceleration equations. However, first we will make a few simplifying assumptions within the scope of our initial homogeneous dust model based on flat spacetime, i.e. [k=0]. So, within the scope of this model, we can convert an energy-density [ρ] back into an effective mass [M] based on the volume implied by the radius [r]:

[3]      3

We can see that the logic of [3] starts with the form of the Friedmann equation, which the classical derivation rooted in the conservation of energy. This is also the starting point of [4], which also makes the assumption that the total energy is zero.

[4]      4  

As such, the results in [3] and [4] would appear to be identical, which is not so surprising, as they are both based on the same conservation of energy premise. However, the implied direction of the velocity in [3] and [4] are reversed. In [3], it is assumed that the velocity relates to the outward expansion, which seems vaguely reflective of the idea of an ‘escape velocity’ with respect to mass [M]. In contrast, [4] is essentially reflecting the ‘free-fall velocity’ under gravitational acceleration. Therefore, we might wish to also compare the acceleration related to [3] and [4]:

[5]      5

The form of [5] is associated with [3] through the Fluid and Acceleration equations, which were rooted in the 1st law of thermodynamics. It is the derivation of the Fluid equation that introduces the negative sign in [5], although as pointed out, this did not implicitly infer either expansion or contraction, only that expansion causes [dρ/dt] to be negative. In contrast, the determination of the gravitational acceleration in [6], associated with [4], has its roots in Newton’s 2nd law of motion:

[6]      6

It may be noted that the direction of the acceleration in [5] and [6] is reversed, but consistent with the premise of [3] and [4] respectively. If we assume the velocity in [3] to be positive, then the negative acceleration in [5] reflects the suggestion that the expansion of the universe is slowing down, unless acted on by some component with negative pressure. In contrast, if we assume the velocity in [4] to be positive, then the positive acceleration simply reflects the increasing velocity under free-fall acceleration.

So how are we to interpret the implications of these equations?

At one level, all the equations seem to be consistent to the conservation of energy, where any change in kinetic energy is matched by a complementary change in the gravitational potential energy. However, we possibly need to be a little more specific as to how gravitation fits into the picture of an expanding universe, at least, as implied by [3]. Of course, the effect of a gravitational force in an expanding universe depends on what assumptions are made about the universe, as a whole, which we might again consider in terms of ‘Newton’s Shells’. For simplicity, we shall reduce all possibilities to just 2 basic forms:

  • Model-1:
    This model assumes the universe to be analogous to a large spherical volume of homogeneous density of radius [r], which exists within an infinite and absolute vacuum. As such, this homogeneous volume will have an effective mass [M] with a defined centre of gravity. The force on an object like a galaxy with mass (m) at a radius [a<r], i.e. within the homogeneous volume, is linearly proportional to [a]. This is because the effective mass [M] is proportional to [a3], as the gravitational effect of the mass outside [a] cancel out. The force on any unspecified object of mass (m) at a radius [a>r], i.e. outside the homogeneous volume, would be subject to the normal inverse square law [1/a2], but also ‘outside’ our local universe.

  • Model-2:
    This model may also assume a universe with a homogeneous density, but now its volume might be thought to conceptually extend to infinity. However, the logic of Newton’s Shells now appears ambiguous as it is difficult to resolve whether any object [m] has a near-zero or near-infinite radius as the centre cannot be specified.

Note: Again, as a general comment, might we consider whether entropy would require a localised universe with higher energy density and pressure to expand into the vacuum of some large quantum universe.

As such, we have defined 2 potential models of the universe, either of which might align to the description of our universe in the present era. Of course, we must also consider the fact that both of these models would have undergone expansion.

  • Model-1:
    At this point, we are only interested in the effects of expansion within the ‘local’ universe, as defined by the homogeneous density. Therefore anything inside this volume, co-moving with the expansion, would also see its radius [a] from the centre of gravity expand. However, the mass [M] contained within any arbitrary radius [a] does not change during this expansion, as the change volume is balanced by the changed density. It is highlighted that any expansion of this model might now be slowed by the gravitational pull towards its centre of gravity.

  • Model-2:
    This model might appear to align to the standard cosmological model, at least in its original form, as it is assumed to have no obvious centre of gravity. If so, there does not appear to be any obvious reason why the expansion of this universe would be slowed by gravity, if the density is homogeneous on the large-scale; although local perturbation might still account for local concentration, i.e. galaxies.

At this point in the discussion, we are only considering potential ideas, which may be later rejected by the rigour of general relativity and/or quantum theory. However, it is worth remembering that if we turn back the clock to when the Big Bang was first conceived, in 1922, linked to the derivation of the Friedmann equations, the concept of inflation and dark energy did not yet exist. As such, it was a model that only contained radiation and matter modelled as dust, i.e. no pressure, and therefore we might yet again be compelled to ask the questions:

What caused the physical expansion of space?

Clearly, there is a suggestion that we still lack a coherent explanation as to how the universe expanded. So, at this point, let us forward some speculative assumptions, which we may have to subsequently modify, to see how they shape up as we delve deeper into the issues:

  1. If the universe is infinite and has no centre of gravity, the expansion of the universe cannot be subject to any gravitational slow down.
  2. If the local universe is finite, within some larger quantum universe, it might have a centre of gravity, which would presumably have some effect on the evolution of the local universe.

  3. The concept of space continuing to expand due to some form of geometric inertia is too vague and is more representative of the effect rather than the cause.

  4.  If (1) and (3) are true, then the expansion of the universe requires a cause, which has to exist throughout the entire timeline of the universe.

In the present context, we are starting to consider the effects of including a preliminary inflation model, as a prerequisite phase, before the start of the standard big-bang model. While the details of the inflation model itself will be the subject of another discussion, we might initially consider the implication of the inflation model on the description of the universe, as a whole. So far, we have defined a ‘visible universe’ simply in terms of the speed of light [c] times the age of the universe. However, we have already shown that this definition is relatively arbitrary and does not necessarily reflect the size of the ‘physical universe’. Of course, without knowing the actual rate of expansion, at all times, we can only guess at the potential size of the physical universe. However, the inclusion of the inflation model introduces two further definitions, which we might refer to as the ‘bubble universe’ and the ‘quantum universe’ that we might then align to either Model-1 or Model-2, as described above. In essence, the bubble universe is analogous to the physical universe, which is assumed to have expanded out of a conceptually infinite quantum universe. While we do not know the size of the bubble universe, it is possibly easier to speculate that it does have a finite size. In this respect, the bubble universe might be said to better align to the description of Model-1 and, as such, it would have a centre of gravity. However, the effect of this centre of gravity on our visible universe would depend on its position within the larger bubble universe.

2

In the diagram above, the total size of the bubble universe is defined by its radius [r]. However, the effective radius used to calculate the gravitational force [F] and the acceleration due to gravity [g] is defined by [a], i.e. the assumed position of our visible universe within the bubble universe. Although we can only speculate as to the size of the bubble universe within this model, we might simply consider an example where [r] is 100x greater than our visible universe. Within this example, mass [m] will be assumed to correspond to a typical galaxy within our visible universe. As such, we can proceed to calculate the gravitational force [F] and the acceleration [g] on [m] by the effective mass [M] corresponding to the density contained within the volume defined by radius [a].

a Radius Volume Density Mass [M] mass [m]
1 1.29E+28 9.03E+84 9.50E-27 8.58E+58 1.94E+41
0.001 1.29E+25 9.03E+75 9.50E-18 8.58E+58 1.94E+41
0.000001 1.29E+22 9.03E+66 9.50E-09 8.58E+58 1.94E+41

a Years F=GMm/a2 g=GMm/a2
1 Now 6.65E+33 3.43E-08
0.001 437,000 6.65E+39 3.43E-02
0.000001 <1 6.65E+45 3.43E+04

The first entry in the table above reflects the example outlined in the present era and while the gravitational force [F] on [m] might appear enormous, the more significant factor is the acceleration [g], which is 9 orders of magnitude smaller than normal Earth gravity. The next 2 entries are reflective of the universe contracting in size as time is reversed back towards the cosmic singularity. However, it is highlighted that the effective mass [M] does not change with the reducing radius [a], as the density increases by a corresponding amount. As such, it can be seen that the effect of gravitational acceleration [g] must become an increasingly significant factor in the early stages of an expanding universe. However, we might make a number of comments based on the results above:

  • The strength of gravitational acceleration [g] would depend on the position of the visible universe within the bubble universe, i.e. it depends on [a].

  • If the visible universe was significantly smaller than the bubble universe, as reflected in the diagram above, the gravitational acceleration would effectively be uniform over the entire visible universe.

  • Based on the table above, the gravitational acceleration [g] might be so small as to be virtually impossible to detect in the present era, especially in comparison to expansion of space.

  • However, the implication of increased gravitational acceleration [g], as the universe is contracted back in time, suggests that the overall expansion of the bubble universe would have had to overcome the inherent probability of gravitational collapse in its earliest stages.

Based on the comments above, this speculative model would require us to modify, and possibly rationalise, the original assumptions made above as follows:

  1. If the universe has a centre of gravity, the expansion of the universe will be subject to a gravitational slow down, but this slow down would weaken with the net expansion

  2. Not only does the expansion of the universe require a cause, which has to exist throughout the entire timeline of the universe, but this cause must be sufficient to overcome the enormous gravitational acceleration in the earliest stages of expansion.

So if the bubble universe associated with the inflation model does have a centre of gravity, the inflation model must also explain how the initial expansion of the bubble universe overcame the inherent probability of gravitational collapse. While the effect of gravitational acceleration [g] on the visible universe might have weaken relatively quickly due to inflation, the subsequent expansion of the universe would still require a cause, e.g. internal pressure relative to quantum vacuum, if we are to  discount the idea of space simply continuing to expansion based on some vague notion of inertial momentum.