Flat Universal Granular Expansion

It’s extremely unlikely that you have read all my posts related to FUGE (including some that are tangentially of relevance), but if you have then you probably have a good idea what I mean by Flat Universal Granular Expansion.  For everyone else, here’s an attempt to explain it in one place.

The FUGE model is, as the name suggests, one of a flat universe that is undergoing granular expansion.  But what is “flat” and what is “granular”?

I am using a standard definition of “flat” from cosmology in that a flat universe is one that has “critical density”, which is sort of like a Goldilocks density – meaning that the universe is neither too dense nor too sparse but rather just dense enough.  Dense enough for what, you might ask.

We can observe that stars around us are red-shifted – and therefore are moving away from us.  The further away they are, the more red-shifted they are, which tells us that it is not that stars themselves that have velocity away from us, but that the universe – as a whole – is expanding.  This expansion has an impact on density of the universe and that density has an impact on what the far future will be like.  There are two ways of thinking of this, either in terms of gravity or in terms of curvature.

A closed universe is expanding insufficiently fast to overcome gravity and it will eventually collapse in on itself.  Or, alternatively, it is positively curved – this is possibly the easiest curvature to comprehend since a sphere is a positively curved surface.  Three points on that surface create the equivalent of a triangle that has internal angles that sum to greater than 180°.

An open universe is expanding at a rate that overcomes its own gravity and results in its eventual dispersal.  Or, alternatively, it is negatively curved.  This is more difficult to visualise, with negative curvature often being represented as a sort of saddle (although missing the vast majority of it since it is open and extends to infinity).  Three points on that surface create the equivalent of a triangle that has internal angles that sum to less than 180°.

A flat universe is in the middle, with zero curvature overall.  And it is finely balanced between collapsing and dispersing.  Flat curvature (or rather a surface without curvature) is represented by a plane on which the sum of the angles within a triangle sum to precisely180°.

Measurements indicate that the universe is flat, or extremely close to flat such that the error bars include precisely flat.  It should be noted that we should only expect the universe to be flat overall, not necessarily where we are and at a resolution that we find easy to observe.

There is a critical density that leads to the universe being flat (the equation for which I have provided elsewhere and can be sought out online) and this is what the FUGE model has as one of its fundamentals – overall, the universe is flat curvature-wise and the density of the universe is equal to the critical density.

Then there is “granular” expansion.  All this is saying is that the expansion of the universe is at a rate of one tiny division of space per tiny division of time.  I prefer to think of this in terms of Planck units, because they appear (at least to me) to be physically significant and so many constants can be resolved to unity using them, but we may not necessary to be limited to this regime – other less comprehensive schemes of natural units are available.  All that must be noted is that whatever tiny division of space and tiny division of time are used, based on whatever scheme of natural units, they must divide so as to equal the speed of light.  As a consequence, the universe is expanding at the speed of light.

Note however that this does not mean that everything in the universe is moving away from everything else at the speed of light.  Instead it could be thought of as grains of space being added stochastically across the entire radius at a very high rate per second such that the radius as a whole increases at the speed of light.  The effect would be equivalent to space being stretched out, only very marginally in local areas, but significantly at intergalactic scales.

Consider the AU, or astronomical unit, a distance equivalent to the average distance at which the Earth orbits around the sun.  The radius of the universe in the FUGE model is 13.77 billion light years, or about 10¹⁵ AU.  This means that, each second, a distance equivalent to that between the Earth and the Sun is stretched by 10²⁹ units of Planck length – or 1.6 micrometres.  Now that does not mean that the Earth is going to move away from the Sun due to expansion, because there’s also a gravitational effect.  We only move away due to the Sun’s loss of mass (by only about fifteen millimetres per year, rather than the 50 metres that the 1.6 micrometres per second would sum to).

Now the keen eyed will note that if the universe is expanding, then density is decreasing, and one may well ask: what about that critical density thing?

The thing is that the critical density is related to the square of the Hubble parameter, which is a measure of the speed at which distant objects appear to be moving away from us divided by their distance from us.  (They aren’t actually moving remember, it’s just that space is expanding between us, so distant objects are said to “recess” rather than “move away” from us.)   The Hubble parameter is inversely related to the age of the universe, meaning that as the universe gets older, the Hubble parameter decreases, and so does the critical density.

However, the critical density decreases with the square of the age, but volume is increasing with the cube of the age.  And for that reason, the other key element of the FUGE model is necessary.

For every tiny division of time, a small amount of mass-energy is added to the universe such that the density of the universe remains critical.  By my calculations, that is the equivalent of half a unit of Planck mass per unit of Planck time during which the radius increases by one unit of Planck length.

There is potential for confusion here, in that one could think that for every unit of Planck time, there is one “grain of space” added with half a unit of Planck mass-energy in it.  That would potentially be observable, since the energy equivalent to a unit of Planck mass is in the order of 2 million kilojoules.  However, it is the radius that is incremented by a unit of Planck length, which means that the volume increases by an amount equivalent to expanding out the entire surface of a sphere by one unit of Planck length, so there would be many “grains of space” added.

The total mass-energy added in this process would be equivalent to half a unit of Planck mass, so currently that would be approximately 10^-9J/m³ each spread across in the order of 10¹²¹ “grains of space” per unit of Planck time.

I remain vague on whether this mass-energy is added to the universe or is created by it because the mechanism remains unclear.  I would much prefer that the mechanism were clear, but I am sanguine about the absence of clarity because the alternative (the standard cosmological model) has numerous aspects for which the mechanisms are unclear: how did inflation start, why did inflation stop when it is hypothesised to have stopped, why does radiation-domination lead to one deceleration rate and matter-domination another, why do either lead to deceleration at all, what is dark matter, what is dark energy, (if dark energy has an invariant energy density) how does dark energy get added to or created by the universe, why does dark energy lead to an accelerated expansion of the universe and what determines the rate of that acceleration?

One of those questions in that list is also a question that remains open in the FUGE model.  There may be hypotheses addressing all of them, but there does not seem to be any grand consensus with respect to the answers to those questions.

My original solution to the mass-energy origin problem, which I am somewhat less confident about now, is that there was a progenitor universe, which collapsed (with the exact mechanism being unclear) and is now “feeding” our universe which is orthogonal to that progenitor universe.  The rate of “feeding” would be limited by physics – in that our universe cannot be denser than a non-rotating black hole, and that there is a fixed rate at which mass-energy can enter our universe linked to the rate at which our universe is “forced” to expand.

Note that this solution seems to just kick the problem back down the road too.  Where did the mass-energy in the progenitor universe come from?

Another solution is that there is something that is driving our universe to expand (with the exact mechanism being unclear) while remaining flat, and mass-energy is created as part of that expansion.  The explanation for this is a little convoluted, but the bottom line is that the expansion itself would constitute negative energy, which would be balanced by the mass-energy that is created in the process of expansion – making our universe zero energy overall. (Update: see The Conservatory - Notes on the Universe.)

This creation of mass-energy is a bit of a sticking point, but we do know that the universe is expanding, and we do know that the density is critical, and we do know that there is mass-energy in the universe.  I have to accept my limitations and leave it to someone else to work out the mechanism (and to the religious to say that their god does it).  (Update: I eventually did come up with something, as hinted at in The Conservatory - Notes on the Universe.)

In conclusion, the FUGE model is one in which the universe is flat (which observations show), is expanding (which observations show) at a linear rate and has a linear increase in the quantity of mass-energy simultaneous with a linear decrease in mass-energy density.  The model results in:

• A Hubble parameter within the measured range
• Density within the measured range
• Radius equivalent to the observed universe (for the current age)
• Ordinary mass equivalent to the observed universe (for the current age)
• No requirement for dark energy (see also Mathematics for Taking Another Look at the Universe)
• No possibility of dark matter (as a literal form of mass-energy)
• No requirement for inflation

The final two in that list are not positive features of the FUGE model per se, but only become a problem if dark matter (as a literal form of mass-energy) is found to exist or if inflation is shown to have happened, noting that inflation is an unexplained and increasingly problematic mechanism evoked to explain a phenomenon (or phenomena) that may not need to be explained or may be explained otherwise.