Big Fat Coincidences and Problems that Don't (Seem to) Exist

If our universe were to be undergoing flat expansion (Flat Uniform Granular Expansion, or FUGE) then, I suggest, a lot of what I have recently identified as “Big Fat Coincidences” would not be coincidences at all but would rather be the natural consequences of the process of that FUGE.  In addition, in the past few posts, I have identified a couple of problems that, given FUGE, aren’t actually problems – namely the Flatness Problem and the Cosmological Constant Problem.  Please note carefully, I am not suggesting that I have solved the problem, I am merely saying that if FUGE is true, then these are not problems.

I am going to summarise how I envisage that FUGE works and try to hit as many big fat coincidences and problems that don’t (seem to) exist as I can on the way.  If I miss any, or it’s just too awkward to address them at the time as they come up, I’ll list them at the end.

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First and foremost, we know that the universe expands.  We can see evidence of that as we look at distant galaxies that are receding from us at a rate proportional to their distance from us.  This rate is what we all the Hubble parameter (H) or, more often, the Hubble constant (H₀) which merely is the value of the Hubble parameter today (meaning that it isn’t really a constant, since it changes with time).

Secondly, our measurements of the curvature of the universe imply that it is flat.  This flatness of the universe (which must be more extreme as we go back in time, per the Flatness Problem) tells us that the density of the universe is critical and that, per the Friedmann equations, is ρ_c = 3H²/8πG.

As can be seen above, curvature of the universe and the Hubble parameter are linked.  In the FUGE model, the universe is expanding and it remains flat throughout that expansion.

The final element of the FUGE model is its granularity.  This granularity is at the Planck scale, which is the scale “below which (or beyond which) the predictions of the Standard Model, quantum field theory and general relativity are no longer reconcilable”.  To be more precisely, in the FUGE model, the universe is granular at the Planck length and Planck time.

To expand, the radius of the universe must increase.  Note however that this is the radius of a 4D shape, specifically a glome.  The 3D universe does not have a radius in the same way that a sphere has a radius, but the volume of the 3D universe is linked to the radius of the glome (r) thus:

V_surface = 2π²r³

It is this radius that increases.  Clearly the universe, to expand, must expand at a rate.  In the FUGE model, the expansion is given by:

Δr/Δt=c

At the granular level, this is equivalent to the radius of the universe (as a glome) expanding by one Planck increment each Planck time.  I say one Planck increment because it could be said that the expansion is time, so the expansion rate is one Planck time per Planck time however given the interchangeability of space and time, this is equivalent to one Planck length per Planck time, which is the value of r that can be used to determine the surface volume of the universe – r = ct, where t is the age of the universe.

As the universe expands, it is filled with Plank atoms – where a Planck atom is the 4D equivalent of Planck volume, so l_pl³.t_pl.  The surface volume of the universe is the current layer of Planck atoms, which appear to us, at the macro level, as 3D space.

Note that as the surface volume of the glome expands, room will become available for more Planck atoms.  In other words, gaps will open up.  These gaps will open up everywhere with a random distribution at a rate proportional to ct.  The consequence of this is that, within the surface volume of the universal glome, for a sphere of r_LH = ct (that is a sphere defined by a radius equal to the light horizon, which is the distance that light could have travelled in the age of the universe, t, to reach an observer in the centre of that sphere), the rate of expansion would be c.  The recession of any object at distance D, as observed from the centre of that light horizon sphere, would be given by:

v = (c/r_LH).D = (1/t).D = H.D

This accounts for one big fat coincidence, namely that the value of the Hubble parameter today, H_o, is the inverse of the age of the universe.  In the FUGE model, the value of the Hubble parameter is always the inverse of the age of the universe.  It also accounts for the fact that the Hubble length (l_H = c/H) is the same as the light horizon.

Note that our observations are based on the light horizon in which we exist.  The volume of that light horizon is V_LH = 4πr³/3, where r=c.t.

As mentioned above, is the universe is flat, then ρ_c = 3H²/8πG.  Given that H = 1/t  and ρ_c = M/V and Δr/Δt = c and V_LH = 4πr³/3, consider the change in mass (ΔM) over a period of Δt:

ΔM/(4π(cΔt)³/3) = 3/8πGΔt²

ΔM/Δt = (c³/G)/2 = (M_pl / t_pl)/2

This is telling us that mass (and thus mass-energy) is increasing within the light horizon at a rate of one half Planck mass per Planck time.  This resolves the flatness problem since, if this is true, the universe will be maintained at precisely critical density forever – and will be flat forever.  Note that this applies to whichever light horizon we choose, from either here in our current spacetime location, or from any other event location.  This implies that the entire universe is increasing in mass at rate that is greater by a factor of 3π/2.

Note that this rate of mass increase is not time constrained, so it would apply even for very small values of t.  This means that there is no singularity problem.  For sure, the universe would have been significantly denser at the beginning, when Hubble parameter values would have been very high, but that density would not have been infinite.

In the FUGE model, the universe does have a size because, while it’s not bounded (in 3D), it’s not infinite.  The volume of the universe is given by 2π²(ct)³ = 4.39x10⁷⁹ m³.  It should be noted that the volume inside a sphere defined by a radius equal to the “comoving distance to the edge of the observable universe” is greater than this.

The radiation that is received by us today as cosmic microwave background radiation has travelled across an expanse of space that now has a comoving distance of 46 billion light years despite the fact that the radiation has travelled at precisely the speed of light for only 13.8 billion years.  Comprehending this difference is a little mind-bending but, in short, it is incorrect to think of a sphere painted with something like the image below receding from us at H_o.cd (where H_o is the current value of the Hubble parameter and cd is the comoving distance to the origin of the cosmic microwave background radiation (not to the “the edge” of the observable universe)).

Instead, think of an expanding sphere, on which there is a “flat” (unbounded but not infinite) 2D surface area and imagine that 2D information travels along the surface in a straight line (or rather a geodesic or a “straight arc”).  Imagine further that the sphere is expanding in such a way that information can’t quite circumnavigate the sphere:

The value r_p is what could be called the “pseudo-radius” in that is the apparent radius of the large circle that the observer (denoted by the star) perceives herself to be the centre of.  Note that there is overlap, in that the observer can spin around to face the other direction and “see” the same expanse again so, to express the area of the apparent surface, we must use only half of the pseudo-radius, r_p/2 – even if the observer would think she is looking at r_p.

As said above, the surface volume of a glome of radius ct is given by 2π²(ct)³.  This can be equated to a sphere with a radius of half the pseudo-radius, or:

    (4π/3).(r_p/2)³ = 2π²(ct)³

    r_p³ = 12π.(ct)³

    r_p = ³√(12π).ct

When ct=13.8Gly, this gives a value of r_p = 46.3Gly, which is (give or take a little) equal to the comoving distance to the “edge” of the observable universe.  Another big fat coincidence.

As discussed above, mass-energy enters the universe at a rate of one half Planck mass per Planck time within the light horizon (where the light horizon can be taken from any location within the surface volume of the universe.  Given that M = ((c³/G)/2).t, and V = (4π/3).(ct) ³, this gives us a mass-energy density of:

      E/V = M.c²/V = ((c⁵/G)/2).t / ((4π/3).(ct) ³) = 3c²/(8πG.t ²)

At the current age of the universe at 13.8 Gy or 4.35x10¹⁷s, that gives us a mass-energy density of 8.48x10^-10 J/m³.  Noting that baryonic and dark matter make up 32% of the mass-energy of the universe, that means 68% is dark energy.  Specifically, that is 68% of 8.48x10^-10 J/m³ which is 5.76x10^-10 J/m³.  According the WMAP Survey the “positive energy density (is) about 6 × 10^-10 joules per cubic meter”.  Another big fat coincidence.

Above I arrived at ΔM/Δt = (c³/G)/2 = (M_pl / t_pl)/2.  It can be seen that his represents the lower bound of the Heisenberg Uncertainty Principle, because:

      (ΔE/Δt) = (ΔM/Δt).c² = (c⁵/G)/2 = (ℏ/2)/t_pl²

Noting that the Planck time is the smallest division of time (equated with the lower limit of Δt), we can multiply through by t_pl² to find:

       ΔE.t_pl = ℏ/2

Given that Δt ≥ t_pl, then we have ΔE.Δt ≥ ℏ/2, which is the Energy-Time variant of the Heisenberg Uncertainty Principle equation.  Another big fat coincidence.

Finally, there are some aspects to the fine-tuned universe argument that are addressed by the FUGE model.  I only want to go into two here, as an example; the value of Ω and the value of Λ as argued by Martin Rees.

Omega (Ω) is the density parameter and, he argued, its value is very close to 1.  If the value of Ω deviated from 1 by any significant margin then gravity would be either too strong or too weak – too strong and the universe would collapse, too weak and stars would not have been able to form.  In the FUGE model, the value of Ω in not “close to 1”, it is precisely 1 as argued above in regard to flatness.

Lambda (Λ) is the cosmological constant and, Rees argued, its value is very, very small.  Making some assumptions, including that the dark energy density is constant, he arrived at 10^-122 as its approximate value which as a very small, but not zero value is curious.  In the FUGE model, however, one of the assumptions made by Rees does not hold.  Dark energy density is not a constant.  Overall mass-density is inversely proportional to time, so density of dark energy is must be decreasing albeit at a slower rate than the density of baryonic and dark matter.

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I’ve not progressed much further than this.  I don’t have anything to say, at this time, about how mass-energy coalesces into baryonic or dark matter.  I have some inkling about how the probability of a Planck atom appearing at any location is inversely proportional to the concentration of mass-energy in that location and the increased likeliness of space to appear where there is already space leads to localised curvature which manifests as gravity.  For the moment, these can remain as projects for the future.