No, Espen Haug, You Cannot Just Magic God Time out of Empty Space

I recently wrote about Eugene Tatum’s assertion that the CMB is related to the Hawking Temperature of the universe.  The coauthor of that particular paper was Espen Gaarder Haug.  I didn’t mention it before, but Tatum is a Doctor, of Anatomic and Clinical Pathology.  Haug is also a Doctor, in the area of finance, specifically quantitative finance.  He got his doctorate in something relevant to options and trading (his doctoral thesis was on that anyway) but I can’t find out what he did his undergraduate in (presumably economics, but not necessarily).  He’s currently a finance professor at the Norwegian University of Life Sciences.

Haug and Tatum are coauthors on a range of papers about the flat cosmological model (which they sometimes refer to as the Haug-Tatum cosmology [HTC] model), while Haug has a quite a few papers of his own about the quantisation of gravity.

During research for No Eugene Tatum, Hawking Temperature is not Related to CMB, I stumbled upon one of Haug’s paper on the quantisation of gravity: God Time = Planck Time: Finally Detected! And Its Relation to Hubble Time.  Note that, like most of Haug’s papers, this one was published at scirp.org, which is a predatory publisher.  That said, being published in an odd place does not necessarily make the content wrong.  It’s just easier to get such publishers to publish bad and low-quality science – and even easier to publish at a blog under a pseudonym.  When you find a paper published by a pay-to-play or predatory publisher, you just need to be more careful and engage your critical thinking more intently than you might otherwise, such as when the source is more reputable.

The very first thing drew my attention was the title.  “God time”?  I wondered if I was looking at the work of someone like Luke Barnes, a person dabbling in physics to support a theological world view.  On closer inspection, and in the context of Haug’s other papers, including two others that mention “god”, it appears that he is just using a hook to draw attention rather than making any serious claim about the existence of an actual god.  Haug does refer to the “the god particle” in at least two of his papers without mentioning that the Higgs boson was know as the “god-damned particle” due to its reticence to reveal itself, which is unfortunate but, in the conclusion to the God Time paper, he indicates that historical references to indivisibility at the smallest scales, including apparent biblical references, are not important.

The basic claim of the God Time paper, and the one that I have a problem with, is that knowledge of G, ħ, and c is unnecessary to establish the value of t_P, the Planck time.  (In another paper, and another, Haug seems most intent on removing the use of G entirely, which is odd.  There are problems in those papers too, but I will try to restrict my efforts here to addressing Haug’s efforts to eliminate fundamental constants.)

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The standard definition of Planck time is:

Given that the defined value consists of G, ħ, and c, and only G, ħ, and c, establishing Planck time without the use of those fundamental constants should be quite a challenge.  Haug does it by using the equation:

Which he later generalises (in a special case) to:

In his numerical example for the first equation, Haug sets object 1 to the Sun and object 2 to the Earth.  The term δ is used to refer to the effect of  gravitational lensing by object 1 (the Sun), expressed in radians.  (Note that in the link provided, the symbol θ is used.) The term g refers to the magnitude of gravitational acceleration at object 2 (the Earth).  Note that the equation at the link has a direction that is implied by the minus sign, it’s not intended to refer to a negative value.  The term λ is a reference to the Compton wavelength of the mass of the relevant object, but in this case it is the reduced variant, in the same way that the reduced Planck constant is given by ħ=h/2π.  So, for clarity, Haug provides the numerical result for this equation (using subscripts E for Earth and S for Sun):

where (per the links above)

If we substitute these into Haug’s equation, we get:

which resolves to

In his generalised version, Haug is talking about a single mass, so no subscripts are required, and:

which also resolves to

So, yes, you can get Planck time using Haug’s equation which does not explicitly use G, ħ and c.  However, there is a problem with his assertion that we can therefore reach a value for Planck time without using any of the fundamental constants because, in order to use his method, we must establish the values for at least gravitational lensing deflection and the reduced Compton wavelength.

The value for gravitational lensing deflection due to the Sun has been measured, first by Eddington in 1919, so while not so easy, this is entirely possible – with some caveats.  The apparent radius r in the equation (per Wikipedia), θ=4GM/c²r, is not necessarily the radius of the body with the mass M, but rather the distance between the centre of the mass and the radiation being deflected.  The value measured by Eddington was for light that grazed the surface of the sun, hence the need to observe a total solar eclipse.  The related deflection is, therefore, at the photosphere, which is considered be the surface of the sun, and r=R_s.

The Compton wavelength, on the other hand, is not something that is measured for, and probably doesn’t even apply to, bodies at the scale of the Earth and the Sun (because it’s a quantum mechanical property).  We can calculate it, sure, but to do so we need to use both ħ and c.  (This could explain Haug’s fixation on eliminating G in his other papers, if he has noted that he can’t get around using ħ and c.  He says it explicitly in section 6 of another paper.)

If we forgive this, then we still have some problems to address.  The accuracy of the value of t_P that can be determined using Haug’s first equation depends on the accuracy of the measurement of:

Deflection due to gravity as light grazes the surface of the sun – 3%

Gravitational acceleration at some point on the Earth with a distance from the centre of R_E (not the defined standard gravity value which is nominal and limited by caveats) – about 0.7% assuming it’s somewhere on the surface

The distance to the centre of the Earth used above (about ±30mm in 6378km) – 0.0000005%

The radius of the photosphere – 0.02%

The mass of the Sun (for calculating the reduced Compton wavelength) – 0.005%

The mass of the Earth (ditto)– 0.01%

We don’t need to worry about the accuracy of h and c, because these values are both defined.  The only inaccuracy in ħ will be due to the approximation to π that is used, but this can be set arbitrarily low by using π to many significant numbers.  So this means the resultant inaccuracy in the measurement of t_P would be, approximately:

3%+(0.7%/2)+0.02%+0.005%+0.0000005%+(0.01%/2)=3.38%

Presumably, from a value of t_P determined this way, we could calculate a value for G using the standard definition of t_P and the defined values of ħ and c.  This value would have an accuracy of approximately 6.76%.

Compare this with the NIST values which have accuracies of 0.0011% for t_P and 0.0022% for G.

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As indicated above, Haug seems to realise that he can’t get around using ħ and c, and is therefore laser focussed on eliminating the use of G, even in the process of determining the value of t_P.

This, to me, is madness.  If expressed in Planck units, all the four values listed above, ħ, c, G and t_P, resolve to unity.  (In addition, m_P, l_P, q_P, v_P, i_P, T_P, E_P, k_B, k_e, m_P, reduced μ₀ and raised ε₀ also resolve to unity.  Even the unitless gravitational and electromagnetic coupling constants resolve to unity when Planck values for mass and charge are used as the reference rather than arbitrary values like a proton or electron mass or the elementary charge.)

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In other papers, like Quantum Gravitational Energy Simplifies Gravitational Physics and Gives a New Einstein Inspired Quantum Field Equation without G and Not Relying on the Newton Gravitational Constant Gives More Accurate Gravitational Predictions, Haug suggests that he’s hit upon some evidence of the quantisation of gravity – all which seem to relate to the notions of “collision length” and “collision time” (which is expanded on in Collision-space-time: Unified quantum gravity – where he assigns photons with a mass (an extremely tiny mass, admittedly)).

In the first two papers above, Haug mentions the “factor”:

Which he refers to as “reduced Compton frequency per Planck time”.  This is an odd way of putting it, for more than one reason.  First, as a simile, we could say that Gm_P²/ħ is “Planck length per Planck time”.  This not completely untrue, because Gm_P²/ħ=c=l_P/t_P, but it’s an odd way of putting it.  The second reason may not be immediately clear, but we can look at Haug’s own words, from Quantum Gravitational Energy Simplifies Gravitational Physics and Gives a New Einstein Inspired Quantum Field Equation without G:

It's not anything “per Planck time”, it’s a value that is multiplied by Planck time, not divided.  If anything, it’s equivalent to “Planck time per reduced Compton period”, noting that there’s another error buried in there.

Haug states that the reduced Compton frequency is the speed of light divided by the reduced Compton wavelength.  Presumably, the vanilla Compton frequency (f_c) is the speed of light (c) divided by the vanilla Compton wavelength (λ_c):

If we implement the reduced Compton wavelength, which is the Compton wavelength divided by 2π, we have (using the bar as an indication of some sort of modification, not necessarily reduction):

So it’s not a reduced Compton frequency, it’s a raised Compton frequency.

He could, however, express things somewhat less awkwardly.  Using the definitions of reduced Compton wavelength and Planck length, we see that:

And, in flat universe, we know that*:

So, Haug’s “factor” is, in fact, simply an expression for the age of the universe (at time t) divided by Planck time (or, to put it another way, the magnitude of the age of the universe when expressed in Planck time).  Or, as I have used frequently elsewhere, ꬱ (first introduced in Avoiding a Contravention of the Extended Consistency Principles).  It’s a useful term in that context, but not so much when mixed up with the notion of a confused (reduced? raised?) Compton frequency.

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I did reach out to Haug to discuss the above but have not, yet, heard back from him.

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* There's a little oversimplification here.  Keen-eyed readers will note that M has not been explained here.  It's not quite the mass of the universe, because in a flat universe, the mass is m_P.ꬱ/2.  To understand what is going on a review of The Conservatory - Notes on the Universe might help.  In brief though, if we think of a standing wave between two nodes (null points), then the minimum full wavelength is twice the minimum distance possible between nodes - because there is a node in the middle of a full wavelength.  The consequence of this, if we have granularity (as in a FUGE Universe), is that when the universe expands by one Planck length, there is one additional node added, allowing for one more half wavelength, or λ(t)=c.t/2.  Each half wavelength corresponds to an extra half a unit of Planck mass/energy, so the M value above (if it were the mass of the universe) would be M(t)=m_P/2.t/t_P.

Note that because, in the FUGE conception, increments are in half wavelengths, Haug’s “factor” would become (noting the caveats above):

A Tiny Error in All Objects and Some Questions

 

In The Mass of Everything, I referred to the paper All objects and some questions and most specifically the image below:

That figure had a long text below it which includes the following statement: “The smallest possible object is a Planck-mass black hole indicated by the white dot labeled ‘instanton’ (Ref. 20). Its mass and size are (m,r)=(m_P,l_P).”  Ref 20 is a paper by Carr and Rees called “The anthropic principle and the structure of the physical world” published by Nature in 1979 and not easily accessed.  I do note that Carr wrote a later paper, Does Compton / Schwarzschild duality in higher dimensions exclude TeV quantum gravity?, in 2018 which includes this image:

The paper also includes this statement: “The Compton and Schwarzschild (radius) lines intersect at around the Planck scales, R_P = √ħG/c³ ≈ 10⁻³³cm, M_P = √ħc/G ≈ 10⁻⁵g”.  Note the lack of precision in Carr’s paper.  This is, as it turns out, fully justified.

The Schwarzschild radius is given by r_S=2GM/c², so where the radius is the Planck r_Pl=√ħG/c³ length we get:

r_S=2GM/c²=√(ħG/c³)

So,

M=√(ħG/c³).c²/2G=√(ħc/G)/2=m_Pl/2

Therefore, the instanton must be (m,r)=(m_P/2,l_P) or, possibly, (m,r)=(m_P,2l_P), but it cannot be (m,r)=(m_P,l_P).

Alternatively, Lineweaver and Patel could have written “The smallest possible object is in the range of a Planck-mass black hole indicated by the white dot labeled ‘instanton’ (Ref. 20). Its mass and size are of the order of the Planck mass and length.”  And the chart would have to be updated to put approximations against the m_P and l_P that intersect on the instanton.

Or, and this would be more appropriate, put a “/2” after m_P in both instances and reword to “The smallest possible black hole has a Planck-length radius indicated by the white dot labeled ‘instanton’ (Ref. 20), and mass of one half of a Planck-mass.”

The Mass of Everything

There is a paper in the American Journal of Physics that got some news attention in mid-October 2023.  The version that I first saw excitingly implied that it provided evidence that the universe is a black hole, which naturally caught my eye. (Anton Petrov also put out a video about it.)

 

This is the image that was being heralded:

 

Up in the top right corner, we can see the Hubble radius, which is the age of the universe times the speed of light (which is also the speed of light divided by the Hubble parameter).  Above that and to the left is a region marked as “forbidden by gravity”, basically indicating that anything in this region of the chart would be denser than the densest of black holes (a non-rotating black hole).

 

We will return to that, but the first thing that leapt out to me about this chart was the fact that atoms, the Covid virus, an unspecified bacterium, an unspecified flea, the (average) human, an unspecified whale, planets, moons, the Earth, main sequence stars and the Sun are all in a straight line.  That line seems to have the gradient log₁₀M/log₁₀r≈3 with little or no offset.  This makes eminent sense since there’s an established relationship between the mass of something and its volume, mediated by its density.  What might not be so immediately obvious is the question of the offset.

 

Let us introduce the density as ρ, and note that we are talking about a radius, such that the associated volume is 4πr³/3, so M=ρ.4πr³/3= (ρ.4π/3).r³ and therefore

 

log₁₀M=log₁₀((ρ.4π/3).r³)=log₁₀(ρ.4π/3)+log₁₀(r³)=3log₁₀r+log₁₀(ρ.4π/3)

 

If there were little or no offset, this would imply that log₁₀(ρ.4π/3)≈0 and so ρ.4π/3≈1, or ρ≈0.25.  However, in the paper, it’s noted that the line is consistent with the density of water (1g/cm³).  This represents an offset of log₁₀(4π/3)=0.6.

 

I looked at the background data (provided in a zip file) and noted that they estimated the radius of spherical humans, blue whales and fleas using the following process – To get the radius, take the length, divide by 2, that gives you a "ball", then divide by 2 again because we want the radius, not the diameter of the ball.  The figures that they used were (where yellow fill means that the values are received, and white means they are calculated, noting that the virion here is a Covid virus particle):

 

I went into a bit more effort and got (using the same colour coding):

 

Even though the virion and human densities is very wrong from their estimate, and the whale was somewhat wrong, there was very little effect on the relationship:

 

 

Note that the blue line in my chart represents black holes of different mass and the thin red line is log₁₀M=3log₁₀r+0.6 with the dots near it coming from the table above.  The other dots, left to right are, the Milky Way galaxy, the current Hubble radius and the “observable universe” (inflated radius of 46.5 ly).

 

We’ll get back to the “observable universe” later.

 

The second thing that leapt out at me was that there are other apparent lines on the chart, albeit shorter. These lines have the same gradient but a different intercept with log₁₀M=0.  There’s a vague line created by globular clusters, a more distinct line for galaxies and clusters of galaxies, and then a line for super-clusters right up against critical density (the density of the Hubble radius).  The offset for galaxies seems to imply a density in the order of about 10^-25g/cm³.  Note that there are voids illustrated that are below the critical density, which makes sense given that if some areas are higher than the critical density overall for a larger region, then some areas must be lower than this density – and these would be voids.

 

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So … the observable universe.  According to calculations performed by Ned Wright (although he might not be the original), the observable universe has a radius of about 47 billion light years – if one includes the assumption of accelerating expansion from about 5 billion years ago, as required to get the appearance, today, of constant expansion at the speed of light since 13.787 billion years ago, due to the disturbances caused by a period of inflation and two different periods of slower expansion. In The Problem(s) with the Standard Cosmological Model, I explained why I have issues with that explanation but if we accept that the observable universe is that big and note that the argument for that radius is based on the assumption of a critical density which is pretty much the density that we observe, then we have a problem.

 

The large orange dot in my chart above shows where the observable universe plots to, given its density and its radius (and thus its volume).  It can be seen to sit above the black hole line and thus in the “forbidden by gravity” zone, or (a little less clearly):

 

 

And, if you read Ned Wright further, you will note that he goes on to say that the universe, as a whole, is more than 20 times the volume of the observable universe.  This is a low-end estimate if space.com is to be believed, given that they report a measurement of 7 trillion light years across, or 3.5 trillion light years as a radius.  They also suggest that another possible figure is 10²³ light years, but this is clearly muddled since the universe would have expanded faster than the speed of light during any inflation event (not at the speed of light).  Plotting these three values as well, maintaining the same critical density (which is an assumption that is common to all three), we get:

 

So, something seems wrong here.  It does make me wonder, given that the paper says “we plot all the composite objects in the Universe: protons, atoms, life forms, asteroids, moons, planets, stars, galaxies, galaxy clusters, giant voids, and the Universe itself”, they don’t plot the universe itself, not even the “observable universe” … unless the authors’ view is that the Hubble radius is the radius of the universe.

 

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One issue raised by various people, including Anton Petrov, is that for the universe to be (inside) a black hole then there could be nothing outside of it, which introduces the issue of how the density would suddenly plummet to zero at the boundary.  Generally, it is thought that the density outside of the Hubble radius is the same as inside (because we are not privileged, our part of space in not special) – and this is the basis on which I chart the “observable universe” above the black hole line.

 

Note that Anton at one point misrepresents the Hubble radius as the radius of the “observable universe”.  The Hubble radius is 13.787 billion light years, not 93 billion light years.

 

Note also that, just after 11:00, he says that when a black hole gets big enough you can technically go inside it and feel nothing, so I am not convinced by the argument that outside the Hubble radius would have to have zero density for us to be in a black hole.

 

If the region inside the Hubble radius constitutes a black hole (which it does, since it has the radius, mass and density of a black hole of its radius, mass and density), then it’s possible to have black holes inside of black holes, since there’s a supermassive black hole at the centre of many galaxies and probably many other, smaller black holes in other locations.  Consider then the possibility that the Hubble radius itself is inside another, larger black hole.

 

Say, for example, that there’s a greater black hole the size of the “observable universe” at 93 billion light years.  What would its density be?  The calculation for the density is ρ=3c²/8Gπr², which for 93 billion light years is 8.3×10^-31g/cm³, or a little under one tenth the critical density of the universe of ρ_c=9.4×10^-30g/cm³ while having a volume that is 38 times greater.

 

Then consider the nesting of many effective black holes between the Hubble radius and the radius of the “observable universe” and think of one that is only slightly larger than the Hubble radius – 15 billion light years.  The density of that black hole would be 8.0×10^-30g/cm³ or about 85% as dense as where we find ourselves, with a total volume that is 30% greater.

 

So then the question is, how dense would it have to be beyond the Hubble radius to still constitute a black hole at 15 billion light years out?  I estimate that it would be in the order of 30% of the critical density.  And for a 93 billion light year radius, the average would be 6%.  The bottom line is that density doesn’t need to suddenly drop to exactly zero, even though we do introduce an apparent problem with privilege since the region inside the Hubble radius is special (due to being towards the centre of a zone that is of higher density than the surrounding area).

 

This is not, however, the only solution.  The other solution, more consistent with the notion of FUGE, and the implication of the paper’s writers, is that there’s nothing outside of the Hubble radius – not even empty space.  The problem with that is that it also seems to indicate that we are in a privileged location, because we appear to be at the centre of such a universe - unless the geometry is such that all points within appear to be central.

MOND, FUGE and Dark Matter Light

In What FUGE does not explain, I make the outrageous claim that dark matter does not exist (at least not in the FUGE model).  This is based on the fact that, in the FUGE model, the mass-energy of the universe at this time is equivalent to 8.77×10⁵²kg.

 

However, it should be noted that the amount ordinary matter in the universe is calculated, per the standard cosmological model, to be 1.46×10⁵³kg, which is higher than I arrive at.  This is because of the assumption of inflation, and the assumption of dark matter and dark energy.  The 1.5×10⁵³kg figure is based on an assumption of total energy density of 9.9×10^-27kg/m³, applied to a universe that is 46.5 billion light years in radius and multiplied by 4.8% (the proportion of ordinary matter in the standard cosmological model).  If we apply that critical density to a FUGE universe, with a radius of 13.77 billion light years, without reducing it, the figure becomes 9.17×10⁵²kg.  Note that my calculated critical density, for 13.77 billion light years, is 9.448×10^-27kg/m³, hence the 8.77×10⁵²kg figure above.

 

The complexity of this, and the outrageousness of my claim, caused me to search for any evidence that dark matter does not exist.  I found that there are indeed people who believe, for other reasons, that dark matter may not be real.  One such person is Pavel Kroupa, a professor of astrophysics at the University of Bonn and the Astronomical Institute of Charles University in Prague.  He claims that his observations falsify dark matter as a hypothesis and favours MOND, or modified Newtonian Dynamics.

 

MOND relies on an acceleration constant, a₀, which the theory’s creator (Mordehai Milgrom) worked out was about 1.2×10^-10m/s².  Basically, the theory posits that gravity works one way in high acceleration scenarios and another way in low acceleration scenarios (where a is much lower than a₀).  Unfortunately, Milgrom worked out the value via a form of numeromancy, taking the data and working out what value of a₀ would make this theory fit.

 

However, it is interesting to note that if we set a₀=c.H₀/2π, where H₀ is the inverse of the age of the universe (1/13.77 billion years = 2.301×10^-18s), we arrive at a₀=1.098×10^-10m/s².  The physical meaning of this would need to be established and note that, if it is a true relationship, then it would imply that a₀ would be a parameter that decreases with the age of the universe.  If so, then it should be possible to see hints of that in the universe today.

 

Another problem with MOND, as detailed by Milgrom himself at Scholarpedia (a location where, I discovered later, the relationship 2πa₀≈c.H₀ was identified), is that:

 

For galaxy clusters, MOND reduces greatly the observed mass discrepancy: from a factor of ∼10, required by standard dynamics, to a factor of about 2. But, this systematically remnant discrepancy is yet to be accounted for. It could be due to, e.g., the presence of some small fraction of the yet undetected, “missing baryons”, which are known to exist (unlike the bulk of the putative “dark matter”, which cannot be made of baryons).

 

Note commentary in What the FUGE model does not explain.

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And yes, I am suggesting that perhaps there *might* be some dark matter, just not as much of it as previously thought.  Call it “dark matter light”.

Note that within the MOND world, there remains a category referred to as "missing baryons" to cover a mass discrepancy, but that is not considered (by Milgrom) to be "dark matter".  However, since "dark matter" is a catch-all term to describe the phenomenon, not necessary matter per se, my pathetic little joke still works.

I should also be noted that Pavel Kroupa seems to be saying there is no dark matter whatsoever, but it's unclear whether this means there is no mass discrepancy.  I have sought clarity on that question.

The Problem(s) with the Standard Cosmological Model

I have talked quite a bit in recent posts about the FUGE model which I see as the simplest model that results in a universe that looks like ours, given certain parameters.

The alternative, the standard cosmological model, which involves a short period of “standard” expansion, then inflation, then two periods of deceleration and finally the current acceleratory period, is fiendishly complicated.

To illustrate, I have put together some graphs that show a comparison between the FUGE model and the standard cosmological model.

There are some assumptions that I have had to make, and I will try to specify them comprehensively here because there are varying assumptions made under the umbrella of the standard cosmological model.  First, however, there are the two (yes, only two) assumptions for the FUGE model, which are that:

• for every unit of Planck time, the radius of the universe increases by one unit of Planck length, and
• for every unit of Planck time, the mass-energy in the universe increases by half a unit of Planck mass.

The assumptions made (by me) about the standard cosmological model are that:

• the density today is equal to the critical density (NASA)
• the radius of the total universe is currently 46.5 billion light years – where this is the minimum size, based on calculations of comoving distance to where the cosmic microwave background originated (wikipedia but also SCSU and UCLA)
• the total mass of ordinary and dark matter is invariant – that is both have been in the universe since before the inflationary period and the total combined mass does not change (Ethan Siegel)
• the density of dark energy is invariant (Ethan Siegel)
• the distribution of ordinary matter, dark matter and dark energy is 4.6%, 24.0% and 71.4% (WMAP)
• critical density is given by ρ_c=3H²/8πG (wikipedia)
• prior to the inflationary period the universe expanded at the speed of light (with an equation of state value of w=-1/3)
• the inflationary period was between t=10^-36 to t=10^-32s (wikipedia), expanding the universe by a factor of 10²⁶ (wikipedia), at a rate that peaked midway (at about t= t=5×10^-33s), the graphs show an notional acceleration and deceleration to achieve the decelerated rate of expansion from t=10^-32s onwards
• during the radiation-dominated era which extended to 47,000 years (wikipedia), the equation of state value was w=1/3
• during the matter-dominated era which extended from 47,000 years to about 9.8 billion years (wikipedia), the equation of state value was w=0, and
• during the current inflationary era, the dark-energy-dominated era (wikipedia), the equation of state value is w=-1.03 (Planck collaboration - via wikipedia).

Note that all the graphs below extend from when the universe was one unit of Planck time old to 328 billion light years and they are in logarithmic scale (on both axes).  Age of the universe is given in seconds.  The right extent brings us to when the Big Rip has started (assuming accelerated expansion at w=-1.03).  The left extent eliminates the need to illustrate a potential infinite density at t=0.  Personally, I don’t think there was ever infinite density*.

The first shows how the radius of the universe has evolved over time.  To emphasise, I have compared the radius to the age of the universe multiplied by the speed of light:

The next shows the quantity of mass-energy, expressed in terms of mass.  To emphasise, I have compared the mass-energy to the age of the universe (in units of Planck time) multiplied by half a unit of Planck mass.  I have zoomed in, eliminating the upper half of the straight line for the standard cosmological model in order to highlight the kink that kicks in during the dark-energy-dominated era.  This should not surprise, because this is the “dark-energy-dominated era”, however the timing of that upward spike should be surprising – right now.  This is, however, just an artefact of the graph.  That bend happens at 9.8 billion years, and there’s very little difference between log₁₀ of 9.8 billion and 13.7 billion (about 2%):

Third is mass-energy density, again expressed in terms of mass density.  To emphasise, I have compared the density to critical density (as it changes over time).

Note that universe has, in the standard cosmological model, had a critical density three times – briefly during the inflationary era, again briefly about one ten thousandths of a second and … right now.

Finally, the Hubble parameter expressed in kilometres per second per megaparsec.  To emphasis, I have compared the Hubble parameter to the inverse of the age of the universe (when expressed in kilometres per second per megaparsec):

Note that the Hubble parameter was equal to the inverse age of the universe three times.  From the beginning until just prior to the inflationary era, at the end of the inflationary era and … right now.

Those standard cosmological model curves are some real crazy shit.  And the assumptions required … well, let’s just say I have my reservations about those too.

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* Even before the FUGE model, I suspected that initial packing of the universe’s mass-energy would have been the equivalent of one unit of Planck mass per Planck volume (notionally a sphere with radius of one unit of Planck length).  I thought that this would have been incredibly hot and unstable, trying desperately to expand while being held in a vice-like grip by gravity (since it was much denser than a black hole of equivalent mass-energy).  When a rupture opened up, I imagined that the universe would have flipped inside out, leading to space expanding rapidly, with the unleashed mass-energy on its tail (figuratively, of course) – and thus the Big Bang.  I no longer think that this is viable, since that density would have been hugely more than that of a non-rotating black hole of the same mass.

Mathematics to Address an Apparent Problem with Imagining a Universe

Imagine a Universe contains only narrative with no equations.  Before I posted that narrative, I posted a piece that explained that I understood that there are at least three apparent problems with the narrative, which I archived for posterity before overlaying it.  Just as with the narrative itself, I tried to minimise the use of equations – which was a little tricky with regard to the glome.  What follows is a very brief explanation as to how mass/energy enters to the inner universe at a rate of one unit of mass/energy per unit of time.

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Below is what I find most problematic to explain without recourse to equations (and even with them, a little):

·        To be nice and neat, it would be great if the inner universe receives one “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length.  That does not initially seem to be the case though, it’s one Planck mass worth of energy for each two units of Planck time during which the universe expands by two Planck lengths, per Hubble volume (which is the sphere defined by the radius of the universe at that time, recalling that universe is a glome).  This gets a little confusing in four dimensions and I am not entirely convinced by people who say they can imagine what a four-dimensional object looks like, so let’s consider a sphere as an analogy.  We can get circles from a sphere by sectioning it.  The greatest circle we can create has the same radius as the sphere itself.  The sectioning effectively creates two hemispheres.  Note that I remain aware that the surface area of the curved section of the hemisphere is not equal to the surface area of the circle created by the section.  By analogy, the universe could be notionally sectioned by a spherical section, creating two halves, meaning two (three-dimensional) Hubble volumes, meaning that the nice one “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length is obtained for the universe as a whole.

I described how, if the universe is spatially flat, the mass (or “mass-energy”) of the universe increases by one unit per two units of time in Mathematics for Imagining a Universe, under the rubric “Critical Density and Expansion”.  In that section I wrote (emphasis added):

Which means that, within a Hubble volume, mass increases at a rate of half a Planck mass per Planck time to maintain critical density.

The challenge is to understand how the universe might have a volume of two Hubble volumes, thus making the mass increase at a rate of one Planck mass per Planck time (to maintain critical density).

What I am describing above with the sphere is the perspective of a 2D character living on the surface of that sphere, let’s call him Fred.  Say that Fred occupies the x and y dimensions, while the sphere occupies the x, y and z dimensions.  The sphere can be described as x² + y² + z² = r², where r is the radius, but Fred cannot perceive the z dimension so as far as he is concerned the relevant equation is x² + y² = r², which is a circle, with himself at the centre – or at coordinates (0,0).  Due to his position and dimensional limitations, however, Fred can only see one half the sphere on which he lives.

Say that Fred is actually at (0,0,r) and that a fellow sphere dweller, Freda, is at (0,0,-r), also perceiving herself to be at (0,0).  Freda too will only perceive a circle, but that circle has no overlap with Fred’s despite also being described, by Freda, as x² + y² = r².  In three dimensions, the two circles are clearly different being (x² + y² = r², z=r) and (x² + y² = r², z=-r) and are descriptions of two separate halves of the sphere, the z-positive hemisphere and the z-negative hemisphere.

Note that any other (non-collocated) 2D observer in that universe will also perceive themselves to be in the centre of a circle that describes half of the sphere, but that hemisphere with overlap with both Fred’s and Freda’s.

Precisely the same logic applies with 3D observers, like ourselves, living in the surface volume of a glome described by w² + x² + y² + z² = r².  We cannot perceive the additional dimension (w), so we see ourselves as being in an apparent sphere (a Hubble volume), but we cannot access the other half (the -w hemiglome, if you like).  The division between positive and negative halves, however, seems irrefutable making the 3D perceivable volume of the universe twice that of the Hubble volume.

Given that the rate of increase in mass is half a Planck mass per Planck time per Hubble volume, then the rate of increase of mass into the universe as a whole is one Planck mass per Planck volume – if the universe is spatially flat.

And is the universe spatially flat?  It very much looks like it.

Apparent Problems with Imagining a Universe

The text below was initially posted where Imagine a Universe is now to be found (see Internet Archive version).  It has been edited slightly to bring it up to date and some retrospective rewording for clarity.

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This text (the content of the post "Imagine a Universe") was replaced shortly after it was posted, as I wanted to record it for posterity at archive.org.  The replacement content is the intended content, which is a narrative about a universe that undergoes expansion, until that expansion stops (for reasons that are unexplained at the time), then gravity takes over and the entirety of the universe eventually ends up in one ginormous black hole, which (effectively) shunts all the universe’s mass/energy into an orthogonal universe.  The incoming mass/energy (effectively) powers that inner universe’s expansion, until the mass/energy of the outer universe is entirely transferred into the inner universe, at which time the inner universe’s expansion stops.

The purpose of this “underlay” to that narrative, if you like, is to note that I am aware of at least five apparent problems with the story:

First, there is an implication of a meta-time.  As described elsewhere I have posited that the expansion of the universe is basically time as experienced in that universe.  However, if the expansion reverses, this is indicative of a sequence associated beyond expansion which in turn implies another sort of time, or a meta-time.  It is possible that this implication is a function more of how our brains work, immersed as they in actual time.  An alternative, at least as I see it, is block time which is linked to a form of hard determinism (or nomological determinism).  Maybe there’s a compromise position in between (that is a universe with no meta-time but without everything being effectively predetermined).

Second, which has two parts (neither of which is really a problem, more of an explanation):

•    The inner universe is a glome which has a “surface volume” which is greater than the volume of a sphere of the same radius.  The idea of critical density that is mentioned is related to the radius of a sphere, specifically the sphere that defined by the distance that light could travel in the time that the universe has been in existence (and, relatedly, since it started expanding).  On the “surface volume” of glome, however, the radius is a little vexed, much the same as it is when considering drawing a (relatively large) circle on the surface of a sphere.  Is the radius that of the flat circle created by sectioning the sphere, or the arc length between the point on the sphere directly above the centre of that section and its outer rim?  That doesn’t really create a circle anyway, and its area is both greater than the circle created by the section and less than a circle as defined by a radius equal to the arc length.  Would two dimensional beings on the surface of such a sphere notice?  I don’t think so, since they would only be sensitive to two dimensions, which would notionally be aligned with a plane passing through the locus of the sphere. 

•    To be nice and neat, it would be great if the inner universe receives one “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length.  That does not initially seem to be the case though, it’s one half Planck mass worth of energy for each unit of Planck time during which the universe expands by two Planck lengths, per Hubble volume (which is the sphere defined by the radius of the universe at that time, recalling that the universe is a glome).  This gets a little confusing in four dimensions and I am not entirely convinced by people who say they can imagine what a four-dimensional object looks like, so let’s consider a sphere as an analogy.  We can get circles from a sphere by sectioning it.  The greatest circle we can create has the same radius as the sphere itself.  The sectioning effectively creates two hemispheres.  Note that I remain aware that the surface area of the curved section of the hemisphere is not equal to the surface area of the circle created by the section.  By analogy, the universe could be notionally sectioned by a spherical section, creating two halves, meaning two (three-dimensional) Hubble volumes, meaning that there is a nice whole single “unit” of energy for each “unit” of time during which the radius increases by one “unit” of length is obtained for the universe as a whole.

Third is the orthogonality.  What I am really talking about here is a orthogonal rotation of spacetime, such that while events in the outer universe could be described by the quaternion P = xi + yj + zk+ t, events in the inner universe would be described by a related quaternion P’ = x’l + y’m + z’n + t’(o).  Where the dimensions represented by i through to (possibly) o are all orthogonal to each of the others.  I say “possibly” about the dimension represented by o, which is linked to time, because of issue described below.

Fourth, and this is probably most obvious, at least to some: in my conception of this whole thing, time is equivalent to expansion.  Once expansion stops then, we have a problem.  Also, I do have a notion of gravity being linked to the expansion, since gravity is what happens where the expansion is resisted by a concentration of mass/energy, rather than being an attractive force per se.  Which means that once expansion stops … either something really bad happens, or nothing else happens.  Unless, that is, the process begins again but in a sort of reverse.  Just how that would play out is not something on which I intend to speculate other than to suggest that, if this is the mechanism, then conceptually the “reservoir” in which mass-energy resided during the inner universe’s expansion, or a similar if not the same reservoir, would begin filling with mass-energy that leaves the inner universe and, in the process, powers a contraction as the critical density is maintained.

The “really bad” is similar to something that I’ve not really considered to be a realistic option, until now – vacuum decay.  The effect, as I have envisioned it, would be to negate the structure of the universe, including not only time (via expansion stopping), but also space, possibly twisting the whole universe in an orthogonal direction and thus seeding a new, inner-inner universe that goes through the same process (with the expansion again being representative of time and thus not requiring the dimension o mentioned above).  I don’t, however, see this event as something that would initially happen in one part of the universe before spreading out at the speed of light, like some sort of cosmic cancer.  It would happen instantaneously across the entire universe as the mass/energy feed ends.  (For fans of the Marvel Universe, imagine a much more substantial snap of the fingers of an uncaring god.)

Fifth, it’s not obvious in the narrative because I’ve deliberately avoided – as much as possible – reference to equations, but the expansion of the universe is related to the critical density of a universe and this is mathematically linked to the mass and radius of a Schwarzschild black hole.  A black hole that contains the final mass of the inner universe will have the final radius of the inner universe.  We would not see, therefore, the entire universe being scrunched down into a tiny black hole.  Instead, as the black hole absorbed all the mass-energy of the universe, it would expand to fill the universe.  At that time though, relative to the outer universe, the entirety of that universe would be smeared over the surface of the universal black hole.

An image might make this slightly more comprehensible.  Say this is a standard universe:

There are some models which expect the universe to stop expanding, then reverse and trigger another universe in the opposite (notionally temporal) direction, setting up an oscillation (or bounce):

There there is a continuously cyclic universe (conformal cyclic cosmology), which is a bit like this:

The universe which is implied in “Imagine a Universe”, would look a bit like this, similar to the oscillating (or bounce) universe, but with the next universe rotating off into an orthogonal direction:

However, what I (currently) have in mind is more like this:

The universe expands out to its maximum, then a “really bad” thing happens (maybe like vacuum decay, maybe the entirety of the universe being absorbed into one final black hole) and the universe becomes, to the next universe in the sequence, a big bang-like event.  The universe that follows in the sequence but is not shown is also orthogonal, so if it helps you can imagine it emerging from the screen, but keep in mind that this is merely a representation of the notion involved.

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In any case, the description of both universes in “Imagine a Universe” is wrong – but it was intentionally, knowingly wrong, while potentially pointing to an underlying truth.  A sort of lie-to-children.