Mathematics for Taking Another Look at the Universe

In Taking Another Look at the Universe, I blithely introduced the equation x'=(ct₀-x).x/ct₀ (where x=ct) without having given any derivation or real explanation as to what the terms mean.  I made the equation very slightly neater than it had been but, in the process, I might have made it less comprehensible for people like myself who like to work from first principles.

The central term is, of course, x – the distance to an event.  The other term, t, is the time of that event – but, in reality, it is more like the difference between the timing of the event and now, so it could be thought of as Δt.  Similarly, x could be more accurately described by Δx – but for reasons that may become clear shortly I dropped the “Δ” from both.

The term t₀ is used such that the subscript aligns with H₀, the current value of the Hubble parameter, and so t₀ is the (current) age of the universe while x₀=ct₀ is the current Hubble length (or the current radius of a FUGE universe).  By analogy, x=ct, where x is the location of an event.

Note that this relationship follows from the notion that we can only observe an event if there has been sufficient time for the photons from that event to reach us.  However, while photons are travelling to us from the event, the space in between is also expanding.  Therefore, for any observed event, there are two components, a temporal one (due to how long ago it happened) and a spatial one (the distance from our location to where the event happened at the time).  The latter is what x' on the vertical axis represents in this chart:

It may get a little complicated here.  I suspect this because I have already explained poorly (possibly more than once and had to start again, such as right now when I am editing), initially due to not getting the conceptualisation quite right and at one point I even got close to persuading myself that the equation must be wrong.

Consider it this way, a maximally distant observed event (MDOEs) in the very distant past was (when it happened) not as distant in space from us (as observers) because the universe had not expanded very much (at that time in the very distant past).  The equation x'=(ct₀-x).x/ct₀ specifically considers MDOEs.  Note that observed events, including any and all MDOEs, have both a space and a time coordinate relative to the eventual observer, basically telling us how distant from the observer the event was at the time and how long ago the event happened relative to the observer who is notionally at rest (relative to the CMB).

The most distant MDOE (in any given direction) would have occurred when the universe was half its current age.  For ease we can call this the ½t₀ event, or “½t₀e” (half-toe). 

Since ½t₀e, all MDOEs have by necessity been less distant because there has been less elapsed time for photons from those events to reach us.  Before ½t₀e, all MDOEs were less distant because the universe was smaller. 

We can consider the universe as being divided into two eras, a pre-½t₀e era and post-½t₀e era, with there being events in both eras that occurred at locations that were equally distant from us (at the time they occurred), meaning that they notionally travelled the same distance in unexpanded space, but photons from the event in the pre-½t₀e era will have experienced more expansion during transit.

A marked-up version of the image above may help to illustrate this fact:

Let us take the most extreme example, the instanton event happened about 13.787 billion years ago.  There is effectively no distance to where that event happened, because the maximum expansion one could consider to have happened at that time is one unit of Planck length.  As a consequence, the entirety of the distance between us and where the location of that event is now is due to expansion.

The next most extreme example illustrated above is an MDOE almost 2 billion years later, by which time the universe had expanded to a radius of 2 billion light years.  Photons from that MDOE were not at the full extent of the universe at the time however but rather at 1.565 billion light years.  Note that the location of that event (following the light green line up to the left) is currently 12 billion light years away, indicating the amount of expansion that has been incurred between our location and the location of the event at that time place is 10.435 billion light years.  Therefore, the time taken for a photon to reach us is 1.565 billion years due to the original separation plus 10.435 billion years due to expansion, or 12 billion years, precisely what we would expect.

The upright light green section can be calculated using the following:

Note that sinϴ=x'/√(x'²+(ct₀-ct)²)=x/√(x²+(ct₀)²), so, noting that x=ct:

x'²/(x'²+(ct₀-x)²)=x²/(x²+(ct₀)²)

x'².(x²+(ct₀)²)=x².(x'²+(ct₀-x)²)

x'².(ct₀)²=x²(ct₀-x)²

x'.(ct₀)= (ct₀-x).x

x'=(ct₀-x).x/ct₀=(ct₀-ct).t/t₀

This should come as no surprise, since this is the equation that I charted.

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Let us consider the (clean) chart again:

The apparent distance to any event is the length of the curve.  If we divide the curve into a relatively large number of finite elements and approximate the curve by summing the length of all those finite elements, we arrive at 15,800 million (light) years.  As mentioned in Taking Another Look at the Universe, this is well within the range used by Lineweaver and Egan (but they got it from integrating a(t) over the age of the universe, if I understand it correctly).

Note also that I asked a question about dark energy in Taking Another Look at the Universe.  We can use these finite elements to take a look at the apparent Hubble parameter value at all points along this curve as the universe expands, and we get a curve that looks like this:

The “apparent” H is based on a set of calculations, using a change in the age of the universe by a millionth of 1% and the consequent change to the values of x'.

The shape of the “Apparent H” curve is of particular interest.  Consider it with respect to the discussion in The Problem(s) with the Standard Cosmological Model and the eras discussed at the Scale Factor page at Wikipedia.

There really are only two eras observable in the chart, from about 7 billion years ago to now (to the left) corresponding to the “dark-energy-dominated era” and the period before that (to the right) corresponding to the “matter-dominated era”.  The radiation-dominated era and the purported era of inflation (plus era that preceded it) are not distinguishable at the scale used.

The chart indicates a very similar situation as that posited with the introduction of dark energy, but without requiring any actual dark energy.  The most recent era appears to have acceleration.  The only times that the apparent Hubble value is equivalent to the inverted age of the universe are at the transition between the “dark-energy-dominated” and the “matter-dominated era" and for a very short period of time a maximally long time ago/(apparently) far away – pre-inflation.

Note that a lack of dark energy is consistent with the mass of the universe being ~10⁵³kg (the mass one would expect in FUGE universe that is 13.787 billion years old).

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I do acknowledge that the “apparent” values of H in the recent past/near vicinity are very high.  This may be worthy of further investigation.

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Update: having done a bit more investigation, I cannot show that this is a factor because the evidence is lacking.  See Apparent Hubble Parameter Value.

Taking Another Look at the Universe

It might be a big claim here, but I suspect that we might be looking at the universe incorrectly.

Generally, we tend to think of the universe a little like this:

I’m not saying that this is entirely wrong (so long as the circle sort of represents a sphere), but this is not how we see the universe.  What we see off at the distance, ~13.787 billion light years away, is the cosmic microwave background that originated no more than 380,000 light years away from us.

It could be more accurate to represent how we view the universe as like this:

That’s not to say that we are outside the universe, per se, but we are certainly not in the universe that we see.  The universe that we see is in the past.  It is merely a trick of perspective that the universe appears to be all around us even though the furthest reaches of what we can see (apparently 13.787 light years away) only arose about 380,000 light years from where we are now.

However, not even this is correct.  There is a relationship between how long it took light from an event to arrive where we are and where it started from and where that location is now.  This relationship is given by the pair of equations: x'=(ct₀-x).x/ct₀, x=ct where t₀ is time since the instanton, t is time since the event, while x and x' is the distance to where the event took place (actual and observed).

This graph illustrates the concept (noting that we are considering a FUGE universe):

Interestingly, the length of that blue curve from intercept to intercept on the horizontal axis … is ~1.148 times that of the length of the horizontal axis between intercepts – if laid flat, that would be 15,800 million years, or 15.8Gyr which, when multiplied by the speed of light, is well within the range of 15.7±0.4 Glyr as used by Lineweaver and Egan (see FUGE Entropy).

So, the question must be asked, is the apparent variability of the scale factor merely an artifact of our observation of the universe?  And if so, does it explain what is currently explained by the introduction of dark energy (noting that small values of Δx translate to values of Δx' that start off at about 1.4 times x and decrease towards the top of the curve to equivalence [at 6.9 billion years ago] and then increase again)? 

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Note the section of the graph that relates to the “dark-energy-dominated era” (using the description from Space Telescope Science Institute's HubbleSite page on dark energy* – “About halfway into the universe’s history — several billion years ago — dark energy became dominant and the expansion accelerated”:

Is it possible that what appears to be dark energy could be an artefact of observation?

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* Note that on the Wikipedia page on the scale factor, the section on the dark-energy-dominated era indicates that this era began when the universe was about 9.8 billion years old, but the reference is from a 2006 book whereas the HubbleSite page was updated in late 2022 so, while it has a more vague reference, it should be considerably more current.  There is also a claim by the Department of Energy (which seems to be paraphrased from a 2022 article by post-doctoral cosmology researcher Luz Ángela García at space.com) that “somewhere between 3 and 7 billion years after the Big Bang, something happened: instead of the expansion slowing down, it sped up. Dark energy started to have a bigger influence than gravity. The expansion has been accelerating ever since”.  Ethan Siegel has the dating at “about 6-to-9 billion years ago”.  Other sources of varying levels of authoritativeness give the figure as about 7 billion years ago (for example Eric Lindner from the Supernova Cosmology Project – but there is no date on the page, so it’s difficult to assess whether this is based on recent work or was just a good guess from as long ago as 2010).

A Dark Question

Dr Becky Smethurst put a video out last week about a possible resolution to the “Hubble Tension”/“Crisis in Cosmology”.  The work has not yet been published, but instead is covered in a talk by Wendy Freedman, but it is interesting to note that the result that the JWST people arrived at is H₀=69.1±1.3km/s/Mpc (which corresponds with a Hubble Time of T_H=14.15+0.27 billion years).

It was quite timely because I was already thinking about expanding on something I was talking with someone about in the past week.

Imagine that soon after Erwin Hubble had identified the redshift of distant objects (in the 1920s), sufficiently advanced telescopes were developed and used to determine the value of the Hubble parameter to be close to 70km/s/Mpc (didn’t happen until the 1990s).  Say then that someone had quickly worked out that ~70km/s/Mpc is the inverse of ~14 billion years (fitting excellently with the age of the oldest known star, although its age was only determined to fit nicely after revision to models in 2015 and 2021).  Then, a short time later, someone else was fiendishly clever enough to use the technology available at the time to measure the geometry of the universe and determine that it is flat, meaning that the density of the universe is critical (this wasn’t really determined until 2000 with analysis of the BOOMERanG experiment results from 1997 and 1998).

So, in this hypothetical world we would have had, in about 1930, all the details necessary to conclude that our universe is a FUGE universe.  A FUGE universe starts out as an “instanton”, effectively a Planck black hole of half a unit of Planck mass-energy with a radius of one unit of Planck length, adding half a unit of Planck mass-energy and expanding its radius by one unit of Planck length every unit of Planck time.  Such a universe has a Hubble parameter that is the inverse of its age and has critical density throughout its life (meaning that it is, has always been and will always be flat).

Now say that in this hypothetical world, about 30 years after the FUGE universe model was established, someone discovers the cosmic microwave background (CMB).  Analysis of this raises bit of a mystery because the CMB has an unexpectedly high level of isotropy.

Under these conditions, would it be reasonable to posit inflation (about 15 years after the discovery of the CMB)?  Note that one of the motivations for inflationary theory would be missing in our hypothetical world, because the flatness problem would not exist – critical density (and thus flatness) of the universe is perfectly explained by the FUGE model.  The other motivations also have other potential explanations: gravity may suffice to explain the homogeneity of the horizon problem and the magnetic-monopole problem only relates to the absence of hypothetical particles (the standard approach, when finding that your hypothesis predicts the existence of some non-existent thing, is to reassess your hypothesis rather than engage in a form of special pleading – especially after 90 years have passed with no observational evidence).

Note also that in a hypothetical world which has accepted the FUGE model, we have a very simple chronology – with smooth expansion of the universe over ~14 billion years to arrive at a Hubble parameter value that is the inverse of ~14 billion years and a density that matches the observed (critical) density.  In order to arrive at the value of the Hubble parameter, after having introduced inflation, we have to posit  a much more complex chronology at least three phases: smooth FUGE-like expansion for a fraction of a second (grand unification epoch), inflationary expansion for a fraction of a second (during which mass-energy would have had to have been added at a much higher rate if critical density were to be maintained) and an approximately 14 billion year-long phase in which the expansion was precisely that necessary to make the universe today look like it had only undergone FUGE-like expansion.

Personally, I don’t think it would be reasonable.

Our situation is actually worse than described above because, in the Standard Model, there are five phases: FUGE-like expansion (grand unification epoch), inflation, two periods of reduced expansion (less than FUGE-like: radiation dominated and matter dominated) and a current period of accelerated expansion (greater than FUGE-like) at a rate necessary to make the universe today look precisely like it had only undergone FUGE-like expansion – a situation that would not have been the case since a fraction of a second after the instanton arose and won’t be the case ever again (because the explanation for observed accelerated expansion is that we are in a dark-energy-dominated era [other explanations are available] and such domination by dark energy is unlikely to suddenly dissipate in order for us to return to FUGE-like expansion on an on-going basis and we are unlikely to return to the conditions of earlier putative eras of reduced expansion [the radiation dominated and matter dominated eras]).

Is it truly reasonable to have such outrageous fiddling of the universe, given the option of the FUGE model (or something like it)?

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.

What FUGE does not explain

I acknowledge that the FUGE concept does not explain two things that contribute to the complexity of the standard cosmological model – the homogeneity/isotropy of the cosmic microwave background (at least not explicitly) and cosmic acceleration (at all).

 

The first is explained in the standard cosmological model by inflation, but there are other explanations other than inflation.  One of the authors of a Scientific American article on inflation together with Alan Guth, Paul Steinhardt, now disowns the theory, going so far as to suggest that inflationary theory “makes no testable predictions”.  The point here is not that Steinhardt’s theory (a cyclic theory of the universe) is necessarily correct either, merely that there are other ways of explaining what inflation set out to explain.

 

While it should be noted that I am not a cosmologist, I am somewhat more sanguine about homogeneity and isotropy.  If physics works the same everywhere in the universe, which seems a reasonable assumption, then if all locations began with the same conditions, in a very much localised area (relative to now), then it should not be surprising that our observations of the cosmic microwave background reveal that all parts of it have evolved over the period of 370,000 years to be pretty similar.

 

That might need a little bit of explanation.  At the beginning, for about 370,000 years, the universe was so hot that it was effectively opaque to photons.  This is known as recombination during the photon epoch – which is to say that none of the photons generated got very far before being absorbed by matter.  The cosmic microwave background is only what we can see from the time that the universe became transparent – we cannot see the Big Bang, we cannot see anything from an inflationary period, and we cannot see anything from a period of about 370,000 years after either of those.

 

Note that during those 370,000 years, the universe was a “hot dense plasma of nuclei, electrons and photons”.  It is pretty difficult to comprehend how a period of inflation of about 10^-32s that occurred 370,000 years previously would be instrumental in ensuring the level of homogeneity and isotropy that we can observe in the cosmic microwave background.  The argument already seems to incorporate the notion that physics would have operated the same way everywhere for the entirety of 370,000 years, leading the universe to evolve into a homogeneous and isotropic state.

 

Note this statement from the wikipedia entry on the photon epoch: “370,000 years after the Big Bang, the temperature of the universe fell to the point where nuclei could combine with electrons to create neutral atoms. As a result, photons no longer interacted frequently with matter, the universe became transparent and the cosmic microwave background radiation was created and then structure formation took place.”  If this is correct, and I have no reason to suspect otherwise, then once the temperature hit a certain point (apparently in the order of 10³K), neutral atoms were created and the universe became transparent.  So we should expect the entirety of the cosmic microwave background to be that temperature divided by the extent of expansion, even if the temperatures were reached at slightly different times.

 

Putting that in figures, the current cosmic microwave background temperature is 2.725K with a variation of 0.0002K between the “hot” and the “cold” regions.  Over the past 13.77 billion years (ish), the temperature has reduced by a factor of about 10³ due to the expansion of the universe (which has expanded by a factor of about 10⁴-10⁵).  The question then is: if different regions cooled down to the temperature required for neutral atoms to form at slightly different times, what effect would that have on temperature observed today?  Using the FUGE values, the universe has expanded by a factor of 3.7×10⁵ since the surface of last scattering when the cosmic microwave background was formed, assuming that it happened at precisely year 370,000 and that this is precisely year 13,770,000,000 (don’t get distracted by all the 3s and 7s, they are just an artefact of using the year as our temporal unit).

 

The standard model describes the CMB as originating from a hydrogen-helium plasma, condensing at a temperature of about 3,000K.  So, assuming this temperature to be precise, together with the 2.7250K value for the “cold” regions, the universe needed to expand by a factor of 3.76162162×10⁵ to reduce the temperature by a factor of 1.10091743×10³.  Assuming that the “hot” regions are precisely 2.7252K, how much later would they have been at 3,000K than the "cool" regions were?  It would require reduction by a factor of 1.100836636×10³, implying expansion of the universe by a factor of 3.721348495×10⁵, indicating that “hot” regions in the cosmic microwave background may have cooled down to 3,000K in the year 370,027.  So … the dappling on the cosmic radiation background that we can see could just be due to variations in the timing of the cooling of the universe by a factor of just under 30 years (or a bit under 0.01%).

 

There’s an additional assumption that can be added to the FUGE model, if one wants to explain homogeneity and isotropy, and that is that mass-energy entering the universe does so in a homogenous and isotropic manner.  This is a direct consequence of the cosmological principle (nowhere in the universe is special), so if energy is entering into or being created by the universe, then this will be happening to the same extent everywhere – similar to the notion of dark energy which involves a consistent density which implies the introduction of (mass-)energy across the universe at a rate equal to the increase in volume.

 

At year 370,000, in the FUGE model, the amount of mass-energy in the universe was equivalent to 2.356×10⁴⁸kg whereas, at 10^-36s, there was only 0.2018kg (noting that the radius at that time was 3.000×10^-28m).  This means that the vast majority of mass-energy in the universe at year 370,000 had entered after the time that, in the standard model, inflation would have commenced.  The only reason why one would consider the distribution of mass-energy at the notional time of inflationary period is that, in the standard model, there would already be 8.08×10⁵³kg of mass-energy in existence at that time.  That is simply not a factor in the FUGE model.  In other words, inflation is a solution to a problem created by the standard model.

 

Note that in the FUGE model there is, today, 8.77×10⁵²kg in the universe (so a density of 9.47×10^-27kg/m³).  The standard model has it that there is 1.5×10⁵³kg of ordinary matter, plus six times that of dark matter and more than twice that again in dark energy – in the observable universe.  This is based on the observable universe being 46.5 billion light years in radius, due to inflation.  The total amount of mass-energy in the Hubble sphere (about 14 billion light years), would be 9.17×10⁵²kg based on a density of 9.9×10^-27kg/m³.  Note that the critical density is related to the Hubble parameter which is not yet nailed down, so there is a range between 8.3×10^-27kg/m³ (Planck Collaboration) and 10.2×10^-27kg/m³ (SHOES) that my calculation comfortably falls into.

 

The standard model has the amount of ordinary matter in a Hubble sphere as 3.9×10⁵¹kg (1.5×10⁵³kg multiplied by the volume of a Hubble sphere, divided by the calculated observable universe volume [so about 2.5%]).  With the assumption that this is 4.8% of the total mass-energy, this is equivalent to 8.12×10⁵²kg in total (within the ballpark of the FUGE model estimate).  However, as the FUGE model does not distinguish between types of mass-energy, it’s worth looking at how much ordinary matter we can see using all the tools available to us (as opposed to how much can be calculated using other assumptions).

 

The observable universe contains about 10²⁴ stars (as is likely “a gross underestimation” and presumably based on an assumption of a density of galaxies and constituent stars applied to a universe of 46.5 light years radius).  According to Kroupa, the average stellar mass sits between 0.20 and 0.38 solar masses.  A solar mass is 1.989×10³⁰kg, so that’s between 3.98×10⁵³kg and 7.56×10⁵³kg in the observable universe, as a gross underestimate.  Given that the Hubble sphere is about 2.5% the volume of an observable universe that is purported to be 46.5 light years in radius, this equates to between 1.0×10⁵²kg and 1.9×10⁵²kg.

 

But this is just stars.  What about cosmic dust?  The intergalactic medium contains about one atom per cubic metre, presumably hydrogen.  The vast majority of space is intergalactic medium, so I am going to use the whole volume of a Hubble sphere for the estimate.

 

A hydrogen atom has a mass of 1.673557×10^-27kg.  So that equates to a total mass for the intergalactic medium of 1.54987×10⁵²kg (in a Hubble sphere), for a total of identified ordinary matter between 2.55×10⁵²kg and 3.45×10⁵²kg.

 

Within a galaxy there is the interstellar medium, and astronomers estimate that, in our galaxy, the mass of that medium is equal to about 15% of the mass contained in stars.  If our galaxy is average, then this is an additional 0.15×10⁵²kg to 0.28×10⁵²kg (for a new total between 2.8×10⁵²kg and 3.7×10⁵²kg).

 

There is also a question about nebulae.  I would not count concentrations of dust (etc) such as the Horsehead Nebula as contributing to the interstellar medium, but perhaps astronomers do.  Nebulae vary greatly in size, for example the Carina Nebula has about 4,300 solar masses while the Cat’s Eye nebula is a planetary nebula and has less than one solar mass.  What the average mass of a nebula is and what is the number of them in each of the galaxies are questions to which I cannot find the answer.

 

Given that we can see only a small proportion of the Milky Way by eye (see image below, from Pablo Carlos Budassi’s image at Wikipedia), and that we can see a number of nebula from where are, my gut feeling is that there is a significant even though relatively small proportion of mass of the galaxy that resides in them.  I suspect that we can safely ignore them.

 

 

Finally, there is the supermassive black hole at the centre of galaxies (assuming that ours is typical).  We have a black hole of about 4 million solar masses.  This is about one to four parts in a hundred thousand and presumably the same could be said for other galaxies, so again, it is in the noise and can be safely ignored.

 

Nevertheless, the identified mass is in the order of half that calculated in the FUGE model.  The comment above, that the number of stars is a gross underestimate, indicates that entirety of mass in the universe could be accounted for by normal matter (stars, planets, dust), or, perhaps, there is scope for a smaller quantity of “dark matter”, in approximately the same order as ordinary matter.  If the former, then an alternative to dark matter would need to be identified.

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The other feature of the standard cosmological model that is not explained by the FUGE model is cosmic acceleration.  I have mentioned this a few times already but the evidence for cosmic acceleration is contentious.  Jacques Colin, Roya Mohayaee, Mohamed Rameez and Subir Sarkar argue that the evidence for cosmic acceleration is lacking.  The explanation, as given a little more clearly by Sabine Hossenfelder, is that the original analysis by Reiss et al. assumed that the cosmological principle applied at the scale at which they were observing supernovae – but that scale is below that at which the concordance model indicates that the cosmological principle applies.  Fundamentally, if you look closely enough, the universe is lumpy (with stellar systems, galaxies, clusters and so on), but if you zoom out and look at averages at the 200-300 megaparsec scale, then the universe is expected to be smooth.  Once you look at the evidence at the appropriate scale, the apparent acceleration goes away.

 

Remember here that cosmic acceleration only exists because of that observation of supernovae.  It doesn’t exist to bring density or Hubble parameter values into alignment with what are currently measured.  So if there were no acceleration and no dark energy, the standard cosmological model would have to be rejigged to result in the values that are reached naturally via the FUGE model (shortened periods of deceleration, redistributed periods of deceleration, reduced rates of deceleration, less inflation, and/or a new period of “standard” expansion with H=1/t).  Consider then the utility of the standard cosmological model if it can be rejigged to get any result we need.  Pretty much zero.  And if it can’t be rejigged to get the result we need.  Precisely zero.

 

It is true that the FUGE model does not explain the observations that lead to dark matter either, but if there are problems with dark matter (and there are) then we already need to look for an alternative solution.  Note that there is no problem in the FUGE model if there is a solution that, under certain circumstances, looks like there is some sort of “dark matter”, but this appearance should not necessarily be taken as meaning that there is literally an additional category of mass-energy.

 

The FUGE model explains only what you need and what you see.  There is no need for an inflaton field (which theoretically drives inflation and for which there is no experimental evidence – so we don’t see it), dark matter (the phenomenon that led to the theory of dark matter was observed in 1993, but as for actual dark matter … there is no experimental evidence – so we don’t see it) or dark energy (for which there is no experimental evidence – note that a phenomenon that leads to a proposed explanation is not evidence of the explanation being real and note also that in the link it is stated that “Currently, the only experimental evidence for dark energy is the accelerating expansion of the universe”.  So, that is not evidence – it is just the phenomenon that dark energy was proposed to explain – and if it is the only experimental evidence, then there is no experimental evidence.  Also see NASA’s comment about the complete mystery involving yet another thing that we don’t see).

 

I should note here that declarations about the balance of mass-energy in the universe (so much ordinary matter, this much dark matter and that much dark energy) are based on assumptions.  According to CERN: “researchers have been able to infer the existence of dark matter only from the gravitational effect it seems to have on visible matter”.  If there’s another mechanism, then dark matter disappears.  According to NASA: “We know how much dark energy there is because we know how it affects the universe's expansion”.  If there’s another mechanism or there is in fact no acceleration, then dark energy disappears.  And then we have just ordinary matter, at a quantity that the FUGE model produces.

Problems with the FUGE cosmological model

After having written much about the problems that I see with the standard cosmological model, I thought it would be fair to talk about the problems with the FUGE (flat uniform granular expansion) model.

Fundamentally, all the FUGE model is saying is 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.

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First off, why the units of Planck time and Planck length and half a unit of Planck mass?  In Half a Problem Solved?, I discuss how our universe could be one of a matched pair.  In the update, I refer to a paper in the Annals of Physics (reported at Live Science) which details a theory involving a mirror universe which runs “backwards in time”.  Each of these mirror universes would receive (or generate) half a unit of Planck mass per delta unit of Planck time (even if these delta units are in opposite directions), summing to a total of one unit of Planck mass per delta unit of Planck time.

And secondly, given that l_P=c.t_P=G.m_P/c², and r_s=2GM/c², which indicates a linear relationship between all the key elements, there is no particular issue if the implied granularity is at the Planck scale, or smaller, or even larger.  I prefer the Planck scale, but I am not irrevocably wedded to it.

Thirdly, in the original meaning of FUGE, it had "universal" in the middle of the definition, which wasn't great.  I actually started with flat, granular and expansion, which suggested FUGE, so the insertion of "universal" is actually somewhat akin to what happens with a backronym.  Recently I realised that FUGE is better rendered as "flat uniform granular expansion", applying when talking about "a FUGE universe" or "the FUGE cosmological model" (as per the title of this post) or something similar.  Note that the uniformity that we are talking about only applies at a sufficiently large scale, as per discussions of homogeneity and isotropy.

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There will be some who will point out issues additional to those that I go through below, of that I am certain, but the major problem that I see is that I have no good explanation for why mass-energy enters the universe.

I have previously suggested that it could be because there was a black hole in a precursor universe, and all the mass-energy from there is entering our universe, but that just kicks the problem down the road – where did the mass-energy from that universe come from?  From an earlier precursor universe perhaps, but that results in an endless regression.  Additionally, an inherent feature of that model is that there are two mirrored universes each going off in different temporal directions (negative and positive) each of which gets half the mass-energy, as mentioned above.  So this sequence of universes implies that they would be halving in mass-energy each time a new universe branches out of an old one.  Do that a few dozen times and you start getting sparsely populated universes (2⁴⁸≈10³⁵).  With the quantity of mass-energy in our universe, there’s a hint that either there was a tremendous amount available at the very beginning or we are one of the very earliest iterations of universes.  Unless, of course, there’s some mechanism by which the mass-energy in both the positive and negative temporal directions recombine when a new pair of universes is generated, in which case a new complication is added because we don’t have anything close to a mechanism for explaining that.

A better explanation, albeit one lacking in key detail, is that expansion itself results in the creation of mass-energy, if the universe is flat.  The tiny detail missing is … what causes the expansion?  The positive aspect here, however, is that we merely have an absence of explanation.  What cannot be denied, at least not reasonably*, is that we observe expansion, even if we may not be able to explain its origin.  The creation of mass-energy is a fundamental requirement of the standard cosmological model even if it is rarely (if ever) stated as such.  The notion of dark energy includes an assumption that there is a background of invariant energy density in the universe, indicating that a universe with increasing mass-energy is not inherently impossible (because, if so, that should have been raised an objection to this explanation for dark energy).

Another problem is that it is not immediately obvious that the universe ought to be flat.  Once we have expansion and the notion that the universe is flat, and therefore has critical density, the quantity of mass-energy entering into, or being created by the expansion of the universe follows naturally.  But why would the universe be flat?

I think it is useful to consider the notion that there are reasons that mitigate against the universe not being flat.

There are only two non-flat options – either the universe could have greater than the critical density, or less than it.

In the first option, the density would be greater than that of a black hole with the radius of the universe.  The densest type of black hole is a non-rotating black hole (like a Schwarzschild black hole) and, if our universe were ever denser than that, then … well, what I would expect to see would be similar to the notion of inflation, massively rapid expansion, until such time as the density was no longer greater than that of a black hole, becoming flat or overshooting into sub-critical density.  While this might sound like an explanation for inflation, we would still have a question as to why the initial density was greater than critical.  And it would also mean that, today, we would only see either a sub-critical or critical density.

Which leaves only the second alternative, the universe having a density that is less than that of a black hole of similar dimensions – or sub-critical density.

The universe as a whole is entirely composed of gravitationally uncoupled systems (uncoupled from each other, or at least only coupled to such a relatively negligible extent that the systems do not collapse into each other).  Each of these systems are less dense than a black hole.  The solar system is an example, or just the Sun itself, or the galaxy, or galaxy cluster (or superclusters).  It is certainly possible to imagine a universe that is less dense than a black hole.

However, remember that we are trying to understand why mass-energy is entering the universe – this is associated with a universe that is flat and we are now considering a universe that would not be flat, so there is no reason to assume that mass-energy would enter it over time.  We have only three options in this sub-critically dense universe:

• there is an invariant quantity of mass-energy in the universe,
• mass-energy is leaving the universe due to some unexplained mechanism,
• or mass-energy is entering the universe at some reduced rate than for a FUGE universe due some other unexplained mechanism.

In the first option, we would have a situation in which – until right now, due to expansion – the density was higher than a Schwarzschild black hole with a radius of 13.77 billion light years, and it will later have a density that is lower.  Nothing is denser than a non-rotating black hole, so we can eliminate this option.

The second option is worse, since in the past the universe would have been even denser, and it too can be eliminated as an option.

The third gets us nowhere, since we still have mass energy entering the universe, we just have a situation in which rate no longer makes any sense.

There is a possible fourth option, being a combination of two or three of the options rejected above, in phases, perhaps also incorporating a flat phase, and a super-critical phase.  Such an option would have the same problems as the Standard Cosmological Model (and indeed could be equivalent to the Standard Model).  

At the risk of sounding Zen, the universe itself seems to be telling us that it is not possible to have less than critical density.

Then there is the fact that the FUGE model deviates from standard cosmology.  I have discussed this in The Problem with the Standard Cosmological Model.  To the extent that there are problems with standard cosmology, the fact that the FUGE model deviates from standard cosmology is not really a problem.  If the FUGE model were shown to not reflect the facts of the universe, then that would be a problem.  But it does not, so far as I can tell, it just results in the universe as it is today, with a lot less faffing about.

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* Maybe there is reasonable denial after all - as reported by Live Science.  However, this paper does not deny the appearance of expansion (per red shift).  Note also that it removes dark matter (as a form of mass-energy) and also dark energy.  I suspect that Lombriser's model introduces gravitational and cosmogenesis-related issues, but given the complexity of the theoretical underpinnings, it's entirely possible that it doesn't and I just cannot see it. 

The Problem(s) with the Standard Cosmological Model - Charts Unmodified

I presented four charts in The Problem(s) with the Standard Cosmological Model, each with a modification for clarity and emphasis.  Some might think that this was unfair, so here are those charts without the clarifying modification.

Again each chart had log values in both axes.

Radius:

The (relatively) strange activity of the universe under the Standard Cosmological Model is still visible, although not as a clearly.

Mass-energy:

In this chart, the only massive oddity that is obvious (pun intended) is the kick upwards at the end, and the lack of correlation which could be expected because the curves relate to different models.

Density:

The bizarre flopping around of values in the Standard Cosmological Model can still be seen.

Hubble Parameter:

In this chart, the use of the logarithmic scale hides the difference in values, except during the funky action at the beginning and the end.  If we zoom in to see the most recent section, and use a linear scale, we see (recalling that we have the age of the universe in seconds):

Of course, all we see here is that there’s a difference between two sets of figures for recent times, which doesn’t in itself tell us which set is correct.  But this is just one of four charts.

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.