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.

---

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).

---

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.

---

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)? 

---

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?

---

* 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)?

Vacuum Energy, Dark Energy and the Units of the Planck Parameter

So, I’ve been asking some questions and getting answers which indicate that the questions are somehow annoying (but which don’t actually address the questions asked).  In the process, the topic of vacuum energy came up, which is something that I had not even considered.

I sat at my desk for a while pondering how I would work out the amount of energy entering the universe at a given time and then get the average amount per cubic metre.  Then I intended to compare that value to the value given for vacuum energy, which I naïvely thought I’d just look up (it’s never that simple).

But as I sat there pondering, I thought: I already have a value that I could work with.  I concluded in Is the Universe Getting More Massive?  (Flatness, not Fatness) that mass-energy enters the universe at a rate of one Planck mass per Planck time.  I worked out that the density of the universe, if flat, after 13.8 billion years of this process would be the critical density at that time, which is approximately 10^-26 kg/m³.  Now we know that E=mc² (it’s really E_o=m_oc², since we need to consider rest mass but I’m sure we can get past that).  Given that I already say mass-energy, I don’t have any problem expressing a mass in terms of its energy equivalent and in this case that is approximately 9x10^-10 J/m³.  According to current estimates, 32% of the universe is either baryonic matter or dark matter, so … if the rest is just dark energy burbling away in “empty space”, then that would 68% of 9x10^-10 J/m³, or about 6x10^-10 J/m³.

At the Wikipedia article on vacuum energy, the first value given for the vacuum energy of free space is 10^-9 J/m³.  This is the value estimated “using the upper limit of the cosmological constant”, and Sean Carroll is cited as the source (via C-SPAN’s Cosmology at Yearly Kos Science Panelbroadcast, Part 1).  The same value is quoted by John Baez, and he goes on to write:

One can know something is very close to zero without knowing whether it is positive, negative or zero. For a long time that's how it was with the cosmological constant. But, recent measurements by the Wilkinson Microwave Anisotropy Probe and many other experiments seem to be converging on a positive cosmological constant, equal to roughly 7 × 10^-27 kilograms per cubic meter. This corresponds to a positive energy density of about 6 × 10^-10 joules per cubic meter.

Interesting, huh?  Another big fat coincidence.

---

In a parallel discussion in which I was accused of saying that there’s a speed (distance/time) associated with the expansion of the universe despite having carefully written, in reference to a hypothetical universe:

The universe expands such that the radius increases by 1 Planck increment every 1 Planck time (possibly with smaller increments depending on at what point the granularity kicks in).

There is a lack of clarity with respect to that statement but I am not saying that the universe expands at any specific rate, I am just saying 1) the universe expands and 2) due to that expansion the radius increases at a rate that looks like it could be a speed.  In reality, I think the universe expands at a rate of 1 Planck time per Planck time, and that’s not a rate at all, it’s dimensionless.  Note that I am not currently thinking of the universe as a simple sphere, but even if the universe were a glome, the surface volume of that universe would still expand in direct proportion to the radius of the glome.  Anyway …

I pointed out to my interlocutor that the Hubble parameter (today) is cited as ~70 because it’s expressed in km/s/Mpc, I assume because these are convenient figures in cosmology.  However, if you express this figure in Hubble lengths (where HL = c/H = 13.8 billion light years) and meters, rather than megaparsecs and kilometres, you get a value of 300,000,000 m/s/HL.  And, to more significant figures than is strictly necessary, this is the speed of light.  So, the expansion of the universe is associated with a very important speed, a speed when expressed in Planck units is 1.  But the expansion itself is not a speed, by its dimensions it’s more of a frequency – once every Planck time.

And the question that arises when thinking of the expansion of the universe as being a frequency is … a frequency of what?  It implies, strongly to me at least, that something is happening to the universe every unit of Planck time.  And for me, the answer is obvious, it’s expanding by an increment (be that a unit of Planck time, or a unit of Planck time multiplied by c, or a Planck length, or however you prefer to think of it).

---

Finally, when looking up the amount of dark energy in the universe, I found the NASA webpage on the issue.  On that page is the following text (for the purposes of transparency I should advise that it is followed immediately by a section of text that I am still a little dubious about although I plan to give it some more thought and I should highlight that, even though from NASA, they are only talking speculatively):

One explanation for dark energy is that it is a property of space. Albert Einstein was the first person to realize that empty space is not nothing. Space has amazing properties, many of which are just beginning to be understood. The first property that Einstein discovered is that it is possible for more space to come into existence. Then one version of Einstein's gravity theory, the version that contains a cosmological constant, makes a second prediction: "empty space" can possess its own energy. Because this energy is a property of space itself, it would not be diluted as space expands. As more space comes into existence, more of this energy-of-space would appear.

I recall reading that, in terms of the FLRW metric, dark energy increases but I can’t find it again.  However, the Wikipedia article on dark energy quite clearly indicates that dark energy increases:

when the volume of the universe doubles, the density of dark matter is halved, but the density of dark energy is nearly unchanged (it is exactly constant in the case of a cosmological constant)

This is entirely consistent with my model – at least now that I have got a better handle on how dark energy might fit in (ie all the energy that is entering the universe today is in the form of dark energy).

Oh, and by the way, I do understand that I am implying that dark energy and vacuum energy might be the same thing.  It’s clearly not outside the realm of possibility though, since actual scientists in the field have made similar claims.

The Dark Energy of Luke Barnes

Maybe I’m being a bit unfair with the title since the articles in question are attributed to more than just Luke, there are eight other authors:  Jaime Salcido, Richard Bower, Geraint Lewis, Pascal Elahi, Tom Theuns, Matthieu Schaller, Robert Crain and Joop Schaye.  However, as far as I know, the others aren’t Templeton Research Fellows – one of the articles has LAB (Luke A Barnes) but only LAB tainted with an association with Templeton.  And I note that there is a statement distancing that paper from Templeton (“The opinions expressed in this publication are those of the author and do not necessarily reflect the views of the John Templeton Foundation”.)  Oddly enough, the other article “The impact of dark energy on galaxy formation. What does the future of our Universe hold?”, despite having the same authors (admittedly in a different order), does not acknowledge that Luke is supported by Templeton.  Perhaps because Luke was not lead and Jaime Salcido didn’t want to have anything to do with Templeton if he could help it?

Anyways, I’m pondering not so much the paper as the raft of articles that are based on the papers, helpfully listed by Luke at Letters to Nature.  The spin of these articles varies somewhat, at least going by their headlines, from “Life in the multiverse could be commonplace” to “Bad information for the multiverse: it is nonetheless not going (sic)”.  Either life is teeming in the multiverse, or there’s no multiverse.

Interestingly, if you look at an example of the former – for example A multiverse may be hospitable to life:study – you will see that the scientist quoted is Jaime Salcido.  But if you take a look at an example of the former – for example Bad news for the multiverse: it's still not likely – the featured scientist is Luke Barnes.  There’s also a third spin, namely that there needs to be a “new law” for dark energy, which means that the article headline writer chose to focus on the words of Richard Bower – for example New Research Questions The Multiverse Theory, Calls For A New Law Of Dark Energy.

It looks very much like Barnes and Salcido have looked at the same data and Barnes has opted to interpret it as negating the multiverse (more on that in a moment) and Salcido has interpreted it as saying that life is probable no matter which universe in the multiverse you are in.  This would mean that we’re not that special and not even our universe is that special, as even one of the more Barnes-centric articles put it “That’s bad news for the multiverse, because it means there’s nothing particularly special about our universe - and no need for a multiverse to explain our existence.”  I’m pretty sure that Barnes wouldn’t be happy with that comment.

Barnes, you see, wants fine tuning, because that supports his god and as I have argued elsewhere, he’s a closeted apologist.  If we don’t need a multiverse to explain our existence, because our universe isn’t particularly special, then we don’t need Barnes’ god either.  If there’s evidence against a multiverse, that’s not going to bother real scientists because real scientists aren’t pushing an agenda.  Not that Barnes et al have come up with evidence against a multiverse, all they have is an argument that you don’t need to call on a multiverse to explain the amount of dark energy we have – as far as the star formation rate goes.  That’s alright, we don’t even know for certain that there is dark energy (see sceptical interpretations of dark energy related measurements – such as this).

Some of the articles present an argument that, if it comes from Barnes and/or Bower and isn’t merely a misinterpretation on the journalist’s part, is worryingly deceptive.  There is a suggestion, for example at Inquisitr, that the multiverse theory is there to explain (in part) the value of dark energy.  This is a ludicrous suggestion given that the multiverse has been suggested since as far back as 1952 and the term “dark energy” was only coined in 1998.  Additionally, it is argued that the multiverse simply falls out of string theory to the extent that if string theory is true then, necessarily, there is a multiverse.  String theory dates back to the 1940s, so again predating dark energy.

Basically, multiverse theory needs a fine-tuned value of dark energy to about the same extent as the value of dark energy needs a multiverse theory – namely, not at all.