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.

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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).

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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.

A Question of Cosmology

There is such a thing as Hubble time, which is simply the inverse of the Hubble parameter (H).  The inverse of the Hubble constant (H₀) is the current Hubble time (because the Hubble constant is the current value of the Hubble parameter).

As I noted back in 2014, in Is the Universe Expanding at the Speed of Light?, the current value of Hubble time is interesting because it’s basically the same as the age of the universe.  Now, because the (current) value of Hubble time is the inverse of the Hubble constant, it varies with measurements of the Hubble constant.

In Is the Universe Expanding at the Speed of Light?, I referred to the values of Hubble constant that were currently available.  These were:

• 2011 (Hubble) ~71.5 to ~76 km/s/Mpc
• 2012 (Spitzer) ~72 to ~76.5 km/s/Mpc
• 2012 (WMAP – after 9 years) 68.52-70.12 km/s/Mpc
• 2013 (Planck – after four years) 67.03 to 68.57 km/s/Mpc

By happy coincidence, Skydive Phil released a video on “The Hubble Tension” at about the same time as my retrospective podcast listening of the Infinite Monkey Cage got me thinking about the Hubble constant and the age of the universe all over again.  What was bothering me was that there were constant references to the acceleration of the expansion of the universe, together with the assertion (claim, reminder, stating) of the fact that the universe is 13 point something (7 or 8) billion years old (might have been raised in this specific episode or this one).

If the rate at which the universe is expanding is accelerating, then surely the age of the universe comes into question.  The value given for age of the universe has not changed significantly since 2014, when I reported it as 13.8 billion years – the current values range from 13.772 (WMAP) to 13.813 (Planck 2015 data) km/s/Mpc.  The value of the Hubble constant on the other hand …

• 2018 (Planck) 67.66 (67.24 to 68.08) km/s/Mpc
• 2018 (Hubble and Gaia) 73.52 (71.90 to 75.14) km/s/Mpc
• 2017 (LIGO and Virgo) 70.0 (62.0 to 82.0) km/s/Mpc
• 2016 (eBOSS – after two years) 67.6 (67.0 to 68.3) km/s/Mpc

Skydive Phil’s video focusses on the difference between the eBOSS (baryonic acoustic oscillation) and Planck measurements and the measurement from Hubble and Gaia collaboration (also known as SH₀ES, sometimes miswritten as SH_oES, or SHoES).  The problem here is that the error bars no longer overlap, which indicates some sort of problem – either they are measuring different things or at least one of them is measuring the wrong way.

You occasionally read that the Hubble time is a useful estimate of the age of the universe.  In that case, ignoring the ridiculously large error bars on the LIGO/Virgo result, the age of the universe is between 13.01 and 14.60 billion years.  In some cases, it is suggested that the Hubble time indicates how long the universe has been expanding – but to all intents and purposes this is what is meant by the age of the universe (much in the same way as a baby is not strictly 0 days old at birth, usually having gestated in a womb for about 9 months, we just pick a nice convenient reference point and count from there).

However, we are now being told that, about 5 billion years ago, the expansion of the universe started accelerating.  If the Hubble time is a useful estimate of the age of the universe and the age of the universe is what we are being told (13.8 billion years, or near enough), then don’t we have a problem?  We can work out the value of the Hubble parameter at a Hubble time of approximately 8.8 billion years (let’s call it H_-5, meaning H at now minus 5 billion years), and it works out to be about 1.5 times that of today – ie about 111.1 km/s/Mpc.  (Perhaps the acceleration started only 4 billion years ago, at the end of the matter-dominated era and the beginning of the dark-energy-dominated era.  The value of the Hubble parameter corresponding with a Hubble time of 9.8 billion years is a bit lower at H_-4=99.8 km/s/Mpc, but still about 40% higher than today.)

What is going on here?

Is the Hubble time only coincidentally a good estimate of the age of the universe at the current time (but won’t be in the future and wasn’t in the past)?  This sounds like it would be a fair addition to the list of fine tunings, and that surely can’t be good.

If the Hubble time isn’t generally a good approximate of the age of the universe, then there’s no reason to suggest that it ought to be a good approximate today and maybe the age of the universe is not 13.8 billion years after all.  Not really, there are a multitude of ways in which the age of the universe is measured, so cosmologists don’t simply rely on inversing the Hubble constant (for example, they look at cosmic background radiation fluctuations).

Another possibility is that the Hubble parameter has been tracking the age of the universe faithfully and 5 billion years ago it actually was H_-5=111.1 km/s/Mpc.  Would that mean that, since that time, Hubble expansion of the universe has decreased and some other expansion of the universe (due to dark-energy, apparently) has got involved?  If so, it still doesn’t really add up.  We have the Hubble constant because that’s what we measure as the current expansion rate of the universe.  If the Hubble component of that is lower than is required to account for that expansion rate, then H₀ < ~70, which increases the estimate for the age of the universe.  Thinking about the notion that the universe has been accelerating in its rate of expansion for the past 5 billion years, let’s say at best case that H_-5 was just a bit lower than today, say just under the error bar for LIGO/Virgo at 61 km/s/Mpc.  That would mean that at that time the Hubble time was 16 billion years, and today the universe would be 21 billion years old.  That’s a big error in measurement.  It just doesn’t seem right.

So, a question for the people in the know … if we were around 5 billion years ago and were measuring the Hubble parameter using the rate at which distant galaxies were receding, approximately what result would we have come up with?

Do I have a solution?  Yes, I think I do.  I just don’t quite understand (yet) why most cosmologists would likely tell me I am wrong.