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.