When Albert Einstein constructed his general theory of relativity he decided to resort to some reverse engineering and introduced a ‘pressure’ term in his equations. The value of this pressure was chosen such that it kept the general relativistic description of the universe stable against the gravitational attraction of the matter filling the universe. Einstein never really liked this fudge factor, but it was the only way to get the equations of general relativity to describe a universe that is static in size.

More than 10 years later, Edwin Hubble’s observations showed that the universe is in fact not static, but rather expanding. With this, the need for the pressure term disappeared. Einstein must have felt floored: if only he would have sticked to the bare equations without the fudge factor, he could have predicted the universe to be non-static. Throughout his later life, Einstein kept referring to the introduction of the pressure term as his ‘biggest blunder’.

Would Einstein have lived till the very end of the 20th century, he would certainly have changed this ordeal. Sure, our universe is expanding, but since the end of the 90’s we know that this expansion is accelerating. Today the universe is expanding faster than yesterday, and tomorrow it will be expanding again faster than today. Without Einstein’s fudge factor, a decelerating expansion is to be expected, and the pressure term is needed to switch from a description yielding a decelerating universe to one that yields an accelerating universe.

What is causing this pressure that is pushing space apart at ever accelerating rates? Cosmologists refer to ‘dark energy’ permeating space as what propels this cosmic acceleration. In order to explain the observed accelerated expansion of the universe, this dark energy should comprise the vast majority of the total energy content in the universe. Recent observations lead to a dark energy density in the universe corresponding roughly to one Planck energy (or equivalently: one Planck mass of about 20 microgram) per 1000 km cubed. The fact that this tiny density constitutes the dominating component of our universe just demonstrates the vast emptiness of space.

But what is this dark energy? No one knows. The most likely explanation is that dark energy is quantum mechanical in origin. In fact, most physicists would probably agree that dark energy results from quantum fluctuations, if only this would lead to predictions of the right magnitude of the dark energy effect. However, the standard quantum field-theoretical (QFT) approach leads to an overestimate of the dark energy density. How much of an over estimate? Well, any statement one can make on this will be an understatement. Applying standard quantum field theory considerations, vacuum fluctuations can be estimated to lead to an energy density of one Planck energy per Planck length cubed. That is a Planck energy per cube with sides of 0.000 000 000 000 000 000 000 000 000 000 000 016 m. A volume a wee bit different from 1000 km cubed.

Where have we gone wrong?

Some simple dimensional analysis hints at a potential solution. There are two key length scales entering the problem: the Planck length ℓ and the cosmic scale L (read: the diameter of the observable universe). The contrast between the two is vast: 61 orders of magnitude. Wouldn’t it be a huge surprise if these two extreme length scales can be combined into a volume of the right size to describe the dark energy density? Well – surprise, surprise – this is easy to achieve. The experimental value of the dark energy density happens to coincide with one Planck quantum per volume of size L^{2}ℓ. Yet, as we saw above, standard quantum field theory predicts a zero point energy density of one Planck quantum per ℓ^{3}. Can we change two of the ℓ’s in this equation into L’s?

Yes we can. Key is to realize that the ℓ^{3} volume enters into the theoretical description because standard QFT assumes one degree of freedom per Planck cube. So according to QFT our universe has a total of (L/ℓ)^{3} degrees of freedom. This however ignores the holographic nature of our universe that was postulated by Gerard ‘t Hooft in 1993. The holographic principle states that standard QFT vastly overestimates the number of degrees of freedom available. More precisely, the holographic principle forbids a system of linear size L to have more than (L/ℓ)^{2} degrees of freedom. So, this in itself already changes one ℓ in the equation for the dark energy density into an L. But there is more. QFT associates a zero-point energy of one Planck unit with each degree of freedom. This does not necessarily carry over into a holographic description. The degrees of freedom in the holographic description are non-local, and the wavelengths corresponding to the zero-point motion can probably be linked to the macroscopic length L, rather then to to the microscopic length ℓ. This effect (embodied in the so-called ‘UV/IR connection’) gives us another swap between ℓ and L in the equation for the dark energy density so that with all holographic effects incorporated we arrive at ℓ/L Planck energies per volume of size ℓ^{2}L, or equivalently, one Planck energy per volume of size L^{2}ℓ.

Is this all the correct way to look at the expansion of our universe? Or is the Planck energy per volume L^{2}ℓ some coincidence? I don’t know the answer. What I do know, is that if the above is in essence correct, holographic considerations will be an integral element of the still elusive theory of quantum gravity. It is also clear that the strict holographic cut-offs to the number of degrees of freedom and the allowed energies per degree of freedom will be of immense help to regularize this theory of quantum gravity. History tells us that experimentally demonstrated discrepancies in our understanding of the fundamental laws of physics never last for more than a few decades. So I dare to make the prediction that in the first half of this century we will witness a revolution in our thinking about the universe in the form of a fully consistent theory of quantum gravity. These are exciting times!

Great entry as always!

Could you give one or two refs about the ir/uv connection? I did not really follow you here though: basically if zero modes are at long length scales L then counting dof for the energy density should scale as (L/L)^3?

Something else: since one major difference about gravity wrt the other forces in QFT is that mass scalles as l^-1, is the holographic principle in essence related to that difference? I still need to learn about it.

I hope my questions make any sense.

Thanks

Cheers

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The UV/IR connection is the result of the quantum length scale hc/E to decrease inversely proportional to energy E, while the gravitational length scale G E/c^4 increases proportional to E. In quantum gravity phenomena, both length scales play a role and one can not count degrees of freedom solely based on the quantum length scale. A consistent picture emerges if one multiplies both length scales, and considers the result hG/c^3 as a minimum area to be used in counting states (holographic principle).

The book “An Introduction To Black Holes, Information and the String Theory Revolution” by Susskind and Lindesay contains a (short) chapter on the UV/IR connection.

Thanks for the kind words!

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Well thank you for your reply, it is an interesting point to know about which I am ignorant of, this UV/IR connection. I believe it has something to do with CFTs ? (Also very ignorant of that field still) But I find your scaling argument at first sight still not very convincing, since ok you are comparing the graviational curvature with the masless DeBroglie wavelength, which precisely yield together Planck’s length scale by combination hG/c^3…

So, from there, how to reach the long range scale L remains unclear intuitively to me at least.

I’ll go have a look to the book you mention by Lenny, if time permits (as we say 😉 )

Your blog is one top notch, you have fans!

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I have added the UV/IR connection on my list ‘ideas for blog posts’. So, if time permits you’ll get a more extensive answer… 😬

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Great

Looking forward to that!🤓

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Would you consider this possibility? https://gregoriobaquero.wordpress.com/2017/01/26/first-blog-post/

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@gregoriobaquero – the standard (LambdaCDM) model of cosmology is relativistically covariant and therefore includes all time dilation effects. Still, the model requires the presence of dark energy (a cosmological factor) and dark matter. Manually adding time dilation effects to account for (parts of) the ‘dark universe’ would result in an erroneous double counting.

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Thank you for reading it.

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Gravitational Time Dilation Explains Dark Matter (https://gregoriobaquero.wordpress.com/2017/01/26/rame-is-dm/)

Please read it. Do not get discouraged because I am not a professional physicist.

Besides requiring a tweak on lambda CDM (vacuum energy density calculated from expansion of the universe to be the effective inflationary component of higher positive and negative energies -maybe the only manifestations of pure energy in the universe)

This model does not suggest any modifications to General Relativity and the particle standard model. It explains the discrepancies of Dark Matter mass when calculated from rotation speeds to that calculated from Gravitational lensing. If the model is run on dwarf galaxies it matches observations were Wimps(CDM) fail.

Give it a try, if the model is right it is the most elegant solution to the Dark Matter mystery.

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Gregorio

It is nice to give a try, to fiddle with numbers and concepts, but you know, credit is awarded only after hard (very hard) work. Gained from indisputable evidence, framework and calculations. Never forgetting the passion and the dreams of course. I feel compelled to tell you this, because reading the very first three sentences of your abstract, they discard all of the remaining and prevent any serious reader to read further.

Rekativity does not explain EM. It is merely the right framework to reconcile EM with mechanics. Energy does not slow down. Energy is something that can increase or decrease and deliver momentum for instance or heat. It is conjugate to time, through Planck constant (an action quantum) and Heisenberg uncertainty …

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You are right.

Most of serious people won’t pass the sentences on the abstract. Your input is greatly welcome.

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I may be right on this occasion, nevertheless don’t give up!

Cheers

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I did not. This is an article I needed to write in order to pin down the idea on my first blog. It is self contained, but if you read it I am sure you would be more likely to read the first one. Hopefully you will enjoy it.

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https://gregoriobaquero.wordpress.com/2017/02/25/on-the-nature-of-dark-energy-and-gravityf/

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