Several folks have asked me how I reacted to the paper

How the huge energy of quantum vacuum gravitates to drive the slow accelerating expansion of the Universe (March 2017)by Qingdi Wang, Zhen Zhu, William G. Unruh (who is famous for the Unruh radiation i.e. the simpler Hawking radiation in the flat Rindler space). They claim that the Hubble constant implied by some vacuum energy density – which arises thanks to the quantum fluctuations in the vacuum – isn't the usual\[

H \sim \frac{\Lambda^2}{M_{Pl}}

\] but instead, it contains some shocking exponentially decreasing factor\[

H \sim \Lambda \exp(-c \Lambda / M_{Pl} ), \quad c\sim O(1).

\] If true, the decreasing exponential could be identified with the tiny factor of \(\exp(-123)\) and this mechanism could explain at least the "old" cosmological constant problem. My first reactions were skeptical or "incomplete" but the more I look into the paper, the more it reminds me of an attempt of mine that I first invented some 15 years ago. And that's an encouraging sign. ;-)

I do believe that the statement about the "totally different formula for the Hubble constant" in a theory with the vacuum energy contradicts some general relationships between the spacetime geometry and the quantum fluctuations e.g. in SUGRA. But the vacuum diagrams are "different" than all other Feynman diagrams, there could be an overlooked exception, and they're arguably least tested. To say the least, the vacuum diagrams are the least successfully tested Feynman diagrams.

Even though they get an exponentially suppressed Ansatz for the Hubble constant, their attempt actually reminds me of some efforts of mine to derive a "seesaw formula" for the cosmological constant,\[

\rho \sim \frac{M_{SUSY}^8}{M_{Planck}^4}

\] by arguing that there's something wrong with the vacuum energy's one-point function and that only its two-point function is nicely calculable and the leading term is of the order written above. Why? Because after SUSY breaking, \(\rho\) is estimated to be of order \(m_{SUSY}^4\), the fourth power of the superpartner masses' differences, but we need to calculate \(\langle\rho^2\rangle\), which is why we get the eighth power, but that has to be divided by the truly fundamental scale, the fourth power of the Planck scale, for dimensional reasons.

Just to be sure, the formula above gives you a pretty good estimate for the observed vacuum energy for the same reason why a similar seesaw formula gives you a good estimate for the neutrino masses: \(M_{SUSY}\) is comparable to \(M_{EW}\) while \(M_{Planck}\) is comparable to \(M_{GUT}\). And yes, the vacuum energy density is comparable to the fourth power of the neutrino masses. I guess that the experts in the cosmological constant problem are aware of all these coincidences.

Similarly, Unruh and his Asian pals claim that the usual formula for the Hubble constant is too classical and naive. The usual power law formula for the Hubble constant starts with imagining the zero-point oscillators of the quantum fields that add some energy density. It has the quantum origin but it's understood to be a classical constant in the spacetime and it's inserted to the classical Einstein equations that are basically trusted at all scales.

Instead, Unruh et al. claim that it's important to treat the effect of the vacuum energy density on the spacetime geometry quantum mechanically, too. In particular, they believe that it's important to stress that the vacuum energy density isn't really a constant as a function of the spacetime. Instead, \(\rho(x,y,z,t)\) is profoundly oscillating as a function of the spacetime coordinates – much like all quantum fields, elementary or composite, are quantum fluctuating in any quantum field theory.

It follows that in some regions, the vacuum energy will be negative, in others, it will be positive. If you look at regions that are smaller than the regions of the size where \(\rho(x,y,z,t)\) likes to change the sign, the usual intuition is said to be basically correct and the effects of the cosmological constant are huge. However, most of these effects are claimed to cancel out. On top of that, they want to believe that due to the asymmetric geometry of the positive and negative \(\rho\), the expanding geometry wins a little bit when you look at very long scales comparable to the size of the Universe and that's why we get some leftover accelerating expansion.

It's cool that they talk about the fluctuations of the stress-energy tensor, especially because of e.g. the equation 29\[

\langle T_{ij}^2 \rangle \sim \langle T_{00}^2 \rangle

\] The expectation values of the squared stress-energy tensor is clearly relevant for the quantification of the quantum oscillations. But in my picture, it was also helpful because I wanted to justify or rationalize a seesaw formula for the vacuum energy. That would be small. They say that the Ansatz for the effective cosmological constant is exponentially small so the parameteric dependence is different but there is at least one shared idea between me and Unruh et al.: To deal with the effects of the vacuum energy density properly, you need to consider not just the one-point function of the stress-energy tensor but the two-point function, too.

The reasons why their Hubble expansion rate has the exponentially small piece remains incomprehensible to me. The paper is based on sort of elementary physics; but it still talks about many things. It surely contains lots of "curved complexities" that don't seem essential to understand why they get the exponential decrease. This sounds very non-Unruhy: Unruh has found the thermal radiation seen by an accelerating observer which was actually a

*simplified*cousin of Hawking's derivation of his radiation. All the curvature problems have been removed. It would have been more logical for the Unruh radiation to be discovered first, and the Hawking radiation of an actual black hole afterwards. History has proceeded illogically: Hawking probably found "his" radiation first because of his excessive brain power. Excessive brain power sometimes makes the people solve harder problems before they solve the more elementary ones. Just to be sure, these compliments aren't meant to contradict the observation that Hawking is an idiot if he believes that the mankind must and will leave Earth in 100 years.

Here, Unruh et al. are deriving something rather complicated to start with but I feel that we don't need all the complicated things and tangents etc.

I am open-minded to the possibility that the paper contains a grain of the truth which would be wonderful. It would be shocking if it were true and if the solution to the old cosmological constant would be found some 17 years after the enthusiasm about the cosmological constant problem peaked. (I have always stressed that I found the claims that "the cosmological constant problem is the deepest mystery of physics" to be exaggerated – and it was always conceivable that the problem was rather small and isolated from the truly deep and universal wisdom of physics.) But it's possible. Even if the toad of truth squatteth in this swamp of unbalanced formulas of the Unruh et al. paper (thanks, Sheldon Cooper), I would love to see that these unorthodox formulae may be compatible with more complete theories of quantum gravity such as string/M-theory, or at least get some "excuse" in supergravity.

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