# QFT vs GR Cosmological Constant

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• dsaun777
In summary, the gap between the theoretical prediction of the cosmological constant and the observed value is because the value is many orders of magnitude off from what is actually observed.

#### dsaun777

I am sorry but I can't seem to find the actual estimated value of the cosmological constant that is predicted by quantum field theory. Can anyone help me and tell me the approximation of that value and/or the value of the approximate observed cosmological constant that physicists today think exists? I am trying to find real numbers for the so-called "vacuum catastrophe."

Moderator's note: Thread moved to the quantum physics forum since the value being asked for is the value predicted by QFT.

topsquark
It basically comes from calculating the expectation value of the Stress-Energy tensor ##T^{\mu\nu}## in the vacuum. There is a way of estimating this value for the Standard Model by using the Minkowski space symmetries that makes each field's contribution differ only by a constant.

The resulting formula is an integral that diverges, but one argues should be cutoff at the Planck scale. It depends exactly where you cut the integral off and exact details of your assumptions for the contribution of each field, but the result is about a factor of ##10^{123}## off the actual value.

A sketch of the calculation is given in Matt Visser "Lorentzian Wormholes", Chapter 8.

dsaun777, vanhees71, dextercioby and 3 others
LittleSchwinger said:
It basically comes from calculating the expectation value of the Stress-Energy tensor ##T^{\mu\nu}## in the vacuum. There is a way of estimating this value for the Standard Model by using the Minkowski space symmetries that makes each field's contribution differ only by a constant.

The resulting formula is an integral that diverges, but one argues should be cutoff at the Planck scale. It depends exactly where you cut the integral off and exact details of your assumptions for the contribution of each field, but the result is about a factor of ##10^{123}## off the actual value.

A sketch of the calculation is given in Matt Visser "Lorentzian Wormholes", Chapter 8.
Does the value have anything to do with Planck density?

dsaun777 said:
Does the value have anything to do with Planck density?
The result comes out as:
##\Lambda = 8\pi G \beta\rho_{P}##
With ##\rho_{P}## being the Planck density. ##\beta## is a constant produced by summing up the contributions of each field. No assumptions on the contributions of each field produce a value for ##\beta## small enough to give the true value.
The Planck Density appears because one has cutoff the integral at the Planck length.

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vanhees71, Demystifier and dsaun777
dsaun777 said:
I am sorry but I can't seem to find the actual estimated value of the cosmological constant that is predicted by quantum field theory.
There isn't one. That is, one cannot write down an equation for it, There are hand-wavy arguments about what it should be, but no calculation.

vanhees71 and strangerep
There isn't one. That is, one cannot write down an equation for it, There are hand-wavy arguments about what it should be, but no calculation.
But it is something of the order of Planck density and that value is many orders of magnitude off from the observed one. Is the one observed more accurate and less hand wavy?

That's a polite way of saying "I don't believe a damn thing you said." Your choice.

vanhees71 and strangerep
That's a polite way of saying "I don't believe a damn thing you said." Your choice.
I believe everything you said. I'm more curious to know about the gap of how the general relativity or classical cosmological constant differs from the qft.

dsaun777 said:
Is the one observed more accurate
Assuming the observations are correct, any actually observed value is more accurate than a theoretical prediction.

dsaun777 said:
and less hand wavy?
How can an actual observation be "hand wavy"?

dsaun777 said:
I'm more curious to know about the gap of how the general relativity or classical cosmological constant differs from the qft.
Classical GR makes no prediction whatever about the value of the cosmological constant; all it tells you is that including the cosmological constant term in the field equation is consistent with all of the other features of the theory.

dsaun777 said:
But it is something of the order of Planck density and that value is many orders of magnitude off from the observed one.
The derivation I mentioned above and covered in Visser's book has many weak points. It's the argument being discussed when people mention the "terrible Cosmological constant prediction of QFT". However there is a large literature disputing the argument.

That's a polite way of saying "I don't believe a damn thing you said." Your choice.
The point of course is that, even in a Dyson-renormalizable theory, as is the Standard Model, you have renormalize the "vacuum energy", i.e., it's a free parameter not determined by the theory. If you naively add up all socalled "zero-point energies" of all fields cut off at the Planck scale, then you get this huge factor compared to what's measured in cosmological observations (redshift-distance relation of SNe, fluctuations of the CMBR), but in fact it's to be renormalized anyway, and thus it's a free parameter.

The issue, however, remains also from the point of view of renormalization theory, because if you renormalize the SM at a small renormalization scale suited to the observations on accelerator experiments with elementary particles and you use the renormalization-group equation to evaluate the zero-point energy at a higher renormalization scale like the GUT scale or even the Planck scale, you get again a huge discrepancy, i.e., you need extreme fine-tuning to get it right, and that's felt to be "unnatural".

otennert, LittleSchwinger and Demystifier
PeterDonis said:
Assuming the observations are correct, any actually observed value is more accurate than a theoretical prediction.

How can an actual observation be "hand wavy"?
It can be hand wavy in the sense that it does not provide the full and necessary observation to make definite conclusion about the actual cosmological constant. The observations are not in depth enough to confidently state the accuracy of the cosmological constant. Despite inconclusive observational evidence it is still presented as if it were the actual cosmological constant in a hand wavy manor.

dsaun777 said:
It can be hand wavy in the sense that it does not provide the full and necessary observation to make definite conclusion about the actual cosmological constant.
I'm not sure what you mean. Every real observation has error bars; we never make observations with infinite accuracy.

dsaun777 said:
Despite inconclusive observational evidence
What observational evidence do you think is inconclusive, and on what basis? Please give specific references.

vanhees71
Well, one quite pressing issue seems to be the "Hubble tension", i.e., determining the Hubble constant from different observables does not lead to the same result. In such cases the question always is, whether it's an observational problem (e.g., the correction due to dust in the determination of the distance of supernovae or the calibration of the light curves to the luminosity, and all that) or whether there's really a problem with our "concordance model", which is at the moment the ##\Lambda \text{CDM}## model.

Here's a pretty recent short review:

https://arxiv.org/abs/2105.09409

LittleSchwinger and dsaun777