Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask)
Jérôme Martin
Abstract and summary · read the original at the source
In one page
Jérôme Martin sets out to explain the cosmological constant problem properly rather than quickly, and along the way he corrects the number everyone quotes. The famous claim is that quantum field theory predicts a vacuum energy 122 orders of magnitude larger than the one the sky shows. Martin argues that figure comes from a shortcut — chopping the sum of zero-point energies off at a cutoff, which is not a legitimate way to regularise the calculation. Do it with the same techniques particle physicists use to compute cross sections and the standard model’s answer comes out around minus two times ten to the eighth in particle-physics units, against a measured vacuum density of about ten to the minus forty-seventh. That is a mismatch of roughly fifty-four orders of magnitude, not a hundred and twenty-two — still an enormous problem, but a different one. He then reviews the evidence that the vacuum fluctuations are real, that they appear to gravitate normally, and what would have to give for the contradiction to go away.
Why it matters hereChapter 2 quotes the vacuum-energy gap constantly, and this is the paper that tells you how to quote it without handing a critic a free win: the gap is real, it is enormous, and the usual hundred-and-twenty-two-order version of it rests on a regularisation nobody would accept anywhere else in physics. Chapter 13 gets both halves of Martin’s ledger — the Lamb shift and the Casimir effect saying the zero-point fluctuations are real, and free-fall tests saying they weigh what they should — which is exactly why the mismatch cannot be waved away.
What it claims
01The often-quoted figure is an artefact of the method. Introducing a cutoff to sum the zero-point energies is not an appropriate regularisation; done with the standard techniques that work when one computes a cross section, the standard model’s predicted vacuum energy density comes out at about minus two times ten to the eighth in units of giga-electronvolts to the fourth power, and as the renormalisation scale is varied it stays, in Martin’s words, very far from the ten to the seventy-second always mentioned in the literature.Section IV A and Section IX; Equations (515) and (516); Figure 5
Published and peer-reviewed02The real mismatch is about fifty-four orders of magnitude. Supernova and flatness data give a measured vacuum density of about ten to the minus forty-seventh in giga-electronvolts to the fourth power, which is a detection and a measurement of the vacuum energy rather than a bound; set against the renormalised prediction this is a gap of something like fifty-four orders of magnitude, and Martin notes in the same sentence that this is much less than the 122 orders of magnitude often quoted in the literature. Something is still clearly wrong, and it is difficult to identify where the mistake is.Section X A; Equation (548) and the remarks following it
Published and peer-reviewed03The zero-point fluctuations are not a bookkeeping artefact of the formalism. Martin reviews the two experiments usually offered as proof that they are physically there — the Lamb shift, which he derives from the jitter the fluctuating electric field imposes on the electron’s position in the Coulomb potential, and the Casimir effect, which he derives from the Green function of a field between two plates.Section XII, Do the Vacuum Fluctuations Really Exist?; Sections XII A and XII B
Settled physics04The fluctuations also appear to gravitate normally. Because the electromagnetic vacuum contributes differently to the binding energy of aluminium and platinum nuclei, Eötvös-type tests of the universality of free fall constrain any anomaly in how that energy falls: with the measured differential acceleration below about ten to the minus twelfth, the anomaly parameter is bounded below roughly 1.7 times ten to the minus third. Martin’s verdict is that it seems difficult to argue that the zero-point fluctuations do not gravitate — which makes the cosmological constant problem more acute, not less.Section XIII, Do the Quantum Fluctuations Gravitate?; Equations (777) and (778)
Published and peer-reviewed05Cosmology remains the only place the constant can actually be measured. Martin works out what the vacuum energy does to a static spherically symmetric metric and then to planetary orbits and to atomic levels: requiring the shift to stay below the Lamb shift bounds the vacuum density only at about 1.3 times ten to the sixteenth in giga-electronvolts to the fourth power, far weaker than the planetary bound and vastly weaker than cosmology. No other experimental context can compete, though these constraints are still worth having because they are independent of cosmological assumptions.Section XI, Measuring the Cosmological Constant Elsewhere; Sections XI B and XI C; Equation (651)
Published and peer-reviewed06Something considered robust has to be abandoned. Martin’s conclusion is that the vacuum fluctuations seem to exist and to have normal gravitational properties, so removing the contradiction means giving up renormalisation in quantum field theory, or the weak equivalence principle, or the high-accuracy measurements of the expansion of the universe. Among the proposals he surveys, he singles out the analogue-gravity models built on superfluid helium, Fermi liquids and Bose-Einstein condensates: there a naive ground-state calculation gives the wrong vacuum energy and only the full microscopic theory gives the right one, which suggests that a correct calculation of the cosmological constant must be based on the fine structure of spacetime itself.Section XV, Conclusions
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Read it · abstract
Abstract
This article aims at discussing the cosmological constant problem at a pedagogical but fully technical level. We review how the vacuum energy can be regularized in flat and curved space-time and how it can be understood in terms of Feynman bubble diagrams. In particular, we show that the properly renormalized value of the zero-point energy density today (for a free theory) is in fact far from being 122 orders of magnitude larger than the critical energy density, as often quoted in the literature. We mainly consider the case of scalar fields but also treat the cases of fermions and gauge bosons which allows us to discuss the question of vacuum energy in super-symmetry. Then, we discuss how the cosmological constant can be measured in cosmology and constrained with experiments such as measurements of planet orbits in our solar system or atomic spectra. We also review why the Lamb shift and the Casimir effect seem to indicate that the quantum zero-point fluctuations are not an artifact of the quantum field theory formalism. We investigate how experiments on the universality of free fall can constrain the gravitational properties of vacuum energy and we discuss the status of the weak equivalence principle in quantum mechanics, in particular the Collela, Overhausser and Werner experiment and the quantum Galileo experiment performed with a Salecker-Wigner-Peres clock. Finally, we briefly conclude with a discussion on the solutions to the cosmological constant problem that have been proposed so far.
Jérôme Martin, Institut d’Astrophysique de Paris, UMR7095-CNRS, Université Pierre et Marie Curie. Preprint arXiv:1205.3365; published in Comptes Rendus Physique 13, 566–665 (2012).
(Abstract only — see the rights note above for why the full text is not reproduced here. The complete 89-page review, with its fourteen figures and its full derivations of the bubble diagrams, the Gaussian effective potential and the curved-space calculation, is at the source.)
Companion sheets on this site: Steven Weinberg’s 1989 review, the canonical statement of the problem Martin is refining, is at /library/stm-5610822bc8; Sean Carroll’s Living Review is at /library/stm-41ce25c10b; Helge Kragh’s history of the identification of zero-point energy with dark energy is at /library/stm-479f3334b8; and Grigory Volovik’s book-length development of the superfluid-vacuum analogue that Martin points to in his conclusions is at /library/stm-6ee45bd8be.
The way in
https://arxiv.org/abs/1205.3365LICENCE. The preprint is arXiv:1205.3365v1, posted 15 May 2012, and the arXiv record carries the arXiv.org perpetual non-exclusive distribution licence rather than a Creative Commons statement — checked on the arXiv abstract page on 2026-09-08. The version of record is Comptes Rendus Physique 13, 566 to 665 (2012), doi 10.1016/j.crhy.2012.04.008, published by Elsevier. So this sheet holds the summary, the claims and the author’s own abstract and sends the reader to the source. The claims below are read against the complete 89-page preprint and the locators use the review’s own section and equation numbering. Jérôme Martin wrote it at the Institut d’Astrophysique de Paris, UMR7095-CNRS, Université Pierre et Marie Curie. REGISTRY NOTE: the record reached the library with the byline spelled Jerome Martin and with chapters ch02, ch03 and ch13; the accented spelling on the paper’s own title page is restored here, and the sheet carries chapters 2 and 13.
How to cite it
Jérôme Martin (2012) Everything You Always Wanted To Know About The Cosmological Constant Problem (But Were Afraid To Ask). doi:10.1016/j.crhy.2012.04.008
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