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Cosmological constant problem

In cosmology, the cosmological constant problem or vacuum catastrophe is the disagreement between the observed values of vacuum energy density (the small value of the cosmological constant) and theoretical large value of zero-point energy suggested by quantum field theory.

Depending on the Planck energy cutoff and other factors, the discrepancy is as high as 120 orders of magnitude,[1] a state of affairs described by physicists as "the largest discrepancy between theory and experiment in all of science"[1] and "the worst theoretical prediction in the history of physics."[2]



The basic problem of a vacuum energy producing a gravitational effect was identified as early as 1916 by Walther Nernst.[3][further explanation needed] The value was predicted[according to whom?] to be either zero or very small[why?], so that the theoretical problem was already apparent, and began to be actively discussed in the 1970s.

With the development of inflationary cosmology in the 1980s, the problem became much more important: as cosmic inflation is driven by vacuum energy, differences in modeling vacuum energy leads to huge differences in the resulting cosmologies.[4][further explanation needed] In February 2018,[5] a solution to the problem was proposed by Christian Henke which is based on a variable cosmological constant term and a carefully analysis of the total energy density of an empty Friedmann universe.

Quantum descriptionEdit

After the development of quantum field theory in the 1940s, the first to address contributions of quantum fluctuations to the cosmological constant was Zel’dovich (1967, 1968).[6][non-primary source needed] In quantum mechanics, the vacuum itself should experience quantum fluctuations. In general relativity, those quantum fluctuations constitute energy that would add to the cosmological constant. However, this calculated vacuum energy density is many orders of magnitude bigger than the observed cosmological constant.[7] Original estimates of the degree of mismatch were as high as 120 orders of magnitude; however, modern research suggests that, when Lorentz invariance is taken into account, the degree of mismatch is closer to 60 orders of magnitude.[8]

The calculated vacuum energy is a positive, rather than negative, contribution to the cosmological constant because the existing vacuum has negative quantum-mechanical pressure, and in general relativity, the gravitational effect of negative pressure is a kind of repulsion. (Pressure here is defined as the flux of quantum-mechanical momentum across a surface.) Roughly, the vacuum energy is calculated by summing over all known quantum-mechanical fields, taking into account interactions and self-interactions between the ground states, and then removing all interactions below a minimum "cutoff" wavelength to reflect that existing theories break down and may fail to be applicable around the cutoff scale. Because the energy is dependent on how fields interact within the current vacuum state, the vacuum energy contribution would have been different in the early universe; for example, the vacuum energy would have been significantly different prior to electroweak symmetry breaking during the quark epoch.[8]


The vacuum energy in quantum field theory can be set to any value by renormalization. This view treats the cosmological constant as simply another fundamental physical constant not predicted or explained by theory.[9] Such a renormalization constant must be chosen very accurately because of the many-orders-of-magnitude discrepancy between theory and observation, and many theorists consider this ad-hoc constant as equivalent to ignoring the problem.[1]

Quantum field theory predictions based on light front quantizationEdit

Light front quantization is a rigorous alternative due to Paul Dirac to the usual second quantization method (instant-form method). Causality and frame-independence (Poincaré invariance) are explicit, contrary to quantization in the instant-form method. The light-front vacuum state is defined as the eigenstate of lowest invariant mass.

Vacuum fluctuations do not appear in the Light-Front vacuum since all particles have positive momenta p+= p0+p3. Since p+ is conserved, particles cannot couple to the light front vacuum since it has p+=0.

These features make the quantum field theory vacuum essentially trivial, with no vacuum dynamics such as condensate (i.e. vacuum expectation value). In contrast, vacuum fluctuations appear in the vacuum of the ordinary instant-form (the lowest energy eigenstate of the instant-form Hamiltonian), but the physical effects depend on the arbitrary choice of Lorentz frame. This fact and the violation of causality indicate that the instant-form vacuum cannot represent of the physical vacuum.

While the features of the LF vacuum have been known for a long time,[10][11] in 2011, Stanley Brodsky and Robert Shrock showed[12] that the absence of condensates implies that in the Standard Model of Particle Physics, there is no contribution from QED, Weak interactions and QCD to the cosmological constant. It is thus predicted to be zero in a flat space-time. This was later validated and developed,[13][14] by other prominent QCD theorists.

In the case of the Higgs mechanism, the usual Higgs vacuum expectation value in the instant-form vacuum is replaced by a constant scalar background field - a "zero mode" with kμ=0. The phenomenological predictions are unchanged using the LF formalism. Since the Higgs zero mode has no energy or momentum density, it does not contribute to the cosmological constant.

The small non-zero value of the cosmological constant must then be attributed to other mechanisms; for example a slight curvature of the shape of the universe (which is not excluded within 0.4% (as of 2017)[15][16][17]) could modify the Higgs field zero-mode, thereby possibly producing a non-zero contribution to the cosmological constant.

See alsoEdit


  1. ^ a b c Adler, Ronald J.; Casey, Brendan; Jacob, Ovid C. (1995). "Vacuum catastrophe: An elementary exposition of the cosmological constant problem". American Journal of Physics. 63 (7): 620–626. doi:10.1119/1.17850. ISSN 0002-9505. 
  2. ^ MP Hobson, GP Efstathiou & AN Lasenby (2006). General Relativity: An introduction for physicists (Reprint ed.). Cambridge University Press. p. 187. ISBN 978-0-521-82951-9. 
  3. ^ W Nernst (1916). "Über einen Versuch von quantentheoretischen Betrachtungen zur Annahme stetiger Energieänderungen zurückzukehren". Verhandlungen der Deutschen Physikalischen Gesellschaft (in German). 18: 83. 
  4. ^ S. Weinberg “The cosmological constant problem”, Review of Modern Physics 61 (1989), 1-23.
  5. ^ C. Henke (2018). "Quantum vacuum energy in general relativity". The European Physical Journal C. 78: 126. doi:10.1140/epjc/s10052-018-5621-7. 
  6. ^ Zel’dovich, Y.B., ‘Cosmological Constant and Elementary Particles’ JETP letters 6 (1967), 316-317 and ‘The Cosmological Constant and the Theory of Elementary Particles’ Soviet Physics Uspekhi 11 (1968), 381-393.
  7. ^ "A simple explanation of mysterious space-stretching 'dark energy?'". Science | AAAS. 10 January 2017. Retrieved 8 October 2017. 
  8. ^ a b Martin, Jerome. "Everything you always wanted to know about the cosmological constant problem (but were afraid to ask)." Comptes Rendus Physique 13.6-7 (2012): 566-665.
  9. ^ Rugh and Zinkernagel (2002), 36ff.
  10. ^ H. Leutwyler, J.R. Klauder, L. Streit. Quantum field theory on lightlike slabs, Nuovo Cim. A66 (1970) 536 DOI: 10.1007/BF02826338
  11. ^ A. Casher and L. Susskind. Chiral magnetism (or magnetohadrochironics) Phys. Rev. D9 (1974) 436 DOI: 10.1103/PhysRevD.9.436
  12. ^ S. J. Brodsky and R. Shrock. Condensates in Quantum Chromodynamics and the Cosmological Constant. Proc.Nat.Acad.Sci. 108 (2011) 45-50, [arXiv:0905.1151].
  13. ^ S. J. Brodsky, C. D. Roberts, R. Shrock and P. C. Tandy. Essence of the vacuum quark condensate. Phys.Rev. C82 (2010) 022201 [arXiv:1005.4610].
  14. ^ S. J. Brodsky, C. D. Roberts, R. Shrock and P. C. Tandy. Confinement contains condensates. Phys.Rev. C85 (2012) 065202 [arXiv:1202.2376]
  15. ^ "Will the Universe expand forever?". NASA. 24 January 2014. Retrieved 16 March 2015. 
  16. ^ "Our universe is Flat". FermiLab/SLAC. 7 April 2015. 
  17. ^ Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. Caltech. LXXIV1: 30.