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] The value was predicted to be either zero or very small, 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]

Quantum description

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).[5] 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.[6] 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.[7]

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.[7]


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.[8] 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]

Proposed solutions

Some physicists propose an anthropic solution, and argue that we live in one region of a vast multiverse that has different regions with different vacuum energies. These anthropic arguments posit that only regions of small vacuum energy such as the one we live in are reasonably capable of supporting intelligent life. Such arguments have existed in some form since at least 1981. Around 1987, Steven Weinberg estimated that the maximum allowable vacuum energy for gravitationally-bound structures to form is problematically large, even given the observational data available in 1987, and concluded the anthropic explanation appears to fail; however, more recent estimates by Weinberg and others, based on other considerations, find the bound to be closer to the actual observed level of dark energy.[9][10] Anthropic arguments gradually gained credibility among many physicists after the discovery of dark energy and the development of the theoretical string theory landscape, but are still derided by a substantial skeptical portion of the scientific community as being problematic to verify. Proponents of anthropic solutions are themselves divided on multiple technical questions surrounding how to calculate the proportion of regions of the universe with various dark energy constants.[9][11]

Other proposals involving modifying gravity to diverge from the general relativity. These proposals face the hurdle that the results of observations and experiments so far have tended to be extremely consistent with general relativity and the ΛCDM model, and inconsistent with thus-far proposed modifications. In addition, some of the proposals are arguably incomplete, because they solve the "new" cosmological constant problem by proposing that the actual cosmological constant is exactly zero rather than a tiny number, but fail to solve the "old" cosmological constant problem of why quantum fluctuations seem to fail to produce substantial vacuum energy in the first place. Nevertheless, many physicists argue that, due in part to a lack of better alternatives, proposals to modify gravity should be considered "one of the most promising routes to tackling" the cosmological constant problem.[11]

Quantum field theory predictions based on light front quantization

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,[12][13] in 2011, Stanley Brodsky and Robert Shrock showed[14] 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,[15][16] 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)[17][18][19]) could modify the Higgs field zero-mode, thereby possibly producing a non-zero contribution to the cosmological constant.

See also


  1. 1 2 3 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. Bibcode:1995AmJPh..63..620A. 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. 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.
  6. "A simple explanation of mysterious space-stretching 'dark energy?'". Science | AAAS. 10 January 2017. Retrieved 8 October 2017.
  7. 1 2 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.
  8. Rugh and Zinkernagel (2002), 36ff.
  9. 1 2 Linde, Andrei. "A brief history of the multiverse." Reports on Progress in Physics 80, no. 2 (2017): 022001.
  10. Martel, Hugo; Shapiro, Paul R.; Weinberg, Steven (January 1998). "Likely Values of the Cosmological Constant". The Astrophysical Journal. 492 (1): 29–40. arXiv:astro-ph/9701099. Bibcode:1998ApJ...492...29M. doi:10.1086/305016.
  11. 1 2 Bull, Philip, Yashar Akrami, Julian Adamek, Tessa Baker, Emilio Bellini, Jose Beltrán Jiménez, Eloisa Bentivegna et al. "Beyond ΛCDM: Problems, solutions, and the road ahead." Physics of the Dark Universe 12 (2016): 56-99.
  12. H. Leutwyler, J.R. Klauder, L. Streit. Quantum field theory on lightlike slabs, Nuovo Cim. A66 (1970) 536 DOI: 10.1007/BF02826338
  13. A. Casher and L. Susskind. Chiral magnetism (or magnetohadrochironics) Phys. Rev. D9 (1974) 436 DOI: 10.1103/PhysRevD.9.436
  14. 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].
  15. 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].
  16. S. J. Brodsky, C. D. Roberts, R. Shrock and P. C. Tandy. Confinement contains condensates. Phys.Rev. C85 (2012) 065202 [arXiv:1202.2376]
  17. "Will the Universe expand forever?". NASA. 24 January 2014. Retrieved 16 March 2015.
  18. "Our universe is Flat". FermiLab/SLAC. 7 April 2015.
  19. Marcus Y. Yoo (2011). "Unexpected connections". Engineering & Science. Caltech. LXXIV1: 30.
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