cosmological constant







In theories of gravity and cosmology a cosmological constant refers to an energy contribution associated with the vacuum itself.

This is called a cosmological constant because the simplest way to see this effect in theory is by a summand to the Einstein-Hilbert action which is proportional to the volume of spacetime, with proportionality factor some “constant” λ\lambda, as in (1) below.

See below

More generally, almost-constant contributions to matter fields may effectively have the same kind of effect.

Specifically in perturbative quantum field theory on curved spacetimes the vacuum expectation value of the stress-energy tensor of the various matter-fields receives constant or essentially constant contributions, notably from renormalization freedom (e.g. Hack 15, section 3.2.1).

See below

In perturbative string theory the string perturbation series associated with a 2d SCFT is (supposedly) UV-finite and hence has “chosen its own renormalization” already, hence here the cosmological constant may in principle be read off from a choice of perturbative string theory vacuum. See below

Observation shows that the effective cosmological constant of the observable universe is comparatively small but positive, see


In classical gravity

In an action functional on a space of pseudo-Riemannian manifolds – such as the Einstein-Hilbert action functional for gravity – a cosmological constant is a term proportional to the volume

S cc:(X,g)λ Xdvol g, S_{cc} \;\colon\; (X,g) \mapsto \lambda \int_X dvol_g \,,

where λ\lambda \in \mathbb{R} is the cosmological constant .

For instance, pure Einstein-Hilbert gravity with cosmological constant (and no other fields) is given by the functional

(1)S EH+S cc:(X,g) XRdvol+λ Xdvol g, S_{EH} + S_{cc} : (X,g) \mapsto \int_X R\, dvol + \lambda \int_X d vol_g \,,

Generically it happens that one considers action functionals where λ\lambda is in fact not a constant, but a function of other fields ϕ\phi on XX.

S:(X,g,ϕ) Xλ(ϕ)dvol g. S \;\colon\; (X,g,\phi) \mapsto \int_X \lambda(\phi) dvol_g \,.

In this context those solutions to the Euler-Lagrange equations are of interest in which λ(ϕ)\lambda(\phi) happens to be exactly or approximately constant. Many such models of not-really-constant-but-effectively-constant terms proportional to the volume are being proposed and considered in attempts to explain observed or speculated dynamics of the cosmos.

See in particular at FRW model for the role of the cosmological constant in homogeneous and isotropic models as in the standard model of cosmology. In that context the cosmological constant is also called the dark energy (density), which makes up about 70% of the energy density of the observable universe (the rest being dark matter) and a comparatively little bit of baryonic matter.

In perturbative quantum gravity

In perturbative quantum field theory on curved spacetimes the cosmological constant receives contributions from the vacuum expectation value of the stress-energy tensor of the matter fields. There is renormalization-freedom in this contribution (Wald 78, Tichy-Flanagan 98 Moretti 01).

Explicitly for FRW models this is discussed in (Dappiaggi-Fredenhagen-Pinamonti 08, Dappiagi-Hack-Moeller-Pinamonti 10). Specifically for the standard model of cosmology see (Hack 13, around (4)).

A useful review is in (Hack 15, section 3.2.1).

This means that apart from the freedom of choosing a classical comsological constant in the Einstein-Hilbert action as above, its perturbative quantization (perturbative quantum gravity) introduces renormalization freedom to the value of the cosmological constant.

The folklore discussion of the “cosmological constant problem” (see the references below) tends not to take this freedom in the theory into account (see the discussion at naturalness).

In inhomogeneous cosmology

It has been suggested the observed cosmological constant/dark energy may be but an artifact of the overly idealistic approximation of cosmic homogeneity, and that a more accurate inhomogeneous cosmology would not need to assume any dark energy (e.g. Buchert 07, Buchert 11, Buchert-Rasanen 11, Scharf 13).

A seminal argument that it is consistent to neglect cosmic inhomogeneity due to (Green-Wald 10, Green-Wald 13), has been called into question in Buchert et al. 15, where it is concluded that the question is more subtle and remains open. Recent review is in Belejko-Korzyński 16.

If the apparent small positive cosmological constant were but an artifact of neglecting backreaction of inhomegeneities, some theoretical puzzlements regarding quantum gravity on de Sitter spacetimes would disappear (see Rajaraman 16 for general discussion and Danielsson-VanRiet 18, p. 27 for discussion of perturbative string theory vacua).


(e.g. Einasto 09, fig 17)



Introduction and review:

  • Konstantinos Xenos, An Introduction to FRW Cosmology and dark energy models (arXiv:2101.06135)




Cautioning against the interpretation of type Ia supernovae as indicative of a small positive cosmological constant (de Sitter spacetime) as in the current standard model of cosmology includes the following (see also at inhomogeneous cosmology and at Witten's Dark Fantasy):


Discussion of the cosmological constant in the rigorous formulation of perturbative AQFT on curved spacetimes includes the following.

The freedom of renormalization of the vacuum expectation value of any stress-energy tensor, hence of the cosmological constant, was discussed in

Notice that (Wald 78) is based on

which claimed no freedom of renormalization, but Wald 78 explains that this was due to a mistake inherited from a citation.

Further development of this includes

  • Wolfgang Tichy, Eanna E. Flanagan, How unique is the expected stress-energy tensor of a massive scalar field?, Phys.Rev. D58 (1998) 124007 (arXiv:gr-qc/9807015)

  • Valter Moretti, Comments on the Stress-Energy Tensor Operator in Curved Spacetime, Commun. Math. Phys. 232 (2003) 189-221 (arXiv:gr-qc/0109048)

A useful review is in

Realization of renormalization of stress-energy/cosmological constant in concrete FRW models is discussed in

Discussion specifically for the standard model of cosmology is in

More on subtleties of renormalization of the cosmological constant:

From higher curvature corrections

Effective dark energy from higher curvature corrections, as in the Starobinsky model of cosmic inflation:

  • Michal Artymowski, Zygmunt Lalak, Inflation and dark energy from f(R)f(R) gravity, JCAP09(2014)036 (arXiv:1405.7818)

  • Michal Artymowski, Zygmunt Lalak, Marek Lewicki, Inflation and dark energy from f(R)f(R) gravity (arXiv:1510.04864)

From inhomogeneous cosmology

Discussion of the cosmological constant as an artefact of inhomogeneous cosmology (see there for more) includes the following

See also at inhomogeneous cosmology.

The “cosmological constant problem”

Discussion of the experimentally observed tiny cosmological constant and the folklore theoretical problem with that includes the following

  • Subir Sarkar, New results in cosmology (arXiv:hep-ph/0201140)

  • Stefanus Nobbenhuis, The cosmological constant problem – an inspiration for new physics PhD thesis (2006) (web pdf)

  • Joan Sola, Cosmological constant and vacuum energy: old and new ideas, J.Phys.Conf.Ser. 453 (2013) 012015 (arXiv:1306.1527)

  • Jaan Einasto, Dark matter (arXiv:0901.0632) 2009

In string theory

Discussion from the point of view of perturbative string theory, where the cosmological constant is fixed by the choice of perturbative string theory vacuum.

For fundamentally vanishing cc (“Witten’s Dark Fantasy”)

An argument for non-perturbative non-supersymmetric 4d string phenomenology with fundamentally vanishing cosmological constant, based on 3d M-theory on 8-manifolds decompactified at strong coupling to 4d via duality between M-theory and type IIA string theory (recall the super 2-brane in 4d):

This suggestion was called Witten's Dark Fantasy in Heckmann-Lawrie-Lin-Zoccarato 19, Section 8, where a concrete realization of this scenario in F-theory is claimed:

For fundamentally non-vanishing cc

But a) there might be a large space of perturbative string theory vacua and b) the de Sitter vacua that seem to correspond to observation tend to exist (only) as metastable vacua:

This observation led to the discussion of the “landscape of string theory vacua”.

For review see

  • Renata Kallosh, section 3 of de Sitter vacua and the landscape of string theory, J. Phys. Conf. Ser. 24 (2005) 87-110 (spire)

However, consistency problems of these arguments are raised in

Last revised on January 18, 2021 at 01:22:03. See the history of this page for a list of all contributions to it.