Contents

Definition

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

$S_{cos} : (X,g) \mapsto \lambda \int_X dvol_g \,,$

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

For instance pure Einstein-Hilbert gravity with cosmological constant (and not other fields) is given by the functional

$S_{EH} + S_{cos} : (X,g) \mapsto \int_X R dvol + \lambda \int_X d vol_g \,,$

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

$S : (X,g,\phi) \mapsto \int_X \lambda(\phi) dvol_g \,.$

In this context those solutions to the Euler-Lagrange equations are of interest in which $\lambda(\phi)$ 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.

(e.g. Einasto 09, fig 17)

References

On the Cosmological constant problem

Discussion of the experimentally observed tiny cosmological constant and the issue with its theoretical explanation includes

• 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 string theory includes

Shortly afterwards it became popular to invoke the landscape of string theory vacua in order to “solve” this state of affairs.