Lagrangian Formulation of a Solution to the Cosmological Constant Problem

A covariant Lagrangian formulation of a solution to the cosmological constant problem, based on vizualising the fluctuations of the vacuum energy as a nonequilibrium process with stochastic behaviour, is presented. The variational principle yields equations of motion for the cosmological “constant” Λ, treated as a dynamical field, together with an equation for a Lagrange multiplier field φ, and the standard Einstein field equations with a variable cosmological constant term. A stochastic model of Λ yields a natural explanation for the smallness or zero value of the constant in the present epoch and its large value in an era of inflation in the early universe. Typeset using REVTEX 1 A recent model for solving the cosmological constant problem has been proposed [1], in which the vacuum energy is treated as a fluctuating environment with stochastic behaviour. In the following, we shall present a covariant formulation based on a Lagrangian density, which yields classical equations incorporating Einstein’s gravitational field equations, upon which a stochastic treatment using a Wiener process can be developed. Although the model uses methods of critical phenomena and non-equilibrium statistical mechanics to model the vacuum energy, it can be considered as a phenomenological description of the kinds of behaviour that could be expected in a more fundamental quantum gravity theory. It is generally agreed that the cosmological constant problem cannot be solved within the context of a purely classical theory of gravity. However, no satisfactory quantum gravity theory has been formulated, so it is hoped that our model can shed light on the solution to the problem without the full apparatus of such a theory. The Lagrangian density is given by L = LR + LΛ + LM , (1) where LR = √ −ggRμν , (2a) LΛ = −2 √ −g[Λ + (Λ,μu − αΛ + Λ)φ], (2b) and Λ = Λ(x) is the variable cosmological “constant”, treated as a dynamical field, φ is a Lagrange multiplier field, u = dx/dτ is an observer’s four-velocity along a world line in spacetime, α is a constant and LM is a matter Lagrangian density. A variation of L with respect to φ and Λ yields the equations of motion: Λ,μu μ − αΛ+ Λ = 0, (3a) 1 √−g ( √ −guφ),μ + (α− 2Λ)φ− 1 = 0. (3b) Varying L with respect to g and using (3a) gives Rμν − 1 2 gμνR + Λgμν = 8πGTμν . (4)