Dark energy as spacetime curvature induced by quantum vacuum fluctuations

Assuming that the vacuum expectation of the quantum vacuum energy-momentum is zero, I show that fluctuations give rise to a smooth curvature of spacetime. That curvature is currently attributed to a uniform energy-momentum tensor, known as “dark energy”.Our calculation leads to an estimate which roughly agrees with observations. An outstanding open problem in cosmology concerns the nature of “dark energy” and “dark matter”, which toghether contribute more than 95% of the total mass-energy of the universe. The present mass density of dark energy, ρDE, and dark matter, ρDM , are known to be [1],[2] ρDE ∼ 10 kg/m, ρDM ∼ 1 3 ρDE , (1) which may be compared with the baryon mass density ρB ∼ 1 5 ρDM ∼ 1 15 ρDE. (2) In the present paper I will be concerned with dark energy only. The current wisdom is to identify dark energy with the cosmological constant introduced by Einstein in 1917. An alternative hypothesis, equivalent in practice, is to postulate a vacuum energy-momentum tensor Tμν = ρDE gμν , (3) 1 gμν being the metric tensor and ρDE a constant. The assumption eq.(3) is appropriate for the vacuum, in Minkowski spacetime, because it is the only energy-momentum tensor which is invariant under Lorentz transformations. A problem appears, however, when one attempts to estimate the value of ρDE as an energy density of the vacuum. In fact if the dark energy is really the quantum vacuum energy it seems difficult to understand why the density is not either strictly zero or of the order of Planck ́s density, that is ρDE ∼ c Gh ∼ 10kg.m. (4) This density is 123 orders of magnitude greater than the observed value. The aim of this paper is to propose a mechanism which may be the origin of what is known as “dark energy”. I start assuming that the quantum vacuum energy is strictly zero, and then I show that fluctuations of the vacuum may give rise to a smooth curvature of spacetime. That curvature happens to be the same which would appear if there existed a uniform energymomentum tensor given by eqs.(3) and (1). In other words there is no dark energy, but there exists a curvature of spacetime induced by the quantum vacuum fluctuation which mimic the effect of a constant energy-momentum tensor. The mechanism is explained in more detail in the following. The quantum vacuum may be described by a set of interacting relativistic quantum fields, likely the fields associated to the standard model of elementary particles. However for our purposes it is not necessary to make any specific hypothesis about the fields. It is enough to assume the existence of an energy-momentum quantum operator, T ν μ , associated to the vacuum. That operator will be a function of the space and time coordinates, but I shall assume that at every point its vacuum expectation value is zero, that is