# Quantum vacuum energy in general relativity

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## Abstract

The paper deals with the scale discrepancy between the observed vacuum energy in cosmology and the theoretical quantum vacuum energy (cosmological constant problem). Here, we demonstrate that Einstein’s equation and an analogy to particle physics leads to the first physical justification of the so-called fine-tuning problem. This fine-tuning could be automatically satisfied with the variable cosmological term \(\varLambda (a)=\varLambda _0+\varLambda _1 a^{-(4-\epsilon )}\), \(0 < \epsilon \ll 1,\) where *a* is the scale factor. As a side effect of our solution of the cosmological constant problem, the dynamical part of the cosmological term generates an attractive force and solves the missing mass problem of dark matter.

## 1 Introduction

The remainder of the paper is organised as follows. In Sect. 2 we review the pressure-free Friedmann equations where the cosmological term \(\varLambda \) is a function of the scale factor *a*. Section 3 is devoted to a Klein-Gordon equation for the scale factor. In order to identify the arrived equation as an Euler-Lagrange equation, the Klein-Gordon equation is transformed to a scale factor independent space-time metric. Then, the Lagrangian density could be determined and the related energy-momentum tensor of an empty Friedmann universe is studied in Sect. 4. This enabled us to establish a canonical decomposition of the total energy into a cosmological and a remainder term. Consequently, the fine-tuning of the energy densities is justified. Moreover, we demonstrate with an application of the model \(\varLambda (a)=\varLambda _0+\varLambda _1 a^{-(4-\epsilon )}\), \(0 < \epsilon \ll 1\) that the total energy density of an empty Friedmann universe equals the quantum zero-point energy. Finally, we show that our solution of the cosmological constant problem explains cosmological observations without the missing mass of dark matter.

## 2 A time-dependent cosmological term

*M*be a 4-dimensional manifold equipped with a metric \(\bar{g}_{\mu \nu },\) which determines the space-time interval as follows:

*k*denotes the curvature parameter of unit \(\text {length}^{-2}.\) Inserting the metric \(\bar{g}_{\mu \nu }\) in Einstein’s field equation, we get Friedmann’s equations for the scaling factor

*a*(

*t*)

## 3 Conformally-related trace equation

In this section we provide the ground for a total energy discussion of an empty Friedmann universe which is compatible with quantum field theory. To do so, we have to establish the notion of total energy on a Lagrangian density such that the Euler–Lagrange equation is consistent with Friedmann’s equations. Therefore, we have to find a space-time metric which is independent of the scale factor. In order to do so, we start from the field equation for the Robertson–Walker metric \(\bar{g}_{\mu \nu }\) and consider the transformed equation for the conformally-related metric \(g_{\mu \nu }.\)

*u*be a strictly positive \(C^\infty (M)-\)function. The metric \(g_{\mu \nu }=u^{-2} \bar{g}_{\mu \nu }\) is said to be conformally-related to \(\bar{g}_{\mu \nu }.\) Introducing the notation \(\nabla _{\mu }\) for the covariant derivative, \(\varDelta =g^{\mu \nu } \nabla _{\mu } \nabla _{\nu }\) and \(|\nabla u|^2=g^{\mu \nu } \nabla _{\mu } u\nabla _{\nu }u,\) we note down the relation for the Ricci scalar (see [6, p. 446])

## 4 Vacuum energy

*a*, Eq. (10) is the Euler–Lagrange equation of the Lagrangian

*V*(

*a*). Using Eq. (1), we get

*a*has an initial singularity if \(\varLambda _1/(3-r)>0\) is satisfied again.

*r*. First, let \(0<r <3.\) Since (18) leads to \(\varLambda _1/(3-r)<0,\) we neglect this case. Further we have \(3<r<4,\) which gives the compatiblility condition \(\varLambda _1/(3-r)>0.\) In [11, 12], this case was excluded from the considerations because it was assumed without further substantiation that \(\varLambda _1\) is always positive. In order to analyse the acceleration behavior, we have to discuss the term

## 5 Concluding remarks

In this paper, the solution of the cosmological constant problem is demonstrated by an application of a variable cosmological term \(\varLambda (a)=\varLambda _0 + \varLambda _1 a^{-(4-\epsilon )}\). It has been shown that the expansion field *a* satisfies the Klein-Gordon equation in a non-dynamical space-time and that a variational cosmological term can realise the usual potentials from particle physics. Further it was confirmed that the total energy density of an empty Friedmann universe is related to the cosmological term such that the fine-tuning problem was avoided by the setting \(\varLambda _1=- \epsilon /2 \pi (4-\epsilon ) l_p^2.\) As a consequence of the constraint \(0< \epsilon <1,\) the initial singularity is guaranteed and the dynamical part of the cosmological term generates the attractive force of dark matter. Finally, the setting of \(\epsilon =9.151 \cdot 10^{-122}\) generates the missing mass of dark matter which constitutes 26% of the matter-energy density.

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