^{1}

^{2}

^{1}

^{2}

^{3}.

In the action formalism variations of metric tensors usually are limited by the Hubble horizon. On the contrary, variations of quantum fields should be extended up to the event horizon, which is the real boundary of the spacetime. As a result the entanglement energy of quantum particles across the apparent horizon is missed in the cosmological equations written for the Hubble volume. We identify this missing boundary term with the dark energy density and express it (using the zero energy assumption for the finite universe) as the critical density multiplied by the ratio of the Hubble and event horizons radii.

Many authors consider so-called emergent theories in which gravity is not a fundamental field, but like thermodynamics or hydrodynamics is defined for the matter in bulk [

Since in General Relativity horizons are unavoidable and horizons block information, entropy and temperature can be introduced for spacetime. One such boundary is the apparent horizon with the radius (for the spatially flat universe),

The concept of entropy is a powerful tool in thermodynamics, information theory, and quantum physics and allows us to study different aspects of physical systems using a similar mathematical framework. In quantum mechanics a measurement is considered as the interaction of three systems: the quantum object, memory (measurement device), and observer. Then the total entropy of the ensemble of all quantum particles, which is formed by the information, statistical (thermodynamic), and quantum (entanglement) components [

In our previous papers it was demonstrated that “world ensemble” approach is compatible with the existing field-theoretical descriptions, as the relativistic [

As it was noted in [

In this paper we want to connect the entanglement energy of quantum particles across the apparent horizon, which is missed in the classical cosmological equations written for the Hubble volume, with the dark energy (DE), origin of which remains a mystery [

Alternatives of the introduction of a cosmological constant are that DE arises from the evolution of dynamical fields of an unknown origin, or modifications of General Relativity. In order to distinguish between these hypotheses, a worldwide effort is ongoing to measure the effective equation of state and clustering properties of DE, using wide field cosmological surveys [

One promising approach for solving the DE puzzle is the Holographic DE model [

We note that, based on spacetime thermodynamics, a proper causal boundary of the classical spacetime is its apparent horizon [

From the other hand, the quantum fluctuations of matter fields should be limited not by the Hubble horizon (

The terms corresponding to entanglement across

Equation (

Connections of the cosmological constant with the boundary conditions and generalized equations of state can be demonstrated also from the cosmological equations. For a homogeneous, isotropic, and flat universe (

If instead of (

To find the value of DE density in our approach let us estimate the entropy input in a region as the sum of the entropy flux (entropy received per unit surface) transferred through the boundary and the entropy supplied by internal sources (entropy generated per unit volume). If we neglect the entropy supplied by internal sources, then according to the Second Law of Thermodynamics the time derivative of the entropy contained within the volume,

For the finite universe limited by the event horizon,

Note that the relation (

To conclude, in this paper we estimate the DE density within the thermodynamic model of gravity using the zero energy condition: the total energy of the universe inside its event horizon is zero. We notice that in the action formalism variations of metric tensor should be limited by the Hubble horizon, which represents a causal boundary of classical spacetime. On the contrary, variations of quantum fields are limited by the event horizon, which is the real boundary of the spacetime. Then the entanglement energy of quantum particles across the apparent horizon is missed in the cosmological equations written for the Hubble volume. We identify this entanglement density with the DE, which can be introduced as a boundary term in the cosmological equations. In our model the DE density equals the critical density reduced by the ratio of the squares of the Hubble and event horizons radii (

The data used to support the findings of this study are available from the corresponding author upon request.

The author declares that there are no conflicts of interest.