Dear Guibert,

The general explanation for dark energy is that it is simply the "cost of having space": that is, a volume of space has some intrinsic, fundamental energy. This is the cosmological constant, sometimes called Lambda  Λ.  Since energy and mass are related by E = mc2,  the general relativity of Einstein's theory predicts that this energy will have a gravitational effect. It is sometimes called a vacuum energy because it is the energy density of empty vacuum. In fact, most theories of particle physics predict vacuum fluctuations that would give the vacuum this sort of energy. This is related to the Casimir effect, in which there is a small suction into regions where virtual particles are geometrically inhibited from forming. The cosmological constant is estimated by cosmologists to be on the order of 10−29 g/cm3, or about 10−120 in reduced Planck units. Particle physics predicts a natural value of 1 in reduced Planck units, leading to a large discrepancy. The cosmological constant has negative pressure equal to its energy density and so causes the expansion of the universe to accelerate. The reason why a cosmological constant has negative pressure can be seen from classical thermodynamics; Energy must be lost from inside a container to do work on the container. A change in volume dV requires work done equal to a change of energy −P dV, where P is the pressure. But the amount of energy in a container full of vacuum actually increases when the volume increases (dV is positive), because the energy is equal to ρV, where ρ is the energy density of the cosmological constant. Therefore, P is negative and, in fact,P = −ρ.

Einstein, A (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitaetstheorie". Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften Berlin. part 1: 142–152.

What is Dark Energy?, Space.com, 1 May 2013

Collaboration, Planck, PAR Ade, N Aghanim, C Armitage-Caplan, M Arnaud, et al., Planck 2015 results. XIII. Cosmological parameters. arXiv preprint 1502.1589v2 , 6 Feb 2015.

Hogan, Jenny (2007). "Welcome to the Dark Side". Nature 448 (7151): 240–245. Bibcode:2007Natur.448..240H. doi:10.1038/448240a. PMID 17637630.

 Urry, Meg (2008). The Mysteries of Dark Energy. Yale Science. Yale University.

 Gamov, George (1970). My World Line. Viking Press. p. 44. ISBN 978-0670503766

Rosen, Rebecca J. "Einstein Likely Never Said One of His Most Oft-Quoted Phrases". The Atlantic. The Atlantic Media Company. Retrieved 10 August 2013.

Hoping this will contribute to the discussion on this topic.

Rafik

Dear Rafik,

Thank you for your answer. Casimir effect has also been evoked to explain the inflationary stage which could have occurred shortly after the Big Bang (see enclosed article of J. C. O'Brien).

So in connexion with the accelerated expansion, I understand that virtual particles (photons ?) are permanently formed from vacuum energy fluctuations in the quantum ground state (zero point energy) resulting in Casimir pressure and accelerated expansion (negative pressure p=-rholambda).

As the Universe is permanently expanding,C cn we then say that there is a permanent increase in the formation of virtual particles from the end of inflation (13.8 billion years ago) in order to hold their average density constant to maintain a negative pressure at cosmic scale ? How can this permanent increase then physically be explained ?

Best regards

Guibert       

Thank you, Ales, for these interesting remarks and the references.

However, this does not help me very much about what is behind the maintained negative pressure at cosmic scale for billion years. So the question remains.

Concerning Dr. Nikodem Popławski’s model, can you explain in what far observations are better fitted with it ? Isn't the standard cosmological model with big bang, dark matter and dark energy (the so-called LambdaCDM model + BBN-data) the simplest model fitting all observational data nowadays ?     

Dear Guibert,

The following paragraphs may be partially address your question:

There are two proposed forms for dark energy: (1) a constant energy density filling space homogeneously and (2) scalar fields such as quintessence or moduli, dynamic quantities whose energy density can vary in time and space.

Contributions from scalar fields that are constant in space are usually also included in the cosmological constant which can be formulated to be equivalent to vacuum energy. Scalar fields that do change in space can be difficult to distinguish from a cosmological constant since the change may be too slow. High-precision measurements of the expansion of the universe are required to understand how the expansion rate changes over time and space. In general relativity, the evolution of the expansion rate is parameterized by the cosmological equation of state; the relationship between temperature, pressure, and combined matter, energy, and vacuum energy density for any region of space. Nowadays, measuring the equation of state for dark energy is one of the great efforts in observational cosmology. Adding the cosmological constant to cosmology's standard FLRW metric leads to the Lambda-CDM model, which has been referred to as the "standard model of cosmology" because of its precise agreement with observations. Dark energy has been used as a crucial ingredient in a recent attempt to formulate a cyclic model for the universe.

http://www.ft.com/intl/cms/s/2/493de45a-8bef-11e0-854c-00144feab49a.html#axzz3m9WSVVkC

Carroll, Sean (2001). "The cosmological constant". Living Reviews in Relativity 4. doi:10.12942/lrr-2001-1. Retrieved 2006-09-28.

Baum, L. and Frampton, P.H. (2007). "Turnaround in Cyclic Cosmology". Physical Review Letters 98 (7): 071301. arXiv:hep-th/0610213. Bibcode:2007PhRvL..98g1301B. doi:10.1103/PhysRevLett.98.071301. PMID 17359014.

Rafik

Rafik,

Thank you for your detailed answer ! I have to study it further as well as the given references.

Ales,

Thank you for these precisions !

Best regards

Guibert