The Physical Properties of the Cosmic Acceleration Spyros Basilakos 1 $^{1}$Academy of Athens, Research Center for Astronomy and Applied Mathematics, Soranou Efesiou 4, 11527, Athens, Greece The observed late-time acceleration of the cosmic expansion constitutes a fundamental problem in modern theoretical physics and cosmology. In an attempt to weight the validity of a large number of dark energy models, I use the recent measurements of the expansion rate of the Universe, the clustering of galaxies the CMB fluctuations as well as the large scale structure formation, to put tight constraints on the different models. Recent studies in observational cosmology lead to the conclusion that the available high quality cosmological data (Type Ia supernovae, CMB, etc.) are well fitted by an emerging “standard model”. This standard model, assuming flatness, is described by the Friedman equation: PACS numbers: 98.80.-k, 95.35.+d, 95.36.+x Keywords: Cosmology; dark matter; dark energy where a ( t ) is the scale factor of the universe, ρ$_{m}$ ( a ) is the density corresponding to the sum of baryonic and cold dark matter, with the latter needed to explain clustering, and an extra component ρ$_{X}$ ( a ) with negative pressure, called dark energy, needed to explain the observed accel- erated cosmic expansion (eg., Davis et al. 2007; Kowalski et al. 2008; Komatsu et al. 2009 and references therein). The nature of the dark energy is one of the most fundamental and difficult problems in physics and cos- mology. Indeed, during the last decade there has been a theoretical debate among the cosmologists regarding the nature of the exotic “dark energy”. Various candi- dates have been proposed in the literature, such as a cos- mological constant Λ (vacuum), time-varying Λ( t ) cos- mologies, quintessence, $^{k}$− essence, vector fields, phan- tom, tachyons, Chaplygin gas and the list goes on (for recent reviews see Peebles & Ratra 2003; Copeland, Sami & Tsujikawa 2006; Caldwell & Kamionkowski 2009). Within this framework, high energy field theories gener- ically indicate that the equation of state of such a dark energy is a function of cosmic time. To identify this type of evolution of the equation of state, a detailed form of the observed H ( a ) is required which may be obtained by a combination of multiple dark energy observational probes. In this cosmological framework, a serious issue is how (and when) the large scale structures form. Galaxies and large-scale structure grew gravitationally from tiny, nearly scale-invariant adiabatic Gaussian fluctuations. In this paper we focus also on galaxy clusters that occupy an eminent position in the structure hierarchy, being the most massive virialized systems known and therefore they appear to be ideal tools for testing theories of struc- ture formation and extracting cosmological information. The exact nature of the dark energy has yet to be found and thus the dark energy equation of state parameter includes our ignorance regarding the physical mechanism which leads to a late cosmic acceleration. On the other hand, it is possible to extent the previous methodology in the framework of modified gravity (see Linder & Jenkins 2003; Linder 2004). In this scenario, it is assumed that the dark energy may be an illusion, indi- cating the need to revise the general relativity and thus also the Friedmann equation. From the mathematical Note, that E ( a ) is the normalized Hubble flow, Ω$_{m}$ is the dimensionless matter density at the present epoch, Ω$_{X}$ = 1 − Ω$_{m}$ is the corresponding dark energy density parameter and w ( a ) is the dark energy equation of state. Inverting the above equation we simply derive In the context of general relativity it is well known that for homogeneous and isotropic flat cosmologies, driven by non-relativistic matter and dark energy with equation of state p$_{X}$ = w ( a ) ρ$_{X}$ , the expansion rate of the Universe can be written as (see eq.1) The structure of the paper is as follows. In section 2 we briefly discuss the dark energy issue. In sections 3 and 4 we present the various dark energy models and we use a joint statistical analysis, in order to place constraints on the main cosmological parameters. In section 5 we present the corresponding theoretical predictions regard- ing the formation of the galaxy clusters. Finally, we draw our conclusions in section 6. Throughout the paper we will use H$_{0}$ ≃ 71km/sec/Mpc. The cluster distribution basically traces scales that have not yet undergone the non-linear phase of gravitation- ally clustering, thus simplifying their connections to the initial conditions of cosmic structure formation. 2. THE DARK ENERGY EQUATION OF STATE 1. INTRODUCTION w ( a ) = − $^{1}$− 2 $_{3}$a d lnE da $^{1}$− Ω$_{m}$ a − $^{3}$E − $^{2}$( a ) . (3) E $^{2}$( a ) = H $^{2}$( a ) H 2 0 = Ω$_{m}$ a − 3 + Ω$_{X}$e 3 ∫ 1 $_{a}$d lny[1+ w ( y )] . (2) H $^{2}$( a ) = ( ˙ a a ) 2 = 8 πG 3 [ ρ$_{m}$ ( a ) + ρ$_{X}$ ( a )] , (1) arXiv:1001.1811v2 [astro-ph.CO] 21 Jan 2010