Virialization of matter overdensities within dark energy subsystems: Special cases

The virialization of matter overdensities within dark energy (i.e. quintessence) subsystems is considered under a number of restrictive assumptions, namely (i) spherical-symmetric density profiles, and (ii) nothing but gravitational interaction between dark energy scalar field and matter. In addition, the quintessence subsystem is conceived as made of “particles” whose mutual interaction has intensity equal to G(1 + 3w) and scales as the inverse square of their distance. The related expression of the self and tidal potential energy and formulation of the virial theorem for subsystems, are found to be consistent with their matter counterparts, passing from −1 ⩽ w < −1/3 to w = 0. The further restriction (iii) time-independent quintessence equation of state parameter, w; is used in getting a simple relation between overdensity configurations at turnaround and virialization. In the special case of fully clustered quintessence, energy conservation is assumed with regard to either the whole system (global energy conservation), or to the matter subsystem within the tidal potential induced by the quintessence subsystem (partial energy conservation). Further investigation is devoted to a few special cases, namely a limiting situation, w = −1/3, and three lower values, w = −1/2, −2/3, −1, where the last one mimics the effect of a cosmological constant. The special case of fully clustered (i.e. collapsing together with the matter) quintessence is studied in detail, using a similar procedure as in Maor and Lahav [Maor, I., Lahav, O., 2005. JCAP 7, 3 (ML05).]. The general case of partially clustered quintessence is considered in terms of a degree of quintessence de-clustering, ζ, 0 ⩽ ζ ⩽ 1, ranging from fully clustered (ζ = 0) to completely de-clustered (ζ = 1) quintessence, respectively. The special case of unclustered (i.e. remaining homogeneous) quintessence is also discussed. The trend exhibited by the fractional (virialization to turnaround) radius, η, as a function of the (i) fractional (quintessence to matter) “mass” ratio at turnaround, m, (ii) degree of quintessence de-clustering, ζ, and (iii) quintessence equation of state parameter, w, is found to be different from its counterparts reported in earlier attempts. In particular, no clear dichotomy with respect to the limiting situation of vanishing quintessence, η = 1/2, is shown when global or partial energy conservation hold in the special case of fully clustered quintessence, with η > 1/2 preferred. The reasons of the above-mentioned discrepancy are recognized as owing to (i) different formulations of the virial theorem for subsystems, and (ii) different descriptions of de-clustered quintessence, with respect to the reference case of fully clustered quintessence. Although typical values of m do not exceed a value of about 0.3 according to Maor and Lahov [Maor, I., Lahav, O., 2005. JCAP 7, 3 (ML05).], the analysis of a larger range shows interesting features, such as the existence of a threshold, m = −1/(1 + 3w), above which no virialized configuration is allowed, and a different behaviour of η as a function of m, for different values of w. The assumption of partial energy conservation yields less bound virialized configurations with respect to the assumption of global energy conservation.