Cosmic structure sizes in generic dark energy models

The maximum allowable size of a spherical cosmic structure as a function of its mass is determined by the maximum turn around radius $R_{\rm TA,max}$, the distance from its centre where the attraction on a radial test particle due to the spherical mass is balanced with the repulsion due to the ambient dark energy. In this work, we extend the existing results in several directions. (a) We first show that for $w\neq -1$, the expression for $R_{\rm TA, max}$ found earlier using the cosmological perturbation theory, can be derived using a static geometry as well. (b) In the generic dark energy model with arbitrary time dependent state parameter $w(t)$, taking into account the effect of inhomogeneities upon the dark energy as well, where it is shown that the data constrain $w(t={\rm today})>-2.3$, and (c) in the quintessence and the generalized Chaplygin gas models, both of which are shown to predict structure sizes consistent with observations.Comment: v2, 19pp; added references and discussions, improved presentation; accepted in EPJ

Set-partition tableaux and representations of diagram algebras

The partition algebra is an associative algebra with a basis of set-partition diagrams and multiplication given by diagram concatenation. It contains as subalgebras a large class of diagram algebras including the Brauer, planar partition, rook monoid, rook-Brauer, Temperley-Lieb, Motzkin, planar rook monoid, and symmetric group algebras. We give a construction of the irreducible modules of these algebras in two isomorphic ways: first, as the span of symmetric diagrams on which the algebra acts by conjugation twisted with an irreducible symmetric group representation and, second, on a basis indexed by set-partition tableaux such that diagrams in the algebra act combinatorially on tableaux. The first representation is analogous to the Gelfand model and the second is a generalization of Young's natural representation of the symmetric group on standard tableaux. The methods of this paper work uniformly for the partition algebra and its diagram subalgebras. As an application, we express the characters of each of these algebras as nonnegative integer combinations of symmetric group characters whose coefficients count fixed points under conjugation