Evolution of spherical overdensities in energy-momentum-squared gravity

Employing the spherical collapse formalism, we investigate the linear evolution of the matter overdensity for energy-momentum-squared gravity (EMSG), which in practical phenomenological terms one may imagine as an extension of the ?CDM model of cosmology. The underlying model, while still having a cosmological constant, is a nonlinear material extension of the general theory of relativity and includes correction terms that are dominant in the high-energy regime, the early Universe. Considering the Friedmann-Robertson-Walker background in the presence of a cosmological constant, we find the effects of the modifications arising from EMSG on the growth of perturbations at the early stages of the Universe. Considering both possible negative and positive values of the model parameter of EMSG, we discuss its role in the evolution of the matter density contrast and growth function in the level of linear perturbations. While EMSG leaves imprints distinguishable from ?CDM, we find that the negative range of the EMSG model parameter is not well behaved, indicating an anomaly in the parameter space of the model. In this regard, for the evaluation of the galaxy cluster number count in the framework of EMSG, we equivalently provide an analysis of the number count of the gravitationally collapsed objects (or the dark matter halos). We show that the galaxy cluster number count decreases compared to the ?CDM model. In agreement with the hierarchical model of structure formation, in EMSG cosmology the more massive structures are less abundant, meaning that they form at later times.