We investigate the effects of a disformal coupling between dark energy and dark matter in the predictions of the spherical collapse and its signatures in galaxy cluster number counts.We find that the disformal coupling has no significant effects on spherical collapse at high redshifts, and in particular during matter domination epoch. However, at lower redshifts, the extrapolated linear density contrast at collapse close to redshift z 1 and overdensity at virialization can be strongly suppressed by a disformal coupling between dark energy and dark matter. We also find that disformal coupling can have different imprints on cluster number counts compared with conformal coupling, such that the disformal coupling can strongly suppress the predicted number of clusters per redshift interval at z > 0.1 while enhance the number of cluster at z < 0.05. Using the specifications of eROSITA survey, we find that the disformal coupling between dark energy and dark matter can be tightly constrained by cluster number counts.

We use a dynamical analysis to study the evolution of the universe at late time for the model in which the interaction between dark energy and dark matter is inspired by a disformal transformation. We extend the analysis in the existing literature by assuming that the disformal coefficient depends both on the scalar field and its kinetic terms. We find that the dependence of the disformal coefficient on the kinetic term of scalar field leads to two classes of the scaling fixed points that can describe the acceleration of the universe at late time. The first class exists only for the case where the disformal coefficient depends on the kinetic terms. The fixed points in this class are saddle points unless the slope of the conformal coefficient is sufficiently large. The second class can be viewed as the generalization of the fixed points studied in the literature. According to the stability analysis of these fixed points, we find that the stable fixed point can take two different physically relevant values for the same value of the parameters of the model. These different values of the fixed points can be reached for different initial conditions for the equation of state parameter of dark energy. We also discuss the situations in which this feature disappears.

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