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We investigate the effects of a disformal coupling between dark energy anddark matter in the predictions of the spherical collapse and its signatures ingalaxy cluster number counts. We find that the disformal coupling has nosignificant effects on spherical collapse at high redshifts, and in particularduring matter domination epoch. However, at lower redshifts, the extrapolatedlinear density contrast at collapse close to redshift $z \lesssim 1$ andoverdensity at virialization can be strongly suppressed by a disformal couplingbetween dark energy and dark matter. We also find that disformal coupling canhave different imprints on cluster number counts compared with conformalcoupling, such that the disformal coupling can strongly suppress the predictednumber of clusters per redshift interval at $z > 0.1$ while enhance the numberof cluster at $z < 0.05$. Using the specifications of eROSITA survey, we findthat the disformal coupling between dark energy and dark matter can be tightlyconstrained by cluster number counts.

Spherical collapse and cluster number counts in dark energy models disformally coupled to dark matter