Halo structure and lensing signatures of a polytropic dark matter fluid

We investigate whether a minimal effective pressure in the dark matter sector can modify nonlinear halo structure while preserving the successful large-scale predictions of the $\Lambda$ cold dark matter ($\Lambda$CDM) model. We consider a barotropic relation $P=K\rho^{\gamma}$ with $\gamma=3/2$, interpreted as an effective coarse-grained closure of the Jeans hierarchy in virialized regions. In this framework, dark matter remains effectively pressureless at cosmological densities while developing a finite effective sound speed inside collapsed halos. For $\gamma=3/2$, equilibrium halo configurations correspond to the $n=2$ Lane--Emden solution, producing finite-radius density profiles with quadratic central flattening. When embedded within the empirical concentration--mass relation of $\Lambda$CDM halos, the resulting core scale exhibits only weak mass dependence across dwarf-to-galaxy mass ranges. For parameter values yielding kiloparsec-scale cores, the background expansion history and linear growth of density perturbations remain observationally indistinguishable from $\Lambda$CDM, while the present-day Jeans scale remains confined to sub-megaparsec lengths. We compute projected surface-density and weak-lensing convergence profiles for the model. Relative to mass-matched Navarro--Frenk--White halos, the model predicts a moderate suppression of the central lensing amplitude, while the convergence power spectrum is modified only at sufficiently high multipoles. The model introduces a single additional parameter controlling nonlinear pressure support and continuously reduces to collision-free cold dark matter in the limit $K\rightarrow0$.