Degeneracy between warm and coupled cold dark matter: A clarifying note

Wei et al [PRD 88, 043510 (2013)] have proposed the existence of a cosmological degeneracy between warm dark matter (WDM), modified gravity and coupled cold dark matter (CDM) cosmologies at both the background expansion and the growth of density perturbation levels, i.e., corresponding cosmological data would not be able to differentiate such scenarios. Here, we will focus on the specific indistinguishability between a warm dark matter plus cosmological constant ($\Lambda$) and coupled scalar field-CDM scenarios. Although the statement of Wei et al is true for very specific conditions we present a more complete discussion on this issue and show in more detail that these models are indeed distinguishable. We show that the degeneracy breaks down since coupled models leave a specific signature in the redshift space distortion data which is absent in the uncoupled warm dark matter cosmologies. Furthermore, we complement our claim by providing the reasons which suggest that even at nonlinear level a breaking of such apparent equivalence is also expected.