Conformal Killing cosmology: Geometry, dark sector, growth of structures, and a big rip

We introduce Sinyukov-like tensors, a special kind of conformal Killing tensors. In Robertson-Walker space-times, they have the perfect-fluid form and only depend on two constants and the scale factor. They are the candidates for the dark term of the newly proposed conformal Killing gravity, by Harada. In addition to ordinary matter, the Friedmann equations contain a dark term and a Λ term that parametrize the Sinyukov-like tensor. The expression of $H(z)$ is tested on cosmological data based on cosmic chronometers or including baryon acoustic oscillations. There is a large incertitude in Ω_Λ and Ωdark that may become negative, but their sum is close to Ω_Λ of Lambda cold dark matter (ΛCDM). In any case, there is a future singularity, that is, a big rip for all positive Ω. We solve the equation for the evolution, in linear approximation, of the density contrast in a matter-dominated universe. The dark sector and the Λ term give no significative deviation from ΛCDM and general relativity results.