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In the standard model ($\Omega_{\rm m}^0 \approx 0.3$, $\Omega_{\Lambda}^0 \approx 0.7$) it is assumed that there is no convertion (energy transfer) between the two components. However, this hypothesis requires observational tests. A general approach to multicomponent FLRW models comes from a new classification (Gromov et al. [CITE]) which naturally follows from two independent model properties: 1) stationarity (or not) of equations of state of component substances and the presence (or absence) of energy transfer between these. The associated one-fluid model becomes characterized by $\gamma = P_{{\rm total}}/{{\cal E}_{{\rm total}}}$. For the case $\gamma = {\rm constant}$ (which for a stationary equation state of dark energy means coherent evolution and is caused by energy transfer), a general integral expression for the analogue of the Mattig equation is derived. For flat geometry, the integral reduces to a simple analytic expression. Comparison with the magnitude-redshift relation for the standard model shows that $\gamma \approx -0.55$ gives the same relation within 0.05 mag in the range $0 γ gives a robust upper limit to w in the equation of state of dark energy.

Limits on dark energy–matter interaction from the Hubble relation for two-fluid FLRW models