A new perspective on the relation between dark energy perturbations and the late-time ISW effect

The effect of quintessence perturbations on the ISW effect is studied for a mixed dynamical scalar field dark energy (DDE) and pressureless perfect fluid dark matter. A new and general methodology is developed to track the growth of the perturbations, which uses only the equation of state (EoS) parameter $w_{\rm DDE} (z) \equiv p_{\rm DDE}/\rho_{\rm DDE}$ of the scalar field DDE, and the initial values of the the relative entropy perturbation (between the matter and DDE) and the intrinsic entropy perturbation of the scalar field DDE as inputs. We also derive a relation between the rest frame sound speed $\hat{c}_{s,{\rm DDE}}^2$ of an arbitrary DDE component and its EoS $w_{\rm DDE} (z)$. We show that the ISW signal differs from that expected in a $\Lambda$CDM cosmology by as much as +20% to -80% for parameterizations of $w_{\rm DDE}$ consistent with SNIa data, and about $\pm$ 20% for parameterizations of $w_{\rm DDE}$ consistent with SNIa+CMB+BAO data, at 95% confidence. Our results indicate that, at least in principle, the ISW effect can be used to phenomenologically distinguish a cosmological constant from DDE.