The Integrated Sachs-Wolfe Effect in Interacting Dark Matter-Dark Energy Models

Interacting dark matter-dark energy (IDMDE) models can be taken to account as one of the present challenges that may affect the cosmic structures. In this work, we study the integrated Sachs-Wolfe (ISW) effect in IDMDE models. To this end, we initially introduce a theoretical framework for IDMDE models. Moreover, we briefly discuss the stability conditions of IDMDE models and by specifying a simple functional form for the energy density transfer rate, we calculate the perturbation equations. In the following, we calculate the amplitude of the matter power spectrum for the IDMDE model and compare it with the corresponding result obtained from the $\Lambda$CDM model. Furthermore, we calculate the amplitude of the ISW auto-power spectrum as a function of multipole order l for the IDMDE model. The results indicate that the amplitude of the ISW auto-power spectrum in the IDMDE model for different phantom dark energy equations of state behaves similar to the one for the $\Lambda$CDM model, whereas, for the quintessence dark energy equations of state, the amplitude of the ISW-auto power spectrum for the IDMDE model should be higher than the one for the $\Lambda$CDM model. Also, it turns out that the corresponding results by different values of the coupling parameter demonstrate that $\xi$ is inversely proportional to the amplitude of the ISW-auto power spectrum in the IDMDE model. Finally, by employing four different surveys, we calculate the amplitude of the ISW-cross power spectrum as a function of multipole order $l$ for the IDMDE model. The results exhibit that the amplitude of the ISW-cross power spectrum for the IDMDE model for all values of $\omega_{\rm x}$ is higher than the one obtained for the $\Lambda$CDM model. Also, it turns out that the amplitude of the ISW-cross power spectrum in the IDMDE model changes inversely with the value of coupling parameter $\xi$.