  This research investigates the impact of the nature of Dark Energy (DE) on
structure formation, focusing on the matter power spectrum and the Integrated
Sachs-Wolfe effect (ISW). By analyzing the matter power spectrum at redshifts
$z = 0$ and $z = 5$, as well as the ISW effect on the scale of $\ell = 10-100$,
the study provides valuable insights into the influence of DE equations of
state (EoS) on structure formation. The findings reveal that dynamical DE
models exhibit a stronger matter power spectrum compared to constant DE models,
with the JBP model demonstrating the highest amplitude and the CPL model the
weakest. Additionally, the study delves into the ISW effect, highlighting the
time evolution of the ISW source term $\mathcal{F}(a)$ and its derivative
$d\mathcal{F}(a)/da$, and demonstrating that models with constant DE EoS
exhibit a stronger amplitude of $\mathcal{F}(a)$ overall, while dynamical
models such as CPL exhibit the highest amplitude among the dynamical models,
whereas JBP has the lowest. The study also explores the ISW auto-correlation
power spectrum and the ISW cross-correlation power spectrum, revealing that
dynamical DE models dominate over those with constant DE EoS across various
surveys. Moreover, it emphasizes the potential of studying the non-linear
matter power spectrum and incorporating datasets from the small scales to
further elucidate the dynamical nature of dark energy. This comprehensive
analysis underscores the significance of both the matter power spectrum and the
ISW signal in discerning the nature of dark energy, paving the way for future
research to explore the matter power spectrum at higher redshifts and in the
non-linear regime, providing deeper insights into the dynamical nature of dark
energy.
