  In this dissertation, the nature of Dark Energy (DE) is examined from both
theoretical and phenomenological perspectives. The possibility of DE being a
dynamic quantity in quantum field theory (QFT) in curved spacetime is studied.
The primary aim is to go beyond the usual approach that relies on ad hoc fields
and instead treat DE as a quantum vacuum under appropriate QFT renormalization.
Specifically, the dynamic behavior of DE could arise from quantum vacuum
fluctuations in the Universe, evolving alongside the background expansion.
Thus, the evolution of the vacuum energy density can be expressed in terms of
the Hubble function and its derivatives, $\rho_{\rm vac} =\rho_{\rm vac}(H)$.
This approach yields a significant revelation: the equation of state of the
quantum vacuum, derived from first principles, deviates from its traditional
constant value of $w_{\rm vac}=-1$. Additionally, a new inflationary mechanism
emerges in this context, rooted in the quantum effects in curved spacetime.
Moreover, the thesis displays a phenomenological exploration of two related
models that go beyond the $\Lambda$CDM model: the Brans-Dicke model with a
cosmological constant and the Running Vacuum Model, which is related to the QFT
calculations. These models have been tested under different datasets and
scenarios to determine the constraints on their free parameters. The results of
the fits are presented and discussed in relation to cosmological tensions
concerning $H_0$ and $\sigma_8$. The conclusions drawn from this thesis
indicate promising signals of the dynamic behavior of quantum vacuum,
potentially impacting the cosmological constant problem and the cosmological
tensions.
