With the observational advance in recent years, primordial gravitational waves (GWs), known as the tensor-mode cosmic perturbations, in the Loop Quantum Cosmology (LQC) are becoming testable and thus require better framework through which to bridge between the observations and the theories. In this work we present a new formalism that employs the transfer functions to bring the GWs from any epoch, even before the quantum bounce, to a later time, including the present. The evolutionary epochs considered here include the possible deflation, quantum bounce, and inflation. This formalism enables us to predict more accurately the GW power spectrum today. With the ADM formalism for the LQC background dynamics, our approach is equivalent to the commonly used Bogoliubov transformations for evolving the primordial GWs, but more transparent for discussions and easier to calculate due to its nature of being linear algebra dealing with linear perturbations. We utilize this advantage to have resolved the IR suppression problem. We also propose the field-free approximation for the effective mass in the quantum bounce epoch to largely improve the accuracy in the predicted GW power spectrum. Our transfer-function formalism is general in dealing with any linear problems, and thus expected to be equally useful under other context with linearity.
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