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Inflationary spectra from a near $Ω$-deformed space-time transition point in Loop Quantum Cosmology

Anomaly-free perturbations of loop quantum cosmology with holonomy corrections reveal an $Ω$ -deformed spacetime structure, $Ω:=1-2ρ/ρ_c$, where $Ω<0$ indicates a Euclidean-like space and $Ω>0$ indicates a Lorentz-like space. It would be reasonable to give the initial value at the spacetime transition point, $ρ=ρ_c/2$, but we find that it is impossible to define a Minkowski-like vacuum even for large $k$ modes at that time. However, if we loosen the condition and give the initial value slightly after $Ω=0$, e.g., $Ω\simeq 0.2$, the vacuum state can be well defined and, furthermore the slow roll approximation also works well in that region. Both scalar and tensor spectra are considered in the framework of loop quantum cosmology with holonomy corrections. We find that if the energy density is not too small in relation to $ρ_c/2$ when the considered $k$ mode crossing the horizon, effective theory can give a much smaller scalar power spectrum than classical theory and the spectrum of tensor perturbations could blueshift. But when compared to other observations, since the energy densities when the modes crossed the horizon were significantly smaller than $ρ_c$,the results we get agree with previous work in the literature and with the classical inflation theory.