Anomaly-free perturbations of loop quantum cosmology with holonomy corrections reveal an $\mathrm{\Omega }$-deformed spacetime structure, $\mathrm{\Omega }≔1-2\rho /{\rho }_c$, where $\mathrm{\Omega }<0$ indicates a Euclidean-like space and $\mathrm{\Omega }>0$ indicates a Lorentz-like space. It would be reasonable to give the initial value at the spacetime transition point, $\rho ={\rho }_c/2$, 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 $\mathrm{\Omega }=0$, e.g., $\mathrm{\Omega }\simeq 0.2$, 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 ${\rho }_c/2$ 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. However, when compared to other observations, since the energy densities when the modes crossed the horizon were significantly smaller than ${\rho }_c$, the results we get agree with previous work in the literature and with the classical inflation theory.

• Received 25 June 2016

DOI:https://doi.org/10.1103/PhysRevD.94.064037