Observational Constraints on Loop Quantum Cosmology

In the inflationary scenario of loop quantum cosmology in the presence of inverse-volume corrections, we give analytic formulas for the power spectra of scalar and tensor perturbations convenient to compare with observations. Since inverse-volume corrections can provide strong contributions to the running spectral indices, inclusion of terms higher than the second-order runnings in the power spectra is crucially important. Using the recent data of cosmic microwave background and other cosmological experiments, we place bounds on the quantum corrections.

Figure 1

Primordial scalar power spectrum ${\mathcal{P}}_s(\ell )$ for the case $n=2$, $\sigma =2$, and ${ϵ}_V(k_0)=0.009$, with three different values of $\delta (k_0)$: 0 (classical case, dotted line), $7\times {10}^{-5}$ (experimental upper bound, solid line), $4.8\times {10}^{-4}$ ($1/10$ of the a priori upper bound, dashed line). Here the pivot wave number is $k_0=0.002{\mathrm{Mpc}}^{-1}$, which corresponds to ${\ell }_0=29$.

Figure 2

2D marginalized distribution for the quantum-gravity parameter $\delta (k_0)={\alpha }_0{\delta }_{\mathrm{Pl}}(k_0)$ and the slow-roll parameter ${ϵ}_V(k_0)$ with the pivot $k_0=0.002{\mathrm{Mpc}}^{-1}$ for $n=2$ and $\sigma =2$. The internal and external solid lines correspond to the 68% and 95% confidence levels, respectively.

Figure 3

2D marginalized distributions as in Fig. 2, but for the case $n=2$ and $\sigma =1$.