Prescriptions in loop quantum cosmology: A comparative analysis

Various prescriptions proposed in the literature to attain the polymeric quantization of a homogeneous and isotropic flat spacetime coupled to a massless scalar field are carefully analyzed in order to discuss their differences. A detailed numerical analysis confirms that, for states which are not deep in the quantum realm, the expectation values and dispersions of some natural observables of interest in cosmology are qualitatively the same for all the considered prescriptions. On the contrary, the amplitude of the wave functions of those states differs considerably at the bounce epoch for these prescriptions. This difference cannot be absorbed by a change of representation. Finally, the prescriptions with simpler superselection sectors are clearly more efficient from the numerical point of view.

Figure 1

Amplitude $|{\Psi }_{\varphi }(v)|$ of the physical wave function corresponding to a state with a logarithmic normal distribution. The amplitude for different prescriptions is compared both away from the bounce (a) and at the bounce (b). The parameters of the profile $\tilde{\Psi }$ of this state are fixed by the conditions $⟨{\widehat{p}}_{\varphi }⟩=100\hslash$ and $\Delta {\widehat{p}}_{\varphi }/⟨{\widehat{p}}_{\varphi }⟩=0.1$. Away from the bounce, the amplitudes are indistinguishable up to numerical noise, whereas at the bounce one can observe differences (phase shift) in the interference pattern. The noise level clearly depends on the used techniques, something which depends in turn on the degeneracy of the spectrum of $\widehat{\Theta }$.

Figure 2

Dynamical trajectories of ${\widehat{H}}_{\varphi }$ (a) and ${\widehat{\rho }}_{\varphi }$ (b), given by the expectation values of these observables on the state of Fig. 1 with $\epsilon =1$ in the APS prescription.

Figure 3

(a) Quantum trajectory of $\ln |\widehat{v}{|}_{\varphi }$ for the same state and the same prescription as in Fig. 2. (b) Uncertainty in $\ln |\widehat{v}{|}_{\varphi }$ for the same state and prescription, compared with the difference between the corresponding expectation values calculated in the APS and MMO prescriptions.

Figure 4

Absolute dispersions (a) and relative dispersions (b) of ${\widehat{H}}_{\varphi }$ for the considered state of Fig. 1, compared with the corresponding difference between the expectation values of ${\widehat{H}}_{\varphi }$ in the APS and MMO prescriptions. For both relative values, one can observe a peak at the bounce owing to the vanishing of $⟨{\widehat{H}}_{\varphi }⟩$; however, the peak in the differences (lower curves) is so sharp that it is placed between probing points.

Figure 5

(a) Relative dispersion of ${\widehat{\rho }}_{\varphi }$ for the same state as in the previous figures, compared with the relative difference between the corresponding expectation values in the APS and MMO prescriptions. (b) Comparison between the relative dispersion of ${\widehat{\rho }}_{\varphi }$ in the APS and MMO prescriptions, for a “generic” superselection sector. Better coherence properties are observed for the state constructed in the MMO prescription.

Figure 6

Expectation values of the observable $({\widehat{\Delta \Theta }}_{\mathit{A}\mathit{B}}{)}^2{|}_{\varphi }$ for $\mathit{A}=\mathrm{APS}$ and $\mathit{B}=\mathrm{MMO}$ (a); as well as for $\mathit{A}=\mathrm{sLQC}$ and $\mathit{B}=\mathrm{sMMO}$ (b). In both cases, $\epsilon =1$ and the observable is evaluated on the state with a logarithmic normal distribution whose parameters are $⟨{\widehat{p}}_{\varphi }⟩=100\hslash$ and $\Delta {\widehat{p}}_{\varphi }/⟨{\widehat{p}}_{\varphi }⟩=0.1$. The state is built in the MMO prescription in case (a), and in the sMMO prescription in case (b). The difference reaches a maximum at the bounce and decays exponentially away from it. The difference between the sLQC and the sMMO prescriptions is many orders of magnitude smaller than the difference between any other pair of prescriptions.