Hybrid loop quantum cosmology and predictions for the cosmic microwave background

We investigate the consequences of the hybrid quantization approach for primordial perturbations in loop quantum cosmology, obtaining predictions for the cosmic microwave background and comparing them with data collected by the Planck mission. In this work, we complete previous studies about the scalar perturbations and incorporate tensor modes. We compute their power spectrum for a variety of vacuum states. We then analyze the tensor-to-scalar ratio and the consistency relation between this quantity and the spectral index of the tensor power spectrum. We also compute the temperature-temperature, electric-electric, temperature-electric, and magnetic-magnetic correlation functions. Finally, we discuss the effects of the quantum geometry in these correlation functions and confront them with observations.

Figure 1

Comparison of the time-dependent mass of the tensor and the scalar perturbations for the hybrid approach and the dressed metric approach in a dynamical trajectory with ${\varphi }_B=0.97$ and $m=1.20\times {10}^{-6}$ in Planck units (kinetically dominated bounce). Here, $s^{(s)}(\eta )$ and $s^{(t)}(\eta )$ are the time-dependent masses of the scalar and the tensor perturbations in our hybrid approach, respectively, while ${\tilde{s}}^{(s)}(\eta )$ and ${\tilde{s}}^{(t)}(\eta )$ are their counterparts in the dressed metric approach. Dashed lines in the right panel indicate negative values of the quantity of which we plot the absolute value.

Figure 2

Comparison between different vacuum prescriptions: functions that determine the initial conditions of the tensor field, for ${\varphi }_B=0.97$ and $m=1.20\times {10}^{-6}$. Left panel: function $D_k$. Note that $D_k({\mathfrak{W}}_k^{(0)})=k=D_k(W_k^{(0)})$. Right panel: function $C_k$. Here, dashed lines indicate negative values of $C_k$. We note that, for the zeroth-order adiabatic initial conditions, $C_k({\mathfrak{W}}_k^{(0)})=0=C_k(W_k^{(0)})$.

Figure 3

Comparison between different vacuum prescriptions for the initial data of the tensor perturbations, for ${\varphi }_B=0.97$ and $m=1.20\times {10}^{-6}$: absolute value of the antilinear coefficients of the Bogoulibov transformation relating the vacuum state of reference, ($r$), with another vacuum, ($n$). Left panel: comparison taking the nonoscillating (no) vacuum as the reference state.

Figure 4

Comparison between power spectra calculated with different strategies in the slow-roll regime. Here, ${\varphi }_B=0.97$ and $m=1.20\times {10}^{-6}$. A dashed line indicates modes that exit the Hubble horizon in an inflationary phase but not in slow-roll regime. Left panel: slow-roll power spectra for the scalar perturbations. Right panel: slow-roll power spectra for the tensor perturbations.

Figure 5

Comparison between primordial power spectra obtained with different sets of solutions for the perturbations. Here, ${\varphi }_B=0.97$ and $m=1.20\times {10}^{-6}$.

Figure 6

Comparison between different vacuum prescriptions: tensor-to-scalar ratio and validity of the consistency relation $r=-8n_t$. In the two upper panels, we show the tensor-to-scalar ratio for the nonoscillating vacuum and for the two fourth-order adiabatic vacua considered in the text. The ratio for these adiabatic vacua has been computed both from the full power spectrum and from its averaged version. Since it turns out that $\overline{r}(W_k^{(4)})\approx \overline{r}({\mathfrak{W}}_k^{(4)})$, we display explicitly only the former of these ratios. In the lowest panel, we plot the quantity $|r+8n_t|/|r-8n_t|$. Dashed (solid) lines indicate negative (positive) values of $r+8n_t$. For both $W_k^{(4)}$ and ${\mathfrak{W}}_k^{(4)}$, we plot the results using the averaged power spectra. Here, we have taken ${\varphi }_B=0.97$ and $m=1.20\times {10}^{-6}$.

Figure 7

$TT$ angular power spectrum provided by Planck best fit and spectra computed for the nonoscillating vacuum with different values of the scalar field at the bounce. Left panel: $m=1.20\times {10}^{-6}$. Right panel: $m=1.18\times {10}^{-6}$. The corresponding cosmological parameters determined by Planck best fit for $\mathrm{TT}+\mathrm{lowP}$ data are given in Ref. [2]. Both panels incorporate lensing corrections.

Figure 8

$EE$ angular power spectrum provided by Planck best fit and spectra computed for the nonoscillating vacuum with different values of the scalar field at the bounce. Left panel: $m=1.20\times {10}^{-6}$. Right panel: $m=1.18\times {10}^{-6}$. The corresponding cosmological parameters determined by Planck best fit for $\mathrm{TT}+\mathrm{lowP}$ data are given in Ref. [2]. Both panels incorporate lensing corrections.

Figure 9

$TE$ angular power spectrum provided by Planck best fit and spectra computed for the nonoscillating vacuum with different values of the scalar field at the bounce. Left panel: $m=1.20\times {10}^{-6}$. Right panel: $m=1.18\times {10}^{-6}$. The corresponding cosmological parameters determined by Planck best fit for $\mathrm{TT}+\mathrm{lowP}$ data are given in Ref. [2]. Both panels incorporate lensing corrections.

Figure 10

$BB$ angular power spectrum provided by Planck best fit and spectra computed for the nonoscillating vacuum with different values of the scalar field at the bounce. Left panel: $m=1.20\times {10}^{-6}$. Right panel: $m=1.18\times {10}^{-6}$. The corresponding cosmological parameters determined by Planck best fit for $\mathrm{TT}+\mathrm{lowP}$ data are given in Ref. [2]. Both panels incorporate lensing corrections.