Comparison of primordial tensor power spectra from the deformed algebra and dressed metric approaches in loop quantum cosmology

Loop quantum cosmology tries to capture the main ideas of loop quantum gravity and to apply them to the Universe as a whole. Two main approaches within this framework have been considered to date for the study of cosmological perturbations: the dressed metric approach and the deformed algebra approach. They both have advantages and drawbacks. In this article, we accurately compare their predictions. In particular, we compute the associated primordial tensor power spectra. We show—numerically and analytically—that the large scale behavior is similar for both approaches and compatible with the usual prediction of general relativity. The small scale behavior is, the other way round, drastically different. Most importantly, we show that in a range of wave numbers explicitly calculated, both approaches do agree on predictions that, in addition, differ from standard general relativity and do not depend on unknown parameters. These features of the power spectrum at intermediate scales might constitute a universal loop quantum cosmology prediction that can hopefully lead to observational tests and constraints. We also present a complete analytical study of the background evolution for the bouncing universe that can be used for other purposes.

Figure 1

Primordial power spectra for tensor modes in the dressed metric approach (upper panel) and in the deformed algebra approach (lower panel) for different values of the mass of the scalar field. The critical energy density is ${\rho }_{\mathrm{c}}=0.41m_{\mathrm{Pl}}^4$ and $\cos {\theta }_{\mathrm{A}}\simeq 1$. The mass of the scalar field takes three values: $m=10^{-3}{m}_{\mathrm{Pl}}$ (triangles), $m=10^{-2.5}{m}_{\mathrm{Pl}}$ (open disks), and $m=10^{-2}{m}_{\mathrm{Pl}}$ (black disks). The dashed vertical line at large $k$, corresponds to $k_{\mathrm{UV}}$ (which does not depend on $m$). The dotted vertical lines at smaller $k$ correspond to $k_{\mathrm{IR}}$ (which scales as $m^{2/3}$).

Figure 2

Primordial power spectra for the tensor modes in the dressed metric approach (upper panel) and in the deformed algebra approach (lower panel) for different values of ${\rho }_{\mathrm{c}}$. The mass of the scalar field is $m=10^{-3}{m}_{\mathrm{Pl}}$ and $\cos {\theta }_{\mathrm{A}}\simeq 1$. The critical energy density is ${\rho }_{\mathrm{c}}=0.0041m_{\mathrm{Pl}}^4$ (triangles), ${\rho }_{\mathrm{c}}=0.041m_{\mathrm{Pl}}^4$ (open disks) and ${\rho }_{\mathrm{c}}=0.41m_{\mathrm{Pl}}^4$ (black disks).

Figure 3

Primordial power spectra for the tensor modes in the dressed metric approach (upper panel), and in the deformed algebra approach (lower panel). The parameter ${\theta }_0$ varies from $(\pi /2-1)\simeq 0.18\times \pi$ to $(\pi /2+1)\simeq 0.81\times \pi$. The exact values are: plain disks for ${\theta }_0=(\pi /2-1)$, open disks for ${\theta }_0=(\pi /2-1/2)$, plain triangles for ${\theta }_0=\pi /2$, open triangles for ${\theta }_0=(\pi /2+1/2)$, and plain stars for ${\theta }_0=(\pi /2+1)$. The mass of the field is $m=10^{-3}{m}_{\mathrm{Pl}}$, and the critical energy density is ${\rho }_{\mathrm{c}}=0.41m_{\mathrm{Pl}}^4$.

Figure 4

The infrared limit (disks) and the ultraviolet limit (triangles) of the primordial power spectrum as a function of ${\theta }_0$. The solid black curve corresponds to the analytical calculations for ${{\mathcal{P}}_{\mathrm{T}}}^{\mathrm{IR}}$, see (45). The IR limit from a numerical simulation is displayed with open disks for the dressed metric approach, and black disks in the deformed algebra approach. The dashed black curve stands for the analytical UV limit in the dressed metric approach, see (52a). The UV limit in the dressed metric approach, as derived from the numerics, corresponds to the black triangles. The mass of the scalar field is $m=10^{-3}{m}_{\mathrm{Pl}}$, and the critical energy density is ${\rho }_{\mathrm{c}}=0.41m_{\mathrm{Pl}}^4$.