Detailed analysis of the predictions of loop quantum cosmology for the primordial power spectra

Ivan Agullo and Noah A. Morris

Phys. Rev. D 92, 124040 – Published 22 December 2015

Abstract

We provide an exhaustive numerical exploration of the predictions of loop quantum cosmology with a postbounce phase of inflation for the primordial power spectrum of scalar and tensor perturbations. We extend previous analysis by characterizing the phenomenologically relevant parameter space and by constraining it using observations. Furthermore, we characterize the shape of loop quantum cosmology corrections to observable quantities across this parameter space. Our analysis provides a framework to contrast more accurately the theory with forthcoming polarization data, and it also paves the road for the computation of other observables beyond the power spectra, such as non-Gaussianity.

Received 23 September 2015

DOI:https://doi.org/10.1103/PhysRevD.92.124040

Authors & Affiliations

Ivan Agullo^* and Noah A. Morris^†

Department of Physics and Astronomy, Louisiana State University, Baton Rouge, Louisiana 70803, USA

^*agullo@lsu.edu
^†nmorri7@lsu.edu

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Vol. 92, Iss. 12 — 15 December 2015

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Images

Figure 1
A typical evolution of the energy density and Hubble rate of the background spacetime (in Planck units). Note that both axes are logarithmically scaled. The LQC modification to the classical Friedmann equation is evident near the bounce (since classically $ρ = \frac{3}{8 π G} H^{2}$ ), but by 2 Planck seconds, the behavior coincides with the classical trajectory.
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Figure 2
The LQC scalar power spectrum for parameter values $ϕ_{B} = 1$ , $m = 1.3 \times 10^{- 6}$ , and preferred instantaneous vacuum [46] initial data for perturbations at initial time $t = - 50000$ . The numerically evolved spectrum, shown in gray, is rapidly oscillatory; its average, shown in black, has an amplitude which is amplified with respect to the standard predictions of slow-roll inflation for modes $k_{I} ≲ k ≲ k_{LQC}$ but agrees with them otherwise.
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Figure 3
The LQC tensor power spectrum for parameter values $ϕ_{B} = 1$ , $m = 1.3 \times 10^{- 6}$ , and preferred instantaneous vacuum [46] initial data for perturbations at initial time $t = - 50000$ . The numerically evolved spectrum is shown in gray and its average in black. The qualitative behavior is similar to the scalar spectrum shown in Fig. 2.
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Figure 4
Regions of the ( $ϕ_{B}$ , $m$ ) parameter space that yield LQC power spectra meeting various criteria. The thick black outline demarcates the points (1) consistent with the Planck 2013 observational constraints on the amplitude and tilt of the scalar power spectrum. The gray region (2) indicates the subset of such points for which the spectra contain potentially observable LQC contributions. Finally, the striped region (3) indicates the points consistent with the constraint on the number of $e$ -folds $N_{⋆}$ . Several values at selected points from this figure (indicated by the black shapes) are given in Table 1. Due to prohibitive constraints on computational resources, the shapes of the regions (1) and (3) have been extrapolated for the largest shown values of $ϕ_{B}$ .
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Figure 5
Averaged LQC scalar power spectrum for two different choices of initial data for perturbations specified at 50000 Planck seconds before the bounce. The two spectra are nearly indistinguishable for observable modes. The background parameters used for both spectra in the figure are $m = 1.3 \times 10^{- 6}$ and $ϕ_{B} = 1$ .
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Figure 6
Averaged LQC scalar power spectrum for Minkowski-like initial data specified at different times. $t = 0$ corresponds to the bounce, and values of time are given in Planck units. The power spectrum changes significantly for very low values of $k$ , indicating that those modes are sensitive to the prebounce evolution. On the other hand, all curves are very similar for observable modes, except for the lowest observable values of $k$ . The background parameters used for both spectra in the figure are $m = 1.3 \times 10^{- 6}$ and $ϕ_{B} = 1$ .
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