Inflationary Cosmological Perturbations of Quantum-Mechanical Origin

TL;DR

This article surveys the inflationary paradigm, detailing the slow-roll background, the transition to reheating, and the generation of cosmological perturbations with a quantum origin. It develops the canonical quantization of both scalar and tensor perturbations, derives their slow-roll power spectra, and compares predictions to CMB observations, highlighting constraints on $H_{inf}$ and the tensor-to-scalar ratio. It further discusses the trans-Planckian problem, presenting frameworks with modified dispersion relations and minimal initial-state modifications, and assesses their potential observational signatures and theoreticalConsistency. The work emphasizes the close interplay between general relativity and quantum field theory in the early Universe and outlines prospects for probing high-energy physics through precision cosmology.

Abstract

This review article aims at presenting the theory of inflation. We first describe the background spacetime behavior during the slow-roll phase and analyze how inflation ends and the Universe reheats. Then, we present the theory of cosmological perturbations with special emphasis on their behavior during inflation. In particular, we discuss the quantum-mechanical nature of the fluctuations and show how the uncertainty principle fixes the amplitude of the perturbations. In a next step, we calculate the inflationary power spectra in the slow-roll approximation and compare these theoretical predictions to the recent high accuracy measurements of the Cosmic Microwave Background radiation (CMBR) anisotropy. We show how these data already constrain the underlying inflationary high energy physics. Finally, we conclude with some speculations about the trans-Planckian problem, arguing that this issue could allow us to open a window on physical phenomena which have never been probed so far.

Figures (3)

• Figure 1: Evolution of the various scales discussed in the text during inflation and the subsequent radiation and matter dominated epochs. In particular, it is apparent that the CMBR measurements only probe the inflationary model when the modes of astrophysical interest today crossed out the horizon during inflation. The small window shows a typical inflationary potential, see Eq. (\ref{['pot']}).
• Figure 2: Evolution of the scalar field during slow-roll inflation and the reheating phase (where the field oscillates) obtained by numerical integration of the equations of motion. The potential is of the type of Eq. (\ref{['pot']}) with $n=2$ and $\lambda _2=(8\pi /6)\times 10^{-10}$. The initial conditions are such that $\varphi _{\rm ini}\simeq 3 m_{_{\rm Pl}}$ leading to $N_{_{\rm T}}\simeq 60$ as confirmed by the plot.
• Figure 3: Angular TT, TE and EE power spectra for two different trans-Planckian models, one with low frequency (LF) superimposed oscillations, the other with high frequency (HF) oscillations, for details see Refs. MR1MR2. A zoom of the temperature multipole moments in the first Doppler peak region is also shown (black curve) and compared with the standard slow-roll prediction (blue curve) calculated with the same cosmological parameters.