Trans-Planckian Physics and the Spectrum of Fluctuations in a Bouncing Universe

TL;DR

The paper investigates how trans-Planckian physics, modeled by a modified dispersion relation omega_phys(k), can modify the spectrum of scalar cosmological fluctuations in a bouncing, asymptotically flat universe. Using the mode equation mu'' + (n_eff^2(n,eta) - a''/a) mu = 0 with n_eff^2(n,eta) = a^2(eta) omega_phys^2[n/a(eta)], and a specific bounce, they analyze regions where the WKB approximation holds or breaks down and identify when non-adiabatic evolution affects the spectrum. Analytic matching of solutions around the non-adiabatic interval yields corrections to the final spectrum, characterized by coefficients C1(n) and C2(n) and controlled by the small parameter epsilon = eta_j/eta_0, producing both amplitude changes and oscillatory modulations atop the standard vacuum spectrum. The results show that trans-Planckian effects can modify the spectrum without invoking imaginary frequencies, with implications for bouncing cosmologies and for interpreting short-distance physics in string-inspired or ekpyrotic scenarios.

Abstract

In this paper, we calculate the spectrum of scalar field fluctuations in a bouncing, asymptotically flat Universe, and investigate the dependence of the result on changes in the physics on length scales shorter than the Planck length which are introduced via modifications of the dispersion relation. In this model, there are no ambiguities concerning the choice of the initial vacuum state. We study an example in which the final spectrum of fluctuations depends sensitively on the modifications of the dispersion relation without needing to invoke complex frequencies. Changes in the amplitude and in the spectral index are possible, in addition to modulations of the spectrum. This strengthens the conclusions of previous work in which the spectrum of cosmological perturbations in expanding inflationary cosmologies was studied, and it was found that, for dispersion relations for which the evolution is not adiabatic, the spectrum changes from the standard prediction of scale-invariance.

Figures (6)

• Figure 1: Corley/Jacobson dispersion relation for $m=1$ and $\vert b_m\vert =1$. In general $\omega _{_{\bf phys}}^2(k)$ vanishes at $k=k_{_{\rm C}}\vert b_m \vert ^{-1/(2m)}$. For $k>k_{_{\rm C}}\vert b_m \vert ^{-1/(2m)}$, the physical frequency becomes imaginary.
• Figure 2: The solid curve is $a"/a$, the dashed one $n^2_{\rm eff}$. The values used are $\ell _0=5000$, $\ell _{\rm b}=10$, $\ell _{_{\rm C}}=1$, $\vert b_m\vert =1$, $m=1$ and $n=60$.
• Figure 3: Absolute value of Q versus conformal time. The values used are $\ell _0=5000$, $\ell _{\rm b}=10$, $\ell _{_{\rm C}}=1$, $\vert b_m\vert =1$, $m=1$ and $n=55$.
• Figure 4: The solid curve represents $a"/a$, the dot-dashed curve its quadratic approximation. The dashed curve represents $n^2_{\rm eff}$, and its quadratic approximation is the fourth curve. Values chosen are $\ell _0=5000$, $\ell _{\rm b}=10$, $\ell _{_{\rm C}}=1$, $\vert b_m\vert =1$, $m=1$ and $n=60$.
• Figure 5: The dependence of the matching time $\eta_{\rm j}(n)$ on the wave number $n$ for the values $\ell _0=5000$, $\ell _{\rm b}=10$, $\ell _{_{\rm C}}=1$, $\vert b_m\vert =1$, $m=1$. The solid curve is the exact result determined numerically, the dotted curve is obtained using the analytical approximations.