On the Dependence of the Spectra of Fluctuations in Inflationary Cosmology on Trans-Planckian Physics

TL;DR

This paper analyzes how trans-Planckian physics can imprint on inflationary perturbations by enforcing mode-by-mode initial conditions at a fixed new-physics scale ℓ_C. It systematically studies minimal trans-Planckian physics across power-law and slow-roll inflation, comparing instantaneous Minkowski, general α-vacua, and Danielsson boundary conditions, and derives how Bogoliubov coefficients modify the mode functions and the resulting power spectra for scalar and tensor perturbations. A key finding is the strong dependence of the corrections on the chosen initial state: instantaneous Minkowski vacuum yields corrections suppressed by σ_0^3 for tensors and σ_0^2 for scalars, while general α-vacua can produce order-unity effects, and Danielsson’s α-vacuum yields linear-in-σ_0 corrections for tensors and quadratic for scalars. The authors also contrast this framework with modified-dispersion-relation approaches, showing that while such models can produce sizeable oscillations, back-reaction constraints typically require fine-tuning of scales, limiting their practical impact.

Abstract

We calculate the power spectrum of metric fluctuations in inflationary cosmology starting with initial conditions which are imposed mode by mode when the wavelength equals some critical length $\ell_{_{\rm C}}$ corresponding to a new energy scale $M_{_{\rm C}}$ at which trans-Planckian physics becomes important. In this case, the power spectrum can differ from what is calculated in the usual framework (which amounts to choosing the adiabatic vacuum state). The fractional difference in the results depends on the ratio $σ_0$ between the Hubble expansion rate $H_{\rm inf}$ during inflation and the new energy scale $M_{_{\rm C}}$. We show how and why different choices of the initial vacuum state (stemming from different assumptions about trans-Planckian physics) lead to fractional differences which depend on different powers of $σ_0$. As we emphasize, the power in general also depends on whether one is calculating the power spectrum of density fluctuations or of gravitational waves.

Figures (6)

• Figure 1: Sketch of the evolution of the Hubble radius vs time comparing how the initial conditions are fixed in the standard procedure and in the framework of the minimal trans-Planckian physics. In the standard procedure, the initial conditions are fixed on a surface of constant time $\eta =\eta _{\rm i}$ with $\eta _{\rm i}\rightarrow -\infty$. In this sense, the initial time does not depend on the wavenumber. On the contrary, in the minimal trans-Planckian physics, the initial time depends on the wavenumber and the Fourier modes are "created" when their wavelength is equal to a characteristic length. They never penetrate into the trans-Planckian region. This causes a modification of the power spectrum which is $k$ dependent since the modification is not the same for all Fourier modes as is apparent from the figure.
• Figure 2: Amplitude of the gravitational wave power spectrum as a function of the parameter $\sigma _0$ in the case where the initial state is taken to be the instantaneous Minkowski state. For small values of $\sigma _0$, the leading order correction is cubic in $\sigma _0$. The amplitude blows up when $\sigma _0=1/\sqrt{2}$.
• Figure 3: Amplitude of the gravitational wave power spectrum as a function of the parameter $\sigma _0$ in the case where the initial state is the one singled out by Danielsson's condition. For small values of $\sigma _0$, the leading order correction is linear in $\sigma _0$.
• Figure 4: Power spectra given by Eq. (\ref{['pssrs2']}). The slow-roll parameters are taken to be $\epsilon=0.10005$ and $\delta =0.2$ which corresponds to a tiny red tilt. The pivot scale is chosen to be $k_*=0.01 \hbox{Mpc}^{-1}$. The time $\eta _0$ is chosen to be the time at which the pivot scale is "created". This corresponds to $k_*/a_0=M_{_{\rm C}}$. The solid line corresponds to $\sigma _0=0$ while the dotted line is for $\sigma _0=0.02$ and the dashed line for $\sigma _0=0.03$.
• Figure 5: Typical example of a dispersion relation which breaks the WKB approximation in the trans-Planckian regime but allows a non-ambiguous definition of the initial state.