Inflation as a Probe of Short Distance Physics

TL;DR

The paper investigates whether trans-Planckian effects, modeled as a string-inspired minimum-length modification to the inflationary perturbation equations, leave observable imprints on the primordial tensor spectrum. It develops a numerical framework to compute the spectrum in a de Sitter background, showing that short-distance physics enters through two coefficients, $D_+$ and $D_-$, which can alter normalization and, in non-de Sitter backgrounds, the spectrum shape. The results demonstrate that for finite \\beta H, the spectrum can deviate from the standard prediction while remaining consistent with particle production bounds, providing a concrete link between Planck-scale physics and cosmological observations. Overall, the work offers a general method for analyzing trans-Planckian modifications and sets the stage for applying similar analyses to scalar perturbations and more realistic inflationary scenarios.

Abstract

We show that a string-inspired Planck scale modification of general relativity can have observable cosmological effects. Specifically, we present a complete analysis of the inflationary perturbation spectrum produced by a phenomenological Lagrangian that has a standard form on large scales but incorporates a string-inspired short distance cutoff, and find a deviation from the standard result. We use the de Sitter calculation as the basis of a qualitative analysis of other inflationary backgrounds, arguing that in these cases the cutoff could have a more pronounced effect, changing the shape of the spectrum. Moreover, the computational approach developed here can be used to provide unambiguous calculations of the perturbation spectrum in other heuristic models that modify trans-Planckian physics and thereby determine their impact on the inflationary perturbation spectrum. Finally, we argue that this model may provide an exception to constraints, recently proposed by Tanaka and Starobinsky, on the ability of Planck-scale physics to modify the cosmological spectrum.

Figures (3)

• Figure 1: The top figure shows the spectrum ($P^{1/2}$) as a function of $\beta$ with $C_{-}/C_{+} = -.5$, while the lower plot shows the spectrum when $C_{-} = 0$. It is only in the latter case the computed value of the spectrum smoothly approaches the usual value in de Sitter space, In this plot $H=.1$, and with $\beta =0$ we would expect $P^{1/2} = 0.0159155$.
• Figure 2: The $\beta$ dependence of the spectrum, $P_k^{1/2}$, is plotted against $\beta$, with $H=0.1$. For a de Sitter background, the spectrum is independent of $k$.
• Figure 3: The values of $D_{+}$ (top) and $D_{-}$ are extracted for the spectrum shown in Fig. 1 with $C_{-} =0$. As $\beta \rightarrow 0$, $D_{+} \rightarrow 1$ and $D_{-} \rightarrow 0$, confirming that as the Hubble length becomes much larger than the minimum length the spectrum approaches the standard result.