Theoretical Uncertainties in Inflationary Predictions

TL;DR

This paper investigates the theoretical uncertainties in inflationary predictions arising from our ignorance of the post-inflation history, notably reheating, and quantifies how the horizon-crossing epoch, encoded in the number of e-folds $N$, propagates into observables like the tensor-to-scalar ratio $r$ and the scalar spectral index $n$. Using the inflationary flow formalism, it derives low-order relations $r \approx 16\epsilon$ and $n-1 \approx \sigma$, then extends to higher orders with a hierarchy of flow parameters, allowing robust estimation of $\Delta r$ and $\Delta n$ due to $\Delta N$. A Monte Carlo exploration of an eight-parameter flow space confirms that typical theoretical uncertainties are of order $\frac{\Delta r}{r} \sim 0.1-1$ and $\frac{\Delta n}{|n-1|} \sim 0.1-1$, and reveals that models can flow between large-field, small-field, and hybrid regions as $N$ varies. The results show that these theoretical errors are competitive with or exceed projected experimental uncertainties, and emphasize the need to incorporate $\Delta N$-driven uncertainties and higher-order flow dynamics into inflationary model testing. The work highlights that a simple one-parameter or boundary-based model classification is insufficient beyond leading order, and that a full flow-based, multi-parameter exploration is essential for reliable interpretation of future CMB data.

Abstract

With present and future observations becoming of higher and higher quality, it is timely and necessary to investigate the most significant theoretical uncertainties in the predictions of inflation. We show that our ignorance of the entire history of the Universe, including the physics of reheating after inflation, translates to considerable errors in observationally relevant parameters. Using the inflationary flow formalism, we estimate that for a spectral index $n$ and tensor/scalar ratio $r$ in the region favored by current observational constraints, the theoretical errors are of order $Δn / | n - 1| \sim 0.1 - 1$ and $Δr /r \sim 0.1 - 1$. These errors represent the dominant theoretical uncertainties in the predictions of inflation, and are generically of the order of or larger than the projected uncertainties in future precision measurements of the Cosmic Microwave Background. We also show that the lowest-order classification of models into small field, large field, and hybrid breaks down when higher order corrections to the dynamics are included. Models can flow from one region to another.

Figures (8)

• Figure 1: Regions in the $r\,-\,n$ plane corresponding to "large field", "small field", and "hybrid" models.
• Figure 2: Predictions of different potentials, including errors due to the uncertainty in $N$, plotted in the region of $n\,-\,\log{r}$ plane favored by current observation. The model curves, from largest $r$ to smallest, are: $V \propto \exp(\phi / \mu)$ (blue, top), $V \propto \phi^p$ (magenta), $V \propto \phi$ (black), $V \propto 1 - (\phi / \mu)^2$ (red), $V \propto 1 - (\phi / \mu)^p\, (p > 2)$ (green, lowest). The horizontal lines labeled with $\Omega_{\rm GW}$ are the expected sensitivities of different proposed configurations of the Big Bang Observer satellite BBO. The hatched error bars are the observational uncertainties for: (a) a cosmic-variance limited temperature-only CMB measurement to $\ell = 1500$ (outer, red), the Planck Surveyor satellite (middle, blue), and a hypothetical CMBPol-like experiment with the same angular resolution as Planck but three times better sensitivity (inner, solid black) kinney98a. The central value for the error bars shown is arbitrary.
• Figure 3: Lowest order estimates (\ref{['eq:lowestordererrors']}) of the theoretical errors $\Delta r / r$ (top) and $\Delta n$ (bottom) as functions of $n$ and $r$, assuming $\Delta N = 14$.
• Figure 4: Flow from $N = 46$ (blue squares) to $N = 60$ (red triangles) for an ensemble of fifty models generated via flow Monte Carlo, plotted in the $n$ - $r$ plane. The path traced by the flow indicates the level of theoretical uncertainty induced by the uncertainty $\Delta N$. The diagonal lines indicate the boundares between small-field and large-field (green, solid) and large-field and hybrid (black, dashed). Models can "shift" class from $N = 46$ to $N = 60$. The error bar at top right shows projected $2\sigma$ measurement uncertainties in $n$ and $r$ for the Planck satellite. The central value for the error bar shown is arbitrary.
• Figure 5: Flow from $N = 46$ (blue squares) to $N = 60$ (red triangles) for an ensemble of fifty models generated via flow Monte Carlo, plotted in the $n$ - $\log{r}$ plane, showing the behavior for small $r$.