Kinematic Constraints to the Key Inflationary Observables

TL;DR

The paper addresses the lack of a model-independent relation between the inflationary observables $T/S$ and $n-1$ by reformulating inflation in terms of flow equations for these quantities. These flow equations produce attractor behavior in the $T/S$--$(n-1)$ plane, with two main attractors at $T/S\approx 0$ and $T/S\approx -5(n-1)$, and an excluded region for $n<1$, especially when the potential curvature $x''$ is small (smooth potentials) leading to a 'favored' region. For sufficiently smooth potentials, the models cluster near these attractors, and as $n$ increases toward unity the allowed $T/S$ grows along the favored region; in particular, if $n>0.85$, then $T/S>10^{-3}$ is expected, increasing the prospects for detecting the gravity-wave signature. Deviations from the favored region require sizable $x''$, producing large running $dn/d\ln k$ and poorly power-law density spectra; two-field models can extend the attractor to $n>1$. Overall, this flow-trajectory framework provides practical guidance for interpreting CMB measurements and testing inflation by linking the scalar tilt to the tensor amplitude.

Abstract

The observables $T/S$ and $n-1$ are key to testing and understanding inflation. ($T$, $S$, and $n-1$ respectively quantify the gravity-wave and density-perturbation contributions to CMB anisotropy and the deviation of the density perturbations from the scale-invariant form.) Absent a standard model, there is no definite prediction for, or relation between, $T/S$ and $n-1$. By reformulating the equations governing inflation we show that models generally predict $T/S \approx -5(n-1)$ or 0, and in particular, if $n>0.85$, $T/S$ is expected to be $>10^{-3}$.

Figures (4)

• Figure 1: Trajectories in the $T/S$ -- $(n-1)$ plane. Squares indicate the initial choices for $T/S$ and $(n-1)$; circles indicate the values 50 e-folds before the end of inflation. A trajectory ends when $T/S$ and/or $|n-1|$ become large; most of inflation occurs when $T/S$ and $|n-1|$ are small. The upper left panel shows a complete trajectory, with ticks indicating e-folds before the end of inflation (from the circle, $50, 49, \cdots , 1$). The other three panels show trajectories in more detail. Note how $T/S$ and $(n-1)$ outside the attractor region are "pulled in" (the attractors are shown as broken lines and the boundary of the excluded region is a solid curve).
• Figure 2: Summary of our model search using the flow equations. The lines $T/S =-5(n-1)$ and $T/S=0$ act as attractors; the dotted curves correspond to $x^{\prime\prime} =0,1,2,5$ (from left to right). We found no model in the excluded region, and we call the region between it and the curve $x^{\prime\prime} =3$ the favored region. Models outside the favored region (upper right part) have large $dn/d\ln k$ and density perturbations that are not well represented by a power law. Diamonds indicate various known inflationary models: chaotic, $V(\phi )=\lambda\phi^n$ for $n=2,3,\cdots$ (diamonds on the diagonal); new inflation ($n=0.94$) and natural inflation (with $n=0.84$). The ellipse is the $2\sigma$ error ellipse "forecasted" for the PLANCK satellite Kinney.
• Figure 3: Same as Fig. \ref{['fig:1fld']}, except with a logarithmic scale for $T/S$ to show more detail. As $n\rightarrow 1$, $T/S$ increases in the favored region. Below and to the left of the broken line ($dn/d\ln k =10^{-2}$), a poor power law does not occur for large $x^{\prime\prime}$ because $dn/d\ln k \propto \sqrt{T/S}\,x^{\prime\prime}$ and $T/S$ is small.
• Figure 4: Summary of two-field models. The filled circles represent the values of $(T/S)_{50}$ and $n_{50}$ for the corresponding one-field models, and the attached curves are the values obtained for inflation ending early due to an auxiliary field. The dashed lines represent fixed points in the $(T/S)$ -- $n$ plane that result from models that do not end without an auxiliary field. In general, two-field models populate the same region as one-field models and extend the $T/S \approx 0$ attractor to $n > 1$.