Inflation and the Scale Dependent Spectral Index: Prospects and Strategies

TL;DR

This paper analyzes how the scale dependence of the primordial power spectrum, encoded in the running $\alpha_s$, can illuminate inflation and the pre-nucleosynthesis history of the universe. Using canonical single-field inflation, the matching equation, and an effective post-inflationary equation of state $\tilde{w}$, it forecasts how $n_s$, $\alpha_s$, and the tensor-to-scalar ratio $r$ depend on the reheating history across several inflationary models (e.g., $\phi^n$, natural, hilltop/inflection). It then assesses the capability of Planck, galaxy surveys, and 21 cm experiments to constrain these parameters via Fisher matrix analyses, highlighting that while $\alpha_s$ could be detected by futuristic surveys, degeneracies with reheating histories complicate model discrimination; crucially, measurements of $n_s$ together with $\alpha_s$ offer a handle on the post-inflationary expansion and reheating mechanisms, potentially probing physics between the TeV and GUT scales. The work emphasizes the potential of upcoming observations to open a window on the primordial dark age, linking inflationary physics to high-energy processes and the reheating phase that followed inflation.

Abstract

We consider the running of the spectral index as a probe of both inflation itself, and of the overall evolution of the very early universe. Surveying a collection of simple single field inflationary models, we confirm that the magnitude of the running is relatively consistent, unlike the tensor amplitude, which varies by orders of magnitude. Given this target, we confirm that the running is potentially detectable by future large scale structure or 21 cm observations, but that only the most futuristic measurements can distinguish between these models on the basis of their running. For any specified inflationary scenario, the combination of the running index and unknown post-inflationary expansion history induces a theoretical uncertainty in the predicted value of the spectral index. This effect can easily dominate the statistical uncertainty with which Planck and its successors are expected to measure the spectral index. More positively, upcoming cosmological experiments thus provide an intriguing probe of physics between TeV and GUT scales by constraining the reheating history associated with any specified inflationary model, opening a window into the "primordial dark age" that follows the end of inflation.

Figures (14)

• Figure 1: The evolution of the Hubble horizon for the inflationary universe is shown in cartoon form. The $x$-axis describes the cosmological scale factor $a(t)$ on an approximately logarithmic scale. The parameter $\tilde{w}$ describes the growth during the epoch between the end of inflation with the standard hot big bang era. Note that for smaller $\tilde{w}$ the value of $N$ at which the pivot leaves the horizon is similarly decreased.
• Figure 2: We illustrate the impact of changing our the assumed value of $\rho_{reh}$. The left hand panel demonstrates that setting a high value of $\rho_{reh}$ reduces the total uncertainty in $N$, whereas the right hand panel illustrates a specific (and baroque) post-inflationary history (solid line) which then fixes the effective expansion history and $\tilde{w}$.
• Figure 3: Top panel: $n_s$ vs. $r$, Bottom, $n_s$ vs. $\alpha_s$, $\alpha_s$ vs. $r$. We plot $\phi^n$ inflation (red), natural inflation (green), inflection (purple) and hilltop (blue). Hilltop and inflection point inflation meet the $\phi^n$ curve at the point where $n=1$; natural inflation is degenerate with $m^2 \phi^2$ inflation in the limit that $f$ is very large.
• Figure 4: Same models as Figure \ref{['fig:instant']}, assuming a period of matter dominated expansion before thermalization, with $T_{reh} >10^3$ GeV. Colors denote the (logarithmic) reheating temperature -- orange is low, and blue is high.
• Figure 5: Same models as Figures \ref{['fig:instant']} and \ref{['fig:general']}, but for the full range of $\tilde{w}$ and $\rho_{reh}^{1/4} = 1$ TeV. Parallel lines for each model denote fixed values of $N$.