Reheating and relic gravitational waves as remedies for degeneracies of non-canonical natural inflation

TL;DR

This work investigates a non-canonical natural inflation model with a power-law kinetic term ${\cal L}(X,\phi)=X(\frac{X}{M^{4}})^{\alpha-1}-V(\phi)$ to address degeneracies in the observable predictions $n_{s}$ and $r$. By deriving the background dynamics, slow-roll parameters, and the sound speed $c_s$, the authors show that $n_{s}$ becomes largely independent of the potential parameter $f$ while $r$ depends on the non-canonical parameter $\alpha$, with $f$ remaining unconstrained by these observables. Reheating constraints on $N_{\rm re}$, $T_{\rm re}$, and $\omega_{\rm re}$ are applied, yielding $47\le N\le56$ but failing to fully lift the degeneracy with respect to $\alpha$ (and leaving $f$ degenerate). Incorporating relic gravitational waves, the study demonstrates that the GW spectrum can break the $\alpha$-degeneracy for certain $N$ and $\alpha$, and forecasts detectable signals by future detectors such as BBO/DECIGO, while the degeneracy with $f$ persists. Overall, the paper highlights the synergistic role of reheating physics and relic GWs in constraining non-canonical inflation and delineating viable regions of parameter space.

Abstract

Here, a natural non-canonical inflationary model based on a power-law Lagrangian is investigated. We analyze the scalar spectral index $n_{\rm s}$ and the tensor-to-scalar ratio $r$ of the model and identify their degeneracies with respect to the free parameters. Notably, $n_{\rm s}$ and $r$ show effective independence from the model parameters due to degeneracies in the slow-roll parameters that leads to unresolved parameter degeneracies. Employing the constraints on reheating parameters such as the reheating duration $N_{\rm{reh}}$, the reheating temperature $T_{\rm{reh}}$, and the equation of state parameter $ω_{\rm{reh}}$, is found to be insufficient to fully break these degeneracies. However, the relic gravitational wave spectrum provides a way to break degeneracy with respect to the non-canonical parameter $α$, degeneracy with respect to the potential parameter $f$ persist. Finally, we specify the allowed ranges for the inflationary duration $N$ and the parameter $α$, in light of the latest observational data. These results highlight the role of relic gravitational waves in refining inflationary models and illustrate the challenges in fully resolving parameter degeneracies.

Figures (6)

• Figure 1: The evolutions of (a) inflaton field $\phi$, (b) Hubble parameter $H$, (c) the first slow-roll parameter $\epsilon$, and (d) the second slow-roll parameter $\eta$ as functions of the $e$-folds number $N$. The dashed and dotted lines correspond to $f=150M_p$ and $f=300M_p$, respectively, while the thick and thin lines represent $\alpha=2$ and $\alpha=80$, respectively. Note that in panel (c), to illustrate the overlap of the curves, $\alpha=2$ and $\alpha=80$ are shown by thick and thin red curves, respectively, confirming the perfect degeneracy.
• Figure 2: The $r-n_{\rm s}$ diagrams for the non-canonical natural inflation for different $e$-folds number $N=60$ (solid curve), $N=56$ (long-dashed curve), $N=54$ (dashed curve), $N=50$ (dash-dotted curve) and $N=47$ (dotted curve). Along each curve, the parameter $\alpha$ varies in the range $2\leq\alpha\leq 130$, from top to bottom. The dark (light) green area represents the 68% (95%) CL constraints from Planck 2018 TT,TE, EE+low E+lensing data. The dark (light) blue region shows the 68% (95%) CL constraints from the joint dataset Planck 2018 TT, TE, EE+low E+lensing+BK18+BAO.
• Figure 3: Variation of the reheating equation of state parameter $\omega _{\rm re}$ with respect to the parameter $\alpha$ for $6\leq\alpha\leq 130$.
• Figure 4: The number of $e$-folds during reheating $N_{\rm re}$ versus (a) the scalar spectral index $n_{\rm s}$ for $\alpha=9$ (solid curve) and $\alpha=130$ (dotted curve) and (b) the parameter $\alpha$ for $N=57$ (dotted curve), $N=56$ (dash-dotted curve), $N=54$ (dashed curve), and $N=50$ (solid curve). In panel (a), the dark and light blue regions show the 68% and 95% CL, respectively, based on Planck 2018 TT,TE, EE + LowE + Lensing + BK18 + BAO data. In panel (b), the blue portions of each curve represent the 68% CL allowed range for $\alpha$ corresponding to each $N$, as listed in Table \ref{['tabalpha']}, while the shaded areas mark the regions excluded by $N_{\rm re}<0$.
• Figure 5: Reheating temperature $T_{\rm re}$ versus (a) $n_{\rm s}$ and (b) $\alpha$ . In panel (a), the shaded gray area at the bottom indicates the region excluded by the BBN constraint, while the horizontal dashed line at the top marks the model-independent bound $T_{\rm re} = 5\times 10^{15} GeV$. The dark (light) blue zones are the same as in Fig. \ref{['fig:1']}.