Reheating constraints to inflationary models (1404.6704v2)

Abstract: Evidence from the BICEP2 experiment for a significant gravitational-wave background has focussed attention on inflaton potentials $V(\phi) \propto \phi^\alpha$ with $\alpha=2$ ("chaotic" or "$m^2\phi^2$" inflation) or with smaller values of $\alpha$, as may arise in axion-monodromy models. Here we show that reheating considerations may provide additional constraints to these models. The reheating phase preceding the radiation era is modeled by an effective equation-of-state parameter $w_{\rm re}$. The canonical reheating scenario is then described by $w_{\rm re}=0$. The simplest $\alpha=2$ models are consistent with $w_{\rm re} = 0$ for values of $n_s$ well within the current $1\sigma$ range. Models with $\alpha=1$ or $\alpha=2/3$ require a more exotic reheating phase, with $-1/3<w_{\rm re}\<0$, unless $n_s$ falls above the current $1\sigma$ range. Likewise, models with $\alpha=4$ require a physically implausible $w_{\rm re}\>1/3$, unless $n_s$ is close to the lower limit of the $2\sigma$ range. For $m^2\phi^2$ inflation and canonical reheating as a benchmark, we derive a relation $\log_{10}\left(T_{\rm re}/10^6\,{\rm GeV} \right) \simeq 2000\,(n_s-0.96)$ between the reheat temperature $T_{\rm re}$ and the scalar spectral index $n_s$. Thus, if $n_s$ is close to its central value, then $T_{\rm re}\lesssim 10^6$~GeV, just above the electroweak scale. If the reheat temperature is higher, as many theorists may prefer, then the scalar spectral index should be closer to $n_s\simeq0.965$ (at the pivot scale $k=0.05\,{\rm Mpc}^{-1}$), near the upper limit of the $1\sigma$ error range. Improved precision in the measurement of $n_s$ should allow $m^2\phi^2$, axion-monodromy, and $\phi^4$ models to be distinguished, even without precise measurement of $r$, and to test the $m^2\phi^2$ expectation of $n_s\simeq0.965$.

• The paper demonstrates that power-law inflation models face significant constraints due to the reheating phase, characterized by an effective equation-of-state parameter.
• It reveals that the m²φ² (α = 2) model aligns with observations under canonical reheating, while other power-law indices require physically challenging conditions unless ns shifts significantly.
• A derived correlation between reheating temperature and the scalar spectral index underscores that precise ns measurements can decisively differentiate viable inflationary scenarios.

Analysis of Reheating Constraints to Inflationary Models

The paper "Reheating constraints to inflationary models" by Liang Dai, Marc Kamionkowski, and Junpu Wang addresses an essential aspect of cosmology that connects inflationary theories with observational data—analyzing the constraints on inflationary models arisen from the dynamics of the reheating period. Utilizing data from the BICEP2 experiment, the authors investigate how the reheating phase following inflation can influence the parameterization and viability of different inflationary potentials.

Key Findings and Methodology

The paper focuses on the implications of potential inflationary models, specifically power-law potentials characterized by the relation $V(\phi) \propto \phi^\alpha$. The paper examines how these models are affected by reheating, which is depicted by an effective equation-of-state parameter $w_{\rm re}$. The canonical value for reheating in this framework is $w_{\rm re}=0$.

The authors demonstrate several points:

• Consistency of $\alpha = 2$: For the $\alpha = 2$ "chaotic" model, consistency is retained with $w_{\rm re} = 0$ for scalar spectral index values $n_s$ within the current $1\sigma$ observational bounds. However, models where $\alpha = 1$ or $\alpha = 2/3$ require a reheating phase characterized by $-1/3 < w_{\rm re} < 0$ unless $n_s$ is higher than its $1\sigma$ range.
• High $\alpha$ Models: Models where $\alpha = 4$ necessitate $w_{\rm re} > 1/3$, a regime considered physically implausible unless $n_s$ lies near the lower limit of the $2\sigma$ range.
• Reheat Temperature and $n_s$ Relationship: For the $\alpha = 2$ inflation model with canonical reheating, a correlation between the reheating temperature $T_{\rm re}$ and $n_s$ is derived: $$\log_{10}(T_{\rm re}/10^6\,{\rm GeV}) \simeq 2000\,(n_s-0.96)$$. A higher $T_{\rm re}$ up to the Grand Unified Theory scale suggests $n_s$ nearing 0.965 at the pivot scale $k = 0.05\,{\rm Mpc}^{-1}$.

The formalism employs equations linking the number of e-folds from the end of inflation to the present-day observation scale with variables characterizing the reheating and radiation-dominated epochs. Based on current measurements, particularly the scalar spectral index $n_s$, the paper delineates the constraints on $N_{\rm re}$, the duration of reheating, and the corresponding reheating temperatures.

Implications

The calculations in this paper reveal critical insights into the nature of power-law potentials for inflaton fields and underline the role of reheating in differentiating amongst these models. The specific case of $m^2\phi^2$ inflation is shown to be reconcilable with simple reheating scenarios, standing favored over other power-law indices in the face of present observational accuracies.

The implications extend to future observational efforts, predominantly focusing on the precise measurement of $n_s$. The refined data could fundamentally aid in distinguishing the most tenable inflation models and enhance understanding of the reheating epoch, offering a complementary avenue to anticipated Cosmic Microwave Background (CMB) observations.

Future Prospects

This work underscores the emerging interplay between theoretical constructs of inflation and the phenomenology of reheating. As empirical techniques advance, the post-inflationary universe, specifically through gravitational wave observations and refined measurement in cosmological parameters such as $n_s$, $r$ (tensor-to-scalar ratio), and $T_{\rm re}$, becomes a promising frontier for frontier cosmological analysis.

In summary, the paper establishes groundwork that bridges inflationary theory and observational constraints through reheating dynamics, guiding prospective research towards scrutinized inflationary models and their respective implications on early universe cosmology.