Effective Bayesian ranking of low order monomial potentials in low temperature warm inflation

Technical Summary: Effective Bayesian Ranking of Low Order Monomial Potentials in Low Temperature Warm Inflation

Problem Statement
In the standard cold single-field slow-roll inflation framework, low-order monomial potentials $V_p(\phi) = \lambda_p \phi^p/p$ (specifically $p=2, 3, 4$) face significant observational tension. While these models are theoretically motivated by chaotic inflation and effective field theory, the cold inflation background predicts a tensor-to-scalar ratio $r$ that scales with the power $p$. For $N_* \approx 50-60$ e-folds, the quartic potential ($p=4$) predicts $r \approx 0.29$, which is in strong conflict with the current observational upper bound $r_{0.05} < 0.036$ (BICEP/Keck 2018). The quadratic ($p=2$) and cubic ($p=3$) cases also struggle to simultaneously satisfy constraints on the scalar spectral index $n_s$ and the tensor ratio $r$. The paper investigates whether the introduction of warm inflation dynamics, specifically within a low-temperature dissipative regime, can resolve this tension and alter the statistical preference among these monomial branches.

Methodology
The authors perform a systematic Bayesian model comparison of the $p=2, 3, 4$ potentials within a fixed low-temperature dissipative effective class. The methodology involves:

1. Theoretical Framework: The study adopts the low-temperature dissipative coefficient $\Upsilon = C_\phi T^3/\phi^2$, valid when the thermal bath temperature $T$ is below the heavy mediator mass scale ($T < m_\chi$) but above the Hubble scale ($T > H$). The background evolution is governed by coupled equations for the inflaton field, radiation bath, and Friedmann constraint, solved numerically without relying on slow-roll approximations for the background trajectory.
2. Primordial Spectrum Calculation: The scalar power spectrum $P_R$ incorporates vacuum fluctuations, thermal occupation effects (Bose-Einstein distribution), and dissipative growth factors. The tensor spectrum $P_T$ remains unenhanced. The tensor-to-scalar ratio is thus reduced primarily through the enhancement of the scalar denominator.
3. Likelihood Construction: A "broadened compressed likelihood" is constructed using the observables $(\ln(10^{10}A_s), n_s, r_{0.05})$. The scalar amplitude and spectral index are treated with Gaussian distributions, while $r$ is treated with a one-sided upper bound penalty function.
4. Prior Settings: Instead of arbitrary uniform hyperpriors, the authors employ "structure-conditioned priors." These priors are calibrated through broad parameter exploration and genetic algorithm (GA) searches to cover the physically viable warm inflation branches for each monomial power. Three prior sets (A, B, C) are tested to ensure the robustness of the ranking against prior volume effects.
5. Evidence Integration: The effective Bayesian evidence $Z_{\text{eff}}$ is computed using nested sampling (dynesty) over the fixed prior regions. This metric integrates the likelihood over the entire parameter space, penalizing models that only fit data in isolated, narrow regions.

Key Results
The analysis yields a clear hierarchy in the effective Bayesian evidence for the three monomial powers:

• Quartic Potential ($p=4$): This model receives the strongest support. A representative trajectory yields $n_s = 0.96420$ and $r_{0.05} = 0.02663$, consistent with observational constraints.
• Cubic Potential ($p=3$): This model occupies an intermediate position but is disfavored relative to the quartic case, with a relative evidence difference of $\Delta \ln Z_{\text{eff}} \approx -6.99$.
• Quadratic Potential ($p=2$): This model is strongly disfavored, with $\Delta \ln Z_{\text{eff}} \approx -32.18$. It produces a blue-tilted spectrum and an excessively large tensor amplitude even within the warm framework.

Physical Mechanism
The paper identifies that the recovery of the quartic potential is driven primarily by Bose-Einstein thermal occupation enhancement rather than strong dissipative friction. In the viable trajectories, the system operates in a weakly dissipative regime ($Q_* \sim 10^{-3}$), but with a significant thermal ratio $T_*/H_* \approx 10.67$. This condition ($T_*/H_* > 1$) enhances the scalar spectrum amplitude, thereby suppressing the tensor-to-scalar ratio $r$. Tests removing the thermal occupation term ($n_* = 0$) result in a dramatic increase in $r$ (to $\sim 0.4$), confirming that thermal occupation, not friction, is the dominant mechanism for the quartic potential's viability.

Robustness and Validity
The hierarchy $p=4 > p=3 \gg p=2$ is shown to be stable under:

• Variations in the pivot e-fold number $N_*$ (50, 55, 60).
• Different prior volume definitions (Sets A, B, and C).
• Independent random seeds for sampling.
• Different smoothing parameters for the $r$ upper bound.
• Numerical stability checks on the spectrum calculation.

The trajectories satisfy all physical validity conditions within the CMB window: $T/H > 1$ (warm inflation), $Q \ll 1$ (weak dissipation), $\rho_R/V \ll 1$ (radiation subdominance), and $T < m_\chi$ (low-temperature validity).

Significance and Claims
The paper claims to reformulate the "tensor-to-scalar ratio tension" of cold monomial inflation as a controlled ranking problem within a fixed low-temperature dissipative effective class. The primary contribution is demonstrating that, when considering the full prior-averaged evidence rather than isolated best-fit points, the quartic potential becomes the most statistically favored low-order model.

The authors explicitly state that this conclusion is an "effective evidence preference" contingent on:

1. The specific low-temperature dissipative form $\Upsilon = C_\phi T^3/\phi^2$.
2. The validity of the thermal occupation assumption ($n_* > 0$).
3. The adopted family of structure-conditioned priors.

The paper does not claim that warm inflation generically favors the quartic potential in all scenarios, but rather that within the specific microphysical regime characterized by weak dissipation and significant thermal occupation, the quartic branch is robustly recovered. The large effective dissipation coefficients ($C_\phi \sim 10^6-10^7$) required are interpreted as effective strengths arising from composite sectors or multiple degrees of freedom, representing a microphysical cost of this realization.