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

Abstract: An effective Bayesian evidence ranking is performed for the monomial potentials (V_p(φ)=λpφ^p/p), with (p=2,3,4), in low temperature warm inflation with the dissipative coefficient fixed as (Υ=CφT^3/φ^2). In cold single field slow roll inflation, these branches are strongly constrained by the observational upper bound on the tensor to scalar ratio (r=\mathcal P_T/\mathcal P_{\mathcal R}), whereas warm inflation can reduce this tension by enhancing the scalar spectrum. The relevant question is therefore which monomial power is favored once (A_s), (n_s), (r_{0.05}), and the viable parameter volume are considered simultaneously. For each branch, the warm background equations including radiation backreaction are solved, and a broadened compressed likelihood for ((A_s,n_s,r_{0.05})) is integrated over the prior volume to obtain (Z_{\rm eff}^{(A_s,n_s,r)}). For (N_=55), (σr=0.005), and structure conditioned priors covering viable warm branches, the quadratic and cubic potentials are disfavored relative to the quartic branch: $$Δ\ln Z</em>{\rm eff}(p=2)=-32.18,~ Δ\ln Z_{\rm eff}(p=3)=-6.99.$$ This hierarchy is stable under changes in (N_), prior ranges, random seeds, and the $r$ bound treatment. A representative quartic trajectory gives (n_s=0.96420), (r_{0.05}=0.02663), (Q_=4.68\times10^{-3}), and (T_/H_=10.67), corresponding to a weakly dissipative but thermally occupied CMB window. Decomposing the primordial spectrum shows that the quartic preference is driven mainly by Bose Einstein occupation enhancement for (T_/H_*>1), not by strong dissipative friction. Within the low temperature dissipative effective class and compressed likelihood adopted here, the evidence hierarchy is (p=4>p=3\gg p=2.)

• The paper establishes an effective Bayesian evidence framework to rank low order monomial potentials in a warm inflation context.
• The authors demonstrate that thermal occupancy via Bose-Einstein statistics enables quartic potentials to overcome cold inflation constraints.
• Robust numerical sampling confirms the quartic model’s preference, highlighting the role of composite dissipative sectors.

Bayesian Evidence Ranking of Low Order Monomial Potentials in Low Temperature Warm Inflation

Introduction and Framework

This study presents a comprehensive Bayesian evidence ranking of single-field inflationary models characterized by low order monomial potentials $V_p(\phi) = \lambda_p \phi^p/p$ ($p=2,3,4$) within the context of low temperature warm inflation, where dissipation is modeled via a cubic temperature-dependent coefficient $\Upsilon = C_\phi T^3/\phi^2$. Contrasting sharply with cold inflationary dynamics where monomial potentials are strongly constrained, especially by the upper limits on the tensor-to-scalar ratio $r$, the warm inflation scenario modifies the scalar power spectrum through dissipative and thermal effects, potentially alleviating those constraints.

Central observables for model discrimination are the amplitude of curvature perturbations $A_s$, scalar spectral index $n_s$, and tensor-to-scalar ratio $r$ at a CMB pivot scale. The key innovation is the establishment of an effective Bayesian evidence $Z_{\rm eff}$, which integrates the likelihood of matching these observables over structure-conditioned, physically motivated prior volumes for each model. All calculations explicitly include backreaction of radiation, verification of the regime's validity (weak dissipation, low temperature), and parameter sampling consistent with the required number of e-folds.

Analytic Cold Inflation Reference and Warm Inflation Modifications

Analytic benchmarks for the monomial models in the cold inflation limit show increasing tension at higher $p$ values: $r$ is directly proportional to $p=2,3,4$0, and the predicted $p=2,3,4$1 too falls outside observational constraints for cubic and quartic terms at typical $p=2,3,4$2. This is summarized by a systematic increase in both $p=2,3,4$3 and the deviation of $p=2,3,4$4 from the observational central value as $p=2,3,4$5 increases, rendering particularly $p=2,3,4$6 (quartic) non-viable in cold single-field slow-roll inflation.

Figure 1: Positions of the cold inflation analytic reference points and the warm representative high likelihood points in the $p=2,3,4$7--$p=2,3,4$8 plane; warm inflation brings quartic models into the observationally viable region, evading the $p=2,3,4$9 constraints.

In warm inflation, however, continuous energy dissipation from the inflaton into a thermal bath—along with an enhanced scalar spectrum via thermal fluctuations and Bose-Einstein occupation—decouples $\Upsilon = C_\phi T^3/\phi^2$0 from the simple cold-inflation slow-roll formula $\Upsilon = C_\phi T^3/\phi^2$1. In this framework, $\Upsilon = C_\phi T^3/\phi^2$2 governs the relative strength of dissipation, and crucially, in the CMB window explored, $\Upsilon = C_\phi T^3/\phi^2$3 but $\Upsilon = C_\phi T^3/\phi^2$4, enabling significant spectral modifications via thermal effects rather than frictional (dissipative) suppression.

Bayesian Evidence Computation and Model Comparison

The paper performs a systematic nested sampling Bayesian evidence computation for each $\Upsilon = C_\phi T^3/\phi^2$5, integrating over five-dimensional prior spaces tailored to each monomial model. Priors are constructed via global parameter space exploration and include only regions compatible with warm inflation's physical requirements. The likelihood function combines Gaussian constraints on $\Upsilon = C_\phi T^3/\phi^2$6, $\Upsilon = C_\phi T^3/\phi^2$7, and an upper bound penalty on $\Upsilon = C_\phi T^3/\phi^2$8, implemented as a softened one-sided constraint.

Strong numerical results are reported:

Model	$\Upsilon = C_\phi T^3/\phi^2$9 (relative to $r$0)
$r$1 (quadratic)	$r$2
$r$3 (cubic)	$r$4
$r$5 (quartic)	$r$6

These substantial effective evidence differences robustly favor the quartic potential among the tested monomial branches within the low temperature warm inflation class. The results are stable against variations in $r$7 (number of e-folds), prior widths, and treatment of the $r$8 upper bound.

Overlaps of high-likelihood regions for the quartic branch, as determined by different inference techniques (MCMC, nested sampling, normalizing flows), corroborate posterior robustness and convergence.

Figure 2: Overlap of the posterior regions for the quartic potential branch in the $r$9-$A_s$0 plane demonstrates consistency across inference methodologies.

Physical Mechanism: Origin of the Quartic Preference

A critical analysis demonstrates that the quartic branch's high evidence does not primarily derive from strong dissipative friction during inflation ($A_s$1) but rather from the enhancement of the scalar perturbation amplitude by thermal occupation effects—specifically, Bose-Einstein statistics when $A_s$2 during the CMB window. Removal of the thermal occupation term ($A_s$3) severely suppresses the scalar spectrum and pushes $A_s$4 well above observational constraints for all $A_s$5, eliminating any advantage for the quartic model.

Thus, the quartic model's reemergence as the statistically preferred monomial in warm inflation is conditioned upon the validity and efficacy of thermal occupancy of inflaton fluctuations during the relevant phase.

Figure 3: Validity diagnostics for the representative trajectory of the quartic potential, confirming the regime of weak dissipation and significant thermal occupancy during the CMB window.

Figure 4: Evolution of the dissipation ratio $A_s$6 for all representative monomial potentials, with $A_s$7 during the CMB window and rapid increase approaching the end of inflation.

Implications and Theoretical Significance

The work's key implication is that within the fixed low temperature, weakly dissipative warm inflation framework, and with observational constraints modeled through a compressed likelihood, the quartic potential—excluded by tensor amplitude measurements in cold inflation—becomes the most statistically supported low order monomial. This has significant consequences for inflationary model building, emphasizing the role of detailed microphysical modeling of dissipation and the necessity for consistent treatment of thermal perturbations.

However, the preference for the quartic branch is conditional: a large effective dissipation parameter ($A_s$8–$A_s$9) is required, interpretable as arising from large numbers of dissipative channels or composite sectors, rather than a single coupling; the thermal occupancy assumption is essential. The result is not generically extensible to all classes of warm inflation, but delineates a well-defined region in the parameter and model space where quartic monomials are viable and even preferred.

The Bayesian model comparison framework established in this study provides a rigorous template for future comparative analyses of inflationary models with more general potential forms or dissipation structures, particularly as cosmological data continue to improve.

Conclusion

The Bayesian ranking of low order monomial inflationary potentials in the low temperature warm inflation scenario, incorporating rigorous backreaction and parameter consistency constraints, finds that the quartic model ($n_s$0) strongly dominates over quadratic and cubic alternatives, contingent on thermally occupied, weakly dissipative dynamics during the CMB window. This outcome strongly contrasts with cold inflation predictions, highlighting the transformative impact of warm inflation microphysics on inflationary model selection and motivating further exploration of composite dissipative sectors and their phenomenological signatures.