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

An effective Bayesian evidence ranking is performed for the monomial potentials \(V_p(\phi)=\lambda_p\phi^p/p\), with \(p=2,3,4\), in low temperature warm inflation with the dissipative coefficient fixed as \(\Upsilon=C_\phi T^3/\phi^2\). 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 \(N_*=55\), \(\sigma_r=0.005\), and structure conditioned priors covering viable warm branches, the quadratic and cubic potentials are disfavored relative to the quartic branch: $\Delta\ln Z_{\rm eff}(p=2)=-32.18,~ \Delta\ln Z_{\rm eff}(p=3)=-6.99.$ 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 \(p=4>p=3\gg p=2.\)