{"id":"1446c0f2-8b4f-4f9a-91aa-43ea9d3b7471","arxiv_id":"2504.17760","paper_version":2,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"WI2easy provides a public Mathematica implementation that computes warm inflation dynamics and curvature perturbation spectra via a deterministic Fokker-Planck approach, and shows that the universality of the G(Q) correction breaks when radiation noise is included.","lead":"WI2easy is a new Mathematica package that solves the background and perturbation equations of warm inflation, producing power spectra and observables like the spectral tilt and tensor-to-scalar ratio. It gives cosmologists a flexible, public tool to test warm inflation models against CMB data without relying on approximate fitting formulas.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The central methodological claim—that the deterministic Fokker-Planck moment equation (3.31) faithfully reproduces the coupled stochastic perturbation system—is asserted but not independently demonstrated; a direct cross-check against an established stochastic solver is needed before the code's…","rationale":"The reader identified the same weakest assumption: the Fokker-Planck moment-equation equivalence and the multi-noise identity (A.5) are cited and claimed but not demonstrated. The paper is a software release whose stated value is high-precision, model-independent power spectra, and the deterministic method is its main selling point over WarmSPy. The code is not independently runnable in this review, and the auxiliary numerical check is asserted in footnote 13 but not shown. However, the paper does provide validation against WarmSPy and earlier fitting functions for G(Q) in the thermal/no-radiation-noise setup, and the Fokker-Planck approach is based on published literature. The concern is not that the method is necessarily wrong, but that the specific implementation is not independently verified. This is a genuine correctness risk warranting a conditional acceptance rather than rejection: if the proposed cross-check passes, the paper's claims stand; if it fails, the derived observables are compromised. The critique is technical and not ad hominem, and the recommendation is proportionate to the unshown verification step.","tokens_in":25053,"tokens_out":1537,"duration_ms":16227,"concrete_test":"Run WI2easy's FindGQ() and ObservationsWI() for the quartic potential with dissipation coefficients Υ = CΥT and Υ = CΥT3/MPl2, then compute the same G(Q) and PR(k) with an independent stochastic solver (e.g., a separate Langevin integration of (3.13)–(3.15) with many realizations, or WarmSPy with identical inputs). Compare G(Q) at Q = 10−4, 1, 102 and PR(k) at the pivot scale; if the fractional difference exceeds a few percent, or if the Q-dependence of G(Q) differs, the deterministic Fokker-Planck implementation is not equivalent to the stochastic system and all derived observables are affected.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central computational innovation is replacing the stochastic perturbation system (3.13)–(3.15) with the deterministic moment equation (3.31) for J = ⟨ΦΦ†⟩, following refs. [40,41]. Every derived observable—G(Q), PR(k), ns, r, αs—flows from this step. The equivalence is not proven in the paper, and the multi-noise identity used in eq. (A.5) is only claimed in footnote 13 to have been 'explicitly verified numerically, using an auxiliary Langevin code,' with no evidence shown. A subtle error in the matrix A (eq. A.4), in the noise matrix B (eq. A.5), or in the mapping from the Fokker-Planck variable Φ to the curvature perturbation R in eq. (3.28) would shift all quoted spectra while internal consistency checks against fitting functions might not reveal it. The paper does not include the auxiliary verification, and the package was not independently run in this review. Since the code's premium over WarmSPy is precisely this deterministic method, the absence of a direct, reproducible cross-check is the most load-bearing gap in the central claim.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper presents WI2easy, a Mathematica package for computing warm-inflation background dynamics and curvature perturbation power spectra. The code solves the deterministic Fokker-Planck moment equation (3.31) for J = <Phi Phi^dagger> rather than integrating the stochastic Langevin system (3.13)-(3.15), following refs. [40,41]. It supports arbitrary single-field potentials and dissipation coefficients, with user options for thermalized (n_BE) and nonthermal inflaton fluctuations and for including the radiation noise term. The package generates initial conditions, evaluates G(Q) from eq. (3.2), normalizes the potential to the Planck amplitude, and outputs P_R(k), n_s, r, and alpha_s. Results are shown for quartic, fibre, axion, hilltop, and hybrid potentials with dissipation coefficients proportional to T, T^3, and T^3/phi^2; comparisons with previous G(Q) fits are given in Fig. 5, and observables for a quartic model with a linear-in-T dissipation coefficient are displayed in Figs. 6-9.","tokens_in":25294,"tokens_out":7902,"duration_ms":81872,"significance":"If the central equivalence is correct, WI2easy is a useful community tool: the deterministic formulation avoids ensemble averaging, the appendix fully specifies the matrix elements entering the solver, and the package is publicly available. The comparison in Fig. 5 reproduces previous G(Q) results in the no-radiation-noise thermal case, and the exploration of radiation-noise effects on the universality of G(Q) is a physically interesting result. The main risk is that the Fokker-Planck moment truncation and the multi-noise amplitude in eq. (A.5) are asserted rather than independently validated in the regimes in which the code makes new predictions.","major_comments":[{"comment":"The central methodological claim is that the deterministic Fokker-Planck moment equation (3.31), with matrix elements (A.4)-(A.6), exactly reproduces the ensemble-averaged power spectrum of the stochastic system (3.13)-(3.15). This equivalence is not demonstrated in the manuscript: it is imported from refs. [40,41], and the multi-noise combination used in b2 of eq. (A.5) is asserted in footnote 13 to have been 'explicitly verified numerically' with an auxiliary Langevin code, but no result of that verification is shown. Because every derived observable (G(Q), P_R(k), n_s, r, alpha_s) flows from J=<Phi Phi^dagger>, an error in A, B, or C would shift all quoted spectra in a way that internal consistency checks against fitting functions might not reveal. Please include either a derivation, a reproducible numerical cross-check against an independent stochastic solver for at least one configuration with the radiation-noise option and one with n_BE != 0, or the auxiliary Langevin verification material itself.","section":"Sec. 3, Eq. (3.31); Appendix A, Eqs. (A.4)-(A.6); footnote 13"},{"comment":"The validation of the code's G(Q) against WarmSPy and earlier fitting functions is restricted to the configuration in which those fits were calibrated: thermal inflaton fluctuations (n* = n_BE) and no radiation noise. The genuinely new options advertised by WI2easy - inclusion of the radiation noise term in eq. (3.14) and nonthermal inflaton fluctuations - are exactly the regimes in which no external comparison is shown. Consequently, the claims about universality breaking (Figs. 3 and 4) rest on the unverified Fokker-Planck implementation. A direct stochastic-solver comparison for at least one model with radiation noise and for one with n* = 0 would close this gap; without it, the new-regime results should be presented as preliminary.","section":"Sec. 6.2, Fig. 5"}],"minor_comments":[{"comment":"The text 'supesymmetry' should read 'supersymmetry'.","section":"Sec. 5"},{"comment":"The phrase 'northermal inflaton fluctuations' appears to be a typo for 'nonthermal inflaton fluctuations'.","section":"Sec. 6.1"},{"comment":"The notation 'e-efolds' is used in places; please standardize to 'e-folds'.","section":"Secs. 4.2.2-4.2.3"},{"comment":"The legend entries 'fit linear T 1610.08758' and 'fit cubic T 2306.16190' are not self-explanatory; please label them with the corresponding reference numbers (e.g., ref. [11] and ref. [39]) in the caption.","section":"Fig. 5 legend"},{"comment":"The claim of 'direct access to the power spectrum ... eliminating reliance on ad hoc approximations' should be qualified: the pipeline uses the analytical proxy, eq. (3.1), multiplied by the code-generated spline of G(Q), so the output is not independent of eq. (3.1).","section":"Secs. 4.2.4 and 7"},{"comment":"The interface field 'Relativistic degrees of fredom' contains a typo ('fredom' should be 'freedom').","section":"Sec. 4.2.1"}],"recommendation":"major_revision","confidential_remarks":"The manuscript is within JCAP scope and the software appears to be a useful contribution. I did not run the package during this review; my recommendation is driven by the absence of a reproducible cross-check of the Fokker-Planck method in the code's new regimes. The requested validation should be feasible without changing the code's design. The authors should also ensure that the GitHub repository makes the auxiliary Langevin verification, mentioned in footnote 13, available."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"WI2easy is a useful, well-documented public code for warm inflation phenomenology, and it deserves peer review. The genuinely new thing here is not the Fokker-Planck moment method itself (that is from Ballesteros et al. [40,41]) but the first general-purpose Mathematica implementation, plus a new numerical result: the near-universality of G(Q) across potentials holds only when the radiation noise term is dropped, and breaks down when it is included. That finding matters because people fit G(Q) from one potential and use it for others.\n\nWhat the paper does well: the equations are fully written out in the appendix, the code interface is clearly documented, and the code is public on GitHub. The validation against earlier fitting functions (Benetti-Ramos, WarmSPy) in fig. 5 is reassuring. The discussion of the controversial thermal term n* and the radiation noise option is honest. As a software contribution, this is solid.\n\nThe soft spots are real but mostly not fatal. The equivalence between the deterministic moment equation (3.31) and the original coupled stochastic system is assumed from [40,41]; the paper doesn't prove it. The multi-noise identity in footnote 13 is claimed to have been verified numerically, but no verification is shown. Since every spectral output flows through that step, the reviewers should ask for a head-to-head comparison of WI2easy's G(Q) with WarmSPy's actual output (not just the fitting formula from WarmSPy) for two or three models. That would settle the question directly.\n\nA second issue, more subtle: the final observables—ns, r, alpha_s, PR(k)—are not produced by integrating the Fokker-Planck system at the chosen Q. They are the analytic proxy of eq. (3.1) times a spline interpolated from the stored G(Q) data. The code uses the FP solver to generate the G(Q) data points, then goes back to the proxy for the observables. That is not circular, and it is consistent with how the field uses G(Q), but it is less direct than the introduction suggests. The abstract says the code solves 'curvature perturbations' and the text emphasizes the 'full system'; readers should be told the pipeline in section 4.2.4 is proxy+spline. If anything, this makes the code more conservative, not less correct.\n\nWe could not run the Mathematica package in review, so independent reproducibility rests on the GitHub release; the authors should include the auxiliary Langevin verification and perhaps an archived notebook output.\n\nWho this is for: anyone doing warm inflation phenomenology—constraining potentials with Planck/BICEP data, or studying scale-dependent spectra for PBH/GW work. It should go to peer review; it will be a standard tool. The requested changes are a direct WarmSPy comparison and clarity on the proxy+spline step.","headline":"Useful code paper whose FP-based observables actually go through the usual G(Q)-corrected proxy; worth refereeing, with a request for a direct cross-check against an existing stochastic solver.","tokens_in":25820,"tokens_out":4509,"would_cite":true,"duration_ms":45737,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A new code computes warm inflation's full perturbation spectra with a deterministic Fokker-Planck equation, replacing stochastic ensemble averaging.","keywords":["warm inflation","Fokker-Planck","curvature power spectrum","dissipation coefficient","G(Q)","CMB observables","Mathematica package"],"falsifier":"Run the original Langevin system (3.13)-(3.15) for a benchmark case such as the quartic potential with $\\Upsilon\\propto T$ at $Q=0.1$, average over many noise realizations, and compare the ensemble power spectrum with WI2easy's deterministic result; any difference beyond numerical tolerance would show the moment equation is not equivalent. Repeating the same comparison at $Q\\gtrsim 100$ tests the regime where the radiation noise term changes the spectrum's shape.","tokens_in":24854,"feed_emoji":"🌌","tokens_out":8448,"duration_ms":76005,"temperature":0.7,"pith_summary":"This paper presents WI2easy, a Mathematica package that computes the full background evolution and curvature perturbation spectra of warm inflation for arbitrary single-field potentials $V(\\phi)$ and dissipation coefficients $\\Upsilon(T,\\phi)$. Its central move is to replace stochastic Langevin simulations of the coupled inflaton-radiation perturbation system by a deterministic Fokker-Planck moment equation, so that the power spectrum $\\mathcal{P}_{\\mathcal{R}}(k)$ and the derived observables $n_s$, $r$, and $\\alpha_s$ are obtained from one coupled ordinary differential system. The paper shows the code reproduces known results from earlier stochastic codes and fitting functions, and it documents how the correction function $G(Q)$ behaves when thermal inflaton fluctuations and the radiation noise term are included or excluded. A sympathetic reader would care because this makes model-by-model confrontation of warm inflation with CMB data routine rather than a numerical tour de force.","feed_headline":"One matrix equation replaces stochastic runs for warm inflation","feed_subtitle":"A new Mathematica package derives power spectra, ns, and r deterministically for any single-field warm inflation model.","key_machinery":"The central object is a four-component perturbation vector $\\Phi=(\\delta\\phi,\\,d\\delta\\phi/dN_e,\\,\\delta\\rho_r,\\,\\Psi_r)^T$. The stochastic system is written as $d\\Phi/dN_e + A\\Phi = B\\xi$, and instead of averaging over $\\xi$ WI2easy evolves $J=\\langle\\Phi\\Phi^\\dagger\\rangle$ through $dJ/dN_e = -A\\cdot J - J\\cdot A^T + B\\cdot B^T$, with the matrices $A$, $B$, and $C$ read from the uniform-curvature-gauge perturbation equations. The function $G(Q)$ is defined as $\\mathcal{P}_{\\mathcal{R}}^{\\rm numerical}/\\mathcal{P}_{\\mathcal{R}}^{\\rm analytic}$ and is interpolated directly from generated data. The code's options to include thermal inflaton fluctuations (through the Bose-Einstein occupation $n_{BE}$) and the radiation noise term are implemented as switchable choices in the same mechanical framework.","core_discovery":"WI2easy claims that the full warm-inflation perturbation problem—the coupled stochastic equations for the inflaton fluctuation $\\delta\\phi$, its $N_e$-derivative, the radiation density fluctuation $\\delta\\rho_r$, and the radiation momentum perturbation $\\Psi_r$—can be replaced by one deterministic matrix differential equation for the two-point correlation matrix $J=\\langle\\Phi\\Phi^\\dagger\\rangle$, namely $dJ/dN_e = -A\\cdot J - J\\cdot A^T + B\\cdot B^T$. Solving this Fokker-Planck moment equation together with the background equations yields the scalar curvature power spectrum $\\mathcal{P}_{\\mathcal{R}}(k)$, the tilt $n_s$, the tensor-to-scalar ratio $r$, and the running $\\alpha_s$ for any single-field potential and dissipation coefficient. The code determines the potential normalization from the Planck amplitude at the pivot scale, and it produces the correction function $G(Q)$—the ratio of the numerical spectrum to the analytic proxy spectrum—by spline interpolation of raw data rather than through a fitting formula. Its numerical results match the earlier stochastic code and the most recent $G(Q)$ fits, and they show that the common assumption of a universal $G(Q)$ across potentials breaks down once the radiation noise term is included.","pith_inferences":["If the Fokker-Planck moment equation is exactly equivalent to the stochastic system, then previously published warm-inflation parameter constraints that used fitting-function forms of $G(Q)$ could shift in the strong-dissipation regime, where the older fits are known to deviate.","The same moment-equation machinery should carry over to multi-field warm inflation and to higher-order statistics, since the equation for second moments can be extended to higher correlators; the paper lists these as planned extensions, not achieved results.","A practical consequence of the deterministic formulation is that dense scans over microphysical coupling constants become cheap, making it feasible to map out the full observationally allowed region of a given warm-inflation model."],"forward_implications":["Observables like $n_s$, $r$, and $\\alpha_s$ can be computed directly from the background and perturbation equations for any defined potential and dissipation coefficient, with the potential normalization fixed by the Planck amplitude at the pivot scale.","The $G(Q)$ correction is generated internally from raw data, so users no longer need to rely on fitting functions calibrated for specific regimes or dissipation forms.","The code reproduces the earlier stochastic results in the weak- and strong-dissipation limits and matches the most recent $G(Q)$ fits, while also showing that earlier fits underestimate $G(Q)$ for $Q \\gtrsim 50$.","Because the computation is deterministic, systematic scans over $Q$ (and hence over the dissipation strength $C_\\Upsilon$) become fast enough to confront models with current and future CMB data.","Including the radiation noise term breaks the potential-universality of $G(Q)$, so precision comparisons require model-by-model evaluation rather than a universal correction."],"supporting_citations":[{"why":"the earlier public warm-inflation code whose numerical $G(Q)$ data serve as the validation baseline for WI2easy's spectra and fitting comparisons.","marker":"[39]"},{"why":"introduced the deterministic Fokker-Planck moment-equation method for warm-inflation perturbation spectra that WI2easy implements.","marker":"[40]"},{"why":"applied the Fokker-Planck approach to monomial warm inflation and supplied the multi-noise treatment of the thermal term adopted here.","marker":"[41]"},{"why":"provides the analytic proxy power spectrum whose ratio with the numerical spectrum defines the correction function $G(Q)$.","marker":"[30]"},{"why":"first showed that coupling between inflaton and radiation perturbations changes the warm-inflation spectrum with temperature-dependent dissipation.","marker":"[36]"},{"why":"derives the full coupled perturbation system and the $G(Q)$ fitting parameterization that WI2easy recovers and extends.","marker":"[37]"},{"why":"establishes the radiation noise term in the radiation perturbation equation and the choices WI2easy lets the user toggle.","marker":"[38]"},{"why":"supplies the BICEP/Keck/Planck constraints on $r$ and $n_s$ against which the code's observable outputs are displayed.","marker":"[64]"},{"why":"supplies the Planck normalization amplitude, pivot scale, and $H_0$ used to set $V_0$ and $N_*$.","marker":"[46]"}],"fun_headline_variants":["Warm inflation spectra without stochastic runs","Matrix equation solves warm inflation in one shot","WI2easy: deterministic warm inflation power spectra","Replace stochastic simulations with a matrix solve","Fast warm inflation: one matrix equation for all spectra"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the deterministic moment equation (3.31) is exactly equivalent to the ensemble statistics of the original stochastic perturbation equations, including the multi-noise identity used for the noise amplitude in eq. (A.5); the paper relies on earlier references and reports a numerical check without presenting it.","fun_headline_variants_meta":{"raw":{"variants":["Warm inflation spectra without stochastic runs","Matrix equation solves warm inflation in one shot","WI2easy: deterministic warm inflation power spectra","Replace stochastic simulations with a matrix solve","Fast warm inflation: one matrix equation for all spectra"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1369,"prompt_tokens":920,"completion_tokens":449,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":536,"completion_tokens_details":{"reasoning_tokens":382}},"tokens_in":536,"tokens_out":449,"duration_ms":4372,"temperature":1.0,"reasoning_tokens":382,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T10:31:21.105311+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Run the original Langevin system (3.13)-(3.15) for a benchmark case such as the quartic potential with $\\Upsilon\\propto T$ at $Q=0.1$, average over many noise realizations, and compare the ensemble power spectrum with WI2easy's deterministic result; any difference beyond numerical tolerance would show the moment equation is not equivalent. Repeating the same comparison at $Q\\gtrsim 100$ tests the regime where the radiation noise term changes the spectrum's shape.","supporting_citations":[],"review_version":1}