Minimal Warm Inflation Models

• MWI is a cosmological paradigm where the inflaton dissipates energy into a thermal bath with minimal additional fields, employing symmetry and supersymmetry to stabilize the potential.
• Its dynamics are characterized by a modified inflaton equation with temperature-dependent dissipation that sustains high temperatures (T > H) and fits CMB observations.
• Variants of MWI use axion-like couplings, Yukawa interactions, and minimal Standard Model extensions to produce distinctive signatures such as suppressed tensor-to-scalar ratios and non-Gaussian patterns.

Minimal Warm Inflation (MWI) is a class of inflationary cosmological models wherein the inflaton field dissipates energy into a coexisting thermal bath, and the particle content or couplings required for this dissipation are reduced to a minimal set. The key feature distinguishing MWI from general warm inflation constructions is its optimization of both the economy of new fields and the control of potentially destabilizing quantum and thermal corrections, often via symmetry arguments or supersymmetry. MWI has been realized in various frameworks, including axion-like couplings to non-Abelian gauge fields, models with pseudo-Nambu–Goldstone boson inflatons, and explicit constructions employing Standard Model interactions.

1. Fundamental Principles and Model Construction

The backbone of MWI is the modification of the inflaton’s equation of motion to include a substantial dissipation term: $$\ddot{\phi} + (3H + \Upsilon(T, \phi))\,\dot{\phi} + V'(\phi) = 0,$$ where $H$ is the Hubble parameter, %%%%1%%%% is the dissipation coefficient (often temperature-dependent), and $V(\phi)$ is the inflaton potential. The simultaneous evolution of the radiation bath obeys

$\dot{\rho}_R + 4H\rho_R = \Upsilon\,\dot{\phi}^2,$

requiring the temperature $T$ to remain above $H$ (i.e., $T > H$), ensuring that the bath is maintained during inflation.

Model minimality is achieved by restricting the couplings and additional field content, for example:

• Axion-like coupling: An inflaton field (often an axion or pseudo-Nambu–Goldstone boson) couples to non-Abelian gauge fields via the Chern-Simons term,

$$\mathcal{L}_{\text{int}} = \frac{\alpha}{16\pi} \frac{\phi}{f} G_{\mu\nu}^a \tilde{G}^{a\mu\nu},$$

where the shift symmetry suppresses large thermal corrections to $V(\phi)$ (Berghaus et al., 2019, Das et al., 2019, Berghaus et al., 24 Mar 2025, Ramos et al., 29 Apr 2025).

• Supersymmetry and symmetry protection: Supersymmetric extensions ensure radiative stability and cancel dangerous loop effects (Mishra et al., 2011).
• Derivative or oscillatory couplings: Dissipative interactions (e.g., inflaton–fermion Yukawa couplings with oscillatory field dependence) are arranged such that corrections to the inflaton mass or potential remain bounded or cancel on average (Bastero-Gil et al., 2018, Ferraz et al., 2023).

The dissipation coefficient, crucial for MWI, is typically sourced by sphaleron transitions in the gauge sector: $\Upsilon(T) \propto \alpha^5\,\frac{T^3}{f^2},$ where $\alpha$ is the gauge coupling and $f$ is the decay constant (Berghaus et al., 2019, Das et al., 2019).

2. The Role of Symmetry, Supersymmetry, and UV Completion

Minimal Warm Inflation typically utilizes symmetry arguments to maintain flatness of the inflaton potential and avoid large quantum or thermal corrections:

• Pseudo-Nambu–Goldstone Boson (PNGB) inflaton: The inflaton potential emerges as $V(\phi) = \Lambda^4 (1 - \cos(\phi/f))$, naturally flat due to the shift symmetry. Derivative interactions with other fields suppress radiative corrections (Mishra et al., 2011).
• Supersymmetrization: When heavy catalyst fields mediate dissipation, supersymmetry ensures loop corrections cancel between bosonic and fermionic partners, preserving the slow-roll conditions (Mishra et al., 2011).
• Axion shift symmetry: In axion-like models, the coupling structure (derivative and Chern-Simons couplings) further protects the inflaton from acquiring a large thermal mass (Berghaus et al., 2019, Das et al., 2019).

These properties allow the construction of models in which large dissipation is achieved—often $\Upsilon \gg H$—without destabilizing the necessary flatness of the potential.

Moreover, these mechanisms enable compatibility with quantum gravity conjectures. For example, strong dissipation allows sub-Planckian inflaton excursions and steep potentials (large $\epsilon_V, \eta_V \sim \mathcal{O}(1)$), which is consistent with the de Sitter swampland bound, Swampland Distance Conjecture, and the Trans-Planckian Censorship Conjecture by virtue of the modified slow-roll parameters,

$$\epsilon_w = \epsilon_V/(1+Q), \qquad \eta_w = \eta_V/(1+Q),$$

with $Q = \Upsilon/(3H)$ and $Q\gg 1$ (Das et al., 2019, Kamali et al., 2021).

3. Dissipative Microphysics and Thermal Bath Dynamics

Dissipation in MWI arises from explicit microphysical processes. Examples include:

• Sphaleron transitions: Non-perturbative gauge field configurations transition among vacua, providing a source of friction for an axion-coupled inflaton (Berghaus et al., 2019, Berghaus et al., 24 Mar 2025). The associated dissipation rate, scaling with $T^3$, ensures $T > H$.
• Yukawa-type couplings: The PNGB or axion-like inflaton couples to heavy or light fermions/fields, with derivative or oscillatory dependence constraining corrections (Bastero-Gil et al., 2018, Ferraz et al., 2023).
• Minimal Standard Model (SM) coupling: The inflaton couples only to SM gluons, with all other dissipative and backreaction effects—such as those from light fermion chiral chemical potentials—controlled by their cosmological dilution (Berghaus et al., 24 Mar 2025, Ramos et al., 29 Apr 2025).

The dynamics are governed by the coupled set of background equations, with the dissipation term entering both the inflaton’s evolution and the growth of the radiation bath. In detailed treatments, the temperature and field dependence of the dissipation coefficient are fully established from underlying quantum field theory, with corrections precisely estimated for parameter ranges of interest (Laine et al., 2021).

4. Phenomenological Predictions and Observational Tests

MWI models predict a range of distinctive cosmological signatures:

• Scalar perturbations and spectral index: The scalar power spectrum obtains thermal and dissipative enhancements, typified by

$$\Delta_\mathcal{R}^2 \propto \left(\frac{H^3 T}{\dot{\phi}^2}\right) (1+Q)^{1/2},$$

with both the amplitude and tilt (spectral index $n_s$) controlled by the dissipation strength $Q$. For strong dissipation, $n_s$ and the amplitude $A_s$ can be tuned to match CMB measurements, e.g., $A_s \simeq 2.38\times10^{-9}$, $n_s \simeq 0.96$ for GUT-scale parameters (Mishra et al., 2011, Das et al., 2019).

• Tensor-to-scalar ratio $r$: In warm inflation, tensor perturbations are not thermally enhanced, leading to strong suppression of $r$,

$$r \propto \frac{\epsilon_V}{Q^{3/2}} \left(\frac{H}{T}\right)[\text{model-dependent factors}],$$

which can approach $r\ll 10^{-3}$, far below the conventional cold inflation predictions (Berghaus et al., 2019).

• Non-Gaussianity: MWI predicts a distinctive "warm" non-Gaussian bispectrum shape with, for instance, a total amplitude $f_{\mathrm{NL}}^{\mathrm{warm}}\sim5$ that splits into equilateral and local components (Berghaus et al., 2019).
• Isocurvature and dark matter: In warm models with symmetries protecting the inflaton relic, the inflaton can serve as dark matter, leading to small, anti-correlated cold dark matter isocurvature perturbations at testable levels (Rosa et al., 2018, Levy et al., 2020).

Recent work has established that MWI scenarios, when analyzed with precision tools such as WI2easy, provide viable fits to the entire range of current CMB constraints for natural parameter choices (such as $f_a \sim 10^{10}$–$10^{13}$ GeV), and that effects previously thought problematic (e.g., fermion-induced suppression of QCD sphaleron dissipation) are alleviated by Hubble dilution (Ramos et al., 29 Apr 2025, Berghaus et al., 24 Mar 2025). Furthermore, generalized methodologies now allow direct, fully numerical mapping from MWI model predictions to the observable comoving scales required by CMB data analyses, sidestepping slow-roll restrictions (Kumar et al., 8 Jul 2024).

5. Model Variants and Theoretical Implications

MWI has been engineered in several variants, each tailored to further model-building aims:

• Warm natural inflation: Employs a PNGB inflaton and derivative couplings to achieve both flatness and large dissipation at f ∼ GUT scale (Mishra et al., 2011).
• Warm Little Inflaton (WLI): Utilizes symmetry-protected oscillatory fermion masses to suppress thermal corrections, with variants ranging from two-fermion interchange-symmetry constructions to more economical single-fermion setups (Bastero-Gil et al., 2018, Ferraz et al., 2023).
• Inflection point models: Warm dissipative dynamics relax the severe fine-tuning of the inflaton potential necessary for successful inflection-point inflation, broadening the parameter space consistent with observation (Cerezo et al., 2012).
• Standard Model-based MWI: Minimal extension via an axion-like inflaton coupled to QCD alone suffices for successful warm inflation; thermal and chemical equilibration issues with light fermions are controlled by cosmological expansion (Berghaus et al., 24 Mar 2025, Ramos et al., 29 Apr 2025).
• Chromo-natural warm inflation: The addition of non-Abelian gauge dynamics to MWI with Chern–Simons couplings results in mixed regimes with both enhanced gravitational wave prospects and strong dissipation (Mukuno et al., 13 Feb 2024).

Importantly, investigations have shown that MWI can be realized for various inflaton potentials, including quartic, hybrid, Higgs-like, and plateau-like types, and retains flexibility for embedding within broader theoretical frameworks such as supersymmetry, extensions of the Standard Model, and quantum gravity-motivated swampland constraints (Levy et al., 2020, Das et al., 2019).

6. Effects on Cosmological Parameter Extraction and CMB Anisotropies

Dissipation in MWI influences both the primordial power spectrum and secondary cosmological observables:

• Altered CMB power spectra: Strong dissipation ($Q=1$–8) can induce observable shifts in the phase and amplitude of the acoustic peaks in the CMB TT mode angular power spectrum, with alternating enhancement or suppression of multipole amplitudes and explicit phase shifts in the oscillatory structure. These shifts originate from the modified form of the primordial power spectrum and are described by modified slow-roll parameters,

$$n_s^{\text{WI}} = 1 - \frac{3}{8 QN} + \frac{2QG'(Q)}{7N G(Q)},$$

accompanied by suppressed tensor-to-scalar ratio,

$$r^{\text{WI}} = \frac{8}{N_c \sqrt{3\pi} Q^{1/4} G(Q)},$$

where details such as the growth factor $G(Q)$ depend nontrivially on the dissipation (B, 16 Oct 2025).

• Interpretation as dynamical dark energy: The presence of a thermal bath during inflation acts as an exotic, time-varying energy component, mimicking a dynamical dark energy (with $w_{\mathrm{DE}} \neq -1$), and can impact the inferred values of cosmological parameters, including the Hubble constant $H_0$, potentially alleviating the Hubble tension.
• Compatibility and constraints: The observed modifications in angular power spectra are not captured by standard $\Lambda$CDM extensions. This suggests that MWI could signal physics beyond $\Lambda$CDM, with implications for both early and late universe measurements (B, 16 Oct 2025).

7. Outlook and Future Prospects

The robust embedding of MWI within both particle physics and cosmological frameworks provides multiple avenues for theoretical development and observational confrontation:

• Parameter space exploration: Ongoing studies are refining the viable ranges of MWI parameters (e.g., dissipation ratios $Q$, decay constants $f$, potential parameters) using CMB and large-scale structure data, with increasing sophistication in the treatment of dissipation microphysics and thermalization (Ramos et al., 29 Apr 2025).
• Experimental connections: MWI models directly relate to axion and axion-like particle searches, as well as sub-eV relic searches, and thus can connect early universe physics with terrestrial experiments (Berghaus et al., 24 Mar 2025).
• Extensions and generalizations: Research continues on constant-roll generalizations, ultrastrong dissipative regimes, SM-only or nearly SM-only variants, and combinations with quantum gravity–motivated effective field theory constraints (Biswas et al., 1 Jun 2024, Das et al., 2019).
• CMB and cosmological tensions: The dynamical dark energy-like phase of MWI and its impact on the CMB hold promise for resolving tensions in $H_0$ and other cosmological measurements, providing a testable prediction for future observational programs (B, 16 Oct 2025).

Minimal Warm Inflation thus represents an economical, theoretically robust, and observationally viable paradigm for early-universe inflation. Its predictions—most notably the suppression of $r$, the modification of the CMB spectrum, and connections to particle physics—provide multiple pathways for experimental scrutiny and model discrimination in the coming years.