Multifield quintessential inflation and Swampland constraints in the IDM

Technical Summary: Multifield Quintessential Inflation and Swampland Constraints in the IDM

Problem Statement
Modern cosmology faces the dual challenge of explaining the accelerated expansion of the Universe during both the primordial inflationary era and the current dark-energy-dominated epoch. While the Inert Doublet Model (IDM) offers a promising extension of the Standard Model capable of unifying inflation, dark matter, and dark energy, its embedding within a consistent quantum gravity framework remains an open question. Specifically, the "Swampland" program posits that low-energy effective field theories compatible with quantum gravity must satisfy certain constraints, such as the de Sitter (dS) conjecture. This conjecture suggests that scalar potentials in a consistent theory of quantum gravity cannot be arbitrarily flat, requiring either a steep gradient ($|\nabla V|/V \gtrsim \mathcal{O}(1)$) or a sufficiently tachyonic direction ($\min(\nabla_i \nabla_j V) \lesssim -\mathcal{O}(1)$).

Standard single-field inflationary models, particularly those with plateau-like potentials (e.g., Starobinsky-like), typically violate these strong dS bounds. This paper investigates whether a multifield realization of quintessential inflation within the IDM, coupled to a light quintessence field via an exponential interaction, can reconcile the observed inflationary phenomenology with Swampland constraints. The central question is whether the non-trivial field-space geometry and multifield dynamics of the IDM can modify the effective Swampland diagnostics or the inflationary observables in a way that resolves the tension with the conjecture.

Methodology
The authors formulate the IDM scalar sector coupled to a quintessence field $\phi$ through an exponential factor $e^{-\beta \phi/M_P}$. The analysis proceeds through the following steps:

1. Frame Transformation and Field-Space Geometry: The model is defined in the Jordan frame with non-minimal gravitational couplings ($\xi_1, \xi_2$) for the active and inert Higgs doublets. A Weyl transformation is performed to the Einstein frame. Unlike canonical single-field reductions, this transformation yields a genuinely non-canonical two-field system evolving on a curved field-space manifold. The authors derive the exact reduced field-space metric $G_{IJ}$ and the Einstein-frame potential $V_E$, retaining the full radial dependence without large-field approximations.
2. Covariant Dynamics: The background evolution and linear perturbations are treated using a fully covariant multifield formalism. The authors solve the exact background equations of motion in terms of the number of e-folds, $N$, without assuming slow-roll initially. They decompose perturbations into adiabatic (curvature, $\mathcal{R}$) and entropy (isocurvature, $S$) modes defined by the instantaneous tangent and normal vectors of the trajectory in field space.
3. Numerical Scan: A parameter scan is performed over the exponential slope $\beta$ (ranging from 0 to 0.020) and the number of e-folds $N_*$ (50, 55, 60), fixing the non-minimal coupling $\xi_2 = 4.9 \times 10^4$. The scalar amplitude $A_s$ is used to normalize the inflationary potential at each scan point, rather than fixing $\beta$.
4. Swampland Diagnostics: The authors compute the covariant dS diagnostics directly from the full multifield potential and metric. They evaluate the effective covariant slope $c_{\text{eff}} = M_P |\nabla V|/V$ and the smallest eigenvalue of the normalized covariant Hessian $\eta_{\min}$ along the inflationary trajectory.
5. Late-Time Consistency: After inflation, the radial direction stabilizes, leaving an effective exponential potential for the quintessence field. The authors analyze the late-time evolution using standard phase-space variables to check consistency with the observed dark energy density and $\Lambda$CDM expansion history.

Key Contributions and Results

• Multifield Phenomenology: The study demonstrates that the inflationary trajectory in the Einstein-frame IDM undergoes a slow but cumulative bending over tens of e-folds. Although the instantaneous turn rate $\eta_\perp$ is small, the integrated effect leads to a significant entropy-to-curvature transfer. This modifies the final scalar power spectrum, rendering the standard single-field relation $r = 16\epsilon_H$ inapplicable. The scalar spectral index $n_s$ shows strong sensitivity to the exponential coupling $\beta$, while the tensor-to-scalar ratio $r$ remains compatible with current CMB limits ($r \lesssim 0.036$).
• Normalization Independence: A crucial finding is that the observed scalar amplitude $A_s$ fixes the overall inflationary normalization (controlled by $\lambda_2/\xi_2^2$) independently of $\beta$. Consequently, $\beta$ cannot be constrained by a single-field normalization argument; it must be constrained through the full multifield evolution and the resulting spectral tilt.
• Swampland Tension: The covariant diagnostics reveal that the effective slope $c_{\text{eff}}$ during inflation is significantly smaller than unity (specifically $c_{\text{eff},*} \lesssim 0.13$ for the scan points compatible with $n_s$). Similarly, the smallest eigenvalue of the Hessian remains parametrically small ($\lambda_{\min} \gtrsim -0.06$). The authors find that the multifield effects, while altering the observable spectrum, do not generate a rapid-turn mechanism capable of supporting inflation on an order-unity potential gradient. Thus, the model does not resolve the tension with a strong interpretation of the dS Swampland conjecture.
• Late-Time Evolution: Once the radial direction stabilizes, the residual quintessence sector behaves as a shallow exponential potential. With appropriate normalization to the observed dark energy density, the late-time expansion history is extremely close to that of a cosmological constant, consistent with current observations.
• Stability: The reduced two-field trajectory is shown to be stable against orthogonal fluctuations of the active Higgs doublet, with mass ratios $m^2/H^2 \gg 1$ for the representative benchmark.

Significance and Claims
The paper claims that its primary contribution is a quantitative determination of how the tension with the dS Swampland conjecture is modified when the full curved-field-space and multifield dynamics of the IDM are taken into account. The authors explicitly state that they do not claim the minimal construction completely resolves the Swampland tension. Instead, they demonstrate that:

1. The entropy sector and trajectory bending are essential for reliable predictions of primordial observables, distinguishing this model from single-field approximations.
2. Despite these multifield effects, the strong order-unity dS criterion remains difficult to satisfy during the observationally relevant stage of inflation.
3. The distinction between the multifield phenomenology (which successfully fits CMB data) and the Swampland consistency (which remains in tension) constitutes a central result.

The work serves as a rigorous covariant analysis of a specific UV-motivated model, highlighting that while multifield dynamics can significantly alter observational predictions, they do not automatically circumvent fundamental quantum gravity constraints on the scalar potential's geometry. The authors note that a definitive assessment would require a dynamical description of reheating, the evolution of entropy perturbations post-inflation, and a full incorporation of renormalization group running and collider constraints, which are left for future work.