Einstein–Cartan pseudoscalaron inflation, reheating and nonthermal leptogenesis

A new theoretical study argues that a peculiar form of gravity—one in which spacetime torsion becomes a dynamical player—can drive cosmic inflation, reheat the Universe, and simultaneously manufacture the matter–antimatter imbalance that made our existence possible, all within a single self-consistent framework. The work, published in The European Physical Journal C, connects an inflationary scenario built on Einstein–Cartan–Holst gravity to a type I seesaw extension of the Standard Model and shows that nonthermal leptogenesis is not merely an option in this setup but a necessity.

The starting point is Einstein–Cartan gravity, an extension of general relativity in which the connection that defines parallel transport is treated as an independent field, à la Palatini, and can carry torsion. In ordinary general relativity, a certain pseudoscalar quantity built from the curvature—the Holst invariant—always vanishes, because the connection reduces to its torsion-free Levi-Civita part. In the class of models studied by Carlo Di Benedetto, Alessandro Di Marco, Emanuele Orazi and Gianfranco Pradisi, however, the torsionful contortion tensor is dynamical, and a nonlinear term in the Holst curvature gives rise to a genuine pseudoscalar degree of freedom. This field, dubbed the pseudoscalaron, is not inserted by hand: it emerges directly from the geometry of spacetime and plays the role of the inflaton.

After a field redefinition, the theory takes the form of a standard single-field slow-roll model with a canonical kinetic term and a potential shaped by two parameters: the exponent p of the nonlinear Holst term and the Barbero–Immirzi parameter γ, a constant that multiplies the Holst invariant. The sign of γ fixes the direction in which the inflaton rolls, while its magnitude controls the height of the inflationary plateau—smaller |γ| means a higher inflation scale and a shorter plateau. For the case p = 2 that the authors focus on, the potential features a flat slow-roll region, a vacuum at zero field value, and an exponential tail at large field amplitudes.

Comparing the model’s predictions for the scalar spectral index n_s and the tensor-to-scalar ratio r against the latest data from Planck, the Atacama Cosmology Telescope, the South Pole Telescope, BICEP/Keck and DESI baryon acoustic oscillations, the team finds broad compatibility. For Barbero–Immirzi parameters in the range roughly γ = −1/80 to −1/150 and 50 to 60 e-folds of inflation, the predicted (n_s, r) pairs sit comfortably within the 68 to 95 percent confidence regions of the observational datasets. For the benchmark γ ≈ −1/100, the model yields n_s ≈ 0.970 and r ≈ 0.004, with the tensor-to-scalar ratio safely below current upper limits. The authors also note a caveat: a mild 2–3 sigma tension exists between CMB and DESI BAO constraints within standard Lambda cold dark matter, so joint constraints should be interpreted with care.

The truly distinctive physics emerges after inflation ends. Because matter fermions are minimally coupled to the full torsionful spin connection, integrating out the contortion produces a universal derivative coupling between the pseudoscalaron and every fermion in the theory, of the form of the pseudoscalaron gradient times the axial fermion bilinear, suppressed by the reduced Planck mass. Crucially, the resulting decay rate of the inflaton into a fermion pair is proportional to the square of the fermion’s mass. Standard Model fermions—effectively massless before electroweak symmetry breaking, and topped by the top quark at roughly 171 GeV afterward—therefore absorb almost none of the inflaton’s energy. Heavy right-handed Majorana neutrinos, by contrast, are ideal decay products.

This mass-squared scaling has a dramatic consequence. In the benchmark scenario, the inflaton mass is of order 10^14 GeV, so only the lightest of the three right-handed neutrinos, N1, with a mass below roughly 5 × 10^13 GeV, is kinematically accessible. The heavier states N2 and N3 play no cosmological role. The inflaton decays almost exclusively into pairs of N1, which are produced nonrelativistically and out of equilibrium. These heavy neutrinos then decay rapidly into Standard Model leptons and Higgs bosons, generating both the hot radiation plasma that ends reheating and a net lepton asymmetry, controlled by the CP-violating parameter ε1, which later electroweak sphaleron processes convert into the baryon asymmetry of the Universe.

The hierarchy of decay rates is striking. With M_N1 ∼ 10^13 GeV and an effective light neutrino mass of about 10^−3 eV, the N1 decay width is around 7 × 10^7 GeV, while the inflaton’s decay width into N1 pairs is only about 7 × 10^−3 GeV—a ratio of roughly 10^−10. The heavy neutrinos therefore decay essentially instantaneously on the timescale set by the inflaton, meaning the inflaton’s lifetime alone dictates the duration of reheating and the reheating temperature, which comes out below about 3 × 10^8 GeV in the favored parameter region. The authors solve the coupled Einstein–Boltzmann equations for the inflaton, heavy-neutrino and radiation energy densities and the lepton asymmetry both numerically and analytically, finding that the approximate solutions reliably capture the late-time evolution toward radiation domination.

The final numbers are encouraging. For γ ≈ −1/100, a lightest right-handed neutrino mass above about 10^13 GeV, and a CP asymmetry parameter of order 3 × 10^−5 in magnitude—well within the Davidson–Ibarra upper bound of roughly 3.4 × 10^−2 set by neutrino oscillation data—the mechanism produces a baryon-to-entropy ratio n_B/s ≈ 8.7 × 10^−11, matching the observed value. Because the mechanism is nonthermal, there is no direct link between the reheating temperature and the heavy-neutrino mass, unlike in vanilla thermal leptogenesis where the lightest right-handed neutrino must weigh at least 10^9 to 10^10 GeV and force a very high reheating temperature. That decoupling helps avoid the gravitino overproduction problem that plagues supersymmetric extensions, and it opens a path toward embedding the scenario in supergravity and its ultraviolet completions such as string theory or M-theory.

The study remains theoretical, and the authors are candid about the limits of predictability. Seesaw models contain 21 free parameters beyond the Standard Model ones, and current neutrino data cannot fix the Yukawa couplings, the Majorana nature of active neutrinos, or the mass hierarchy of the heavy states. What the paper establishes is a consistency proof: over a wide window of the Barbero–Immirzi parameter, the same geometric inflaton that satisfies CMB constraints necessarily reheats the Universe through heavy-neutrino production, and that channel automatically yields the observed matter asymmetry. The team points toward future work that would add dark matter and dark energy to the picture, and toward the next generation of experiments that may finally constrain the seesaw parameter space. For now, the result stands as an unusually tight weave of spacetime geometry, inflation and the origin of matter—three puzzles usually treated separately, here resolved by a single torsion-driven field.

Subject of Research: Einstein–Cartan pseudoscalaron inflation, reheating and nonthermal leptogenesis in a type I seesaw extension of the Standard Model

Article Title: Einstein–Cartan pseudoscalaron inflation, reheating and nonthermal leptogenesis

Article References: Di Benedetto, C., Di Marco, A., Orazi, E., & Pradisi, G. (2026). Einstein–Cartan pseudoscalaron inflation, reheating and nonthermal leptogenesis. The European Physical Journal C, 86(10), Article 1206. https://doi.org/10.1140/epjc/s10052-026-16310-x

Image Credits: AI Generated

DOI: 10.1140/epjc/s10052-026-16310-x

Keywords: cosmic inflation, Einstein–Cartan gravity, torsion, Barbero–Immirzi parameter, leptogenesis, right-handed neutrinos, seesaw mechanism, reheating, baryon asymmetry, cosmic microwave background, neutrino physics, theoretical physics