⚛️ general relativity

Inflation and Reheating by Dynamical Torsion

This paper investigates inflation driven by dynamical torsion in gravity theories, demonstrating that anomaly-induced decays into gauge bosons and Higgs-involved three-body processes provide a non-vanishing inflaton decay width, thereby establishing a natural lower bound on the reheating temperature.

Original authors: Xiaolin Ma, Kamil Mudrunka, Kazunori Nakayama

Published 2026-08-03

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CC BY 4.0

Original authors: Xiaolin Ma, Kamil Mudrunka, Kazunori Nakayama

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine the universe as a giant, expanding balloon. A long time ago, this balloon didn't just grow; it inflated at a speed so fast it defied common sense, stretching from the size of a grain of sand to something larger than our entire observable universe in a tiny fraction of a second. This cosmic "inflation" is the leading story for how our universe began, but it leaves us with a big mystery: what actually drove the balloon to blow up, and how did it stop? The thing that did the blowing is called the "inflaton," a mysterious field that acted like a cosmic engine. Once the engine stopped, the universe was cold and empty, filled only with the energy of the inflaton itself. To get the universe we see today—full of stars, planets, and us—this energy had to be dumped into creating particles. This messy, energetic process of turning the inflaton's energy into matter is called "reheating." Scientists have been trying to figure out exactly how this engine works and how it dumps its energy for decades, because the details determine what the universe looks like today.

Now, enter a new idea about what that engine might be. Instead of being a simple, smooth scalar field (like a rolling ball), what if the engine was actually a twist in the very fabric of space-time itself? In standard physics, space-time is like a smooth sheet, but in some advanced theories, it can have a "twist" or "torsion," much like a screw has a spiral groove. This paper explores a universe where this twisting force, called dynamical torsion, is the inflaton. The authors, Xiaolin Ma, Kamil Mudrunka, and Kazunori Nakayama, wanted to solve a specific puzzle: if the inflaton is this twisting force, how does it dump its energy to create the universe?

Here is the tricky part: because this "twist" connects to matter in a very specific, mathematical way (it couples to the "axial current" of particles), the usual way it should dump energy doesn't work. Imagine trying to push a swing by pushing it exactly when it's at the top of its arc; you'd get nowhere. In this model, the inflaton tries to decay into pairs of particles (like electrons and their partners), but because of the twisty nature of the connection, this "two-body" decay vanishes if the particles are massless. Since most particles in the early hot universe were effectively massless, it looked like the engine might just stall, leaving the universe cold and dead.

However, the authors discovered that the engine doesn't stall; it just uses a different gear. They calculated that even though the direct "two-body" push fails, the inflaton can still dump its energy through two clever backdoors. First, it can decay into a trio of particles: a pair of fermions plus a Higgs boson. Think of this as the engine not pushing the swing directly, but instead throwing a heavy weight (the Higgs) at the swing, which then knocks the swing into motion. Second, and perhaps more surprisingly, the laws of quantum physics allow the inflaton to decay directly into pairs of force-carrying particles (gauge bosons) through a phenomenon called a "quantum anomaly." This is like the engine finding a hidden shortcut through a wall that shouldn't exist, thanks to the weird rules of the quantum world.

The paper finds that these two processes—the three-body decay and the anomaly-induced two-body decay—are strong enough to successfully reheat the universe. This is a big deal because it means we don't need to invent new, complicated rules to make the universe work; the minimal, twisty model works on its own. But there is a catch: because these decay paths are a bit more complicated than the standard ones, the universe heats up to a lower temperature than in other popular models. The authors calculate that the reheating temperature is likely around $10^5$ GeV (which is incredibly hot, but much cooler than the $5 \times 10^9$ GeV predicted by the famous Starobinsky model).

This difference in temperature is the paper's "smoking gun." It suggests that if we look closely at the cosmic microwave background (the afterglow of the Big Bang) or if we can detect primordial gravitational waves (ripples from the very beginning of time), we might be able to tell if our universe was driven by a twisting force or a standard scalar field. The authors predict that the "twist" model would leave a specific fingerprint on the gravitational waves, potentially detectable by future space-based detectors like DECIGO. So, while the paper doesn't prove that torsion is the inflaton, it shows that it is a viable, self-consistent engine that can build a universe, and it gives us a clear way to test if that's actually what happened.

Technical Summary: Inflation and Reheating by Dynamical Torsion

Problem Statement
While Starobinsky inflation remains a viable and observationally consistent model, there is ongoing interest in identifying the inflaton as a dynamical degree of freedom within the gravity sector itself, specifically in theories extending General Relativity to include torsion (Metric-Affine and Einstein-Cartan gravity). A critical challenge in these "torsion-driven" inflation models is the reheating mechanism. Unlike the scalaron in Starobinsky inflation, the torsion inflaton (often termed a "pseudo-scalaron") couples derivatively to the axial current of Standard Model (SM) fermions. In the limit of massless fermions—which is appropriate for the early universe—this derivative coupling suppresses the standard two-body decay into fermion pairs ( $\phi \to \psi\bar{\psi}$ ). Previous studies suggested that successful reheating in such models might require non-minimal couplings (e.g., direct coupling to the Higgs) or the existence of heavy right-handed neutrinos. This paper investigates whether reheating can occur successfully using only the minimal derivative couplings inherent to the torsion-matter interaction.

Methodology
The authors analyze the dynamics of inflation and reheating within Metric-Affine and Einstein-Cartan gravity frameworks, where the torsion vector field $\hat{T}_\mu$ becomes dynamical.

Inflationary Dynamics: They consider an action involving a function $f(\hat{R})$ of the Holst term $\hat{R}$ . By introducing an auxiliary field and solving the equations of motion for the non-dynamical components of the distortion tensor, they derive an effective action for a canonical scalar field $\phi$ . In the limit of large parameters ( $\beta \gg 1$ ), the resulting scalar potential $V(\phi)$ reproduces the Starobinsky plateau potential, ensuring consistency with Cosmic Microwave Background (CMB) observations regarding the scalar spectral index ( $n_s$ ) and tensor-to-scalar ratio ( $r$ ).
Reheating Mechanism: The core of the analysis focuses on the decay of the inflaton $\phi$ into SM particles. The authors utilize the derivative coupling Lagrangian $\mathcal{L} \sim \frac{1}{f_{eff}} (\partial_\mu \phi) J^\mu_5$ , where $J^\mu_5$ is the axial current.
Decay Channels: Recognizing that the tree-level two-body decay to massless fermions vanishes, the authors calculate decay rates arising from:
- Anomaly-induced two-body decays: Utilizing the chiral anomaly, they derive effective couplings between $\phi$ and gauge bosons ($gg, WW, BB$).
- Yukawa-induced three-body decays: They calculate the decay into a fermion pair and a Higgs boson ( $\phi \to f\bar{f}H$ ), dominated by the top-quark Yukawa coupling.
- Loop calculations: Detailed calculations in the appendices verify the equivalence between the derivative coupling basis and the non-derivative basis (via chiral rotation), confirming the non-vanishing nature of these channels. They also explicitly demonstrate that three-body decays into fermion pairs plus gauge bosons vanish in the massless fermion limit.

Key Contributions and Results

Non-Vanishing Decay Widths: The paper demonstrates that even with minimal couplings, the inflaton possesses a non-zero total decay width ( $\Gamma_{tot}$ ). The dominant channels are the anomaly-induced decay into gauge boson pairs and the top-Yukawa-induced three-body decay into a fermion pair plus a Higgs boson.
Decay Coefficients: The total decay width is expressed as $\Gamma_{tot} = A_{tot} \frac{m_\phi^3}{f_{eff}^2}$ $Γ_{t o t} = A_{t o t} \frac{m _{ϕ}^{3}}{f _{e f f}^{2}}$ . The authors calculate the coefficient $A_{tot} \approx 4.49 \times 10^{-4}$ $A_{t o t} \approx 4.49 \times 1 0^{- 4}$ , derived from the sum of contributions:
- Three-body decay ( $f\bar{f}H$ ): $A_{f\bar{f}H} \approx 2.52 \times 10^{-4}$ (dominated by top quark).
- Two-body anomaly decays: $A_{gg} \approx 1.31 \times 10^{-4}$ , $A_{WW} \approx 6.17 \times 10^{-5}$ , $A_{BB} \approx 4.81 \times 10^{-6}$ .
- The branching ratios are roughly comparable between the three-body and two-body channels.
Reheating Temperature ( $T_{RH}$ ): The derived decay width leads to a reheating temperature typically in the range of $T_{RH} \sim 10^5 - 10^7$ GeV, depending on the model parameters $\beta$ and $\gamma$ .
Distinction from Starobinsky Inflation: The authors highlight a crucial phenomenological difference. While the inflationary potential in the large $\beta$ limit mimics Starobinsky inflation, the reheating dynamics are distinct. Standard Starobinsky inflation predicts $T_{RH} \sim 5 \times 10^9$ GeV due to unsuppressed two-body Higgs decays. In contrast, the torsion-driven model predicts a significantly lower reheating temperature due to the derivative coupling suppression and the reliance on anomaly/Yukawa channels.

Significance and Claims
The paper claims that successful reheating in torsion-driven inflation does not require the introduction of non-minimal couplings or specific heavy neutrino masses; the minimal derivative coupling to the SM axial current is sufficient.

Observational Distinguishability: The lower reheating temperature ( $T_{RH} \lesssim 10^7$ GeV) in torsion models, compared to the $\sim 10^9$ GeV in Starobinsky models, leads to different predictions for the scalar spectral index ( $n_s$ ) and the tensor-to-scalar ratio ( $r$ ) when matched to CMB data. Furthermore, the lower $T_{RH}$ shifts the cutoff frequency of the primordial gravitational wave spectrum, potentially making these models distinguishable by future space-based interferometers like DECIGO.
Leptogenesis Implications: The authors note that the relatively low reheating temperature poses challenges for thermal leptogenesis unless right-handed neutrino masses are degenerate. However, they suggest that non-thermal leptogenesis via inflaton decay into right-handed neutrinos remains a viable possibility, provided the branching ratio is sufficient.
General Applicability: The calculations regarding derivative couplings and anomaly-induced decays are presented as generally applicable to other derivatively coupled scalar particles, such as axion-like dark matter candidates.

In conclusion, the paper establishes that dynamical torsion provides a viable inflationary scenario with a robust reheating mechanism driven by quantum anomalies and Yukawa interactions, offering a testable alternative to standard scalaron inflation through precision measurements of the reheating temperature and gravitational wave spectra.

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