Geometric Reheating in Early Universe

• Geometric reheating is a mechanism in early-universe cosmology where energy from the inflaton field transfers to matter solely through gravitational couplings to spacetime curvature.
• The process exploits non-minimal couplings (e.g., ξRχ²) that trigger tachyonic instabilities or non-adiabatic amplification, with efficiency sensitive to the inflaton potential and coupling strength.
• Advanced simulations and analytical models indicate that this mechanism can achieve reheating temperatures up to 10¹⁵ GeV and leave distinctive signatures such as a blue-tilted gravitational wave spectrum.

Geometric reheating refers to a class of mechanisms in early-universe cosmology where post-inflationary energy transfer from the inflaton sector to other particle species occurs exclusively, or dominantly, via gravitational effects—particularly through couplings to spacetime curvature—rather than through direct, model-dependent couplings. Geometric reheating processes leverage the time-dependent dynamics of background geometry or gravitational couplings (e.g., $\xi R|\chi|^2$ or curvature-induced mass terms) to convert the vacuum energy of inflation into a thermal bath, thus bridging inflation with standard hot big bang cosmology even in highly minimalistic, sequestered, or non-interacting models.

1. Essential Mechanisms of Geometric Reheating

Core geometric reheating scenarios exploit the rapid post-inflationary evolution of the Ricci scalar $R$ and the non-minimal curvature couplings of scalar fields. The archetypal Lagrangian includes terms such as

$\mathcal{L} \supset -\frac{1}{2}\, \xi R \chi^2,$

where $\chi$ is a spectator field and $\xi$ is a dimensionless coupling. After inflation, $R$ typically acquires a highly nontrivial, oscillatory or sign-changing time profile due to the inflaton's coherent oscillations or a rapid equation-of-state transition. If $\xi \gtrsim O(10)$, the effective mass squared $m_{\rm eff}^2 = \xi R$ can briefly become negative and induce explosive (tachyonic) particle production, or at minimum, allow for intense non-adiabatic mode amplification. This process may be purely gravitational or may involve specific structures such as geometric tachyons in D-brane constructions (Bhattacharjee et al., 2013).

Variants include:

• Tachyonic Ricci reheating: The curvature-induced instability grows $\chi$ fluctuations exponentially when $R<0$ and $\xi>0$ (Laverda et al., 2023, Figueroa et al., 2024).
• Curvature-driven particle production: Even when $m_{\rm eff}^2$ is not tachyonic, the oscillatory $R(t)$ acts as a gravitational "pump," periodically modulating the effective masses of all non-minimally coupled fields and creating particles via non-adiabatic transitions (Shah et al., 25 Dec 2025).
• Geometric tachyon warm inflation: In D-brane setups, the geometrical structure of the compact space generates an inflaton potential; energy dissipation into light degrees of freedom is driven by brane kinematics, with reheating occurring in a smooth, continuous, geometric manner (Bhattacharjee et al., 2013).

These scenarios permit efficient or partial reheating even with infinitesimal or vanishing direct inflaton–matter couplings, in models compatible with both minimal and extended field content.

2. Field-Theoretic Realizations and Dynamical Equations

Geometric reheating is realized in models where the action contains the following ingredients:

• Inflaton potential $V_{\rm inf}(\phi)$, often monomial or of $\alpha$-attractor type;
• Spectator or daughter field $\chi$ with non-minimal coupling $\xi R \chi^2$, and possibly self-interaction $V_{\text{NMC}}(\chi) = \lambda \chi^4/4$;
• Einstein–Hilbert gravitational sector.

The coupled system in a spatially flat FLRW background is described by \begin{align*} S &= \int d^4x \sqrt{-g} \Bigg[ \frac{1}{2} m_p^2 R - \frac{1}{2} (\partial \phi)^2 - V_{\rm inf}(\phi) \ &\qquad\qquad - \frac{1}{2} (\partial \chi)^2 - \frac{1}{2}\xi R \chi^2 - V_{\text{NMC}}(\chi) \Bigg] \end{align*} with field equations (schematically, in cosmic/conformal time) (Figueroa et al., 2024, Laverda et al., 2023):

• For $\chi$:

$$\chi'' + 3H \chi' - \frac{\nabla^2}{a^2}\chi + \xi R \chi + \partial_\chi V_{\text{NMC}} = 0,$$

and for each Fourier mode in conformal time,

$$\tilde{\chi}_k'' + \left(k^2 + a^2(\xi - 1/6)R \right) \tilde{\chi}_k = 0.$$

• For the inflaton:

$$\phi'' + 3H \phi' - \frac{\nabla^2}{a^2}\phi + V'_{\rm inf}(\phi) = 0.$$

Backreaction from $\chi$ can eventually shut off the tachyonic growth when its energy density becomes significant.

A geometric reheating event is defined operationally as the epoch at which the radiation-like field $\chi$ comes to dominate the energy budget, i.e., $\rho_\chi = \rho_\phi$ or when the universe enters the radiation-dominated regime.

3. Parameter Dependence and Efficiency Conditions

The efficiency and qualitative behavior of geometric reheating are controlled by:

• The inflaton potential steepness $p$ if $V_{\rm inf} \sim |\phi|^p$;
• The strength of non-minimal coupling $\xi$;
• The inflationary scale $\Lambda$ (or $H_{\rm end}$), and the initial amplitude of the inflaton;
• Self-interaction strength $\lambda$ of $\chi$.

The key analytic results (Figueroa et al., 2024, Laverda et al., 2023) can be summarized as:

• Quadratic inflaton ($p=2$): $\langle R \rangle > 0$ always, so no sustained tachyonic phase; geometric reheating fails.
• Quartic inflaton ($p=4$): $\langle R \rangle \sim 0$ (short negative dips only); partial reheating is possible only at large $\xi$ and large inflaton amplitude, but $\chi$ never fully dominates.
• Sextic or steeper ($p=6$): $\langle R \rangle < 0$ robustly, enabling exponential growth of $\chi$ and full energy transfer within 1–4 e-folds if $\xi \gtrsim 50$ and inflaton amplitude $M \gtrsim 0.1m_p$.
• Self-interaction $\lambda$: If $\lambda \gtrsim 10^{-6}$, rescattering screens the tachyon too rapidly and $\chi$ remains subdominant.
• Curvature-pump regimes ($\xi_\chi$ arbitrary): For minimal coupling, particle production is much weaker, but sufficient $\xi_\chi$ strongly amplifies both number and energy density of produced particles (Shah et al., 25 Dec 2025).

The process ceases to be efficient if the inflaton fragments before $\chi$ backreaction.

The following table summarizes the regime of efficiency for geometric reheating with monomial potentials (Figueroa et al., 2024):

Potential index $p$	$\langle R \rangle$ tachyonic?	Efficient reheating?
2	No	No
4	Marginal/No	Partial (large $\xi$ only)
6	Yes	Yes (for $\xi \gtrsim 50$, $M\gtrsim 0.1\,m_p$)

4. Lattice Simulations and Nonlinear Dynamics

Fully capturing geometric reheating requires treatment of the non-linear $\phi$–$\chi$–gravity system. State-of-the-art simulations utilize codes such as CosmoLattice, evolving field inhomogeneities, backreaction, and rescattering processes in 3+1D. Key findings from these simulations (Laverda et al., 2023, Figueroa et al., 2024) include:

• The onset of tachyonic amplification can be analytically estimated but its shutdown (by self-interaction or gravitational fragmentation) is non-linear and highly sensitive to model parameters.
• Reheating efficiency ($\Theta_{ht}$) and temperature ($T_{ht}$) can be fitted by accurate parametric formulas, depending on $\xi$, $\lambda$, $H_{\rm kin}$.
• Heating efficiencies span $\Theta_{ht} \sim 10^{-16}–10^{-2}$; reheating temperatures $T_{ht} \sim 10^5 – 10^{15}\,\text{GeV}$, depending on the scenario.

These results delineate the precise boundaries of parameter space for success or failure of geometric reheating, including the role of cosmic expansion in shutting off non-adiabaticity and the timings for radiation domination.

5. Phenomenological Implications and Observational Signatures

Geometric reheating provides a minimal, model-independent channel bridging inflation with standard hot big bang physics, crucial for scenarios where direct inflaton–matter couplings are forbidden, vanishing, or Planck-suppressed (Shah et al., 25 Dec 2025). Notably, for successful geometric reheating:

• The universe rapidly reheats to $T_{RH} \gtrsim 10^{12-15}\,\text{GeV}$, comfortably above BBN thresholds.
• The gravitational wave background inherits a unique blue tilt for modes re-entering during the stiff equation-of-state phase, with the tilt $n_{\rm GW} = (6 w - 2)/(1 + 3w)$; these predictions are potentially within reach of CMB-S4 or GW detectors (Haque et al., 2023).
• In type-I seesaw extensions, geometric reheating via gravitational neutrino production connects lepton-number violation, the baryon asymmetry, and a non-vanishing lightest active neutrino mass to early-universe dynamics (Haque et al., 2023).
• The parameter $\Theta_{ht}$ alters the number of inflationary e-folds and thus shifts the spectral index and tensor-to-scalar ratio; this effect is critical for precision CMB forecasting (Laverda et al., 2023).

Geometric reheating mechanisms can seamlessly coexist with slow-roll or warm inflation scenarios (e.g., D3–NS5 systems, where radiation is generated throughout inflation via geometric tachyon dissipation) (Bhattacharjee et al., 2013). In such constructions, the transition to thermaldomination occurs gradually, with observational predictions $n_s \approx 0.98$, $r \approx 10^{-3}$–$10^{-2}$, and non-Gaussianity within observational bounds.

6. Variations: D-brane, Neutrino, and Higgs Sector Scenarios

• Geometric tachyon and brane-driven reheating: In D3–NS5 ring geometries, geometric tachyons drive inflation with a cosine potential. Dissipative friction maintains a radiation bath during inflation, leading to a smooth handover into radiation dominance without a reheating “jump,” and drastically reduces the required number of background branes for observational consistency (Bhattacharjee et al., 2013).
• Purity of gravitational channels: In models where the inflaton only couples gravitationally to fields such as right-handed neutrinos, reheating proceeds through the expansion-driven gravitational production of heavy Majorana neutrinos, followed by their decay into the standard model bath. The final reheat temperature, non-vanishing lowest neutrino mass, and baryon asymmetry are set by the geometric reheating dynamics (Haque et al., 2023).
• Higgs–inflaton models: In Higgs inflation and its extensions, Bogoliubov calculations confirm that the oscillatory $R$ after inflation is sufficient to reheat Standard Model degrees of freedom with no direct couplings, yielding $T_{\rm RH} \sim 10^{8-9}\,\text{GeV}$ for reasonable $H_{\rm end}$ and $\xi$ (Shah et al., 25 Dec 2025).

7. Constraints, Contingencies, and Model Limitations

Several critical limitations and subtleties delimit the effectiveness of geometric reheating:

• Success is not guaranteed for all inflaton potentials; $p>4$ monomials, high enough $H_{\rm end}$, and large $\xi$ are typically required (Figueroa et al., 2024).
• Robust reheating is blocked if self-interactions in $\chi$ are not extremely small, or if the inflaton condensate fragments into inhomogeneities before $\chi$ absorbs enough energy.
• For moderate $\xi$, the efficiency may be insufficient to yield prompt radiation domination; other mechanisms or delayed transitions may be necessary (Laverda et al., 2023).
• Models are constrained by BBN, overproduction of gravitational waves, and requirements on initial field fluctuations (Laverda et al., 2023, Figueroa et al., 2024).

This suggests that geometric reheating is best implemented in models designed for large non-minimal couplings, sufficiently stiff post-inflationary EoS, and negligible spectator self-interaction, or as a subdominant reheating channel in more complex inflationary frameworks.

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References: (Bhattacharjee et al., 2013, Haque et al., 2023, Shah et al., 25 Dec 2025, Laverda et al., 2023, Figueroa et al., 2024)