Hydrodynamic models of Reheating

TL;DR

The paper develops a causal relativistic hydrodynamic description of reheating after inflation by coupling a homogeneous inflaton condensate to a relativistic fluid that represents fluctuations, within a divergence-type theory derived from kinetic theory. It introduces two nonequilibrium tensors to encode dissipative and tensor-fluctuation dynamics and maps the tensor sector to the viscous part of the fluid energy-momentum tensor, enabling a tractable macroscopic treatment. A parametric resonance in the tensor sector amplifies viscous stresses, generating gravitational waves with a characteristic spectral peak that aligns with CosmoLattice simulations, while remaining computationally efficient. This framework provides a bridge between microscopic field dynamics and cosmological observables and can be extended to incorporate backreaction, gauge fields, and full thermalization, offering a versatile tool for exploring stochastic gravitational-wave production in the early Universe.

Abstract

We develop a causal hydrodynamic model that provides an effective macroscopic description of the field-theoretic dynamics during the early stages of reheating. The inflaton condensate is treated as a homogeneous background coupled to a relativistic fluid that represents its inhomogeneous fluctuations. Within the divergence-type theory framework derived from kinetic considerations, the model captures essential dissipative and non-equilibrium effects while remaining stable and causal. We find that the coupling between the oscillating condensate and the fluid induces a parametric resonance in the tensor sector, leading to the amplification of the viscous stress tensor and the generation of gravitational waves with a characteristic spectral peak. The predicted spectrum agrees with lattice simulations performed with CosmoLattice. This hydrodynamic approach offers an effective bridge between microscopic field dynamics and macroscopic cosmological observables.

Figures (5)

• Figure 1: Graph of the function $G(z)$, defined in \ref{['G(z) function']}, that appears in the background equation of the fluid \ref{['Fluid background equation simplified']}. It exhibits two simple asymptotic regimes: for $z\ll1$ we have $G(z)\approx z/2$ and, for $z\gg1$, $G(z)\approx 3/2$. In between these two regimes there is a fast transition.
• Figure 2: (Left) Instability regions and Floquet exponents of the Mathieu-like equation \ref{['Mathieu-like equation']} for $y_\kappa$. The resonance band is centered at $\kappa^2=7r$, with maximum amplification at $\delta=1/2$. White dashed lines mark the instability boundaries from \ref{['Analytical Floquet exponent']}, in excellent agreement with the numerical results. (Right) Numerical (dashed lines) and analytical (points) Floquet exponents $\tilde{\mu}_\kappa$ and $\mu_\kappa$ for $y_\kappa$ and $\zeta_\kappa$, respectively. The parameters used were: $\delta=0.11$, $z=20$, $q=1$, $\tilde{\tau}=85$.
• Figure 3: Evolution of the fluid energy fraction spectrum for several times, up to $\theta_{\text{nl}}\approx400$. We see that there is an exponential amplification of the fluid in the resonance band (highlighted in green in the background) and that modes outside it decay exponentially due to the usual dissipation in a theory with relativistic real fluids.
• Figure 4: Evolution of the gravitational-wave energy fraction spectrum at several times, up to $\theta_{\text{nl}}\approx400$. The main amplification occurs within the parametric resonance band (highlighted in green), while outside this region a rapid growth due to standard resonance takes place, which quickly saturates.
• Figure 5: Gravitational-wave energy fraction spectrum obtained with CosmoLattice for the model $V(\phi)=\lambda\phi^4/4$ with $\lambda=10^{-14}$ and initial field $\phi_0=2m_p$. The position of the resonance band (highlighted in green) coincides with that found in the hydrodynamical model, and the peak of the spectrum appears at the same location, although CosmoLattice yields a broader distribution.