Comprehensive analysis of dissipative effects in the induced gravitational waves

Dissipation is an intrinsic property of the cosmic fluid, leading to the damping of curvature perturbations at small scales. In this paper, we comprehensively study dissipative effects in gravitational waves induced by curvature perturbations, known as induced gravitational waves (IGWs). We find dissipative effects become especially significant at wavenumber $k \sim k_{\mathcal{H},\mathrm{dec}}$, where $k_{\mathcal{H},\mathrm{dec}}$ corresponds to the horizon scale at the decoupling of weakly-interacting particles. They can leave characteristic features on the IGW spectrum, including a notable suppression with a ``double-valley’’ structure at $k \sim k_{\mathcal{H},\mathrm{dec}}$ and a modified infrared behavior without logarithmic running at $k \lesssim k_{\mathcal{H},\mathrm{dec}}$. Within the Standard Model of particle physics, dissipative effects caused by neutrinos at the nanohertz frequencies can be important in the analysis of pulsar timing array data. Furthermore, dissipation-induced features associated with possible new weakly-interacting particles can be detectable by a wide range of gravitational-wave experiments, serving as a promising probe of new physics at extremely high energy scales. As an extension, we also discuss dissipative effects in the presence of primordial non-Gaussianity and their impacts on the anisotropies of IGWs and the poltergeist mechanism. These dissipative effects not only provide a more realistic description of IGWs but also exhibit rich phenomenology and profound physical implications, opening a new window into understanding the early Universe and fundamental physics.

💡 Research Summary

The paper presents a comprehensive study of dissipative effects in induced gravitational waves (IGWs), emphasizing that the cosmic fluid in the early radiation‑dominated era is not a perfect fluid but possesses shear viscosity, bulk viscosity, and heat conduction. By incorporating the dissipative term ΔT^{μν}(η, ξ, χ) into the energy‑momentum tensor, the authors derive a modified evolution equation for the scalar perturbation ϕ. The key result is a transfer function Φ(k, τ) that includes an exponential damping factor exp(−k²/k_D²), where the damping scale k_D is set by the shear viscosity η, itself determined by the mean free time t_i and interaction cross‑sections ⟨σ_i v⟩ of the constituent particles (photons, neutrinos, and possible beyond‑Standard‑Model (BSM) weakly‑interacting particles X).

The paper analytically relates k_D to the temperature T via an integral over the cosmic history (Eq. 2.9) and shows that k_D is generally much larger than the Hubble scale k_H, but approaches k_H near the decoupling of the most weakly interacting species. This crossing point, denoted k_{H,dec}, is where dissipative effects become most pronounced. The authors compute k_D(T) for the Standard Model (SM) where neutrino scattering dominates, finding k_D≈10⁵ Mpc⁻¹ at the neutrino decoupling temperature T≈1.5 MeV. For hypothetical X particles with weaker couplings, k_D can be shifted to much higher temperatures, corresponding to frequencies well above the nanohertz band.

Using three representative primordial curvature power spectra—scale‑invariant, monochromatic, and log‑normal—the authors numerically evaluate the resulting GW energy‑density spectrum Ω_gw(k). The main phenomenological features are:

1. Double‑valley suppression: Around k≈k_{H,dec} the spectrum exhibits two adjacent dips, a “double‑valley” structure, caused by the rapid transition of η as particles decouple.
2. Modified infrared behavior: For k≲k_{H,dec} the usual logarithmic running (Ω_gw∝k³ ln k) disappears; the spectrum follows a pure k³ scaling, reflecting the loss of resonant enhancement when the source perturbations are damped.
3. Frequency‑dependent damping: The strength and width of the suppression depend on the magnitude of η (or equivalently on the interaction strength and particle number density). Stronger dissipation leads to broader and deeper valleys, potentially affecting frequencies from nanohertz (PTA) up to millihertz (LISA) and beyond.

The authors discuss observational implications. In the SM, neutrino‑induced dissipation reduces Ω_gw by ∼10 % in the nanohertz band, a correction that must be accounted for when interpreting the stochastic GW background reported by PTA collaborations (NANOGrav, EPTA, PPTA, CPTA). For BSM scenarios, the presence of X particles decoupling at higher energies would imprint analogous features at higher GW frequencies, making them observable by space‑based interferometers (LISA, DECIGO, TianQin) and future ground‑based detectors (Einstein Telescope, Cosmic Explorer). Detecting the double‑valley pattern would thus provide a novel probe of weakly interacting particles at energy scales far beyond the reach of colliders.

The paper also extends the analysis to non‑Gaussian initial conditions. Primordial non‑Gaussianity (parameterized by f_NL) modifies the second‑order source term for IGWs; when combined with dissipation, the resulting anisotropic GW background acquires distinctive angular patterns. Moreover, the authors introduce the “poltergeist mechanism,” wherein a sudden increase in viscosity (e.g., due to a rapid phase transition) briefly amplifies the source before an abrupt decay, leaving a transient spike‑and‑drop signature in the time‑frequency domain of the GW background.

In summary, the work demonstrates that incorporating realistic dissipative physics is essential for accurate predictions of IGWs. It reveals rich phenomenology—double‑valley suppression, altered infrared scaling, and anisotropic signatures—that can be leveraged to extract information about particle interactions in the early universe, test models of primordial non‑Gaussianity, and potentially discover new weakly interacting particles through upcoming gravitational‑wave observations.