inflation

The non-linear nature of the Einstein’s equations of general relativity suggests that space-time can be turbulent. Such a turbulence is expected during the primordial universe (first second) when gravitational waves (GW) have been excited through eg. the merger of primordial black holes. The analytical theory of weak GW turbulence, published in 2017 [1], is built from a diagonal space-time metric reduced to the variables t, x and y [2]. The theory predicts the existence of a dual cascade driven by 4–wave interactions with a direct cascade of energy and an inverse cascade of wave action. In the latter case, the isotropic Kolmogorov-Zakharov spectrum N(k) has the power law index -2/3 involving an explosive phenomenon. In this context, we developed a fourth-order and a second-order nonlinear diffusion models in spectral space to describe GW turbulence in the approximation of strongly local interactions [3]. We showed analytically that the model equations satisfy the conservation of energy and wave action, and reproduce the power law solutions previously derived from the kinetic equations. We show numerically by computing the second-order diffusion model that, in the non-stationary regime, the isotropic wave action spectrum N(k) presents an anomalous scaling which is understood as a self-similar solution of the second kind. The regime of weak GW turbulence is actually limited to a narrow wavenumber window and turbulence is expected to become strong at larger scales. Then the phenomenology of critical balance can be used. The formation of a condensate may happen and its rapid growth can be interpreted as an accelerated expansion of the universe that could be at the origin of the cosmic inflation. We can show with this scenario that the fossil spectrum obtained after inflation is compatible with the latest data obtained with the Planck/ESA satellite [4].

[1] Galtier & Nazarenko, Turbulence of weak gravitational waves in the early universe, Phys. Rev. Lett. 119, 221101 (2017).

[2] Hadad & Zakharov, Transparency of strong gravitational waves, J. Geom. Phys. 80, 37 (2014).

[3] Galtier, Nazarenko, Buchlin & Thalabard, Nonlinear diffusion models for gravitational wave turbulence Physica D 390, 84 (2019).

[4] Galtier, Nazarenko & Laurie, Cosmic inflation driven by space-time turbulence (2019).