Talks

The Lagrangian dynamics of a fluid element within a self-gravitational matter field is intrinsically nonlocal due to the presence of the tidal force. Instead of searching for local approximations, we provide a statistical solution that could decouple the evolution of the fluid parcel from the surrounding environment. Given the probability distribution of the matter field, the method produces a set of ordinary differential equations to be solved locally. Mathematically, it corresponds to the characteristic curve of the transport equation of the density-weighted probability density function (ρPDF). Physically, it describes the mean evolution of the element with specific density and shape averaged over various environments. Furthermore, it is guaranteed that the one-point ρPDF would be preserved if one evolves these local, yet nonlinear, curves with the same set of initial data as the real system. This PDF based method, which is well developed in turbulence and other fields, provides a new perspective for understanding the non-linear structure formation in cosmology, e.g. the halo formation and the evolution of cosmic web. For Gaussian distributed dynamical variables, we demonstrate that the localized mean tidal tensor is proportional to the shear tensor, and the coefficient would recover the prediction of Zel’dovich approximation (ZA) with the further assumption of the linearized continuity equation. For Weakly non-Gaussian field, the averaged tidal tensor could then be expressed as polynomial of other variables.  Moreover, one could further generalize this concept of the mean evolution of the fluid element to incorporate some stochastic contributions, which we suggest would be valuable in describing a variety of processes in cosmology, such as the shell-crossing and realistic halo formation.

We consider the spectral action as an action functional for modified gravity on a spacetime that exhibits a fractal structure modeled on an Apollonian packing of 3-spheres (packed swiss cheese) or on a fractal arrangements of dodecahedral spaces. The contributions in the asymptotic expansion of the spectral action, that arise from the real poles of the zeta function, include the Einstein-Hilbert action with cosmological term and conformal and Gauss-Bonnet gravity terms. We show that these contributions are affected by the presence of fractality, which modifies the corresponding effective gravitational and cosmological constants, while an additional term appears in the action, which is entirely due to fractality. This term is further affected by a contribution of oscillatory terms coming from the poles of the zeta function that are off the real line, which are also a property specific to fractals. We show that the shape of the slow-roll potential obtained by scalar perturbations of the Dirac operators is also affected by the presence of fractality. The talk is based on joint work with Adam Ball.