Periodic Reporting for period 1 - Cosmo-Blow-Up (Deflating the blow-up: controlling infinities in cosmic fluid descriptions)
Reporting period: 2018-12-01 to 2020-11-30
The main objective of this action was to mathematically analyse shell-crossings in our Universe, which is the instance when dark matter particle trajectories cross for the first time. This instance is accompanied with formally infinite matter densities and is thus highly relevant for the process of cosmic structure formation; resolving these infinities in a tractable manner constitutes a major challenge in the field of mathematical cosmology.
The scientific tasks were essentially three-fold, namely
(1) to provide the theoretical framework that allows to solve the underlying cosmic fluid equations to arbitrary high precision,
(2) to numerically implement these theoretical findings, thereby establishing a simulation of the dark matter fluid for random initial conditions, from which shell-crossings are detected, and
(3) to further exploit the theoretical findings by establishing exact analytical shell-crossing solutions for simplified initial conditions.
All those three tasks were successfully executed and lead to the anticipated deliverables and milestones, which are scientific papers. Specifically, building up on previous works of myself, we were able to complete the underlying theory. Thanks to the expertise provided at the Lagrange Institute, we implemented these findings in a computationally highly efficient way (full support of parallel computing), from which we were able to detect shell-crossings to extremely high accuracy.
This numerical implementation is publicly available and can be directly exploited by the wider cosmological audience, e.g. for generating highly accurate initial conditions of heavily used cosmological N-body simulations.
I discovered also new exact analytical solutions for nonlinear matter collapse models, and we were able to exploit the matter collapse in the presence of massive neutrinos. As a side product, we performed highly efficient two-fluid simulations that include neutrino clustering.
The deliverables and milestones have been published in peer reviewed papers with high impact factor.
The scientific tasks were essentially three-fold, namely
(1) to provide the theoretical framework that allows to solve the underlying cosmic fluid equations to arbitrary high precision,
(2) to numerically implement these theoretical findings, thereby establishing a simulation of the dark matter fluid for random initial conditions, from which shell-crossings are detected, and
(3) to further exploit the theoretical findings by establishing exact analytical shell-crossing solutions for simplified initial conditions.
All those three tasks were successfully executed and lead to the anticipated deliverables and milestones, which are scientific papers. Specifically, building up on previous works of myself, we were able to complete the underlying theory. Thanks to the expertise provided at the Lagrange Institute, we implemented these findings in a computationally highly efficient way (full support of parallel computing), from which we were able to detect shell-crossings to extremely high accuracy.
This numerical implementation is publicly available and can be directly exploited by the wider cosmological audience, e.g. for generating highly accurate initial conditions of heavily used cosmological N-body simulations.
I discovered also new exact analytical solutions for nonlinear matter collapse models, and we were able to exploit the matter collapse in the presence of massive neutrinos. As a side product, we performed highly efficient two-fluid simulations that include neutrino clustering.
The deliverables and milestones have been published in peer reviewed papers with high impact factor.
The main goal of this action was the implementation of a novel numerical simulation technique for the dark matter fluid until shell-crossing, which is superior in terms of efficiency as compared to standard (symplectic) numerical integrating schemes. At the heart of this implementation is a rigorous mathematical description of the dark matter trajectories. While it was known that this description is indeed mathematically convergent, only lower bounds on the radius of convergence were reported in the literature. During the action, we obtained strong evidence that the radius of convergence surpasses the first instance of shell-crossing, implying for the fluid simulation that only a single time step is required to evolve the fluid from initial time to the time of shell-crossing. This came as a surprise, especially considering that the shell-crossing problem is highly non-linear. Obviously, a single time step simulation is to preferred to a multi time step simulation as for the latter case numerical errors could easily accumulate during run time. Furthermore, the underlying theoretical description of the numerical algorithm simplified.
The numerical implementation of the fluid simulation was embedded into the publicly available initial condition generator monofonIC for cosmological simulations. This implementation essentially evaluates the dark-matter trajectories in terms of a Taylor-series representation to arbitrary high Taylor orders in a recursive manner, and is fully parallelised. By performing mathematical convergence tests up to the 40th order (which is 37 orders higher as the previous state-of-the-art), we were able to resolve shell-crossings to high accuracy and furthermore we were able to categorise the nature of the convergence-limiting singularities, a milestone in mathematical cosmology.
At the same time, the numerical implementation enabled us to further exploit our findings, by dissecting numerical and theoretical (truncation errors in the Taylor series) that are inherent to numerical simulations. This led to a series of papers in journals with high impact factor, which all received much attention in the wider cosmological community. Another direct exploitation of our results for the dark-matter fluid was a semi-classical application. For this we formulated a fluid propagator that allows us to evolve an associated wave function from initial to final state. All observables, such as the fluid density and velocity can be determined very accurately from this wave function, which we have also exploited to generate very accurate initial conditions for simulations that require Eulerian input.
I have also discovered an exact analytical shell-crossing solution to the cosmological fluid equations, namely for a collapse that predominantly has a spherically symmetric signature but allows small deviations from spherically symmetry. Being able to incorporate such deviations is beneficial for at least two reasons, namely (1) that perfect spherical collapse has zero probability in a Universe with random initial conditions and thus, the improved model is more realistic; and (2) the model allows to determine explicitly the radius of convergence of the Taylor-series representation of the trajectories.
We have analysed the formation of collapsed matter structures in the presence of massive neutrinos, by employing novel techniques within the theory of General Relativity. Such avenues are especially important in the context of efficiently pinning down the allowed parameter space of cosmological observations.
We have also provided a novel theoretical model of the dark matter fluid that is valid even shortly after shell-crossing. In there we analysed the phase-space of the dark matter particles at shell-crossing locations adopting methods originally developed in optics and turbulence, providing us deep insight in the underlying “Vlasov-Poisson” description. In particular we identified several non-differential features in the particle acceleration due to secondary gravitational infall.
The numerical implementation of the fluid simulation was embedded into the publicly available initial condition generator monofonIC for cosmological simulations. This implementation essentially evaluates the dark-matter trajectories in terms of a Taylor-series representation to arbitrary high Taylor orders in a recursive manner, and is fully parallelised. By performing mathematical convergence tests up to the 40th order (which is 37 orders higher as the previous state-of-the-art), we were able to resolve shell-crossings to high accuracy and furthermore we were able to categorise the nature of the convergence-limiting singularities, a milestone in mathematical cosmology.
At the same time, the numerical implementation enabled us to further exploit our findings, by dissecting numerical and theoretical (truncation errors in the Taylor series) that are inherent to numerical simulations. This led to a series of papers in journals with high impact factor, which all received much attention in the wider cosmological community. Another direct exploitation of our results for the dark-matter fluid was a semi-classical application. For this we formulated a fluid propagator that allows us to evolve an associated wave function from initial to final state. All observables, such as the fluid density and velocity can be determined very accurately from this wave function, which we have also exploited to generate very accurate initial conditions for simulations that require Eulerian input.
I have also discovered an exact analytical shell-crossing solution to the cosmological fluid equations, namely for a collapse that predominantly has a spherically symmetric signature but allows small deviations from spherically symmetry. Being able to incorporate such deviations is beneficial for at least two reasons, namely (1) that perfect spherical collapse has zero probability in a Universe with random initial conditions and thus, the improved model is more realistic; and (2) the model allows to determine explicitly the radius of convergence of the Taylor-series representation of the trajectories.
We have analysed the formation of collapsed matter structures in the presence of massive neutrinos, by employing novel techniques within the theory of General Relativity. Such avenues are especially important in the context of efficiently pinning down the allowed parameter space of cosmological observations.
We have also provided a novel theoretical model of the dark matter fluid that is valid even shortly after shell-crossing. In there we analysed the phase-space of the dark matter particles at shell-crossing locations adopting methods originally developed in optics and turbulence, providing us deep insight in the underlying “Vlasov-Poisson” description. In particular we identified several non-differential features in the particle acceleration due to secondary gravitational infall.
Our numerical implementation of the fluid simulation has an unprecedented level of accuracy, which allowed us to pin down the instance of shell-crossing on a rigorous mathematical level.
Furthermore, analytical solutions to fully non-linear problems are very rare, also in the field of cosmological structure formation. The new analytical shell-crossing solution that we have discovered is of direct relevance for approaches that aim to extract cosmological parameters from observations.
Finally, our discovery of non-differential features in the phase-space of dark matter indicates the existence of singularities in the Vlasov-Poisson equations. Addressing questions of singularities are some of the most challenging ones in the field of mathematical physics (cf. MIllenium Clay Puzzle of Navier Stokes).
Furthermore, analytical solutions to fully non-linear problems are very rare, also in the field of cosmological structure formation. The new analytical shell-crossing solution that we have discovered is of direct relevance for approaches that aim to extract cosmological parameters from observations.
Finally, our discovery of non-differential features in the phase-space of dark matter indicates the existence of singularities in the Vlasov-Poisson equations. Addressing questions of singularities are some of the most challenging ones in the field of mathematical physics (cf. MIllenium Clay Puzzle of Navier Stokes).