arXiv subjects

Maximilian Ofner

Publications and source records attributed to Maximilian Ofner.

7 recordsLinked to original sources

arXiv2025

A critical threshold for the cosmological Euler-Poisson system

We consider the gravitational Euler-Poisson system with a linear equation of state on an expanding cosmological model of the Universe. The expansion of the spatial sections introduces an additional dissipating effect in the Euler equation. We prescribe the expansion rate of space by a scale factor $a(t)=t^\alpha$ with $\alpha\in(0,1)$, which describes the growth of length scales over time. This model is regularly applied in cosmology to study classical fluids in an expanding Universe. We study the behaviour of solutions to this system arising from small, near-homogeneous initial data and discover a \emph{critical} change of behaviour near the expansion rate $\alpha=2/3$, which corresponds to the matter-dominated regime in cosmology. In particular, we prove that for $\alpha>2/3$ the fluid variables are global in time and remain small provided they are sufficiently small in a suitable norm initially. In the complementary regime $\alpha\leq2/3$, we present numerical evidence for shock formation of solutions to the Euler equation for arbitrarily small initial data. In combination, this establishes the existence of a critical stability threshold for barotropic fluids in expanding domains. In contrast to our previous work on the corresponding relativistic system, the threshold in the classical system considered here is independent of the speed of sound of the fluid. This establishes that fluids in cosmology behave fundamentally different in the non-relativistic regime than in the relativistic one.

math.AP↗

arXiv2025

Testing the Missing Completely at Random Assumption for Functional Data

We consider functional data which have only been observed on a subset of their domain. This paper aims to develop statistical tests to determine whether the function and the domain over which it is observed are independent. The assumption that data are missing completely at random (MCAR) is essential for many functional data methods handling incomplete observations. However, no general testing procedures have been established to validate this assumption. We address this critical gap by introducing a testing framework which is generally based on a partition of the observation patterns. Besides deterministic partitions, we also consider a systematic approach based on clustering. We establish asymptotic results for our tests and illustrate the methodology in several real data applications.

stat.ME↗

arXiv2025

Stability of fluids in spacetimes with decelerated expansion

We prove the nonlinear stability of homogeneous barotropic perfect fluid solutions in fixed cosmological spacetimes undergoing decelerated expansion. The results hold provided a specific inequality between the speed of sound of the fluid and the expansion rate of spacetime is valid. Numerical studies in our earlier complementary paper provide strong evidence that the aforementioned condition is sharp, i.e. that instabilities occur when the inequality is violated. In this regard, our present result covers the regime of slowest possible expansion which allows for fluids to stabilize, depending on their speed of sound. Our proof relies on an energy functional which is universal in the sense that it also applies to the case of linear expansion and enables a significantly simplified proof of bounds for fluids on linearly expanding spacetimes. Finally, we consider the special cases of dust and radiation fluids in the decelerated regime and prove shock formation for arbitrarily small perturbations of homogeneous solutions.

gr-qc↗

arXiv2024

Phase transition between shock formation and stability in cosmological fluids

We demonstrate a novel phase transition from stable to unstable fluid behaviour for fluid-filled cosmological spacetimes undergoing decelerated expansion. This transition occurs when the fluid speed of sound $c_S$ exceeds a critical value relative to the expansion rate $a(t) = t^\alpha$ of spacetime. We present an explicit relationship between $\alpha$ and $c_S$ , which subdivides the $(\alpha,c_S)$-parameter space into two regions. Using rigorous techniques, we establish stability of quiet fluid solutions in the first stable region. Numerical experiments reveal that the complement of the stable region consists of unstable solutions, implying sharpness of our stability result. We provide a definitive analytical bound and high-precision numerical evidence for the exact location of the critical line separating the stable from the unstable region.

gr-qc↗

arXiv2023

Covariate-informed reconstruction of partially observed functional data via factor models

This paper studies linear reconstruction of partially observed functional data which are recorded on a discrete grid. We propose a novel estimation approach based on approximate factor models with increasing rank taking into account potential covariate information. Whereas alternative reconstruction procedures commonly involve some preliminary smoothing, our method separates the signal from noise and reconstructs missing fragments at once. We establish uniform convergence rates of our estimator and introduce a new method for constructing simultaneous prediction bands for the missing trajectories. A simulation study examines the performance of the proposed methods in finite samples. Finally, a real data application of temperature curves demonstrates that our theory provides a simple and effective method to recover missing fragments.

math.ST↗

arXiv2023

The Stability of Relativistic Fluids in Linearly Expanding Cosmologies

In this paper we study cosmological solutions to the Einstein--Euler equations. We first establish the future stability of nonlinear perturbations of a class of homogeneous solutions to the relativistic Euler equations on fixed linearly expanding cosmological spacetimes with a linear equation of state $p=K \rho$ for the parameter values $K \in (0,1/3)$. This removes the restriction to irrotational perturbations in earlier work, and relies on a novel transformation of the fluid variables that is well-adapted to Fuchsian methods. We then apply this new transformation to show the global regularity and stability of the Milne spacetime under the coupled Einstein--Euler equations, again with a linear equation of state $p=K \rho$, $K \in (0,1/3)$. Our proof requires a correction mechanism to account for the spatially curved geometry. In total, this is indicative that structure formation in cosmological fluid-filled spacetimes requires an epoch of decelerated expansion.

math.AP↗

arXiv2021

Slowly expanding stable dust spacetimes

We establish the future nonlinear stability of a large class of FLRW models as solutions to the Einstein-Dust system. We consider the case of a vanishing cosmological constant, which in particular implies that the expansion rate of the respective models is linear i.e. has zero acceleration. The resulting spacetimes are future globally regular. These solutions constitute the first generic class of future regular Einstein-Dust spacetimes not undergoing accelerated expansion and are thereby the slowest expanding generic family of future complete Einstein-Dust spacetimes currently known.

gr-qc↗