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David Fajman

Publications and source records attributed to David Fajman.

At least 19 recordsLinked to original sources
arXiv2026

Nonlinear stability of Einstein-de Sitter universes

The Einstein-de Sitter universe is the prevailing model used in cosmology to describe the cold dark matter-dominated epoch of the universe. This model is a spatially homogeneous and isotropic spacetime undergoing decelerated expansion, and is linearly unstable under the Einstein-Euler equations with a pressureless fluid equation of state. We show that every initial data set for the Einstein-Euler equations on $\mathbb{T}^3$ with a near-flat metric and positive fluid energy density converges to a flat metric under the Einstein-Euler flow with a polytropic equation of state. This means the metric asymptotes to an Einstein-de Sitter spacetime. In particular, this settles the question of whether the Einstein-de Sitter model can be nonlinearly stable for an appropriate matter model.

gr-qc↗
arXiv2026

Non-linear stability of the matter dominated universe

We numerically study non-linear perturbations of the Einstein-de Sitter spacetime as a solution to the Gowdy-symmetric Einstein-Euler system for a polytropic equation of state. The results suggest that the Einstein-de Sitter spacetime is stable for sufficiently small but otherwise generic perturbations. This is in stark contrast to the well known instability of this spacetime when the matter model is dust. Moreover, this indicates a previously unknown stable regime of the Einstein-Euler equations with direct implications for cosmology.

gr-qc↗
arXiv2026

Landau damping on expanding backgrounds

We analyse the effect of expansion in Newtonian cosmology on the asymptotic behaviour of charged self-interacting plasmas close to Poisson equilibria. To this end, we study the Vlasov-Poisson system on the phase space of a $3$-torus which is expanding with respect to the scale factor $a(t)$. We show that, for $a(t)=t^q$ with $q\in(0,\frac12)$, solutions to this system exhibit nonlinear Landau damping for initial data that is small with respect to a suitably strong Gevrey class, i.e., the charge density contrast of the plasma decays superpolynomially. For larger choices of $q$ within this range, the initial data requirements become stricter while the decay weakens. To our knowledge, this is the first result showing Landau damping in a cosmological setting.

math.AP↗
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

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↗
arXiv2024

On the past maximal development of near-FLRW data for the Einstein scalar-field Vlasov system

We show that the maximal globally hyperbolic development of near-FLRW initial data for the Einstein scalar-field Vlasov system exhibits stable Big Bang formation in the collapsing direction. The solutions exhibit stable Kretschmann scalar blow-up, causing the spacetime to become causally geodesically past incomplete, and are asymptotically velocity term dominated. This is the first stability result for the Einstein equations in the collapsing spacetime direction in presence of Vlasov matter that does not rely on any symmetry assumptions. Furthermore, the Vlasov distribution remains close to that of the FLRW solution as a function on the co-mass shell, and so does its momentum support if one assumes it to be close to that of the FLRW distribution initially. On the other hand, the leading order terms in components of the Vlasov energy-momentum tensor exbihit an offset in asymptotic order controlled by the perturbation size, and when viewed on the mass shell, the distribution asymptotically concentrates in certain preferred velocity directions. To ensure that this behaviour is sufficiently mitigated by the scalar field, we crucially exploit a scaling hierarchy between horizontal and vertical derivatives in the commuted Vlasov equation.

gr-qc↗
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↗
arXiv2022

Cosmic Censorship near FLRW spacetimes with negative spatial curvature

We consider general initial data for the Einstein scalar-field system on a closed $3$-manifold $(M,\gamma)$ which is close to data for a Friedman-Lema\^itre-Robertson-Walker solution with homogeneous scalar field matter and a negative Einstein metric $\gamma$ as spatial geometry. We prove that the maximal globally hyperbolic development of such initial data in the Einstein scalar-field system is past incomplete in the contracting direction and exhibits stable collapse into a Big Bang curvature singularity. Under an additional condition on the first positive eigenvalue of $-\Delta_\gamma$ satisfied, for example, by closed hyperbolic 3-manifolds of small diameter, we prove that the data evolves to a future complete spacetime in the expanding direction which asymptotes to a vacuum Friedman solution with $(M,\gamma)$ as the expansion normalized spatial geometry. In particular, the Strong Cosmic Censorship conjecture holds for this class of solutions in the $C^{2}$-sense.

gr-qc↗
arXiv2022

Recollapsing spacetimes with $\Lambda<0$

We show that any homogeneous initial data set with $\Lambda<0$ on a product 3-manifold of the orthogonal form $(F\times \mathbb S^1,a_0^2dz^2+b_0^2\sigma^2,c_0dz^2+d_0\sigma)$, where $(F,\sigma)$ is a closed 2-surface of constant curvature and $a_0,..., d_0$ are suitable constants, recollapses under the Einstein-flow with a negative cosmological constant and forms crushing singularities at the big bang and the big crunch, respectively. Towards certain singularities among those the Kretschmann scalar remains bounded, hence these are not curvature singularities. We then show that the presence of a massless scalar field causes the Kretschmann scalar to blow-up towards both ends of spacetime for all solutions in the corresponding class. By standard arguments this recollapsing behaviour extends to an open neighborhood in the set of initial data sets and is in this sense generic close to the homogeneous regime.

gr-qc↗
arXiv2021

Blow-up of waves on singular spacetimes with generic spatial metrics

We study the asymptotic behaviour of solutions to the linear wave equation on cosmological spacetimes with Big Bang singularities and show that appropriately rescaled waves converge against a blow-up profile. Our class of spacetimes includes Friedman-Lema\^itre-Robertson-Walker (FLRW) spacetimes with negative sectional curvature that solve the Einstein equations in presence of a perfect irrotational fluid with $p=(\gamma-1)\rho$. As such, these results are closely related to the still open problem of past nonlinear stability of such FLRW spacetimes within the Einstein scalar field equations. In contrast to earlier works, our results hold for spatial metrics of arbitrary geometry, hence indicating that the matter blow-up in the aforementioned problem is not dependent on spatial geometry. Additionally, we use the energy estimates derived in the proof in order to formulate open conditions on the initial data that ensure a non-trivial blow-up profile, for initial data sufficiently close to the Big Bang singularity and with less harsh assumptions for $\gamma<2$.

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↗
arXiv2021

Averaging with a time-dependent perturbation parameter

Motivated by recent problems in mathematical cosmology, in which temporal averaging methods are applied in order to analyze the future asymptotics of models which exhibit oscillatory behavior, we provide a theorem concerning the large-time behavior for solutions of a general class of systems. We thus propose our result to be applicable to a wide range of problems in spatially homogenous cosmology with oscillatory behavior. Mathematically the theorem builds upon the standard theory of averaging in non-linear dynamical systems.

math.DS↗
arXiv2020

Stable cosmologies with collisionless charged matter

It is shown that Milne models (a subclass of FLRW spacetimes with negative spatial curvature) are nonlinearly stable in the set of solutions to the Einstein-Vlasov-Maxwell system, describing universes with ensembles of collisionless self-gravitating, charged particles. The system contains various slowly decaying borderline terms in the mutually coupled equations describing the propagation of particles and Maxwell fields. The effects of those terms are controlled using a suitable hierarchy based on the energy density of the matter fields.

math-ph↗
arXiv2020

On the oscillations and future asymptotics of locally rotationally symmetric Bianchi type III cosmologies with a massive scalar field

We analyse spatially homogenous cosmological models of locally rotationally symmetric Bianchi type III with a massive scalar field as matter model. Our main result concerns the future asymptotics of these spacetimes and gives the dominant time behaviour of the metric and the scalar field for all solutions for late times. This metric is forever expanding in all directions, however in one spatial direction only at a logarithmic rate, while at a power-law rate in the other two. Although the energy density goes to zero, it is matter dominated in the sense that the metric components differ qualitatively from the corresponding vacuum future asymptotics. Our results rely on a conjecture for which we give strong analytical and numerical support. For this we apply methods from the theory of averaging in nonlinear dynamical systems. This allows us to control the oscillations entering the system through the scalar field by the Klein-Gordon equation in a perturbative approach.

gr-qc↗
arXiv2020

Asymptotic Stability of Minkowski Space-Time with non-compactly supported massless Vlasov matter

We prove the global asymptotic stability of the Minkowski space for the massless Einstein-Vlasov system in wave coordinates. In contrast with previous work on the subject, no compact support assumptions on the initial data of the Vlasov field in space or the momentum variables are required. In fact, the initial decay in $v$ is optimal. The present proof is based on vector field and weighted vector field techniques for Vlasov fields, as developed in previous work of Fajman, Joudioux, and Smulevici, and heavily relies on several structural properties of the massless Vlasov equation, similar to the null and weak null conditions. To deal with the weak decay rate of the metric, we propagate well-chosen hierarchized weighted energy norms which reflect the strong decay properties satisfied by the particle density far from the light cone. A particular analytical difficulty arises at top order, when we do not have access to improved pointwise decay estimates for certain metric components. This difficulty is resolved using a novel hierarchy in the massless Einstein-Vlasov system, which exploits the propagation of different growth rates for the energy norms of different metric components.

math.AP↗
arXiv2020

Stabilizing relativistic fluids on spacetimes with non-accelerated expansion

We establish global regularity and stability for the irrotational relativistic Euler equations with equation of state $\overline{p}=K\overlineρ$, where $0<K<1/3$, for small initial data in the expanding direction of FLRW spacetimes of the form $(\mathbb R\times\mathbb T^3,-d\tb^2+\tb^2δ_{ij} dx^i dx^j)$. This provides the first case of non-dust fluid stabilization by spacetime expansion where the expansion rate is of power law type but non-accelerated. In particular, the time integral of the inverse scale factor diverges as $t\rightarrow\infty$.

gr-qc↗