Numerical Construction of initial data sets for inhomogeneous cosmological space-times with spatial topology of $\mathbb{T}^3$

TL;DR

This work investigates applying the Algebraic–Hyperbolic Formulation (AHF) of Einstein constraints to cosmological settings with a compact $\,\mathbb{T}^3$ topology, using a pseudo-spectral Fourier solver to evolve the hyperbolic constraint system for $(X,Y_i)$ and reconstruct the extrinsic curvature. It demonstrates the method's ability to reproduce analytic solutions for $\,\mathbb{T}^3$ Gowdy and perturbed FLRW spacetimes, but reveals stability limitations tied to the hyperbolicity condition and the spectral discretization, notably for PFLRW and certain Gowdy regimes. The authors analyze stability via eigenvalues and $\epsilon$-pseudospectra, and propose two data-construction strategies (restricting $Y_i$ or enforcing maximal foliations with parabolic relaxation) to generate new initial data while mitigating instabilities. Overall, the work provides a critical assessment of hyperbolic constraint formulations in cosmology, offering concrete methods and caveats for building periodic, fully relativistic initial data in compact-topology universes and outlining avenues for future refinement.

Abstract

In this work, we study the viability of the algebraic-hyperbolic formulation of the Einstein's constraint equations to construct initial data sets for inhomogeneous cosmological space-times with $\mathbb{T}^3$ topology. To do so, we implement a pseudo-spectral approach based on the discrete Fourier transform for numerically and explore the advantages and disadvantages of this method by comparing the numerical solutions with known analytical initial data sets. Additionally, we perform an stability analysis of the system to gain deeper understanding on the limitations of the proposed scheme. Finally, we numerically obtain new families of initial data sets through manipulation of the original system by imposing restrictions on some variables.

Figures (18)

• Figure 1: Hamiltonian constraint evaluation depending on the number of nodes considered in a squared grid. Gowdy eq. (\ref{['Gowdy_Metric']}), Minkowsky (MXY) eq. (\ref{['MXY_Metric']}), Gowdy gauge wave (GRX) eq. (\ref{['GRX_Metric']}) and PFRW eq. (\ref{['SPFRW_Metric']}).
• Figure 2: First (left) and second (right) components of the Momentum constraint evaluation depending on the number of nodes of the grid. Gowdy eq. (\ref{['Gowdy_Metric']}), Minkowsky (MXY) eq. (\ref{['MXY_Metric']}), Gowdy gauge wave (GRX) eq. (\ref{['GRX_Metric']}) and PFRW eq. (\ref{['SPFRW_Metric']}).
• Figure 5: Eigenvalues of $\Delta r$ times the Matrix $\boldsymbol{L}$, for $L=0.5$, superimposed on the stability region of (from left to right) Crank Nicholson and Implicit Euler. The eigenvalues were scaled by a factor $\theta=10^{2}$ for visualization purposes.
• Figure 6: Left: Eigenvalues of $\Delta r$ times the matrix $\boldsymbol{L}$ with coefficients frozen by taking their maximum in the mesh. Right: Eigenvalues of $\Delta r$ times the matrix $\boldsymbol{L}$ with coefficients frozen by taking their average in the mesh. In both cases, the spectrum is superimposed on the stability region of CN.
• Figure 7: Behaviour of the numerical solution of the linearized AHF equations using CN for PFLRW spacetime under initial condition modifications of the first component, for N=32 nodes and $\Delta r= 10^{-2}/N$. (a) Solution for the first component $\delta X(r,x)$. (b) Solution for the second component $\delta Y_{1}(r,x)$.