Stability of Expanding Cosmologies

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Cosmological expansion can damp perturbations even when there is no asymptotically flat spatial exterior through which energy disperses. In de Sitter-like solutions, the scale factor grows exponentially, spatial derivatives are redshifted, and suitable energies acquire a coercive damping term. Nonlinear stability then follows from a bootstrap whose mechanism is temporal expansion rather than radial escape to infinity.

Required background. Analysis for Mathematical Relativity supplies rescaled energies; The Einstein Equation supplies the cosmological term and constraints; PDEs and Green Functions for Gravitation supplies damped-wave estimates.

Helpful background. Causality and Global Structure clarifies future completeness and conformal infinity.

Expansion as a damping mechanism

In flat slicing, de Sitter spacetime has

g=-dt^2+e^{2Ht}\delta_{ij}dx^idx^j, \qquad H=\sqrt{\frac{\Lambda}{3}}.

For the future normal $n=\partial_t$ , this volume’s convention gives

K_{ij}=-h_i{}^\mu h_j{}^\nu\nabla_\mu n_\nu =-Hh_{ij}.

A scalar wave obeys

\phi_{tt}+3H\phi_t-e^{-2Ht}\Delta\phi=0.

The term $3H\phi_t$ damps time derivatives, while the coefficient $e^{-2Ht}$ suppresses physical spatial gradients. In an energy identity, the expanding volume alone is not enough; variables or weights must be chosen so that the deformation terms have the favorable sign.

For example, on a flat torus, use the fixed coordinate measure $d^3x$ and define

E(t)=\frac12\int_{\mathbb T^3} \left(\phi_t^2+e^{-2Ht}|\nabla_x\phi|^2\right)d^3x.

Multiplication by $\phi_t$ and spatial integration by parts give the exact identity

E'(t)=-3H\int\phi_t^2\,d^3x -H e^{-2Ht}\int|\nabla_x\phi|^2\,d^3x \le-2HE(t).

Hence $E(t)\le E(0)e^{-2Ht}$ . The cross term from differentiating the spatial energy cancels the integrated Laplacian term. This is a rescaled derivative energy, not energy integrated against the growing physical volume $e^{3Ht}d^3x$ ; confusing the two measures changes the damping statement.

For an FLRW solution with scale factor $a(t)$ , spatial curvature $k$ , matter density $\rho$ , and cosmological constant,

\left(\frac{\dot a}{a}\right)^2 =\frac{8\pi G}{3}\rho +\frac{\Lambda}{3} -\frac{k}{a^2}.

Positive $\Lambda$ drives $\dot a/a$ toward a positive constant when the matter decays in the theorem’s regime. The resulting redshift can dominate quadratic perturbative errors.

De Sitter stability

Friedrich’s conformal formulation gives a geometric stability theorem for smooth vacuum data with $\Lambda\gt0$ on $S^3$ , sufficiently close in the theorem’s Sobolev topology to data on a global de Sitter Cauchy slice and satisfying the constraints. Their maximal developments have smooth spacelike conformal boundaries in both time directions and are asymptotically simple and causally geodesically complete. The proof uses regular conformal field equations on a compactified domain; see Friedrich 1986, de Sitter stability theorem, pp. 587–609.

This is stronger than bounded coordinate evolution. It also concerns global $S^3$ slices, whereas the flat slicing above covers only a de Sitter patch. Reversing an expanding flat patch changes damping to antidamping; it does not contradict the two-sided stability theorem for complete global de Sitter data.

Expanding matter cosmologies

Matter perturbations introduce acoustic characteristics, vorticity, and possible shocks. Speck’s theorem concerns $3+1$ Euler–Einstein solutions on $[0,\infty)\times\mathbb T^3$ , with $\Lambda\gt0$ , positive fluid density, and equation of state

p=c_s^2\rho,\qquad 0\lt c_s^2\lt\frac13.

Here $c_s$ is the constant sound speed. Sufficiently small constrained data near an expanding flat FLRW solution, with spatial metric in $H^{N+1}$ and extrinsic curvature and fluid data in $H^N$ , $N\ge3$ , evolve to a future causally complete solution. Vorticity is allowed. See Speck 2012, Theorem 11.1 and Remark 11.2.1, Open PDF pp. 65–66.

The sound-speed interval, torus topology, regularity, and constrained smallness are hypotheses of this result. The endpoints $c_s^2=0$ and $c_s^2=1/3$ are outside this theorem; they require separate arguments, rather than being declared unstable. The result also does not cover arbitrary equations of state, vanishing $\Lambda$ , large density contrast, or contracting data.

Other expanding backgrounds use different mechanisms. For negatively curved Milne-like vacuum solutions, expansion is polynomial and spatial curvature participates in coercivity. For scalar-field or kinetic matter models, the decay hierarchy depends on the matter equation. “Cosmological expansion stabilizes” is therefore a program, not a theorem without a named background and matter class.

A typical energy closure

After fixing a wave, CMC, or conformal gauge, introduce perturbation variables rescaled by powers of $a(t)$ . A schematic high-order energy satisfies

\frac{dE}{dt} \le-cH E+C E^{3/2}+R(t)E,

where $c>0$ and $R(t)$ is integrable or decaying. Under the bootstrap $E\le4\varepsilon^2$ , choose $\varepsilon$ so that $C E^{1/2}\le cH/2$ . Then

\frac{dE}{dt} \le-\frac{cH}{2}E+R(t)E,

and Grönwall gives uniform decay or boundedness with a strict bootstrap improvement. Elliptic lapse/shift estimates and the constraints are coupled to this energy, just as in asymptotically flat stability.

The limiting data need not be a member of the finite-dimensional homogeneous FLRW family. In Speck’s sharper asymptotic theorem, for $N\ge5$ , the rescaled spatial metric approaches a generally spatially varying metric $g^{(\infty)}_{ij}(x)$ on $\mathbb T^3$ ; rescaled fluid fields likewise retain profiles. These are controlled frozen asymptotic data, as stated in Speck 2012, Theorem 12.1, Open PDF pp. 70–71. Decay of physical curvature and velocity perturbations therefore must not be replaced by a claim that every rescaled component approaches a homogeneous FLRW value.

Limitations and interpretation

A future-stability theorem alone is not a past-stability theorem. Many expanding matter solutions are incomplete or singular toward the past; the global de Sitter result above is a distinct two-sided theorem. A cosmic-no-hair conclusion applies only in its data class and asymptotic topology; it does not classify all solutions with $\Lambda\gt0$ .

Exponential redshift also changes observable radiation questions. Perturbations can freeze to nonzero spatial profiles even while their physical derivatives decay. Convergence of curvature or rescaled variables need not mean every coordinate component tends to zero.

Accordingly, each theorem must state the topology, matter regularity, expansion rate, smallness norm, future time orientation, and precise modulated limit.

Solved diagnostic: homogeneous de Sitter damping

Solve the scalar equation for spatially homogeneous data and determine whether decay of derivative energy forces the field itself to vanish.

Solution

For a spatially homogeneous scalar, the wave equation reduces to

\phi_{tt}+3H\phi_t=0.

Integrating gives

\phi(t)=\phi_\infty+C e^{-3Ht}.

Thus $\phi_t$ decays exponentially while $\phi$ approaches a generally nonzero constant. This demonstrates both the damping mechanism and the need to state the norm and limiting family: derivative energy decays to zero, but the field itself need not.

The final page contrasts these perturbative theorems with large-data and threshold regimes, where expansion or dispersion may not dominate.

References

Friedrich, Helmut. “On the Existence of $n$ -Geodesically Complete or Future Complete Solutions of Einstein’s Field Equations with Smooth Asymptotic Structure.” Communications in Mathematical Physics 107 (1986): 587–609. DOI.
Speck, Jared. “The Nonlinear Future Stability of the FLRW Family of Solutions to the Euler–Einstein System with a Positive Cosmological Constant.” Selecta Mathematica 18 (2012): 633–715. DOI. Open PDF.