For asymptotically de Sitter and anti-de Sitter black hole spacetimes, the problem of defining the quasinormal frequencies has been satisfactorily resolved based on making use of a hyperboloidal foliation of the spacetime [1, 2].¹ For asymptotically flat black hole spacetimes, the situation is not as fully developed, but nevertheless in many cases a suitably robust mathematical definition exists either through casting the problem in terms of scattering resonances and making use of the method of complex scaling [3, 5] or through using a hyperboloidal slicing [4, 6, 7]. In all cases, the quasinormal frequencies can ultimately be understood as eigenvalues of some operator which is not self-adjoint.

A feature of operators which are not self-adjoint is that their spectra can be unstable to “small” perturbations. For a simple example in finite dimensions, consider the matrices

Clearly for 0 < ϵ ≪1, A′−A is “small” by any reasonable notion of smallness; however, A has a repeated eigenvalue at 0, while A′ has eigenvalues ±ϵ⁻¹, so the spectra diverge as ϵ → 0.

In the context of black hole quasinormal spectra, it was noticed already by Aguirregabiria–Vishveshwara Nollert–Price in the 90's [8–11] that seemingly innocuous changes to the operators used in defining quasinormal modes could have dramatic effects on the spectrum. Motivated in part by mathematical [2, 4, 6, 12] and numerical [13–15] studies which cast the problem of finding the quasinormal spectrum as an eigenvalue problem for the time evolution operator on a hyperboloidal foliation, there has been a resurgence of interest in the problem of quasinormal spectral stability, see [16, 17] in the specific context of instability arising from non-self-adjoint operators as well as [18–23] and references therein for many other studies.

In this short article, we shall consider the problem of the conformal wave equation on the static patch of de Sitter space. The high degree of symmetry enjoyed by the de Sitter spacetime means that the problem of determining the quasinormal spectrum is completely solvable, and the various objects involved can be computed explicitly. This makes this a helpful test-bed for understanding the effects on the spectrum of small perturbations. The perturbations we consider consist of stationary modifications to the potential. In a more physically motivated situation, we should consider the linearized gravitational field (rather than a conformal scalar field) and permit perturbations to the geometry of the background rather than just a potential. Our methods can, in principle, be applied in this situation, but for simplicity, we focus on the toy model.²

We are able to compute exactly the first order correction to each quasinormal frequency in terms of the perturbing potential. We find that the answer to the question of whether an individual quasinormal frequency is stable to “small” perturbations depends sensitively on what is meant by “small” [cf [25, footnote 18]]. In particular, the relevant notion of smallness varies depending on which modes we are considering, and those representing more rapidly decaying modes require a more stringent notion of smallness. One may alternatively view this by first fixing the notion of smallness considered and then one observes that the more rapidly decaying modes are more unstable to small perturbations, consistent with expectations going back to [9].³

To confirm the analytic computations, we also perform some numerics. For this, we make use of a spectral method on a hyperboloidal (or null) slicing, similar to that used in [16], but applied to an enlarged system obtained by differentiating the equation by hand k-times motivated by the analysis of [2]. This has a doubly beneficial effect—First, it stabilizes the numerical computation of quasinormal frequencies; second, it permits us to stably compute a family of pseudospectra that we define, associated with the problem, which allow the stability properties to be directly visualized. In this context, we should also mention the forthcoming study [26] which also provides a numerically stable computation of pseudospectra.

We consider the static patch of the de Sitter spacetime, written in coordinates that are regular at the future horizon. This is a metric on ℝ⁴ = {(t, x):t ∈ ℝ, x ∈ ℝ³}

with δ_ij the usual Kronecker delta and κ > 0 a constant. The static patch is the region R=ℝt×B where B = {x ∈ ℝ³:|x| < κ⁻¹} is the ball of radius κ⁻¹, and the future cosmological horizon is H+=ℝt×∂B. This metric is Einstein with cosmological constant Λ = 3κ². We will keep track of κ for later discussion, but nothing is lost by setting κ = 1 throughout.

The wave operator in these coordinates takes the form

We shall consider the following family of equations on this background

Here, V_h is a time-independent potential depending on some small parameter |h| < ϵ, and we assume that the map (h, x) ↦ V_h(x) is smooth on (−ϵ, ϵ) × ℝ³. We are interested in particular in the quasinormal ring-down behavior of solutions to this equation. To discuss this, we introduce the Laplace transformed operator which acts on functions u:B → ℂ

We define the quasinormal frequencies through the solvability properties of this operator. More precisely, for k = 0, 1, 2, … we define an inner product and norm on functions u, w:B → ℂ by

Here, ∇^(l)u is the rank l-tensor ∇_i1⋯∇_ilu, and · means contraction on all indices.⁴ Notice that (u, w)₀ is the usual L²−inner product. We define H^k, the Sobolev space of order k, to consist of those functions u:B → ℂ with ||u||_k < ∞. This is a Hilbert space with the corresponding inner product. We define the domain of L_s to be

It can be shown that H^k+2 ⊂ D^k ⊂ H^k+1, so that u∈Dk⇒L^(s,h)u∈Hk for all s, h.

With this definition in hand, we can state the basic theorem we shall require, which follows straightforwardly from Vasy, Warnick, and Hintz and Xie [1, 2, 27, 28]:

Theorem 1. Fix |h| < ϵ, k ∈ ℕ and let Uk={z∈ℂ:ℜ(z)>-(k+12)κ}. Then, the operator L^(s,h):Dk→Hk is invertible for s ∈ U_k, except at a discrete set Λ_k(h) ⊂ U_k. Moreover, for each σ ∈ Λ_k(h) there is an integer d>0 such that:

It follows from the characterization of points in Λ_k(h) that Λ_k+1(h) ∩ U_k = Λ_k(h). We call any σ ∈ Λ_k(h) for some k a quasinormal frequency of L(h), with geometric multiplicity d. A corresponding smooth solution to L^(σ,h)w=0 is a quasinormal mode, and a distribution X satisfying ii) above we call a co-mode. Notice that the condition on X implies that X is supported in B¯ and so can be uniquely extended to act on test functions in C∞(B¯).

The residue of L^(s,h)-1 at s = σ is a finite rank operator, and we identify the rank of this residue with the algebraic multiplicity of σ. As in the familiar case of matrices, the algebraic multiplicity is an upper bound for the geometric multiplicity. We say that a quasinormal frequency σ ∈ Λ_k(h) is simple if it has algebraic multiplicity one.

The result above holds for h fixed. The question we shall consider in this study, that of quasinormal spectral instability, amounts to trying to understand how the set Λ_k(h) changes as h varies.

In Jaramillo et al. [16] and subsequently [see Boyanov et al. [23] and references therein], the instability of the quasinormal spectrum has been investigated using the notion of pseudospectrum, comparing the results from this approach to computations with explicit perturbations. Recall that for a matrix A we can define the ϵ−pseudospectrum to be⁶

where we define ||(A-sι)-1||ℝn→ℝn=∞ whenever A−sι is not invertible. It can be shown [30–32] that Λ^ϵ corresponds to the set of points which can appear in the spectrum of A + δA, where δA is a perturbation satisfying ||δA||ℝn→ℝn<ϵ.

This notion generalizes to operators on infinite dimensional spaces in the obvious way. However, this definition cannot immediately be applied to our problem above because L^(s,h) is not of the form A−sI for some operator A. There are two possible approaches to resolve this. The approach taken by Jaramillo et al. [16] is to follow [2, 13, 14] and recast the problem of finding the quasinormal frequencies as a genuine eigenvalue problem by writing

where L_j(h) is a differential operator of order j. Then, we can verify that L^(s,h)w=0 has a solution if and only if

has a smooth solution. Thus, the set Λ_k(h) can be identified with the part of the spectrum of

in U_k, where L(h) is thought of as a closed unbounded operator on Hk:=Hk+1×Hk. This motivates one definition of the ϵ−pseudospectrum⁷ as

This has the advantage of being the standard definition applied to L, but the disadvantage that in numerical computations one has to double the dimension of the approximation space to account for the two functions w, v. Moreover, since L does not have compact resolvent, approximation by matrices can be more challenging.

An alternative approach is to generalize the notion of ϵ−pseudospectrum by declaring

This has the advantage that L^(s,0)-1:Hk→Hk is compact, but the disadvantage that since it is not the standard definition of pseudospectrum, one cannot readily make use of existing numerical libraries. We note as an aside that we could also consider the H^k → D^k norm in place of the H^k → H^k norm in Equation 9, but it will not make a significant difference for the type of perturbations we consider.

We shall take Equation 9 as our definition of the pseudospectrum for the rest of the study [see Besson et al. [26] for an alternative approach]. A modification of the usual arguments for pseudospectra [31, 32] shows that Λkϵ is precisely the set of points in U_k which can occur as quasinormal frequencies of L(s, 0)+E for some operator E:H^k → H^k satisfying ||E||Hk→Hk<ϵ. One can verify that the fact that s does not appear linearly in L(s, 0) does not affect this argument. In particular, provided ||Vh||Ck<ϵ/κ2 we have Λk(h)⊂Λkϵ.

We note that this definition agrees with that for the null slicing in Cownden et al. and Boyanov et al. [22, 23]; however, we do not assume that the slicing is everywhere null.

To give a concrete demonstration of the ideas above, we will work in a setting where the quasinormal frequencies, modes, and co-modes of the operator are known explicitly at h = 0. In particular, from now on we assume that we perturb about the conformal wave operator on de Sitter, in our language:

Under this assumption, we have [27, 28]:

Lemma 2. Suppose V₀(x) = 2. Then:

While it is possible to specify the constants An,li explicitly, see Hintz and Xie; Joykutty [27, 28, 33], for the purposes of our results below it is more computationally efficient to find X_{n, l, m} for any particular choice of n, l, m by simply using Equation 10 as an ansatz in Equation 5 and solving the resulting linear system for An,li. Doing so using Mathematica to perform the computations, we find the results in Table 1.

Since for σ ≠ −κ the quasinormal frequencies are not simple, to make use of Equation 7 to estimate the change in the QNF, we shall make the additional assumption that the potential V_h is spherically symmetric. Under this assumption, the QNFs are simple once we restrict our attention to a single fixed angular mode. If we fix l, m with |m| ≤ l, then for k ≥ l, the unperturbed quasinormal spectrum restricted to the l, m angular mode is Λkl,m(0)={-lκ,…,-kκ} and all the QNFs are simple. We can compute the rate of change of the QNF at −κ n by

To use this formula, we also need ∂L^∂s and ∂L^∂h. For the particular case of interest, with L(h) given by Equation 2, we have

so that

and

where we introduce W=∂Vh∂h|h=0, the first order perturbation to the potential.

We now have all that is required to compute σn,l,m′(0). In view of the structure of the operator X_{n, l, m}, we can write

for some constants Bn,li. Note that this is independent of m due to the spherical symmetry of the perturbing potential. We can again use Mathematica to compute these constants and present the results for the first few modes in the l = 0, 1, 2 angular sectors in Table 2.

Picking two cases as examples, we can read off from the tables that

We see very explicitly here and from Table 8 that the rate of change of the quasinormal frequency depends on higher derivatives of the perturbing potential, and the larger n, that is, the deeper into the stable plane we go, the more derivatives that are required. Equivalently, the deeper into the stable plane, the more control we require on the high wavenumber component of our perturbation. The increasing order of the operator norm that appears on the right-hand side of Equation 8 as we probe deeper into the plane is not simply an artifact of our framework but is necessary.

To see why it is necessary to use higher order norms to constrain the perturbations, let us consider the case κ = 1 and consider a family of perturbations W(r)=ϵ3exp(-r2ϵ2). We clearly have that

so in particular as ϵ → 0, we see that in the ‘energy norm', that is, the operator norm associated to the H¹ norm we have that the perturbation tends to zero. However, W″(1) = (4ϵ⁻¹ − 2ϵ)exp(−1/ϵ²) ~ ϵ⁻¹ as ϵ → 0, so that using the expression above we see that the l = m = 1, n = 3 mode is displaced (to first order) by a term proportional to ϵ⁻¹. Hence, smallness of the perturbation in the energy norm is no guarantee of stability of the quasinormal modes lying sufficiently deep in the stable half-plane.

For the choice of potential Vh(r)=2+hexp(-r2), with κ = 1, which we study numerically below, we have computed |σn,l,m′(0)| for n ≤ 20, l ≤ 2 and presented the results graphically in Figure 1. Noting the logarithmic scale on the y−axis, we see that for this choice of perturbing potential |σn,l,m′(0)| grows roughly exponentially in n, consistent with our expectation that modes deeper in the stable plane become more and more unstable.

To test numerically the computations above, we have computed the quasinormal frequencies for the choice Vh(r)=2+hexp(-r2). We first present the results and then comment on the methods used below.

We have investigated the stability of the quasinormal spectrum of the conformal wave equation on the static patch of de Sitter. We find that the quasinormal frequencies are stable, provided the perturbing potential is small in a sufficiently high regularity norm. Conversely, one could instead interpret this as a spectral instability for perturbing potentials which are not sufficiently regular at the cosmological horizon. We numerically verify our computations using a spectral method and propose a definition for a family of pseudospectra that demonstrate good convergence properties and capture the (in)stability of the quasinormal frequencies.