Coherent spin states and emergent de Sitter quasinormal modes

As a toy model for the microscopic description of matter in de Sitter space, we consider a Hamiltonian acting on the spin-j representation of SU(2). This is a model with a finite-dimensional Hilbert space, from which quasinormal modes emerge in the large-spin limit. The path integral over coherent spin states can be evaluated at the semiclassical level and from it we find the single-particle de Sitter density of states, including 1/j corrections. Along the way, we discuss the use of quasinormal modes in quantum mechanics, starting from the paradigmatic upside-down harmonic oscillator.

The time is near, in which the cosmological constant, minute but strictly positive, will come to dominate the evolution of our cosmos.The resulting accelerated expansion will drive the observable universe towards a semiclassical equilibrium state: our cosmic habitat will become asymptotically indistinguishable from a de Sitter (dS) static patch [1][2][3][4].Interpreting the Gibbons-Hawking entropy [5] to imply a finite Hilbert space, the universe as we know it may well be a small coherent fluctuation in an enormous but finite quantum system [6].
A rather pertinent question is: which finite quantum system?In the absence of a precise formulation of quantum gravity in dS, this question remains open, although various proposals have been outlined in the literature [7][8][9][10][11].Often, but not always, spinors serve as pixel operators on the (stretched) horizon, which then appears as a discrete quantum system to an outside observer [12][13][14].Other works have focused more directly on this observer, rather than the horizon, arguing that their static patch experience may be reproduced by a (gauged) worldline quantum mechanics at large N [15][16][17].Both perspectives are expected to be connected by RG flow, with the observer worldline and cosmological horizon respectively resembling the boundary and black hole horizon in an AdS setup [15,18,19].
An equally crucial question regards the type of output one can even expect from such a microscopic dS description [20].Precise observables typically require measurements at the boundary of spacetime, but in dS one cannot take the detector beyond the cosmological horizon of the experimental system [21].It then seems like a standard S-matrix is excluded, even in QFT.At least for light fields, wave packets simply do not separate as they would in flat space, but instead freeze out on super-horizon scales.That the precision of measurements is fundamentally limited also follows from finiteness of the entropy, which requires to give up the exact SOp1, Dq symmetry of dS D and limits its lifetime to the Poincaré recurrence time [22,23].One can then imagine a universality class of dS theories which may make different predictions for unmeasurable quantities (beyond Poincaré), but which all agree, within the fundamental bounds on precision, on the early-time local behavior [21].
Within this broader context, we want to ask the following simple question: how would a finite quantum system reproduce the dS quasinormal modes (QNMs)?In this paper we will therefore explore non-relativistic quantum systems with a finite number of degrees of freedom and the following striking feature: the emergence of dS-like QNMs in the limit where the Hilbert space dimension N grows large.At finite N , these exponentially decaying modes cannot really exist as eigenfunctions of a Hermitian Hamiltonian.Nonetheless, they do capture the macroscopic effective dynamics with increasing accuracy as N Ñ 8.  Our focus on QNMs originates with static patch excitations being at most resonances, due to the presence of the cosmological horizon [21].As fig.1.1 illustrates, fluctuations in AdS and dS lead rather different lives.The same is expected to hold for the bulk manifestation of simple microscopic operators.While in AdS holography they describe boundary perturbations, in dS they are likely to describe horizon disturbances: the low-lying QNMs [24].
Moreover, the QNM basis is manifestly dS-invariant, and it is hard to imagine a microscopic model reproducing their spectrum without also to some extent the dynamics, and vice versa.
The issues with defining a dS S-matrix are reminiscent of the situation in CFT, since both involve a continuous density of states.What sets the dS spectrum apart are its particular QNM poles [4,25].Although these do not fit into a standard Hilbert space framework, it is not inconceivable for them to take over the role usually played by asympotic single-particle states.Quantizing QNMs requires a non-standard norm -the KG-norm being infinite [26] but does lead to a natural definition of the Euclidean vacuum [27].This was used in [28] to argue for dS emerging from two entangled boundary CFTs, as already advocated for in [29].
QNM-oriented approaches have indeed been fruitful before.For instance, as Fourier transform of the density of states, the character counts QNM degeneracies, and efficiently encodes the 1-loop partition function [4].There is of course a vast literature on QNMs [30,31], but in the current context let us just emphasize that their role in organizing the (thermo)dynamics is not limited to dS [32,33] or partition functions alone [34][35][36].
Plan of the paper: In sec.2, the aim is to demystify QNMs a bit more, explaining their role even in ordinary quantum mechanics.We will see that in the simplest example, the upside-down (or inverted) harmonic oscillator, inserting the QNM completeness relation is equivalent to doing a Taylor series expansion.It also allows for a most convenient evaluation of the character tr e ´itH .Another application consists of perturbatively determining resonance spectra.Next, in sec.3, we describe the phase space of a massive particle moving in dS, deferring details to app.A.3.From it, we are naturally led to a conformal boundary quantum mechanics that reproduces two copies of the inverted oscillator QNMs; one each for the northern and southern observers.The combination of these yields the principal series character.In sec.4, we present a toy model for the microscopic description of matter in dS.
The model consists of a Hermitian Hamiltonian acting on the spin-j representation of SUp2q.
It shares key features with the boundary Hamiltonian, and at large spin we expect it to yield the dS spectrum.This convergence is confirmed by numerical diagonalization.In sec.5, we proceed analytically, using coherent spin states and holomorphic wave functions.These let us characterize the exact spin-model spectrum in terms of Heun polynomials, and shed light on the emergent QNMs.Moreover, they allow us to understand the large-j limit as a semiclassical one, in which we can evaluate the path integral for the coarse-grained character.
From it, we retrieve both the leading dS result and first 1{j correction.Finally, in sec.6, we mention several directions for future research, including possible generalizations towards interacting multi-particle microscopic models for matter in dS.

Quasinormal quantum mechanics
The upshot of this section is that resonances are as fundamental to the quantum mechanics of the upside-down harmonic oscillator as energy eigenstates are to the ordinary oscillator.
Despite early studies of its propagator and density of states [37], as well as several general results on QNM completeness [38][39][40][41][42], it seems like the upside-down oscillator has remained somewhat underappreciated.This point was also made a few years ago by [43,44], with emphasis on its broad relevance as a prototype to study physics near unstable equilibrium, and with applications ranging from black holes to quantum Hall physics.Time-dependent variations, which arise for instance in the description of superhorizon modes of light fields in Mukhanov-Sasaki variables, were studied in [45].The upside-down oscillator also appeared very recently as toy model in the context of double-cone wormholes [46].
Let us begin our discussion by defining the character, a quantity which will be of interest throughout the rest of the paper.For any given Hamiltonian H, it is defined as χptq " tr e ´itH . (2.1) For elliptic/stable systems like an ordinary harmonic oscillator it can be calculated simply by summing over the discrete energy spectrum (with a suitable iϵ prescription if needed).
For a single oscillator with frequency ω " 1, taking ω Ñ ω ´iϵ for t ą 0, one finds χptq " e ´pϵ`iqt{2 1 ´e´pϵ`iqt pt ą 0q . (2.2) For ordinary oscillators, this Fourier transformed density of states is not particularly convenient.On the other hand, hyperbolic/unstable systems, like the upside-down oscillator, have a continuous energy spectrum, and calculating the trace by integrating over the spectral density seems trickier.However, the final expression will simply take the form of (2.2) with imaginary frequency ω Ñ ´iω.Indeed, the main result of this section is an orthonormal pairing of resonances |ψ n y with anti-resonances | ψn y, leading to completeness relation (2.18).
This trivializes the calculation of the character, as follows, for t ą 0: and similarly for t ă 0: The different insertions for t ą 0 and t ă 0 are needed because forward (positive-t) time evolution converges on the resonance expansion, while backward (negative-t) time evolution converges on the anti-resonance expansion.

Upside-down harmonic oscillator
The character of the ordinary harmonic oscillator (2.2), and its generalization to stable systems with a spectrum of normal modes ω n ą 0, has the obvious interpretation of counting energy eigenstates.For unstable systems like the upside-down oscillator, we expect to count resonances instead, as manifested by (2.3).To arrive at this point, we will begin by taking a closer look at what happens to the construction of the harmonic oscillator energy eigenstates upon analytic continuation ω Ñ ˘iω.

Analytic continuation of the harmonic oscillator
The upside-down harmonic oscillator has a Hamiltonian which can be obtained from the usual oscillator by ω Ñ ˘iω.Similarly, we can continue the creation and annihilation operators.If we pick ω Ñ ´iω, we obtain for instance We then define the primary resonance |ψ 0 y to be the state annihilated by b ´: where we fixed the prefactor by continuation from the harmonic oscillator.More explicitly, putting ω " 1, the position space wave functions are given by where H n is the n-th Hermite polynomial.For n " 1, 2, 3, 4 this gives 6 p4x 4 ´12ix 2 ´3q . (2.12)

Decomposition of the identity
Given (2.8) and (2.10), it follows that the resonance raising (lowering) operator is the antiresonance lowering (raising) operator.The two types of modes are mapped into each other by ω Ñ ´ω.For the ordinary oscillator, the analogous sign flip maps e ´ωx 2 {2 Ñ e `ωx 2 {2 , which is non-normalizable and therefore discarded.In our case, both towers are equally non-normalizable, so there is no reason to keep one and not the other.
In fact, one might therefore think that we should keep neither.However, despite their non-normalizability, these resonances do govern the dynamics of normalizable wave packets moving in the upside-down potential, much like the energy eigenstates do for the ordinary oscillator.The time-evolution kernel for the Hamiltonian ( it nonetheless does in a distributional sense.Inserting a convergence factor like e ´ϵ x 2 , calculating the integral, and then taking the limit ϵ Ñ 0, leads unambiguously to the finite result above.Clearly then, there is a sense in which the standard orthonormal decomposition in energy eigenstates of the ordinary oscillator generalizes to an analogous decomposition into (anti-)resonances of the upside-down oscillator: with the decomposition on the left applicable to forward time evolution and the one on the right to backward time evolution in the setup described above.A somewhat more general argument is presented in app.A.1.For a mathematically more precise account of resonances and rigged Hilbert spaces we refer the reader to [38].
Given the completeness relation (2.18) and time-dependence (2.9), it now becomes clear how the calculation of the character χptq " tr e ´itH gets trivialized as claimed in (2.3) and (2.4) in the introduction of this section.

Resonance expansion and Taylor series
As an illuminating example of the more general decompositions in app.A.1, consider Like the b ˘in (2.6), the c ˘are proportional to Hermitian operators.In fact, they can be understood as resonance raising and lowering operators for the Hamiltonian which we recognize as Taylor expanding βpxq at x " 0 and exchanging integral and sum.
Whether this exchange is allowed and the decomposition converges depends on α and β.
The precise conditions are more subtle than for the standard decomposition into a discrete set of normalizable states.For example, when αpxq " e ´α x 2 , βpxq " e ´β x 2 , xα|βy " with α, β ą 0, the decomposition suggested by (2.24) is which converges to (2.25) only for α ą β.On the other hand, the analogous converges to (2.25) only when α ă β.As in (2.18), the understanding is that although these decompositions formally look equivalent, mapped into each other by Hermitian conjugation, convergence of one versus the other depends on what they are sandwiched between.For the wave packet time evolution discussed previously, the criterion was whether t ą 0 or not.
Since time evolution spreads out wave packets, we can understand why in the example above this criterion translates into whether α ą β or not.

Coherent state basis
As a slight detour, let us consider the coherent state basis defined in app.A.4.1, which will prove to be of use in sec. 5. On holomorphic wave functions (A.34), the Hamiltonian (2.20) acts as while resonance raising and lowering operators are represented by One then obtains the following towers of (anti-)resonances: As before, the (anti-)resonances by themselves are non-normalizable under the inner product (A.35).However, orthonormality in the sense x ψn |ψ m y " δ nm still holds, and the advantage of coherent states is that the required integrals can be calculated without need for an ϵ-prescription.
In fact, coherent states allow one to directly evaluate the character as the trace of the time-evolution kernel for the Hamiltonian (2.28), see also [46]: where in holomorphic polarization the kernel is given by so that (2.32) becomes a simple Gaussian integral.
Finally, it is instructive to obtain (2.33) from the path integral (A.38): The classical solution satisfying upT q " u f and ūp0q " ūi is given by uptq " ´ū i sinh t`pu f `ū i sinh T q cosh t cosh T , ūptq " ūi cosh t´pu f `ū i sinh T q sinh t cosh T .(2.35)This illustrates that paths generically get complexified: u and ū are not each other's complex conjugate.The on-shell action comes from the boundary terms alone: Since the Hamiltonian is quadratic, the semiclassical Morette-Van-Hove formula [47,48] gives the exact result (2.33).The semiclassical time-evolution of coherent states is rather intuitive [49,50].The normalized overlap |pv|e ´itH |ūq| 2 {pv|vq peaks at v " u cosh t ´ū sinh t, which is the classical path when both u, ū are specified at the initial time.During this Ehrenfest evolution, the wave packets are sheared along the classical hyperbolic trajectories.

Upside-down oscillator in a magnetic field
For any system with a quadratic Hamiltonian, the Heisenberg equations of motion are linear, and their solution can be written as a sum of exponential eigenmodes: So far we considered examples with ω α P R (harmonic oscillator) or ω α P iR (upside-down oscillator).In general, ω α will be a generic complex number.Consider for example a particle on the plane, in a magnetic field and upside-down harmonic potential: (2.39 The system is characterized by four frequencies: in terms of which the Hamiltonian (2.39) can be expressed as We can then contruct a tower of modes as before: ) The |ψ n,n 1 y are eigenmodes of the Hamiltonian (2.42) with eigenvalues  and k 2 .The orthonormality and completeness relations can be used to calculate the character, as in (2.3): χptq " tr e ´itH " e ´iω `t{2 1 ´e´iω `t ¨e´iω 1 `t{2 which is real and even under t Ñ ´t when k 2 ą b 2 .In fact, in the most general case, finding the exponents ω α in (2.38) amounts to diagonalizing a matrix M in the Lie algebra spp2d, Rq.The spectrum of such M is invariant under both ω Ñ ´ω and ω Ñ ´ω˚.This means that, whenever the ω are not purely real, we can split them into two disjoint sets Ω with Ω ˘" ´Ω˚" ´Ω¯, each containing d eigenvalues.Then we can write χptq " which is once again real and even under time reflection.

Pöschl-Teller perturbatively
In previous works [39][40][41][42], the assumption that the potential falls off sufficiently fast allows for a more rigorous treatment of QNM completeness.In some cases though, such potentials can be approximated near their maximum by an upside-down harmonic oscillator (2.7), and one might wonder whether their resonance spectrum can then be determined order by order in perturbation theory, similar to the standard approach for bound state spectra.This idea will prove to be quite useful in sec.5.4.To illustrate it, consider the Schrödinger problem for the inverted Pöschl-Teller potential: x `sech 2 λxq ψpxq " E ψpxq . (2.50) Our goal now is to see how we can reproduce the exact resonance spectrum (A.7) in perturbation theory.Expanding the potential in (2.50) near its maximum we can treat the total Hamiltonian as an unperturbed H p0q plus small corrections H piq : Here, H p0q is the inverted oscillator analyzed in sec.2.1 with resonance spectrum (2.53) The unperturbed (anti-)resonances ψ n and ψn are those in (2.11), up to a rescaling x Ñ ?λx.
Keeping in mind this rescaling, and making use of the completeness relation (2.18), we can now determine the first perturbative correction: which is precisely the first term in the λn ! 1 expansion (A.8) of the exact spectrum.The second correction is in turn given by the combination of H p1q , evaluated to second order in perturbation theory, and H p2q evaluated to first order.
In calculating the matrix elements, it is useful to note that only terms with n ´m " ˘2, ˘4 are non-vanishing.The result indeed matches the second correction in (A.8).

Quasinormal de Sitter
It is time to clarify the relevance of the upside-down harmonic oscillator to de Sitter physics.
In app.A.3 we obtain the phase space of a single particle in de Sitter space, described as a constrained system in embedding space.Phase space can be thought of as the space of particle trajectories.The static patch Hamiltonian acts on their asymptotic direction p precisely like the inverted oscillator (2.20).Northern and southern descriptions are related by a symplectomorphism, which flips the sign of the Hamiltonian.In this section we will see that at the quantum level, this translates into the existence of two QNM towers.Both appear in the character, which is therefore sensitive to the global structure of phase space.

From phase space to boundary quantum mechanics
Locally, the phase space of a relativistic particle in dS 2 is that of a non-relativistic particle on R, as derived in app. A. H " ´i ppB p `∆q . (3.1) Note that it is essentially the same as the upside-down oscillator Hamiltonian (2.20).Here ∆ " 1 2 `iν is the conformal weight appropriate to principal-series unitary irreps of SOp1, 2q.The coefficient ν P R determines the mass of the particle.In a similar way, the translation and special conformal generators in (A.17) become Together they satisfy the sop1, 2q Euclidean conformal algebra For the isometries to act unitarily, we further need to impose fall-offs with c ψ P C constant.The complex vector space F ∆ of infinitely differentiable functions ψ satisfying the above condition then furnishes a representation of SOp1, 2q [51].Under conformal transformations such wave functions will transform as primaries Globally, northern and southern patches are glued together by the inversion (A.22) which maps p Ñ ´1{p.This interchanges Q Ø K and flips the sign of H as in (A.21).As argued at the end of app.A.3, we should therefore really deal with wave functions on the boundary circle.Secretly, this is already imprinted in (3.4).Indeed, mapping between planar and global coordinates (A.31) amounts to where we omitted the prime on the transformed wave function to avoid clutter.Then, whenever ψpφq is a smooth function on the circle, with ψpπq " c ψ , we will have ψppq in planar coordinates satisfying (3.4).
In global coordinates, the conformal generators are Here, the inversion simply acts as φ Ñ φ `π.The global angular momentum J is conjugate to φ, and as in (A.31), depending on the coordinate system, it takes the form Its eigenstates form a discrete basis of F ∆ , namely the Fourier basis: The Hilbert space is then technically the L 2 -completion of F ∆ with the standard norm xξ|ψy " under which the states (3.9) are of course orthogonal.

Two towers and the principal-series character
Although (3.1) looks like a simple upside-down harmonic oscillator, we really are dealing with a different physical system, since it is SOp1, 2q rather than the Heisenberg algebra which acts unitarily on it 3 .An important consequence is the presence of two QNM towers in de Sitter, whereas for the upside-down oscillator we only found one.
The two resonance towers ψ n and χ n are found by first obtaining the primaries annihilated by Q, and then acting repeatedly with K.For the anti-resonances ψn and χn , we follow the same procedure with K and Q interchanged: The (anti-)resonances of each tower have complex-conjugated frequencies Since the inversion p Ñ ´p´1 interchanges K Ø Q and flips H Ñ ´H as in (A.21), we have that southern resonances must be northern anti-resonances and vice versa.This inversion acts on the wave functions as (3.5), and we can see from (3.11) that it indeed interchanges ψ n Ø χn and χ n Ø ψn , which gives further intuition for the appearance of the two towers.
The role of χ-tower will be to do the same at p " 8.This is needed since we are allowed to consider Gaussian wave packets centered at the north pole.Having vanishing derivatives at p " 0, these were absent from the discussion for the upside-down oscillator in sec.2.1.3.
The completeness relation should then include both towers: where p is such that the Taylor series converges.For instance for the angular momentum eigenstates (3.9) the radius of convergence is p " 1.On the other hand, the time-evolved wave function e ´itH f n ppq " e ´t∆ f n pe ´tpq will have a rescaled radius of convergence p " e t , as is visualized in fig.

Flipping the Hamiltonian and evaluating the character
The north-south map (A.21) flips the sign of H. Indeed, boosts translate time in different directions on the northern and southern patches.On the other hand, the physics in either patch looks exactly the same.This symmetry implies that the character χptq " tr e ´itH must be even under time-reflection, and therefore real.We can see this explicitly in the angular momentum basis (3.9).The north-south map φ Ñ φ `π acts as e ´iπJ |ny " p´1q n |ny and e ´iπJ H e iπJ " ´H, so that: Here it was important that J is a well-defined, self-adjoint operator, and that is where (3.4) comes into play.The argument does not work when restricting to Gaussian wave packets on the real line, as we did for the upside-down oscillator.
To actually compute the character, we can insert the decomposition (3.13) to arrive at which is indeed the dS 2 Harish-Chandra character [53], featuring both QNM towers.
Alternatively, using position eigenstates |φy, one can directly calculate [54] χptq " which localizes onto the fixed points of H, φ " 0 and π, each of which yields a QNM tower.

Quasinormal emergence
We can now apply what we learned in the previous section to the construction of toy versions of microscopic toy models, along the lines of sec. 1.When exploring microscopic models, there are a few de Sitter avatars that one could look for : 1. Excitations that locally behave like quasinormal modes, for a certain amount of time, which increases as the Hilbert space dimension grows large, until recurrences kick in.
2. Logarithmic tails in ρpωq, particular to the equidistant dS resonance spectrum.
3. A symmetry H Ñ ´H which flips the sign of the Hamiltonian, interpreted as a northsouth map, and which also gives rise to a second "shadow" tower of QNMs.
Our aspiration then boils down to finding a finite quantum mechanical system within which dS-like resonances emerge in the limit of large Hilbert space dimension N .As a diagnostic for this we will use the character χptq or equivalently the density of states ρpωq of the quantum system, and compare it with the exact dS results.Agreement of these quantities at the single-particle level then opens up possibilities towards the construction of multi-particle interacting microscopic models, as we will briefly discuss in sec.6.

Spin resonances mimic de Sitter
A first ambition would be to describe massive matter in dS 2 .For our single-particle model, we will consider states in the spin-j representation of SUp2q, so that the dimension of the Hilbert space is finite: dim H " 2j `1.By J i we denote the sup2q generators in the spin-j representation, satisfying Defining then J ˘" J 1 ˘iJ 2 , the Hamiltonian of interest will be: One can immediately notice two properties of H j crucial to mimic matter excitations in dS 2 .
First, close to maximal spin, sup2q behaves like the Heisenberg algebra: so that the first term of (4.2), being proportional to J 1 J 2 `J2 J 1 , behaves as the pB p in (3.1), while the second one, with prefactor ν, serves as the mass term.We thus expect resonance  peaks that are at least qualitatively similar to those in dS 2 ; the flipping of a spin mimicking a particle rolling down an upside-down harmonic potential.This analogy between J 3 and the boundary position is illustrated in fig.4.1, and will be made precise in the next subsection.
The other relevant property is that H j possesses a time-reversal symmetry: conjugation by e iπJ 2 maps H j Ñ ´Hj and serves as a discrete analog of the north-south map.
The Hamiltonian (4.2) has a simple structure.In the J 3 -eigenbasis pJ 3 q n,n " j ´n, where n runs from 0 to 2j, its matrix elements are: pH j q n,n " ν p1 ´n j q , pH j q n`2,n " ´pH j q n,n`2 " i apn `1q , (4.4) where apnq " 1 4j a npn `1qp2j ´nqp2j ´n `1q .(4.5) We will numerically diagonalize H j in sec.4.3.The surprise is that the full model, not limited to near-maximal spin, reproduces the exact dS 2 density of states in the large-j limit.
Finally, at the level of dynamics, one observes that Gaussian spin wave packets move towards lower and lower spin as time moves on, before eventually bouncing back, as shown in fig.4.1.This generically happens on a timescale which grows like log j.At finite ν{j, the wave turns back slightly before reaching other side.The precise turning point depends on the choice of initial wave packet and size of ν{j, and the effect goes away in the large-j limit.
These properties of H j can be understood analytically using the methods in sec. 5.

From spin to boundary Hamiltonian
The analogy, illustrated in fig. 4 The above is now seen to be the continuum limit of after identifying y " nN ´1 for n " 0, 1, . . ., N .The S ˘are shift operators acting on the states |ny.Like its continuum counterpart, H N is Hermitian.At this point the relation to the spin model (4.2) is clear: at large N " 2j the matrix elements (4.4) of H j are essentially the same as those of H N .The main difference is that H j only shifts n by multiples of two, meaning that the even-and odd-spin subspaces are invariant.
A caveat is that not every naive discretization of (4.8) reproduces the dS 2 character.The resonance peaks in particular are very sensitive to the behavior near y " 0 and y " 1.In a way, the discretized model should not want to push wave packets beyond those points.

Numerical diagonalization and large-spin limit
We will now numerically diagonalize the spin model Hamiltonian (4.2).The comparison of its spectrum with that of the dS 2 Hamiltonian will be done at the level of the character χptq, as given in (3.15), and equivalently, at the level of the dS 2 density of states [4]: where ψpxq " Γ 1 pxq{Γpxq is the digamma function and γ the Euler-Mascheroni constant.
The UV-regulator Λ shifts ρ without affecting its shape; it is a constant to be matched5 .
The poles of ρ are the quasinormal frequencies.Finally, it will be useful to recall that so that we can split each ψ in (4.10) into contributions coming from even and odd resonances: .12)

Peak splitting and averaging
Grouping the ordered eigenvalues of H j into ω e,i and ω o,i , according to the invariant subspaces of even and odd spin, we define the discretized densities to be the inverse level spacing: ρ even `1 2 pω e,i`1 `ωe,i q ˘" pω e,i`1 ´ωe,i q ´1 , ρ odd `1 2 pω o,i`1 `ωo,i q ˘" pω o,i`1 ´ωo,i q ´1 .(4.13) Fig. 4.2 shows that, in the large-j limit, these match rather seamlessly onto the contributions to ρ dS 2 coming from even and odd resonances.Given the identity (4.12), the full dS 2 result (4.10) will then be obtained by considering the sum ρ `1 4 pω e,i`1 `ωe,i `ωo,i`1 `ωo,i q ˘" pω e,i`1 ´ωe,i q ´1 `pω o,i`1 ´ωo,i q ´1 .( A glance at fig. 4.3 reveals that this prescription works very well indeed.

Capturing overtones
The previous figures focused on the behavior close to the resonance peaks.To get a sense for how well the overtones are captured, we should look at the logarithmic tails of ρ dS 2 , since these arise due to the equidistant dS 2 resonance spectrum.Increasing j in the spin model increases the value of |ω| at which deviations from this tail become visible.
In fact, we can truncate the sum (4.11) in ρ dS 2 to include only those contributions from the lowest j overtones.In fig.4.4 we can see that it is this quantity which the spin model appears to reproduce.This makes it tempting to conclude that in the large-j limit, the spin model will eventually resolve all resonances, and we will demonstrate this in sec. 5.  from even and odd resonances, according to the split (4.12).The sharpest peaks are due to the even ones.In magenta is the numerical ρ even (4.13)coming from the even-spin eigenvalues of the spin model.In cyan we see ρ odd .For larger ω these overlap almost perfectly, while for ω À ν they appear nicely riffled.In the examples above we took ν " 2. The spin-model spectrum matches very well with that of the dS 2 principal series.The graphs thus indicate that H j captures roughly the first j overtones.

Convergence of the coarse-grained character
The comparison to the dS 2 results can also be done at the level of the character.For the spin model it is simply a sum χ j ptq " ř e ´iω i t over all eigenvalues ω i of H j .As such, it avoids having to think about even-and odd-spin invariant subspace subtleties.At first sight χ j ptq is rapidly oscillating and rather unwieldy, but we should keep in mind that it will be neither possible nor desirable to resolve arbitrarily small energy differences, or be sensitive to arbitrary UV-states.In fact this was already implicit when defining the discretized ρ as the inverse level spacing, rather than a sum over δ-spikes.
In other words, test function overlaps ´dωρpωqf pωq should only agree when f does not vary significantly over scales smaller than the eigenvalue spacings.For the character, this provides an IR-cutoff t IR .Since the spectrum of H j is bounded, reasonable f should also have compact support |ω| ă ϵ ´1 for some UV-cutoff ϵ.At the character level, this translates to a smearing in time of order t UV " ϵ.This leads us to define the coarse-grained character: It is this quantity which at large j, and t UV ă t ă t IR , converges to the dS 2 character (3.15).
Numerical evidence of this provided in fig.4.5.An analytic demonstration, and more precise understanding of the timescales t UV and t IR , will be the topic of the next section.

Quasinormal semiclassics
In the previous section we found strong numerical hints that the spectrum of the spin model converges in the large-j limit to that of a massive particle in dS 2 .To get an analytic handle on this problem, the coherent spin state formalism is ideally suited.It is reviewed in app.A.4.2.
In app.A.5 we recall how the large-j limit can then be understood as a semiclassical one.
Starting from the associated phase space path integral, we will prove the claims made in sec.4.3, and find the 1{j corrections.

Spin-model spectrum and Heun polynomials
The 2j `1 eigenvalues of the spin Hamiltonian H j in (4.2) are in principle found as roots of its characteristic polynomial 6 .Since in our case it does not take on any particularly pleasant form, we will proceed differently.In holomorphic polarization, see app.A.4.2, H j acts as Solutions of the finite-j eigenvalue equation are then polynomials ppzq satisfying H j ppzq " λ ppzq , degppq ď 2j . (5.2) In fact, degppq must be either 2j or 2j ´1; otherwise one finds from (5.2) that ppzq vanishes in its entirity.Not restricting to polynomials, one finds eigenfunctions for every λ: ´j `iν, z 2 q . (5.3) In the above, Hℓ is the Heun function.The finite-j eigenvalues are then those λ for which one of these functions reduces to a polynomial of degree 2j or 2j ´1.Moreover, these two options correspond to the even-and odd-spin subspaces discussed in sec.4.Although this will be of limited practical use, it is good to have at least a characterization of the exact spin-model spectrum.It is also noteworthy that in the large-j limit, it relates the dS 2 density of states to a distribution of Heun polynomials.

Large-spin emergent phase space and density of states
The real value of coherent spin states lies in them making manifest the emergence of a classical S 2 phase space in the large-j limit [56].This is reviewed in app.A.5.
This emergent classical dynamics on the spin sphere, with symplectic form (A.53), is governed by the coherent state expectation value, see (A.47), of the Hamiltonian H j : H j pz, zq " i pj ´1 2 q pz 2 ´z2 q p1 `z zq 2 `ν z z ´1 1 `z z . (5.4) In the current section we will only consider the leading large-j behavior, so for now we can drop the ´1 2 above.The classical Hamiltonian then takes the elegant form in Cartesian coordinates.Phase space orbits for different values of ν j are shown in fig.5.1.For small ν j , there are 2 hyperbolic and 4 elliptic fixed points.The energy of the elliptic ones grows with j.The hyperbolic ones, at the north and south poles, instead have fixed energy ˘ν.Close to the poles, the orbits look just like those of the upside-down harmonic oscillator in fig.2.1d.The hyperbolic fixed points are therefore responsible for the emergent de-Sitter-like dynamics in the spin model at large j.In the figure one also sees that for larger ν j the orbits start crossing over and eventually we are left with 2 elliptic fixed points at the poles.That the number of elliptic minus hyperbolic fixed points always equals 2, the Euler characteristic of the sphere, is a cute instance of Morse theory.) generates orbits on the compact phase space S 2 , which we can think of as the spin sphere.We see that (a) at ν j " 0 phase space is split into 4 regions, (b) when ν ‰ 0 these regions merge in pairs as orbits start crossing over, (c) eventually the hyperbolic fixed points disappear and (d) when ν " j the orbits become circles at fixed J 3 .Now we can try to find the logarithmic tails of (4.10) in semiclassical quantization 7 , where the density of states at energy ω is given in terms of the periods as: A factor 2 was included since there will be two distinct orbits contributing the same T pωq.
To calculate the periods, we can use angular coordinates ϕ P r0, 2πq, φ P r0, πs on the sphere.
(5. 7) In what follows we will put ℏ " 1 again.The Hamilton equations are 9 φ " sin φ cos 2ϕ, 9 ϕ " ´cos φ sin 2ϕ `ν j . (5.8) Since the energy H j " ω is conserved, we can further elimate φ: in terms of the elliptic K-function 9 .We are interested in the high-energy tails of the spectrum at large j, meaning j " ω " ν.Then it follows from (5.10) that sin a Ñ ν 2 2jω and sin b Ñ 2ω j .Combining this with (5.6) and (5.11) yields the difference in density of states compared to a reference scale ω.This matches the large-ω behavior of the exact (4.10).
The goal now is to also get analytic control near the resonance peaks.As explained in sec.2.3, for potentials which approximate an upside-down harmonic oscillator, changes in the resonance spectrum can be calculated perturbatively.This will prove to be powerful in combination with the coherent spin path integral, as we shall discuss next.

Coherent spin path integral and saddle points
We will now compute the large-j limit of the coarse-grained character (4.15).Inserting the completeness relation (A.43) for coherent spin states, it takes the form χ j,ϵ ptq " 1 ?
2πϵ ˆ8 ´8 dt 1 e ´pt´t 1 q 2 {2ϵ 2 ˆd2 z µpz, zq xz| e ´it 1 H j |zy . (5.13) To see the semiclassical limit, we can use the path integral representation [57][58][59] pz f |e ´iT H |z i q " ˆrDzDz µpz, zqs e iSpz,zq , ( with the measure µpz, zq defined in (A.43), and the action: The symbol H j pz, zq was given in (5.4) and the boundary term takes the form S bdy " ij `logp1 `zp0qz i q `logp1 `zpT qz f q ˘(5.16) The boundary conditions are zp0q " zi and zpT q " z f .It is instructive to compare the above expressions to (A.38) for the standard coherent states, especially in the limit (A.46).To evaluate how fluctuations around the z " 0 saddle contribute to the character, we go to Darboux coordinates u (A.54), which trivialize the measure.Zooming in near z " 0 is done by keeping u fixed and sending j Ñ 8.As shown in [63] and reviewed in app.A.4.2, this also turns coherent spin states into harmonic oscillator coherent states.The leading limit of the symbol H j pz, zq Ñ i 2 pū 2 ´u2 q ´ν, is then recognized as the symbol of the upside-down harmonic oscillator, which we already solved in sec.2.1.4.Alternatively, we could see this by scaling z Ñ u ?2j in (5.1), leading at large j to the "free" Hamiltonian Thus, the two hyperbolic saddles in the large-j limit of the spin model yield the dS 2 character.
It remains to discuss the coarse-graining in (5.13).Besides exponentially suppressing the other saddles, it introduces a correction to the above hyperbolic contribution, since for ϵ !1: For (5.19), the relative size of this correction becomes of order one on scales t À t UV " ϵ.
Further, at a much smaller timescale t À j ´1 the saddle-point approximation breaks down.
In particular, at t " 0 the entire phase contributes: the fine-grained character χ j p0q " 2j `1 then simply counts the total number of states.Finally, recall that periodic orbits enter the picture when t Á t IR " log ϵj.These different effects can be seen in figs.5.2 and 5.3a.
The take-away message is then that for t UV À t À t IR , the large-j limit of the coarsegrained spin model character (5.13) will be dominated by the contribution coming from the hyperbolic fixed points, which, as we just showed, converges to the dS 2 result (3.15).Moreover, taking ϵ " j ´α with α P p0, 1q, we get an increasingly large window of convergence.Finally, we can ask which spin states are the emergent QNMs?Manipulating (C.8) in [63] for the associated Legendre polynomials P m l , we find that, again with z " u ?2j : where ψ n puq is the QNM defined in (2.30).This follows from the Rodrigues formulas In (a) we take j " 25 and ν " 0, plotting the character (4.15) with coarsegraining ϵ " 0, 0.25, 0.5 in cyan, magenta and red.For ϵ " 0 the state count at t " 0 is correct, but oscillations around the dS 2 result in black are not suppressed.Non-zero ϵ tames these oscillations, at the expense of modifying the behavior for t ă t UV , which scales linearly with ϵ.In (b) we see how for t ą t IR periodic orbit contributions kick in, leading to rapid oscillations.We show j " 500 with ϵ " 0.02, 0.02e in black and yellow, and j " t500eu with the same ϵ in orange and red.The results are consistent with t IR " log ϵj.this quantity, which matches the numerical result very well.In (a) we took a rather large ϵ " 0.05, for which the coarse-graining error (5.20) dominates.In (b) ϵ " 0.008 so that the intrinsic correction (5.28) is the larger one.The errors vanish when both ϵ Ñ 0 and ϵj Ñ 8.

Finding the 1{j corrections
To find the first subleading correction to the hyperbolic saddles, we again expand (5.1) in Darboux coordinates, now keeping the 1{j terms: z Ñ u ?2j p1 `uBu 4j q and B z Ñ p1 ´uBu 4j q ?2jB u .This has the following effect on the spin operators (A.45): which is the same as found by expanding the Holstein-Primakoff operators [64] to this order.
Plugging this into (5.1)then gives where H 0 was found in (5.18) and the first subleading correction takes the form This H p1q int introduces a correction to the spectrum, which we want to evaluate perturbatively.As in sec.2.3, we will do so by making use of the QNM basis.This is further simplified by noting that, with c ˘as defined in (2.29),To this order then, adding also the ν Ñ ´ν contribution of the z " 8 saddle, the overall effect on the character is a shift in its time-dependence: δχ j ptq " tr e ´itpH 0 `Hp1q int q ´tr e ´itH 0 " χptp1 `1 2j qq ´χptq « t 2j χ 1 ptq . (5.28) This intrinsic 1{j correction is (sub)dominant compared to the coarse-graining one (5.20),depending on whether pϵ ą j ´1{2 q or ϵ ă j ´1{2 .By varying this relative size, both effects can be distinguished in the numerical results, as shown in fig.5.3.Since the final result has the simple effect of rescaling time by 1 `1 2j , one may wonder if there exists a simpler way of arriving at this conclusion.At least at the level of (5.25) this does not appear trivial12 .

Quasinormal discussion
We have encountered a rather simple and finite quantum mechanical model with Hermitian Hamiltonian (4.2) acting on the spin-j representation of SUp2q, and showed that its coarsegrained character converges at large j to that of a massive particle in dS 2 , indicating the emergence of de Sitter quasinormal modes.
In fact, it turns out 13 that in the quantum optics literature, the Hamiltonian (4.2), at least when ν " 0, is known as the two-axis twisting Hamiltonian.It can be used to generate squeezed states in the sense of [66].Thinking of experimental realizations, it is interesting to note a relatively recent proposal for engineering two-axis twisting in cavity QED [67].
As the introduction suggested, we are ultimately in pursuit of de Sitter quantum gravity.
It will then not have escaped the reader's notice that we have yet to talk about black holes or indeed gravity.The idea is that perhaps our toy model could arise as a matter subsector in a more mature microscopic description.As a first step in that direction, we could begin by associating a pair of fermionic creation and annihilation operators c ω , c : ω to any of the 2j `1 spin-model frequencies.This leads to finite-dimensional multi-particle Hilbert space.
Along the lines of [29], we can then split these oscillators into southern ones with ω ą 0 and northern ones with ω ă 0, and further impose the constraint H " 0 on the physical states.Similar steps were recently taken for the double-scaled SYK model [68], reproducing QNMs for the complementary rather than principal series.It will be interesting to see whether, building further on our ideas, we end up having to go in a similar direction, or not.
Ideally, the picture outlined above would provide a unified description of particles and black holes in the spirit of [10].Black holes in de Sitter are not stable, so we might expect also these to eventually show up as huge resonances in the microscopic model.
Besides the generalization to other fields, the treatment of the higher-dimensional case also remains an open question.Moreover, since the analysis in sec.5 suggests that the crux lies in the emergence of two hyperbolic fixed points, the result does not appear terribly sensitive to other details, and one may wonder what happens when considering ensembles of Hamiltonians like (4.2).All of these are ongoing efforts on which we hope to report before disappearing in the horizon [69]. 13I am thankful to Ana Asenjo-Garcia and Stuart Masson for bringing this to my attention.

Figure 1 . 1 :
Figure 1.1: Small excitations in AdS are trapped in a harmonic oscillator potential and described by oscillating normal modes.Fluctuations in dS instead experience an inverted potential, that of an upside-down harmonic oscillator.Here, quasinormal modes play a more natural role, describing dissipation towards the cosmological horizon.

Figure 2 . 1 :
Figure 2.1: The standard harmonic oscillator (a) has a normalizable ground state, and contours of constant energy in phase space (b) correspond to elliptic orbits.The inverted oscillator (c) on the other hand, has a non-normalizable primary resonance (2.7), the real part of which is plotted above.It is purely outgoing on either side of the potential hill.The contours of constant energy in phase space (d) correspond to hyperbolic orbits.

. 40 )
The magnetic field dominates when b 2 ą k 2 .In this case ω α P R: the classical trajectories are oscillatory, leading to normalizable energy eigenstates at the quantum level.On the other hand, when k 2 ą b 2 we find ω α P C: the classical trajectories are spiraling orbits, described by a discrete set of resonances at the quantum level, see fig.2.2.Their construction proceeds again as in app.A.1, now with two pairs of raising and lowering operators c ˘and c 1 ˘:

Figure 2 . 2 :
Figure 2.2: Consider the first descendant resonance ψ 1,0 when k 2 ą b 2 .In (a) we see the radial potential for angular momentum n ´n1 " 1 with the real part of the resonance wave function, and in (b) its phase, together with a quantum trajectory starting at the red dot.
3. The quantum description is thus in terms of wave functions ψppq.As shown in fig.A.1, p P R is a stereographic coordinate on the future conformal boundary circle, indicating the asymptotic direction of a particle trajectory.It is the boost generator H (A.17), generating time-translations in the southern static patch, which serves as Hamiltonian for the boundary quantum mechanics.It is represented by a Hermitian operator acting on the wave functions 2 :

3 . 1 .ă 1 p ą 1 řFigure 3 . 1 :
Figure 3.1: Illustration of (3.13) on the complex p-sphere.Southern excitations localized near p " 0 are expanded in terms of the resonances ψ n , and northern ones near p " 8 in terms of the shadow resonances χ n .The radius of convergence of the expansions depends on time since e ´itH ψppq " e ´t∆ ψpe ´tpq.

Figure 4 . 1 :
Figure 4.1: In the spin model with Hamiltonian (4.2) we think of J 3 as labeling the position on the boundary circle.In this interpretation, wave packets keep moving towards the north pole until they notice that dim H " 2j `1, and bounce back after a time of order log j.

Figure 4 . 2 :
Figure 4.2: In black are the contributions to the exact density of states ρ dS 2 (4.10) coming

Figure 4 . 3 :
Figure 4.3: The total density of states (4.14) found by numerically diagonalizing the spin model at N " 1001 is in excellent agreement with the analytic result (4.10) in black.

Figure 4 . 4 :
Figure 4.4: In black is the exact density of states ρ dS 2 (4.10) for ν " 0, while the dashed lines include only those contributions coming from the first 25 and 125 resonances respectively.The dots in magenta instead represent the averaged density of states (4.14) obtained by numerically diagonalizing the spin model Hamiltonian H j at N " 2j " 50 and 250 respectively.

Figure 4 . 5 :
Figure 4.5: The dashed red line is the spin model character (4.15) for j " 500 with smearing ϵ " 0.02.Agreement with the dS 2 Harish-Chandra character (3.15) in black is excellent, as long as t UV ă t ă t IR .These cutoffs are discussed more in fig.5.2.The ϵ 2 and 1{j deviations are shown in fig.5.3 and are obtained analytically in the main text.

Figure 5 . 1 :
Figure 5.1: The Hamiltonian (5.5) generates orbits on the compact phase space S 2 , which we

26 )
Since c ˘are the QNM raising and lowering operators, only the first term contributes to the 1{j correction x ψn |H p1q int |ψ n y, which is then simply proportional to the original spectrum: On top of |ψ 0 y, ´t{2 e ix 2 {2 e iy 2 {2 `1 ´2i e ´t xy ´1 2 e ´2t p2x 2 ´iqp2y 2 ´iq `¨¨¨˘.(2.14) .1, between J 3 and position on the future conformal boundary of dS 2 , can be made more precise.To this end, let us define a map4 .18)For the z " 8 saddle H0 " ´H0 : the now familiar sign flip in the north-south map (A.21).