Dynamical Evolution of Boson Stars II: Excited States and Self-Interacting Fields

Edward Seidel; Jayashree Balakrishna; Wai-Mo Suen

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Dynamical Evolution of Boson Stars II: Excited States and Self-Interacting Fields

Jayashree Balakrishna, Edward Seidel, Wai-Mo Suen

General Relativity and Quantum Cosmology · Submitted 15 Dec 1997 · 16 references

Pith's verdict: The excited-state instability result is solid and worth having; the 'cascade' framing in the abstract leans harder on Fig. 11 than the figure can carry.

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CitationEdward Seidel, Jayashree Balakrishna, Wai-Mo Suen. "Dynamical Evolution of Boson Stars II: Excited States and Self-Interacting Fields." arXiv:gr-qc/9712064 (1997). https://pith.science/paper/J4ZNKL2B

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Editor's note

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Two things you should take from this. First, the substantive result: excited-state boson stars are dynamically unstable on both sides of the mass peak, not just on the U branch, and the paper maps the instability time scales across the S branch, the U branch, and the M > Nm dispersal regime. That is a real correction to how people read the mass-versus-central-density curve for excited configurations, and it is backed by a lot of explicit evolutions with stated central densities. Second, the paper is honest about the one place it could have picked a fight: it separates its generic-perturbation instability from Jetzer's stability under infinitesimal perturbations that strictly conserve M and N, rather than pretending the tension isn't there. That is the right way to handle it. The quasinormal mode curves for Lambda = 5 through 200 are new measurements and will get used. Appendix A is the best part methodologically: the chi ~ 1/sqrt(Lambda) scaling is derived from the reduced equations and then checked against Lambda = 800, 1200, 1600 in Table I, with the residual error shrinking as Lambda grows. That is a prediction followed by a test, not a fit dressed up as one. Note the authors themselves say the truncation holds only for ground states, so it does not underwrite the excited-state claims; they flag that, correctly. Where I'd push back is the abstract's headline, [abstract:s7]. The evidence is Fig. 11: one five-node run that ends in a black hole, and a side-by-side overlay of the t=1000 snapshot against an equilibrium four-node profile at sigma = 0.06, justified by 'the maxima are at similar radii.' Matching peak radii is weak. A configuration contracting toward horizon formation pulls its density maxima inward and suppresses the outer ones, which can produce a similar pattern without ever resembling a four-node eigenstate. The paper concedes the decaying star is 'roughly a combination of lower excited configurations' with no exact superposition. So the qualitative observation in Fig. 11b, that node structure is lost progressively, is fine and interesting. 'Cascade of intermediate states similar to atomic transitions' is a stronger claim than a single collapsing trajectory supports. The sponge boundary is a real systematic too, but it is acknowledged in Section II.B and it is generic; the cascade framing is specific to the headline and is not acknowledged. Minor: no error bars on frequencies or collapse times, convergence deferred to Paper I. Who should read it: anyone doing numerical relativity with scalar fields, and anyone quoting excited-state boson star lifetimes. Send it out. The instability result deserves referee time; ask the referee to press on Fig. 11.

Referee report

Significance

The paper's strengths are real and worth naming. Every qualitative claim is backed by explicit evolutions with stated central densities, masses and Λ values, and the paper is careful about a genuine potential conflict: it distinguishes its generic-perturbation instability of S-branch excited states from Jetzer's [4] stability under infinitesimal perturbations that strictly conserve M and N, and aligns itself with Lee & Pang [12]. The instability-time-scale study produces falsifiable thresholds (σ₁ = 0.021, σ₂ = 0.048, σ(0) = 0.541 for M = Nm) that a later calculation could confirm or refute. Appendix A is the strongest single element: an analytic reduction that predicts χ ∝ 1/√Λ and is then tested against three independent numerical evolutions in Table I, with the residual error decreasing as Λ increases, exactly as the approximation requires. The astrophysical motivation (M ~ 0.1 M_⊙ for λ = 0.01, relevant to microlensing) is stated concretely rather than gestured at. If the qualitative picture holds, the practical consequence is significant: the mass-versus-central-density curve of an excited boson star is not a stability diagram, and excited configurations are transient stages.

Major objections (3)

[§II.B, §III.B, §III.C] The outer boundary treatment is load-bearing for the quantitative claims and is not validated. §II.B states plainly that 'there is no perfect algorithm for the implementation of the outgoing wave condition,' and adopts a sponge of adjustable width D plus an outer condition exact only for m = 0. §III.B then concedes that 'exact details of the curves have some dependence on the sponge parameters but the basic results are the same,' without quantifying that dependence anywhere in the paper. This matters because the mass-loss rate through that boundary is precisely what determines the settling mass in Fig. 3c, the migration endpoint in Figs. 5–6, the extrapolated final mass of 0.76 m_Pl²/m in Fig. 4b, and the critical density σ₂ = 0.048 that separates decay to the ground state from collapse in Fig. 9b. A parameter study varying the sponge width D and the outer boundary location, showing that the final masses and the threshold densities move by less than the quoted precision, is needed before these numbers can be read as physical. If the thresholds shift appreciably with D, the paper should say so and downgrade them to order-of-magnitude statements.
[§IV, Fig. 9b] The instability time scales in Fig. 9b are presented as a quantitative result ('we carried out a detailed study of the instability time scales'), but the seed perturbation is the discretization error of the scheme, and no resolution-convergence study is reported for the improved code. §II explicitly declines to repeat convergence tests on the grounds that they are 'similar to those reported in Paper I,' yet the same paragraph states that the numerical treatment has 'various improvements' over Paper I, so the earlier tests do not automatically transfer. The normalization actually performed ('we cover the radius of the star in all cases by the same number of grid points') equalizes grid points per stellar radius, which means different physical resolutions for different configurations; it is not a convergence test. Since the quoted decay time is a total time to collapse, it depends on the seed amplitude logarithmically even when the underlying growth rate is amplitude-independent, so the monotonic trend of decay time with σ(0) across the S branch and into the U branch could be partly a resolution artifact. Please show at least two resolutions for three or four representative configurations in Fig. 9b and demonstrate that the ordering and approximate values of the time scales survive.
[§IV, Fig. 11] The cascade interpretation of Fig. 11 rests on a weaker identification than the abstract's wording implies. The evidence that the five-node star passes through a four-node state is Fig. 11c, which places the t = 1000 snapshot beside an equilibrium four-node profile at σ = 0.06 and asserts that 'the maxima are at similar radii (although the size of the peaks are somewhat different).' Matching the radii of density maxima is a weak criterion: a configuration contracting toward a black hole will pull its density maxima inward and suppress the outer ones, which can reproduce a similar peak pattern without the field resembling a four-node eigenstate. The paper itself concedes that the decaying star is 'roughly a combination of lower excited configurations' and that 'there is no exact superposition due to the intrinsic non-linearity,' so the cascade is asserted at exactly the level of precision the authors decline to quantify. A projection of the snapshot onto the equilibrium one-, two-, three- and four-node families (an overlap integral against the eigenstates, even if only approximate and even if it cannot be exact) would convert the visual comparison into a measurement. Failing that, the abstract and §IV should say 'loses node structure progressively, passing near a four-node configuration' rather than 'cascade of intermediate states,' since 'cascade' implies a sequence of quasi-equilibrium occupations that a single collapsing trajectory does not establish.

Minor objections (9)

[§II.B] The sentence 'Its value at the origin is determined by integrating Eq. (2.15) inward from the outer boundary' appears to cite the wrong equation: (2.15) is the Hamiltonian constraint, which the text says is not solved during evolution, whereas the lapse is fixed by the polar slicing condition (2.12). Please correct the reference.
[Eq. (2.16)] The mass formula M = ½ r[1 − 1/g²(∞)] leaves the prefactor r ambiguous; presumably this is the outer grid radius r_N or r evaluated at infinity. Please write it unambiguously.
[Table I and Appendix A] The table caption reads 'is compared to Λ0.540 (which is the predicted ratio for large Λ),' which appears to be a garbled Λ^0.5; please fix. The text states the comparison is made for Λ = 800, 1200 and 1600, but the table also lists Λ = 600, which is not mentioned in the text.
[§III.B] The frequency extraction 'f = 1/(199N(∞)) = 4.7 × 10⁻³' should be spelled out: state which observable the period is read from (the peak of g_rr in Fig. 3b, presumably), how N(∞) enters the conversion from code time to physical frequency, and whether reading the period from a different observable (e.g. the central field amplitude) gives the same value within the quoted precision.
[§III.B, Fig. 4b] The final mass of about 0.76 m_Pl²/m is obtained 'by extrapolating this line to where it meets the Λ = 10 curve' rather than by running the evolution to a fixed point. Please state whether the run at t = 4800 had in fact settled, and if so report the measured final mass alongside the extrapolated one.
[§IV] The critical density for dispersal is quoted as σ(0) > 0.54 in one place and σ(0) = 0.541 in another; please use consistent precision, and state the uncertainty on σ₂ = 0.048 given the finite sampling of central densities.
[Appendix A] The text correctly notes that the high-Λ reduction is valid only for ground states because derivative terms cannot be dropped at a node. It would help to state explicitly in the abstract or conclusion that no high-Λ excited-state results are claimed, so the appendix is not read as covering the node configurations of §IV.
[References] Reference [2], cited for the Jeans instability mechanism, is a numerical-relativity paper by two of the present authors; please check that this is the intended citation for that mechanism. Reference [13] should read 'Bonazzola' rather than 'Bonozzola.'
[Figures] There are several typographical errors in the figure captions and body text, including 'exctited' in the Fig. 9 caption, 'magniture' in §III.B, and 'thereﬂection' in §II.B. The author affiliation line also renders 'f ¨ur' incorrectly.

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The paper

The paper claims

Excited-state boson stars are unstable on both sides of their mass peak, and the most highly excited ones decay by stepping down through lower-node states before collapsing to black holes.

The reading

This paper asks what happens when a gravitating cloud of scalar particles is not in its ground state, and when the particles push back on each other through a $λφ^4$ self-coupling. The authors evolve spherically symmetric boson star configurations numerically, solving the coupled Einstein and Klein-Gordon equations, and track how the field, the spatial metric, and the total mass change in time. They confirm that self-coupling preserves the familiar stable/unstable branch structure of ground states while raising the maximum mass, and they measure the characteristic oscillation frequencies of stable stars. The headline result is about excited states: unlike ground states, configurations with nodes are unstable whether they sit to the left or the right of the mass peak, differing only in how long they last, and a five-node star visibly passes through a four-node stage on its way to a black hole. If this is right, an excited boson star's mass-versus-central-density curve is not a stability diagram at all, and excited configurations are transient way stations rather than places a star can live.

What carries the argument

The machinery is a spherically symmetric numerical evolution of the Einstein-Klein-Gordon system in radial gauge with polar slicing, where the complex scalar field is split into real and imaginary parts scaled by radius, and the lapse is made to collapse rapidly as an apparent horizon is approached, giving a clean signal for black hole formation. Equilibrium configurations are generated first as an eigenvalue problem: for each central field value $σ(0)$, a regular isolated solution exists only for particular values of the central lapse, and different eigenvalues correspond to different node counts. The node count is the diagnostic that carries the cascade claim, since the density function $ρr^2$ has $n+1$ maxima for an $n$-node star, so watching maxima disappear is watching the star change state. A separate high-coupling appendix rescales $r$ and $σ$ by powers of $Λ$ to get approximate equations exact as $Λ → ∞$, and argues from those that quasinormal frequencies scale as $1/√Λ$, checked numerically at $Λ = 600, 800, 1200, 1600$.

Extended reading notes

Core claim

The paper's central claim is that excited boson star configurations are intrinsically unstable on both the stable ($S$) and unstable ($U$) branches of their mass-versus-central-density curve, and that the instability time scales differ sharply between them. The authors evolve configurations with one, two, three and five nodes, perturbed only by finite-difference error or by deliberate additions and removals of scalar field. Low-density excited stars lose enough mass through scalar radiation to reach a ground state; denser ones collapse to a black hole, with the collapse time shrinking as central density rises until one approaches the zero-binding-energy point $M = Nm$, past which stars disperse to infinity instead. The five-node star of central density $σ(0) = 0.075$ and mass $3.07 m_{Pl}^2/m$ is shown cascading into a four-node configuration near $t = 1000$ before going on to form a black hole, which the authors read as the gravitational analogue of an atom stepping down through intermediate levels. The paper is careful to say this does not contradict earlier work finding $S$-branch excited states stable under infinitesimal perturbations that strictly conserve mass and particle number; the instability here is under generic perturbations.

Load-bearing premise

All the mass-loss, migration, and settling results depend on how much scalar radiation actually escapes through the outer boundary, and the authors state outright that no perfect outgoing-wave condition exists for a massive field, using instead a damping sponge plus an approximation exact only for a massless field, with the further concession that curve details depend on the sponge parameters in a way they do not quantify.

Editorial extensions

If this is right

Ground-state boson stars with self-coupling keep the stable/unstable branch structure, so the standard stability criterion based on where the mass curve peaks carries over to interacting fields.
Stable self-coupled stars have measurable quasinormal oscillation frequencies that rise with mass and drop toward zero at the branch transition, giving an observational signature that distinguishes coupling strengths.
A perturbed unstable-branch star either migrates to a lower-mass stable configuration or collapses to a black hole; which one happens is set by whether it can shed enough mass.
Excited configurations cannot be long-lived endpoints, so any formation channel that produces them yields either a ground-state star, a black hole, or dispersal, on a computable time scale.
For very large self-coupling the star develops a sharp surface layer like a neutron star, and its oscillation frequency falls off as $1/√Λ$, making such stars progressively harder to evolve numerically.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

If excited states are always transient, then a population of excited boson stars could only be observed as an ongoing decay, so the observable consequence is a radiation signature tied to the node-change events rather than a static object.
The cascade picture suggests a testable selection structure: measuring which intermediate node states appear and how long each persists would probe whether the decay prefers neighbouring node numbers, something the authors explicitly decline to formalize because of the nonlinearity.
The same instability logic should apply to any gravitating scalar configuration with internal radial structure, so the result is a caution for proposals that treat excited solitonic states as dark matter candidates.
Because the collapse times are measured against a sponge boundary that absorbs outgoing radiation imperfectly, an independent check with a larger domain or an exact massive-field outgoing condition would be the cheapest way to confirm the reported time scales.

Original abstract

The dynamical evolution of self-gravitating scalar field configurations in numerical relativity is studied. The previous analysis on ground state boson stars of non-interacting fields is extended to excited states and to fields with self couplings. Self couplings can significantly change the physical dimensions of boson stars, making them much more astrophysically interesting (e.g., having mass of order 0.1 solar mass). The stable ($S$) and unstable ($U$) branches of equilibrium configurations of boson stars of self-interacting fields are studied; their behavior under perturbations and their quasi-normal oscillation frequencies are determined and compared to the non-interacting case. Excited states of boson stars with and without self-couplings are studied and compared. Excited states also have equilibrium configurations with $S$ and $U$ branch structures; both branches are intrinsically unstable under a generic perturbation but have very different instability time scales. We carried out a detailed study of the instability time scales of these configurations. It is found that highly excited states spontaneously decay through a cascade of intermediate states similar to atomic transitions.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13-20260917 via OpenRouter, question bank abstract-p1-20260923, 2026-09-23: 1 request, 6,248 input tokens, $0.0003 reported.

What the abstract itself states, as probabilities: claims first 0.04 · states limitation 0.11 · human participants 0.01 · controlled comparison 0.73 · randomized assignment 0.01 · states quantitative result 0.81 · compares with prior results 0.93 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; show every role.

The dynamical evolution of self-gravitating scalar field configurations in numerical relativity is studied. The previous analysis on ground state boson stars of non-interacting fields is extended to excited states and to fields with self couplings. Self couplings can significantly change the physical dimensions of boson stars, making them much more astrophysically interesting (e.g., having mass of order 0.1 solar mass). The stable ($S$) and unstable ($U$) branches of equilibrium configurations of boson stars of self-interacting fields are studied; their behavior under perturbations and their quasi-normal oscillation frequencies are determined and compared to the non-interacting case. Excited states of boson stars with and without self-couplings are studied and compared. Excited states also have equilibrium configurations with $S$ and $U$ branch structures; both branches are intrinsically unstable under a generic perturbation but have very different instability time scales. We carried out a detailed study of the instability time scales of these configurations. It is found that highly excited states spontaneously decay through a cascade of intermediate states similar to atomic transitions.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#	sentence	role	the authors' own work	states the main conclusion	strength
s0	The dynamical evolution of self-gravitating scalar field configurations in numerical relativity is studied.	contribution 0.75	0.85	0.02	asserted 0.97
s1	The previous analysis on ground state boson stars of non-interacting fields is extended to excited states and to fields with self couplings.	contribution 0.76	0.57	0.05	asserted 0.98
s2	Self couplings can significantly change the physical dimensions of boson stars, making them much more astrophysically interesting (e.g., having mass of order 0.1 solar mass).	interpretation 0.66	0.42	0.14	asserted 0.70
s3	The stable ($S$) and unstable ($U$) branches of equilibrium configurations of boson stars of self-interacting fields are studied; their behavior under perturbations and their quasi-normal oscillation frequencies are determined and compared to the non-interacting case.	contribution 0.94	0.95	0.13	asserted 1.00
s4	Excited states of boson stars with and without self-couplings are studied and compared.	contribution 1.00	0.87	0.04	asserted 0.95
s5	Excited states also have equilibrium configurations with $S$ and $U$ branch structures; both branches are intrinsically unstable under a generic perturbation but have very different instability time scales.	result 0.98	0.87	0.19	asserted 1.00
s6	We carried out a detailed study of the instability time scales of these configurations.	contribution 0.93	0.95	0.06	asserted 0.79
s7	It is found that highly excited states spontaneously decay through a cascade of intermediate states similar to atomic transitions.	result 1.00	0.94	0.42	asserted 1.00

What kind of paper the classifier reports this is: simulation study 1.00 · evidence from simulation or computation 1.00.

What was examined, and what was not

sources read

arXiv listing abstract v1

not examined

full text
figures and tables
references and citing sentences
relations to other papers

levels run

P abstract

not run

S sections
T body sentences
C citations
R cross-paper relations

Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures

**Figure 1.** FIG. 1. The mass profiles of ground state boson stars for differ

**Figure 4.** FIG. 4. (a)The oscillation frequencies of different ground s

**Figure 5.** FIG. 5. (a) The radial metric

**Figure 6.** FIG. 6. The migration of the

**Figure 10.** FIG. 10. (a) The evolution of the metric function

**Figure 11.** FIG. 11. (a) The initial field configuration of a 5-node star. T

**Figure 12.** FIG. 12. The equilibrium profiles of a Λ = 800 star with central

Showing 7 of the paper's 12 figures. See all figures in the PDF ↗ Figures reproduced from arXiv:gr-qc/9712064 by the authors.

References

Works cited (16)

[4]

P.Jetzer, Phys.\ Rep. 220 , 183 (1992)

View on Pith 1992
[12]

T. D. Lee and Y. Pang, Nucl.\ Phys. B315 , 477 (1989)

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[1]

J.R. Primack, D. Seckel and B. Sadoulet, Annu.\ Rev.\ Nucl.\ Part.\ Sci. 38 , 751 (1988)

View on Pith 1988
[2]

E. Seidel and W-M. Suen, Phys.\ Rev.\ Lett. 66 , 2516 (1993)

View on Pith 1993
[3]

T. D. Lee and Y. Pang, Phys.\ Rep. 221 , 251 (1992) and A. R. Liddle and M. S. Madsen , Int. J. Mod. Phys. D, 1 , 101 (1992)

View on Pith 1992
[5]

E. Seidel and W-M. Suen, Phys.\ Rev.\ D 42 , 384 (1990)

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[6]

M. Colpi, S.L. Shapiro and I. Wasserman, Phys.\ Rev.\ Lett. 57 , 2485 (1986)

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J.D. Breit, S. Gupta and A. Zaks, Phys.\ Lett. 140B , 329 (1984)

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M. Gleiser, Phys.\ Rev.\ D 38 , 2376 (1988)

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[9]

P. Jetzer, Fermilab Report No. Conf-88/88-A (unpublished)

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[10]

T. D. Lee, Phys.\ Rev.\ D 35 , 3637 (1987)

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[11]

M. Gleiser and R. Watkins, Nucl.\ Phys. B319 , 733 (1989)

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[16]

C. Bona, Private Communications

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How the literature uses it

37 arXiv papers cite this in 58 sentences

Each citing sentence is taken from the citing paper's LaTeX source and typed by Pith's classifier. A label appears at probability 0.8 or more (0.95 for support), a contest at 0.7 or more; the quoted words are the citing authors'. How each label was checked · JSON

Papers citing it · 40 sentences without a confident label

Uses its method or tool 1 paper

For that we make use of techniques and methods that parallel those used in boson star studies and for non-interacting oscillatons [ref] , which will ease the comparison between the two types of scalar objects.

$\Phi^{4}$ Oscillatons · Introduction · 0.98

Compares with it 4 papers

In the absence of self-interactions ($Λ=0$), we recover the result first discussed in [ref] that the excited stars in the (candidate) stable branch decay to a non-excited star without a node.

Self-interactions can stabilize excited boson stars · Introduction · 0.81
Our results are similar to results found for boson stars [ref] , oscillatons [ref] , and Proca stars [ref] .

Perturbing the ground state of Dirac stars · Introduction · 1.00
This behaviour is similar to that of neutron and BSs, where the maximum mass configuration divides the solution curve into the stable and the unstable branches [ref] .

Horndeski stars · Outline of the paper and main results · 0.92

Cites it as support 2 papers

Such putatively stable configurations turn out to be dynamically unstable, either decaying to the ground state or collapsing into a BH [ref] .

Radial stability of spherical bosonic stars and critical points · Conclusion · 0.96
Direct studies of fundamental boson stars in various theories [ref] support this intuition.

All holographic systems have scar states · Boson stars · 1.00

Cites it as background 10 papers

Here we give a brief description of a gravitational bounded system of self-gravitating scalar field particles following the argumentation in the seminal paper [ref] , see also [ref] .

Bosonic gas as a Galactic Dark Matter Halo · Mathematical Background · 0.89
The stability of these families of solutions has been a central focus of discussion for some time [ref] .

Full 3D nonlinear dynamics of charged and magnetized boson stars · Introduction · 0.99
Their isolated structure, stability, and phenomenology have been investigated in a wide range of models [ref] , while their binary dynamics have revealed a rich variety of nonlinear behaviours [ref] .

Massive boson stars: Waveform-based branch diagnosis with neural reconstruction · Introduction · 0.97

Discussion

Disagree with the review, or know something it missed? Leave a comment, highlight any passage above to comment on it, or challenge this review formally.

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