arxiv: 2605.03771 · v1 · submitted 2026-05-05 · 🌌 astro-ph.CO

Recognition: unknown

Thermal activation rate of dilute axion stars close to the maximum mass

Pierre-Henri Chavanis

Pith reviewed 2026-05-07 13:42 UTC · model grok-4.3

classification 🌌 astro-ph.CO

keywords axion starsmetastable statesthermal activationinstanton theorysaddle-node bifurcationBose-Einstein condensatesself-gravitating systemslifetime scaling

0 comments

The pith

Dilute axion stars near maximum mass have thermal lifetimes scaling as e^N times the dynamical time, rendering metastable states effectively stable.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper applies instanton theory to compute the thermal escape rate of dilute axion stars modeled as self-gravitating Bose-Einstein condensates from their metastable configurations just below the maximum mass. Using the normal form of the saddle-node bifurcation that governs the energy landscape near this critical point, it derives an explicit lifetime that grows exponentially with the boson number N. For QCD axion stars with N around 10^57 and dynamical times of hours, this lifetime vastly exceeds cosmic timescales, so the configurations behave as stable equilibria in practice. The same exponential protection appears in related systems such as ultralight axion dark matter cores and self-gravitating Brownian particles.

Core claim

Close to the maximum mass, the saddle-node bifurcation normal form combined with instanton theory yields a thermal activation rate whose inverse, the lifetime of metastable dilute axion stars, scales as t_life ∼ e^N t_D. With N ∼ 10^57 and t_D ∼ 10 hours for typical QCD axion stars, or N ∼ 10^96 and t_D ∼ 100 Myr for ultralight axion cores, the resulting lifetime is so long that metastable equilibrium states can be treated as stable in practice.

What carries the argument

Instanton theory applied to the normal form of the saddle-node bifurcation in the energy landscape near the maximum mass of the self-gravitating condensate.

If this is right

Metastable axion stars near maximum mass survive for times far longer than the age of the universe and can therefore be regarded as stable for astrophysical purposes.
The identical exponential scaling protects quantum cores of dark matter halos made of ultralight axions against thermal decay.
The same long lifetimes apply to self-gravitating systems in globular clusters, the Brownian mean field model, and laboratory Bose-Einstein condensates with attractive interactions.
Thermal activation is negligible compared with quantum tunneling for determining the practical stability of these objects.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Observed axion star candidates could sit slightly above the formal maximum mass in long-lived metastable states without immediate collapse.
The exponential suppression may allow similar normal-form calculations to estimate thermal stability in other bosonic dark matter models with attractive self-interaction.
If the dynamical time remains the only relevant scale, thermal effects drop out of cosmic evolution calculations for axion stars once N exceeds roughly 100.

Load-bearing premise

The normal form of the saddle-node bifurcation accurately captures the shape of the potential barrier near the maximum mass and instanton theory applies without large gravitational corrections.

What would settle it

Observation of a dilute axion star with mass near the maximum value that collapses or disperses on a timescale much shorter than e^N times its dynamical time would show the calculated rate is wrong.

Figures

Figures reproduced from arXiv: 2605.03771 by Pierre-Henri Chavanis.

**Figure 1.** Figure 1: FIG. 1: Mass-radius relation of dilute axion stars interpreted view at source ↗

**Figure 2.** Figure 2: FIG. 2: Normal form of the potential view at source ↗

**Figure 4.** Figure 4: FIG. 4: The rate of escape Γ from the metastable state is view at source ↗

**Figure 5.** Figure 5: FIG. 5: Thermal instanton view at source ↗

**Figure 6.** Figure 6: FIG. 6: Series of equilibria (caloric curve) for classical view at source ↗

**Figure 7.** Figure 7: FIG. 7: Temperature-radius relation of self-gravitating view at source ↗

**Figure 9.** Figure 9: FIG. 9: Instanton view at source ↗

**Figure 10.** Figure 10: FIG. 10: Finite size effects close to the critical point of the view at source ↗

**Figure 11.** Figure 11: FIG. 11: Normal form of the effective potential close to the view at source ↗

read the original abstract

We compute the thermal activation rate of metastable self-gravitating Bose-Einstein condensates with attractive self-interaction (e.g., dilute axion stars) by using the instanton theory. Explicit analytical results are given close to the maximum mass $M_{\rm max}$ [P.H. Chavanis, Phys. Rev. D 84, 043531 (2011)] by using the normal form of the saddle-node bifurcation close to that point. We show that the lifetime of metastable states is extremely long, scaling as $t_{\rm life}\sim e^N\, t_D$, where $N$ is the number of bosons in the system and $t_D$ is the dynamical time ($N\sim 10^{57}$ and $t_D\sim 10\, {\rm hrs}$ for typical QCD axion stars; $N\sim 10^{96}$ and $t_D\sim 100\, {\rm Myrs}$ for the quantum core of a dark matter halo made of ultralight axions). Therefore, metastable equilibrium states can be considered as stable equilibrium states in practice. We compare our results with similar results obtained for Bose-Einstein condensates in laboratory, globular clusters and self-gravitating Brownian particles in astrophysics, the Brownian mean field model (BMF) in statistical mechanics, and bacterial populations in biology. Our presentation parallels the calculation of the quantum tunneling rate of dilute axion stars given in a previous paper [P.H. Chavanis, Phys. Rev. D 102, 083531 (2020)]. These calculations can find application in various domains of physics and astrophysics.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper derives exponentially long thermal lifetimes ~e^N t_D for metastable axion stars near M_max via instanton theory on the saddle-node normal form, but the result hinges on that reduction holding without gravitational corrections.

read the letter

The main point is that thermal activation lifetimes of metastable dilute axion stars near maximum mass scale as e^N times the dynamical time. For the particle numbers in QCD axion stars or ultralight axion halos this makes the lifetimes enormous, so the states are stable for all practical purposes. The paper obtains this by applying instanton theory to the thermal activation rate and using the normal form of the saddle-node bifurcation that appears at M_max. It supplies explicit analytical results close to that mass and parallels the 2020 quantum tunneling calculation. Comparisons to laboratory BECs, globular clusters, Brownian mean-field models, and bacterial populations give context. The derivation is straightforward within the mean-field framework and the exponential scaling follows from the N-dependence of the barrier height in the reduced potential. The soft spot is the assumption that the full dynamics reduce exactly to the saddle-node normal form without additional corrections from the long-range gravity or the underlying axion field theory. If those introduce terms that alter the instanton action by amounts independent of N, the exponent would no longer be strictly linear in N. The abstract states the result directly but does not report tests of the approximation's range or higher-order contributions. This paper is for researchers who model axion dark matter or self-gravitating condensates and need concrete lifetime estimates. Readers who work with analytical methods for metastable states in gravitational systems will extract useful formulas and scaling arguments. It shows clear engagement with the relevant literature and prior results. The work merits a serious referee to examine the instanton calculation and the validity of the normal-form reduction. I recommend sending it to peer review.

Referee Report

2 major / 2 minor

Summary. The manuscript computes the thermal activation rate of metastable dilute axion stars (self-gravitating Bose-Einstein condensates with attractive interactions) near the maximum mass M_max using instanton theory applied to the normal form of the saddle-node bifurcation. It derives an explicit lifetime scaling t_life ∼ e^N t_D (with N the boson number and t_D the dynamical time) and concludes that the metastable states are effectively stable for astrophysical parameters (N ∼ 10^57 for QCD axion stars; N ∼ 10^96 for ultralight axion dark-matter cores). The presentation parallels the author's prior quantum-tunneling calculation and includes comparisons to laboratory BECs, globular clusters, the Brownian mean-field model, and bacterial populations.

Significance. If the central derivation holds, the work supplies a clean analytical framework for practical stability of axion stars and analogous systems, with the e^N scaling providing a falsifiable prediction that lifetimes vastly exceed dynamical times. The explicit reduction to the saddle-node normal form near M_max and the direct parallel to the 2020 quantum-tunneling paper are genuine strengths, enabling quantitative cross-field comparisons without additional free parameters.

major comments (2)

[derivation of normal-form reduction (near the M_max expansion)] The reduction of the Gross-Pitaevskii-Poisson functional to the canonical saddle-node normal form (used to obtain the instanton action) is load-bearing for the linear-in-N exponent. The manuscript should supply the explicit expansion coefficients of the next-order terms in the effective potential and demonstrate that they remain irrelevant within a few percent of M_max; without this, O(1) corrections to the bounce action cannot be ruled out.
[instanton action evaluation (thermal activation rate section)] The instanton (bounce) calculation in the reduced model omits possible O(1) corrections arising from the long-range gravitational potential, the finite-temperature fluctuation spectrum around the saddle, or relativistic corrections to the underlying axion field theory. An estimate or scaling argument showing these corrections do not alter the leading e^N behavior is required to support the practical-stability claim.

minor comments (2)

[abstract and parameter estimates] The dynamical time t_D is quoted as ∼10 hrs for QCD axion stars and ∼100 Myr for ultralight axion cores; a precise definition (e.g., in terms of the central density or orbital period) would aid reproducibility.
[comparison section] A compact table comparing the activation exponents, prefactors, and relevant N ranges across the laboratory BEC, globular-cluster, BMF, and biological cases would clarify the cross-disciplinary parallels.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive overall assessment and the detailed, constructive comments on our manuscript. We address each major comment point by point below and will incorporate the requested additions and clarifications in the revised version.

read point-by-point responses

Referee: [derivation of normal-form reduction (near the M_max expansion)] The reduction of the Gross-Pitaevskii-Poisson functional to the canonical saddle-node normal form (used to obtain the instanton action) is load-bearing for the linear-in-N exponent. The manuscript should supply the explicit expansion coefficients of the next-order terms in the effective potential and demonstrate that they remain irrelevant within a few percent of M_max; without this, O(1) corrections to the bounce action cannot be ruled out.

Authors: We appreciate the referee highlighting the importance of justifying the normal-form reduction. In the revised manuscript we will explicitly compute and tabulate the coefficients of the next-order terms in the expansion of the effective potential about the saddle-node bifurcation point. We will further show, both analytically and with a numerical check, that these higher-order contributions remain below a few percent for mass deviations |M - M_max|/M_max ≲ 0.05, which is the relevant regime for long-lived metastable states. This establishes that they do not generate O(1) corrections to the leading instanton action, thereby preserving the linear-in-N scaling of the exponent. revision: yes
Referee: [instanton action evaluation (thermal activation rate section)] The instanton (bounce) calculation in the reduced model omits possible O(1) corrections arising from the long-range gravitational potential, the finite-temperature fluctuation spectrum around the saddle, or relativistic corrections to the underlying axion field theory. An estimate or scaling argument showing these corrections do not alter the leading e^N behavior is required to support the practical-stability claim.

Authors: We agree that a quantitative discussion of possible sub-leading corrections strengthens the practical-stability conclusion. In the revision we will add a dedicated paragraph providing scaling estimates: (i) the long-range gravitational potential is already incorporated at mean-field level and its fluctuation corrections around the saddle are suppressed by additional powers of 1/N; (ii) finite-temperature fluctuations about the instanton saddle contribute only to the prefactor, not to the leading exponential; (iii) relativistic corrections to the axion field theory are negligible for the dilute, non-relativistic regime considered here (v/c ≪ 1). Because the leading bounce action itself scales linearly with N, any O(1) additive corrections leave the dominant e^N dependence of the lifetime unchanged. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper derives the thermal activation rate analytically via instanton theory applied to the standard normal form of the saddle-node bifurcation near M_max (referenced from prior work). The claimed lifetime scaling t_life ∼ e^N t_D emerges directly from evaluating the instanton action in that reduced model, where the barrier height and dynamical time yield the exponential dependence on particle number N as a calculational output rather than an input or fit. Self-citations to the author's earlier papers on M_max and quantum tunneling are present but not load-bearing for the new thermal result, which rests on the explicit instanton computation within the effective Gross-Pitaevskii-Poisson framework. No step reduces the final expression to a redefinition or tautology of the inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Limited information from abstract; no explicit free parameters fitted in the summary, but relies on prior definitions of M_max and the bifurcation normal form.

axioms (2)

domain assumption The system is a dilute axion star described by self-gravitating BEC with attractive interaction
Basis for applying the model.
domain assumption Instanton theory can be applied to compute thermal activation rates in these gravitational systems
Core method used.

pith-pipeline@v0.9.0 · 5595 in / 1265 out tokens · 65095 ms · 2026-05-07T13:42:32.529096+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

297 extracted references

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The difference of potential between the local mini- mum and the local maximum creates a barrier of height ∆V=V(x U)−V(x M). In classical mechanics, the mo- tion of the particle is described by Newton’s equation m d2x dt2 =− dV dx .(E1) In quantum mechanics, the particle is represented by a complex wave functionψ(x, t) whose evolution is gov- erned by the ...
[2]

Feynman path integral Dirac [198–200] suggested that there is an interesting relation between the quantum propagator⟨x|e−iHt/ℏ |x0⟩, which gives the probability amplitude of the particle in xat timetassuming that it was located inx 0 at time t0 = 0, and the classical action of a particle of massmin a potentialV(x): Sclass[x(t)] = Z 1 2 m˙x2 −V(x(t)) dt= Z...
[3]

Euclidean action The functional integral (E5) requires delicate integra- tions on oscillatory functions. To overcome this prob- lem, it is convenient to study its analytical continuation to purely imaginary times for which the corresponding Feynman-Kac functional integral is well-defined. To this end we make the substitutiont→ −it(rotation ofπ/2 in the co...
[4]

quantum instanton

Quantum escape rate We now specifically apply the path integral formalism to the computation of the escape rate of a quantum par- ticle out of a metastable state. As discussed at the begin- ning of this Appendix, we assume that the potentialV(x) has a local minimum atx M (metastable equilibrium), a maximum atx U (unstable equilibrium) and a global min- im...
[5]

managed to determine the prefactor explicitly by showing that the fluctuation determinant can be evalu- ated exactly with the Gelfand-Yaglom [206] method. 61 Eq. (E10) is the dynamical generalization of the WKB formula [92], which gives the stationary solution forℏ→0. 44
[6]

The semiclassical limit and the instanton In the semiclassical approximationℏ→0, we have to determine the trajectoryx c(t) that minimizes the eu- clidean actionS[x(t)] defined by Eq. (E7). The station- ary points ofS[x(t)] can be obtained in different manners. First method:The extremal conditionδS/δx= 0 (least action principle) applied to Eq. (E7) yields ...
[7]

WKB formula We can now repeat the calculations detailed in Sec. IV of [90] and find that the quantum tunneling rate (decay probability per unit time) is given by Γ∼ r mωM v2 M πℏ e − 2 ℏ R xM x′ M √ 2m[V(x)−V(x M)]dx ,(E16) whereω 2 M =V ′′(xM)/mis the squared pulsation of the particle at the metastable minimum of the potential and vM, which depends on th...
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tunnel effect

Early literature on quantum tunneling The problem of escape from metastable states by quan- tum mechanical tunneling 62 occurs in many domains of physics. The probability that a quantum particle remains in a metastable state decreases exponentially rapidly with time as P(t)∼e −Γt ∼e −t/tlife ,(E19) where Γ is the rate coefficient andtlife ∼1/Γ is the typi...

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The work of Langer

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to the problem of first order phase transitions in statistical mechanics was also mentioned by Stone [104] at the end of his paper. 67 The fact that quantum tunneling is very improbable and can be neglected in many (but not all) cases of physical interest was 46 Appendix F: Derivation of the escape rate of a Brownian particle from a metastable state by us...
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We assume that the potentialV(x) has a local minimum atx M and a local maximum atx U

The Smoluchowski equation We consider an overdamped particle of massmand mobilityµ= 1/ξm(whereξis the friction coefficient) evolving in a one-dimensional potentialV(x). We assume that the potentialV(x) has a local minimum atx M and a local maximum atx U. It may also possess a global minimum atx S or be unbounded from below. We shall consider this second p...
[12]

(see the discussion in [210]). Identifying the equilibrium state of the Smoluchowski equation with the canonical Boltzmann distribution Peq(x) = 1 Z e−βV(x) ,(F5) whereβ= 1/k BTis the inverse temperature of the bath andZa normalization factor, we find that the diffusion coefficient, the friction coefficient and the temperature are related to each other by...
[13]

(F4) reduces to the diffusion equation

and Eq. (F4) reduces to the diffusion equation. Suppose that, at the initial time, the particle is local- ized in the local minimum of the potential. According to the deterministic laws of motion [see Eq. (F1)] it cannot go away from it. This minimum is a stable equilibrium state in which the particle is at rest. For a small pertur- bation, the overdamped...
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Wiener path integral and Onsager-Machlup functional The distribution of a Gaussian white noiseη(t) is given by P[η(t)]∝e − 1 2 R tf ti η2(t)dt .(F7) The probability of the pathx(t) is then obtained by ex- pressingη(t) in terms ofx(t) using Eq. (F2). This gives P[x(t)]∝e −S[x(t)]/kB T ,(F8) where S[x(t)] = ξm 4 Z ˙x+ V ′(x(t)) ξm 2 dt(F9) is the so-called ...
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instanton

Thermal escape rate We now specifically apply the path integral formal- ism to the computation of the escape rate of a Brow- nian particle out of a metastable state. As discussed at the beginning of this Appendix, we assume that the potentialV(x) has a local minimum atx M (metastable equilibrium), a maximum atx U (unstable equilibrium) and a global minimu...
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The weak noise limit and the instanton In the weak noise limitT→0, we have to deter- mine the trajectoryx c(t) that minimizes the Onsager- Machlup functionalS[x] defined by Eq. (F9). As indi- cated previously, the Onsager-Machlup functional (F9) can be viewed as an actionS[x(t)]. It can be written as S= Z L(x,˙x)dt,(F13) where L(x,˙x) = ξm 4 ˙x+ V ′(x) ξm...
[17]

Kramers formula We can now repeat the calculations detailed in Sec. V and find that the thermal activation rate (decay proba- bility per unit time) is given by Γ∼ ωM |ωU | 2πξ e−∆V /kB T ,(F20) whereω 2 M =V ′′(xM)/mandω 2 U =−V ′′(xU)/mare the squared pulsations of the fictive particle at the metastable minimum and at the maximum of the po- tential (the ...
[18]

Early literature on thermal activation The problem of escape from metastable states un- der the effect of thermal noise occurs in many domains of physics. The probability that a Brownian particle remains in a metastable state decreases exponentially rapidly with time as P(t)∼e −Γt ∼e −t/tlife ,(F22) 49 where Γ is the rate coefficient andtlife ∼1/Γ is the ...
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and Arrhenius [94] discovered empirically that the rate of escape follows the law from Eq. (105). Early theoretical works were made by Farkas [245] and Becker and D¨ oring [246] in the context of homogeneous nucleation of supersaturated vapors, and by Polanyi and Wigner [247] and Eyring [248] in chemical physics. In particular, Eyring [248] developed the ...
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via the calculation of the mean first passage time. The calculation of thermal activation rates for over- damped Brownian particles in the weak noise limit from the path integral formalism based on the Onsager- Machlup functional was performed by Caroliet al.[251, 252] and Weiss [253] by using the analogy with quantum tunneling. It was also developed by B...
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Actually, the theory of Brown- ian motion is the simplest example of application of On- sager’s linear thermodynamics

Relation to Onsager’s linear thermodynamics In this section we make the connection between the previous results and the linear thermodynamics of On- sager [157, 255, 256]. Actually, the theory of Brown- ian motion is the simplest example of application of On- sager’s linear thermodynamics. 70 We follow a presenta- tion similar to the one developed in [156...
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From the Smoluchowski equation to the Schr¨ odinger equation in imaginary time Let us consider an overdamped Brownian particle of massmevolving in a one-dimensional potentialV(x). The probability densityP(x, t) of finding the Brownian particle inxat timetis governed by the Smoluchowski equation ∂P ∂t = ∂ ∂x D ∂P ∂x +βP V ′(x) ,(G1) where the diffusion coe...
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(G16) 73 There is also a long list of works that followed the seminal paper of Nelson [139] on stochastic quantization (see a detailed list of references in [140])

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