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Setting up stasis with gravitational interactions

T0 review · 3 major / 4 minor · reviewed 2026-08-07 · deepseek-v4-flash

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Pith's one-line read Gravity alone can set up stasis in the early universe.

desk verdict A solid, narrow result: PBH evaporation can seed stasis towers with the right mass scalings, with one unquantified no-thermalization assumption that is probably safe. read the letter →

arxiv 2506.04502 v1 pith:ARI4K5FC submitted 2025-06-04 hep-ph

Andrew J. Long , Barmak Shams Es Haghi , Moira Venegas This is my paper
classification hep-ph PACS 98.80.Cq04.70.-s
keywords cosmologicalstasisprimordialblackholeevaporationHawkingradiationgravitationalparticleproductiontowerpower-lawrelicabundancesearlyuniversecosmologyBoltzmannequation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper claims that the initial condition needed for cosmological stasis—a long epoch when matter and radiation keep fixed energy fractions because a tower of heavy particles decays gradually into radiation—can be produced by gravitational interactions alone. It studies two purely gravitational production channels: Hawking evaporation of primordial black holes (PBHs) and cosmological gravitational particle production (CGPP) during inflation. For PBH evaporation, the abundance of particles on tower level $l$ scales as $\Omega_l \propto m_l^{+1}$ for particles lighter than the initial black hole temperature and $\Omega_l \propto m_l^{-1}$ for heavier particles. For CGPP, the paper reports distinct spin-dependent scalings $\Omega_l \propto m_l^{2}$ for spin-$1/2$ and $\Omega_l \propto m_l^{0}, m_l^{1/2}$ for bosons. Because these exponents can satisfy the stasis inequalities for reasonable tower parameters, the paper concludes that the stasis initial condition is reasonably generic rather than an engineered coincidence.

What carries the argument

The central object is the tower of $N$ unstable particle species with masses $m_l = m_1 + (l-1)^\delta \Delta m$ and decay rates $\Gamma_l = (m_l/m_1)^\gamma \Gamma_1$, whose initial abundances must follow $\Omega_l \propto m_l^{\alpha}$ with $-1/\delta < \alpha \le \gamma/2 - 1/\delta$ for stasis to occur. In the PBH scenario, the mechanism is the broken mass scaling of Hawking-emission number densities: particles lighter than the initial black hole temperature are emitted with $n_l \propto m_l^{0}$, while heavier particles are emitted with $n_l \propto m_l^{-2}$; after cosmological redshifting makes the particles non-relativistic and the tower dominates radiation, these translate into $\Omega_l \propto m_l^{+1}$ and $\Omega_l \propto m_l^{-1}$. The paper carries the argument with a Boltzmann equation in which the evaporating PBHs act as a source term, and it identifies the break scale $p_{*,l}$ in the comoving-momentum spectrum that separates the rising and falling branches of the distribution. In the CGPP scenario, the mechanism is the spin-dependent spectrum of particles produced by the expanding spacetime background during and after inflation.

What would settle it

A direct calculation of the scattering or annihilation rate of the emitted tower particles with the radiation bath would settle the claim: if any number-changing reaction remains in equilibrium after PBH evaporation, so that $\Gamma_{\rm scatter} > H$ at $t_{\rm ev}$, then the comoving density conservation assumed in eq. (2.24) fails and the broken power law $\Omega_l \propto m_l^{\pm 1}$ would be replaced by a thermal spectrum.

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Extended reading notes

Core claim

On the paper's own terms, the discovery is that two minimal gravitational mechanisms naturally produce the power-law abundance spectrum that cosmological stasis requires. For a monochromatic population of PBHs evaporating in a radiation-dominated universe, the solution of the momentum-dependent Boltzmann equation with a Hawking source term gives final energy fractions scaling as $\Omega_l \propto m_l$ for $m_l < T_0$ and $\Omega_l \propto m_l^{-1}$ for $m_l > T_0$, where $T_0$ is the initial black hole temperature. The paper validates these analytic scalings by direct numerical integration, and shows that both cases can be compatible with the stasis condition $-1/\delta < \alpha \le \gamma/2 - 1/\delta$ depending on the tower parameters. For CGPP, using abundance formulas from the literature, the paper finds $\Omega_l \propto m_l^{\alpha}$ with $\alpha=2$ for spin-$1/2$ particles and $\alpha=0$ or $1/2$ for spin-$0$ and spin-$1$ particles, which are again compatible with stasis for suitable $\delta$ and $\gamma$. The paper's central conclusion is that gravitational particle production, in either channel, can fill the tower with the right relative abundances for stasis to begin.

Load-bearing premise

The load-bearing premise is that the particles emitted by the black holes neither scatter, annihilate, nor decay until well after evaporation, so their comoving number density is conserved and the predicted $\Omega_l \propto m_l^{\pm 1}$ scalings survive to the stasis epoch.

Editorial extensions

If this is right

  • If PBH evaporation populates the tower with particles lighter than $T_0$, the natural stasis exponent is $\alpha = +1$, which satisfies the stasis inequality for the fiducial choice $\delta=1$, $\gamma=5$.
  • If the tower masses all exceed $T_0$, the natural exponent is $\alpha = -1$, which is viable for stasis when $0 < \delta < 1$ for $\gamma=5$.
  • The decay rates $\Gamma_l$ can be chosen so that the tower dominates, stasis proceeds, and the universe returns to radiation domination above the MeV scale, preserving big bang nucleosynthesis.
  • Under the most optimistic CGPP parameters, gravitationally produced particles can dominate as early as $T \approx 1$ TeV, leaving ample time for a stasis phase before nucleosynthesis.
  • Because PBH evaporation and CGPP predict different exponents ($\alpha = \pm1$ versus $\alpha = 2, 0, 1/2$), a detected stasis epoch would carry a fingerprint of which gravitational production mechanism populated the tower.

Reading between the lines

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

  • If a tower straddles $m_l = T_0$, the predicted $\Omega_l$ is a broken power law rather than a single monomial, so extending stasis analyses to non-monomial abundance spectra would be a natural next step; the paper notes this possibility but does not develop it.
  • The setup requires that emitted tower particles neither scatter, annihilate, nor decay until after they become non-relativistic; translating this into explicit upper bounds on the tower particles' gauge or Yukawa couplings would provide a concrete, testable constraint on the scenario.
  • The CGPP spin-$1/2$ result $\alpha = 2$ becomes compatible with stasis if $\gamma = 7$, which suggests connecting the tower decay operator's dimension to the stasis viability condition in explicit particle models.
  • In theories with compact extra dimensions, PBH evaporation can be modified when the black hole radius approaches the compactification scale, potentially changing the predicted exponents; the paper flags this as an avenue for future work.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 4 minor

Summary. The paper asks whether the initial conditions required for cosmological stasis—a tower of non-relativistic, decaying particles with abundances Ω_l ∝ m_l^α—can arise from purely gravitational particle production. For PBH evaporation, the authors model Hawking emission with a geometric-optics greybody factor, derive analytic scalings for the total emitted number N_l and the resulting energy fractions under a step-function approximation for the emission rates, and then solve the Boltzmann equation numerically for the phase-space distribution. They find Ω_l ∝ m_l for m_l < T_0 and Ω_l ∝ m_l^{-1} for m_l > T_0, i.e. α = +1 or −1, and they verify compatibility with the stasis inequalities (1.4). They also construct the required cosmological time ordering and map viable parameter regions in Fig. 8. For CGPP, they quote standard results from the literature and obtain α = 2 for spin-1/2 particles and α = 0 or 1/2 for spin-0 and spin-1 particles. The overall conclusion is that both gravitational mechanisms can supply the stasis initial condition, with distinct, mechanism-dependent exponents.

Significance. If the central result holds, the paper identifies a minimal and fairly generic gravitational origin for the non-thermal initial condition required by stasis, which is a valuable addition to the stasis program. The main strengths are the clean analytic scalings in Sec. 2, the direct numerical solution of the Boltzmann equation that validates the step-function and total-number approximations, the explicit broken power-law prediction Ω_l ∝ m_l^{±1} around m_l = T_0, and the concrete parameter-space map in Fig. 8 that encodes the necessary time-ordering constraints. The CGPP comparison is brief but provides falsifiable, mechanism-distinguishing exponents. The central caveat is that the PBH-derived scaling relies on the emitted tower particles being collisionless from evaporation until decay; this assumption is stated but not quantified, and the same interactions that make the tower decay also mediate scattering and annihilation.

major comments (3)
  1. The derivation of Ω_l ∝ m_l^{±1} assumes that the comoving number density of each emitted species is conserved from t_ev until its decay, as stated in Eq. (2.24), and the Boltzmann treatment includes only the PBH source term. The paper acknowledges this only qualitatively ('provided that the particles in the tower interact very weakly'), but it never quantifies the condition. The couplings required for the tower to decay through Eq. (1.2) with γ = 5 also mediate elastic scattering and annihilation with the radiation bath and among tower states. If any of these rates exceeds H before the species decays, the spectrum re-thermalizes or depletes and the broken power law in Eq. (2.45) no longer follows. The authors should compute or bound Γ_scat/H and Γ_ann/H for the benchmark point and for the allowed region of Fig. 8, or derive an explicit condition on the underlying coupling/scale Λ that guarantees the collisionless regime. This is load-bearing because the stasis initial condition is precisely the power-law abundance spectrum.
  2. The prediction α = ±1 rests on the geometric-optics approximation to the greybody factor, b_S^2 = (27/4) R_S^2, used in both the analytic estimates and the numerical Boltzmann solution. The numerical validation therefore does not test this approximation. For initially heavy particles (m_l > T_0), emission occurs near the kinematic threshold, where the true greybody factors are spin- and momentum-dependent and can differ substantially from geometric optics. The authors assert that these factors 'would not significantly impact the total number of emitted particles,' but no estimate is given. Since the central claim is the mass scaling of the abundances, the robustness of α = ±1 to actual greybody factors should be demonstrated or at least estimated, for example by comparing with Page's spin-dependent emission rates for a representative heavy mass.
  3. The CGPP scenario inherits the same collisionless assumption as the PBH scenario: the gravitationally produced particles are assumed to preserve their number density until they decay and drive stasis. The paper does not check whether the couplings that provide the decay rates Γ_l in Eq. (1.2) also thermalize or annihilate the CGPP products during reheating or radiation domination. Because the quoted formulas for Ω_l assume free propagation after production, the prediction α = 2, 0, 1/2 is only valid if scattering/annihilation is negligible over the relevant cosmic time; this condition should be stated and quantified, at least parametrically, for the fiducial H_e and T_rh values.
minor comments (4)
  1. The name 'Schwartzchild' appears several times; it should be 'Schwarzschild'.
  2. The symbol P(t) in Eq. (2.2) is defined as a sum over species, but P_l(t) appears later in Eq. (2.9); the distinction between the per-species and total power should be made explicit at first use.
  3. For m_l > T_0 the statement a(t_nr,l)/a(t_ev) ~ 'few' is vague; since p_*,l and p_rms,l are computed in Fig. 4, specifying the corresponding numerical ratio (e.g., using p_* or p_rms) would make the estimate concrete.
  4. The dashed guide lines labeled ρ_l ∝ m_l and ρ_l ∝ m_l^{-1} are helpful, but in Fig. 5 the slope labels are easy to miss because they overlap the data; placing the labels in empty regions or using distinct colors would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the PBH scaling is derived from the Hawking spectrum, and the CGPP formulas are cited from an independent review rather than fitted to the stasis target.

full rationale

No circular derivation is present. The PBH branch (the core new result) starts from the Hawking spectrum in eq. (2.7), integrates it to the closed-form yield eq. (2.22) under the stated step-function approximation eq. (2.20), and obtains the stasis scaling by evaluating Omega_l = m_l n_l / sum at late times; the exponents alpha = +1 and alpha = -1 follow algebraically from n_l proportional to m_l^0 for m_l < T0 and n_l proportional to m_l^{-2} for m_l > T0. No quantity is fitted to make eq. (2.45) match eq. (1.3), and the stasis inequalities are taken from ref. [4], whose authors do not overlap with the present paper. The CGPP scaling eq. (3.3) is quoted from ref. [44], a review co-authored by A.J. Long, but those formulas are parameter-free published calculations whose assumptions (inflationary expansion, spin-dependent couplings to gravity) do not include the stasis initial condition; this is a self-citation used as standard literature, not a load-bearing circular input. The one clearly flagged limitation, the assumption that emitted tower particles preserve comoving number density until decay, eq. (2.24), and the neglect of scattering and decay terms in the collision operator ('provided that the particles in the tower interact very weakly'), is an unquantified physical robustness condition, not a circularity: the derivation is explicitly conditional on that assumption and is checked numerically via the Boltzmann equation. The paper is therefore self-contained against external Hawking-radiation benchmarks and receives a score of 0.

Assumptions & free parameters 5 free parameters · 7 assumptions · 0 invented entities

The paper introduces no new particles, forces, or dimensions. The assumptions are dominated by standard early-universe cosmology and the prior stasis framework. The only ad hoc modeling choice specific to this paper is the step-function emission approximation, which is checked numerically. The main free parameters are PBH properties, tower mass and decay parameters, and CGPP environmental scales.

free parameters (5)
  • PBH initial mass M0 = 1 gram in fiducial examples, scanned from 10^-3 to 10^3 grams in Fig. 8
    Sets the initial PBH temperature T0 = M_Pl^2 / (8π M0) and the boundary m_l = T0 in the broken power law. Chosen by hand.
  • PBH energy fraction at formation β = 10^-8 in fiducial examples, scanned from 10^-12 to 1 in Fig. 8
    Controls the PBH number density n_pbh,0 and the overall relic abundance of emitted particles. Chosen by hand.
  • Tower mass spectrum parameters m1, Δm, δ, N = m1 = Δm = 10^-3 T0 (always light) or 10^2 T0 (initially heavy), δ = 1, N = 100
    Defines the mass tower in eq. (1.1) whose abundances are the stasis initial condition. These values are illustrative.
  • Decay rate normalization Γ1 and index γ = Γ1 = 10^-40 M_Pl and γ = 5 in the fiducial model
    The decay hierarchy in eq. (1.2) makes stasis possible; in a complete model γ would be fixed by an operator dimension, but here it is chosen.
  • CGPP environment parameters H_e and T_rh = H_e = 10^14 GeV and T_rh = 10^15 GeV in the optimistic scenario
    The CGPP abundance formulas (3.1)-(3.2) depend on the end-of-inflation Hubble scale and reheating temperature; the optimistic values are at the high end allowed by CMB constraints.
assumptions (7)
  • domain assumption Hawking radiation is described by a blackbody spectrum with a geometric-optics greybody factor b_S = 3√3 R_S/2 (eq. 2.7).
    Standard result from Page used to compute all emission rates; not re-derived in this paper.
  • domain assumption The early universe is radiation-dominated at PBH formation and evaporation, with H = 1/(2t) (eq. 2.11).
    Used throughout for t_M, t_nr and redshift factors; the paper restricts to β < β_c so PBHs evaporate before domination.
  • domain assumption All PBHs are monochromatic, non-spinning, uncharged, and do not merge or accrete appreciably.
    Simplifies PBH evolution; extensions to spinning or mass-distributed PBHs are not performed.
  • domain assumption Tower particles do not thermalize, annihilate, or scatter appreciably, so comoving number density is conserved from evaporation until decay (eq. 2.24).
    If this fails, n_l ∝ a^{-3} breaks down and the Ω_l ∝ m_l^{±1} scaling is not guaranteed.
  • ad hoc to paper The step-function approximation f_N ≈ f_E ≈ θ(T_pbh - m_l) (eq. 2.20) captures the mass scaling of emitted number densities.
    Introduced for analytic tractability; the paper checks it against numerical integration, but it is not a rigorous bound.
  • domain assumption The stasis existence conditions (1.4)-(1.5) and the power-law tower parametrization (1.1)-(1.3) from ref. [4] are valid.
    The paper builds on prior stasis work rather than re-deriving the stasis dynamics.
  • domain assumption The CGPP abundance formulas (3.1)-(3.2) from ref. [44] are correct and apply to the tower states.
    The CGPP section is a literature-based estimate, not a new computation.

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Cite this review

Pith. "Pith review of Setting up stasis with gravitational interactions." pith.science (2026). https://pith.science/paper/ARI4K5FC

@misc{pith2026250604502,
  author       = {Pith},
  title        = {Pith review of: Setting up stasis with gravitational interactions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ARI4K5FC}},
  note         = {Machine review of arXiv:2506.04502}
}
abstract

An epoch known as cosmological stasis may have taken place in the early Universe. During matter-radiation stasis, a population of non-relativistic particles with different masses gradually decay into relativistic particles, and the effective equation of state $w$ remains approximately constant at a value between that of matter ($w=0$) and that of radiation ($w=1/3$). In this work, we investigate how to set up the appropriate initial conditions for stasis using gravitational interactions. We consider two scenarios: that the tower of non-relativistic particles is populated by the evaporation of primordial black holes (PBHs) and that the tower is populated by cosmological gravitational particle production (CGPP) during inflation. We calculate the abundance of particles on different levels of the tower to assess whether stasis is viable. We find that both scenarios can provide the needed initial conditions for stasis, and that they predict distinctive scaling exponents $\Omega_l \propto m_l^\alpha$ with mass $m_l$.

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Forward citations

Cited by 3 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Detecting Cosmological Stasis with Future Gravitational Wave Observatories

    hep-ph 2026-07 conditional novelty 6.0 of 10

    Future detectors, BBO in particular, could detect the gravitational-wave imprint of a cosmological stasis epoch across much of its parameter space; known particle-physics transitions would add fixed calibration steps.

  2. The Pull of Stasis: A Study of the Dynamics of the Thermal Stasis Attractor

    astro-ph.CO 2026-07 conditional novelty 5.0 of 10

    An explicit three-scalar model realizes a thermal stasis attractor with a finite lifetime, whose fast and slow trajectories make the duration of stasis strongly initial-condition dependent.

  3. Gravitational Wave Signatures of Cosmological Stasis: A Unified Spectral Template

    hep-ph 2026-07 conditional novelty 5.0 of 10

    Any constant-w stasis epoch imprints a closed-form IGWB template whose tilt α and amplitude step C² must lie on a single falsifiable consistency curve C²(α), resolvable by BBO/DECIGO.

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Reviewed August 7, 2026 · model on record in the stance chip above.