Pith. sign in

REVIEW 3 major objections 8 minor 106 references

Reviewed by Pith at T0; open to challenge.

T0 means a machine referee read the full paper against a public rubric. The mark states how deep the mechanical check went, never who wrote it. the ladder, T0–T4 →

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

T0 review · glm-5.2

Lorentz-violating gravity boosts primordial black holes but breaks its own rules

2026-07-09 21:37 UTC pith:4EFQ23BU

load-bearing objection The tachyonic inconsistency result is clean and likely correct, but the derivation of the mass matrix M₂₂ is underdetermined, and the PBH abundance numbers are built on foundations the paper itself proves inconsistent. the 3 major comments →

arxiv 2607.07011 v1 pith:4EFQ23BU submitted 2026-07-08 gr-qc hep-th

Primordial black hole in Lorentz-violating theories: Insights from Bumblebee gravity

Mohsen Khodadi , Javad T. Firouzjaee This is my paper
classification gr-qc hep-th
keywords bumblebeelorentz-violatingvectorblackbreakingcosmologicalfieldfoundation
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper investigates whether primordial black holes (PBHs) could constitute dark matter in a modified gravity framework called Bumblebee gravity, where a vector field spontaneously breaks Lorentz symmetry by acquiring a vacuum expectation value and coupling to spacetime curvature. The authors derive the full set of modified cosmological perturbation equations and find that the Lorentz-violating coupling parameter lambda produces three independent enhancements to PBH abundance: it alters the cosmic expansion rate, lowers the density threshold for gravitational collapse, and amplifies the primordial power spectrum of density fluctuations. For modest coupling values (lambda ~ 0.1) and observationally consistent inflationary parameters, these combined effects raise PBH production by roughly two orders of magnitude relative to general relativity, potentially making PBHs a viable dark matter candidate in the asteroid-mass window (10^17 to 10^23 grams) where observational constraints remain permissive. However, the authors then construct the complete quadratic action for perturbations in the uniform inflaton gauge and identify two intrinsic instabilities in the vector sector. First, the longitudinal perturbation of the timelike vector field carries a wrong-sign kinetic term (a ghost), which originates from the Maxwell-form kinetic term lacking the degeneracy structure needed to eliminate extra propagating degrees of freedom. Second, and more fundamentally, the potential that enforces spontaneous Lorentz symmetry breaking must have positive curvature at its minimum (V''(0) > 0) to be a stable equilibrium, but this same positive curvature gives the vector perturbation a negative mass squared, producing a tachyonic instability that disrupts the cosmological background on a timescale of roughly 10^{-43} seconds. Conversely, choosing V''(0) < 0 to avoid the tachyon eliminates the minimum, so the symmetry-breaking vacuum is not dynamically maintained. The paper concludes that within the minimal Bumblebee framework, the mathematical conditions required for spontaneous symmetry breaking and those required for cosmological stability are mutually exclusive, rendering the model internally inconsistent and its PBH predictions illustrative rather than physically realized.

Core claim

The central finding is a no-go result: in the minimal Bumblebee gravity model with a timelike vector vacuum expectation value, the condition V''(0) > 0 that defines a stable symmetry-breaking minimum simultaneously produces a tachyonic mass squared m^2 = -4 b_0^2 V''(0) / M_p^2 < 0 for the vector perturbation, while the requirement V''(0) < 0 needed to avoid this tachyon destroys the minimum itself. This creates a logical contradiction at the level of the model's definition, independent of parameter choices. The ghost instability in the vector sector is a separate pathology arising from the Maxwell kinetic term's incompatibility with timelike condensates. Together, these instabilities mean B

What carries the argument

The argument proceeds through three layers. At the background level, the non-minimal coupling xi B^mu B^nu R_{mu nu} modifies the Friedmann equations via the combination lambda = xi b_0^2, changing the radiation-era scale factor to a(t) ~ t^{1/2 - lambda/8} and the effective gravitational constant to G_eff = 2G/(2+lambda). At the perturbation level, the modified growth exponent n = 2 + lambda for superhorizon density contrasts suppresses the collapse threshold as delta_c^{BG} ~ delta_c^{GR} exp(-lambda N), while the curvature power spectrum acquires an enhancement factor E(lambda) ~ 1 + 9 lambda / (16 epsilon). At the stability level, the quadratic action in uniform inflaton gauge yields a 2

Load-bearing premise

The ghost instability in the vector perturbation sector is asserted to be a physical pathology rather than an artifact of gauge choice or incomplete constraint analysis. This rests on the claim that all constraint equations in the uniform inflaton gauge have been correctly identified and solved, so that the wrong-sign kinetic term for the longitudinal vector mode survives all field redefinitions and cannot be eliminated.

What would settle it

A more careful or complete treatment of the full constraint structure of the Bumblebee model could potentially remove the ghost degree of freedom through a secondary constraint or degeneracy relation not captured in the quadratic action as written. Additionally, if the potential can be constructed such that the symmetry-breaking condition is enforced through a mechanism other than a local minimum (e.g., a non-analytic constraint or a non-standard kinetic structure), the tachyonic instability might be avoided while preserving the vacuum expectation value.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • If a ghost-free, non-tachyonic extension of Lorentz-violating vector gravity exists (e.g., via generalized Proca or Horndeski-type couplings with appropriate degeneracy conditions), the same three enhancement mechanisms could operate without theoretical inconsistency, potentially producing observable PBH abundances testable against gravitational-wave and microlensing data.
  • The no-go result provides a concrete diagnostic: any Lorentz-violating model using a timelike vector VEV with a standard potential must demonstrate that its symmetry-breaking minimum and its perturbation mass spectrum are simultaneously stable, a requirement that constrains model-building beyond mere phenomenological viability.
  • The mapping between the Lorentz-violating parameter lambda and PBH abundance means that future PBH searches (or non-detections) in the asteroid-mass window can be translated into bounds on Lorentz-violating couplings, even though the minimal model itself is inconsistent.
  • The tachyonic timescale tau ~ M_p^{-1} ~ 10^{-43} s sets a hard upper bound on the validity of any cosmological calculation in this framework: no inflationary or radiation-dominated epoch can last long enough for the background to be physically meaningful.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 8 minor

Summary. The manuscript investigates primordial black hole (PBH) formation in Bumblebee gravity (BG), a Lorentz-violating extension of GR with a vector field non-minimally coupled to curvature. The paper pursues two objectives: (1) deriving the phenomenological consequences for PBH abundance—modified Friedmann equations, modified density contrast evolution, modified power spectrum, and the resulting PBH mass function—and (2) assessing the theoretical consistency of the minimal BG model via a quadratic action analysis. The authors find that modest Lorentz-violating couplings ($0 < λ < 2/3$) enhance PBH production by 2–4 orders of magnitude, potentially making PBHs viable dark matter candidates in the asteroid-mass window. However, they also identify a tachyonic instability: the condition $V''(0) > 0$ required for a stable symmetry-breaking minimum simultaneously produces $m^2_{ghost} = -4b_0^2 V''(0)/M_p^2 < 0$, rendering the model cosmologically unviable. The authors conclude that the phenomenological enhancements are symptoms of an underlying theoretical inconsistency.

Significance. The paper tackles a timely and important question—whether Lorentz-violating gravity can enhance PBH production—and provides a candid assessment of the theoretical price. The systematic derivation of modified Friedmann equations (Eqs. 2.21), the density contrast evolution (Eq. 3.28), and the curvature power spectrum (Eq. 4.65) constitutes a useful contribution. The central inconsistency argument—that a stable minimum requires $V''(0)>0$ while stability of the $δb$ perturbation requires $V''(0)<0$—is logically compelling if the mass matrix element $M_{22}$ is correctly derived. The numerical PBH abundance calculations (Tables I–III, Fig. 1) provide a falsifiable mapping between Lorentz-violating parameters and PBH observables. The paper's honesty in presenting both the phenomenological promise and the theoretical pathology is commendable.

major comments (3)
  1. §IV.C, Eq. (4.31): The mass matrix element $M_{22} ≈ -2M_p^2 b_0^2 V''(0)$ is the load-bearing quantity for the paper's central claim of tachyonic instability. Yet its derivation is not shown. The constraint equation (4.24) is solved only in the subhorizon limit (Eq. 4.25: $δb_0 = δ̇b - ξb_0 α$), but $M_{22}$ is a mass term (no spatial derivatives) whose value should not depend on the subhorizon approximation. Tracing the origin of a bare $δb^2$ mass term proportional to $V''(0)$ is non-trivial: the potential term $L_V^{(2)}$ (Eq. 4.12–4.13) gives gradient terms via $V'(0)·a^{-2}(∂δb)^2$ and $(δb_0)^2$ terms via $V''(0)·(2b_0 δb_0)^2$, but after substituting the constraint $δb_0 = δ̇b - ξb_0 α$ and $α = ζ̇/H$, these become kinetic and mixing terms, not a bare $δb^2$ mass. The intermediate steps from the full quadratic action to Eq. (4.31) must be provided, or an explicit argument must be
  2. §IV.C, Eq. (4.31): The expression for $M_{22}$ contains a correction factor $(1 - λ^2/(2M_p^2(1+λ)))$ that appears dimensionally inconsistent: $λ$ is dimensionless while $M_p^2$ carries dimensions of mass$^2$, so $λ^2/M_p^2$ cannot be subtracted from 1. This needs correction or clarification. Given that $M_{22}$ is the quantity from which the tachyonic mass $m^2_{ghost} = M_{22}/G_{22}$ is derived (Eq. 4.38), this dimensional issue directly affects the quantitative claim $m^2_{ghost} = -4b_0^2 V''(0)/M_p^2$.
  3. §IV.C, Eq. (4.11) and surrounding text: The paper asserts that the ghost in the $δb$ sector is physical ('This ghost is not a gauge artifact; it is a physical pathology'). However, the subsequent analysis shows that after constraint elimination, $G_{22} = M_p^2/2·[1+λ] > 0$ (Eq. 4.31) and $det G > 0$ is satisfiable (Eq. 4.36). The paper itself concedes the ghost is 'mixed away into a heavy eigenstate.' This internal tension should be resolved: either the ghost is physical and cannot be removed (in which case $det G < 0$ should be unavoidable), or it is removable (in which case the framing in §IV.C and the abstract overstates the issue). The authors acknowledge in §IV.F that the ghost is 'secondary to the tachyon issue,' but the abstract and introduction still present it as an independent fatal pathology.
minor comments (8)
  1. §V.A: The choice $σ_{GR}(M) = 0.1$ is used throughout the numerical analysis but is not motivated. In standard cosmology, $σ ∼ 10^{-5}$ on CMB scales; $σ = 0.1$ implies a power spectrum enhancement of $P_ζ ∼ 10^{-2}$, which is enormous. The paper should clarify whether this represents a pre-enhanced spectrum (i.e., already tuned for PBH production in GR) or a specific model prediction.
  2. §III.B, Eq. (3.50): The expression $δ_c^{BG} ≈ δ_c^{GR} exp[-λN]$ is derived heuristically. The relation $a_h/a_i ∼ e^N$ is stated without specifying what $N$ represents precisely (e-folds from perturbation generation to horizon crossing). For PBH formation during radiation domination, this should be the number of e-folds between horizon exit during inflation and horizon re-entry, which depends on the comoving scale. This should be clarified.
  3. §II, Eq. (2.7): The text states $B_μ = (b(t), 0, 0, 0)$ but then says $b(t) = b_0$ (constant). The notation $b(t)$ is misleading if $b$ is constant; consider writing $B_μ = (b_0, 0, 0, 0)$ directly.
  4. §IV.E, Eq. (4.65): The power spectrum $P_ζ(k) = H^2/(16π^2 ε M_p^2 (1+λ)) · [1 + 9λ/(8ε(1+λ))]^{1/2}$ reduces to the standard result when $λ = 0$. However, the enhancement factor $E(λ) ≈ 1 + 9λ/(16ε)$ (Eq. 4.70) diverges as $ε → 0$, which would be problematic near the end of inflation. The paper should comment on the regime of validity.
  5. Table I–III: The enhancement factor $R$ is reported as nearly mass-independent (varying between 64.2 and 64.5 for $N=30$). This is stated to reflect 'scale-invariant' dependence, but the mass-dependence enters through $σ_{GR}(M)$, which is held fixed at 0.1. If $σ_{GR}$ varies with mass (as it should physically), the enhancement would acquire mass dependence. This caveat should be noted.
  6. §IV.B, Eq. (4.19): The expression $∇^2ψ/a^2 ≈ -3ζ̇ - (9/2)Hζ + O(ε, λ)$ is stated without derivation. A brief indication of how this follows from the Hamiltonian constraint would help.
  7. The reference list includes several 2025 and 2026 dated papers (e.g., Refs. [28–32], [57–60], [66], [75], [76]). The authors should verify these are correctly cited and that preprint versions are stable.
  8. §IV.C: The text uses both $b_0$ and $b$ for the background VEV (e.g., Eq. 4.7 uses $b_0$, Eq. 4.8 uses $b_0$, but Eq. 4.37 uses $b_0$ while the surrounding text sometimes refers to $b$). Consistent notation would improve readability.

Circularity Check

0 steps flagged

No significant circularity: the central inconsistency follows from independent mathematical requirements, and the PBH enhancement is derived from first-principles action variation, not fitted to data.

full rationale

The paper's central claim—that the minimal Bumblebee model is internally inconsistent because a stable symmetry-breaking minimum requires V''(0) > 0 while stability of the perturbation δb requires V''(0) < 0—is not circular. It follows from two independent mathematical facts: (1) the calculus definition of a local minimum (V'' > 0 at the minimum), and (2) the mass formula m²_ghost = M₂₂/G₂₂ = -4b₀²V''(0)/M²ₚ derived from the quadratic action (Eq. 4.38). Neither input is defined in terms of the other. The PBH enhancement factor E(λ) ≈ 1 + 9λ/(16ε) (Eq. 4.70) is derived from the sound speed c²_s (Eq. 4.45) and the kinetic matrix G_ij (Eq. 4.31), which come from expanding the action (2.1) to quadratic order—this is a first-principles derivation from the Lagrangian, not a fit to data renamed as a prediction. The phenomenological parameters (λ, ε, N, σ_GR) are inputs with stated physical values, not fitted outputs. The paper does cite prior work by the same authors (Refs. [50], [54], [55], [68], [69]) on Bumblebee gravity phenomenology, but these citations provide background context, not the load-bearing mathematical argument. The inconsistency proof is self-contained within this paper's equations. The skeptic's concern that the derivation of M₂₂ is underdetermined (constraint solved only in subhorizon limit) is a correctness/completeness issue, not a circularity issue—M₂₂ is not defined in terms of the conclusion it supports. No step in the derivation chain reduces to its own inputs by construction.

Axiom & Free-Parameter Ledger

6 free parameters · 5 axioms · 0 invented entities

No new entities are invented. The Bumblebee vector field and its non-minimal coupling are from prior literature [47,48].

free parameters (6)
  • λ = 0 to 0.1 (scanned)
    Dimensionless Lorentz-violating coupling λ=ξb²₀, treated as a free parameter scanned over [0, 0.1]
  • b₀ = ~M_p (assumed)
    Bumblebee VEV, assumed to be of order Planck scale; only enters through λ combination
  • ε = 0.006
    Slow-roll parameter fixed at CMB-compatible value
  • N = 30, 40
    Number of e-folds between perturbation generation and horizon crossing, chosen for illustration
  • σ_GR = 0.1
    GR variance, set by hand for numerical illustration
  • δ_c^GR = 0.414
    GR critical collapse threshold, taken from refs [14,15]
axioms (5)
  • domain assumption Maxwell kinetic term -1/4 B_μν B^μν for the Bumblebee vector field
    Eq. 2.1; this is the standard Bumblebee model definition from [47,48], but the Maxwell form is known to be problematic for timelike VEVs
  • domain assumption Timelike VEV B_μ = (b₀, 0, 0, 0) in flat FLRW
    Eq. 2.7; chosen to maintain spatial isotropy while breaking Lorentz symmetry
  • standard math Press-Schechter formalism for PBH mass fraction
    Eq. 5.1-5.3; standard cosmological assumption for PBH abundance calculation
  • domain assumption Slow-roll inflation with canonical scalar field
    Section II.A; assumed for background evolution and power spectrum derivation
  • domain assumption Gaussian primordial perturbations
    Eq. 5.2; standard assumption for PBH formation statistics

pith-pipeline@v1.1.0-glm · 28279 in / 4231 out tokens · 184533 ms · 2026-07-09T21:37:48.990350+00:00 · methodology

0 comments
read the original abstract

The Bumblebee gravity (BG) model, featuring spontaneous Lorentz symmetry breaking via a vector field non-minimally coupled to curvature, has been widely used to explore Lorentz-violating effects in cosmology. We investigate primordial black hole (PBH) formation within this framework, deriving the complete set of modified perturbation equations. We demonstrate that BG, sourced by a timelike vector field, introduces three distinct enhancements to PBH abundance--modified expansion history, suppressed collapse threshold, and amplified power spectrum--which together render PBHs viable dark matter candidates across the asteroid-mass window for modest Lorentz-violating couplings. However, a systematic analysis of the quadratic action reveals that these phenomenological consequences emerge from a theoretically pathological foundation. The vector sector exhibits an intrinsic ghost instability, while the requirement of a stable symmetry-breaking minimum simultaneously induces a tachyonic instability on timescales far below cosmological scales. The model thus suffers from a fundamental inconsistency: the conditions for cosmological viability and spontaneous symmetry breaking are mutually exclusive within the minimal Bumblebee framework. Our results illustrate both the notable power of Lorentz violation to influence early Universe observables and the necessity of a consistent theoretical foundation for such predictions.

Figures

Figures reproduced from arXiv: 2607.07011 by Javad T. Firouzjaee, Mohsen Khodadi.

Figure 1
Figure 1. Figure 1: FIG. 1. Density plot showing the full [PITH_FULL_IMAGE:figures/full_fig_p015_1.png] view at source ↗

discussion (0)

Sign in with ORCID, Apple, or X to comment. Anyone can read and Pith papers without signing in.

Reference graph

Works this paper leans on

106 extracted references · 106 canonical work pages · 73 internal anchors

  1. [1]

    This would mean the Lorentz- violating couplingξ= λ b2 0 should be positive for consistency with inflation

    is the physi- cally reasonable regime for slow-roll inflation in BG so that λ < 2 3 comes from requiring the kinetic energy to remain positive during slow-roll. This would mean the Lorentz- violating couplingξ= λ b2 0 should be positive for consistency with inflation. III. MODIFYING DENSITY CONTRAST EVOLUTION EQUATION We serve the Newtonian gauge ds2 =−(1...

    work page

  2. [2]

    2H ∇2α a2 −2 ∇2 ˙ζ a2 # ,(4.15) δLN M δψ =ξM 2 p a3

    Momentum Constraint (δS/δψ= 0) δLEH δψ =M 2 p a3 " 2H ∇2α a2 −2 ∇2 ˙ζ a2 # ,(4.15) δLN M δψ =ξM 2 p a3 " b2 0H ∇2α a2 −b 2 0 ∇2 ˙ζ a2 # .(4.16) Summing: M2 p a3(1 +λ) " 2H ∇2α a2 −2 ∇2 ˙ζ a2 # = 0.(4.17) Cancelling the non-zero factor2M 2 p a3(1 +λ)∇ 2/a2: Hα− ˙ζ= 0⇒α= ˙ζ H .(4.18) This is exact at linear order, no approximations

    work page

  3. [3]

    For subhori- zon modes (k≫aH), the∇ 2ψterms dominate

    Hamiltonian constraint (δS/δα= 0) The full Hamiltonian constraint is lengthy. For subhori- zon modes (k≫aH), the∇ 2ψterms dominate. Using the standard solution forψfrom the momentum constraint and spatial gauge condition ∇2ψ a2 ≈ −3 ˙ζ− 9 2 Hζ+O(ϵ, λ).(4.19) Substituting into the EH+NM part and keeping leading terms: M2 p a3(1 +λ) −12H2α−6H ˙ζ+ 2H ∇2ψ a2 ...

    work page

  4. [4]

    Linearizing: ∇2 a2 ( ˙δb−δb 0)−3ξb 0H2α−ξb 0 ∇2α a2 −4b 2 0V ′′(0)δb0 = 0

    Bumblebee temporal Constraint (δS/δ(δb 0) = 0) δLN M δ(δb0) =ξM 2 p a3 −3H2b0α−b 0 ∇2α a2 ,(4.21) δLB,kin δ(δb0) =a ∇2 a2 ( ˙δb−δb 0),(4.22) δLV δ(δb0) = 2a3b0V ′(0)−4a 3b2 0V ′′(0)δb0.(4.23) The background term2a 3b0V ′(0)cancels with the back- ground equation of motion. Linearizing: ∇2 a2 ( ˙δb−δb 0)−3ξb 0H2α−ξb 0 ∇2α a2 −4b 2 0V ′′(0)δb0 = 0. (4.24) Fo...

    work page

  5. [5]

    VaryingL B,kin: δLB,kin δ(δb) = d dt a(1 +α+ 3ζ) ∇2 a2 ( ˙δb−δb 0) (4.27) +∂ i h a(1 +α+ 3ζ)∂ i( ˙δb−δb 0) i +· · · (4.28) Contributions fromL N M andL V give mass and mixing terms

    Bumblebee spatial constraint (δS/δ(δb) = 0) This is the dynamical equation of motion forδb, not a con- straint. VaryingL B,kin: δLB,kin δ(δb) = d dt a(1 +α+ 3ζ) ∇2 a2 ( ˙δb−δb 0) (4.27) +∂ i h a(1 +α+ 3ζ)∂ i( ˙δb−δb 0) i +· · · (4.28) Contributions fromL N M andL V give mass and mixing terms. The full linearized equation is: d dt a ∇2 a2 ( ˙δb−δb 0) + ∇2 ...

    work page

  6. [6]

    (4.42) For smallλ,G 12 ∼λ 2 is negligible compared to the diagonal entries. Expanding to leading order: c2 s,+ ≈ F11 G11 = 2ϵ(1 +λ) 2ϵ(1 +λ) + 9 4 λ = 1 1 + 9λ 8ϵ(1+λ) ,(4.43) c2 s,− ≈ F22 G22 = 1.(4.44) Both sound speeds are positive for0≤λ <2/3andϵ >0, i.e., no gradient instability. Keeping the exact expression for later use (c2 s)ζ = 1 1 + 9λ 8ϵ(1+λ) ....

    work page

  7. [7]

    The Bumblebee VEVb 0 is sufficiently large to make the kinetic matrix positive definite (detG>0)

    work page

  8. [8]

    The potential curvatureV ′′(0)is negative, giving the ghost a positive mass squared

    work page

  9. [9]

    We now critically examine the physical plausibility of the three stability assumptions underlying our PBH calculations

    The magnitude ofV ′′(0)is sufficiently large that the ghost mass exceeds the Hubble scale during inflation, allowing it to be integrated out in an effective field the- ory. We now critically examine the physical plausibility of the three stability assumptions underlying our PBH calculations. This scrutiny reveals that the Bumblebee model, in its sim- ples...

    work page

  10. [10]

    A local minimum requires pos- itive second derivative

    At the mini- mum, by definition:V(0) = 0 =V ′(0), andV ′′(0)> 0(for a local minimum). A local minimum requires pos- itive second derivative. This is a fundamental theorem of calculus. A point withV ′′(0)<0is a local maximum, not a minimum. As a result, The assumptionV ′′(0)<0is math- ematically inconsistent with the requirement thatB µBµ = −b2 0 is a stab...

    work page

  11. [11]

    A ghost in theδbsector (wrong-sign kinetic term)

    work page

  12. [12]

    A tachyonic instability in theδbsector if the potential has a stable minimum

    work page

  13. [13]

    asteroid-mass window

    An inconsistent minimum if one attempts to cure the tachyon. The PBH abundance calculations in the next section are therefore not predictions of a physically consistent theory, rather they are illustrations of how dramatically Lorentz vi- olation could affect PBH formation, if a stable, ghost-free, non-tachyonic completion of BG could be found. V . THE CO...

    work page

  14. [14]

    Methodology 1.Parameter ranges:We takeλ∈[0,0.1]with step size∆λ= 10 −4

    For numerical illustration we re- strict toλ≤0.1, ensuring the small–λexpansion remains valid. Methodology 1.Parameter ranges:We takeλ∈[0,0.1]with step size∆λ= 10 −4. The slow–roll parameter is fixed at the CMB–compatible valueϵ= 0.006[18]. The variance in GR is set toσGR(M) = 0.1and the critical thresholdδ GR c = 0.414[14, 15]. 2.Function evaluation:For ...

    work page

  15. [15]

    Particle Dark Matter: Evidence, Candidates and Constraints

    G. Bertone, D. Hooper and J. Silk, Phys. Rept.405(2005), 279-390 [arXiv:hep-ph/0404175 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  16. [16]

    A History of Dark Matter

    G. Bertone and D. Hooper, Rev. Mod. Phys.90(2018) no.4, 045002 [arXiv:1605.04909 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  17. [17]

    B. J. Carr and S. W. Hawking, Mon. Not. Roy. Astron. Soc. 168(1974), 399-415

    work page 1974

  18. [18]

    B. Carr, K. Kohri, Y . Sendouda and J. Yokoyama, Rept. Prog. Phys.84(2021) no.11, 116902 [arXiv:2002.12778 [astro- ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  19. [19]

    Primordial Black Holes as Dark Matter Candidates

    B. Carr and F. Kuhnel, SciPost Phys. Lect. Notes48(2022), 1 [arXiv:2110.02821 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  20. [20]

    B. Carr, S. Clesse, J. Garcia-Bellido, M. Hawkins and F. Kuh- nel, Phys. Rept.1054(2024), 1-68 [arXiv:2306.03903 [astro- ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  21. [21]

    Hawking, Mon

    S. Hawking, Mon. Not. Roy. Astron. Soc.152(1971), 75

    work page 1971

  22. [22]

    B. J. Carr, Astrophys. J.201(1975), 1-19

    work page 1975

  23. [23]

    B. P. Abbottet al.[LIGO Scientific and Virgo], Phys. Rev. Lett.116(2016) no.6, 061102 [arXiv:1602.03837 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  24. [24]

    Primordial Black Hole Scenario for the Gravitational-Wave Event GW150914

    M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, Phys. Rev. Lett.117(2016) no.6, 061101 [erratum: Phys. Rev. Lett.121 (2018) no.5, 059901] [arXiv:1603.08338 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  25. [25]

    S. Bird, I. Cholis, J. B. Mu ˜noz, Y . Ali-Ha ¨ımoud, M. Kamionkowski, E. D. Kovetz, A. Raccanelli and A. G. Riess, Phys. Rev. Lett.116(2016) no.20, 201301 [arXiv:1603.00464 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2016

  26. [26]

    The clustering of massive Primordial Black Holes as Dark Matter: measuring their mass distribution with Advanced LIGO

    S. Clesse and J. Garc ´ıa-Bellido, Phys. Dark Univ.15(2017), 142-147 [arXiv:1603.05234 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  27. [27]

    Primordial black hole formation in the radiative era: investigation of the critical nature of the collapse

    I. Musco, J. C. Miller and A. G. Polnarev, Class. Quant. Grav. 26(2009), 235001 [arXiv:0811.1452 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  28. [28]

    Primordial black hole formation in the early universe: critical behaviour and self-similarity

    I. Musco and J. C. Miller, Class. Quant. Grav.30(2013), 145009 [arXiv:1201.2379 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  29. [29]

    Threshold of primordial black hole formation

    T. Harada, C. M. Yoo and K. Kohri, Phys. Rev. D88(2013) no.8, 084051 [erratum: Phys. Rev. D89(2014) no.2, 029903] [arXiv:1309.4201 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  30. [31]

    Planck 2018 results. VI. Cosmological parameters

    N. Aghanimet al.[Planck], Astron. Astrophys.641 (2020), A6 [erratum: Astron. Astrophys.652(2021), C4] [arXiv:1807.06209 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  31. [32]

    Planck 2018 results. X. Constraints on inflation

    Y . Akramiet al.[Planck], Astron. Astrophys.641(2020), A10 [arXiv:1807.06211 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2020

  32. [33]

    A. S. Josan, A. M. Green and K. A. Malik, Phys. Rev. D79 (2009), 103520 [arXiv:0903.3184 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  33. [34]

    B. J. Carr, K. Kohri, Y . Sendouda and J. Yokoyama, Phys. Rev. D81(2010), 104019 [arXiv:0912.5297 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2010

  34. [35]

    B. Carr, M. Raidal, T. Tenkanen, V . Vaskonen and H. Veerm¨ae, Phys. Rev. D96(2017) no.2, 023514 [arXiv:1705.05567 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  35. [36]

    Density Perturbations and Black Hole Formation in Hybrid Inflation

    J. Garcia-Bellido, A. D. Linde and D. Wands, Phys. Rev. D54 (1996), 6040-6058 [arXiv:astro-ph/9605094 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 1996

  36. [37]

    Primordial Black Holes and Slow-Roll Violation

    H. Motohashi and W. Hu, Phys. Rev. D96(2017) no.6, 063503 [arXiv:1706.06784 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2017

  37. [38]

    C. T. Byrnes, P. S. Cole and S. P. Patil, JCAP06(2019), 028 [arXiv:1811.11158 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  38. [39]

    B. A. Bassett, S. Tsujikawa and D. Wands, Rev. Mod. Phys. 78(2006), 537-589 [arXiv:astro-ph/0507632 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  39. [40]

    A. G. Polnarev and I. Musco, Class. Quant. Grav.24(2007), 1405-1432 [arXiv:gr-qc/0605122 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2007

  40. [41]

    Primordial Black Holes and Scalar-induced Gravitational Waves in Radiative Hybrid Inflation

    A. Afzal and A. Ghoshal, Eur. Phys. J. C84(2024) no.9, 983 [arXiv:2402.06613 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  41. [42]

    P. C. W. Davies, D. A. Easson and P. B. Levin, Phys. Rev. D 111(2025) no.10, 103512 [arXiv:2410.21577 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  42. [43]

    Primordial regular black holes as all the dark matter. I. Time-radial-symmetric metrics

    M. Calz `a, D. Pedrotti and S. Vagnozzi, Phys. Rev. D111 (2025) no.2, 024009 [arXiv:2409.02804 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  43. [44]

    Primordial regular black holes as all the dark matter. II. Non-time-radial-symmetric and loop quantum gravity-inspired metrics

    M. Calz `a, D. Pedrotti and S. Vagnozzi, Phys. Rev. D111 (2025) no.2, 024010 [arXiv:2409.02807 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  44. [45]

    Calz `a, D

    M. Calz `a, D. Pedrotti, G. W. Yuan and S. Vagnozzi, Phys. Rev. D112(2025) no.12, 124015 [arXiv:2507.02396 [gr-qc]]

    work page arXiv 2025

  45. [46]

    N. P. D. Loc, [arXiv:2510.09395 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv

  46. [47]

    J. A. d. Pacheco, Universe4(2018) no.5, 62 [arXiv:1805.03053 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  47. [48]

    Khodadi, Phys

    M. Khodadi, Phys. Lett. B870(2025), 139922 [arXiv:2509.20483 [gr-qc]]

    work page arXiv 2025

  48. [49]

    Dialektopoulos, T

    K. Dialektopoulos, T. Papanikolaou and V . Zarikas, Phys. Lett. B870(2025), 139948 [arXiv:2502.18352 [gr-qc]]

    work page arXiv 2025

  49. [50]

    Primordial Black Holes - Perspectives in Gravitational Wave Astronomy -

    M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, Class. Quant. Grav.35(2018) no.6, 063001 [arXiv:1801.05235 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  50. [51]

    A. M. Green and B. J. Kavanagh, J. Phys. G48(2021) no.4, 043001 [arXiv:2007.10722 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  51. [52]

    Inflation and Primordial Black Holes

    O. ¨Ozsoy and G. Tasinato, Universe9(2023) no.5, 203 [arXiv:2301.03600 [astro-ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  52. [53]

    V . A. Kostelecky, Phys. Rev. D69(2004), 105009 [arXiv:hep- th/0312310 [hep-th]]

    work page arXiv 2004

  53. [54]

    Mattingly, Living Rev

    D. Mattingly, Living Rev. Rel.8(2005), 5 [arXiv:gr- qc/0502097 [gr-qc]]

    work page arXiv 2005

  54. [55]

    Tests of Lorentz invariance: a 2013 update

    S. Liberati, Class. Quant. Grav.30(2013), 133001 [arXiv:1304.5795 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2013

  55. [56]

    Lorentz violation at high energy: concepts, phenomena and astrophysical constraints

    T. Jacobson, S. Liberati and D. Mattingly, Annals Phys.321 (2006), 150-196 [arXiv:astro-ph/0505267 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2006

  56. [57]

    Lorentz Violation: Motivation and new constraints

    S. Liberati and L. Maccione, Ann. Rev. Nucl. Part. Sci.59 (2009), 245-267 [arXiv:0906.0681 [astro-ph.HE]]

    work page internal anchor Pith review Pith/arXiv arXiv 2009

  57. [58]

    V . A. Kostelecky and N. Russell, Rev. Mod. Phys.83(2011), 11-31 [arXiv:0801.0287 [hep-ph]]

    work page arXiv 2011

  58. [59]

    M. A. Hohensee, S. Chu, A. Peters and H. Mueller, Phys. Rev. Lett.106(2011), 151102 [arXiv:1102.4362 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2011

  59. [60]

    Cosmological CPT violating effect on CMB polarization

    M. Li and X. Zhang, Phys. Rev. D78(2008), 103516 [arXiv:0810.0403 [astro-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2008

  60. [61]

    Spontaneous Lorentz Violation, Nambu-Goldstone Modes, and Gravity

    R. Bluhm and V . A. Kostelecky, Phys. Rev. D71(2005), 065008 [arXiv:hep-th/0412320 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  61. [62]

    The flight of the bumblebee: vacuum solutions of a gravity model with vector-induced spontaneous Lorentz symmetry breaking

    O. Bertolami and J. Paramos, Phys. Rev. D72(2005), 044001 [arXiv:hep-th/0504215 [hep-th]]

    work page internal anchor Pith review Pith/arXiv arXiv 2005

  62. [63]

    R. V . Maluf and J. C. S. Neves, Phys. Rev. D103(2021) no.4, 044002 [arXiv:2011.12841 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  63. [64]

    Black Hole Superradiance in the Presence of Lorentz Symmetry Violation

    M. Khodadi, Phys. Rev. D103(2021) no.6, 064051 [arXiv:2103.03611 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

  64. [65]

    An exact Schwarzschild-like solution in a bumblebee gravity model

    R. Casana, A. Cavalcante, F. P. Poulis and E. B. Santos, Phys. Rev. D97(2018) no.10, 104001 [arXiv:1711.02273 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2018

  65. [66]

    Z. F. Mai, R. Xu, D. Liang and L. Shao, Phys. Rev. D109 (2024) no.8, 084076 [arXiv:2401.07757 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  66. [67]

    R. C. Pantig, S. Kala, A. ¨Ovg¨un and N. J. L. S. Lobos, [arXiv:2410.13661 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  67. [68]

    Probing the Lorentz Symmetry Violation Using the First Image of Sagittarius A*: Constraints on Standard-Model Extension Coefficients

    M. Khodadi and G. Lambiase, Phys. Rev. D106(2022) no.10, 104050 [arXiv:2206.08601 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  68. [69]

    Spontaneous Lorentz symmetry breaking effects on GRBs jets arising from neutrino pair annihilation process near a black hole

    M. Khodadi, G. Lambiase and L. Mastrototaro, Eur. Phys. J. C83(2023) no.3, 239 [arXiv:2302.14200 [hep-ph]]

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  69. [70]

    Probing Schwarzschild-like Black Holes in Metric-Affine Bumblebee Gravity with Accretion Disk, Deflection Angle, Greybody Bounds, and Neutrino Propagation

    G. Lambiase, L. Mastrototaro, R. C. Pantig and A. Ovgun, JCAP12(2023), 026 [arXiv:2309.13594 [gr-qc]]. 17

    work page internal anchor Pith review Pith/arXiv arXiv 2023

  70. [71]

    A. A. Ara ´ujo Filho, J. A. A. S. Reis and A. ¨Ovg¨un, Eur. Phys. J. C85(2025) no.1, 83 [arXiv:2409.17351 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  71. [72]

    C. Ding, C. Liu, Y . Xiao and J. Chen, Eur. Phys. J. C85(2025) no.1, 73 [arXiv:2407.16916 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  72. [73]

    J. Z. Liu, W. D. Guo, S. W. Wei and Y . X. Liu, Eur. Phys. J. C 85(2025) no.2, 145 [arXiv:2407.08396 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  73. [74]

    A. A. Ara ´ujo Filho, N. Heidari, I. P. Lobo and V . B. Bezerra, [arXiv:2511.12839 [gr-qc]]

    work page arXiv

  74. [75]

    P. Ji, Z. Li, L. Yang, R. Xu, Z. Hu and L. Shao, Phys. Rev. D 110(2024) no.10, 104057 [arXiv:2409.04805 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2024

  75. [76]

    Y . LIU, A. F. SANG, W. Y ANG and Y . P. HU, Sci. Sin. Phys. Mech. Astro.54(2024) no.9, 290411

    work page 2024

  76. [77]

    Z. Luo, S. Li and H. Yu, [arXiv:2601.07196 [gr-qc]]

    work page arXiv

  77. [78]

    Polarizations of Gravitational Waves in the Bumblebee Gravity Model

    D. Liang, R. Xu, X. Lu and L. Shao, Phys. Rev. D106(2022) no.12, 124019 [arXiv:2207.14423 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2022

  78. [79]

    Gravitational-wave generation in the presence of Lorentz invariance violation

    S. Aoulad Lafkih, M. C. Angonin, C. Le Poncin-Lafitte and N. A. Nilsson, [arXiv:2506.08859 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv

  79. [80]

    Primordial gravitational waves from spontaneous Lorentz symmetry breaking

    M. Khodadi, G. Lambiase, L. Mastrototaro and T. K. Poddar, Phys. Lett. B867(2025), 139597 [arXiv:2501.14395 [astro- ph.CO]]

    work page internal anchor Pith review Pith/arXiv arXiv 2025

  80. [81]

    R. V . Maluf and J. C. S. Neves, JCAP10(2021), 038 [arXiv:2105.08659 [gr-qc]]

    work page internal anchor Pith review Pith/arXiv arXiv 2021

Showing first 80 references.