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This paper argues that when the initial state of a gravitationally produced relic is not the Bunch–Davies vacuum, the final abundance changes in a non-additive way, and for the longitudinal mode of a massive spin-1 field this effect can shi

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load-bearing objection The framework is good, but the thermal-scenario headline contradicts the paper's own bound; the two-stage part is the reliable core. the 3 major comments →

arxiv 2603.03430 v2 pith:DZ2JCPL6 submitted 2026-03-03 gr-qc astro-ph.HEhep-ph

The effects of non Bunch-Davies initial conditions on gravitationally produced relics

Enrico Bertuzzo , Gabriel M. Salla , Andrea Tesi This is my paper
classification gr-qc astro-ph.HEhep-ph MSC 83C4781T20 PACS 95.35.+d98.80.Cq
keywords gravitational particle productiondark matterBunch-Davies vacuuminitial conditionsBogoliubov transformationmassive spin-1two-stage inflationthermal initial state
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 relaxes the standard assumption that gravitational particle production starts from the Bunch–Davies vacuum — the state with no particles — and asks how a pre-existing population of relics changes the final abundance. It derives a general formula in which the final number density is not the initial density plus the vacuum production: initial quanta are stimulated or blocked by the gravitational amplification, and correlated pairs in the initial state contribute with interference. For fields whose only departure from conformality is a mass term (conformally coupled scalars, fermions, transverse vector polarizations), the initial state has little effect because the initial population redshifts away before production begins. For the longitudinal mode of a massive spin-1 field the effect is large: with a pre-inflationary thermal population, the mass that reproduces all of the dark matter can range from m ≲ 10^-17 GeV up to m ≲ H_inf, compared with the single Bunch–Davies value m ≃ 10^-13 GeV. A two-stage inflationary history with an intermediate radiation phase shifts the allowed masses upward, to m ≳ 6×10^-6 eV, while allowing the correct abundance for much smaller Hubble parameters and reheating temperatures.

Core claim

The central claim is that the final comoving number density of a gravitationally produced relic is governed by n_k = |β_k|^2 + (1 ± 2|β_k|^2)⟨n_k^in⟩ − (α_k^* β_k^* ξ_k^* ⟨c_k^in⟩ + c.c.), where |β_k|^2 is the usual vacuum pair-production coefficient, ⟨n_k^in⟩ is the initial occupation number, and ⟨c_k^in⟩ is the initial pair-correlation amplitude. This formula shows that a non-empty initial state does not simply add to the vacuum abundance: initial particles are stimulated or blocked by the gravitational amplification, and correlated pairs interfere with the production. For the longitudinal mode of a massive spin-1 field, the paper finds this effect is numerically large. With a pre-inflatio

What carries the argument

The central object is the Bogoliubov transformation between early-time and late-time mode functions of the Weyl-rescaled field, together with the two initial-state correlation functions ⟨n_k^in⟩ (occupation number) and ⟨c_k^in⟩ (pair amplitude). The identity at the heart of the paper, eq. (18), packages the vacuum production |β_k|^2, the stimulated-emission/Pauli-blocking term, and the pair-correlation interference term; eqs. (17) and (22) give equivalent routes through the Hamiltonian or the power spectrum. For the longitudinal mode of a massive vector, the transfer function T(k,η), which grows as a^2, is constant, then falls as a^-1 across different regimes, controls where the spectral pea

Load-bearing premise

The computation assumes the relic is a free, non-interacting field for the whole evolution, and in the thermal scenario that the interactions that created the thermal bath are switched off at the moment initial conditions are imposed; if self-interactions or couplings to a bath act during inflation, the mode functions and the simple final-state formula (17)–(18) would not capture the abundance.

What would settle it

The decisive check is a direct calculation or simulation of the late-time vector spectrum starting from a specified thermal initial state with the interactions kept on during inflation. If, for masses below the Bunch–Davies value, the abundance does not exceed the vacuum result by the factor ~T/k_* used in eq. (35), or if the spectral peak shifts away from k_*, the central claim fails. More concretely, for fixed m and T_RH, the total-abundance curve as a function of initial temperature should be flat at low m and then drop sharply at the Bunch–Davies mass; a curve that tracks the vacuum predic

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

If this is right

  • For spin-1 relics, the standard single-mass prediction m ≃ 10^-13 GeV is not a robust consequence of GPP; the same observed dark-matter abundance can be obtained for masses orders of magnitude lighter or heavier once the initial state is specified.
  • For conformally coupled scalars and spin-1/2 fermions, the Bunch–Davies computation is robust: non-vacuum initial populations redshift away before production, so existing predictions and bounds for those fields remain unchanged.
  • In the thermal-initial-state scenario the spectral peak stays at k_*, but the overall abundance is enhanced by roughly T/k_* relative to vacuum production, which is why lower masses can reach the observed abundance.
  • In the two-stage inflation scenario, a second spectral peak appears at k_dR for masses above m_dR, the spectrum develops oscillations from modes that exit, re-enter, and re-exit the horizon, and the total abundance can be obtained for final Hubble parameters and reheating temperatures much smaller than in the single-stage case.
  • The paper's final formulas apply to any initial state with integrable, sufficiently small initial energy density, so the same method can be used to evaluate abundances for arbitrary non-Bunch–Davies initial conditions.

Where Pith is reading between the lines

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

  • A natural extension is to minimally coupled scalars that evade isocurvature bounds: the same enhancement mechanism could shift their allowed mass ranges if a pre-inflationary population is present, though the paper does not quantify that case.
  • The pair-correlation term ⟨c_k^in⟩ is a distinctive signature: an initial squeezed state would imprint oscillations on the late-time spectrum at fixed k, which neither the vacuum nor thermal cases produce; a future measurement of such oscillations would point to the production history.
  • The thermal example assumes the bath interactions switch off before inflation; including a finite coupling during inflation would likely interpolate between the Bunch–Davies and thermal results, and is the clearest testable modification of the calculation.
  • In the two-stage scenario, the shift of the spectral peak toward k_dR means superradiance and isocurvature bounds should be re-examined on a model-by-model basis, since the peak momentum, not just the mass, controls those constraints.

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 / 4 minor

Summary. The paper develops a general Bogoliubov/field-power-spectrum framework for gravitational particle production (GPP) from non-Bunch–Davies initial states, with the central result being the final comoving number-density formula of Eqs. (17)–(18). This formula contains stimulated-emission/Pauli-blocking terms proportional to the initial occupation ⟨n_in⟩ and pair-correlation terms proportional to ⟨c_in⟩. The authors argue that fields whose only conformal-symmetry breaking is a mass term are largely insensitive to the initial state, whereas the longitudinal mode of a massive spin-1 field is sensitive. Two concrete scenarios are studied: an initial thermal state (Sec. III A) and a two-stage inflationary history with an intermediate radiation phase (Sec. III B). The paper claims that in the thermal case the observed dark-matter abundance can be obtained for masses from m ≲ 10^{-17} GeV up to m ≲ H_inf, and in the two-stage case for masses m ≳ 6×10^{-6} eV.

Significance. If the main claims hold, the paper would provide a useful generalization of GPP to arbitrary initial conditions and would sharpen the parameter space of spin-1 dark matter. The general formulas (17)–(18) are cleanly derived and the paper correctly emphasizes that the final abundance is not obtained by simply adding the initial population to the Bunch–Davies result. The analytical estimates in App. C are consistent with the numerical spectra shown in Fig. 5, and the two-stage scenario gives concrete, falsifiable predictions. However, the thermal-scenario headline mass range is not supported by the paper's own consistency bound, Eq. (37); this weakens the paper's most prominent phenomenological conclusion. The two-stage part of the paper is more robust and is a valuable contribution even if the thermal claim is revised.

major comments (3)
  1. [Sec. III A, Eq. (37), Figs. 2–3 and Abstract] The claimed thermal-scenario mass range, 'from m ≲ 10^{-17} GeV up to m ≲ H_inf' (Sec. III A and Abstract), is contradicted by the authors' own constraint. In the low-mass branch, Eq. (35) gives Ω ∝ Ω_BD T / k⋆, with Ω_BD / k⋆ approximately independent of m for fixed H_inf and T_RH, so the required temperature is nearly constant. Equation (37), on the other hand, bounds T / (a_e H_inf) by (k⋆/(a_e H_inf)) sqrt(10^{13} GeV/H_inf), which scales as (m/H_inf)^{1/2} (or m^{1/3} in the reheating regime). The intersection cuts off the horizontal curves at a mass far above 10^{-17} GeV for the high-T_RH cases. The text itself states that 'most of the low mass regime is excluded' and that for T_RH = 10^{-1} GeV 'all points are disfavored.' The abstract and Sec. III A should be revised to the surviving branch; the mass-range claim is not supported by the paper's own formulas.
  2. [Sec. III A, paragraph after Eq. (37)] The sentence 'we can obtain the correct abundance with masses that go from m ≲ 10^{-17} GeV up to m ≲ H_inf' is inconsistent with the immediately preceding discussion of Eq. (37). The surviving thermal parameter region appears to be a set of narrow windows around the Bunch–Davies mass for each T_RH (roughly where Ω_BD is within a factor of a few of Ω_CDM), not a broad continuous range. The authors should quantify the allowed region after imposing Eq. (37) and re-state the claim in terms of the actual surviving masses. This is a load-bearing point because the wide-mass-range claim is the main phenomenological headline of the thermal section.
  3. [Sec. III A, Eq. (33) and free-field assumption] The thermal example assumes that 'the interactions that generated the thermal bath are switched off at the time we impose initial conditions.' This is essential because Eqs. (17)–(18) and the mode equations (4) describe a free field. No concrete mechanism or estimate is given for how such a switch-off can occur during inflation without leaving residual self-interactions or couplings. This is not an internal inconsistency, but it is a physical limitation that should be stated more prominently in the abstract/conclusions, since the thermal scenario's applicability depends on this assumption.
minor comments (4)
  1. [Eq. (37)] The notation sqrt(3√10/π M_P/H_inf) is ambiguous: the argument of the square root should be enclosed in parentheses. This is cosmetic but could confuse readers checking the bound.
  2. [Fig. 3 caption and text] The yellow region is described as a 'band around the curves where the constraint is violated,' but Eq. (37) defines a half-plane in the (m, T) plane. The caption should state that the excluded region is the half-plane above the line T/(a_e H_inf) = (k⋆/(a_e H_inf)) sqrt(10^13 GeV/H_inf).
  3. [Sec. II A and Sec. III A] The statements 'we have checked for several cases' and 'we have explicitly checked' (near Eqs. (17)–(18) and after Eq. (35)) are presented without any numerical or analytical demonstration. Since the code is not shipped, these checks should either be shown in an appendix or replaced by a direct algebraic statement of the equivalence.
  4. [Sec. III A, final paragraph] The claim that the correct abundance can be obtained for masses 'both smaller and larger' than the Bunch–Davies value is ambiguous: it is only meaningful if T_RH is allowed to vary, and for T_RH = 10^{-1} GeV the text says the branch is disfavored. Please clarify which T_RH values support which side of the comparison.

Circularity Check

0 steps flagged

No significant circularity: final-abundance formulas are derived from standard QFT in curved spacetime, and benchmark parameters are inputs rather than fitted outputs.

full rationale

The paper's central result, eqs. (17)-(18), is obtained by matching the one-loop Hamiltonian expectation value (eq. (5)) to the late-time adiabatic Hamiltonian (eq. (12)) via eq. (13), with Omega_k and F_k computed from free-field mode functions in appendix A. No output quantity is used to define an input: for the thermal scenario, <n_in> = 1/(e^{k/T}-1) and <c_in>=0 are assumed, and the abundance contours in fig. 3 are computed, not fitted, for fixed H_inf = 10^13 GeV and scanned T and T_RH. In the two-stage scenario, H_I, N_dR, a_e/a_dR are independent inputs leading to eqs. (39)-(41) and fig. 6 via transfer functions; no parameter is tuned to the target Omega_CDM. Self-citations (refs. 28, 42, 49-50) are used as background context (effective-action method, non-standard cosmological evolution) or as one of several citations for the standard Omega proportional to sqrt(m) behavior, which the paper rederives explicitly in eqs. (30)-(31); none of these carries the new non-Bunch-Davies effect. The apparent tension between the abstract/Sec. III A claim of masses down to 10^-17 GeV and the energy-density bound eq. (37) is a consistency/correctness concern, not a case of a prediction reducing to its input, so it is not scored here.

Axiom & Free-Parameter Ledger

5 free parameters · 8 axioms · 0 invented entities

The central claim relies on standard QFT-in-curved-spacetime tools and a set of clearly-stated cosmological modeling assumptions. No new particles or interactions are introduced; the non-BD states are either postulated thermal states or generated by a two-stage inflationary history. The free parameters are scanned model inputs, not fitted constants.

free parameters (5)
  • Initial comoving temperature T (thermal scenario) = Scanned across ~6 orders of magnitude in fig. 3
    Characterizes the non-Bunch–Davies thermal initial state (eq. (33)); free input, not fitted to data; central to showing mass range.
  • N_dR (duration of intermediate radiation phase in two-stage inflation) = 1.3–15 e-folds (figs. 5-6)
    Controls H_e relative to H_I via eq. (38); scanned to match Ω_DM contours.
  • H_e (Hubble at end of second inflation stage) = 10^0–10^14 GeV (fig. 6)
    Sets the production scale; scanned for Ω_vector=Ω_CDM.
  • m (relic mass) = 10^-25–10^8 GeV across figures
    Independent variable of the abundance contours; not fitted.
  • T_RH (reheating temperature) = Benchmark values 0.1, 10^2, 10^8, 10^12 GeV
    Sets entropy/dilution and constraints; chosen by hand for the plots.
axioms (8)
  • standard math Standard QFT in curved spacetime: mode functions, Bogoliubov transformations, adiabatic vacuum.
    Used throughout Sec. II to derive eqs. (14)-(18).
  • domain assumption Relic field is free and non-interacting during GPP evolution.
    Assumed in Sec. III A: interactions generating the thermal bath are switched off; free EoMs (4) used.
  • domain assumption Initial state in thermal scenario is Gaussian with ⟨c_in⟩=0.
    Eq. (33) sets ⟨n_in⟩=1/(e^{k/T}-1) and ⟨c_in⟩=0; no pair coherence.
  • domain assumption In the two-stage scenario, the state at the start of the first dS stage is Bunch–Davies; non-BD is generated by the intermediate radiation phase.
    Sec. III B: 'far in the past the vacuum state for DM was still the Bunch-Davies one'.
  • domain assumption Analytic spectra use pure dS stages and instantaneous reheating/instantaneous transitions.
    Sec. II B and App. C; transfer functions (A37) and formulas (39)-(41) rely on this.
  • domain assumption Quadratic (chaotic) inflation model used for numerics, despite being excluded by Planck data.
    Footnote 4; argued insensitive for m<H_inf.
  • domain assumption Isocurvature constraints assume uncorrelated 'axion I' perturbations and Gaussian density contrasts.
    App. D 1, citing Planck; used to set bounds in eqs. (D4)-(D8).
  • domain assumption Massive gauge field has no non-minimal couplings to curvature; longitudinal mode decomposition as in app. A 3.
    Eq. (A33) ignores non-minimal couplings; referenced in Sec. II B 1.

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read the original abstract

Typical gravitational production of relics from amplification of inflationary perturbations assumes Bunch-Davies initial conditions, i.e. a vacuum with initially no particles. In this paper we investigate the impact of non Bunch-Davies initial conditions to the final abundance of relics, with particular attention to the parameter space where the total dark matter abundance is reproduced. We present a general framework for any initial condition, through which we show their non-trivial effect on both spectrum and late-time abundance. We argue that for particles whose source of conformal symmetry breaking comes only from a mass term (spin-1/2 fermions and conformally coupled scalars), the choice of initial conditions has little impact on the mass range relevant to dark matter. For other particles, e.g. the longitudinal mode of spin-1, we see a large deviation from the standard computation. We exemplify and quantify our results with an initial thermal state and a two-stage inflation scenario, highlighting that the total dark matter can be obtained for a wide range of masses.

Figures

Figures reproduced from arXiv: 2603.03430 by Andrea Tesi, Enrico Bertuzzo, Gabriel M. Salla.

Figure 1
Figure 1. Figure 1: FIG. 1. Sketch showing the behavior of the transfer func [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. Abundance levels for spin-1 DM in the scenario with [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Comoving horizon for a two-stage inflation with an in [PITH_FULL_IMAGE:figures/full_fig_p010_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Comoving momentum spectra for a spin-1 relic with a low mass [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Curves in which Ω [PITH_FULL_IMAGE:figures/full_fig_p012_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Lyman- [PITH_FULL_IMAGE:figures/full_fig_p018_7.png] view at source ↗

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Reference graph

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