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Analytic Gravitational Wave Spectrum in Next-to-Minimal Bouncing Cosmology

T0 review · 2 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read This paper derives a closed-form broken power-law gravitational-wave background for a five-phase bouncing cosmology and shows that the ΔNeff bound automatically forces the bounce energy scale below the Planck mass.

desk verdict Solid analytic SGWB template for five-phase bounces, but the abstract's 'all NMBC models' overclaims: ΔNeff bounds amplitude, not tilt, so the sub-Planckian conclusion is unproven for blue-tilted spectra. read the letter →

arxiv 2507.12968 v1 pith:SKTSOVJK submitted 2025-07-17 astro-ph.CO gr-qchep-ph

Changhong Li This is my paper
classification astro-ph.COgr-qchep-ph
keywords bouncingcosmologystochasticgravitational-wavebackgroundprimordialgravitationalwavestrans-PlanckianproblembrokenpowerlawmatrixpropagationmethodΔNeffboundpivotfrequency
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

Bouncing cosmologies replace the initial singularity with a finite minimum scale, but the minimal four-phase version predicts a featureless power-law gravitational-wave background that is hard to observe and often requires super-Planckian bounce energies. This paper adds one early contraction phase, making a next-to-minimal bouncing cosmology (NMBC) whose gravitational-wave spectrum is a broken power law: high frequencies match the minimal model, while low frequencies carry a new tilt set by the extra phase. Using a matrix-propagation method built from an inequality algebra, the author derives a closed-form expression for the spectrum and shows that the CMB/BBN bound $\Omega_{\rm GW}h^2<1.7\times10^{-6}$ forces the bounce energy scale below $\rho_{s\downarrow}^{1/4}=0.79\,m_{\rm pl}$. The result matters because it makes bouncing cosmologies with no trans-Planckian physics potentially detectable across CMB, pulsar-timing, interferometer, and laboratory gravitational-wave experiments.

What carries the argument

The carrying object is the PGW propagation kernel $N^{(0)}_{22}(\{\tilde\nu_i\},\{\eta_{i\downarrow/\uparrow}\})$, the (2,2) entry of $X^{(0)\dagger}X^{(0)}$, where $X^{(0)}$ is a product of phase transformation matrices $T_i$ and boundary-matching matrices $M_{i\uparrow/\downarrow}$ that propagate Bunch-Davies vacuum amplitudes across the five phases. An inequality-based algebra—$\chi_i=0$ or $\bar\chi_i=0$ depending on whether $\nu_i>1/2$ or $\nu_i\le1/2$—collapses the matching matrices into two branches and lets the kernel be evaluated in closed form. The extra phase 0 enters only for $f<f_\star$ through the factor $T_1^{-1}M_{0\downarrow}T_0$, producing the broken power law; the pivot frequency $f_\star=(2\pi a_0\eta_{0\downarrow})^{-1}$ is set by the conformal time at the phase-0→1 transition.

What would settle it

Numerically integrate the tensor perturbation equation across the five phases for the paper's four example parameter sets without imposing $k\eta_{s\downarrow}\ll k\eta_{0\downarrow}\ll1$ or the $k\eta_{0\downarrow}\ll1$ approximation, and compare the exact spectrum with Eqs. (22) and (28); any material departure on the $f\ge f_\star$ branch would shift the $0.79\,m_{\rm pl}$ intersection and weaken the automatic sub-Planckian conclusion.

Watch

Extended reading notes

Core claim

The central claim is that adding a fifth, early contraction phase (phase 0) to the minimal bouncing cosmology produces a stochastic gravitational-wave background with an analytically computed broken power law, with pivot frequency $f_\star=(2\pi a_0\eta_{0\downarrow})^{-1}$. The amplitude matrix $X^{(0)}$ equals the MBC matrix for $f>f_\star$ and acquires a phase-0 factor $T_1^{-1}M_{0\downarrow}T_0$ for $f<f_\star$; in the long-phase-1 limit the low-frequency spectrum reduces to a closed form with tilt $n_T^{(0)}=3\mp2\tilde\nu_0$ and coefficient $C^{(0)}$, while the high-frequency branch is exactly the earlier minimal-bouncing result. Imposing the $\Delta N_{\rm eff}$ bound $\Omega_{\rm GW}h^2\lesssim1.7\times10^{-6}$ at the five pivot frequencies $7.75\times10^{-17}$, $10^{-8}$, $30$, $10^7$, and $10^9$ Hz forces $w_1\le0$; at $w_1=0$ all constraint curves intersect at $\rho_{s\downarrow}^{1/4}=0.79\,m_{\rm pl}$, so every viable NMBC model bounces below the Planck scale. Four worked parameter sets reproduce features such as the PTA best fit and lie within projected sensitivities of CMB, space, ground, and laboratory detectors.

Load-bearing premise

The load-bearing premise is that the new early contraction phase changes only low-frequency modes, leaving the high-frequency spectrum exactly equal to the minimal bouncing result; if phase-0 effects leak above the pivot frequency $f_\star$, or the deep-bounce approximation $k\eta_{s\downarrow}\ll1$ fails near $f_\star$, the automatic sub-Planckian conclusion changes.

Editorial extensions

If this is right

  • Every NMBC model satisfying the current ΔNeff bound has a red or scale-invariant high-frequency tilt, so the bounce energy scale is automatically sub-Planckian (≤ 0.79 Planck mass), removing the trans-Planckian problem without fine-tuning.
  • The high-frequency branch inherits the full minimal bouncing cosmology spectrum and its constraints, so all existing MBC bounds continue to apply unchanged to those modes.
  • The broken power law with pivot frequency set by the phase-0 transition opens broad observational windows; the four example spectra sit within projected CMB, PTA, LVK, and superconducting circuit/cavity sensitivities.
  • The closed-form kernel makes parameter inversion direct: specified low- and high-frequency tilts, pivot frequency, and amplitude uniquely determine the equation-of-state parameters, the bounce scale, and the cutoff frequency.
  • The matrix-propagation algebra extends to other multi-phase early-universe histories, giving closed-form SGWB predictions that complement numerical approaches.

Reading between the lines

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

  • If the matrix algebra generalizes as the paper suggests, closed-form spectra could be written for any multi-phase history—kination, reheating, or dark phase transitions—turning parameter scans into analytic exercises rather than numerical grids.
  • The case of zero high-frequency tilt acts as a selection rule for model builders: targeting a red or scale-invariant high-frequency tilt keeps the bounce sub-Planckian, while blue tilts push the required bounce energy past the Planck scale.
  • A future broken power-law detection would effectively measure two pre-bounce conformal-time scales and two equation-of-state parameters, turning the framework into an observational probe of the contraction phase.
  • The small discontinuity at the pivot frequency is a direct diagnostic of the approximation used; precise numerical spectra near the pivot could test whether phase-0 leakage is genuinely negligible for the sub-Planckian bound.
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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

2 major / 4 minor

Summary. This paper introduces the next-to-minimal bouncing cosmology (NMBC), a five-phase bouncing model obtained by adding an early contraction phase (phase 0) to the minimal bouncing cosmology (MBC). The authors extend their earlier matrix-propagation method to derive a closed-form, broken power-law stochastic gravitational-wave background (SGWB), with the high-frequency branch identical to the MBC result and the low-frequency branch modified by phase 0. They then claim that the current ΔNeff bound Ω_GW h²(f) < 1.7×10⁻⁶ forces w1 ≤ 0, and hence that all viable NMBC models have bounce energy scale ρ_{s↓}^{1/4} < 0.79 m_pl, automatically avoiding the trans-Planckian problem. Four illustrative NMBC spectra are presented as examples.

Significance. The paper provides a genuinely useful analytic tool for SGWB predictions in multi-phase bouncing cosmologies: the matrix method yields closed-form spectra, the two internal consistency checks in Eqs. (31)-(32) are sensible, and the four examples are explicit, falsifiable, and tied to concrete detector sensitivities. If the sub-Planckian claim were fully established, it would be an important result connecting gravitational-wave observatories to the bounce energy scale. However, the central inference from the ΔNeff bound to w1 ≤ 0 is not logically valid, so the headline claim is not currently supported; the strength of the paper lies in the analytic machinery rather than in the demonstrated universality of the sub-Planckian conclusion.

major comments (2)
  1. [Constraint on ρ_{s↓}^{1/4} from ΔNeff, Eqs. (24)-(25)] The inference that the ΔNeff upper limit Ω_GW h²(f) < 1.7×10⁻⁶ requires a red or scale-invariant high-frequency SGWB, and hence w1 ≤ 0, is a non-sequitur. An upper limit on the amplitude does not restrict the tilt: for any w1 > 0 (n_T > 0), the spectrum is blue, but the amplitude factor in Eq. (19) contains (ρ_{s↓}^{1/4})^{4-n_T}, so the amplitude can be made arbitrarily small by choosing ρ_{s↓}^{1/4} small enough. No independent normalization fixes this amplitude, and the bound at any single frequency (or even the integrated BBN bound) can be satisfied by a sufficiently low bounce scale. Consequently, the subsequent derivation of the 0.79 m_pl bound applies only to the w1 ≤ 0 branch shown in Fig. 2, not to all NMBC models. The abstract's claim that 'all NMBC models satisfying the current ΔNeff bound automatically avoid the trans-Planckian problem' is therefore not established. Please either extend the calculation to w1 > 0, demonstrating (if true) that the amplitude constraint alone still forces sub-Planckian ρ_{s↓}^{1/4}, or explicitly restrict the claim to the w1 ≤ 0 branch. A concrete test would be to evaluate Eq. (19) at f = 10⁻⁷ Hz with, say, w1 = 1/3 and choose ρ_{s↓}^{1/4} so that Ω_GW h² = 10⁻⁷; this directly contradicts the asserted exclusion of w1 > 0.
  2. [Cosmological Applications and Supplemental Material Sec. VII] The four illustrative examples are intended to validate the analytic framework, but the parameters given for Examples 2 and 3 are internally inconsistent. In the main text, Eq. (36) states ρ_{s↓}^{1/4} = 0.06 m_pl for Example 2, while SM Sec. VII gives 0.06×10⁻⁷ m_pl, a discrepancy of seven orders of magnitude. For Example 3, Eq. (37) gives w0 = 1.3×10⁵, while SM Sec. VII gives w0 = 4×10⁵/3; moreover, neither value is consistent with the stated low-frequency tilt n_T^{(0)}(f < f⋆) = 4−10⁻⁵ through Eq. (30) for the ν0 ≤ 1/2 branch, which would require w0 ≈ 1. These inconsistencies undermine the claim that the examples validate the analytic formulas, even though they are illustrative rather than central to the sub-Planckian argument.
minor comments (4)
  1. [Supplemental Material, Eq. (S25)] The initial Bunch-Davies vacuum (A0, B0) for phase 0 is written with exp(−i(ν̃1 π/2 + π/4)), but the phase should presumably involve ν̃0, not ν̃1.
  2. [Constraint on ρ_{s↓}^{1/4}, after Eq. (24)] Equation (24) is stated for f ≳ 10⁻⁷ Hz, but the text then evaluates it at f = 7.75×10⁻¹⁷ Hz. Please clarify which experimental bound is being used at that frequency and whether the ΔNeff bound actually applies there.
  3. [Constraint on ρ_{s↓}^{1/4}, paragraph after Eq. (25)] The sentence 'which is belong to the branch ν1 > 1/2' is grammatically incorrect; please rephrase.
  4. [Relation between MBC and NMBC, Eqs. (31)-(32)] The reduction to MBC is verified only for the full kernel in the SM, while the simplified long-phase-1 formulas in Eqs. (28)-(30) do not apply in the η0↓ = ηs↓ limit. Please state this explicitly in the main text so that readers do not attempt to use the simplified formulas for the consistency check.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the NMBC spectrum is derived from vacuum initial conditions, and the sub-Planckian bound is an inversion of an external observational constraint rather than a fitted input.

full rationale

The central derivation is self-contained. The NMBC SGWB spectrum, Eq. (28), follows from solving the tensor perturbation equation, Eq. (9), with Bunch-Davies initial conditions and matching across phase boundaries; no parameter is fitted to the quantity that is later claimed as a prediction. The high-frequency branch, Eq. (22), coincides with the MBC spectrum by construction through Eq. (18), which sets X^(0) = X^(1) for f > f⋆; this is an explicit structural reduction, not a hidden equivalence. The result ρ_{s↓}^{1/4} < 0.79 m_pl is obtained by evaluating the external ΔNeff bound, Eq. (24), against the derived spectrum, Eq. (19), and solving for the bounce energy scale; that is a constraint inversion, not a fitted parameter renamed as a prediction. The four illustrative examples are explicitly reverse-engineered from chosen spectral shapes and amplitudes, and the Supplemental Material describes how the fundamental parameters are determined from those phenomenological inputs, so they are not presented as independent predictions. The paper does rely heavily on the author's prior MBC derivation in Refs. [49] and [7], but that prior work is a parameter-free analytic derivation with stated assumptions that do not include the present paper's ΔNeff bound, so it constitutes real evidence rather than a circular self-citation chain. The apparent logical gap in the argument that an amplitude bound requires a red or scale-invariant spectrum concerns the validity of the 'all NMBC models' claim, but it is not a circularity and is not scored here.

Assumptions & free parameters 4 free parameters · 7 assumptions · 0 invented entities

The central results rest on the five-phase parameterization, standard tensor perturbation theory, the Bunch-Davies vacuum, the deep-bounce and long-phase-1 limits, and the external ΔNeff bound. The free parameters w0, w1, ρ_s and f⋆ are model inputs chosen by hand for the illustrative examples; they are not constrained by data in this paper.

free parameters (4)
  • w0 (phase 0 equation of state) = examples: 1/9, 3/11, 4e5/3, 1/9
    Sets the low-frequency tilt n_T(f<f⋆) through ν0. It is a model input chosen by hand for the illustrative examples.
  • w1 (phase 1 equation of state) = examples: 0, -1/51, 0, -1/15
    Sets the high-frequency tilt n_T(f≥f⋆) through ν1. It is a model input chosen by hand for the illustrative examples.
  • ρ_s^{1/4} (bounce energy scale) = examples: 0.39, 0.06, 0.012, 0.042 m_pl
    Controls the overall SGWB amplitude and the trans-Planckian bound. The paper inverts the ΔNeff constraint to bound it, and chooses specific values for the examples.
  • f⋆ (pivot frequency, from η0↓) = examples: 1e-6, 1e-7, 1e-10, 1e6 Hz
    Marks where the spectrum breaks from the MBC-like branch to the phase-0 branch. Chosen by hand for the four examples.
assumptions (7)
  • standard math Tensor perturbations obey Eq. (9) on a perturbed FLRW background with scale factor a(η)=a_i |η|^{ν_i} in each phase.
    Basis of the PGW calculation, imported from standard cosmological perturbation theory and Ref. [49].
  • domain assumption Initial state is Bunch-Davies vacuum in phase 1 and in the new phase 0 (SM Eqs. S23, S25).
    Sets the mode normalization; alternative vacua would change the spectrum and the 0.79 m_pl bound.
  • domain assumption Deep-bounce limit kη_s↓ << 1 (Eq. 5) holds for all relevant modes.
    Used to obtain the power-law forms and the trans-Planckian bound.
  • domain assumption Long-phase-1 limit Eq. (27): kη_s↓ << kη0↓ << 1 and (kη_s↓)^{2ν̃1} << (kη0↓)^{2ν̃0+2ν̃1}.
    Needed for the simplified low-frequency closed form Eq. (28); the full kernel in the SM is more complex.
  • domain assumption Constant equations of state per phase and instantaneous transitions, with symmetric bounce w2=w3=-∞ and η1↓=η3↓.
    Defines the MBC/NMBC models; non-instantaneous transitions would require extra phases.
  • domain assumption Phase 4 is radiation dominated with standard transfer function Teq and constants Ωγ0 h²=2.474e-5, h=0.677.
    Converts the PGW spectrum to today's SGWB; values are from standard cosmology.
  • domain assumption ΔNeff bound Ω_GW h²(f)<1.7e-6 for f≳1e-7 Hz, taken from Refs. [50,51].
    External input used to derive the 0.79 m_pl maximum bounce scale.

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Pith. "Pith review of Analytic Gravitational Wave Spectrum in Next-to-Minimal Bouncing Cosmology." pith.science (2026). https://pith.science/paper/SKTSOVJK

@misc{pith2026250712968,
  author       = {Pith},
  title        = {Pith review of: Analytic Gravitational Wave Spectrum in Next-to-Minimal Bouncing Cosmology},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/SKTSOVJK}},
  note         = {Machine review of arXiv:2507.12968}
}
abstract

Bouncing cosmology offers a singularity-free alternative to inflation, but its minimal realization-comprising only four cosmic phases-predicts a simple power-law stochastic gravitational-wave background (SGWB) with a narrow observational window. We introduce the next-to-minimal bouncing cosmology (NMBC), which adds an extra early contraction phase that imprints a broken power-law feature in the SGWB spectrum, enhancing detectability. Using our matrix-representation method grounded in an inequality algebra, we derive a closed-form expression for the NMBC SGWB spectrum. From this analytical result, we show that all NMBC models satisfying the current \(\Delta N_{\rm eff}\) bound \(\Omega_{\rm GW}h^2(f)<1.7\times10^{-6}\) automatically avoid the trans-Planckian problem, \(\rho_{s\downarrow}^{1/4}<0.79\,m_{\rm pl}\). These findings establish the NMBC as a self-consistent, self-contained framework capable of generating a potentially detectable SGWB in both astrophysical and laboratory searches, and demonstrate the broad utility of our matrix-representation method for future SGWB analyses in multi-phase cosmologies.

Figures

Figures reproduced from arXiv: 2507.12968 by the authors.

Figure 1
Figure 1. FIG. 1. Schematic of the effective Hubble radius [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Contour curves where [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Illustrative SGWB spectra for four NMBC examples [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗

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Works this paper leans on

94 extracted references · 34 canonical work pages

  1. [49]

    Papanikolaou, S

    T. Papanikolaou, S. Banerjee, Y .-F. Cai, S. Capozziello, and E. N. Saridakis (2024), 2404.03779

    arXiv 2024

  2. [1]

    Substituting ν0 = ν1 into Eq

    ν0 = ν1 (w0 = w1) limit: Phases 0 and 1 coincide, so NMBC → MBC. Substituting ν0 = ν1 into Eq. (S27) gives N (0) 22 (ν0 = ν1, ν1 > 1 2 ) = N (1) 22 (ν1 > 1 2 ) , (31) as expected

  3. [2]

    Setting η0↓ = ηs↓ in Eq

    η0↓ = ηs↓ limit: Phase 0 vanishes, restoring the MBC. Setting η0↓ = ηs↓ in Eq. (S27) yields N (0) 22 (ν0, ν1 > 1 2 , η0↓ = ηs↓) = N (1) 22 (ν0) , (32) as expected. Here one must use the full form of N (0) 22 (Eqs. S33–S36 in the Sec.V of the SM), since the long-phase-1 approximation (Eqs. (28)–(30) following Eq. (27)) does not apply. Eqs. (31) and (32) co...

  4. [3]

    Analytically, this enhancement vanishes when phase 1 is removed, as expected

    relative to the usual n(1) T = 3 − 2˜ν1 modes. Analytically, this enhancement vanishes when phase 1 is removed, as expected. Cosmological Applications. In Fig. 3 we showcase four illus- trative NMBC spectra chosen to lie near current observational limits: • Example 1 (red): n(0) T (f < f⋆) = 1, n(0) T (f ≥ f⋆) = 0, f⋆ = 10 −6 Hz, Ω(0) GW(f⋆)h2 = 10 −7, wh...

  5. [4]

    Novello and S

    M. Novello and S. E. P. Bergliaffa, Phys. Rept.463, 127 (2008), 0802.1634

    arXiv 2008

  6. [5]

    Brandenberger and P

    R. Brandenberger and P. Peter, Found. Phys. 47, 797 (2017), 1603.05834

    arXiv 2017

  7. [6]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and V . K. Oikonomou, Phys. Rept. 692, 1 (2017), 1705.11098

    arXiv 2017

  8. [7]

    S. D. Odintsov, V . K. Oikonomou, I. Giannakoudi, F. P. Fron- imos, and E. C. Lymperiadou, Symmetry 15, 1701 (2023), 2307.16308

    arXiv 2023

Show all 94 references
  1. [8]

    Caprini and D

    C. Caprini and D. G. Figueroa, Class. Quant. Grav. 35, 163001 (2018), 1801.04268

    2018 arXiv

  2. [9]

    C. Li, J. Lai, J. Xiang, and C. Wu, JHEP 09, 138 (2024), 2405.15889

    2024 arXiv

  3. [10]

    Li, Phys

    C. Li, Phys. Rev. D 111, 103517 (2025), 2502.19124

    2025 arXiv

  4. [11]

    Lai and C

    J. Lai and C. Li, Eur. Phys. J. C 85, 714 (2025), 2504.19251

    2025 arXiv

  5. [12]

    Li (2025), 2506.00684

    C. Li (2025), 2506.00684

    2025 arXiv

  6. [13]

    Khoury, B

    J. Khoury, B. A. Ovrut, P. J. Steinhardt, and N. Turok, Phys. Rev. D 64, 123522 (2001), hep-th/0103239

    2001 arXiv

  7. [14]

    Gasperini and G

    M. Gasperini and G. Veneziano, Phys. Rept.373, 1 (2003), hep- th/0207130

    2003

  8. [15]

    Creminelli, M

    P. Creminelli, M. A. Luty, A. Nicolis, and L. Senatore, JHEP 12, 080 (2006), hep-th/0606090

    2006 arXiv

  9. [16]

    Peter, E

    P. Peter, E. J. C. Pinho, and N. Pinto-Neto, Phys. Rev. D 75, 023516 (2007), hep-th/0610205

    2007 arXiv

  10. [17]

    Y .-F. Cai, T. Qiu, Y .-S. Piao, M. Li, and X. Zhang, JHEP 10, 071 (2007), 0704.1090

    2007 arXiv

  11. [18]

    Cai, T.-t

    Y .-F. Cai, T.-t. Qiu, R. Brandenberger, and X.-m. Zhang, Phys. Rev. D 80, 023511 (2009), 0810.4677

    2009 arXiv

  12. [19]

    Saidov and A

    T. Saidov and A. Zhuk, Phys. Rev. D 81, 124002 (2010), 1002.4138

    2010 arXiv

  13. [20]

    C. Li, L. Wang, and Y .-K. E. Cheung, Phys. Dark Univ. 3, 18 (2014), 1101.0202

    2014 arXiv

  14. [21]

    Cai, S.-H

    Y .-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta, and E. N. Saridakis, Class. Quant. Grav. 28, 215011 (2011), 1104.4349

    2011 arXiv

  15. [22]

    D. A. Easson, I. Sawicki, and A. Vikman, JCAP11, 021 (2011), 1109.1047

    2011 arXiv

  16. [23]

    Bhattacharya, Y .-F

    K. Bhattacharya, Y .-F. Cai, and S. Das, Phys. Rev. D87, 083511 (2013), 1301.0661

    2013 arXiv

  17. [24]

    Qiu and Y .-T

    T. Qiu and Y .-T. Wang, JHEP04, 130 (2015), 1501.03568

    2015 arXiv

  18. [25]

    J. D. Barrow and C. Ganguly, Phys. Rev. D 95, 083515 (2017), 1703.05969

    2017 arXiv

  19. [26]

    de Haro and J

    J. de Haro and J. Amor ´os, Phys. Rev. D 97, 064014 (2018), 1712.08399

    2018 arXiv

  20. [27]

    Ijjas and P

    A. Ijjas and P. J. Steinhardt, Class. Quant. Grav. 35, 135004 (2018), 1803.01961

    2018 arXiv

  21. [28]

    S. S. Boruah, H. J. Kim, M. Rouben, and G. Geshnizjani, JCAP 08, 031 (2018), 1802.06818

    2018 arXiv

  22. [29]

    Nojiri, S

    S. Nojiri, S. D. Odintsov, and E. N. Saridakis, Nucl. Phys. B 949, 114790 (2019), 1908.00389

    2019 arXiv

  23. [30]

    C. A. S. Silva, Eur. Phys. J. C 78, 409 (2018), 1503.00559

    2018 arXiv

  24. [31]

    Silva, Phys

    C. Silva, Phys. Rev. D 102, 046001 (2020), 2008.07279

    2020 arXiv

  25. [32]

    Silva, Nucl

    C. Silva, Nucl. Phys. B 998, 116402 (2024), 2312.05260

    2024 arXiv

  26. [33]

    Nayeri, R

    A. Nayeri, R. H. Brandenberger, and C. Vafa, Phys. Rev. Lett. 97, 021302 (2006), hep-th/0511140

    2006 arXiv

  27. [34]

    R. H. Brandenberger, A. Nayeri, S. P. Patil, and C. Vafa, Phys. Rev. Lett. 98, 231302 (2007), hep-th/0604126

    2007 arXiv

  28. [35]

    Fischler and L

    W. Fischler and L. Susskind (1998), hep-th/9806039

    1998 arXiv

  29. [36]

    Y .-F. Cai, W. Xue, R. Brandenberger, and X.-m. Zhang, JCAP 06, 037 (2009), 0903.4938

    2009 arXiv

  30. [37]

    C. Li, R. H. Brandenberger, and Y .-K. E. Cheung, Phys. Rev. D 90, 123535 (2014), 1403.5625

    2014 arXiv

  31. [38]

    Y .-K. E. Cheung, J. U. Kang, and C. Li, JCAP 11, 001 (2014), 1408.4387

    2014 arXiv

  32. [39]

    Li, Phys

    C. Li, Phys. Rev. D 92, 063513 (2015), 1404.4012

    2015 arXiv

  33. [40]

    Li, JCAP 09, 038 (2016), 1512.06794

    C. Li, JCAP 09, 038 (2016), 1512.06794

    2016 arXiv

  34. [41]

    Li, Phys

    C. Li, Phys. Rev. D 102, 123530 (2020), 2008.10264

    2020 arXiv

  35. [42]

    L. A. Boyle, P. J. Steinhardt, and N. Turok, Phys. Rev. D 70, 023504 (2004), hep-th/0403026

    2004 arXiv

  36. [43]

    Piao and Y .-Z

    Y .-S. Piao and Y .-Z. Zhang, Phys. Rev. D 70, 043516 (2004), astro-ph/0403671

    2004 arXiv

  37. [44]

    Cai, Sci

    Y .-F. Cai, Sci. China Phys. Mech. Astron. 57, 1414 (2014), 1405.1369

    2014 arXiv

  38. [45]

    Chowdhury, V

    D. Chowdhury, V . Sreenath, and L. Sriramkumar, JCAP11, 002 (2015), 1506.06475

    2015 arXiv

  39. [46]

    Y .-F. Cai, A. Marciano, D.-G. Wang, and E. Wilson-Ewing, Universe 3, 1 (2016), 1610.00938

    2016 arXiv

  40. [47]

    Zhang and Y .-K

    N. Zhang and Y .-K. E. Cheung, Eur. Phys. J. C80, 100 (2020), 1901.06423

    2020 arXiv

  41. [48]

    M. Zhu, G. Ye, and Y . Cai, Eur. Phys. J. C 83, 816 (2023), 2307.16211

    2023 arXiv

  42. [50]

    Ben-Dayan, G

    I. Ben-Dayan, G. Calcagni, M. Gasperini, A. Mazum- dar, E. Pavone, U. Thattarampilly, and A. Verma (2024), 2406.13521

    2024 arXiv

  43. [51]

    Qiu and M

    T. Qiu and M. Zhu (2024), 2408.06582

    2024 arXiv

  44. [52]

    Li, Phys

    C. Li, Phys. Rev. D 110, 083535 (2024), 2407.10071

    2024 arXiv

  45. [53]

    T. L. Smith, E. Pierpaoli, and M. Kamionkowski, Phys. Rev. Lett. 97, 021301 (2006), astro-ph/0603144

    2006 arXiv

  46. [54]

    Pagano, L

    L. Pagano, L. Salvati, and A. Melchiorri, Phys. Lett. B760, 823 (2016), 1508.02393

    2016 arXiv

  47. [55]

    Schmitz, JHEP 01, 097 (2021), 2002.04615

    K. Schmitz, JHEP 01, 097 (2021), 2002.04615

    2021 arXiv

  48. [56]

    Annis, J

    J. Annis, J. A. Newman, and A. Slosar (2022), 2209.08049

    2022 arXiv

  49. [57]

    Bi, Y .-M

    Y .-C. Bi, Y .-M. Wu, Z.-C. Chen, and Q.-G. Huang, Sci. China Phys. Mech. Astron. 66, 120402 (2023), 2307.00722

    2023 arXiv

  50. [58]

    Y . Chen, C. Li, Y . Liu, J. Shu, Y . Yang, and Y . Zeng, Rept. Prog. Phys. 88, 057601 (2025), 2309.12387

    2025 arXiv

  51. [59]

    Y .-F. Cai, E. N. Saridakis, M. R. Setare, and J.-Q. Xia, Phys. Rept. 493, 1 (2010), 0909.2776

    2010 arXiv

  52. [60]

    Martin and R

    J. Martin and R. H. Brandenberger, Phys. Rev. D 63, 123501 (2001), hep-th/0005209

    2001 arXiv

  53. [61]

    R. H. Brandenberger and J. Martin, Mod. Phys. Lett. A 16, 999 (2001), astro-ph/0005432

    2001 arXiv

  54. [62]

    R. H. Brandenberger and J. Martin, Class. Quant. Grav. 30, 113001 (2013), 1211.6753

    2013 arXiv

  55. [63]

    Kaloper, M

    N. Kaloper, M. Kleban, A. Lawrence, S. Shenker, and L. Susskind, JHEP 11, 037 (2002), hep-th/0209231

    2002 arXiv

  56. [64]

    Easther, B

    R. Easther, B. R. Greene, W. H. Kinney, and G. Shiu, Phys. Rev. D 66, 023518 (2002), hep-th/0204129

    2002 arXiv

  57. [65]

    Bedroya and C

    A. Bedroya and C. Vafa, JHEP 09, 123 (2020), 1909.11063

    2020 arXiv

  58. [66]

    Bedroya, R

    A. Bedroya, R. Brandenberger, M. Loverde, and C. Vafa, Phys. Rev. D 101, 103502 (2020), 1909.11106

    2020 arXiv

  59. [67]

    Benetti, L

    M. Benetti, L. L. Graef, and S. Vagnozzi, Phys. Rev. D 105, 043520 (2022), 2111.04758

    2022 arXiv

  60. [68]

    Bedroya, JHEP 06, 016 (2024), 2211.09128

    A. Bedroya, JHEP 06, 016 (2024), 2211.09128

    2024 arXiv

  61. [69]

    Bedroya, Q

    A. Bedroya, Q. Lu, and P. Steinhardt (2024), 2407.08793

    2024 arXiv

  62. [70]

    Bedroya, H

    A. Bedroya, H. Lee, and P. Steinhardt (2025), 2504.13260

    2025 arXiv

  63. [71]

    Vagnozzi, JHEAp 39, 81 (2023), 2306.16912

    S. Vagnozzi, JHEAp 39, 81 (2023), 2306.16912

    2023 arXiv

  64. [72]

    Aggarwal et al., Living Rev

    N. Aggarwal et al., Living Rev. Rel. 24, 4 (2021), 2011.12414

    2021 arXiv

  65. [73]

    Bian et al

    L. Bian et al. (2025), 2505.19747. Supplemental Material I. Cosmic History of the Minimal and Next-to-Minimal Bouncing Cosmologies A. Minimal Bouncing Cosmology (MBC) The MBC consists of four successive phases:

    2025

  66. [74]

    Phase 1: Collapsing contraction ( ˙a <0, ¨a <0) with kη → 0 and w1 ≥ −1 3

  67. [75]

    Phase 2: Bounce contraction ( ˙a <0, ¨a >0) with kη → ∞and w2 < − 1 3

  68. [76]

    Phase 3: Bounce expansion ( ˙a >0, ¨a >0) with kη → 0 and w3 < − 1 3

  69. [77]

    Phase 4: Post-bounce expansion ( ˙a >0, ¨a >0) with kη → ∞and w4 ≥ −1 3. B. Next-to-Minimal Bouncing Cosmology (NMBC) The NMBC adds an extra pre-bounce contraction phase:

  70. [78]

    Phase 0: Early collapsing contraction ( ˙a <0, ¨a <0) with kη → kη0↓ and w0 ≥ −1

  71. [79]

    In the main text we assume Phase 4 corresponds to the standard radiation-dominated era, and subsequent matter- and dark-energy-dominated phases follow the usual treatment [5]

    In general w0 ̸= w1. In the main text we assume Phase 4 corresponds to the standard radiation-dominated era, and subsequent matter- and dark-energy-dominated phases follow the usual treatment [5]. II. Transformation and Boundary Matrices A. Transformation Matrix In Ref. [49], ...

  72. [80]

    (S2)) into the matching conditions yields Ai+1 Bi+1 = Mi↑ Ai Bi , (S7) with Mi↑ = e−i(˜νi−˜νi+1)π/2 0 0 e i(˜νi−˜νi+1)π/2

    Sub-horizon matching Mi↑ : For modes with kηi↑ ≫ 1, substituting the sub-horizon solution (Eq. (S2)) into the matching conditions yields Ai+1 Bi+1 = Mi↑ Ai Bi , (S7) with Mi↑ = e−i(˜νi−˜νi+1)π/2 0 0 e i(˜νi−˜νi+1)π/2 . (S8)

  73. [81]

    Super-horizon matching Mi↓ : For modes with kηi↓ ≪ 1, using the super-horizon solution (Eq. (S3)) gives Ei+1 Fi+1 = Mi↓ Ei Fi , (S9) where Mi↓ = 1 ¯χi+1 − χi+1 ( ¯χi+1 − χi) (kηi↓)˜νi−˜νi+1 ( ¯χi+1 − ¯χi) (kηi↓)−˜νi−˜νi+1 (χi − χi+1) (kηi↓)˜νi+˜νi+1 ( ¯χi − χi+1) (kηi↓)−˜νi+˜ν...

  74. [82]

    Values of χi and ¯χi: χi = 0, ¯χi = −2˜νi, ν i > 1 2 ; (S12) χi = 2˜νi, ¯χi = 0, ν i ≤ 1 2 . (S13)

  75. [83]

    Orthogonality: χi ¯χi = 0 for all νi. (S14)

  76. [84]

    (S16) This inequality-based algebra greatly simplifies the boundary-matching matrices (Eq

    Squares: χiχi = 0, ¯χi ¯χi = 4˜ν2 i , ν i > 1 2 ; (S15) χiχi = 4˜ν2 i , ¯χi ¯χi = 0, ν i ≤ 1 2 . (S16) This inequality-based algebra greatly simplifies the boundary-matching matrices (Eq. (S10)) and highlights the two distinct branches νi > 1 2 and νi ≤ 1 2 in the final SGWB s...

  77. [85]

    (S22) Here (A1, B1) encode the Bunch–Davies vacuum at phase 1: A1 B1 = 0√π 2 e−i( ˜ν1 π 2 + π 4 ) !

    High frequency modes f > f⋆ : These modes exit during phase 1, so A4 B4 = T −1 4 E4 F4 = T −1 4 M3↓ E3 F3 = T −1 4 M3↓T3 A3 B3 = T −1 4 M3↓T3M2↑ A2 B2 = T −1 4 M3↓T3M2↑T −1 2 E2 F2 = T −1 4 M3↓T3M2↑T −1 2 M1↓ E1 F1 = T −1 4 M3↓T3M2↑T −1 2 M1↓T1 A1 B1 , f > f⋆. (S22) Here (A1, ...

  78. [86]

    (S24) Here (A0, B0) encode the Bunch–Davies vacuum at phase 0: A0 B0 = 0√π 2 e−i( ˜ν1 π 2 + π 4 ) !

    Low frequency modes f < f⋆ : These modes exit during phase 0, so A4 B4 = T −1 4 E4 F4 = T −1 4 M3↓ E3 F3 = T −1 4 M3↓T3 A3 B3 = T −1 4 M3↓T3M2↑ A2 B2 = T −1 4 M3↓T3M2↑T −1 2 E2 F2 = T −1 4 M3↓T3M2↑T −1 2 M1↓ E1 F1 = T −1 4 M3↓T3M2↑T −1 2 M1↓M0↓ E0 F0 = T −1 4 M3↓T3M2↑T −1 2 M1...

  79. [87]

    (30) and (21) of the main text

    Solve n(0) T (f < f⋆) and n(0) T (f ≥ f⋆) for w0 and w1 using Eqs. (30) and (21) of the main text

  80. [88]

    Compute η0↓ from f⋆ = (2πa0η0↓)−1, and ηs↓ from ηs↓ = H −1 0 (ρc0/Ωγ0)/ρs↓ 1/4

  81. [89]

    (22) and (19) of the main text)

    Determine ρ1/4 s↓ by inverting Ω(0) GW(f⋆)h2 with w1 (Eqs. (22) and (19) of the main text)

  82. [90]

    These values are then used in the main text to generate the analytic NMBC spectra

    Evaluate the cutoff frequency fcut = (2πa0ηs↓)−1; modes with f > fcut never exit the horizon. These values are then used in the main text to generate the analytic NMBC spectra. In particular,

  83. [91]

    This yields {wi} = 1 9 , 0, −∞, −∞, 1 3 , {ηi↓/↑} = 1.6 × 105, 1.38 × 10−11, ∞, 1.38 × 10−11 Hz−1, ρ1/4 s↓ = 0.39 mpl, f cut = 1.15 × 1010 Hz

    Example 1 (red): Phenomenological inputs: n(0) T (f < f⋆) = 1, n (0) T (f ≥ f⋆) = 0, f ⋆ = 10−6 Hz, Ω(0) GW(f⋆)h2 = 10−7. This yields {wi} = 1 9 , 0, −∞, −∞, 1 3 , {ηi↓/↑} = 1.6 × 105, 1.38 × 10−11, ∞, 1.38 × 10−11 Hz−1, ρ1/4 s↓ = 0.39 mpl, f cut = 1.15 × 1010 Hz

  84. [92]

    This yields {wi} = 3 11 , − 1 51 , −∞, −∞, 1 3 , {ηi↓/↑} = 1.59 × 106, 8.96 × 10−11, ∞, 8.96 × 10−11 Hz−1, ρ1/4 s↓ = 0.06 × 10−7 mpl, f cut = 1.7 × 109 Hz

    Example 2 (green) Phenomenological inputs: n(0) T (f < f⋆) = 1.8, n (0) T (f ≥ f⋆) = − 1 4 , f ⋆ = 10−7 Hz, Ω(0) GW(f⋆)h2 = 10−6. This yields {wi} = 3 11 , − 1 51 , −∞, −∞, 1 3 , {ηi↓/↑} = 1.59 × 106, 8.96 × 10−11, ∞, 8.96 × 10−11 Hz−1, ρ1/4 s↓ = 0.06 × 10−7 mpl, f cut = 1.7 × 109 Hz

  85. [93]

    This yields {wi} = 4 × 105/3, 0, −∞, −∞, 1 3 , {ηi↓/↑} = 1.6 × 109, 4.3 × 10−10, ∞, 4.3 × 10−10 Hz−1, ρ1/4 s↓ = 0.012 mpl, f cut = 3.65 × 108 Hz

    Example 3 (blue) Phenomenological inputs: n(0) T (f < f⋆) = 4 − 10−5, n (0) T (f ≥ f⋆) = 0, f ⋆ = 10−10 Hz, Ω(0) GW(f⋆)h2 = 10−13. This yields {wi} = 4 × 105/3, 0, −∞, −∞, 1 3 , {ηi↓/↑} = 1.6 × 109, 4.3 × 10−10, ∞, 4.3 × 10−10 Hz−1, ρ1/4 s↓ = 0.012 mpl, f cut = 3.65 × 108 Hz

  86. [94]

    This yields {wi} = 1 9 , − 1 15 , −∞, −∞, 1 3 , {ηi↓/↑} = 1.6 × 10−7, 1.27 × 10−10, ∞, 1.27 × 10−10 Hz−1, ρ1/4 s↓ = 0.042 mpl, f cut = 1.25 × 109 Hz

    Example 4 (orange) Phenomenological inputs: n(0) T (f < f⋆) = 1, n (0) T (f ≥ f⋆) = −1, f ⋆ = 106 Hz, Ω(0) GW(f⋆)h2 = 10−7. This yields {wi} = 1 9 , − 1 15 , −∞, −∞, 1 3 , {ηi↓/↑} = 1.6 × 10−7, 1.27 × 10−10, ∞, 1.27 × 10−10 Hz−1, ρ1/4 s↓ = 0.042 mpl, f cut = 1.25 × 109 Hz

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.