The DHOST bounce scenario of Zhu, Ilyas, Zheng, Cai and Saridakis (JCAP 11 (2021) 045) realizes a ghost-free nonsingular bouncing universe within degenerate higher-order scalar-tensor (DHOST) theories and derives its scalar and tensor perturbation predictions analytically, under a short-bounce approximation (H≃ΓtH\simeq\Gamma t) with the scale-dependent cT2k2c_T^2k^2 terms neglected during the bounce; the tensor modes are found to pass almost unchanged through the bounce, giving a blue tensor spectrum and a strongly suppressed tensor-to-scalar ratio. The paper explicitly leaves open the imprint of the null-energy-condition (NEC) violating bounce phase on primordial gravitational waves. The open problem is to compute the relic gravitational-wave spectrum of this specific model — evolving the tensor modes through the Ekpyrotic contraction, the NEC-violating DHOST bounce, and the fast-roll expansion into the radiation era, with the model's actual background and DHOST tensor sector and without the simplifying approximations — to determine whether the bounce phase leaves distinctive, possibly frequency-dependent signatures in ΩGW(f)\Omega_{\rm GW}(f), to assess their detectability against CMB, pulsar-timing-array, interferometer and big-bang-nucleosynthesis bounds, and to confirm or correct the analytic tensor-transfer conclusions. Subsequent work has computed gravitational-wave signatures only for phenomenological bounce parametrizations or sibling theory classes, so the model-specific question remains open.

Background

Inflation is the standard paradigm for the early universe, but alternatives in which the Big Bang is replaced by a nonsingular bounce have been pursued for decades. A contracting phase must violate the null energy condition (NEC), the condition ρ+p≥0\rho+p\geq 0 on the energy density and pressure that in general relativity forbids the Hubble parameter HH from growing, in order for HH to pass continuously from negative (contraction) to positive (expansion) through a bounce. Because a naive NEC-violating phase typically excites a ghost (negative-kinetic-energy) mode, realizations are usually built in modified gravity. DHOST theories (degenerate higher-order scalar-tensor theories) are the most general four-dimensional scalar-tensor theories containing up to cubic powers of the second derivatives of the scalar field whose degeneracy conditions eliminate the Ostrogradsky ghost, propagating at most three degrees of freedom; Horndeski and Galileon theories are subclasses. Ilyas, Zhu, Zheng, Cai and Saridakis (JCAP 09 (2020) 002) constructed a healthy DHOST bounce free of ghost, gradient and Belinski-Khalatnikov-Lifshitz (BKL) instabilities (the BKL instability is the disruptive growth of anisotropy, which scales as a−6a^{-6} during contraction and must be diluted by a stiff contraction), and Zhu, Ilyas, Zheng, Cai and Saridakis (JCAP 11 (2021) 045, arXiv:2108.01339) — the source of this problem — analyzed its cosmological perturbations through three phases: an Ekpyrotic contraction (a stiff contraction with effective equation-of-state parameter w>1w>1, parametrized by qq with a(τ)∝(τ−τ−)qa(\tau)\propto(\tau-\tau_-)^q, 0<q<1/30<q<1/3), a short NEC-violating bounce phase approximated by H≃ΓtH\simeq\Gamma t, and a fast-roll expansion into the standard radiation era, with junction (matching) conditions connecting the perturbation solutions across phase boundaries. Both scalar and tensor modes obey Mukhanov-Sasaki-type equations, μk′′+(c2(τ)k2−a′′/a)μk=0\mu_k''+(c^2(\tau)k^2-a''/a)\mu_k=0 in conformal time τ\tau (primes denoting d/dτd/d\tau), where csc_s and cTc_T are the scalar and tensor propagation speeds set by the DHOST Lagrangian; the tensor variable μk=aγk/2\mu_k=a\gamma_k/2 describes the two gravitational-wave polarizations γij\gamma_{ij}. Three sets of initial conditions were studied: Ekpyrotic vacuum fluctuations (which give a blue scalar spectrum, ruled out), matter vacuum fluctuations (a preceding matter-dominated contraction, w=0w=0, can give a nearly scale-invariant scalar spectrum), and thermal fluctuations of a hot gas present at the bounce, the most successful case, which reproduces the observed scalar tilt ns≃0.96n_s\simeq0.96 and a tensor-to-scalar ratio rr suppressed by many orders of magnitude because the tensor spectrum is blue (tensor index nt≃2.48n_t\simeq2.48) and the bounce amplifies only the scalar mode, by a nearly scale-independent factor Fs∼104F_s\sim10^4, while leaving the tensor modes almost invariant. These conclusions rest on the approximations the authors themselves list: the linear fast-bounce form H≃ΓtH\simeq\Gamma t with small-argument Bessel expansions (Γ τB±∼10−2\sqrt{\Gamma}\,\tau_{B\pm}\sim10^{-2}), neglect of the scale-dependent cs2k2c_s^2k^2 and cT2k2c_T^2k^2 terms during the bounce, and restriction to adiabatic perturbations. The paper closes by noting, in the inflationary context, that NEC-violating phases can produce observable gravitational-wave signals (Tahara and Kobayashi, Phys. Rev. D 102 (2020) 123533; Cai and Piao, Phys. Rev. D 103 (2021) 083521), and leaves the imprint on primordial gravitational waves from the NEC-violating bounce phase of the DHOST model as an open direction. The observational context has since sharpened considerably: pulsar timing arrays report a nanohertz stochastic background; proposed detectors (LISA at millihertz, DECIGO/BBO in the decihertz band, and high-frequency proposals) extend coverage over many decades of frequency; and the total stochastic background is bounded by big-bang-nucleosynthesis and cosmic-microwave-background (CMB) constraints on ∫f ΩGW(f) df\int f\,\Omega_{\rm GW}(f)\,df. Related but distinct work frames why the model-specific computation is nontrivial and not yet done: Zhu and Cai (Astrophys. J. Lett. 1004 (2026) L34, arXiv:2603.13924) showed that in nonsingular bounces the effective potential MPl2a′′/aM_{\rm Pl}^2a''/a develops two peaks between which relic gravitational waves interfere, producing oscillatory high-frequency features — but for a phenomenological parametrized background with standard tensor propagation, not the DHOST model; Qiu and Zhu (Phys. Rev. D 111 (2025) 043508) fit pulsar-timing-array data with the blue tensor spectrum of a sibling Ekpyrosis-bounce model built from k-essence/ghost-condensate NEC violation plus a curvaton-like field; Li (Phys. Rev. D 110 (2024) 083535) derived analytic gravitational-wave spectra for a generic parametrized four-phase big bounce; and An, Kang, Kim, Mun and Ri (JHEP 05 (2025) 005) constructed a fully viable two-field DHOST bounce with entropic generation of curvature perturbations but did not compute its gravitational-wave spectrum. What is missing, and what this problem targets, is the gravitational-wave prediction of the DHOST bounce model itself, including its DHOST tensor sector and its specific background, and the validation of the source paper's analytic tensor-transfer results without the simplifying approximations.

Problem Statement

For the DHOST bounce cosmology of Zhu, Ilyas, Zheng, Cai and Saridakis (JCAP 11 (2021) 045) — with its Ekpyrotic contraction, short NEC-violating bounce phase, and fast-roll expansion matched onto the standard radiation era — determine the relic primordial gravitational-wave prediction of the model. This requires, as one jointly supporting objective: (i) evolving the tensor mode equation μk′′+(cT2(τ)k2−a′′/a)μk=0\mu_k''+(c_T^2(\tau)k^2-a''/a)\mu_k=0 (with the DHOST-determined tensor speed and friction, i.e. the quadratic tensor action of the theory) on the model's actual background solution, across the comoving wavenumber range relevant to present and planned observations, without the source paper's simplifying approximations (the linear fast-bounce form H≃ΓtH\simeq\Gamma t with small-argument expansions, and the neglect of the scale-dependent cT2k2c_T^2k^2 term during the bounce); (ii) deriving the resulting tensor power spectrum and gravitational-wave energy-density spectrum ΩGW(f)\Omega_{\rm GW}(f) for the surviving viable initial conditions (in particular the thermal-fluctuation case with its blue tensor index nt≃2.48n_t\simeq2.48), identifying any distinctive, possibly frequency-dependent imprint of the NEC-violating bounce phase — such as interference-induced oscillatory features — that would distinguish the model from inflation and from other bounce scenarios, and assessing its amplitude against CMB anisotropy bounds, pulsar-timing-array observations, LISA/DECIGO-class interferometer sensitivities, and the big-bang-nucleosynthesis/CMB bound on the integrated gravitational-wave energy density; and (iii) establishing whether the paper's analytic conclusions — that the tensor perturbations remain almost invariant during the bounce phase while scalar perturbations are amplified by a nearly scale-independent factor, so that the tensor-to-scalar ratio is strongly suppressed — are confirmed or corrected by this full treatment. The problem is solved when the model's gravitational-wave spectrum is quantitatively established (analytically and/or numerically), its observable consequences are stated with detectability or bounds, and the validity of the analytic tensor-transfer results is determined.

References

  1. Mian Zhu, Amara Ilyas, Yunlong Zheng, Yi-Fu Cai, Emmanuel N. Saridakis, "Scalar and tensor perturbations in DHOST bounce cosmology", JCAP 11 (2021) 045. DOI: 10.1088/1475-7516/2021/11/045, arXiv:2108.01339, https://arxiv.org/abs/2108.01339 (source paper)
  2. Amara Ilyas, Mian Zhu, Yunlong Zheng, Yi-Fu Cai, Emmanuel N. Saridakis, "DHOST Bounce", JCAP 09 (2020) 002. DOI: 10.1088/1475-7516/2020/09/002, arXiv:2002.08269, https://arxiv.org/abs/2002.08269
  3. Mian Zhu, Yi-Fu Cai, "Smoking-gun signatures of bounce cosmology from echoes of relic gravitational waves", Astrophys. J. Lett. 1004 (2026) L34. DOI: 10.3847/2041-8213/ae6f10, arXiv:2603.13924, https://arxiv.org/abs/2603.13924
  4. Taotao Qiu, Mian Zhu, "Interpreting pulsar timing array data of gravitational waves with Ekpyrosis-bouncing cosmology", Phys. Rev. D 111 (2025) 043508. DOI: 10.1103/PhysRevD.111.043508, arXiv:2408.06582, https://arxiv.org/abs/2408.06582
  5. Ok Song An, Jin U. Kang, Yong Jin Kim, Ui Ri Mun, Un Gyong Ri, "Fully viable DHOST bounce with extra scalar", JHEP 05 (2025) 005. DOI: 10.1007/JHEP05(2025)005, arXiv:2501.09985, https://arxiv.org/abs/2501.09985
  6. Changhong Li, "Primordial gravitational waves of big bounce cosmology in light of stochastic gravitational wave background", Phys. Rev. D 110 (2024) 083535. DOI: 10.1103/PhysRevD.110.083535, arXiv:2407.10071, https://arxiv.org/abs/2407.10071
  7. Philippe Brax, Andrei Lazanu, "Primordial gravitational waves in DHOST inflation", Mon. Not. R. Astron. Soc. Lett. 540 (2025) L98. DOI: 10.1093/mnrasl/slaf040, arXiv:2501.13210, https://arxiv.org/abs/2501.13210
  8. Igor de O. C. Pedreira, Amara Ilyas, Ziwei Wang, Leila L. Graef, Yi-Fu Cai, "Sound speed resonance in the gravitational wave background as a probe for non-standard early universe cosmologies", JCAP 07 (2026) 011. DOI: 10.1088/1475-7516/2026/07/011, arXiv:2603.12580, https://arxiv.org/abs/2603.12580
  9. V. E. Volkova, S. A. Mironov, "Nonsingular cosmological scenarios in scalar-tensor theories and their stability" (review), Phys. Usp. 68 (2024) 163 [Uspekhi Fizicheskikh Nauk 195 (2024) 172]. DOI: 10.3367/UFNe.2024.12.039826, arXiv:2409.16108, https://arxiv.org/abs/2409.16108
  10. Hiroaki W. H. Tahara, Tsutomu Kobayashi, "Nanohertz gravitational waves from a null-energy-condition violation in the early universe", Phys. Rev. D 102 (2020) 123533. DOI: 10.1103/PhysRevD.102.123533, arXiv:2011.01605, https://arxiv.org/abs/2011.01605
  11. Yong Cai, Yun-Song Piao, "Intermittent null energy condition violations during inflation and primordial gravitational waves", Phys. Rev. D 103 (2021) 083521. DOI: 10.1103/PhysRevD.103.083521, arXiv:2012.11304, https://arxiv.org/abs/2012.11304