Primordial black holes and induced gravitational waves in non-singular matter bouncing cosmology

Author(s)

Papanikolaou, Theodoros, Banerjee, Shreya, Cai, Yi-Fu, Capozziello, Salvatore, Saridakis, Emmanuel N.

Abstract

We present a novel model-independent generic mechanism for primordial black hole formation within the context of non-singular matter bouncing cosmology. In particular, considering a short duration transition from the matter contracting phase to the Hot Big Bang expanding Universe, we find naturally enhanced curvature perturbations on very small scales which can collapse and form primordial black holes. Interestingly, the primordial black hole masses that we find can lie within the observationally unconstrained asteroid-mass window, potentially explaining the totality of dark matter. Remarkably, the enhanced curvature perturbations, collapsing to primordial black holes, can induce as well a stochastic gravitational-wave background, being potentially detectable by future experiments, in particular by SKA, PTAs, LISA and ET, hence serving as a new portal to probe the potential bouncing nature of the initial conditions prevailing in the early Universe.

Figures

\it{The solid blue curve corresponds to the full curvature power spectrum, for $H_{+} = 10^{-10}\Mp$, $H_{-} = 6\times 10^{-11}\Mp$ and $Y = 1.7345\times 10^{-10}\Mp^2$. The dashed green curve corresponds to the analytic approximation for $\mathcal{P}_{\mathcal{R}}(k)$ up to linear order in $k$, while the dashed red curve  depicts the  analytic approximation for $\mathcal{P}_{\mathcal{R}}(k)$ up $\mathcal{O}(k^{9/2})$.}

\it{The solid blue curve corresponds to the full curvature power spectrum, for $H_{+} = 10^{-10}\Mp$, $H_{-} = 6\times 10^{-11}\Mp$ and $Y = 1.7345\times 10^{-10}\Mp^2$. The dashed green curve corresponds to the analytic approximation for $\mathcal{P}_{\mathcal{R}}(k)$ up to linear order in $k$, while the dashed red curve depicts the analytic approximation for $\mathcal{P}_{\mathcal{R}}(k)$ up $\mathcal{O}(k^{9/2})$.}


\it{Left Panel: The curvature power spectra for different fiducial values of the parameters $H_{+}$, $H_{-}$ and $Y$. Right Panel: The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}.}

\it{Left Panel: The curvature power spectra for different fiducial values of the parameters $H_{+}$, $H_{-}$ and $Y$. Right Panel: The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}.}


\it{Left Panel: The curvature power spectra for different fiducial values of the parameters $H_{+}$, $H_{-}$ and $Y$. Right Panel: The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}.}

\it{Left Panel: The curvature power spectra for different fiducial values of the parameters $H_{+}$, $H_{-}$ and $Y$. Right Panel: The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}.}


\it{The scalar-induced gravitational-wave spectra for different values of the parameters $H_{+}$, $H_{-}$ and $Y$. On   top of the   GW spectra we present   the sensitivity curves of SKA~\cite{Janssen:2014dka}, LISA~\cite{LISACosmologyWorkingGroup:2022jok}, BBO~\cite{Harry:2006fi} and ET~\cite{Maggiore:2019uih} GW experiments.}

\it{The scalar-induced gravitational-wave spectra for different values of the parameters $H_{+}$, $H_{-}$ and $Y$. On top of the GW spectra we present the sensitivity curves of SKA~\cite{Janssen:2014dka}, LISA~\cite{LISACosmologyWorkingGroup:2022jok}, BBO~\cite{Harry:2006fi} and ET~\cite{Maggiore:2019uih} GW experiments.}


\it{Left Panel:  The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}. Right Panel: The scalar-induced gravitational-wave spectrum for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$, in comparison with the NANOGrav GW data  \cite{NANOGrav:2023gor}. On   top of our GW spectra we additionally present  the sensitivity curves of SKA~\cite{Janssen:2014dka}, LISA~\cite{LISACosmologyWorkingGroup:2022jok}, BBO~\cite{Harry:2006fi} and ET~\cite{Maggiore:2019uih} GW experiments.}

\it{Left Panel: The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}. Right Panel: The scalar-induced gravitational-wave spectrum for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$, in comparison with the NANOGrav GW data \cite{NANOGrav:2023gor}. On top of our GW spectra we additionally present the sensitivity curves of SKA~\cite{Janssen:2014dka}, LISA~\cite{LISACosmologyWorkingGroup:2022jok}, BBO~\cite{Harry:2006fi} and ET~\cite{Maggiore:2019uih} GW experiments.}


\it{Left Panel:  The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}. Right Panel: The scalar-induced gravitational-wave spectrum for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$, in comparison with the NANOGrav GW data  \cite{NANOGrav:2023gor}. On   top of our GW spectra we additionally present  the sensitivity curves of SKA~\cite{Janssen:2014dka}, LISA~\cite{LISACosmologyWorkingGroup:2022jok}, BBO~\cite{Harry:2006fi} and ET~\cite{Maggiore:2019uih} GW experiments.}

\it{Left Panel: The fraction of dark matter in terms of PBHs denoted as $f_\mathrm{PBH}=\Omega_\mathrm{PBH,0}/\Omega_\mathrm{DM,0}$ as a function of the PBH mass for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$. The colored regions are excluded from evaporation (blue region), microlensing (red region), gravitational-wave (green region) and CMB (violet region) observational probes concerning the PBH abundances. The data for the constraints on $f_\mathrm{PBH}$ from the different observational probes were obtained from~\cite{Green:2020jor}. Right Panel: The scalar-induced gravitational-wave spectrum for $H_+=8\times 10^{-3}\Mp$, $H_-=4\times 10^{-3}\Mp$ and $\Upsilon = 9\times 10^{-12}\Mp^2$, in comparison with the NANOGrav GW data \cite{NANOGrav:2023gor}. On top of our GW spectra we additionally present the sensitivity curves of SKA~\cite{Janssen:2014dka}, LISA~\cite{LISACosmologyWorkingGroup:2022jok}, BBO~\cite{Harry:2006fi} and ET~\cite{Maggiore:2019uih} GW experiments.}


References
  • [1] M. S. Turner, The Hot big bang and beyond, AIP Conf. Proc. 342 (1995) 43–62, [astro-ph/9503017].
  • [2] A. A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99–102.
  • [3] A. H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys.Rev. D23 (1981) 347–356.
  • [4] A. D. Linde, A New Inflationary Universe Scenario: A Possible Solution of the Horizon, Flatness, Homogeneity, Isotropy and Primordial Monopole Problems, Phys.Lett. B108 (1982) 389–393.
  • [5] A. Albrecht and P. J. Steinhardt, Cosmology for Grand Unified Theories with Radiatively Induced Symmetry Breaking, Phys.Rev.Lett. 48 (1982) 1220–1223.
  • [6] A. D. Linde, Chaotic Inflation, Phys.Lett. B129 (1983) 177–181.
  • [7] V. F. Mukhanov and R. H. Brandenberger, A Nonsingular universe, Phys. Rev. Lett. 68 (1992) 1969–1972.
  • [8] R. H. Brandenberger, V. F. Mukhanov and A. Sornborger, A Cosmological theory without singularities, Phys. Rev. D 48 (1993) 1629–1642, [gr-qc/9303001].
  • [9] A. Borde and A. Vilenkin, Singularities in inflationary cosmology: A Review, Int. J. Mod. Phys. D 5 (1996) 813–824, [gr-qc/9612036].
  • [10] M. Novello and S. E. P. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463 (2008) 127–213, [0802.1634].
  • [11] M. Lilley and P. Peter, Bouncing alternatives to inflation, Comptes Rendus Physique 16 (2015) 1038–1047, [1503.06578].
  • [12] D. Battefeld and P. Peter, A Critical Review of Classical Bouncing Cosmologies, Phys. Rept. 571 (2015) 1–66, [1406.2790].
  • [13] P. Peter and N. Pinto-Neto, Cosmology without inflation, Phys. Rev. D 78 (2008) 063506, [0809.2022].
  • [14] Y.-F. Cai, Exploring Bouncing Cosmologies with Cosmological Surveys, Sci. China Phys. Mech. Astron. 57 (2014) 1414–1430, [1405.1369].
  • [15] Y.-F. Cai, J. Quintin, E. N. Saridakis and E. Wilson-Ewing, Nonsingular bouncing cosmologies in light of BICEP2, JCAP 07 (2014) 033, [1404.4364].
  • [16] CANTATA collaboration, E. N. Saridakis et al., Modified Gravity and Cosmology: An Update by the CANTATA Network, 2105.12582.
  • [17] S. Nojiri and S. D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, eConf C0602061 (2006) 06, [hep-th/0601213].
  • [18] S. Capozziello and M. De Laurentis, Extended Theories of Gravity, Phys. Rept. 509 (2011) 167–321, [1108.6266].
  • [19] G. Veneziano, Scale factor duality for classical and quantum strings, Phys. Lett. B 265 (1991) 287–294.
  • [20] J. Khoury, B. A. Ovrut, P. J. Steinhardt and N. Turok, The Ekpyrotic universe: Colliding branes and the origin of the hot big bang, Phys. Rev. D 64 (2001) 123522, [hep-th/0103239].
  • [21] J. Khoury, B. A. Ovrut, N. Seiberg, P. J. Steinhardt and N. Turok, From big crunch to big bang, Phys. Rev. D 65 (2002) 086007, [hep-th/0108187].
  • [22] T. Biswas, A. Mazumdar and W. Siegel, Bouncing universes in string-inspired gravity, JCAP 03 (2006) 009, [hep-th/0508194].
  • [23] S. Nojiri and E. N. Saridakis, Phantom without ghost, Astrophys. Space Sci. 347 (2013) 221–226, [1301.2686].
  • [24] M. Miranda, D. Vernieri, S. Capozziello and F. S. N. Lobo, Bouncing Cosmology in Fourth-Order Gravity, Universe 8 (2022) 161, [2203.04918].
  • [25] K. Bamba, A. N. Makarenko, A. N. Myagky, S. Nojiri and S. D. Odintsov, Bounce cosmology from F(R) gravity and F(R) bigravity, JCAP 01 (2014) 008, [1309.3748].
  • [26] S. Nojiri and S. D. Odintsov, Mimetic F(R) gravity: inflation, dark energy and bounce, 1408.3561.
  • [27] Y.-F. Cai, S.-H. Chen, J. B. Dent, S. Dutta and E. N. Saridakis, Matter Bounce Cosmology with the f(T) Gravity, Class. Quant. Grav. 28 (2011) 215011, [1104.4349].
  • [28] F. Bajardi, D. Vernieri and S. Capozziello, Bouncing Cosmology in f(Q) Symmetric Teleparallel Gravity, Eur. Phys. J. Plus 135 (2020) 912, [2011.01248].
  • [29] Y.-F. Cai and E. N. Saridakis, Non-singular cosmology in a model of non-relativistic gravity, JCAP 10 (2009) 020, [0906.1789].
  • [30] E. N. Saridakis, Horava-Lifshitz Dark Energy, Eur. Phys. J. C 67 (2010) 229–235, [0905.3532].
  • [31] Y.-F. Cai, C. Gao and E. N. Saridakis, Bounce and cyclic cosmology in extended nonlinear massive gravity, JCAP 10 (2012) 048, [1207.3786].
  • [32] Y. Shtanov and V. Sahni, Bouncing brane worlds, Phys. Lett. B 557 (2003) 1–6, [gr-qc/0208047].
  • [33] E. N. Saridakis, Cyclic Universes from General Collisionless Braneworld Models, Nucl. Phys. B 808 (2009) 224–236, [0710.5269].
  • [34] J.-L. Lehners, Ekpyrotic and Cyclic Cosmology, Phys. Rept. 465 (2008) 223–263, [0806.1245].
  • [35] S. Banerjee and E. N. Saridakis, Bounce and cyclic cosmology in weakly broken galileon theories, Phys. Rev. D 95 (2017) 063523, [1604.06932].
  • [36] E. N. Saridakis, S. Banerjee and R. Myrzakulov, Bounce and cyclic cosmology in new gravitational scalar-tensor theories, Phys. Rev. D 98 (2018) 063513, [1807.00346].
  • [37] A. Ilyas, M. Zhu, Y. Zheng, Y.-F. Cai and E. N. Saridakis, DHOST Bounce, JCAP 09 (2020) 002, [2002.08269].
  • [38] A. Ilyas, M. Zhu, Y. Zheng and Y.-F. Cai, Emergent Universe and Genesis from the DHOST Cosmology, JHEP 01 (2021) 141, [2009.10351].
  • [39] M. Zhu, A. Ilyas, Y. Zheng, Y.-F. Cai and E. N. Saridakis, Scalar and tensor perturbations in DHOST bounce cosmology, JCAP 11 (2021) 045, [2108.01339].
  • [40] Y. B. Zel’dovich and I. D. Novikov, The Hypothesis of Cores Retarded during Expansion and the Hot Cosmological Model, Soviet Astronomy 10 (Feb., 1967) 602.
  • [41] B. J. Carr and S. W. Hawking, Black holes in the early Universe, Mon. Not. Roy. Astron. Soc. 168 (1974) 399–415.
  • [42] B. J. Carr, The primordial black hole mass spectrum, ApJ 201 (Oct., 1975) 1–19.
  • [43] I. D. Novikov, A. G. Polnarev, A. A. Starobinskii and I. B. Zeldovich, Primordial black holes, Astronomy and Astrophysics 80 (Nov., 1979) 104–109.
  • [44] M. Y. Khlopov, Primordial Black Holes, Res. Astron. Astrophys. 10 (2010) 495–528, [0801.0116].
  • [45] B. Carr, K. Kohri, Y. Sendouda and J. Yokoyama, Constraints on primordial black holes, Rept. Prog. Phys. 84 (2021) 116902, [2002.12778].
  • [46] G. F. Chapline, Cosmological effects of primordial black holes, Nature 253 (1975) 251–252.
  • [47] K. M. Belotsky, A. D. Dmitriev, E. A. Esipova, V. A. Gani, A. V. Grobov, M. Y. Khlopov et al., Signatures of primordial black hole dark matter, Mod. Phys. Lett. A 29 (2014) 1440005, [1410.0203].
  • [48] P. Meszaros, Primeval black holes and galaxy formation, Astron. Astrophys. 38 (1975) 5–13.
  • [49] N. Afshordi, P. McDonald and D. Spergel, Primordial black holes as dark matter: The Power spectrum and evaporation of early structures, Astrophys. J. Lett. 594 (2003) L71–L74, [astro-ph/0302035].
  • [50] B. J. Carr and M. J. Rees, How large were the first pregalactic objects?, Monthly Notices of Royal Academy of Science 206 (Jan., 1984) 315–325.
  • [51] R. Bean and J. Magueijo, Could supermassive black holes be quintessential primordial black holes?, Phys. Rev. D 66 (2002) 063505, [astro-ph/0204486].
  • [52] M. Sasaki, T. Suyama, T. Tanaka and S. Yokoyama, Primordial black holes—perspectives in gravitational wave astronomy, Class. Quant. Grav. 35 (2018) 063001, [1801.05235].
  • [53] LISA Cosmology Working Group collaboration, E. Bagui et al., Primordial black holes and their gravitational-wave signatures, 2310.19857.
  • [54] B. Carr, S. Clesse, J. Garcia-Bellido, M. Hawkins and F. Kuhnel, Observational evidence for primordial black holes: A positivist perspective, Phys. Rept. 1054 (2024) 1–68, [2306.03903].
  • [55] B. J. Carr and A. A. Coley, Persistence of black holes through a cosmological bounce, Int. J. Mod. Phys. D 20 (2011) 2733–2738, [1104.3796].
  • [56] J. Quintin and R. H. Brandenberger, Black hole formation in a contracting universe, JCAP 11 (2016) 029, [1609.02556].
  • [57] J.-W. Chen, J. Liu, H.-L. Xu and Y.-F. Cai, Tracing Primordial Black Holes in Nonsingular Bouncing Cosmology, Phys. Lett. B 769 (2017) 561–568, [1609.02571].
  • [58] T. Clifton, B. Carr and A. Coley, Persistent Black Holes in Bouncing Cosmologies, Class. Quant. Grav. 34 (2017) 135005, [1701.05750].
  • [59] J.-W. Chen, M. Zhu, S.-F. Yan, Q.-Q. Wang and Y.-F. Cai, Enhance primordial black hole abundance through the non-linear processes around bounce point, JCAP 01 (2023) 015, [2207.14532].
  • [60] S. Banerjee, T. Papanikolaou and E. N. Saridakis, Constraining F(R) bouncing cosmologies through primordial black holes, Phys. Rev. D 106 (2022) 124012, [2206.01150].
  • [61] G. Domènech, Scalar Induced Gravitational Waves Review, Universe 7 (2021) 398, [2109.01398].
  • [62] Y.-F. Cai, D. A. Easson and R. Brandenberger, Towards a Nonsingular Bouncing Cosmology, JCAP 08 (2012) 020, [1206.2382].
  • [63] Y.-F. Cai, E. McDonough, F. Duplessis and R. H. Brandenberger, Two Field Matter Bounce Cosmology, JCAP 10 (2013) 024, [1305.5259].
  • [64] P. S. Cole, A. D. Gow, C. T. Byrnes and S. P. Patil, Primordial black holes from single-field inflation: a fine-tuning audit, JCAP 08 (2023) 031, [2304.01997].
  • [65] Planck collaboration, N. Aghanim et al., Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6, [1807.06209].
  • [66] A. A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117 (1982) 175–178.
  • [67] D. Wands, K. A. Malik, D. H. Lyth and A. R. Liddle, A New approach to the evolution of cosmological perturbations on large scales, Phys.Rev. D62 (2000) 043527, [astro-ph/0003278].
  • [68] S. Young, C. T. Byrnes and M. Sasaki, Calculating the mass fraction of primordial black holes, JCAP 1407 (2014) 045, [1405.7023].
  • [69] V. De Luca, G. Franciolini, A. Kehagias, M. Peloso, A. Riotto and C. Ünal, The Ineludible non-Gaussianity of the Primordial Black Hole Abundance, JCAP 07 (2019) 048, [1904.00970].
  • [70] S. Young, I. Musco and C. T. Byrnes, Primordial black hole formation and abundance: contribution from the non-linear relation between the density and curvature perturbation, JCAP 11 (2019) 012, [1904.00984].
  • [71] J. C. Niemeyer and K. Jedamzik, Near-critical gravitational collapse and the initial mass function of primordial black holes, Phys. Rev. Lett. 80 (1998) 5481–5484, [astro-ph/9709072].
  • [72] J. C. Niemeyer and K. Jedamzik, Dynamics of primordial black hole formation, Phys. Rev. D 59 (1999) 124013, [astro-ph/9901292].
  • [73] I. Musco, J. C. Miller and A. G. Polnarev, Primordial black hole formation in the radiative era: Investigation of the critical nature of the collapse, Class. Quant. Grav. 26 (2009) 235001, [0811.1452].
  • [74] I. Musco and J. C. Miller, Primordial black hole formation in the early universe: critical behaviour and self-similarity, Class. Quant. Grav. 30 (2013) 145009, [1201.2379].
  • [75] I. Musco, Threshold for primordial black holes: Dependence on the shape of the cosmological perturbations, Phys. Rev. D 100 (2019) 123524, [1809.02127].
  • [76] I. Musco, V. De Luca, G. Franciolini and A. Riotto, Threshold for primordial black holes. II. A simple analytic prescription, Phys. Rev. D 103 (2021) 063538, [2011.03014].
  • [77] T. Harada, C.-M. Yoo and K. Kohri, Threshold of primordial black hole formation, Phys. Rev. D88 (2013) 084051, [1309.4201].
  • [78] A. Escrivà, C. Germani and R. K. Sheth, Analytical thresholds for black hole formation in general cosmological backgrounds, JCAP 01 (2021) 030, [2007.05564].
  • [79] T. Papanikolaou, Toward the primordial black hole formation threshold in a time-dependent equation-of-state background, Phys. Rev. D 105 (2022) 124055, [2205.07748].
  • [80] I. Musco and T. Papanikolaou, Primordial black hole formation for an anisotropic perfect fluid: Initial conditions and estimation of the threshold, Phys. Rev. D 106 (2022) 083017, [2110.05982].
  • [81] C.-M. Yoo, T. Harada and H. Okawa, Threshold of Primordial Black Hole Formation in Nonspherical Collapse, Phys. Rev. D 102 (2020) 043526, [2004.01042].
  • [82] C.-M. Yoo, Primordial black hole formation from a nonspherical density profile with a misaligned deformation tensor, 2403.11147.
  • [83] J. M. Bardeen, J. R. Bond, N. Kaiser and A. S. Szalay, The Statistics of Peaks of Gaussian Random Fields, Astrophys. J. 304 (1986) 15–61.
  • [84] E. W. Kolb and M. S. Turner, The Early Universe, vol. 69. 1990.
  • [85] A. M. Green and B. J. Kavanagh, Primordial Black Holes as a dark matter candidate, J. Phys. G 48 (2021) 043001, [2007.10722].
  • [86] S. Matarrese, O. Pantano and D. Saez, A General relativistic approach to the nonlinear evolution of collisionless matter, Phys. Rev. D 47 (1993) 1311–1323.
  • [87] S. Matarrese, O. Pantano and D. Saez, General relativistic dynamics of irrotational dust: Cosmological implications, Phys. Rev. Lett. 72 (1994) 320–323, [astro-ph/9310036].
  • [88] S. Matarrese, S. Mollerach and M. Bruni, Second order perturbations of the Einstein-de Sitter universe, Phys. Rev. D 58 (1998) 043504, [astro-ph/9707278].
  • [89] S. Mollerach, D. Harari and S. Matarrese, CMB polarization from secondary vector and tensor modes, Phys. Rev. D 69 (2004) 063002, [astro-ph/0310711].
  • [90] J.-C. Hwang, D. Jeong and H. Noh, Gauge dependence of gravitational waves generated from scalar perturbations, Astrophys. J. 842 (2017) 46, [1704.03500].
  • [91] K. Tomikawa and T. Kobayashi, Gauge dependence of gravitational waves generated at second order from scalar perturbations, Phys. Rev. D 101 (2020) 083529, [1910.01880].
  • [92] V. De Luca, G. Franciolini, A. Kehagias and A. Riotto, On the Gauge Invariance of Cosmological Gravitational Waves, JCAP 03 (2020) 014, [1911.09689].
  • [93] C. Yuan, Z.-C. Chen and Q.-G. Huang, Scalar induced gravitational waves in different gauges, Phys. Rev. D 101 (2020) 063018, [1912.00885].
  • [94] K. Inomata and T. Terada, Gauge Independence of Induced Gravitational Waves, Phys. Rev. D 101 (2020) 023523, [1912.00785].
  • [95] G. Domènech and M. Sasaki, Approximate gauge independence of the induced gravitational wave spectrum, Phys. Rev. D 103 (2021) 063531, [2012.14016].
  • [96] Z. Chang, S. Wang and Q.-H. Zhu, Note on gauge invariance of second order cosmological perturbations, Chin. Phys. C 45 (2021) 095101, [2009.11025].
  • [97] K. N. Ananda, C. Clarkson and D. Wands, The Cosmological gravitational wave background from primordial density perturbations, Phys. Rev. D75 (2007) 123518, [gr-qc/0612013].
  • [98] D. Baumann, P. J. Steinhardt, K. Takahashi and K. Ichiki, Gravitational Wave Spectrum Induced by Primordial Scalar Perturbations, Phys. Rev. D76 (2007) 084019, [hep-th/0703290].
  • [99] K. Kohri and T. Terada, Semianalytic calculation of gravitational wave spectrum nonlinearly induced from primordial curvature perturbations, Phys. Rev. D97 (2018) 123532, [1804.08577].
  • [100] J. R. Espinosa, D. Racco and A. Riotto, A Cosmological Signature of the SM Higgs Instability: Gravitational Waves, JCAP 1809 (2018) 012, [1804.07732].
  • [101] R. A. Isaacson, Gravitational Radiation in the Limit of High Frequency. II. Nonlinear Terms and the Ef fective Stress Tensor, Phys. Rev. 166 (1968) 1272–1279.
  • [102] M. Maggiore, Gravitational wave experiments and early universe cosmology, Phys. Rept. 331 (2000) 283–367, [gr-qc/9909001].
  • [103] P. J. E. Peebles, The large-scale structure of the universe. 1980.
  • [104] J. Silk, Cosmic Black-Body Radiation and Galaxy Formation, Astrophysics Journal 151 (Feb., 1968) 459.
  • [105] V. F. Mukhanov, H. A. Feldman and R. H. Brandenberger, Theory of cosmological perturbations. Part 1. Classical perturbations. Part 2. Quantum theory of perturbations. Part 3. Extensions, Phys. Rept. 215 (1992) 203–333.
  • [106] G. Janssen et al., Gravitational wave astronomy with the SKA, PoS AASKA14 (2015) 037, [1501.00127].
  • [107] LISA Cosmology Working Group collaboration, P. Auclair et al., Cosmology with the Laser Interferometer Space Antenna, Living Rev. Rel. 26 (2023) 5, [2204.05434].
  • [108] G. M. Harry, P. Fritschel, D. A. Shaddock, W. Folkner and E. S. Phinney, Laser interferometry for the big bang observer, Class. Quant. Grav. 23 (2006) 4887–4894.
  • [109] M. Maggiore et al., Science Case for the Einstein Telescope, JCAP 03 (2020) 050, [1912.02622].
  • [110] N. Fernandez, J. W. Foster, B. Lillard and J. Shelton, Stochastic Gravitational Waves from Early Structure Formation, 2312.12499.
  • [111] NANOGrav collaboration, G. Agazie et al., The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background, Astrophys. J. Lett. 951 (2023) L8, [2306.16213].
  • [112] T. Papanikolaou, C. Tzerefos, S. Basilakos and E. N. Saridakis, Scalar induced gravitational waves from primordial black hole Poisson fluctuations in f(R) gravity, JCAP 10 (2022) 013, [2112.15059].
  • [113] T. Papanikolaou, C. Tzerefos, S. Basilakos and E. N. Saridakis, No constraints for f(T) gravity from gravitational waves induced from primordial black hole fluctuations, 2205.06094.
  • [114] T. Papanikolaou, Primordial black holes in loop quantum cosmology: the effect on the threshold, Class. Quant. Grav. 40 (2023) 134001, [2301.11439].