## Author(s)

Valbusa Dall'Armi, Lorenzo, Mierna, Alina, Matarrese, Sabino, Ricciardone, Angelo## Abstract

The initial conditions on the anisotropies of the stochastic gravitational-wave background of cosmological origin (CGWB) largely depend on the mechanism that generates the gravitational waves. Since the CGWB is expected to be non-thermal, the computation of the initial conditions could be more challenging w.r.t. the Cosmic Microwave Background (CMB), whose interactions with other particles in the early Universe lead to a blackbody spectrum. In this paper, we show that the initial conditions for the cosmological background generated by quantum fluctuations of the metric during inflation deviate from adiabaticity. These primordial gravitational waves are indeed generated by quantum fluctuations of two independent degrees of freedom (the two polarization states of the gravitons). Furthermore, the CGWB plays a negligible role in the Einstein's equations, because its energy density is subdominant w.r.t. ordinary matter. Therefore, the only possible way to compute the initial conditions for inflationary gravitons is to perturb the energy-momentum tensor of the gravitational field defined in terms of the small-scale tensor perturbation of the metric. This new and self-consistent approach shows that a large, non-adiabatic initial condition is present even during the single-field inflation. Such a contribution enhances the total angular power spectrum of the CGWB compared to the standard adiabatic case, increasing also the sensitivity of the anisotropies to the presence of relativistic and decoupled particles in the early Universe. In this work we have also proved that our findings are quite general and apply to both single-field inflation and other scenarios in which the CGWB is generated by the quantum fluctuations of the metric, like the curvaton.

## Figures

Left: plot of the angular power spectrum of the CGWB for adiabatic initial conditions (AD) and for inflationary initial conditions (IIC) with $n_{\rm gwb} = 0.35$. Right: plot of the tensor contributions to the angular power spectrum for AD and IIC.

Left: plot of the angular power spectrum of the CGWB for adiabatic initial conditions (AD) and for inflationary initial conditions (IIC) with $n_{\rm gwb} = 0.35$. Right: plot of the tensor contributions to the angular power spectrum for AD and IIC.

Plot of the correlation between the CMB and the CGWB at different multipoles for adiabatic initial conditions (AD) and for inflationary initial conditions (IIC) with $n_{\rm gwb} = 0.35$.

## References

- [1] G. Agazie et al., “The NANOGrav 15 yr Data Set: Evidence for a Gravitational-wave Background,” Astrophys. J. Lett., vol. 951, no. 1, p. L8, 2023.
- [2] J. Antoniadis et al., “The second data release from the European Pulsar Timing Array - III. Search for gravitational wave signals,” Astron. Astrophys., vol. 678, p. A50, 2023.
- [3] D. J. Reardon et al., “Search for an Isotropic Gravitational-wave Background with the Parkes Pulsar Timing Array,” Astrophys. J. Lett., vol. 951, no. 1, p. L6, 2023.
- [4] H. Xu et al., “Searching for the Nano-Hertz Stochastic Gravitational Wave Background with the Chinese Pulsar Timing Array Data Release I,” Res. Astron. Astrophys., vol. 23, no. 7, p. 075024, 2023.
- [5] A. Afzal et al., “The NANOGrav 15 yr Data Set: Search for Signals from New Physics,” Astrophys. J. Lett., vol. 951, no. 1, p. L11, 2023.
- [6] G. Agazie et al., “The NANOGrav 15 yr Data Set: Constraints on Supermassive Black Hole Binaries from the Gravitational-wave Background,” Astrophys. J. Lett., vol. 952, no. 2, p. L37, 2023.
- [7] J. Antoniadis et al., “The second data release from the European Pulsar Timing Array: V. Implications for massive black holes, dark matter and the early Universe,” 6 2023.
- [8] G. Franciolini, A. Iovino, Junior., V. Vaskonen, and H. Veermae, “Recent Gravitational Wave Observation by Pulsar Timing Arrays and Primordial Black Holes: The Importance of Non-Gaussianities,” Phys. Rev. Lett., vol. 131, no. 20, p. 201401, 2023.
- [9] J. Ellis, M. Lewicki, C. Lin, and V. Vaskonen, “Cosmic superstrings revisited in light of NANOGrav 15-year data,” Phys. Rev. D, vol. 108, no. 10, p. 103511, 2023.
- [10] G. Franciolini, D. Racco, and F. Rompineve, “Footprints of the QCD Crossover on Cosmological Gravitational Waves at Pulsar Timing Arrays,” 6 2023.
- [11] S. Vagnozzi, “Inflationary interpretation of the stochastic gravitational wave background signal detected by pulsar timing array experiments,” JHEAp, vol. 39, pp. 81–98, 2023.
- [12] D. G. Figueroa, M. Pieroni, A. Ricciardone, and P. Simakachorn, “Cosmological Background Interpretation of Pulsar Timing Array Data,” 7 2023.
- [13] L. Liu, Z.-C. Chen, and Q.-G. Huang, “Implications for the non-Gaussianity of curvature perturbation from pulsar timing arrays,” 7 2023.
- [14] P. Amaro-Seoane et al., “Laser Interferometer Space Antenna,” 2 2017.
- [15] G. Orlando, M. Pieroni, and A. Ricciardone, “Measuring Parity Violation in the Stochastic Gravitational Wave Background with the LISA-Taiji network,” JCAP, vol. 03, p. 069, 2021.
- [16] V. Corbin and N. J. Cornish, “Detecting the cosmic gravitational wave background with the big bang observer,” Class. Quant. Grav., vol. 23, pp. 2435–2446, 2006.
- [17] S. Kawamura et al., “The Japanese space gravitational wave antenna DECIGO,” Class. Quant. Grav., vol. 23, pp. S125–S132, 2006.
- [18] M. Punturo et al., “The Einstein Telescope: A third-generation gravitational wave observatory,” Class. Quant. Grav., vol. 27, p. 194002, 2010.
- [19] M. Maggiore et al., “Science Case for the Einstein Telescope,” JCAP, vol. 03, p. 050, 2020.
- [20] M. Branchesi et al., “Science with the Einstein Telescope: a comparison of different designs,” JCAP, vol. 07, p. 068, 2023.
- [21] D. Reitze et al., “Cosmic Explorer: The U.S. Contribution to Gravitational-Wave Astronomy beyond LIGO,” Bull. Am. Astron. Soc., vol. 51, no. 7, p. 035, 2019.
- [22] B. P. Abbott et al., “Exploring the Sensitivity of Next Generation Gravitational Wave Detectors,” Class. Quant. Grav., vol. 34, no. 4, p. 044001, 2017.
- [23] M. C. Guzzetti, N. Bartolo, M. Liguori, and S. Matarrese, “Gravitational waves from inflation,” Riv. Nuovo Cim., vol. 39, no. 9, pp. 399–495, 2016.
- [24] N. Bartolo et al., “Science with the space-based interferometer LISA. IV: Probing inflation with gravitational waves,” JCAP, vol. 12, p. 026, 2016.
- [25] C. Caprini and D. G. Figueroa, “Cosmological Backgrounds of Gravitational Waves,” Class. Quant. Grav., vol. 35, no. 16, p. 163001, 2018.
- [26] C. Caprini, D. G. Figueroa, R. Flauger, G. Nardini, M. Peloso, M. Pieroni, A. Ricciardone, and G. Tasinato, “Reconstructing the spectral shape of a stochastic gravitational wave background with LISA,” JCAP, vol. 11, p. 017, 2019.
- [27] R. Flauger, N. Karnesis, G. Nardini, M. Pieroni, A. Ricciardone, and J. Torrado, “Improved reconstruction of a stochastic gravitational wave background with LISA,” JCAP, vol. 01, p. 059, 2021.
- [28] E. S. Phinney, “A Practical theorem on gravitational wave backgrounds,” 7 2001.
- [29] N. Bartolo, D. Bertacca, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, and G. Tasinato, “Anisotropies and non-Gaussianity of the Cosmological Gravitational Wave Background,” Phys. Rev. D, vol. 100, no. 12, p. 121501, 2019.
- [30] N. Bartolo, D. Bertacca, S. Matarrese, M. Peloso, A. Ricciardone, A. Riotto, and G. Tasinato, “Characterizing the cosmological gravitational wave background: Anisotropies and non-Gaussianity,” Phys. Rev. D, vol. 102, no. 2, p. 023527, 2020.
- [31] Y. Cui, S. Kumar, R. Sundrum, and Y. Tsai, “Unraveling Cosmological Anisotropies within Stochastic Gravitational Wave Backgrounds,” 7 2023.
- [32] C. R. Contaldi, G. Mentasti, and M. Peloso, “Probing the galactic and extragalactic gravitational wave backgrounds with space-based interferometers,” 12 2023.
- [33] S. Dodelson, Modern Cosmology. Amsterdam: Academic Press, 2003.
- [34] C. R. Contaldi, “Anisotropies of Gravitational Wave Backgrounds: A Line Of Sight Approach,” Phys. Lett. B, vol. 771, pp. 9–12, 2017.
- [35] W. Hu, D. N. Spergel, and M. J. White, “Distinguishing causal seeds from inflation,” Phys. Rev. D, vol. 55, pp. 3288–3302, 1997.
- [36] 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. D, vol. 62, p. 043527, 2000.
- [37] R. A. Isaacson, Gravitational Radiation in the Limit of High Frequency. PhD thesis, Maryland U., 1967.
- [38] L. D. Landau and E. M. Lifschits, The Classical Theory of Fields, vol. Volume 2 of Course of Theoretical Physics. Oxford: Pergamon Press, 1975.
- [39] S. Mollerach, “Isocurvature Baryon Perturbations and Inflation,” Phys. Rev. D, vol. 42, pp. 313–325, 1990.
- [40] G. Mentasti, C. R. Contaldi, and M. Peloso, “Intrinsic Limits on the Detection of the Anisotropies of the Stochastic Gravitational Wave Background,” Phys. Rev. Lett., vol. 131, no. 22, p. 221403, 2023.
- [41] G. Mentasti, C. Contaldi, and M. Peloso, “Prospects for detecting anisotropies and polarization of the stochastic gravitational wave background with ground-based detectors,” JCAP, vol. 08, p. 053, 2023.
- [42] L. Valbusa Dall’Armi, A. Ricciardone, N. Bartolo, D. Bertacca, and S. Matarrese, “Imprint of relativistic particles on the anisotropies of the stochastic gravitational-wave background,” Phys. Rev. D, vol. 103, no. 2, p. 023522, 2021.
- [43] A. Ricciardone, L. V. Dall’Armi, N. Bartolo, D. Bertacca, M. Liguori, and S. Matarrese, “Cross-Correlating Astrophysical and Cosmological Gravitational Wave Backgrounds with the Cosmic Microwave Background,” Phys. Rev. Lett., vol. 127, no. 27, p. 271301, 2021.
- [44] M. Braglia and S. Kuroyanagi, “Probing prerecombination physics by the cross-correlation of stochastic gravitational waves and CMB anisotropies,” Phys. Rev. D, vol. 104, no. 12, p. 123547, 2021.
- [45] L. Valbusa Dall’Armi, A. Mierna, S. Matarrese, and A. Ricciardone, “Adiabatic or Non-Adiabatic? Unraveling the Nature of Initial Conditions in the Cosmological Gravitational Wave Background,” 7 2023.
- [46] C. W. Misner, K. S. Thorne, and J. A. Wheeler, Gravitation. San Francisco: W. H. Freeman, 1973.
- [47] N. Bartolo, A. Hoseinpour, G. Orlando, S. Matarrese, and M. Zarei, “Photon-graviton scattering: A new way to detect anisotropic gravitational waves?,” Phys. Rev. D, vol. 98, no. 2, p. 023518, 2018.
- [48] K. A. Malik, D. Wands, and C. Ungarelli, “Large scale curvature and entropy perturbations for multiple interacting fluids,” Phys. Rev. D, vol. 67, p. 063516, 2003.
- [49] S. Weinberg, “Adiabatic modes in cosmology,” Phys. Rev. D, vol. 67, p. 123504, 2003.
- [50] S. Weinberg, “Can non-adiabatic perturbations arise after single-field inflation?,” Phys. Rev. D, vol. 70, p. 043541, 2004.
- [51] A. R. Liddle, D. H. Lyth, K. A. Malik, and D. Wands, “Superhorizon perturbations and preheating,” Phys. Rev. D, vol. 61, p. 103509, 2000.
- [52] F. Schulze, L. Valbusa Dall’Armi, J. Lesgourgues, A. Ricciardone, N. Bartolo, D. Bertacca, C. Fidler, and S. Matarrese, “GW_CLASS: Cosmological Gravitational Wave Background in the cosmic linear anisotropy solving system,” JCAP, vol. 10, p. 025, 2023.
- [53] M. Giovannini, “Effective energy density of relic gravitons,” Phys. Rev. D, vol. 100, no. 8, p. 083531, 2019.
- [54] L. H. Ford and L. Parker, “Quantized Gravitational Wave Perturbations in Robertson-Walker Universes,” Phys. Rev. D, vol. 16, pp. 1601–1608, 1977.
- [55] L. H. Ford and L. Parker, “Infrared Divergences in a Class of Robertson-Walker Universes,” Phys. Rev. D, vol. 16, pp. 245–250, 1977.
- [56] S. V. Babak and L. P. Grishchuk, “The Energy momentum tensor for the gravitational field,” Phys. Rev. D, vol. 61, p. 024038, 2000.
- [57] C. Pitrou, X. Roy, and O. Umeh, “xPand: An algorithm for perturbing homogeneous cosmologies,” Class. Quant. Grav., vol. 30, p. 165002, 2013.
- [58] K. Tomita, “Non-Linear Theory of Gravitational Instability in the Expanding Universe. II,” Progress of Theoretical Physics, vol. 45, pp. 1747–1762, June 1971.
- [59] S. Matarrese, O. Pantano, and D. Saez, “A General relativistic approach to the nonlinear evolution of collisionless matter,” Phys. Rev. D, vol. 47, pp. 1311–1323, 1993.
- [60] S. Matarrese, O. Pantano, and D. Saez, “General relativistic dynamics of irrotational dust: Cosmological implications,” Phys. Rev. Lett., vol. 72, pp. 320–323, 1994.
- [61] S. Matarrese, S. Mollerach, and M. Bruni, “Second order perturbations of the Einstein-de Sitter universe,” Phys. Rev. D, vol. 58, p. 043504, 1998.
- [62] A. Malhotra, E. Dimastrogiovanni, G. Domènech, M. Fasiello, and G. Tasinato, “New universal property of cosmological gravitational wave anisotropies,” Phys. Rev. D, vol. 107, no. 10, p. 103502, 2023.
- [63] Y. Akrami et al., “Planck 2018 results. X. Constraints on inflation,” Astron. Astrophys., vol. 641, p. A10, 2020.
- [64] M. Tristram et al., “Improved limits on the tensor-to-scalar ratio using BICEP and Planck data,” Phys. Rev. D, vol. 105, no. 8, p. 083524, 2022.
- [65] G. Galloni, N. Bartolo, S. Matarrese, M. Migliaccio, A. Ricciardone, and N. Vittorio, “Updated constraints on amplitude and tilt of the tensor primordial spectrum,” JCAP, vol. 04, p. 062, 2023.
- [66] D. R. Brill and J. B. Hartle, “Method of the Self-Consistent Field in General Relativity and its Application to the Gravitational Geon,” Phys. Rev., vol. 135, pp. B271–B278, 1964.