Quadratic gravity

Short description: Theory extending Einstein gravity

Quadratic gravity (QG) is an extension of general relativity obtained by adding all local quadratic-in-curvature terms to the Einstein–Hilbert Lagrangian.[1] Doing this makes the theory renormalizable.[1] This is one of numerous alternatives to general relativity.[2]: 63 It has been suggested that consistency with quantum chromodynamics requires these additional quadratic terms.[3]

The theory was originally proved to be renormalizable by Kellogg Stelle in 1977,[4] but had difficulty being accepted as a viable theory because of its introduction of a massive spin-2 ghost particle.[5][1] Aside from the massive ghost, the theory also predicts the existence of an additional scalar boson and the massless graviton[5], ensuring that general relativity is recovered at low energies. The additional scalar particle appears also in Alexei Starobinsky's work of 1980 on the early universe. In Starobinsky inflation, the scalar particle is responsible for cosmic inflation.[5][1]

QG, besides being renormalizable, has also been shown to feature a non-perturbative ultraviolet fixed point.[6] Unitarity, essential to a theory of quantum gravity, has been established in appropriate constructions,[7][8][9] where the appropriate inner product to compute probabilities through the Born rule is constructed. In scattering processes the unitarity of the S-matrix has been proved using the instability of the massive spin-2 particle.[10]

The relevant solutions are not unstable but metastable: when the energies are much below a threshold (that is high enough to describe the whole cosmology) runaways are avoided and the possible instability occurring when the bound is violated not only is compatible with cosmology but would also explain why we live in a homogeneous and isotropic universe.[9][11]

John Donoghue believes quadratic gravity could be a viable theory of quantum gravity.[5] He has reinterpreted its ghost particles as time-reversed unstable particles.[12][8]

Work has been directed towards using the Event Horizon Telescope to test for the possibility of QG being a valid theory.[13]

## References

1. ↑ 1.0 1.1 1.2 1.3 Salvio, Alberto (2018). "Quadratic Gravity". Frontiers in Physics 6 (77). doi:10.3389/fphy.2018.00077. Bibcode: 2018FrP.....6...77S.
2. ↑ Clifton, Timothy; Ferreira, Pedro G.; Padilla, Antonio; Skordis, Constantinos (March 1, 2012). "Modified gravity and cosmology". Physics Reports 513 (1): 1–189. doi:10.1016/j.physrep.2012.01.001. ISSN 0370-1573. Bibcode: 2012PhR...513....1C. https://www.sciencedirect.com/science/article/pii/S0370157312000105.
3. ↑ Salvio, Alberto (2021). "Dimensional Transmutation in Gravity and Cosmology". Int. J. Mod. Phys. A 36 (8n09, 2130006): 2130006–2130831. doi:10.1142/S0217751X21300064. Bibcode: 2021IJMPA..3630006S.
4. ↑ Stelle, K. S. (August 15, 1977). "Renormalization of higher-derivative quantum gravity" (in en). Physical Review D 16 (4): 953–969. doi:10.1103/PhysRevD.16.953. ISSN 0556-2821. Bibcode: 1977PhRvD..16..953S. https://link.aps.org/doi/10.1103/PhysRevD.16.953.
5. ↑ 5.0 5.1 5.2 5.3 Wood, Charlie (2025-11-17). "Old 'Ghost' Theory of Quantum Gravity Makes a Comeback" (in en). https://www.quantamagazine.org/old-ghost-theory-of-quantum-gravity-makes-a-comeback-20251117/.
6. ↑ Falls, Kevin; Ohta, Nobuyoshi; Percacci, Roberto (2020). "Towards the determination of the dimension of the critical surface in asymptotically safe gravity". Physics Letters B 810 (135773). doi:10.1016/j.physletb.2020.135773. Bibcode: 2020PhLB..81035773F.
7. ↑ Salvio, Alberto (July 1, 2019). "Quasi-Conformal Models and the Early Universe" (in en). The European Physical Journal C 79 (9). doi:10.1140/epjc/s10052-019-7267-5. Bibcode: 2019EPJC...79..750S. https://inspirehep.net/literature/1742317.
8. ↑ 8.0 8.1 Kubo, Jisuke; Kuntz, Jeffrey (December 21, 2022). "Spontaneous conformal symmetry breaking and quantum quadratic gravity" (in en). Physical Review D 106 (12). doi:10.1103/PhysRevD.106.126015. ISSN 2470-0010. Bibcode: 2022PhRvD.106l6015K. https://link.aps.org/doi/10.1103/PhysRevD.106.126015.
9. ↑ 9.0 9.1 Salvio, Alberto (April 11, 2024). "A non-perturbative and background-independent formulation of quadratic gravity" (in en). Journal of Cosmology and Astroparticle Physics 07 (92): 092. doi:10.1088/1475-7516/2024/07/092. Bibcode: 2024JCAP...07..092S. https://inspirehep.net/literature/2776893.
10. ↑ Donoghue, John F.; Menezes, Gabriel (2019). "Unitarity, stability and loops of unstable ghosts". Physical Review D 100 (105006). doi:10.1103/PhysRevD.100.105006. Bibcode: 2019PhRvD.100j5006D.
11. ↑ Salvio, Alberto (February 25, 2019). "Metastability in Quadratic Gravity" (in en). Physical Review D 99 (10). doi:10.1103/PhysRevD.99.103507. Bibcode: 2019PhRvD..99j3507S. https://inspirehep.net/literature/1722053.
12. ↑ Donoghue, John F.; Menezes, Gabriel (2022-05-03). "On quadratic gravity". Il Nuovo Cimento C 45 (2): 1–11. doi:10.1393/ncc/i2022-22026-7. https://doi.org/10.1393/ncc/i2022-22026-7.
13. ↑ Daas, Jesse; Kuijpers, Kolja; Saueressig, Frank; Wondrak, Michael F.; Falcke, Heino (May 2023). "Probing quadratic gravity with the Event Horizon Telescope". Astronomy & Astrophysics 673: A53. doi:10.1051/0004-6361/202244080. ISSN 0004-6361. Bibcode: 2023A&A...673A..53D. https://www.aanda.org/10.1051/0004-6361/202244080.

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