Abstract
Higher derivative scalar interactions can give rise to interesting cosmological scenarios. We present a complete classification of such operators that can yield sizeable effects without introducing ghosts and, at the same time, define an effective field theory robust under the inclusion of quantum corrections. A set of rules to power count consistently the coefficients of the resulting Lagrangian is provided by the presence of an approximate global symmetry. The interactions that we derive in this way contain a subset of the so-called Horndeski and beyond Horndeski theories. Our construction therefore provides a structurally robust context to study their phenomenology. Applications to dark energy/modified gravity and geodesically complete cosmologies are briefly discussed.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Nicolis, R. Rattazzi and E. Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 (2009) 064036 [arXiv:0811.2197] [INSPIRE].
E. Silverstein and D. Tong, Scalar speed limits and cosmology: Acceleration from D-cceleration, Phys. Rev. D 70 (2004) 103505 [hep-th/0310221] [INSPIRE].
M. Alishahiha, E. Silverstein and D. Tong, DBI in the sky, Phys. Rev. D 70 (2004) 123505 [hep-th/0404084] [INSPIRE].
M.A. Luty, M. Porrati and R. Rattazzi, Strong interactions and stability in the DGP model, JHEP 09 (2003) 029 [hep-th/0303116] [INSPIRE].
D. Pirtskhalava, L. Santoni, E. Trincherini and F. Vernizzi, Weakly Broken Galileon Symmetry, JCAP 09 (2015) 007 [arXiv:1505.00007] [INSPIRE].
G.W. Horndeski, Second-order scalar-tensor field equations in a four-dimensional space, Int. J. Theor. Phys. 10 (1974) 363 [INSPIRE].
C. Deffayet, X. Gao, D.A. Steer and G. Zahariade, From k-essence to generalised Galileons, Phys. Rev. D 84 (2011) 064039 [arXiv:1103.3260] [INSPIRE].
J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Healthy theories beyond Horndeski, Phys. Rev. Lett. 114 (2015) 211101 [arXiv:1404.6495] [INSPIRE].
M. Zumalacárregui and J. García-Bellido, Transforming gravity: from derivative couplings to matter to second-order scalar-tensor theories beyond the Horndeski Lagrangian, Phys. Rev. D 89 (2014) 064046 [arXiv:1308.4685] [INSPIRE].
P. Creminelli and F. Vernizzi, Dark Energy after GW170817 and GRB170817A, Phys. Rev. Lett. 119 (2017) 251302 [arXiv:1710.05877] [INSPIRE].
T. Baker, E. Bellini, P.G. Ferreira, M. Lagos, J. Noller and I. Sawicki, Strong constraints on cosmological gravity from GW170817 and GRB 170817A, Phys. Rev. Lett. 119 (2017) 251301 [arXiv:1710.06394] [INSPIRE].
J.M. Ezquiaga and M. Zumalacárregui, Dark Energy After GW170817: Dead Ends and the Road Ahead, Phys. Rev. Lett. 119 (2017) 251304 [arXiv:1710.05901] [INSPIRE].
J. Sakstein and B. Jain, Implications of the Neutron Star Merger GW170817 for Cosmological Scalar-Tensor Theories, Phys. Rev. Lett. 119 (2017) 251303 [arXiv:1710.05893] [INSPIRE].
P. Creminelli, D. Pirtskhalava, L. Santoni and E. Trincherini, Stability of Geodesically Complete Cosmologies, JCAP 11 (2016) 047 [arXiv:1610.04207] [INSPIRE].
Y. Cai, Y. Wan, H.-G. Li, T. Qiu and Y.-S. Piao, The Effective Field Theory of nonsingular cosmology, JHEP 01 (2017) 090 [arXiv:1610.03400] [INSPIRE].
B. Finelli, G. Goon, E. Pajer and L. Santoni, Soft Theorems For Shift-Symmetric Cosmologies, Phys. Rev. D 97 (2018) 063531 [arXiv:1711.03737] [INSPIRE].
B. Finelli, G. Goon, E. Pajer and L. Santoni, The Effective Theory of Shift-Symmetric Cosmologies, JCAP 05 (2018) 060 [arXiv:1802.01580] [INSPIRE].
D. Pirtskhalava, L. Santoni, E. Trincherini and F. Vernizzi, Large Non-Gaussianity in Slow-Roll Inflation, JHEP 04 (2016) 117 [arXiv:1506.06750] [INSPIRE].
D. Pirtskhalava, L. Santoni and E. Trincherini, Constraints on Single-Field Inflation, JCAP 06 (2016) 051 [arXiv:1511.01817] [INSPIRE].
C. Deffayet, O. Pujolas, I. Sawicki and A. Vikman, Imperfect Dark Energy from Kinetic Gravity Braiding, JCAP 10 (2010) 026 [arXiv:1008.0048] [INSPIRE].
J. Ben Achour, D. Langlois and K. Noui, Degenerate higher order scalar-tensor theories beyond Horndeski and disformal transformations, Phys. Rev. D 93 (2016) 124005 [arXiv:1602.08398] [INSPIRE].
M. Trodden and K. Hinterbichler, Generalizing Galileons, Class. Quant. Grav. 28 (2011) 204003 [arXiv:1104.2088] [INSPIRE].
G. Goon, K. Hinterbichler, A. Joyce and M. Trodden, Aspects of Galileon Non-Renormalization, JHEP 11 (2016) 100 [arXiv:1606.02295] [INSPIRE].
D. Langlois and K. Noui, Degenerate higher derivative theories beyond Horndeski: evading the Ostrogradski instability, JCAP 02 (2016) 034 [arXiv:1510.06930] [INSPIRE].
D. Langlois and K. Noui, Hamiltonian analysis of higher derivative scalar-tensor theories, JCAP 07 (2016) 016 [arXiv:1512.06820] [INSPIRE].
J.D. Bekenstein, The Relation between physical and gravitational geometry, Phys. Rev. D 48 (1993) 3641 [gr-qc/9211017] [INSPIRE].
J. Gleyzes, D. Langlois, F. Piazza and F. Vernizzi, Exploring gravitational theories beyond Horndeski, JCAP 02 (2015) 018 [arXiv:1408.1952] [INSPIRE].
M. Crisostomi, K. Koyama and G. Tasinato, Extended Scalar-Tensor Theories of Gravity, JCAP 04 (2016) 044 [arXiv:1602.03119] [INSPIRE].
M. Crisostomi, M. Hull, K. Koyama and G. Tasinato, Horndeski: beyond, or not beyond?, JCAP 03 (2016) 038 [arXiv:1601.04658] [INSPIRE].
Virgo, LIGO Scientific collaboration, B. Abbott et al., GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral, Phys. Rev. Lett. 119 (2017) 161101 [arXiv:1710.05832] [INSPIRE].
A. Goldstein et al., An Ordinary Short Gamma-Ray Burst with Extraordinary Implications: Fermi-GBM Detection of GRB 170817A, Astrophys. J. 848 (2017) L14 [arXiv:1710.05446] [INSPIRE].
Virgo, Fermi-GBM, INTEGRAL, LIGO Scientific collaboration, B.P. Abbott et al., Gravitational Waves and Gamma-rays from a Binary Neutron Star Merger: GW170817 and GRB 170817A, Astrophys. J. 848 (2017) L13 [arXiv:1710.05834] [INSPIRE].
L. Lombriser and A. Taylor, Breaking a Dark Degeneracy with Gravitational Waves, JCAP 03 (2016) 031 [arXiv:1509.08458] [INSPIRE].
P. Creminelli, G. D’Amico, J. Norena and F. Vernizzi, The Effective Theory of Quintessence: the w < −1 Side Unveiled, JCAP 02 (2009) 018 [arXiv:0811.0827] [INSPIRE].
G. Gubitosi, F. Piazza and F. Vernizzi, The Effective Field Theory of Dark Energy, JCAP 02 (2013) 032 [arXiv:1210.0201] [INSPIRE].
A. Borde, A.H. Guth and A. Vilenkin, Inflationary space-times are incompletein past directions, Phys. Rev. Lett. 90 (2003) 151301 [gr-qc/0110012] [INSPIRE].
P. Creminelli, A. Nicolis and E. Trincherini, Galilean Genesis: An Alternative to inflation, JCAP 11 (2010) 021 [arXiv:1007.0027] [INSPIRE].
M. Novello and S.E.P. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463 (2008) 127 [arXiv:0802.1634] [INSPIRE].
R. Brandenberger and P. Peter, Bouncing Cosmologies: Progress and Problems, Found. Phys. 47 (2017) 797 [arXiv:1603.05834] [INSPIRE].
A. Nicolis, R. Rattazzi and E. Trincherini, Energy’s and amplitudes’ positivity, JHEP 05 (2010) 095 [Erratum ibid. 1111 (2011) 128] [arXiv:0912.4258] [INSPIRE].
M. Libanov, S. Mironov and V. Rubakov, Generalized Galileons: instabilities of bouncing and Genesis cosmologies and modified Genesis, JCAP 08 (2016) 037 [arXiv:1605.05992] [INSPIRE].
T. Kobayashi, Generic instabilities of nonsingular cosmologies in Horndeski theory: A no-go theorem, Phys. Rev. D 94 (2016) 043511 [arXiv:1606.05831] [INSPIRE].
D. Pirtskhalava, L. Santoni, E. Trincherini and P. Uttayarat, Inflation from Minkowski Space, JHEP 12 (2014) 151 [arXiv:1410.0882] [INSPIRE].
D.A. Easson, I. Sawicki and A. Vikman, G-Bounce, JCAP 11 (2011) 021 [arXiv:1109.1047] [INSPIRE].
A. Ijjas and P.J. Steinhardt, Classically stable nonsingular cosmological bounces, Phys. Rev. Lett. 117 (2016) 121304 [arXiv:1606.08880] [INSPIRE].
A. Ijjas and P.J. Steinhardt, Fully stable cosmological solutions with a non-singular classical bounce, Phys. Lett. B 764 (2017) 289 [arXiv:1609.01253] [INSPIRE].
D.A. Dobre, A.V. Frolov, J.T.G. Ghersi, S. Ramazanov and A. Vikman, Unbraiding the Bounce: Superluminality around the Corner, JCAP 03 (2018) 020 [arXiv:1712.10272] [INSPIRE].
P. Creminelli, M.A. Luty, A. Nicolis and L. Senatore, Starting the Universe: Stable Violation of the Null Energy Condition and Non-standard Cosmologies, JHEP 12 (2006) 080 [hep-th/0606090] [INSPIRE].
C. Cheung, P. Creminelli, A.L. Fitzpatrick, J. Kaplan and L. Senatore, The Effective Field Theory of Inflation, JHEP 03 (2008) 014 [arXiv:0709.0293] [INSPIRE].
Y. Cai and Y.-S. Piao, A covariant Lagrangian for stable nonsingular bounce, JHEP 09 (2017) 027 [arXiv:1705.03401] [INSPIRE].
R. Kolevatov, S. Mironov, N. Sukhov and V. Volkova, Cosmological bounce and Genesis beyond Horndeski, JCAP 08 (2017) 038 [arXiv:1705.06626] [INSPIRE].
J. Ben Achour, M. Crisostomi, K. Koyama, D. Langlois, K. Noui and G. Tasinato, Degenerate higher order scalar-tensor theories beyond Horndeski up to cubic order, JHEP 12 (2016) 100 [arXiv:1608.08135] [INSPIRE].
A. Adams, N. Arkani-Hamed, S. Dubovsky, A. Nicolis and R. Rattazzi, Causality, analyticity and an IR obstruction to UV completion, JHEP 10 (2006) 014 [hep-th/0602178] [INSPIRE].
C. de Rham, S. Melville, A.J. Tolley and S.-Y. Zhou, Massive Galileon Positivity Bounds, JHEP 09 (2017) 072 [arXiv:1702.08577] [INSPIRE].
B. Bellazzini, F. Riva, J. Serra and F. Sgarlata, Beyond Positivity Bounds and the Fate of Massive Gravity, Phys. Rev. Lett. 120 (2018) 161101 [arXiv:1710.02539] [INSPIRE].
Open Access
This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1806.10073
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made.
The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.
To view a copy of this licence, visit https://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Santoni, L., Trincherini, E. & Trombetta, L.G. Behind Horndeski: structurally robust higher derivative EFTs. J. High Energ. Phys. 2018, 118 (2018). https://doi.org/10.1007/JHEP08(2018)118
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP08(2018)118