Abstract
In scale-invariant models of fundamental physics all mass scales are generated via spontaneous symmetry breaking. In this work, we study inflation in scale-invariant quadratic gravity, in which the Planck mass is generated classically by a scalar field, which evolves from an unstable fixed point to a stable one thus breaking scale-invariance. We investigate the dynamics by means of dynamical system standard techniques. By computing the spectral indices and comparing them with data, we put some constraints on the three dimensionless parameters of the theory. We show that certain regions of the parameter space will be within the range of future CMB missions like CMB-S4, LiteBIRD and STPol. The second half of the paper is dedicated to the analysis of inflationary first-order tensor perturbations and the calculation of the power spectrum of the gravitational waves. We comment on our results and compare them with the ones of mixed Starobinsky-Higgs inflation.
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References
F. Cooper and G. Venturi, Cosmology and broken scale invariance, Phys. Rev. D 24 (1981) 3338 [INSPIRE].
S.L. Adler, Einstein gravity as a symmetry-breaking effect in quantum field theory, Rev. Mod. Phys. 54 (1982) 729.
S. Coleman and E. Weinberg, Radiative corrections as the origin of spontaneous symmetry breaking, Phys. Rev. D 7 (1973) 1888.
A. Salvio and A. Strumia, Agravity, JHEP 06 (2014) 080 [arXiv:1403.4226] [INSPIRE].
M.B. Einhorn and D.R.T. Jones, Naturalness and dimensional transmutation in classically scale-invariant gravity, JHEP 03 (2015) 047 [arXiv:1410.8513] [INSPIRE].
M.B. Einhorn and D.R.T. Jones, Induced gravity I: real scalar field, JHEP 01 (2016) 019 [arXiv:1511.01481] [INSPIRE].
M.B. Einhorn and D.R.T. Jones, Induced gravity II: grand unification, JHEP 05 (2016) 185 [arXiv:1602.06290] [INSPIRE].
A. Edery and Y. Nakayama, Generating Einstein gravity, cosmological constant and Higgs mass from restricted Weyl invariance, Mod. Phys. Lett. A 30 (2015) 1550152 [arXiv:1502.05932] [INSPIRE].
P.G. Ferreira, C.T. Hill, J. Noller and G.G. Ross, Inflation in a scale invariant universe, Phys. Rev. D 97 (2018) 123516 [arXiv:1802.06069] [INSPIRE].
P.G. Ferreira, C.T. Hill, J. Noller and G.G. Ross, Scale-independent R2 inflation, Phys. Rev. D 100 (2019) 123516 [arXiv:1906.03415] [INSPIRE].
A. Salvio, Dimensional transmutation in gravity and cosmology, Int. J. Mod. Phys. A 36 (2021) 2130006 [arXiv:2012.11608] [INSPIRE].
P.G. Ferreira, C.T. Hill, J. Noller and G.G. Ross, R2/Higgs inflation and the hierarchy problem, arXiv:2108.06095 [INSPIRE].
V.V. Khoze, Inflation and dark matter in the Higgs portal of classically scale invariant standard model, JHEP 11 (2013) 215 [arXiv:1308.6338] [INSPIRE].
K. Kannike, A. Racioppi and M. Raidal, Embedding inflation into the standard model — more evidence for classical scale invariance, JHEP 06 (2014) 154 [arXiv:1405.3987] [INSPIRE].
M. Rinaldi, G. Cognola, L. Vanzo and S. Zerbini, Inflation in scale-invariant theories of gravity, Phys. Rev. D 91 (2015) 123527 [arXiv:1410.0631] [INSPIRE].
K. Kannike et al., Dynamically induced Planck scale and inflation, JHEP 05 (2015) 065 [arXiv:1502.01334] [INSPIRE].
N.D. Barrie, A. Kobakhidze and S. Liang, Natural inflation with hidden scale invariance, Phys. Lett. B 756 (2016) 390 [arXiv:1602.04901] [INSPIRE].
G. Tambalo and M. Rinaldi, Inflation and reheating in scale-invariant scalar-tensor gravity, Gen. Rel. Grav. 49 (2017) 52 [arXiv:1610.06478] [INSPIRE].
T. Hambye and A. Strumia, Dynamical generation of the weak and dark matter scale, Phys. Rev. D 88 (2013) 055022 [arXiv:1306.2329] [INSPIRE].
A. Karam and K. Tamvakis, Dark matter and neutrino masses from a scale-invariant multi-Higgs portal, Phys. Rev. D 92 (2015) 075010 [arXiv:1508.03031] [INSPIRE].
K. Kannike, G.M. Pelaggi, A. Salvio and A. Strumia, The Higgs of the Higgs and the diphoton channel, JHEP 07 (2016) 101 [arXiv:1605.08681] [INSPIRE].
A. Karam and K. Tamvakis, Dark matter from a classically scale-invariant SU(3)X, Phys. Rev. D 94 (2016) 055004 [arXiv:1607.01001] [INSPIRE].
A.D. Plascencia, Classical scale invariance in the inert doublet model, JHEP 09 (2015) 026 [arXiv:1507.04996] [INSPIRE].
V.V. Khoze and A.D. Plascencia, Dark matter and leptogenesis linked by classical scale invariance, JHEP 11 (2016) 025 [arXiv:1605.06834] [INSPIRE].
W.A. Bardeen, On naturalness in the standard model, in the proceedings of the Ontake summer institute on particle physics, (1995).
R. Foot, A. Kobakhidze, K.L. McDonald and R.R. Volkas, A solution to the hierarchy problem from an almost decoupled hidden sector within a classically scale invariant theory, Phys. Rev. D 77 (2008) 035006 [arXiv:0709.2750] [INSPIRE].
L. Alexander-Nunneley and A. Pilaftsis, The minimal scale invariant extension of the standard model, JHEP 09 (2010) 021 [arXiv:1006.5916] [INSPIRE].
C. Englert, J. Jaeckel, V.V. Khoze and M. Spannowsky, Emergence of the electroweak scale through the Higgs portal, JHEP 04 (2013) 060 [arXiv:1301.4224] [INSPIRE].
A. Farzinnia, H.-J. He and J. Ren, Natural electroweak symmetry breaking from scale invariant Higgs mechanism, Phys. Lett. B 727 (2013) 141 [arXiv:1308.0295] [INSPIRE].
W. Altmannshofer et al., Light dark matter, naturalness, and the radiative origin of the electroweak scale, JHEP 01 (2015) 032 [arXiv:1408.3429] [INSPIRE].
M. Holthausen, J. Kubo, K.S. Lim and M. Lindner, Electroweak and conformal symmetry breaking by a strongly coupled hidden sector, JHEP 12 (2013) 076 [arXiv:1310.4423] [INSPIRE].
A. Farzinnia and S. Kouwn, Classically scale invariant inflation, supermassive WIMPs, and adimensional gravity, Phys. Rev. D 93 (2016) 063528 [arXiv:1512.05890] [INSPIRE].
A. Ghoshal, A. Mazumdar, N. Okada and D. Villalba, Stability of infinite derivative Abelian Higgs models, Phys. Rev. D 97 (2018) 076011 [arXiv:1709.09222] [INSPIRE].
A. Ghoshal, Scalar dark matter probes the scale of nonlocality, Int. J. Mod. Phys. A 34 (2019) 1950130 [arXiv:1812.02314] [INSPIRE].
M. Frasca and A. Ghoshal, Diluted mass gap in strongly coupled non-local Yang-Mills, JHEP 21 (2020) 226 [arXiv:2102.10665] [INSPIRE].
A. Ghoshal, A. Mazumdar, N. Okada and D. Villalba, Nonlocal non-Abelian gauge theory: conformal invariance and β-function, Phys. Rev. D 104 (2021) 015003 [arXiv:2010.15919] [INSPIRE].
M. Frasca and A. Ghoshal, Mass gap in strongly coupled infinite derivative non-local Higgs: Dyson-Schwinger approach, Class. Quant. Grav. 38 (2021) 175013 [arXiv:2011.10586] [INSPIRE].
B. Barman and A. Ghoshal, Scale invariant FIMP miracle, JCAP 03 (2022) 003 [arXiv:2109.03259] [INSPIRE].
M. Frasca, A. Ghoshal and N. Okada, Fate of false vacuum in non-perturbative regimes, arXiv:2201.12267 [INSPIRE].
A. Ghoshal, N. Okada and A. Paul, Radiative plateau inflation with conformal invariance: dynamical generation of electroweak and seesaw scales, Phys. Rev. D 106 (2022) 055024 [arXiv:2203.00677] [INSPIRE].
B. Barman and A. Ghoshal, Probing pre-BBN era with scale invariant FIMP, JCAP 10 (2022) 082 [arXiv:2203.13269] [INSPIRE].
N. Bernal, A. Ghoshal, F. Hajkarim and G. Lambiase, Primordial gravitational wave signals in modified cosmologies, JCAP 11 (2020) 051 [arXiv:2008.04959] [INSPIRE].
S.D. Odintsov, V.K. Oikonomou and R. Myrzakulov, Spectrum of primordial gravitational waves in modified gravities: a short overview, Symmetry 14 (2022) 729 [arXiv:2204.00876] [INSPIRE].
S.D. Odintsov, V.K. Oikonomou and F.P. Fronimos, Quantitative predictions for f (R) gravity primordial gravitational waves, Phys. Dark Univ. 35 (2022) 100950 [arXiv:2108.11231] [INSPIRE].
A. Ghoshal and G. Lambiase, Constraints on Tsallis cosmology from big bang nucleosynthesis and dark matter freeze-out, arXiv:2104.11296 [INSPIRE].
J. Jaeckel, V.V. Khoze and M. Spannowsky, Hearing the signal of dark sectors with gravitational wave detectors, Phys. Rev. D 94 (2016) 103519 [arXiv:1602.03901] [INSPIRE].
L. Marzola, A. Racioppi and V. Vaskonen, Phase transition and gravitational wave phenomenology of scalar conformal extensions of the standard model, Eur. Phys. J. C 77 (2017) 484 [arXiv:1704.01034] [INSPIRE].
S. Iso, P.D. Serpico and K. Shimada, QCD-electroweak first-order phase transition in a supercooled universe, Phys. Rev. Lett. 119 (2017) 141301 [arXiv:1704.04955] [INSPIRE].
I. Baldes and C. Garcia-Cely, Strong gravitational radiation from a simple dark matter model, JHEP 05 (2019) 190 [arXiv:1809.01198] [INSPIRE].
T. Prokopec, J. Rezacek and B. Świeżewska, Gravitational waves from conformal symmetry breaking, JCAP 02 (2019) 009 [arXiv:1809.11129] [INSPIRE].
V. Brdar, A.J. Helmboldt and J. Kubo, Gravitational waves from first-order phase transitions: LIGO as a window to unexplored seesaw scales, JCAP 02 (2019) 021 [arXiv:1810.12306] [INSPIRE].
C. Marzo, L. Marzola and V. Vaskonen, Phase transition and vacuum stability in the classically conformal B-L model, Eur. Phys. J. C 79 (2019) 601 [arXiv:1811.11169] [INSPIRE].
A. Ghoshal and A. Salvio, Gravitational waves from fundamental axion dynamics, JHEP 12 (2020) 049 [arXiv:2007.00005] [INSPIRE].
C. Dioguardi and M. Rinaldi, A note on the linear stability of black holes in quadratic gravity, Eur. Phys. J. Plus 135 (2020) 920 [arXiv:2007.11468] [INSPIRE].
G. Cognola, M. Rinaldi and L. Vanzo, Scale-invariant rotating black holes in quadratic gravity, Entropy 17 (2015) 5145 [arXiv:1506.07096] [INSPIRE].
G. Cognola, M. Rinaldi, L. Vanzo and S. Zerbini, Thermodynamics of topological black holes in R2 gravity, Phys. Rev. D 91 (2015) 104004 [arXiv:1503.05151] [INSPIRE].
U. Aydemir, B. Holdom and J. Ren, Not quite black holes as dark matter, Phys. Rev. D 102 (2020) 024058 [arXiv:2003.10682] [INSPIRE].
A. Zee, A broken symmetric theory of gravity, Phys. Rev. Lett. 42 (1979) 417 [INSPIRE].
A.A. Starobinsky, A new type of isotropic cosmological models without singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
D. Gorbunov and A. Tokareva, Scale-invariance as the origin of dark radiation?, Phys. Lett. B 739 (2014) 50 [arXiv:1307.5298] [INSPIRE].
G. Turchetti and G. Venturi, Gravitation and broken scale invariance, Nuovo Cim. A 66 (1981) 221.
M. Rinaldi, L. Vanzo, S. Zerbini and G. Venturi, Inflationary quasiscale-invariant attractors, Phys. Rev. D 93 (2016) 024040 [arXiv:1505.03386] [INSPIRE].
M. Rinaldi and L. Vanzo, Inflation and reheating in theories with spontaneous scale invariance symmetry breaking, Phys. Rev. D 94 (2016) 024009 [arXiv:1512.07186] [INSPIRE].
S. Vicentini, L. Vanzo and M. Rinaldi, Scale-invariant inflation with one-loop quantum corrections, Phys. Rev. D 99 (2019) 103516 [arXiv:1902.04434] [INSPIRE].
A. Salvio, Inflationary perturbations in no-scale theories, Eur. Phys. J. C 77 (2017) 267 [arXiv:1703.08012] [INSPIRE].
Y.S. Myung and T. Moon, Primordial gravitational waves from conformal gravity, arXiv:1407.0441 [INSPIRE].
A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
J. Martin, C. Ringeval and V. Vennin, Encyclopædia inflationaris, Phys. Dark Univ. 5-6 (2014) 75 [arXiv:1303.3787] [INSPIRE].
Planck collaboration, Planck 2018 results. VI. Cosmological parameters, Astron. Astrophys. 641 (2020) A6 [Erratum ibid. 652 (2021) C4] [arXiv:1807.06209] [INSPIRE].
BICEP and Keck collaborations, Improved constraints on primordial gravitational waves using Planck, WMAP, and BICEP/Keck observations through the 2018 observing season, Phys. Rev. Lett. 127 (2021) 151301 [arXiv:2110.00483] [INSPIRE].
BICEP and Keck collaborations, The latest constraints on inflationary B-modes from the BICEP/Keck telescopes, in the proceedings of the 56th Rencontres de Moriond on cosmology, (2022) [arXiv:2203.16556] [INSPIRE].
P. Campeti and E. Komatsu, New constraint on the tensor-to-scalar ratio from the Planck and BICEP/Keck array data using the profile likelihood, Astrophys. J. 941 (2022) 110 [arXiv:2205.05617] [INSPIRE].
Simons Observatory collaboration, The Simons Observatory: science goals and forecasts, JCAP 02 (2019) 056 [arXiv:1808.07445] [INSPIRE].
LiteBIRD collaboration, Probing cosmic inflation with the LiteBIRD cosmic microwave background polarization survey, Prog. Theor. Exper. Phys. 2023 (2022) 042F01 [arXiv:2202.02773] [INSPIRE].
D. Baumann, Inflation, in the proceedings of the Theoretical Advanced Study Institute in Elementary Particle Physics: physics of the large and the small, (2011), p. 523 [https://doi.org/10.1142/9789814327183_0010] [arXiv:0907.5424] [INSPIRE].
D. Adak et al., B-mode forecast of \( CMB- Bh\overline{a} rat \), Mon. Not. Roy. Astron. Soc. 514 (2022) 3002 [arXiv:2110.12362] [INSPIRE].
M. He, A.A. Starobinsky and J. Yokoyama, Inflation in the mixed Higgs-R2 model, JCAP 05 (2018) 064 [arXiv:1804.00409] [INSPIRE].
J.F. Donoghue and G. Menezes, Inducing the Einstein action in QCD-like theories, Phys. Rev. D 97 (2018) 056022 [arXiv:1712.04468] [INSPIRE].
L.-H. Liu, T. Prokopec and A.A. Starobinsky, Inflation in an effective gravitational model and asymptotic safety, Phys. Rev. D 98 (2018) 043505 [arXiv:1806.05407] [INSPIRE].
C. Laporte, A.D. Pereira, F. Saueressig and J. Wang, Scalar-tensor theories within asymptotic safety, JHEP 12 (2021) 001 [arXiv:2110.09566] [INSPIRE].
Acknowledgments
We would like to thank C. Cecchini, A. Salvio for numerous discussions over the course of this work. DM would like to thank Aparajita Sen for illuminating discussions regard- ing LiteBIRD, Simons Obervatory and CMB-Bharat observational data. AG thanks P. Michalak for reading manuscripts and comments. DM also acknowledges the hospitality of IISER Bhopal while this work was undergoing. DM’s work is supported in part by the SERB core research grant CRG/2018/002373.
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Ghoshal, A., Mukherjee, D. & Rinaldi, M. Inflation and primordial gravitational waves in scale-invariant quadratic gravity with Higgs. J. High Energ. Phys. 2023, 23 (2023). https://doi.org/10.1007/JHEP05(2023)023
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DOI: https://doi.org/10.1007/JHEP05(2023)023