Abstract
We present an introduction to cosmic inflation in the context of Palatini gravity, which is an interesting alternative to the usual metric theory of gravity. In the latter case only the metric \(g_{\mu \nu }\) determines the geometry of space-time, whereas in the former case both the metric and the space-time connection \(\varGamma ^\lambda _{\mu \nu }\) are a priori independent variables—a choice which can lead to a theory of gravity different from the metric one. In scenarios where the field(s) responsible for cosmic inflation are coupled non-minimally to gravity or the gravitational sector is otherwise extended, assumptions of the underlying gravitational degrees of freedom can have a big impact on the observational consequences of inflation. We demonstrate this explicitly by reviewing several interesting and well-motivated scenarios including Higgs inflation, \(R^2\) inflation, and \(\xi \)-attractor models. We also discuss some prospects for future research and argue why \(r=10^{-3}\) is a particularly important goal for future missions that search for signatures of primordial gravitational waves.
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Notes
For clarity, we note that sometimes the notation \(\{^\sigma _{\mu \nu }\}\) is used instead of \({\bar{\varGamma }}^\sigma _{\mu \nu }\), and the name “Christoffel symbol” or “Riemannian connection” instead of the “Levi-Civita connection” we use in this paper.
Parallel transporting a tensor T along the path \(x^\mu (\lambda )\) parameterized by \(\lambda \) means that the covariant derivative of T along the path vanishes,
Also this quantity depends, a priori, only on the connection \(\varGamma ^\sigma _{\mu \nu }\). Note, however, that if the connection is not metric-compatible, parallel transport does not necessarily preserve the norm of vectors, a feature often taken as granted.
Derivation of the equations of motion for the metric case requires adding a so-called Gibbons–Hawking–York boundary term to the action to cancel a total derivative term that depends on the second derivatives of the metric [71, 153]. For the possibility of adding a (non-covariant) boundary term that only depends on the first derivatives of the metric, see e.g. Ref. [48].
In principle, because the SM field content and couplings are known, the details of (p)reheating can be calculated exactly, which then gives the total number of e-folds between the end of inflation and horizon exit of the scale where measurements are made [22, 65, 128, 142]. However, in practice the SM couplings are not exactly known, and neither is the Beyond-the-Standard-Model (BSM) physics which accommodates e.g. dark matter or baryogenesis, and which may affect the renormalization group running of the SM couplings up to the scales where (p)reheating (or inflation) occurs.
This is not a coincidence, as we will discuss in Sect. 3.2.3.
In principle, the SM Higgs should not be forgotten, and one can ask what is its effect on the inflationary dynamics. It can be shown that the presence of the Higgs alongside the \(R^2\) term leads to multifield inflation in metric gravity, as recently studied in Refs. [36, 37, 50, 54, 69, 77, 80, 102, 131, 151].
One can still ask how quickly the slow-roll regime is reached if the non-canonical terms are important in the beginning of inflation. This was recently addressed in Ref. [148].
References
Abazajian, K.N., et al.: CMB-S4 Science Book, 1st Edn (2016)
Accetta, F.S., Zoller, D.J., Turner, M.S.: Induced gravity inflation. Phys. Rev. D 31, 3046 (1985). https://doi.org/10.1103/PhysRevD.31.3046
Ade, P., et al.: The Simons observatory: science goals and forecasts (2018). https://doi.org/10.1088/1475-7516/2019/02/056
Ade, P.A.R., et al.: BICEP2/Keck Array x: constraints on primordial gravitational waves using Planck, WMAP, and New BICEP2/Keck Observations through the 2015 Season. Phys. Rev. Lett. (Submitted to) (2018). https://doi.org/10.1103/PhysRevLett.121.221301
Aghanim, N., et al.: Planck 2018 results. VI, Cosmological parameters (2018). https://arxiv.org/abs/1807.06209
Akrami, Y., et al.: Planck 2018 results. X. Constraints on inflation (2018). https://arxiv.org/abs/1807.06211
Allahverdi, R., Brandenberger, R., Cyr-Racine, F.Y., Mazumdar, A.: Reheating in inflationary cosmology: theory and applications. Ann. Rev. Nucl. Part. Sci. 60, 27–51 (2010). https://doi.org/10.1146/annurev.nucl.012809.104511
Allemandi, G., Borowiec, A., Francaviglia, M.: Accelerated cosmological models in first order nonlinear gravity. Phys. Rev. D 70, 043524 (2004). https://doi.org/10.1103/PhysRevD.70.043524
Almeida, J.P.B., Bernal, N., Rubio, J., Tenkanen, T.: Hidden inflaton dark matter. JCAP 1903, 012 (2019). https://doi.org/10.1088/1475-7516/2019/03/012
Amendola, L., Litterio, M., Occhionero, F.: The Phase space view of inflation. 1: The nonminimally coupled scalar field. Int. J. Mod. Phys. A 5, 3861–3886 (1990). https://doi.org/10.1142/S0217751X90001653
Amin, M.A., Hertzberg, M.P., Kaiser, D.I., Karouby, J.: Nonperturbative dynamics of reheating after inflation: a review. Int. J. Mod. Phys. D 24, 1530003 (2014). https://doi.org/10.1142/S0218271815300037
Antoniadis, I., Karam, A., Lykkas, A., Pappas, T., Tamvakis, K.: Rescuing quartic and natural inflation in the palatini formalism. JCAP 1903, 005 (2019). https://doi.org/10.1088/1475-7516/2019/03/005
Antoniadis, I., Karam, A., Lykkas, A., Tamvakis, K.: Palatini inflation in models with an \(R^2\) term. JCAP 1811(11), 028 (2018). https://doi.org/10.1088/1475-7516/2018/11/028
Aoki, K., Shimada, K.: Galileon and generalized Galileon with projective invariance in metric-affine formalism. Phys. Rev. D 98(4), 044038 (2018). https://doi.org/10.1103/PhysRevD.98.044038
Azri, H.: Are there really conformal frames? Uniqueness of affine inflation. Int. J. Mod. Phys. D 27(09), 1830006 (2018). https://doi.org/10.1142/S0218271818300069
Azri, H., Demir, D.: Affine inflation. Phys. Rev. D 95(12), 124007 (2017). https://doi.org/10.1103/PhysRevD.95.124007
Bastero-Gil, M., Cerezo, R., Rosa, J.G.: Inflaton dark matter from incomplete decay. Phys. Rev. D 93(10), 103531 (2016). https://doi.org/10.1103/PhysRevD.93.103531
Bauer, F., Demir, D.A.: Inflation with non-minimal coupling: metric versus Palatini formulations. Phys. Lett. B 665, 222–226 (2008). https://doi.org/10.1016/j.physletb.2008.06.014
Bauer, F., Demir, D.A.: Higgs-Palatini inflation and unitarity. Phys. Lett. B698, 425–429 (2011). https://doi.org/10.1016/j.physletb.2011.03.042
Baumann, D.: Inflation, pp. 523–686 (2011). https://doi.org/10.1142/9789814327183/0010
Benisty, D., Guendelman, E., Nissimov, E., Pacheva, S.: Dynamically generated inflation from non-Riemannian volume forms. Eur. Phys. J. C 79(9), 806 (2019). https://doi.org/10.1140/epjc/s10052-019-7310-6
Bezrukov, F., Gorbunov, D., Shaposhnikov, M.: On initial conditions for the Hot Big Bang. JCAP 0906, 029 (2009). https://doi.org/10.1088/1475-7516/2009/06/029
Bezrukov, F., Pauly, M., Rubio, J.: On the robustness of the primordial power spectrum in renormalized Higgs inflation. JCAP 1802(02), 040 (2018). https://doi.org/10.1088/1475-7516/2018/02/040
Bezrukov, F., Rubio, J., Shaposhnikov, M.: Living beyond the edge: Higgs inflation and vacuum metastability. Phys. Rev. D 92(8), 083512 (2015). https://doi.org/10.1103/PhysRevD.92.083512
Bezrukov, F., Shaposhnikov, M.: Standard model Higgs boson mass from inflation: two loop analysis. JHEP 07, 089 (2009). https://doi.org/10.1088/1126-6708/2009/07/089
Bezrukov, F., Shaposhnikov, M.: Higgs inflation at the critical point. Phys. Lett. B 734, 249–254 (2014). https://doi.org/10.1016/j.physletb.2014.05.074
Bezrukov, F.L., Gorbunov, D.S.: Distinguishing between \(\text{ R }^2\)-inflation and Higgs-inflation. Phys. Lett. B 713, 365–368 (2012). https://doi.org/10.1016/j.physletb.2012.06.040
Bezrukov, F.L., Shaposhnikov, M.: The standard model Higgs boson as the inflaton. Phys. Lett. B 659, 703–706 (2008). https://doi.org/10.1016/j.physletb.2007.11.072
Bilandzic, A., Prokopec, T.: Quantum radiative corrections to slow-roll inflation. Phys. Rev. D 76, 103507 (2007). https://doi.org/10.1103/PhysRevD.76.103507
Birrell, N.D., Davies, P.C.W.: Quantum Fields in Curved Space. Cambridge Monographs on Mathematical Physics. Cambridge Univ. Press, Cambridge (1984). https://doi.org/10.1017/CBO9780511622632
Borowiec, A., Kamionka, M., Kurek, A., Szydlowski, M.: Cosmic acceleration from modified gravity with Palatini formalism. JCAP 1202, 027 (2012). https://doi.org/10.1088/1475-7516/2012/02/027
Borowiec, A., Stachowski, A., Szydłowski, M., Wojnar, A.: Inflationary cosmology with Chaplygin gas in Palatini formalism. JCAP 1601(01), 040 (2016). https://doi.org/10.1088/1475-7516/2016/01/040
Bostan, N.: Non-minimally coupled quartic inflation with Coleman–Weinberg one-loop corrections in the Palatini formulation (2019). https://arxiv.org/abs/1907.13235
Bostan, N.: Preheating in radiative corrections to \(\phi ^4\) inflation with non-minimal coupling in Palatini formulation (2019). https://arxiv.org/abs/1912.12977
Bostan, N.: Quadratic, Higgs and hilltop potentials in the Palatini gravity (2019). https://arxiv.org/abs/1908.09674
Calmet, X., Kuntz, I.: Higgs Starobinsky inflation. Eur. Phys. J. C76(5), 289 (2016). https://doi.org/10.1140/epjc/s10052-016-4136-3
Canko, D.D., Gialamas, I.D., Kodaxis, G.P.: A simple \(F({{{\cal{R}}}},\phi )\) deformation of Starobinsky inflationary model (2019). https://arxiv.org/abs/1901.06296
Capozziello, S., Francaviglia, M.: Extended theories of gravity and their cosmological and astrophysical applications. Gen. Relativ. Gravit. 40, 357–420 (2008). https://doi.org/10.1007/s10714-007-0551-y
Capozziello, S., Harko, T., Koivisto, T.S., Lobo, F.S.N., Olmo, G.J.: Hybrid metric-Palatini gravity. Universe 1(2), 199–238 (2015). https://doi.org/10.3390/universe1020199
Carrilho, P., Mulryne, D., Ronayne, J., Tenkanen, T.: Attractor behaviour in multifield inflation. JCAP 1806(06), 032 (2018). https://doi.org/10.1088/1475-7516/2018/06/032
Carroll, S.M.: Spacetime and Geometry. Cambridge University Press (2019). http://www.slac.stanford.edu/spires/find/books/www?cl=QC6:C37:2004
Cervantes-Cota, J.L., Dehnen, H.: Induced gravity inflation in the standard model of particle physics. Nucl. Phys. B 442, 391–412 (1995). https://doi.org/10.1016/0550-3213(95)00128-X
Cook, J.L., Krauss, L.M., Long, A.J., Sabharwal, S.: Is Higgs inflation ruled out? Phys. Rev. D 89(10), 103525 (2014). https://doi.org/10.1103/PhysRevD.89.103525
De Simone, A., Hertzberg, M.P., Wilczek, F.: Running inflation in the standard model. Phys. Lett. B 678, 1–8 (2009). https://doi.org/10.1016/j.physletb.2009.05.054
Demir, D., Pulice, B.: Geometric Dark Matter (2020). https://arxiv.org/abs/2001.06577
Dvali, G., Gruzinov, A., Zaldarriaga, M.: A new mechanism for generating density perturbations from inflation. Phys. Rev. D 69, 023505 (2004). https://doi.org/10.1103/PhysRevD.69.023505
Dvali, G.R., Zaldarriaga, M.: Changing alpha with time: implications for fifth force type experiments and quintessence. Phys. Rev. Lett. 88, 091303 (2002). https://doi.org/10.1103/PhysRevLett.88.091303
Dyer, E., Hinterbichler, K.: Boundary terms, variational principles and higher derivative modified gravity. Phys. Rev. D 79, 024028 (2009). https://doi.org/10.1103/PhysRevD.79.024028
Einstein, A.: Einheitliche Feldtheorie von Gravitation und Elektrizität. Verlag der Koeniglich-Preussichen Akademie der Wissenschaften 22, 414–419 (1925)
Ema, Y.: Higgs Scalaron mixed inflation. Phys. Lett. B770, 403–411 (2017). https://doi.org/10.1016/j.physletb.2017.04.060
Enckell, V.M., Enqvist, K., Nurmi, S.: Observational signatures of Higgs inflation. JCAP 1607(07), 047 (2016). https://doi.org/10.1088/1475-7516/2016/07/047
Enckell, V.M., Enqvist, K., Rasanen, S., Tomberg, E.: Higgs inflation at the hilltop. JCAP 1806(06), 005 (2018). https://doi.org/10.1088/1475-7516/2018/06/005
Enckell, V.M., Enqvist, K., Rasanen, S., Wahlman, L.P.: Inflation with \(R^2\) term in the Palatini formalism. JCAP 1902, 022 (2019). https://doi.org/10.1088/1475-7516/2019/02/022
Enckell, V.M., Enqvist, K., Rasanen, S., Wahlman, L.P.: Higgs-\(R^2\) inflation-full slow-roll study at tree-level. JCAP 2001, 041 (2020). https://doi.org/10.1088/1475-7516/2020/01/041
Enqvist, K., Koivisto, T., Rigopoulos, G.: Non-metric chaotic inflation. JCAP 1205, 023 (2012). https://doi.org/10.1088/1475-7516/2012/05/023
Enqvist, K., Sloth, M.S.: Adiabatic CMB perturbations in pre-big bang string cosmology. Nucl. Phys. B 626, 395–409 (2002). https://doi.org/10.1016/S0550-3213(02)00043-3
Fakir, R., Unruh, W.G.: Improvement on cosmological chaotic inflation through nonminimal coupling. Phys. Rev. D 41, 1783–1791 (1990). https://doi.org/10.1103/PhysRevD.41.1783
Ferrara, S., Kallosh, R., Linde, A., Porrati, M.: Minimal supergravity models of inflation. Phys. Rev. D 88(8), 085038 (2013). https://doi.org/10.1103/PhysRevD.88.085038
Ferraris, M., Francaviglia, M., Reina, C.: Einheitliche Feldtheorie von Gravitation und Elektrizität. Gen. Relativ. Gravit. 14(3), 243–254 (1982). https://doi.org/10.1007/BF00756060
Fu, C., Wu, P., Yu, H.: Inflationary dynamics and preheating of the nonminimally coupled inflaton field in the metric and Palatini formalisms. Phys. Rev. D 96(10), 103542 (2017). https://doi.org/10.1103/PhysRevD.96.103542
Fumagalli, J.: Renormalization group independence of cosmological attractors. Phys. Lett. B 769, 451–459 (2017). https://doi.org/10.1016/j.physletb.2017.04.017
Fumagalli, J., Postma, M.: UV (in)sensitivity of Higgs inflation. JHEP 05, 049 (2016). https://doi.org/10.1007/JHEP05(2016)049
Futamase, T.: Maeda, Ki: Chaotic inflationary scenario in models having nonminimal coupling with curvature. Phys. Rev. D 39, 399–404 (1989). https://doi.org/10.1103/PhysRevD.39.399
Galante, M., Kallosh, R., Linde, A., Roest, D.: Unity of cosmological inflation attractors. Phys. Rev. Lett. 114(14), 141302 (2015). https://doi.org/10.1103/PhysRevLett.114.141302
Garcia-Bellido, J., Figueroa, D.G., Rubio, J.: Preheating in the standard model with the Higgs-inflaton coupled to gravity. Phys. Rev. D 79, 063531 (2009). https://doi.org/10.1103/PhysRevD.79.063531
George, D.P., Mooij, S., Postma, M.: Quantum corrections in Higgs inflation: the real scalar case. JCAP 1402, 024 (2014). https://doi.org/10.1088/1475-7516/2014/02/024
George, D.P., Mooij, S., Postma, M.: Quantum corrections in Higgs inflation: the standard model case. JCAP 1604(04), 006 (2016). https://doi.org/10.1088/1475-7516/2016/04/006
Germani, C., Kehagias, A.: New model of inflation with non-minimal derivative coupling of standard model Higgs boson to gravity. Phys. Rev. Lett. 105, 011302 (2010). https://doi.org/10.1103/PhysRevLett.105.011302
Ghilencea, D.M.: Two-loop corrections to Starobinsky-Higgs inflation. Phys. Rev. D 98(10), 103524 (2018). https://doi.org/10.1103/PhysRevD.98.103524
Gialamas, I.D., Lahanas, A.B.: Reheating in \(R^2\) Palatini inflationary models (2019). https://arxiv.org/abs/1911.11513
Gibbons, G.W., Hawking, S.W.: Action integrals and partition functions in quantum gravity. Phys. Rev. D 15, 2752–2756 (1977). https://doi.org/10.1103/PhysRevD.15.2752
Giovannini, M.: Post-inflationary phases stiffer than radiation and Palatini formulation. Class. Quant. Gravit. 36(23), 235017 (2019). https://doi.org/10.1088/1361-6382/ab52a8
Gorbunov, D.S., Panin, A.G.: Scalaron the mighty: producing dark matter and baryon asymmetry at reheating. Phys. Lett. B 700, 157–162 (2011). https://doi.org/10.1016/j.physletb.2011.04.067
Greenwood, R.N., Kaiser, D.I., Sfakianakis, E.I.: Multifield dynamics of Higgs inflation. Phys. Rev. D 87, 064021 (2013). https://doi.org/10.1103/PhysRevD.87.064021
Guendelman, E.I.: Scale invariance, inflation and the present vacuum energy of the universe. In: Proceedings, 35th Rencontres de Moriond, pp. 37–40 (2002). http://moriond.in2p3.fr/J00/ProcMJ2000/guendel/abstract.html
Guendelman, E.I., Kaganovich, A.B.: Gravity, cosmology and particle physics without the cosmological constant problem. Mod. Phys. Lett. A 13, 1583–1586 (1998). https://doi.org/10.1142/S0217732398001662
Gundhi, A., Steinwachs, C.F.: Scalaron-Higgs inflation (2018). https://arxiv.org/abs/1810.10546
Hamada, Y., Kawai, H., Oda, Ky, Park, S.C.: Higgs inflation is still alive after the results from BICEP2. Phys. Rev. Lett. 112(24), 241301 (2014). https://doi.org/10.1103/PhysRevLett.112.241301
Hanany, S., et al.: PICO: Probe of Inflation and Cosmic Origins (2019). https://arxiv.org/abs/1902.10541
He, M., Starobinsky, A.A., Yokoyama, J.: Inflation in the mixed Higgs-\(R^2\) model. JCAP 1805(05), 064 (2018). https://doi.org/10.1088/1475-7516/2018/05/064
Herranen, M., Hohenegger, A., Osland, A., Tranberg, A.: Quantum corrections to inflation: the importance of RG-running and choosing the optimal RG-scale. Phys. Rev. D 95(2), 023525 (2017). https://doi.org/10.1103/PhysRevD.95.023525
Hertzberg, M.P.: On inflation with non-minimal coupling. JHEP 11, 023 (2010). https://doi.org/10.1007/JHEP11(2010)023
Heurtier, L.: The inflaton portal to dark matter. JHEP 12, 072 (2017). https://doi.org/10.1007/JHEP12(2017)072
Hooper, D., Krnjaic, G., Long, A.J., Mcdermott, S.D.: Can the inflaton also be a weakly interacting massive particle? Phys. Rev. Lett. 122(9), 091802 (2019). https://doi.org/10.1103/PhysRevLett.122.091802
Jarv, L., Racioppi, A., Tenkanen, T.: Palatini side of inflationary attractors. Phys. Rev. D 97(8), 083513 (2018). https://doi.org/10.1103/PhysRevD.97.083513
Jinno, R., Kaneta, K.: Oda, Ky: Hill-climbing Higgs inflation. Phys. Rev. D 97(2), 023523 (2018). https://doi.org/10.1103/PhysRevD.97.023523
Hillclimbing inflation in metric and Palatini formulations: Jinno, R., Kaneta, K., Oda, Ky, Park, S.C. Phys. Lett. B 791, 396–402 (2019). https://doi.org/10.1016/j.physletb.2019.03.012
Jinno, R., Kubota, M., Oda, K.y., Park, S.C.: Higgs inflation in metric and Palatini formalisms: required suppression of higher dimensional operators (2019). https://doi.org/10.1088/1475-7516/2020/03/063
Kaganovich, A.B.: Field theory model giving rise to ’quintessential inflation’ without the cosmological constant and other fine tuning problems. Phys. Rev. D 63, 025022 (2001). https://doi.org/10.1103/PhysRevD.63.025022
Kahlhoefer, F., McDonald, J.: WIMP dark matter and unitarity-conserving inflation via a gauge singlet scalar. JCAP 1511(11), 015 (2015). https://doi.org/10.1088/1475-7516/2015/11/015
Kaiser, D.I.: Constraints in the context of induced gravity inflation. Phys. Rev. D 49, 6347–6353 (1994). https://doi.org/10.1103/PhysRevD.49.6347
Kaiser, D.I.: Induced gravity inflation and the density perturbation spectrum. Phys. Lett. B 340, 23–28 (1994). https://doi.org/10.1016/0370-2693(94)91292-0
Kaiser, D.I.: Primordial spectral indices from generalized Einstein theories. Phys. Rev. D 52, 4295–4306 (1995). https://doi.org/10.1103/PhysRevD.52.4295
Kaiser, D.I., Sfakianakis, E.I.: Multifield inflation after planck: the case for nonminimal couplings. Phys. Rev. Lett. 112(1), 011302 (2014). https://doi.org/10.1103/PhysRevLett.112.011302
Kallosh, R., Linde, A.: B-mode targets. Phys. Lett. B798, 134970 (2019). https://doi.org/10.1016/j.physletb.2019.134970
Kallosh, R., Linde, A.: CMB targets after the latest \(Planck\) data release. Phys. Rev. D 100(12), 123523 (2019). https://doi.org/10.1103/PhysRevD.100.123523
Kallosh, R., Linde, A., Roest, D.: Superconformal inflationary \(\alpha \)-attractors. JHEP 11, 198 (2013). https://doi.org/10.1007/JHEP11(2013)198
Kallosh, R., Linde, A., Roest, D.: Universal attractor for inflation at strong coupling. Phys. Rev. Lett. 112(1), 011303 (2014). https://doi.org/10.1103/PhysRevLett.112.011303
Kamada, K., Kobayashi, T., Takahashi, T., Yamaguchi, M., Yokoyama, J.: Generalized Higgs inflation. Phys. Rev. D 86, 023504 (2012). https://doi.org/10.1103/PhysRevD.86.023504
Kamada, K., Kobayashi, T., Yamaguchi, M., Yokoyama, J.: Higgs G-inflation. Phys. Rev. D 83, 083515 (2011). https://doi.org/10.1103/PhysRevD.83.083515
Kannike, K., Kubarski, A., Marzola, L., Racioppi, A.: A minimal model of inflation and dark radiation. Phys. Lett. B 792, 74–80 (2019). https://doi.org/10.1016/j.physletb.2019.03.025
Karam, A., Pappas, T., Tamvakis, K.: Nonminimal Coleman-Weinberg inflation with an \(R^2\) term. JCAP 1902, 006 (2019). https://doi.org/10.1088/1475-7516/2019/02/006
Kofman, L.: Probing string theory with modulated cosmological fluctuations (2003). https://arxiv.org/abs/astro-ph/0303614
Koivisto, T., Kurki-Suonio, H.: Cosmological perturbations in the palatini formulation of modified gravity. Class. Quant. Gravit. 23, 2355–2369 (2006). https://doi.org/10.1088/0264-9381/23/7/009
Komatsu, E., Futamase, T.: Complete constraints on a nonminimally coupled chaotic inflationary scenario from the cosmic microwave background. Phys. Rev. D 59, 064029 (1999). https://doi.org/10.1103/PhysRevD.59.064029
Lerner, R.N., McDonald, J.: Gauge singlet scalar as inflaton and thermal relic dark matter. Phys. Rev. D 80, 123507 (2009). https://doi.org/10.1103/PhysRevD.80.123507
Lerner, R.N., McDonald, J.: Distinguishing Higgs inflation and its variants. Phys. Rev. D 83, 123522 (2011). https://doi.org/10.1103/PhysRevD.83.123522
Liddle, A.R., Leach, S.M.: How long before the end of inflation were observable perturbations produced? Phys. Rev. D 68, 103503 (2003). https://doi.org/10.1103/PhysRevD.68.103503
Lyth, D.H., Riotto, A.: Particle physics models of inflation and the cosmological density perturbation. Phys. Rept. 314, 1–146 (1999). https://doi.org/10.1016/S0370-1573(98)00128-8
Lyth, D.H., Wands, D.: Generating the curvature perturbation without an inflaton. Phys. Lett. B 524, 5–14 (2002). https://doi.org/10.1016/S0370-2693(01)01366-1
Markkanen, T., Tenkanen, T., Vaskonen, V., Veermäe, H.: Quantum corrections to quartic inflation with a non-minimal coupling: metric vs. Palatini. JCAP 1803(03), 029 (2018). https://doi.org/10.1088/1475-7516/2018/03/029
Matsumura, T., et al.: Mission design of LiteBIRD (2013) [J. Low. Temp. Phys. 176, 733 (2014)]. https://doi.org/10.1007/s10909-013-0996-1
Meng, X.H., Wang, P.: Palatini formulation of modified gravity with squared scalar curvature. Gen. Relativ. Gravit. 36, 2673 (2004). https://doi.org/10.1023/B:GERG.0000048981.40061.63
Meng, X.H., Wang, P.: R**2 corrections to the cosmological dynamics of inflation in the Palatini formulation. Class. Quant. Gravit. 21, 2029–2036 (2004). https://doi.org/10.1088/0264-9381/21/8/008
Mooij, S., Postma, M.: Goldstone bosons and a dynamical Higgs field. JCAP 1109, 006 (2011). https://doi.org/10.1088/1475-7516/2011/09/006
Moroi, T., Takahashi, T.: Effects of cosmological moduli fields on cosmic microwave background. Phys. Lett. B 522, 215–221 (2001). https://doi.org/10.1016/S0370-2693(02)02070-1 [Erratum: Phys. Lett. B 539, 303 (2002)] https://doi.org/10.1016/S0370-2693(01)01295-3
Nakayama, K., Takahashi, F.: Running kinetic inflation. JCAP 1011, 009 (2010). https://doi.org/10.1088/1475-7516/2010/11/009
Palatini, A.: Deduzione invariantiva delle equazioni gravitazionali dal principio di Hamilton. Rendiconti del Circolo Matematico di Palermo 43(1), 203–212 (1919). https://doi.org/10.1007/BF03014670
Poplawski, N.J.: Acceleration of the universe in the Einstein frame of a metric-affine f(R) gravity. Class. Quant. Gravit. 23, 2011–2020 (2006). https://doi.org/10.1088/0264-9381/23/6/011
Raatikainen, S., Rasanen, S.: Higgs inflation and teleparallel gravity. JCAP 1912(12), 021 (2019). https://doi.org/10.1088/1475-7516/2019/12/021
Racioppi, A.: Coleman-Weinberg linear inflation: metric vs. Palatini formulation. JCAP 1712(12), 041 (2017). https://doi.org/10.1088/1475-7516/2017/12/041
Racioppi, A.: New universal attractor in nonminimally coupled gravity: linear inflation. Phys. Rev. D 97(12), 123514 (2018). https://doi.org/10.1103/PhysRevD.97.123514
Racioppi, A.: Non-minimal (self-)running inflation: metric vs. Palatini formulation (2019). https://arxiv.org/abs/1912.10038
Rasanen, S.: Higgs inflation in the Palatini formulation with kinetic terms for the metric. Open J, Astrophys (2018). https://doi.org/10.21105/astro.1811.09514
Rasanen, S., Tomberg, E.: Planck scale black hole dark matter from Higgs inflation. JCAP 1901(01), 038 (2019). https://doi.org/10.1088/1475-7516/2019/01/038
Rasanen, S., Wahlman, P.: Higgs inflation with loop corrections in the Palatini formulation. JCAP 1711(11), 047 (2017). https://doi.org/10.1088/1475-7516/2017/11/047
Rubio, J.: Higgs inflation. Front. Astron. Space Sci. 5, 50 (2019). https://doi.org/10.3389/fspas.2018.00050
Rubio, J., Tomberg, E.S.: Preheating in Palatini Higgs inflation. JCAP 1904(04), 021 (2019). https://doi.org/10.1088/1475-7516/2019/04/021
Salopek, D.S., Bond, J.R., Bardeen, J.M.: Designing density fluctuation spectra in inflation. Phys. Rev. D 40, 1753 (1989). https://doi.org/10.1103/PhysRevD.40.1753
Saltas, I.D.: Higgs inflation and quantum gravity: an exact renormalisation group approach. JCAP 1602, 048 (2016). https://doi.org/10.1088/1475-7516/2016/02/048
Salvio, A., Mazumdar, A.: Classical and quantum initial conditions for higgs inflation. Phys. Lett. B 750, 194–200 (2015). https://doi.org/10.1016/j.physletb.2015.09.020
Shaposhnikov, M., Shkerin, A., Zell, S.: Standard Model Meets Gravity: Electroweak Symmetry Breaking and Inflation (2020). https://arxiv.org/abs/2001.09088
Shimada, K., Aoki, K.: Maeda, Ki: Metric-affine gravity and inflation. Phys. Rev. D 99(10), 104020 (2019). https://doi.org/10.1103/PhysRevD.99.104020
Sotiriou, T.P.: Constraining f(R) gravity in the Palatini formalism. Class. Quant. Gravit. 23, 1253–1267 (2006). https://doi.org/10.1088/0264-9381/23/4/012
Sotiriou, T.P.: Unification of inflation and cosmic acceleration in the Palatini formalism. Phys. Rev. D 73, 063515 (2006). https://doi.org/10.1103/PhysRevD.73.063515
Sotiriou, T.P., Faraoni, V.: f(R) theories of gravity. Rev. Mod. Phys. 82, 451–497 (2010). https://doi.org/10.1103/RevModPhys.82.451
Sotiriou, T.P., Liberati, S.: Metric-affine f(R) theories of gravity. Ann. Phys. 322, 935–966 (2007). https://doi.org/10.1016/j.aop.2006.06.002
Spokoiny, B.L.: Inflation and generation of perturbations in broken symmetric theory of gravity. Phys. Lett. 147B, 39–43 (1984). https://doi.org/10.1016/0370-2693(84)90587-2
Stachowski, A., Szydłowski, M., Borowiec, A.: Starobinsky cosmological model in Palatini formalism. Eur. Phys. J. C 77(6), 406 (2017). https://doi.org/10.1140/epjc/s10052-017-4981-8
Starobinsky, A.A.: A new type of isotropic cosmological models without singularity. Phys. Lett. B 91, 99–102 (1980). https://doi.org/10.1016/0370-2693(80)90670-X.[771(1980)]
Szydłowski, M., Stachowski, A., Borowiec, A., Wojnar, A.: Do sewn up singularities falsify the Palatini cosmology? Eur. Phys. J. C 76(10), 567 (2016). https://doi.org/10.1140/epjc/s10052-016-4426-9
Takahashi, T., Tenkanen, T.: Towards distinguishing variants of non-minimal inflation. JCAP 1904, 035 (2019). https://doi.org/10.1088/1475-7516/2019/04/035
Tamanini, N., Contaldi, C.R.: Inflationary perturbations in palatini generalised gravity. Phys. Rev. D 83, 044018 (2011). https://doi.org/10.1103/PhysRevD.83.044018
Tenkanen, T.: Feebly interacting dark matter particle as the inflaton. JHEP 09, 049 (2016). https://doi.org/10.1007/JHEP09(2016)049
Tenkanen, T.: Resurrecting quadratic inflation with a non-minimal coupling to gravity. JCAP 1712(12), 001 (2017). https://doi.org/10.1088/1475-7516/2017/12/001
Tenkanen, T.: Minimal Higgs inflation with an \(R^2\) term in Palatini gravity. Phys. Rev. D 99(6), 063528 (2019). https://doi.org/10.1103/PhysRevD.99.063528
Tenkanen, T.: Trans-Planckian censorship, inflation and dark matter. Phys. Rev. D 101(6), 063517 (2020). https://doi.org/10.1103/PhysRevD.101.063517
Tenkanen, T., Tomberg, E.: Initial conditions for plateau inflation (2020). https://arxiv.org/abs/2002.02420
Tenkanen, T., Tuominen, K., Vaskonen, V.: A Strong Electroweak Phase Transition from the Inflaton Field. JCAP 1609(09), 037 (2016). https://doi.org/10.1088/1475-7516/2016/09/037
Tenkanen, T., Visinelli, L.: Axion dark matter from Higgs inflation with an intermediate \(H_*\). JCAP 1908, 033 (2019). https://doi.org/10.1088/1475-7516/2019/08/033
Wang, Y.C., Wang, T.: Primordial perturbations generated by Higgs field and \(R^2\) operator. Phys. Rev. D 96(12), 123506 (2017). https://doi.org/10.1103/PhysRevD.96.123506
Wu, W.L.K., et al.: Initial performance of BICEP3: a degree angular scale 95 GHz band polarimeter. J. Low. Temp. Phys. 184(3–4), 765–771 (2016). https://doi.org/10.1007/s10909-015-1403-x
York Jr., J.W.: Role of conformal three geometry in the dynamics of gravitation. Phys. Rev. Lett. 28, 1082–1085 (1972). https://doi.org/10.1103/PhysRevLett.28.1082
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I thank Félix-Louis Julié and Ryan McManus for useful discussions and the Simons foundation for funding.
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Tenkanen, T. Tracing the high energy theory of gravity: an introduction to Palatini inflation. Gen Relativ Gravit 52, 33 (2020). https://doi.org/10.1007/s10714-020-02682-2
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DOI: https://doi.org/10.1007/s10714-020-02682-2