Abstract
The inflationary model based on a single scalar field with a quadratic potential \(V(\phi )\sim \phi ^{2}\) is disfavored by the recent Planck constraints on the values of scalar spectral index, and the tensor-to-scalar ratio for cosmological density perturbations. In order to overcome the discrepancies we study the scalar field equations based on the minimally coupled and canonical Lagrangian densities with negative quadratic potential and modified Higgs-like potential. We use a Lie symmetry-based approach to study the homogeneous scalar field equations for both cases. In particular, we have investigated the Lie symmetries of the scalar field equations and use them to find the exact analytical solutions. New exact analytical solution is obtained for an inflationary model with the modified Higgs-like potential. We have calculated the values of the inflationary parameters, namely the amplitude of scalar power-spectrum \((\mathcal {P}_{S})\), scalar spectral index (nS), its running (nSrun), tensor-to-scalar ratio (r) and non-Gaussinity parameters. We make useful checks whether the obtained parameters are supported by the observational constraints set by the Planck2018 data. We find that the model equation with negative quadratic potential is disfavored by the recent Planck2018 constraints on the tensor-to-scalar ratio for cosmological density perturbations but the scalar field model equation with the modified Higgs-like potential resolves the problem. We have predicted the values of Higgs self-coupling constant (λ) that lies within the \(3.924\times 10^{-14}\lesssim \lambda \lesssim 6.56\times 10^{-14}\) and also the values of vacuum expectation value of the scalar inflaton field lies in the range \(17.17\lesssim \phi _{0}\lesssim 23.54\) which give best fit the Planck2018 measured range of the scalar spectral index nS and upper bound of tensor-to-scalar ratio r. The small value of λ indicates that the inflaton field is extremely weakly coupled and gives the necessary condition for successful inflation. The values of tensor-to-scalar ratio and all the other important observable indices obtained using the model with the modified Higgs-like potential lie well inside the limits set in by the Planck2018 data. But we have checked that both the models generate very small values of non-Gaussianity parameters. We also explain the exit from the inflationary phase in both the cases successfully.
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References
Weinberg, S.: Cosmology. Oxford University Press Inc, New York (2008)
Rocher, J., Sakellariadou, M.: . J. Cosmol. Astropart. Phys. 03, 004 (2005)
Guth, A.H.: . Phys. Rev. D 23, 347 (1981)
Kolb, E.W., Turner, M.S.: The Early Universe. Addison-Wesley, New York (1990)
Bairagi, M., Choudhuri, A.: . Eur. Phys. J. Plus 133, 545 (2018)
Maartens, R., Taylor, D.R., Roussos, N.: . Phys. Rev. D 52, 3358 (1995)
Linde, A.D.: . Phys. Lett. B 108, 389 (1982)
Albrecht, A., Steinhardt, P.: . Phys. Rev. Lett. 48, 1220 (1982)
Coleman, S., Weinberg, S.: . Phys. Rev. D 7, 1888 (1973)
Linde, A.D.: . Phys. Lett. B 129, 177 (1983)
Kolb, E.W.: First-order inflation. FNAL-CONF-90/195 (1990)
Dodelson, S.: Modern Cosmology. Academic Press, San Diego (2003)
Mukhanov, V.F.: Physical Foundations of Cosmology. Cambridge University Press, Cambridge (2005)
Turner, M.S.: . Acta Phys. Polon. B18, 813 (1987)
Kallosh, R., Linde, A.: arXiv:1909.04687v2 [hep-th] 26 Sep (2019)
Guth, A.H., Kaiser, D.I., Nomura, Y.: . Phys. Lett. B 733, 112 (2014)
Halliwell, J.J.: . Phys. Lett. B 185, 341 (1987)
Tsamparlis, M., Paliathanasis, A.: . Gen. Rel. Grav. 42, 2957 (2010)
Tsamparlis, M., Paliathanasis, A.: . J. Phys. A 44, 175202 (2011)
Szydlowski, M., Hrycyna, O., Stachowski, A.: . IJGMMP 11, 1460012 (2014)
Fedler, G., Frolov, A., Kofman, L., Linde, A.D.: Cosmology with Negative potentials. Phys. Rev. D 66, 023507 (2002). arXiv:0202017
Ellis, J., Fairbairn, M., Sueiroa, M.: . JCAP 02, 044 (2014)
Urena-Lopez, L.A., Reyes-Ibarra, M.J.: . Int. J. Mod. Phys. D 18, 621 (2009)
Urena-Lopez, L.A.: . J. Phys.: Conf. Ser. 761, 012076 (2016)
Oda, I., Tomoyose, T.: . Adv. Studies Theor. Phys. 8, 551 (2014)
Harigayaa, K., Ibe, M., Kawasaki, M., Yanagida, T.T.: . Phys. Lett. B 756, 113 (2016)
Wen-Fu, W.: . Chin. Phys. Lett. 20, 593 (2003)
Steinhardt, P.J., Turok, N.: arXiv:0111030 (2020)
Komatsu, E., et al: arXiv:1001.4538 [astro-ph.CO] (2020)
Linde, A.D.: arXiv:1402.0526v2 [hep-th] 9 Mar (2014)
Okada, N., Rehman, M.U., Sha, Q.: . Phys. Rev. D 82, 043502 (2010). arXiv:1005.5161 [hep-ph]
Fomin, I.V., Chervon, S.V.: . J. Phys. Conf. Ser. 1557, 012020 (2020)
Aad, G., et al: [ATLAS Collaboration]. Phys. Lett. B 716, 1 (2012). arXiv:1207.7214 [hep-ex]
Chatrchyan, S., et al: [CMS Collaboration]. Phys. Lett. B 716, 30 (2012). arXiv:1207.7235 [hep-ex]
Linde, A.: arXiv:1710.04278v1 [hep-th] 11 Oct (2017)
Olver, P.J.: Applications of Lie Groups to Differential Equations. Springer, New York (1993)
Stephani, H. In: MacCallum, M. (ed.) : Differential Equations: Their Solution Using Symmetries. Cambridge, Cambridge University Press (1990)
Bluman, G.W., Kumei, S.: Symmetries and Differential Equations. Springer, New York (1989)
Choudhuri, A.: . Physica Scripta 90, 055004 (2015)
Andriopoulos, K., Leach, P.G.L.: . Cent. Eur. J. Phys. 6, 469 (2008)
Choudhuri, A., Ganguly, A.: . Found. Phys. 1, 49 (2019)
Ganguly, A., Choudhuri, A.: . Gravity and Cosmology 26, 228 (2020)
Chervon, S.V., Fomin, I.V., Beesham, A.: . Eur. Phys. J. C 78, 301 (2018)
Ivanov, G.G., Chervon, S.V., Khapaeva, A.V.: . Space, Time and Fundamental Interactions 3, 66 (2020)
Muslimov, A.G.: . Class. Quant. Grav. 7, 231 (1990)
Lucchin, F., Matarrese, S.: . Phys. Rev. D 32, 1316 (1985)
Ellis, G.F.R., Madsen, M.S.: . Class. Quant. Grav. 8, 667 (1991)
Barrow, J.D.: . Phys. Lett. B 187, 12 (1987)
Bairagi, M., Choudhuri, A.: . Gravit. Cosmol. 26, 326 (2020)
Bairagi, M., Choudhuri, A.: . Gen. Relativ. Gravit. 53, 1 (2021)
Akrami, Y., et al.: Planck 2018 results. X. Constraints on inflation arXiv:1807.06211v1 [astro-ph.CO] 17 Jul (2018)
Choudhuri, A.: Nonlinear evolution equations: lagrangian approach LAP LAMBERT academic publishing (2011)
Bassett, B.A., Tsujikawa, S., Wands, D.: . Rev. Mod. Phys. 78, 537 (2006)
Freese, K., Frieman, J.A., Orinto, A.V.: . Phys. Rev. Lett. 65, 3233 (1990)
Coone, D., Roest, D., Vennin, V.: . JCAP 1511, 010 (2015). arXiv:1507.00096
Schwarz, D.J., Terrero-Escalante, C.A., Garcia, A.A.: . Phys. Lett. B 517, 243 (2001). arXiv:0106020
Baumann, D.: The Physics of Inflation ICTS course (2011)
Riotto, A.: Inflation and the Theory of Cosmological Perturbations. arXiv:0210162 (2002)
Croon, D., Gonzalo, T.E., Graf, L., Košnik, N., White, G.: . Frontiers in Physics 7, 1 (2019)
Jimenez, J.B., Musso, M., Ringeval, C.: . Phys. Rev. D 88, 043524 (2013). arXiv:1303.2788 [astro-ph.CO]
Wagenaa, L.: Inflation, Quantum Fluctuations and Gravitational Waves. University of Amsterdam, Thesis (2016)
Kosowsky, A., Turner, M.S.: . Phys. Rev. D 52, 1739 (1995)
Maldacena, J.: . JHEP 0305, 013 (2003)
Komatsu, E., Spergel, D.N.: . Phys. Rev. D 063002, 63 (2001)
Asadi, K., Nozari, K.: The size of local bispectrum and trispectrum in a Non-Minimal inflation, nucl. Phys. B 934, 118 (2018)
Byrnes, C.T., Sasaki, M., Wands, D.: Primordial trispectrum from inflation. Phys. Rev. D 74, 123519 (2006). arXiv:0611075
Bunn, E.F., Liddle, A.R., White, M.J.: . Phys. Rev. D 54, R5917 (1996)
Barry, D.J., Culligen-Hensley, P.J., Barry, S.J.: Real Values of the W Function. ACM Trans. Math. Software 21, 161–171 (1995)
Guth, A.: . Proc. Natl. Acad. Sci. USA 90, 4871 (1993)
Chakravarty, G.K., Das, S., Lambiase, G., Mohanty, S.: . Phys. Rev. D 94, 023521 (2016)
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AC acknowledges UGC, The Government of India, for financial support through Project No.F.30-302/2016(BSR).
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Choudhuri, A., Bairagi, M. The Minimally Coupled and Canonical Scalar Field Inflationary Cosmology with Negative Quadratic and Modified Higgs-like Potentials: A Symmetry Based Approach. Int J Theor Phys 61, 158 (2022). https://doi.org/10.1007/s10773-022-05146-2
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DOI: https://doi.org/10.1007/s10773-022-05146-2