Abstract
We study the early inflationary phase of the universe driven by noncanonical scalar field models using an exponential potential. The noncanonical scalar field models are represented by Lagrangian densities containing square and square-root kinetic corrections to the canonical Lagrangian density. We investigate the Lie symmetry of the homogeneous scalar field equations obtained from noncanonical Lagrangian densities and find only a one-parameter Lie point symmetry for both canonical and noncanonical scalar field equations. We use the Lie symmetry generator to obtain an exact analytical group-invariant solution of the homogeneous scalar field equations from an invariant curve condition. The solutions obtained are consistent and satisfy the Friedmann equations subject to constraint conditions on the potential parameter \(\lambda\) for the canonical case and on the parameter \(\mu\) for the noncanonical case. In this scenario, we obtain the values of various inflationary parameters and make useful checks on the observational constraints on the parameters from Planck data by imposing a set of bounds on the parameters \(\lambda\) and \(\mu\). The results for the scalar spectral index (\(n_{S}\)) and the tensor-to-scalar ratio (\(r\)) are presented in the \((n_{S},r)\) plane in the background of Planck-2015 and Planck-2018 data for noncanonical cases and are in good agreement with cosmological observations. For theoretical completeness of the noncanonical models, we verify that the noncanonical models under consideration are free from ghosts and Laplacian instabilities. We also treat the noncanonical scalar field model equations for two power-law (kinetic) corrections by the dynamical system theory. We provide useful checks on the stability of the critical points for both cases and show that the group-invariant analytical noncanonical inflation solutions are stable attractors in phase space.
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REFERENCES
J. Martin, C. Ringeval, and V. Vennin, arXiv: 1303.3787v3.
E. W. Kolb and M. S. Turner, The Early Universe (New York: Addison-Wesley, 1990).
A. D. Linde, Particle Physics and Inflationary Cosmology (Harwood Academic Publishers, Chur, Switzerland, 1990); arXiv: hep-th/0503203.
A. Liddle and D. Lyth, Cosmological Inflation and Large-Scale Structure (Cambridge University Press, 2000).
D. H. Lyth and A. R. Liddle, The Primordial Density Perturbation (Cambridge University Press, 2009).
P. A. R. Ade et al., arXiv: 1502.01592.
Planck 2015 results. XX. Constraints on inflation, arxiv: 1502.02114.
Planck 2018 results. X. Constraints on inflation, arXiv: 1807.06211.
M. Bairagi and A. Choudhuri, Eur. Phys. J. Plus 133, 545 (2018).
A. H. Chamseddine, A. Connes, and M. Marcolli, Adv. Theor. Math. Phys. 11, 991 (2007).
A. Paliathanasis, S. Pan, and S. Pramanik, Class. Quantum Grav. 32, 245006 (2015);
A. Paliathanasis, S. Pan, and S. Pramanik, Class. Quantum Grav. 32, 245006 (2015); S. Das and E. C. Vagenas, Phys. Rev. Lett. 101, 221301 (2008).
E. Silverstein and D. Tong, Phys. Rev. D 70, 103505 (2004).
M. Alishahiha, E. Silverstein and D. Tong, Phys. Rev. D 70, 123505 (2004).
C. Armendariz-Picon, T. Damour, and V. F. Mukhanov, Phys. Lett. B 458, 209 (1999).
R. Gwyn, M. Rummel and A. Westphal, JCAP 12, 010 (2013).
C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt, Phys. Rev. D 63, 103510 (2001).
S. Weinberg, Cosmology (Oxford University Press Inc., New York, 2008).
A. H. Guth, Phys. Rev. D 23, 347 (1981).
V. F. Mukhanov, Physical Foundations of Cosmology (Cambridge University Press, 2005).
D. H. Lyth and A. Riotto, Phys. Rep. 314, 1 (1999).
J. A. Stein-Schabas, Phys. Rev. D 35, 2345 (1987).
S. Unnikrishnan, V. Sahni, and A. Toporensky, JCAP 08, 018 (2012).
S. Unnikrishnan and V. Sahni, JCAP 10, 063 (2013).
S. V. Sushkov, Phys. Rev. D 85, 123520 (2012).
Planck 2015 results. XIII. Cosmological parameters, arxiv:1502.01589.
K. Rezazadeh and K. Karami, JCAP 09, 053 (2015).
S. V. Sushkov, Phys. Rev. D 80, 103505 (2009).
E. N. Saridakis and S. V. Sushkov, Phys. Rev. D 81, 083510 (2010).
S. Sushkov and R. Korolev, Class. Quant. Grav. 29, 085008 (2012).
M. Gasperini and G. Veneziano, Phys. Rept. 373, 1 (2003).
F. Piazza and S. Tsujikawa, JCAP 0407, 004 (2004).
S. Tsujikawa and M. Sami, Phys. Lett. B 603, 113 (2004).
N. Tamanini, Phys. Rev. D 89, 083521 (2014).
Yi-Fu Cai, J. B. Dent, and D. A. Easson, Phys. Rev. D 83, 101301(R) (2011).
M. Szydlowski, O. Hrycyna and A. Stachowski, IJGMMP 11, 1460012, (2014).
B. Chetry, J. Dutta, and W. Khyllep, Int. J. Mod. Phys. D 28, 1950163 (2019),
F. Lucchin and S. Matarrese, Phys. Rev. D 32, 1316 (1985).
L. F. Abbott and M. B. Wise, Nucl. Phys. B244, 541 (1987).
J. D. Barrow, Phys. Lett. B 235, 40 (1990).
J. D. Barrow and A. R. Liddle, Phys. Rev. D 47, 5219 (1993).
A. Linde, Phys. Rev. D 49, 748 (1994).
J. J. Halliwell, Phys. Lett. B 185, 341 (1987).
J. D. Barrow, Phys. Lett. B 187, 12 (1987).
P. J. Olver, Applications of Lie Groups to Differential Equations (New York: Springer, 1993); Amitava Choudhuri, Physica Scripta 90, 055004 (2015);
K. Andriopoulos and P. G. L. Leach, Cent. Eur. J. Phys. 6, 469 (2008);
P. J. Olver, Applications of Lie Groups to Differential Equations (New York: Springer, 1993); Amitava Choudhuri, Physica Scripta 90, 055004 (2015); K. Andriopoulos and P. G. L. Leach, Cent. Eur. J. Phys. 6, 469 (2008); Amitava Choudhuri, Nonlinear Evolution Equations: Lagrangian Approach (LAMBERT Academic Publishing, 2011).
H. Stephani, Differential Equations: Their Solution Using Symmetries, Ed. M. MacCallum (CUP, Cambridge, 1990); G.W. Bluman and S. Kumei, Symmetries and Differential Equations (Springer NY, 1989).
X. Chen, M. Huang, S. Kachru and G. Shiu, JCAP 0701, 002 (2007).
V. Mukhanov and A. Vikman, JCAP 02, 004 (2006).
D. Blokhinzev, Space and Tme in Micro-World (Nauka, Moscow, 1970, in Russian).
J. Garriga and V. F. Mukhanov, Phys. Lett. B 458, 219 (1999).
E. F. Bunn, A. R. Liddle, and M. J. White, Phys. Rev. D 54, R5917 (1996).
B. A. Bassett et al., Rev. Mod. Phys. 78, 537 (2006).
U. Seljak et al., Phys. Rev. D 71, 103515 (2005).
I. Torres, J. C. Fabris, and O. F. Piattella, Phys. Lett. B 798, 135003 (2019).
A. De Felice and S. Tsujikawac, JCAP 02 007 (2012).
V. Mukhanov and G. Chibisov, JETP Lett. 33, 532 (1981).
A. A. Starobinsky, JETP Lett. 30, 682 (1979).
R. H. Brandenberger and A. R. Zhitnitsky, Phys. Rev. D 55, 4640 (1997).
A. Bedroya and C. Vafa, arXiv: 1909.11063.
A. Bedroya, R. Brandenberger, M. Loverde, and C. Vafa, Phys. Rev. D 101, 103502 (2020).
S. Brahma, Phys. Rev. D 101, 023526 (2020).
R. H. Brandenberger and C. Vafa, Nucl. Phys. B 316, 391 (1989);
R. H. Brandenberger and C. Vafa, Nucl. Phys. B 316, 391 (1989); S. Laliberte and R. H. Brandenberger, arXiv: 1911.00199.
W. L. K. Wu et al., J. Low. Temp. Phys. 184, 765 (2016).
T. Matsumura et al., J. Low. Temp. Phys. 176, 733 (2014).
R. Brandenberger and E. Wilson-Ewing, JCAP 03, 047 (2020).
P. Agrawal, G. Obied, P. J. Steinhardt, and C. Vafa, Phys. Lett. B 784, 271 (2018).
R. R. Caldwell, M. Kamionkowski, and N. N. Weinberg, Phys. Rev. Lett. 91, 071301 (2003).
J. Maldacena, JHEP 0305, 013 (2003).
F. Oliveri, Symmetry 2, 658 (2010).
P. E. Hydon, Symmetry Methods for Differential Equations (CUP, Cambridge, 2000).
K. T. Alligood, T. Sauer, and J. A. Yorke, Chaos: An Introduction to Dynamical Systems (Springer-Verlag, NewYork, 1997).
A. A. Coley: arXiv: gr-qc/9910074.
J. Matsumoto and S. V. Sushkov, JCAP 01, 040 (2018).
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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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Bairagi, M., Choudhuri, A. Early Inflationary Phase with Canonical and Noncanonical Scalar Fields: A Symmetry-Based Approach. Gravit. Cosmol. 26, 326–350 (2020). https://doi.org/10.1134/S0202289320040027
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DOI: https://doi.org/10.1134/S0202289320040027