Abstract
We consider a method for seeking exact solutions of the equation of a nonlocal scalar field in a nonflat metric. In the Friedmann-Robertson-Walker metric, the proposed method can be used in the case of an arbitrary potential except linear and quadratic potentials, and it allows obtaining solutions in quadratures depending on two arbitrary parameters. We find exact solutions for an arbitrary cubic potential, which consideration is motivated by string field theory, and also for exponential, logarithmic, and power potentials. We show that the k-essence field can be added to the model to obtain exact solutions satisfying all the Einstein equations.
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References
K. Ohmori, “A review on tachyon condensation in open string field theories,” arXiv:hep-th/0102085v4 (2001); I. Ya. Aref’eva, D. M. Belov, A. A. Giryavets, A. S. Koshelev, and P. B. Medvedev, “Noncommutative field theories and (super)string field theories,” in: Particles and Fields (G. A. Alves, O. J. P. Éboli, and V. O. Rivelles, eds.), World Scientific, River Edge, N. J. (2002), pp. 1–163; arXiv:hep-th/0111208v2 (2001); W. Taylor, “Lectures on D-branes, tachyon condensation, and string field theory,” in: Lectures on Quantum Gravity (A. Gomberoff and D. Marolf, eds.), Springer, New York (2005), pp. 151–206; arXiv:hep-th/0301094v1 (2003).
I. Ya. Aref’eva, Theor. Math. Phys., 163, 697–704 (2010).
L. Brekke, P. G. O. Freund, M. Olson, and E. Witten, Nucl. Phys. B, 302, 365–402 (1988); P. H. Frampton and Y. Okada, Phys. Rev. D, 37, 3077–3079 (1988); V. S. Vladimirov, I. V. Volovich, and E. I. Zelenov, p-Adic Analysis and Mathematical Physics (Ser. Sov. East European Math., Vol. 10), World Scientific, Singapore (1994); B. Dragovich, A. Yu. Khrennikov, S. V. Kozyrev, and I. V. Volovich, Anal. Appl., 1, 1–17 (2009); arXiv:0904.4205v1 [math-ph] (2009).
I. Ya. Aref’eva, “Nonlocal string tachyon as a model for cosmological dark energy,” in: p-Adic Mathematical Physics (A. Yu. Khrennikov, Z. Rakić, and I. V. Volovich, eds.) (AIP Conf. Proc., Vol. 826), AIP, Melville, N. Y. (2006), pp. 301–311; arXiv:astro-ph/0410443v2 (2004); “Stringy model of cosmological dark energy,” in: Particles, Strings, and Cosmology (AIP Conf. Proc., Vol. 957, A. Rajantie, P. Dauncey, C. Contaldi, and H. Stoica, eds.), AIP, Melville, N. Y. (2007), pp. 297–300; arXiv:0710.3017v1 [hep-th] (2007).
G. Calcagni, JHEP, 0605, 012 (2006); arXiv:hep-th/0512259v4 (2005).
N. Barnaby, T. Biswas, and J. M. Cline, JHEP, 0704, 056 (2007); arXiv:hep-th/0612230v3 (2006); N. Barnaby and J. M. Cline, J. Cosmol. Astropart. Phys., 0707, 017 (2007); arXiv:0704.3426v2 [hep-th] (2007); N. Barnaby, Can. J. Phys., 87, 189–194 (2009); arXiv:0811.0814v1 [hep-th] (2008).
A. S. Koshelev, JHEP, 0704, 029 (2007); arXiv:hep-th/0701103v2 (2007).
I. Ya. Aref’eva, L. V. Joukovskaya, and S. Yu. Vernov, JHEP, 0707, 087 (2007); arXiv:hep-th/0701184v4 (2007).
I. Ya. Aref’eva and I. V. Volovich, Int. J. Geom. Meth. Mod. Phys., 4, 881–895 (2007); arXiv:hep-th/0701284v2 (2007).
J. E. Lidsey, Phys. Rev. D, 76, 043511 (2007); arXiv:hep-th/0703007v1 (2007).
G. Calcagni, M. Montobbio, and G. Nardelli, Phys. Rev. D, 76, 126001 (2007); arXiv:0705.3043v3 [hep-th] (2007); G. Calcagni and G. Nardelli, Phys. Rev. D, 78, 126010 (2008); arXiv:0708.0366v4 [hep-th] (2007); G. Calcagni, M. Montobbio, and G. Nardelli, Phys. Lett. B, 662, 285–289 (2008); arXiv:0712.2237v2 [hep-th] (2007); G. Calcagni and G. Nardelli, Phys. Lett. B, 669, 102–106 (2008); arXiv:0802.4395v2 [hep-th] (2008).
L. V. Joukovskaya, Phys. Rev. D, 76, 105007 (2007); arXiv:0707.1545v3 [hep-th] (2007); “Rolling tachyon in nonlocal cosmology,” in: Particles, Strings, and Cosmology (AIP Conf. Proc., Vol. 957, A. Rajantie, P. Dauncey, C. Contaldi, and H. Stoica, eds.), AIP, Melville, N. Y. (2007), pp. 325–328; arXiv:0710.0404v1 [hep-th] (2007).
N. Barnaby and N. Kamran, JHEP, 0802, 008 (2008); arXiv:0709.3968v3 [hep-th] (2007); JHEP, 0812, 022 (2008); arXiv:0809.4513v1 [hep-th] (2008).
I. Ya. Aref’eva, L. V. Joukovskaya, and S. Yu. Vernov, J. Phys. A, 41, 304003 (2008); arXiv:0711.1364v3 [hep-th] (2007).
D. J. Mulryne and N. J. Nunes, Phys. Rev. D, 78, 063519 (2008); arXiv:0805.0449v2 [hep-th] (2008); “Nonlinear non-local cosmology,” in: The Dark Side of the Universe (AIP Conf. Proc., Vol. 1115, Kh. Shaaban, ed.), Springer, Berlin (2009), pp. 329–334; arXiv:0810.5471v1 [astro-ph] (2008).
L. V. Joukovskaya, JHEP, 0902, 045 (2009); arXiv:0807.2065v1 [hep-th] (2008).
A. S. Koshelev and S. Yu. Vernov, “Cosmological perturbations in SFT inspired non-local scalar field models,” arXiv:0903.5176v2 [hep-th] (2009).
G. Calcagni and G. Nardelli, Internat. J. Mod. Phys. D, 19, 329–338 (2010); arXiv:0904.4245v2 [hep-th] (2009).
S. Yu. Vernov, Class. Q. Grav., 27, 035006 (2010); arXiv:0907.0468v4 [hep-th] (2009); “Localization of the SFT inspired nonlocal linear models and exact solutions,” Phys. Part. Nucl. Lett., 8, No. 3 (2011 in press); arXiv:1005.0372v1 [astro-ph.CO] (2010).
G. Calcagni and G. Nardelli, Phys. Rev. D, 82, 123518 (2010); arXiv:1004.5144v2 [hep-th] (2010).
A. S. Koshelev and S. Yu. Vernov, “Analysis of scalar perturbations in cosmological models with a non-local scalar field,” arXiv:1009.0746v1 [hep-th] (2010).
F. Galli and A. S. Koshelev, Theor. Math. Phys., 164, 1169–1175 (2010); arXiv:1010.1773v1 [hep-th] (2010).
I. Ya. Aref’eva and I. V. Volovich, Theor. Math. Phys., 155, 503–511 (2008); arXiv:hep-th/0612098v1 (2006).
R. Kallosh, J. U. Kang, A. Linde, and V. Mukhanov, J. Cosmol. Astropart. Phys., 0804, 018 (2008); arXiv:0712.2040v3 [hep-th] (2007).
S. Weinberg, Phys. Rev. D, 77, 123541 (2008); arXiv:0804.4291v2 [hep-th] (2008).
J. Z. Simon, Phys. Rev. D, 41, 3720–3733 (1990).
P. Creminelli, G. D’Amico, J. Noreña, and F. Vernizzi, J. Cosmol. Astropart. Phys., 0902, 018 (2009); arXiv:0811.0827v3 [astro-ph] (2008).
I. Ya. Aref’eva, S. Yu. Vernov, and A. S. Koshelev, Theor. Math. Phys., 148, 895–909 (2006); arXiv:astroph/0412619v5 (2004); I. Ya. Aref’eva, A. S. Koshelev, and S. Yu. Vernov, Phys. Lett. B, 628, 1–10 (2005); arXiv:astro-ph/0505605v2 (2005); Phys. Rev. D, 72, 064017 (2005); arXiv:astro-ph/0507067v2 (2005).
I. Ya. Aref’eva and L. V. Joukovskaya, JHEP, 0510, 087 (2005); arXiv:hep-th/0504200v2 (2005).
S. Yu. Vernov, Theor. Math. Phys., 155, 544–556 (2008); arXiv:astro-ph/0612487v3 (2006).
I. Ya. Aref’eva, A. S. Koshelev, and L. V. Joukovskaya, JHEP, 0309, 012 (2003); arXiv:hep-th/0301137v2 (2003).
G. Calcagni and G. Nardelli, Nucl. Phys. B, 823, 234–253 (2009); arXiv:0904.3744v2 [hep-th] (2009).
A. Erdélyi, W. Magnus, F. Oberhettinger, and F. Tricomi, Higher Transcendental Functions, Vol. 3, McGraw-Hill, New York (1955).
C. Armendáriz-Picón, T. Damour, and V. Mukhanov, Phys. Lett. B, 458, 209–218 (1999); arXiv:hep-th/9904075v1 (1999); J. Garriga and V. F. Mukhanov, Phys. Lett. B, 458, 219–225 (1999); arXiv:hep-th/9904176v1 (1999).
N. Arkani-Hamed, H.-C. Cheng, M. A. Luty, and S. Mukohyama, JHEP, 0405, 074 (2004); arXiv:hepth/0312099v1 (2003); N. Arkani-Hamed, P. Creminelli, Sh. Mukohyama, and M. Zaldarriaga, J. Cosmol. Astropart. Phys., 0404, 001 (2004); arXiv:hep-th/0312100v2 (2003).
P. Creminelli, M. A. Luty, Al. Nicolis, and L. Senatore, JHEP, 0612, 080 (2006); arXiv:hep-th/0606090v2 (2006).
C. Armendariz-Picon, V. Mukhanov, and P. J. Steinhardt, Phys. Rev. Lett., 85, 4438–4441 (2000); arXiv:astroph/0004134v1 (2000); Phys. Rev. D, 63, 103510 (2001); arXiv:astro-ph/0006373v1 (2000).
A. D. Rendall, Class. Q. Grav., 23, 1557–1569 (2006); arXiv:gr-qc/0511158v1 (2005).
J. U. Kang, V. Vanchurin, and S. Winitzki, Phys. Rev. D, 76, 083511 (2007); arXiv:0706.3994v2 [gr-qc] (2007).
E. Babichev, V. Mukhanov, and A. Vikman, JHEP, 0802, 101 (2008); arXiv:0708.0561v1 [hep-th] (2007).
A. Vikman, “k-Essence: Cosmology, causality, and emergent geometry,” Doctoral dissertation, edoc.ub.unimuenchen. de/7761, LMU, Munich (2007).
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Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 166, No. 3, pp. 452–464, March, 2011.
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Vernov, S.Y. Exact solutions of nonlocal nonlinear field equations in cosmology. Theor Math Phys 166, 392–402 (2011). https://doi.org/10.1007/s11232-011-0031-0
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DOI: https://doi.org/10.1007/s11232-011-0031-0