Stable exact solutions in cosmological models with two scalar fields

I. Ya. Aref'eva; N. V. Bulatov; S. Yu. Vernov

I. Ya. Aref'eva ; N. V. Bulatov ; S. Yu. Vernov

Teoretičeskaâ i matematičeskaâ fizika, Tome 163 (2010) no. 3, pp. 475-494 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

Résumé

We consider the stability of isotropic solutions for two-field models in the Bianchi I metric. We prove that the sufficient conditions for Lyapunov stability in the Friedmann–Robertson–Walker metric ensure the stability under anisotropic perturbations in the Bianchi I metric and also under perturbations of the energy density for cold dark matter. We find sufficient conditions for the Lyapunov stability of isotropic fixed points for the system of Einstein equations. We use the superpotential method to construct stable kink-type solutions and obtain sufficient conditions on the superpotential for the Lyapunov stability of the corresponding exact solutions. We analyze the stability of isotropic kink-type solutions for models related to string field theory.

Keywords: cosmology, stability, Bianchi metric.

@article{TMF_2010_163_3_a11,
     author = {I. Ya. Aref'eva and N. V. Bulatov and S. Yu. Vernov},
     title = {Stable exact solutions in cosmological models with two scalar fields},
     journal = {Teoreti\v{c}eska\^a i matemati\v{c}eska\^a fizika},
     pages = {475--494},
     year = {2010},
     volume = {163},
     number = {3},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TMF_2010_163_3_a11/}
}

TY  - JOUR
AU  - I. Ya. Aref'eva
AU  - N. V. Bulatov
AU  - S. Yu. Vernov
TI  - Stable exact solutions in cosmological models with two scalar fields
JO  - Teoretičeskaâ i matematičeskaâ fizika
PY  - 2010
SP  - 475
EP  - 494
VL  - 163
IS  - 3
UR  - http://geodesic.mathdoc.fr/item/TMF_2010_163_3_a11/
LA  - ru
ID  - TMF_2010_163_3_a11
ER  -

%0 Journal Article
%A I. Ya. Aref'eva
%A N. V. Bulatov
%A S. Yu. Vernov
%T Stable exact solutions in cosmological models with two scalar fields
%J Teoretičeskaâ i matematičeskaâ fizika
%D 2010
%P 475-494
%V 163
%N 3
%U http://geodesic.mathdoc.fr/item/TMF_2010_163_3_a11/
%G ru
%F TMF_2010_163_3_a11

I. Ya. Aref'eva; N. V. Bulatov; S. Yu. Vernov. Stable exact solutions in cosmological models with two scalar fields. Teoretičeskaâ i matematičeskaâ fizika, Tome 163 (2010) no. 3, pp. 475-494. http://geodesic.mathdoc.fr/item/TMF_2010_163_3_a11/

Bibliographie
Cité par

[1] A. G. Riess et al. (Team Collab.), J. Astrophys., 607:2 (2004), 665–687 ; ; M. Tegmark et al. (SDSS Collab.), J. Astrophys., 606:2 (2004), 702–740 ; ; E. Komatsu et al. (WMAP Collab.), J. Astrophys. Suppl., 180:2 (2009), 330–376 ; ; P. Astier et al., Astron. Astrophys., 447:1 (2006), 31–48 ; ; Shirley Ho, Ch. Hirata, N. Padmanabhan, U. Seljak, N. Bahcall, Phys. Rev. D, 78:4 (2008), 043519 ; ; W. M. Wood-Vasey et al. (ESSENCE Collab.), J. Astrophys., 666:2 (2007), 694–715 ; ; D. Baumann et al. (Team Collab.), “Probing inflation with CMB polarization”, CMB Polarization workshop: theory and foregrounds: CMBPol mission concept, AIP Conf. Proc., 1141, eds. S. Dodelson, D. Baumann, A. Cooray et al., AIP, Melville, NY, 2009, 10–120 ; arXiv: astro-ph/0402512 arXiv: astro-ph/0310725arXiv: 0803.0547arXiv: astro-ph/0510447arXiv: 0801.0642 arXiv: astro-ph/0701041 arXiv: 0811.3919 | DOI | DOI | DOI | DOI | DOI | DOI | DOI

[2] J. D. Barrow, Phys. Rev. D, 55:12 (1997), 7451–7460 ; ; A. Bernui, B. Mota, M. J. Rebouças, R. Tavakol, Astron. Astrophys., 464:2 (2007), 479–485 ; arXiv: gr-qc/9701038arXiv: astro-ph/0511666 | DOI | DOI

[3] U. Alam, V. Sahni, T. D. Saina, A. A. Starobinsky, Mon. Not. Roy. Astron. Soc., 354:1 (2004), 275–291 ; arXiv: astro-ph/0311364 | DOI

[4] S. Nesseris, L. Perivolaropoulos, J. Cosmol. Astropart. Phys., 01 (2007), 018 ; ; Jingfei Zhang, Yuan-Xing Gui, Reconstructing quintom from WMAP 5-year observations: Generalized ghost condensate, arXiv: astro-ph/0610092 arXiv: 0910.1200 | DOI | MR

[5] Yi-Fu Cai, E. N. Saridakis, M. R. Setare, Jun-Qing Xia, Quintom cosmology: theoretical implications and observations, arXiv: 0909.2776 | MR

[6] S. Khoking, Dzh. Ellis, Krupnomasshtabnaya struktura prostranstva-vremeni, Mir, M., 1977 | MR | Zbl

[7] P. J. Steinhardt, N. Turok, Phys. Rev. D, 65:12 (2002), 126003 ; ; M. Gasperini, G. Veneziano, Phys. Rept., 373:1–2 (2003), 1–212 ; arXiv: hep-th/0111098 arXiv: hep-th/0207130 | DOI | DOI | MR

[8] Zong-Kuan Guo, Yun-Song Piao, Xinmin Zhang, Yuan-Zhong Zhang, Phys. Lett. B, 608:3–4 (2005), 177–182 ; arXiv: astro-ph/0410654 | DOI

[9] I. Ya. Aref'eva, A. S. Koshelev, S. Yu. Vernov, Phys. Rev. D, 72:6 (2005), 064017 ; arXiv: astro-ph/0507067 | DOI

[10] R. Lazkoz, G. León, I. Quiros, Phys. Lett. B, 649:2–3 (2007), 103–110 ; arXiv: astro-ph/0701353 | DOI | Zbl | MR

[11] R. Lazkoz, G. León, Phys. Lett. B, 638:4 (2006), 303–309 ; ; G. León, R. Cardenas, J. L. Morales, Equilibrium sets in quintom cosmologies: the past asymptotic dynamics, arXiv: astro-ph/0602590arXiv: 0812.0830 | DOI | MR

[12] J. Sadeghi, M. R. Setare, A. Banijamali, Phys. Lett. B, 678:2 (2009), 164–167 ; ; M. R. Setare, E. N. Saridakis, Phys. Rev. D, 79:4 (2009), 043005 ; ; M. R. Setare, E. N. Saridakis, Internat. J. Modern Phys. D, 18:4 (2009), 549–557 ; ; H. M. Sadjadi, M. Alimohammadi, Phys. Rev. D, 74:4 (2006), 043506 ; ; M. Alimohammadi, Gen. Rel. Grav., 40:1 (2008), 107–115 ; ; Wen Zhao, Yang Zhang, Phys. Rev. D, 73:12 (2006), 123509 ; ; Zong-Kuan Guo, Yun-Song Piao, Xinmin Zhang, Yuan-Zhong Zhang, Phys. Lett. B, 608:3–4 (2005), 177–182 ; ; Xiao-Fei Zhang, Hong Li, Xinmin Zhang, Yun-Song Piao, Modern Phys. Lett. A, 21:3 (2006), 231–241 ; ; H. Štefančić, Phys. Rev. D, 71:12 (2005), 124036 ; ; E. Elizalde, Sh. Nojiri, S. D. Odintsov, Phys. Rev. D, 70:4 (2004), 043539 ; ; Yi-Fu Cai, Hong Li, Yun-Song Piao, Xinmin Zhang, Phys. Lett. B, 646:4 (2007), 141–144 ; ; Yi-Fu Cai, Taotao Qiu, Yun-Song Piao, Mingzhe Li, Xinmin Zhang, JHEP, 10 (2007), 071 ; ; Xin Zhang, Phys. Rev. D, 74:10 (2006), 103505 ; arXiv: 0903.4073arXiv: 0810.4775arXiv: 0807.3807arXiv: gr-qc/0605143arXiv: 0706.1360arXiv: astro-ph/0604460arXiv: astro-ph/0410654arXiv: astro-ph/0501652arXiv: astro-ph/0504518arXiv: hep-th/0405034arXiv: gr-qc/0609039arXiv: 0704.1090arXiv: astro-ph/0609699 | DOI | MR | DOI | MR | DOI | Zbl | DOI | DOI | MR | Zbl | DOI | DOI | DOI | DOI | DOI | DOI | MR | Zbl | DOI | DOI

[13] I. Ya. Arefeva, I. V. Volovich, TMF, 155:1 (2008), 3–12 ; ; R. Kallosh, J. U. Kang, A. Linde, V. Mukhanov, J. Cosmol. Astropart. Phys., 04 (2008), 018 ; arXiv: hep-th/0612098 arXiv: 0712.2040 | DOI | MR | Zbl | DOI | MR

[14] S. Weinberg, Phys. Rev. D, 77:12 (2008), 123541 ; ; J. Z. Simon, Phys. Rev. D, 41:12 (1990), 3720–3733 ; P. Creminelli, G. D'Amico, J. Noreña, F. Vernizzi, J. Cosmol. Astropart. Phys., 02 (2009), 018 ; arXiv: 0804.4291 arXiv: 0811.0827 | DOI | MR | DOI | MR | DOI

[15] Z.-K. Guo, Y.-S. Piao, Y.-Zh. Zhang, Phys. Lett. B, 594:3–4 (2004), 247–251 ; arXiv: astro-ph/0404225 | DOI

[16] I. Ya. Aref'eva, A. S. Koshelev, S. Yu. Vernov, Phys. Lett. B, 628:1–2 (2005), 1–10 ; arXiv: astro-ph/0505605 | DOI

[17] A. A. Starobinskii, Pisma v ZhETF, 37:1 (1983), 55–58 ; V. Müller, H. J. Schmidt, A. A. Starobinsky, Class. Quant. Grav., 7:7 (1990), 1163–1168 | DOI | MR

[18] L. Bianchi, Mem. Soc. Ital. delle Scienze (3), 11 (1898), 267–352 ; Gen. Rel. Grav., 33:12 (2001), 2171–2253 ; L. Bianchi, Lezioni sulla teoria dei gruppi continui finiti di trasformazioni, Spoerri, Pisa, 1918 | Zbl | DOI | MR | Zbl | Zbl

[19] G. F. R. Ellis, H. van Elst, “Cosmological models”, Theoretical and Observational Cosmology. Proceedings of the NATO ASI, NATO Adv. Study Inst. Ser. C, 541, ed. M. Lachièze-Rey, Kluwer, Boston, 1999, 1–116 ; ; G. F. R. Ellis, Gen. Rel. Grav., 38:6 (2006), 1003–1015 arXiv: gr-qc/9812046 | Zbl | DOI | MR | Zbl

[20] J. Wainwright, G. F. R. Ellis (eds.), Dynamical Systems in Cosmology, Cambridge Univ. Press, Cambridge, 1997 | MR | Zbl

[21] C. Germani, A. Kehagias, J. Cosmol. Astropart. Phys., 03 (2009), 028 ; ; T. S. Koivisto, D. F. Mota, C. Pitrou, JHEP, 09 (2009), 092 ; arXiv: 0902.3667arXiv: 0903.4158 | DOI | DOI

[22] R. M. Wald, Phys. Rev. D, 28:8 (1983), 2118–2120 | DOI | MR | Zbl

[23] I. Moss, V. Sahni, Phys. Lett. B, 178:2–3 (1986), 159–162 ; J. Wainwright, L. Hsu, Class. Quant. Grav., 6:10 (1989), 1409–1431 ; A. A. Coley, J. Wainwright, L. Hsu, Class. Quant. Grav., 9:3 (1992), 651–665 ; Yu. Kitada, K. Maeda, Phys. Rev. D, 45:4 (1992), 1416–1419 ; Yu. Kitada, K. Maeda, Class. Quant. Grav., 10:4 (1993), 703–734 ; A. D. Rendall, “Mathematical properties of cosmological models with accelerated expansion”, Analytical and Numerical Approaches to Mathematical Relativity, Lecture Notes Phys., 692, eds. J. Frauendiener, D. J. W. Giulini, V. Perlick, Springer, Berlin, 2006, 141–155 ; arXiv: gr-qc/0408053 | DOI | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl | DOI | MR | Zbl

[24] T. S. Pereira, C. Pitrou, J.-P. Uzan, J. Cosmol. Astropart. Phys., 09 (2007), 006 ; arXiv: 0707.0736 | DOI

[25] I. Ya. Aref'eva, N. V. Bulatov, L. V. Joukovskaya, S. Yu. Vernov, Phys. Rev. D, 80:8 (2009), 083532 ; arXiv: 0903.5264 | DOI | MR

[26] A. M. Lyapunov, Obschaya zadacha ob ustoichivosti dvizheniya, GITTL, M.–L., 1950 ; Л. С. Понтрягин, Обыкновенные дифференциальные уравнения, Наука, М., 1982 | MR | MR | Zbl | MR | Zbl

[27] S. Yu. Vernov, TMF, 155:1 (2008), 47–61 ; arXiv: astro-ph/0612487 | DOI | MR | Zbl

[28] J. Wainwright, W. C. Lim, J. Hyperbolic Differ. Equat., 2:2 (2005), 437–469 ; arXiv: gr-qc/0409082 | DOI | MR | Zbl

[29] O. DeWolfe, D. Z. Freedman, S. S. Gubser, A. Karch, Phys. Rev. D, 62:4 (2000), 046008 ; arXiv: hep-th/9909134 | DOI | MR

[30] A. G. Muslimov, Class. Quant. Grav., 7:2 (1990), 231–237 ; D. S. Salopek, J. R. Bond, Phys. Rev. D, 42:12 (1990), 3936–3962 ; A. R. Liddle, D. H. Lyth, Cosmological Inflation and Large-Scale Structure, Cambridge Univ. Press, Cambridge, 2000 ; D. Bazeia, M. J. dos Santos, R. F. Ribeiro, Phys. Lett. A, 208:1–2 (1995), 84–88 ; ; A. Brandhuber, K. Sfetsos, JHEP, 10 (1999), 013 ; ; D. Bazeia, C. B. Gomes, L. Losano, R. Menezes, Phys. Lett. B, 633:4–5 (2006), 415–419 ; ; K. Skenderis, P. K. Townsend, Phys. Rev. D, 74:12 (2006), 125008 ; ; D. Bazeia, L. Losano, J. J. Rodrigues, First-order formalism for scalar field in cosmology, ; A. S. Mikhailov, Yu. S. Mikhailov, M. N. Smolyakov, I. P. Volobuev, Class. Quant. Grav., 24:1 (2007), 231–242 ; ; P. K. Townsend, Class. Quant. Grav., 25:4 (2008), 045017 ; arXiv: hep-th/0311265 arXiv: hep-th/9908116 arXiv: astro-ph/0512197 arXiv: hep-th/0609056 arXiv: hep-th/0610028 arXiv: hep-th/0602143 arXiv: 0710.5178 | DOI | MR | DOI | MR | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | Zbl | DOI | MR | MR | DOI | MR | Zbl | DOI | MR | Zbl

[31] I. Ya. Arefeva, S. Yu. Vernov, A. S. Koshelev, TMF, 148:1 (2006), 23–41 ; arXiv: astro-ph/0412619 | DOI | MR | Zbl

[32] I. Ya. Aref'eva, “Nonlocal string tachyon as a model for cosmological dark energy”, $p$-Adic Mathematical Physics, AIP Conf. Proc., 826, eds. A. Yu. Khrennikov, Z. Rakić, I. V. Volovich, AIP, Melville, NY, 2006, 301–311 ; ; I. Ya. Aref'eva, “Stringy model of cosmological dark energy”, Particles, Strings, and Cosmology, AIP Conf. Proc., 957, eds. A. Rajantie, P. Dauncey, C. Contaldi, H. Stoica, AIP, Melville, NY, 2007, 297–300 ; arXiv: astro-ph/0410443 arXiv: 0710.3017 | DOI | MR | Zbl | DOI | MR | Zbl

[33] I. Ya. Arefeva, TMF, 163:3 (2010), 355–365 | DOI | MR | Zbl

[34] I. Ya. Aref'eva, L. V. Joukovskaya, JHEP, 10 (2005), 087 ; arXiv: hep-th/0504200 | DOI | MR

[35] I. Ya. Aref'eva, A. S. Koshelev, D. M. Belov, P. B. Medvedev, Nucl. Phys. B, 638:1–2 (2002), 3–20 ; arXiv: hep-th/0011117 | DOI | MR | Zbl

[36] I. Ya. Aref'eva, D. M. Belov, A. A. Giryavets, A. S. Koshelev, P. B. Medvedev, “Noncommutative field theories and (super)string field theories”, Particles and Fields, eds. G. A. Alves, O. J. P. Eboli, V. O. Rivelles, World Sci., Singapore, 2002, 1–163 ; arXiv: hep-th/0111208 | DOI | MR | Zbl

[37] I. Ya. Aref'eva, L. V. Joukovskaya, S. Yu. Vernov, J. Phys. A, 41:30 (2008), 304003 ; ; D. J. Mulryne, N. J. Nunes, Phys. Rev. D, 78:6 (2008), 063519 ; ; S. Yu. Vernov, Class. Quant. Grav., 27:3 (2010), 035006 ; arXiv: 0711.1364 arXiv: 0805.0449 arXiv: 0907.0468 | DOI | MR | Zbl | DOI | MR | DOI | MR | Zbl

[38] I. Ya. Aref'eva, A. S. Koshelev, L. V. Joukovskaya, JHEP, 09 (2003), 012 ; arXiv: hep-th/0301137 | DOI | MR

[39] K. Ohmori, Phys. Rev. D, 69:2 (2004), 026008 ; arXiv: hep-th/0306096 | DOI | MR

[40] B. Zwiebach, Ann. Phys., 267:2 (1998), 193–248 ; arXiv: hep-th/9705241 | DOI | MR | Zbl

Parcourir par

Geodesic

Parcourir par