Abstract
In this work, we study cosmological solutions of the 8-dimensional Einstein Yang-Mills theory coupled to a perfect-fluid matter. A Yang-Mills instanton of extra dimensions causes a 4-dimensional expanding universe with dynamical compactification of the extra dimensions. To construct physically reliable situations, we impose the null energy condition on the matter. This energy condition is affected by the extra dimensions. Then, we consider cosmological constant to grasp the structure of the solution space. Even in this simple case, we find several interesting solutions, such as bouncing universes and oscillatory solutions, eventually arriving at a de Sitter universe with stabilized compact dimensions. In addition, we consider a class of matters whose energy density depends on the volume of the extra dimensions. This case shows another set of bouncing universes. Also, a real scalar with potential is taken into account. The scalar field model admits de Sitter solutions due to the choice of potential, and we demonstrate how potentials can be constructed using flow equations. Thus, what we discuss in this work is based on the 8-dimensional Einstein frame, which corresponds to the 4-dimensional Jordan frame by dimensional reduction. Consequently, the results are derived in the 4-dimensional Jordan frame, not in the 4-dimensional Einstein frame.
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References
J. Polchinski, String theory. Vol. 1: An introduction to the bosonic string, Cambridge University Press (2007) [https://doi.org/10.1017/CBO9780511816079] [INSPIRE].
M.B. Green, J.H. Schwarz and E. Witten, Superstring theory. Vol. 1: introduction, Cambridge University Press (1988) [INSPIRE].
N. Arkani-Hamed, S. Dimopoulos and G.R. Dvali, Phenomenology, astrophysics and cosmology of theories with submillimeter dimensions and TeV scale quantum gravity, Phys. Rev. D 59 (1999) 086004 [hep-ph/9807344] [INSPIRE].
C. Csaki, M. Graesser, L. Randall and J. Terning, Cosmology of brane models with radion stabilization, Phys. Rev. D 62 (2000) 045015 [hep-ph/9911406] [INSPIRE].
S.-W. Kim, J. Nishimura and A. Tsuchiya, Expanding universe as a classical solution in the Lorentzian matrix model for nonperturbative superstring theory, Phys. Rev. D 86 (2012) 027901 [arXiv:1110.4803] [INSPIRE].
S.-W. Kim, J. Nishimura and A. Tsuchiya, Late time behaviors of the expanding universe in the IIB matrix model, JHEP 10 (2012) 147 [arXiv:1208.0711] [INSPIRE].
T. Aoki et al., On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model, PTEP 2019 (2019) 093B03 [arXiv:1904.05914] [INSPIRE].
J. Nishimura, New perspectives on the emergence of (3+1)D expanding space-time in the Lorentzian type IIB matrix model, PoS CORFU2019 (2020) 178 [arXiv:2006.00768] [INSPIRE].
K.K. Kim, S. Koh and H.S. Yang, Expanding Universe and Dynamical Compactification Using Yang-Mills Instantons, JHEP 12 (2018) 085 [arXiv:1810.12291] [INSPIRE].
Q. Shafi and C. Wetterich, Cosmology from Higher Dimensional Gravity, Phys. Lett. B 129 (1983) 387 [INSPIRE].
S. Randjbar-Daemi, A. Salam and J.A. Strathdee, On Kaluza-Klein Cosmology, Phys. Lett. B 135 (1984) 388 [INSPIRE].
E.W. Kolb, D. Lindley and D. Seckel, More Dimensions - Less Entropy, Phys. Rev. D 30 (1984) 1205 [INSPIRE].
Y. Okada, Inflation in Kaluza-Klein Cosmology, Phys. Lett. B 150 (1985) 103 [INSPIRE].
K.-I. Maeda, Cosmological Solutions With Calabi-yau Compactification, Phys. Lett. B 166 (1986) 59 [INSPIRE].
F.S. Accetta, M. Gleiser, R. Holman and E.W. Kolb, Stable Compactifications, Nucl. Phys. B 276 (1986) 501 [INSPIRE].
E.W. Kolb and M.S. Turner, The Early Universe, Front. Phys. 69 (1990) 1 [INSPIRE].
A. Mazumdar, Extra dimensions and inflation, Phys. Lett. B 469 (1999) 55 [hep-ph/9902381] [INSPIRE].
P.J. Steinhardt and D. Wesley, Dark Energy, Inflation and Extra Dimensions, Phys. Rev. D 79 (2009) 104026 [arXiv:0811.1614] [INSPIRE].
Z. Horvath and L. Palla, Spontaneous Compactification and ‘Monopoles’ in Higher Dimensions, Nucl. Phys. B 142 (1978) 327 [INSPIRE].
S. Randjbar-Daemi, A. Salam and J.A. Strathdee, Spontaneous Compactification in Six-Dimensional Einstein-Maxwell Theory, Nucl. Phys. B 214 (1983) 491 [INSPIRE].
A. Salam and E. Sezgin, Chiral Compactification on Minkowski x S2 of N=2 Einstein-Maxwell Supergravity in Six-Dimensions, Phys. Lett. B 147 (1984) 47 [INSPIRE].
S. Randjbar-Daemi, A. Salam and J.A. Strathdee, Instanton Induced Compactification and Fermion Chirality, Phys. Lett. B 132 (1983) 56 [INSPIRE].
P.H. Frampton, P.J. Moxhay and K. Yamamoto, Results on stability of instanton-induced compactification in eight dimensions, Phys. Lett. B 144 (1984) 354.
H. Kihara et al., Dynamical Compactification and Inflation in Einstein-Yang-Mills Theory with Higher Derivative Coupling, Phys. Rev. D 80 (2009) 066004 [arXiv:0906.4493] [INSPIRE].
E. O Colgain and I. Zaballa, Compactification driven Hilltop Inflation in Einstein-Yang-Mills, Phys. Rev. D 81 (2010) 083504 [arXiv:0912.3349] [INSPIRE].
K.K. Kim, S. Koh, J. Ho and H.S. Yang, to appear.
J.M. Maldacena and C. Nunez, Supergravity description of field theories on curved manifolds and a no go theorem, Int. J. Mod. Phys. A 16 (2001) 822 [hep-th/0007018] [INSPIRE].
C.G. Boehmer and N. Chan, Dynamical systems in cosmology, arXiv:1409.5585 [https://doi.org/10.1142/9781786341044_0004] [INSPIRE].
M. Novello and S.E.P. Bergliaffa, Bouncing Cosmologies, Phys. Rept. 463 (2008) 127 [arXiv:0802.1634] [INSPIRE].
D. Battefeld and P. Peter, A Critical Review of Classical Bouncing Cosmologies, Phys. Rept. 571 (2015) 1 [arXiv:1406.2790] [INSPIRE].
M. Lilley and P. Peter, Bouncing alternatives to inflation, Comptes Rendus Physique 16 (2015) 1038 [arXiv:1503.06578] [INSPIRE].
R. Brandenberger and P. Peter, Bouncing Cosmologies: Progress and Problems, Found. Phys. 47 (2017) 797 [arXiv:1603.05834] [INSPIRE].
K.K. Kim, S. Koh and G. Tumurtushaa, in progress.
Acknowledgments
We thank Hyun Seok Yang, Seokcheon Lee, Miok Park, and Yunseok Seo for their helpful discussions. This work is supported by Mid-career Research Program through NRF grant No. NRF-2019R1A2C1007396 (K. Kim) and No. NRF-2021R1A2C1005748 (S. Koh and G. Tumurtushaa). This work is also supported by Basic Research Program through NRF grant No. 2022R1I1A1A01053784 (G. Tumurtushaa).
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Kim, K.K., Koh, S. & Tumurtushaa, G. Dynamical Compactification with Matter. J. High Energ. Phys. 2023, 181 (2023). https://doi.org/10.1007/JHEP06(2023)181
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DOI: https://doi.org/10.1007/JHEP06(2023)181