Abstract
The IKKT matrix model yields an emergent space-time. We further develop these ideas and give a proposal for an emergent metric. Based on previous numerical studies of this model, we provide evidence that the emergent space-time is continuous and infinite in extent, both in space and in time, and that the metric is spatially flat. The time evolution describes the transition from a string-theoretic emergent phase to a phase in which the SO(9) symmetry of the model is spontaneously broken to SO(6) × SO(3), with three dimensions of space expanding, becoming classical and at later times evolving like in a radiation-dominated universe, and the remaining six dimensions of space stabilized at the string scale. We speculate on how this analysis can be extended to yield an early universe cosmology which, in addition to the above-mentioned properties, also leads to a roughly scale-invariant spectrum of cosmological fluctuations and gravitational waves.
Article PDF
Similar content being viewed by others
References
N. Weiss, Constraints on Hamiltonian Lattice Formulations of Field Theories in an Expanding Universe, Phys. Rev. D 32 (1985) 3228 [INSPIRE].
J. Cotler and A. Strominger, The Universe as a Quantum Encoder, arXiv:2201.11658 [INSPIRE].
A. Bedroya and C. Vafa, Trans-Planckian Censorship and the Swampland, JHEP 09 (2020) 123 [arXiv:1909.11063] [INSPIRE].
A. Bedroya, R. Brandenberger, M. Loverde and C. Vafa, Trans-Planckian Censorship and Inflationary Cosmology, Phys. Rev. D 101 (2020) 103502 [arXiv:1909.11106] [INSPIRE].
R. Brandenberger, String cosmology and the breakdown of local effective field theory, Nuovo Cim. C 45 (2022) 40 [arXiv:2112.04082] [INSPIRE].
R. Brandenberger and V. Kamali, Unitarity Problems for an Effective Field Theory Description of Early Universe Cosmology, arXiv:2203.11548 [INSPIRE].
A.H. Guth, The Inflationary Universe: A Possible Solution to the Horizon and Flatness Problems, Phys. Rev. D 23 (1981) 347 [INSPIRE].
R. Brout, F. Englert and E. Gunzig, The Creation of the Universe as a Quantum Phenomenon, Annals Phys. 115 (1978) 78 [INSPIRE].
A.A. Starobinsky, A New Type of Isotropic Cosmological Models Without Singularity, Phys. Lett. B 91 (1980) 99 [INSPIRE].
K. Sato, First Order Phase Transition of a Vacuum and Expansion of the Universe, Mon. Not. Roy. Astron. Soc. 195 (1981) 467 [INSPIRE].
R.H. Brandenberger, Alternatives to the inflationary paradigm of structure formation, Int. J. Mod. Phys. Conf. Ser. 01 (2011) 67 [arXiv:0902.4731] [INSPIRE].
R. Brandenberger and P. Peter, Bouncing Cosmologies: Progress and Problems, Found. Phys. 47 (2017) 797 [arXiv:1603.05834] [INSPIRE].
R.H. Brandenberger, String Gas Cosmology, [arXiv:0808.0746] [INSPIRE].
N. Ishibashi, H. Kawai, Y. Kitazawa and A. Tsuchiya, A Large N reduced model as superstring, Nucl. Phys. B 498 (1997) 467 [hep-th/9612115] [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].
J. Nishimura, Signature change of the emergent space-time in the IKKT matrix model, in 21st Hellenic School and Workshops on Elementary Particle Physics and Gravity, Corfu Greece, August 29 – September 8 2021 [arXiv:2205.04726] [INSPIRE].
B. Ydri, Review of M(atrix)-Theory, Type IIB Matrix Model and Matrix String Theory, arXiv:1708.00734 [INSPIRE].
M. Hirasawa et al., A new phase in the Lorentzian type IIB matrix model and the emergence of continuous space-time, PoS LATTICE2021 (2022) 428 [arXiv:2112.15390] [INSPIRE].
K. Hatakeyama et al., Relationship between the Euclidean and Lorentzian versions of the type IIB matrix model, PoS LATTICE2021 (2022) 341 [arXiv:2112.15368] [INSPIRE].
K. Hatakeyama et al., Complex Langevin studies of the emergent space-time in the type IIB matrix model, (2022) [DOI] [arXiv:2201.13200] [INSPIRE].
T. Banks, W. Fischler, S.H. Shenker and L. Susskind, M theory as a matrix model: A Conjecture, Phys. Rev. D 55 (1997) 5112 [hep-th/9610043] [INSPIRE].
S. R. Das and A. Jevicki, String Field Theory and Physical Interpretation of D = 1 Strings, Mod. Phys. Lett. A 5 (1990) 1639.
R. Dijkgraaf, E.P. Verlinde and H.L. Verlinde, Matrix string theory, Nucl. Phys. B 500 (1997) 43 [hep-th/9703030] [INSPIRE].
B. Craps, S. Sethi and E.P. Verlinde, A Matrix big bang, JHEP 10 (2005) 005 [hep-th/0506180] [INSPIRE].
M. Blau and M. O’Loughlin, DLCQ and Plane Wave Matrix Big Bang Models, JHEP 09 (2008) 097 [arXiv:0806.3255] [INSPIRE].
D.E. Berenstein, J.M. Maldacena and H.S. Nastase, Strings in flat space and pp waves from N = 4 superYang-Mills, JHEP 04 (2002) 013 [hep-th/0202021] [INSPIRE].
J.M. Maldacena and C. Núñez, Towards the large N limit of pure N = 1 superYang-Mills, Phys. Rev. Lett. 86 (2001) 588 [hep-th/0008001] [INSPIRE].
L. Smolin, M theory as a matrix extension of Chern-Simons theory, Nucl. Phys. B 591 (2000) 227 [hep-th/0002009] [INSPIRE].
Y. Yargic, J. Lanier, L. Smolin and D. Wecker, A Cubic Matrix Action for the Standard Model and Beyond, arXiv:2201.04183 [INSPIRE].
J.Hoppe, Quantum Theory of a Massless Relativistic Surface, Ph.D. Thesis, Department of Physics, Massachusetts Institute of Technology, Cambridge, U.S.A. (1982) [http://dspace.mit.edu/handle/1721.1/15717].
B. de Wit, J. Hoppe and H. Nicolai, On the Quantum Mechanics of Supermembranes, Nucl. Phys. B 305 (1988) 545 [INSPIRE].
S. Brahma, R. Brandenberger and S. Laliberte, Emergent cosmology from matrix theory, JHEP 03 (2022) 067 [arXiv:2107.11512] [INSPIRE].
Y. Ito, J. Nishimura and A. Tsuchiya, Power-law expansion of the Universe from the bosonic Lorentzian type IIB matrix model, JHEP 11 (2015) 070 [arXiv:1506.04795] [INSPIRE].
K. Hatakeyama, A. Matsumoto, J. Nishimura, A. Tsuchiya and A. Yosprakob, The emergence of expanding space-time and intersecting D-branes from classical solutions in the Lorentzian type IIB matrix model, PTEP 2020 (2020) 043B10 [arXiv:1911.08132] [INSPIRE].
H. Aoki, S. Iso, H. Kawai, Y. Kitazawa and T. Tada, Space-time structures from IIB matrix model, Prog. Theor. Phys. 99 (1998) 713 [hep-th/9802085] [INSPIRE].
S.-W. Kim, J. Nishimura and A. Tsuchiya, Expanding (3 + 1)-dimensional universe from a Lorentzian matrix model for superstring theory in (9 + 1)-dimensions, Phys. Rev. Lett. 108 (2012) 011601 [arXiv:1108.1540] [INSPIRE].
T. Hotta, J. Nishimura and A. Tsuchiya, Dynamical aspects of large N reduced models, Nucl. Phys. B 545 (1999) 543 [hep-th/9811220] [INSPIRE].
J. Nishimura and G. Vernizzi, Spontaneous breakdown of Lorentz invariance in IIB matrix model, JHEP 04 (2000) 015 [hep-th/0003223] [INSPIRE].
J. Ambjørn, K.N. Anagnostopoulos, W. Bietenholz, T. Hotta and J. Nishimura, Monte Carlo studies of the IIB matrix model at large N , JHEP 07 (2000) 011 [hep-th/0005147] [INSPIRE].
J. Nishimura and F. Sugino, Dynamical generation of four-dimensional space-time in the IIB matrix model, JHEP 05 (2002) 001 [hep-th/0111102] [INSPIRE].
H. Kawai, S. Kawamoto, T. Kuroki, T. Matsuo and S. Shinohara, Mean field approximation of IIB matrix model and emergence of four-dimensional space-time, Nucl. Phys. B 647 (2002) 153 [hep-th/0204240] [INSPIRE].
T. Aoyama, J. Nishimura and T. Okubo, Spontaneous breaking of the rotational symmetry in dimensionally reduced super Yang-Mills models, Prog. Theor. Phys. 125 (2011) 537 [arXiv:1007.0883] [INSPIRE].
J. Nishimura, T. Okubo and F. Sugino, Systematic study of the SO(10) symmetry breaking vacua in the matrix model for type IIB superstrings, JHEP 10 (2011) 135 [arXiv:1108.1293] [INSPIRE].
R.H. Brandenberger and C. Vafa, Superstrings in the Early Universe, Nucl. Phys. B 316 (1989) 391 [INSPIRE].
T. Aoki, M. Hirasawa, Y. Ito, J. Nishimura and A. Tsuchiya, On the structure of the emergent 3d expanding space in the Lorentzian type IIB matrix model, PTEP 2019 (2019) 093B03 [arXiv:1904.05914] [INSPIRE].
R. Hagedorn, Statistical thermodynamics of strong interactions at high-energies, Nuovo Cim. Suppl. 3 (1965) 147 [INSPIRE].
P. Agrawal, S. Gukov, G. Obied and C. Vafa, Topological Gravity as the Early Phase of Our Universe, arXiv:2009.10077 [INSPIRE].
W. Siegel, Two vierbein formalism for string inspired axionic gravity, Phys. Rev. D 47 (1993) 5453 [hep-th/9302036] [INSPIRE].
W. Siegel, Superspace duality in low-energy superstrings, Phys. Rev. D 48 (1993) 2826 [hep-th/9305073] [INSPIRE].
W. Siegel, Manifest duality in low-energy superstrings, in International Conference on Strings 93, Berkeley U.S.A., May 24–29 1993 [hep-th/9308133] [INSPIRE].
C. Hull and B. Zwiebach, Double Field Theory, JHEP 09 (2009) 099 [arXiv:0904.4664] [INSPIRE].
H. Bernardo, R. Brandenberger and G. Franzmann, String cosmology backgrounds from classical string geometry, Phys. Rev. D 103 (2021) 043540 [arXiv:2005.08324] [INSPIRE].
H. Bernardo, R. Brandenberger and G. Franzmann, Solution of the Size and Horizon Problems from Classical String Geometry, JHEP 10 (2020) 155 [arXiv:2007.14096] [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].
N. Kawahara, J. Nishimura and S. Takeuchi, High temperature expansion in supersymmetric matrix quantum mechanics, JHEP 12 (2007) 103 [arXiv:0710.2188] [INSPIRE].
A. Nayeri, R.H. Brandenberger and C. Vafa, Producing a scale-invariant spectrum of perturbations in a Hagedorn phase of string cosmology, Phys. Rev. Lett. 97 (2006) 021302 [hep-th/0511140] [INSPIRE].
R.H. Brandenberger, A. Nayeri, S.P. Patil and C. Vafa, Tensor Modes from a Primordial Hagedorn Phase of String Cosmology, Phys. Rev. Lett. 98 (2007) 231302 [hep-th/0604126] [INSPIRE].
R.H. Brandenberger, A. Nayeri and S.P. Patil, Closed String Thermodynamics and a Blue Tensor Spectrum, Phys. Rev. D 90 (2014) 067301 [arXiv:1403.4927] [INSPIRE].
S. Brahma, S. Laliberte and R. Brandenberger, Spontaneous symmetry breaking in the BFSS model: Analytical results using the Gaussian expansion method, to be published.
F.R. Klinkhamer, On the emergence of an expanding universe from a Lorentzian matrix model, PTEP 2020 (2019) 103 [arXiv:1912.12229] [INSPIRE].
F.R. Klinkhamer, IIB matrix model: Emergent spacetime from the master field, PTEP 2021 (2021) 013B04 [arXiv:2007.08485] [INSPIRE].
F. R. Klinkhamer, On the emergence of an expanding universe from a Lorentzian matrix model, PTEP 2020 (2020) 103B03.
F.R. Klinkhamer, Towards a Numerical Solution of the Bosonic Master-Field Equation of the IIB Matrix Model, Acta Phys. Polon. B 53 (2022) 5 [arXiv:2110.15309] [INSPIRE].
H. Steinacker, Emergent Gravity from Noncommutative Gauge Theory, JHEP 12 (2007) 049 [arXiv:0708.2426] [INSPIRE].
H. Grosse, H. Steinacker and M. Wohlgenannt, Emergent Gravity, Matrix Models and UV/IR Mixing, JHEP 04 (2008) 023 [arXiv:0802.0973] [INSPIRE].
H. Steinacker, Emergent Gravity and Noncommutative Branes from Yang-Mills Matrix Models, Nucl. Phys. B 810 (2009) 1 [arXiv:0806.2032] [INSPIRE].
H. Steinacker, Emergent Geometry and Gravity from Matrix Models: an Introduction, Class. Quant. Grav. 27 (2010) 133001 [arXiv:1003.4134] [INSPIRE].
H.C. Steinacker, Gravity as a quantum effect on quantum space-time, Phys. Lett. B 827 (2022) 136946 [arXiv:2110.03936] [INSPIRE].
D.Z. Freedman, G.W. Gibbons and M. Schnabl, Matrix cosmology, AIP Conf. Proc. 743 (2004) 286 [hep-th/0411119] [INSPIRE].
J. Hoppe, Some classical solutions of membrane matrix model equations, NATO Sci. Ser. C 520 (1999) 423 [hep-th/9702169] [INSPIRE].
J. Hoppe and H. Nicolai, Relativistic Minimal Surfaces, Phys. Lett. B 196 (1987) 451.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher’s Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
ArXiv ePrint: 2206.12468
Rights and permissions
Open Access . This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.
About this article
Cite this article
Brahma, S., Brandenberger, R. & Laliberte, S. Emergent metric space-time from matrix theory. J. High Energ. Phys. 2022, 31 (2022). https://doi.org/10.1007/JHEP09(2022)031
Received:
Accepted:
Published:
Version of record:
DOI: https://doi.org/10.1007/JHEP09(2022)031