Abstract
We analyze flat FRW cosmologies and hyperscaling-violating geometries by emphasizing the analytic continuation between them and their scale covariance. We exhibit two main calculations where this point of view is useful. First, based on the scale covariance, we show that the structure of higher curvature corrections to Einstein’s equation is very simple. Second, in the context of accelerated FRW cosmologies, also known as Q-space, we begin by calculating the Bunch-Davies wavefunctional for a massless scalar field and considering its interpretation as a generating functional of correlation functions of a holographic dual. We use this to conjecture a Q-space/QFT duality, a natural extension of dS/CFT, and argue that the Euclidean dual theory violates hyperscaling. This proposal, when extended to epochs in our own cosmological history like matter or radiation domination, suggests a holographically dual description via RG phases which violate hyper-scaling. We further use the wavefunctional to compute Anninos-Denef overlaps and show that the ultrametric structure discovered for de Sitter becomes sharper in accelerated FRW cosmologies as the acceleration slows. The substitution d → d eff = d − θ permeates and illuminates the discussion of wavefunctionals and overlaps in FRW cosmologies, allowing one to predict the sharpened structure. We conjecture that the sharpening of ultrametricity is holographically manifested by the growth of the effective dimensionality of the dual theory. We try to find an alternate manifestation of this ultrametric structure by studying the connection of the θ → −∞ background to little string theory.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
J.M. Maldacena, The Large-N limit of superconformal field theories and supergravity, Adv. Theor. Math. Phys. 2 (1998) 231 [Int. J. Theor. Phys. 38 (1999) 1113] [hep-th/9711200] [INSPIRE].
S.A. Hartnoll, Horizons, holography and condensed matter, arXiv:1106.4324 [INSPIRE].
S. Kachru, X. Liu and M. Mulligan, Gravity duals of Lifshitz-like fixed points, Phys. Rev. D 78 (2008) 106005 [arXiv:0808.1725] [INSPIRE].
D.T. Son, Toward an AdS/cold atoms correspondence: A Geometric realization of the Schrödinger symmetry, Phys. Rev. D 78 (2008) 046003 [arXiv:0804.3972] [INSPIRE].
K. Balasubramanian and J. McGreevy, Gravity duals for non-relativistic CFTs, Phys. Rev. Lett. 101 (2008) 061601 [arXiv:0804.4053] [INSPIRE].
A. Adams, A. Maloney, A. Sinha and S.E. Vazquez, 1/N Effects in Non-Relativistic Gauge-Gravity Duality, JHEP 03 (2009) 097 [arXiv:0812.0166] [INSPIRE].
D. Anninos and F. Denef, Cosmic Clustering, arXiv:1111.6061 [INSPIRE].
L. Huijse, S. Sachdev and B. Swingle, Hidden Fermi surfaces in compressible states of gauge-gravity duality, Phys. Rev. B 85 (2012) 035121 [arXiv:1112.0573] [INSPIRE].
B. Gouteraux and E. Kiritsis, Generalized Holographic Quantum Criticality at Finite Density, JHEP 12 (2011) 036 [arXiv:1107.2116] [INSPIRE].
E. Shaghoulian, Holographic Entanglement Entropy and Fermi Surfaces, JHEP 05 (2012) 065 [arXiv:1112.2702] [INSPIRE].
K. Copsey and R. Mann, Singularities in Hyperscaling Violating Spacetimes, JHEP 04 (2013) 079 [arXiv:1210.1231] [INSPIRE].
G.T. Horowitz and B. Way, Lifshitz Singularities, Phys. Rev. D 85 (2012) 046008 [arXiv:1111.1243] [INSPIRE].
N. Bao, X. Dong, S. Harrison and E. Silverstein, The Benefits of Stress: Resolution of the Lifshitz Singularity, Phys. Rev. D 86 (2012) 106008 [arXiv:1207.0171] [INSPIRE].
S. Harrison, S. Kachru and H. Wang, Resolving Lifshitz Horizons, arXiv:1202.6635 [INSPIRE].
J. Bhattacharya, S. Cremonini and A. Sinkovics, On the IR completion of geometries with hyperscaling violation, JHEP 02 (2013) 147 [arXiv:1208.1752] [INSPIRE].
G. Knodel and J.T. Liu, Higher derivative corrections to Lifshitz backgrounds, JHEP 10 (2013) 002 [arXiv:1305.3279] [INSPIRE].
N. Ogawa, T. Takayanagi and T. Ugajin, Holographic Fermi Surfaces and Entanglement Entropy, JHEP 01 (2012) 125 [arXiv:1111.1023] [INSPIRE].
S. Ryu and T. Takayanagi, Holographic derivation of entanglement entropy from AdS/CFT, Phys. Rev. Lett. 96 (2006) 181602 [hep-th/0603001] [INSPIRE].
M.M. Wolf, Violation of the entropic area law for Fermions, Phys. Rev. Lett. 96 (2006) 010404 [quant-ph/0503219] [INSPIRE].
D. Gioev and I. Klich, Entanglement Entropy of Fermions in Any Dimension and the Widom Conjecture, Phys. Rev. Lett. 96 (2006) 100503 [INSPIRE].
S.A. Hartnoll and E. Shaghoulian, Spectral weight in holographic scaling geometries, JHEP 07 (2012) 078 [arXiv:1203.4236] [INSPIRE].
M.A. Metlitski and S. Sachdev, Quantum phase transitions of metals in two spatial dimensions: I. Ising-nematic order, Phys. Rev. B 82 (2010) 075127 [arXiv:1001.1153] [INSPIRE].
S.C. Thier and W. Metzner, Singular order parameter interaction at the nematic quantum critical point in two-dimensional electron systems, Phys. Rev. B 84 (2011) 155133 [arXiv:1108.1929].
S. Kachru, personal correspondence.
K. Skenderis and P.K. Townsend, Hidden supersymmetry of domain walls and cosmologies, Phys. Rev. Lett. 96 (2006) 191301 [hep-th/0602260] [INSPIRE].
K. Skenderis and P.K. Townsend, Pseudo-Supersymmetry and the Domain-Wall/Cosmology Correspondence, J. Phys. A 40 (2007) 6733 [hep-th/0610253] [INSPIRE].
K. Skenderis, P.K. Townsend and A. Van Proeyen, Domain-wall/cosmology correspondence in AdS/dS supergravity, JHEP 08 (2007) 036 [arXiv:0704.3918] [INSPIRE].
E. Kiritsis, Asymptotic freedom, asymptotic flatness and cosmology, JCAP 11 (2013) 011 [arXiv:1307.5873] [INSPIRE].
T.P. Sotiriou and V. Faraoni, f(R) Theories Of Gravity, Rev. Mod. Phys. 82 (2010) 451 [arXiv:0805.1726] [INSPIRE].
A. De Felice and S. Tsujikawa, f(R) theories, Living Rev. Rel. 13 (2010) 3 [arXiv:1002.4928] [INSPIRE].
S.’i. Nojiri and S.D. Odintsov, Introduction to modified gravity and gravitational alternative for dark energy, eConf C 0602061 (2006) 06 [Int. J. Geom. Meth. Mod. Phys. 4 (2007) 115] [hep-th/0601213] [INSPIRE] [INSPIRE].
S. Fay, S. Nesseris and L. Perivolaropoulos, Can f(R) Modified Gravity Theories Mimic a LCDM Cosmology?, Phys. Rev. D 76 (2007) 063504 [gr-qc/0703006] [INSPIRE].
S. Capozziello, S. Nojiri, S.D. Odintsov and A. Troisi, Cosmological viability of f(R)-gravity as an ideal fluid and its compatibility with a matter dominated phase, Phys. Lett. B 639 (2006) 135 [astro-ph/0604431] [INSPIRE].
L.G. Jaime, L. Patino and M. Salgado, f(R) Cosmology revisited, arXiv:1206.1642 [INSPIRE].
V. Faraoni, f(R) gravity: Successes and challenges, arXiv:0810.2602 [INSPIRE].
H. Gil-Marin, F. Schmidt, W. Hu, R. Jimenez and L. Verde, The Bispectrum of f(R) Cosmologies, JCAP 11 (2011) 019 [arXiv:1109.2115] [INSPIRE].
S.’i. Nojiri and S.D. Odintsov, Modified gravity with negative and positive powers of the curvature: Unification of the inflation and of the cosmic acceleration, Phys. Rev. D 68 (2003) 123512 [hep-th/0307288] [INSPIRE].
N. Ohta, Accelerating cosmologies and inflation from M/superstring theories, Int. J. Mod. Phys. A 20 (2005) 1 [hep-th/0411230] [INSPIRE].
K.-i. Maeda and N. Ohta, Inflation from superstring /M theory compactification with higher order corrections. 1., Phys. Rev. D 71 (2005) 063520 [hep-th/0411093] [INSPIRE].
I.H. Brevik, Crossing of the W = −1 barrier in two-fluid viscous modified gravity, Gen. Rel. Grav. 38 (2006) 1317 [gr-qc/0603025] [INSPIRE].
S.’i. Nojiri and S.D. Odintsov, Can F(R)-gravity be a viable model: the universal unification scenario for inflation, dark energy and dark matter, arXiv:0801.4843 [INSPIRE].
S.’i. Nojiri and S.D. Odintsov, Ghost-free F (R) bigravity and accelerating cosmology, Phys. Lett. B 716 (2012) 377 [arXiv:1207.5106] [INSPIRE].
S. Hellerman, N. Kaloper and L. Susskind, String theory and quintessence, JHEP 06 (2001) 003 [hep-th/0104180] [INSPIRE].
R. Bousso, Cosmology and the S-matrix, Phys. Rev. D 71 (2005) 064024 [hep-th/0412197] [INSPIRE].
J.B. Hartle and S.W. Hawking, Wave Function of the Universe, Phys. Rev. D 28 (1983) 2960 [INSPIRE].
T.S. Bunch and P.C.W. Davies, Quantum Field Theory in de Sitter Space: Renormalization by Point Splitting, Proc. Roy. Soc. Lond. A 360 (1978) 117 [INSPIRE].
X. Dong, S. Harrison, S. Kachru, G. Torroba and H. Wang, Aspects of holography for theories with hyperscaling violation, JHEP 06 (2012) 041 [arXiv:1201.1905] [INSPIRE].
L.H. Ford and L. Parker, Infrared Divergences in a Class of Robertson-Walker Universes, Phys. Rev. D 16 (1977) 245 [INSPIRE].
D.Z. Freedman, K. Johnson and J.I. Latorre, Differential regularization and renormalization: A New method of calculation in quantum field theory, Nucl. Phys. B 371 (1992) 353 [INSPIRE].
D.Z. Freedman, K. Johnson, R. Muñoz-Tapia and X. Vilasis-Cardona, A Cutoff procedure and counterterms for differential renormalization, Nucl. Phys. B 395 (1993) 454 [hep-th/9206028] [INSPIRE].
D. Harlow and D. Stanford, Operator Dictionaries and Wave Functions in AdS/CFT and dS/CFT, arXiv:1104.2621 [INSPIRE].
D. Anninos, T. Hartman and A. Strominger, Higher Spin Realization of the dS/CFT Correspondence, arXiv:1108.5735 [INSPIRE].
D. Anninos, F. Denef and D. Harlow, The Wave Function of Vasiliev’s Universe - A Few Slices Thereof, Phys. Rev. D 88 (2013) 084049 [arXiv:1207.5517] [INSPIRE].
D. Anninos, F. Denef, G. Konstantinidis and E. Shaghoulian, Higher Spin de Sitter Holography from Functional Determinants, arXiv:1305.6321 [INSPIRE].
G.S. Ng and A. Strominger, State/Operator Correspondence in Higher-Spin dS/CFT, Class. Quant. Grav. 30 (2013) 104002 [arXiv:1204.1057] [INSPIRE].
D. Das, S.R. Das, A. Jevicki and Q. Ye, Bi-local Construction of Sp(2N)/dS Higher Spin Correspondence, JHEP 01 (2013) 107 [arXiv:1205.5776] [INSPIRE].
A. Karch and C.F. Uhlemann, Higher-spin realization of a dS static patch/cut-off CFT correspondence, Phys. Rev. D 88 (2013) 046001 [arXiv:1306.0582] [INSPIRE].
S. Banerjee et al., Topology of Future Infinity in dS/CFT, JHEP 11 (2013) 026 [arXiv:1306.6629] [INSPIRE].
A.A. Starobinsky, Dynamics of Phase Transition in the New Inflationary Universe Scenario and Generation of Perturbations, Phys. Lett. B 117 (1982) 175 [INSPIRE].
A. Strominger, The dS/CFT correspondence, JHEP 10 (2001) 034 [hep-th/0106113] [INSPIRE].
A. Strominger, Inflation and the dS/CFT correspondence, JHEP 11 (2001) 049 [hep-th/0110087] [INSPIRE].
E. Witten, Quantum gravity in de Sitter space, hep-th/0106109 [INSPIRE].
J.M. Maldacena, Non-Gaussian features of primordial fluctuations in single field inflationary models, JHEP 05 (2003) 013 [astro-ph/0210603] [INSPIRE].
R. Bousso, The Holographic principle, Rev. Mod. Phys. 74 (2002) 825 [hep-th/0203101] [INSPIRE].
R. Bousso, The Holographic principle for general backgrounds, Class. Quant. Grav. 17 (2000) 997 [hep-th/9911002] [INSPIRE].
R. Bousso, A Covariant entropy conjecture, JHEP 07 (1999) 004 [hep-th/9905177] [INSPIRE].
J.D. Brown and J.W. York, Quasilocal energy and conserved charges derived from the gravitational action, Phys. Rev. D 47 (1993) 1407 [gr-qc/9209012] [INSPIRE].
M. Ammon, M. Kaminski and A. Karch, Hyperscaling-Violation on Probe D-branes, JHEP 11 (2012) 028 [arXiv:1207.1726] [INSPIRE].
D. Anninos, de Sitter Musings, Int. J. Mod. Phys. A 27 (2012) 1230013 [arXiv:1205.3855] [INSPIRE].
Planck collaboration, P.A.R. Ade et al., Planck 2013 results. XVI. Cosmological parameters, arXiv:1303.5076 [INSPIRE].
A. Jevicki and T. Yoneya, Space-time uncertainty principle and conformal symmetry in D particle dynamics, Nucl. Phys. B 535 (1998) 335 [hep-th/9805069] [INSPIRE].
A. Jevicki, Y. Kazama and T. Yoneya, Generalized conformal symmetry in D-brane matrix models, Phys. Rev. D 59 (1999) 066001 [hep-th/9810146] [INSPIRE].
H.J. Boonstra, K. Skenderis and P.K. Townsend, The domain wall/QFT correspondence, JHEP 01 (1999) 003 [hep-th/9807137] [INSPIRE].
I. Kanitscheider, K. Skenderis and M. Taylor, Precision holography for non-conformal branes, JHEP 09 (2008) 094 [arXiv:0807.3324] [INSPIRE].
D. Harlow and L. Susskind, Crunches, Hats and a Conjecture, arXiv:1012.5302 [INSPIRE].
D. Anninos, G.S. Ng and A. Strominger, Future Boundary Conditions in de Sitter Space, JHEP 02 (2012) 032 [arXiv:1106.1175] [INSPIRE].
M. Mezard, G. Parisi, and M.A. Virasoro, Spin Glass Theory and Beyond, vol. 9, Lecture Notes in Physics, World Scientic (1987) [ISBN 9971501155] [Google books].
G. Parisi, An Infinite Number of Order Parameters for Spin Glasses, Phys. Rev. Lett. 43 (1979) 1754 [INSPIRE].
G. Parisi, A Sequence of Approximated Solutions to the SK Model for Spin Glasses, J. Phys. A 13 (1980) L115 [INSPIRE].
G. Parisi, Order parameter for spin-glasses, Phys. Rev. Lett. 50 (1983) 1946 [INSPIRE].
D. Sherrington and S. Kirkpatrick, Solvable Model of a Spin-Glass, Phys. Rev. Lett. 35 (1975) 1792 [INSPIRE].
R. Rammal, G. Toulouse and M.A. Virasoro, Ultrametricity for physicists, Rev. Mod. Phys. 58 (1986) 765 [INSPIRE].
F. Denef, TASI lectures on complex structures, arXiv:1104.0254 [INSPIRE].
D. Anninos, Classical and Quantum Symmetries of De Sitter Space, Ph.D. Thesis, Harvard University (May 2011).
M.K. Benna, De (Baby) Sitter Overlaps, Nucl. Phys. B 867 (2013) 82 [arXiv:1111.4195] [INSPIRE].
D.A. Roberts and D. Stanford, On memory in exponentially expanding spaces, JHEP 06 (2013) 042 [arXiv:1210.5238] [INSPIRE].
T. Castellani and A. Cavagna, Spin-glass theory for pedestrians , J. Stat. Mech. (2005) P05012 [cond-mat/0505032].
S.S. Gubser and F.D. Rocha, Peculiar properties of a charged dilatonic black hole in AdS 5, Phys. Rev. D 81 (2010) 046001 [arXiv:0911.2898] [INSPIRE].
R.J. Anantua, S.A. Hartnoll, V.L. Martin and D.M. Ramirez, The Pauli exclusion principle at strong coupling: Holographic matter and momentum space, JHEP 03 (2013) 104 [arXiv:1210.1590] [INSPIRE].
D. Kutasov, Introduction to little string theory, Spring School on Superstrings and Related Matters, 2-10 April 2001, Trieste, Italy.
S. Ryu and T. Takayanagi, Aspects of Holographic Entanglement Entropy, JHEP 08 (2006) 045 [hep-th/0605073] [INSPIRE].
M. Kulaxizi, A. Parnachev and K. Schalm, On Holographic Entanglement Entropy of Charged Matter, JHEP 10 (2012) 098 [arXiv:1208.2937] [INSPIRE].
J.L.F. Barbon and C.A. Fuertes, Holographic entanglement entropy probes (non)locality, JHEP 04 (2008) 096 [arXiv:0803.1928] [INSPIRE].
W. Li and T. Takayanagi, Holography and Entanglement in Flat Spacetime, Phys. Rev. Lett. 106 (2011) 141301 [arXiv:1010.3700] [INSPIRE].
D.S. Fisher and D.A. Huse, Ordered Phase of Short-Range Ising Spin-Glasses, Phys. Rev. Lett. 56 (1986) 1601 [INSPIRE].
V.A. Belinsky, I.M. Khalatnikov and E.M. Lifshitz, Oscillatory approach to a singular point in the relativistic cosmology, Adv. Phys. 19 (1970) 525 [INSPIRE].
V.A. Belinski and I.M. Khalatnikov, Effect of Scalar and Vector Fields on the Nature of the Cosmological Singularity, Sov. Phys. JETP 36 (1973) 591 [INSPIRE].
M. Henneaux, D. Persson and P. Spindel, Spacelike Singularities and Hidden Symmetries of Gravity, Living Rev. Rel. 11 (2008) 1 [arXiv:0710.1818] [INSPIRE].
A. Ashtekar and E. Wilson-Ewing, Loop quantum cosmology of Bianchi I models, Phys. Rev. D 79 (2009) 083535 [arXiv:0903.3397] [INSPIRE].
Y. Korovin, K. Skenderis and M. Taylor, Lifshitz as a deformation of Anti-de Sitter, JHEP 08 (2013) 026 [arXiv:1304.7776] [INSPIRE].
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
ArXiv ePrint: 1308.1095
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (https://creativecommons.org/licenses/by/4.0), which permits use, duplication, adaptation, distribution, and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Shaghoulian, E. FRW cosmologies and hyperscaling-violating geometries: higher curvature corrections, ultrametricity, Q-space/QFT duality, and a little string theory. J. High Energ. Phys. 2014, 11 (2014). https://doi.org/10.1007/JHEP03(2014)011
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/JHEP03(2014)011