Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Lattice Holographic Cosmology Kostas Skenderis STAG R R E S E S E E A A R R C H C H CENTER CENTER C E N T E R Numerical approaches to the holographic principle, quantum gravity and cosmology Kyoto, Japan, 22 July 2015 Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Outline Introduction 1 Holographic cosmology 2 Perturbative QFT 3 Holographic Lattice Cosmology 4 Conclusions 5 Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Introduction ➢ The scale of inflation could be as high as 10 14 GeV and as such it is highest energy scale which is directly observable. ➢ This is much larger than any energy scale we could achieve with accelerators. ➠ The physics of the Early Universe is a unique probe of physics beyond the standard model . Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Inflation ➢ The leading theoretical paradigm for the very early universe is the theory of inflation. ➢ It describes remarkably well existing observational data. ➢ It is based on gravity coupled to scalar field(s), perturbatively quantized around an accelerating FRW background. Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Inflation: problems ➢ Despite its successes the theory of inflation still has a number of shortcomings: fine tuning, trans-Planckian issues etc. ➢ This description breaks down at some point since the background has a curvature singularity: the theory has to be embedded in a "UV complete theory" (the "initial singularity problem"). ➢ However, it has been very difficult to embed inflation in fundamental theory (such as string theory). Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Holographic cosmology ➢ Holography provides a new framework that can accommodate: ➠ conventional inflation: strongly coupled dual QFT ➠ qualitatively new models for the very early Universe : QFT at weak and intermediate coupling. ➢ The new framework gives new insight into conventional inflation. ➢ The new models are falsifiable with current data. Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions References ➢ A holographic framework for cosmology was put forward in works with Paul McFadden and Adam Bzowski (2009 - on-going). ➢ Related work: [Hull (1998)] ... (E-branes) [Witten (2001)] [Strominger (2001)] ... (dS/CFT correspondence) [Maldacena (2002)] ... (wavefunction of the universe) [Hartle, Hawking, Hertog (2012)] ... (quantum cosmology) [Trivedi et al][Garriga et al] [Coriano et al] .... [Arkani-Hamed, Maldacena] Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions References In this talk I will discuss new work on ➢ further developing the models based on perturbative QFT. This part is based on work in progress with ➢ Claudio Coriano and Luigi Delle Rose ➢ using Lattice methods to construct models valid for any value of the coupling constant. This part is based on work in progress with ➢ Evan Berkovitz, Philip Powel, Enrico Rinaldi, Pavlos Vranas (Lawrence Livermore National Laboratory) ➢ Masanori Hanada (YITP Kyoto and SITP Stanford) ➢ Andreas Jüttner, Antonin Portelli, Francesco Sanfilippo (SHEP , Southampton) Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Outline Introduction 1 Holographic cosmology 2 Perturbative QFT 3 Holographic Lattice Cosmology 4 Conclusions 5 Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Outline Introduction 1 Holographic cosmology 2 Perturbative QFT 3 Holographic Lattice Cosmology 4 Conclusions 5 Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Holographic cosmology in a nutshell In holographic cosmology one relates: ➢ cosmological observable such as the power spectra and non-gaussianities ➢ to correlation functions of the the energy momentum tensor of the dual QFT, upon a specific analytic continuation. Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Holographic cosmology: the dual QFT ➢ The dual QFT is 3d QFT that admits a large N limit and our results apply to two classes of theories: ➠ QFTs with a non-trivial UV fixed point. ➠ A class of super-renormalizable QFTs. ➢ The results hold perturbatively in 1 / N 2 . It is not clear whether these dualities hold non-perturbatively in 1 / N 2 . Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Holographic cosmology: the bulk These results hold for spacetimes that at late times approach ➠ de Sitter spacetime, ds 2 → ds 2 = − dt 2 + e 2 t dx i dx i , t → ∞ as ➠ power-law scaling solutions, ds 2 → ds 2 = − dt 2 + t 2 n dx i dx i , ( n > 1 ) t → ∞ as Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Holographic formula for the scalar power spectrum ➢ The scalar power spectrum is given by R ( q ) = − q 3 1 ∆ 2 Im � T ( q ) T ( − q ) � , 4 π 2 where T = T i i is the trace of the energy momentum tensor T ij and we Fourier transformed to momentum space. The imaginary part is taken after the analytic continuation, q → − iq , N → − iN ➢ The power spectrum of tensors is related with the 2-point of the traceless part of T and non-gausiaities are related with higher-point functions of T ij . Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Sketch of derivation I The underlying framework is gravity coupled to a scalar field Φ with a potential V (Φ) , d 4 x √− g ( R − ( ∂ Φ) 2 − 2 κ 2 V (Φ)) � 1 S = 2 κ 2 There is 1-1 correspondence [KS, Tonwsend (2006)] between: Domain-wall solutions dr 2 + e 2 A ( r ) dx i dx i ds 2 = Φ = Φ( r ) FRW spacetimes − dt 2 + a 2 ( t ) dx i dx i ds 2 = Φ = Φ( t ) Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Domain-wall/Cosmology correspondence Domain-wall/Cosmology correspondence FRW solutions of Domain-wall solutions of ↔ the theory with potential V (Φ) the theory with potential − V (Φ) . This correspondence can be understood as analytic continuation. An example of this correspondence is the analytic continuation from de Sitter to Anti de Sitter. This theorem shows that this relation is not accidental. Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Inflation/holographic RG correspondence ➢ A special case of the correspondence is that between inflationary backgrounds and holographic RG flow spacetimes. ➢ Inflationary spacetimes are mapped to ➢ asympotically Anti-de Sitter spacetime, ds 2 → ds 2 = dr 2 + e 2 r dx i dx i , as r → ∞ ➢ power-law scaling solutions, ds 2 → ds 2 = dr 2 + r 2 n dx i dx i , ( n > 1 ) as r → ∞ For special values of n these backgrounds are related to non-conformal branes. ➢ For these backgrounds there is an established holographic dictionary. Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Holographic formulae for cosmology [McFadden, KS] ➢ Given an FRW, compute cosmological observables using standard cosmological perturbation theory. ➢ Corresponding to this FRW there is a domain-wall. ➢ Use holography to compute energy-momentum tensor correlators for the QFT dual to the domain-wall. ➠ Comparing the two results leads to the holographic formulae. Kostas Skenderis Holographic cosmology meets lattice gauge theory
Introduction Holographic cosmology Perturbative QFT Holographic Lattice Cosmology Conclusions Remarks ➢ This derivation holds in the regime the gravity approximation is valid. ➠ Conventional inflation is holographic. ➢ We will next present an alternative derivation which does not make this assumption but postulates the form of the duality. Kostas Skenderis Holographic cosmology meets lattice gauge theory
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